LMFDB - Embedded newform 2550.2.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(1,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.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:	$20.3618525154$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.363328$ of defining polynomial
Character	$\chi$	$=$	2550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.50466 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.726656 q^{11} +1.00000 q^{12} +2.00000 q^{13} -1.50466 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +2.72666 q^{19} -1.50466 q^{21} -0.726656 q^{22} +7.78734 q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} -1.50466 q^{28} -2.05135 q^{29} +3.22199 q^{31} +1.00000 q^{32} -0.726656 q^{33} -1.00000 q^{34} +1.00000 q^{36} +8.23132 q^{37} +2.72666 q^{38} +2.00000 q^{39} +3.55602 q^{41} -1.50466 q^{42} -1.27334 q^{43} -0.726656 q^{44} +7.78734 q^{46} -1.55602 q^{47} +1.00000 q^{48} -4.73599 q^{49} -1.00000 q^{51} +2.00000 q^{52} +9.00933 q^{53} +1.00000 q^{54} -1.50466 q^{56} +2.72666 q^{57} -2.05135 q^{58} -1.71733 q^{59} +11.2406 q^{61} +3.22199 q^{62} -1.50466 q^{63} +1.00000 q^{64} -0.726656 q^{66} +8.56534 q^{67} -1.00000 q^{68} +7.78734 q^{69} -10.3340 q^{71} +1.00000 q^{72} -15.2920 q^{73} +8.23132 q^{74} +2.72666 q^{76} +1.09337 q^{77} +2.00000 q^{78} +4.77801 q^{79} +1.00000 q^{81} +3.55602 q^{82} +1.27334 q^{83} -1.50466 q^{84} -1.27334 q^{86} -2.05135 q^{87} -0.726656 q^{88} +2.28267 q^{89} -3.00933 q^{91} +7.78734 q^{92} +3.22199 q^{93} -1.55602 q^{94} +1.00000 q^{96} +12.2827 q^{97} -4.73599 q^{98} -0.726656 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9} + 2 q^{11} + 3 q^{12} + 6 q^{13} + 6 q^{14} + 3 q^{16} - 3 q^{17} + 3 q^{18} + 4 q^{19} + 6 q^{21} + 2 q^{22} - 4 q^{23} + 3 q^{24}+ \cdots + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9 + 2 * q^11 + 3 * q^12 + 6 * q^13 + 6 * q^14 + 3 * q^16 - 3 * q^17 + 3 * q^18 + 4 * q^19 + 6 * q^21 + 2 * q^22 - 4 * q^23 + 3 * q^24 + 6 * q^26 + 3 * q^27 + 6 * q^28 - 4 * q^29 + 16 * q^31 + 3 * q^32 + 2 * q^33 - 3 * q^34 + 3 * q^36 + 10 * q^37 + 4 * q^38 + 6 * q^39 - 2 * q^41 + 6 * q^42 - 8 * q^43 + 2 * q^44 - 4 * q^46 + 8 * q^47 + 3 * q^48 + 11 * q^49 - 3 * q^51 + 6 * q^52 + 6 * q^53 + 3 * q^54 + 6 * q^56 + 4 * q^57 - 4 * q^58 - 22 * q^59 - 2 * q^61 + 16 * q^62 + 6 * q^63 + 3 * q^64 + 2 * q^66 - 8 * q^67 - 3 * q^68 - 4 * q^69 - 12 * q^71 + 3 * q^72 - 8 * q^73 + 10 * q^74 + 4 * q^76 + 20 * q^77 + 6 * q^78 + 8 * q^79 + 3 * q^81 - 2 * q^82 + 8 * q^83 + 6 * q^84 - 8 * q^86 - 4 * q^87 + 2 * q^88 - 10 * q^89 + 12 * q^91 - 4 * q^92 + 16 * q^93 + 8 * q^94 + 3 * q^96 + 20 * q^97 + 11 * q^98 + 2 * q^99

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−1.50466	−0.568710	−0.284355	−	0.958719i	$-0.591779\pi$
$7$	−1.50466	−0.568710	−0.284355	+	0.958719i	$0.591779\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.726656	−0.219095	−0.109548	−	0.993982i	$-0.534940\pi$
$11$	−0.726656	−0.219095	−0.109548	+	0.993982i	$0.534940\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.50466	−0.402138
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.00000	0.235702
$19$	2.72666	0.625538	0.312769	−	0.949829i	$-0.398743\pi$
$19$	2.72666	0.625538	0.312769	+	0.949829i	$0.398743\pi$
$20$	0	0
$21$	−1.50466	−0.328345
$22$	−0.726656	−0.154924
$23$	7.78734	1.62377	0.811886	−	0.583816i	$-0.198441\pi$
$23$	7.78734	1.62377	0.811886	+	0.583816i	$0.198441\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	−1.50466	−0.284355
$29$	−2.05135	−0.380926	−0.190463	−	0.981694i	$-0.560999\pi$
$29$	−2.05135	−0.380926	−0.190463	+	0.981694i	$0.560999\pi$
$30$	0	0
$31$	3.22199	0.578687	0.289343	−	0.957225i	$-0.406563\pi$
$31$	3.22199	0.578687	0.289343	+	0.957225i	$0.406563\pi$
$32$	1.00000	0.176777
$33$	−0.726656	−0.126495
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	8.23132	1.35322	0.676610	−	0.736341i	$-0.263448\pi$
$37$	8.23132	1.35322	0.676610	+	0.736341i	$0.263448\pi$
$38$	2.72666	0.442322
$39$	2.00000	0.320256
$40$	0	0
$41$	3.55602	0.555356	0.277678	−	0.960674i	$-0.410435\pi$
$41$	3.55602	0.555356	0.277678	+	0.960674i	$0.410435\pi$
$42$	−1.50466	−0.232175
$43$	−1.27334	−0.194183	−0.0970915	−	0.995275i	$-0.530954\pi$
$43$	−1.27334	−0.194183	−0.0970915	+	0.995275i	$0.530954\pi$
$44$	−0.726656	−0.109548
$45$	0	0
$46$	7.78734	1.14818
$47$	−1.55602	−0.226968	−0.113484	−	0.993540i	$-0.536201\pi$
$47$	−1.55602	−0.226968	−0.113484	+	0.993540i	$0.536201\pi$
$48$	1.00000	0.144338
$49$	−4.73599	−0.676569
$50$	0	0
$51$	−1.00000	−0.140028
$52$	2.00000	0.277350
$53$	9.00933	1.23753	0.618763	−	0.785578i	$-0.287634\pi$
$53$	9.00933	1.23753	0.618763	+	0.785578i	$0.287634\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−1.50466	−0.201069
$57$	2.72666	0.361154
$58$	−2.05135	−0.269356
$59$	−1.71733	−0.223577	−0.111789	−	0.993732i	$-0.535658\pi$
$59$	−1.71733	−0.223577	−0.111789	+	0.993732i	$0.535658\pi$
$60$	0	0
$61$	11.2406	1.43922	0.719609	−	0.694380i	$-0.244321\pi$
$61$	11.2406	1.43922	0.719609	+	0.694380i	$0.244321\pi$
$62$	3.22199	0.409193
$63$	−1.50466	−0.189570
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.726656	−0.0894452
$67$	8.56534	1.04642	0.523212	−	0.852203i	$-0.324734\pi$
$67$	8.56534	1.04642	0.523212	+	0.852203i	$0.324734\pi$
$68$	−1.00000	−0.121268
$69$	7.78734	0.937485
$70$	0	0
$71$	−10.3340	−1.22642	−0.613211	−	0.789919i	$-0.710123\pi$
