LMFDB - Embedded newform 8001.2.a.z.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.904420 q^{2} -1.18202 q^{4} +0.974726 q^{5} -1.00000 q^{7} -2.87789 q^{8} +O(q^{10})$	\(q+0.904420 q^{2} -1.18202 q^{4} +0.974726 q^{5} -1.00000 q^{7} -2.87789 q^{8} +0.881562 q^{10} -1.96544 q^{11} +2.74141 q^{13} -0.904420 q^{14} -0.238773 q^{16} +1.83682 q^{17} +2.02640 q^{19} -1.15215 q^{20} -1.77759 q^{22} +4.95844 q^{23} -4.04991 q^{25} +2.47939 q^{26} +1.18202 q^{28} -5.22491 q^{29} -1.18264 q^{31} +5.53982 q^{32} +1.66126 q^{34} -0.974726 q^{35} -0.458876 q^{37} +1.83272 q^{38} -2.80515 q^{40} -4.63274 q^{41} -9.80048 q^{43} +2.32320 q^{44} +4.48451 q^{46} +1.05030 q^{47} +1.00000 q^{49} -3.66282 q^{50} -3.24041 q^{52} -0.354557 q^{53} -1.91577 q^{55} +2.87789 q^{56} -4.72552 q^{58} -2.99975 q^{59} +10.7918 q^{61} -1.06960 q^{62} +5.48787 q^{64} +2.67212 q^{65} -11.3466 q^{67} -2.17117 q^{68} -0.881562 q^{70} +11.1482 q^{71} +15.2419 q^{73} -0.415017 q^{74} -2.39526 q^{76} +1.96544 q^{77} -10.8394 q^{79} -0.232738 q^{80} -4.18994 q^{82} -2.78288 q^{83} +1.79040 q^{85} -8.86376 q^{86} +5.65632 q^{88} -5.79864 q^{89} -2.74141 q^{91} -5.86099 q^{92} +0.949912 q^{94} +1.97519 q^{95} +14.0797 q^{97} +0.904420 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) $ 32 * q + 30 * q^4 - 32 * q^7	$ 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) $ 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

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 $

$2$	0.904420	0.639522	0.319761	−	0.947498i	$-0.396397\pi$
$2$	0.904420	0.639522	0.319761	+	0.947498i	$0.396397\pi$
$3$	0	0
$3$	0	0
$4$	−1.18202	−0.591012
$5$	0.974726	0.435911	0.217955	−	0.975959i	$-0.430061\pi$
$5$	0.974726	0.435911	0.217955	+	0.975959i	$0.430061\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.87789	−1.01749
$9$	0	0
$10$	0.881562	0.278774
$11$	−1.96544	−0.592603	−0.296302	−	0.955094i	$-0.595753\pi$
$11$	−1.96544	−0.592603	−0.296302	+	0.955094i	$0.595753\pi$
$12$	0	0
$13$	2.74141	0.760330	0.380165	−	0.924919i	$-0.375867\pi$
$13$	2.74141	0.760330	0.380165	+	0.924919i	$0.375867\pi$
$14$	−0.904420	−0.241717
$15$	0	0
$16$	−0.238773	−0.0596931
$17$	1.83682	0.445495	0.222748	−	0.974876i	$-0.428497\pi$
$17$	1.83682	0.445495	0.222748	+	0.974876i	$0.428497\pi$
$18$	0	0
$19$	2.02640	0.464889	0.232444	−	0.972610i	$-0.425328\pi$
$19$	2.02640	0.464889	0.232444	+	0.972610i	$0.425328\pi$
$20$	−1.15215	−0.257628
$21$	0	0
$22$	−1.77759	−0.378983
$23$	4.95844	1.03391	0.516953	−	0.856014i	$-0.327066\pi$
$23$	4.95844	1.03391	0.516953	+	0.856014i	$0.327066\pi$
$24$	0	0
$25$	−4.04991	−0.809982
$26$	2.47939	0.486248
$27$	0	0
$28$	1.18202	0.223381
$29$	−5.22491	−0.970242	−0.485121	−	0.874447i	$-0.661224\pi$
$29$	−5.22491	−0.970242	−0.485121	+	0.874447i	$0.661224\pi$
$30$	0	0
$31$	−1.18264	−0.212408	−0.106204	−	0.994344i	$-0.533870\pi$
$31$	−1.18264	−0.212408	−0.106204	+	0.994344i	$0.533870\pi$
$32$	5.53982	0.979312
$33$	0	0
$34$	1.66126	0.284904
$35$	−0.974726	−0.164759
$36$	0	0
$37$	−0.458876	−0.0754388	−0.0377194	−	0.999288i	$-0.512009\pi$
$37$	−0.458876	−0.0754388	−0.0377194	+	0.999288i	$0.512009\pi$
$38$	1.83272	0.297307
$39$	0	0
$40$	−2.80515	−0.443533
$41$	−4.63274	−0.723512	−0.361756	−	0.932273i	$-0.617823\pi$
$41$	−4.63274	−0.723512	−0.361756	+	0.932273i	$0.617823\pi$
$42$	0	0
$43$	−9.80048	−1.49456	−0.747280	−	0.664509i	$-0.768641\pi$
$43$	−9.80048	−1.49456	−0.747280	+	0.664509i	$0.768641\pi$
$44$	2.32320	0.350235
$45$	0	0
$46$	4.48451	0.661205
$47$	1.05030	0.153202	0.0766010	−	0.997062i	$-0.475593\pi$
$47$	1.05030	0.153202	0.0766010	+	0.997062i	$0.475593\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.66282	−0.518001
$51$	0	0
$52$	−3.24041	−0.449364
$53$	−0.354557	−0.0487021	−0.0243511	−	0.999703i	$-0.507752\pi$
$53$	−0.354557	−0.0487021	−0.0243511	+	0.999703i	$0.507752\pi$
$54$	0	0
$55$	−1.91577	−0.258322
$56$	2.87789	0.384574
$57$	0	0
$58$	−4.72552	−0.620491
$59$	−2.99975	−0.390534	−0.195267	−	0.980750i	$-0.562557\pi$
$59$	−2.99975	−0.390534	−0.195267	+	0.980750i	$0.562557\pi$
$60$	0	0
$61$	10.7918	1.38174	0.690872	−	0.722977i	$-0.257227\pi$
$61$	10.7918	1.38174	0.690872	+	0.722977i	$0.257227\pi$
$62$	−1.06960	−0.135839
$63$	0	0
$64$	5.48787	0.685984
$65$	2.67212	0.331436
$66$	0	0
$67$	−11.3466	−1.38621	−0.693104	−	0.720837i	$-0.743757\pi$
$67$	−11.3466	−1.38621	−0.693104	+	0.720837i	$0.743757\pi$
$68$	−2.17117	−0.263293
$69$	0	0
$70$	−0.881562	−0.105367
$71$	11.1482	1.32305	0.661525	−	0.749923i	$-0.269909\pi$
$71$	11.1482	1.32305	0.661525	+	0.749923i	$0.269909\pi$
$72$	0	0
$73$	15.2419	1.78393	0.891966	−	0.452102i	$-0.149326\pi$
$73$	15.2419	1.78393	0.891966	+	0.452102i	$0.149326\pi$
$74$	−0.415017	−0.0482447
$75$	0	0
$76$	−2.39526	−0.274755
$77$	1.96544	0.223983
$78$	0	0
$79$	−10.8394	−1.21953	−0.609764	−	0.792583i	$-0.708736\pi$
