LMFDB - Embedded newform 8379.2.a.bq.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8379,2,Mod(1,8379)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8379.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8379, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,0,9,0,0,0,9,0,-10,-4,0,-2,0,0,13,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.9066518536$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.86620$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.34889 q^{2} +3.51730 q^{4} -0.517304 q^{5} -3.56399 q^{8} +1.21509 q^{10} +3.73240 q^{11} -5.73240 q^{13} +1.33682 q^{16} +4.51730 q^{17} +1.00000 q^{19} -1.81952 q^{20} -8.76700 q^{22} +7.73240 q^{23} -4.73240 q^{25} +13.4648 q^{26} +6.18048 q^{29} +6.69779 q^{31} +3.98793 q^{32} -10.6107 q^{34} +9.46479 q^{37} -2.34889 q^{38} +1.84366 q^{40} +2.26760 q^{41} +1.30221 q^{43} +13.1280 q^{44} -18.1626 q^{46} -4.18048 q^{47} +11.1159 q^{50} -20.1626 q^{52} -13.6453 q^{53} -1.93078 q^{55} -14.5173 q^{58} +6.06922 q^{59} +5.46479 q^{61} -15.7324 q^{62} -12.0409 q^{64} +2.96539 q^{65} +6.43018 q^{67} +15.8887 q^{68} -0.947489 q^{71} -3.03461 q^{73} -22.2318 q^{74} +3.51730 q^{76} +6.06922 q^{79} -0.691542 q^{80} -5.32636 q^{82} +6.24970 q^{83} -2.33682 q^{85} -3.05876 q^{86} -13.3022 q^{88} -7.80161 q^{89} +27.1972 q^{92} +9.81952 q^{94} -0.517304 q^{95} -15.3956 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 9 q^{4} + 9 q^{8} - 10 q^{10} - 4 q^{11} - 2 q^{13} + 13 q^{16} + 12 q^{17} + 3 q^{19} - 16 q^{20} - 8 q^{22} + 8 q^{23} + q^{25} + 10 q^{26} + 8 q^{29} + 8 q^{31} + 27 q^{32} + 6 q^{34}+ \cdots - 22 q^{97}+O(q^{100}) $ 3 * q - q^2 + 9 * q^4 + 9 * q^8 - 10 * q^10 - 4 * q^11 - 2 * q^13 + 13 * q^16 + 12 * q^17 + 3 * q^19 - 16 * q^20 - 8 * q^22 + 8 * q^23 + q^25 + 10 * q^26 + 8 * q^29 + 8 * q^31 + 27 * q^32 + 6 * q^34 - 2 * q^37 - q^38 - 14 * q^40 + 22 * q^41 + 16 * q^43 - 12 * q^46 - 2 * q^47 + 9 * q^50 - 18 * q^52 - 12 * q^55 - 42 * q^58 + 12 * q^59 - 14 * q^61 - 32 * q^62 + 29 * q^64 + 12 * q^65 - 8 * q^67 + 52 * q^68 + 26 * q^71 - 6 * q^73 - 18 * q^74 + 9 * q^76 + 12 * q^79 - 52 * q^80 + 2 * q^82 + 2 * q^83 - 16 * q^85 + 24 * q^86 - 52 * q^88 - 2 * q^89 + 36 * q^92 + 40 * q^94 - 22 * q^97

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 $

$2$	−2.34889	−1.66092	−0.830460	−	0.557079i	$-0.811922\pi$
$2$	−2.34889	−1.66092	−0.830460	+	0.557079i	$0.811922\pi$
$3$	0	0
$3$	0	0
$4$	3.51730	1.75865
$5$	−0.517304	−0.231345	−0.115673	−	0.993287i	$-0.536902\pi$
$5$	−0.517304	−0.231345	−0.115673	+	0.993287i	$0.536902\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−3.56399	−1.26006
$9$	0	0
$10$	1.21509	0.384246
$11$	3.73240	1.12536	0.562680	−	0.826675i	$-0.309770\pi$
$11$	3.73240	1.12536	0.562680	+	0.826675i	$0.309770\pi$
$12$	0	0
$13$	−5.73240	−1.58988	−0.794940	−	0.606688i	$-0.792498\pi$
$13$	−5.73240	−1.58988	−0.794940	+	0.606688i	$0.792498\pi$
$14$	0	0
$15$	0	0
$16$	1.33682	0.334205
$17$	4.51730	1.09561	0.547804	−	0.836607i	$-0.315464\pi$
$17$	4.51730	1.09561	0.547804	+	0.836607i	$0.315464\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.81952	−0.406856
$21$	0	0
$22$	−8.76700	−1.86913
$23$	7.73240	1.61232	0.806158	−	0.591700i	$-0.201543\pi$
$23$	7.73240	1.61232	0.806158	+	0.591700i	$0.201543\pi$
$24$	0	0
$25$	−4.73240	−0.946479
$26$	13.4648	2.64066
$27$	0	0
$28$	0	0
$29$	6.18048	1.14769	0.573844	−	0.818965i	$-0.305452\pi$
$29$	6.18048	1.14769	0.573844	+	0.818965i	$0.305452\pi$
$30$	0	0
$31$	6.69779	1.20296	0.601479	−	0.798888i	$-0.294578\pi$
$31$	6.69779	1.20296	0.601479	+	0.798888i	$0.294578\pi$
$32$	3.98793	0.704972
$33$	0	0
$34$	−10.6107	−1.81971
$35$	0	0
$36$	0	0
$37$	9.46479	1.55600	0.778001	−	0.628263i	$-0.216234\pi$
$37$	9.46479	1.55600	0.778001	+	0.628263i	$0.216234\pi$
$38$	−2.34889	−0.381041
$39$	0	0
$40$	1.84366	0.291509
$41$	2.26760	0.354140	0.177070	−	0.984198i	$-0.443338\pi$
$41$	2.26760	0.354140	0.177070	+	0.984198i	$0.443338\pi$
$42$	0	0
$43$	1.30221	0.198585	0.0992927	−	0.995058i	$-0.468342\pi$
$43$	1.30221	0.198585	0.0992927	+	0.995058i	$0.468342\pi$
$44$	13.1280	1.97912
$45$	0	0
$46$	−18.1626	−2.67793
$47$	−4.18048	−0.609786	−0.304893	−	0.952387i	$-0.598621\pi$
$47$	−4.18048	−0.609786	−0.304893	+	0.952387i	$0.598621\pi$
$48$	0	0
$49$	0	0
$50$	11.1159	1.57203
$51$	0	0
$52$	−20.1626	−2.79605
$53$	−13.6453	−1.87432	−0.937162	−	0.348896i	$-0.886557\pi$
$53$	−13.6453	−1.87432	−0.937162	+	0.348896i	$0.886557\pi$
$54$	0	0
$55$	−1.93078	−0.260347
$56$	0	0
$57$	0	0
$58$	−14.5173	−1.90622
$59$	6.06922	0.790145	0.395072	−	0.918650i	$-0.370720\pi$
$59$	6.06922	0.790145	0.395072	+	0.918650i	$0.370720\pi$
$60$	0	0
$61$	5.46479	0.699695	0.349848	−	0.936807i	$-0.386233\pi$
$61$	5.46479	0.699695	0.349848	+	0.936807i	$0.386233\pi$
$62$	−15.7324	−1.99802
$63$	0	0
