LMFDB - Embedded newform 5355.2.a.bx.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5355,2,Mod(1,5355)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5355, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5355.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5355 = 3^{2} \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5355.a (trivial)

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:	yes
Analytic conductor:	$42.7598902824$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 12x^{5} + 8x^{4} + 43x^{3} - 15x^{2} - 44x + 8 $ x^7 - x^6 - 12x^5 + 8x^4 + 43x^3 - 15x^2 - 44*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.69076$ of defining polynomial
Character	$\chi$	$=$	5355.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.69076 q^{2} +0.858669 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.92972 q^{8} +O(q^{10})$	\(q+1.69076 q^{2} +0.858669 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.92972 q^{8} -1.69076 q^{10} +1.62048 q^{11} -5.42161 q^{13} +1.69076 q^{14} -4.98003 q^{16} -1.00000 q^{17} -1.08805 q^{19} -0.858669 q^{20} +2.73984 q^{22} +4.50807 q^{23} +1.00000 q^{25} -9.16665 q^{26} +0.858669 q^{28} +7.85204 q^{29} +6.05090 q^{31} -4.56060 q^{32} -1.69076 q^{34} -1.00000 q^{35} -4.04447 q^{37} -1.83964 q^{38} +1.92972 q^{40} +10.3292 q^{41} +1.76010 q^{43} +1.39145 q^{44} +7.62207 q^{46} -5.77515 q^{47} +1.00000 q^{49} +1.69076 q^{50} -4.65537 q^{52} +6.01240 q^{53} -1.62048 q^{55} -1.92972 q^{56} +13.2759 q^{58} +0.989598 q^{59} +14.1177 q^{61} +10.2306 q^{62} +2.24918 q^{64} +5.42161 q^{65} +7.38152 q^{67} -0.858669 q^{68} -1.69076 q^{70} +1.62048 q^{71} -12.8299 q^{73} -6.83823 q^{74} -0.934278 q^{76} +1.62048 q^{77} +15.0920 q^{79} +4.98003 q^{80} +17.4642 q^{82} -17.1699 q^{83} +1.00000 q^{85} +2.97591 q^{86} -3.12706 q^{88} +0.0644430 q^{89} -5.42161 q^{91} +3.87094 q^{92} -9.76439 q^{94} +1.08805 q^{95} +18.7340 q^{97} +1.69076 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) $ 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8	$ 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8} + q^{10} - 6 q^{11} + 16 q^{13} - q^{14} + 11 q^{16} - 7 q^{17} + 6 q^{19} - 11 q^{20} - 6 q^{22} - 14 q^{23} + 7 q^{25} - 16 q^{26} + 11 q^{28} - 12 q^{29} + 12 q^{31} - 29 q^{32} + q^{34} - 7 q^{35} + 4 q^{37} + 30 q^{38} + 9 q^{40} + 36 q^{41} + 6 q^{43} + 18 q^{44} + 14 q^{46} + 8 q^{47} + 7 q^{49} - q^{50} + 44 q^{52} + 18 q^{53} + 6 q^{55} - 9 q^{56} + 48 q^{58} - 12 q^{59} + 38 q^{61} + 24 q^{62} + 31 q^{64} - 16 q^{65} + 26 q^{67} - 11 q^{68} + q^{70} - 6 q^{71} - 8 q^{73} - 22 q^{74} - 30 q^{76} - 6 q^{77} + 6 q^{79} - 11 q^{80} - 18 q^{82} + 7 q^{85} + 78 q^{86} - 102 q^{88} + 14 q^{89} + 16 q^{91} - 10 q^{92} - 14 q^{94} - 6 q^{95} + 20 q^{97} - q^{98}+O(q^{100}) $ 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8 + q^10 - 6 * q^11 + 16 * q^13 - q^14 + 11 * q^16 - 7 * q^17 + 6 * q^19 - 11 * q^20 - 6 * q^22 - 14 * q^23 + 7 * q^25 - 16 * q^26 + 11 * q^28 - 12 * q^29 + 12 * q^31 - 29 * q^32 + q^34 - 7 * q^35 + 4 * q^37 + 30 * q^38 + 9 * q^40 + 36 * q^41 + 6 * q^43 + 18 * q^44 + 14 * q^46 + 8 * q^47 + 7 * q^49 - q^50 + 44 * q^52 + 18 * q^53 + 6 * q^55 - 9 * q^56 + 48 * q^58 - 12 * q^59 + 38 * q^61 + 24 * q^62 + 31 * q^64 - 16 * q^65 + 26 * q^67 - 11 * q^68 + q^70 - 6 * q^71 - 8 * q^73 - 22 * q^74 - 30 * q^76 - 6 * q^77 + 6 * q^79 - 11 * q^80 - 18 * q^82 + 7 * q^85 + 78 * q^86 - 102 * q^88 + 14 * q^89 + 16 * q^91 - 10 * q^92 - 14 * q^94 - 6 * q^95 + 20 * q^97 - q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.69076	1.19555	0.597774	−	0.801665i	$-0.296052\pi$
$2$	1.69076	1.19555	0.597774	+	0.801665i	$0.296052\pi$
$3$	0	0
$3$	0	0
$4$	0.858669	0.429335
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.92972	−0.682258
$9$	0	0
$10$	−1.69076	−0.534665
$11$	1.62048	0.488592	0.244296	−	0.969701i	$-0.421443\pi$
$11$	1.62048	0.488592	0.244296	+	0.969701i	$0.421443\pi$
$12$	0	0
$13$	−5.42161	−1.50369	−0.751843	−	0.659343i	$-0.770835\pi$
$13$	−5.42161	−1.50369	−0.751843	+	0.659343i	$0.770835\pi$
$14$	1.69076	0.451875
$15$	0	0
$16$	−4.98003	−1.24501
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−1.08805	−0.249617	−0.124808	−	0.992181i	$-0.539832\pi$
$19$	−1.08805	−0.249617	−0.124808	+	0.992181i	$0.539832\pi$
$20$	−0.858669	−0.192004
$21$	0	0
$22$	2.73984	0.584135
$23$	4.50807	0.939998	0.469999	−	0.882667i	$-0.344254\pi$
$23$	4.50807	0.939998	0.469999	+	0.882667i	$0.344254\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−9.16665	−1.79773
$27$	0	0
$28$	0.858669	0.162273
$29$	7.85204	1.45809	0.729043	−	0.684468i	$-0.239965\pi$
$29$	7.85204	1.45809	0.729043	+	0.684468i	$0.239965\pi$
$30$	0	0
$31$	6.05090	1.08677	0.543387	−	0.839482i	$-0.317142\pi$
$31$	6.05090	1.08677	0.543387	+	0.839482i	$0.317142\pi$
$32$	−4.56060	−0.806207
$33$	0	0
$34$	−1.69076	−0.289963
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−4.04447	−0.664907	−0.332453	−	0.943120i	$-0.607876\pi$
$37$	−4.04447	−0.664907	−0.332453	+	0.943120i	$0.607876\pi$
$38$	−1.83964	−0.298429
$39$	0	0
$40$	1.92972	0.305115
$41$	10.3292	1.61315	0.806574	−	0.591133i	$-0.201319\pi$
$41$	10.3292	1.61315	0.806574	+	0.591133i	$0.201319\pi$
$42$	0	0
$43$	1.76010	0.268413	0.134206	−	0.990953i	$-0.457151\pi$