$71$	−10.3340	−1.22642	−0.613211	+	0.789919i	$0.710123\pi$
$72$	1.00000	0.117851
$73$	−15.2920	−1.78979	−0.894897	−	0.446273i	$-0.852751\pi$
$73$	−15.2920	−1.78979	−0.894897	+	0.446273i	$0.852751\pi$
$74$	8.23132	0.956872
$75$	0	0
$76$	2.72666	0.312769
$77$	1.09337	0.124602
$78$	2.00000	0.226455
$79$	4.77801	0.537568	0.268784	−	0.963200i	$-0.413378\pi$
$79$	4.77801	0.537568	0.268784	+	0.963200i	$0.413378\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.55602	0.392696
$83$	1.27334	0.139768	0.0698838	−	0.997555i	$-0.477737\pi$
$83$	1.27334	0.139768	0.0698838	+	0.997555i	$0.477737\pi$
$84$	−1.50466	−0.164172
$85$	0	0
$86$	−1.27334	−0.137308
$87$	−2.05135	−0.219928
$88$	−0.726656	−0.0774618
$89$	2.28267	0.241963	0.120981	−	0.992655i	$-0.461396\pi$
$89$	2.28267	0.241963	0.120981	+	0.992655i	$0.461396\pi$
$90$	0	0
$91$	−3.00933	−0.315463
$92$	7.78734	0.811886
$93$	3.22199	0.334105
$94$	−1.55602	−0.160491
$95$	0	0
$96$	1.00000	0.102062
$97$	12.2827	1.24712	0.623558	−	0.781777i	$-0.285686\pi$
$97$	12.2827	1.24712	0.623558	+	0.781777i	$0.285686\pi$
$98$	−4.73599	−0.478407
$99$	−0.726656	−0.0730317
$100$	0	0
$101$	−3.71733	−0.369888	−0.184944	−	0.982749i	$-0.559210\pi$
$101$	−3.71733	−0.369888	−0.184944	+	0.982749i	$0.559210\pi$
$102$	−1.00000	−0.0990148
$103$	−4.72666	−0.465731	−0.232866	−	0.972509i	$-0.574810\pi$
$103$	−4.72666	−0.465731	−0.232866	+	0.972509i	$0.574810\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	9.00933	0.875063
$107$	−0.102703	−0.00992865	−0.00496433	−	0.999988i	$-0.501580\pi$
$107$	−0.102703	−0.00992865	−0.00496433	+	0.999988i	$0.501580\pi$
$108$	1.00000	0.0962250
$109$	−15.2406	−1.45979	−0.729895	−	0.683560i	$-0.760431\pi$
$109$	−15.2406	−1.45979	−0.729895	+	0.683560i	$0.760431\pi$
$110$	0	0
$111$	8.23132	0.781282
$112$	−1.50466	−0.142177
$113$	−5.00933	−0.471238	−0.235619	−	0.971846i	$-0.575712\pi$
$113$	−5.00933	−0.471238	−0.235619	+	0.971846i	$0.575712\pi$
$114$	2.72666	0.255375
$115$	0	0
$116$	−2.05135	−0.190463
$117$	2.00000	0.184900
$118$	−1.71733	−0.158093
$119$	1.50466	0.137932
$120$	0	0
$121$	−10.4720	−0.951997
$122$	11.2406	1.01768
$123$	3.55602	0.320635
$124$	3.22199	0.289343
$125$	0	0
$126$	−1.50466	−0.134046
$127$	−3.73599	−0.331515	−0.165758	−	0.986167i	$-0.553007\pi$
$127$	−3.73599	−0.331515	−0.165758	+	0.986167i	$0.553007\pi$
$128$	1.00000	0.0883883
$129$	−1.27334	−0.112112
$130$	0	0
$131$	0.726656	0.0634883	0.0317441	−	0.999496i	$-0.489894\pi$
$131$	0.726656	0.0634883	0.0317441	+	0.999496i	$0.489894\pi$
$132$	−0.726656	−0.0632473
$133$	−4.10270	−0.355749
$134$	8.56534	0.739933
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	0.264015	0.0225563	0.0112782	−	0.999936i	$-0.496410\pi$
$137$	0.264015	0.0225563	0.0112782	+	0.999936i	$0.496410\pi$
$138$	7.78734	0.662902
$139$	19.5747	1.66030	0.830151	−	0.557539i	$-0.188254\pi$
$139$	19.5747	1.66030	0.830151	+	0.557539i	$0.188254\pi$
$140$	0	0
$141$	−1.55602	−0.131040
$142$	−10.3340	−0.867212
$143$	−1.45331	−0.121532
$144$	1.00000	0.0833333
$145$	0	0
$146$	−15.2920	−1.26558
$147$	−4.73599	−0.390617
$148$	8.23132	0.676610
$149$	−23.2920	−1.90816	−0.954078	−	0.299560i	$-0.903160\pi$
$149$	−23.2920	−1.90816	−0.954078	+	0.299560i	$0.903160\pi$
$150$	0	0
$151$	2.54669	0.207246	0.103623	−	0.994617i	$-0.466956\pi$
$151$	2.54669	0.207246	0.103623	+	0.994617i	$0.466956\pi$
$152$	2.72666	0.221161
$153$	−1.00000	−0.0808452
$154$	1.09337	0.0881066
$155$	0	0
$156$	2.00000	0.160128
$157$	2.99067	0.238682	0.119341	−	0.992853i	$-0.461922\pi$
$157$	2.99067	0.238682	0.119341	+	0.992853i	$0.461922\pi$
$158$	4.77801	0.380118
$159$	9.00933	0.714486
$160$	0	0
$161$	−11.7173	−0.923455
$162$	1.00000	0.0785674
$163$	−5.45331	−0.427136	−0.213568	−	0.976928i	$-0.568509\pi$
$163$	−5.45331	−0.427136	−0.213568	+	0.976928i	$0.568509\pi$
$164$	3.55602	0.277678
$165$	0	0
$166$	1.27334	0.0988306
$167$	21.8060	1.68740	0.843699	−	0.536816i	$-0.180373\pi$
$167$	21.8060	1.68740	0.843699	+	0.536816i	$0.180373\pi$
$168$	−1.50466	−0.116087
$169$	−9.00000	−0.692308
$170$	0	0
$171$	2.72666	0.208513
$172$	−1.27334	−0.0970915
$173$	3.76868	0.286527	0.143264	−	0.989685i	$-0.454240\pi$
$173$	3.76868	0.286527	0.143264	+	0.989685i	$0.454240\pi$
$174$	−2.05135	−0.155513
$175$	0	0
$176$	−0.726656	−0.0547738
$177$	−1.71733	−0.129082
$178$	2.28267	0.171094
$179$	−21.5747	−1.61257	−0.806283	−	0.591529i	$-0.798524\pi$
$179$	−21.5747	−1.61257	−0.806283	+	0.591529i	$0.798524\pi$
$180$	0	0
$181$	−17.7873	−1.32212	−0.661061	−	0.750332i	$-0.729894\pi$
$181$	−17.7873	−1.32212	−0.661061	+	0.750332i	$0.729894\pi$
$182$	−3.00933	−0.223066
$183$	11.2406	0.830933
$184$	7.78734	0.574090
$185$	0	0
$186$	3.22199	0.236248
$187$	0.726656	0.0531384
$188$	−1.55602	−0.113484
$189$	−1.50466	−0.109448
$190$	0	0
$191$	−13.5560	−0.980879	−0.490439	−	0.871475i	$-0.663164\pi$
$191$	−13.5560	−0.980879	−0.490439	+	0.871475i	$0.663164\pi$
$192$	1.00000	0.0721688
$193$	2.82936	0.203662	0.101831	−	0.994802i	$-0.467530\pi$
$193$	2.82936	0.203662	0.101831	+	0.994802i	$0.467530\pi$
$194$	12.2827	0.881844
$195$	0	0
$196$	−4.73599	−0.338285
$197$	14.7780	1.05289	0.526445	−	0.850209i	$-0.323525\pi$
$197$	14.7780	1.05289	0.526445	+	0.850209i	$0.323525\pi$
$198$	−0.726656	−0.0516412
$199$	−4.77801	−0.338704	−0.169352	−	0.985556i	$-0.554167\pi$
$199$	−4.77801	−0.338704	−0.169352	+	0.985556i	$0.554167\pi$
$200$	0	0
