LMFDB - Embedded newform 5225.2.a.bc.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] 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("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [30,0,0,42,0,12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.45151 q^{2} +0.0791235 q^{3} +0.106883 q^{4} -0.114849 q^{6} -2.96150 q^{7} +2.74788 q^{8} -2.99374 q^{9} +1.00000 q^{11} +0.00845698 q^{12} -3.94825 q^{13} +4.29865 q^{14} -4.20234 q^{16} -4.79134 q^{17} +4.34544 q^{18} +1.00000 q^{19} -0.234324 q^{21} -1.45151 q^{22} +5.69554 q^{23} +0.217422 q^{24} +5.73093 q^{26} -0.474246 q^{27} -0.316535 q^{28} +2.19295 q^{29} -8.56978 q^{31} +0.603987 q^{32} +0.0791235 q^{33} +6.95468 q^{34} -0.319981 q^{36} -6.70306 q^{37} -1.45151 q^{38} -0.312399 q^{39} +5.78933 q^{41} +0.340124 q^{42} -11.8680 q^{43} +0.106883 q^{44} -8.26713 q^{46} -3.44076 q^{47} -0.332504 q^{48} +1.77048 q^{49} -0.379107 q^{51} -0.422002 q^{52} +2.73870 q^{53} +0.688373 q^{54} -8.13784 q^{56} +0.0791235 q^{57} -3.18309 q^{58} -1.87690 q^{59} -13.7308 q^{61} +12.4391 q^{62} +8.86596 q^{63} +7.52799 q^{64} -0.114849 q^{66} -10.1502 q^{67} -0.512114 q^{68} +0.450651 q^{69} -5.59507 q^{71} -8.22643 q^{72} +3.73181 q^{73} +9.72957 q^{74} +0.106883 q^{76} -2.96150 q^{77} +0.453451 q^{78} +16.6160 q^{79} +8.94369 q^{81} -8.40327 q^{82} -9.90171 q^{83} -0.0250453 q^{84} +17.2266 q^{86} +0.173514 q^{87} +2.74788 q^{88} -1.30858 q^{89} +11.6927 q^{91} +0.608757 q^{92} -0.678071 q^{93} +4.99431 q^{94} +0.0477895 q^{96} -9.85288 q^{97} -2.56986 q^{98} -2.99374 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9} + 30 q^{11} - 4 q^{14} + 66 q^{16} + 30 q^{19} + 14 q^{21} + 22 q^{24} + 30 q^{29} + 26 q^{31} + 12 q^{34} + 78 q^{36} + 64 q^{39} + 22 q^{41} + 42 q^{44} + 28 q^{46}+ \cdots + 40 q^{99}+O(q^{100}) $ 30 * q + 42 * q^4 + 12 * q^6 + 40 * q^9 + 30 * q^11 - 4 * q^14 + 66 * q^16 + 30 * q^19 + 14 * q^21 + 22 * q^24 + 30 * q^29 + 26 * q^31 + 12 * q^34 + 78 * q^36 + 64 * q^39 + 22 * q^41 + 42 * q^44 + 28 * q^46 + 60 * q^49 + 64 * q^51 + 62 * q^54 - 32 * q^56 - 14 * q^59 + 78 * q^61 + 90 * q^64 + 12 * q^66 - 28 * q^69 + 20 * q^71 + 42 * q^74 + 42 * q^76 + 102 * q^79 + 42 * q^81 + 98 * q^84 - 52 * q^86 - 8 * q^89 + 56 * q^91 + 40 * q^94 - 74 * q^96 + 40 * q^99

Coefficient data

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

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

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

$2$	−1.45151	−1.02637	−0.513187	−	0.858277i	$-0.671535\pi$
$2$	−1.45151	−1.02637	−0.513187	+	0.858277i	$0.671535\pi$
$3$	0.0791235	0.0456820	0.0228410	−	0.999739i	$-0.492729\pi$
$3$	0.0791235	0.0456820	0.0228410	+	0.999739i	$0.492729\pi$
$4$	0.106883	0.0534416
$5$	0	0
$5$	0	0
$6$	−0.114849	−0.0468868
$7$	−2.96150	−1.11934	−0.559671	−	0.828715i	$-0.689072\pi$
$7$	−2.96150	−1.11934	−0.559671	+	0.828715i	$0.689072\pi$
$8$	2.74788	0.971522
$9$	−2.99374	−0.997913
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.00845698	0.00244132
$13$	−3.94825	−1.09505	−0.547524	−	0.836790i	$-0.684429\pi$
$13$	−3.94825	−1.09505	−0.547524	+	0.836790i	$0.684429\pi$
$14$	4.29865	1.14886
$15$	0	0
$16$	−4.20234	−1.05059
$17$	−4.79134	−1.16207	−0.581035	−	0.813879i	$-0.697352\pi$
$17$	−4.79134	−1.16207	−0.581035	+	0.813879i	$0.697352\pi$
$18$	4.34544	1.02423
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.234324	−0.0511337
$22$	−1.45151	−0.309463
$23$	5.69554	1.18760	0.593801	−	0.804612i	$-0.297627\pi$
$23$	5.69554	1.18760	0.593801	+	0.804612i	$0.297627\pi$
$24$	0.217422	0.0443811
$25$	0	0
$26$	5.73093	1.12393
$27$	−0.474246	−0.0912686
$28$	−0.316535	−0.0598194
$29$	2.19295	0.407221	0.203611	−	0.979052i	$-0.434732\pi$
$29$	2.19295	0.407221	0.203611	+	0.979052i	$0.434732\pi$
$30$	0	0
$31$	−8.56978	−1.53918	−0.769589	−	0.638539i	$-0.779539\pi$
$31$	−8.56978	−1.53918	−0.769589	+	0.638539i	$0.779539\pi$
$32$	0.603987	0.106771
$33$	0.0791235	0.0137736
$34$	6.95468	1.19272
$35$	0	0
$36$	−0.319981	−0.0533301
$37$	−6.70306	−1.10198	−0.550988	−	0.834513i	$-0.685749\pi$
$37$	−6.70306	−1.10198	−0.550988	+	0.834513i	$0.685749\pi$
$38$	−1.45151	−0.235466
$39$	−0.312399	−0.0500239
$40$	0	0
$41$	5.78933	0.904141	0.452071	−	0.891982i	$-0.350686\pi$
$41$	5.78933	0.904141	0.452071	+	0.891982i	$0.350686\pi$
$42$	0.340124	0.0524823
$43$	−11.8680	−1.80986	−0.904930	−	0.425560i	$-0.860077\pi$
$43$	−11.8680	−1.80986	−0.904930	+	0.425560i	$0.860077\pi$
$44$	0.106883	0.0161133
$45$	0	0
$46$	−8.26713	−1.21892
$47$	−3.44076	−0.501887	−0.250944	−	0.968002i	$-0.580741\pi$
$47$	−3.44076	−0.501887	−0.250944	+	0.968002i	$0.580741\pi$
$48$	−0.332504	−0.0479928
$49$	1.77048	0.252925
$50$	0	0
$51$	−0.379107	−0.0530857
$52$	−0.422002	−0.0585211
$53$	2.73870	0.376190	0.188095	−	0.982151i	$-0.439769\pi$
$53$	2.73870	0.376190	0.188095	+	0.982151i	$0.439769\pi$
$54$	0.688373	0.0936757
$55$	0	0
$56$	−8.13784	−1.08746
$57$	0.0791235	0.0104802
$58$	−3.18309	−0.417961
$59$	−1.87690	−0.244351	−0.122176	−	0.992508i	$-0.538987\pi$
$59$	−1.87690	−0.244351	−0.122176	+	0.992508i	$0.538987\pi$
$60$	0	0
$61$	−13.7308	−1.75804	−0.879021	−	0.476782i	$-0.841803\pi$
$61$	−13.7308	−1.75804	−0.879021	+	0.476782i	$0.841803\pi$
$62$	12.4391	1.57977
$63$	8.86596	1.11701
$64$	7.52799	0.940999
$65$	0	0
$66$	−0.114849	−0.0141369
$67$	−10.1502	−1.24005	−0.620025	−	0.784582i	$-0.712877\pi$