$43$	1.76010	0.268413	0.134206	+	0.990953i	$0.457151\pi$
$44$	1.39145	0.209769
$45$	0	0
$46$	7.62207	1.12381
$47$	−5.77515	−0.842392	−0.421196	−	0.906970i	$-0.638390\pi$
$47$	−5.77515	−0.842392	−0.421196	+	0.906970i	$0.638390\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.69076	0.239110
$51$	0	0
$52$	−4.65537	−0.645584
$53$	6.01240	0.825866	0.412933	−	0.910761i	$-0.364504\pi$
$53$	6.01240	0.825866	0.412933	+	0.910761i	$0.364504\pi$
$54$	0	0
$55$	−1.62048	−0.218505
$56$	−1.92972	−0.257869
$57$	0	0
$58$	13.2759	1.74321
$59$	0.989598	0.128835	0.0644173	−	0.997923i	$-0.479481\pi$
$59$	0.989598	0.128835	0.0644173	+	0.997923i	$0.479481\pi$
$60$	0	0
$61$	14.1177	1.80759	0.903796	−	0.427963i	$-0.140769\pi$
$61$	14.1177	1.80759	0.903796	+	0.427963i	$0.140769\pi$
$62$	10.2306	1.29929
$63$	0	0
$64$	2.24918	0.281147
$65$	5.42161	0.672468
$66$	0	0
$67$	7.38152	0.901796	0.450898	−	0.892575i	$-0.351104\pi$
$67$	7.38152	0.901796	0.450898	+	0.892575i	$0.351104\pi$
$68$	−0.858669	−0.104129
$69$	0	0
$70$	−1.69076	−0.202084
$71$	1.62048	0.192315	0.0961576	−	0.995366i	$-0.469345\pi$
$71$	1.62048	0.192315	0.0961576	+	0.995366i	$0.469345\pi$
$72$	0	0
$73$	−12.8299	−1.50162	−0.750812	−	0.660516i	$-0.770338\pi$
$73$	−12.8299	−1.50162	−0.750812	+	0.660516i	$0.770338\pi$
$74$	−6.83823	−0.794928
$75$	0	0
$76$	−0.934278	−0.107169
$77$	1.62048	0.184670
$78$	0	0
$79$	15.0920	1.69799	0.848994	−	0.528403i	$-0.177209\pi$
$79$	15.0920	1.69799	0.848994	+	0.528403i	$0.177209\pi$
$80$	4.98003	0.556784
$81$	0	0
$82$	17.4642	1.92860
$83$	−17.1699	−1.88464	−0.942319	−	0.334715i	$-0.891360\pi$
$83$	−17.1699	−1.88464	−0.942319	+	0.334715i	$0.891360\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	2.97591	0.320900
$87$	0	0
$88$	−3.12706	−0.333346
$89$	0.0644430	0.00683095	0.00341547	−	0.999994i	$-0.498913\pi$
$89$	0.0644430	0.00683095	0.00341547	+	0.999994i	$0.498913\pi$
$90$	0	0
$91$	−5.42161	−0.568340
$92$	3.87094	0.403574
$93$	0	0
$94$	−9.76439	−1.00712
$95$	1.08805	0.111632
$96$	0	0
$97$	18.7340	1.90215	0.951073	−	0.308966i	$-0.0999829\pi$
$97$	18.7340	1.90215	0.951073	+	0.308966i	$0.0999829\pi$
$98$	1.69076	0.170793
$99$	0	0
$100$	0.858669	0.0858669
$101$	−14.2006	−1.41301	−0.706506	−	0.707707i	$-0.749730\pi$
$101$	−14.2006	−1.41301	−0.706506	+	0.707707i	$0.749730\pi$
$102$	0	0
$103$	12.1990	1.20200	0.601002	−	0.799248i	$-0.294768\pi$
$103$	12.1990	1.20200	0.601002	+	0.799248i	$0.294768\pi$
$104$	10.4622	1.02590
$105$	0	0
$106$	10.1655	0.987363
$107$	6.41027	0.619704	0.309852	−	0.950785i	$-0.399720\pi$
$107$	6.41027	0.619704	0.309852	+	0.950785i	$0.399720\pi$
$108$	0	0
$109$	7.62142	0.730000	0.365000	−	0.931008i	$-0.381069\pi$
$109$	7.62142	0.730000	0.365000	+	0.931008i	$0.381069\pi$
$110$	−2.73984	−0.261233
$111$	0	0
$112$	−4.98003	−0.470568
$113$	−12.9277	−1.21613	−0.608067	−	0.793885i	$-0.708055\pi$
$113$	−12.9277	−1.21613	−0.608067	+	0.793885i	$0.708055\pi$
$114$	0	0
$115$	−4.50807	−0.420380
$116$	6.74230	0.626007
$117$	0	0
$118$	1.67317	0.154028
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−8.37406	−0.761278
$122$	23.8697	2.16106
$123$	0	0
$124$	5.19572	0.466589
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.76199	0.511294	0.255647	−	0.966770i	$-0.417712\pi$
$127$	5.76199	0.511294	0.255647	+	0.966770i	$0.417712\pi$
$128$	12.9240	1.14233
$129$	0	0
$130$	9.16665	0.803968
$131$	7.55680	0.660240	0.330120	−	0.943939i	$-0.392911\pi$
$131$	7.55680	0.660240	0.330120	+	0.943939i	$0.392911\pi$
$132$	0	0
$133$	−1.08805	−0.0943462
$134$	12.4804	1.07814
$135$	0	0
$136$	1.92972	0.165472
$137$	6.01240	0.513674	0.256837	−	0.966455i	$-0.417320\pi$
$137$	6.01240	0.513674	0.256837	+	0.966455i	$0.417320\pi$
$138$	0	0
$139$	22.3544	1.89608	0.948040	−	0.318152i	$-0.103062\pi$
$139$	22.3544	1.89608	0.948040	+	0.318152i	$0.103062\pi$
$140$	−0.858669	−0.0725708
$141$	0	0
$142$	2.73984	0.229922
$143$	−8.78560	−0.734689
$144$	0	0
$145$	−7.85204	−0.652076
$146$	−21.6923	−1.79526
$147$	0	0
$148$	−3.47286	−0.285467
$149$	14.3995	1.17965	0.589825	−	0.807531i	$-0.299197\pi$
$149$	14.3995	1.17965	0.589825	+	0.807531i	$0.299197\pi$
$150$	0	0
$151$	−18.9649	−1.54334	−0.771669	−	0.636024i	$-0.780578\pi$
$151$	−18.9649	−1.54334	−0.771669	+	0.636024i	$0.780578\pi$
$152$	2.09964	0.170303
$153$	0	0
$154$	2.73984	0.220782
$155$	−6.05090	−0.486020
$156$	0	0
$157$	−1.55792	−0.124335	−0.0621677	−	0.998066i	$-0.519801\pi$
$157$	−1.55792	−0.124335	−0.0621677	+	0.998066i	$0.519801\pi$
$158$	25.5170	2.03003
$159$	0	0
$160$	4.56060	0.360547
$161$	4.50807	0.355286
$162$	0	0
$163$	6.23327	0.488227	0.244114	−	0.969747i	$-0.421503\pi$
$163$	6.23327	0.488227	0.244114	+	0.969747i	$0.421503\pi$
$164$	8.86935	0.692580
$165$	0	0
$166$	−29.0301	−2.25318
$167$	0.835822	0.0646779	0.0323389	−	0.999477i	$-0.489704\pi$
$167$	0.835822	0.0646779	0.0323389	+	0.999477i	$0.489704\pi$
$168$	0	0
$169$	16.3939	1.26107
$170$	1.69076	0.129675
$171$	0	0
$172$	1.51134	0.115239
$173$	−15.9504	−1.21268	−0.606341	−	0.795204i	$-0.707364\pi$
$173$	−15.9504	−1.21268	−0.606341	+	0.795204i	$0.707364\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−8.07001	−0.608300
$177$	0	0
$178$	0.108958	0.00816673
$179$	−4.33756	−0.324204	−0.162102	−	0.986774i	$-0.551827\pi$
$179$	−4.33756	−0.324204	−0.162102	+	0.986774i	$0.551827\pi$
