LMFDB - Embedded newform 725.2.b.f.349.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [725,2,Mod(349,725)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("725.349"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	725.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-8,0,-2,0,0,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.59416223908864.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 14x^{8} + 71x^{6} + 159x^{4} + 151x^{2} + 49 $ x^10 + 14x^8 + 71x^6 + 159x^4 + 151x^2 + 49
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.5
Root			$-0.838718i$ of defining polynomial
Character	$\chi$	$=$	725.349
Dual form			725.2.b.f.349.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.838718i q^{2} -1.90376i q^{3} +1.29655 q^{4} -1.59672 q^{6} -2.46832i q^{7} -2.76488i q^{8} -0.624302 q^{9} -1.49072 q^{11} -2.46832i q^{12} +0.929772i q^{13} -2.07023 q^{14} +0.274153 q^{16} -2.80583i q^{17} +0.523614i q^{18} +0.0373385 q^{19} -4.69910 q^{21} +1.25030i q^{22} +6.79703i q^{23} -5.26366 q^{24} +0.779816 q^{26} -4.52276i q^{27} -3.20031i q^{28} -1.00000 q^{29} +2.15767 q^{31} -5.75969i q^{32} +2.83798i q^{33} -2.35330 q^{34} -0.809441 q^{36} -7.43182i q^{37} -0.0313164i q^{38} +1.77006 q^{39} -1.97399 q^{41} +3.94122i q^{42} +9.61550i q^{43} -1.93280 q^{44} +5.70079 q^{46} +5.17430i q^{47} -0.521922i q^{48} +0.907373 q^{49} -5.34162 q^{51} +1.20550i q^{52} -1.33750i q^{53} -3.79332 q^{54} -6.82462 q^{56} -0.0710835i q^{57} +0.838718i q^{58} -4.57758 q^{59} +12.1506 q^{61} -1.80968i q^{62} +1.54098i q^{63} -4.28245 q^{64} +2.38027 q^{66} +0.291367i q^{67} -3.63790i q^{68} +12.9399 q^{69} -10.7486 q^{71} +1.72612i q^{72} +0.0141028i q^{73} -6.23320 q^{74} +0.0484113 q^{76} +3.67959i q^{77} -1.48458i q^{78} -5.37881 q^{79} -10.4832 q^{81} +1.65562i q^{82} +10.1503i q^{83} -6.09263 q^{84} +8.06469 q^{86} +1.90376i q^{87} +4.12167i q^{88} +6.88136 q^{89} +2.29498 q^{91} +8.81271i q^{92} -4.10769i q^{93} +4.33978 q^{94} -10.9651 q^{96} +9.42386i q^{97} -0.761030i q^{98} +0.930663 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{4} - 2 q^{6} - 22 q^{9} - 4 q^{11} - 22 q^{14} - 20 q^{16} - 4 q^{19} - 2 q^{21} - 26 q^{24} - 32 q^{26} - 10 q^{29} - 2 q^{31} - 6 q^{34} - 26 q^{36} + 12 q^{39} + 10 q^{41} - 14 q^{44} - 40 q^{46}+ \cdots + 52 q^{99}+O(q^{100}) $ 10 * q - 8 * q^4 - 2 * q^6 - 22 * q^9 - 4 * q^11 - 22 * q^14 - 20 * q^16 - 4 * q^19 - 2 * q^21 - 26 * q^24 - 32 * q^26 - 10 * q^29 - 2 * q^31 - 6 * q^34 - 26 * q^36 + 12 * q^39 + 10 * q^41 - 14 * q^44 - 40 * q^46 + 6 * q^49 + 10 * q^51 - 20 * q^54 - 32 * q^56 + 22 * q^59 - 10 * q^61 + 10 * q^64 - 54 * q^66 + 52 * q^69 - 10 * q^71 - 16 * q^74 + 32 * q^76 + 20 * q^79 + 98 * q^81 - 64 * q^84 - 26 * q^86 + 36 * q^89 + 2 * q^91 + 26 * q^94 - 76 * q^96 + 52 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$.

$n$	$176$	$552$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 0.838718i	− 0.593063i	−0.955023	−	0.296532i	$-0.904170\pi$
$2$	− 0.838718i	− 0.593063i	0.955023	−	0.296532i	$-0.0958300\pi$
$3$	− 1.90376i	− 1.09914i	−0.835449	−	0.549568i	$-0.814792\pi$
$3$	− 1.90376i	− 1.09914i	0.835449	−	0.549568i	$-0.185208\pi$
$4$	1.29655	0.648276
$5$	0	0
$5$	0	0
$6$	−1.59672	−0.651857
$7$	− 2.46832i	− 0.932939i	−0.884537	−	0.466470i	$-0.845526\pi$
$7$	− 2.46832i	− 0.932939i	0.884537	−	0.466470i	$-0.154474\pi$
$8$	− 2.76488i	− 0.977532i
$9$	−0.624302	−0.208101
$10$	0	0
$11$	−1.49072	−0.449470	−0.224735	−	0.974420i	$-0.572152\pi$
$11$	−1.49072	−0.449470	−0.224735	+	0.974420i	$0.572152\pi$
$12$	− 2.46832i	− 0.712544i
$13$	0.929772i	0.257872i	0.991653	+	0.128936i	$0.0411562\pi$
$13$	0.929772i	0.257872i	−0.991653	+	0.128936i	$0.958844\pi$
$14$	−2.07023	−0.553292
$15$	0	0
$16$	0.274153	0.0685383
$17$	− 2.80583i	− 0.680513i	−0.940333	−	0.340257i	$-0.889486\pi$
$17$	− 2.80583i	− 0.680513i	0.940333	−	0.340257i	$-0.110514\pi$
$18$	0.523614i	0.123417i
$19$	0.0373385	0.00856603	0.00428302	−	0.999991i	$-0.498637\pi$
$19$	0.0373385	0.00856603	0.00428302	+	0.999991i	$0.498637\pi$
$20$	0	0
$21$	−4.69910	−1.02543
$22$	1.25030i	0.266564i
$23$	6.79703i	1.41728i	0.705571	+	0.708639i	$0.250691\pi$
$23$	6.79703i	1.41728i	−0.705571	+	0.708639i	$0.749309\pi$
$24$	−5.26366	−1.07444
$25$	0	0
$26$	0.779816	0.152935
$27$	− 4.52276i	− 0.870405i
$28$	− 3.20031i	− 0.604802i
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	2.15767	0.387529	0.193764	−	0.981048i	$-0.437930\pi$
$31$	2.15767	0.387529	0.193764	+	0.981048i	$0.437930\pi$
$32$	− 5.75969i	− 1.01818i
$33$	2.83798i	0.494029i
$34$	−2.35330	−0.403587
$35$	0	0
$36$	−0.809441	−0.134907
$37$	− 7.43182i	− 1.22178i	−0.791714	−	0.610892i	$-0.790811\pi$
$37$	− 7.43182i	− 1.22178i	0.791714	−	0.610892i	$-0.209189\pi$
$38$	− 0.0313164i	− 0.00508020i
$39$	1.77006	0.283437
$40$	0	0
$41$	−1.97399	−0.308285	−0.154143	−	0.988049i	$-0.549262\pi$
$41$	−1.97399	−0.308285	−0.154143	+	0.988049i	$0.549262\pi$
$42$	3.94122i	0.608143i
$43$	9.61550i	1.46635i	0.680040	+	0.733175i	$0.261963\pi$
$43$	9.61550i	1.46635i	−0.680040	+	0.733175i	$0.738037\pi$
$44$	−1.93280	−0.291381
$45$	0	0
$46$	5.70079	0.840536
$47$	5.17430i	0.754749i	0.926061	+	0.377375i	$0.123173\pi$
$47$	5.17430i	0.754749i	−0.926061	+	0.377375i	$0.876827\pi$
$48$	− 0.521922i	− 0.0753329i
$49$	0.907373	0.129625
$50$	0	0
$51$	−5.34162	−0.747977
$52$	1.20550i	0.167173i
$53$	− 1.33750i	− 0.183720i	−0.995772	−	0.0918601i	$-0.970719\pi$
$53$	− 1.33750i	− 0.183720i	0.995772	−	0.0918601i	$-0.0292813\pi$
$54$	−3.79332	−0.516205
$55$	0	0
$56$	−6.82462	−0.911978
$57$	− 0.0710835i	− 0.00941524i
$58$	0.838718i	0.110129i
