LMFDB - Embedded newform 2401.2.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2401,2,Mod(1,2401)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2401, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2401.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2401 = 7^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2401.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:	$19.1720815253$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	2401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.02147 q^{2} -1.40055 q^{3} +2.08635 q^{4} -2.27513 q^{5} +2.83116 q^{6} -0.174553 q^{8} -1.03847 q^{9} +4.59911 q^{10} +1.90811 q^{11} -2.92203 q^{12} +4.76338 q^{13} +3.18642 q^{15} -3.81985 q^{16} +4.04848 q^{17} +2.09924 q^{18} +0.437865 q^{19} -4.74671 q^{20} -3.85720 q^{22} +6.82567 q^{23} +0.244469 q^{24} +0.176204 q^{25} -9.62905 q^{26} +5.65606 q^{27} -8.50425 q^{29} -6.44126 q^{30} -0.818801 q^{31} +8.07082 q^{32} -2.67240 q^{33} -8.18388 q^{34} -2.16661 q^{36} -3.72507 q^{37} -0.885133 q^{38} -6.67134 q^{39} +0.397129 q^{40} +10.7964 q^{41} -8.07673 q^{43} +3.98099 q^{44} +2.36265 q^{45} -13.7979 q^{46} -1.53673 q^{47} +5.34987 q^{48} -0.356192 q^{50} -5.67008 q^{51} +9.93808 q^{52} -0.827527 q^{53} -11.4336 q^{54} -4.34120 q^{55} -0.613251 q^{57} +17.1911 q^{58} -3.05424 q^{59} +6.64798 q^{60} +0.0583083 q^{61} +1.65518 q^{62} -8.67524 q^{64} -10.8373 q^{65} +5.40219 q^{66} -12.1294 q^{67} +8.44654 q^{68} -9.55967 q^{69} +4.90313 q^{71} +0.181268 q^{72} +10.8389 q^{73} +7.53013 q^{74} -0.246782 q^{75} +0.913540 q^{76} +13.4859 q^{78} +2.45993 q^{79} +8.69063 q^{80} -4.80616 q^{81} -21.8246 q^{82} -4.61877 q^{83} -9.21080 q^{85} +16.3269 q^{86} +11.9106 q^{87} -0.333066 q^{88} +4.06662 q^{89} -4.77604 q^{90} +14.2407 q^{92} +1.14677 q^{93} +3.10646 q^{94} -0.996200 q^{95} -11.3035 q^{96} -4.05436 q^{97} -1.98152 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - q^{2} + 7 q^{3} + 23 q^{4} + 14 q^{5} + 14 q^{6} - 3 q^{8} + 15 q^{9} + 14 q^{10} - 4 q^{11} + 14 q^{12} + 14 q^{13} + 15 q^{15} + 17 q^{16} + 28 q^{17} - 2 q^{18} + 21 q^{19} + 42 q^{20} + 4 q^{22}+ \cdots - 8 q^{99}+O(q^{100}) $ 24 * q - q^2 + 7 * q^3 + 23 * q^4 + 14 * q^5 + 14 * q^6 - 3 * q^8 + 15 * q^9 + 14 * q^10 - 4 * q^11 + 14 * q^12 + 14 * q^13 + 15 * q^15 + 17 * q^16 + 28 * q^17 - 2 * q^18 + 21 * q^19 + 42 * q^20 + 4 * q^22 - 8 * q^23 + 21 * q^24 + 4 * q^25 + 28 * q^26 + 28 * q^27 - 8 * q^29 - 11 * q^30 + 35 * q^31 + 18 * q^32 - 7 * q^33 + q^36 - 8 * q^37 + 14 * q^38 - 14 * q^39 + 7 * q^40 + 35 * q^41 - 8 * q^43 - 20 * q^44 + 42 * q^45 - 6 * q^46 + 70 * q^47 + 42 * q^48 + 20 * q^50 + 24 * q^51 - 21 * q^52 + 24 * q^53 + 14 * q^54 + 56 * q^55 - 20 * q^57 + 19 * q^58 + 84 * q^59 + 56 * q^62 - 9 * q^64 + 14 * q^65 - 11 * q^67 + 77 * q^68 + 42 * q^69 - 8 * q^71 + 40 * q^72 - 7 * q^73 + 43 * q^74 + 49 * q^75 + 7 * q^76 - 15 * q^79 + 70 * q^80 - 36 * q^81 - 56 * q^82 + 56 * q^83 + 36 * q^85 - 86 * q^86 - 70 * q^88 + 70 * q^89 - 56 * q^90 - 68 * q^92 - 11 * q^93 + 84 * q^94 + 54 * q^95 + 56 * q^96 - 8 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.02147	−1.42940	−0.714698	−	0.699433i	$-0.753436\pi$
$2$	−2.02147	−1.42940	−0.714698	+	0.699433i	$0.753436\pi$
$3$	−1.40055	−0.808605	−0.404303	−	0.914625i	$-0.632486\pi$
$3$	−1.40055	−0.808605	−0.404303	+	0.914625i	$0.632486\pi$
$4$	2.08635	1.04317
$5$	−2.27513	−1.01747	−0.508734	−	0.860924i	$-0.669886\pi$
$5$	−2.27513	−1.01747	−0.508734	+	0.860924i	$0.669886\pi$
$6$	2.83116	1.15582
$7$	0	0
$7$	0	0
$8$	−0.174553	−0.0617136
$9$	−1.03847	−0.346157
$10$	4.59911	1.45437
$11$	1.90811	0.575318	0.287659	−	0.957733i	$-0.407123\pi$
$11$	1.90811	0.575318	0.287659	+	0.957733i	$0.407123\pi$
$12$	−2.92203	−0.843517
$13$	4.76338	1.32113	0.660563	−	0.750771i	$-0.270318\pi$
$13$	4.76338	1.32113	0.660563	+	0.750771i	$0.270318\pi$
$14$	0	0
$15$	3.18642	0.822730
$16$	−3.81985	−0.954961
$17$	4.04848	0.981900	0.490950	−	0.871188i	$-0.336650\pi$
$17$	4.04848	0.981900	0.490950	+	0.871188i	$0.336650\pi$
$18$	2.09924	0.494796
$19$	0.437865	0.100453	0.0502266	−	0.998738i	$-0.484006\pi$
$19$	0.437865	0.100453	0.0502266	+	0.998738i	$0.484006\pi$
$20$	−4.74671	−1.06140
$21$	0	0
$22$	−3.85720	−0.822358
$23$	6.82567	1.42325	0.711626	−	0.702559i	$-0.247959\pi$
$23$	6.82567	1.42325	0.711626	+	0.702559i	$0.247959\pi$
$24$	0.244469	0.0499020
$25$	0.176204	0.0352409
$26$	−9.62905	−1.88841
$27$	5.65606	1.08851
$28$	0	0
$29$	−8.50425	−1.57920	−0.789600	−	0.613622i	$-0.789712\pi$
$29$	−8.50425	−1.57920	−0.789600	+	0.613622i	$0.789712\pi$
$30$	−6.44126	−1.17601
$31$	−0.818801	−0.147061	−0.0735305	−	0.997293i	$-0.523427\pi$
$31$	−0.818801	−0.147061	−0.0735305	+	0.997293i	$0.523427\pi$
$32$	8.07082	1.42673
$33$	−2.67240	−0.465205
$34$	−8.18388	−1.40352
$35$	0	0
$36$	−2.16661	−0.361102
$37$	−3.72507	−0.612398	−0.306199	−	0.951967i	$-0.599057\pi$
$37$	−3.72507	−0.612398	−0.306199	+	0.951967i	$0.599057\pi$
$38$	−0.885133	−0.143588
$39$	−6.67134	−1.06827
$40$	0.397129	0.0627916
$41$	10.7964	1.68612	0.843058	−	0.537823i	$-0.180753\pi$
$41$	10.7964	1.68612	0.843058	+	0.537823i	$0.180753\pi$
$42$	0	0
$43$	−8.07673	−1.23169	−0.615845	−	0.787867i	$-0.711185\pi$
$43$	−8.07673	−1.23169	−0.615845	+	0.787867i	$0.711185\pi$
$44$	3.98099	0.600157
$45$	2.36265	0.352204
$46$	−13.7979	−2.03439
$47$	−1.53673	−0.224156	−0.112078	−	0.993699i	$-0.535751\pi$
$47$	−1.53673	−0.224156	−0.112078	+	0.993699i	$0.535751\pi$
$48$	5.34987	0.772187
$49$	0	0
$50$	−0.356192	−0.0503732
$51$	−5.67008	−0.793970
$52$	9.93808	1.37816
$53$	−0.827527	−0.113670	−0.0568348	−	0.998384i	$-0.518101\pi$
$53$	−0.827527	−0.113670	−0.0568348	+	0.998384i	$0.518101\pi$
