LMFDB - Embedded newform 775.2.b.g.249.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [775,2,Mod(249,775)] 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("775.249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(775, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	775.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-12,0,-2,0,0,-12] 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:	$6.18840615665$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.21295345337344.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 12x^{8} + 48x^{6} + 73x^{4} + 34x^{2} + 1 $ x^10 + 12x^8 + 48x^6 + 73x^4 + 34x^2 + 1
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			249.10
Root			$0.871612i$ of defining polynomial
Character	$\chi$	$=$	775.249
Dual form			775.2.b.g.249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.58398i q^{2} -2.24029i q^{3} -4.67698 q^{4} +5.78888 q^{6} -2.11190i q^{7} -6.91727i q^{8} -2.01891 q^{9} -4.70783 q^{11} +10.4778i q^{12} +3.53665i q^{13} +5.45713 q^{14} +8.52017 q^{16} +6.45560i q^{17} -5.21684i q^{18} -0.766680 q^{19} -4.73128 q^{21} -12.1650i q^{22} +5.38481i q^{23} -15.4967 q^{24} -9.13865 q^{26} -2.19793i q^{27} +9.87733i q^{28} -8.03414 q^{29} -1.00000 q^{31} +8.18144i q^{32} +10.5469i q^{33} -16.6812 q^{34} +9.44240 q^{36} +11.5690i q^{37} -1.98109i q^{38} +7.92313 q^{39} -6.47040 q^{41} -12.2256i q^{42} +5.80082i q^{43} +22.0184 q^{44} -13.9143 q^{46} -8.68724i q^{47} -19.0877i q^{48} +2.53986 q^{49} +14.4624 q^{51} -16.5408i q^{52} -4.05063i q^{53} +5.67941 q^{54} -14.6086 q^{56} +1.71759i q^{57} -20.7601i q^{58} -9.36743 q^{59} +3.40129 q^{61} -2.58398i q^{62} +4.26375i q^{63} -4.10039 q^{64} -27.2531 q^{66} +7.82514i q^{67} -30.1927i q^{68} +12.0635 q^{69} -3.15106 q^{71} +13.9654i q^{72} -13.4359i q^{73} -29.8942 q^{74} +3.58574 q^{76} +9.94249i q^{77} +20.4733i q^{78} +11.6812 q^{79} -10.9807 q^{81} -16.7194i q^{82} -9.79474i q^{83} +22.1281 q^{84} -14.9892 q^{86} +17.9988i q^{87} +32.5653i q^{88} +0.211117 q^{89} +7.46907 q^{91} -25.1846i q^{92} +2.24029i q^{93} +22.4477 q^{94} +18.3288 q^{96} +7.13932i q^{97} +6.56296i q^{98} +9.50469 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 12 q^{4} - 2 q^{6} - 12 q^{9} - 4 q^{11} - 4 q^{14} + 8 q^{16} - 16 q^{19} - 30 q^{21} - 52 q^{24} - 32 q^{26} - 12 q^{29} - 10 q^{31} - 62 q^{34} - 26 q^{36} + 26 q^{39} - 4 q^{41} + 8 q^{44} - 22 q^{46}+ \cdots + 66 q^{99}+O(q^{100}) $ 10 * q - 12 * q^4 - 2 * q^6 - 12 * q^9 - 4 * q^11 - 4 * q^14 + 8 * q^16 - 16 * q^19 - 30 * q^21 - 52 * q^24 - 32 * q^26 - 12 * q^29 - 10 * q^31 - 62 * q^34 - 26 * q^36 + 26 * q^39 - 4 * q^41 + 8 * q^44 - 22 * q^46 + 18 * q^49 + 10 * q^51 - 20 * q^54 - 18 * q^56 + 8 * q^59 - 10 * q^61 + 10 * q^64 - 50 * q^66 - 20 * q^69 + 12 * q^71 - 8 * q^74 - 10 * q^76 + 12 * q^79 - 14 * q^81 + 30 * q^84 - 14 * q^86 + 62 * q^89 + 16 * q^91 + 28 * q^94 + 20 * q^96 + 66 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$.

$n$	$251$	$652$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.58398i	1.82715i	0.406667	+	0.913577i	$0.366691\pi$
$2$	2.58398i	1.82715i	−0.406667	+	0.913577i	$0.633309\pi$
$3$	− 2.24029i	− 1.29343i	−0.762730	−	0.646717i	$-0.776142\pi$
$3$	− 2.24029i	− 1.29343i	0.762730	−	0.646717i	$-0.223858\pi$
$4$	−4.67698	−2.33849
$5$	0	0
$5$	0	0
$6$	5.78888	2.36330
$7$	− 2.11190i	− 0.798225i	−0.916902	−	0.399112i	$-0.869318\pi$
$7$	− 2.11190i	− 0.798225i	0.916902	−	0.399112i	$-0.130682\pi$
$8$	− 6.91727i	− 2.44562i
$9$	−2.01891	−0.672971
$10$	0	0
$11$	−4.70783	−1.41946	−0.709732	−	0.704472i	$-0.751184\pi$
$11$	−4.70783	−1.41946	−0.709732	+	0.704472i	$0.751184\pi$
$12$	10.4778i	3.02468i
$13$	3.53665i	0.980890i	0.871472	+	0.490445i	$0.163166\pi$
$13$	3.53665i	0.980890i	−0.871472	+	0.490445i	$0.836834\pi$
$14$	5.45713	1.45848
$15$	0	0
$16$	8.52017	2.13004
$17$	6.45560i	1.56571i	0.622203	+	0.782856i	$0.286238\pi$
$17$	6.45560i	1.56571i	−0.622203	+	0.782856i	$0.713762\pi$
$18$	− 5.21684i	− 1.22962i
$19$	−0.766680	−0.175888	−0.0879442	−	0.996125i	$-0.528030\pi$
$19$	−0.766680	−0.175888	−0.0879442	+	0.996125i	$0.528030\pi$