$67$	−10.1502	−1.24005	−0.620025	+	0.784582i	$0.712877\pi$
$68$	−0.512114	−0.0621029
$69$	0.450651	0.0542520
$70$	0	0
$71$	−5.59507	−0.664013	−0.332006	−	0.943277i	$-0.607726\pi$
$71$	−5.59507	−0.664013	−0.332006	+	0.943277i	$0.607726\pi$
$72$	−8.22643	−0.969495
$73$	3.73181	0.436775	0.218388	−	0.975862i	$-0.429920\pi$
$73$	3.73181	0.436775	0.218388	+	0.975862i	$0.429920\pi$
$74$	9.72957	1.13104
$75$	0	0
$76$	0.106883	0.0122603
$77$	−2.96150	−0.337494
$78$	0.453451	0.0513432
$79$	16.6160	1.86944	0.934722	−	0.355379i	$-0.115648\pi$
$79$	16.6160	1.86944	0.934722	+	0.355379i	$0.115648\pi$
$80$	0	0
$81$	8.94369	0.993744
$82$	−8.40327	−0.927986
$83$	−9.90171	−1.08685	−0.543427	−	0.839456i	$-0.682874\pi$
$83$	−9.90171	−1.08685	−0.543427	+	0.839456i	$0.682874\pi$
$84$	−0.0250453	−0.00273267
$85$	0	0
$86$	17.2266	1.85759
$87$	0.173514	0.0186027
$88$	2.74788	0.292925
$89$	−1.30858	−0.138709	−0.0693546	−	0.997592i	$-0.522094\pi$
$89$	−1.30858	−0.138709	−0.0693546	+	0.997592i	$0.522094\pi$
$90$	0	0
$91$	11.6927	1.22573
$92$	0.608757	0.0634674
$93$	−0.678071	−0.0703127
$94$	4.99431	0.515123
$95$	0	0
$96$	0.0477895	0.00487750
$97$	−9.85288	−1.00041	−0.500204	−	0.865908i	$-0.666742\pi$
$97$	−9.85288	−1.00041	−0.500204	+	0.865908i	$0.666742\pi$
$98$	−2.56986	−0.259595
$99$	−2.99374	−0.300882
$100$	0	0
$101$	18.1171	1.80272	0.901358	−	0.433076i	$-0.142572\pi$
$101$	18.1171	1.80272	0.901358	+	0.433076i	$0.142572\pi$
$102$	0.550279	0.0544857
$103$	−14.2177	−1.40091	−0.700455	−	0.713696i	$-0.747020\pi$
$103$	−14.2177	−1.40091	−0.700455	+	0.713696i	$0.747020\pi$
$104$	−10.8493	−1.06386
$105$	0	0
$106$	−3.97526	−0.386111
$107$	1.93848	0.187400	0.0937001	−	0.995600i	$-0.470131\pi$
$107$	1.93848	0.187400	0.0937001	+	0.995600i	$0.470131\pi$
$108$	−0.0506889	−0.00487754
$109$	−10.8606	−1.04026	−0.520130	−	0.854087i	$-0.674117\pi$
$109$	−10.8606	−1.04026	−0.520130	+	0.854087i	$0.674117\pi$
$110$	0	0
$111$	−0.530370	−0.0503405
$112$	12.4452	1.17596
$113$	16.1744	1.52156	0.760782	−	0.649008i	$-0.224816\pi$
$113$	16.1744	1.52156	0.760782	+	0.649008i	$0.224816\pi$
$114$	−0.114849	−0.0107566
$115$	0	0
$116$	0.234390	0.0217626
$117$	11.8200	1.09276
$118$	2.72434	0.250796
$119$	14.1895	1.30075
$120$	0	0
$121$	1.00000	0.0909091
$122$	19.9303	1.80441
$123$	0.458072	0.0413030
$124$	−0.915966	−0.0822562
$125$	0	0
$126$	−12.8690	−1.14646
$127$	13.0331	1.15650	0.578251	−	0.815859i	$-0.303736\pi$
$127$	13.0331	1.15650	0.578251	+	0.815859i	$0.303736\pi$
$128$	−12.1349	−1.07259
$129$	−0.939042	−0.0826780
$130$	0	0
$131$	13.8516	1.21022	0.605111	−	0.796141i	$-0.293129\pi$
$131$	13.8516	1.21022	0.605111	+	0.796141i	$0.293129\pi$
$132$	0.00845698	0.000736085 0
$133$	−2.96150	−0.256795
$134$	14.7332	1.27275
$135$	0	0
$136$	−13.1660	−1.12898
$137$	−7.87545	−0.672845	−0.336422	−	0.941711i	$-0.609217\pi$
$137$	−7.87545	−0.672845	−0.336422	+	0.941711i	$0.609217\pi$
$138$	−0.654125	−0.0556828
$139$	−18.7282	−1.58850	−0.794251	−	0.607590i	$-0.792136\pi$
$139$	−18.7282	−1.58850	−0.794251	+	0.607590i	$0.792136\pi$
$140$	0	0
$141$	−0.272245	−0.0229272
$142$	8.12131	0.681525
$143$	−3.94825	−0.330169
$144$	12.5807	1.04839
$145$	0	0
$146$	−5.41676	−0.448294
$147$	0.140086	0.0115541
$148$	−0.716445	−0.0588914
$149$	−5.28231	−0.432744	−0.216372	−	0.976311i	$-0.569422\pi$
$149$	−5.28231	−0.432744	−0.216372	+	0.976311i	$0.569422\pi$
$150$	0	0
$151$	15.4741	1.25926	0.629631	−	0.776895i	$-0.283206\pi$
$151$	15.4741	1.25926	0.629631	+	0.776895i	$0.283206\pi$
$152$	2.74788	0.222882
$153$	14.3440	1.15964
$154$	4.29865	0.346395
$155$	0	0
$156$	−0.0333903	−0.00267336
$157$	−19.8551	−1.58461	−0.792303	−	0.610128i	$-0.791118\pi$
$157$	−19.8551	−1.58461	−0.792303	+	0.610128i	$0.791118\pi$
$158$	−24.1183	−1.91875
$159$	0.216696	0.0171851
$160$	0	0
$161$	−16.8673	−1.32933
$162$	−12.9819	−1.01995
$163$	−0.0217138	−0.00170076	−0.000850380	−	1.00000i	$-0.500271\pi$
$163$	−0.0217138	−0.00170076	−0.000850380		1.00000i	$0.500271\pi$
$164$	0.618782	0.0483188
$165$	0	0
$166$	14.3724	1.11552
$167$	1.69054	0.130818	0.0654088	−	0.997859i	$-0.479165\pi$
$167$	1.69054	0.130818	0.0654088	+	0.997859i	$0.479165\pi$
$168$	−0.643895	−0.0496775
$169$	2.58868	0.199129
$170$	0	0
$171$	−2.99374	−0.228937
$172$	−1.26850	−0.0967219
$173$	−24.1483	−1.83596	−0.917982	−	0.396623i	$-0.870182\pi$
$173$	−24.1483	−1.83596	−0.917982	+	0.396623i	$0.870182\pi$
$174$	−0.251858	−0.0190933
$175$	0	0
$176$	−4.20234	−0.316763
$177$	−0.148507	−0.0111625
$178$	1.89942	0.142367
$179$	4.79370	0.358298	0.179149	−	0.983822i	$-0.442666\pi$
$179$	4.79370	0.358298	0.179149	+	0.983822i	$0.442666\pi$
$180$	0	0
$181$	14.6891	1.09183	0.545917	−	0.837839i	$-0.316181\pi$
$181$	14.6891	1.09183	0.545917	+	0.837839i	$0.316181\pi$
$182$	−16.9721	−1.25806
$183$	−1.08643	−0.0803109
$184$	15.6506	1.15378
$185$	0	0
$186$	0.984228	0.0721671
$187$	−4.79134	−0.350377
$188$	−0.367760	−0.0268217
$189$	1.40448	0.102161
$190$	0	0
$191$	25.2511	1.82711	0.913553	−	0.406719i	$-0.133327\pi$
$191$	25.2511	1.82711	0.913553	+	0.406719i	$0.133327\pi$
$192$	0.595641	0.0429867
$193$	3.62179	0.260702	0.130351	−	0.991468i	$-0.458390\pi$
$193$	3.62179	0.260702	0.130351	+	0.991468i	$0.458390\pi$
$194$	14.3016	1.02679
$195$	0	0
$196$	0.189234	0.0135167
$197$	−7.51197	−0.535206	−0.267603	−	0.963529i	$-0.586232\pi$