$180$	0	0
$181$	−16.3155	−1.21272	−0.606359	−	0.795191i	$-0.707371\pi$
$181$	−16.3155	−1.21272	−0.606359	+	0.795191i	$0.707371\pi$
$182$	−9.16665	−0.679477
$183$	0	0
$184$	−8.69930	−0.641321
$185$	4.04447	0.297355
$186$	0	0
$187$	−1.62048	−0.118501
$188$	−4.95894	−0.361668
$189$	0	0
$190$	1.83964	0.133461
$191$	8.75689	0.633626	0.316813	−	0.948488i	$-0.397387\pi$
$191$	8.75689	0.633626	0.316813	+	0.948488i	$0.397387\pi$
$192$	0	0
$193$	−6.37282	−0.458726	−0.229363	−	0.973341i	$-0.573664\pi$
$193$	−6.37282	−0.458726	−0.229363	+	0.973341i	$0.573664\pi$
$194$	31.6746	2.27411
$195$	0	0
$196$	0.858669	0.0613335
$197$	−0.0133415	−0.000950539 0	−0.000475270	−	1.00000i	$-0.500151\pi$
$197$	−0.0133415	−0.000950539 0	−0.000475270		1.00000i	$0.500151\pi$
$198$	0	0
$199$	−0.544175	−0.0385756	−0.0192878	−	0.999814i	$-0.506140\pi$
$199$	−0.544175	−0.0385756	−0.0192878	+	0.999814i	$0.506140\pi$
$200$	−1.92972	−0.136452
$201$	0	0
$202$	−24.0098	−1.68932
$203$	7.85204	0.551105
$204$	0	0
$205$	−10.3292	−0.721422
$206$	20.6256	1.43705
$207$	0	0
$208$	26.9998	1.87210
$209$	−1.76317	−0.121961
$210$	0	0
$211$	10.8648	0.747962	0.373981	−	0.927436i	$-0.377993\pi$
$211$	10.8648	0.747962	0.373981	+	0.927436i	$0.377993\pi$
$212$	5.16266	0.354573
$213$	0	0
$214$	10.8382	0.740886
$215$	−1.76010	−0.120038
$216$	0	0
$217$	6.05090	0.410762
$218$	12.8860	0.872749
$219$	0	0
$220$	−1.39145	−0.0938118
$221$	5.42161	0.364697
$222$	0	0
$223$	−23.9501	−1.60382	−0.801908	−	0.597447i	$-0.796182\pi$
$223$	−23.9501	−1.60382	−0.801908	+	0.597447i	$0.796182\pi$
$224$	−4.56060	−0.304718
$225$	0	0
$226$	−21.8576	−1.45395
$227$	10.4716	0.695026	0.347513	−	0.937675i	$-0.387026\pi$
$227$	10.4716	0.695026	0.347513	+	0.937675i	$0.387026\pi$
$228$	0	0
$229$	−1.92134	−0.126966	−0.0634828	−	0.997983i	$-0.520221\pi$
$229$	−1.92134	−0.126966	−0.0634828	+	0.997983i	$0.520221\pi$
$230$	−7.62207	−0.502584
$231$	0	0
$232$	−15.1522	−0.994791
$233$	−13.7057	−0.897893	−0.448946	−	0.893559i	$-0.648201\pi$
$233$	−13.7057	−0.897893	−0.448946	+	0.893559i	$0.648201\pi$
$234$	0	0
$235$	5.77515	0.376729
$236$	0.849737	0.0553132
$237$	0	0
$238$	−1.69076	−0.109596
$239$	−25.5474	−1.65253	−0.826263	−	0.563285i	$-0.809537\pi$
$239$	−25.5474	−1.65253	−0.826263	+	0.563285i	$0.809537\pi$
$240$	0	0
$241$	3.07052	0.197789	0.0988947	−	0.995098i	$-0.468469\pi$
$241$	3.07052	0.197789	0.0988947	+	0.995098i	$0.468469\pi$
$242$	−14.1585	−0.910144
$243$	0	0
$244$	12.1225	0.776062
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	5.89901	0.375345
$248$	−11.6765	−0.741460
$249$	0	0
$250$	−1.69076	−0.106933
$251$	25.1731	1.58891	0.794456	−	0.607321i	$-0.207756\pi$
$251$	25.1731	1.58891	0.794456	+	0.607321i	$0.207756\pi$
$252$	0	0
$253$	7.30523	0.459276
$254$	9.74214	0.611276
$255$	0	0
$256$	17.3530	1.08457
$257$	−24.8856	−1.55232	−0.776162	−	0.630534i	$-0.782836\pi$
$257$	−24.8856	−1.55232	−0.776162	+	0.630534i	$0.782836\pi$
$258$	0	0
$259$	−4.04447	−0.251311
$260$	4.65537	0.288714
$261$	0	0
$262$	12.7767	0.789349
$263$	12.3378	0.760783	0.380391	−	0.924826i	$-0.375789\pi$
$263$	12.3378	0.760783	0.380391	+	0.924826i	$0.375789\pi$
$264$	0	0
$265$	−6.01240	−0.369339
$266$	−1.83964	−0.112795
$267$	0	0
$268$	6.33828	0.387172
$269$	29.0307	1.77003	0.885016	−	0.465562i	$-0.154148\pi$
$269$	29.0307	1.77003	0.885016	+	0.465562i	$0.154148\pi$
$270$	0	0
$271$	−14.0488	−0.853404	−0.426702	−	0.904392i	$-0.640325\pi$
$271$	−14.0488	−0.853404	−0.426702	+	0.904392i	$0.640325\pi$
$272$	4.98003	0.301958
$273$	0	0
$274$	10.1655	0.614122
$275$	1.62048	0.0977184
$276$	0	0
$277$	25.2678	1.51820	0.759098	−	0.650977i	$-0.225640\pi$
$277$	25.2678	1.51820	0.759098	+	0.650977i	$0.225640\pi$
$278$	37.7960	2.26685
$279$	0	0
$280$	1.92972	0.115323
$281$	4.21308	0.251331	0.125666	−	0.992073i	$-0.459893\pi$
$281$	4.21308	0.251331	0.125666	+	0.992073i	$0.459893\pi$
$282$	0	0
$283$	28.0669	1.66841	0.834203	−	0.551458i	$-0.185928\pi$
$283$	28.0669	1.66841	0.834203	+	0.551458i	$0.185928\pi$
$284$	1.39145	0.0825676
$285$	0	0
$286$	−14.8543	−0.878355
$287$	10.3292	0.609713
$288$	0	0
$289$	1.00000	0.0588235
$290$	−13.2759	−0.779588
$291$	0	0
$292$	−11.0166	−0.644699
$293$	−32.0355	−1.87153	−0.935766	−	0.352620i	$-0.885291\pi$
$293$	−32.0355	−1.87153	−0.935766	+	0.352620i	$0.885291\pi$
$294$	0	0
$295$	−0.989598	−0.0576166
$296$	7.80468	0.453638
$297$	0	0
$298$	24.3460	1.41033
$299$	−24.4410	−1.41346
$300$	0	0
$301$	1.76010	0.101451
$302$	−32.0650	−1.84514
$303$	0	0
$304$	5.41854	0.310774
$305$	−14.1177	−0.808380
$306$	0	0
$307$	12.2257	0.697757	0.348879	−	0.937168i	$-0.386563\pi$
$307$	12.2257	0.697757	0.348879	+	0.937168i	$0.386563\pi$
$308$	1.39145	0.0792854
$309$	0	0
$310$	−10.2306	−0.581060
$311$	2.44908	0.138875	0.0694373	−	0.997586i	$-0.477880\pi$
$311$	2.44908	0.138875	0.0694373	+	0.997586i	$0.477880\pi$
$312$	0	0
$313$	−13.1392	−0.742673	−0.371337	−	0.928498i	$-0.621100\pi$
$313$	−13.1392	−0.742673	−0.371337	+	0.928498i	$0.621100\pi$
$314$	−2.63407	−0.148649
$315$	0	0
$316$	12.9591	0.729005
$317$	−32.9715	−1.85186	−0.925932	−	0.377691i	$-0.876718\pi$
$317$	−32.9715	−1.85186	−0.925932	+	0.377691i	$0.876718\pi$
$318$	0	0
$319$	12.7240	0.712409
$320$	−2.24918	−0.125733
$321$	0	0
$322$	7.62207	0.424761
$323$	1.08805	0.0605409