$20$	0	0
$21$	−4.73128	−1.03245
$22$	− 12.1650i	− 2.59358i
$23$	5.38481i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$23$	5.38481i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$24$	−15.4967	−3.16325
$25$	0	0
$26$	−9.13865	−1.79224
$27$	− 2.19793i	− 0.422991i
$28$	9.87733i	1.86664i
$29$	−8.03414	−1.49190	−0.745951	−	0.666000i	$-0.768005\pi$
$29$	−8.03414	−1.49190	−0.745951	+	0.666000i	$0.768005\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	8.18144i	1.44629i
$33$	10.5469i	1.83598i
$34$	−16.6812	−2.86080
$35$	0	0
$36$	9.44240	1.57373
$37$	11.5690i	1.90194i	0.309287	+	0.950969i	$0.399910\pi$
$37$	11.5690i	1.90194i	−0.309287	+	0.950969i	$0.600090\pi$
$38$	− 1.98109i	− 0.321375i
$39$	7.92313	1.26872
$40$	0	0
$41$	−6.47040	−1.01051	−0.505254	−	0.862971i	$-0.668601\pi$
$41$	−6.47040	−1.01051	−0.505254	+	0.862971i	$0.668601\pi$
$42$	− 12.2256i	− 1.88645i
$43$	5.80082i	0.884617i	0.896863	+	0.442309i	$0.145840\pi$
$43$	5.80082i	0.884617i	−0.896863	+	0.442309i	$0.854160\pi$
$44$	22.0184	3.31940
$45$	0	0
$46$	−13.9143	−2.05155
$47$	− 8.68724i	− 1.26716i	−0.773675	−	0.633582i	$-0.781584\pi$
$47$	− 8.68724i	− 1.26716i	0.773675	−	0.633582i	$-0.218416\pi$
$48$	− 19.0877i	− 2.75507i
$49$	2.53986	0.362837
$50$	0	0
$51$	14.4624	2.02514
$52$	− 16.5408i	− 2.29380i
$53$	− 4.05063i	− 0.556396i	−0.960524	−	0.278198i	$-0.910263\pi$
$53$	− 4.05063i	− 0.556396i	0.960524	−	0.278198i	$-0.0897372\pi$
$54$	5.67941	0.772869
$55$	0	0
$56$	−14.6086	−1.95216
$57$	1.71759i	0.227500i
$58$	− 20.7601i	− 2.72593i
$59$	−9.36743	−1.21954	−0.609768	−	0.792580i	$-0.708737\pi$
$59$	−9.36743	−1.21954	−0.609768	+	0.792580i	$0.708737\pi$
$60$	0	0
$61$	3.40129	0.435491	0.217745	−	0.976006i	$-0.430130\pi$
$61$	3.40129	0.435491	0.217745	+	0.976006i	$0.430130\pi$
$62$	− 2.58398i	− 0.328166i
$63$	4.26375i	0.537182i
$64$	−4.10039	−0.512548
$65$	0	0
$66$	−27.2531	−3.35462
$67$	7.82514i	0.955993i	0.878362	+	0.477997i	$0.158637\pi$
$67$	7.82514i	0.955993i	−0.878362	+	0.477997i	$0.841363\pi$
$68$	− 30.1927i	− 3.66140i
$69$	12.0635	1.45228
$70$	0	0
$71$	−3.15106	−0.373962	−0.186981	−	0.982363i	$-0.559870\pi$
$71$	−3.15106	−0.373962	−0.186981	+	0.982363i	$0.559870\pi$
$72$	13.9654i	1.64583i
$73$	− 13.4359i	− 1.57255i	−0.617874	−	0.786277i	$-0.712006\pi$
$73$	− 13.4359i	− 1.57255i	0.617874	−	0.786277i	$-0.287994\pi$
$74$	−29.8942	−3.47513
$75$	0	0
$76$	3.58574	0.411313
$77$	9.94249i	1.13305i
$78$	20.4733i	2.31814i
$79$	11.6812	1.31423	0.657117	−	0.753789i	$-0.271776\pi$
$79$	11.6812	1.31423	0.657117	+	0.753789i	$0.271776\pi$
$80$	0	0
$81$	−10.9807	−1.22008
$82$	− 16.7194i	− 1.84635i
$83$	− 9.79474i	− 1.07511i	−0.843228	−	0.537556i	$-0.819347\pi$
$83$	− 9.79474i	− 1.07511i	0.843228	−	0.537556i	$-0.180653\pi$
$84$	22.1281	2.41438
$85$	0	0
$86$	−14.9892	−1.61633
$87$	17.9988i	1.92968i
$88$	32.5653i	3.47148i
$89$	0.211117	0.0223784	0.0111892	−	0.999937i	$-0.496438\pi$
$89$	0.211117	0.0223784	0.0111892	+	0.999937i	$0.496438\pi$
$90$	0	0
$91$	7.46907	0.782971
$92$	− 25.1846i	− 2.62568i
$93$	2.24029i	0.232308i
$94$	22.4477	2.31530
$95$	0	0
$96$	18.3288	1.87068
$97$	7.13932i	0.724888i	0.932006	+	0.362444i	$0.118058\pi$
$97$	7.13932i	0.724888i	−0.932006	+	0.362444i	$0.881942\pi$
$98$	6.56296i	0.662959i
$99$	9.50469	0.955257
$100$	0	0
$101$	−8.60641	−0.856370	−0.428185	−	0.903691i	$-0.640847\pi$
$101$	−8.60641	−0.856370	−0.428185	+	0.903691i	$0.640847\pi$
$102$	37.3707i	3.70025i
$103$	− 1.57511i	− 0.155200i	−0.996985	−	0.0776002i	$-0.975274\pi$
$103$	− 1.57511i	− 0.155200i	0.996985	−	0.0776002i	$-0.0247258\pi$
$104$	24.4640	2.39889
$105$	0	0
$106$	10.4668	1.01662
$107$	1.24980i	0.120823i	0.998174	+	0.0604116i	$0.0192413\pi$
$107$	1.24980i	0.120823i	−0.998174	+	0.0604116i	$0.980759\pi$
$108$	10.2796i	0.989160i
$109$	−10.6818	−1.02313	−0.511566	−	0.859244i	$-0.670934\pi$
$109$	−10.6818	−1.02313	−0.511566	+	0.859244i	$0.670934\pi$
$110$	0	0
$111$	25.9180	2.46003
$112$	− 17.9938i	− 1.70025i
$113$	− 0.755521i	− 0.0710734i	−0.999368	−	0.0355367i	$-0.988686\pi$
$113$	− 0.755521i	− 0.0710734i	0.999368	−	0.0355367i	$-0.0113141\pi$
