LMFDB - Embedded newform 2601.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.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:	$20.7690895657$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.765367$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.765367 q^{2} -1.41421 q^{4} -0.0823922 q^{5} -2.76537 q^{7} +2.61313 q^{8} +O(q^{10})$	$q-0.765367 q^{2} -1.41421 q^{4} -0.0823922 q^{5} -2.76537 q^{7} +2.61313 q^{8} +0.0630603 q^{10} +0.668179 q^{11} -3.28130 q^{13} +2.11652 q^{14} +0.828427 q^{16} +3.64047 q^{19} +0.116520 q^{20} -0.511402 q^{22} +9.30864 q^{23} -4.99321 q^{25} +2.51140 q^{26} +3.91082 q^{28} +6.24264 q^{29} -5.04373 q^{31} -5.86030 q^{32} +0.227845 q^{35} -2.40621 q^{37} -2.78629 q^{38} -0.215301 q^{40} -0.480217 q^{41} +8.27452 q^{43} -0.944947 q^{44} -7.12453 q^{46} +8.88311 q^{47} +0.647254 q^{49} +3.82164 q^{50} +4.64047 q^{52} -11.3524 q^{53} -0.0550527 q^{55} -7.22625 q^{56} -4.77791 q^{58} -9.59379 q^{59} +2.58701 q^{61} +3.86030 q^{62} +2.82843 q^{64} +0.270354 q^{65} -0.944947 q^{67} -0.174385 q^{70} -3.28809 q^{71} -15.6819 q^{73} +1.84163 q^{74} -5.14840 q^{76} -1.84776 q^{77} +8.37170 q^{79} -0.0682559 q^{80} +0.367542 q^{82} -0.899869 q^{83} -6.33304 q^{86} +1.74603 q^{88} -5.64431 q^{89} +9.07401 q^{91} -13.1644 q^{92} -6.79884 q^{94} -0.299946 q^{95} +10.9723 q^{97} -0.495387 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 8 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 - 8 * q^7	$ 4 q + 4 q^{5} - 8 q^{7} - 8 q^{10} + 4 q^{11} - 4 q^{13} + 8 q^{14} - 8 q^{16} - 12 q^{19} + 8 q^{22} + 12 q^{23} + 8 q^{29} - 8 q^{31} - 16 q^{35} - 24 q^{37} - 16 q^{38} + 12 q^{41} + 4 q^{43} + 8 q^{44} - 8 q^{46} - 8 q^{47} - 4 q^{49} - 16 q^{50} - 8 q^{52} - 8 q^{53} - 12 q^{55} - 8 q^{56} - 24 q^{59} - 24 q^{61} - 8 q^{62} + 12 q^{65} + 8 q^{67} + 24 q^{70} - 24 q^{71} - 8 q^{73} - 8 q^{74} - 8 q^{76} - 8 q^{80} + 8 q^{82} - 8 q^{83} + 16 q^{86} - 16 q^{89} + 8 q^{91} - 16 q^{94} - 12 q^{95} + 16 q^{97} - 32 q^{98}+O(q^{100}) $ 4 * q + 4 * q^5 - 8 * q^7 - 8 * q^10 + 4 * q^11 - 4 * q^13 + 8 * q^14 - 8 * q^16 - 12 * q^19 + 8 * q^22 + 12 * q^23 + 8 * q^29 - 8 * q^31 - 16 * q^35 - 24 * q^37 - 16 * q^38 + 12 * q^41 + 4 * q^43 + 8 * q^44 - 8 * q^46 - 8 * q^47 - 4 * q^49 - 16 * q^50 - 8 * q^52 - 8 * q^53 - 12 * q^55 - 8 * q^56 - 24 * q^59 - 24 * q^61 - 8 * q^62 + 12 * q^65 + 8 * q^67 + 24 * q^70 - 24 * q^71 - 8 * q^73 - 8 * q^74 - 8 * q^76 - 8 * q^80 + 8 * q^82 - 8 * q^83 + 16 * q^86 - 16 * q^89 + 8 * q^91 - 16 * q^94 - 12 * q^95 + 16 * q^97 - 32 * 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 $

$2$	−0.765367	−0.541196	−0.270598	−	0.962692i	$-0.587221\pi$
$2$	−0.765367	−0.541196	−0.270598	+	0.962692i	$0.587221\pi$
$3$	0	0
$3$	0	0
$4$	−1.41421	−0.707107
$5$	−0.0823922	−0.0368469	−0.0184235	−	0.999830i	$-0.505865\pi$
$5$	−0.0823922	−0.0368469	−0.0184235	+	0.999830i	$0.505865\pi$
$6$	0	0
$7$	−2.76537	−1.04521	−0.522605	−	0.852575i	$-0.675040\pi$
$7$	−2.76537	−1.04521	−0.522605	+	0.852575i	$0.675040\pi$
$8$	2.61313	0.923880
$9$	0	0
$10$	0.0630603	0.0199414
$11$	0.668179	0.201463	0.100732	−	0.994914i	$-0.467882\pi$
$11$	0.668179	0.201463	0.100732	+	0.994914i	$0.467882\pi$
$12$	0	0
$13$	−3.28130	−0.910070	−0.455035	−	0.890474i	$-0.650373\pi$
$13$	−3.28130	−0.910070	−0.455035	+	0.890474i	$0.650373\pi$
$14$	2.11652	0.565664
$15$	0	0
$16$	0.828427	0.207107
$17$	0	0
$17$	0	0
$18$	0	0
$19$	3.64047	0.835180	0.417590	−	0.908636i	$-0.362875\pi$
$19$	3.64047	0.835180	0.417590	+	0.908636i	$0.362875\pi$
$20$	0.116520	0.0260547
$21$	0	0
$22$	−0.511402	−0.109031
$23$	9.30864	1.94099	0.970493	−	0.241128i	$-0.0775175\pi$
$23$	9.30864	1.94099	0.970493	+	0.241128i	$0.0775175\pi$
$24$	0	0
$25$	−4.99321	−0.998642
$26$	2.51140	0.492526
$27$	0	0
$28$	3.91082	0.739075
$29$	6.24264	1.15923	0.579615	−	0.814891i	$-0.303203\pi$
$29$	6.24264	1.15923	0.579615	+	0.814891i	$0.303203\pi$
$30$	0	0
$31$	−5.04373	−0.905880	−0.452940	−	0.891541i	$-0.649625\pi$
$31$	−5.04373	−0.905880	−0.452940	+	0.891541i	$0.649625\pi$
$32$	−5.86030	−1.03596
$33$	0	0
$34$	0	0
$35$	0.227845	0.0385128
$36$	0	0
$37$	−2.40621	−0.395578	−0.197789	−	0.980245i	$-0.563376\pi$
$37$	−2.40621	−0.395578	−0.197789	+	0.980245i	$0.563376\pi$
$38$	−2.78629	−0.451996
$39$	0	0
$40$	−0.215301	−0.0340421
$41$	−0.480217	−0.0749973	−0.0374986	−	0.999297i	$-0.511939\pi$
$41$	−0.480217	−0.0749973	−0.0374986	+	0.999297i	$0.511939\pi$
$42$	0	0
$43$	8.27452	1.26185	0.630926	−	0.775843i	$-0.282675\pi$
$43$	8.27452	1.26185	0.630926	+	0.775843i	$0.282675\pi$
$44$	−0.944947	−0.142456
$45$	0	0
$46$	−7.12453	−1.05045
$47$	8.88311	1.29573	0.647867	−	0.761753i	$-0.275661\pi$
$47$	8.88311	1.29573	0.647867	+	0.761753i	$0.275661\pi$
$48$	0	0
$49$	0.647254	0.0924648
$50$	3.82164	0.540461
$51$	0	0
$52$	4.64047	0.643517
$53$	−11.3524	−1.55937	−0.779684	−	0.626173i	$-0.784620\pi$
$53$	−11.3524	−1.55937	−0.779684	+	0.626173i	$0.784620\pi$
$54$	0	0
$55$	−0.0550527	−0.00742331
$56$	−7.22625	−0.965649
$57$	0	0
$58$	−4.77791	−0.627370
$59$	−9.59379	−1.24901	−0.624503	−	0.781023i	$-0.714698\pi$
$59$	−9.59379	−1.24901	−0.624503	+	0.781023i	$0.714698\pi$
$60$	0	0
$61$	2.58701	0.331232	0.165616	−	0.986190i	$-0.447039\pi$