$201$	8.56534	0.604153
$202$	−3.71733	−0.261550
$203$	3.08660	0.216637
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	−4.72666	−0.329322
$207$	7.78734	0.541257
$208$	2.00000	0.138675
$209$	−1.98134	−0.137052
$210$	0	0
$211$	25.0280	1.72300	0.861499	−	0.507760i	$-0.169526\pi$
$211$	25.0280	1.72300	0.861499	+	0.507760i	$0.169526\pi$
$212$	9.00933	0.618763
$213$	−10.3340	−0.708076
$214$	−0.102703	−0.00702062
$215$	0	0
$216$	1.00000	0.0680414
$217$	−4.84802	−0.329105
$218$	−15.2406	−1.03223
$219$	−15.2920	−1.03334
$220$	0	0
$221$	−2.00000	−0.134535
$222$	8.23132	0.552450
$223$	11.8387	0.792777	0.396389	−	0.918083i	$-0.370263\pi$
$223$	11.8387	0.792777	0.396389	+	0.918083i	$0.370263\pi$
$224$	−1.50466	−0.100535
$225$	0	0
$226$	−5.00933	−0.333216
$227$	−12.5653	−0.833991	−0.416996	−	0.908909i	$-0.636917\pi$
$227$	−12.5653	−0.833991	−0.416996	+	0.908909i	$0.636917\pi$
$228$	2.72666	0.180577
$229$	−18.5653	−1.22683	−0.613416	−	0.789760i	$-0.710205\pi$
$229$	−18.5653	−1.22683	−0.613416	+	0.789760i	$0.710205\pi$
$230$	0	0
$231$	1.09337	0.0719387
$232$	−2.05135	−0.134678
$233$	4.44398	0.291135	0.145568	−	0.989348i	$-0.453499\pi$
$233$	4.44398	0.291135	0.145568	+	0.989348i	$0.453499\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−1.71733	−0.111789
$237$	4.77801	0.310365
$238$	1.50466	0.0975329
$239$	−19.5747	−1.26618	−0.633090	−	0.774078i	$-0.718214\pi$
$239$	−19.5747	−1.26618	−0.633090	+	0.774078i	$0.718214\pi$
$240$	0	0
$241$	−3.55602	−0.229063	−0.114532	−	0.993420i	$-0.536537\pi$
$241$	−3.55602	−0.229063	−0.114532	+	0.993420i	$0.536537\pi$
$242$	−10.4720	−0.673164
$243$	1.00000	0.0641500
$244$	11.2406	0.719609
$245$	0	0
$246$	3.55602	0.226723
$247$	5.45331	0.346986
$248$	3.22199	0.204597
$249$	1.27334	0.0806949
$250$	0	0
$251$	−19.9160	−1.25708	−0.628542	−	0.777776i	$-0.716348\pi$
$251$	−19.9160	−1.25708	−0.628542	+	0.777776i	$0.716348\pi$
$252$	−1.50466	−0.0947849
$253$	−5.65872	−0.355761
$254$	−3.73599	−0.234417
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.7360	−0.981584	−0.490792	−	0.871277i	$-0.663292\pi$
$257$	−15.7360	−0.981584	−0.490792	+	0.871277i	$0.663292\pi$
$258$	−1.27334	−0.0792749
$259$	−12.3854	−0.769590
$260$	0	0
$261$	−2.05135	−0.126975
$262$	0.726656	0.0448930
$263$	−5.55602	−0.342599	−0.171299	−	0.985219i	$-0.554797\pi$
$263$	−5.55602	−0.342599	−0.171299	+	0.985219i	$0.554797\pi$
$264$	−0.726656	−0.0447226
$265$	0	0
$266$	−4.10270	−0.251553
$267$	2.28267	0.139697
$268$	8.56534	0.523212
$269$	−27.9673	−1.70520	−0.852598	−	0.522567i	$-0.824975\pi$
$269$	−27.9673	−1.70520	−0.852598	+	0.522567i	$0.824975\pi$
$270$	0	0
$271$	0.565344	0.0343422	0.0171711	−	0.999853i	$-0.494534\pi$
$271$	0.565344	0.0343422	0.0171711	+	0.999853i	$0.494534\pi$
$272$	−1.00000	−0.0606339
$273$	−3.00933	−0.182133
$274$	0.264015	0.0159497
$275$	0	0
$276$	7.78734	0.468743
$277$	5.89004	0.353898	0.176949	−	0.984220i	$-0.443377\pi$
$277$	5.89004	0.353898	0.176949	+	0.984220i	$0.443377\pi$
$278$	19.5747	1.17401
$279$	3.22199	0.192896
$280$	0	0
$281$	22.3200	1.33150	0.665749	−	0.746175i	$-0.268112\pi$
$281$	22.3200	1.33150	0.665749	+	0.746175i	$0.268112\pi$
$282$	−1.55602	−0.0926594
$283$	23.5747	1.40137	0.700684	−	0.713471i	$-0.252878\pi$
$283$	23.5747	1.40137	0.700684	+	0.713471i	$0.252878\pi$
$284$	−10.3340	−0.613211
$285$	0	0
$286$	−1.45331	−0.0859362
$287$	−5.35061	−0.315837
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	12.2827	0.720023
$292$	−15.2920	−0.894897
$293$	−18.0373	−1.05375	−0.526876	−	0.849942i	$-0.676637\pi$
$293$	−18.0373	−1.05375	−0.526876	+	0.849942i	$0.676637\pi$
$294$	−4.73599	−0.276208
$295$	0	0
$296$	8.23132	0.478436
$297$	−0.726656	−0.0421649
$298$	−23.2920	−1.34927
$299$	15.5747	0.900707
$300$	0	0
$301$	1.91595	0.110434
$302$	2.54669	0.146545
$303$	−3.71733	−0.213555
$304$	2.72666	0.156384
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	9.27334	0.529258	0.264629	−	0.964350i	$-0.414751\pi$
$307$	9.27334	0.529258	0.264629	+	0.964350i	$0.414751\pi$
$308$	1.09337	0.0623008
$309$	−4.72666	−0.268890
$310$	0	0
$311$	−7.22199	−0.409522	−0.204761	−	0.978812i	$-0.565642\pi$
$311$	−7.22199	−0.409522	−0.204761	+	0.978812i	$0.565642\pi$
$312$	2.00000	0.113228
$313$	6.30133	0.356172	0.178086	−	0.984015i	$-0.443009\pi$
$313$	6.30133	0.356172	0.178086	+	0.984015i	$0.443009\pi$
$314$	2.99067	0.168773
$315$	0	0
$316$	4.77801	0.268784
$317$	−10.8807	−0.611122	−0.305561	−	0.952173i	$-0.598844\pi$
$317$	−10.8807	−0.611122	−0.305561	+	0.952173i	$0.598844\pi$
$318$	9.00933	0.505218
$319$	1.49063	0.0834591
$320$	0	0
$321$	−0.102703	−0.00573231
$322$	−11.7173	−0.652981
$323$	−2.72666	−0.151715
$324$	1.00000	0.0555556
$325$	0	0
$326$	−5.45331	−0.302031
$327$	−15.2406	−0.842810
$328$	3.55602	0.196348
$329$	2.34128	0.129079
$330$	0	0
$331$	2.54669	0.139979	0.0699893	−	0.997548i	$-0.477703\pi$
$331$	2.54669	0.139979	0.0699893	+	0.997548i	$0.477703\pi$
$332$	1.27334	0.0698838
$333$	8.23132	0.451074
$334$	21.8060	1.19317
$335$	0	0
$336$	−1.50466	−0.0820862
$337$	29.3107	1.59665	0.798327	−	0.602225i	$-0.205719\pi$
$337$	29.3107	1.59665	0.798327	+	0.602225i	$0.205719\pi$
$338$	−9.00000	−0.489535
$339$	−5.00933	−0.272069
$340$	0	0
$341$	−2.34128	−0.126787
$342$	2.72666	0.147441
$343$	17.6587	0.953481
$344$	−1.27334	−0.0686541
$345$	0	0
$346$	3.76868	0.202605
$347$	−26.0187	−1.39675	−0.698377	−	0.715730i	$-0.746094\pi$