$54$	−11.4336	−1.55591
$55$	−4.34120	−0.585368
$56$	0	0
$57$	−0.613251	−0.0812270
$58$	17.1911	2.25730
$59$	−3.05424	−0.397628	−0.198814	−	0.980037i	$-0.563709\pi$
$59$	−3.05424	−0.397628	−0.198814	+	0.980037i	$0.563709\pi$
$60$	6.64798	0.858251
$61$	0.0583083	0.00746561	0.00373280	−	0.999993i	$-0.498812\pi$
$61$	0.0583083	0.00746561	0.00373280	+	0.999993i	$0.498812\pi$
$62$	1.65518	0.210208
$63$	0	0
$64$	−8.67524	−1.08440
$65$	−10.8373	−1.34420
$66$	5.40219	0.664963
$67$	−12.1294	−1.48184	−0.740918	−	0.671595i	$-0.765610\pi$
$67$	−12.1294	−1.48184	−0.740918	+	0.671595i	$0.765610\pi$
$68$	8.44654	1.02429
$69$	−9.55967	−1.15085
$70$	0	0
$71$	4.90313	0.581895	0.290947	−	0.956739i	$-0.406030\pi$
$71$	4.90313	0.581895	0.290947	+	0.956739i	$0.406030\pi$
$72$	0.181268	0.0213626
$73$	10.8389	1.26860	0.634299	−	0.773088i	$-0.281289\pi$
$73$	10.8389	1.26860	0.634299	+	0.773088i	$0.281289\pi$
$74$	7.53013	0.875360
$75$	−0.246782	−0.0284960
$76$	0.913540	0.104790
$77$	0	0
$78$	13.4859	1.52698
$79$	2.45993	0.276764	0.138382	−	0.990379i	$-0.455810\pi$
$79$	2.45993	0.276764	0.138382	+	0.990379i	$0.455810\pi$
$80$	8.69063	0.971642
$81$	−4.80616	−0.534018
$82$	−21.8246	−2.41013
$83$	−4.61877	−0.506976	−0.253488	−	0.967339i	$-0.581578\pi$
$83$	−4.61877	−0.506976	−0.253488	+	0.967339i	$0.581578\pi$
$84$	0	0
$85$	−9.21080	−0.999051
$86$	16.3269	1.76057
$87$	11.9106	1.27695
$88$	−0.333066	−0.0355050
$89$	4.06662	0.431061	0.215530	−	0.976497i	$-0.430852\pi$
$89$	4.06662	0.431061	0.215530	+	0.976497i	$0.430852\pi$
$90$	−4.77604	−0.503439
$91$	0	0
$92$	14.2407	1.48470
$93$	1.14677	0.118914
$94$	3.10646	0.320407
$95$	−0.996200	−0.102208
$96$	−11.3035	−1.15366
$97$	−4.05436	−0.411658	−0.205829	−	0.978588i	$-0.565989\pi$
$97$	−4.05436	−0.411658	−0.205829	+	0.978588i	$0.565989\pi$
$98$	0	0
$99$	−1.98152	−0.199151
$100$	0.367624	0.0367624
$101$	3.59474	0.357690	0.178845	−	0.983877i	$-0.442764\pi$
$101$	3.59474	0.357690	0.178845	+	0.983877i	$0.442764\pi$
$102$	11.4619	1.13490
$103$	−13.7197	−1.35184	−0.675922	−	0.736973i	$-0.736254\pi$
$103$	−13.7197	−1.35184	−0.675922	+	0.736973i	$0.736254\pi$
$104$	−0.831461	−0.0815314
$105$	0	0
$106$	1.67282	0.162479
$107$	3.04603	0.294471	0.147235	−	0.989102i	$-0.452963\pi$
$107$	3.04603	0.294471	0.147235	+	0.989102i	$0.452963\pi$
$108$	11.8005	1.13551
$109$	3.30774	0.316824	0.158412	−	0.987373i	$-0.449363\pi$
$109$	3.30774	0.316824	0.158412	+	0.987373i	$0.449363\pi$
$110$	8.77562	0.836723
$111$	5.21714	0.495189
$112$	0	0
$113$	−2.83044	−0.266265	−0.133133	−	0.991098i	$-0.542504\pi$
$113$	−2.83044	−0.266265	−0.133133	+	0.991098i	$0.542504\pi$
$114$	1.23967	0.116106
$115$	−15.5293	−1.44811
$116$	−17.7428	−1.64738
$117$	−4.94664	−0.457317
$118$	6.17406	0.568368
$119$	0	0
$120$	−0.556198	−0.0507737
$121$	−7.35910	−0.669009
$122$	−0.117869	−0.0106713
$123$	−15.1209	−1.36340
$124$	−1.70830	−0.153410
$125$	10.9747	0.981611
$126$	0	0
$127$	5.84079	0.518286	0.259143	−	0.965839i	$-0.416560\pi$
$127$	5.84079	0.518286	0.259143	+	0.965839i	$0.416560\pi$
$128$	1.39512	0.123312
$129$	11.3118	0.995951
$130$	21.9073	1.92140
$131$	4.46291	0.389926	0.194963	−	0.980811i	$-0.437541\pi$
$131$	4.46291	0.389926	0.194963	+	0.980811i	$0.437541\pi$
$132$	−5.57556	−0.485291
$133$	0	0
$134$	24.5192	2.11813
$135$	−12.8683	−1.10752
$136$	−0.706672	−0.0605966
$137$	4.35685	0.372231	0.186115	−	0.982528i	$-0.440410\pi$
$137$	4.35685	0.372231	0.186115	+	0.982528i	$0.440410\pi$
$138$	19.3246	1.64502
$139$	−7.83351	−0.664430	−0.332215	−	0.943204i	$-0.607796\pi$
$139$	−7.83351	−0.664430	−0.332215	+	0.943204i	$0.607796\pi$
$140$	0	0
$141$	2.15226	0.181253
$142$	−9.91154	−0.831758
$143$	9.08909	0.760067
$144$	3.96680	0.330567
$145$	19.3482	1.60678
$146$	−21.9105	−1.81333
$147$	0	0
$148$	−7.77181	−0.638838
$149$	6.72339	0.550802	0.275401	−	0.961329i	$-0.411189\pi$
$149$	6.72339	0.550802	0.275401	+	0.961329i	$0.411189\pi$
$150$	0.498864	0.0407320
$151$	7.21250	0.586945	0.293473	−	0.955967i	$-0.405189\pi$
$151$	7.21250	0.586945	0.293473	+	0.955967i	$0.405189\pi$
$152$	−0.0764305	−0.00619933
$153$	−4.20423	−0.339892
$154$	0	0
$155$	1.86288	0.149630
$156$	−13.9187	−1.11439
$157$	−24.4317	−1.94986	−0.974932	−	0.222504i	$-0.928577\pi$
$157$	−24.4317	−1.94986	−0.974932	+	0.222504i	$0.928577\pi$
$158$	−4.97268	−0.395605
$159$	1.15899	0.0919138
$160$	−18.3621	−1.45165
$161$	0	0
$162$	9.71552	0.763324
$163$	22.5452	1.76588	0.882939	−	0.469487i	$-0.155561\pi$
$163$	22.5452	1.76588	0.882939	+	0.469487i	$0.155561\pi$
$164$	22.5251	1.75891
$165$	6.08006	0.473332
$166$	9.33671	0.724669
$167$	10.5173	0.813850	0.406925	−	0.913462i	$-0.366601\pi$
$167$	10.5173	0.813850	0.406925	+	0.913462i	$0.366601\pi$
$168$	0	0
$169$	9.68984	0.745372
$170$	18.6194	1.42804
$171$	−0.454711	−0.0347726
$172$	−16.8509	−1.28487
$173$	−4.68134	−0.355916	−0.177958	−	0.984038i	$-0.556949\pi$
$173$	−4.68134	−0.355916	−0.177958	+	0.984038i	$0.556949\pi$
$174$	−24.0769	−1.82527
$175$	0	0
$176$	−7.28870	−0.549407
$177$	4.27760	0.321524
$178$	−8.22056	−0.616157
$179$	4.97864	0.372121	0.186061	−	0.982538i	$-0.440428\pi$
$179$	4.97864	0.372121	0.186061	+	0.982538i	$0.440428\pi$
$180$	4.92932	0.367410
$181$	22.1654	1.64754	0.823770	−	0.566924i	$-0.191867\pi$
$181$	22.1654	1.64754	0.823770	+	0.566924i	$0.191867\pi$
$182$	0	0
$183$	−0.0816634	−0.00603673
$184$	−1.19144	−0.0878340
$185$	8.47502	0.623096
$186$	−2.31816	−0.169976
$187$	7.72496	0.564905
$188$	−3.20616	−0.233833
$189$	0	0
$190$	2.01379	0.146096
$191$	13.0651	0.945356	0.472678	−	0.881235i	$-0.343287\pi$
$191$	13.0651	0.945356	0.472678	+	0.881235i	$0.343287\pi$
$192$	12.1501	0.876856
$193$	−10.7528	−0.774005	−0.387003	−	0.922079i	$-0.626490\pi$