$61$	2.58701	0.331232	0.165616	+	0.986190i	$0.447039\pi$
$62$	3.86030	0.490259
$63$	0	0
$64$	2.82843	0.353553
$65$	0.270354	0.0335333
$66$	0	0
$67$	−0.944947	−0.115444	−0.0577218	−	0.998333i	$-0.518384\pi$
$67$	−0.944947	−0.115444	−0.0577218	+	0.998333i	$0.518384\pi$
$68$	0	0
$69$	0	0
$70$	−0.174385	−0.0208430
$71$	−3.28809	−0.390225	−0.195112	−	0.980781i	$-0.562507\pi$
$71$	−3.28809	−0.390225	−0.195112	+	0.980781i	$0.562507\pi$
$72$	0	0
$73$	−15.6819	−1.83543	−0.917716	−	0.397237i	$-0.869969\pi$
$73$	−15.6819	−1.83543	−0.917716	+	0.397237i	$0.869969\pi$
$74$	1.84163	0.214085
$75$	0	0
$76$	−5.14840	−0.590561
$77$	−1.84776	−0.210572
$78$	0	0
$79$	8.37170	0.941890	0.470945	−	0.882162i	$-0.343913\pi$
$79$	8.37170	0.941890	0.470945	+	0.882162i	$0.343913\pi$
$80$	−0.0682559	−0.00763125
$81$	0	0
$82$	0.367542	0.0405882
$83$	−0.899869	−0.0987734	−0.0493867	−	0.998780i	$-0.515727\pi$
$83$	−0.899869	−0.0987734	−0.0493867	+	0.998780i	$0.515727\pi$
$84$	0	0
$85$	0	0
$86$	−6.33304	−0.682909
$87$	0	0
$88$	1.74603	0.186128
$89$	−5.64431	−0.598296	−0.299148	−	0.954207i	$-0.596702\pi$
$89$	−5.64431	−0.598296	−0.299148	+	0.954207i	$0.596702\pi$
$90$	0	0
$91$	9.07401	0.951215
$92$	−13.1644	−1.37248
$93$	0	0
$94$	−6.79884	−0.701246
$95$	−0.299946	−0.0307738
$96$	0	0
$97$	10.9723	1.11407	0.557033	−	0.830490i	$-0.311940\pi$
$97$	10.9723	1.11407	0.557033	+	0.830490i	$0.311940\pi$
$98$	−0.495387	−0.0500416
$99$	0	0
$100$	7.06147	0.706147
$101$	3.27677	0.326051	0.163025	−	0.986622i	$-0.447875\pi$
$101$	3.27677	0.326051	0.163025	+	0.986622i	$0.447875\pi$
$102$	0	0
$103$	−12.0228	−1.18464	−0.592321	−	0.805702i	$-0.701788\pi$
$103$	−12.0228	−1.18464	−0.592321	+	0.805702i	$0.701788\pi$
$104$	−8.57446	−0.840795
$105$	0	0
$106$	8.68873	0.843924
$107$	−8.72927	−0.843891	−0.421945	−	0.906621i	$-0.638653\pi$
$107$	−8.72927	−0.843891	−0.421945	+	0.906621i	$0.638653\pi$
$108$	0	0
$109$	4.26998	0.408990	0.204495	−	0.978868i	$-0.434445\pi$
$109$	4.26998	0.408990	0.204495	+	0.978868i	$0.434445\pi$
$110$	0.0421355	0.00401746
$111$	0	0
$112$	−2.29090	−0.216470
$113$	13.8558	1.30344	0.651720	−	0.758459i	$-0.274048\pi$
$113$	13.8558	1.30344	0.651720	+	0.758459i	$0.274048\pi$
$114$	0	0
$115$	−0.766960	−0.0715194
$116$	−8.82843	−0.819699
$117$	0	0
$118$	7.34277	0.675957
$119$	0	0
$120$	0	0
$121$	−10.5535	−0.959412
$122$	−1.98001	−0.179262
$123$	0	0
$124$	7.13291	0.640554
$125$	0.823363	0.0736438
$126$	0	0
$127$	0.633677	0.0562297	0.0281149	−	0.999605i	$-0.491050\pi$
$127$	0.633677	0.0562297	0.0281149	+	0.999605i	$0.491050\pi$
$128$	9.55582	0.844623
$129$	0	0
$130$	−0.206920	−0.0181481
$131$	−15.2081	−1.32874	−0.664371	−	0.747403i	$-0.731300\pi$
$131$	−15.2081	−1.32874	−0.664371	+	0.747403i	$0.731300\pi$
$132$	0	0
$133$	−10.0672	−0.872939
$134$	0.723231	0.0624777
$135$	0	0
$136$	0	0
$137$	−2.38847	−0.204060	−0.102030	−	0.994781i	$-0.532534\pi$
$137$	−2.38847	−0.204060	−0.102030	+	0.994781i	$0.532534\pi$
$138$	0	0
$139$	−1.61472	−0.136959	−0.0684793	−	0.997653i	$-0.521815\pi$
$139$	−1.61472	−0.136959	−0.0684793	+	0.997653i	$0.521815\pi$
$140$	−0.322221	−0.0272326
$141$	0	0
$142$	2.51660	0.211188
$143$	−2.19250	−0.183346
$144$	0	0
$145$	−0.514345	−0.0427140
$146$	12.0024	0.993329
$147$	0	0
$148$	3.40289	0.279716
$149$	−10.8082	−0.885442	−0.442721	−	0.896660i	$-0.645987\pi$
$149$	−10.8082	−0.885442	−0.442721	+	0.896660i	$0.645987\pi$
$150$	0	0
$151$	−21.7566	−1.77053	−0.885264	−	0.465089i	$-0.846022\pi$
$151$	−21.7566	−1.77053	−0.885264	+	0.465089i	$0.846022\pi$
$152$	9.51299	0.771606
$153$	0	0
$154$	1.41421	0.113961
$155$	0.415564	0.0333789
$156$	0	0
$157$	−19.8535	−1.58448	−0.792241	−	0.610208i	$-0.791086\pi$
$157$	−19.8535	−1.58448	−0.792241	+	0.610208i	$0.791086\pi$
$158$	−6.40743	−0.509747
$159$	0	0
$160$	0.482843	0.0381721
$161$	−25.7418	−2.02874
$162$	0	0
$163$	1.75511	0.137471	0.0687353	−	0.997635i	$-0.478104\pi$
$163$	1.75511	0.137471	0.0687353	+	0.997635i	$0.478104\pi$
$164$	0.679129	0.0530311
$165$	0	0
$166$	0.688730	0.0534558
$167$	−8.05921	−0.623641	−0.311820	−	0.950141i	$-0.600939\pi$
$167$	−8.05921	−0.623641	−0.311820	+	0.950141i	$0.600939\pi$
$168$	0	0
$169$	−2.23304	−0.171772
$170$	0	0
$171$	0	0
$172$	−11.7019	−0.892264
$173$	5.75282	0.437379	0.218690	−	0.975794i	$-0.429822\pi$
$173$	5.75282	0.437379	0.218690	+	0.975794i	$0.429822\pi$
$174$	0	0
$175$	13.8081	1.04379
$176$	0.553537	0.0417244
$177$	0	0
$178$	4.31997	0.323795
$179$	−14.6173	−1.09255	−0.546274	−	0.837607i	$-0.683954\pi$
$179$	−14.6173	−1.09255	−0.546274	+	0.837607i	$0.683954\pi$
$180$	0	0
$181$	−14.1000	−1.04804	−0.524022	−	0.851704i	$-0.675569\pi$
$181$	−14.1000	−1.04804	−0.524022	+	0.851704i	$0.675569\pi$
$182$	−6.94495	−0.514794
$183$	0	0
$184$	24.3247	1.79324
$185$	0.198253	0.0145758
$186$	0	0
$187$	0	0
$188$	−12.5626	−0.916222
$189$	0	0
$190$	0.229569	0.0166547
$191$	−16.0167	−1.15893	−0.579464	−	0.814998i	$-0.696738\pi$
$191$	−16.0167	−1.15893	−0.579464	+	0.814998i	$0.696738\pi$
$192$	0	0
$193$	4.59539	0.330783	0.165392	−	0.986228i	$-0.447111\pi$
$193$	4.59539	0.330783	0.165392	+	0.986228i	$0.447111\pi$
$194$	−8.39782	−0.602929
$195$	0	0
$196$	−0.915355	−0.0653825
$197$	21.1689	1.50822	0.754112	−	0.656745i	$-0.228067\pi$