$347$	−26.0187	−1.39675	−0.698377	+	0.715730i	$0.746094\pi$
$348$	−2.05135	−0.109964
$349$	28.1214	1.50530	0.752651	−	0.658420i	$-0.228775\pi$
$349$	28.1214	1.50530	0.752651	+	0.658420i	$0.228775\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	−0.726656	−0.0387309
$353$	31.1307	1.65692	0.828460	−	0.560049i	$-0.189218\pi$
$353$	31.1307	1.65692	0.828460	+	0.560049i	$0.189218\pi$
$354$	−1.71733	−0.0912749
$355$	0	0
$356$	2.28267	0.120981
$357$	1.50466	0.0796353
$358$	−21.5747	−1.14026
$359$	31.1493	1.64400	0.822000	−	0.569488i	$-0.192858\pi$
$359$	31.1493	1.64400	0.822000	+	0.569488i	$0.192858\pi$
$360$	0	0
$361$	−11.5653	−0.608702
$362$	−17.7873	−0.934882
$363$	−10.4720	−0.549636
$364$	−3.00933	−0.157732
$365$	0	0
$366$	11.2406	0.587558
$367$	−34.6354	−1.80795	−0.903975	−	0.427585i	$-0.859365\pi$
$367$	−34.6354	−1.80795	−0.903975	+	0.427585i	$0.859365\pi$
$368$	7.78734	0.405943
$369$	3.55602	0.185119
$370$	0	0
$371$	−13.5560	−0.703793
$372$	3.22199	0.167053
$373$	−12.0187	−0.622302	−0.311151	−	0.950360i	$-0.600715\pi$
$373$	−12.0187	−0.622302	−0.311151	+	0.950360i	$0.600715\pi$
$374$	0.726656	0.0375745
$375$	0	0
$376$	−1.55602	−0.0802454
$377$	−4.10270	−0.211300
$378$	−1.50466	−0.0773916
$379$	7.11203	0.365321	0.182660	−	0.983176i	$-0.441529\pi$
$379$	7.11203	0.365321	0.182660	+	0.983176i	$0.441529\pi$
$380$	0	0
$381$	−3.73599	−0.191400
$382$	−13.5560	−0.693586
$383$	−18.1214	−0.925958	−0.462979	−	0.886369i	$-0.653220\pi$
$383$	−18.1214	−0.925958	−0.462979	+	0.886369i	$0.653220\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	2.82936	0.144011
$387$	−1.27334	−0.0647277
$388$	12.2827	0.623558
$389$	−20.7453	−1.05183	−0.525915	−	0.850537i	$-0.676277\pi$
$389$	−20.7453	−1.05183	−0.525915	+	0.850537i	$0.676277\pi$
$390$	0	0
$391$	−7.78734	−0.393823
$392$	−4.73599	−0.239203
$393$	0.726656	0.0366550
$394$	14.7780	0.744505
$395$	0	0
$396$	−0.726656	−0.0365159
$397$	8.23132	0.413118	0.206559	−	0.978434i	$-0.433773\pi$
$397$	8.23132	0.413118	0.206559	+	0.978434i	$0.433773\pi$
$398$	−4.77801	−0.239500
$399$	−4.10270	−0.205392
$400$	0	0
$401$	−30.6680	−1.53149	−0.765745	−	0.643145i	$-0.777629\pi$
$401$	−30.6680	−1.53149	−0.765745	+	0.643145i	$0.777629\pi$
$402$	8.56534	0.427201
$403$	6.44398	0.320998
$404$	−3.71733	−0.184944
$405$	0	0
$406$	3.08660	0.153185
$407$	−5.98134	−0.296484
$408$	−1.00000	−0.0495074
$409$	12.8294	0.634371	0.317185	−	0.948364i	$-0.397262\pi$
$409$	12.8294	0.634371	0.317185	+	0.948364i	$0.397262\pi$
$410$	0	0
$411$	0.264015	0.0130229
$412$	−4.72666	−0.232866
$413$	2.58400	0.127150
$414$	7.78734	0.382727
$415$	0	0
$416$	2.00000	0.0980581
$417$	19.5747	0.958576
$418$	−1.98134	−0.0969106
$419$	−4.30133	−0.210134	−0.105067	−	0.994465i	$-0.533506\pi$
$419$	−4.30133	−0.210134	−0.105067	+	0.994465i	$0.533506\pi$
$420$	0	0
$421$	−16.4440	−0.801431	−0.400715	−	0.916203i	$-0.631238\pi$
$421$	−16.4440	−0.801431	−0.400715	+	0.916203i	$0.631238\pi$
$422$	25.0280	1.21834
$423$	−1.55602	−0.0756561
$424$	9.00933	0.437532
$425$	0	0
$426$	−10.3340	−0.500685
$427$	−16.9134	−0.818497
$428$	−0.102703	−0.00496433
$429$	−1.45331	−0.0701666
$430$	0	0
$431$	−2.23132	−0.107479	−0.0537395	−	0.998555i	$-0.517114\pi$
$431$	−2.23132	−0.107479	−0.0537395	+	0.998555i	$0.517114\pi$
$432$	1.00000	0.0481125
$433$	−0.0772666	−0.00371320	−0.00185660	−	0.999998i	$-0.500591\pi$
$433$	−0.0772666	−0.00371320	−0.00185660	+	0.999998i	$0.500591\pi$
$434$	−4.84802	−0.232712
$435$	0	0
$436$	−15.2406	−0.729895
$437$	21.2334	1.01573
$438$	−15.2920	−0.730680
$439$	−26.3713	−1.25864	−0.629318	−	0.777148i	$-0.716666\pi$
$439$	−26.3713	−1.25864	−0.629318	+	0.777148i	$0.716666\pi$
$440$	0	0
$441$	−4.73599	−0.225523
$442$	−2.00000	−0.0951303
$443$	26.0187	1.23618	0.618092	−	0.786106i	$-0.287906\pi$
$443$	26.0187	1.23618	0.618092	+	0.786106i	$0.287906\pi$
$444$	8.23132	0.390641
$445$	0	0
$446$	11.8387	0.560578
$447$	−23.2920	−1.10167
$448$	−1.50466	−0.0710887
$449$	26.4626	1.24885	0.624425	−	0.781085i	$-0.285334\pi$
$449$	26.4626	1.24885	0.624425	+	0.781085i	$0.285334\pi$
$450$	0	0
$451$	−2.58400	−0.121676
$452$	−5.00933	−0.235619
$453$	2.54669	0.119654
$454$	−12.5653	−0.589721
$455$	0	0
$456$	2.72666	0.127687
$457$	−6.82936	−0.319464	−0.159732	−	0.987160i	$-0.551063\pi$
$457$	−6.82936	−0.319464	−0.159732	+	0.987160i	$0.551063\pi$
$458$	−18.5653	−0.867502
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−18.8294	−0.876971	−0.438485	−	0.898738i	$-0.644485\pi$
$461$	−18.8294	−0.876971	−0.438485	+	0.898738i	$0.644485\pi$
$462$	1.09337	0.0508684
$463$	−25.4974	−1.18496	−0.592482	−	0.805583i	$-0.701852\pi$
$463$	−25.4974	−1.18496	−0.592482	+	0.805583i	$0.701852\pi$
$464$	−2.05135	−0.0952316
$465$	0	0
$466$	4.44398	0.205864
$467$	16.0373	0.742118	0.371059	−	0.928609i	$-0.378995\pi$
$467$	16.0373	0.742118	0.371059	+	0.928609i	$0.378995\pi$
$468$	2.00000	0.0924500
$469$	−12.8880	−0.595111
$470$	0	0
$471$	2.99067	0.137803
$472$	−1.71733	−0.0790464
$473$	0.925283	0.0425446
$474$	4.77801	0.219461
$475$	0	0
$476$	1.50466	0.0689662
$477$	9.00933	0.412509
$478$	−19.5747	−0.895325
$479$	24.3527	1.11270	0.556351	−	0.830947i	$-0.312201\pi$
$479$	24.3527	1.11270	0.556351	+	0.830947i	$0.312201\pi$
$480$	0	0
$481$	16.4626	0.750632
$482$	−3.55602	−0.161972
$483$	−11.7173	−0.533157
$484$	−10.4720	−0.475999
$485$	0	0
$486$	1.00000	0.0453609
$487$	−24.5513	−1.11253	−0.556263	−	0.831006i	$-0.687765\pi$