$114$	−4.43822	−0.415677
$115$	0	0
$116$	37.5755	3.48880
$117$	− 7.14018i	− 0.660110i
$118$	− 24.2053i	− 2.22828i
$119$	13.6336	1.24979
$120$	0	0
$121$	11.1637	1.01488
$122$	8.78888i	0.795708i
$123$	14.4956i	1.30702i
$124$	4.67698	0.420005
$125$	0	0
$126$	−11.0175	−0.981514
$127$	− 7.11477i	− 0.631334i	−0.948870	−	0.315667i	$-0.897772\pi$
$127$	− 7.11477i	− 0.631334i	0.948870	−	0.315667i	$-0.102228\pi$
$128$	5.76754i	0.509784i
$129$	12.9955	1.14419
$130$	0	0
$131$	−6.39521	−0.558752	−0.279376	−	0.960182i	$-0.590128\pi$
$131$	−6.39521	−0.558752	−0.279376	+	0.960182i	$0.590128\pi$
$132$	− 49.3277i	− 4.29342i
$133$	1.61915i	0.140398i
$134$	−20.2200	−1.74675
$135$	0	0
$136$	44.6551	3.82914
$137$	13.6264i	1.16418i	0.813125	+	0.582089i	$0.197764\pi$
$137$	13.6264i	1.16418i	−0.813125	+	0.582089i	$0.802236\pi$
$138$	31.1720i	2.65354i
$139$	0.912415	0.0773900	0.0386950	−	0.999251i	$-0.487680\pi$
$139$	0.912415	0.0773900	0.0386950	+	0.999251i	$0.487680\pi$
$140$	0	0
$141$	−19.4620	−1.63899
$142$	− 8.14230i	− 0.683287i
$143$	− 16.6499i	− 1.39234i
$144$	−17.2015	−1.43346
$145$	0	0
$146$	34.7182	2.87330
$147$	− 5.69003i	− 0.469306i
$148$	− 54.1081i	− 4.44766i
$149$	6.88466	0.564013	0.282007	−	0.959412i	$-0.409000\pi$
$149$	6.88466	0.564013	0.282007	+	0.959412i	$0.409000\pi$
$150$	0	0
$151$	−15.4481	−1.25715	−0.628575	−	0.777749i	$-0.716362\pi$
$151$	−15.4481	−1.25715	−0.628575	+	0.777749i	$0.716362\pi$
$152$	5.30333i	0.430157i
$153$	− 13.0333i	− 1.05368i
$154$	−25.6912	−2.07026
$155$	0	0
$156$	−37.0563	−2.96688
$157$	2.73611i	0.218365i	0.994022	+	0.109183i	$0.0348233\pi$
$157$	2.73611i	0.218365i	−0.994022	+	0.109183i	$0.965177\pi$
$158$	30.1840i	2.40131i
$159$	−9.07459	−0.719662
$160$	0	0
$161$	11.3722	0.896255
$162$	− 28.3740i	− 2.22928i
$163$	− 6.17569i	− 0.483718i	−0.970311	−	0.241859i	$-0.922243\pi$
$163$	− 6.17569i	− 0.483718i	0.970311	−	0.241859i	$-0.0777571\pi$
$164$	30.2619	2.36306
$165$	0	0
$166$	25.3095	1.96440
$167$	17.1559i	1.32756i	0.747926	+	0.663782i	$0.231050\pi$
$167$	17.1559i	1.32756i	−0.747926	+	0.663782i	$0.768950\pi$
$168$	32.7276i	2.52499i
$169$	0.492102	0.0378540
$170$	0	0
$171$	1.54786	0.118368
$172$	− 27.1303i	− 2.06867i
$173$	6.07994i	0.462250i	0.972924	+	0.231125i	$0.0742406\pi$
$173$	6.07994i	0.462250i	−0.972924	+	0.231125i	$0.925759\pi$
$174$	−46.5087	−3.52582
$175$	0	0
$176$	−40.1115	−3.02352
$177$	20.9858i	1.57739i
$178$	0.545524i	0.0408888i
$179$	−17.8901	−1.33717	−0.668583	−	0.743638i	$-0.733099\pi$
$179$	−17.8901	−1.33717	−0.668583	+	0.743638i	$0.733099\pi$
$180$	0	0
$181$	−18.3051	−1.36061	−0.680303	−	0.732931i	$-0.738152\pi$
$181$	−18.3051	−1.36061	−0.680303	+	0.732931i	$0.738152\pi$
$182$	19.3000i	1.43061i
$183$	− 7.61989i	− 0.563278i
$184$	37.2482	2.74597
$185$	0	0
$186$	−5.78888	−0.424461
$187$	− 30.3918i	− 2.22247i
$188$	40.6300i	2.96325i
$189$	−4.64181	−0.337642
$190$	0	0
$191$	1.09600	0.0793041	0.0396521	−	0.999214i	$-0.487375\pi$
$191$	1.09600	0.0793041	0.0396521	+	0.999214i	$0.487375\pi$
$192$	9.18607i	0.662947i
$193$	15.5155i	1.11683i	0.829562	+	0.558414i	$0.188590\pi$
$193$	15.5155i	1.11683i	−0.829562	+	0.558414i	$0.811410\pi$
$194$	−18.4479	−1.32448
$195$	0	0
$196$	−11.8789	−0.848490
$197$	12.2418i	0.872194i	0.899900	+	0.436097i	$0.143640\pi$
$197$	12.2418i	0.872194i	−0.899900	+	0.436097i	$0.856360\pi$
$198$	24.5600i	1.74540i
$199$	6.49517	0.460430	0.230215	−	0.973140i	$-0.426057\pi$
$199$	6.49517	0.460430	0.230215	+	0.973140i	$0.426057\pi$
$200$	0	0
$201$	17.5306	1.23651
$202$	− 22.2388i	− 1.56472i
$203$	16.9673i	1.19087i
$204$	−67.6405	−4.73578
$205$	0	0
$206$	4.07006	0.283575
$207$	− 10.8714i	− 0.755618i
$208$	30.1329i	2.08934i
$209$	3.60940	0.249667
$210$	0	0
$211$	8.40626	0.578711	0.289355	−	0.957222i	$-0.406559\pi$
$211$	8.40626	0.578711	0.289355	+	0.957222i	$0.406559\pi$
$212$	18.9447i	1.30113i
$213$	7.05931i	0.483696i
$214$	−3.22947	−0.220762
$215$	0	0
$216$	−15.2036	−1.03448
$217$	2.11190i	0.143365i
$218$	− 27.6016i	− 1.86942i
$219$	−30.1004	−2.03399
$220$	0	0
$221$	−22.8312	−1.53579
$222$	66.9718i	4.49485i
$223$	− 10.1283i	− 0.678238i	−0.940743	−	0.339119i	$-0.889871\pi$