$324$	0	0
$325$	−5.42161	−0.300737
$326$	10.5390	0.583699
$327$	0	0
$328$	−19.9324	−1.10058
$329$	−5.77515	−0.318394
$330$	0	0
$331$	32.0116	1.75952	0.879760	−	0.475419i	$-0.157703\pi$
$331$	32.0116	1.75952	0.879760	+	0.475419i	$0.157703\pi$
$332$	−14.7432	−0.809141
$333$	0	0
$334$	1.41317	0.0773255
$335$	−7.38152	−0.403296
$336$	0	0
$337$	10.4908	0.571472	0.285736	−	0.958308i	$-0.407762\pi$
$337$	10.4908	0.571472	0.285736	+	0.958308i	$0.407762\pi$
$338$	27.7181	1.50767
$339$	0	0
$340$	0.858669	0.0465679
$341$	9.80534	0.530989
$342$	0	0
$343$	1.00000	0.0539949
$344$	−3.39650	−0.183127
$345$	0	0
$346$	−26.9682	−1.44982
$347$	6.74160	0.361908	0.180954	−	0.983492i	$-0.442081\pi$
$347$	6.74160	0.361908	0.180954	+	0.983492i	$0.442081\pi$
$348$	0	0
$349$	24.3133	1.30146	0.650732	−	0.759308i	$-0.274462\pi$
$349$	24.3133	1.30146	0.650732	+	0.759308i	$0.274462\pi$
$350$	1.69076	0.0903749
$351$	0	0
$352$	−7.39034	−0.393906
$353$	26.6521	1.41855	0.709274	−	0.704933i	$-0.249023\pi$
$353$	26.6521	1.41855	0.709274	+	0.704933i	$0.249023\pi$
$354$	0	0
$355$	−1.62048	−0.0860060
$356$	0.0553353	0.00293276
$357$	0	0
$358$	−7.33377	−0.387602
$359$	18.5859	0.980924	0.490462	−	0.871463i	$-0.336828\pi$
$359$	18.5859	0.980924	0.490462	+	0.871463i	$0.336828\pi$
$360$	0	0
$361$	−17.8161	−0.937692
$362$	−27.5855	−1.44986
$363$	0	0
$364$	−4.65537	−0.244008
$365$	12.8299	0.671547
$366$	0	0
$367$	−0.705542	−0.0368290	−0.0184145	−	0.999830i	$-0.505862\pi$
$367$	−0.705542	−0.0368290	−0.0184145	+	0.999830i	$0.505862\pi$
$368$	−22.4503	−1.17030
$369$	0	0
$370$	6.83823	0.355502
$371$	6.01240	0.312148
$372$	0	0
$373$	35.0193	1.81323	0.906616	−	0.421956i	$-0.138656\pi$
$373$	35.0193	1.81323	0.906616	+	0.421956i	$0.138656\pi$
$374$	−2.73984	−0.141674
$375$	0	0
$376$	11.1444	0.574728
$377$	−42.5707	−2.19250
$378$	0	0
$379$	32.2155	1.65480	0.827399	−	0.561615i	$-0.189820\pi$
$379$	32.2155	1.65480	0.827399	+	0.561615i	$0.189820\pi$
$380$	0.934278	0.0479275
$381$	0	0
$382$	14.8058	0.757531
$383$	−1.17052	−0.0598110	−0.0299055	−	0.999553i	$-0.509521\pi$
$383$	−1.17052	−0.0598110	−0.0299055	+	0.999553i	$0.509521\pi$
$384$	0	0
$385$	−1.62048	−0.0825871
$386$	−10.7749	−0.548429
$387$	0	0
$388$	16.0863	0.816657
$389$	−21.4561	−1.08787	−0.543935	−	0.839128i	$-0.683066\pi$
$389$	−21.4561	−1.08787	−0.543935	+	0.839128i	$0.683066\pi$
$390$	0	0
$391$	−4.50807	−0.227983
$392$	−1.92972	−0.0974654
$393$	0	0
$394$	−0.0225572	−0.00113642
$395$	−15.0920	−0.759363
$396$	0	0
$397$	−4.74095	−0.237941	−0.118971	−	0.992898i	$-0.537959\pi$
$397$	−4.74095	−0.237941	−0.118971	+	0.992898i	$0.537959\pi$
$398$	−0.920070	−0.0461189
$399$	0	0
$400$	−4.98003	−0.249001
$401$	−23.0659	−1.15186	−0.575928	−	0.817500i	$-0.695359\pi$
$401$	−23.0659	−1.15186	−0.575928	+	0.817500i	$0.695359\pi$
$402$	0	0
$403$	−32.8056	−1.63417
$404$	−12.1936	−0.606655
$405$	0	0
$406$	13.2759	0.658872
$407$	−6.55397	−0.324868
$408$	0	0
$409$	−15.0193	−0.742658	−0.371329	−	0.928501i	$-0.621098\pi$
$409$	−15.0193	−0.742658	−0.371329	+	0.928501i	$0.621098\pi$
$410$	−17.4642	−0.862494
$411$	0	0
$412$	10.4749	0.516062
$413$	0.989598	0.0486949
$414$	0	0
$415$	17.1699	0.842836
$416$	24.7258	1.21228
$417$	0	0
$418$	−2.98109	−0.145810
$419$	39.8046	1.94458	0.972291	−	0.233776i	$-0.0751081\pi$
$419$	39.8046	1.94458	0.972291	+	0.233776i	$0.0751081\pi$
$420$	0	0
$421$	−10.1319	−0.493799	−0.246900	−	0.969041i	$-0.579412\pi$
$421$	−10.1319	−0.493799	−0.246900	+	0.969041i	$0.579412\pi$
$422$	18.3697	0.894224
$423$	0	0
$424$	−11.6022	−0.563454
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	14.1177	0.683206
$428$	5.50430	0.266060
$429$	0	0
$430$	−2.97591	−0.143511
$431$	0.281719	0.0135699	0.00678496	−	0.999977i	$-0.497840\pi$
$431$	0.281719	0.0135699	0.00678496	+	0.999977i	$0.497840\pi$
$432$	0	0
$433$	8.21174	0.394631	0.197316	−	0.980340i	$-0.436778\pi$
$433$	8.21174	0.394631	0.197316	+	0.980340i	$0.436778\pi$
$434$	10.2306	0.491085
$435$	0	0
$436$	6.54428	0.313414
$437$	−4.90503	−0.234639
$438$	0	0
$439$	−8.22156	−0.392394	−0.196197	−	0.980565i	$-0.562859\pi$
$439$	−8.22156	−0.392394	−0.196197	+	0.980565i	$0.562859\pi$
$440$	3.12706	0.149077
$441$	0	0
$442$	9.16665	0.436013
$443$	26.1735	1.24354	0.621770	−	0.783200i	$-0.286414\pi$
$443$	26.1735	1.24354	0.621770	+	0.783200i	$0.286414\pi$
$444$	0	0
$445$	−0.0644430	−0.00305489
$446$	−40.4939	−1.91744
$447$	0	0
$448$	2.24918	0.106264
$449$	5.35784	0.252852	0.126426	−	0.991976i	$-0.459649\pi$
$449$	5.35784	0.252852	0.126426	+	0.991976i	$0.459649\pi$
$450$	0	0
$451$	16.7382	0.788171
$452$	−11.1006	−0.522129
$453$	0	0
$454$	17.7050	0.830937
$455$	5.42161	0.254169
$456$	0	0
$457$	17.9996	0.841986	0.420993	−	0.907064i	$-0.361682\pi$
$457$	17.9996	0.841986	0.420993	+	0.907064i	$0.361682\pi$
$458$	−3.24852	−0.151794
$459$	0	0
$460$	−3.87094	−0.180484
$461$	−2.87946	−0.134110	−0.0670549	−	0.997749i	$-0.521360\pi$
$461$	−2.87946	−0.134110	−0.0670549	+	0.997749i	$0.521360\pi$
$462$	0	0
$463$	−6.67598	−0.310259	−0.155129	−	0.987894i	$-0.549579\pi$
$463$	−6.67598	−0.310259	−0.155129	+	0.987894i	$0.549579\pi$
$464$	−39.1033	−1.81533
$465$	0	0
$466$	−23.1731	−1.07347
$467$	11.7976	0.545927	0.272963	−	0.962024i	$-0.411996\pi$
$467$	11.7976	0.545927	0.272963	+	0.962024i	$0.411996\pi$
$468$	0	0