$59$	−4.57758	−0.595951	−0.297975	−	0.954574i	$-0.596311\pi$
$59$	−4.57758	−0.595951	−0.297975	+	0.954574i	$0.596311\pi$
$60$	0	0
$61$	12.1506	1.55572	0.777860	−	0.628437i	$-0.216305\pi$
$61$	12.1506	1.55572	0.777860	+	0.628437i	$0.216305\pi$
$62$	− 1.80968i	− 0.229829i
$63$	1.54098i	0.194145i
$64$	−4.28245	−0.535306
$65$	0	0
$66$	2.38027	0.292990
$67$	0.291367i	0.0355961i	0.999842	+	0.0177981i	$0.00566560\pi$
$67$	0.291367i	0.0355961i	−0.999842	+	0.0177981i	$0.994334\pi$
$68$	− 3.63790i	− 0.441161i
$69$	12.9399	1.55778
$70$	0	0
$71$	−10.7486	−1.27563	−0.637813	−	0.770191i	$-0.720161\pi$
$71$	−10.7486	−1.27563	−0.637813	+	0.770191i	$0.720161\pi$
$72$	1.72612i	0.203425i
$73$	0.0141028i	0.00165061i	1.00000		0.000825303i	$0.000262702\pi$
$73$	0.0141028i	0.00165061i	−1.00000		0.000825303i	$0.999737\pi$
$74$	−6.23320	−0.724595
$75$	0	0
$76$	0.0484113	0.00555315
$77$	3.67959i	0.419328i
$78$	− 1.48458i	− 0.168096i
$79$	−5.37881	−0.605163	−0.302582	−	0.953123i	$-0.597848\pi$
$79$	−5.37881	−0.605163	−0.302582	+	0.953123i	$0.597848\pi$
$80$	0	0
$81$	−10.4832	−1.16479
$82$	1.65562i	0.182833i
$83$	10.1503i	1.11414i	0.830464	+	0.557072i	$0.188075\pi$
$83$	10.1503i	1.11414i	−0.830464	+	0.557072i	$0.811925\pi$
$84$	−6.09263	−0.664760
$85$	0	0
$86$	8.06469	0.869638
$87$	1.90376i	0.204105i
$88$	4.12167i	0.439371i
$89$	6.88136	0.729423	0.364711	−	0.931121i	$-0.381168\pi$
$89$	6.88136	0.729423	0.364711	+	0.931121i	$0.381168\pi$
$90$	0	0
$91$	2.29498	0.240579
$92$	8.81271i	0.918788i
$93$	− 4.10769i	− 0.425947i
$94$	4.33978	0.447614
$95$	0	0
$96$	−10.9651	−1.11912
$97$	9.42386i	0.956848i	0.878129	+	0.478424i	$0.158792\pi$
$97$	9.42386i	0.956848i	−0.878129	+	0.478424i	$0.841208\pi$
$98$	− 0.761030i	− 0.0768756i
$99$	0.930663	0.0935351
$100$	0	0
$101$	11.0204	1.09657	0.548284	−	0.836293i	$-0.315281\pi$
$101$	11.0204	1.09657	0.548284	+	0.836293i	$0.315281\pi$
$102$	4.48012i	0.443598i
$103$	− 4.51423i	− 0.444801i	−0.974955	−	0.222400i	$-0.928611\pi$
$103$	− 4.51423i	− 0.444801i	0.974955	−	0.222400i	$-0.0713892\pi$
$104$	2.57071	0.252078
$105$	0	0
$106$	−1.12179	−0.108958
$107$	0.277920i	0.0268675i	0.999910	+	0.0134338i	$0.00427623\pi$
$107$	0.277920i	0.0268675i	−0.999910	+	0.0134338i	$0.995724\pi$
$108$	− 5.86399i	− 0.564263i
$109$	16.6659	1.59630	0.798150	−	0.602459i	$-0.205812\pi$
$109$	16.6659	1.59630	0.798150	+	0.602459i	$0.205812\pi$
$110$	0	0
$111$	−14.1484	−1.34291
$112$	− 0.676699i	− 0.0639420i
$113$	− 14.9017i	− 1.40183i	−0.713244	−	0.700916i	$-0.752775\pi$
$113$	− 14.9017i	− 1.40183i	0.713244	−	0.700916i	$-0.247225\pi$
$114$	−0.0596190	−0.00558383
$115$	0	0
$116$	−1.29655	−0.120382
$117$	− 0.580459i	− 0.0536635i
$118$	3.83930i	0.353436i
$119$	−6.92570	−0.634877
$120$	0	0
$121$	−8.77774	−0.797977
$122$	− 10.1909i	− 0.922640i
$123$	3.75800i	0.338847i
$124$	2.79753	0.251226
$125$	0	0
$126$	1.29245	0.115140
$127$	− 14.2274i	− 1.26248i	−0.775587	−	0.631240i	$-0.782546\pi$
$127$	− 14.2274i	− 1.26248i	0.775587	−	0.631240i	$-0.217454\pi$
$128$	− 7.92762i	− 0.700709i
$129$	18.3056	1.61172
$130$	0	0
$131$	19.7992	1.72987	0.864933	−	0.501888i	$-0.167361\pi$
$131$	19.7992	1.72987	0.864933	+	0.501888i	$0.167361\pi$
$132$	3.67959i	0.320267i
$133$	− 0.0921635i	− 0.00799159i
$134$	0.244374	0.0211107
$135$	0	0
$136$	−7.75777	−0.665223
$137$	0.954829i	0.0815765i	0.999168	+	0.0407883i	$0.0129869\pi$
$137$	0.954829i	0.0815765i	−0.999168	+	0.0407883i	$0.987013\pi$
$138$	− 10.8529i	− 0.923863i
$139$	22.5382	1.91166	0.955831	−	0.293916i	$-0.0949585\pi$
$139$	22.5382	1.91166	0.955831	+	0.293916i	$0.0949585\pi$
$140$	0	0
$141$	9.85063	0.829572
$142$	9.01506i	0.756527i
$143$	− 1.38603i	− 0.115906i
$144$	−0.171154	−0.0142629
$145$	0	0
$146$	0.0118282	0.000978913 0
$147$	− 1.72742i	− 0.142475i
$148$	− 9.63575i	− 0.792054i
$149$	−8.56867	−0.701972	−0.350986	−	0.936381i	$-0.614154\pi$
$149$	−8.56867	−0.701972	−0.350986	+	0.936381i	$0.614154\pi$
$150$	0	0
$151$	13.2588	1.07898	0.539491	−	0.841991i	$-0.318617\pi$
$151$	13.2588	1.07898	0.539491	+	0.841991i	$0.318617\pi$
$152$	− 0.103236i	− 0.00837357i
$153$	1.75169i	0.141615i
$154$	3.08614	0.248688
$155$	0	0
$156$	2.29498	0.183745
$157$	− 14.1001i	− 1.12531i	−0.826691	−	0.562656i	$-0.809780\pi$
$157$	− 14.1001i	− 1.12531i	0.826691	−	0.562656i	$-0.190220\pi$
$158$	4.51130i	0.358900i
$159$	−2.54629	−0.201934
$160$	0	0
$161$	16.7773	1.32223
$162$	8.79241i	0.690797i
$163$	17.2267i	1.34930i	0.738138	+	0.674650i	$0.235705\pi$
$163$	17.2267i	1.34930i	−0.738138	+	0.674650i	$0.764295\pi$
$164$	−2.55938	−0.199854
$165$	0	0
$166$	8.51326	0.660757
$167$	14.4358i	1.11707i	0.829480	+	0.558537i	$0.188637\pi$
$167$	14.4358i	1.11707i	−0.829480	+	0.558537i	$0.811363\pi$
$168$	12.9924i	1.00239i
$169$	12.1355	0.933502
$170$	0	0
$171$	−0.0233105	−0.00178260
$172$	12.4670i	0.950600i
$173$	− 15.5680i	− 1.18362i	−0.806079	−	0.591808i	$-0.798414\pi$
$173$	− 15.5680i	− 1.18362i	0.806079	−	0.591808i	$-0.201586\pi$
$174$	1.59672	0.121047
$175$	0	0
$176$	−0.408687	−0.0308059
$177$	8.71462i	0.655031i
$178$	− 5.77152i	− 0.432594i
$179$	−20.5247	−1.53409	−0.767044	−	0.641594i	$-0.778273\pi$
$179$	−20.5247	−1.53409	−0.767044	+	0.641594i	$0.778273\pi$
$180$	0	0
$181$	17.5111	1.30159	0.650794	−	0.759255i	$-0.274436\pi$
$181$	17.5111	1.30159	0.650794	+	0.759255i	$0.274436\pi$
$182$	− 1.92484i	− 0.142679i
$183$	− 23.1318i	− 1.70995i
$184$	18.7930	1.38543
$185$	0	0
$186$	−3.44519	−0.252613
$187$	4.18272i	0.305870i
$188$	6.70875i	0.489286i
$189$	−11.1636	−0.812035
$190$	0	0
$191$	15.5726	1.12680	0.563398	−	0.826186i	$-0.309494\pi$
$191$	15.5726	1.12680	0.563398	+	0.826186i	$0.309494\pi$
$192$	8.15276i	0.588375i