$223$	− 10.1283i	− 0.678238i	0.940743	−	0.339119i	$-0.110129\pi$
$224$	17.2784	1.15446
$225$	0	0
$226$	1.95225	0.129862
$227$	− 3.73800i	− 0.248100i	−0.992276	−	0.124050i	$-0.960412\pi$
$227$	− 3.73800i	− 0.248100i	0.992276	−	0.124050i	$-0.0395883\pi$
$228$	− 8.03312i	− 0.532006i
$229$	19.7631	1.30598	0.652990	−	0.757367i	$-0.273515\pi$
$229$	19.7631	1.30598	0.652990	+	0.757367i	$0.273515\pi$
$230$	0	0
$231$	22.2741	1.46553
$232$	55.5743i	3.64863i
$233$	− 26.5881i	− 1.74184i	−0.491423	−	0.870921i	$-0.663523\pi$
$233$	− 26.5881i	− 1.74184i	0.491423	−	0.870921i	$-0.336477\pi$
$234$	18.4501	1.20612
$235$	0	0
$236$	43.8113	2.85187
$237$	− 26.1692i	− 1.69987i
$238$	35.2290i	2.28356i
$239$	16.6224	1.07522	0.537608	−	0.843195i	$-0.319328\pi$
$239$	16.6224	1.07522	0.537608	+	0.843195i	$0.319328\pi$
$240$	0	0
$241$	−0.0903393	−0.00581926	−0.00290963	−	0.999996i	$-0.500926\pi$
$241$	−0.0903393	−0.00581926	−0.00290963	+	0.999996i	$0.500926\pi$
$242$	28.8467i	1.85434i
$243$	18.0063i	1.15510i
$244$	−15.9078	−1.01839
$245$	0	0
$246$	−37.4564	−2.38813
$247$	− 2.71148i	− 0.172527i
$248$	6.91727i	0.439247i
$249$	−21.9431	−1.39059
$250$	0	0
$251$	3.67391	0.231895	0.115948	−	0.993255i	$-0.463010\pi$
$251$	3.67391	0.231895	0.115948	+	0.993255i	$0.463010\pi$
$252$	− 19.9415i	− 1.25619i
$253$	− 25.3507i	− 1.59379i
$254$	18.3845	1.15354
$255$	0	0
$256$	−23.1040	−1.44400
$257$	− 31.5960i	− 1.97090i	−0.169960	−	0.985451i	$-0.554364\pi$
$257$	− 31.5960i	− 1.97090i	0.169960	−	0.985451i	$-0.445636\pi$
$258$	33.5803i	2.09062i
$259$	24.4327	1.51817
$260$	0	0
$261$	16.2202	1.00401
$262$	− 16.5251i	− 1.02093i
$263$	− 20.6444i	− 1.27299i	−0.771281	−	0.636495i	$-0.780384\pi$
$263$	− 20.6444i	− 1.27299i	0.771281	−	0.636495i	$-0.219616\pi$
$264$	72.9559	4.49012
$265$	0	0
$266$	−4.18387	−0.256530
$267$	− 0.472965i	− 0.0289450i
$268$	− 36.5980i	− 2.23558i
$269$	21.8419	1.33172	0.665862	−	0.746075i	$-0.268064\pi$
$269$	21.8419	1.33172	0.665862	+	0.746075i	$0.268064\pi$
$270$	0	0
$271$	9.08392	0.551809	0.275905	−	0.961185i	$-0.411023\pi$
$271$	9.08392	0.551809	0.275905	+	0.961185i	$0.411023\pi$
$272$	55.0028i	3.33503i
$273$	− 16.7329i	− 1.01272i
$274$	−35.2103	−2.12713
$275$	0	0
$276$	−56.4209	−3.39614
$277$	− 22.4910i	− 1.35135i	−0.737199	−	0.675676i	$-0.763852\pi$
$277$	− 22.4910i	− 1.35135i	0.737199	−	0.675676i	$-0.236148\pi$
$278$	2.35767i	0.141403i
$279$	2.01891	0.120869
$280$	0	0
$281$	−9.88551	−0.589720	−0.294860	−	0.955540i	$-0.595273\pi$
$281$	−9.88551	−0.589720	−0.294860	+	0.955540i	$0.595273\pi$
$282$	− 50.2894i	− 2.99469i
$283$	4.56044i	0.271090i	0.990771	+	0.135545i	$0.0432785\pi$
$283$	4.56044i	0.271090i	−0.990771	+	0.135545i	$0.956722\pi$
$284$	14.7375	0.874507
$285$	0	0
$286$	43.0232	2.54402
$287$	13.6649i	0.806612i
$288$	− 16.5176i	− 0.973309i
$289$	−24.6747	−1.45145
$290$	0	0
$291$	15.9942	0.937595
$292$	62.8394i	3.67740i
$293$	32.9820i	1.92683i	0.268015	+	0.963415i	$0.413632\pi$
$293$	32.9820i	1.92683i	−0.268015	+	0.963415i	$0.586368\pi$
$294$	14.7029	0.857493
$295$	0	0
$296$	80.0261	4.65143
$297$	10.3475i	0.600420i
$298$	17.7899i	1.03054i
$299$	−19.0442	−1.10135
$300$	0	0
$301$	12.2508	0.706123
$302$	− 39.9177i	− 2.29701i
$303$	19.2809i	1.10766i
$304$	−6.53224	−0.374650
$305$	0	0
$306$	33.6778	1.92523
$307$	− 11.2816i	− 0.643877i	−0.946761	−	0.321939i	$-0.895665\pi$
$307$	− 11.2816i	− 0.643877i	0.946761	−	0.321939i	$-0.104335\pi$
$308$	− 46.5008i	− 2.64963i
$309$	−3.52871	−0.200741
$310$	0	0
$311$	−12.2410	−0.694126	−0.347063	−	0.937842i	$-0.612821\pi$
$311$	−12.2410	−0.694126	−0.347063	+	0.937842i	$0.612821\pi$
$312$	− 54.8065i	− 3.10280i
$313$	10.0264i	0.566728i	0.959012	+	0.283364i	$0.0914504\pi$
$313$	10.0264i	0.566728i	−0.959012	+	0.283364i	$0.908550\pi$
$314$	−7.07006	−0.398987
$315$	0	0
$316$	−54.6325	−3.07332
$317$	2.30927i	0.129702i	0.997895	+	0.0648509i	$0.0206572\pi$
$317$	2.30927i	0.129702i	−0.997895	+	0.0648509i	$0.979343\pi$
$318$	− 23.4486i	− 1.31493i
$319$	37.8234	2.11770
$320$	0	0
$321$	2.79993	0.156277
$322$	29.3856i	1.63759i
$323$	− 4.94937i	− 0.275391i