$469$	7.38152	0.340847
$470$	9.76439	0.450398
$471$	0	0
$472$	−1.90964	−0.0878984
$473$	2.85220	0.131144
$474$	0	0
$475$	−1.08805	−0.0499233
$476$	−0.858669	−0.0393570
$477$	0	0
$478$	−43.1946	−1.97567
$479$	−6.16642	−0.281751	−0.140875	−	0.990027i	$-0.544992\pi$
$479$	−6.16642	−0.281751	−0.140875	+	0.990027i	$0.544992\pi$
$480$	0	0
$481$	21.9275	0.999810
$482$	5.19151	0.236467
$483$	0	0
$484$	−7.19054	−0.326843
$485$	−18.7340	−0.850666
$486$	0	0
$487$	8.83593	0.400394	0.200197	−	0.979756i	$-0.435842\pi$
$487$	8.83593	0.400394	0.200197	+	0.979756i	$0.435842\pi$
$488$	−27.2432	−1.23324
$489$	0	0
$490$	−1.69076	−0.0763807
$491$	−41.2335	−1.86084	−0.930420	−	0.366494i	$-0.880558\pi$
$491$	−41.2335	−1.86084	−0.930420	+	0.366494i	$0.880558\pi$
$492$	0	0
$493$	−7.85204	−0.353638
$494$	9.97380	0.448743
$495$	0	0
$496$	−30.1336	−1.35304
$497$	1.62048	0.0726883
$498$	0	0
$499$	−24.2674	−1.08636	−0.543180	−	0.839616i	$-0.682780\pi$
$499$	−24.2674	−1.08636	−0.543180	+	0.839616i	$0.682780\pi$
$500$	−0.858669	−0.0384009
$501$	0	0
$502$	42.5617	1.89962
$503$	−6.76304	−0.301549	−0.150775	−	0.988568i	$-0.548177\pi$
$503$	−6.76304	−0.301549	−0.150775	+	0.988568i	$0.548177\pi$
$504$	0	0
$505$	14.2006	0.631918
$506$	12.3514	0.549086
$507$	0	0
$508$	4.94764	0.219516
$509$	−34.9069	−1.54722	−0.773611	−	0.633661i	$-0.781551\pi$
$509$	−34.9069	−1.54722	−0.773611	+	0.633661i	$0.781551\pi$
$510$	0	0
$511$	−12.8299	−0.567561
$512$	3.49181	0.154318
$513$	0	0
$514$	−42.0756	−1.85588
$515$	−12.1990	−0.537553
$516$	0	0
$517$	−9.35849	−0.411586
$518$	−6.83823	−0.300454
$519$	0	0
$520$	−10.4622	−0.458797
$521$	−11.8554	−0.519394	−0.259697	−	0.965690i	$-0.583623\pi$
$521$	−11.8554	−0.519394	−0.259697	+	0.965690i	$0.583623\pi$
$522$	0	0
$523$	−20.8484	−0.911638	−0.455819	−	0.890072i	$-0.650654\pi$
$523$	−20.8484	−0.911638	−0.455819	+	0.890072i	$0.650654\pi$
$524$	6.48879	0.283464
$525$	0	0
$526$	20.8603	0.909552
$527$	−6.05090	−0.263581
$528$	0	0
$529$	−2.67727	−0.116403
$530$	−10.1655	−0.441562
$531$	0	0
$532$	−0.934278	−0.0405061
$533$	−56.0009	−2.42567
$534$	0	0
$535$	−6.41027	−0.277140
$536$	−14.2442	−0.615257
$537$	0	0
$538$	49.0839	2.11616
$539$	1.62048	0.0697989
$540$	0	0
$541$	−25.6969	−1.10479	−0.552397	−	0.833581i	$-0.686287\pi$
$541$	−25.6969	−1.10479	−0.552397	+	0.833581i	$0.686287\pi$
$542$	−23.7532	−1.02029
$543$	0	0
$544$	4.56060	0.195534
$545$	−7.62142	−0.326466
$546$	0	0
$547$	13.0776	0.559158	0.279579	−	0.960123i	$-0.409805\pi$
$547$	13.0776	0.559158	0.279579	+	0.960123i	$0.409805\pi$
$548$	5.16266	0.220538
$549$	0	0
$550$	2.73984	0.116827
$551$	−8.54344	−0.363963
$552$	0	0
$553$	15.0920	0.641779
$554$	42.7218	1.81508
$555$	0	0
$556$	19.1951	0.814052
$557$	25.5840	1.08403	0.542015	−	0.840369i	$-0.317661\pi$
$557$	25.5840	1.08403	0.542015	+	0.840369i	$0.317661\pi$
$558$	0	0
$559$	−9.54259	−0.403608
$560$	4.98003	0.210444
$561$	0	0
$562$	7.12331	0.300479
$563$	9.60151	0.404655	0.202328	−	0.979318i	$-0.435149\pi$
$563$	9.60151	0.404655	0.202328	+	0.979318i	$0.435149\pi$
$564$	0	0
$565$	12.9277	0.543872
$566$	47.4544	1.99466
$567$	0	0
$568$	−3.12706	−0.131209
$569$	−10.8330	−0.454142	−0.227071	−	0.973878i	$-0.572915\pi$
$569$	−10.8330	−0.454142	−0.227071	+	0.973878i	$0.572915\pi$
$570$	0	0
$571$	−31.9564	−1.33734	−0.668668	−	0.743561i	$-0.733135\pi$
$571$	−31.9564	−1.33734	−0.668668	+	0.743561i	$0.733135\pi$
$572$	−7.54392	−0.315427
$573$	0	0
$574$	17.4642	0.728941
$575$	4.50807	0.188000
$576$	0	0
$577$	11.7556	0.489394	0.244697	−	0.969600i	$-0.421312\pi$
$577$	11.7556	0.489394	0.244697	+	0.969600i	$0.421312\pi$
$578$	1.69076	0.0703263
$579$	0	0
$580$	−6.74230	−0.279959
$581$	−17.1699	−0.712327
$582$	0	0
$583$	9.74295	0.403512
$584$	24.7580	1.02450
$585$	0	0
$586$	−54.1643	−2.23751
$587$	−33.5758	−1.38582	−0.692912	−	0.721023i	$-0.743672\pi$
$587$	−33.5758	−1.38582	−0.692912	+	0.721023i	$0.743672\pi$
$588$	0	0
$589$	−6.58370	−0.271277
$590$	−1.67317	−0.0688834
$591$	0	0
$592$	20.1416	0.827813
$593$	−10.6476	−0.437245	−0.218623	−	0.975809i	$-0.570156\pi$
$593$	−10.6476	−0.437245	−0.218623	+	0.975809i	$0.570156\pi$
$594$	0	0
$595$	1.00000	0.0409960
$596$	12.3644	0.506465
$597$	0	0
$598$	−41.3239	−1.68986
$599$	21.5281	0.879613	0.439807	−	0.898093i	$-0.355047\pi$
$599$	21.5281	0.879613	0.439807	+	0.898093i	$0.355047\pi$
$600$	0	0
$601$	−10.9790	−0.447841	−0.223921	−	0.974607i	$-0.571886\pi$
$601$	−10.9790	−0.447841	−0.223921	+	0.974607i	$0.571886\pi$
$602$	2.97591	0.121289
$603$	0	0
$604$	−16.2845	−0.662609
$605$	8.37406	0.340454
$606$	0	0
$607$	25.9087	1.05160	0.525800	−	0.850608i	$-0.323766\pi$
$607$	25.9087	1.05160	0.525800	+	0.850608i	$0.323766\pi$
$608$	4.96217	0.201243
$609$	0	0
$610$	−23.8697	−0.966457
$611$	31.3106	1.26669
$612$	0	0
$613$	42.8240	1.72964	0.864822	−	0.502079i	$-0.167431\pi$
$613$	42.8240	1.72964	0.864822	+	0.502079i	$0.167431\pi$
$614$	20.6707	0.834202
$615$	0	0
$616$	−3.12706	−0.125993
$617$	−10.5080	−0.423036	−0.211518	−	0.977374i	$-0.567841\pi$
$617$	−10.5080	−0.423036	−0.211518	+	0.977374i	$0.567841\pi$
$618$	0	0
$619$	−22.5767	−0.907436	−0.453718	−	0.891145i	$-0.649903\pi$
$619$	−22.5767	−0.907436	−0.453718	+	0.891145i	$0.649903\pi$
$620$	−5.19572	−0.208665
$621$	0	0
$622$	4.14081	0.166031
$623$	0.0644430	0.00258186