$193$	− 8.44449i	− 0.607847i	−0.952696	−	0.303924i	$-0.901703\pi$
$193$	− 8.44449i	− 0.607847i	0.952696	−	0.303924i	$-0.0982968\pi$
$194$	7.90396	0.567471
$195$	0	0
$196$	1.17646	0.0840326
$197$	10.1937i	0.726272i	0.931736	+	0.363136i	$0.118294\pi$
$197$	10.1937i	0.726272i	−0.931736	+	0.363136i	$0.881706\pi$
$198$	− 0.780563i	− 0.0554722i
$199$	−6.54034	−0.463633	−0.231816	−	0.972760i	$-0.574467\pi$
$199$	−6.54034	−0.463633	−0.231816	+	0.972760i	$0.574467\pi$
$200$	0	0
$201$	0.554692	0.0391250
$202$	− 9.24297i	− 0.650333i
$203$	2.46832i	0.173242i
$204$	−6.92570	−0.484896
$205$	0	0
$206$	−3.78617	−0.263795
$207$	− 4.24340i	− 0.294937i
$208$	0.254900i	0.0176741i
$209$	−0.0556614	−0.00385018
$210$	0	0
$211$	15.5988	1.07386	0.536932	−	0.843625i	$-0.319583\pi$
$211$	15.5988	1.07386	0.536932	+	0.843625i	$0.319583\pi$
$212$	− 1.73414i	− 0.119101i
$213$	20.4628i	1.40209i
$214$	0.233096	0.0159341
$215$	0	0
$216$	−12.5049	−0.850849
$217$	− 5.32583i	− 0.361541i
$218$	− 13.9780i	− 0.946707i
$219$	0.0268483	0.00181424
$220$	0	0
$221$	2.60878	0.175486
$222$	11.8665i	0.796429i
$223$	23.7662i	1.59150i	0.605623	+	0.795751i	$0.292924\pi$
$223$	23.7662i	1.59150i	−0.605623	+	0.795751i	$0.707076\pi$
$224$	−14.2168	−0.949899
$225$	0	0
$226$	−12.4983	−0.831375
$227$	9.10250i	0.604154i	0.953284	+	0.302077i	$0.0976800\pi$
$227$	9.10250i	0.604154i	−0.953284	+	0.302077i	$0.902320\pi$
$228$	− 0.0921635i	− 0.00610367i
$229$	−24.3822	−1.61122	−0.805611	−	0.592445i	$-0.798163\pi$
$229$	−24.3822	−1.61122	−0.805611	+	0.592445i	$0.798163\pi$
$230$	0	0
$231$	7.00506	0.460899
$232$	2.76488i	0.181523i
$233$	18.6324i	1.22065i	0.792151	+	0.610325i	$0.208961\pi$
$233$	18.6324i	1.22065i	−0.792151	+	0.610325i	$0.791039\pi$
$234$	−0.486841	−0.0318258
$235$	0	0
$236$	−5.93508	−0.386341
$237$	10.2400i	0.665157i
$238$	5.80870i	0.376522i
$239$	−1.69469	−0.109620	−0.0548101	−	0.998497i	$-0.517455\pi$
$239$	−1.69469	−0.109620	−0.0548101	+	0.998497i	$0.517455\pi$
$240$	0	0
$241$	8.57531	0.552384	0.276192	−	0.961102i	$-0.410927\pi$
$241$	8.57531	0.552384	0.276192	+	0.961102i	$0.410927\pi$
$242$	7.36205i	0.473250i
$243$	6.38914i	0.409863i
$244$	15.7538	1.00854
$245$	0	0
$246$	3.15190	0.200958
$247$	0.0347163i	0.00220894i
$248$	− 5.96569i	− 0.378822i
$249$	19.3238	1.22460
$250$	0	0
$251$	−27.6230	−1.74355	−0.871774	−	0.489909i	$-0.837030\pi$
$251$	−27.6230	−1.74355	−0.871774	+	0.489909i	$0.837030\pi$
$252$	1.99796i	0.125860i
$253$	− 10.1325i	− 0.637025i
$254$	−11.9328	−0.748731
$255$	0	0
$256$	−15.2139	−0.950871
$257$	− 6.02021i	− 0.375530i	−0.982214	−	0.187765i	$-0.939876\pi$
$257$	− 6.02021i	− 0.375530i	0.982214	−	0.187765i	$-0.0601244\pi$
$258$	− 15.3532i	− 0.955851i
$259$	−18.3442	−1.13985
$260$	0	0
$261$	0.624302	0.0386434
$262$	− 16.6060i	− 1.02592i
$263$	8.13397i	0.501562i	0.968044	+	0.250781i	$0.0806874\pi$
$263$	8.13397i	0.501562i	−0.968044	+	0.250781i	$0.919313\pi$
$264$	7.84667	0.482929
$265$	0	0
$266$	−0.0772991	−0.00473951
$267$	− 13.1005i	− 0.801735i
$268$	0.377772i	0.0230761i
$269$	3.43905	0.209682	0.104841	−	0.994489i	$-0.466567\pi$
$269$	3.43905	0.209682	0.104841	+	0.994489i	$0.466567\pi$
$270$	0	0
$271$	−10.4822	−0.636748	−0.318374	−	0.947965i	$-0.603137\pi$
$271$	−10.4822	−0.636748	−0.318374	+	0.947965i	$0.603137\pi$
$272$	− 0.769227i	− 0.0466412i
$273$	− 4.36909i	− 0.264429i
$274$	0.800832	0.0483800
$275$	0	0
$276$	16.7773	1.00987
$277$	− 18.9262i	− 1.13716i	−0.822627	−	0.568582i	$-0.807492\pi$
$277$	− 18.9262i	− 1.13716i	0.822627	−	0.568582i	$-0.192508\pi$
$278$	− 18.9032i	− 1.13374i
$279$	−1.34704	−0.0806451
$280$	0	0
$281$	−24.6790	−1.47223	−0.736113	−	0.676859i	$-0.763341\pi$
$281$	−24.6790	−1.47223	−0.736113	+	0.676859i	$0.763341\pi$
$282$	− 8.26190i	− 0.491989i
$283$	3.12685i	0.185872i	0.995672	+	0.0929361i	$0.0296252\pi$
$283$	3.12685i	0.185872i	−0.995672	+	0.0929361i	$0.970375\pi$
$284$	−13.9361	−0.826958
$285$	0	0
$286$	−1.16249	−0.0687395
$287$	4.87244i	0.287611i
$288$	3.59579i	0.211884i
$289$	9.12733	0.536902
$290$	0	0
$291$	17.9408	1.05171
$292$	0.0182850i	0.00107005i
$293$	− 6.95025i	− 0.406038i	−0.979175	−	0.203019i	$-0.934925\pi$
$293$	− 6.95025i	− 0.406038i	0.979175	−	0.203019i	$-0.0650753\pi$
$294$	−1.44882	−0.0844968
$295$	0	0
$296$	−20.5481	−1.19433
$297$	6.74218i	0.391221i
$298$	7.18669i	0.416314i
$299$	−6.31969	−0.365477
$300$	0	0
$301$	23.7342	1.36802
$302$	− 11.1203i	− 0.639904i
$303$	− 20.9801i	− 1.20528i
$304$	0.0102365	0.000587101 0
$305$	0	0
$306$	1.46917	0.0839869
$307$	− 15.4392i	− 0.881160i	−0.897713	−	0.440580i	$-0.854773\pi$
$307$	− 15.4392i	− 0.881160i	0.897713	−	0.440580i	$-0.145227\pi$
$308$	4.77078i	0.271841i
$309$	−8.59402	−0.488896
$310$	0	0
$311$	−12.9889	−0.736534	−0.368267	−	0.929720i	$-0.620049\pi$
$311$	−12.9889	−0.736534	−0.368267	+	0.929720i	$0.620049\pi$
$312$	− 4.89401i	− 0.277069i
$313$	24.8765i	1.40611i	0.711138	+	0.703053i	$0.248180\pi$
$313$	24.8765i	1.40611i	−0.711138	+	0.703053i	$0.751820\pi$
$314$	−11.8260	−0.667381
$315$	0	0
$316$	−6.97391	−0.392313
$317$	17.1132i	0.961176i	0.876947	+	0.480588i	$0.159577\pi$
$317$	17.1132i	0.961176i	−0.876947	+	0.480588i	$0.840423\pi$
$318$	2.13562i	0.119759i
$319$	1.49072	0.0834645
$320$	0	0
$321$	0.529093	0.0295311
$322$	− 14.0714i	− 0.784169i
$323$	− 0.104765i	− 0.00582930i
$324$	−13.5920	−0.755109
$325$	0	0
$326$	14.4483	0.800219
$327$	− 31.7278i	− 1.75455i
$328$	5.45783i	0.301359i
$329$	12.7719	0.704135
$330$	0	0
$331$	−25.9203	−1.42471	−0.712353	−	0.701821i	$-0.752371\pi$
$331$	−25.9203	−1.42471	−0.712353	+	0.701821i	$0.752371\pi$
$332$	13.1604i	0.722273i
$333$	4.63971i	0.254254i