$79$	−10.8394	−1.21953	−0.609764	+	0.792583i	$0.708736\pi$
$80$	−0.232738	−0.0260209
$81$	0	0
$82$	−4.18994	−0.462702
$83$	−2.78288	−0.305461	−0.152730	−	0.988268i	$-0.548807\pi$
$83$	−2.78288	−0.305461	−0.152730	+	0.988268i	$0.548807\pi$
$84$	0	0
$85$	1.79040	0.194196
$86$	−8.86376	−0.955804
$87$	0	0
$88$	5.65632	0.602966
$89$	−5.79864	−0.614655	−0.307327	−	0.951604i	$-0.599435\pi$
$89$	−5.79864	−0.614655	−0.307327	+	0.951604i	$0.599435\pi$
$90$	0	0
$91$	−2.74141	−0.287378
$92$	−5.86099	−0.611051
$93$	0	0
$94$	0.949912	0.0979760
$95$	1.97519	0.202650
$96$	0	0
$97$	14.0797	1.42957	0.714787	−	0.699343i	$-0.246524\pi$
$97$	14.0797	1.42957	0.714787	+	0.699343i	$0.246524\pi$
$98$	0.904420	0.0913603
$99$	0	0
$100$	4.78709	0.478709
$101$	18.6908	1.85980	0.929901	−	0.367809i	$-0.119892\pi$
$101$	18.6908	1.85980	0.929901	+	0.367809i	$0.119892\pi$
$102$	0	0
$103$	−11.2067	−1.10423	−0.552114	−	0.833768i	$-0.686179\pi$
$103$	−11.2067	−1.10423	−0.552114	+	0.833768i	$0.686179\pi$
$104$	−7.88947	−0.773626
$105$	0	0
$106$	−0.320668	−0.0311461
$107$	0.851347	0.0823028	0.0411514	−	0.999153i	$-0.486897\pi$
$107$	0.851347	0.0823028	0.0411514	+	0.999153i	$0.486897\pi$
$108$	0	0
$109$	5.23394	0.501320	0.250660	−	0.968075i	$-0.419352\pi$
$109$	5.23394	0.501320	0.250660	+	0.968075i	$0.419352\pi$
$110$	−1.73266	−0.165203
$111$	0	0
$112$	0.238773	0.0225619
$113$	−20.4549	−1.92424	−0.962118	−	0.272634i	$-0.912105\pi$
$113$	−20.4549	−1.92424	−0.962118	+	0.272634i	$0.912105\pi$
$114$	0	0
$115$	4.83312	0.450690
$116$	6.17597	0.573424
$117$	0	0
$118$	−2.71304	−0.249755
$119$	−1.83682	−0.168381
$120$	0	0
$121$	−7.13704	−0.648822
$122$	9.76030	0.883656
$123$	0	0
$124$	1.39790	0.125536
$125$	−8.82118	−0.788990
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−6.11630	−0.540610
$129$	0	0
$130$	2.41672	0.211961
$131$	−4.73386	−0.413600	−0.206800	−	0.978383i	$-0.566305\pi$
$131$	−4.73386	−0.413600	−0.206800	+	0.978383i	$0.566305\pi$
$132$	0	0
$133$	−2.02640	−0.175712
$134$	−10.2621	−0.886511
$135$	0	0
$136$	−5.28617	−0.453285
$137$	2.27099	0.194024	0.0970118	−	0.995283i	$-0.469072\pi$
$137$	2.27099	0.194024	0.0970118	+	0.995283i	$0.469072\pi$
$138$	0	0
$139$	−1.52891	−0.129680	−0.0648402	−	0.997896i	$-0.520654\pi$
$139$	−1.52891	−0.129680	−0.0648402	+	0.997896i	$0.520654\pi$
$140$	1.15215	0.0973743
$141$	0	0
$142$	10.0827	0.846120
$143$	−5.38808	−0.450574
$144$	0	0
$145$	−5.09286	−0.422939
$146$	13.7851	1.14086
$147$	0	0
$148$	0.542402	0.0445852
$149$	−3.83317	−0.314025	−0.157013	−	0.987597i	$-0.550186\pi$
$149$	−3.83317	−0.314025	−0.157013	+	0.987597i	$0.550186\pi$
$150$	0	0
$151$	−14.1993	−1.15553	−0.577763	−	0.816205i	$-0.696074\pi$
$151$	−14.1993	−1.15553	−0.577763	+	0.816205i	$0.696074\pi$
$152$	−5.83176	−0.473018
$153$	0	0
$154$	1.77759	0.143242
$155$	−1.15275	−0.0925908
$156$	0	0
$157$	5.23858	0.418084	0.209042	−	0.977907i	$-0.432965\pi$
$157$	5.23858	0.418084	0.209042	+	0.977907i	$0.432965\pi$
$158$	−9.80338	−0.779915
$159$	0	0
$160$	5.39981	0.426892
$161$	−4.95844	−0.390780
$162$	0	0
$163$	3.56149	0.278957	0.139479	−	0.990225i	$-0.455457\pi$
$163$	3.56149	0.278957	0.139479	+	0.990225i	$0.455457\pi$
$164$	5.47601	0.427604
$165$	0	0
$166$	−2.51689	−0.195349
$167$	−15.4593	−1.19628	−0.598140	−	0.801392i	$-0.704093\pi$
$167$	−15.4593	−1.19628	−0.598140	+	0.801392i	$0.704093\pi$
$168$	0	0
$169$	−5.48467	−0.421898
$170$	1.61927	0.124193
$171$	0	0
$172$	11.5844	0.883303
$173$	−13.8233	−1.05097	−0.525484	−	0.850804i	$-0.676116\pi$
$173$	−13.8233	−1.05097	−0.525484	+	0.850804i	$0.676116\pi$
$174$	0	0
$175$	4.04991	0.306144
$176$	0.469294	0.0353743
$177$	0	0
$178$	−5.24441	−0.393085
$179$	−24.2437	−1.81206	−0.906028	−	0.423218i	$-0.860901\pi$
$179$	−24.2437	−1.81206	−0.906028	+	0.423218i	$0.860901\pi$
$180$	0	0
$181$	−19.1972	−1.42692	−0.713460	−	0.700696i	$-0.752873\pi$
$181$	−19.1972	−1.42692	−0.713460	+	0.700696i	$0.752873\pi$
$182$	−2.47939	−0.183784
$183$	0	0
$184$	−14.2698	−1.05199
$185$	−0.447278	−0.0328846
$186$	0	0
$187$	−3.61017	−0.264002
$188$	−1.24148	−0.0905442
$189$	0	0
$190$	1.78640	0.129599
$191$	7.51446	0.543727	0.271864	−	0.962336i	$-0.412360\pi$
$191$	7.51446	0.543727	0.271864	+	0.962336i	$0.412360\pi$
$192$	0	0
$193$	−4.90984	−0.353418	−0.176709	−	0.984263i	$-0.556545\pi$
$193$	−4.90984	−0.353418	−0.176709	+	0.984263i	$0.556545\pi$
$194$	12.7339	0.914243
$195$	0	0
$196$	−1.18202	−0.0844303
$197$	−23.6073	−1.68195	−0.840975	−	0.541074i	$-0.818018\pi$
$197$	−23.6073	−1.68195	−0.840975	+	0.541074i	$0.818018\pi$
$198$	0	0
$199$	−6.60082	−0.467920	−0.233960	−	0.972246i	$-0.575168\pi$
$199$	−6.60082	−0.467920	−0.233960	+	0.972246i	$0.575168\pi$
$200$	11.6552	0.824146
$201$	0	0
$202$	16.9043	1.18938
$203$	5.22491	0.366717
$204$	0	0
$205$	−4.51565	−0.315387
$206$	−10.1356	−0.706178
$207$	0	0
$208$	−0.654573	−0.0453865