$193$	−10.7528	−0.774005	−0.387003	+	0.922079i	$0.626490\pi$
$194$	8.19577	0.588422
$195$	15.1781	1.08693
$196$	0	0
$197$	−5.67753	−0.404507	−0.202254	−	0.979333i	$-0.564827\pi$
$197$	−5.67753	−0.404507	−0.202254	+	0.979333i	$0.564827\pi$
$198$	4.00559	0.284665
$199$	−13.6219	−0.965629	−0.482815	−	0.875723i	$-0.660386\pi$
$199$	−13.6219	−0.965629	−0.482815	+	0.875723i	$0.660386\pi$
$200$	−0.0307569	−0.00217484
$201$	16.9877	1.19822
$202$	−7.26667	−0.511281
$203$	0	0
$204$	−11.8298	−0.828249
$205$	−24.5632	−1.71557
$206$	27.7340	1.93232
$207$	−7.08827	−0.492669
$208$	−18.1954	−1.26162
$209$	0.835498	0.0577926
$210$	0	0
$211$	16.1337	1.11069	0.555344	−	0.831621i	$-0.312587\pi$
$211$	16.1337	1.11069	0.555344	+	0.831621i	$0.312587\pi$
$212$	−1.72651	−0.118577
$213$	−6.86706	−0.470523
$214$	−6.15746	−0.420915
$215$	18.3756	1.25321
$216$	−0.987280	−0.0671759
$217$	0	0
$218$	−6.68651	−0.452868
$219$	−15.1804	−1.02579
$220$	−9.05727	−0.610641
$221$	19.2845	1.29721
$222$	−10.5463	−0.707821
$223$	7.46799	0.500093	0.250047	−	0.968234i	$-0.419554\pi$
$223$	7.46799	0.500093	0.250047	+	0.968234i	$0.419554\pi$
$224$	0	0
$225$	−0.182983	−0.0121989
$226$	5.72165	0.380599
$227$	9.62855	0.639069	0.319535	−	0.947575i	$-0.396473\pi$
$227$	9.62855	0.639069	0.319535	+	0.947575i	$0.396473\pi$
$228$	−1.27946	−0.0847340
$229$	13.9322	0.920668	0.460334	−	0.887746i	$-0.347730\pi$
$229$	13.9322	0.920668	0.460334	+	0.887746i	$0.347730\pi$
$230$	31.3920	2.06993
$231$	0	0
$232$	1.48444	0.0974581
$233$	−6.48414	−0.424790	−0.212395	−	0.977184i	$-0.568126\pi$
$233$	−6.48414	−0.424790	−0.212395	+	0.977184i	$0.568126\pi$
$234$	9.99949	0.653687
$235$	3.49626	0.228071
$236$	−6.37221	−0.414795
$237$	−3.44524	−0.223793
$238$	0	0
$239$	−17.1211	−1.10747	−0.553737	−	0.832691i	$-0.686799\pi$
$239$	−17.1211	−1.10747	−0.553737	+	0.832691i	$0.686799\pi$
$240$	−12.1716	−0.785675
$241$	−1.58604	−0.102166	−0.0510828	−	0.998694i	$-0.516267\pi$
$241$	−1.58604	−0.102166	−0.0510828	+	0.998694i	$0.516267\pi$
$242$	14.8762	0.956279
$243$	−10.2369	−0.656700
$244$	0.121651	0.00778793
$245$	0	0
$246$	30.5664	1.94884
$247$	2.08572	0.132711
$248$	0.142924	0.00907567
$249$	6.46879	0.409943
$250$	−22.1851	−1.40311
$251$	16.7437	1.05685	0.528425	−	0.848980i	$-0.322783\pi$
$251$	16.7437	1.05685	0.528425	+	0.848980i	$0.322783\pi$
$252$	0	0
$253$	13.0242	0.818823
$254$	−11.8070	−0.740836
$255$	12.9001	0.807838
$256$	14.5303	0.908143
$257$	29.3473	1.83063	0.915317	−	0.402735i	$-0.131940\pi$
$257$	29.3473	1.83063	0.915317	+	0.402735i	$0.131940\pi$
$258$	−22.8666	−1.42361
$259$	0	0
$260$	−22.6104	−1.40224
$261$	8.83142	0.546651
$262$	−9.02165	−0.557359
$263$	10.5462	0.650305	0.325153	−	0.945662i	$-0.394584\pi$
$263$	10.5462	0.650305	0.325153	+	0.945662i	$0.394584\pi$
$264$	0.466474	0.0287095
$265$	1.88273	0.115655
$266$	0	0
$267$	−5.69549	−0.348558
$268$	−25.3061	−1.54581
$269$	10.5327	0.642188	0.321094	−	0.947047i	$-0.395950\pi$
$269$	10.5327	0.642188	0.321094	+	0.947047i	$0.395950\pi$
$270$	26.0128	1.58309
$271$	24.8424	1.50907	0.754534	−	0.656261i	$-0.227863\pi$
$271$	24.8424	1.50907	0.754534	+	0.656261i	$0.227863\pi$
$272$	−15.4646	−0.937676
$273$	0	0
$274$	−8.80725	−0.532066
$275$	0.336218	0.0202747
$276$	−19.9448	−1.20054
$277$	13.1810	0.791969	0.395985	−	0.918257i	$-0.370403\pi$
$277$	13.1810	0.791969	0.395985	+	0.918257i	$0.370403\pi$
$278$	15.8352	0.949733
$279$	0.850301	0.0509062
$280$	0	0
$281$	−20.3525	−1.21413	−0.607066	−	0.794652i	$-0.707653\pi$
$281$	−20.3525	−1.21413	−0.607066	+	0.794652i	$0.707653\pi$
$282$	−4.35074	−0.259083
$283$	−23.0501	−1.37019	−0.685094	−	0.728455i	$-0.740239\pi$
$283$	−23.0501	−1.37019	−0.685094	+	0.728455i	$0.740239\pi$
$284$	10.2296	0.607018
$285$	1.39522	0.0826459
$286$	−18.3733	−1.08644
$287$	0	0
$288$	−8.38131	−0.493874
$289$	−0.609839	−0.0358729
$290$	−39.1119	−2.29673
$291$	5.67831	0.332869
$292$	22.6137	1.32337
$293$	11.8715	0.693539	0.346769	−	0.937950i	$-0.387279\pi$
$293$	11.8715	0.693539	0.346769	+	0.937950i	$0.387279\pi$
$294$	0	0
$295$	6.94878	0.404574
$296$	0.650221	0.0377933
$297$	10.7924	0.626240
$298$	−13.5912	−0.787314
$299$	32.5133	1.88029
$300$	−0.514874	−0.0297263
$301$	0	0
$302$	−14.5799	−0.838977
$303$	−5.03460	−0.289230
$304$	−1.67258	−0.0959290
$305$	−0.132659	−0.00759602
$306$	8.49873	0.485840
$307$	14.7347	0.840957	0.420478	−	0.907303i	$-0.361862\pi$
$307$	14.7347	0.840957	0.420478	+	0.907303i	$0.361862\pi$
$308$	0	0
$309$	19.2151	1.09311
$310$	−3.76575	−0.213880
$311$	27.6333	1.56694	0.783470	−	0.621429i	$-0.213448\pi$
$311$	27.6333	1.56694	0.783470	+	0.621429i	$0.213448\pi$
$312$	1.16450	0.0659268
$313$	25.1189	1.41980	0.709901	−	0.704302i	$-0.248740\pi$
$313$	25.1189	1.41980	0.709901	+	0.704302i	$0.248740\pi$
$314$	49.3880	2.78713
$315$	0	0
$316$	5.13227	0.288713
$317$	−21.9118	−1.23069	−0.615344	−	0.788259i	$-0.710983\pi$
$317$	−21.9118	−1.23069	−0.615344	+	0.788259i	$0.710983\pi$
$318$	−2.34286	−0.131381
$319$	−16.2271	−0.908542
$320$	19.7373	1.10335
$321$	−4.26610	−0.238110
$322$	0	0
$323$	1.77269	0.0986350
$324$	−10.0273	−0.557074
$325$	0.839329	0.0465576
$326$	−45.5746	−2.52414
$327$	−4.63265	−0.256186
$328$	−1.88454	−0.104056
$329$	0	0
$330$	−12.2907	−0.676579
$331$	12.7105	0.698631	0.349316	−	0.937005i	$-0.386414\pi$
$331$	12.7105	0.698631	0.349316	+	0.937005i	$0.386414\pi$
$332$	−9.63636	−0.528864
$333$	3.86838	0.211986
$334$	−21.2604	−1.16331
$335$	27.5958	1.50772
$336$	0	0
$337$	9.94181	0.541565	0.270782	−	0.962641i	$-0.412718\pi$
$337$	9.94181	0.541565	0.270782	+	0.962641i	$0.412718\pi$
$338$	−19.5877	−1.06543
$339$	3.96416	0.215304
$340$	−19.2169	−1.04219
$341$	−1.56237	−0.0846069
$342$	0.919185	0.0497038