$334$	12.1075	0.662495
$335$	0	0
$336$	−1.28827	−0.0702810
$337$	18.9557i	1.03258i	0.856414	+	0.516290i	$0.172687\pi$
$337$	18.9557i	1.03258i	−0.856414	+	0.516290i	$0.827313\pi$
$338$	− 10.1783i	− 0.553625i
$339$	−28.3692	−1.54081
$340$	0	0
$341$	−3.21649	−0.174183
$342$	0.0195509i	0.00105719i
$343$	− 19.5180i	− 1.05387i
$344$	26.5857	1.43340
$345$	0	0
$346$	−13.0572	−0.701959
$347$	− 7.70815i	− 0.413795i	−0.978363	−	0.206898i	$-0.933663\pi$
$347$	− 7.70815i	− 0.413795i	0.978363	−	0.206898i	$-0.0663367\pi$
$348$	2.46832i	0.132316i
$349$	15.0015	0.803014	0.401507	−	0.915856i	$-0.368487\pi$
$349$	15.0015	0.803014	0.401507	+	0.915856i	$0.368487\pi$
$350$	0	0
$351$	4.20513	0.224453
$352$	8.58611i	0.457641i
$353$	− 16.9486i	− 0.902085i	−0.892502	−	0.451043i	$-0.851052\pi$
$353$	− 16.9486i	− 0.902085i	0.892502	−	0.451043i	$-0.148948\pi$
$354$	7.30911	0.388475
$355$	0	0
$356$	8.92205	0.472867
$357$	13.1849i	0.697817i
$358$	17.2144i	0.909811i
$359$	5.23876	0.276491	0.138246	−	0.990398i	$-0.455854\pi$
$359$	5.23876	0.276491	0.138246	+	0.990398i	$0.455854\pi$
$360$	0	0
$361$	−18.9986	−0.999927
$362$	− 14.6868i	− 0.771923i
$363$	16.7107i	0.877085i
$364$	2.97556	0.155962
$365$	0	0
$366$	−19.4010	−1.01411
$367$	− 15.2817i	− 0.797701i	−0.917016	−	0.398850i	$-0.869409\pi$
$367$	− 15.2817i	− 0.797701i	0.917016	−	0.398850i	$-0.130591\pi$
$368$	1.86343i	0.0971378i
$369$	1.23237	0.0641544
$370$	0	0
$371$	−3.30139	−0.171400
$372$	− 5.32583i	− 0.276131i
$373$	24.3674i	1.26169i	0.775907	+	0.630847i	$0.217292\pi$
$373$	24.3674i	1.26169i	−0.775907	+	0.630847i	$0.782708\pi$
$374$	3.50812	0.181400
$375$	0	0
$376$	14.3063	0.737791
$377$	− 0.929772i	− 0.0478857i
$378$	9.36314i	0.481588i
$379$	12.1058	0.621832	0.310916	−	0.950437i	$-0.399364\pi$
$379$	12.1058	0.621832	0.310916	+	0.950437i	$0.399364\pi$
$380$	0	0
$381$	−27.0856	−1.38764
$382$	− 13.0610i	− 0.668261i
$383$	21.0250i	1.07432i	0.843479	+	0.537162i	$0.180504\pi$
$383$	21.0250i	1.07432i	−0.843479	+	0.537162i	$0.819496\pi$
$384$	−15.0923	−0.770175
$385$	0	0
$386$	−7.08254	−0.360492
$387$	− 6.00298i	− 0.305149i
$388$	12.2185i	0.620302i
$389$	10.3647	0.525509	0.262755	−	0.964863i	$-0.415369\pi$
$389$	10.3647	0.525509	0.262755	+	0.964863i	$0.415369\pi$
$390$	0	0
$391$	19.0713	0.964477
$392$	− 2.50877i	− 0.126712i
$393$	− 37.6929i	− 1.90136i
$394$	8.54964	0.430725
$395$	0	0
$396$	1.20665	0.0606366
$397$	17.7864i	0.892673i	0.894865	+	0.446337i	$0.147272\pi$
$397$	17.7864i	0.892673i	−0.894865	+	0.446337i	$0.852728\pi$
$398$	5.48550i	0.274963i
$399$	−0.175457	−0.00878384
$400$	0	0
$401$	−29.9868	−1.49747	−0.748734	−	0.662870i	$-0.769338\pi$
$401$	−29.9868	−1.49747	−0.748734	+	0.662870i	$0.769338\pi$
$402$	− 0.465230i	− 0.0232036i
$403$	2.00614i	0.0999330i
$404$	14.2885	0.710878
$405$	0	0
$406$	2.07023	0.102744
$407$	11.0788i	0.549156i
$408$	14.7689i	0.731171i
$409$	−30.6717	−1.51662	−0.758309	−	0.651896i	$-0.773974\pi$
$409$	−30.6717	−1.51662	−0.758309	+	0.651896i	$0.773974\pi$
$410$	0	0
$411$	1.81776	0.0896637
$412$	− 5.85294i	− 0.288354i
$413$	11.2990i	0.555986i
$414$	−3.55902	−0.174916
$415$	0	0
$416$	5.35520	0.262560
$417$	− 42.9073i	− 2.10118i
$418$	0.0466842i	0.00228340i
$419$	15.0395	0.734729	0.367364	−	0.930077i	$-0.380260\pi$
$419$	15.0395	0.734729	0.367364	+	0.930077i	$0.380260\pi$
$420$	0	0
$421$	−31.3625	−1.52852	−0.764258	−	0.644911i	$-0.776894\pi$
$421$	−31.3625	−1.52852	−0.764258	+	0.644911i	$0.776894\pi$
$422$	− 13.0830i	− 0.636869i
$423$	− 3.23033i	− 0.157064i
$424$	−3.69803	−0.179592
$425$	0	0
$426$	17.1625	0.831526
$427$	− 29.9915i	− 1.45139i
$428$	0.360338i	0.0174176i
$429$	−2.63868	−0.127396
$430$	0	0
$431$	−28.7116	−1.38299	−0.691495	−	0.722381i	$-0.743048\pi$
$431$	−28.7116	−1.38299	−0.691495	+	0.722381i	$0.743048\pi$
$432$	− 1.23993i	− 0.0596561i
$433$	30.9086i	1.48537i	0.669640	+	0.742686i	$0.266448\pi$
$433$	30.9086i	1.48537i	−0.669640	+	0.742686i	$0.733552\pi$
$434$	−4.46687	−0.214417
$435$	0	0
$436$	21.6082	1.03484
$437$	0.253791i	0.0121405i
$438$	− 0.0225181i	− 0.00107596i
$439$	−18.5828	−0.886907	−0.443454	−	0.896297i	$-0.646247\pi$
$439$	−18.5828	−0.886907	−0.443454	+	0.896297i	$0.646247\pi$
$440$	0	0
$441$	−0.566475	−0.0269750
$442$	− 2.18803i	− 0.104074i
$443$	− 32.4127i	− 1.53997i	−0.638061	−	0.769986i	$-0.720263\pi$
$443$	− 32.4127i	− 1.53997i	0.638061	−	0.769986i	$-0.279737\pi$
$444$	−18.3442	−0.870575
$445$	0	0
$446$	19.9331	0.943862
$447$	16.3127i	0.771563i
$448$	10.5705i	0.499408i
$449$	12.7486	0.601645	0.300822	−	0.953680i	$-0.402739\pi$
$449$	12.7486	0.601645	0.300822	+	0.953680i	$0.402739\pi$
$450$	0	0
$451$	2.94267	0.138565
$452$	− 19.3208i	− 0.908775i
$453$	− 25.2415i	− 1.18595i
$454$	7.63443	0.358301
$455$	0	0
$456$	−0.196537	−0.00920369
$457$	20.5383i	0.960742i	0.877065	+	0.480371i	$0.159498\pi$
$457$	20.5383i	0.960742i	−0.877065	+	0.480371i	$0.840502\pi$
$458$	20.4498i	0.955556i
$459$	−12.6901	−0.592322
$460$	0	0
$461$	15.0965	0.703114	0.351557	−	0.936167i	$-0.385652\pi$
$461$	15.0965	0.703114	0.351557	+	0.936167i	$0.385652\pi$
$462$	− 5.87527i	− 0.273342i
$463$	− 1.19501i	− 0.0555367i	−0.999614	−	0.0277684i	$-0.991160\pi$
$463$	− 1.19501i	− 0.0555367i	0.999614	−	0.0277684i	$-0.00884008\pi$
$464$	−0.274153	−0.0127272
$465$	0	0
$466$	15.6273	0.723922
$467$	7.08659i	0.327929i	0.986466	+	0.163964i	$0.0524282\pi$
$467$	7.08659i	0.327929i	−0.986466	+	0.163964i	$0.947572\pi$
$468$	− 0.752596i	− 0.0347887i
$469$	0.719188	0.0332090
$470$	0	0
$471$	−26.8432	−1.23687
$472$	12.6565i	0.582561i
$473$	− 14.3341i	− 0.659081i
$474$	8.58844	0.394480
$475$	0	0
$476$	−8.97953	−0.411576