$324$	51.3566	2.85315
$325$	0	0
$326$	15.9579	0.883826
$327$	23.9304i	1.32335i
$328$	44.7575i	2.47132i
$329$	−18.3466	−1.01148
$330$	0	0
$331$	9.04859	0.497356	0.248678	−	0.968586i	$-0.420004\pi$
$331$	9.04859	0.497356	0.248678	+	0.968586i	$0.420004\pi$
$332$	45.8098i	2.51414i
$333$	− 23.3569i	− 1.27995i
$334$	−44.3306	−2.42566
$335$	0	0
$336$	−40.3113	−2.19916
$337$	26.4888i	1.44294i	0.692448	+	0.721468i	$0.256532\pi$
$337$	26.4888i	1.44294i	−0.692448	+	0.721468i	$0.743468\pi$
$338$	1.27158i	0.0691651i
$339$	−1.69259	−0.0919287
$340$	0	0
$341$	4.70783	0.254943
$342$	3.99964i	0.216276i
$343$	− 20.1473i	− 1.08785i
$344$	40.1259	2.16344
$345$	0	0
$346$	−15.7105	−0.844601
$347$	37.0463i	1.98875i	0.105927	+	0.994374i	$0.466219\pi$
$347$	37.0463i	1.98875i	−0.105927	+	0.994374i	$0.533781\pi$
$348$	− 84.1801i	− 4.51253i
$349$	17.5804	0.941059	0.470529	−	0.882384i	$-0.344063\pi$
$349$	17.5804	0.941059	0.470529	+	0.882384i	$0.344063\pi$
$350$	0	0
$351$	7.77329	0.414908
$352$	− 38.5168i	− 2.05295i
$353$	− 24.2026i	− 1.28817i	−0.764952	−	0.644087i	$-0.777237\pi$
$353$	− 24.2026i	− 1.28817i	0.764952	−	0.644087i	$-0.222763\pi$
$354$	−54.2269	−2.88213
$355$	0	0
$356$	−0.987391	−0.0523316
$357$	− 30.5433i	− 1.61652i
$358$	− 46.2276i	− 2.44321i
$359$	−13.0616	−0.689366	−0.344683	−	0.938719i	$-0.612014\pi$
$359$	−13.0616	−0.689366	−0.344683	+	0.938719i	$0.612014\pi$
$360$	0	0
$361$	−18.4122	−0.969063
$362$	− 47.3000i	− 2.48604i
$363$	− 25.0098i	− 1.31268i
$364$	−34.9327	−1.83097
$365$	0	0
$366$	19.6897	1.02920
$367$	− 21.4926i	− 1.12191i	−0.827848	−	0.560953i	$-0.810435\pi$
$367$	− 21.4926i	− 1.12191i	0.827848	−	0.560953i	$-0.189565\pi$
$368$	45.8795i	2.39163i
$369$	13.0632	0.680042
$370$	0	0
$371$	−8.55454	−0.444129
$372$	− 10.4778i	− 0.543249i
$373$	− 7.79176i	− 0.403442i	−0.979443	−	0.201721i	$-0.935347\pi$
$373$	− 7.79176i	− 0.403442i	0.979443	−	0.201721i	$-0.0646534\pi$
$374$	78.5321	4.06080
$375$	0	0
$376$	−60.0920	−3.09901
$377$	− 28.4140i	− 1.46339i
$378$	− 11.9944i	− 0.616924i
$379$	−23.8722	−1.22623	−0.613116	−	0.789993i	$-0.710084\pi$
$379$	−23.8722	−1.22623	−0.613116	+	0.789993i	$0.710084\pi$
$380$	0	0
$381$	−15.9392	−0.816589
$382$	2.83206i	0.144901i
$383$	8.53896i	0.436321i	0.975913	+	0.218160i	$0.0700056\pi$
$383$	8.53896i	0.436321i	−0.975913	+	0.218160i	$0.929994\pi$
$384$	12.9210	0.659371
$385$	0	0
$386$	−40.0918	−2.04062
$387$	− 11.7113i	− 0.595321i
$388$	− 33.3904i	− 1.69514i
$389$	−2.02322	−0.102581	−0.0512906	−	0.998684i	$-0.516333\pi$
$389$	−2.02322	−0.102581	−0.0512906	+	0.998684i	$0.516333\pi$
$390$	0	0
$391$	−34.7621	−1.75800
$392$	− 17.5689i	− 0.887363i
$393$	14.3271i	0.722709i
$394$	−31.6327	−1.59363
$395$	0	0
$396$	−44.4532	−2.23386
$397$	− 8.41708i	− 0.422441i	−0.977438	−	0.211220i	$-0.932256\pi$
$397$	− 8.41708i	− 0.422441i	0.977438	−	0.211220i	$-0.0677438\pi$
$398$	16.7834i	0.841277i
$399$	3.62738	0.181596
$400$	0	0
$401$	9.32972	0.465904	0.232952	−	0.972488i	$-0.425161\pi$
$401$	9.32972	0.465904	0.232952	+	0.972488i	$0.425161\pi$
$402$	45.2988i	2.25930i
$403$	− 3.53665i	− 0.176173i
$404$	40.2520	2.00261
$405$	0	0
$406$	−43.8434	−2.17591
$407$	− 54.4650i	− 2.69973i
$408$	− 100.041i	− 4.95274i
$409$	−22.2516	−1.10027	−0.550136	−	0.835075i	$-0.685424\pi$
$409$	−22.2516	−1.10027	−0.550136	+	0.835075i	$0.685424\pi$
$410$	0	0
$411$	30.5270	1.50579
$412$	7.36676i	0.362934i
$413$	19.7831i	0.973463i
$414$	28.0917	1.38063
$415$	0	0
$416$	−28.9349	−1.41865
$417$	− 2.04408i	− 0.100099i
$418$	9.32663i	0.456180i
$419$	6.87270	0.335753	0.167877	−	0.985808i	$-0.446309\pi$
$419$	6.87270	0.335753	0.167877	+	0.985808i	$0.446309\pi$
$420$	0	0
$421$	20.9630	1.02168	0.510838	−	0.859677i	$-0.329335\pi$
$421$	20.9630	1.02168	0.510838	+	0.859677i	$0.329335\pi$
$422$	21.7216i	1.05739i
$423$	17.5388i	0.852764i
$424$	−28.0193	−1.36074
$425$	0	0
$426$	−18.2411	−0.883786
$427$	− 7.18320i	− 0.347619i
$428$	− 5.84531i	− 0.282544i
$429$	−37.3008	−1.80090
$430$	0	0
$431$	33.2926	1.60365	0.801825	−	0.597559i	$-0.203863\pi$
$431$	33.2926	1.60365	0.801825	+	0.597559i	$0.203863\pi$
$432$	− 18.7267i	− 0.900988i