$209$	−3.98278	−0.275495
$210$	0	0
$211$	−15.9501	−1.09805	−0.549024	−	0.835806i	$-0.685001\pi$
$211$	−15.9501	−1.09805	−0.549024	+	0.835806i	$0.685001\pi$
$212$	0.419094	0.0287835
$213$	0	0
$214$	0.769976	0.0526345
$215$	−9.55278	−0.651494
$216$	0	0
$217$	1.18264	0.0802826
$218$	4.73368	0.320605
$219$	0	0
$220$	2.26448	0.152671
$221$	5.03549	0.338724
$222$	0	0
$223$	10.7130	0.717399	0.358699	−	0.933453i	$-0.383220\pi$
$223$	10.7130	0.717399	0.358699	+	0.933453i	$0.383220\pi$
$224$	−5.53982	−0.370145
$225$	0	0
$226$	−18.4998	−1.23059
$227$	3.94057	0.261545	0.130772	−	0.991412i	$-0.458254\pi$
$227$	3.94057	0.261545	0.130772	+	0.991412i	$0.458254\pi$
$228$	0	0
$229$	−18.2625	−1.20682	−0.603409	−	0.797432i	$-0.706191\pi$
$229$	−18.2625	−1.20682	−0.603409	+	0.797432i	$0.706191\pi$
$230$	4.37117	0.288226
$231$	0	0
$232$	15.0367	0.987208
$233$	−8.66383	−0.567587	−0.283793	−	0.958885i	$-0.591593\pi$
$233$	−8.66383	−0.567587	−0.283793	+	0.958885i	$0.591593\pi$
$234$	0	0
$235$	1.02375	0.0667823
$236$	3.54578	0.230810
$237$	0	0
$238$	−1.66126	−0.107684
$239$	−3.60516	−0.233198	−0.116599	−	0.993179i	$-0.537199\pi$
$239$	−3.60516	−0.233198	−0.116599	+	0.993179i	$0.537199\pi$
$240$	0	0
$241$	−6.19669	−0.399164	−0.199582	−	0.979881i	$-0.563958\pi$
$241$	−6.19669	−0.399164	−0.199582	+	0.979881i	$0.563958\pi$
$242$	−6.45488	−0.414936
$243$	0	0
$244$	−12.7561	−0.816628
$245$	0.974726	0.0622729
$246$	0	0
$247$	5.55520	0.353469
$248$	3.40350	0.216122
$249$	0	0
$250$	−7.97805	−0.504576
$251$	−0.903566	−0.0570326	−0.0285163	−	0.999593i	$-0.509078\pi$
$251$	−0.903566	−0.0570326	−0.0285163	+	0.999593i	$0.509078\pi$
$252$	0	0
$253$	−9.74552	−0.612696
$254$	−0.904420	−0.0567484
$255$	0	0
$256$	−16.5075	−1.03172
$257$	8.84869	0.551966	0.275983	−	0.961162i	$-0.410997\pi$
$257$	8.84869	0.551966	0.275983	+	0.961162i	$0.410997\pi$
$258$	0	0
$259$	0.458876	0.0285132
$260$	−3.15851	−0.195883
$261$	0	0
$262$	−4.28140	−0.264506
$263$	−1.87830	−0.115821	−0.0579104	−	0.998322i	$-0.518444\pi$
$263$	−1.87830	−0.115821	−0.0579104	+	0.998322i	$0.518444\pi$
$264$	0	0
$265$	−0.345595	−0.0212298
$266$	−1.83272	−0.112371
$267$	0	0
$268$	13.4120	0.819266
$269$	27.5980	1.68268	0.841339	−	0.540507i	$-0.181768\pi$
$269$	27.5980	1.68268	0.841339	+	0.540507i	$0.181768\pi$
$270$	0	0
$271$	−1.69809	−0.103151	−0.0515757	−	0.998669i	$-0.516424\pi$
$271$	−1.69809	−0.103151	−0.0515757	+	0.998669i	$0.516424\pi$
$272$	−0.438583	−0.0265930
$273$	0	0
$274$	2.05393	0.124082
$275$	7.95986	0.479998
$276$	0	0
$277$	−7.81718	−0.469689	−0.234844	−	0.972033i	$-0.575458\pi$
$277$	−7.81718	−0.469689	−0.234844	+	0.972033i	$0.575458\pi$
$278$	−1.38278	−0.0829334
$279$	0	0
$280$	2.80515	0.167640
$281$	−27.2136	−1.62343	−0.811713	−	0.584056i	$-0.801465\pi$
$281$	−27.2136	−1.62343	−0.811713	+	0.584056i	$0.801465\pi$
$282$	0	0
$283$	−28.2507	−1.67933	−0.839663	−	0.543107i	$-0.817248\pi$
$283$	−28.2507	−1.67933	−0.839663	+	0.543107i	$0.817248\pi$
$284$	−13.1775	−0.781939
$285$	0	0
$286$	−4.87309	−0.288152
$287$	4.63274	0.273462
$288$	0	0
$289$	−13.6261	−0.801534
$290$	−4.60608	−0.270478
$291$	0	0
$292$	−18.0163	−1.05433
$293$	28.5553	1.66822	0.834108	−	0.551601i	$-0.185983\pi$
$293$	28.5553	1.66822	0.834108	+	0.551601i	$0.185983\pi$
$294$	0	0
$295$	−2.92393	−0.170238
$296$	1.32059	0.0767579
$297$	0	0
$298$	−3.46679	−0.200826
$299$	13.5931	0.786110
$300$	0	0
$301$	9.80048	0.564891
$302$	−12.8422	−0.738984
$303$	0	0
$304$	−0.483850	−0.0277507
$305$	10.5190	0.602317
$306$	0	0
$307$	27.6944	1.58060	0.790301	−	0.612719i	$-0.209924\pi$
$307$	27.6944	1.58060	0.790301	+	0.612719i	$0.209924\pi$
$308$	−2.32320	−0.132377
$309$	0	0
$310$	−1.04257	−0.0592139
$311$	−9.66622	−0.548121	−0.274061	−	0.961712i	$-0.588367\pi$
$311$	−9.66622	−0.548121	−0.274061	+	0.961712i	$0.588367\pi$
$312$	0	0
$313$	5.52701	0.312405	0.156203	−	0.987725i	$-0.450075\pi$
$313$	5.52701	0.312405	0.156203	+	0.987725i	$0.450075\pi$
$314$	4.73788	0.267374
$315$	0	0
$316$	12.8124	0.720756
$317$	−11.9503	−0.671193	−0.335597	−	0.942006i	$-0.608938\pi$
$317$	−11.9503	−0.671193	−0.335597	+	0.942006i	$0.608938\pi$
$318$	0	0
$319$	10.2693	0.574968
$320$	5.34917	0.299028
$321$	0	0
$322$	−4.48451	−0.249912
$323$	3.72215	0.207106
$324$	0	0
$325$	−11.1025	−0.615854
$326$	3.22108	0.178399
$327$	0	0
$328$	13.3325	0.736164
$329$	−1.05030	−0.0579049
$330$	0	0
$331$	−15.3085	−0.841432	−0.420716	−	0.907192i	$-0.638221\pi$
$331$	−15.3085	−0.841432	−0.420716	+	0.907192i	$0.638221\pi$
$332$	3.28943	0.180531
$333$	0	0
$334$	−13.9817	−0.765047
$335$	−11.0598	−0.604263
$336$	0	0
$337$	−24.6748	−1.34412	−0.672062	−	0.740495i	$-0.734591\pi$
$337$	−24.6748	−1.34412	−0.672062	+	0.740495i	$0.734591\pi$
$338$	−4.96045	−0.269813
$339$	0	0
$340$	−2.11629	−0.114772
$341$	2.32440	0.125874
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	28.2047	1.52069