$64$	−12.0409	−1.50511
$65$	2.96539	0.367812
$66$	0	0
$67$	6.43018	0.785572	0.392786	−	0.919630i	$-0.371511\pi$
$67$	6.43018	0.785572	0.392786	+	0.919630i	$0.371511\pi$
$68$	15.8887	1.92679
$69$	0	0
$70$	0	0
$71$	−0.947489	−0.112446	−0.0562231	−	0.998418i	$-0.517906\pi$
$71$	−0.947489	−0.112446	−0.0562231	+	0.998418i	$0.517906\pi$
$72$	0	0
$73$	−3.03461	−0.355174	−0.177587	−	0.984105i	$-0.556829\pi$
$73$	−3.03461	−0.355174	−0.177587	+	0.984105i	$0.556829\pi$
$74$	−22.2318	−2.58439
$75$	0	0
$76$	3.51730	0.403462
$77$	0	0
$78$	0	0
$79$	6.06922	0.682840	0.341420	−	0.939911i	$-0.389092\pi$
$79$	6.06922	0.682840	0.341420	+	0.939911i	$0.389092\pi$
$80$	−0.691542	−0.0773168
$81$	0	0
$82$	−5.32636	−0.588198
$83$	6.24970	0.685994	0.342997	−	0.939337i	$-0.388558\pi$
$83$	6.24970	0.685994	0.342997	+	0.939337i	$0.388558\pi$
$84$	0	0
$85$	−2.33682	−0.253464
$86$	−3.05876	−0.329834
$87$	0	0
$88$	−13.3022	−1.41802
$89$	−7.80161	−0.826969	−0.413485	−	0.910511i	$-0.635688\pi$
$89$	−7.80161	−0.826969	−0.413485	+	0.910511i	$0.635688\pi$
$90$	0	0
$91$	0	0
$92$	27.1972	2.83550
$93$	0	0
$94$	9.81952	1.01281
$95$	−0.517304	−0.0530743
$96$	0	0
$97$	−15.3956	−1.56318	−0.781592	−	0.623790i	$-0.785592\pi$
$97$	−15.3956	−1.56318	−0.781592	+	0.623790i	$0.785592\pi$
$98$	0	0
$99$	0	0
$100$	−16.6453	−1.66453
$101$	−11.9821	−1.19226	−0.596132	−	0.802887i	$-0.703296\pi$
$101$	−11.9821	−1.19226	−0.596132	+	0.802887i	$0.703296\pi$
$102$	0	0
$103$	−15.4648	−1.52379	−0.761896	−	0.647700i	$-0.775731\pi$
$103$	−15.4648	−1.52379	−0.761896	+	0.647700i	$0.775731\pi$
$104$	20.4302	2.00334
$105$	0	0
$106$	32.0513	3.11310
$107$	14.5173	1.40344	0.701720	−	0.712452i	$-0.252416\pi$
$107$	14.5173	1.40344	0.701720	+	0.712452i	$0.252416\pi$
$108$	0	0
$109$	−6.49940	−0.622530	−0.311265	−	0.950323i	$-0.600753\pi$
$109$	−6.49940	−0.622530	−0.311265	+	0.950323i	$0.600753\pi$
$110$	4.53521	0.432415
$111$	0	0
$112$	0	0
$113$	2.18048	0.205123	0.102561	−	0.994727i	$-0.467296\pi$
$113$	2.18048	0.205123	0.102561	+	0.994727i	$0.467296\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	21.7386	2.01838
$117$	0	0
$118$	−14.2559	−1.31237
$119$	0	0
$120$	0	0
$121$	2.93078	0.266435
$122$	−12.8362	−1.16214
$123$	0	0
$124$	23.5582	2.11559
$125$	5.03461	0.450309
$126$	0	0
$127$	18.4302	1.63541	0.817707	−	0.575634i	$-0.195245\pi$
$127$	18.4302	1.63541	0.817707	+	0.575634i	$0.195245\pi$
$128$	20.3068	1.79489
$129$	0	0
$130$	−6.96539	−0.610905
$131$	14.6107	1.27654	0.638270	−	0.769813i	$-0.279650\pi$
$131$	14.6107	1.27654	0.638270	+	0.769813i	$0.279650\pi$
$132$	0	0
$133$	0	0
$134$	−15.1038	−1.30477
$135$	0	0
$136$	−16.0996	−1.38053
$137$	3.39558	0.290104	0.145052	−	0.989424i	$-0.453665\pi$
$137$	3.39558	0.290104	0.145052	+	0.989424i	$0.453665\pi$
$138$	0	0
$139$	−2.96539	−0.251521	−0.125761	−	0.992061i	$-0.540137\pi$
$139$	−2.96539	−0.251521	−0.125761	+	0.992061i	$0.540137\pi$
$140$	0	0
$141$	0	0
$142$	2.22555	0.186764
$143$	−21.3956	−1.78919
$144$	0	0
$145$	−3.19719	−0.265512
$146$	7.12797	0.589915
$147$	0	0
$148$	33.2906	2.73647
$149$	10.3610	0.848804	0.424402	−	0.905474i	$-0.360484\pi$
$149$	10.3610	0.848804	0.424402	+	0.905474i	$0.360484\pi$
$150$	0	0
$151$	13.8950	1.13076	0.565379	−	0.824831i	$-0.308730\pi$
$151$	13.8950	1.13076	0.565379	+	0.824831i	$0.308730\pi$
$152$	−3.56399	−0.289077
$153$	0	0
$154$	0	0
$155$	−3.46479	−0.278299
$156$	0	0
$157$	8.06922	0.643994	0.321997	−	0.946741i	$-0.395646\pi$
$157$	8.06922	0.643994	0.321997	+	0.946741i	$0.395646\pi$
$158$	−14.2559	−1.13414
$159$	0	0
$160$	−2.06297	−0.163092
$161$	0	0
$162$	0	0
$163$	−8.76700	−0.686685	−0.343342	−	0.939210i	$-0.611559\pi$
$163$	−8.76700	−0.686685	−0.343342	+	0.939210i	$0.611559\pi$
$164$	7.97585	0.622809
$165$	0	0
$166$	−14.6799	−1.13938
$167$	−14.5686	−1.12735	−0.563677	−	0.825995i	$-0.690614\pi$
$167$	−14.5686	−1.12735	−0.563677	+	0.825995i	$0.690614\pi$
$168$	0	0
$169$	19.8604	1.52772
$170$	5.48894	0.420983
$171$	0	0
$172$	4.58027	0.349243
$173$	−3.80161	−0.289031	−0.144516	−	0.989503i	$-0.546162\pi$
$173$	−3.80161	−0.289031	−0.144516	+	0.989503i	$0.546162\pi$
$174$	0	0
$175$	0	0
$176$	4.98954	0.376101
$177$	0	0
$178$	18.3252	1.37353
$179$	3.55191	0.265482	0.132741	−	0.991151i	$-0.457622\pi$
$179$	3.55191	0.265482	0.132741	+	0.991151i	$0.457622\pi$
$180$	0	0
$181$	−19.3956	−1.44166	−0.720831	−	0.693111i	$-0.756240\pi$
$181$	−19.3956	−1.44166	−0.720831	+	0.693111i	$0.756240\pi$
$182$	0	0
$183$	0	0
$184$	−27.5582	−2.03161
$185$	−4.89618	−0.359974
$186$	0	0
$187$	16.8604	1.23295
$188$	−14.7040	−1.07240
$189$	0	0
$190$	1.21509	0.0881521
$191$	9.80161	0.709220	0.354610	−	0.935014i	$-0.384614\pi$
$191$	9.80161	0.709220	0.354610	+	0.935014i	$0.384614\pi$
$192$	0	0
$193$	−14.8604	−1.06967	−0.534836	−	0.844956i	$-0.679627\pi$