$433$	0.696968i	0.0334941i	0.999860	+	0.0167471i	$0.00533101\pi$
$433$	0.696968i	0.0334941i	−0.999860	+	0.0167471i	$0.994669\pi$
$434$	−5.45713	−0.261951
$435$	0	0
$436$	49.9586	2.39258
$437$	− 4.12842i	− 0.197489i
$438$	− 77.7789i	− 3.71642i
$439$	17.6634	0.843030	0.421515	−	0.906821i	$-0.361498\pi$
$439$	17.6634	0.843030	0.421515	+	0.906821i	$0.361498\pi$
$440$	0	0
$441$	−5.12775	−0.244179
$442$	− 58.9955i	− 2.80613i
$443$	30.2630i	1.43784i	0.695093	+	0.718920i	$0.255363\pi$
$443$	30.2630i	1.43784i	−0.695093	+	0.718920i	$0.744637\pi$
$444$	−121.218	−5.75275
$445$	0	0
$446$	26.1713	1.23925
$447$	− 15.4237i	− 0.729514i
$448$	8.65963i	0.409129i
$449$	−0.907674	−0.0428358	−0.0214179	−	0.999771i	$-0.506818\pi$
$449$	−0.907674	−0.0428358	−0.0214179	+	0.999771i	$0.506818\pi$
$450$	0	0
$451$	30.4616	1.43438
$452$	3.53355i	0.166204i
$453$	34.6083i	1.62604i
$454$	9.65893	0.453316
$455$	0	0
$456$	11.8810	0.556379
$457$	31.6382i	1.47997i	0.672622	+	0.739987i	$0.265168\pi$
$457$	31.6382i	1.47997i	−0.672622	+	0.739987i	$0.734832\pi$
$458$	51.0674i	2.38622i
$459$	14.1889	0.662282
$460$	0	0
$461$	−4.12145	−0.191955	−0.0959775	−	0.995384i	$-0.530598\pi$
$461$	−4.12145	−0.191955	−0.0959775	+	0.995384i	$0.530598\pi$
$462$	57.5559i	2.67774i
$463$	1.55269i	0.0721598i	0.999349	+	0.0360799i	$0.0114871\pi$
$463$	1.55269i	0.0721598i	−0.999349	+	0.0360799i	$0.988513\pi$
$464$	−68.4522	−3.17782
$465$	0	0
$466$	68.7031	3.18261
$467$	− 0.497828i	− 0.0230367i	−0.999934	−	0.0115184i	$-0.996334\pi$
$467$	− 0.497828i	− 0.0230367i	0.999934	−	0.0115184i	$-0.00366649\pi$
$468$	33.3945i	1.54366i
$469$	16.5260	0.763098
$470$	0	0
$471$	6.12968	0.282441
$472$	64.7970i	2.98253i
$473$	− 27.3093i	− 1.25568i
$474$	67.6209	3.10593
$475$	0	0
$476$	−63.7641	−2.92262
$477$	8.17785i	0.374438i
$478$	42.9521i	1.96459i
$479$	−16.6728	−0.761802	−0.380901	−	0.924616i	$-0.624386\pi$
$479$	−16.6728	−0.761802	−0.380901	+	0.924616i	$0.624386\pi$
$480$	0	0
$481$	−40.9156	−1.86559
$482$	− 0.233435i	− 0.0106327i
$483$	− 25.4771i	− 1.15925i
$484$	−52.2122	−2.37328
$485$	0	0
$486$	−46.5279	−2.11055
$487$	1.58398i	0.0717772i	0.999356	+	0.0358886i	$0.0114262\pi$
$487$	1.58398i	0.0717772i	−0.999356	+	0.0358886i	$0.988574\pi$
$488$	− 23.5276i	− 1.06505i
$489$	−13.8354	−0.625657
$490$	0	0
$491$	21.1723	0.955494	0.477747	−	0.878498i	$-0.341454\pi$
$491$	21.1723	0.955494	0.477747	+	0.878498i	$0.341454\pi$
$492$	− 67.7956i	− 3.05646i
$493$	− 51.8652i	− 2.33589i
$494$	7.00642	0.315234
$495$	0	0
$496$	−8.52017	−0.382567
$497$	6.65475i	0.298506i
$498$	− 56.7006i	− 2.54082i
$499$	−8.52633	−0.381691	−0.190845	−	0.981620i	$-0.561123\pi$
$499$	−8.52633	−0.381691	−0.190845	+	0.981620i	$0.561123\pi$
$500$	0	0
$501$	38.4342	1.71711
$502$	9.49333i	0.423708i
$503$	27.2917i	1.21688i	0.793601	+	0.608439i	$0.208204\pi$
$503$	27.2917i	1.21688i	−0.793601	+	0.608439i	$0.791796\pi$
$504$	29.4935	1.31375
$505$	0	0
$506$	65.5059	2.91209
$507$	− 1.10245i	− 0.0489616i
$508$	33.2756i	1.47637i
$509$	−27.4555	−1.21694	−0.608472	−	0.793576i	$-0.708217\pi$
$509$	−27.4555	−1.21694	−0.608472	+	0.793576i	$0.708217\pi$
$510$	0	0
$511$	−28.3753	−1.25525
$512$	− 48.1654i	− 2.12863i
$513$	1.68510i	0.0743992i
$514$	81.6435	3.60114
$515$	0	0
$516$	−60.7798	−2.67568
$517$	40.8980i	1.79869i
$518$	63.1337i	2.77394i
$519$	13.6209	0.597889
$520$	0	0
$521$	31.8719	1.39633	0.698167	−	0.715935i	$-0.253999\pi$
$521$	31.8719	1.39633	0.698167	+	0.715935i	$0.253999\pi$
$522$	41.9128i	1.83447i
$523$	25.7877i	1.12762i	0.825906	+	0.563808i	$0.190664\pi$
$523$	25.7877i	1.12762i	−0.825906	+	0.563808i	$0.809336\pi$
$524$	29.9103	1.30664
$525$	0	0
$526$	53.3449	2.32595
$527$	− 6.45560i	− 0.281210i
$528$	89.8615i	3.91072i
$529$	−5.99614	−0.260702
$530$	0	0
$531$	18.9120	0.820711
$532$	− 7.57275i	− 0.328320i
$533$	− 22.8836i	− 0.991197i
$534$	1.22213	0.0528869
$535$	0	0
$536$	54.1286	2.33800
$537$	40.0790i	1.72953i
$538$	56.4392i	2.43327i
$539$	−11.9572	−0.515034
$540$	0	0
$541$	−33.7515	−1.45109	−0.725546	−	0.688174i	$-0.758413\pi$
$541$	−33.7515	−1.45109	−0.725546	+	0.688174i	$0.758413\pi$
$542$	23.4727i	1.00824i
