LMFDB - Embedded newform 88.4.i.a.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [88,4,Mod(9,88)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(88, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 6]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("88.9");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 88 = 2^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	88.i (of order $5$, degree $4$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.19216808051$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{5})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 60 x^{14} - 83 x^{13} + 1685 x^{12} - 14618 x^{11} + 106543 x^{10} - 521269 x^{9} + \cdots + 2025 $ x^16 - 4x^15 + 60x^14 - 83x^13 + 1685x^12 - 14618x^11 + 106543x^10 - 521269x^9 + 3907139x^8 - 15948298x^7 + 37290102x^6 - 37052438x^5 + 40742731x^4 + 4262985x^3 + 9787185x^2 + 227475*x + 2025
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			49.3
Root			$2.60776 - 1.89465i$ of defining polynomial
Character	$\chi$	$=$	88.49
Dual form			88.4.i.a.9.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.57022 + 3.32046i) q^{3} +(-3.76204 + 11.5784i) q^{5} +(-27.3815 + 19.8938i) q^{7} +(1.51800 + 4.67192i) q^{9} +(34.2797 - 12.4861i) q^{11} +(-6.38348 - 19.6463i) q^{13} +(-55.6388 + 40.4240i) q^{15} +(-18.5932 + 57.2239i) q^{17} +(85.6982 + 62.2634i) q^{19} -191.196 q^{21} +186.999 q^{23} +(-18.7786 - 13.6434i) q^{25} +(38.5577 - 118.668i) q^{27} +(-109.554 + 79.5957i) q^{29} +(27.6973 + 85.2435i) q^{31} +(198.125 + 56.7601i) q^{33} +(-127.328 - 391.874i) q^{35} +(148.834 - 108.135i) q^{37} +(36.0609 - 110.984i) q^{39} +(-301.585 - 219.114i) q^{41} +294.257 q^{43} -59.8040 q^{45} +(-6.86032 - 4.98432i) q^{47} +(247.988 - 763.230i) q^{49} +(-274.985 + 199.788i) q^{51} +(-67.9504 - 209.130i) q^{53} +(15.6071 + 443.876i) q^{55} +(184.916 + 569.114i) q^{57} +(-210.773 + 153.136i) q^{59} +(-159.109 + 489.688i) q^{61} +(-134.507 - 97.7252i) q^{63} +251.487 q^{65} -263.041 q^{67} +(854.625 + 620.922i) q^{69} +(-95.6197 + 294.287i) q^{71} +(170.648 - 123.983i) q^{73} +(-40.5198 - 124.707i) q^{75} +(-690.232 + 1023.84i) q^{77} +(-288.634 - 888.325i) q^{79} +(677.554 - 492.271i) q^{81} +(-73.5873 + 226.478i) q^{83} +(-592.612 - 430.558i) q^{85} -764.980 q^{87} -620.760 q^{89} +(565.629 + 410.954i) q^{91} +(-156.465 + 481.549i) q^{93} +(-1043.31 + 758.008i) q^{95} +(372.130 + 1145.30i) q^{97} +(110.371 + 141.198i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - q^{3} - q^{5} - 13 q^{7} + 7 q^{9} + 83 q^{11} + 69 q^{13} + 93 q^{15} - 217 q^{17} + 126 q^{19} + 34 q^{21} - 92 q^{23} + 307 q^{25} + 158 q^{27} - 553 q^{29} + 205 q^{31} - 198 q^{33} + 7 q^{35}+ \cdots - 3265 q^{99}+O(q^{100}) $ 16 * q - q^3 - q^5 - 13 * q^7 + 7 * q^9 + 83 * q^11 + 69 * q^13 + 93 * q^15 - 217 * q^17 + 126 * q^19 + 34 * q^21 - 92 * q^23 + 307 * q^25 + 158 * q^27 - 553 * q^29 + 205 * q^31 - 198 * q^33 + 7 * q^35 + 127 * q^37 - 629 * q^39 - 553 * q^41 + 70 * q^43 - 1132 * q^45 + 779 * q^47 + 847 * q^49 - 2176 * q^51 + 1401 * q^53 - 385 * q^55 - 488 * q^57 + 1354 * q^59 - 1623 * q^61 + 256 * q^63 + 1674 * q^65 - 1658 * q^67 + 5470 * q^69 + 1985 * q^71 - 1245 * q^73 + 4114 * q^75 - 2645 * q^77 - 2007 * q^79 + 1966 * q^81 - 2946 * q^83 - 1543 * q^85 - 262 * q^87 - 6410 * q^89 + 3937 * q^91 + 2221 * q^93 - 8359 * q^95 + 6566 * q^97 - 3265 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/88\mathbb{Z}\right)^\times$.

$n$	$23$	$45$	$57$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{5}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	4.57022	+	3.32046i	0.879539	+	0.639023i	0.933129	−	0.359540i	$-0.117067\pi$
$3$	4.57022	+	3.32046i	0.879539	+	0.639023i	−0.0535904	+	0.998563i	$0.517067\pi$
$4$		0			0
$5$	−3.76204	+	11.5784i	−0.336487	+	1.03560i	0.629498	+	0.777002i	$0.283261\pi$
$5$	−3.76204	+	11.5784i	−0.336487	+	1.03560i	−0.965985	+	0.258599i	$0.916739\pi$
$6$		0			0
$7$	−27.3815	+	19.8938i	−1.47846	+	1.07416i	−0.500411	+	0.865788i	$0.666818\pi$
$7$	−27.3815	+	19.8938i	−1.47846	+	1.07416i	−0.978049	+	0.208376i	$0.933182\pi$
$8$		0			0
$9$	1.51800	+	4.67192i	0.0562222	+	0.173034i
$10$		0			0
$11$	34.2797	−	12.4861i	0.939611	−	0.342246i
$11$	34.2797	−	12.4861i	0.939611	−	0.342246i
$12$		0			0
$13$	−6.38348	−	19.6463i	−0.136189	−	0.419147i	0.859584	−	0.510995i	$-0.170723\pi$
$13$	−6.38348	−	19.6463i	−0.136189	−	0.419147i	−0.995773	+	0.0918474i	$0.970723\pi$
$14$		0			0
$15$	−55.6388	+	40.4240i	−0.957726	+	0.695829i
$16$		0			0
$17$	−18.5932	+	57.2239i	−0.265265	+	0.816403i	0.726367	+	0.687307i	$0.241207\pi$
$17$	−18.5932	+	57.2239i	−0.265265	+	0.816403i	−0.991632	+	0.129095i	$0.958793\pi$
$18$		0			0
$19$	85.6982	+	62.2634i	1.03476	+	0.751800i	0.969257	−	0.246052i	$-0.0791335\pi$
$19$	85.6982	+	62.2634i	1.03476	+	0.751800i	0.0655071	+	0.997852i	$0.479133\pi$
$20$		0			0
$21$	−191.196			−1.98678
$22$		0			0
$23$	186.999			1.69530			0.847651	−	0.530555i	$-0.178016\pi$
$23$	186.999			1.69530			0.847651	+	0.530555i	$0.178016\pi$
$24$		0			0
$25$	−18.7786	−	13.6434i	−0.150229	−	0.109148i
$26$		0			0
$27$	38.5577	−	118.668i	0.274831	−	0.845843i
$28$		0			0
$29$	−109.554	+	79.5957i	−0.701506	+	0.509674i	−0.880422	−	0.474190i	$-0.842741\pi$
$29$	−109.554	+	79.5957i	−0.701506	+	0.509674i	0.178916	+	0.983864i	$0.442741\pi$
$30$		0			0
$31$	27.6973	+	85.2435i	0.160470	+	0.493877i	0.998674	−	0.0514804i	$-0.0163940\pi$
$31$	27.6973	+	85.2435i	0.160470	+	0.493877i	−0.838204	+	0.545357i	$0.816394\pi$
$32$		0			0
$33$	198.125	+	56.7601i	1.04513	+	0.299414i
$34$		0			0
$35$	−127.328	−	391.874i	−0.614922	−	1.89254i
$36$		0			0
$37$	148.834	−	108.135i	0.661303	−	0.480465i	−0.205799	−	0.978594i	$-0.565979\pi$
$37$	148.834	−	108.135i	0.661303	−	0.480465i	0.867103	+	0.498129i	$0.165979\pi$
$38$		0			0
$39$	36.0609	−	110.984i	0.148061	−	0.455684i
$40$		0			0
$41$	−301.585	−	219.114i	−1.14877	−	0.834632i	−0.160455	−	0.987043i	$-0.551296\pi$
$41$	−301.585	−	219.114i	−1.14877	−	0.834632i	−0.988317	+	0.152412i	$0.951296\pi$
$42$		0			0
$43$	294.257			1.04358			0.521789	−	0.853075i	$-0.325265\pi$
$43$	294.257			1.04358			0.521789	+	0.853075i	$0.325265\pi$
$44$		0			0
$45$	−59.8040			−0.198112
$46$		0			0
$47$	−6.86032	−	4.98432i	−0.0212911	−	0.0154689i	0.577089	−	0.816681i	$-0.304189\pi$
$47$	−6.86032	−	4.98432i	−0.0212911	−	0.0154689i	−0.598380	+	0.801213i	$0.704189\pi$
$48$		0			0
$49$	247.988	−	763.230i	0.722998	−	2.22516i
$50$		0			0
$51$	−274.985	+	199.788i	−0.755011	+	0.548548i
$52$		0			0
$53$	−67.9504	−	209.130i	−0.176108	−	0.542004i	0.823575	−	0.567208i	$-0.191976\pi$
$53$	−67.9504	−	209.130i	−0.176108	−	0.542004i	−0.999682	+	0.0252042i	$0.991976\pi$
$54$		0			0
$55$	15.6071	+	443.876i	0.0382630	+	1.08822i
$56$		0			0
$57$	184.916	+	569.114i	0.429698	+	1.32247i