$193$	−14.8604	−1.06967	−0.534836	+	0.844956i	$0.679627\pi$
$194$	36.1626	2.59632
$195$	0	0
$196$	0	0
$197$	−9.32636	−0.664476	−0.332238	−	0.943196i	$-0.607804\pi$
$197$	−9.32636	−0.664476	−0.332238	+	0.943196i	$0.607804\pi$
$198$	0	0
$199$	8.36097	0.592693	0.296347	−	0.955080i	$-0.404232\pi$
$199$	8.36097	0.592693	0.296347	+	0.955080i	$0.404232\pi$
$200$	16.8662	1.19262
$201$	0	0
$202$	28.1447	1.98025
$203$	0	0
$204$	0	0
$205$	−1.17304	−0.0819287
$206$	36.3252	2.53089
$207$	0	0
$208$	−7.66318	−0.531346
$209$	3.73240	0.258175
$210$	0	0
$211$	−13.5340	−0.931720	−0.465860	−	0.884859i	$-0.654255\pi$
$211$	−13.5340	−0.931720	−0.465860	+	0.884859i	$0.654255\pi$
$212$	−47.9946	−3.29628
$213$	0	0
$214$	−34.0996	−2.33100
$215$	−0.673639	−0.0459418
$216$	0	0
$217$	0	0
$218$	15.2664	1.03397
$219$	0	0
$220$	−6.79115	−0.457859
$221$	−25.8950	−1.74188
$222$	0	0
$223$	−14.1626	−0.948397	−0.474198	−	0.880418i	$-0.657262\pi$
$223$	−14.1626	−0.948397	−0.474198	+	0.880418i	$0.657262\pi$
$224$	0	0
$225$	0	0
$226$	−5.12173	−0.340692
$227$	−19.8258	−1.31588	−0.657941	−	0.753069i	$-0.728572\pi$
$227$	−19.8258	−1.31588	−0.657941	+	0.753069i	$0.728572\pi$
$228$	0	0
$229$	18.3610	1.21333	0.606663	−	0.794959i	$-0.292508\pi$
$229$	18.3610	1.21333	0.606663	+	0.794959i	$0.292508\pi$
$230$	9.39558	0.619526
$231$	0	0
$232$	−22.0272	−1.44615
$233$	−9.10382	−0.596411	−0.298206	−	0.954502i	$-0.596388\pi$
$233$	−9.10382	−0.596411	−0.298206	+	0.954502i	$0.596388\pi$
$234$	0	0
$235$	2.16258	0.141071
$236$	21.3473	1.38959
$237$	0	0
$238$	0	0
$239$	20.2318	1.30869	0.654343	−	0.756198i	$-0.272945\pi$
$239$	20.2318	1.30869	0.654343	+	0.756198i	$0.272945\pi$
$240$	0	0
$241$	−7.39558	−0.476391	−0.238195	−	0.971217i	$-0.576556\pi$
$241$	−7.39558	−0.476391	−0.238195	+	0.971217i	$0.576556\pi$
$242$	−6.88410	−0.442527
$243$	0	0
$244$	19.2213	1.23052
$245$	0	0
$246$	0	0
$247$	−5.73240	−0.364744
$248$	−23.8708	−1.51580
$249$	0	0
$250$	−11.8258	−0.747927
$251$	22.7491	1.43591	0.717955	−	0.696089i	$-0.245078\pi$
$251$	22.7491	1.43591	0.717955	+	0.696089i	$0.245078\pi$
$252$	0	0
$253$	28.8604	1.81444
$254$	−43.2906	−2.71629
$255$	0	0
$256$	−23.6169	−1.47606
$257$	19.6274	1.22432	0.612161	−	0.790733i	$-0.290300\pi$
$257$	19.6274	1.22432	0.612161	+	0.790733i	$0.290300\pi$
$258$	0	0
$259$	0	0
$260$	10.4302	0.646853
$261$	0	0
$262$	−34.3189	−2.12023
$263$	18.6978	1.15296	0.576478	−	0.817113i	$-0.304427\pi$
$263$	18.6978	1.15296	0.576478	+	0.817113i	$0.304427\pi$
$264$	0	0
$265$	7.05876	0.433616
$266$	0	0
$267$	0	0
$268$	22.6169	1.38155
$269$	−19.6274	−1.19670	−0.598351	−	0.801234i	$-0.704177\pi$
$269$	−19.6274	−1.19670	−0.598351	+	0.801234i	$0.704177\pi$
$270$	0	0
$271$	11.4648	0.696437	0.348218	−	0.937413i	$-0.386787\pi$
$271$	11.4648	0.696437	0.348218	+	0.937413i	$0.386787\pi$
$272$	6.03882	0.366157
$273$	0	0
$274$	−7.97585	−0.481839
$275$	−17.6632	−1.06513
$276$	0	0
$277$	−3.66318	−0.220099	−0.110050	−	0.993926i	$-0.535101\pi$
$277$	−3.66318	−0.220099	−0.110050	+	0.993926i	$0.535101\pi$
$278$	6.96539	0.417756
$279$	0	0
$280$	0	0
$281$	−23.2151	−1.38490	−0.692448	−	0.721468i	$-0.743468\pi$
$281$	−23.2151	−1.38490	−0.692448	+	0.721468i	$0.743468\pi$
$282$	0	0
$283$	−29.8950	−1.77707	−0.888536	−	0.458807i	$-0.848277\pi$
$283$	−29.8950	−1.77707	−0.888536	+	0.458807i	$0.848277\pi$
$284$	−3.33261	−0.197754
$285$	0	0
$286$	50.2559	2.97170
$287$	0	0
$288$	0	0
$289$	3.40604	0.200355
$290$	7.50986	0.440994
$291$	0	0
$292$	−10.6736	−0.624627
$293$	14.2676	0.833522	0.416761	−	0.909016i	$-0.363165\pi$
$293$	14.2676	0.833522	0.416761	+	0.909016i	$0.363165\pi$
$294$	0	0
$295$	−3.13963	−0.182796
$296$	−33.7324	−1.96066
$297$	0	0
$298$	−24.3368	−1.40979
$299$	−44.3252	−2.56339
$300$	0	0
$301$	0	0
$302$	−32.6378	−1.87810
$303$	0	0
$304$	1.33682	0.0766719
$305$	−2.82696	−0.161871
$306$	0	0
$307$	4.76700	0.272067	0.136034	−	0.990704i	$-0.456564\pi$
$307$	4.76700	0.272067	0.136034	+	0.990704i	$0.456564\pi$
$308$	0	0
$309$	0	0
$310$	8.13843	0.462232
$311$	11.1459	0.632025	0.316012	−	0.948755i	$-0.397656\pi$
$311$	11.1459	0.632025	0.316012	+	0.948755i	$0.397656\pi$
$312$	0	0
$313$	20.7912	1.17519	0.587593	−	0.809157i	$-0.300076\pi$
$313$	20.7912	1.17519	0.587593	+	0.809157i	$0.300076\pi$
$314$	−18.9537	−1.06962
$315$	0	0
$316$	21.3473	1.20088
$317$	6.71569	0.377191	0.188595	−	0.982055i	$-0.439607\pi$
$317$	6.71569	0.377191	0.188595	+	0.982055i	$0.439607\pi$
$318$	0	0
$319$	23.0680	1.29156
$320$	6.22878	0.348200
$321$	0	0
$322$	0	0
$323$	4.51730	0.251350
$324$	0	0
$325$	27.1280	1.50479
$326$	20.5928	1.14053
$327$	0	0
$328$	−8.08171	−0.446238
$329$	0	0
$330$	0	0
$331$	10.4302	0.573295	0.286647	−	0.958036i	$-0.407459\pi$
$331$	10.4302	0.573295	0.286647	+	0.958036i	$0.407459\pi$
$332$	21.9821	1.20642
$333$	0	0
$334$	34.2201	1.87244
$335$	−3.32636	−0.181738