$487$	−24.5513	−1.11253	−0.556263	+	0.831006i	$0.687765\pi$
$488$	11.2406	0.508840
$489$	−5.45331	−0.246607
$490$	0	0
$491$	40.4813	1.82690	0.913448	−	0.406956i	$-0.133410\pi$
$491$	40.4813	1.82690	0.913448	+	0.406956i	$0.133410\pi$
$492$	3.55602	0.160318
$493$	2.05135	0.0923882
$494$	5.45331	0.245356
$495$	0	0
$496$	3.22199	0.144672
$497$	15.5492	0.697479
$498$	1.27334	0.0570599
$499$	23.2147	1.03923	0.519617	−	0.854399i	$-0.326075\pi$
$499$	23.2147	1.03923	0.519617	+	0.854399i	$0.326075\pi$
$500$	0	0
$501$	21.8060	0.974220
$502$	−19.9160	−0.888893
$503$	−42.1659	−1.88009	−0.940043	−	0.341056i	$-0.889215\pi$
$503$	−42.1659	−1.88009	−0.940043	+	0.341056i	$0.889215\pi$
$504$	−1.50466	−0.0670231
$505$	0	0
$506$	−5.65872	−0.251561
$507$	−9.00000	−0.399704
$508$	−3.73599	−0.165758
$509$	−24.3854	−1.08086	−0.540431	−	0.841388i	$-0.681739\pi$
$509$	−24.3854	−1.08086	−0.540431	+	0.841388i	$0.681739\pi$
$510$	0	0
$511$	23.0093	1.01787
$512$	1.00000	0.0441942
$513$	2.72666	0.120385
$514$	−15.7360	−0.694085
$515$	0	0
$516$	−1.27334	−0.0560558
$517$	1.13069	0.0497276
$518$	−12.3854	−0.544182
$519$	3.76868	0.165427
$520$	0	0
$521$	−20.4813	−0.897302	−0.448651	−	0.893707i	$-0.648095\pi$
$521$	−20.4813	−0.897302	−0.448651	+	0.893707i	$0.648095\pi$
$522$	−2.05135	−0.0897852
$523$	32.5653	1.42398	0.711992	−	0.702188i	$-0.247793\pi$
$523$	32.5653	1.42398	0.711992	+	0.702188i	$0.247793\pi$
$524$	0.726656	0.0317441
$525$	0	0
$526$	−5.55602	−0.242254
$527$	−3.22199	−0.140352
$528$	−0.726656	−0.0316237
$529$	37.6426	1.63664
$530$	0	0
$531$	−1.71733	−0.0745257
$532$	−4.10270	−0.177875
$533$	7.11203	0.308056
$534$	2.28267	0.0987809
$535$	0	0
$536$	8.56534	0.369967
$537$	−21.5747	−0.931016
$538$	−27.9673	−1.20576
$539$	3.44143	0.148233
$540$	0	0
$541$	22.3527	0.961017	0.480508	−	0.876990i	$-0.340452\pi$
$541$	22.3527	0.961017	0.480508	+	0.876990i	$0.340452\pi$
$542$	0.565344	0.0242836
$543$	−17.7873	−0.763328
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−3.00933	−0.128787
$547$	5.55602	0.237558	0.118779	−	0.992921i	$-0.462102\pi$
$547$	5.55602	0.237558	0.118779	+	0.992921i	$0.462102\pi$
$548$	0.264015	0.0112782
$549$	11.2406	0.479739
$550$	0	0
$551$	−5.59333	−0.238284
$552$	7.78734	0.331451
$553$	−7.18930	−0.305720
$554$	5.89004	0.250244
$555$	0	0
$556$	19.5747	0.830151
$557$	−7.91595	−0.335410	−0.167705	−	0.985837i	$-0.553636\pi$
$557$	−7.91595	−0.335410	−0.167705	+	0.985837i	$0.553636\pi$
$558$	3.22199	0.136398
$559$	−2.54669	−0.107713
$560$	0	0
$561$	0.726656	0.0306795
$562$	22.3200	0.941512
$563$	−42.1986	−1.77846	−0.889230	−	0.457460i	$-0.848759\pi$
$563$	−42.1986	−1.77846	−0.889230	+	0.457460i	$0.848759\pi$
$564$	−1.55602	−0.0655201
$565$	0	0
$566$	23.5747	0.990917
$567$	−1.50466	−0.0631900
$568$	−10.3340	−0.433606
$569$	7.13069	0.298934	0.149467	−	0.988767i	$-0.452244\pi$
$569$	7.13069	0.298934	0.149467	+	0.988767i	$0.452244\pi$
$570$	0	0
$571$	−6.01866	−0.251873	−0.125936	−	0.992038i	$-0.540194\pi$
$571$	−6.01866	−0.251873	−0.125936	+	0.992038i	$0.540194\pi$
$572$	−1.45331	−0.0607661
$573$	−13.5560	−0.566311
$574$	−5.35061	−0.223330
$575$	0	0
$576$	1.00000	0.0416667
$577$	−38.5840	−1.60627	−0.803137	−	0.595795i	$-0.796837\pi$
$577$	−38.5840	−1.60627	−0.803137	+	0.595795i	$0.796837\pi$
$578$	1.00000	0.0415945
$579$	2.82936	0.117584
$580$	0	0
$581$	−1.91595	−0.0794872
$582$	12.2827	0.509133
$583$	−6.54669	−0.271136
$584$	−15.2920	−0.632788
$585$	0	0
$586$	−18.0373	−0.745115
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	−4.73599	−0.195309
$589$	8.78527	0.361991
$590$	0	0
$591$	14.7780	0.607886
$592$	8.23132	0.338305
$593$	8.62395	0.354143	0.177072	−	0.984198i	$-0.443338\pi$
$593$	8.62395	0.354143	0.177072	+	0.984198i	$0.443338\pi$
$594$	−0.726656	−0.0298151
$595$	0	0
$596$	−23.2920	−0.954078
$597$	−4.77801	−0.195551
$598$	15.5747	0.636896
$599$	−0.359939	−0.0147067	−0.00735335	−	0.999973i	$-0.502341\pi$
$599$	−0.359939	−0.0147067	−0.00735335	+	0.999973i	$0.502341\pi$
$600$	0	0
$601$	−40.0560	−1.63392	−0.816959	−	0.576696i	$-0.804342\pi$
$601$	−40.0560	−1.63392	−0.816959	+	0.576696i	$0.804342\pi$
$602$	1.91595	0.0780885
$603$	8.56534	0.348808
$604$	2.54669	0.103623
$605$	0	0
$606$	−3.71733	−0.151006
$607$	45.9673	1.86576	0.932878	−	0.360193i	$-0.117289\pi$
$607$	45.9673	1.86576	0.932878	+	0.360193i	$0.117289\pi$
$608$	2.72666	0.110581
$609$	3.08660	0.125075
$610$	0	0
$611$	−3.11203	−0.125899
$612$	−1.00000	−0.0404226
$613$	−11.5560	−0.466743	−0.233372	−	0.972388i	$-0.574976\pi$
$613$	−11.5560	−0.466743	−0.233372	+	0.972388i	$0.574976\pi$
$614$	9.27334	0.374242
$615$	0	0
$616$	1.09337	0.0440533
$617$	−49.0093	−1.97304	−0.986521	−	0.163637i	$-0.947677\pi$
$617$	−49.0093	−1.97304	−0.986521	+	0.163637i	$0.947677\pi$
$618$	−4.72666	−0.190134
$619$	6.34128	0.254878	0.127439	−	0.991846i	$-0.459324\pi$
$619$	6.34128	0.254878	0.127439	+	0.991846i	$0.459324\pi$
$620$	0	0
$621$	7.78734	0.312495
$622$	−7.22199	−0.289576
$623$	−3.43466	−0.137607
$624$	2.00000	0.0800641
$625$	0	0
$626$	6.30133	0.251852
$627$	−1.98134	−0.0791272
$628$	2.99067	0.119341
$629$	−8.23132	−0.328204
$630$	0	0
$631$	20.4626	0.814605	0.407302	−	0.913293i	$-0.366469\pi$
$631$	20.4626	0.814605	0.407302	+	0.913293i	$0.366469\pi$
$632$	4.77801	0.190059
$633$	25.0280	0.994773
$634$	−10.8807	−0.432128
$635$	0	0
$636$	9.00933	0.357243
$637$	−9.47197	−0.375293