$58$		0			0
$59$	−210.773	+	153.136i	−0.465091	+	0.337908i	−0.795525	−	0.605921i	$-0.792805\pi$
$59$	−210.773	+	153.136i	−0.465091	+	0.337908i	0.330434	+	0.943829i	$0.392805\pi$
$60$		0			0
$61$	−159.109	+	489.688i	−0.333965	+	1.02784i	0.633265	+	0.773935i	$0.281714\pi$
$61$	−159.109	+	489.688i	−0.333965	+	1.02784i	−0.967230	+	0.253902i	$0.918286\pi$
$62$		0			0
$63$	−134.507	−	97.7252i	−0.268989	−	0.195432i
$64$		0			0
$65$	251.487			0.479895
$66$		0			0
$67$	−263.041			−0.479635			−0.239818	−	0.970818i	$-0.577088\pi$
$67$	−263.041			−0.479635			−0.239818	+	0.970818i	$0.577088\pi$
$68$		0			0
$69$	854.625	+	620.922i	1.49108	+	1.08334i
$70$		0			0
$71$	−95.6197	+	294.287i	−0.159831	+	0.491908i	−0.998618	−	0.0525507i	$-0.983265\pi$
$71$	−95.6197	+	294.287i	−0.159831	+	0.491908i	0.838788	+	0.544459i	$0.183265\pi$
$72$		0			0
$73$	170.648	−	123.983i	0.273601	−	0.198783i	−0.442520	−	0.896758i	$-0.645916\pi$
$73$	170.648	−	123.983i	0.273601	−	0.198783i	0.716122	+	0.697975i	$0.245916\pi$
$74$		0			0
$75$	−40.5198	−	124.707i	−0.0623843	−	0.191999i
$76$		0			0
$77$	−690.232	+	1023.84i	−1.02155	+	1.51529i
$78$		0			0
$79$	−288.634	−	888.325i	−0.411062	−	1.26512i	−0.915726	−	0.401803i	$-0.868384\pi$
$79$	−288.634	−	888.325i	−0.411062	−	1.26512i	0.504664	−	0.863316i	$-0.331616\pi$
$80$		0			0
$81$	677.554	−	492.271i	0.929429	−	0.675269i
$82$		0			0
$83$	−73.5873	+	226.478i	−0.0973163	+	0.299509i	−0.987850	−	0.155408i	$-0.950331\pi$
$83$	−73.5873	+	226.478i	−0.0973163	+	0.299509i	0.890534	+	0.454917i	$0.150331\pi$
$84$		0			0
$85$	−592.612	−	430.558i	−0.756209	−	0.549418i
$86$		0			0
$87$	−764.980			−0.942695
$88$		0			0
$89$	−620.760			−0.739331			−0.369666	−	0.929165i	$-0.620528\pi$
$89$	−620.760			−0.739331			−0.369666	+	0.929165i	$0.620528\pi$
$90$		0			0
$91$	565.629	+	410.954i	0.651583	+	0.473403i
$92$		0			0
$93$	−156.465	+	481.549i	−0.174459	+	0.536928i
$94$		0			0
$95$	−1043.31	+	758.008i	−1.12675	+	0.818631i
$96$		0			0
$97$	372.130	+	1145.30i	0.389527	+	1.19884i	0.933143	+	0.359506i	$0.117055\pi$
$97$	372.130	+	1145.30i	0.389527	+	1.19884i	−0.543616	+	0.839334i	$0.682945\pi$
$98$		0			0
$99$	110.371	+	141.198i	0.112047	+	0.143343i
$100$		0			0
$101$	−318.523	−	980.312i	−0.313804	−	0.965789i	−0.976244	−	0.216674i	$-0.930479\pi$
$101$	−318.523	−	980.312i	−0.313804	−	0.965789i	0.662440	−	0.749115i	$-0.269521\pi$
$102$		0			0
$103$	1232.35	−	895.352i	1.17890	−	0.856521i	0.186853	−	0.982388i	$-0.440171\pi$
$103$	1232.35	−	895.352i	1.17890	−	0.856521i	0.992047	+	0.125867i	$0.0401712\pi$
$104$		0			0
$105$	719.286	−	2213.74i	0.668525	−	2.05751i
$106$		0			0
$107$	625.775	+	454.652i	0.565383	+	0.410775i	0.833425	−	0.552633i	$-0.186377\pi$
$107$	625.775	+	454.652i	0.565383	+	0.410775i	−0.268042	+	0.963407i	$0.586377\pi$
$108$		0			0
$109$	403.012			0.354143			0.177072	−	0.984198i	$-0.443338\pi$
$109$	403.012			0.354143			0.177072	+	0.984198i	$0.443338\pi$
$110$		0			0
$111$	1039.26			0.888670
$112$		0			0
$113$	967.999	+	703.293i	0.805856	+	0.585489i	0.912626	−	0.408795i	$-0.134051\pi$
$113$	967.999	+	703.293i	0.805856	+	0.585489i	−0.106770	+	0.994284i	$0.534051\pi$
$114$		0			0
$115$	−703.497	+	2165.14i	−0.570447	+	1.75566i
$116$		0			0
$117$	82.0960	−	59.6462i	0.0648699	−	0.0471307i
$118$		0			0
$119$	−629.293	−	1936.76i	−0.484766	−	1.49196i
$120$		0			0
$121$	1019.19	−	856.040i	0.765736	−	0.643155i
$122$		0			0
$123$	−650.749	−	2002.80i	−0.477041	−	1.46818i
$124$		0			0
$125$	−1002.53	+	728.380i	−0.717351	+	0.521186i
$126$		0			0
$127$	170.598	−	525.046i	0.119198	−	0.366852i	−0.873602	−	0.486641i	$-0.838222\pi$
$127$	170.598	−	525.046i	0.119198	−	0.366852i	0.992799	+	0.119789i	$0.0382218\pi$
$128$		0			0
$129$	1344.82	+	977.069i	0.917867	+	0.666869i
$130$		0			0
$131$	−2719.39			−1.81369			−0.906847	−	0.421460i	$-0.861518\pi$
$131$	−2719.39			−1.81369			−0.906847	+	0.421460i	$0.861518\pi$
$132$		0			0
$133$	−3585.20			−2.33741
$134$		0			0
$135$	1228.93	+	892.871i	0.783478	+	0.569230i
$136$		0			0
$137$	594.730	−	1830.39i	0.370885	−	1.14147i	−0.575328	−	0.817923i	$-0.695126\pi$
$137$	594.730	−	1830.39i	0.370885	−	1.14147i	0.946213	−	0.323544i	$-0.104874\pi$
$138$		0			0
$139$	908.016	−	659.712i	0.554079	−	0.402562i	−0.275208	−	0.961385i	$-0.588747\pi$
$139$	908.016	−	659.712i	0.554079	−	0.402562i	0.829287	+	0.558823i	$0.188747\pi$
$140$		0			0
$141$	−14.8030	−	45.5588i	−0.00884138	−	0.0272110i
$142$		0			0
$143$	−464.130	−	593.766i	−0.271416	−	0.347225i
$144$		0			0
$145$	−509.442	−	1567.90i	−0.291771	−	0.897979i
$146$		0			0
$147$	3667.63	−	2664.69i	2.05783	−	1.49510i
$148$		0			0
$149$	682.011	−	2099.01i	0.374983	−	1.15408i	−0.568507	−	0.822678i	$-0.692479\pi$
$149$	682.011	−	2099.01i	0.374983	−	1.15408i	0.943490	−	0.331401i	$-0.107521\pi$
$150$		0			0
$151$	2738.16	+	1989.39i	1.47569	+	1.07215i	0.978915	+	0.204268i	$0.0654814\pi$
$151$	2738.16	+	1989.39i	1.47569	+	1.07215i	0.496772	+	0.867881i	$0.334519\pi$
$152$		0			0
$153$	−295.570			−0.156179
$154$		0			0
$155$	−1091.18			−0.565456
$156$		0			0
$157$	1070.89	+	778.045i	0.544370	+	0.395508i	0.825705	−	0.564101i	$-0.190777\pi$
$157$	1070.89	+	778.045i	0.544370	+	0.395508i	−0.281335	+	0.959610i	$0.590777\pi$
$158$		0			0
$159$	383.859	−	1181.40i	0.191459	−	0.589250i
$160$		0			0
$161$	−5120.30	+	3720.11i	−2.50643	+	1.82103i
$162$		0			0
$163$	1033.88	+	3181.96i	0.496808	+	1.52902i	0.814119	+	0.580698i	$0.197220\pi$
$163$	1033.88	+	3181.96i	0.496808	+	1.52902i	−0.317311	+	0.948322i	$0.602780\pi$
$164$		0			0
$165$	−1402.54	+	2080.43i	−0.661745	+	0.981585i
$166$		0			0
$167$	497.523	+	1531.22i	0.230536	+	0.709517i	0.997682	+	0.0680446i	$0.0216760\pi$
$167$	497.523	+	1531.22i	0.230536	+	0.709517i	−0.767146	+	0.641472i	$0.778324\pi$
$168$		0			0
$169$	1432.18	−	1040.54i	0.651880	−	0.473619i
$170$		0			0
$171$	−160.800	+	494.891i	−0.0719103	+	0.221317i
$172$		0			0
$173$	−1433.45	−	1041.46i	−0.629959	−	0.457692i	0.226427	−	0.974028i	$-0.427295\pi$
$173$	−1433.45	−	1041.46i	−0.629959	−	0.457692i	−0.856386	+	0.516336i	$0.827295\pi$
$174$		0			0
$175$	785.605			0.339349
$176$		0			0
$177$	−1471.76			−0.624997
$178$		0			0
$179$	−904.917	−	657.461i	−0.377858	−	0.274530i	0.382604	−	0.923913i	$-0.375028\pi$
$179$	−904.917	−	657.461i	−0.377858	−	0.274530i	−0.760462	+	0.649382i	$0.775028\pi$
$180$		0			0
$181$	−800.805	+	2464.62i	−0.328858	+	1.01212i	0.640810	+	0.767699i	$0.278598\pi$
$181$	−800.805	+	2464.62i	−0.328858	+	1.01212i	−0.969669	+	0.244423i	$0.921402\pi$
$182$		0			0
$183$	−2353.15	+	1709.66i	−0.950546	+	0.690612i
$184$		0			0
$185$	692.101	+	2130.07i	0.275050	+	0.846517i
$186$		0			0
$187$	77.1354	+	2193.78i	0.0301642	+	0.857886i
$188$		0			0
$189$	1305.00	+	4016.37i	0.502247	+	1.54576i
$190$		0			0