$336$	0	0
$337$	20.4302	1.11290	0.556452	−	0.830880i	$-0.312162\pi$
$337$	20.4302	1.11290	0.556452	+	0.830880i	$0.312162\pi$
$338$	−46.6499	−2.53742
$339$	0	0
$340$	−8.21931	−0.445754
$341$	24.9988	1.35376
$342$	0	0
$343$	0	0
$344$	−4.64106	−0.250229
$345$	0	0
$346$	8.92959	0.480058
$347$	15.2330	0.817750	0.408875	−	0.912590i	$-0.365921\pi$
$347$	15.2330	0.817750	0.408875	+	0.912590i	$0.365921\pi$
$348$	0	0
$349$	0.465991	0.0249439	0.0124720	−	0.999922i	$-0.496030\pi$
$349$	0.465991	0.0249439	0.0124720	+	0.999922i	$0.496030\pi$
$350$	0	0
$351$	0	0
$352$	14.8845	0.793348
$353$	10.5865	0.563464	0.281732	−	0.959493i	$-0.409091\pi$
$353$	10.5865	0.563464	0.281732	+	0.959493i	$0.409091\pi$
$354$	0	0
$355$	0.490140	0.0260139
$356$	−27.4406	−1.45435
$357$	0	0
$358$	−8.34307	−0.440945
$359$	22.6620	1.19605	0.598027	−	0.801476i	$-0.295952\pi$
$359$	22.6620	1.19605	0.598027	+	0.801476i	$0.295952\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	45.5582	2.39448
$363$	0	0
$364$	0	0
$365$	1.56982	0.0821679
$366$	0	0
$367$	7.82576	0.408501	0.204251	−	0.978919i	$-0.434524\pi$
$367$	7.82576	0.408501	0.204251	+	0.978919i	$0.434524\pi$
$368$	10.3368	0.538844
$369$	0	0
$370$	11.5006	0.597888
$371$	0	0
$372$	0	0
$373$	−28.7912	−1.49075	−0.745375	−	0.666646i	$-0.767729\pi$
$373$	−28.7912	−1.49075	−0.745375	+	0.666646i	$0.767729\pi$
$374$	−39.6032	−2.04783
$375$	0	0
$376$	14.8992	0.768367
$377$	−35.4290	−1.82469
$378$	0	0
$379$	−35.9642	−1.84736	−0.923678	−	0.383169i	$-0.874833\pi$
$379$	−35.9642	−1.84736	−0.923678	+	0.383169i	$0.874833\pi$
$380$	−1.81952	−0.0933392
$381$	0	0
$382$	−23.0230	−1.17796
$383$	19.9642	1.02012	0.510061	−	0.860138i	$-0.329623\pi$
$383$	19.9642	1.02012	0.510061	+	0.860138i	$0.329623\pi$
$384$	0	0
$385$	0	0
$386$	34.9054	1.77664
$387$	0	0
$388$	−54.1509	−2.74910
$389$	7.03461	0.356669	0.178334	−	0.983970i	$-0.442929\pi$
$389$	7.03461	0.356669	0.178334	+	0.983970i	$0.442929\pi$
$390$	0	0
$391$	34.9296	1.76647
$392$	0	0
$393$	0	0
$394$	21.9066	1.10364
$395$	−3.13963	−0.157972
$396$	0	0
$397$	26.9988	1.35503	0.677516	−	0.735508i	$-0.263057\pi$
$397$	26.9988	1.35503	0.677516	+	0.735508i	$0.263057\pi$
$398$	−19.6390	−0.984416
$399$	0	0
$400$	−6.32636	−0.316318
$401$	−19.7145	−0.984495	−0.492247	−	0.870455i	$-0.663824\pi$
$401$	−19.7145	−0.984495	−0.492247	+	0.870455i	$0.663824\pi$
$402$	0	0
$403$	−38.3944	−1.91256
$404$	−42.1447	−2.09678
$405$	0	0
$406$	0	0
$407$	35.3264	1.75106
$408$	0	0
$409$	19.1280	0.945817	0.472909	−	0.881111i	$-0.343204\pi$
$409$	19.1280	0.945817	0.472909	+	0.881111i	$0.343204\pi$
$410$	2.75535	0.136077
$411$	0	0
$412$	−54.3944	−2.67982
$413$	0	0
$414$	0	0
$415$	−3.23300	−0.158702
$416$	−22.8604	−1.12082
$417$	0	0
$418$	−8.76700	−0.428808
$419$	13.0409	0.637087	0.318544	−	0.947908i	$-0.396806\pi$
$419$	13.0409	0.637087	0.318544	+	0.947908i	$0.396806\pi$
$420$	0	0
$421$	−25.1521	−1.22584	−0.612920	−	0.790145i	$-0.710005\pi$
$421$	−25.1521	−1.22584	−0.612920	+	0.790145i	$0.710005\pi$
$422$	31.7900	1.54751
$423$	0	0
$424$	48.6316	2.36176
$425$	−21.3777	−1.03697
$426$	0	0
$427$	0	0
$428$	51.0618	2.46816
$429$	0	0
$430$	1.58231	0.0763056
$431$	−22.4815	−1.08290	−0.541448	−	0.840734i	$-0.682124\pi$
$431$	−22.4815	−1.08290	−0.541448	+	0.840734i	$0.682124\pi$
$432$	0	0
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	0	0
$436$	−22.8604	−1.09481
$437$	7.73240	0.369891
$438$	0	0
$439$	33.4889	1.59834	0.799170	−	0.601105i	$-0.205273\pi$
$439$	33.4889	1.59834	0.799170	+	0.601105i	$0.205273\pi$
$440$	6.88129	0.328053
$441$	0	0
$442$	60.8246	2.89313
$443$	−23.6966	−1.12586	−0.562930	−	0.826505i	$-0.690326\pi$
$443$	−23.6966	−1.12586	−0.562930	+	0.826505i	$0.690326\pi$
$444$	0	0
$445$	4.03581	0.191316
$446$	33.2664	1.57521
$447$	0	0
$448$	0	0
$449$	−7.38933	−0.348724	−0.174362	−	0.984682i	$-0.555786\pi$
$449$	−7.38933	−0.348724	−0.174362	+	0.984682i	$0.555786\pi$
$450$	0	0
$451$	8.46360	0.398535
$452$	7.66943	0.360739
$453$	0	0
$454$	46.5686	2.18557
$455$	0	0
$456$	0	0
$457$	−18.2676	−0.854522	−0.427261	−	0.904128i	$-0.640522\pi$
$457$	−18.2676	−0.854522	−0.427261	+	0.904128i	$0.640522\pi$
$458$	−43.1280	−2.01524
$459$	0	0
$460$	−14.0692	−0.655981
$461$	−32.8425	−1.52963	−0.764813	−	0.644252i	$-0.777169\pi$
$461$	−32.8425	−1.52963	−0.764813	+	0.644252i	$0.777169\pi$
$462$	0	0
$463$	−13.9308	−0.647418	−0.323709	−	0.946157i	$-0.604930\pi$
$463$	−13.9308	−0.647418	−0.323709	+	0.946157i	$0.604930\pi$
$464$	8.26219	0.383563
$465$	0	0
$466$	21.3839	0.990591
$467$	−6.78491	−0.313968	−0.156984	−	0.987601i	$-0.550177\pi$
$467$	−6.78491	−0.313968	−0.156984	+	0.987601i	$0.550177\pi$
$468$	0	0
$469$	0	0
$470$	−5.07968	−0.234308
$471$	0	0
$472$	−21.6306	−0.995629
$473$	4.86037	0.223480
$474$	0	0
$475$	−4.73240	−0.217137
$476$	0	0
$477$	0	0
$478$	−47.5224	−2.17362