$343$	0	0
$344$	1.40981	0.0760121
$345$	21.7495	1.17095
$346$	9.46320	0.508745
$347$	−16.6833	−0.895608	−0.447804	−	0.894132i	$-0.647794\pi$
$347$	−16.6833	−0.895608	−0.447804	+	0.894132i	$0.647794\pi$
$348$	24.8497	1.33208
$349$	0.832391	0.0445569	0.0222784	−	0.999752i	$-0.492908\pi$
$349$	0.832391	0.0445569	0.0222784	+	0.999752i	$0.492908\pi$
$350$	0	0
$351$	26.9420	1.43806
$352$	15.4000	0.820825
$353$	−27.7609	−1.47756	−0.738782	−	0.673945i	$-0.764599\pi$
$353$	−27.7609	−1.47756	−0.738782	+	0.673945i	$0.764599\pi$
$354$	−8.64705	−0.459586
$355$	−11.1552	−0.592059
$356$	8.48439	0.449672
$357$	0	0
$358$	−10.0642	−0.531909
$359$	25.5205	1.34692	0.673461	−	0.739222i	$-0.264807\pi$
$359$	25.5205	1.34692	0.673461	+	0.739222i	$0.264807\pi$
$360$	−0.412407	−0.0217358
$361$	−18.8083	−0.989909
$362$	−44.8067	−2.35499
$363$	10.3068	0.540964
$364$	0	0
$365$	−24.6599	−1.29076
$366$	0.165080	0.00862888
$367$	−21.9400	−1.14526	−0.572630	−	0.819814i	$-0.694077\pi$
$367$	−21.9400	−1.14526	−0.572630	+	0.819814i	$0.694077\pi$
$368$	−26.0730	−1.35915
$369$	−11.2118	−0.583661
$370$	−17.1320	−0.890651
$371$	0	0
$372$	2.39256	0.124048
$373$	29.7677	1.54131	0.770657	−	0.637251i	$-0.219928\pi$
$373$	29.7677	1.54131	0.770657	+	0.637251i	$0.219928\pi$
$374$	−15.6158	−0.807473
$375$	−15.3706	−0.793736
$376$	0.268241	0.0138335
$377$	−40.5090	−2.08632
$378$	0	0
$379$	7.05411	0.362346	0.181173	−	0.983451i	$-0.442011\pi$
$379$	7.05411	0.362346	0.181173	+	0.983451i	$0.442011\pi$
$380$	−2.07842	−0.106621
$381$	−8.18029	−0.419089
$382$	−26.4107	−1.35129
$383$	13.4581	0.687676	0.343838	−	0.939029i	$-0.388273\pi$
$383$	13.4581	0.687676	0.343838	+	0.939029i	$0.388273\pi$
$384$	−1.95393	−0.0997109
$385$	0	0
$386$	21.7365	1.10636
$387$	8.38746	0.426358
$388$	−8.45881	−0.429431
$389$	11.9510	0.605940	0.302970	−	0.953000i	$-0.402022\pi$
$389$	11.9510	0.605940	0.302970	+	0.953000i	$0.402022\pi$
$390$	−30.6822	−1.55365
$391$	27.6336	1.39749
$392$	0	0
$393$	−6.25051	−0.315297
$394$	11.4770	0.578201
$395$	−5.59665	−0.281598
$396$	−4.13415	−0.207749
$397$	5.16368	0.259158	0.129579	−	0.991569i	$-0.458637\pi$
$397$	5.16368	0.259158	0.129579	+	0.991569i	$0.458637\pi$
$398$	27.5362	1.38027
$399$	0	0
$400$	−0.673073	−0.0336537
$401$	17.1854	0.858199	0.429100	−	0.903257i	$-0.358831\pi$
$401$	17.1854	0.858199	0.429100	+	0.903257i	$0.358831\pi$
$402$	−34.3402	−1.71273
$403$	−3.90026	−0.194286
$404$	7.49988	0.373133
$405$	10.9346	0.543346
$406$	0	0
$407$	−7.10787	−0.352324
$408$	0.989726	0.0489987
$409$	−21.7134	−1.07366	−0.536830	−	0.843690i	$-0.680378\pi$
$409$	−21.7134	−1.07366	−0.536830	+	0.843690i	$0.680378\pi$
$410$	49.6538	2.45223
$411$	−6.10197	−0.300988
$412$	−28.6241	−1.41021
$413$	0	0
$414$	14.3287	0.704219
$415$	10.5083	0.515831
$416$	38.4444	1.88489
$417$	10.9712	0.537261
$418$	−1.68894	−0.0826085
$419$	10.4332	0.509693	0.254847	−	0.966982i	$-0.417975\pi$
$419$	10.4332	0.509693	0.254847	+	0.966982i	$0.417975\pi$
$420$	0	0
$421$	−11.1129	−0.541609	−0.270805	−	0.962634i	$-0.587290\pi$
$421$	−11.1129	−0.541609	−0.270805	+	0.962634i	$0.587290\pi$
$422$	−32.6138	−1.58761
$423$	1.59585	0.0775930
$424$	0.144447	0.00701496
$425$	0.713359	0.0346030
$426$	13.8816	0.672564
$427$	0	0
$428$	6.35508	0.307184
$429$	−12.7297	−0.614595
$430$	−37.1457	−1.79133
$431$	−22.8167	−1.09904	−0.549520	−	0.835481i	$-0.685189\pi$
$431$	−22.8167	−1.09904	−0.549520	+	0.835481i	$0.685189\pi$
$432$	−21.6053	−1.03949
$433$	−15.2658	−0.733628	−0.366814	−	0.930294i	$-0.619551\pi$
$433$	−15.2658	−0.733628	−0.366814	+	0.930294i	$0.619551\pi$
$434$	0	0
$435$	−27.0981	−1.29925
$436$	6.90111	0.330503
$437$	2.98873	0.142970
$438$	30.6867	1.46627
$439$	8.56528	0.408798	0.204399	−	0.978888i	$-0.434476\pi$
$439$	8.56528	0.408798	0.204399	+	0.978888i	$0.434476\pi$
$440$	0.757768	0.0361252
$441$	0	0
$442$	−38.9830	−1.85423
$443$	−7.75645	−0.368520	−0.184260	−	0.982878i	$-0.558989\pi$
$443$	−7.75645	−0.368520	−0.184260	+	0.982878i	$0.558989\pi$
$444$	10.8848	0.516568
$445$	−9.25208	−0.438590
$446$	−15.0963	−0.714832
$447$	−9.41642	−0.445381
$448$	0	0
$449$	1.63850	0.0773254	0.0386627	−	0.999252i	$-0.487690\pi$
$449$	1.63850	0.0773254	0.0386627	+	0.999252i	$0.487690\pi$
$450$	0.369895	0.0174370
$451$	20.6008	0.970053
$452$	−5.90529	−0.277761
$453$	−10.1014	−0.474607
$454$	−19.4638	−0.913483
$455$	0	0
$456$	0.107044	0.00501282
$457$	−18.1708	−0.849993	−0.424997	−	0.905195i	$-0.639725\pi$
$457$	−18.1708	−0.849993	−0.424997	+	0.905195i	$0.639725\pi$
$458$	−28.1636	−1.31600
$459$	22.8984	1.06881
$460$	−32.3995	−1.51063
$461$	−31.2115	−1.45367	−0.726833	−	0.686815i	$-0.759008\pi$
$461$	−31.2115	−1.45367	−0.726833	+	0.686815i	$0.759008\pi$
$462$	0	0
$463$	32.2429	1.49846	0.749228	−	0.662312i	$-0.230425\pi$
$463$	32.2429	1.49846	0.749228	+	0.662312i	$0.230425\pi$
$464$	32.4849	1.50807
$465$	−2.60904	−0.120992
$466$	13.1075	0.607194
$467$	20.7707	0.961154	0.480577	−	0.876952i	$-0.340427\pi$
$467$	20.7707	0.961154	0.480577	+	0.876952i	$0.340427\pi$
$468$	−10.3204	−0.477061
$469$	0	0
$470$	−7.06760	−0.326004
$471$	34.2177	1.57667
$472$	0.533125	0.0245391
$473$	−15.4113	−0.708614
$474$	6.96446	0.319888
$475$	0.0771538	0.00354006
$476$	0	0
$477$	0.859363	0.0393475
$478$	34.6099	1.58302
$479$	23.7179	1.08370	0.541850	−	0.840475i	$-0.317724\pi$
$479$	23.7179	1.08370	0.541850	+	0.840475i	$0.317724\pi$
$480$	25.7170	1.17382
$481$	−17.7440	−0.809055
$482$	3.20613	0.146035
$483$	0	0
$484$	−15.3536	−0.697893
$485$	9.22418	0.418848
$486$	20.6937	0.938685
$487$	−23.6896	−1.07348	−0.536740	−	0.843748i	$-0.680344\pi$
$487$	−23.6896	−1.07348	−0.536740	+	0.843748i	$0.680344\pi$
$488$	−0.0101779	−0.000460730 0
$489$	−31.5756	−1.42790