$191$	1057.53	−	768.341i	0.400629	−	0.291074i	−0.369168	−	0.929363i	$-0.620357\pi$
$191$	1057.53	−	768.341i	0.400629	−	0.291074i	0.769797	+	0.638288i	$0.220357\pi$
$192$		0			0
$193$	628.268	−	1933.61i	0.234320	−	0.721163i	−0.762891	−	0.646527i	$-0.776221\pi$
$193$	628.268	−	1933.61i	0.234320	−	0.721163i	0.997211	−	0.0746355i	$-0.0237793\pi$
$194$		0			0
$195$	1149.35	+	835.054i	0.422086	+	0.306664i
$196$		0			0
$197$	206.856			0.0748115			0.0374057	−	0.999300i	$-0.488091\pi$
$197$	206.856			0.0748115			0.0374057	+	0.999300i	$0.488091\pi$
$198$		0			0
$199$	−4390.77			−1.56409			−0.782044	−	0.623224i	$-0.785823\pi$
$199$	−4390.77			−1.56409			−0.782044	+	0.623224i	$0.785823\pi$
$200$		0			0
$201$	−1202.15	−	873.416i	−0.421858	−	0.306498i
$202$		0			0
$203$	1416.29	−	4358.89i	0.489675	−	1.50707i
$204$		0			0
$205$	3671.56	−	2667.54i	1.25089	−	0.908826i
$206$		0			0
$207$	283.864	+	873.643i	0.0953135	+	0.293345i
$208$		0			0
$209$	3715.13	+	1064.33i	1.22958	+	0.352256i
$210$		0			0
$211$	−1068.19	−	3287.56i	−0.348518	−	1.07263i	−0.959673	−	0.281118i	$-0.909295\pi$
$211$	−1068.19	−	3287.56i	−0.348518	−	1.07263i	0.611155	−	0.791511i	$-0.290705\pi$
$212$		0			0
$213$	−1414.17	+	1027.46i	−0.454917	+	0.330517i
$214$		0			0
$215$	−1107.01	+	3407.02i	−0.351150	+	1.08073i
$216$		0			0
$217$	−2454.21	−	1783.09i	−0.767754	−	0.557806i
$218$		0			0
$219$	1191.58			0.367670
$220$		0			0
$221$	1242.93			0.378319
$222$		0			0
$223$	737.404	+	535.756i	0.221436	+	0.160883i	0.692973	−	0.720964i	$-0.256301\pi$
$223$	737.404	+	535.756i	0.221436	+	0.160883i	−0.471537	+	0.881846i	$0.656301\pi$
$224$		0			0
$225$	35.2352	−	108.443i	0.0104401	−	0.0321312i
$226$		0			0
$227$	181.762	−	132.058i	0.0531451	−	0.0386122i	−0.560896	−	0.827887i	$-0.689543\pi$
$227$	181.762	−	132.058i	0.0531451	−	0.0386122i	0.614041	+	0.789274i	$0.289543\pi$
$228$		0			0
$229$	−1574.52	−	4845.86i	−0.454353	−	1.39836i	−0.871893	−	0.489697i	$-0.837107\pi$
$229$	−1574.52	−	4845.86i	−0.454353	−	1.39836i	0.417539	−	0.908659i	$-0.362893\pi$
$230$		0			0
$231$	−6554.13	+	2387.29i	−1.86680	+	0.679966i
$232$		0			0
$233$	107.964	+	332.278i	0.0303560	+	0.0934261i	0.965087	−	0.261931i	$-0.0843593\pi$
$233$	107.964	+	332.278i	0.0303560	+	0.0934261i	−0.934731	+	0.355357i	$0.884359\pi$
$234$		0			0
$235$	83.5191	−	60.6802i	0.0231838	−	0.0168440i
$236$		0			0
$237$	1630.52	−	5018.24i	0.446894	−	1.37540i
$238$		0			0
$239$	−1720.16	−	1249.77i	−0.465556	−	0.338246i	0.330151	−	0.943928i	$-0.392900\pi$
$239$	−1720.16	−	1249.77i	−0.465556	−	0.338246i	−0.795707	+	0.605682i	$0.792900\pi$
$240$		0			0
$241$	−2080.20			−0.556007			−0.278004	−	0.960580i	$-0.589673\pi$
$241$	−2080.20			−0.556007			−0.278004	+	0.960580i	$0.589673\pi$
$242$		0			0
$243$	1362.20			0.359610
$244$		0			0
$245$	7904.01	+	5742.60i	2.06110	+	1.49748i
$246$		0			0
$247$	676.194	−	2081.11i	0.174191	−	0.536105i
$248$		0			0
$249$	−1088.32	+	790.713i	−0.276986	+	0.201242i
$250$		0			0
$251$	−1410.50	−	4341.06i	−0.354700	−	1.09165i	−0.956183	−	0.292769i	$-0.905423\pi$
$251$	−1410.50	−	4341.06i	−0.354700	−	1.09165i	0.601483	−	0.798885i	$-0.294577\pi$
$252$		0			0
$253$	6410.26	−	2334.89i	1.59292	−	0.580210i
$254$		0			0
$255$	−1278.72	−	3935.49i	−0.314025	−	0.966469i
$256$		0			0
$257$	1937.75	−	1407.86i	0.470324	−	0.341711i	−0.327243	−	0.944940i	$-0.606120\pi$
$257$	1937.75	−	1407.86i	0.470324	−	0.341711i	0.797568	+	0.603229i	$0.206120\pi$
$258$		0			0
$259$	−1924.10	+	5921.76i	−0.461612	+	1.42070i
$260$		0			0
$261$	−538.168	−	391.002i	−0.127631	−	0.0927295i
$262$		0			0
$263$	4960.41			1.16301			0.581505	−	0.813543i	$-0.302464\pi$
$263$	4960.41			1.16301			0.581505	+	0.813543i	$0.302464\pi$
$264$		0			0
$265$	2677.01			0.620557
$266$		0			0
$267$	−2837.01	−	2061.21i	−0.650271	−	0.472449i
$268$		0			0
$269$	255.831	−	787.367i	0.0579862	−	0.178463i	−0.917868	−	0.396885i	$-0.870091\pi$
$269$	255.831	−	787.367i	0.0579862	−	0.178463i	0.975854	+	0.218422i	$0.0700910\pi$
$270$		0			0
$271$	−1085.93	+	788.976i	−0.243416	+	0.176852i	−0.702804	−	0.711384i	$-0.748069\pi$
$271$	−1085.93	+	788.976i	−0.243416	+	0.176852i	0.459388	+	0.888236i	$0.348069\pi$
$272$		0			0
$273$	1220.49	+	3756.30i	0.270578	+	0.832752i
$274$		0			0
$275$	−814.077	−	233.222i	−0.178512	−	0.0511410i
$276$		0			0
$277$	−2346.73	−	7222.51i	−0.509031	−	1.56664i	−0.793887	−	0.608066i	$-0.791946\pi$
$277$	−2346.73	−	7222.51i	−0.509031	−	1.56664i	0.284856	−	0.958570i	$-0.408054\pi$
$278$		0			0
$279$	−356.206	+	258.799i	−0.0764355	+	0.0555336i
$280$		0			0
$281$	−1863.97	+	5736.71i	−0.395712	+	1.21788i	0.532693	+	0.846309i	$0.321180\pi$
$281$	−1863.97	+	5736.71i	−0.395712	+	1.21788i	−0.928405	+	0.371569i	$0.878820\pi$
$282$		0			0
$283$	−1507.05	−	1094.93i	−0.316554	−	0.229990i	0.418150	−	0.908378i	$-0.362679\pi$
$283$	−1507.05	−	1094.93i	−0.316554	−	0.229990i	−0.734704	+	0.678388i	$0.762679\pi$
$284$		0			0
$285$	−7285.08			−1.51414
$286$		0			0
$287$	12616.8			2.59494
$288$		0			0
$289$	1045.83	+	759.838i	0.212869	+	0.154659i
$290$		0			0
$291$	−2102.20	+	6469.91i	−0.423482	+	1.30334i
$292$		0			0
$293$	3136.46	−	2278.77i	0.625371	−	0.454359i	−0.229423	−	0.973327i	$-0.573684\pi$
$293$	3136.46	−	2278.77i	0.625371	−	0.454359i	0.854793	+	0.518968i	$0.173684\pi$
$294$		0			0
$295$	−980.126	−	3016.52i	−0.193441	−	0.595350i
$296$		0			0
$297$	−159.960	−	4549.35i	−0.0312519	−	0.888822i
$298$		0			0
$299$	−1193.70	−	3673.84i	−0.230882	−	0.710581i
$300$		0			0
$301$	−8057.19	+	5853.89i	−1.54289	+	1.12097i
$302$		0			0
$303$	1799.37	−	5537.88i	0.341158	−	1.04998i
$304$		0			0
$305$	−5071.21	−	3684.45i	−0.952055	−	0.691708i
$306$		0			0
$307$	3787.90			0.704192			0.352096	−	0.935964i	$-0.385469\pi$
$307$	3787.90			0.704192			0.352096	+	0.935964i	$0.385469\pi$
$308$		0			0
$309$	8605.07			1.58423
$310$		0			0
$311$	−5305.29	−	3854.52i	−0.967317	−	0.702797i	−0.0124785	−	0.999922i	$-0.503972\pi$
$311$	−5305.29	−	3854.52i	−0.967317	−	0.702797i	−0.954839	+	0.297125i	$0.903972\pi$
$312$		0			0
$313$	−2691.42	+	8283.34i	−0.486032	+	1.49585i	0.344448	+	0.938805i	$0.388066\pi$
$313$	−2691.42	+	8283.34i	−0.486032	+	1.49585i	−0.830480	+	0.557048i	$0.811934\pi$
$314$		0			0
$315$	1637.52	−	1189.73i	0.292901	−	0.212805i
$316$		0			0
$317$	−1037.83	−	3194.11i	−0.183881	−	0.565928i	0.816046	−	0.577987i	$-0.196161\pi$
$317$	−1037.83	−	3194.11i	−0.183881	−	0.565928i	−0.999927	+	0.0120587i	$0.996161\pi$
$318$		0			0
$319$	−2761.64	+	4096.42i	−0.484709	+	0.718983i
$320$		0			0
$321$	1350.28	+	4155.72i	0.234782	+	0.722585i
$322$		0			0
$323$	−5156.36	+	3746.31i	−0.888258	+	0.645357i
$324$		0			0
$325$	−148.171	+	456.023i	−0.0252894	+	0.0778326i
$326$		0			0
$327$	1841.85	+	1338.19i	0.311483	+	0.226305i
$328$		0			0
$329$	287.003			0.0480941
$330$		0			0
$331$	−3380.77			−0.561402			−0.280701	−	0.959795i	$-0.590567\pi$