$197$	−7.51197	−0.535206	−0.267603	+	0.963529i	$0.586232\pi$
$198$	4.34544	0.308817
$199$	3.89016	0.275766	0.137883	−	0.990448i	$-0.455970\pi$
$199$	3.89016	0.275766	0.137883	+	0.990448i	$0.455970\pi$
$200$	0	0
$201$	−0.803123	−0.0566479
$202$	−26.2971	−1.85026
$203$	−6.49443	−0.455819
$204$	−0.0405202	−0.00283698
$205$	0	0
$206$	20.6371	1.43786
$207$	−17.0510	−1.18512
$208$	16.5919	1.15044
$209$	1.00000	0.0691714
$210$	0	0
$211$	3.21646	0.221430	0.110715	−	0.993852i	$-0.464686\pi$
$211$	3.21646	0.221430	0.110715	+	0.993852i	$0.464686\pi$
$212$	0.292722	0.0201042
$213$	−0.442702	−0.0303334
$214$	−2.81373	−0.192343
$215$	0	0
$216$	−1.30317	−0.0886695
$217$	25.3794	1.72287
$218$	15.7643	1.06770
$219$	0.295274	0.0199528
$220$	0	0
$221$	18.9174	1.27252
$222$	0.769838	0.0516681
$223$	−10.4914	−0.702553	−0.351277	−	0.936272i	$-0.614252\pi$
$223$	−10.4914	−0.702553	−0.351277	+	0.936272i	$0.614252\pi$
$224$	−1.78871	−0.119513
$225$	0	0
$226$	−23.4774	−1.56169
$227$	0.798523	0.0529999	0.0264999	−	0.999649i	$-0.491564\pi$
$227$	0.798523	0.0529999	0.0264999	+	0.999649i	$0.491564\pi$
$228$	0.00845698	0.000560077 0
$229$	−16.3354	−1.07947	−0.539735	−	0.841835i	$-0.681476\pi$
$229$	−16.3354	−1.07947	−0.539735	+	0.841835i	$0.681476\pi$
$230$	0	0
$231$	−0.234324	−0.0154174
$232$	6.02597	0.395624
$233$	14.9185	0.977340	0.488670	−	0.872469i	$-0.337482\pi$
$233$	14.9185	0.977340	0.488670	+	0.872469i	$0.337482\pi$
$234$	−17.1569	−1.12158
$235$	0	0
$236$	−0.200609	−0.0130585
$237$	1.31472	0.0853999
$238$	−20.5963	−1.33506
$239$	−23.5350	−1.52235	−0.761177	−	0.648544i	$-0.775378\pi$
$239$	−23.5350	−1.52235	−0.761177	+	0.648544i	$0.775378\pi$
$240$	0	0
$241$	24.7890	1.59680	0.798399	−	0.602129i	$-0.205681\pi$
$241$	24.7890	1.59680	0.798399	+	0.602129i	$0.205681\pi$
$242$	−1.45151	−0.0933066
$243$	2.13039	0.136665
$244$	−1.46759	−0.0939527
$245$	0	0
$246$	−0.664896	−0.0423922
$247$	−3.94825	−0.251221
$248$	−23.5487	−1.49535
$249$	−0.783458	−0.0496497
$250$	0	0
$251$	−5.81152	−0.366820	−0.183410	−	0.983036i	$-0.558714\pi$
$251$	−5.81152	−0.366820	−0.183410	+	0.983036i	$0.558714\pi$
$252$	0.947622	0.0596946
$253$	5.69554	0.358075
$254$	−18.9177	−1.18700
$255$	0	0
$256$	2.55800	0.159875
$257$	26.8322	1.67375	0.836873	−	0.547397i	$-0.184381\pi$
$257$	26.8322	1.67375	0.836873	+	0.547397i	$0.184381\pi$
$258$	1.36303	0.0848585
$259$	19.8511	1.23349
$260$	0	0
$261$	−6.56513	−0.406371
$262$	−20.1058	−1.24214
$263$	24.5162	1.51173	0.755865	−	0.654727i	$-0.227216\pi$
$263$	24.5162	1.51173	0.755865	+	0.654727i	$0.227216\pi$
$264$	0.217422	0.0133814
$265$	0	0
$266$	4.29865	0.263567
$267$	−0.103539	−0.00633651
$268$	−1.08489	−0.0662702
$269$	15.0720	0.918956	0.459478	−	0.888189i	$-0.348037\pi$
$269$	15.0720	0.918956	0.459478	+	0.888189i	$0.348037\pi$
$270$	0	0
$271$	−18.3558	−1.11503	−0.557517	−	0.830165i	$-0.688246\pi$
$271$	−18.3558	−1.11503	−0.557517	+	0.830165i	$0.688246\pi$
$272$	20.1348	1.22085
$273$	0.925171	0.0559939
$274$	11.4313	0.690590
$275$	0	0
$276$	0.0481670	0.00289931
$277$	11.1338	0.668964	0.334482	−	0.942402i	$-0.391439\pi$
$277$	11.1338	0.668964	0.334482	+	0.942402i	$0.391439\pi$
$278$	27.1841	1.63040
$279$	25.6557	1.53597
$280$	0	0
$281$	3.26607	0.194837	0.0974187	−	0.995243i	$-0.468941\pi$
$281$	3.26607	0.194837	0.0974187	+	0.995243i	$0.468941\pi$
$282$	0.395167	0.0235319
$283$	−25.0622	−1.48979	−0.744897	−	0.667180i	$-0.767501\pi$
$283$	−25.0622	−1.48979	−0.744897	+	0.667180i	$0.767501\pi$
$284$	−0.598020	−0.0354859
$285$	0	0
$286$	5.73093	0.338877
$287$	−17.1451	−1.01204
$288$	−1.80818	−0.106548
$289$	5.95691	0.350407
$290$	0	0
$291$	−0.779594	−0.0457006
$292$	0.398868	0.0233420
$293$	21.9171	1.28041	0.640205	−	0.768204i	$-0.278849\pi$
$293$	21.9171	1.28041	0.640205	+	0.768204i	$0.278849\pi$
$294$	−0.203337	−0.0118588
$295$	0	0
$296$	−18.4192	−1.07059
$297$	−0.474246	−0.0275185
$298$	7.66733	0.444157
$299$	−22.4874	−1.30048
$300$	0	0
$301$	35.1472	2.02585
$302$	−22.4608	−1.29247
$303$	1.43349	0.0823516
$304$	−4.20234	−0.241021
$305$	0	0
$306$	−20.8205	−1.19023
$307$	16.0425	0.915593	0.457796	−	0.889057i	$-0.348639\pi$
$307$	16.0425	0.915593	0.457796	+	0.889057i	$0.348639\pi$
$308$	−0.316535	−0.0180362
$309$	−1.12495	−0.0639964
$310$	0	0
$311$	14.9546	0.847999	0.424000	−	0.905662i	$-0.360626\pi$
$311$	14.9546	0.847999	0.424000	+	0.905662i	$0.360626\pi$
$312$	−0.858436	−0.0485994
$313$	−34.3903	−1.94385	−0.971927	−	0.235282i	$-0.924399\pi$
$313$	−34.3903	−1.94385	−0.971927	+	0.235282i	$0.924399\pi$
$314$	28.8198	1.62640
$315$	0	0
$316$	1.77597	0.0999061
$317$	7.59079	0.426341	0.213171	−	0.977015i	$-0.431621\pi$
$317$	7.59079	0.426341	0.213171	+	0.977015i	$0.431621\pi$
$318$	−0.314536	−0.0176383
$319$	2.19295	0.122782
$320$	0	0
$321$	0.153380	0.00856081
$322$	24.4831	1.36439
$323$	−4.79134	−0.266597
$324$	0.955931	0.0531073
$325$	0	0
$326$	0.0315179	0.00174561
$327$	−0.859332	−0.0475212
$328$	15.9084	0.878393
$329$	10.1898	0.561783
$330$	0	0
$331$	−28.0896	−1.54395	−0.771973	−	0.635655i	$-0.780730\pi$
$331$	−28.0896	−1.54395	−0.771973	+	0.635655i	$0.780730\pi$
$332$	−1.05833	−0.0580832
$333$	20.0672	1.09968
$334$	−2.45383	−0.134268
$335$	0	0
$336$	0.984711	0.0537204
$337$	20.8306	1.13472	0.567358	−	0.823472i	$-0.307966\pi$
$337$	20.8306	1.13472	0.567358	+	0.823472i	$0.307966\pi$
$338$	−3.75750	−0.204381