$197$	21.1689	1.50822	0.754112	+	0.656745i	$0.228067\pi$
$198$	0	0
$199$	2.32541	0.164844	0.0824219	−	0.996598i	$-0.473735\pi$
$199$	2.32541	0.164844	0.0824219	+	0.996598i	$0.473735\pi$
$200$	−13.0479	−0.922625
$201$	0	0
$202$	−2.50793	−0.176457
$203$	−17.2632	−1.21164
$204$	0	0
$205$	0.0395661	0.00276342
$206$	9.20186	0.641124
$207$	0	0
$208$	−2.71832	−0.188482
$209$	2.43248	0.168258
$210$	0	0
$211$	−7.58541	−0.522201	−0.261101	−	0.965312i	$-0.584085\pi$
$211$	−7.58541	−0.522201	−0.261101	+	0.965312i	$0.584085\pi$
$212$	16.0547	1.10264
$213$	0	0
$214$	6.68110	0.456710
$215$	−0.681756	−0.0464953
$216$	0	0
$217$	13.9478	0.946836
$218$	−3.26810	−0.221344
$219$	0	0
$220$	0.0778563	0.00524907
$221$	0	0
$222$	0	0
$223$	−15.6624	−1.04883	−0.524415	−	0.851463i	$-0.675716\pi$
$223$	−15.6624	−1.04883	−0.524415	+	0.851463i	$0.675716\pi$
$224$	16.2059	1.08280
$225$	0	0
$226$	−10.6047	−0.705417
$227$	14.6314	0.971122	0.485561	−	0.874203i	$-0.338615\pi$
$227$	14.6314	0.971122	0.485561	+	0.874203i	$0.338615\pi$
$228$	0	0
$229$	16.3451	1.08011	0.540056	−	0.841629i	$-0.318403\pi$
$229$	16.3451	1.08011	0.540056	+	0.841629i	$0.318403\pi$
$230$	0.587006	0.0387060
$231$	0	0
$232$	16.3128	1.07099
$233$	−0.357811	−0.0234409	−0.0117205	−	0.999931i	$-0.503731\pi$
$233$	−0.357811	−0.0234409	−0.0117205	+	0.999931i	$0.503731\pi$
$234$	0	0
$235$	−0.731899	−0.0477438
$236$	13.5677	0.883180
$237$	0	0
$238$	0	0
$239$	9.71153	0.628187	0.314093	−	0.949392i	$-0.398299\pi$
$239$	9.71153	0.628187	0.314093	+	0.949392i	$0.398299\pi$
$240$	0	0
$241$	−9.36173	−0.603042	−0.301521	−	0.953460i	$-0.597494\pi$
$241$	−9.36173	−0.603042	−0.301521	+	0.953460i	$0.597494\pi$
$242$	8.07733	0.519230
$243$	0	0
$244$	−3.65858	−0.234216
$245$	−0.0533287	−0.00340704
$246$	0	0
$247$	−11.9455	−0.760072
$248$	−13.1799	−0.836924
$249$	0	0
$250$	−0.630175	−0.0398557
$251$	−3.28196	−0.207156	−0.103578	−	0.994621i	$-0.533029\pi$
$251$	−3.28196	−0.207156	−0.103578	+	0.994621i	$0.533029\pi$
$252$	0	0
$253$	6.21984	0.391038
$254$	−0.484995	−0.0304313
$255$	0	0
$256$	−12.9706	−0.810660
$257$	−3.54500	−0.221131	−0.110566	−	0.993869i	$-0.535266\pi$
$257$	−3.54500	−0.221131	−0.110566	+	0.993869i	$0.535266\pi$
$258$	0	0
$259$	6.65404	0.413462
$260$	−0.382338	−0.0237116
$261$	0	0
$262$	11.6398	0.719110
$263$	−13.5133	−0.833265	−0.416632	−	0.909075i	$-0.636790\pi$
$263$	−13.5133	−0.833265	−0.416632	+	0.909075i	$0.636790\pi$
$264$	0	0
$265$	0.935347	0.0574579
$266$	7.70512	0.472431
$267$	0	0
$268$	1.33636	0.0816310
$269$	5.16895	0.315156	0.157578	−	0.987507i	$-0.449631\pi$
$269$	5.16895	0.315156	0.157578	+	0.987507i	$0.449631\pi$
$270$	0	0
$271$	−6.68592	−0.406141	−0.203070	−	0.979164i	$-0.565092\pi$
$271$	−6.68592	−0.406141	−0.203070	+	0.979164i	$0.565092\pi$
$272$	0	0
$273$	0	0
$274$	1.82805	0.110437
$275$	−3.33636	−0.201190
$276$	0	0
$277$	−2.90159	−0.174340	−0.0871699	−	0.996193i	$-0.527782\pi$
$277$	−2.90159	−0.174340	−0.0871699	+	0.996193i	$0.527782\pi$
$278$	1.23585	0.0741215
$279$	0	0
$280$	0.595387	0.0355812
$281$	22.9207	1.36734	0.683668	−	0.729793i	$-0.260384\pi$
$281$	22.9207	1.36734	0.683668	+	0.729793i	$0.260384\pi$
$282$	0	0
$283$	−27.7209	−1.64784	−0.823918	−	0.566709i	$-0.808216\pi$
$283$	−27.7209	−1.64784	−0.823918	+	0.566709i	$0.808216\pi$
$284$	4.65007	0.275931
$285$	0	0
$286$	1.67807	0.0992261
$287$	1.32798	0.0783879
$288$	0	0
$289$	0	0
$290$	0.393663	0.0231167
$291$	0	0
$292$	22.1776	1.29785
$293$	−11.4677	−0.669949	−0.334974	−	0.942227i	$-0.608728\pi$
$293$	−11.4677	−0.669949	−0.334974	+	0.942227i	$0.608728\pi$
$294$	0	0
$295$	0.790454	0.0460220
$296$	−6.28772	−0.365466
$297$	0	0
$298$	8.27223	0.479198
$299$	−30.5445	−1.76643
$300$	0	0
$301$	−22.8821	−1.31890
$302$	16.6518	0.958203
$303$	0	0
$304$	3.01586	0.172971
$305$	−0.213149	−0.0122049
$306$	0	0
$307$	10.5245	0.600663	0.300332	−	0.953835i	$-0.402903\pi$
$307$	10.5245	0.600663	0.300332	+	0.953835i	$0.402903\pi$
$308$	2.61313	0.148897
$309$	0	0
$310$	−0.318059	−0.0180645
$311$	−5.41421	−0.307012	−0.153506	−	0.988148i	$-0.549056\pi$
$311$	−5.41421	−0.307012	−0.153506	+	0.988148i	$0.549056\pi$
$312$	0	0
$313$	0.951362	0.0537742	0.0268871	−	0.999638i	$-0.491441\pi$
$313$	0.951362	0.0537742	0.0268871	+	0.999638i	$0.491441\pi$
$314$	15.1952	0.857516
$315$	0	0
$316$	−11.8394	−0.666017
$317$	−21.5647	−1.21119	−0.605596	−	0.795772i	$-0.707065\pi$
$317$	−21.5647	−1.21119	−0.605596	+	0.795772i	$0.707065\pi$
$318$	0	0
$319$	4.17120	0.233542
$320$	−0.233040	−0.0130274
$321$	0	0
$322$	19.7019	1.09795
$323$	0	0
$324$	0	0
$325$	16.3842	0.908835
$326$	−1.34330	−0.0743985
$327$	0	0
$328$	−1.25487	−0.0692885
$329$	−24.5650	−1.35431
$330$	0	0
$331$	−20.7366	−1.13979	−0.569894	−	0.821718i	$-0.693016\pi$
$331$	−20.7366	−1.13979	−0.569894	+	0.821718i	$0.693016\pi$
$332$	1.27261	0.0698434
$333$	0	0
$334$	6.16826	0.337512
$335$	0.0778563	0.00425374
$336$	0	0
$337$	13.9275	0.758681	0.379340	−	0.925257i	$-0.376151\pi$
$337$	13.9275	0.758681	0.379340	+	0.925257i	$0.376151\pi$
$338$	1.70910	0.0929625
$339$	0	0
$340$	0	0
$341$	−3.37011	−0.182502
$342$	0	0
$343$	17.5677	0.948565
$344$	21.6224	1.16580
$345$	0	0
$346$	−4.40302	−0.236708
$347$	16.0894	0.863722	0.431861	−	0.901940i	$-0.357857\pi$