$331$	−3380.77			−0.561402			−0.280701	+	0.959795i	$0.590567\pi$
$332$		0			0
$333$	731.126	+	531.194i	0.120317	+	0.0874152i
$334$		0			0
$335$	989.570	−	3045.58i	0.161391	−	0.496711i
$336$		0			0
$337$	−304.122	+	220.957i	−0.0491589	+	0.0357161i	−0.612093	−	0.790785i	$-0.709672\pi$
$337$	−304.122	+	220.957i	−0.0491589	+	0.0357161i	0.562934	+	0.826502i	$0.309672\pi$
$338$		0			0
$339$	2088.72	+	6428.40i	0.334641	+	1.02992i
$340$		0			0
$341$	2013.81	+	2576.29i	0.319807	+	0.409132i
$342$		0			0
$343$	4805.89	+	14791.0i	0.756541	+	2.32839i
$344$		0			0
$345$	−10404.4	+	7559.23i	−1.62363	+	1.17964i
$346$		0			0
$347$	−3090.54	+	9511.71i	−0.478124	+	1.47151i	0.363574	+	0.931565i	$0.381557\pi$
$347$	−3090.54	+	9511.71i	−0.478124	+	1.47151i	−0.841698	+	0.539949i	$0.818443\pi$
$348$		0			0
$349$	4064.35	+	2952.92i	0.623380	+	0.452912i	0.854101	−	0.520108i	$-0.174108\pi$
$349$	4064.35	+	2952.92i	0.623380	+	0.452912i	−0.230720	+	0.973020i	$0.574108\pi$
$350$		0			0
$351$	−2577.53			−0.391962
$352$		0			0
$353$	−3533.41			−0.532760			−0.266380	−	0.963868i	$-0.585828\pi$
$353$	−3533.41			−0.532760			−0.266380	+	0.963868i	$0.585828\pi$
$354$		0			0
$355$	−3047.64	−	2214.24i	−0.455639	−	0.331041i
$356$		0			0
$357$	3554.94	−	10941.0i	0.527023	−	1.62201i
$358$		0			0
$359$	1806.59	−	1312.57i	0.265594	−	0.192965i	−0.447016	−	0.894526i	$-0.647513\pi$
$359$	1806.59	−	1312.57i	0.265594	−	0.192965i	0.712610	+	0.701561i	$0.247513\pi$
$360$		0			0
$361$	1347.90	+	4148.42i	0.196516	+	0.604814i
$362$		0			0
$363$	7500.39	−	528.096i	1.08449	−	0.0763577i
$364$		0			0
$365$	793.539	+	2442.26i	0.113797	+	0.350230i
$366$		0			0
$367$	10883.1	−	7907.06i	1.54794	−	1.12465i	0.602844	−	0.797859i	$-0.294034\pi$
$367$	10883.1	−	7907.06i	1.54794	−	1.12465i	0.945098	−	0.326787i	$-0.105966\pi$
$368$		0			0
$369$	565.879	−	1741.60i	0.0798332	−	0.245701i
$370$		0			0
$371$	6020.97	+	4374.49i	0.842569	+	0.612162i
$372$		0			0
$373$	−939.175			−0.130372			−0.0651859	−	0.997873i	$-0.520764\pi$
$373$	−939.175			−0.130372			−0.0651859	+	0.997873i	$0.520764\pi$
$374$		0			0
$375$	−7000.33			−0.963988
$376$		0			0
$377$	2263.10	+	1644.24i	0.309166	+	0.224622i
$378$		0			0
$379$	77.9211	−	239.817i	0.0105608	−	0.0325028i	−0.945637	−	0.325223i	$-0.894561\pi$
$379$	77.9211	−	239.817i	0.0105608	−	0.0325028i	0.956198	+	0.292720i	$0.0945605\pi$
$380$		0			0
$381$	2523.06	−	1833.11i	0.339266	−	0.246491i
$382$		0			0
$383$	−1522.94	−	4687.13i	−0.203182	−	0.625329i	−0.999783	−	0.0208240i	$-0.993371\pi$
$383$	−1522.94	−	4687.13i	−0.203182	−	0.625329i	0.796602	−	0.604505i	$-0.206629\pi$
$384$		0			0
$385$	−9257.73	−	11843.5i	−1.22550	−	1.56779i
$386$		0			0
$387$	446.682	+	1374.75i	0.0586722	+	0.180574i
$388$		0			0
$389$	−711.690	+	517.073i	−0.0927612	+	0.0673950i	−0.633199	−	0.773989i	$-0.718259\pi$
$389$	−711.690	+	517.073i	−0.0927612	+	0.0673950i	0.540438	+	0.841384i	$0.318259\pi$
$390$		0			0
$391$	−3476.90	+	10700.8i	−0.449705	+	1.38405i
$392$		0			0
$393$	−12428.2	−	9029.61i	−1.59521	−	1.15899i
$394$		0			0
$395$	11371.2			1.44848
$396$		0			0
$397$	−9864.12			−1.24702			−0.623509	−	0.781816i	$-0.714293\pi$
$397$	−9864.12			−1.24702			−0.623509	+	0.781816i	$0.714293\pi$
$398$		0			0
$399$	−16385.1	−	11904.5i	−2.05585	−	1.49366i
$400$		0			0
$401$	875.773	−	2695.35i	0.109062	−	0.335659i	−0.881600	−	0.471997i	$-0.843533\pi$
$401$	875.773	−	2695.35i	0.109062	−	0.335659i	0.990662	+	0.136338i	$0.0435332\pi$
$402$		0			0
$403$	1497.92	−	1088.30i	0.185153	−	0.134521i
$404$		0			0
$405$	3150.72	+	9696.91i	0.386569	+	1.18974i
$406$		0			0
$407$	3751.82	−	5565.18i	0.456931	−	0.677778i
$408$		0			0
$409$	2919.05	+	8983.90i	0.352904	+	1.08613i	0.957215	+	0.289377i	$0.0934482\pi$
$409$	2919.05	+	8983.90i	0.352904	+	1.08613i	−0.604312	+	0.796748i	$0.706552\pi$
$410$		0			0
$411$	8795.79	−	6390.52i	1.05563	−	0.766961i
$412$		0			0
$413$	2724.83	−	8386.17i	0.324649	−	0.999168i
$414$		0			0
$415$	−2345.41	−	1704.04i	−0.277426	−	0.201562i
$416$		0			0
$417$	6340.38			0.744580
$418$		0			0
$419$	10842.0			1.26411			0.632057	−	0.774921i	$-0.282211\pi$
$419$	10842.0			1.26411			0.632057	+	0.774921i	$0.282211\pi$
$420$		0			0
$421$	−13683.9	−	9941.91i	−1.58411	−	1.15092i	−0.911785	−	0.410668i	$-0.865295\pi$
$421$	−13683.9	−	9941.91i	−1.58411	−	1.15092i	−0.672326	−	0.740255i	$-0.734705\pi$
$422$		0			0
$423$	12.8724	−	39.6171i	0.00147961	−	0.00455378i
$424$		0			0
$425$	1129.89	−	820.910i	0.128959	−	0.0936940i
$426$		0			0
$427$	−5385.10	−	16573.6i	−0.610313	−	1.87835i
$428$		0			0
$429$	−149.602	−	4254.76i	−0.0168365	−	0.478839i
$430$		0			0
$431$	3069.95	+	9448.33i	0.343096	+	1.05594i	0.962595	+	0.270943i	$0.0873354\pi$
$431$	3069.95	+	9448.33i	0.343096	+	1.05594i	−0.619500	+	0.784997i	$0.712665\pi$
$432$		0			0
$433$	−5558.38	+	4038.40i	−0.616903	+	0.448206i	−0.851838	−	0.523805i	$-0.824512\pi$
$433$	−5558.38	+	4038.40i	−0.616903	+	0.448206i	0.234936	+	0.972011i	$0.424512\pi$
$434$		0			0
$435$	2877.89	−	8857.23i	0.317205	−	0.976256i
$436$		0			0
$437$	16025.5	+	11643.2i	1.75424	+	1.27453i
$438$		0			0
$439$	7685.97			0.835606			0.417803	−	0.908538i	$-0.362800\pi$
$439$	7685.97			0.835606			0.417803	+	0.908538i	$0.362800\pi$
$440$		0			0
$441$	3942.19			0.425677
$442$		0			0
$443$	−6498.22	−	4721.23i	−0.696929	−	0.506349i	0.182002	−	0.983298i	$-0.441742\pi$
$443$	−6498.22	−	4721.23i	−0.696929	−	0.506349i	−0.878931	+	0.476950i	$0.841742\pi$
$444$		0			0
$445$	2335.33	−	7187.39i	0.248775	−	0.765652i
$446$		0			0
$447$	10086.6	−	7328.36i	1.06729	−	0.775435i
$448$		0			0
$449$	934.240	+	2875.29i	0.0981949	+	0.302213i	0.988073	−	0.153985i	$-0.0492109\pi$
$449$	934.240	+	2875.29i	0.0981949	+	0.302213i	−0.889878	+	0.456198i	$0.849211\pi$
$450$		0			0
$451$	−13074.1	−	3745.55i	−1.36505	−	0.391066i
$452$		0			0
$453$	5908.32	+	18183.9i	0.612797	+	1.88599i
$454$		0			0
$455$	−6886.09	+	5003.04i	−0.709506	+	0.515486i
$456$		0			0
$457$	2322.41	−	7147.63i	0.237719	−	0.731624i	−0.759030	−	0.651056i	$-0.774326\pi$
$457$	2322.41	−	7147.63i	0.237719	−	0.731624i	0.996749	−	0.0805683i	$-0.0256735\pi$
$458$		0			0
$459$	6073.77	+	4412.85i	0.617645	+	0.448746i
$460$		0			0
$461$	4085.19			0.412725			0.206363	−	0.978476i	$-0.433837\pi$
$461$	4085.19			0.412725			0.206363	+	0.978476i	$0.433837\pi$
$462$		0			0
$463$	−970.931			−0.0974579			−0.0487290	−	0.998812i	$-0.515517\pi$
$463$	−970.931			−0.0974579			−0.0487290	+	0.998812i	$0.515517\pi$
$464$		0			0
$465$	−4986.93	−	3623.21i	−0.497340	−	0.361339i
$466$		0			0
$467$	−3193.85	+	9829.67i	−0.316475	+	0.974011i	0.658668	+	0.752434i	$0.271120\pi$
$467$	−3193.85	+	9829.67i	−0.316475	+	0.974011i	−0.975143	+	0.221577i	$0.928880\pi$
$468$		0			0
$469$	7202.44	−	5232.88i	0.709121	−	0.515207i
$470$		0			0