$339$	1.27978	0.0695081
$340$	0	0
$341$	−8.56978	−0.464080
$342$	4.34544	0.234975
$343$	15.4872	0.836232
$344$	−32.6120	−1.75832
$345$	0	0
$346$	35.0516	1.88438
$347$	−14.9351	−0.801760	−0.400880	−	0.916130i	$-0.631296\pi$
$347$	−14.9351	−0.801760	−0.400880	+	0.916130i	$0.631296\pi$
$348$	0.0185458	0.000994157 0
$349$	36.0679	1.93067	0.965335	−	0.261013i	$-0.0840567\pi$
$349$	36.0679	1.93067	0.965335	+	0.261013i	$0.0840567\pi$
$350$	0	0
$351$	1.87244	0.0999435
$352$	0.603987	0.0321926
$353$	28.0400	1.49242	0.746208	−	0.665713i	$-0.231872\pi$
$353$	28.0400	1.49242	0.746208	+	0.665713i	$0.231872\pi$
$354$	0.215559	0.0114568
$355$	0	0
$356$	−0.139865	−0.00741284
$357$	1.12273	0.0594210
$358$	−6.95811	−0.367747
$359$	1.28132	0.0676256	0.0338128	−	0.999428i	$-0.489235\pi$
$359$	1.28132	0.0676256	0.0338128	+	0.999428i	$0.489235\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−21.3214	−1.12063
$363$	0.0791235	0.00415291
$364$	1.24976	0.0655051
$365$	0	0
$366$	1.57696	0.0824289
$367$	2.01037	0.104940	0.0524702	−	0.998622i	$-0.483291\pi$
$367$	2.01037	0.104940	0.0524702	+	0.998622i	$0.483291\pi$
$368$	−23.9346	−1.24768
$369$	−17.3317	−0.902254
$370$	0	0
$371$	−8.11067	−0.421085
$372$	−0.0724745	−0.00375763
$373$	−2.59241	−0.134230	−0.0671150	−	0.997745i	$-0.521379\pi$
$373$	−2.59241	−0.134230	−0.0671150	+	0.997745i	$0.521379\pi$
$374$	6.95468	0.359618
$375$	0	0
$376$	−9.45480	−0.487594
$377$	−8.65833	−0.445926
$378$	−2.03862	−0.104855
$379$	14.3543	0.737330	0.368665	−	0.929562i	$-0.379815\pi$
$379$	14.3543	0.737330	0.368665	+	0.929562i	$0.379815\pi$
$380$	0	0
$381$	1.03123	0.0528313
$382$	−36.6523	−1.87529
$383$	−8.17879	−0.417917	−0.208958	−	0.977925i	$-0.567007\pi$
$383$	−8.17879	−0.417917	−0.208958	+	0.977925i	$0.567007\pi$
$384$	−0.960159	−0.0489979
$385$	0	0
$386$	−5.25706	−0.267577
$387$	35.5298	1.80608
$388$	−1.05311	−0.0534634
$389$	−19.7560	−1.00167	−0.500833	−	0.865544i	$-0.666973\pi$
$389$	−19.7560	−1.00167	−0.500833	+	0.865544i	$0.666973\pi$
$390$	0	0
$391$	−27.2892	−1.38008
$392$	4.86505	0.245722
$393$	1.09599	0.0552854
$394$	10.9037	0.549321
$395$	0	0
$396$	−0.319981	−0.0160796
$397$	−12.6658	−0.635678	−0.317839	−	0.948145i	$-0.602957\pi$
$397$	−12.6658	−0.635678	−0.317839	+	0.948145i	$0.602957\pi$
$398$	−5.64661	−0.283039
$399$	−0.234324	−0.0117309
$400$	0	0
$401$	−36.6271	−1.82907	−0.914534	−	0.404509i	$-0.867442\pi$
$401$	−36.6271	−1.82907	−0.914534	+	0.404509i	$0.867442\pi$
$402$	1.16574	0.0581419
$403$	33.8357	1.68547
$404$	1.93641	0.0963400
$405$	0	0
$406$	9.42673	0.467841
$407$	−6.70306	−0.332258
$408$	−1.04174	−0.0515739
$409$	−8.99494	−0.444771	−0.222386	−	0.974959i	$-0.571384\pi$
$409$	−8.99494	−0.444771	−0.222386	+	0.974959i	$0.571384\pi$
$410$	0	0
$411$	−0.623133	−0.0307369
$412$	−1.51963	−0.0748669
$413$	5.55843	0.273513
$414$	24.7496	1.21638
$415$	0	0
$416$	−2.38469	−0.116919
$417$	−1.48184	−0.0725659
$418$	−1.45151	−0.0709957
$419$	27.2758	1.33251	0.666254	−	0.745725i	$-0.267897\pi$
$419$	27.2758	1.33251	0.666254	+	0.745725i	$0.267897\pi$
$420$	0	0
$421$	33.0585	1.61117	0.805587	−	0.592477i	$-0.201850\pi$
$421$	33.0585	1.61117	0.805587	+	0.592477i	$0.201850\pi$
$422$	−4.66873	−0.227270
$423$	10.3008	0.500840
$424$	7.52563	0.365477
$425$	0	0
$426$	0.642586	0.0311334
$427$	40.6636	1.96785
$428$	0.207191	0.0100150
$429$	−0.312399	−0.0150828
$430$	0	0
$431$	−13.7567	−0.662635	−0.331317	−	0.943519i	$-0.607493\pi$
$431$	−13.7567	−0.662635	−0.331317	+	0.943519i	$0.607493\pi$
$432$	1.99294	0.0958855
$433$	16.8098	0.807828	0.403914	−	0.914797i	$-0.367650\pi$
$433$	16.8098	0.807828	0.403914	+	0.914797i	$0.367650\pi$
$434$	−36.8385	−1.76830
$435$	0	0
$436$	−1.16082	−0.0555932
$437$	5.69554	0.272454
$438$	−0.428593	−0.0204790
$439$	−6.24019	−0.297828	−0.148914	−	0.988850i	$-0.547578\pi$
$439$	−6.24019	−0.297828	−0.148914	+	0.988850i	$0.547578\pi$
$440$	0	0
$441$	−5.30034	−0.252397
$442$	−27.4588	−1.30608
$443$	24.8505	1.18068	0.590341	−	0.807154i	$-0.298993\pi$
$443$	24.8505	1.18068	0.590341	+	0.807154i	$0.298993\pi$
$444$	−0.0566877	−0.00269028
$445$	0	0
$446$	15.2283	0.721081
$447$	−0.417955	−0.0197686
$448$	−22.2941	−1.05330
$449$	−0.426567	−0.0201309	−0.0100655	−	0.999949i	$-0.503204\pi$
$449$	−0.426567	−0.0201309	−0.0100655	+	0.999949i	$0.503204\pi$
$450$	0	0
$451$	5.78933	0.272609
$452$	1.72878	0.0813148
$453$	1.22436	0.0575255
$454$	−1.15907	−0.0543976
$455$	0	0
$456$	0.217422	0.0101817
$457$	17.3831	0.813146	0.406573	−	0.913618i	$-0.366724\pi$
$457$	17.3831	0.813146	0.406573	+	0.913618i	$0.366724\pi$
$458$	23.7109	1.10794
$459$	2.27227	0.106061
$460$	0	0
$461$	−24.8094	−1.15549	−0.577745	−	0.816218i	$-0.696067\pi$
$461$	−24.8094	−1.15549	−0.577745	+	0.816218i	$0.696067\pi$
$462$	0.340124	0.0158240
$463$	−7.55986	−0.351337	−0.175668	−	0.984449i	$-0.556209\pi$
$463$	−7.55986	−0.351337	−0.175668	+	0.984449i	$0.556209\pi$
$464$	−9.21554	−0.427821
$465$	0	0
$466$	−21.6543	−1.00312
$467$	11.6238	0.537884	0.268942	−	0.963156i	$-0.413326\pi$
$467$	11.6238	0.537884	0.268942	+	0.963156i	$0.413326\pi$
$468$	1.26336	0.0583990
$469$	30.0599	1.38804
$470$	0	0
$471$	−1.57100	−0.0723879
$472$	−5.15749	−0.237393
$473$	−11.8680	−0.545694
$474$	−1.90832	−0.0876522
$475$	0	0
$476$	1.51662	0.0695143
$477$	−8.19897	−0.375405
$478$	34.1614	1.56250
$479$	25.9454	1.18548	0.592739	−	0.805395i	$-0.298047\pi$