$490$	0	0
$491$	2.93971	0.132667	0.0663337	−	0.997797i	$-0.478870\pi$
$491$	2.93971	0.132667	0.0663337	+	0.997797i	$0.478870\pi$
$492$	−31.5474	−1.42227
$493$	−34.4293	−1.55062
$494$	−4.21623	−0.189697
$495$	4.50822	0.202629
$496$	3.12769	0.140438
$497$	0	0
$498$	−13.0765	−0.585971
$499$	−2.03583	−0.0911363	−0.0455682	−	0.998961i	$-0.514510\pi$
$499$	−2.03583	−0.0911363	−0.0455682	+	0.998961i	$0.514510\pi$
$500$	22.8972	1.02399
$501$	−14.7299	−0.658084
$502$	−33.8468	−1.51066
$503$	8.01457	0.357352	0.178676	−	0.983908i	$-0.442819\pi$
$503$	8.01457	0.357352	0.178676	+	0.983908i	$0.442819\pi$
$504$	0	0
$505$	−8.17849	−0.363938
$506$	−26.3280	−1.17042
$507$	−13.5711	−0.602712
$508$	12.1859	0.540663
$509$	−16.9055	−0.749324	−0.374662	−	0.927161i	$-0.622241\pi$
$509$	−16.9055	−0.749324	−0.374662	+	0.927161i	$0.622241\pi$
$510$	−26.0773	−1.15472
$511$	0	0
$512$	−32.1628	−1.42141
$513$	2.47660	0.109344
$514$	−59.3247	−2.61670
$515$	31.2141	1.37546
$516$	23.6004	1.03895
$517$	−2.93226	−0.128961
$518$	0	0
$519$	6.55643	0.287795
$520$	1.89168	0.0829556
$521$	9.50919	0.416605	0.208303	−	0.978064i	$-0.433206\pi$
$521$	9.50919	0.416605	0.208303	+	0.978064i	$0.433206\pi$
$522$	−17.8525	−0.781381
$523$	−21.0898	−0.922194	−0.461097	−	0.887350i	$-0.652544\pi$
$523$	−21.0898	−0.922194	−0.461097	+	0.887350i	$0.652544\pi$
$524$	9.31119	0.406761
$525$	0	0
$526$	−21.3188	−0.929544
$527$	−3.31490	−0.144399
$528$	10.2082	0.444253
$529$	23.5898	1.02564
$530$	−3.80588	−0.165317
$531$	3.17174	0.137642
$532$	0	0
$533$	51.4274	2.22757
$534$	11.5133	0.498228
$535$	−6.93010	−0.299614
$536$	2.11721	0.0914495
$537$	−6.97282	−0.300899
$538$	−21.2915	−0.917941
$539$	0	0
$540$	−26.8477	−1.15534
$541$	−21.9564	−0.943981	−0.471990	−	0.881604i	$-0.656464\pi$
$541$	−21.9564	−0.943981	−0.471990	+	0.881604i	$0.656464\pi$
$542$	−50.2182	−2.15706
$543$	−31.0436	−1.33221
$544$	32.6745	1.40091
$545$	−7.52554	−0.322359
$546$	0	0
$547$	−31.9769	−1.36723	−0.683617	−	0.729841i	$-0.739594\pi$
$547$	−31.9769	−1.36723	−0.683617	+	0.729841i	$0.739594\pi$
$548$	9.08991	0.388302
$549$	−0.0605515	−0.00258427
$550$	−0.679656	−0.0289806
$551$	−3.72372	−0.158636
$552$	1.66866	0.0710231
$553$	0	0
$554$	−26.6450	−1.13204
$555$	−11.8697	−0.503839
$556$	−16.3434	−0.693116
$557$	26.0094	1.10206	0.551028	−	0.834487i	$-0.314236\pi$
$557$	26.0094	1.10206	0.551028	+	0.834487i	$0.314236\pi$
$558$	−1.71886	−0.0727652
$559$	−38.4726	−1.62722
$560$	0	0
$561$	−10.8192	−0.456785
$562$	41.1421	1.73547
$563$	−8.64103	−0.364176	−0.182088	−	0.983282i	$-0.558286\pi$
$563$	−8.64103	−0.364176	−0.182088	+	0.983282i	$0.558286\pi$
$564$	4.49038	0.189079
$565$	6.43961	0.270916
$566$	46.5952	1.95854
$567$	0	0
$568$	−0.855854	−0.0359108
$569$	−8.91670	−0.373808	−0.186904	−	0.982378i	$-0.559845\pi$
$569$	−8.91670	−0.373808	−0.186904	+	0.982378i	$0.559845\pi$
$570$	−2.82041	−0.118134
$571$	27.3305	1.14374	0.571872	−	0.820343i	$-0.306217\pi$
$571$	27.3305	1.14374	0.571872	+	0.820343i	$0.306217\pi$
$572$	18.9630	0.792883
$573$	−18.2982	−0.764420
$574$	0	0
$575$	1.20271	0.0501566
$576$	9.00899	0.375374
$577$	44.6496	1.85879	0.929394	−	0.369089i	$-0.120330\pi$
$577$	44.6496	1.85879	0.929394	+	0.369089i	$0.120330\pi$
$578$	1.23277	0.0512766
$579$	15.0598	0.625865
$580$	40.3672	1.67616
$581$	0	0
$582$	−11.4786	−0.475801
$583$	−1.57902	−0.0653962
$584$	−1.89196	−0.0782898
$585$	11.2542	0.465305
$586$	−23.9979	−0.991342
$587$	2.64680	0.109245	0.0546226	−	0.998507i	$-0.482604\pi$
$587$	2.64680	0.109245	0.0546226	+	0.998507i	$0.482604\pi$
$588$	0	0
$589$	−0.358525	−0.0147728
$590$	−14.0468	−0.578296
$591$	7.95164	0.327087
$592$	14.2292	0.584817
$593$	−28.7786	−1.18179	−0.590897	−	0.806747i	$-0.701226\pi$
$593$	−28.7786	−1.18179	−0.590897	+	0.806747i	$0.701226\pi$
$594$	−21.8166	−0.895145
$595$	0	0
$596$	14.0273	0.574582
$597$	19.0781	0.780813
$598$	−65.7248	−2.68769
$599$	−14.4588	−0.590769	−0.295385	−	0.955378i	$-0.595448\pi$
$599$	−14.4588	−0.590769	−0.295385	+	0.955378i	$0.595448\pi$
$600$	0.0430765	0.00175859
$601$	25.8450	1.05424	0.527121	−	0.849790i	$-0.323272\pi$
$601$	25.8450	1.05424	0.527121	+	0.849790i	$0.323272\pi$
$602$	0	0
$603$	12.5960	0.512948
$604$	15.0478	0.612286
$605$	16.7429	0.680695
$606$	10.1773	0.413424
$607$	−26.0394	−1.05691	−0.528454	−	0.848962i	$-0.677228\pi$
$607$	−26.0394	−1.05691	−0.528454	+	0.848962i	$0.677228\pi$
$608$	3.53393	0.143320
$609$	0	0
$610$	0.268166	0.0108577
$611$	−7.32005	−0.296138
$612$	−8.77149	−0.354566
$613$	−21.3201	−0.861112	−0.430556	−	0.902564i	$-0.641682\pi$
$613$	−21.3201	−0.861112	−0.430556	+	0.902564i	$0.641682\pi$
$614$	−29.7859	−1.20206
$615$	34.4019	1.38722
$616$	0	0
$617$	48.0475	1.93432	0.967159	−	0.254172i	$-0.0818029\pi$
$617$	48.0475	1.93432	0.967159	+	0.254172i	$0.0818029\pi$
$618$	−38.8428	−1.56249
$619$	−37.2547	−1.49739	−0.748696	−	0.662913i	$-0.769320\pi$
$619$	−37.2547	−1.49739	−0.748696	+	0.662913i	$0.769320\pi$
$620$	3.88661	0.156090
$621$	38.6065	1.54922
$622$	−55.8600	−2.23978
$623$	0	0
$624$	25.4835	1.02016
$625$	−25.8500	−1.03400
$626$	−50.7771	−2.02946
$627$	−1.17015	−0.0467314
$628$	−50.9731	−2.03405
$629$	−15.0809	−0.601314
$630$	0	0
$631$	−21.4591	−0.854272	−0.427136	−	0.904187i	$-0.640477\pi$
$631$	−21.4591	−0.854272	−0.427136	+	0.904187i	$0.640477\pi$
$632$	−0.429387	−0.0170801
$633$	−22.5959	−0.898108
$634$	44.2940	1.75914
$635$	−13.2885	−0.527339
$636$	2.41806	0.0958821
$637$	0	0
$638$	32.8026	1.29867
$639$	−5.09176	−0.201427
$640$	−3.17407	−0.125466
$641$	29.6865	1.17254	0.586272	−	0.810114i	$-0.300595\pi$
$641$	29.6865	1.17254	0.586272	+	0.810114i	$0.300595\pi$
$642$	8.62380	0.340354
$643$	−20.9202	−0.825011	−0.412506	−	0.910955i	$-0.635346\pi$