$477$	0.835007i	0.0382323i
$478$	1.42136i	0.0650117i
$479$	−38.8768	−1.77633	−0.888163	−	0.459528i	$-0.848019\pi$
$479$	−38.8768	−1.77633	−0.888163	+	0.459528i	$0.848019\pi$
$480$	0	0
$481$	6.90990	0.315064
$482$	− 7.19226i	− 0.327599i
$483$	− 31.9399i	− 1.45332i
$484$	−11.3808	−0.517309
$485$	0	0
$486$	5.35868	0.243075
$487$	− 30.3083i	− 1.37340i	−0.726940	−	0.686701i	$-0.759058\pi$
$487$	− 30.3083i	− 1.37340i	0.726940	−	0.686701i	$-0.240942\pi$
$488$	− 33.5948i	− 1.52077i
$489$	32.7955	1.48306
$490$	0	0
$491$	−9.84537	−0.444315	−0.222158	−	0.975011i	$-0.571310\pi$
$491$	−9.84537	−0.444315	−0.222158	+	0.975011i	$0.571310\pi$
$492$	4.87244i	0.219667i
$493$	2.80583i	0.126368i
$494$	0.0291171	0.00131004
$495$	0	0
$496$	0.591532	0.0265606
$497$	26.5311i	1.19008i
$498$	− 16.2072i	− 0.726262i
$499$	−20.9719	−0.938829	−0.469415	−	0.882978i	$-0.655535\pi$
$499$	−20.9719	−0.938829	−0.469415	+	0.882978i	$0.655535\pi$
$500$	0	0
$501$	27.4822	1.22782
$502$	23.1679i	1.03403i
$503$	42.2017i	1.88168i	0.338849	+	0.940841i	$0.389963\pi$
$503$	42.2017i	1.88168i	−0.338849	+	0.940841i	$0.610037\pi$
$504$	4.26062	0.189783
$505$	0	0
$506$	−8.49831	−0.377796
$507$	− 23.1031i	− 1.02605i
$508$	− 18.4466i	− 0.818436i
$509$	−7.00080	−0.310305	−0.155153	−	0.987891i	$-0.549587\pi$
$509$	−7.00080	−0.310305	−0.155153	+	0.987891i	$0.549587\pi$
$510$	0	0
$511$	0.0348102	0.00153991
$512$	− 3.09504i	− 0.136783i
$513$	− 0.168873i	− 0.00745592i
$514$	−5.04925	−0.222713
$515$	0	0
$516$	23.7342	1.04484
$517$	− 7.71345i	− 0.339237i
$518$	15.3856i	0.676003i
$519$	−29.6378	−1.30096
$520$	0	0
$521$	12.0767	0.529091	0.264545	−	0.964373i	$-0.414778\pi$
$521$	12.0767	0.529091	0.264545	+	0.964373i	$0.414778\pi$
$522$	− 0.523614i	− 0.0229179i
$523$	30.9879i	1.35500i	0.735521	+	0.677502i	$0.236938\pi$
$523$	30.9879i	1.35500i	−0.735521	+	0.677502i	$0.763062\pi$
$524$	25.6707	1.12143
$525$	0	0
$526$	6.82211	0.297458
$527$	− 6.05405i	− 0.263719i
$528$	0.778041i	0.0338599i
$529$	−23.1996	−1.00868
$530$	0	0
$531$	2.85780	0.124018
$532$	− 0.119495i	− 0.00518075i
$533$	− 1.83536i	− 0.0794982i
$534$	−10.9876	−0.475479
$535$	0	0
$536$	0.805593	0.0347963
$537$	39.0741i	1.68617i
$538$	− 2.88439i	− 0.124355i
$539$	−1.35264	−0.0582624
$540$	0	0
$541$	−5.40732	−0.232479	−0.116240	−	0.993221i	$-0.537084\pi$
$541$	−5.40732	−0.232479	−0.116240	+	0.993221i	$0.537084\pi$
$542$	8.79161i	0.377632i
$543$	− 33.3369i	− 1.43062i
$544$	−16.1607	−0.692885
$545$	0	0
$546$	−3.66443	−0.156823
$547$	0.766804i	0.0327862i	0.999866	+	0.0163931i	$0.00521832\pi$
$547$	0.766804i	0.0327862i	−0.999866	+	0.0163931i	$0.994782\pi$
$548$	1.23799i	0.0528841i
$549$	−7.58563	−0.323747
$550$	0	0
$551$	−0.0373385	−0.00159067
$552$	− 35.7773i	− 1.52278i
$553$	13.2766i	0.564580i
$554$	−15.8737	−0.674410
$555$	0	0
$556$	29.2219	1.23929
$557$	38.2414i	1.62034i	0.586195	+	0.810170i	$0.300625\pi$
$557$	38.2414i	1.62034i	−0.586195	+	0.810170i	$0.699375\pi$
$558$	1.12979i	0.0478276i
$559$	−8.94023	−0.378131
$560$	0	0
$561$	7.96289	0.336193
$562$	20.6987i	0.873123i
$563$	36.0317i	1.51855i	0.650768	+	0.759277i	$0.274447\pi$
$563$	36.0317i	1.51855i	−0.650768	+	0.759277i	$0.725553\pi$
$564$	12.7719	0.537792
$565$	0	0
$566$	2.62255	0.110234
$567$	25.8758i	1.08668i
$568$	29.7186i	1.24697i
$569$	17.6536	0.740077	0.370038	−	0.929016i	$-0.379345\pi$
$569$	17.6536	0.740077	0.370038	+	0.929016i	$0.379345\pi$
$570$	0	0
$571$	−3.66464	−0.153360	−0.0766801	−	0.997056i	$-0.524432\pi$
$571$	−3.66464	−0.153360	−0.0766801	+	0.997056i	$0.524432\pi$
$572$	− 1.79707i	− 0.0751391i
$573$	− 29.6466i	− 1.23850i
$574$	4.08661	0.170572
$575$	0	0
$576$	2.67354	0.111398
$577$	25.9865i	1.08183i	0.841076	+	0.540917i	$0.181923\pi$
$577$	25.9865i	1.08183i	−0.841076	+	0.540917i	$0.818077\pi$
$578$	− 7.65525i	− 0.318417i
$579$	−16.0763	−0.668107
$580$	0	0
$581$	25.0543	1.03943
$582$	− 15.0472i	− 0.623728i
$583$	1.99385i	0.0825768i
$584$	0.0389924	0.00161352
$585$	0	0
$586$	−5.82930	−0.240806
$587$	26.3857i	1.08905i	0.838744	+	0.544527i	$0.183291\pi$
$587$	26.3857i	1.08905i	−0.838744	+	0.544527i	$0.816709\pi$
$588$	− 2.23969i	− 0.0923633i
$589$	0.0805641	0.00331958
$590$	0	0
$591$	19.4064	0.798272
$592$	− 2.03746i	− 0.0837390i
$593$	− 37.1560i	− 1.52581i	−0.646508	−	0.762907i	$-0.723771\pi$
$593$	− 37.1560i	− 1.52581i	0.646508	−	0.762907i	$-0.276229\pi$
$594$	5.65479	0.232019
$595$	0	0
$596$	−11.1097	−0.455072
$597$	12.4512i	0.509596i
$598$	5.30044i	0.216751i
$599$	5.96103	0.243561	0.121781	−	0.992557i	$-0.461140\pi$
$599$	5.96103	0.243561	0.121781	+	0.992557i	$0.461140\pi$
$600$	0	0
$601$	9.40636	0.383693	0.191847	−	0.981425i	$-0.438552\pi$
$601$	9.40636	0.383693	0.191847	+	0.981425i	$0.438552\pi$
$602$	− 19.9063i	− 0.811320i
$603$	− 0.181901i	− 0.00740758i
$604$	17.1907	0.699478
$605$	0	0
$606$	−17.5964	−0.714805
$607$	11.5121i	0.467263i	0.972325	+	0.233632i	$0.0750610\pi$
$607$	11.5121i	0.467263i	−0.972325	+	0.233632i	$0.924939\pi$
$608$	− 0.215058i	− 0.00872176i
$609$	4.69910	0.190417
$610$	0	0
$611$	−4.81092	−0.194629
$612$	2.27115i	0.0918059i
$613$	− 33.9655i	− 1.37185i	−0.727670	−	0.685927i	$-0.759397\pi$
$613$	− 33.9655i	− 1.37185i	0.727670	−	0.685927i	$-0.240603\pi$
$614$	−12.9491	−0.522583
$615$	0	0
$616$	10.1736	0.409907
$617$	− 18.1241i	− 0.729650i	−0.931076	−	0.364825i	$-0.881129\pi$
$617$	− 18.1241i	− 0.729650i	0.931076	−	0.364825i	$-0.118871\pi$
$618$	7.20795i	0.289946i
$619$	43.7822	1.75976	0.879878	−	0.475200i	$-0.157624\pi$
$619$	43.7822	1.75976	0.879878	+	0.475200i	$0.157624\pi$
$620$	0	0
$621$	30.7413	1.23361
$622$	10.8940i	0.436811i
$623$	− 16.9854i	− 0.680507i
$624$	0.485268	0.0194263