$479$	25.9454	1.18548	0.592739	+	0.805395i	$0.298047\pi$
$480$	0	0
$481$	26.4654	1.20672
$482$	−35.9815	−1.63891
$483$	−1.33460	−0.0607265
$484$	0.106883	0.00485833
$485$	0	0
$486$	−3.09229	−0.140269
$487$	8.22122	0.372539	0.186270	−	0.982499i	$-0.440360\pi$
$487$	8.22122	0.372539	0.186270	+	0.982499i	$0.440360\pi$
$488$	−37.7305	−1.70798
$489$	−0.00171808	−7.76941e−5 0
$490$	0	0
$491$	8.68356	0.391884	0.195942	−	0.980616i	$-0.437224\pi$
$491$	8.68356	0.391884	0.195942	+	0.980616i	$0.437224\pi$
$492$	0.0489602	0.00220730
$493$	−10.5072	−0.473219
$494$	5.73093	0.257847
$495$	0	0
$496$	36.0132	1.61704
$497$	16.5698	0.743257
$498$	1.13720	0.0509591
$499$	−22.3175	−0.999068	−0.499534	−	0.866294i	$-0.666495\pi$
$499$	−22.3175	−0.999068	−0.499534	+	0.866294i	$0.666495\pi$
$500$	0	0
$501$	0.133761	0.00597601
$502$	8.43549	0.376494
$503$	18.6767	0.832754	0.416377	−	0.909192i	$-0.363300\pi$
$503$	18.6767	0.832754	0.416377	+	0.909192i	$0.363300\pi$
$504$	24.3626	1.08520
$505$	0	0
$506$	−8.26713	−0.367519
$507$	0.204826	0.00909662
$508$	1.39302	0.0618053
$509$	−27.7521	−1.23009	−0.615045	−	0.788492i	$-0.710862\pi$
$509$	−27.7521	−1.23009	−0.615045	+	0.788492i	$0.710862\pi$
$510$	0	0
$511$	−11.0518	−0.488901
$512$	20.5569	0.908495
$513$	−0.474246	−0.0209385
$514$	−38.9472	−1.71789
$515$	0	0
$516$	−0.100368	−0.00441845
$517$	−3.44076	−0.151325
$518$	−28.8141	−1.26602
$519$	−1.91070	−0.0838704
$520$	0	0
$521$	−10.3834	−0.454905	−0.227452	−	0.973789i	$-0.573040\pi$
$521$	−10.3834	−0.454905	−0.227452	+	0.973789i	$0.573040\pi$
$522$	9.52936	0.417089
$523$	−22.1998	−0.970729	−0.485365	−	0.874312i	$-0.661313\pi$
$523$	−22.1998	−0.970729	−0.485365	+	0.874312i	$0.661313\pi$
$524$	1.48051	0.0646763
$525$	0	0
$526$	−35.5855	−1.55160
$527$	41.0607	1.78863
$528$	−0.332504	−0.0144704
$529$	9.43914	0.410398
$530$	0	0
$531$	5.61895	0.243842
$532$	−0.316535	−0.0137235
$533$	−22.8577	−0.990078
$534$	0.150289	0.00650362
$535$	0	0
$536$	−27.8916	−1.20474
$537$	0.379294	0.0163678
$538$	−21.8772	−0.943191
$539$	1.77048	0.0762598
$540$	0	0
$541$	37.8116	1.62565	0.812824	−	0.582510i	$-0.197929\pi$
$541$	37.8116	1.62565	0.812824	+	0.582510i	$0.197929\pi$
$542$	26.6436	1.14444
$543$	1.16226	0.0498772
$544$	−2.89390	−0.124075
$545$	0	0
$546$	−1.34290	−0.0574706
$547$	−11.2780	−0.482212	−0.241106	−	0.970499i	$-0.577510\pi$
$547$	−11.2780	−0.482212	−0.241106	+	0.970499i	$0.577510\pi$
$548$	−0.841753	−0.0359579
$549$	41.1063	1.75437
$550$	0	0
$551$	2.19295	0.0934229
$552$	1.23833	0.0527070
$553$	−49.2082	−2.09255
$554$	−16.1608	−0.686606
$555$	0	0
$556$	−2.00173	−0.0848921
$557$	7.20468	0.305272	0.152636	−	0.988282i	$-0.451224\pi$
$557$	7.20468	0.305272	0.152636	+	0.988282i	$0.451224\pi$
$558$	−37.2395	−1.57647
$559$	46.8580	1.98188
$560$	0	0
$561$	−0.379107	−0.0160059
$562$	−4.74073	−0.199976
$563$	−1.81101	−0.0763248	−0.0381624	−	0.999272i	$-0.512150\pi$
$563$	−1.81101	−0.0763248	−0.0381624	+	0.999272i	$0.512150\pi$
$564$	−0.0290985	−0.00122527
$565$	0	0
$566$	36.3781	1.52908
$567$	−26.4867	−1.11234
$568$	−15.3746	−0.645103
$569$	12.4854	0.523415	0.261707	−	0.965147i	$-0.415714\pi$
$569$	12.4854	0.523415	0.261707	+	0.965147i	$0.415714\pi$
$570$	0	0
$571$	−37.5830	−1.57280	−0.786400	−	0.617718i	$-0.788057\pi$
$571$	−37.5830	−1.57280	−0.786400	+	0.617718i	$0.788057\pi$
$572$	−0.422002	−0.0176448
$573$	1.99796	0.0834659
$574$	24.8863	1.03873
$575$	0	0
$576$	−22.5368	−0.939035
$577$	7.68994	0.320136	0.160068	−	0.987106i	$-0.448829\pi$
$577$	7.68994	0.320136	0.160068	+	0.987106i	$0.448829\pi$
$578$	−8.64652	−0.359648
$579$	0.286568	0.0119094
$580$	0	0
$581$	29.3239	1.21656
$582$	1.13159	0.0469059
$583$	2.73870	0.113426
$584$	10.2546	0.424337
$585$	0	0
$586$	−31.8129	−1.31418
$587$	8.55695	0.353183	0.176592	−	0.984284i	$-0.443493\pi$
$587$	8.55695	0.353183	0.176592	+	0.984284i	$0.443493\pi$
$588$	0.0149729	0.000617471 0
$589$	−8.56978	−0.353112
$590$	0	0
$591$	−0.594374	−0.0244493
$592$	28.1686	1.15772
$593$	17.3812	0.713761	0.356881	−	0.934150i	$-0.383840\pi$
$593$	17.3812	0.713761	0.356881	+	0.934150i	$0.383840\pi$
$594$	0.688373	0.0282443
$595$	0	0
$596$	−0.564591	−0.0231265
$597$	0.307803	0.0125976
$598$	32.6407	1.33478
$599$	−15.2377	−0.622594	−0.311297	−	0.950313i	$-0.600763\pi$
$599$	−15.2377	−0.622594	−0.311297	+	0.950313i	$0.600763\pi$
$600$	0	0
$601$	7.32582	0.298826	0.149413	−	0.988775i	$-0.452262\pi$
$601$	7.32582	0.298826	0.149413	+	0.988775i	$0.452262\pi$
$602$	−51.0166	−2.07928
$603$	30.3872	1.23746
$604$	1.65392	0.0672970
$605$	0	0
$606$	−2.08072	−0.0845235
$607$	46.8213	1.90042	0.950208	−	0.311615i	$-0.100870\pi$
$607$	46.8213	1.90042	0.950208	+	0.311615i	$0.100870\pi$
$608$	0.603987	0.0244949
$609$	−0.513862	−0.0208227
$610$	0	0
$611$	13.5850	0.549590
$612$	1.53313	0.0619733
$613$	−0.581511	−0.0234870	−0.0117435	−	0.999931i	$-0.503738\pi$
$613$	−0.581511	−0.0234870	−0.0117435	+	0.999931i	$0.503738\pi$
$614$	−23.2858	−0.939740
$615$	0	0
$616$	−8.13784	−0.327883
$617$	31.3211	1.26094	0.630470	−	0.776214i	$-0.282862\pi$
$617$	31.3211	1.26094	0.630470	+	0.776214i	$0.282862\pi$
$618$	1.63288	0.0656841
$619$	4.35588	0.175078	0.0875388	−	0.996161i	$-0.472100\pi$
$619$	4.35588	0.175078	0.0875388	+	0.996161i	$0.472100\pi$
$620$	0	0
$621$	−2.70108	−0.108391
$622$	−21.7068	−0.870364
$623$	3.87536	0.155263