$643$	−20.9202	−0.825011	−0.412506	+	0.910955i	$0.635346\pi$
$644$	0	0
$645$	−25.7359	−1.01335
$646$	−3.58344	−0.140989
$647$	18.5687	0.730010	0.365005	−	0.931006i	$-0.381067\pi$
$647$	18.5687	0.730010	0.365005	+	0.931006i	$0.381067\pi$
$648$	0.838928	0.0329562
$649$	−5.82784	−0.228763
$650$	−1.69668	−0.0665493
$651$	0	0
$652$	47.0372	1.84212
$653$	−17.4082	−0.681236	−0.340618	−	0.940202i	$-0.610636\pi$
$653$	−17.4082	−0.681236	−0.340618	+	0.940202i	$0.610636\pi$
$654$	9.36476	0.366191
$655$	−10.1537	−0.396738
$656$	−41.2406	−1.61017
$657$	−11.2559	−0.439134
$658$	0	0
$659$	20.2798	0.789988	0.394994	−	0.918684i	$-0.370747\pi$
$659$	20.2798	0.789988	0.394994	+	0.918684i	$0.370747\pi$
$660$	12.6851	0.493768
$661$	7.76359	0.301969	0.150984	−	0.988536i	$-0.451756\pi$
$661$	7.76359	0.301969	0.150984	+	0.988536i	$0.451756\pi$
$662$	−25.6939	−0.998621
$663$	−27.0088	−1.04893
$664$	0.806217	0.0312873
$665$	0	0
$666$	−7.81983	−0.303012
$667$	−58.0472	−2.24760
$668$	21.9427	0.848988
$669$	−10.4593	−0.404378
$670$	−55.7842	−2.15513
$671$	0.111259	0.00429510
$672$	0	0
$673$	40.0940	1.54551	0.772756	−	0.634703i	$-0.218878\pi$
$673$	40.0940	1.54551	0.772756	+	0.634703i	$0.218878\pi$
$674$	−20.0971	−0.774111
$675$	0.996623	0.0383600
$676$	20.2164	0.777553
$677$	−30.8143	−1.18429	−0.592144	−	0.805832i	$-0.701718\pi$
$677$	−30.8143	−1.18429	−0.592144	+	0.805832i	$0.701718\pi$
$678$	−8.01344	−0.307754
$679$	0	0
$680$	1.60777	0.0616551
$681$	−13.4852	−0.516755
$682$	3.15828	0.120937
$683$	28.5143	1.09107	0.545535	−	0.838088i	$-0.316326\pi$
$683$	28.5143	1.09107	0.545535	+	0.838088i	$0.316326\pi$
$684$	−0.948686	−0.0362739
$685$	−9.91239	−0.378733
$686$	0	0
$687$	−19.5127	−0.744457
$688$	30.8519	1.17622
$689$	−3.94183	−0.150172
$690$	−43.9659	−1.67375
$691$	14.0399	0.534103	0.267051	−	0.963682i	$-0.413951\pi$
$691$	14.0399	0.534103	0.267051	+	0.963682i	$0.413951\pi$
$692$	−9.76691	−0.371282
$693$	0	0
$694$	33.7249	1.28018
$695$	17.8222	0.676036
$696$	−2.07902	−0.0788052
$697$	43.7090	1.65560
$698$	−1.68266	−0.0636895
$699$	9.08134	0.343488
$700$	0	0
$701$	−29.9152	−1.12988	−0.564941	−	0.825131i	$-0.691101\pi$
$701$	−29.9152	−1.12988	−0.564941	+	0.825131i	$0.691101\pi$
$702$	−54.4625	−2.05556
$703$	−1.63108	−0.0615174
$704$	−16.5533	−0.623878
$705$	−4.89668	−0.184419
$706$	56.1179	2.11202
$707$	0	0
$708$	8.92457	0.335406
$709$	10.0782	0.378495	0.189247	−	0.981929i	$-0.439395\pi$
$709$	10.0782	0.378495	0.189247	+	0.981929i	$0.439395\pi$
$710$	22.5500	0.846287
$711$	−2.55457	−0.0958037
$712$	−0.709838	−0.0266023
$713$	−5.58887	−0.209305
$714$	0	0
$715$	−20.6788	−0.773344
$716$	10.3872	0.388188
$717$	23.9789	0.895510
$718$	−51.5891	−1.92529
$719$	26.8180	1.00014	0.500071	−	0.865984i	$-0.333307\pi$
$719$	26.8180	1.00014	0.500071	+	0.865984i	$0.333307\pi$
$720$	−9.02498	−0.336341
$721$	0	0
$722$	38.0204	1.41497
$723$	2.22132	0.0826117
$724$	46.2447	1.71867
$725$	−1.49849	−0.0556524
$726$	−20.8348	−0.773252
$727$	18.4884	0.685698	0.342849	−	0.939391i	$-0.388608\pi$
$727$	18.4884	0.685698	0.342849	+	0.939391i	$0.388608\pi$
$728$	0	0
$729$	28.7558	1.06503
$730$	49.8493	1.84500
$731$	−32.6985	−1.20940
$732$	−0.170378	−0.00629736
$733$	52.1580	1.92650	0.963250	−	0.268608i	$-0.0865635\pi$
$733$	52.1580	1.92650	0.963250	+	0.268608i	$0.0865635\pi$
$734$	44.3511	1.63703
$735$	0	0
$736$	55.0888	2.03060
$737$	−23.1442	−0.852528
$738$	22.6643	0.834283
$739$	28.9360	1.06443	0.532214	−	0.846610i	$-0.321360\pi$
$739$	28.9360	1.06443	0.532214	+	0.846610i	$0.321360\pi$
$740$	17.6818	0.649998
$741$	−2.92115	−0.107311
$742$	0	0
$743$	−13.0264	−0.477894	−0.238947	−	0.971033i	$-0.576802\pi$
$743$	−13.0264	−0.477894	−0.238947	+	0.971033i	$0.576802\pi$
$744$	−0.200171	−0.00733863
$745$	−15.2966	−0.560423
$746$	−60.1746	−2.20315
$747$	4.79646	0.175493
$748$	16.1170	0.589294
$749$	0	0
$750$	31.0713	1.13456
$751$	−3.56512	−0.130093	−0.0650466	−	0.997882i	$-0.520720\pi$
$751$	−3.56512	−0.130093	−0.0650466	+	0.997882i	$0.520720\pi$
$752$	5.87008	0.214060
$753$	−23.4503	−0.854575
$754$	81.8878	2.98218
$755$	−16.4094	−0.597198
$756$	0	0
$757$	4.69874	0.170779	0.0853893	−	0.996348i	$-0.472787\pi$
$757$	4.69874	0.170779	0.0853893	+	0.996348i	$0.472787\pi$
$758$	−14.2597	−0.517935
$759$	−18.2409	−0.662104
$760$	0.173889	0.00630762
$761$	−6.09978	−0.221117	−0.110558	−	0.993870i	$-0.535264\pi$
$761$	−6.09978	−0.221117	−0.110558	+	0.993870i	$0.535264\pi$
$762$	16.5362	0.599044
$763$	0	0
$764$	27.2583	0.986171
$765$	9.56515	0.345829
$766$	−27.2051	−0.982962
$767$	−14.5485	−0.525316
$768$	−20.3503	−0.734329
$769$	−8.59341	−0.309886	−0.154943	−	0.987923i	$-0.549519\pi$
$769$	−8.59341	−0.309886	−0.154943	+	0.987923i	$0.549519\pi$
$770$	0	0
$771$	−41.1022	−1.48026
$772$	−22.4341	−0.807423
$773$	−45.5089	−1.63684	−0.818421	−	0.574619i	$-0.805150\pi$
$773$	−45.5089	−1.63684	−0.818421	+	0.574619i	$0.805150\pi$
$774$	−16.9550	−0.609435
$775$	−0.144276	−0.00518256
$776$	0.707698	0.0254049
$777$	0	0
$778$	−24.1586	−0.866129
$779$	4.72737	0.169376
$780$	31.6669	1.13386
$781$	9.35574	0.334775
$782$	−55.8605	−1.99757
$783$	−48.1006	−1.71897
$784$	0	0
$785$	55.5853	1.98392
$786$	12.6352	0.450684
$787$	45.9650	1.63847	0.819237	−	0.573455i	$-0.194397\pi$
$787$	45.9650	1.63847	0.819237	+	0.573455i	$0.194397\pi$
$788$	−11.8453	−0.421971
$789$	−14.7704	−0.525841
$790$	11.3135	0.402515
$791$	0	0
$792$	0.345880	0.0122903
$793$	0.277745	0.00986300
$794$	−10.4382	−0.370439
$795$	−2.63685	−0.0935193
$796$	−28.4200	−1.00732
$797$	−23.4197	−0.829568	−0.414784	−	0.909920i	$-0.636143\pi$
$797$	−23.4197	−0.829568	−0.414784	+	0.909920i	$0.636143\pi$
$798$	0	0
$799$	−6.22143	−0.220098