$347$	16.0894	0.863722	0.431861	+	0.901940i	$0.357857\pi$
$348$	0	0
$349$	9.30051	0.497845	0.248922	−	0.968523i	$-0.419924\pi$
$349$	9.30051	0.497845	0.248922	+	0.968523i	$0.419924\pi$
$350$	−10.5682	−0.564896
$351$	0	0
$352$	−3.91573	−0.208709
$353$	27.8142	1.48040	0.740201	−	0.672385i	$-0.234730\pi$
$353$	27.8142	1.48040	0.740201	+	0.672385i	$0.234730\pi$
$354$	0	0
$355$	0.270913	0.0143786
$356$	7.98226	0.423059
$357$	0	0
$358$	11.1876	0.591282
$359$	−4.64022	−0.244902	−0.122451	−	0.992475i	$-0.539075\pi$
$359$	−4.64022	−0.244902	−0.122451	+	0.992475i	$0.539075\pi$
$360$	0	0
$361$	−5.74701	−0.302474
$362$	10.7917	0.567198
$363$	0	0
$364$	−12.8326	−0.672610
$365$	1.29207	0.0676300
$366$	0	0
$367$	−4.32119	−0.225564	−0.112782	−	0.993620i	$-0.535976\pi$
$367$	−4.32119	−0.225564	−0.112782	+	0.993620i	$0.535976\pi$
$368$	7.71153	0.401991
$369$	0	0
$370$	−0.151736	−0.00788838
$371$	31.3935	1.62987
$372$	0	0
$373$	−8.44020	−0.437017	−0.218509	−	0.975835i	$-0.570119\pi$
$373$	−8.44020	−0.437017	−0.218509	+	0.975835i	$0.570119\pi$
$374$	0	0
$375$	0	0
$376$	23.2127	1.19710
$377$	−20.4840	−1.05498
$378$	0	0
$379$	7.65616	0.393271	0.196635	−	0.980477i	$-0.436998\pi$
$379$	7.65616	0.393271	0.196635	+	0.980477i	$0.436998\pi$
$380$	0.424188	0.0217604
$381$	0	0
$382$	12.2587	0.627207
$383$	−18.5264	−0.946654	−0.473327	−	0.880887i	$-0.656947\pi$
$383$	−18.5264	−0.946654	−0.473327	+	0.880887i	$0.656947\pi$
$384$	0	0
$385$	0.152241	0.00775892
$386$	−3.51716	−0.179019
$387$	0	0
$388$	−15.5172	−0.787764
$389$	−26.8124	−1.35944	−0.679720	−	0.733472i	$-0.737899\pi$
$389$	−26.8124	−1.35944	−0.679720	+	0.733472i	$0.737899\pi$
$390$	0	0
$391$	0	0
$392$	1.69136	0.0854264
$393$	0	0
$394$	−16.2020	−0.816245
$395$	−0.689763	−0.0347058
$396$	0	0
$397$	−3.14026	−0.157605	−0.0788025	−	0.996890i	$-0.525110\pi$
$397$	−3.14026	−0.157605	−0.0788025	+	0.996890i	$0.525110\pi$
$398$	−1.77979	−0.0892128
$399$	0	0
$400$	−4.13651	−0.206826
$401$	26.4666	1.32168	0.660840	−	0.750526i	$-0.270200\pi$
$401$	26.4666	1.32168	0.660840	+	0.750526i	$0.270200\pi$
$402$	0	0
$403$	16.5500	0.824415
$404$	−4.63405	−0.230553
$405$	0	0
$406$	13.2127	0.655734
$407$	−1.60778	−0.0796945
$408$	0	0
$409$	−31.0443	−1.53504	−0.767521	−	0.641024i	$-0.778510\pi$
$409$	−31.0443	−1.53504	−0.767521	+	0.641024i	$0.778510\pi$
$410$	−0.0302826	−0.00149555
$411$	0	0
$412$	17.0028	0.837668
$413$	26.5304	1.30547
$414$	0	0
$415$	0.0741422	0.00363950
$416$	19.2294	0.942801
$417$	0	0
$418$	−1.86174	−0.0910607
$419$	−38.7394	−1.89255	−0.946273	−	0.323370i	$-0.895184\pi$
$419$	−38.7394	−1.89255	−0.946273	+	0.323370i	$0.895184\pi$
$420$	0	0
$421$	−19.0379	−0.927851	−0.463926	−	0.885874i	$-0.653560\pi$
$421$	−19.0379	−0.927851	−0.463926	+	0.885874i	$0.653560\pi$
$422$	5.80562	0.282613
$423$	0	0
$424$	−29.6652	−1.44067
$425$	0	0
$426$	0	0
$427$	−7.15402	−0.346207
$428$	12.3451	0.596721
$429$	0	0
$430$	0.521793	0.0251631
$431$	−11.9054	−0.573462	−0.286731	−	0.958011i	$-0.592569\pi$
$431$	−11.9054	−0.573462	−0.286731	+	0.958011i	$0.592569\pi$
$432$	0	0
$433$	−21.3675	−1.02686	−0.513428	−	0.858133i	$-0.671625\pi$
$433$	−21.3675	−1.02686	−0.513428	+	0.858133i	$0.671625\pi$
$434$	−10.6752	−0.512424
$435$	0	0
$436$	−6.03866	−0.289200
$437$	33.8878	1.62107
$438$	0	0
$439$	−21.1872	−1.01121	−0.505606	−	0.862764i	$-0.668731\pi$
$439$	−21.1872	−1.01121	−0.505606	+	0.862764i	$0.668731\pi$
$440$	−0.143860	−0.00685824
$441$	0	0
$442$	0	0
$443$	21.5467	1.02372	0.511858	−	0.859070i	$-0.328957\pi$
$443$	21.5467	1.02372	0.511858	+	0.859070i	$0.328957\pi$
$444$	0	0
$445$	0.465047	0.0220454
$446$	11.9875	0.567623
$447$	0	0
$448$	−7.82164	−0.369538
$449$	36.1769	1.70729	0.853646	−	0.520854i	$-0.174386\pi$
$449$	36.1769	1.70729	0.853646	+	0.520854i	$0.174386\pi$
$450$	0	0
$451$	−0.320871	−0.0151092
$452$	−19.5950	−0.921672
$453$	0	0
$454$	−11.1984	−0.525567
$455$	−0.747628	−0.0350493
$456$	0	0
$457$	28.2159	1.31988	0.659942	−	0.751316i	$-0.270581\pi$
$457$	28.2159	1.31988	0.659942	+	0.751316i	$0.270581\pi$
$458$	−12.5100	−0.584552
$459$	0	0
$460$	1.08464	0.0505718
$461$	28.4543	1.32525	0.662623	−	0.748953i	$-0.269443\pi$
$461$	28.4543	1.32525	0.662623	+	0.748953i	$0.269443\pi$
$462$	0	0
$463$	−6.05277	−0.281296	−0.140648	−	0.990060i	$-0.544919\pi$
$463$	−6.05277	−0.281296	−0.140648	+	0.990060i	$0.544919\pi$
$464$	5.17157	0.240084
$465$	0	0
$466$	0.273856	0.0126861
$467$	32.1428	1.48739	0.743696	−	0.668518i	$-0.233071\pi$
$467$	32.1428	1.48739	0.743696	+	0.668518i	$0.233071\pi$
$468$	0	0
$469$	2.61313	0.120663
$470$	0.560171	0.0258388
$471$	0	0
$472$	−25.0698	−1.15393
$473$	5.52885	0.254217
$474$	0	0
$475$	−18.1776	−0.834046
$476$	0	0
$477$	0	0
$478$	−7.43289	−0.339972
$479$	−15.5513	−0.710556	−0.355278	−	0.934761i	$-0.615614\pi$
$479$	−15.5513	−0.710556	−0.355278	+	0.934761i	$0.615614\pi$
$480$	0	0
$481$	7.89549	0.360004
$482$	7.16516	0.326364
$483$	0	0
$484$	14.9250	0.678407
$485$	−0.904031	−0.0410499
$486$	0	0
$487$	20.0717	0.909537	0.454768	−	0.890610i	$-0.349722\pi$
$487$	20.0717	0.909537	0.454768	+	0.890610i	$0.349722\pi$
$488$	6.76017	0.306019
$489$	0	0
$490$	0.0408160	0.00184388
$491$	−5.37086	−0.242383	−0.121192	−	0.992629i	$-0.538672\pi$