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.2153	−0.887901
$627$	0	0
$628$	−1.33774	−0.0533815
$629$	4.04447	0.161264
$630$	0	0
$631$	2.78794	0.110986	0.0554930	−	0.998459i	$-0.482327\pi$
$631$	2.78794	0.110986	0.0554930	+	0.998459i	$0.482327\pi$
$632$	−29.1234	−1.15847
$633$	0	0
$634$	−55.7469	−2.21399
$635$	−5.76199	−0.228657
$636$	0	0
$637$	−5.42161	−0.214812
$638$	21.5133	0.851720
$639$	0	0
$640$	−12.9240	−0.510866
$641$	41.2155	1.62792	0.813958	−	0.580924i	$-0.197309\pi$
$641$	41.2155	1.62792	0.813958	+	0.580924i	$0.197309\pi$
$642$	0	0
$643$	47.7104	1.88151	0.940757	−	0.339080i	$-0.110116\pi$
$643$	47.7104	1.88151	0.940757	+	0.339080i	$0.110116\pi$
$644$	3.87094	0.152537
$645$	0	0
$646$	1.83964	0.0723796
$647$	−24.6250	−0.968107	−0.484053	−	0.875038i	$-0.660836\pi$
$647$	−24.6250	−0.968107	−0.484053	+	0.875038i	$0.660836\pi$
$648$	0	0
$649$	1.60362	0.0629476
$650$	−9.16665	−0.359546
$651$	0	0
$652$	5.35231	0.209613
$653$	22.4576	0.878832	0.439416	−	0.898284i	$-0.355185\pi$
$653$	22.4576	0.878832	0.439416	+	0.898284i	$0.355185\pi$
$654$	0	0
$655$	−7.55680	−0.295268
$656$	−51.4396	−2.00838
$657$	0	0
$658$	−9.76439	−0.380656
$659$	−47.5036	−1.85048	−0.925240	−	0.379383i	$-0.876136\pi$
$659$	−47.5036	−1.85048	−0.925240	+	0.379383i	$0.876136\pi$
$660$	0	0
$661$	−4.57327	−0.177880	−0.0889398	−	0.996037i	$-0.528348\pi$
$661$	−4.57327	−0.177880	−0.0889398	+	0.996037i	$0.528348\pi$
$662$	54.1240	2.10359
$663$	0	0
$664$	33.1330	1.28581
$665$	1.08805	0.0421929
$666$	0	0
$667$	35.3976	1.37060
$668$	0.717695	0.0277684
$669$	0	0
$670$	−12.4804	−0.482159
$671$	22.8775	0.883175
$672$	0	0
$673$	−20.6892	−0.797511	−0.398755	−	0.917057i	$-0.630558\pi$
$673$	−20.6892	−0.797511	−0.398755	+	0.917057i	$0.630558\pi$
$674$	17.7375	0.683222
$675$	0	0
$676$	14.0769	0.541421
$677$	−50.0899	−1.92511	−0.962556	−	0.271083i	$-0.912618\pi$
$677$	−50.0899	−1.92511	−0.962556	+	0.271083i	$0.912618\pi$
$678$	0	0
$679$	18.7340	0.718944
$680$	−1.92972	−0.0740012
$681$	0	0
$682$	16.5785	0.634823
$683$	−20.7030	−0.792177	−0.396088	−	0.918212i	$-0.629633\pi$
$683$	−20.7030	−0.792177	−0.396088	+	0.918212i	$0.629633\pi$
$684$	0	0
$685$	−6.01240	−0.229722
$686$	1.69076	0.0645535
$687$	0	0
$688$	−8.76535	−0.334176
$689$	−32.5969	−1.24184
$690$	0	0
$691$	−50.6828	−1.92806	−0.964032	−	0.265786i	$-0.914368\pi$
$691$	−50.6828	−1.92806	−0.964032	+	0.265786i	$0.914368\pi$
$692$	−13.6961	−0.520647
$693$	0	0
$694$	11.3984	0.432678
$695$	−22.3544	−0.847952
$696$	0	0
$697$	−10.3292	−0.391246
$698$	41.1080	1.55596
$699$	0	0
$700$	0.858669	0.0324546
$701$	10.3674	0.391573	0.195786	−	0.980647i	$-0.437274\pi$
$701$	10.3674	0.391573	0.195786	+	0.980647i	$0.437274\pi$
$702$	0	0
$703$	4.40060	0.165972
$704$	3.64474	0.137366
$705$	0	0
$706$	45.0623	1.69594
$707$	−14.2006	−0.534069
$708$	0	0
$709$	36.3595	1.36551	0.682754	−	0.730648i	$-0.260782\pi$
$709$	36.3595	1.36551	0.682754	+	0.730648i	$0.260782\pi$
$710$	−2.73984	−0.102824
$711$	0	0
$712$	−0.124357	−0.00466047
$713$	27.2779	1.02157
$714$	0	0
$715$	8.78560	0.328563
$716$	−3.72453	−0.139192
$717$	0	0
$718$	31.4242	1.17274
$719$	−40.4560	−1.50875	−0.754377	−	0.656441i	$-0.772061\pi$
$719$	−40.4560	−1.50875	−0.754377	+	0.656441i	$0.772061\pi$
$720$	0	0
$721$	12.1990	0.454315
$722$	−30.1228	−1.12106
$723$	0	0
$724$	−14.0096	−0.520662
$725$	7.85204	0.291617
$726$	0	0
$727$	12.4785	0.462800	0.231400	−	0.972859i	$-0.425669\pi$
$727$	12.4785	0.462800	0.231400	+	0.972859i	$0.425669\pi$
$728$	10.4622	0.387754
$729$	0	0
$730$	21.6923	0.802866
$731$	−1.76010	−0.0650997
$732$	0	0
$733$	−6.21268	−0.229471	−0.114735	−	0.993396i	$-0.536602\pi$
$733$	−6.21268	−0.229471	−0.114735	+	0.993396i	$0.536602\pi$
$734$	−1.19290	−0.0440308
$735$	0	0
$736$	−20.5595	−0.757833
$737$	11.9616	0.440610
$738$	0	0
$739$	−26.0604	−0.958646	−0.479323	−	0.877639i	$-0.659118\pi$
$739$	−26.0604	−0.958646	−0.479323	+	0.877639i	$0.659118\pi$
$740$	3.47286	0.127665
$741$	0	0
$742$	10.1655	0.373188
$743$	17.5058	0.642225	0.321113	−	0.947041i	$-0.395943\pi$
$743$	17.5058	0.642225	0.321113	+	0.947041i	$0.395943\pi$
$744$	0	0
$745$	−14.3995	−0.527556
$746$	59.2093	2.16781
$747$	0	0
$748$	−1.39145	−0.0508766
$749$	6.41027	0.234226
$750$	0	0
$751$	−51.3508	−1.87382	−0.936909	−	0.349574i	$-0.886326\pi$
$751$	−51.3508	−1.87382	−0.936909	+	0.349574i	$0.886326\pi$
$752$	28.7604	1.04878
$753$	0	0
$754$	−71.9769	−2.62124
$755$	18.9649	0.690202
$756$	0	0
$757$	21.2497	0.772334	0.386167	−	0.922429i	$-0.373799\pi$
$757$	21.2497	0.772334	0.386167	+	0.922429i	$0.373799\pi$
$758$	54.4686	1.97839
$759$	0	0
$760$	−2.09964	−0.0761618
$761$	−19.6504	−0.712326	−0.356163	−	0.934424i	$-0.615915\pi$
$761$	−19.6504	−0.712326	−0.356163	+	0.934424i	$0.615915\pi$
$762$	0	0
$763$	7.62142	0.275914
$764$	7.51927	0.272038
$765$	0	0
$766$	−1.97907	−0.0715069
$767$	−5.36522	−0.193727
$768$	0	0
$769$	−37.2847	−1.34452	−0.672261	−	0.740314i	$-0.734677\pi$
$769$	−37.2847	−1.34452	−0.672261	+	0.740314i	$0.734677\pi$
$770$	−2.73984	−0.0987369
$771$	0	0
$772$	−5.47215	−0.196947
$773$	−47.9916	−1.72614	−0.863069	−	0.505086i	$-0.831461\pi$
$773$	−47.9916	−1.72614	−0.863069	+	0.505086i	$0.831461\pi$
$774$	0	0
$775$	6.05090	0.217355
$776$	−36.1512	−1.29775
$777$	0	0
$778$	−36.2772	−1.30060
$779$	−11.2387	−0.402669