$345$	0	0
$346$	−12.5021	−0.672117
$347$	−16.4238	−0.881674	−0.440837	−	0.897587i	$-0.645318\pi$
$347$	−16.4238	−0.881674	−0.440837	+	0.897587i	$0.645318\pi$
$348$	0	0
$349$	−14.5558	−0.779156	−0.389578	−	0.920994i	$-0.627379\pi$
$349$	−14.5558	−0.779156	−0.389578	+	0.920994i	$0.627379\pi$
$350$	3.66282	0.195786
$351$	0	0
$352$	−10.8882	−0.580343
$353$	6.68260	0.355679	0.177839	−	0.984060i	$-0.443089\pi$
$353$	6.68260	0.355679	0.177839	+	0.984060i	$0.443089\pi$
$354$	0	0
$355$	10.8665	0.576732
$356$	6.85413	0.363268
$357$	0	0
$358$	−21.9265	−1.15885
$359$	31.4312	1.65888	0.829439	−	0.558598i	$-0.188660\pi$
$359$	31.4312	1.65888	0.829439	+	0.558598i	$0.188660\pi$
$360$	0	0
$361$	−14.8937	−0.783878
$362$	−17.3624	−0.912547
$363$	0	0
$364$	3.24041	0.169844
$365$	14.8567	0.777635
$366$	0	0
$367$	−33.1243	−1.72907	−0.864537	−	0.502570i	$-0.832388\pi$
$367$	−33.1243	−1.72907	−0.864537	+	0.502570i	$0.832388\pi$
$368$	−1.18394	−0.0617171
$369$	0	0
$370$	−0.404528	−0.0210304
$371$	0.354557	0.0184077
$372$	0	0
$373$	−14.3474	−0.742883	−0.371441	−	0.928456i	$-0.621136\pi$
$373$	−14.3474	−0.742883	−0.371441	+	0.928456i	$0.621136\pi$
$374$	−3.26511	−0.168835
$375$	0	0
$376$	−3.02264	−0.155881
$377$	−14.3236	−0.737704
$378$	0	0
$379$	10.7475	0.552061	0.276031	−	0.961149i	$-0.410981\pi$
$379$	10.7475	0.552061	0.276031	+	0.961149i	$0.410981\pi$
$380$	−2.33472	−0.119769
$381$	0	0
$382$	6.79623	0.347725
$383$	15.3578	0.784746	0.392373	−	0.919806i	$-0.371654\pi$
$383$	15.3578	0.784746	0.392373	+	0.919806i	$0.371654\pi$
$384$	0	0
$385$	1.91577	0.0976365
$386$	−4.44056	−0.226019
$387$	0	0
$388$	−16.6425	−0.844895
$389$	26.9989	1.36890	0.684449	−	0.729061i	$-0.260043\pi$
$389$	26.9989	1.36890	0.684449	+	0.729061i	$0.260043\pi$
$390$	0	0
$391$	9.10778	0.460600
$392$	−2.87789	−0.145355
$393$	0	0
$394$	−21.3509	−1.07564
$395$	−10.5654	−0.531605
$396$	0	0
$397$	9.42370	0.472962	0.236481	−	0.971636i	$-0.424006\pi$
$397$	9.42370	0.472962	0.236481	+	0.971636i	$0.424006\pi$
$398$	−5.96992	−0.299245
$399$	0	0
$400$	0.967007	0.0483504
$401$	−27.0326	−1.34994	−0.674972	−	0.737843i	$-0.735845\pi$
$401$	−27.0326	−1.34994	−0.674972	+	0.737843i	$0.735845\pi$
$402$	0	0
$403$	−3.24209	−0.161500
$404$	−22.0929	−1.09917
$405$	0	0
$406$	4.72552	0.234523
$407$	0.901894	0.0447052
$408$	0	0
$409$	−1.53002	−0.0756546	−0.0378273	−	0.999284i	$-0.512044\pi$
$409$	−1.53002	−0.0756546	−0.0378273	+	0.999284i	$0.512044\pi$
$410$	−4.08405	−0.201697
$411$	0	0
$412$	13.2466	0.652612
$413$	2.99975	0.147608
$414$	0	0
$415$	−2.71254	−0.133153
$416$	15.1869	0.744600
$417$	0	0
$418$	−3.60211	−0.176185
$419$	24.5561	1.19964	0.599821	−	0.800134i	$-0.295239\pi$
$419$	24.5561	1.19964	0.599821	+	0.800134i	$0.295239\pi$
$420$	0	0
$421$	−17.0319	−0.830082	−0.415041	−	0.909803i	$-0.636233\pi$
$421$	−17.0319	−0.830082	−0.415041	+	0.909803i	$0.636233\pi$
$422$	−14.4256	−0.702226
$423$	0	0
$424$	1.02037	0.0495538
$425$	−7.43897	−0.360843
$426$	0	0
$427$	−10.7918	−0.522250
$428$	−1.00631	−0.0486420
$429$	0	0
$430$	−8.63973	−0.416645
$431$	19.1165	0.920807	0.460404	−	0.887710i	$-0.347705\pi$
$431$	19.1165	0.920807	0.460404	+	0.887710i	$0.347705\pi$
$432$	0	0
$433$	−20.3990	−0.980314	−0.490157	−	0.871634i	$-0.663061\pi$
$433$	−20.3990	−0.980314	−0.490157	+	0.871634i	$0.663061\pi$
$434$	1.06960	0.0513425
$435$	0	0
$436$	−6.18664	−0.296286
$437$	10.0478	0.480651
$438$	0	0
$439$	25.7152	1.22732	0.613661	−	0.789570i	$-0.289696\pi$
$439$	25.7152	1.22732	0.613661	+	0.789570i	$0.289696\pi$
$440$	5.51336	0.262839
$441$	0	0
$442$	4.55420	0.216621
$443$	−1.86722	−0.0887145	−0.0443572	−	0.999016i	$-0.514124\pi$
$443$	−1.86722	−0.0887145	−0.0443572	+	0.999016i	$0.514124\pi$
$444$	0	0
$445$	−5.65209	−0.267935
$446$	9.68910	0.458792
$447$	0	0
$448$	−5.48787	−0.259278
$449$	31.0418	1.46495	0.732477	−	0.680792i	$-0.238364\pi$
$449$	31.0418	1.46495	0.732477	+	0.680792i	$0.238364\pi$
$450$	0	0
$451$	9.10538	0.428756
$452$	24.1782	1.13725
$453$	0	0
$454$	3.56393	0.167264
$455$	−2.67212	−0.125271
$456$	0	0
$457$	−6.64753	−0.310958	−0.155479	−	0.987839i	$-0.549692\pi$
$457$	−6.64753	−0.310958	−0.155479	+	0.987839i	$0.549692\pi$
$458$	−16.5170	−0.771787
$459$	0	0
$460$	−5.71286	−0.266363
$461$	39.1504	1.82342	0.911709	−	0.410837i	$-0.134763\pi$
$461$	39.1504	1.82342	0.911709	+	0.410837i	$0.134763\pi$
$462$	0	0
$463$	3.32152	0.154364	0.0771821	−	0.997017i	$-0.475408\pi$
$463$	3.32152	0.154364	0.0771821	+	0.997017i	$0.475408\pi$
$464$	1.24757	0.0579168
$465$	0	0
$466$	−7.83575	−0.362984
$467$	−4.67881	−0.216509	−0.108255	−	0.994123i	$-0.534526\pi$
$467$	−4.67881	−0.216509	−0.108255	+	0.994123i	$0.534526\pi$
$468$	0	0
$469$	11.3466	0.523938
$470$	0.925904	0.0427088
$471$	0	0
$472$	8.63294	0.397363
$473$	19.2623	0.885681
$474$	0	0
$475$	−8.20675	−0.376552
$476$	2.17117	0.0995154