$800$	1.42211	0.0502793
$801$	−4.22307	−0.149215
$802$	−34.7399	−1.22671
$803$	20.6819	0.729847
$804$	35.4423	1.24995
$805$	0	0
$806$	7.88427	0.277712
$807$	−14.7515	−0.519276
$808$	−0.627471	−0.0220743
$809$	−4.99112	−0.175479	−0.0877393	−	0.996143i	$-0.527964\pi$
$809$	−4.99112	−0.175479	−0.0877393	+	0.996143i	$0.527964\pi$
$810$	−22.1041	−0.776657
$811$	0.695321	0.0244160	0.0122080	−	0.999925i	$-0.496114\pi$
$811$	0.695321	0.0244160	0.0122080	+	0.999925i	$0.496114\pi$
$812$	0	0
$813$	−34.7929	−1.22024
$814$	14.3684	0.503611
$815$	−51.2933	−1.79673
$816$	21.6588	0.758210
$817$	−3.53652	−0.123727
$818$	43.8931	1.53469
$819$	0	0
$820$	−51.2474	−1.78964
$821$	12.0283	0.419789	0.209895	−	0.977724i	$-0.432688\pi$
$821$	12.0283	0.419789	0.209895	+	0.977724i	$0.432688\pi$
$822$	12.3350	0.430231
$823$	42.4226	1.47876	0.739379	−	0.673289i	$-0.235119\pi$
$823$	42.4226	1.47876	0.739379	+	0.673289i	$0.235119\pi$
$824$	2.39481	0.0834272
$825$	−0.470889	−0.0163942
$826$	0	0
$827$	−4.56866	−0.158868	−0.0794339	−	0.996840i	$-0.525311\pi$
$827$	−4.56866	−0.158868	−0.0794339	+	0.996840i	$0.525311\pi$
$828$	−14.7886	−0.513939
$829$	1.31532	0.0456830	0.0228415	−	0.999739i	$-0.492729\pi$
$829$	1.31532	0.0456830	0.0228415	+	0.999739i	$0.492729\pi$
$830$	−21.2422	−0.737328
$831$	−18.4606	−0.640391
$832$	−41.3235	−1.43263
$833$	0	0
$834$	−22.1779	−0.767960
$835$	−23.9281	−0.828066
$836$	1.74314	0.0602878
$837$	−4.63119	−0.160077
$838$	−21.0903	−0.728553
$839$	8.28117	0.285898	0.142949	−	0.989730i	$-0.454342\pi$
$839$	8.28117	0.285898	0.142949	+	0.989730i	$0.454342\pi$
$840$	0	0
$841$	43.3223	1.49387
$842$	22.4644	0.774174
$843$	28.5047	0.981753
$844$	33.6605	1.15864
$845$	−22.0456	−0.758392
$846$	−3.22597	−0.110911
$847$	0	0
$848$	3.16102	0.108550
$849$	32.2828	1.10794
$850$	−1.44204	−0.0494614
$851$	−25.4261	−0.871597
$852$	−14.3271	−0.490838
$853$	−6.47317	−0.221637	−0.110819	−	0.993841i	$-0.535347\pi$
$853$	−6.47317	−0.221637	−0.110819	+	0.993841i	$0.535347\pi$
$854$	0	0
$855$	1.03453	0.0353800
$856$	−0.531692	−0.0181728
$857$	35.6767	1.21869	0.609347	−	0.792903i	$-0.291432\pi$
$857$	35.6767	1.21869	0.609347	+	0.792903i	$0.291432\pi$
$858$	25.7327	0.878500
$859$	28.1159	0.959304	0.479652	−	0.877459i	$-0.340763\pi$
$859$	28.1159	0.959304	0.479652	+	0.877459i	$0.340763\pi$
$860$	38.3379	1.30731
$861$	0	0
$862$	46.1232	1.57096
$863$	46.4267	1.58038	0.790191	−	0.612861i	$-0.209981\pi$
$863$	46.4267	1.58038	0.790191	+	0.612861i	$0.209981\pi$
$864$	45.6491	1.55301
$865$	10.6506	0.362133
$866$	30.8594	1.04864
$867$	0.854108	0.0290070
$868$	0	0
$869$	4.69383	0.159227
$870$	54.7781	1.85715
$871$	−57.7768	−1.95769
$872$	−0.577375	−0.0195524
$873$	4.21033	0.142498
$874$	−6.04163	−0.204361
$875$	0	0
$876$	−31.6716	−1.07008
$877$	−31.7326	−1.07153	−0.535767	−	0.844366i	$-0.679977\pi$
$877$	−31.7326	−1.07153	−0.535767	+	0.844366i	$0.679977\pi$
$878$	−17.3145	−0.584335
$879$	−16.6265	−0.560799
$880$	16.5827	0.559004
$881$	39.5493	1.33245	0.666225	−	0.745750i	$-0.267909\pi$
$881$	39.5493	1.33245	0.666225	+	0.745750i	$0.267909\pi$
$882$	0	0
$883$	−8.59085	−0.289105	−0.144553	−	0.989497i	$-0.546174\pi$
$883$	−8.59085	−0.289105	−0.144553	+	0.989497i	$0.546174\pi$
$884$	40.2341	1.35322
$885$	−9.73209	−0.327141
$886$	15.6794	0.526761
$887$	−26.8277	−0.900787	−0.450393	−	0.892830i	$-0.648716\pi$
$887$	−26.8277	−0.900787	−0.450393	+	0.892830i	$0.648716\pi$
$888$	−0.910664	−0.0305599
$889$	0	0
$890$	18.7028	0.626920
$891$	−9.17071	−0.307230
$892$	15.5808	0.521685
$893$	−0.672882	−0.0225171
$894$	19.0350	0.636627
$895$	−11.3271	−0.378622
$896$	0	0
$897$	−45.5364	−1.52042
$898$	−3.31217	−0.110529
$899$	6.96329	0.232239
$900$	−0.381767	−0.0127256
$901$	−3.35022	−0.111612
$902$	−41.6439	−1.38659
$903$	0	0
$904$	0.494060	0.0164322
$905$	−50.4291	−1.67632
$906$	20.4198	0.678402
$907$	28.7924	0.956036	0.478018	−	0.878350i	$-0.341355\pi$
$907$	28.7924	0.956036	0.478018	+	0.878350i	$0.341355\pi$
$908$	20.0885	0.666661
$909$	−3.73303	−0.123817
$910$	0	0
$911$	−49.9036	−1.65338	−0.826690	−	0.562657i	$-0.809779\pi$
$911$	−49.9036	−1.65338	−0.826690	+	0.562657i	$0.809779\pi$
$912$	2.34252	0.0775687
$913$	−8.81314	−0.291672
$914$	36.7317	1.21498
$915$	0.185795	0.00614218
$916$	29.0675	0.960417
$917$	0	0
$918$	−46.2886	−1.52775
$919$	31.1797	1.02852	0.514262	−	0.857633i	$-0.328066\pi$
$919$	31.1797	1.02852	0.514262	+	0.857633i	$0.328066\pi$
$920$	2.71067	0.0893683
$921$	−20.6367	−0.680002
$922$	63.0932	2.07786
$923$	23.3555	0.768756
$924$	0	0
$925$	−0.656374	−0.0215815
$926$	−65.1782	−2.14189
$927$	14.2475	0.467950
$928$	−68.6362	−2.25309
$929$	−4.11331	−0.134953	−0.0674767	−	0.997721i	$-0.521495\pi$
$929$	−4.11331	−0.134953	−0.0674767	+	0.997721i	$0.521495\pi$
$930$	5.27411	0.172945
$931$	0	0
$932$	−13.5282	−0.443130
$933$	−38.7017	−1.26704
$934$	−41.9874	−1.37387
$935$	−17.5753	−0.574773
$936$	0.863448	0.0282227
$937$	−53.0120	−1.73183	−0.865913	−	0.500195i	$-0.833262\pi$
$937$	−53.0120	−1.73183	−0.865913	+	0.500195i	$0.833262\pi$
$938$	0	0
$939$	−35.1801	−1.14806
$940$	7.29443	0.237918
$941$	57.3024	1.86801	0.934003	−	0.357266i	$-0.116291\pi$
$941$	57.3024	1.86801	0.934003	+	0.357266i	$0.116291\pi$
$942$	−69.1702	−2.25369
$943$	73.6927	2.39977
$944$	11.6667	0.379719
$945$	0	0
$946$	31.1536	1.01289
$947$	35.8075	1.16359	0.581793	−	0.813337i	$-0.302351\pi$
$947$	35.8075	1.16359	0.581793	+	0.813337i	$0.302351\pi$
$948$	−7.18798	−0.233455
$949$	51.6299	1.67598
$950$	−0.155964	−0.00506015
$951$	30.6884	0.995140
$952$	0	0
$953$	−46.7461	−1.51426	−0.757128	−	0.653267i	$-0.773398\pi$
$953$	−46.7461	−1.51426	−0.757128	+	0.653267i	$0.773398\pi$
$954$	−1.73718	−0.0562432
$955$	−29.7247	−0.961869