$479$	3.11007	0.142103	0.0710514	−	0.997473i	$-0.477365\pi$
$479$	3.11007	0.142103	0.0710514	+	0.997473i	$0.477365\pi$
$480$	0	0
$481$	−54.2559	−2.47386
$482$	17.3714	0.791247
$483$	0	0
$484$	10.3085	0.468566
$485$	7.96419	0.361635
$486$	0	0
$487$	−35.7900	−1.62180	−0.810899	−	0.585186i	$-0.801021\pi$
$487$	−35.7900	−1.62180	−0.810899	+	0.585186i	$0.801021\pi$
$488$	−19.4764	−0.881657
$489$	0	0
$490$	0	0
$491$	15.2330	0.687455	0.343728	−	0.939069i	$-0.388310\pi$
$491$	15.2330	0.687455	0.343728	+	0.939069i	$0.388310\pi$
$492$	0	0
$493$	27.9191	1.25741
$494$	13.4648	0.605810
$495$	0	0
$496$	8.95374	0.402035
$497$	0	0
$498$	0	0
$499$	28.1867	1.26181	0.630906	−	0.775860i	$-0.282683\pi$
$499$	28.1867	1.26181	0.630906	+	0.775860i	$0.282683\pi$
$500$	17.7082	0.791937
$501$	0	0
$502$	−53.4352	−2.38493
$503$	6.47224	0.288583	0.144291	−	0.989535i	$-0.453910\pi$
$503$	6.47224	0.288583	0.144291	+	0.989535i	$0.453910\pi$
$504$	0	0
$505$	6.19839	0.275825
$506$	−67.7900	−3.01363
$507$	0	0
$508$	64.8246	2.87612
$509$	−43.6274	−1.93375	−0.966875	−	0.255252i	$-0.917842\pi$
$509$	−43.6274	−1.93375	−0.966875	+	0.255252i	$0.917842\pi$
$510$	0	0
$511$	0	0
$512$	14.8600	0.656723
$513$	0	0
$514$	−46.1026	−2.03350
$515$	8.00000	0.352522
$516$	0	0
$517$	−15.6032	−0.686229
$518$	0	0
$519$	0	0
$520$	−10.5686	−0.463465
$521$	38.9537	1.70659	0.853297	−	0.521425i	$-0.174599\pi$
$521$	38.9537	1.70659	0.853297	+	0.521425i	$0.174599\pi$
$522$	0	0
$523$	−37.7658	−1.65138	−0.825692	−	0.564122i	$-0.809215\pi$
$523$	−37.7658	−1.65138	−0.825692	+	0.564122i	$0.809215\pi$
$524$	51.3902	2.24499
$525$	0	0
$526$	−43.9191	−1.91496
$527$	30.2559	1.31797
$528$	0	0
$529$	36.7900	1.59956
$530$	−16.5803	−0.720201
$531$	0	0
$532$	0	0
$533$	−12.9988	−0.563041
$534$	0	0
$535$	−7.50986	−0.324680
$536$	−22.9171	−0.989868
$537$	0	0
$538$	46.1026	1.98763
$539$	0	0
$540$	0	0
$541$	20.6527	0.887930	0.443965	−	0.896044i	$-0.353571\pi$
$541$	20.6527	0.887930	0.443965	+	0.896044i	$0.353571\pi$
$542$	−26.9296	−1.15672
$543$	0	0
$544$	18.0147	0.772373
$545$	3.36217	0.144019
$546$	0	0
$547$	10.7912	0.461396	0.230698	−	0.973025i	$-0.425899\pi$
$547$	10.7912	0.461396	0.230698	+	0.973025i	$0.425899\pi$
$548$	11.9433	0.510191
$549$	0	0
$550$	41.4889	1.76909
$551$	6.18048	0.263297
$552$	0	0
$553$	0	0
$554$	8.60442	0.365567
$555$	0	0
$556$	−10.4302	−0.442338
$557$	−38.8246	−1.64505	−0.822525	−	0.568729i	$-0.807435\pi$
$557$	−38.8246	−1.64505	−0.822525	+	0.568729i	$0.807435\pi$
$558$	0	0
$559$	−7.46479	−0.315727
$560$	0	0
$561$	0	0
$562$	54.5298	2.30020
$563$	−1.56982	−0.0661598	−0.0330799	−	0.999453i	$-0.510532\pi$
$563$	−1.56982	−0.0661598	−0.0330799	+	0.999453i	$0.510532\pi$
$564$	0	0
$565$	−1.12797	−0.0474542
$566$	70.2201	2.95157
$567$	0	0
$568$	3.37684	0.141689
$569$	−16.2139	−0.679722	−0.339861	−	0.940476i	$-0.610380\pi$
$569$	−16.2139	−0.679722	−0.339861	+	0.940476i	$0.610380\pi$
$570$	0	0
$571$	46.2559	1.93575	0.967876	−	0.251430i	$-0.0809007\pi$
$571$	46.2559	1.93575	0.967876	+	0.251430i	$0.0809007\pi$
$572$	−75.2547	−3.14656
$573$	0	0
$574$	0	0
$575$	−36.5928	−1.52602
$576$	0	0
$577$	−1.86157	−0.0774981	−0.0387490	−	0.999249i	$-0.512337\pi$
$577$	−1.86157	−0.0774981	−0.0387490	+	0.999249i	$0.512337\pi$
$578$	−8.00042	−0.332774
$579$	0	0
$580$	−11.2455	−0.466943
$581$	0	0
$582$	0	0
$583$	−50.9296	−2.10929
$584$	10.8153	0.447540
$585$	0	0
$586$	−33.5131	−1.38441
$587$	11.9579	0.493557	0.246779	−	0.969072i	$-0.420628\pi$
$587$	11.9579	0.493557	0.246779	+	0.969072i	$0.420628\pi$
$588$	0	0
$589$	6.69779	0.275978
$590$	7.37466	0.303610
$591$	0	0
$592$	12.6527	0.520024
$593$	−16.8425	−0.691637	−0.345819	−	0.938301i	$-0.612399\pi$
$593$	−16.8425	−0.691637	−0.345819	+	0.938301i	$0.612399\pi$
$594$	0	0
$595$	0	0
$596$	36.4427	1.49275
$597$	0	0
$598$	104.115	4.25758
$599$	35.8771	1.46590	0.732949	−	0.680284i	$-0.238143\pi$
$599$	35.8771	1.46590	0.732949	+	0.680284i	$0.238143\pi$
$600$	0	0
$601$	−20.0692	−0.818640	−0.409320	−	0.912391i	$-0.634234\pi$
$601$	−20.0692	−0.818640	−0.409320	+	0.912391i	$0.634234\pi$
$602$	0	0
$603$	0	0
$604$	48.8729	1.98861
$605$	−1.51611	−0.0616385
$606$	0	0
$607$	−16.5352	−0.671143	−0.335572	−	0.942015i	$-0.608929\pi$
$607$	−16.5352	−0.671143	−0.335572	+	0.942015i	$0.608929\pi$
$608$	3.98793	0.161732
$609$	0	0
$610$	6.64023	0.268855
$611$	23.9642	0.969488
$612$	0	0
$613$	24.1984	0.977364	0.488682	−	0.872462i	$-0.337478\pi$
$613$	24.1984	0.977364	0.488682	+	0.872462i	$0.337478\pi$
$614$	−11.1972	−0.451882
$615$	0	0
$616$	0	0
$617$	3.93078	0.158247	0.0791237	−	0.996865i	$-0.474788\pi$
$617$	3.93078	0.158247	0.0791237	+	0.996865i	$0.474788\pi$
$618$	0	0
$619$	5.03461	0.202358	0.101179	−	0.994868i	$-0.467739\pi$
$619$	5.03461	0.202358	0.101179	+	0.994868i	$0.467739\pi$
$620$	−12.1867	−0.489431