$638$	1.49063	0.0590145
$639$	−10.3340	−0.408808
$640$	0	0
$641$	−27.5933	−1.08987	−0.544936	−	0.838478i	$-0.683446\pi$
$641$	−27.5933	−1.08987	−0.544936	+	0.838478i	$0.683446\pi$
$642$	−0.102703	−0.00405336
$643$	−9.13069	−0.360079	−0.180040	−	0.983659i	$-0.557623\pi$
$643$	−9.13069	−0.360079	−0.180040	+	0.983659i	$0.557623\pi$
$644$	−11.7173	−0.461727
$645$	0	0
$646$	−2.72666	−0.107279
$647$	−27.0466	−1.06331	−0.531657	−	0.846960i	$-0.678430\pi$
$647$	−27.0466	−1.06331	−0.531657	+	0.846960i	$0.678430\pi$
$648$	1.00000	0.0392837
$649$	1.24791	0.0489846
$650$	0	0
$651$	−4.84802	−0.190009
$652$	−5.45331	−0.213568
$653$	−23.3434	−0.913496	−0.456748	−	0.889596i	$-0.650986\pi$
$653$	−23.3434	−0.913496	−0.456748	+	0.889596i	$0.650986\pi$
$654$	−15.2406	−0.595957
$655$	0	0
$656$	3.55602	0.138839
$657$	−15.2920	−0.596598
$658$	2.34128	0.0912727
$659$	−8.26401	−0.321920	−0.160960	−	0.986961i	$-0.551459\pi$
$659$	−8.26401	−0.321920	−0.160960	+	0.986961i	$0.551459\pi$
$660$	0	0
$661$	14.6680	0.570521	0.285260	−	0.958450i	$-0.407920\pi$
$661$	14.6680	0.570521	0.285260	+	0.958450i	$0.407920\pi$
$662$	2.54669	0.0989798
$663$	−2.00000	−0.0776736
$664$	1.27334	0.0494153
$665$	0	0
$666$	8.23132	0.318957
$667$	−15.9746	−0.618538
$668$	21.8060	0.843699
$669$	11.8387	0.457710
$670$	0	0
$671$	−8.16809	−0.315326
$672$	−1.50466	−0.0580437
$673$	−18.8294	−0.725818	−0.362909	−	0.931824i	$-0.618216\pi$
$673$	−18.8294	−0.725818	−0.362909	+	0.931824i	$0.618216\pi$
$674$	29.3107	1.12900
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−34.8153	−1.33806	−0.669031	−	0.743235i	$-0.733291\pi$
$677$	−34.8153	−1.33806	−0.669031	+	0.743235i	$0.733291\pi$
$678$	−5.00933	−0.192382
$679$	−18.4813	−0.709247
$680$	0	0
$681$	−12.5653	−0.481505
$682$	−2.34128	−0.0896523
$683$	9.45331	0.361721	0.180860	−	0.983509i	$-0.442112\pi$
$683$	9.45331	0.361721	0.180860	+	0.983509i	$0.442112\pi$
$684$	2.72666	0.104256
$685$	0	0
$686$	17.6587	0.674213
$687$	−18.5653	−0.708312
$688$	−1.27334	−0.0485458
$689$	18.0187	0.686456
$690$	0	0
$691$	50.6867	1.92821	0.964107	−	0.265516i	$-0.0855422\pi$
$691$	50.6867	1.92821	0.964107	+	0.265516i	$0.0855422\pi$
$692$	3.76868	0.143264
$693$	1.09337	0.0415338
$694$	−26.0187	−0.987655
$695$	0	0
$696$	−2.05135	−0.0777563
$697$	−3.55602	−0.134694
$698$	28.1214	1.06441
$699$	4.44398	0.168087
$700$	0	0
$701$	4.64261	0.175349	0.0876745	−	0.996149i	$-0.472056\pi$
$701$	4.64261	0.175349	0.0876745	+	0.996149i	$0.472056\pi$
$702$	2.00000	0.0754851
$703$	22.4440	0.846491
$704$	−0.726656	−0.0273869
$705$	0	0
$706$	31.1307	1.17162
$707$	5.59333	0.210359
$708$	−1.71733	−0.0645411
$709$	15.4461	0.580089	0.290044	−	0.957013i	$-0.406330\pi$
$709$	15.4461	0.580089	0.290044	+	0.957013i	$0.406330\pi$
$710$	0	0
$711$	4.77801	0.179189
$712$	2.28267	0.0855468
$713$	25.0907	0.939655
$714$	1.50466	0.0563106
$715$	0	0
$716$	−21.5747	−0.806283
$717$	−19.5747	−0.731030
$718$	31.1493	1.16248
$719$	7.11929	0.265505	0.132752	−	0.991149i	$-0.457619\pi$
$719$	7.11929	0.265505	0.132752	+	0.991149i	$0.457619\pi$
$720$	0	0
$721$	7.11203	0.264866
$722$	−11.5653	−0.430418
$723$	−3.55602	−0.132250
$724$	−17.7873	−0.661061
$725$	0	0
$726$	−10.4720	−0.388651
$727$	30.6426	1.13647	0.568236	−	0.822866i	$-0.307626\pi$
$727$	30.6426	1.13647	0.568236	+	0.822866i	$0.307626\pi$
$728$	−3.00933	−0.111533
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.27334	0.0470963
$732$	11.2406	0.415466
$733$	32.0560	1.18401	0.592007	−	0.805933i	$-0.298336\pi$
$733$	32.0560	1.18401	0.592007	+	0.805933i	$0.298336\pi$
$734$	−34.6354	−1.27841
$735$	0	0
$736$	7.78734	0.287045
$737$	−6.22406	−0.229266
$738$	3.55602	0.130899
$739$	−21.2733	−0.782553	−0.391276	−	0.920273i	$-0.627966\pi$
$739$	−21.2733	−0.782553	−0.391276	+	0.920273i	$0.627966\pi$
$740$	0	0
$741$	5.45331	0.200332
$742$	−13.5560	−0.497657
$743$	−51.7219	−1.89749	−0.948747	−	0.316036i	$-0.897648\pi$
$743$	−51.7219	−1.89749	−0.948747	+	0.316036i	$0.897648\pi$
$744$	3.22199	0.118124
$745$	0	0
$746$	−12.0187	−0.440034
$747$	1.27334	0.0465892
$748$	0.726656	0.0265692
$749$	0.154533	0.00564652
$750$	0	0
$751$	51.9274	1.89486	0.947428	−	0.319969i	$-0.103672\pi$
$751$	51.9274	1.89486	0.947428	+	0.319969i	$0.103672\pi$
$752$	−1.55602	−0.0567421
$753$	−19.9160	−0.725778
$754$	−4.10270	−0.149412
$755$	0	0
$756$	−1.50466	−0.0547241
$757$	19.0280	0.691584	0.345792	−	0.938311i	$-0.387610\pi$
$757$	19.0280	0.691584	0.345792	+	0.938311i	$0.387610\pi$
$758$	7.11203	0.258321
$759$	−5.65872	−0.205398
$760$	0	0
$761$	−7.09337	−0.257135	−0.128567	−	0.991701i	$-0.541038\pi$
$761$	−7.09337	−0.257135	−0.128567	+	0.991701i	$0.541038\pi$
$762$	−3.73599	−0.135340
$763$	22.9321	0.830196
$764$	−13.5560	−0.490439
$765$	0	0
$766$	−18.1214	−0.654751
$767$	−3.43466	−0.124018
$768$	1.00000	0.0360844
$769$	−25.2666	−0.911136	−0.455568	−	0.890201i	$-0.650564\pi$
$769$	−25.2666	−0.911136	−0.455568	+	0.890201i	$0.650564\pi$
$770$	0	0
$771$	−15.7360	−0.566718
$772$	2.82936	0.101831
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−1.27334	−0.0457694
$775$	0	0
$776$	12.2827	0.440922
$777$	−12.3854	−0.444323
$778$	−20.7453	−0.743756
$779$	9.69603	0.347396
$780$	0	0
$781$	7.50929	0.268703
$782$	−7.78734	−0.278475
$783$	−2.05135	−0.0733093
$784$	−4.73599	−0.169142
$785$	0	0
$786$	0.726656	0.0259190
$787$	25.5560	0.910974	0.455487	−	0.890243i	$-0.349465\pi$