$543$	41.0087i	1.75985i
$544$	−52.8161	−2.26447
$545$	0	0
$546$	43.2376	1.85040
$547$	− 4.64965i	− 0.198805i	−0.995047	−	0.0994023i	$-0.968307\pi$
$547$	− 4.64965i	− 0.198805i	0.995047	−	0.0994023i	$-0.0316931\pi$
$548$	− 63.7301i	− 2.72242i
$549$	−6.86690	−0.293072
$550$	0	0
$551$	6.15961	0.262408
$552$	− 83.4468i	− 3.55173i
$553$	− 24.6695i	− 1.04905i
$554$	58.1163	2.46913
$555$	0	0
$556$	−4.26734	−0.180976
$557$	21.1741i	0.897175i	0.893739	+	0.448587i	$0.148073\pi$
$557$	21.1741i	0.897175i	−0.893739	+	0.448587i	$0.851927\pi$
$558$	5.21684i	0.220846i
$559$	−20.5155	−0.867712
$560$	0	0
$561$	−68.0866	−2.87462
$562$	− 25.5440i	− 1.07751i
$563$	17.4396i	0.734993i	0.930025	+	0.367497i	$0.119785\pi$
$563$	17.4396i	0.734993i	−0.930025	+	0.367497i	$0.880215\pi$
$564$	91.0232	3.83277
$565$	0	0
$566$	−11.7841	−0.495322
$567$	23.1903i	0.973899i
$568$	21.7968i	0.914572i
$569$	3.10033	0.129973	0.0649864	−	0.997886i	$-0.479300\pi$
$569$	3.10033	0.129973	0.0649864	+	0.997886i	$0.479300\pi$
$570$	0	0
$571$	−0.0734149	−0.00307232	−0.00153616	−	0.999999i	$-0.500489\pi$
$571$	−0.0734149	−0.00307232	−0.00153616	+	0.999999i	$0.500489\pi$
$572$	77.8714i	3.25597i
$573$	− 2.45537i	− 0.102575i
$574$	−35.3098	−1.47380
$575$	0	0
$576$	8.27832	0.344930
$577$	− 37.6211i	− 1.56619i	−0.621903	−	0.783094i	$-0.713640\pi$
$577$	− 37.6211i	− 1.56619i	0.621903	−	0.783094i	$-0.286360\pi$
$578$	− 63.7591i	− 2.65203i
$579$	34.7592	1.44454
$580$	0	0
$581$	−20.6856	−0.858182
$582$	41.3287i	1.71313i
$583$	19.0697i	0.789784i
$584$	−92.9398	−3.84588
$585$	0	0
$586$	−85.2250	−3.52061
$587$	1.80472i	0.0744887i	0.999306	+	0.0372443i	$0.0118580\pi$
$587$	1.80472i	0.0744887i	−0.999306	+	0.0372443i	$0.988142\pi$
$588$	26.6121i	1.09747i
$589$	0.766680	0.0315905
$590$	0	0
$591$	27.4253	1.12812
$592$	98.5701i	4.05121i
$593$	11.8238i	0.485544i	0.970083	+	0.242772i	$0.0780567\pi$
$593$	11.8238i	0.485544i	−0.970083	+	0.242772i	$0.921943\pi$
$594$	−26.7377	−1.09706
$595$	0	0
$596$	−32.1994	−1.31894
$597$	− 14.5511i	− 0.595536i
$598$	− 49.2099i	− 2.01234i
$599$	−2.94551	−0.120350	−0.0601751	−	0.998188i	$-0.519166\pi$
$599$	−2.94551	−0.120350	−0.0601751	+	0.998188i	$0.519166\pi$
$600$	0	0
$601$	23.0418	0.939896	0.469948	−	0.882694i	$-0.344273\pi$
$601$	23.0418	0.939896	0.469948	+	0.882694i	$0.344273\pi$
$602$	31.6558i	1.29020i
$603$	− 15.7983i	− 0.643355i
$604$	72.2505	2.93983
$605$	0	0
$606$	−49.8215	−2.02386
$607$	40.1456i	1.62946i	0.579840	+	0.814731i	$0.303115\pi$
$607$	40.1456i	1.62946i	−0.579840	+	0.814731i	$0.696885\pi$
$608$	− 6.27254i	− 0.254385i
$609$	38.0118	1.54032
$610$	0	0
$611$	30.7237	1.24295
$612$	60.9564i	2.46401i
$613$	− 1.94946i	− 0.0787378i	−0.999225	−	0.0393689i	$-0.987465\pi$
$613$	− 1.94946i	− 0.0787378i	0.999225	−	0.0393689i	$-0.0125347\pi$
$614$	29.1516	1.17646
$615$	0	0
$616$	68.7749	2.77102
$617$	47.6120i	1.91679i	0.285449	+	0.958394i	$0.407857\pi$
$617$	47.6120i	1.91679i	−0.285449	+	0.958394i	$0.592143\pi$
$618$	− 9.11813i	− 0.366785i
$619$	−18.8037	−0.755785	−0.377893	−	0.925849i	$-0.623351\pi$
$619$	−18.8037	−0.755785	−0.377893	+	0.925849i	$0.623351\pi$
$620$	0	0
$621$	11.8354	0.474938
$622$	− 31.6307i	− 1.26827i
$623$	− 0.445860i	− 0.0178630i
$624$	67.5064	2.70242
$625$	0	0
$626$	−25.9082	−1.03550
$627$	− 8.08610i	− 0.322928i
$628$	− 12.7967i	− 0.510645i
$629$	−74.6850	−2.97789
$630$	0	0
$631$	−27.4357	−1.09220	−0.546099	−	0.837721i	$-0.683888\pi$
$631$	−27.4357	−1.09220	−0.546099	+	0.837721i	$0.683888\pi$
$632$	− 80.8018i	− 3.21412i
$633$	− 18.8325i	− 0.748524i
$634$	−5.96713	−0.236985
$635$	0	0
$636$	42.4416	1.68292
$637$	8.98259i	0.355903i
$638$	97.7350i	3.86937i
$639$	6.36172	0.251666
$640$	0	0
$641$	−28.9531	−1.14358	−0.571790	−	0.820400i	$-0.693751\pi$
$641$	−28.9531	−1.14358	−0.571790	+	0.820400i	$0.693751\pi$
$642$	7.23497i	0.285541i
$643$	30.1906i	1.19060i	0.803502	+	0.595301i	$0.202967\pi$
$643$	30.1906i	1.19060i	−0.803502	+	0.595301i	$0.797033\pi$
$644$	−53.1875	−2.09588
$645$	0	0
$646$	12.7891	0.503181
$647$	18.3546i	0.721595i	0.932644	+	0.360797i	$0.117495\pi$
$647$	18.3546i	0.721595i	−0.932644	+	0.360797i	$0.882505\pi$