$621$	0	0
$622$	−26.1805	−1.04974
$623$	0	0
$624$	0	0
$625$	21.0576	0.842302
$626$	−48.8362	−1.95189
$627$	0	0
$628$	28.3819	1.13256
$629$	42.7553	1.70477
$630$	0	0
$631$	−23.3714	−0.930402	−0.465201	−	0.885205i	$-0.654018\pi$
$631$	−23.3714	−0.930402	−0.465201	+	0.885205i	$0.654018\pi$
$632$	−21.6306	−0.860419
$633$	0	0
$634$	−15.7744	−0.626483
$635$	−9.53401	−0.378346
$636$	0	0
$637$	0	0
$638$	−54.1843	−2.14518
$639$	0	0
$640$	−10.5048	−0.415239
$641$	13.1459	0.519231	0.259615	−	0.965712i	$-0.416404\pi$
$641$	13.1459	0.519231	0.259615	+	0.965712i	$0.416404\pi$
$642$	0	0
$643$	−6.96539	−0.274688	−0.137344	−	0.990523i	$-0.543857\pi$
$643$	−6.96539	−0.274688	−0.137344	+	0.990523i	$0.543857\pi$
$644$	0	0
$645$	0	0
$646$	−10.6107	−0.417471
$647$	22.6107	0.888917	0.444459	−	0.895799i	$-0.353396\pi$
$647$	22.6107	0.888917	0.444459	+	0.895799i	$0.353396\pi$
$648$	0	0
$649$	22.6527	0.889197
$650$	−63.7207	−2.49933
$651$	0	0
$652$	−30.8362	−1.20764
$653$	30.8604	1.20766	0.603830	−	0.797113i	$-0.293641\pi$
$653$	30.8604	1.20766	0.603830	+	0.797113i	$0.293641\pi$
$654$	0	0
$655$	−7.55816	−0.295322
$656$	3.03138	0.118355
$657$	0	0
$658$	0	0
$659$	−16.4481	−0.640727	−0.320363	−	0.947295i	$-0.603805\pi$
$659$	−16.4481	−0.640727	−0.320363	+	0.947295i	$0.603805\pi$
$660$	0	0
$661$	35.9883	1.39978	0.699892	−	0.714249i	$-0.253231\pi$
$661$	35.9883	1.39978	0.699892	+	0.714249i	$0.253231\pi$
$662$	−24.4994	−0.952196
$663$	0	0
$664$	−22.2738	−0.864393
$665$	0	0
$666$	0	0
$667$	47.7900	1.85043
$668$	−51.2423	−1.98262
$669$	0	0
$670$	7.81327	0.301853
$671$	20.3968	0.787409
$672$	0	0
$673$	−2.99880	−0.115595	−0.0577977	−	0.998328i	$-0.518408\pi$
$673$	−2.99880	−0.115595	−0.0577977	+	0.998328i	$0.518408\pi$
$674$	−47.9883	−1.84844
$675$	0	0
$676$	69.8550	2.68673
$677$	−14.7670	−0.567542	−0.283771	−	0.958892i	$-0.591586\pi$
$677$	−14.7670	−0.567542	−0.283771	+	0.958892i	$0.591586\pi$
$678$	0	0
$679$	0	0
$680$	8.32839	0.319379
$681$	0	0
$682$	−58.7195	−2.24849
$683$	40.4123	1.54633	0.773166	−	0.634203i	$-0.218672\pi$
$683$	40.4123	1.54633	0.773166	+	0.634203i	$0.218672\pi$
$684$	0	0
$685$	−1.75655	−0.0671142
$686$	0	0
$687$	0	0
$688$	1.74082	0.0663682
$689$	78.2201	2.97995
$690$	0	0
$691$	27.3264	1.03954	0.519772	−	0.854305i	$-0.326017\pi$
$691$	27.3264	1.03954	0.519772	+	0.854305i	$0.326017\pi$
$692$	−13.3714	−0.508305
$693$	0	0
$694$	−35.7807	−1.35822
$695$	1.53401	0.0581883
$696$	0	0
$697$	10.2435	0.387999
$698$	−1.09456	−0.0414298
$699$	0	0
$700$	0	0
$701$	−12.4660	−0.470834	−0.235417	−	0.971894i	$-0.575646\pi$
$701$	−12.4660	−0.470834	−0.235417	+	0.971894i	$0.575646\pi$
$702$	0	0
$703$	9.46479	0.356971
$704$	−44.9412	−1.69379
$705$	0	0
$706$	−24.8666	−0.935867
$707$	0	0
$708$	0	0
$709$	−19.4048	−0.728764	−0.364382	−	0.931250i	$-0.618720\pi$
$709$	−19.4048	−0.728764	−0.364382	+	0.931250i	$0.618720\pi$
$710$	−1.15129	−0.0432070
$711$	0	0
$712$	27.8048	1.04203
$713$	51.7900	1.93955
$714$	0	0
$715$	11.0680	0.413920
$716$	12.4932	0.466891
$717$	0	0
$718$	−53.2306	−1.98655
$719$	28.3189	1.05612	0.528059	−	0.849208i	$-0.322920\pi$
$719$	28.3189	1.05612	0.528059	+	0.849208i	$0.322920\pi$
$720$	0	0
$721$	0	0
$722$	−2.34889	−0.0874168
$723$	0	0
$724$	−68.2201	−2.53538
$725$	−29.2485	−1.08626
$726$	0	0
$727$	−26.6527	−0.988495	−0.494247	−	0.869321i	$-0.664556\pi$
$727$	−26.6527	−0.988495	−0.494247	+	0.869321i	$0.664556\pi$
$728$	0	0
$729$	0	0
$730$	−3.68733	−0.136474
$731$	5.88249	0.217572
$732$	0	0
$733$	19.7565	0.729725	0.364862	−	0.931061i	$-0.381116\pi$
$733$	19.7565	0.729725	0.364862	+	0.931061i	$0.381116\pi$
$734$	−18.3819	−0.678488
$735$	0	0
$736$	30.8362	1.13664
$737$	24.0000	0.884051
$738$	0	0
$739$	36.0934	1.32772	0.663858	−	0.747859i	$-0.268918\pi$
$739$	36.0934	1.32772	0.663858	+	0.747859i	$0.268918\pi$
$740$	−17.2213	−0.633069
$741$	0	0
$742$	0	0
$743$	33.1700	1.21689	0.608445	−	0.793596i	$-0.291794\pi$
$743$	33.1700	1.21689	0.608445	+	0.793596i	$0.291794\pi$
$744$	0	0
$745$	−5.35977	−0.196367
$746$	67.6274	2.47601
$747$	0	0
$748$	59.3030	2.16833
$749$	0	0
$750$	0	0
$751$	36.5477	1.33364	0.666822	−	0.745217i	$-0.267654\pi$
$751$	36.5477	1.33364	0.666822	+	0.745217i	$0.267654\pi$
$752$	−5.58855	−0.203794
$753$	0	0
$754$	83.2189	3.03066
$755$	−7.18793	−0.261595
$756$	0	0
$757$	−44.7912	−1.62796	−0.813981	−	0.580891i	$-0.802704\pi$
$757$	−44.7912	−1.62796	−0.813981	+	0.580891i	$0.802704\pi$
$758$	84.4761	3.06831
$759$	0	0
$760$	1.84366	0.0668767
$761$	−10.7250	−0.388779	−0.194390	−	0.980924i	$-0.562273\pi$
$761$	−10.7250	−0.388779	−0.194390	+	0.980924i	$0.562273\pi$
$762$	0	0
$763$	0	0
$764$	34.4753	1.24727
$765$	0	0
$766$	−46.8938	−1.69434
$767$	−34.7912	−1.25624
$768$	0	0
$769$	−16.0692	−0.579471	−0.289735	−	0.957107i	$-0.593567\pi$
$769$	−16.0692	−0.579471	−0.289735	+	0.957107i	$0.593567\pi$