$477$	0	0
$478$	−3.26058	−0.149135
$479$	−13.8988	−0.635054	−0.317527	−	0.948249i	$-0.602852\pi$
$479$	−13.8988	−0.635054	−0.317527	+	0.948249i	$0.602852\pi$
$480$	0	0
$481$	−1.25797	−0.0573584
$482$	−5.60442	−0.255274
$483$	0	0
$484$	8.43615	0.383461
$485$	13.7238	0.623166
$486$	0	0
$487$	40.1874	1.82106	0.910532	−	0.413439i	$-0.135673\pi$
$487$	40.1874	1.82106	0.910532	+	0.413439i	$0.135673\pi$
$488$	−31.0575	−1.40591
$489$	0	0
$490$	0.881562	0.0398249
$491$	19.5893	0.884052	0.442026	−	0.897002i	$-0.354260\pi$
$491$	19.5893	0.884052	0.442026	+	0.897002i	$0.354260\pi$
$492$	0	0
$493$	−9.59724	−0.432238
$494$	5.02424	0.226051
$495$	0	0
$496$	0.282381	0.0126793
$497$	−11.1482	−0.500066
$498$	0	0
$499$	−12.0588	−0.539826	−0.269913	−	0.962885i	$-0.586995\pi$
$499$	−12.0588	−0.539826	−0.269913	+	0.962885i	$0.586995\pi$
$500$	10.4268	0.466303
$501$	0	0
$502$	−0.817203	−0.0364736
$503$	−16.7958	−0.748889	−0.374444	−	0.927249i	$-0.622167\pi$
$503$	−16.7958	−0.748889	−0.374444	+	0.927249i	$0.622167\pi$
$504$	0	0
$505$	18.2184	0.810707
$506$	−8.81405	−0.391832
$507$	0	0
$508$	1.18202	0.0524438
$509$	−3.01771	−0.133758	−0.0668789	−	0.997761i	$-0.521304\pi$
$509$	−3.01771	−0.133758	−0.0668789	+	0.997761i	$0.521304\pi$
$510$	0	0
$511$	−15.2419	−0.674263
$512$	−2.69708	−0.119195
$513$	0	0
$514$	8.00294	0.352994
$515$	−10.9235	−0.481345
$516$	0	0
$517$	−2.06430	−0.0907880
$518$	0.415017	0.0182348
$519$	0	0
$520$	−7.69007	−0.337232
$521$	39.2726	1.72056	0.860281	−	0.509819i	$-0.170288\pi$
$521$	39.2726	1.72056	0.860281	+	0.509819i	$0.170288\pi$
$522$	0	0
$523$	−8.32811	−0.364163	−0.182081	−	0.983283i	$-0.558283\pi$
$523$	−8.32811	−0.364163	−0.182081	+	0.983283i	$0.558283\pi$
$524$	5.59554	0.244442
$525$	0	0
$526$	−1.69877	−0.0740700
$527$	−2.17230	−0.0946267
$528$	0	0
$529$	1.58611	0.0689615
$530$	−0.312564	−0.0135769
$531$	0	0
$532$	2.39526	0.103848
$533$	−12.7002	−0.550108
$534$	0	0
$535$	0.829830	0.0358767
$536$	32.6542	1.41045
$537$	0	0
$538$	24.9602	1.07611
$539$	−1.96544	−0.0846576
$540$	0	0
$541$	25.5876	1.10010	0.550048	−	0.835133i	$-0.314609\pi$
$541$	25.5876	1.10010	0.550048	+	0.835133i	$0.314609\pi$
$542$	−1.53579	−0.0659676
$543$	0	0
$544$	10.1757	0.436279
$545$	5.10165	0.218531
$546$	0	0
$547$	−9.33215	−0.399014	−0.199507	−	0.979896i	$-0.563934\pi$
$547$	−9.33215	−0.399014	−0.199507	+	0.979896i	$0.563934\pi$
$548$	−2.68436	−0.114670
$549$	0	0
$550$	7.19906	0.306969
$551$	−10.5878	−0.451055
$552$	0	0
$553$	10.8394	0.460939
$554$	−7.07001	−0.300376
$555$	0	0
$556$	1.80721	0.0766426
$557$	−29.0421	−1.23055	−0.615276	−	0.788312i	$-0.710955\pi$
$557$	−29.0421	−1.23055	−0.615276	+	0.788312i	$0.710955\pi$
$558$	0	0
$559$	−26.8671	−1.13636
$560$	0.232738	0.00983496
$561$	0	0
$562$	−24.6125	−1.03822
$563$	42.9533	1.81027	0.905134	−	0.425127i	$-0.139771\pi$
$563$	42.9533	1.81027	0.905134	+	0.425127i	$0.139771\pi$
$564$	0	0
$565$	−19.9379	−0.838795
$566$	−25.5505	−1.07397
$567$	0	0
$568$	−32.0833	−1.34619
$569$	−17.8917	−0.750060	−0.375030	−	0.927013i	$-0.622368\pi$
$569$	−17.8917	−0.750060	−0.375030	+	0.927013i	$0.622368\pi$
$570$	0	0
$571$	16.4317	0.687646	0.343823	−	0.939035i	$-0.388278\pi$
$571$	16.4317	0.687646	0.343823	+	0.939035i	$0.388278\pi$
$572$	6.36884	0.266295
$573$	0	0
$574$	4.18994	0.174885
$575$	−20.0812	−0.837445
$576$	0	0
$577$	−44.9708	−1.87216	−0.936079	−	0.351789i	$-0.885574\pi$
$577$	−44.9708	−1.87216	−0.936079	+	0.351789i	$0.885574\pi$
$578$	−12.3237	−0.512598
$579$	0	0
$580$	6.01988	0.249962
$581$	2.78288	0.115453
$582$	0	0
$583$	0.696861	0.0288610
$584$	−43.8646	−1.81513
$585$	0	0
$586$	25.8260	1.06686
$587$	−6.71570	−0.277186	−0.138593	−	0.990349i	$-0.544258\pi$
$587$	−6.71570	−0.277186	−0.138593	+	0.990349i	$0.544258\pi$
$588$	0	0
$589$	−2.39650	−0.0987461
$590$	−2.64446	−0.108871
$591$	0	0
$592$	0.109567	0.00450318
$593$	−16.3420	−0.671085	−0.335543	−	0.942025i	$-0.608920\pi$
$593$	−16.3420	−0.671085	−0.335543	+	0.942025i	$0.608920\pi$
$594$	0	0
$595$	−1.79040	−0.0733992
$596$	4.53089	0.185593
$597$	0	0
$598$	12.2939	0.502735
$599$	−18.6301	−0.761207	−0.380603	−	0.924738i	$-0.624284\pi$
$599$	−18.6301	−0.761207	−0.380603	+	0.924738i	$0.624284\pi$
$600$	0	0
$601$	19.0537	0.777217	0.388608	−	0.921403i	$-0.372956\pi$
$601$	19.0537	0.777217	0.388608	+	0.921403i	$0.372956\pi$
$602$	8.86376	0.361260
$603$	0	0
$604$	16.7839	0.682929
$605$	−6.95665	−0.282828
$606$	0	0
$607$	−45.4627	−1.84527	−0.922636	−	0.385672i	$-0.873970\pi$
$607$	−45.4627	−1.84527	−0.922636	+	0.385672i	$0.873970\pi$
$608$	11.2259	0.455271
$609$	0	0
$610$	9.51361	0.385195
$611$	2.87930	0.116484
$612$	0	0
$613$	41.9261	1.69338	0.846689	−	0.532088i	$-0.178592\pi$
$613$	41.9261	1.69338	0.846689	+	0.532088i	$0.178592\pi$
$614$	25.0474	1.01083
$615$	0	0
$616$	−5.65632	−0.227900
$617$	26.3445	1.06059	0.530295	−	0.847813i	$-0.322081\pi$