$491$	−5.37086	−0.242383	−0.121192	+	0.992629i	$0.538672\pi$
$492$	0	0
$493$	0	0
$494$	9.14267	0.411348
$495$	0	0
$496$	−4.17836	−0.187614
$497$	9.09278	0.407867
$498$	0	0
$499$	−28.0658	−1.25640	−0.628199	−	0.778053i	$-0.716207\pi$
$499$	−28.0658	−1.25640	−0.628199	+	0.778053i	$0.716207\pi$
$500$	−1.16441	−0.0520740
$501$	0	0
$502$	2.51191	0.112112
$503$	8.76921	0.391000	0.195500	−	0.980704i	$-0.437367\pi$
$503$	8.76921	0.391000	0.195500	+	0.980704i	$0.437367\pi$
$504$	0	0
$505$	−0.269980	−0.0120140
$506$	−4.76046	−0.211628
$507$	0	0
$508$	−0.896155	−0.0397604
$509$	13.5248	0.599475	0.299737	−	0.954022i	$-0.403101\pi$
$509$	13.5248	0.599475	0.299737	+	0.954022i	$0.403101\pi$
$510$	0	0
$511$	43.3663	1.91841
$512$	−9.18440	−0.405897
$513$	0	0
$514$	2.71323	0.119675
$515$	0.990585	0.0436504
$516$	0	0
$517$	5.93550	0.261043
$518$	−5.09278	−0.223764
$519$	0	0
$520$	0.706469	0.0309807
$521$	−9.32360	−0.408474	−0.204237	−	0.978921i	$-0.565471\pi$
$521$	−9.32360	−0.408474	−0.204237	+	0.978921i	$0.565471\pi$
$522$	0	0
$523$	−17.0634	−0.746129	−0.373065	−	0.927805i	$-0.621693\pi$
$523$	−17.0634	−0.746129	−0.373065	+	0.927805i	$0.621693\pi$
$524$	21.5076	0.939562
$525$	0	0
$526$	10.3426	0.450960
$527$	0	0
$528$	0	0
$529$	63.6509	2.76743
$530$	−0.715884	−0.0310960
$531$	0	0
$532$	14.2372	0.617261
$533$	1.57574	0.0682528
$534$	0	0
$535$	0.719224	0.0310948
$536$	−2.46927	−0.106656
$537$	0	0
$538$	−3.95614	−0.170561
$539$	0.432481	0.0186283
$540$	0	0
$541$	−3.81082	−0.163840	−0.0819200	−	0.996639i	$-0.526105\pi$
$541$	−3.81082	−0.163840	−0.0819200	+	0.996639i	$0.526105\pi$
$542$	5.11718	0.219802
$543$	0	0
$544$	0	0
$545$	−0.351813	−0.0150700
$546$	0	0
$547$	−26.0859	−1.11535	−0.557677	−	0.830058i	$-0.688307\pi$
$547$	−26.0859	−1.11535	−0.557677	+	0.830058i	$0.688307\pi$
$548$	3.37780	0.144293
$549$	0	0
$550$	2.55354	0.108883
$551$	22.7261	0.968165
$552$	0	0
$553$	−23.1508	−0.984474
$554$	2.22078	0.0943520
$555$	0	0
$556$	2.28356	0.0968444
$557$	−2.00763	−0.0850662	−0.0425331	−	0.999095i	$-0.513543\pi$
$557$	−2.00763	−0.0850662	−0.0425331	+	0.999095i	$0.513543\pi$
$558$	0	0
$559$	−27.1512	−1.14837
$560$	0.188753	0.00797626
$561$	0	0
$562$	−17.5428	−0.739997
$563$	27.9031	1.17598	0.587988	−	0.808869i	$-0.299920\pi$
$563$	27.9031	1.17598	0.587988	+	0.808869i	$0.299920\pi$
$564$	0	0
$565$	−1.14161	−0.0480278
$566$	21.2167	0.891802
$567$	0	0
$568$	−8.59220	−0.360521
$569$	22.8126	0.956356	0.478178	−	0.878263i	$-0.341297\pi$
$569$	22.8126	0.956356	0.478178	+	0.878263i	$0.341297\pi$
$570$	0	0
$571$	15.7098	0.657435	0.328718	−	0.944428i	$-0.393384\pi$
$571$	15.7098	0.657435	0.328718	+	0.944428i	$0.393384\pi$
$572$	3.10066	0.129645
$573$	0	0
$574$	−1.01639	−0.0424233
$575$	−46.4800	−1.93835
$576$	0	0
$577$	29.1062	1.21171	0.605853	−	0.795577i	$-0.292832\pi$
$577$	29.1062	1.21171	0.605853	+	0.795577i	$0.292832\pi$
$578$	0	0
$579$	0	0
$580$	0.727394	0.0302034
$581$	2.48847	0.103239
$582$	0	0
$583$	−7.58541	−0.314156
$584$	−40.9789	−1.69572
$585$	0	0
$586$	8.77698	0.362574
$587$	−19.0186	−0.784983	−0.392492	−	0.919756i	$-0.628387\pi$
$587$	−19.0186	−0.784983	−0.392492	+	0.919756i	$0.628387\pi$
$588$	0	0
$589$	−18.3615	−0.756573
$590$	−0.604987	−0.0249069
$591$	0	0
$592$	−1.99337	−0.0819269
$593$	10.4425	0.428820	0.214410	−	0.976744i	$-0.431217\pi$
$593$	10.4425	0.428820	0.214410	+	0.976744i	$0.431217\pi$
$594$	0	0
$595$	0	0
$596$	15.2851	0.626102
$597$	0	0
$598$	23.3777	0.955987
$599$	22.1338	0.904361	0.452180	−	0.891927i	$-0.350646\pi$
$599$	22.1338	0.904361	0.452180	+	0.891927i	$0.350646\pi$
$600$	0	0
$601$	−2.96951	−0.121129	−0.0605643	−	0.998164i	$-0.519290\pi$
$601$	−2.96951	−0.121129	−0.0605643	+	0.998164i	$0.519290\pi$
$602$	17.5132	0.713784
$603$	0	0
$604$	30.7685	1.25195
$605$	0.869529	0.0353514
$606$	0	0
$607$	−29.6467	−1.20332	−0.601661	−	0.798752i	$-0.705494\pi$
$607$	−29.6467	−1.20332	−0.601661	+	0.798752i	$0.705494\pi$
$608$	−21.3342	−0.865217
$609$	0	0
$610$	0.163137	0.00660523
$611$	−29.1482	−1.17921
$612$	0	0
$613$	30.3538	1.22598	0.612988	−	0.790092i	$-0.289967\pi$
$613$	30.3538	1.22598	0.612988	+	0.790092i	$0.289967\pi$
$614$	−8.05508	−0.325077
$615$	0	0
$616$	−4.82843	−0.194543
$617$	35.4772	1.42826	0.714129	−	0.700014i	$-0.246823\pi$
$617$	35.4772	1.42826	0.714129	+	0.700014i	$0.246823\pi$
$618$	0	0
$619$	26.2024	1.05316	0.526582	−	0.850124i	$-0.323473\pi$
$619$	26.2024	1.05316	0.526582	+	0.850124i	$0.323473\pi$
$620$	−0.587696	−0.0236024
$621$	0	0
$622$	4.14386	0.166154
$623$	15.6086	0.625345
$624$	0	0
$625$	24.8982	0.995929
$626$	−0.728141	−0.0291024
$627$	0	0
$628$	28.0771	1.12040
$629$	0	0
$630$	0	0
$631$	−0.438772	−0.0174672	−0.00873362	−	0.999962i	$-0.502780\pi$
$631$	−0.438772	−0.0174672	−0.00873362	+	0.999962i	$0.502780\pi$
$632$	21.8763	0.870193
$633$	0	0
$634$	16.5049	0.655493
$635$	−0.0522100	−0.00207189
$636$	0	0
$637$	−2.12384	−0.0841495
$638$	−3.19250	−0.126392
$639$	0	0
$640$	−0.787325	−0.0311218
$641$	4.78793	0.189112	0.0945559	−	0.995520i	$-0.469857\pi$
$641$	4.78793	0.189112	0.0945559	+	0.995520i	$0.469857\pi$
$642$	0	0
$643$	20.5340	0.809782	0.404891	−	0.914365i	$-0.367310\pi$
$643$	20.5340	0.809782	0.404891	+	0.914365i	$0.367310\pi$
$644$	36.4044	1.43454