$956$	−35.7207	−1.15529
$957$	22.7268	0.734652
$958$	−47.9451	−1.54904
$959$	0	0
$960$	−27.6430	−0.892172
$961$	−30.3296	−0.978373
$962$	35.8689	1.15646
$963$	−3.16321	−0.101933
$964$	−3.30903	−0.106577
$965$	24.4640	0.787525
$966$	0	0
$967$	43.5139	1.39931	0.699656	−	0.714480i	$-0.253337\pi$
$967$	43.5139	1.39931	0.699656	+	0.714480i	$0.253337\pi$
$968$	1.28455	0.0412870
$969$	−2.48273	−0.0797568
$970$	−18.6464	−0.598700
$971$	17.1391	0.550021	0.275011	−	0.961441i	$-0.411319\pi$
$971$	17.1391	0.550021	0.275011	+	0.961441i	$0.411319\pi$
$972$	−21.3578	−0.685053
$973$	0	0
$974$	47.8879	1.53443
$975$	−1.17552	−0.0376467
$976$	−0.222729	−0.00712937
$977$	33.1795	1.06151	0.530753	−	0.847527i	$-0.321909\pi$
$977$	33.1795	1.06151	0.530753	+	0.847527i	$0.321909\pi$
$978$	63.8293	2.04103
$979$	7.75958	0.247997
$980$	0	0
$981$	−3.43500	−0.109671
$982$	−5.94254	−0.189634
$983$	60.3160	1.92378	0.961890	−	0.273435i	$-0.0881599\pi$
$983$	60.3160	1.92378	0.961890	+	0.273435i	$0.0881599\pi$
$984$	2.63938	0.0841405
$985$	12.9171	0.411573
$986$	69.5978	2.21644
$987$	0	0
$988$	4.35154	0.138441
$989$	−55.1291	−1.75300
$990$	−9.11323	−0.289638
$991$	44.8493	1.42468	0.712342	−	0.701832i	$-0.247634\pi$
$991$	44.8493	1.42468	0.712342	+	0.701832i	$0.247634\pi$
$992$	−6.60839	−0.209817
$993$	−17.8016	−0.564917
$994$	0	0
$995$	30.9915	0.982497
$996$	13.4962	0.427642
$997$	1.15357	0.0365339	0.0182669	−	0.999833i	$-0.494185\pi$
$997$	1.15357	0.0365339	0.0182669	+	0.999833i	$0.494185\pi$
$998$	4.11538	0.130270
$999$	−21.0693	−0.666602

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2401.2.a.i.1.5		24
7.6	odd	2		2401.2.a.h.1.5		24
49.2	even	21		343.2.g.h.263.1		48
49.9	even	21		343.2.g.g.116.1		48
49.11	even	21		343.2.g.g.275.1		48
49.13	odd	14		343.2.e.d.148.7		48
49.15	even	7		343.2.e.c.197.7		48
49.24	odd	42		343.2.g.i.30.1		48
49.25	even	21		343.2.g.h.30.1		48
49.34	odd	14		343.2.e.d.197.7		48
49.36	even	7		343.2.e.c.148.7		48
49.38	odd	42		49.2.g.a.23.1	✓	48
49.40	odd	42		49.2.g.a.32.1	yes	48
49.47	odd	42		343.2.g.i.263.1		48
147.38	even	42		441.2.bb.d.415.4		48
147.89	even	42		441.2.bb.d.424.4		48
196.87	even	42		784.2.bg.c.513.1		48
196.187	even	42		784.2.bg.c.81.1		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.2.g.a.23.1	✓	48	49.38	odd	42
49.2.g.a.32.1	yes	48	49.40	odd	42
343.2.e.c.148.7		48	49.36	even	7
343.2.e.c.197.7		48	49.15	even	7
343.2.e.d.148.7		48	49.13	odd	14
343.2.e.d.197.7		48	49.34	odd	14
343.2.g.g.116.1		48	49.9	even	21
343.2.g.g.275.1		48	49.11	even	21
343.2.g.h.30.1		48	49.25	even	21
343.2.g.h.263.1		48	49.2	even	21
343.2.g.i.30.1		48	49.24	odd	42
343.2.g.i.263.1		48	49.47	odd	42
441.2.bb.d.415.4		48	147.38	even	42
441.2.bb.d.424.4		48	147.89	even	42
784.2.bg.c.81.1		48	196.187	even	42
784.2.bg.c.513.1		48	196.87	even	42
2401.2.a.h.1.5		24	7.6	odd	2
2401.2.a.i.1.5		24	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:	\( N \)	\(=\)	\( 2401 = 7^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2401.a (trivial)

\(f(q)\)	\(=\)	\(q-2.02147 q^{2} -1.40055 q^{3} +2.08635 q^{4} -2.27513 q^{5} +2.83116 q^{6} -0.174553 q^{8} -1.03847 q^{9} +4.59911 q^{10} +1.90811 q^{11} -2.92203 q^{12} +4.76338 q^{13} +3.18642 q^{15} -3.81985 q^{16} +4.04848 q^{17} +2.09924 q^{18} +0.437865 q^{19} -4.74671 q^{20} -3.85720 q^{22} +6.82567 q^{23} +0.244469 q^{24} +0.176204 q^{25} -9.62905 q^{26} +5.65606 q^{27} -8.50425 q^{29} -6.44126 q^{30} -0.818801 q^{31} +8.07082 q^{32} -2.67240 q^{33} -8.18388 q^{34} -2.16661 q^{36} -3.72507 q^{37} -0.885133 q^{38} -6.67134 q^{39} +0.397129 q^{40} +10.7964 q^{41} -8.07673 q^{43} +3.98099 q^{44} +2.36265 q^{45} -13.7979 q^{46} -1.53673 q^{47} +5.34987 q^{48} -0.356192 q^{50} -5.67008 q^{51} +9.93808 q^{52} -0.827527 q^{53} -11.4336 q^{54} -4.34120 q^{55} -0.613251 q^{57} +17.1911 q^{58} -3.05424 q^{59} +6.64798 q^{60} +0.0583083 q^{61} +1.65518 q^{62} -8.67524 q^{64} -10.8373 q^{65} +5.40219 q^{66} -12.1294 q^{67} +8.44654 q^{68} -9.55967 q^{69} +4.90313 q^{71} +0.181268 q^{72} +10.8389 q^{73} +7.53013 q^{74} -0.246782 q^{75} +0.913540 q^{76} +13.4859 q^{78} +2.45993 q^{79} +8.69063 q^{80} -4.80616 q^{81} -21.8246 q^{82} -4.61877 q^{83} -9.21080 q^{85} +16.3269 q^{86} +11.9106 q^{87} -0.333066 q^{88} +4.06662 q^{89} -4.77604 q^{90} +14.2407 q^{92} +1.14677 q^{93} +3.10646 q^{94} -0.996200 q^{95} -11.3035 q^{96} -4.05436 q^{97} -1.98152 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{2} + 7 q^{3} + 23 q^{4} + 14 q^{5} + 14 q^{6} - 3 q^{8} + 15 q^{9} + 14 q^{10} - 4 q^{11} + 14 q^{12} + 14 q^{13} + 15 q^{15} + 17 q^{16} + 28 q^{17} - 2 q^{18} + 21 q^{19} + 42 q^{20} + 4 q^{22}+ \cdots - 8 q^{99}+O(q^{100}) \) 24 * q - q^2 + 7 * q^3 + 23 * q^4 + 14 * q^5 + 14 * q^6 - 3 * q^8 + 15 * q^9 + 14 * q^10 - 4 * q^11 + 14 * q^12 + 14 * q^13 + 15 * q^15 + 17 * q^16 + 28 * q^17 - 2 * q^18 + 21 * q^19 + 42 * q^20 + 4 * q^22 - 8 * q^23 + 21 * q^24 + 4 * q^25 + 28 * q^26 + 28 * q^27 - 8 * q^29 - 11 * q^30 + 35 * q^31 + 18 * q^32 - 7 * q^33 + q^36 - 8 * q^37 + 14 * q^38 - 14 * q^39 + 7 * q^40 + 35 * q^41 - 8 * q^43 - 20 * q^44 + 42 * q^45 - 6 * q^46 + 70 * q^47 + 42 * q^48 + 20 * q^50 + 24 * q^51 - 21 * q^52 + 24 * q^53 + 14 * q^54 + 56 * q^55 - 20 * q^57 + 19 * q^58 + 84 * q^59 + 56 * q^62 - 9 * q^64 + 14 * q^65 - 11 * q^67 + 77 * q^68 + 42 * q^69 - 8 * q^71 + 40 * q^72 - 7 * q^73 + 43 * q^74 + 49 * q^75 + 7 * q^76 - 15 * q^79 + 70 * q^80 - 36 * q^81 - 56 * q^82 + 56 * q^83 + 36 * q^85 - 86 * q^86 - 70 * q^88 + 70 * q^89 - 56 * q^90 - 68 * q^92 - 11 * q^93 + 84 * q^94 + 54 * q^95 + 56 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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