$617$	26.3445	1.06059	0.530295	+	0.847813i	$0.322081\pi$
$618$	0	0
$619$	15.1455	0.608748	0.304374	−	0.952553i	$-0.401553\pi$
$619$	15.1455	0.608748	0.304374	+	0.952553i	$0.401553\pi$
$620$	1.36257	0.0547223
$621$	0	0
$622$	−8.74233	−0.350536
$623$	5.79864	0.232318
$624$	0	0
$625$	11.6513	0.466053
$626$	4.99874	0.199790
$627$	0	0
$628$	−6.19213	−0.247093
$629$	−0.842875	−0.0336076
$630$	0	0
$631$	37.6323	1.49812	0.749060	−	0.662502i	$-0.230506\pi$
$631$	37.6323	1.49812	0.749060	+	0.662502i	$0.230506\pi$
$632$	31.1946	1.24085
$633$	0	0
$634$	−10.8081	−0.429243
$635$	−0.974726	−0.0386808
$636$	0	0
$637$	2.74141	0.108619
$638$	9.28773	0.367705
$639$	0	0
$640$	−5.96172	−0.235657
$641$	−16.6216	−0.656513	−0.328256	−	0.944589i	$-0.606461\pi$
$641$	−16.6216	−0.656513	−0.328256	+	0.944589i	$0.606461\pi$
$642$	0	0
$643$	−39.1067	−1.54222	−0.771109	−	0.636703i	$-0.780298\pi$
$643$	−39.1067	−1.54222	−0.771109	+	0.636703i	$0.780298\pi$
$644$	5.86099	0.230955
$645$	0	0
$646$	3.36639	0.132449
$647$	42.9546	1.68872	0.844360	−	0.535776i	$-0.179981\pi$
$647$	42.9546	1.68872	0.844360	+	0.535776i	$0.179981\pi$
$648$	0	0
$649$	5.89584	0.231432
$650$	−10.0413	−0.393852
$651$	0	0
$652$	−4.20976	−0.164867
$653$	23.8090	0.931719	0.465859	−	0.884859i	$-0.345745\pi$
$653$	23.8090	0.931719	0.465859	+	0.884859i	$0.345745\pi$
$654$	0	0
$655$	−4.61422	−0.180292
$656$	1.10617	0.0431887
$657$	0	0
$658$	−0.949912	−0.0370314
$659$	−30.9550	−1.20583	−0.602917	−	0.797804i	$-0.705995\pi$
$659$	−30.9550	−1.20583	−0.602917	+	0.797804i	$0.705995\pi$
$660$	0	0
$661$	−29.0899	−1.13147	−0.565733	−	0.824588i	$-0.691407\pi$
$661$	−29.0899	−1.13147	−0.565733	+	0.824588i	$0.691407\pi$
$662$	−13.8453	−0.538114
$663$	0	0
$664$	8.00881	0.310802
$665$	−1.97519	−0.0765945
$666$	0	0
$667$	−25.9074	−1.00314
$668$	18.2733	0.707015
$669$	0	0
$670$	−10.0027	−0.386439
$671$	−21.2106	−0.818826
$672$	0	0
$673$	−3.04793	−0.117489	−0.0587446	−	0.998273i	$-0.518710\pi$
$673$	−3.04793	−0.117489	−0.0587446	+	0.998273i	$0.518710\pi$
$674$	−22.3164	−0.859597
$675$	0	0
$676$	6.48301	0.249347
$677$	29.7290	1.14258	0.571289	−	0.820749i	$-0.306444\pi$
$677$	29.7290	1.14258	0.571289	+	0.820749i	$0.306444\pi$
$678$	0	0
$679$	−14.0797	−0.540328
$680$	−5.15257	−0.197592
$681$	0	0
$682$	2.10224	0.0804989
$683$	28.7088	1.09851	0.549256	−	0.835654i	$-0.314911\pi$
$683$	28.7088	1.09851	0.549256	+	0.835654i	$0.314911\pi$
$684$	0	0
$685$	2.21359	0.0845770
$686$	−0.904420	−0.0345309
$687$	0	0
$688$	2.34009	0.0892149
$689$	−0.971985	−0.0370297
$690$	0	0
$691$	−36.9473	−1.40554	−0.702772	−	0.711416i	$-0.748054\pi$
$691$	−36.9473	−1.40554	−0.702772	+	0.711416i	$0.748054\pi$
$692$	16.3395	0.621134
$693$	0	0
$694$	−14.8540	−0.563850
$695$	−1.49027	−0.0565290
$696$	0	0
$697$	−8.50953	−0.322321
$698$	−13.1646	−0.498287
$699$	0	0
$700$	−4.78709	−0.180935
$701$	3.28373	0.124025	0.0620123	−	0.998075i	$-0.480248\pi$
$701$	3.28373	0.124025	0.0620123	+	0.998075i	$0.480248\pi$
$702$	0	0
$703$	−0.929868	−0.0350707
$704$	−10.7861	−0.406516
$705$	0	0
$706$	6.04388	0.227464
$707$	−18.6908	−0.702939
$708$	0	0
$709$	13.0439	0.489874	0.244937	−	0.969539i	$-0.421233\pi$
$709$	13.0439	0.489874	0.244937	+	0.969539i	$0.421233\pi$
$710$	9.82784	0.368832
$711$	0	0
$712$	16.6878	0.625403
$713$	−5.86403	−0.219610
$714$	0	0
$715$	−5.25190	−0.196410
$716$	28.6566	1.07095
$717$	0	0
$718$	28.4271	1.06089
$719$	22.6400	0.844329	0.422165	−	0.906519i	$-0.361270\pi$
$719$	22.6400	0.844329	0.422165	+	0.906519i	$0.361270\pi$
$720$	0	0
$721$	11.2067	0.417359
$722$	−13.4702	−0.501307
$723$	0	0
$724$	22.6916	0.843327
$725$	21.1604	0.785878
$726$	0	0
$727$	−13.0887	−0.485435	−0.242717	−	0.970097i	$-0.578039\pi$
$727$	−13.0887	−0.485435	−0.242717	+	0.970097i	$0.578039\pi$
$728$	7.88947	0.292403
$729$	0	0
$730$	13.4367	0.497315
$731$	−18.0018	−0.665819
$732$	0	0
$733$	1.28284	0.0473828	0.0236914	−	0.999719i	$-0.492458\pi$
$733$	1.28284	0.0473828	0.0236914	+	0.999719i	$0.492458\pi$
$734$	−29.9583	−1.10578
$735$	0	0
$736$	27.4689	1.01252
$737$	22.3011	0.821471
$738$	0	0
$739$	−25.5084	−0.938341	−0.469171	−	0.883108i	$-0.655447\pi$
$739$	−25.5084	−0.938341	−0.469171	+	0.883108i	$0.655447\pi$
$740$	0.528693	0.0194352
$741$	0	0
$742$	0.320668	0.0117721
$743$	−7.17039	−0.263056	−0.131528	−	0.991312i	$-0.541988\pi$
$743$	−7.17039	−0.263056	−0.131528	+	0.991312i	$0.541988\pi$
$744$	0	0
$745$	−3.73629	−0.136887
$746$	−12.9761	−0.475090
$747$	0	0
$748$	4.26731	0.156028
$749$	−0.851347	−0.0311076
$750$	0	0
$751$	−3.47503	−0.126805	−0.0634027	−	0.997988i	$-0.520195\pi$
$751$	−3.47503	−0.126805	−0.0634027	+	0.997988i	$0.520195\pi$
$752$	−0.250783	−0.00914510
$753$	0	0
$754$	−12.9546	−0.471778
$755$	−13.8405	−0.503706
$756$	0	0
$757$	24.2740	0.882254	0.441127	−	0.897445i	$-0.354579\pi$
$757$	24.2740	0.882254	0.441127	+	0.897445i	$0.354579\pi$