$645$	0	0
$646$	0	0
$647$	−18.7594	−0.737508	−0.368754	−	0.929527i	$-0.620216\pi$
$647$	−18.7594	−0.737508	−0.368754	+	0.929527i	$0.620216\pi$
$648$	0	0
$649$	−6.41037	−0.251629
$650$	−12.5400	−0.491858
$651$	0	0
$652$	−2.48210	−0.0972064
$653$	14.2176	0.556379	0.278189	−	0.960526i	$-0.410266\pi$
$653$	14.2176	0.556379	0.278189	+	0.960526i	$0.410266\pi$
$654$	0	0
$655$	1.25303	0.0489600
$656$	−0.397825	−0.0155324
$657$	0	0
$658$	18.8013	0.732950
$659$	−34.5061	−1.34417	−0.672084	−	0.740475i	$-0.734601\pi$
$659$	−34.5061	−1.34417	−0.672084	+	0.740475i	$0.734601\pi$
$660$	0	0
$661$	12.7758	0.496922	0.248461	−	0.968642i	$-0.420075\pi$
$661$	12.7758	0.496922	0.248461	+	0.968642i	$0.420075\pi$
$662$	15.8711	0.616849
$663$	0	0
$664$	−2.35147	−0.0912547
$665$	0.829461	0.0321651
$666$	0	0
$667$	58.1105	2.25005
$668$	11.3975	0.440981
$669$	0	0
$670$	−0.0595886	−0.00230211
$671$	1.72858	0.0667312
$672$	0	0
$673$	−33.8641	−1.30536	−0.652682	−	0.757632i	$-0.726356\pi$
$673$	−33.8641	−1.30536	−0.652682	+	0.757632i	$0.726356\pi$
$674$	−10.6597	−0.410595
$675$	0	0
$676$	3.15800	0.121461
$677$	−35.6921	−1.37176	−0.685880	−	0.727714i	$-0.740583\pi$
$677$	−35.6921	−1.37176	−0.685880	+	0.727714i	$0.740583\pi$
$678$	0	0
$679$	−30.3424	−1.16443
$680$	0	0
$681$	0	0
$682$	2.57937	0.0987692
$683$	−22.2664	−0.851999	−0.426000	−	0.904723i	$-0.640078\pi$
$683$	−22.2664	−0.851999	−0.426000	+	0.904723i	$0.640078\pi$
$684$	0	0
$685$	0.196791	0.00751900
$686$	−13.4457	−0.513360
$687$	0	0
$688$	6.85483	0.261338
$689$	37.2506	1.41913
$690$	0	0
$691$	29.0193	1.10395	0.551973	−	0.833862i	$-0.313875\pi$
$691$	29.0193	1.10395	0.551973	+	0.833862i	$0.313875\pi$
$692$	−8.13572	−0.309274
$693$	0	0
$694$	−12.3143	−0.467443
$695$	0.133040	0.00504650
$696$	0	0
$697$	0	0
$698$	−7.11830	−0.269432
$699$	0	0
$700$	−19.5275	−0.738072
$701$	33.5632	1.26766	0.633832	−	0.773471i	$-0.281481\pi$
$701$	33.5632	1.26766	0.633832	+	0.773471i	$0.281481\pi$
$702$	0	0
$703$	−8.75971	−0.330379
$704$	1.88989	0.0712281
$705$	0	0
$706$	−21.2881	−0.801188
$707$	−9.06147	−0.340792
$708$	0	0
$709$	41.7240	1.56698	0.783489	−	0.621405i	$-0.213438\pi$
$709$	41.7240	1.56698	0.783489	+	0.621405i	$0.213438\pi$
$710$	−0.207348	−0.00778163
$711$	0	0
$712$	−14.7493	−0.552753
$713$	−46.9503	−1.75830
$714$	0	0
$715$	0.180645	0.00675573
$716$	20.6720	0.772548
$717$	0	0
$718$	3.55147	0.132540
$719$	−50.6576	−1.88921	−0.944604	−	0.328212i	$-0.893554\pi$
$719$	−50.6576	−1.88921	−0.944604	+	0.328212i	$0.893554\pi$
$720$	0	0
$721$	33.2475	1.23820
$722$	4.39857	0.163698
$723$	0	0
$724$	19.9404	0.741080
$725$	−31.1708	−1.15766
$726$	0	0
$727$	11.4948	0.426317	0.213159	−	0.977018i	$-0.431625\pi$
$727$	11.4948	0.426317	0.213159	+	0.977018i	$0.431625\pi$
$728$	23.7115	0.878808
$729$	0	0
$730$	−0.988907	−0.0366011
$731$	0	0
$732$	0	0
$733$	−48.4680	−1.79021	−0.895103	−	0.445859i	$-0.852898\pi$
$733$	−48.4680	−1.79021	−0.895103	+	0.445859i	$0.852898\pi$
$734$	3.30729	0.122074
$735$	0	0
$736$	−54.5515	−2.01079
$737$	−0.631394	−0.0232577
$738$	0	0
$739$	2.50512	0.0921523	0.0460761	−	0.998938i	$-0.485328\pi$
$739$	2.50512	0.0921523	0.0460761	+	0.998938i	$0.485328\pi$
$740$	−0.280372	−0.0103067
$741$	0	0
$742$	−24.0275	−0.882078
$743$	−14.1003	−0.517289	−0.258645	−	0.965973i	$-0.583276\pi$
$743$	−14.1003	−0.517289	−0.258645	+	0.965973i	$0.583276\pi$
$744$	0	0
$745$	0.890511	0.0326258
$746$	6.45985	0.236512
$747$	0	0
$748$	0	0
$749$	24.1396	0.882043
$750$	0	0
$751$	22.3611	0.815969	0.407984	−	0.912989i	$-0.366232\pi$
$751$	22.3611	0.815969	0.407984	+	0.912989i	$0.366232\pi$
$752$	7.35901	0.268355
$753$	0	0
$754$	15.6778	0.570951
$755$	1.79258	0.0652385
$756$	0	0
$757$	−9.15121	−0.332606	−0.166303	−	0.986075i	$-0.553183\pi$
$757$	−9.15121	−0.332606	−0.166303	+	0.986075i	$0.553183\pi$
$758$	−5.85977	−0.212837
$759$	0	0
$760$	−0.783797	−0.0284313
$761$	29.3561	1.06416	0.532079	−	0.846694i	$-0.321411\pi$
$761$	29.3561	1.06416	0.532079	+	0.846694i	$0.321411\pi$
$762$	0	0
$763$	−11.8081	−0.427481
$764$	22.6510	0.819486
$765$	0	0
$766$	14.1795	0.512325
$767$	31.4802	1.13668
$768$	0	0
$769$	−17.3301	−0.624939	−0.312470	−	0.949928i	$-0.601156\pi$
$769$	−17.3301	−0.624939	−0.312470	+	0.949928i	$0.601156\pi$
$770$	−0.116520	−0.00419910
$771$	0	0
$772$	−6.49886	−0.233899
$773$	−45.5488	−1.63828	−0.819139	−	0.573595i	$-0.805548\pi$
$773$	−45.5488	−1.63828	−0.819139	+	0.573595i	$0.805548\pi$
$774$	0	0
$775$	25.1844	0.904650
$776$	28.6720	1.02926
$777$	0	0
$778$	20.5213	0.735724
$779$	−1.74821	−0.0626362
$780$	0	0
$781$	−2.19703	−0.0786160
$782$	0	0
$783$	0	0
$784$	0.536203	0.0191501
$785$	1.63577	0.0583833
$786$	0	0
$787$	33.4248	1.19147	0.595733	−	0.803183i	$-0.296862\pi$
$787$	33.4248	1.19147	0.595733	+	0.803183i	$0.296862\pi$
$788$	−29.9374	−1.06648
$789$	0	0
$790$	0.527922	0.0187826
$791$	−38.3163	−1.36237
$792$	0	0
$793$	−8.48875	−0.301444
$794$	2.40345	0.0852952
$795$	0	0
$796$	−3.28862	−0.116562
$797$	9.90666	0.350912	0.175456	−	0.984487i	$-0.443860\pi$
$797$	9.90666	0.350912	0.175456	+	0.984487i	$0.443860\pi$
$798$	0	0
$799$	0	0
$800$	29.2617	1.03456
$801$	0	0
$802$	−20.2567	−0.715289
$803$	−10.4783	−0.369773
$804$	0	0
$805$	2.12092	0.0747528