$787$	25.5560	0.910974	0.455487	+	0.890243i	$0.349465\pi$
$788$	14.7780	0.526445
$789$	−5.55602	−0.197799
$790$	0	0
$791$	7.53736	0.267998
$792$	−0.726656	−0.0258206
$793$	22.4813	0.798334
$794$	8.23132	0.292119
$795$	0	0
$796$	−4.77801	−0.169352
$797$	−17.0093	−0.602501	−0.301251	−	0.953545i	$-0.597404\pi$
$797$	−17.0093	−0.602501	−0.301251	+	0.953545i	$0.597404\pi$
$798$	−4.10270	−0.145234
$799$	1.55602	0.0550479
$800$	0	0
$801$	2.28267	0.0806543
$802$	−30.6680	−1.08293
$803$	11.1120	0.392135
$804$	8.56534	0.302076
$805$	0	0
$806$	6.44398	0.226980
$807$	−27.9673	−0.984496
$808$	−3.71733	−0.130775
$809$	−31.9160	−1.12211	−0.561053	−	0.827780i	$-0.689603\pi$
$809$	−31.9160	−1.12211	−0.561053	+	0.827780i	$0.689603\pi$
$810$	0	0
$811$	10.5840	0.371655	0.185827	−	0.982582i	$-0.440504\pi$
$811$	10.5840	0.371655	0.185827	+	0.982582i	$0.440504\pi$
$812$	3.08660	0.108318
$813$	0.565344	0.0198275
$814$	−5.98134	−0.209646
$815$	0	0
$816$	−1.00000	−0.0350070
$817$	−3.47197	−0.121469
$818$	12.8294	0.448568
$819$	−3.00933	−0.105154
$820$	0	0
$821$	−42.4113	−1.48016	−0.740082	−	0.672517i	$-0.765213\pi$
$821$	−42.4113	−1.48016	−0.740082	+	0.672517i	$0.765213\pi$
$822$	0.264015	0.00920857
$823$	−14.0700	−0.490450	−0.245225	−	0.969466i	$-0.578862\pi$
$823$	−14.0700	−0.490450	−0.245225	+	0.969466i	$0.578862\pi$
$824$	−4.72666	−0.164661
$825$	0	0
$826$	2.58400	0.0899089
$827$	28.1400	0.978524	0.489262	−	0.872137i	$-0.337266\pi$
$827$	28.1400	0.978524	0.489262	+	0.872137i	$0.337266\pi$
$828$	7.78734	0.270629
$829$	−49.1120	−1.70573	−0.852866	−	0.522130i	$-0.825137\pi$
$829$	−49.1120	−1.70573	−0.852866	+	0.522130i	$0.825137\pi$
$830$	0	0
$831$	5.89004	0.204323
$832$	2.00000	0.0693375
$833$	4.73599	0.164092
$834$	19.5747	0.677815
$835$	0	0
$836$	−1.98134	−0.0685262
$837$	3.22199	0.111368
$838$	−4.30133	−0.148587
$839$	41.3807	1.42862	0.714310	−	0.699830i	$-0.246741\pi$
$839$	41.3807	1.42862	0.714310	+	0.699830i	$0.246741\pi$
$840$	0	0
$841$	−24.7920	−0.854895
$842$	−16.4440	−0.566697
$843$	22.3200	0.768741
$844$	25.0280	0.861499
$845$	0	0
$846$	−1.55602	−0.0534969
$847$	15.7568	0.541410
$848$	9.00933	0.309382
$849$	23.5747	0.809081
$850$	0	0
$851$	64.1001	2.19732
$852$	−10.3340	−0.354038
$853$	28.6940	0.982463	0.491231	−	0.871029i	$-0.336547\pi$
$853$	28.6940	0.982463	0.491231	+	0.871029i	$0.336547\pi$
$854$	−16.9134	−0.578765
$855$	0	0
$856$	−0.102703	−0.00351031
$857$	−13.4720	−0.460194	−0.230097	−	0.973168i	$-0.573904\pi$
$857$	−13.4720	−0.460194	−0.230097	+	0.973168i	$0.573904\pi$
$858$	−1.45331	−0.0496153
$859$	21.9813	0.749994	0.374997	−	0.927026i	$-0.377644\pi$
$859$	21.9813	0.749994	0.374997	+	0.927026i	$0.377644\pi$
$860$	0	0
$861$	−5.35061	−0.182348
$862$	−2.23132	−0.0759991
$863$	−38.2614	−1.30243	−0.651216	−	0.758892i	$-0.725741\pi$
$863$	−38.2614	−1.30243	−0.651216	+	0.758892i	$0.725741\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−0.0772666	−0.00262563
$867$	1.00000	0.0339618
$868$	−4.84802	−0.164552
$869$	−3.47197	−0.117779
$870$	0	0
$871$	17.1307	0.580451
$872$	−15.2406	−0.516114
$873$	12.2827	0.415705
$874$	21.2334	0.718230
$875$	0	0
$876$	−15.2920	−0.516669
$877$	6.35268	0.214515	0.107257	−	0.994231i	$-0.465793\pi$
$877$	6.35268	0.214515	0.107257	+	0.994231i	$0.465793\pi$
$878$	−26.3713	−0.889990
$879$	−18.0373	−0.608384
$880$	0	0
$881$	14.5653	0.490719	0.245359	−	0.969432i	$-0.421094\pi$
$881$	14.5653	0.490719	0.245359	+	0.969432i	$0.421094\pi$
$882$	−4.73599	−0.159469
$883$	−27.6774	−0.931418	−0.465709	−	0.884938i	$-0.654201\pi$
$883$	−27.6774	−0.931418	−0.465709	+	0.884938i	$0.654201\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	26.0187	0.874114
$887$	15.5819	0.523190	0.261595	−	0.965178i	$-0.415752\pi$
$887$	15.5819	0.523190	0.261595	+	0.965178i	$0.415752\pi$
$888$	8.23132	0.276225
$889$	5.62140	0.188536
$890$	0	0
$891$	−0.726656	−0.0243439
$892$	11.8387	0.396389
$893$	−4.24272	−0.141977
$894$	−23.2920	−0.779001
$895$	0	0
$896$	−1.50466	−0.0502673
$897$	15.5747	0.520023
$898$	26.4626	0.883070
$899$	−6.60944	−0.220437
$900$	0	0
$901$	−9.00933	−0.300144
$902$	−2.58400	−0.0860379
$903$	1.91595	0.0637590
$904$	−5.00933	−0.166608
$905$	0	0
$906$	2.54669	0.0846080
$907$	52.5000	1.74323	0.871616	−	0.490189i	$-0.163072\pi$
$907$	52.5000	1.74323	0.871616	+	0.490189i	$0.163072\pi$
$908$	−12.5653	−0.416996
$909$	−3.71733	−0.123296
$910$	0	0
$911$	−17.3807	−0.575847	−0.287924	−	0.957653i	$-0.592965\pi$
$911$	−17.3807	−0.575847	−0.287924	+	0.957653i	$0.592965\pi$
$912$	2.72666	0.0902886
$913$	−0.925283	−0.0306224
$914$	−6.82936	−0.225895
$915$	0	0
$916$	−18.5653	−0.613416
$917$	−1.09337	−0.0361064
$918$	−1.00000	−0.0330049
$919$	−11.3174	−0.373328	−0.186664	−	0.982424i	$-0.559768\pi$
$919$	−11.3174	−0.373328	−0.186664	+	0.982424i	$0.559768\pi$
$920$	0	0
$921$	9.27334	0.305567
$922$	−18.8294	−0.620112
$923$	−20.6680	−0.680297
$924$	1.09337	0.0359694
$925$	0	0
$926$	−25.4974	−0.837897
$927$	−4.72666	−0.155244
$928$	−2.05135	−0.0673389
$929$	56.1214	1.84128	0.920641	−	0.390410i	$-0.127667\pi$
$929$	56.1214	1.84128	0.920641	+	0.390410i	$0.127667\pi$
$930$	0	0
$931$	−12.9134	−0.423220
$932$	4.44398	0.145568
$933$	−7.22199	−0.236437
$934$	16.0373	0.524757
$935$	0	0
$936$	2.00000	0.0653720
$937$	−26.8667	−0.877696	−0.438848	−	0.898561i	$-0.644613\pi$
$937$	−26.8667	−0.877696	−0.438848	+	0.898561i	$0.644613\pi$
$938$	−12.8880	−0.420807
$939$	6.30133	0.205636