$780$	0	0
$781$	2.62594	0.0939636
$782$	−7.62207	−0.272565
$783$	0	0
$784$	−4.98003	−0.177858
$785$	1.55792	0.0556045
$786$	0	0
$787$	−44.6932	−1.59314	−0.796570	−	0.604546i	$-0.793354\pi$
$787$	−44.6932	−1.59314	−0.796570	+	0.604546i	$0.793354\pi$
$788$	−0.0114559	−0.000408099 0
$789$	0	0
$790$	−25.5170	−0.907855
$791$	−12.9277	−0.459656
$792$	0	0
$793$	−76.5410	−2.71805
$794$	−8.01581	−0.284470
$795$	0	0
$796$	−0.467266	−0.0165618
$797$	18.0210	0.638335	0.319168	−	0.947698i	$-0.396597\pi$
$797$	18.0210	0.638335	0.319168	+	0.947698i	$0.396597\pi$
$798$	0	0
$799$	5.77515	0.204310
$800$	−4.56060	−0.161241
$801$	0	0
$802$	−38.9989	−1.37710
$803$	−20.7905	−0.733682
$804$	0	0
$805$	−4.50807	−0.158889
$806$	−55.4665	−1.95372
$807$	0	0
$808$	27.4031	0.964039
$809$	38.0397	1.33740	0.668702	−	0.743531i	$-0.266850\pi$
$809$	38.0397	1.33740	0.668702	+	0.743531i	$0.266850\pi$
$810$	0	0
$811$	6.64290	0.233264	0.116632	−	0.993175i	$-0.462790\pi$
$811$	6.64290	0.233264	0.116632	+	0.993175i	$0.462790\pi$
$812$	6.74230	0.236608
$813$	0	0
$814$	−11.0812	−0.388395
$815$	−6.23327	−0.218342
$816$	0	0
$817$	−1.91508	−0.0670003
$818$	−25.3941	−0.887884
$819$	0	0
$820$	−8.86935	−0.309731
$821$	17.2460	0.601888	0.300944	−	0.953642i	$-0.402698\pi$
$821$	17.2460	0.601888	0.300944	+	0.953642i	$0.402698\pi$
$822$	0	0
$823$	9.45086	0.329436	0.164718	−	0.986341i	$-0.447329\pi$
$823$	9.45086	0.329436	0.164718	+	0.986341i	$0.447329\pi$
$824$	−23.5406	−0.820077
$825$	0	0
$826$	1.67317	0.0582171
$827$	6.38677	0.222090	0.111045	−	0.993815i	$-0.464580\pi$
$827$	6.38677	0.222090	0.111045	+	0.993815i	$0.464580\pi$
$828$	0	0
$829$	43.9066	1.52494	0.762470	−	0.647024i	$-0.223987\pi$
$829$	43.9066	1.52494	0.762470	+	0.647024i	$0.223987\pi$
$830$	29.0301	1.00765
$831$	0	0
$832$	−12.1942	−0.422757
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−0.835822	−0.0289248
$836$	−1.51398	−0.0523619
$837$	0	0
$838$	67.3000	2.32484
$839$	−33.6367	−1.16127	−0.580634	−	0.814165i	$-0.697195\pi$
$839$	−33.6367	−1.16127	−0.580634	+	0.814165i	$0.697195\pi$
$840$	0	0
$841$	32.6545	1.12602
$842$	−17.1306	−0.590361
$843$	0	0
$844$	9.32925	0.321126
$845$	−16.3939	−0.563967
$846$	0	0
$847$	−8.37406	−0.287736
$848$	−29.9419	−1.02821
$849$	0	0
$850$	−1.69076	−0.0579926
$851$	−18.2328	−0.625011
$852$	0	0
$853$	23.8336	0.816045	0.408023	−	0.912972i	$-0.366218\pi$
$853$	23.8336	0.816045	0.408023	+	0.912972i	$0.366218\pi$
$854$	23.8697	0.816805
$855$	0	0
$856$	−12.3700	−0.422798
$857$	−41.7174	−1.42504	−0.712520	−	0.701652i	$-0.752446\pi$
$857$	−41.7174	−1.42504	−0.712520	+	0.701652i	$0.752446\pi$
$858$	0	0
$859$	−9.06004	−0.309125	−0.154562	−	0.987983i	$-0.549397\pi$
$859$	−9.06004	−0.309125	−0.154562	+	0.987983i	$0.549397\pi$
$860$	−1.51134	−0.0515364
$861$	0	0
$862$	0.476319	0.0162235
$863$	10.7594	0.366253	0.183126	−	0.983089i	$-0.441378\pi$
$863$	10.7594	0.366253	0.183126	+	0.983089i	$0.441378\pi$
$864$	0	0
$865$	15.9504	0.542328
$866$	13.8841	0.471800
$867$	0	0
$868$	5.19572	0.176354
$869$	24.4563	0.829623
$870$	0	0
$871$	−40.0198	−1.35602
$872$	−14.7072	−0.498048
$873$	0	0
$874$	−8.29322	−0.280522
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	33.3120	1.12487	0.562434	−	0.826842i	$-0.309865\pi$
$877$	33.3120	1.12487	0.562434	+	0.826842i	$0.309865\pi$
$878$	−13.9007	−0.469126
$879$	0	0
$880$	8.07001	0.272040
$881$	45.0324	1.51718	0.758591	−	0.651568i	$-0.225888\pi$
$881$	45.0324	1.51718	0.758591	+	0.651568i	$0.225888\pi$
$882$	0	0
$883$	16.2185	0.545795	0.272897	−	0.962043i	$-0.412018\pi$
$883$	16.2185	0.545795	0.272897	+	0.962043i	$0.412018\pi$
$884$	4.65537	0.156577
$885$	0	0
$886$	44.2531	1.48671
$887$	10.6266	0.356806	0.178403	−	0.983958i	$-0.442907\pi$
$887$	10.6266	0.356806	0.178403	+	0.983958i	$0.442907\pi$
$888$	0	0
$889$	5.76199	0.193251
$890$	−0.108958	−0.00365227
$891$	0	0
$892$	−20.5652	−0.688574
$893$	6.28367	0.210275
$894$	0	0
$895$	4.33756	0.144989
$896$	12.9240	0.431761
$897$	0	0
$898$	9.05881	0.302297
$899$	47.5119	1.58461
$900$	0	0
$901$	−6.01240	−0.200302
$902$	28.3003	0.942296
$903$	0	0
$904$	24.9468	0.829717
$905$	16.3155	0.542344
$906$	0	0
$907$	22.4341	0.744912	0.372456	−	0.928050i	$-0.378516\pi$
$907$	22.4341	0.744912	0.372456	+	0.928050i	$0.378516\pi$
$908$	8.99167	0.298399
$909$	0	0
$910$	9.16665	0.303871
$911$	−21.3389	−0.706990	−0.353495	−	0.935436i	$-0.615007\pi$
$911$	−21.3389	−0.706990	−0.353495	+	0.935436i	$0.615007\pi$
$912$	0	0
$913$	−27.8234	−0.920819
$914$	30.4330	1.00663
$915$	0	0
$916$	−1.64979	−0.0545107
$917$	7.55680	0.249547
$918$	0	0
$919$	29.9266	0.987187	0.493594	−	0.869693i	$-0.335683\pi$
$919$	29.9266	0.987187	0.493594	+	0.869693i	$0.335683\pi$
$920$	8.69930	0.286808
$921$	0	0
$922$	−4.86848	−0.160335
$923$	−8.78560	−0.289181
$924$	0	0
$925$	−4.04447	−0.132981
$926$	−11.2875	−0.370929
$927$	0	0
$928$	−35.8100	−1.17552
$929$	−10.8491	−0.355948	−0.177974	−	0.984035i	$-0.556954\pi$
$929$	−10.8491	−0.355948	−0.177974	+	0.984035i	$0.556954\pi$
$930$	0	0
$931$	−1.08805	−0.0356595
$932$	−11.7687	−0.385496
$933$	0	0
$934$	19.9469	0.652681
$935$	1.62048	0.0529952
$936$	0	0
$937$	−23.9251	−0.781600	−0.390800	−	0.920476i	$-0.627802\pi$
$937$	−23.9251	−0.781600	−0.390800	+	0.920476i	$0.627802\pi$
$938$	12.4804	0.407499
$939$	0	0