$770$	0	0
$771$	0	0
$772$	−52.2684	−1.88118
$773$	9.05876	0.325821	0.162910	−	0.986641i	$-0.447912\pi$
$773$	9.05876	0.325821	0.162910	+	0.986641i	$0.447912\pi$
$774$	0	0
$775$	−31.6966	−1.13858
$776$	54.8696	1.96970
$777$	0	0
$778$	−16.5236	−0.592398
$779$	2.26760	0.0812453
$780$	0	0
$781$	−3.53640	−0.126543
$782$	−82.0459	−2.93396
$783$	0	0
$784$	0	0
$785$	−4.17424	−0.148985
$786$	0	0
$787$	37.7658	1.34621	0.673103	−	0.739549i	$-0.264961\pi$
$787$	37.7658	1.34621	0.673103	+	0.739549i	$0.264961\pi$
$788$	−32.8036	−1.16858
$789$	0	0
$790$	7.37466	0.262379
$791$	0	0
$792$	0	0
$793$	−31.3264	−1.11243
$794$	−63.4173	−2.25060
$795$	0	0
$796$	29.4081	1.04234
$797$	30.7312	1.08855	0.544277	−	0.838905i	$-0.316804\pi$
$797$	30.7312	1.08855	0.544277	+	0.838905i	$0.316804\pi$
$798$	0	0
$799$	−18.8845	−0.668086
$800$	−18.8724	−0.667242
$801$	0	0
$802$	46.3073	1.63517
$803$	−11.3264	−0.399699
$804$	0	0
$805$	0	0
$806$	90.1843	3.17661
$807$	0	0
$808$	42.7040	1.50232
$809$	−40.1175	−1.41046	−0.705228	−	0.708980i	$-0.749156\pi$
$809$	−40.1175	−1.41046	−0.705228	+	0.708980i	$0.749156\pi$
$810$	0	0
$811$	40.3252	1.41601	0.708004	−	0.706208i	$-0.249596\pi$
$811$	40.3252	1.41601	0.708004	+	0.706208i	$0.249596\pi$
$812$	0	0
$813$	0	0
$814$	−82.9779	−2.90837
$815$	4.53521	0.158861
$816$	0	0
$817$	1.30221	0.0455586
$818$	−44.9296	−1.57093
$819$	0	0
$820$	−4.12594	−0.144084
$821$	−23.3598	−0.815262	−0.407631	−	0.913147i	$-0.633645\pi$
$821$	−23.3598	−0.815262	−0.407631	+	0.913147i	$0.633645\pi$
$822$	0	0
$823$	−14.3010	−0.498502	−0.249251	−	0.968439i	$-0.580184\pi$
$823$	−14.3010	−0.498502	−0.249251	+	0.968439i	$0.580184\pi$
$824$	55.1163	1.92007
$825$	0	0
$826$	0	0
$827$	−6.73984	−0.234367	−0.117184	−	0.993110i	$-0.537387\pi$
$827$	−6.73984	−0.234367	−0.117184	+	0.993110i	$0.537387\pi$
$828$	0	0
$829$	28.6044	0.993473	0.496736	−	0.867901i	$-0.334532\pi$
$829$	28.6044	0.993473	0.496736	+	0.867901i	$0.334532\pi$
$830$	7.59396	0.263590
$831$	0	0
$832$	69.0230	2.39294
$833$	0	0
$834$	0	0
$835$	7.53640	0.260808
$836$	13.1280	0.454040
$837$	0	0
$838$	−30.6316	−1.05815
$839$	20.1384	0.695256	0.347628	−	0.937633i	$-0.386987\pi$
$839$	20.1384	0.695256	0.347628	+	0.937633i	$0.386987\pi$
$840$	0	0
$841$	9.19839	0.317186
$842$	59.0797	2.03602
$843$	0	0
$844$	−47.6032	−1.63857
$845$	−10.2738	−0.353431
$846$	0	0
$847$	0	0
$848$	−18.2413	−0.626408
$849$	0	0
$850$	50.2139	1.72232
$851$	73.1855	2.50877
$852$	0	0
$853$	−10.3252	−0.353527	−0.176763	−	0.984253i	$-0.556563\pi$
$853$	−10.3252	−0.353527	−0.176763	+	0.984253i	$0.556563\pi$
$854$	0	0
$855$	0	0
$856$	−51.7395	−1.76842
$857$	−15.2664	−0.521490	−0.260745	−	0.965408i	$-0.583968\pi$
$857$	−15.2664	−0.521490	−0.260745	+	0.965408i	$0.583968\pi$
$858$	0	0
$859$	8.67364	0.295941	0.147970	−	0.988992i	$-0.452726\pi$
$859$	8.67364	0.295941	0.147970	+	0.988992i	$0.452726\pi$
$860$	−2.36939	−0.0807957
$861$	0	0
$862$	52.8067	1.79860
$863$	6.55311	0.223070	0.111535	−	0.993760i	$-0.464423\pi$
$863$	6.55311	0.223070	0.111535	+	0.993760i	$0.464423\pi$
$864$	0	0
$865$	1.96659	0.0668661
$866$	42.2801	1.43674
$867$	0	0
$868$	0	0
$869$	22.6527	0.768441
$870$	0	0
$871$	−36.8604	−1.24897
$872$	23.1638	0.784425
$873$	0	0
$874$	−18.1626	−0.614358
$875$	0	0
$876$	0	0
$877$	30.9988	1.04676	0.523378	−	0.852101i	$-0.324672\pi$
$877$	30.9988	1.04676	0.523378	+	0.852101i	$0.324672\pi$
$878$	−78.6620	−2.65471
$879$	0	0
$880$	−2.58111	−0.0870092
$881$	8.51730	0.286955	0.143478	−	0.989654i	$-0.454171\pi$
$881$	8.51730	0.286955	0.143478	+	0.989654i	$0.454171\pi$
$882$	0	0
$883$	23.3264	0.784995	0.392497	−	0.919753i	$-0.371611\pi$
$883$	23.3264	0.784995	0.392497	+	0.919753i	$0.371611\pi$
$884$	−91.0805	−3.06337
$885$	0	0
$886$	55.6608	1.86996
$887$	19.9642	0.670332	0.335166	−	0.942159i	$-0.391208\pi$
$887$	19.9642	0.670332	0.335166	+	0.942159i	$0.391208\pi$
$888$	0	0
$889$	0	0
$890$	−9.47968	−0.317760
$891$	0	0
$892$	−49.8141	−1.66790
$893$	−4.18048	−0.139895
$894$	0	0
$895$	−1.83742	−0.0614181
$896$	0	0
$897$	0	0
$898$	17.3568	0.579202
$899$	41.3956	1.38062
$900$	0	0
$901$	−61.6399	−2.05352
$902$	−19.8801	−0.661935
$903$	0	0
$904$	−7.77122	−0.258467
$905$	10.0334	0.333522
$906$	0	0
$907$	7.86157	0.261039	0.130520	−	0.991446i	$-0.458335\pi$
$907$	7.86157	0.261039	0.130520	+	0.991446i	$0.458335\pi$
$908$	−69.7332	−2.31418
$909$	0	0
$910$	0	0
$911$	32.4123	1.07387	0.536933	−	0.843625i	$-0.319583\pi$
$911$	32.4123	1.07387	0.536933	+	0.843625i	$0.319583\pi$
$912$	0	0
$913$	23.3264	0.771990
$914$	42.9087	1.41929
$915$	0	0
$916$	64.5811	2.13382
$917$	0	0
$918$	0	0
$919$	53.8592	1.77665	0.888325	−	0.459215i	$-0.151869\pi$
$919$	53.8592	1.77665	0.888325	+	0.459215i	$0.151869\pi$
$920$	14.2559	0.470005
$921$	0	0
$922$	77.1435	2.54059
$923$	5.43138	0.178776