$471$	2310.72	+	7111.67i	0.226056	+	0.695730i
$472$		0			0
$473$	10087.1	−	3674.13i	0.980556	−	0.357160i
$474$		0			0
$475$	−759.804	−	2338.44i	−0.0733941	−	0.225884i
$476$		0			0
$477$	873.889	−	634.918i	0.0838839	−	0.0609452i
$478$		0			0
$479$	−194.181	+	597.629i	−0.0185227	+	0.0570070i	−0.959891	−	0.280375i	$-0.909541\pi$
$479$	−194.181	+	597.629i	−0.0185227	+	0.0570070i	0.941368	+	0.337382i	$0.109541\pi$
$480$		0			0
$481$	−3074.53	−	2233.78i	−0.291448	−	0.211749i
$482$		0			0
$483$	−35753.4			−3.36819
$484$		0			0
$485$	−14660.7			−1.37259
$486$		0			0
$487$	14233.9	+	10341.5i	1.32443	+	0.962255i	0.999866	+	0.0163922i	$0.00521804\pi$
$487$	14233.9	+	10341.5i	1.32443	+	0.962255i	0.324565	+	0.945863i	$0.394782\pi$
$488$		0			0
$489$	−5840.50	+	17975.2i	−0.540115	+	1.66230i
$490$		0			0
$491$	−765.687	+	556.304i	−0.0703767	+	0.0511317i	−0.622417	−	0.782686i	$-0.713849\pi$
$491$	−765.687	+	556.304i	−0.0703767	+	0.0511317i	0.552041	+	0.833817i	$0.313849\pi$
$492$		0			0
$493$	−2517.82	−	7749.05i	−0.230014	−	0.707911i
$494$		0			0
$495$	−2050.06	+	746.719i	−0.186148	+	0.0678030i
$496$		0			0
$497$	−3236.28	−	9960.25i	−0.292087	−	0.898950i
$498$		0			0
$499$	−13523.7	+	9825.58i	−1.21324	+	0.881469i	−0.995521	−	0.0945424i	$-0.969861\pi$
$499$	−13523.7	+	9825.58i	−1.21324	+	0.881469i	−0.217718	+	0.976012i	$0.569861\pi$
$500$		0			0
$501$	−2810.56	+	8650.01i	−0.250632	+	0.771365i
$502$		0			0
$503$	1315.06	+	955.444i	0.116571	+	0.0846941i	0.644544	−	0.764568i	$-0.277047\pi$
$503$	1315.06	+	955.444i	0.116571	+	0.0846941i	−0.527972	+	0.849262i	$0.677047\pi$
$504$		0			0
$505$	12548.7			1.10576
$506$		0			0
$507$	10000.4			0.876007
$508$		0			0
$509$	−8241.22	−	5987.60i	−0.717654	−	0.521406i	0.167980	−	0.985790i	$-0.446276\pi$
$509$	−8241.22	−	5987.60i	−0.717654	−	0.521406i	−0.885634	+	0.464385i	$0.846276\pi$
$510$		0			0
$511$	−2206.10	+	6789.69i	−0.190983	+	0.587785i
$512$		0			0
$513$	10693.0	−	7768.94i	0.920290	−	0.668629i
$514$		0			0
$515$	5730.58	+	17636.9i	0.490329	+	1.50908i
$516$		0			0
$517$	−297.405	−	85.2021i	−0.0252995	−	0.00724794i
$518$		0			0
$519$	−3093.04	−	9519.39i	−0.261598	−	0.805115i
$520$		0			0
$521$	9802.50	−	7121.94i	0.824291	−	0.598882i	−0.0936475	−	0.995605i	$-0.529853\pi$
$521$	9802.50	−	7121.94i	0.824291	−	0.598882i	0.917938	+	0.396723i	$0.129853\pi$
$522$		0			0
$523$	2969.13	−	9138.05i	0.248243	−	0.764013i	−0.746843	−	0.665000i	$-0.768431\pi$
$523$	2969.13	−	9138.05i	0.248243	−	0.764013i	0.995086	−	0.0990131i	$-0.0315686\pi$
$524$		0			0
$525$	3590.39	+	2608.57i	0.298471	+	0.216852i
$526$		0			0
$527$	−5392.95			−0.445770
$528$		0			0
$529$	22801.5			1.87405
$530$		0			0
$531$	−1035.39	−	752.256i	−0.0846181	−	0.0614786i
$532$		0			0
$533$	−2379.63	+	7323.75i	−0.193383	+	0.595172i
$534$		0			0
$535$	−7618.32	+	5535.04i	−0.615643	+	0.447291i
$536$		0			0
$537$	−1952.60	−	6009.48i	−0.156910	−	0.482920i
$538$		0			0
$539$	−1028.80	−	29259.7i	−0.0822145	−	2.33823i
$540$		0			0
$541$	2366.58	+	7283.59i	0.188073	+	0.578828i	0.999988	−	0.00494752i	$-0.00157485\pi$
$541$	2366.58	+	7283.59i	0.188073	+	0.578828i	−0.811915	+	0.583775i	$0.801575\pi$
$542$		0			0
$543$	−11843.5	+	8604.83i	−0.936012	+	0.680053i
$544$		0			0
$545$	−1516.15	+	4666.23i	−0.119165	+	0.366751i
$546$		0			0
$547$	1459.52	+	1060.40i	0.114085	+	0.0828877i	0.643365	−	0.765560i	$-0.277538\pi$
$547$	1459.52	+	1060.40i	0.114085	+	0.0828877i	−0.529280	+	0.848447i	$0.677538\pi$
$548$		0			0
$549$	−2529.31			−0.196627
$550$		0			0
$551$	−14344.5			−1.10907
$552$		0			0
$553$	25575.4	+	18581.6i	1.96668	+	1.42888i
$554$		0			0
$555$	−3909.75	+	12033.0i	−0.299026	+	0.920308i
$556$		0			0
$557$	−7180.49	+	5216.93i	−0.546225	+	0.396856i	−0.826392	−	0.563096i	$-0.809610\pi$
$557$	−7180.49	+	5216.93i	−0.546225	+	0.396856i	0.280167	+	0.959951i	$0.409610\pi$
$558$		0			0
$559$	−1878.39	−	5781.08i	−0.142124	−	0.437412i
$560$		0			0
$561$	−6931.82	+	10282.2i	−0.521678	+	0.773820i
$562$		0			0
$563$	−1051.54	−	3236.30i	−0.0787159	−	0.242263i	0.903953	−	0.427632i	$-0.140652\pi$
$563$	−1051.54	−	3236.30i	−0.0787159	−	0.242263i	−0.982669	+	0.185369i	$0.940652\pi$
$564$		0			0
$565$	−11784.6	+	8562.04i	−0.877493	+	0.637536i
$566$		0			0
$567$	−8759.26	+	26958.2i	−0.648773	+	1.99672i
$568$		0			0
$569$	−14255.4	−	10357.1i	−1.05029	−	0.763083i	−0.0780255	−	0.996951i	$-0.524862\pi$
$569$	−14255.4	−	10357.1i	−1.05029	−	0.763083i	−0.972268	+	0.233868i	$0.924862\pi$
$570$		0			0
$571$	9250.73			0.677988			0.338994	−	0.940789i	$-0.389913\pi$
$571$	9250.73			0.677988			0.338994	+	0.940789i	$0.389913\pi$
$572$		0			0
$573$	7384.39			0.538372
$574$		0			0
$575$	−3511.57	−	2551.31i	−0.254683	−	0.185038i
$576$		0			0
$577$	−7065.29	+	21744.7i	−0.509761	+	1.56888i	0.282856	+	0.959162i	$0.408718\pi$
$577$	−7065.29	+	21744.7i	−0.509761	+	1.56888i	−0.792617	+	0.609720i	$0.791282\pi$
$578$		0			0
$579$	9291.80	−	6750.89i	0.666933	−	0.484555i
$580$		0			0
$581$	−2490.59	−	7665.24i	−0.177843	−	0.547346i
$582$		0			0
$583$	−4940.54	−	6320.47i	−0.350971	−	0.449000i
$584$		0			0
$585$	381.758	+	1174.93i	0.0269807	+	0.0830382i
$586$		0			0
$587$	−7162.81	+	5204.09i	−0.503647	+	0.365921i	−0.810408	−	0.585865i	$-0.800755\pi$
$587$	−7162.81	+	5204.09i	−0.503647	+	0.365921i	0.306761	+	0.951787i	$0.400755\pi$
$588$		0			0
$589$	−2933.94	+	9029.74i	−0.205248	+	0.631687i
$590$		0			0
$591$	945.376	+	686.856i	0.0657996	+	0.0478062i
$592$		0			0
$593$	12748.1			0.882801			0.441401	−	0.897310i	$-0.354482\pi$
$593$	12748.1			0.882801			0.441401	+	0.897310i	$0.354482\pi$
$594$		0			0
$595$	24792.0			1.70819
$596$		0			0
$597$	−20066.8	−	14579.4i	−1.37568	−	0.999487i
$598$		0			0
$599$	4042.90	−	12442.8i	0.275773	−	0.848743i	−0.713240	−	0.700920i	$-0.752773\pi$
$599$	4042.90	−	12442.8i	0.275773	−	0.848743i	0.989014	−	0.147824i	$-0.0472269\pi$
$600$		0			0
$601$	6511.48	−	4730.87i	0.441945	−	0.321092i	−0.344462	−	0.938800i	$-0.611939\pi$
$601$	6511.48	−	4730.87i	0.441945	−	0.321092i	0.786407	+	0.617708i	$0.211939\pi$
$602$		0			0
$603$	−399.296	−	1228.91i	−0.0269661	−	0.0829932i
$604$		0			0
$605$	6077.29	+	15021.1i	0.408392	+	1.00941i
$606$		0			0
$607$	−3374.16	−	10384.6i	−0.225622	−	0.694394i	−0.998228	−	0.0595079i	$-0.981047\pi$
$607$	−3374.16	−	10384.6i	−0.225622	−	0.694394i	0.772605	−	0.634887i	$-0.218953\pi$
$608$		0			0
$609$	20946.3	−	15218.4i	1.39374	−	1.01261i
$610$		0			0
$611$	−54.1308	+	166.598i	−0.00358412	+	0.0110308i
$612$		0			0
$613$	−10058.7	−	7308.07i	−0.662752	−	0.481517i	0.204839	−	0.978796i	$-0.434333\pi$
$613$	−10058.7	−	7308.07i	−0.662752	−	0.481517i	−0.867591	+	0.497278i	$0.834333\pi$
$614$		0			0
$615$	25637.3			1.68097
$616$		0			0
$617$	−25435.2			−1.65961			−0.829807	−	0.558051i	$-0.811549\pi$
$617$	−25435.2			−1.65961			−0.829807	+	0.558051i	$0.811549\pi$