$625$	0	0
$626$	20.8644	0.833909
$627$	0.105966i	0.00423187i
$628$	− 18.2815i	− 0.729513i
$629$	−20.8524	−0.831440
$630$	0	0
$631$	−31.8836	−1.26927	−0.634633	−	0.772814i	$-0.718849\pi$
$631$	−31.8836	−1.26927	−0.634633	+	0.772814i	$0.718849\pi$
$632$	14.8717i	0.591566i
$633$	− 29.6963i	− 1.18032i
$634$	14.3532	0.570038
$635$	0	0
$636$	−3.30139	−0.130909
$637$	0.843650i	0.0334266i
$638$	− 1.25030i	− 0.0494997i
$639$	6.71039	0.265459
$640$	0	0
$641$	−7.56055	−0.298624	−0.149312	−	0.988790i	$-0.547706\pi$
$641$	−7.56055	−0.298624	−0.149312	+	0.988790i	$0.547706\pi$
$642$	− 0.443759i	− 0.0175138i
$643$	− 6.78946i	− 0.267750i	−0.990998	−	0.133875i	$-0.957258\pi$
$643$	− 6.78946i	− 0.267750i	0.990998	−	0.133875i	$-0.0427421\pi$
$644$	21.7526	0.857173
$645$	0	0
$646$	−0.0878685	−0.00345714
$647$	− 32.4336i	− 1.27510i	−0.770410	−	0.637549i	$-0.779949\pi$
$647$	− 32.4336i	− 1.27510i	0.770410	−	0.637549i	$-0.220051\pi$
$648$	28.9846i	1.13862i
$649$	6.82391	0.267862
$650$	0	0
$651$	−10.1391	−0.397383
$652$	22.3353i	0.874718i
$653$	20.3067i	0.794661i	0.917676	+	0.397331i	$0.130063\pi$
$653$	20.3067i	0.794661i	−0.917676	+	0.397331i	$0.869937\pi$
$654$	−26.6107	−1.04056
$655$	0	0
$656$	−0.541175	−0.0211293
$657$	− 0.00880440i	0 0.000343492i
$658$	− 10.7120i	− 0.417597i
$659$	47.3789	1.84562	0.922809	−	0.385257i	$-0.125887\pi$
$659$	47.3789	1.84562	0.922809	+	0.385257i	$0.125887\pi$
$660$	0	0
$661$	23.1097	0.898862	0.449431	−	0.893315i	$-0.351627\pi$
$661$	23.1097	0.898862	0.449431	+	0.893315i	$0.351627\pi$
$662$	21.7398i	0.844941i
$663$	− 4.96649i	− 0.192883i
$664$	28.0644	1.08911
$665$	0	0
$666$	3.89140	0.150789
$667$	− 6.79703i	− 0.263182i
$668$	18.7167i	0.724172i
$669$	45.2452	1.74928
$670$	0	0
$671$	−18.1131	−0.699250
$672$	27.0654i	1.04407i
$673$	0.691021i	0.0266369i	0.999911	+	0.0133184i	$0.00423952\pi$
$673$	0.691021i	0.0266369i	−0.999911	+	0.0133184i	$0.995760\pi$
$674$	15.8984	0.612385
$675$	0	0
$676$	15.7343	0.605167
$677$	− 10.2253i	− 0.392989i	−0.980505	−	0.196494i	$-0.937044\pi$
$677$	− 10.2253i	− 0.392989i	0.980505	−	0.196494i	$-0.0629557\pi$
$678$	23.7938i	0.913795i
$679$	23.2611	0.892681
$680$	0	0
$681$	17.3290	0.664048
$682$	2.69773i	0.103301i
$683$	− 9.74838i	− 0.373012i	−0.982454	−	0.186506i	$-0.940284\pi$
$683$	− 9.74838i	− 0.373012i	0.982454	−	0.186506i	$-0.0597163\pi$
$684$	−0.0302233	−0.00115562
$685$	0	0
$686$	−16.3701	−0.625012
$687$	46.4179i	1.77095i
$688$	2.63612i	0.100501i
$689$	1.24357	0.0473764
$690$	0	0
$691$	−32.7638	−1.24639	−0.623197	−	0.782065i	$-0.714167\pi$
$691$	−32.7638	−1.24639	−0.623197	+	0.782065i	$0.714167\pi$
$692$	− 20.1848i	− 0.767311i
$693$	− 2.29718i	− 0.0872626i
$694$	−6.46496	−0.245407
$695$	0	0
$696$	5.26366	0.199519
$697$	5.53867i	0.209792i
$698$	− 12.5821i	− 0.476238i
$699$	35.4716	1.34166
$700$	0	0
$701$	−14.7894	−0.558588	−0.279294	−	0.960206i	$-0.590100\pi$
$701$	−14.7894	−0.558588	−0.279294	+	0.960206i	$0.590100\pi$
$702$	− 3.52692i	− 0.133115i
$703$	− 0.277493i	− 0.0104658i
$704$	6.38395	0.240604
$705$	0	0
$706$	−14.2151	−0.534993
$707$	− 27.2018i	− 1.02303i
$708$	11.2990i	0.424641i
$709$	−42.1665	−1.58360	−0.791799	−	0.610782i	$-0.790855\pi$
$709$	−42.1665	−1.58360	−0.791799	+	0.610782i	$0.790855\pi$
$710$	0	0
$711$	3.35800	0.125935
$712$	− 19.0261i	− 0.713034i
$713$	14.6657i	0.549236i
$714$	11.0584	0.413849
$715$	0	0
$716$	−26.6114	−0.994513
$717$	3.22628i	0.120488i
$718$	− 4.39384i	− 0.163977i
$719$	16.2819	0.607211	0.303606	−	0.952798i	$-0.401809\pi$
$719$	16.2819	0.607211	0.303606	+	0.952798i	$0.401809\pi$
$720$	0	0
$721$	−11.1426	−0.414972
$722$	15.9345i	0.593020i
$723$	− 16.3253i	− 0.607146i
$724$	22.7040	0.843788
$725$	0	0
$726$	14.0156	0.520167
$727$	− 10.2431i	− 0.379894i	−0.981794	−	0.189947i	$-0.939168\pi$
$727$	− 10.2431i	− 0.379894i	0.981794	−	0.189947i	$-0.0608317\pi$
$728$	− 6.34534i	− 0.235174i
$729$	−19.2861	−0.714299
$730$	0	0
$731$	26.9795	0.997871
$732$	− 29.9915i	− 1.10852i
$733$	1.34776i	0.0497807i	0.999690	+	0.0248904i	$0.00792367\pi$
$733$	1.34776i	0.0497807i	−0.999690	+	0.0248904i	$0.992076\pi$
$734$	−12.8171	−0.473087
$735$	0	0
$736$	39.1488	1.44304
$737$	− 0.434347i	− 0.0159994i
$738$	− 1.03361i	− 0.0380476i
$739$	33.2111	1.22169	0.610844	−	0.791751i	$-0.290830\pi$
$739$	33.2111	1.22169	0.610844	+	0.791751i	$0.290830\pi$
$740$	0	0
$741$	0.0660914	0.00242793
$742$	2.76894i	0.101651i
$743$	− 36.6826i	− 1.34576i	−0.739754	−	0.672878i	$-0.765058\pi$
$743$	− 36.6826i	− 1.34576i	0.739754	−	0.672878i	$-0.234942\pi$
$744$	−11.3572	−0.416377
$745$	0	0
$746$	20.4373	0.748264
$747$	− 6.33688i	− 0.231854i
$748$	5.42311i	0.198289i
$749$	0.685996	0.0250658
$750$	0	0
$751$	−16.0304	−0.584959	−0.292480	−	0.956272i	$-0.594480\pi$
$751$	−16.0304	−0.584959	−0.292480	+	0.956272i	$0.594480\pi$
$752$	1.41855i	0.0517292i
$753$	52.5875i	1.91640i
$754$	−0.779816	−0.0283992
$755$	0	0
$756$	−14.4742	−0.526423
$757$	42.4991i	1.54466i	0.635224	+	0.772328i	$0.280908\pi$
$757$	42.4991i	1.54466i	−0.635224	+	0.772328i	$0.719092\pi$
$758$	− 10.1533i	− 0.368786i
$759$	−19.2898	−0.700177
$760$	0	0
$761$	24.7756	0.898114	0.449057	−	0.893503i	$-0.351760\pi$
$761$	24.7756	0.898114	0.449057	+	0.893503i	$0.351760\pi$
$762$	22.7172i	0.822957i
$763$	− 41.1368i	− 1.48925i
$764$	20.1907	0.730475
$765$	0	0
$766$	17.6340	0.637142
$767$	− 4.25611i	− 0.153679i
$768$	28.9637i	1.04514i
$769$	−22.9249	−0.826694	−0.413347	−	0.910573i	$-0.635640\pi$
$769$	−22.9249	−0.826694	−0.413347	+	0.910573i	$0.635640\pi$
$770$	0	0
$771$	−11.4610	−0.412759
$772$	− 10.9487i	− 0.394053i
$773$	− 46.2921i	− 1.66501i	−0.554016	−	0.832506i	$-0.686905\pi$