$624$	1.31281	0.0525544
$625$	0	0
$626$	49.9179	1.99512
$627$	0.0791235	0.00315989
$628$	−2.12217	−0.0846839
$629$	32.1166	1.28057
$630$	0	0
$631$	9.65769	0.384467	0.192233	−	0.981349i	$-0.438427\pi$
$631$	9.65769	0.384467	0.192233	+	0.981349i	$0.438427\pi$
$632$	45.6587	1.81621
$633$	0.254498	0.0101154
$634$	−11.0181	−0.437585
$635$	0	0
$636$	0.0231612	0.000918400 0
$637$	−6.99028	−0.276965
$638$	−3.18309	−0.126020
$639$	16.7502	0.662627
$640$	0	0
$641$	33.4025	1.31932	0.659661	−	0.751564i	$-0.270700\pi$
$641$	33.4025	1.31932	0.659661	+	0.751564i	$0.270700\pi$
$642$	−0.222632	−0.00878659
$643$	23.3959	0.922646	0.461323	−	0.887232i	$-0.347375\pi$
$643$	23.3959	0.922646	0.461323	+	0.887232i	$0.347375\pi$
$644$	−1.80283	−0.0710416
$645$	0	0
$646$	6.95468	0.273628
$647$	−25.8707	−1.01708	−0.508540	−	0.861038i	$-0.669815\pi$
$647$	−25.8707	−1.01708	−0.508540	+	0.861038i	$0.669815\pi$
$648$	24.5762	0.965444
$649$	−1.87690	−0.0736747
$650$	0	0
$651$	2.00811	0.0787039
$652$	−0.00232085	−9.08914e−5 0
$653$	9.99941	0.391307	0.195654	−	0.980673i	$-0.437317\pi$
$653$	9.99941	0.391307	0.195654	+	0.980673i	$0.437317\pi$
$654$	1.24733	0.0487744
$655$	0	0
$656$	−24.3287	−0.949878
$657$	−11.1721	−0.435864
$658$	−14.7906	−0.576599
$659$	15.8834	0.618729	0.309365	−	0.950944i	$-0.399884\pi$
$659$	15.8834	0.618729	0.309365	+	0.950944i	$0.399884\pi$
$660$	0	0
$661$	20.4289	0.794590	0.397295	−	0.917691i	$-0.369949\pi$
$661$	20.4289	0.794590	0.397295	+	0.917691i	$0.369949\pi$
$662$	40.7724	1.58466
$663$	1.49681	0.0581313
$664$	−27.2087	−1.05590
$665$	0	0
$666$	−29.1278	−1.12868
$667$	12.4900	0.483616
$668$	0.180690	0.00699111
$669$	−0.830113	−0.0320940
$670$	0	0
$671$	−13.7308	−0.530070
$672$	−0.141529	−0.00545959
$673$	8.43312	0.325073	0.162536	−	0.986703i	$-0.448032\pi$
$673$	8.43312	0.325073	0.162536	+	0.986703i	$0.448032\pi$
$674$	−30.2358	−1.16464
$675$	0	0
$676$	0.276687	0.0106418
$677$	−0.570325	−0.0219194	−0.0109597	−	0.999940i	$-0.503489\pi$
$677$	−0.570325	−0.0219194	−0.0109597	+	0.999940i	$0.503489\pi$
$678$	−1.85761	−0.0713412
$679$	29.1793	1.11980
$680$	0	0
$681$	0.0631820	0.00242114
$682$	12.4391	0.476319
$683$	−15.1949	−0.581415	−0.290708	−	0.956812i	$-0.593891\pi$
$683$	−15.1949	−0.581415	−0.290708	+	0.956812i	$0.593891\pi$
$684$	−0.319981	−0.0122348
$685$	0	0
$686$	−22.4799	−0.858286
$687$	−1.29251	−0.0493123
$688$	49.8736	1.90141
$689$	−10.8131	−0.411946
$690$	0	0
$691$	3.49109	0.132807	0.0664036	−	0.997793i	$-0.478848\pi$
$691$	3.49109	0.132807	0.0664036	+	0.997793i	$0.478848\pi$
$692$	−2.58105	−0.0981169
$693$	8.86596	0.336790
$694$	21.6785	0.822905
$695$	0	0
$696$	0.476796	0.0180729
$697$	−27.7386	−1.05068
$698$	−52.3529	−1.98159
$699$	1.18040	0.0446468
$700$	0	0
$701$	2.17137	0.0820116	0.0410058	−	0.999159i	$-0.486944\pi$
$701$	2.17137	0.0820116	0.0410058	+	0.999159i	$0.486944\pi$
$702$	−2.71787	−0.102579
$703$	−6.70306	−0.252811
$704$	7.52799	0.283722
$705$	0	0
$706$	−40.7003	−1.53178
$707$	−53.6537	−2.01785
$708$	−0.0158729	−0.000596540 0
$709$	−3.77213	−0.141665	−0.0708326	−	0.997488i	$-0.522566\pi$
$709$	−3.77213	−0.141665	−0.0708326	+	0.997488i	$0.522566\pi$
$710$	0	0
$711$	−49.7439	−1.86554
$712$	−3.59582	−0.134759
$713$	−48.8095	−1.82793
$714$	−1.62965	−0.0609881
$715$	0	0
$716$	0.512366	0.0191480
$717$	−1.86217	−0.0695442
$718$	−1.85985	−0.0694091
$719$	−13.6203	−0.507952	−0.253976	−	0.967211i	$-0.581738\pi$
$719$	−13.6203	−0.507952	−0.253976	+	0.967211i	$0.581738\pi$
$720$	0	0
$721$	42.1057	1.56810
$722$	−1.45151	−0.0540196
$723$	1.96139	0.0729449
$724$	1.57002	0.0583494
$725$	0	0
$726$	−0.114849	−0.00426243
$727$	1.60527	0.0595361	0.0297681	−	0.999557i	$-0.490523\pi$
$727$	1.60527	0.0595361	0.0297681	+	0.999557i	$0.490523\pi$
$728$	32.1302	1.19083
$729$	−26.6625	−0.987501
$730$	0	0
$731$	56.8638	2.10318
$732$	−0.116121	−0.00429194
$733$	21.6583	0.799968	0.399984	−	0.916522i	$-0.369016\pi$
$733$	21.6583	0.799968	0.399984	+	0.916522i	$0.369016\pi$
$734$	−2.91807	−0.107708
$735$	0	0
$736$	3.44003	0.126801
$737$	−10.1502	−0.373889
$738$	25.1572	0.926050
$739$	−6.77955	−0.249390	−0.124695	−	0.992195i	$-0.539795\pi$
$739$	−6.77955	−0.249390	−0.124695	+	0.992195i	$0.539795\pi$
$740$	0	0
$741$	−0.312399	−0.0114763
$742$	11.7727	0.432190
$743$	26.6272	0.976857	0.488429	−	0.872604i	$-0.337570\pi$
$743$	26.6272	0.976857	0.488429	+	0.872604i	$0.337570\pi$
$744$	−1.86326	−0.0683104
$745$	0	0
$746$	3.76291	0.137770
$747$	29.6431	1.08459
$748$	−0.512114	−0.0187247
$749$	−5.74082	−0.209765
$750$	0	0
$751$	32.6441	1.19120	0.595600	−	0.803281i	$-0.296914\pi$
$751$	32.6441	1.19120	0.595600	+	0.803281i	$0.296914\pi$
$752$	14.4593	0.527275
$753$	−0.459828	−0.0167571
$754$	12.5677	0.457687
$755$	0	0
$756$	0.150115	0.00545964
$757$	−0.751499	−0.0273137	−0.0136569	−	0.999907i	$-0.504347\pi$
$757$	−0.751499	−0.0273137	−0.0136569	+	0.999907i	$0.504347\pi$
$758$	−20.8354	−0.756775
$759$	0.450651	0.0163576
$760$	0	0
$761$	−17.5654	−0.636746	−0.318373	−	0.947966i	$-0.603136\pi$
$761$	−17.5654	−0.636746	−0.318373	+	0.947966i	$0.603136\pi$
$762$	−1.49683	−0.0542246
$763$	32.1638	1.16441
$764$	2.69892	0.0976435
$765$	0	0
$766$	11.8716	0.428938
$767$	7.41047	0.267576
$768$	0.202398	0.00730341
$769$	−35.7457	−1.28902	−0.644512	−	0.764595i	$-0.722939\pi$
$769$	−35.7457	−1.28902	−0.644512	+	0.764595i	$0.722939\pi$