$940$	0	0
$941$	22.0514	0.718854	0.359427	−	0.933173i	$-0.382972\pi$
$941$	22.0514	0.718854	0.359427	+	0.933173i	$0.382972\pi$
$942$	2.99067	0.0974413
$943$	27.6919	0.901772
$944$	−1.71733	−0.0558943
$945$	0	0
$946$	0.925283	0.0300836
$947$	30.6867	0.997184	0.498592	−	0.866837i	$-0.333851\pi$
$947$	30.6867	0.997184	0.498592	+	0.866837i	$0.333851\pi$
$948$	4.77801	0.155182
$949$	−30.5840	−0.992799
$950$	0	0
$951$	−10.8807	−0.352831
$952$	1.50466	0.0487665
$953$	43.6960	1.41545	0.707727	−	0.706486i	$-0.249721\pi$
$953$	43.6960	1.41545	0.707727	+	0.706486i	$0.249721\pi$
$954$	9.00933	0.291688
$955$	0	0
$956$	−19.5747	−0.633090
$957$	1.49063	0.0481852
$958$	24.3527	0.786799
$959$	−0.397254	−0.0128280
$960$	0	0
$961$	−20.6188	−0.665122
$962$	16.4626	0.530777
$963$	−0.102703	−0.00330955
$964$	−3.55602	−0.114532
$965$	0	0
$966$	−11.7173	−0.376999
$967$	−28.7267	−0.923787	−0.461893	−	0.886935i	$-0.652830\pi$
$967$	−28.7267	−0.923787	−0.461893	+	0.886935i	$0.652830\pi$
$968$	−10.4720	−0.336582
$969$	−2.72666	−0.0875928
$970$	0	0
$971$	−14.4881	−0.464945	−0.232472	−	0.972603i	$-0.574681\pi$
$971$	−14.4881	−0.464945	−0.232472	+	0.972603i	$0.574681\pi$
$972$	1.00000	0.0320750
$973$	−29.4533	−0.944230
$974$	−24.5513	−0.786675
$975$	0	0
$976$	11.2406	0.359804
$977$	−19.1706	−0.613323	−0.306662	−	0.951819i	$-0.599212\pi$
$977$	−19.1706	−0.613323	−0.306662	+	0.951819i	$0.599212\pi$
$978$	−5.45331	−0.174378
$979$	−1.65872	−0.0530129
$980$	0	0
$981$	−15.2406	−0.486596
$982$	40.4813	1.29181
$983$	−27.7219	−0.884193	−0.442096	−	0.896968i	$-0.645765\pi$
$983$	−27.7219	−0.884193	−0.442096	+	0.896968i	$0.645765\pi$
$984$	3.55602	0.113362
$985$	0	0
$986$	2.05135	0.0653283
$987$	2.34128	0.0745238
$988$	5.45331	0.173493
$989$	−9.91595	−0.315309
$990$	0	0
$991$	37.7033	1.19768	0.598842	−	0.800867i	$-0.295628\pi$
$991$	37.7033	1.19768	0.598842	+	0.800867i	$0.295628\pi$
$992$	3.22199	0.102298
$993$	2.54669	0.0808167
$994$	15.5492	0.493192
$995$	0	0
$996$	1.27334	0.0403474
$997$	42.6099	1.34947	0.674735	−	0.738060i	$-0.264258\pi$
$997$	42.6099	1.34947	0.674735	+	0.738060i	$0.264258\pi$
$998$	23.2147	0.734850
$999$	8.23132	0.260427

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.a.bo.1.1		3
3.2	odd	2		7650.2.a.dl.1.1		3
5.2	odd	4		510.2.d.d.409.6	yes	6
5.3	odd	4		510.2.d.d.409.3	✓	6
5.4	even	2		2550.2.a.bn.1.3		3
15.2	even	4		1530.2.d.i.919.1		6
15.8	even	4		1530.2.d.i.919.4		6
15.14	odd	2		7650.2.a.dm.1.3		3
20.3	even	4		4080.2.m.p.2449.3		6
20.7	even	4		4080.2.m.p.2449.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.d.d.409.3	✓	6	5.3	odd	4
510.2.d.d.409.6	yes	6	5.2	odd	4
1530.2.d.i.919.1		6	15.2	even	4
1530.2.d.i.919.4		6	15.8	even	4
2550.2.a.bn.1.3		3	5.4	even	2
2550.2.a.bo.1.1		3	1.1	even	1	trivial
4080.2.m.p.2449.3		6	20.3	even	4
4080.2.m.p.2449.6		6	20.7	even	4
7650.2.a.dl.1.1		3	3.2	odd	2
7650.2.a.dm.1.3		3	15.14	odd	2

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:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.50466 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.726656 q^{11} +1.00000 q^{12} +2.00000 q^{13} -1.50466 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +2.72666 q^{19} -1.50466 q^{21} -0.726656 q^{22} +7.78734 q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} -1.50466 q^{28} -2.05135 q^{29} +3.22199 q^{31} +1.00000 q^{32} -0.726656 q^{33} -1.00000 q^{34} +1.00000 q^{36} +8.23132 q^{37} +2.72666 q^{38} +2.00000 q^{39} +3.55602 q^{41} -1.50466 q^{42} -1.27334 q^{43} -0.726656 q^{44} +7.78734 q^{46} -1.55602 q^{47} +1.00000 q^{48} -4.73599 q^{49} -1.00000 q^{51} +2.00000 q^{52} +9.00933 q^{53} +1.00000 q^{54} -1.50466 q^{56} +2.72666 q^{57} -2.05135 q^{58} -1.71733 q^{59} +11.2406 q^{61} +3.22199 q^{62} -1.50466 q^{63} +1.00000 q^{64} -0.726656 q^{66} +8.56534 q^{67} -1.00000 q^{68} +7.78734 q^{69} -10.3340 q^{71} +1.00000 q^{72} -15.2920 q^{73} +8.23132 q^{74} +2.72666 q^{76} +1.09337 q^{77} +2.00000 q^{78} +4.77801 q^{79} +1.00000 q^{81} +3.55602 q^{82} +1.27334 q^{83} -1.50466 q^{84} -1.27334 q^{86} -2.05135 q^{87} -0.726656 q^{88} +2.28267 q^{89} -3.00933 q^{91} +7.78734 q^{92} +3.22199 q^{93} -1.55602 q^{94} +1.00000 q^{96} +12.2827 q^{97} -4.73599 q^{98} -0.726656 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9} + 2 q^{11} + 3 q^{12} + 6 q^{13} + 6 q^{14} + 3 q^{16} - 3 q^{17} + 3 q^{18} + 4 q^{19} + 6 q^{21} + 2 q^{22} - 4 q^{23} + 3 q^{24}+ \cdots + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9 + 2 * q^11 + 3 * q^12 + 6 * q^13 + 6 * q^14 + 3 * q^16 - 3 * q^17 + 3 * q^18 + 4 * q^19 + 6 * q^21 + 2 * q^22 - 4 * q^23 + 3 * q^24 + 6 * q^26 + 3 * q^27 + 6 * q^28 - 4 * q^29 + 16 * q^31 + 3 * q^32 + 2 * q^33 - 3 * q^34 + 3 * q^36 + 10 * q^37 + 4 * q^38 + 6 * q^39 - 2 * q^41 + 6 * q^42 - 8 * q^43 + 2 * q^44 - 4 * q^46 + 8 * q^47 + 3 * q^48 + 11 * q^49 - 3 * q^51 + 6 * q^52 + 6 * q^53 + 3 * q^54 + 6 * q^56 + 4 * q^57 - 4 * q^58 - 22 * q^59 - 2 * q^61 + 16 * q^62 + 6 * q^63 + 3 * q^64 + 2 * q^66 - 8 * q^67 - 3 * q^68 - 4 * q^69 - 12 * q^71 + 3 * q^72 - 8 * q^73 + 10 * q^74 + 4 * q^76 + 20 * q^77 + 6 * q^78 + 8 * q^79 + 3 * q^81 - 2 * q^82 + 8 * q^83 + 6 * q^84 - 8 * q^86 - 4 * q^87 + 2 * q^88 - 10 * q^89 + 12 * q^91 - 4 * q^92 + 16 * q^93 + 8 * q^94 + 3 * q^96 + 20 * q^97 + 11 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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