$773$	− 46.2921i	− 1.66501i	0.554016	−	0.832506i	$-0.313095\pi$
$774$	−5.03481	−0.180972
$775$	0	0
$776$	26.0558	0.935349
$777$	34.9229i	1.25285i
$778$	− 8.69302i	− 0.311660i
$779$	−0.0737057	−0.00264078
$780$	0	0
$781$	16.0232	0.573356
$782$	− 15.9954i	− 0.571996i
$783$	4.52276i	0.161630i
$784$	0.248759	0.00888425
$785$	0	0
$786$	−31.6137	−1.12762
$787$	− 3.89849i	− 0.138966i	−0.997583	−	0.0694832i	$-0.977865\pi$
$787$	− 3.89849i	− 0.138966i	0.997583	−	0.0694832i	$-0.0221350\pi$
$788$	13.2167i	0.470825i
$789$	15.4851	0.551285
$790$	0	0
$791$	−36.7822	−1.30782
$792$	− 2.57317i	− 0.0914336i
$793$	11.2973i	0.401177i
$794$	14.9178	0.529411
$795$	0	0
$796$	−8.47990	−0.300562
$797$	− 21.1877i	− 0.750506i	−0.926922	−	0.375253i	$-0.877556\pi$
$797$	− 21.1877i	− 0.750506i	0.926922	−	0.375253i	$-0.122444\pi$
$798$	0.147159i	0.00520937i
$799$	14.5182	0.513617
$800$	0	0
$801$	−4.29605	−0.151793
$802$	25.1504i	0.888093i
$803$	− 0.0210233i	0 0.000741898i
$804$	0.719188	0.0253638
$805$	0	0
$806$	1.68259	0.0592666
$807$	− 6.54712i	− 0.230470i
$808$	− 30.4699i	− 1.07193i
$809$	−23.9959	−0.843651	−0.421826	−	0.906677i	$-0.638611\pi$
$809$	−23.9959	−0.843651	−0.421826	+	0.906677i	$0.638611\pi$
$810$	0	0
$811$	−8.15390	−0.286322	−0.143161	−	0.989699i	$-0.545727\pi$
$811$	−8.15390	−0.286322	−0.143161	+	0.989699i	$0.545727\pi$
$812$	3.20031i	0.112309i
$813$	19.9556i	0.699873i
$814$	9.29198	0.325684
$815$	0	0
$816$	−1.46442	−0.0512651
$817$	0.359028i	0.0125608i
$818$	25.7249i	0.899450i
$819$	−1.43276	−0.0500647
$820$	0	0
$821$	−8.76657	−0.305955	−0.152978	−	0.988230i	$-0.548886\pi$
$821$	−8.76657	−0.305955	−0.152978	+	0.988230i	$0.548886\pi$
$822$	− 1.52459i	− 0.0531762i
$823$	49.1112i	1.71191i	0.517050	+	0.855955i	$0.327030\pi$
$823$	49.1112i	1.71191i	−0.517050	+	0.855955i	$0.672970\pi$
$824$	−12.4813	−0.434807
$825$	0	0
$826$	9.47664	0.329734
$827$	− 34.1055i	− 1.18597i	−0.805215	−	0.592983i	$-0.797950\pi$
$827$	− 34.1055i	− 1.18597i	0.805215	−	0.592983i	$-0.202050\pi$
$828$	− 5.50179i	− 0.191201i
$829$	−1.20117	−0.0417183	−0.0208591	−	0.999782i	$-0.506640\pi$
$829$	−1.20117	−0.0417183	−0.0208591	+	0.999782i	$0.506640\pi$
$830$	0	0
$831$	−36.0309	−1.24990
$832$	− 3.98170i	− 0.138041i
$833$	− 2.54593i	− 0.0882113i
$834$	−35.9871	−1.24613
$835$	0	0
$836$	−0.0721679	−0.00249598
$837$	− 9.75862i	− 0.337307i
$838$	− 12.6139i	− 0.435740i
$839$	1.87849	0.0648526	0.0324263	−	0.999474i	$-0.489677\pi$
$839$	1.87849	0.0648526	0.0324263	+	0.999474i	$0.489677\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	26.3043i	0.906506i
$843$	46.9829i	1.61818i
$844$	20.2246	0.696161
$845$	0	0
$846$	−2.70933	−0.0931488
$847$	21.6663i	0.744463i
$848$	− 0.366681i	− 0.0125919i
$849$	5.95278	0.204299
$850$	0	0
$851$	50.5143	1.73161
$852$	26.5311i	0.908940i
$853$	− 33.0917i	− 1.13304i	−0.824049	−	0.566519i	$-0.808290\pi$
$853$	− 33.0917i	− 1.13304i	0.824049	−	0.566519i	$-0.191710\pi$
$854$	−25.1544	−0.860767
$855$	0	0
$856$	0.768414	0.0262639
$857$	47.6859i	1.62892i	0.580220	+	0.814460i	$0.302966\pi$
$857$	47.6859i	1.62892i	−0.580220	+	0.814460i	$0.697034\pi$
$858$	2.21310i	0.0755541i
$859$	−0.746091	−0.0254563	−0.0127281	−	0.999919i	$-0.504052\pi$
$859$	−0.746091	−0.0254563	−0.0127281	+	0.999919i	$0.504052\pi$
$860$	0	0
$861$	9.27596	0.316124
$862$	24.0810i	0.820201i
$863$	− 46.8245i	− 1.59392i	−0.604030	−	0.796962i	$-0.706439\pi$
$863$	− 46.8245i	− 1.59392i	0.604030	−	0.796962i	$-0.293561\pi$
$864$	−26.0497	−0.886229
$865$	0	0
$866$	25.9236	0.880919
$867$	− 17.3762i	− 0.590128i
$868$	− 6.90522i	− 0.234378i
$869$	8.01832	0.272003
$870$	0	0
$871$	−0.270905	−0.00917925
$872$	− 46.0791i	− 1.56043i
$873$	− 5.88334i	− 0.199121i
$874$	0.212859	0.00720005
$875$	0	0
$876$	0.0348102	0.00117613
$877$	− 0.970257i	− 0.0327632i	−0.999866	−	0.0163816i	$-0.994785\pi$
$877$	− 0.970257i	− 0.0327632i	0.999866	−	0.0163816i	$-0.00521466\pi$
$878$	15.5857i	0.525992i
$879$	−13.2316	−0.446291
$880$	0	0
$881$	0.755979	0.0254696	0.0127348	−	0.999919i	$-0.495946\pi$
$881$	0.755979	0.0254696	0.0127348	+	0.999919i	$0.495946\pi$
$882$	0.475113i	0.0159979i
$883$	15.8839i	0.534536i	0.963622	+	0.267268i	$0.0861209\pi$
$883$	15.8839i	0.534536i	−0.963622	+	0.267268i	$0.913879\pi$
$884$	3.38242	0.113763
$885$	0	0
$886$	−27.1851	−0.913300
$887$	12.5061i	0.419915i	0.977710	+	0.209958i	$0.0673326\pi$
$887$	12.5061i	0.419915i	−0.977710	+	0.209958i	$0.932667\pi$
$888$	39.1186i	1.31273i
$889$	−35.1179	−1.17782
$890$	0	0
$891$	15.6275	0.523541
$892$	30.8141i	1.03173i
$893$	0.193200i	0.00646521i
$894$	13.6817	0.457586
$895$	0	0
$896$	−19.5679	−0.653719
$897$	12.0312i	0.401709i
$898$	− 10.6925i	− 0.356813i
$899$	−2.15767	−0.0719623
$900$	0	0
$901$	−3.75281	−0.125024
$902$	− 2.46807i	− 0.0821778i
$903$	− 45.1842i	− 1.50364i
$904$	−41.2013	−1.37034
$905$	0	0
$906$	−21.1705	−0.703342
$907$	− 0.966094i	− 0.0320786i	−0.999871	−	0.0160393i	$-0.994894\pi$
$907$	− 0.966094i	− 0.0320786i	0.999871	−	0.0160393i	$-0.00510569\pi$
$908$	11.8019i	0.391659i
$909$	−6.88004	−0.228197
$910$	0	0
$911$	−21.2288	−0.703340	−0.351670	−	0.936124i	$-0.614386\pi$
$911$	−21.2288	−0.703340	−0.351670	+	0.936124i	$0.614386\pi$
$912$	− 0.0194878i	0 0.000645304i
$913$	− 15.1313i	− 0.500774i
$914$	17.2259	0.569781
$915$	0	0
$916$	−31.6128	−1.04452
$917$	− 48.8709i	− 1.61386i
$918$	10.6434i	0.351284i
$919$	−16.9547	−0.559284	−0.279642	−	0.960104i	$-0.590216\pi$
$919$	−16.9547	−0.559284	−0.279642	+	0.960104i	$0.590216\pi$
$920$	0	0
$921$	−29.3925	−0.968515
$922$	− 12.6617i	− 0.416991i
$923$	− 9.99377i	− 0.328949i
$924$	9.08243	0.298790
$925$	0	0
$926$	−1.00227	−0.0329368