$924$	0	0
$925$	−44.7912	−1.47272
$926$	32.7219	1.07531
$927$	0	0
$928$	24.6473	0.809088
$929$	45.7028	1.49946	0.749731	−	0.661743i	$-0.230183\pi$
$929$	45.7028	1.49946	0.749731	+	0.661743i	$0.230183\pi$
$930$	0	0
$931$	0	0
$932$	−32.0209	−1.04888
$933$	0	0
$934$	15.9370	0.521476
$935$	−8.72194	−0.285238
$936$	0	0
$937$	−13.0680	−0.426914	−0.213457	−	0.976953i	$-0.568472\pi$
$937$	−13.0680	−0.426914	−0.213457	+	0.976953i	$0.568472\pi$
$938$	0	0
$939$	0	0
$940$	7.60646	0.248095
$941$	50.6954	1.65262	0.826311	−	0.563214i	$-0.190435\pi$
$941$	50.6954	1.65262	0.826311	+	0.563214i	$0.190435\pi$
$942$	0	0
$943$	17.5340	0.570986
$944$	8.11345	0.264070
$945$	0	0
$946$	−11.4165	−0.371182
$947$	23.7807	0.772769	0.386384	−	0.922338i	$-0.373724\pi$
$947$	23.7807	0.772769	0.386384	+	0.922338i	$0.373724\pi$
$948$	0	0
$949$	17.3956	0.564684
$950$	11.1159	0.360647
$951$	0	0
$952$	0	0
$953$	−5.32011	−0.172335	−0.0861677	−	0.996281i	$-0.527462\pi$
$953$	−5.32011	−0.172335	−0.0861677	+	0.996281i	$0.527462\pi$
$954$	0	0
$955$	−5.07041	−0.164075
$956$	71.1614	2.30152
$957$	0	0
$958$	−7.30523	−0.236021
$959$	0	0
$960$	0	0
$961$	13.8604	0.447109
$962$	127.441	4.10888
$963$	0	0
$964$	−26.0125	−0.837806
$965$	7.68733	0.247464
$966$	0	0
$967$	−3.41973	−0.109971	−0.0549855	−	0.998487i	$-0.517511\pi$
$967$	−3.41973	−0.109971	−0.0549855	+	0.998487i	$0.517511\pi$
$968$	−10.4453	−0.335724
$969$	0	0
$970$	−18.7070	−0.600647
$971$	29.1855	0.936608	0.468304	−	0.883567i	$-0.344865\pi$
$971$	29.1855	0.936608	0.468304	+	0.883567i	$0.344865\pi$
$972$	0	0
$973$	0	0
$974$	84.0668	2.69367
$975$	0	0
$976$	7.30544	0.233842
$977$	3.93703	0.125957	0.0629784	−	0.998015i	$-0.479940\pi$
$977$	3.93703	0.125957	0.0629784	+	0.998015i	$0.479940\pi$
$978$	0	0
$979$	−29.1187	−0.930638
$980$	0	0
$981$	0	0
$982$	−35.7807	−1.14181
$983$	44.8604	1.43082	0.715412	−	0.698703i	$-0.246239\pi$
$983$	44.8604	1.43082	0.715412	+	0.698703i	$0.246239\pi$
$984$	0	0
$985$	4.82456	0.153723
$986$	−65.5791	−2.08846
$987$	0	0
$988$	−20.1626	−0.641457
$989$	10.0692	0.320182
$990$	0	0
$991$	−42.7912	−1.35931	−0.679653	−	0.733534i	$-0.737870\pi$
$991$	−42.7912	−1.35931	−0.679653	+	0.733534i	$0.737870\pi$
$992$	26.7103	0.848052
$993$	0	0
$994$	0	0
$995$	−4.32516	−0.137117
$996$	0	0
$997$	−19.7207	−0.624562	−0.312281	−	0.949990i	$-0.601093\pi$
$997$	−19.7207	−0.624562	−0.312281	+	0.949990i	$0.601093\pi$
$998$	−66.2076	−2.09577
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bq.1.1		3
3.2	odd	2		2793.2.a.w.1.3		3
7.6	odd	2		1197.2.a.m.1.1		3
21.20	even	2		399.2.a.e.1.3	✓	3
84.83	odd	2		6384.2.a.bu.1.2		3
105.104	even	2		9975.2.a.x.1.1		3
399.398	odd	2		7581.2.a.l.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.e.1.3	✓	3	21.20	even	2
1197.2.a.m.1.1		3	7.6	odd	2
2793.2.a.w.1.3		3	3.2	odd	2
6384.2.a.bu.1.2		3	84.83	odd	2
7581.2.a.l.1.1		3	399.398	odd	2
8379.2.a.bq.1.1		3	1.1	even	1	trivial
9975.2.a.x.1.1		3	105.104	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q-2.34889 q^{2} +3.51730 q^{4} -0.517304 q^{5} -3.56399 q^{8} +1.21509 q^{10} +3.73240 q^{11} -5.73240 q^{13} +1.33682 q^{16} +4.51730 q^{17} +1.00000 q^{19} -1.81952 q^{20} -8.76700 q^{22} +7.73240 q^{23} -4.73240 q^{25} +13.4648 q^{26} +6.18048 q^{29} +6.69779 q^{31} +3.98793 q^{32} -10.6107 q^{34} +9.46479 q^{37} -2.34889 q^{38} +1.84366 q^{40} +2.26760 q^{41} +1.30221 q^{43} +13.1280 q^{44} -18.1626 q^{46} -4.18048 q^{47} +11.1159 q^{50} -20.1626 q^{52} -13.6453 q^{53} -1.93078 q^{55} -14.5173 q^{58} +6.06922 q^{59} +5.46479 q^{61} -15.7324 q^{62} -12.0409 q^{64} +2.96539 q^{65} +6.43018 q^{67} +15.8887 q^{68} -0.947489 q^{71} -3.03461 q^{73} -22.2318 q^{74} +3.51730 q^{76} +6.06922 q^{79} -0.691542 q^{80} -5.32636 q^{82} +6.24970 q^{83} -2.33682 q^{85} -3.05876 q^{86} -13.3022 q^{88} -7.80161 q^{89} +27.1972 q^{92} +9.81952 q^{94} -0.517304 q^{95} -15.3956 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 9 q^{4} + 9 q^{8} - 10 q^{10} - 4 q^{11} - 2 q^{13} + 13 q^{16} + 12 q^{17} + 3 q^{19} - 16 q^{20} - 8 q^{22} + 8 q^{23} + q^{25} + 10 q^{26} + 8 q^{29} + 8 q^{31} + 27 q^{32} + 6 q^{34}+ \cdots - 22 q^{97}+O(q^{100}) \) 3 * q - q^2 + 9 * q^4 + 9 * q^8 - 10 * q^10 - 4 * q^11 - 2 * q^13 + 13 * q^16 + 12 * q^17 + 3 * q^19 - 16 * q^20 - 8 * q^22 + 8 * q^23 + q^25 + 10 * q^26 + 8 * q^29 + 8 * q^31 + 27 * q^32 + 6 * q^34 - 2 * q^37 - q^38 - 14 * q^40 + 22 * q^41 + 16 * q^43 - 12 * q^46 - 2 * q^47 + 9 * q^50 - 18 * q^52 - 12 * q^55 - 42 * q^58 + 12 * q^59 - 14 * q^61 - 32 * q^62 + 29 * q^64 + 12 * q^65 - 8 * q^67 + 52 * q^68 + 26 * q^71 - 6 * q^73 - 18 * q^74 + 9 * q^76 + 12 * q^79 - 52 * q^80 + 2 * q^82 + 2 * q^83 - 16 * q^85 + 24 * q^86 - 52 * q^88 - 2 * q^89 + 36 * q^92 + 40 * q^94 - 22 * q^97

Introduction

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