$806$	−12.6668	−0.446170
$807$	0	0
$808$	8.56261	0.301232
$809$	4.21493	0.148189	0.0740945	−	0.997251i	$-0.476393\pi$
$809$	4.21493	0.148189	0.0740945	+	0.997251i	$0.476393\pi$
$810$	0	0
$811$	25.0270	0.878816	0.439408	−	0.898288i	$-0.355188\pi$
$811$	25.0270	0.878816	0.439408	+	0.898288i	$0.355188\pi$
$812$	24.4138	0.856758
$813$	0	0
$814$	1.23054	0.0431303
$815$	−0.144607	−0.00506537
$816$	0	0
$817$	30.1231	1.05387
$818$	23.7603	0.830758
$819$	0	0
$820$	−0.0559550	−0.00195403
$821$	−21.1135	−0.736867	−0.368433	−	0.929654i	$-0.620106\pi$
$821$	−21.1135	−0.736867	−0.368433	+	0.929654i	$0.620106\pi$
$822$	0	0
$823$	25.2967	0.881786	0.440893	−	0.897560i	$-0.354662\pi$
$823$	25.2967	0.881786	0.440893	+	0.897560i	$0.354662\pi$
$824$	−31.4171	−1.09447
$825$	0	0
$826$	−20.3055	−0.706517
$827$	17.8491	0.620675	0.310338	−	0.950626i	$-0.399558\pi$
$827$	17.8491	0.620675	0.310338	+	0.950626i	$0.399558\pi$
$828$	0	0
$829$	8.81114	0.306023	0.153012	−	0.988224i	$-0.451103\pi$
$829$	8.81114	0.306023	0.153012	+	0.988224i	$0.451103\pi$
$830$	−0.0567460	−0.00196968
$831$	0	0
$832$	−9.28093	−0.321758
$833$	0	0
$834$	0	0
$835$	0.664016	0.0229792
$836$	−3.44005	−0.118977
$837$	0	0
$838$	29.6499	1.02424
$839$	9.83630	0.339587	0.169793	−	0.985480i	$-0.445690\pi$
$839$	9.83630	0.339587	0.169793	+	0.985480i	$0.445690\pi$
$840$	0	0
$841$	9.97056	0.343813
$842$	14.5710	0.502149
$843$	0	0
$844$	10.7274	0.369252
$845$	0.183985	0.00632928
$846$	0	0
$847$	29.1844	1.00279
$848$	−9.40461	−0.322956
$849$	0	0
$850$	0	0
$851$	−22.3985	−0.767811
$852$	0	0
$853$	−52.8949	−1.81109	−0.905543	−	0.424253i	$-0.860537\pi$
$853$	−52.8949	−1.81109	−0.905543	+	0.424253i	$0.860537\pi$
$854$	5.47545	0.187366
$855$	0	0
$856$	−22.8107	−0.779653
$857$	−48.0500	−1.64136	−0.820678	−	0.571392i	$-0.806404\pi$
$857$	−48.0500	−1.64136	−0.820678	+	0.571392i	$0.806404\pi$
$858$	0	0
$859$	18.0931	0.617329	0.308665	−	0.951171i	$-0.400118\pi$
$859$	18.0931	0.617329	0.308665	+	0.951171i	$0.400118\pi$
$860$	0.964148	0.0328772
$861$	0	0
$862$	9.11198	0.310355
$863$	−29.4477	−1.00241	−0.501206	−	0.865328i	$-0.667110\pi$
$863$	−29.4477	−1.00241	−0.501206	+	0.865328i	$0.667110\pi$
$864$	0	0
$865$	−0.473988	−0.0161161
$866$	16.3540	0.555730
$867$	0	0
$868$	−19.7251	−0.669514
$869$	5.59379	0.189756
$870$	0	0
$871$	3.10066	0.105062
$872$	11.1580	0.377857
$873$	0	0
$874$	−25.9366	−0.877318
$875$	−2.27690	−0.0769733
$876$	0	0
$877$	−35.7165	−1.20606	−0.603030	−	0.797719i	$-0.706040\pi$
$877$	−35.7165	−1.20606	−0.603030	+	0.797719i	$0.706040\pi$
$878$	16.2160	0.547264
$879$	0	0
$880$	−0.0456072	−0.00153742
$881$	−8.14664	−0.274467	−0.137234	−	0.990539i	$-0.543821\pi$
$881$	−8.14664	−0.274467	−0.137234	+	0.990539i	$0.543821\pi$
$882$	0	0
$883$	28.3255	0.953228	0.476614	−	0.879113i	$-0.341864\pi$
$883$	28.3255	0.953228	0.476614	+	0.879113i	$0.341864\pi$
$884$	0	0
$885$	0	0
$886$	−16.4912	−0.554032
$887$	−40.2883	−1.35275	−0.676375	−	0.736558i	$-0.736450\pi$
$887$	−40.2883	−1.35275	−0.676375	+	0.736558i	$0.736450\pi$
$888$	0	0
$889$	−1.75235	−0.0587719
$890$	−0.355932	−0.0119309
$891$	0	0
$892$	22.1499	0.741635
$893$	32.3386	1.08217
$894$	0	0
$895$	1.20435	0.0402570
$896$	−26.4253	−0.882809
$897$	0	0
$898$	−27.6886	−0.923980
$899$	−31.4862	−1.05012
$900$	0	0
$901$	0	0
$902$	0.245584	0.00817705
$903$	0	0
$904$	36.2069	1.20422
$905$	1.16173	0.0386172
$906$	0	0
$907$	43.4425	1.44248	0.721242	−	0.692683i	$-0.243572\pi$
$907$	43.4425	1.44248	0.721242	+	0.692683i	$0.243572\pi$
$908$	−20.6920	−0.686687
$909$	0	0
$910$	0.572209	0.0189686
$911$	−2.22475	−0.0737091	−0.0368546	−	0.999321i	$-0.511734\pi$
$911$	−2.22475	−0.0737091	−0.0368546	+	0.999321i	$0.511734\pi$
$912$	0	0
$913$	−0.601273	−0.0198992
$914$	−21.5955	−0.714316
$915$	0	0
$916$	−23.1154	−0.763754
$917$	42.0561	1.38881
$918$	0	0
$919$	−47.4595	−1.56554	−0.782772	−	0.622308i	$-0.786195\pi$
$919$	−47.4595	−1.56554	−0.782772	+	0.622308i	$0.786195\pi$
$920$	−2.00416	−0.0660753
$921$	0	0
$922$	−21.7779	−0.717218
$923$	10.7892	0.355132
$924$	0	0
$925$	12.0147	0.395041
$926$	4.63259	0.152236
$927$	0	0
$928$	−36.5838	−1.20092
$929$	−32.0598	−1.05185	−0.525924	−	0.850532i	$-0.676280\pi$
$929$	−32.0598	−1.05185	−0.525924	+	0.850532i	$0.676280\pi$
$930$	0	0
$931$	2.35631	0.0772248
$932$	0.506021	0.0165753
$933$	0	0
$934$	−24.6011	−0.804971
$935$	0	0
$936$	0	0
$937$	16.9740	0.554517	0.277258	−	0.960795i	$-0.410574\pi$
$937$	16.9740	0.554517	0.277258	+	0.960795i	$0.410574\pi$
$938$	−2.00000	−0.0653023
$939$	0	0
$940$	1.03506	0.0337600
$941$	−14.2357	−0.464070	−0.232035	−	0.972707i	$-0.574538\pi$
$941$	−14.2357	−0.464070	−0.232035	+	0.972707i	$0.574538\pi$
$942$	0	0
$943$	−4.47017	−0.145569
$944$	−7.94776	−0.258678
$945$	0	0
$946$	−4.23160	−0.137581
$947$	−38.5283	−1.25200	−0.626001	−	0.779823i	$-0.715309\pi$
$947$	−38.5283	−1.25200	−0.626001	+	0.779823i	$0.715309\pi$
$948$	0	0
$949$	51.4572	1.67037
$950$	13.9125	0.451383
$951$	0	0
$952$	0	0
$953$	−17.8637	−0.578663	−0.289331	−	0.957229i	$-0.593433\pi$
$953$	−17.8637	−0.578663	−0.289331	+	0.957229i	$0.593433\pi$
$954$	0	0
$955$	1.31965	0.0427029
$956$	−13.7342	−0.444195
$957$	0	0
$958$	11.9024	0.384550
$959$	6.60499	0.213286
$960$	0	0
$961$	−5.56080	−0.179381