$770$	0	0
$771$	2.12306	0.0764600
$772$	0.387108	0.0139323
$773$	−14.7295	−0.529783	−0.264891	−	0.964278i	$-0.585336\pi$
$773$	−14.7295	−0.529783	−0.264891	+	0.964278i	$0.585336\pi$
$774$	−51.5720	−1.85372
$775$	0	0
$776$	−27.0745	−0.971919
$777$	1.57069	0.0563482
$778$	28.6760	1.02808
$779$	5.78933	0.207424
$780$	0	0
$781$	−5.59507	−0.200207
$782$	39.6106	1.41647
$783$	−1.04000	−0.0371665
$784$	−7.44014	−0.265719
$785$	0	0
$786$	−1.59084	−0.0567434
$787$	−40.2297	−1.43403	−0.717016	−	0.697056i	$-0.754493\pi$
$787$	−40.2297	−1.43403	−0.717016	+	0.697056i	$0.754493\pi$
$788$	−0.802904	−0.0286023
$789$	1.93980	0.0690589
$790$	0	0
$791$	−47.9006	−1.70315
$792$	−8.22643	−0.292314
$793$	54.2125	1.92514
$794$	18.3845	0.652442
$795$	0	0
$796$	0.415793	0.0147374
$797$	4.53756	0.160729	0.0803643	−	0.996766i	$-0.474392\pi$
$797$	4.53756	0.160729	0.0803643	+	0.996766i	$0.474392\pi$
$798$	0.340124	0.0120403
$799$	16.4859	0.583228
$800$	0	0
$801$	3.91755	0.138420
$802$	53.1646	1.87731
$803$	3.73181	0.131693
$804$	−0.0858404	−0.00302736
$805$	0	0
$806$	−49.1128	−1.72992
$807$	1.19255	0.0419797
$808$	49.7835	1.75138
$809$	1.86885	0.0657052	0.0328526	−	0.999460i	$-0.489541\pi$
$809$	1.86885	0.0657052	0.0328526	+	0.999460i	$0.489541\pi$
$810$	0	0
$811$	26.5983	0.933994	0.466997	−	0.884259i	$-0.345336\pi$
$811$	26.5983	0.933994	0.466997	+	0.884259i	$0.345336\pi$
$812$	−0.694145	−0.0243597
$813$	−1.45237	−0.0509370
$814$	9.72957	0.341021
$815$	0	0
$816$	1.59314	0.0557710
$817$	−11.8680	−0.415211
$818$	13.0563	0.456501
$819$	−35.0050	−1.22317
$820$	0	0
$821$	11.4661	0.400171	0.200086	−	0.979778i	$-0.435878\pi$
$821$	11.4661	0.400171	0.200086	+	0.979778i	$0.435878\pi$
$822$	0.904484	0.0315475
$823$	13.8917	0.484233	0.242117	−	0.970247i	$-0.422158\pi$
$823$	13.8917	0.484233	0.242117	+	0.970247i	$0.422158\pi$
$824$	−39.0685	−1.36102
$825$	0	0
$826$	−8.06813	−0.280726
$827$	53.0943	1.84627	0.923134	−	0.384478i	$-0.125618\pi$
$827$	53.0943	1.84627	0.923134	+	0.384478i	$0.125618\pi$
$828$	−1.82246	−0.0633349
$829$	−0.910926	−0.0316378	−0.0158189	−	0.999875i	$-0.505036\pi$
$829$	−0.910926	−0.0316378	−0.0158189	+	0.999875i	$0.505036\pi$
$830$	0	0
$831$	0.880944	0.0305596
$832$	−29.7224	−1.03044
$833$	−8.48294	−0.293917
$834$	2.15090	0.0744797
$835$	0	0
$836$	0.106883	0.00369663
$837$	4.06418	0.140479
$838$	−39.5911	−1.36765
$839$	−45.6602	−1.57636	−0.788182	−	0.615442i	$-0.788977\pi$
$839$	−45.6602	−1.57636	−0.788182	+	0.615442i	$0.788977\pi$
$840$	0	0
$841$	−24.1910	−0.834171
$842$	−47.9848	−1.65367
$843$	0.258423	0.00890056
$844$	0.343786	0.0118336
$845$	0	0
$846$	−14.9517	−0.514048
$847$	−2.96150	−0.101758
$848$	−11.5090	−0.395220
$849$	−1.98301	−0.0680567
$850$	0	0
$851$	−38.1775	−1.30871
$852$	−0.0473174	−0.00162107
$853$	−15.5961	−0.534001	−0.267000	−	0.963696i	$-0.586033\pi$
$853$	−15.5961	−0.534001	−0.267000	+	0.963696i	$0.586033\pi$
$854$	−59.0237	−2.01975
$855$	0	0
$856$	5.32672	0.182063
$857$	−3.21516	−0.109828	−0.0549138	−	0.998491i	$-0.517488\pi$
$857$	−3.21516	−0.109828	−0.0549138	+	0.998491i	$0.517488\pi$
$858$	0.453451	0.0154806
$859$	20.1995	0.689198	0.344599	−	0.938750i	$-0.388015\pi$
$859$	20.1995	0.689198	0.344599	+	0.938750i	$0.388015\pi$
$860$	0	0
$861$	−1.35658	−0.0462321
$862$	19.9679	0.680111
$863$	−3.58013	−0.121869	−0.0609345	−	0.998142i	$-0.519408\pi$
$863$	−3.58013	−0.121869	−0.0609345	+	0.998142i	$0.519408\pi$
$864$	−0.286438	−0.00974482
$865$	0	0
$866$	−24.3996	−0.829133
$867$	0.471332	0.0160073
$868$	2.71263	0.0920728
$869$	16.6160	0.563659
$870$	0	0
$871$	40.0757	1.35791
$872$	−29.8437	−1.01064
$873$	29.4970	0.998321
$874$	−8.26713	−0.279640
$875$	0	0
$876$	0.0315598	0.00106631
$877$	1.12880	0.0381169	0.0190585	−	0.999818i	$-0.493933\pi$
$877$	1.12880	0.0381169	0.0190585	+	0.999818i	$0.493933\pi$
$878$	9.05771	0.305683
$879$	1.73416	0.0584917
$880$	0	0
$881$	−9.42205	−0.317437	−0.158719	−	0.987324i	$-0.550736\pi$
$881$	−9.42205	−0.317437	−0.158719	+	0.987324i	$0.550736\pi$
$882$	7.69350	0.259054
$883$	−33.9568	−1.14274	−0.571368	−	0.820694i	$-0.693587\pi$
$883$	−33.9568	−1.14274	−0.571368	+	0.820694i	$0.693587\pi$
$884$	2.02195	0.0680056
$885$	0	0
$886$	−36.0708	−1.21182
$887$	23.9547	0.804321	0.402161	−	0.915569i	$-0.368259\pi$
$887$	23.9547	0.804321	0.402161	+	0.915569i	$0.368259\pi$
$888$	−1.45739	−0.0489069
$889$	−38.5975	−1.29452
$890$	0	0
$891$	8.94369	0.299625
$892$	−1.12135	−0.0375456
$893$	−3.44076	−0.115141
$894$	0.606666	0.0202900
$895$	0	0
$896$	35.9376	1.20059
$897$	−1.77928	−0.0594085
$898$	0.619167	0.0206619
$899$	−18.7931	−0.626786
$900$	0	0
$901$	−13.1221	−0.437159
$902$	−8.40327	−0.279798
$903$	2.78097	0.0925449
$904$	44.4454	1.47823
$905$	0	0
$906$	−1.77717	−0.0590427
$907$	−20.1340	−0.668538	−0.334269	−	0.942478i	$-0.608489\pi$
$907$	−20.1340	−0.668538	−0.334269	+	0.942478i	$0.608489\pi$
$908$	0.0853488	0.00283240
$909$	−54.2378	−1.79895
$910$	0	0
$911$	−51.7375	−1.71414	−0.857070	−	0.515201i	$-0.827717\pi$
$911$	−51.7375	−1.71414	−0.857070	+	0.515201i	$0.827717\pi$
$912$	−0.332504	−0.0110103
$913$	−9.90171	−0.327699
$914$	−25.2317	−0.834591
$915$	0	0
$916$	−1.74598	−0.0576886
$917$	−41.0216	−1.35465
$918$	−3.29823	−0.108858
$919$	19.5840	0.646017	0.323009	−	0.946396i	$-0.395306\pi$
$919$	19.5840	0.646017	0.323009	+	0.946396i	$0.395306\pi$
$920$	0	0