$758$	9.72025	0.353055
$759$	0	0
$760$	−5.68437	−0.206194
$761$	18.2719	0.662355	0.331178	−	0.943568i	$-0.392554\pi$
$761$	18.2719	0.662355	0.331178	+	0.943568i	$0.392554\pi$
$762$	0	0
$763$	−5.23394	−0.189481
$764$	−8.88227	−0.321349
$765$	0	0
$766$	13.8899	0.501862
$767$	−8.22355	−0.296935
$768$	0	0
$769$	10.0830	0.363601	0.181801	−	0.983335i	$-0.441807\pi$
$769$	10.0830	0.363601	0.181801	+	0.983335i	$0.441807\pi$
$770$	1.73266	0.0624407
$771$	0	0
$772$	5.80355	0.208874
$773$	−17.1958	−0.618490	−0.309245	−	0.950982i	$-0.600076\pi$
$773$	−17.1958	−0.618490	−0.309245	+	0.950982i	$0.600076\pi$
$774$	0	0
$775$	4.78957	0.172047
$776$	−40.5197	−1.45457
$777$	0	0
$778$	24.4183	0.875440
$779$	−9.38780	−0.336353
$780$	0	0
$781$	−21.9112	−0.784044
$782$	8.23726	0.294564
$783$	0	0
$784$	−0.238773	−0.00852759
$785$	5.10618	0.182247
$786$	0	0
$787$	6.09963	0.217428	0.108714	−	0.994073i	$-0.465327\pi$
$787$	6.09963	0.217428	0.108714	+	0.994073i	$0.465327\pi$
$788$	27.9044	0.994053
$789$	0	0
$790$	−9.55561	−0.339973
$791$	20.4549	0.727293
$792$	0	0
$793$	29.5847	1.05058
$794$	8.52299	0.302470
$795$	0	0
$796$	7.80233	0.276546
$797$	−12.7977	−0.453318	−0.226659	−	0.973974i	$-0.572780\pi$
$797$	−12.7977	−0.453318	−0.226659	+	0.973974i	$0.572780\pi$
$798$	0	0
$799$	1.92922	0.0682507
$800$	−22.4358	−0.793225
$801$	0	0
$802$	−24.4489	−0.863319
$803$	−29.9571	−1.05716
$804$	0	0
$805$	−4.83312	−0.170345
$806$	−2.93222	−0.103283
$807$	0	0
$808$	−53.7900	−1.89232
$809$	8.95423	0.314814	0.157407	−	0.987534i	$-0.449687\pi$
$809$	8.95423	0.314814	0.157407	+	0.987534i	$0.449687\pi$
$810$	0	0
$811$	−1.49580	−0.0525247	−0.0262623	−	0.999655i	$-0.508361\pi$
$811$	−1.49580	−0.0525247	−0.0262623	+	0.999655i	$0.508361\pi$
$812$	−6.17597	−0.216734
$813$	0	0
$814$	0.815692	0.0285900
$815$	3.47147	0.121600
$816$	0	0
$817$	−19.8597	−0.694804
$818$	−1.38378	−0.0483828
$819$	0	0
$820$	5.33761	0.186397
$821$	53.6744	1.87325	0.936625	−	0.350334i	$-0.113932\pi$
$821$	53.6744	1.87325	0.936625	+	0.350334i	$0.113932\pi$
$822$	0	0
$823$	−9.21268	−0.321134	−0.160567	−	0.987025i	$-0.551332\pi$
$823$	−9.21268	−0.321134	−0.160567	+	0.987025i	$0.551332\pi$
$824$	32.2516	1.12354
$825$	0	0
$826$	2.71304	0.0943986
$827$	−47.7919	−1.66189	−0.830944	−	0.556356i	$-0.812199\pi$
$827$	−47.7919	−1.66189	−0.830944	+	0.556356i	$0.812199\pi$
$828$	0	0
$829$	12.8026	0.444654	0.222327	−	0.974972i	$-0.428635\pi$
$829$	12.8026	0.444654	0.222327	+	0.974972i	$0.428635\pi$
$830$	−2.45328	−0.0851546
$831$	0	0
$832$	15.0445	0.521575
$833$	1.83682	0.0636422
$834$	0	0
$835$	−15.0686	−0.521471
$836$	4.70774	0.162821
$837$	0	0
$838$	22.2090	0.767197
$839$	−26.8790	−0.927965	−0.463982	−	0.885844i	$-0.653580\pi$
$839$	−26.8790	−0.927965	−0.463982	+	0.885844i	$0.653580\pi$
$840$	0	0
$841$	−1.70029	−0.0586307
$842$	−15.4040	−0.530856
$843$	0	0
$844$	18.8534	0.648960
$845$	−5.34605	−0.183910
$846$	0	0
$847$	7.13704	0.245232
$848$	0.0846584	0.00290718
$849$	0	0
$850$	−6.72796	−0.230767
$851$	−2.27531	−0.0779966
$852$	0	0
$853$	−1.36355	−0.0466870	−0.0233435	−	0.999728i	$-0.507431\pi$
$853$	−1.36355	−0.0466870	−0.0233435	+	0.999728i	$0.507431\pi$
$854$	−9.76030	−0.333991
$855$	0	0
$856$	−2.45008	−0.0837421
$857$	−25.6174	−0.875075	−0.437537	−	0.899200i	$-0.644149\pi$
$857$	−25.6174	−0.875075	−0.437537	+	0.899200i	$0.644149\pi$
$858$	0	0
$859$	−25.9662	−0.885956	−0.442978	−	0.896532i	$-0.646078\pi$
$859$	−25.9662	−0.885956	−0.442978	+	0.896532i	$0.646078\pi$
$860$	11.2916	0.385041
$861$	0	0
$862$	17.2893	0.588876
$863$	28.3650	0.965554	0.482777	−	0.875743i	$-0.339628\pi$
$863$	28.3650	0.965554	0.482777	+	0.875743i	$0.339628\pi$
$864$	0	0
$865$	−13.4739	−0.458128
$866$	−18.4493	−0.626932
$867$	0	0
$868$	−1.39790	−0.0474480
$869$	21.3042	0.722696
$870$	0	0
$871$	−31.1057	−1.05398
$872$	−15.0627	−0.510087
$873$	0	0
$874$	9.08744	0.307387
$875$	8.82118	0.298210
$876$	0	0
$877$	12.1836	0.411411	0.205705	−	0.978614i	$-0.434051\pi$
$877$	12.1836	0.411411	0.205705	+	0.978614i	$0.434051\pi$
$878$	23.2574	0.784899
$879$	0	0
$880$	0.457432	0.0154200
$881$	−14.3304	−0.482804	−0.241402	−	0.970425i	$-0.577607\pi$
$881$	−14.3304	−0.482804	−0.241402	+	0.970425i	$0.577607\pi$
$882$	0	0
$883$	−51.3214	−1.72710	−0.863552	−	0.504260i	$-0.831765\pi$
$883$	−51.3214	−1.72710	−0.863552	+	0.504260i	$0.831765\pi$
$884$	−5.95207	−0.200190
$885$	0	0
$886$	−1.68876	−0.0567348
$887$	−24.4353	−0.820458	−0.410229	−	0.911982i	$-0.634551\pi$
$887$	−24.4353	−0.820458	−0.410229	+	0.911982i	$0.634551\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−5.11186	−0.171350
$891$	0	0
$892$	−12.6631	−0.423991
$893$	2.12833	0.0712219
$894$	0	0
$895$	−23.6309	−0.789894
$896$	6.11630	0.204331
$897$	0	0
$898$	28.0748	0.936869
$899$	6.17917	0.206087
$900$	0	0
$901$	−0.651258	−0.0216966
$902$	8.23509	0.274199