$962$	−6.04295	−0.194833
$963$	0	0
$964$	13.2395	0.426415
$965$	−0.378624	−0.0121883
$966$	0	0
$967$	22.2654	0.716008	0.358004	−	0.933720i	$-0.383457\pi$
$967$	22.2654	0.716008	0.358004	+	0.933720i	$0.383457\pi$
$968$	−27.5777	−0.886382
$969$	0	0
$970$	0.691915	0.0222161
$971$	4.38252	0.140642	0.0703209	−	0.997524i	$-0.477598\pi$
$971$	4.38252	0.140642	0.0703209	+	0.997524i	$0.477598\pi$
$972$	0	0
$973$	4.46529	0.143151
$974$	−15.3622	−0.492238
$975$	0	0
$976$	2.14315	0.0686004
$977$	27.5043	0.879940	0.439970	−	0.898013i	$-0.354989\pi$
$977$	27.5043	0.879940	0.439970	+	0.898013i	$0.354989\pi$
$978$	0	0
$979$	−3.77141	−0.120535
$980$	0.0754181	0.00240914
$981$	0	0
$982$	4.11068	0.131177
$983$	31.1372	0.993123	0.496562	−	0.868001i	$-0.334596\pi$
$983$	31.1372	0.993123	0.496562	+	0.868001i	$0.334596\pi$
$984$	0	0
$985$	−1.74416	−0.0555734
$986$	0	0
$987$	0	0
$988$	16.8935	0.537452
$989$	77.0245	2.44924
$990$	0	0
$991$	0.764676	0.0242907	0.0121454	−	0.999926i	$-0.496134\pi$
$991$	0.764676	0.0242907	0.0121454	+	0.999926i	$0.496134\pi$
$992$	29.5578	0.938460
$993$	0	0
$994$	−6.95932	−0.220736
$995$	−0.191595	−0.00607398
$996$	0	0
$997$	1.19938	0.0379849	0.0189924	−	0.999820i	$-0.493954\pi$
$997$	1.19938	0.0379849	0.0189924	+	0.999820i	$0.493954\pi$
$998$	21.4806	0.679957
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bd.1.2		4
3.2	odd	2		867.2.a.n.1.3		4
17.5	odd	16		153.2.l.e.127.2		8
17.7	odd	16		153.2.l.e.100.2		8
17.16	even	2		2601.2.a.bc.1.2		4
51.2	odd	8		867.2.e.h.616.3		8
51.5	even	16		51.2.h.a.25.1	✓	8
51.8	odd	8		867.2.e.i.829.2		8
51.11	even	16		867.2.h.b.733.2		8
51.14	even	16		867.2.h.b.757.2		8
51.20	even	16		867.2.h.f.757.2		8
51.23	even	16		867.2.h.f.733.2		8
51.26	odd	8		867.2.e.h.829.2		8
51.29	even	16		867.2.h.g.688.1		8
51.32	odd	8		867.2.e.i.616.3		8
51.38	odd	4		867.2.d.e.577.3		8
51.41	even	16		51.2.h.a.49.1	yes	8
51.44	even	16		867.2.h.g.712.1		8
51.47	odd	4		867.2.d.e.577.4		8
51.50	odd	2		867.2.a.m.1.3		4
204.107	odd	16		816.2.bq.a.433.2		8
204.143	odd	16		816.2.bq.a.49.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.h.a.25.1	✓	8	51.5	even	16
51.2.h.a.49.1	yes	8	51.41	even	16
153.2.l.e.100.2		8	17.7	odd	16
153.2.l.e.127.2		8	17.5	odd	16
816.2.bq.a.49.2		8	204.143	odd	16
816.2.bq.a.433.2		8	204.107	odd	16
867.2.a.m.1.3		4	51.50	odd	2
867.2.a.n.1.3		4	3.2	odd	2
867.2.d.e.577.3		8	51.38	odd	4
867.2.d.e.577.4		8	51.47	odd	4
867.2.e.h.616.3		8	51.2	odd	8
867.2.e.h.829.2		8	51.26	odd	8
867.2.e.i.616.3		8	51.32	odd	8
867.2.e.i.829.2		8	51.8	odd	8
867.2.h.b.733.2		8	51.11	even	16
867.2.h.b.757.2		8	51.14	even	16
867.2.h.f.733.2		8	51.23	even	16
867.2.h.f.757.2		8	51.20	even	16
867.2.h.g.688.1		8	51.29	even	16
867.2.h.g.712.1		8	51.44	even	16
2601.2.a.bc.1.2		4	17.16	even	2
2601.2.a.bd.1.2		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q-0.765367 q^{2} -1.41421 q^{4} -0.0823922 q^{5} -2.76537 q^{7} +2.61313 q^{8} +O(q^{10})\)	\(q-0.765367 q^{2} -1.41421 q^{4} -0.0823922 q^{5} -2.76537 q^{7} +2.61313 q^{8} +0.0630603 q^{10} +0.668179 q^{11} -3.28130 q^{13} +2.11652 q^{14} +0.828427 q^{16} +3.64047 q^{19} +0.116520 q^{20} -0.511402 q^{22} +9.30864 q^{23} -4.99321 q^{25} +2.51140 q^{26} +3.91082 q^{28} +6.24264 q^{29} -5.04373 q^{31} -5.86030 q^{32} +0.227845 q^{35} -2.40621 q^{37} -2.78629 q^{38} -0.215301 q^{40} -0.480217 q^{41} +8.27452 q^{43} -0.944947 q^{44} -7.12453 q^{46} +8.88311 q^{47} +0.647254 q^{49} +3.82164 q^{50} +4.64047 q^{52} -11.3524 q^{53} -0.0550527 q^{55} -7.22625 q^{56} -4.77791 q^{58} -9.59379 q^{59} +2.58701 q^{61} +3.86030 q^{62} +2.82843 q^{64} +0.270354 q^{65} -0.944947 q^{67} -0.174385 q^{70} -3.28809 q^{71} -15.6819 q^{73} +1.84163 q^{74} -5.14840 q^{76} -1.84776 q^{77} +8.37170 q^{79} -0.0682559 q^{80} +0.367542 q^{82} -0.899869 q^{83} -6.33304 q^{86} +1.74603 q^{88} -5.64431 q^{89} +9.07401 q^{91} -13.1644 q^{92} -6.79884 q^{94} -0.299946 q^{95} +10.9723 q^{97} -0.495387 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 8 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 - 8 * q^7	\( 4 q + 4 q^{5} - 8 q^{7} - 8 q^{10} + 4 q^{11} - 4 q^{13} + 8 q^{14} - 8 q^{16} - 12 q^{19} + 8 q^{22} + 12 q^{23} + 8 q^{29} - 8 q^{31} - 16 q^{35} - 24 q^{37} - 16 q^{38} + 12 q^{41} + 4 q^{43} + 8 q^{44} - 8 q^{46} - 8 q^{47} - 4 q^{49} - 16 q^{50} - 8 q^{52} - 8 q^{53} - 12 q^{55} - 8 q^{56} - 24 q^{59} - 24 q^{61} - 8 q^{62} + 12 q^{65} + 8 q^{67} + 24 q^{70} - 24 q^{71} - 8 q^{73} - 8 q^{74} - 8 q^{76} - 8 q^{80} + 8 q^{82} - 8 q^{83} + 16 q^{86} - 16 q^{89} + 8 q^{91} - 16 q^{94} - 12 q^{95} + 16 q^{97} - 32 q^{98}+O(q^{100}) \) 4 * q + 4 * q^5 - 8 * q^7 - 8 * q^10 + 4 * q^11 - 4 * q^13 + 8 * q^14 - 8 * q^16 - 12 * q^19 + 8 * q^22 + 12 * q^23 + 8 * q^29 - 8 * q^31 - 16 * q^35 - 24 * q^37 - 16 * q^38 + 12 * q^41 + 4 * q^43 + 8 * q^44 - 8 * q^46 - 8 * q^47 - 4 * q^49 - 16 * q^50 - 8 * q^52 - 8 * q^53 - 12 * q^55 - 8 * q^56 - 24 * q^59 - 24 * q^61 - 8 * q^62 + 12 * q^65 + 8 * q^67 + 24 * q^70 - 24 * q^71 - 8 * q^73 - 8 * q^74 - 8 * q^76 - 8 * q^80 + 8 * q^82 - 8 * q^83 + 16 * q^86 - 16 * q^89 + 8 * q^91 - 16 * q^94 - 12 * q^95 + 16 * q^97 - 32 * q^98

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