$618$		0			0
$619$	15148.9	+	11006.3i	0.983660	+	0.714671i	0.958524	−	0.285013i	$-0.0919981\pi$
$619$	15148.9	+	11006.3i	0.983660	+	0.714671i	0.0251362	+	0.999684i	$0.491998\pi$
$620$		0			0
$621$	7210.24	−	22190.9i	0.465921	−	1.43396i
$622$		0			0
$623$	16997.3	−	12349.3i	1.09307	−	0.794163i
$624$		0			0
$625$	−5558.50	−	17107.3i	−0.355744	−	1.09487i
$626$		0			0
$627$	13444.9	+	17200.2i	0.856360	+	1.09555i
$628$		0			0
$629$	3420.58	+	10527.5i	0.216832	+	0.667341i
$630$		0			0
$631$	7931.28	−	5762.41i	0.500379	−	0.363547i	−0.308783	−	0.951133i	$-0.599922\pi$
$631$	7931.28	−	5762.41i	0.500379	−	0.363547i	0.809162	+	0.587586i	$0.199922\pi$
$632$		0			0
$633$	6034.33	−	18571.7i	0.378899	−	1.16613i
$634$		0			0
$635$	5437.38	+	3950.49i	0.339804	+	0.246882i
$636$		0			0
$637$	−16577.7			−1.03113
$638$		0			0
$639$	−1520.04			−0.0941028
$640$		0			0
$641$	−23470.8	−	17052.6i	−1.44624	−	1.05076i	−0.986691	−	0.162607i	$-0.948010\pi$
$641$	−23470.8	−	17052.6i	−1.44624	−	1.05076i	−0.459553	−	0.888150i	$-0.651990\pi$
$642$		0			0
$643$	4432.05	−	13640.5i	0.271824	−	0.836590i	−0.718218	−	0.695818i	$-0.755042\pi$
$643$	4432.05	−	13640.5i	0.271824	−	0.836590i	0.990042	−	0.140771i	$-0.0449582\pi$
$644$		0			0
$645$	−16372.1	+	11895.1i	−0.999461	+	0.726151i
$646$		0			0
$647$	2014.23	+	6199.15i	0.122392	+	0.376683i	0.993417	−	0.114556i	$-0.0365444\pi$
$647$	2014.23	+	6199.15i	0.122392	+	0.376683i	−0.871025	+	0.491238i	$0.836544\pi$
$648$		0			0
$649$	−5313.18	+	7881.19i	−0.321357	+	0.476678i
$650$		0			0
$651$	−5295.60	−	16298.2i	−0.318819	−	0.981224i
$652$		0			0
$653$	−7850.09	+	5703.42i	−0.470441	+	0.341795i	−0.797613	−	0.603170i	$-0.793904\pi$
$653$	−7850.09	+	5703.42i	−0.470441	+	0.341795i	0.327172	+	0.944965i	$0.393904\pi$
$654$		0			0
$655$	10230.4	−	31486.1i	0.610285	−	1.87826i
$656$		0			0
$657$	838.284	+	609.049i	0.0497787	+	0.0361663i
$658$		0			0
$659$	−26277.8			−1.55332			−0.776661	−	0.629919i	$-0.783088\pi$
$659$	−26277.8			−1.55332			−0.776661	+	0.629919i	$0.783088\pi$
$660$		0			0
$661$	17275.4			1.01654			0.508271	−	0.861197i	$-0.330285\pi$
$661$	17275.4			1.01654			0.508271	+	0.861197i	$0.330285\pi$
$662$		0			0
$663$	5680.46	+	4127.10i	0.332746	+	0.241754i
$664$		0			0
$665$	13487.7	−	41510.7i	0.786509	−	2.42063i
$666$		0			0
$667$	−20486.5	+	14884.3i	−1.18926	+	0.864051i
$668$		0			0
$669$	1591.14	+	4897.04i	0.0919540	+	0.283005i
$670$		0			0
$671$	660.078	+	18773.0i	0.0379762	+	1.08006i
$672$		0			0
$673$	1237.78	+	3809.50i	0.0708960	+	0.218195i	0.980226	−	0.197879i	$-0.0634055\pi$
$673$	1237.78	+	3809.50i	0.0708960	+	0.218195i	−0.909330	+	0.416075i	$0.863405\pi$
$674$		0			0
$675$	−2343.11	+	1702.37i	−0.133609	+	0.0970727i
$676$		0			0
$677$	−2317.43	+	7132.33i	−0.131560	+	0.404901i	−0.995039	−	0.0994843i	$-0.968281\pi$
$677$	−2317.43	+	7132.33i	−0.131560	+	0.404901i	0.863479	+	0.504385i	$0.168281\pi$
$678$		0			0
$679$	−32973.8	−	23956.9i	−1.86365	−	1.35402i
$680$		0			0
$681$	1269.18			0.0714173
$682$		0			0
$683$	−3368.22			−0.188699			−0.0943496	−	0.995539i	$-0.530077\pi$
$683$	−3368.22			−0.188699			−0.0943496	+	0.995539i	$0.530077\pi$
$684$		0			0
$685$	18955.6	+	13772.0i	1.05731	+	0.768178i
$686$		0			0
$687$	8894.60	−	27374.8i	0.493959	−	1.52025i
$688$		0			0
$689$	−3674.88	+	2669.95i	−0.203195	+	0.147630i
$690$		0			0
$691$	−4784.92	−	14726.5i	−0.263425	−	0.810739i	−0.992052	−	0.125829i	$-0.959841\pi$
$691$	−4784.92	−	14726.5i	−0.263425	−	0.810739i	0.728627	−	0.684911i	$-0.240159\pi$
$692$		0			0
$693$	−5831.07	−	1670.52i	−0.319631	−	0.0915696i
$694$		0			0
$695$	4222.40	+	12995.2i	0.230453	+	0.709261i
$696$		0			0
$697$	18146.0	−	13183.8i	0.986125	−	0.716462i
$698$		0			0
$699$	−609.898	+	1877.07i	−0.0330021	+	0.101570i
$700$		0			0
$701$	−2355.83	−	1711.61i	−0.126931	−	0.0922206i	0.522508	−	0.852634i	$-0.324996\pi$
$701$	−2355.83	−	1711.61i	−0.126931	−	0.0922206i	−0.649439	+	0.760414i	$0.724996\pi$
$702$		0			0
$703$	19487.7			1.04551
$704$		0			0
$705$	583.187			0.0311547
$706$		0			0
$707$	28223.7	+	20505.7i	1.50136	+	1.09080i
$708$		0			0
$709$	−2476.01	+	7620.37i	−0.131154	+	0.403652i	−0.994972	−	0.100153i	$-0.968067\pi$
$709$	−2476.01	+	7620.37i	−0.131154	+	0.403652i	0.863818	+	0.503805i	$0.168067\pi$
$710$		0			0
$711$	3712.04	−	2696.95i	0.195798	−	0.142255i
$712$		0			0
$713$	5179.36	+	15940.4i	0.272046	+	0.837270i
$714$		0			0
$715$	8620.91	−	3140.10i	0.450914	−	0.164242i
$716$		0			0
$717$	−3711.70	−	11423.4i	−0.193328	−	0.595002i
$718$		0			0
$719$	7830.73	−	5689.36i	0.406171	−	0.295101i	−0.365879	−	0.930663i	$-0.619232\pi$
$719$	7830.73	−	5689.36i	0.406171	−	0.295101i	0.772050	+	0.635562i	$0.219232\pi$
$720$		0			0
$721$	−15931.5	+	49032.1i	−0.822912	+	2.53266i
$722$		0			0
$723$	−9506.99	−	6907.23i	−0.489030	−	0.355301i
$724$		0			0
$725$	3143.23			0.161016
$726$		0			0
$727$	−12376.6			−0.631395			−0.315698	−	0.948860i	$-0.602239\pi$
$727$	−12376.6			−0.631395			−0.315698	+	0.948860i	$0.602239\pi$
$728$		0			0
$729$	−12068.4	−	8768.20i	−0.613138	−	0.445471i
$730$		0			0
$731$	−5471.18	+	16838.6i	−0.276825	+	0.851979i
$732$		0			0
$733$	−5559.86	+	4039.48i	−0.280161	+	0.203549i	−0.718988	−	0.695023i	$-0.755394\pi$
$733$	−5559.86	+	4039.48i	−0.280161	+	0.203549i	0.438826	+	0.898572i	$0.355394\pi$
$734$		0			0
$735$	17055.0	+	52489.9i	0.855896	+	2.63418i
$736$		0			0
$737$	−9016.96	+	3284.36i	−0.450670	+	0.164153i
$738$		0			0
$739$	6178.91	+	19016.7i	0.307571	+	0.946605i	0.978705	+	0.205270i	$0.0658072\pi$
$739$	6178.91	+	19016.7i	0.307571	+	0.946605i	−0.671135	+	0.741335i	$0.734193\pi$
$740$		0			0
$741$	10000.6	−	7265.86i	0.495791	−	0.360213i
$742$		0			0
$743$	−1064.71	+	3276.86i	−0.0525715	+	0.161798i	−0.973895	−	0.226998i	$-0.927109\pi$
$743$	−1064.71	+	3276.86i	−0.0525715	+	0.161798i	0.921324	+	0.388796i	$0.127109\pi$
$744$		0			0
$745$	21737.4	+	15793.1i	1.06899	+	0.776666i
$746$		0			0
$747$	−1169.79			−0.0572966
$748$		0			0
$749$	−26179.4			−1.27713
$750$		0			0
$751$	−21795.3	−	15835.2i	−1.05902	−	0.769420i	−0.0851097	−	0.996372i	$-0.527124\pi$
$751$	−21795.3	−	15835.2i	−1.05902	−	0.769420i	−0.973906	+	0.226952i	$0.927124\pi$
$752$		0			0
$753$	7968.03	−	24523.1i	0.385619	−	1.18681i
$754$		0			0
$755$	−33335.0	+	24219.3i	−1.60687	+	1.16746i
$756$		0			0
$757$	2368.39	+	7289.16i	0.113713	+	0.349972i	0.991676	−	0.128755i	$-0.0410982\pi$
$757$	2368.39	+	7289.16i	0.113713	+	0.349972i	−0.877963	+	0.478728i	$0.841098\pi$
$758$		0			0
$759$	37049.2	+	10614.1i	1.77180	+	0.507597i
$760$		0			0
$761$	−9432.42	−	29030.0i	−0.449310	−	1.38283i	−0.877687	−	0.479234i	$-0.840915\pi$
$761$	−9432.42	−	29030.0i	−0.449310	−	1.38283i	0.428377	−	0.903600i	$-0.359085\pi$
$762$		0			0
$763$	−11035.1	+	8017.45i	−0.523586	+	0.380408i
$764$		0			0