$940$	4.95894	0.161743
$941$	13.9999	0.456383	0.228192	−	0.973616i	$-0.426719\pi$
$941$	13.9999	0.456383	0.228192	+	0.973616i	$0.426719\pi$
$942$	0	0
$943$	46.5647	1.51636
$944$	−4.92822	−0.160400
$945$	0	0
$946$	4.82239	0.156789
$947$	−48.4210	−1.57347	−0.786735	−	0.617291i	$-0.788230\pi$
$947$	−48.4210	−1.57347	−0.786735	+	0.617291i	$0.788230\pi$
$948$	0	0
$949$	69.5587	2.25797
$950$	−1.83964	−0.0596857
$951$	0	0
$952$	1.92972	0.0625425
$953$	−14.3412	−0.464557	−0.232279	−	0.972649i	$-0.574618\pi$
$953$	−14.3412	−0.464557	−0.232279	+	0.972649i	$0.574618\pi$
$954$	0	0
$955$	−8.75689	−0.283366
$956$	−21.9368	−0.709487
$957$	0	0
$958$	−10.4259	−0.336847
$959$	6.01240	0.194150
$960$	0	0
$961$	5.61337	0.181077
$962$	37.0742	1.19532
$963$	0	0
$964$	2.63656	0.0849178
$965$	6.37282	0.205148
$966$	0	0
$967$	9.86697	0.317300	0.158650	−	0.987335i	$-0.449286\pi$
$967$	9.86697	0.317300	0.158650	+	0.987335i	$0.449286\pi$
$968$	16.1596	0.519388
$969$	0	0
$970$	−31.6746	−1.01701
$971$	−12.2328	−0.392568	−0.196284	−	0.980547i	$-0.562888\pi$
$971$	−12.2328	−0.392568	−0.196284	+	0.980547i	$0.562888\pi$
$972$	0	0
$973$	22.3544	0.716651
$974$	14.9394	0.478690
$975$	0	0
$976$	−70.3067	−2.25046
$977$	32.7565	1.04797	0.523986	−	0.851727i	$-0.324444\pi$
$977$	32.7565	1.04797	0.523986	+	0.851727i	$0.324444\pi$
$978$	0	0
$979$	0.104428	0.00333755
$980$	−0.858669	−0.0274292
$981$	0	0
$982$	−69.7159	−2.22472
$983$	2.76609	0.0882247	0.0441124	−	0.999027i	$-0.485954\pi$
$983$	2.76609	0.0882247	0.0441124	+	0.999027i	$0.485954\pi$
$984$	0	0
$985$	0.0133415	0.000425094 0
$986$	−13.2759	−0.422791
$987$	0	0
$988$	5.06530	0.161149
$989$	7.93466	0.252308
$990$	0	0
$991$	−13.1787	−0.418634	−0.209317	−	0.977848i	$-0.567124\pi$
$991$	−13.1787	−0.418634	−0.209317	+	0.977848i	$0.567124\pi$
$992$	−27.5957	−0.876164
$993$	0	0
$994$	2.73984	0.0869023
$995$	0.544175	0.0172515
$996$	0	0
$997$	−32.6091	−1.03274	−0.516370	−	0.856366i	$-0.672717\pi$
$997$	−32.6091	−1.03274	−0.516370	+	0.856366i	$0.672717\pi$
$998$	−41.0304	−1.29880
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5355.2.a.bx.1.6	✓	7
3.2	odd	2		5355.2.a.by.1.2	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5355.2.a.bx.1.6	✓	7	1.1	even	1	trivial
5355.2.a.by.1.2	yes	7	3.2	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 5355 = 3^{2} \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5355.a (trivial)

\(f(q)\)	\(=\)	\(q+1.69076 q^{2} +0.858669 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.92972 q^{8} +O(q^{10})\)	\(q+1.69076 q^{2} +0.858669 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.92972 q^{8} -1.69076 q^{10} +1.62048 q^{11} -5.42161 q^{13} +1.69076 q^{14} -4.98003 q^{16} -1.00000 q^{17} -1.08805 q^{19} -0.858669 q^{20} +2.73984 q^{22} +4.50807 q^{23} +1.00000 q^{25} -9.16665 q^{26} +0.858669 q^{28} +7.85204 q^{29} +6.05090 q^{31} -4.56060 q^{32} -1.69076 q^{34} -1.00000 q^{35} -4.04447 q^{37} -1.83964 q^{38} +1.92972 q^{40} +10.3292 q^{41} +1.76010 q^{43} +1.39145 q^{44} +7.62207 q^{46} -5.77515 q^{47} +1.00000 q^{49} +1.69076 q^{50} -4.65537 q^{52} +6.01240 q^{53} -1.62048 q^{55} -1.92972 q^{56} +13.2759 q^{58} +0.989598 q^{59} +14.1177 q^{61} +10.2306 q^{62} +2.24918 q^{64} +5.42161 q^{65} +7.38152 q^{67} -0.858669 q^{68} -1.69076 q^{70} +1.62048 q^{71} -12.8299 q^{73} -6.83823 q^{74} -0.934278 q^{76} +1.62048 q^{77} +15.0920 q^{79} +4.98003 q^{80} +17.4642 q^{82} -17.1699 q^{83} +1.00000 q^{85} +2.97591 q^{86} -3.12706 q^{88} +0.0644430 q^{89} -5.42161 q^{91} +3.87094 q^{92} -9.76439 q^{94} +1.08805 q^{95} +18.7340 q^{97} +1.69076 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) \) 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8	\( 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8} + q^{10} - 6 q^{11} + 16 q^{13} - q^{14} + 11 q^{16} - 7 q^{17} + 6 q^{19} - 11 q^{20} - 6 q^{22} - 14 q^{23} + 7 q^{25} - 16 q^{26} + 11 q^{28} - 12 q^{29} + 12 q^{31} - 29 q^{32} + q^{34} - 7 q^{35} + 4 q^{37} + 30 q^{38} + 9 q^{40} + 36 q^{41} + 6 q^{43} + 18 q^{44} + 14 q^{46} + 8 q^{47} + 7 q^{49} - q^{50} + 44 q^{52} + 18 q^{53} + 6 q^{55} - 9 q^{56} + 48 q^{58} - 12 q^{59} + 38 q^{61} + 24 q^{62} + 31 q^{64} - 16 q^{65} + 26 q^{67} - 11 q^{68} + q^{70} - 6 q^{71} - 8 q^{73} - 22 q^{74} - 30 q^{76} - 6 q^{77} + 6 q^{79} - 11 q^{80} - 18 q^{82} + 7 q^{85} + 78 q^{86} - 102 q^{88} + 14 q^{89} + 16 q^{91} - 10 q^{92} - 14 q^{94} - 6 q^{95} + 20 q^{97} - q^{98}+O(q^{100}) \) 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8 + q^10 - 6 * q^11 + 16 * q^13 - q^14 + 11 * q^16 - 7 * q^17 + 6 * q^19 - 11 * q^20 - 6 * q^22 - 14 * q^23 + 7 * q^25 - 16 * q^26 + 11 * q^28 - 12 * q^29 + 12 * q^31 - 29 * q^32 + q^34 - 7 * q^35 + 4 * q^37 + 30 * q^38 + 9 * q^40 + 36 * q^41 + 6 * q^43 + 18 * q^44 + 14 * q^46 + 8 * q^47 + 7 * q^49 - q^50 + 44 * q^52 + 18 * q^53 + 6 * q^55 - 9 * q^56 + 48 * q^58 - 12 * q^59 + 38 * q^61 + 24 * q^62 + 31 * q^64 - 16 * q^65 + 26 * q^67 - 11 * q^68 + q^70 - 6 * q^71 - 8 * q^73 - 22 * q^74 - 30 * q^76 - 6 * q^77 + 6 * q^79 - 11 * q^80 - 18 * q^82 + 7 * q^85 + 78 * q^86 - 102 * q^88 + 14 * q^89 + 16 * q^91 - 10 * q^92 - 14 * q^94 - 6 * q^95 + 20 * q^97 - q^98

Introduction

L-functions

Modular forms

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