$921$	1.26934	0.0418261
$922$	36.0111	1.18596
$923$	22.0907	0.727126
$924$	−0.0250453	−0.000823931 0
$925$	0	0
$926$	10.9732	0.360602
$927$	42.5640	1.39799
$928$	1.32451	0.0434793
$929$	−26.0616	−0.855055	−0.427527	−	0.904002i	$-0.640615\pi$
$929$	−26.0616	−0.855055	−0.427527	+	0.904002i	$0.640615\pi$
$930$	0	0
$931$	1.77048	0.0580250
$932$	1.59453	0.0522307
$933$	1.18326	0.0387383
$934$	−16.8720	−0.552069
$935$	0	0
$936$	32.4800	1.06164
$937$	−37.3846	−1.22130	−0.610650	−	0.791900i	$-0.709092\pi$
$937$	−37.3846	−1.22130	−0.610650	+	0.791900i	$0.709092\pi$
$938$	−43.6323	−1.42465
$939$	−2.72108	−0.0887991
$940$	0	0
$941$	−33.0380	−1.07701	−0.538504	−	0.842623i	$-0.681010\pi$
$941$	−33.0380	−1.07701	−0.538504	+	0.842623i	$0.681010\pi$
$942$	2.28033	0.0742970
$943$	32.9733	1.07376
$944$	7.88737	0.256712
$945$	0	0
$946$	17.2266	0.560085
$947$	−47.6863	−1.54960	−0.774799	−	0.632208i	$-0.782149\pi$
$947$	−47.6863	−1.54960	−0.774799	+	0.632208i	$0.782149\pi$
$948$	0.140521	0.00456391
$949$	−14.7341	−0.478290
$950$	0	0
$951$	0.600610	0.0194761
$952$	38.9911	1.26371
$953$	−22.0635	−0.714708	−0.357354	−	0.933969i	$-0.616321\pi$
$953$	−22.0635	−0.714708	−0.357354	+	0.933969i	$0.616321\pi$
$954$	11.9009	0.385306
$955$	0	0
$956$	−2.51550	−0.0813571
$957$	0.173514	0.00560892
$958$	−37.6601	−1.21674
$959$	23.3231	0.753143
$960$	0	0
$961$	42.4412	1.36907
$962$	−38.4148	−1.23854
$963$	−5.80331	−0.187009
$964$	2.64953	0.0853355
$965$	0	0
$966$	1.93719	0.0623280
$967$	53.0016	1.70442	0.852209	−	0.523201i	$-0.175262\pi$
$967$	53.0016	1.70442	0.852209	+	0.523201i	$0.175262\pi$
$968$	2.74788	0.0883202
$969$	−0.379107	−0.0121787
$970$	0	0
$971$	−10.0046	−0.321062	−0.160531	−	0.987031i	$-0.551321\pi$
$971$	−10.0046	−0.321062	−0.160531	+	0.987031i	$0.551321\pi$
$972$	0.227703	0.00730359
$973$	55.4634	1.77808
$974$	−11.9332	−0.382364
$975$	0	0
$976$	57.7013	1.84697
$977$	−0.508472	−0.0162675	−0.00813373	−	0.999967i	$-0.502589\pi$
$977$	−0.508472	−0.0162675	−0.00813373	+	0.999967i	$0.502589\pi$
$978$	0.00249381	7.97431e−5 0
$979$	−1.30858	−0.0418224
$980$	0	0
$981$	32.5139	1.03809
$982$	−12.6043	−0.402219
$983$	−22.0873	−0.704476	−0.352238	−	0.935910i	$-0.614579\pi$
$983$	−22.0873	−0.704476	−0.352238	+	0.935910i	$0.614579\pi$
$984$	1.25873	0.0401267
$985$	0	0
$986$	15.2513	0.485700
$987$	0.806254	0.0256634
$988$	−0.422002	−0.0134257
$989$	−67.5949	−2.14939
$990$	0	0
$991$	−48.5368	−1.54182	−0.770911	−	0.636943i	$-0.780199\pi$
$991$	−48.5368	−1.54182	−0.770911	+	0.636943i	$0.780199\pi$
$992$	−5.17603	−0.164339
$993$	−2.22255	−0.0705305
$994$	−24.0512	−0.762859
$995$	0	0
$996$	−0.0837386	−0.00265336
$997$	−20.4836	−0.648723	−0.324362	−	0.945933i	$-0.605149\pi$
$997$	−20.4836	−0.648723	−0.324362	+	0.945933i	$0.605149\pi$
$998$	32.3941	1.02542
$999$	3.17890	0.100576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.bc.1.10		30
5.2	odd	4		1045.2.b.e.419.10	✓	30
5.3	odd	4		1045.2.b.e.419.21	yes	30
5.4	even	2	inner	5225.2.a.bc.1.21		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.e.419.10	✓	30	5.2	odd	4
1045.2.b.e.419.21	yes	30	5.3	odd	4
5225.2.a.bc.1.10		30	1.1	even	1	trivial
5225.2.a.bc.1.21		30	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q-1.45151 q^{2} +0.0791235 q^{3} +0.106883 q^{4} -0.114849 q^{6} -2.96150 q^{7} +2.74788 q^{8} -2.99374 q^{9} +1.00000 q^{11} +0.00845698 q^{12} -3.94825 q^{13} +4.29865 q^{14} -4.20234 q^{16} -4.79134 q^{17} +4.34544 q^{18} +1.00000 q^{19} -0.234324 q^{21} -1.45151 q^{22} +5.69554 q^{23} +0.217422 q^{24} +5.73093 q^{26} -0.474246 q^{27} -0.316535 q^{28} +2.19295 q^{29} -8.56978 q^{31} +0.603987 q^{32} +0.0791235 q^{33} +6.95468 q^{34} -0.319981 q^{36} -6.70306 q^{37} -1.45151 q^{38} -0.312399 q^{39} +5.78933 q^{41} +0.340124 q^{42} -11.8680 q^{43} +0.106883 q^{44} -8.26713 q^{46} -3.44076 q^{47} -0.332504 q^{48} +1.77048 q^{49} -0.379107 q^{51} -0.422002 q^{52} +2.73870 q^{53} +0.688373 q^{54} -8.13784 q^{56} +0.0791235 q^{57} -3.18309 q^{58} -1.87690 q^{59} -13.7308 q^{61} +12.4391 q^{62} +8.86596 q^{63} +7.52799 q^{64} -0.114849 q^{66} -10.1502 q^{67} -0.512114 q^{68} +0.450651 q^{69} -5.59507 q^{71} -8.22643 q^{72} +3.73181 q^{73} +9.72957 q^{74} +0.106883 q^{76} -2.96150 q^{77} +0.453451 q^{78} +16.6160 q^{79} +8.94369 q^{81} -8.40327 q^{82} -9.90171 q^{83} -0.0250453 q^{84} +17.2266 q^{86} +0.173514 q^{87} +2.74788 q^{88} -1.30858 q^{89} +11.6927 q^{91} +0.608757 q^{92} -0.678071 q^{93} +4.99431 q^{94} +0.0477895 q^{96} -9.85288 q^{97} -2.56986 q^{98} -2.99374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9} + 30 q^{11} - 4 q^{14} + 66 q^{16} + 30 q^{19} + 14 q^{21} + 22 q^{24} + 30 q^{29} + 26 q^{31} + 12 q^{34} + 78 q^{36} + 64 q^{39} + 22 q^{41} + 42 q^{44} + 28 q^{46}+ \cdots + 40 q^{99}+O(q^{100}) \) 30 * q + 42 * q^4 + 12 * q^6 + 40 * q^9 + 30 * q^11 - 4 * q^14 + 66 * q^16 + 30 * q^19 + 14 * q^21 + 22 * q^24 + 30 * q^29 + 26 * q^31 + 12 * q^34 + 78 * q^36 + 64 * q^39 + 22 * q^41 + 42 * q^44 + 28 * q^46 + 60 * q^49 + 64 * q^51 + 62 * q^54 - 32 * q^56 - 14 * q^59 + 78 * q^61 + 90 * q^64 + 12 * q^66 - 28 * q^69 + 20 * q^71 + 42 * q^74 + 42 * q^76 + 102 * q^79 + 42 * q^81 + 98 * q^84 - 52 * q^86 - 8 * q^89 + 56 * q^91 + 40 * q^94 - 74 * q^96 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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