$765$	1111.95	−	3422.22i	0.0525523	−	0.161739i
$766$		0			0
$767$	4354.03	+	3163.39i	0.204974	+	0.148922i
$768$		0			0
$769$	6866.20			0.321979			0.160989	−	0.986956i	$-0.448532\pi$
$769$	6866.20			0.321979			0.160989	+	0.986956i	$0.448532\pi$
$770$		0			0
$771$	13530.7			0.632030
$772$		0			0
$773$	−184.203	−	133.831i	−0.00857090	−	0.00622712i	0.583492	−	0.812119i	$-0.301686\pi$
$773$	−184.203	−	133.831i	−0.00857090	−	0.00622712i	−0.592062	+	0.805892i	$0.701686\pi$
$774$		0			0
$775$	642.899	−	1978.64i	0.0297982	−	0.0917094i
$776$		0			0
$777$	−28456.5	+	20674.9i	−1.31386	+	0.954577i
$778$		0			0
$779$	−12202.5	−	37555.4i	−0.561232	−	1.72729i
$780$		0			0
$781$	396.687	+	11282.0i	0.0181749	+	0.516903i
$782$		0			0
$783$	5221.34	+	16069.6i	0.238309	+	0.733438i
$784$		0			0
$785$	−13037.2	+	9472.09i	−0.592762	+	0.430667i
$786$		0			0
$787$	5416.02	−	16668.8i	0.245312	−	0.754991i	−0.750273	−	0.661128i	$-0.770078\pi$
$787$	5416.02	−	16668.8i	0.245312	−	0.754991i	0.995585	−	0.0938638i	$-0.0299218\pi$
$788$		0			0
$789$	22670.1	+	16470.8i	1.02291	+	0.743190i
$790$		0			0
$791$	−40496.4			−1.82034
$792$		0			0
$793$	10636.2			0.476297
$794$		0			0
$795$	12234.5	+	8888.92i	0.545804	+	0.396550i
$796$		0			0
$797$	−11809.2	+	36344.9i	−0.524846	+	1.61531i	0.239774	+	0.970829i	$0.422927\pi$
$797$	−11809.2	+	36344.9i	−0.524846	+	1.61531i	−0.764620	+	0.644481i	$0.777073\pi$
$798$		0			0
$799$	412.778	−	299.901i	0.0182766	−	0.0132787i
$800$		0			0
$801$	−942.313	−	2900.14i	−0.0415668	−	0.127929i
$802$		0			0
$803$	4301.71	−	6380.85i	0.189046	−	0.280417i
$804$		0			0
$805$	−23810.1	−	73279.9i	−1.04248	−	3.20842i
$806$		0			0
$807$	3783.62	−	2748.96i	0.165043	−	0.119911i
$808$		0			0
$809$	−926.503	+	2851.48i	−0.0402646	+	0.123922i	−0.969168	−	0.246400i	$-0.920752\pi$
$809$	−926.503	+	2851.48i	−0.0402646	+	0.123922i	0.928904	+	0.370321i	$0.120752\pi$
$810$		0			0
$811$	13642.3	+	9911.68i	0.590684	+	0.429157i	0.842560	−	0.538603i	$-0.181048\pi$
$811$	13642.3	+	9911.68i	0.590684	+	0.429157i	−0.251876	+	0.967759i	$0.581048\pi$
$812$		0			0
$813$	−7582.71			−0.327106
$814$		0			0
$815$	−40731.4			−1.75062
$816$		0			0
$817$	25217.3	+	18321.5i	1.07986	+	0.784561i
$818$		0			0
$819$	−1061.32	+	3266.40i	−0.0452814	+	0.139362i
$820$		0			0
$821$	8878.41	−	6450.55i	0.377416	−	0.274209i	−0.382863	−	0.923805i	$-0.625062\pi$
$821$	8878.41	−	6450.55i	0.377416	−	0.274209i	0.760280	+	0.649596i	$0.225062\pi$
$822$		0			0
$823$	−8583.43	−	26417.1i	−0.363548	−	1.11888i	−0.950886	−	0.309543i	$-0.899824\pi$
$823$	−8583.43	−	26417.1i	−0.363548	−	1.11888i	0.587338	−	0.809342i	$-0.300176\pi$
$824$		0			0
$825$	−2946.11	−	3768.98i	−0.124328	−	0.159054i
$826$		0			0
$827$	−5555.24	−	17097.3i	−0.233585	−	0.718900i	−0.997306	−	0.0733539i	$-0.976630\pi$
$827$	−5555.24	−	17097.3i	−0.233585	−	0.718900i	0.763721	−	0.645546i	$-0.223370\pi$
$828$		0			0
$829$	8425.01	−	6121.13i	0.352971	−	0.256448i	−0.397144	−	0.917757i	$-0.629998\pi$
$829$	8425.01	−	6121.13i	0.352971	−	0.256448i	0.750114	+	0.661308i	$0.229998\pi$
$830$		0			0
$831$	13256.9	−	40800.7i	0.553403	−	1.70320i
$832$		0			0
$833$	39064.1	+	28381.7i	1.62484	+	1.18052i
$834$		0			0
$835$	−19600.7			−0.812349
$836$		0			0
$837$	11183.7			0.461844
$838$		0			0
$839$	11419.3	+	8296.63i	0.469892	+	0.341396i	0.797399	−	0.603452i	$-0.206209\pi$
$839$	11419.3	+	8296.63i	0.469892	+	0.341396i	−0.327507	+	0.944849i	$0.606209\pi$
$840$		0			0
$841$	−1869.99	+	5755.25i	−0.0766737	+	0.235977i
$842$		0			0
$843$	−27567.3	+	20028.8i	−1.12630	+	0.818302i
$844$		0			0
$845$	6659.84	+	20496.9i	0.271131	+	0.834454i
$846$		0			0
$847$	−10877.2	+	43715.3i	−0.441256	+	1.77340i
$848$		0			0
$849$	−3251.86	−	10008.2i	−0.131453	−	0.404570i
$850$		0			0
$851$	27831.8	−	20221.0i	1.12111	−	0.814533i
$852$		0			0
$853$	−1773.98	+	5459.74i	−0.0712073	+	0.219154i	−0.980327	−	0.197382i	$-0.936756\pi$
$853$	−1773.98	+	5459.74i	−0.0712073	+	0.219154i	0.909119	+	0.416536i	$0.136756\pi$
$854$		0			0
$855$	−5125.09	−	3723.60i	−0.204999	−	0.148941i
$856$		0			0
$857$	−37005.0			−1.47499			−0.737495	−	0.675353i	$-0.763991\pi$
$857$	−37005.0			−1.47499			−0.737495	+	0.675353i	$0.763991\pi$
$858$		0			0
$859$	−7255.76			−0.288199			−0.144100	−	0.989563i	$-0.546029\pi$
$859$	−7255.76			−0.288199			−0.144100	+	0.989563i	$0.546029\pi$
$860$		0			0
$861$	57661.8	+	41893.7i	2.28235	+	1.65823i
$862$		0			0
$863$	−6039.18	+	18586.7i	−0.238211	+	0.733138i	0.758468	+	0.651710i	$0.225948\pi$
$863$	−6039.18	+	18586.7i	−0.238211	+	0.733138i	−0.996679	+	0.0814283i	$0.974052\pi$
$864$		0			0
$865$	17451.1	−	12679.0i	0.685959	−	0.498378i
$866$		0			0
$867$	2256.65	+	6945.25i	0.0883965	+	0.272057i
$868$		0			0
$869$	−20986.0	−	26847.6i	−0.819219	−	1.04803i
$870$		0			0
$871$	1679.12	+	5167.79i	0.0653211	+	0.201038i
$872$		0			0
$873$	−4785.85	+	3477.12i	−0.185540	+	0.134803i
$874$		0			0
$875$	12960.5	−	39888.2i	0.500736	−	1.54111i
$876$		0			0
$877$	−29172.7	−	21195.2i	−1.12325	−	0.816091i	−0.138554	−	0.990355i	$-0.544245\pi$
$877$	−29172.7	−	21195.2i	−1.12325	−	0.816091i	−0.984699	+	0.174264i	$0.944245\pi$
$878$		0			0
$879$	21900.8			0.840384
$880$		0			0
$881$	43821.4			1.67580			0.837900	−	0.545823i	$-0.183783\pi$
$881$	43821.4			1.67580			0.837900	+	0.545823i	$0.183783\pi$
$882$		0			0
$883$	6408.32	+	4655.92i	0.244232	+	0.177445i	0.703167	−	0.711025i	$-0.251769\pi$
$883$	6408.32	+	4655.92i	0.244232	+	0.177445i	−0.458934	+	0.888470i	$0.651769\pi$
$884$		0			0
$885$	5536.83	−	17040.6i	0.210303	−	0.647247i
$886$		0			0
$887$	−13299.1	+	9662.37i	−0.503428	+	0.365762i	−0.810325	−	0.585981i	$-0.800709\pi$
$887$	−13299.1	+	9662.37i	−0.503428	+	0.365762i	0.306897	+	0.951743i	$0.400709\pi$
$888$		0			0
$889$	5773.94	+	17770.3i	0.217831	+	0.670414i
$890$		0			0
$891$	17079.8	−	25334.9i	0.642193	−	0.952583i
$892$		0			0
$893$	−277.577	−	854.294i	−0.0104017	−	0.0320133i
$894$		0			0
$895$	11016.7	−	8004.07i	0.411448	−	0.298935i
$896$		0			0
$897$	6743.35	−	20753.9i	0.251008	−	0.772522i
$898$		0			0
$899$	−9819.37	−	7134.19i	−0.364287	−	0.264670i
$900$		0			0
$901$	13230.7			0.489209
$902$		0			0
$903$	−56260.8			−2.07336
$904$		0			0
$905$	−25523.7	−	18544.0i	−0.937498	−	0.681132i
$906$		0			0
$907$	−11391.7	+	35059.9i	−0.417038	+	1.28351i	0.493377	+	0.869816i	$0.335762\pi$
$907$	−11391.7	+	35059.9i	−0.417038	+	1.28351i	−0.910415	+	0.413696i	$0.864238\pi$
$908$		0			0
$909$	4096.42	−	2976.22i	0.149472	−	0.108597i
$910$		0			0
$911$	3507.69	+	10795.6i	0.127569	+	0.392616i	0.994360	−	0.106055i	$-0.0338218\pi$
$911$	3507.69	+	10795.6i	0.127569	+	0.392616i	−0.866792	+	0.498670i	$0.833822\pi$
$912$		0			0
$913$	305.283	+	8682.43i	0.0110662	+	0.314728i
$914$		0			0
$915$	−10942.5	−	33677.5i	−0.395352	−	1.21677i
$916$		0			0
$917$	74460.8	−	54098.9i	2.68147	−	1.94820i