$927$	2.81825i	0.0925634i
$928$	5.75969i	0.189071i
$929$	26.0792	0.855632	0.427816	−	0.903866i	$-0.359283\pi$
$929$	26.0792	0.855632	0.427816	+	0.903866i	$0.359283\pi$
$930$	0	0
$931$	0.0338799	0.00111037
$932$	24.1579i	0.791318i
$933$	24.7278i	0.809552i
$934$	5.94365	0.194482
$935$	0	0
$936$	−1.60490	−0.0524577
$937$	− 6.07903i	− 0.198593i	−0.995058	−	0.0992967i	$-0.968341\pi$
$937$	− 6.07903i	− 0.198593i	0.995058	−	0.0992967i	$-0.0316593\pi$
$938$	− 0.603196i	− 0.0196950i
$939$	47.3590	1.54550
$940$	0	0
$941$	49.5166	1.61419	0.807097	−	0.590419i	$-0.201037\pi$
$941$	49.5166	1.61419	0.807097	+	0.590419i	$0.201037\pi$
$942$	22.5139i	0.733542i
$943$	− 13.4173i	− 0.436926i
$944$	−1.25496	−0.0408454
$945$	0	0
$946$	−12.0222	−0.390877
$947$	26.9125i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$947$	26.9125i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$948$	13.2766i	0.431205i
$949$	−0.0131124	−0.000425645 0
$950$	0	0
$951$	32.5795	1.05646
$952$	19.1487i	0.620613i
$953$	30.8022i	0.997781i	0.866665	+	0.498891i	$0.166259\pi$
$953$	30.8022i	0.997781i	−0.866665	+	0.498891i	$0.833741\pi$
$954$	0.700335	0.0226742
$955$	0	0
$956$	−2.19725	−0.0710642
$957$	− 2.83798i	− 0.0917389i
$958$	32.6067i	1.05347i
$959$	2.35683	0.0761059
$960$	0	0
$961$	−26.3445	−0.849821
$962$	− 5.79546i	− 0.186853i
$963$	− 0.173506i	− 0.00559115i
$964$	11.1183	0.358098
$965$	0	0
$966$	−26.7886	−0.861908
$967$	− 50.1110i	− 1.61146i	−0.592283	−	0.805730i	$-0.701773\pi$
$967$	− 50.1110i	− 1.61146i	0.592283	−	0.805730i	$-0.298227\pi$
$968$	24.2694i	0.780047i
$969$	−0.199448	−0.00640719
$970$	0	0
$971$	−13.0301	−0.418156	−0.209078	−	0.977899i	$-0.567046\pi$
$971$	−13.0301	−0.418156	−0.209078	+	0.977899i	$0.567046\pi$
$972$	8.28385i	0.265705i
$973$	− 55.6315i	− 1.78346i
$974$	−25.4201	−0.814514
$975$	0	0
$976$	3.33111	0.106626
$977$	3.50116i	0.112012i	0.998430	+	0.0560060i	$0.0178366\pi$
$977$	3.50116i	0.112012i	−0.998430	+	0.0560060i	$0.982163\pi$
$978$	− 27.5062i	− 0.879550i
$979$	−10.2582	−0.327854
$980$	0	0
$981$	−10.4045	−0.332191
$982$	8.25748i	0.263507i
$983$	− 41.2281i	− 1.31497i	−0.753467	−	0.657486i	$-0.771620\pi$
$983$	− 41.2281i	− 1.31497i	0.753467	−	0.657486i	$-0.228380\pi$
$984$	10.3904	0.331234
$985$	0	0
$986$	2.35330	0.0749443
$987$	− 24.3145i	− 0.773941i
$988$	0.0450115i	0.00143201i
$989$	−65.3569	−2.07823
$990$	0	0
$991$	3.17997	0.101015	0.0505076	−	0.998724i	$-0.483916\pi$
$991$	3.17997	0.101015	0.0505076	+	0.998724i	$0.483916\pi$
$992$	− 12.4275i	− 0.394574i
$993$	49.3460i	1.56595i
$994$	22.2521	0.705793
$995$	0	0
$996$	25.0543	0.793876
$997$	− 33.8545i	− 1.07218i	−0.844160	−	0.536092i	$-0.819900\pi$
$997$	− 33.8545i	− 1.07218i	0.844160	−	0.536092i	$-0.180100\pi$
$998$	17.5895i	0.556785i
$999$	−33.6123	−1.06345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.b.f.349.5		10
5.2	odd	4		725.2.a.k.1.3	yes	5
5.3	odd	4		725.2.a.h.1.3	✓	5
5.4	even	2	inner	725.2.b.f.349.6		10
15.2	even	4		6525.2.a.bm.1.3		5
15.8	even	4		6525.2.a.bq.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
725.2.a.h.1.3	✓	5	5.3	odd	4
725.2.a.k.1.3	yes	5	5.2	odd	4
725.2.b.f.349.5		10	1.1	even	1	trivial
725.2.b.f.349.6		10	5.4	even	2	inner
6525.2.a.bm.1.3		5	15.2	even	4
6525.2.a.bq.1.3		5	15.8	even	4

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:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	725.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.838718i q^{2} -1.90376i q^{3} +1.29655 q^{4} -1.59672 q^{6} -2.46832i q^{7} -2.76488i q^{8} -0.624302 q^{9} -1.49072 q^{11} -2.46832i q^{12} +0.929772i q^{13} -2.07023 q^{14} +0.274153 q^{16} -2.80583i q^{17} +0.523614i q^{18} +0.0373385 q^{19} -4.69910 q^{21} +1.25030i q^{22} +6.79703i q^{23} -5.26366 q^{24} +0.779816 q^{26} -4.52276i q^{27} -3.20031i q^{28} -1.00000 q^{29} +2.15767 q^{31} -5.75969i q^{32} +2.83798i q^{33} -2.35330 q^{34} -0.809441 q^{36} -7.43182i q^{37} -0.0313164i q^{38} +1.77006 q^{39} -1.97399 q^{41} +3.94122i q^{42} +9.61550i q^{43} -1.93280 q^{44} +5.70079 q^{46} +5.17430i q^{47} -0.521922i q^{48} +0.907373 q^{49} -5.34162 q^{51} +1.20550i q^{52} -1.33750i q^{53} -3.79332 q^{54} -6.82462 q^{56} -0.0710835i q^{57} +0.838718i q^{58} -4.57758 q^{59} +12.1506 q^{61} -1.80968i q^{62} +1.54098i q^{63} -4.28245 q^{64} +2.38027 q^{66} +0.291367i q^{67} -3.63790i q^{68} +12.9399 q^{69} -10.7486 q^{71} +1.72612i q^{72} +0.0141028i q^{73} -6.23320 q^{74} +0.0484113 q^{76} +3.67959i q^{77} -1.48458i q^{78} -5.37881 q^{79} -10.4832 q^{81} +1.65562i q^{82} +10.1503i q^{83} -6.09263 q^{84} +8.06469 q^{86} +1.90376i q^{87} +4.12167i q^{88} +6.88136 q^{89} +2.29498 q^{91} +8.81271i q^{92} -4.10769i q^{93} +4.33978 q^{94} -10.9651 q^{96} +9.42386i q^{97} -0.761030i q^{98} +0.930663 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{4} - 2 q^{6} - 22 q^{9} - 4 q^{11} - 22 q^{14} - 20 q^{16} - 4 q^{19} - 2 q^{21} - 26 q^{24} - 32 q^{26} - 10 q^{29} - 2 q^{31} - 6 q^{34} - 26 q^{36} + 12 q^{39} + 10 q^{41} - 14 q^{44} - 40 q^{46}+ \cdots + 52 q^{99}+O(q^{100}) \) 10 * q - 8 * q^4 - 2 * q^6 - 22 * q^9 - 4 * q^11 - 22 * q^14 - 20 * q^16 - 4 * q^19 - 2 * q^21 - 26 * q^24 - 32 * q^26 - 10 * q^29 - 2 * q^31 - 6 * q^34 - 26 * q^36 + 12 * q^39 + 10 * q^41 - 14 * q^44 - 40 * q^46 + 6 * q^49 + 10 * q^51 - 20 * q^54 - 32 * q^56 + 22 * q^59 - 10 * q^61 + 10 * q^64 - 54 * q^66 + 52 * q^69 - 10 * q^71 - 16 * q^74 + 32 * q^76 + 20 * q^79 + 98 * q^81 - 64 * q^84 - 26 * q^86 + 36 * q^89 + 2 * q^91 + 26 * q^94 - 76 * q^96 + 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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