$903$	0	0
$904$	58.8669	1.95788
$905$	−18.7120	−0.622010
$906$	0	0
$907$	28.7487	0.954584	0.477292	−	0.878745i	$-0.341618\pi$
$907$	28.7487	0.954584	0.477292	+	0.878745i	$0.341618\pi$
$908$	−4.65785	−0.154576
$909$	0	0
$910$	−2.41672	−0.0801136
$911$	27.4863	0.910662	0.455331	−	0.890322i	$-0.349521\pi$
$911$	27.4863	0.910662	0.455331	+	0.890322i	$0.349521\pi$
$912$	0	0
$913$	5.46959	0.181017
$914$	−6.01216	−0.198865
$915$	0	0
$916$	21.5867	0.713244
$917$	4.73386	0.156326
$918$	0	0
$919$	−15.9691	−0.526771	−0.263385	−	0.964691i	$-0.584839\pi$
$919$	−15.9691	−0.526771	−0.263385	+	0.964691i	$0.584839\pi$
$920$	−13.9092	−0.458572
$921$	0	0
$922$	35.4085	1.16612
$923$	30.5618	1.00596
$924$	0	0
$925$	1.85841	0.0611040
$926$	3.00405	0.0987193
$927$	0	0
$928$	−28.9451	−0.950169
$929$	−12.7716	−0.419023	−0.209511	−	0.977806i	$-0.567187\pi$
$929$	−12.7716	−0.419023	−0.209511	+	0.977806i	$0.567187\pi$
$930$	0	0
$931$	2.02640	0.0664127
$932$	10.2409	0.335450
$933$	0	0
$934$	−4.23161	−0.138462
$935$	−3.51893	−0.115081
$936$	0	0
$937$	52.7627	1.72368	0.861842	−	0.507178i	$-0.169311\pi$
$937$	52.7627	1.72368	0.861842	+	0.507178i	$0.169311\pi$
$938$	10.2621	0.335069
$939$	0	0
$940$	−1.21010	−0.0394692
$941$	51.2315	1.67010	0.835049	−	0.550176i	$-0.185439\pi$
$941$	51.2315	1.67010	0.835049	+	0.550176i	$0.185439\pi$
$942$	0	0
$943$	−22.9712	−0.748044
$944$	0.716258	0.0233122
$945$	0	0
$946$	17.4212	0.566412
$947$	2.89717	0.0941455	0.0470727	−	0.998891i	$-0.485011\pi$
$947$	2.89717	0.0941455	0.0470727	+	0.998891i	$0.485011\pi$
$948$	0	0
$949$	41.7844	1.35638
$950$	−7.42236	−0.240813
$951$	0	0
$952$	5.28617	0.171326
$953$	−50.7174	−1.64290	−0.821448	−	0.570283i	$-0.806834\pi$
$953$	−50.7174	−1.64290	−0.821448	+	0.570283i	$0.806834\pi$
$954$	0	0
$955$	7.32454	0.237016
$956$	4.26138	0.137823
$957$	0	0
$958$	−12.5704	−0.406131
$959$	−2.27099	−0.0733341
$960$	0	0
$961$	−29.6014	−0.954883
$962$	−1.13773	−0.0366819
$963$	0	0
$964$	7.32464	0.235911
$965$	−4.78575	−0.154059
$966$	0	0
$967$	0.226419	0.00728114	0.00364057	−	0.999993i	$-0.498841\pi$
$967$	0.226419	0.00728114	0.00364057	+	0.999993i	$0.498841\pi$
$968$	20.5396	0.660167
$969$	0	0
$970$	12.4121	0.398528
$971$	18.1946	0.583894	0.291947	−	0.956435i	$-0.405697\pi$
$971$	18.1946	0.583894	0.291947	+	0.956435i	$0.405697\pi$
$972$	0	0
$973$	1.52891	0.0490146
$974$	36.3463	1.16461
$975$	0	0
$976$	−2.57678	−0.0824807
$977$	35.6302	1.13991	0.569956	−	0.821675i	$-0.306960\pi$
$977$	35.6302	1.13991	0.569956	+	0.821675i	$0.306960\pi$
$978$	0	0
$979$	11.3969	0.364246
$980$	−1.15215	−0.0368040
$981$	0	0
$982$	17.7170	0.565371
$983$	−30.5746	−0.975179	−0.487589	−	0.873073i	$-0.662124\pi$
$983$	−30.5746	−0.975179	−0.487589	+	0.873073i	$0.662124\pi$
$984$	0	0
$985$	−23.0106	−0.733180
$986$	−8.67994	−0.276426
$987$	0	0
$988$	−6.56638	−0.208904
$989$	−48.5951	−1.54523
$990$	0	0
$991$	29.5461	0.938561	0.469281	−	0.883049i	$-0.344513\pi$
$991$	29.5461	0.938561	0.469281	+	0.883049i	$0.344513\pi$
$992$	−6.55160	−0.208014
$993$	0	0
$994$	−10.0827	−0.319803
$995$	−6.43399	−0.203971
$996$	0	0
$997$	−7.43799	−0.235563	−0.117782	−	0.993040i	$-0.537578\pi$
$997$	−7.43799	−0.235563	−0.117782	+	0.993040i	$0.537578\pi$
$998$	−10.9062	−0.345230
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.20	yes	32
3.2	odd	2	inner	8001.2.a.z.1.13	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.13	✓	32	3.2	odd	2	inner
8001.2.a.z.1.20	yes	32	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+0.904420 q^{2} -1.18202 q^{4} +0.974726 q^{5} -1.00000 q^{7} -2.87789 q^{8} +O(q^{10})\)	\(q+0.904420 q^{2} -1.18202 q^{4} +0.974726 q^{5} -1.00000 q^{7} -2.87789 q^{8} +0.881562 q^{10} -1.96544 q^{11} +2.74141 q^{13} -0.904420 q^{14} -0.238773 q^{16} +1.83682 q^{17} +2.02640 q^{19} -1.15215 q^{20} -1.77759 q^{22} +4.95844 q^{23} -4.04991 q^{25} +2.47939 q^{26} +1.18202 q^{28} -5.22491 q^{29} -1.18264 q^{31} +5.53982 q^{32} +1.66126 q^{34} -0.974726 q^{35} -0.458876 q^{37} +1.83272 q^{38} -2.80515 q^{40} -4.63274 q^{41} -9.80048 q^{43} +2.32320 q^{44} +4.48451 q^{46} +1.05030 q^{47} +1.00000 q^{49} -3.66282 q^{50} -3.24041 q^{52} -0.354557 q^{53} -1.91577 q^{55} +2.87789 q^{56} -4.72552 q^{58} -2.99975 q^{59} +10.7918 q^{61} -1.06960 q^{62} +5.48787 q^{64} +2.67212 q^{65} -11.3466 q^{67} -2.17117 q^{68} -0.881562 q^{70} +11.1482 q^{71} +15.2419 q^{73} -0.415017 q^{74} -2.39526 q^{76} +1.96544 q^{77} -10.8394 q^{79} -0.232738 q^{80} -4.18994 q^{82} -2.78288 q^{83} +1.79040 q^{85} -8.86376 q^{86} +5.65632 q^{88} -5.79864 q^{89} -2.74141 q^{91} -5.86099 q^{92} +0.949912 q^{94} +1.97519 q^{95} +14.0797 q^{97} +0.904420 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 32 * q + 30 * q^4 - 32 * q^7	\( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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