$918$		0			0
$919$	−5108.20	+	15721.4i	−0.183356	+	0.564311i	−0.999916	−	0.0129488i	$-0.995878\pi$
$919$	−5108.20	+	15721.4i	−0.183356	+	0.564311i	0.816560	+	0.577260i	$0.195878\pi$
$920$		0			0
$921$	17311.5	+	12577.6i	0.619365	+	0.449995i
$922$		0			0
$923$	6392.05			0.227949
$924$		0			0
$925$	−4270.23			−0.151788
$926$		0			0
$927$	6053.71	+	4398.28i	0.214488	+	0.155834i
$928$		0			0
$929$	−5474.81	+	16849.7i	−0.193350	+	0.595072i	0.806641	+	0.591041i	$0.201283\pi$
$929$	−5474.81	+	16849.7i	−0.193350	+	0.595072i	−0.999992	+	0.00403046i	$0.998717\pi$
$930$		0			0
$931$	68773.4	−	49966.8i	2.42101	−	1.75896i
$932$		0			0
$933$	−11447.6	−	35232.0i	−0.401690	−	1.23627i
$934$		0			0
$935$	−25690.5	−	7359.97i	−0.898578	−	0.257430i
$936$		0			0
$937$	−2512.98	−	7734.15i	−0.0876151	−	0.269652i	0.897644	−	0.440722i	$-0.145277\pi$
$937$	−2512.98	−	7734.15i	−0.0876151	−	0.269652i	−0.985259	+	0.171070i	$0.945277\pi$
$938$		0			0
$939$	−39804.9	+	28919.9i	−1.38337	+	1.00508i
$940$		0			0
$941$	14663.6	−	45130.0i	0.507992	−	1.56344i	−0.287691	−	0.957723i	$-0.592888\pi$
$941$	14663.6	−	45130.0i	0.507992	−	1.56344i	0.795682	−	0.605714i	$-0.207112\pi$
$942$		0			0
$943$	−56396.0	−	40974.1i	−1.94751	−	1.41495i
$944$		0			0
$945$	−51412.5			−1.76979
$946$		0			0
$947$	−6370.98			−0.218616			−0.109308	−	0.994008i	$-0.534863\pi$
$947$	−6370.98			−0.218616			−0.109308	+	0.994008i	$0.534863\pi$
$948$		0			0
$949$	−3525.15	−	2561.17i	−0.120581	−	0.0876071i
$950$		0			0
$951$	5862.81	−	18043.9i	0.199910	−	0.615260i
$952$		0			0
$953$	−23929.8	+	17386.0i	−0.813390	+	0.590963i	−0.914812	−	0.403881i	$-0.867661\pi$
$953$	−23929.8	+	17386.0i	−0.813390	+	0.590963i	0.101421	+	0.994844i	$0.467661\pi$
$954$		0			0
$955$	4917.66	+	15135.0i	0.166630	+	0.512835i
$956$		0			0
$957$	−26223.3	+	9551.62i	−0.885767	+	0.322633i
$958$		0			0
$959$	20128.9	+	61950.2i	0.677784	+	2.08600i
$960$		0			0
$961$	17602.1	−	12788.7i	0.590853	−	0.429280i
$962$		0			0
$963$	−1174.17	+	3613.73i	−0.0392909	+	0.120925i
$964$		0			0
$965$	20024.5	+	14548.6i	0.667991	+	0.485324i
$966$		0			0
$967$	44254.9			1.47171			0.735854	−	0.677141i	$-0.236781\pi$
$967$	44254.9			1.47171			0.735854	+	0.677141i	$0.236781\pi$
$968$		0			0
$969$	−36005.2			−1.19366
$970$		0			0
$971$	26389.2	+	19172.9i	0.872162	+	0.633663i	0.931166	−	0.364595i	$-0.118793\pi$
$971$	26389.2	+	19172.9i	0.872162	+	0.633663i	−0.0590042	+	0.998258i	$0.518793\pi$
$972$		0			0
$973$	−11738.6	+	36127.8i	−0.386766	+	1.19034i
$974$		0			0
$975$	−2191.38	+	1592.13i	−0.0719798	+	0.0522964i
$976$		0			0
$977$	−9812.65	−	30200.2i	−0.321325	−	0.988937i	−0.973072	−	0.230501i	$-0.925964\pi$
$977$	−9812.65	−	30200.2i	−0.321325	−	0.988937i	0.651747	−	0.758436i	$-0.274036\pi$
$978$		0			0
$979$	−21279.5	+	7750.88i	−0.694683	+	0.253033i
$980$		0			0
$981$	611.772	+	1882.84i	0.0199107	+	0.0612788i
$982$		0			0
$983$	28463.8	−	20680.2i	0.923555	−	0.671002i	−0.0208517	−	0.999783i	$-0.506638\pi$
$983$	28463.8	−	20680.2i	0.923555	−	0.671002i	0.944406	+	0.328781i	$0.106638\pi$
$984$		0			0
$985$	−778.199	+	2395.05i	−0.0251731	+	0.0774748i
$986$		0			0
$987$	1311.67	+	952.980i	0.0423007	+	0.0307332i
$988$		0			0
$989$	55025.7			1.76918
$990$		0			0
$991$	−34640.2			−1.11038			−0.555188	−	0.831725i	$-0.687354\pi$
$991$	−34640.2			−1.11038			−0.555188	+	0.831725i	$0.687354\pi$
$992$		0			0
$993$	−15450.9	−	11225.7i	−0.493775	−	0.358749i
$994$		0			0
$995$	16518.3	−	50838.0i	0.526295	−	1.61977i
$996$		0			0
$997$	−2686.27	+	1951.69i	−0.0853309	+	0.0619965i	−0.629633	−	0.776893i	$-0.716795\pi$
$997$	−2686.27	+	1951.69i	−0.0853309	+	0.0619965i	0.544302	+	0.838889i	$0.316795\pi$
$998$		0			0
$999$	−7093.44	−	21831.4i	−0.224651	−	0.691405i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	88.4.i.a.49.3	yes	16
4.3	odd	2		176.4.m.e.49.2		16
11.3	even	5		968.4.a.n.1.4		8
11.8	odd	10		968.4.a.o.1.4		8
11.9	even	5	inner	88.4.i.a.9.3	✓	16
44.3	odd	10		1936.4.a.bw.1.5		8
44.19	even	10		1936.4.a.bv.1.5		8
44.31	odd	10		176.4.m.e.97.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.a.9.3	✓	16	11.9	even	5	inner
88.4.i.a.49.3	yes	16	1.1	even	1	trivial
176.4.m.e.49.2		16	4.3	odd	2
176.4.m.e.97.2		16	44.31	odd	10
968.4.a.n.1.4		8	11.3	even	5
968.4.a.o.1.4		8	11.8	odd	10
1936.4.a.bv.1.5		8	44.19	even	10
1936.4.a.bw.1.5		8	44.3	odd	10

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	88.i (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(4.57022 + 3.32046i) q^{3} +(-3.76204 + 11.5784i) q^{5} +(-27.3815 + 19.8938i) q^{7} +(1.51800 + 4.67192i) q^{9} +(34.2797 - 12.4861i) q^{11} +(-6.38348 - 19.6463i) q^{13} +(-55.6388 + 40.4240i) q^{15} +(-18.5932 + 57.2239i) q^{17} +(85.6982 + 62.2634i) q^{19} -191.196 q^{21} +186.999 q^{23} +(-18.7786 - 13.6434i) q^{25} +(38.5577 - 118.668i) q^{27} +(-109.554 + 79.5957i) q^{29} +(27.6973 + 85.2435i) q^{31} +(198.125 + 56.7601i) q^{33} +(-127.328 - 391.874i) q^{35} +(148.834 - 108.135i) q^{37} +(36.0609 - 110.984i) q^{39} +(-301.585 - 219.114i) q^{41} +294.257 q^{43} -59.8040 q^{45} +(-6.86032 - 4.98432i) q^{47} +(247.988 - 763.230i) q^{49} +(-274.985 + 199.788i) q^{51} +(-67.9504 - 209.130i) q^{53} +(15.6071 + 443.876i) q^{55} +(184.916 + 569.114i) q^{57} +(-210.773 + 153.136i) q^{59} +(-159.109 + 489.688i) q^{61} +(-134.507 - 97.7252i) q^{63} +251.487 q^{65} -263.041 q^{67} +(854.625 + 620.922i) q^{69} +(-95.6197 + 294.287i) q^{71} +(170.648 - 123.983i) q^{73} +(-40.5198 - 124.707i) q^{75} +(-690.232 + 1023.84i) q^{77} +(-288.634 - 888.325i) q^{79} +(677.554 - 492.271i) q^{81} +(-73.5873 + 226.478i) q^{83} +(-592.612 - 430.558i) q^{85} -764.980 q^{87} -620.760 q^{89} +(565.629 + 410.954i) q^{91} +(-156.465 + 481.549i) q^{93} +(-1043.31 + 758.008i) q^{95} +(372.130 + 1145.30i) q^{97} +(110.371 + 141.198i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{3} - q^{5} - 13 q^{7} + 7 q^{9} + 83 q^{11} + 69 q^{13} + 93 q^{15} - 217 q^{17} + 126 q^{19} + 34 q^{21} - 92 q^{23} + 307 q^{25} + 158 q^{27} - 553 q^{29} + 205 q^{31} - 198 q^{33} + 7 q^{35}+ \cdots - 3265 q^{99}+O(q^{100}) \) 16 * q - q^3 - q^5 - 13 * q^7 + 7 * q^9 + 83 * q^11 + 69 * q^13 + 93 * q^15 - 217 * q^17 + 126 * q^19 + 34 * q^21 - 92 * q^23 + 307 * q^25 + 158 * q^27 - 553 * q^29 + 205 * q^31 - 198 * q^33 + 7 * q^35 + 127 * q^37 - 629 * q^39 - 553 * q^41 + 70 * q^43 - 1132 * q^45 + 779 * q^47 + 847 * q^49 - 2176 * q^51 + 1401 * q^53 - 385 * q^55 - 488 * q^57 + 1354 * q^59 - 1623 * q^61 + 256 * q^63 + 1674 * q^65 - 1658 * q^67 + 5470 * q^69 + 1985 * q^71 - 1245 * q^73 + 4114 * q^75 - 2645 * q^77 - 2007 * q^79 + 1966 * q^81 - 2946 * q^83 - 1543 * q^85 - 262 * q^87 - 6410 * q^89 + 3937 * q^91 + 2221 * q^93 - 8359 * q^95 + 6566 * q^97 - 3265 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more