$648$	75.9567i	2.98386i
$649$	44.1002	1.73109
$650$	0	0
$651$	4.73128	0.185434
$652$	28.8836i	1.13117i
$653$	− 22.9005i	− 0.896166i	−0.893992	−	0.448083i	$-0.852107\pi$
$653$	− 22.9005i	− 0.896166i	0.893992	−	0.448083i	$-0.147893\pi$
$654$	−61.8357	−2.41797
$655$	0	0
$656$	−55.1289	−2.15242
$657$	27.1259i	1.05828i
$658$	− 47.4074i	− 1.84813i
$659$	0.736981	0.0287087	0.0143544	−	0.999897i	$-0.495431\pi$
$659$	0.736981	0.0287087	0.0143544	+	0.999897i	$0.495431\pi$
$660$	0	0
$661$	−24.4038	−0.949200	−0.474600	−	0.880202i	$-0.657407\pi$
$661$	−24.4038	−0.949200	−0.474600	+	0.880202i	$0.657407\pi$
$662$	23.3814i	0.908745i
$663$	51.1486i	1.98645i
$664$	−67.7529	−2.62932
$665$	0	0
$666$	60.3538	2.33866
$667$	− 43.2623i	− 1.67512i
$668$	− 80.2378i	− 3.10449i
$669$	−22.6903	−0.877256
$670$	0	0
$671$	−16.0127	−0.618163
$672$	− 38.7087i	− 1.49322i
$673$	18.9137i	0.729070i	0.931190	+	0.364535i	$0.118772\pi$
$673$	18.9137i	0.729070i	−0.931190	+	0.364535i	$0.881228\pi$
$674$	−68.4466	−2.63647
$675$	0	0
$676$	−2.30155	−0.0885212
$677$	9.95364i	0.382550i	0.981537	+	0.191275i	$0.0612622\pi$
$677$	9.95364i	0.382550i	−0.981537	+	0.191275i	$0.938738\pi$
$678$	− 4.37362i	− 0.167968i
$679$	15.0776	0.578624
$680$	0	0
$681$	−8.37421	−0.320901
$682$	12.1650i	0.465820i
$683$	21.3600i	0.817319i	0.912687	+	0.408660i	$0.134004\pi$
$683$	21.3600i	0.817319i	−0.912687	+	0.408660i	$0.865996\pi$
$684$	−7.23930	−0.276802
$685$	0	0
$686$	52.0602	1.98767
$687$	− 44.2750i	− 1.68920i
$688$	49.4240i	1.88427i
$689$	14.3256	0.545764
$690$	0	0
$691$	−42.4198	−1.61372	−0.806862	−	0.590739i	$-0.798836\pi$
$691$	−42.4198	−1.61372	−0.806862	+	0.590739i	$0.798836\pi$
$692$	− 28.4358i	− 1.08097i
$693$	− 20.0730i	− 0.762510i
$694$	−95.7270	−3.63375
$695$	0	0
$696$	124.503	4.71927
$697$	− 41.7703i	− 1.58216i
$698$	45.4276i	1.71946i
$699$	−59.5650	−2.25296
$700$	0	0
$701$	11.7949	0.445486	0.222743	−	0.974877i	$-0.428499\pi$
$701$	11.7949	0.445486	0.222743	+	0.974877i	$0.428499\pi$
$702$	20.0861i	0.758100i
$703$	− 8.86974i	− 0.334529i
$704$	19.3039	0.727544
$705$	0	0
$706$	62.5392	2.35369
$707$	18.1759i	0.683576i
$708$	− 98.1500i	− 3.68870i
$709$	−40.8193	−1.53300	−0.766500	−	0.642245i	$-0.778003\pi$
$709$	−40.8193	−1.53300	−0.766500	+	0.642245i	$0.778003\pi$
$710$	0	0
$711$	−23.5832	−0.884441
$712$	− 1.46036i	− 0.0547292i
$713$	− 5.38481i	− 0.201663i
$714$	78.9233	2.95363
$715$	0	0
$716$	83.6714	3.12695
$717$	− 37.2391i	− 1.39072i
$718$	− 33.7510i	− 1.25958i
$719$	40.5659	1.51285	0.756426	−	0.654080i	$-0.226944\pi$
$719$	40.5659	1.51285	0.756426	+	0.654080i	$0.226944\pi$
$720$	0	0
$721$	−3.32649	−0.123885
$722$	− 47.5769i	− 1.77063i
$723$	0.202386i	0.00752683i
$724$	85.6124	3.18176
$725$	0	0
$726$	64.6251	2.39846
$727$	− 20.5810i	− 0.763306i	−0.924306	−	0.381653i	$-0.875355\pi$
$727$	− 20.5810i	− 0.763306i	0.924306	−	0.381653i	$-0.124645\pi$
$728$	− 51.6656i	− 1.91485i
$729$	7.39714	0.273968
$730$	0	0
$731$	−37.4478	−1.38506
$732$	35.6380i	1.31722i
$733$	25.8781i	0.955831i	0.878406	+	0.477916i	$0.158608\pi$
$733$	25.8781i	0.955831i	−0.878406	+	0.477916i	$0.841392\pi$
$734$	55.5366	2.04989
$735$	0	0
$736$	−44.0555	−1.62391
$737$	− 36.8394i	− 1.35700i
$738$	33.7550i	1.24254i
$739$	26.4783	0.974021	0.487011	−	0.873396i	$-0.338087\pi$
$739$	26.4783	0.974021	0.487011	+	0.873396i	$0.338087\pi$
$740$	0	0
$741$	−6.07450	−0.223152
$742$	− 22.1048i	− 0.811492i
$743$	− 5.00968i	− 0.183787i	−0.995769	−	0.0918936i	$-0.970708\pi$
$743$	− 5.00968i	− 0.183787i	0.995769	−	0.0918936i	$-0.0292920\pi$
$744$	15.4967	0.568137
$745$	0	0
$746$	20.1338	0.737151
$747$	19.7747i	0.723519i
$748$	142.142i	5.19723i
$749$	2.63947	0.0964440
$750$	0	0
$751$	−21.0437	−0.767895	−0.383947	−	0.923355i	$-0.625436\pi$
$751$	−21.0437	−0.767895	−0.383947	+	0.923355i	$0.625436\pi$
$752$	− 74.0168i	− 2.69911i
$753$	− 8.23064i	− 0.299941i
$754$	73.4212	2.67384
$755$	0	0
$756$	21.7096	0.789572
$757$	− 32.6609i	− 1.18708i	−0.804804	−	0.593541i	$-0.797730\pi$
$757$	− 32.6609i	− 1.18708i	0.804804	−	0.593541i	$-0.202270\pi$
$758$	− 61.6854i	− 2.24051i
$759$	−56.7931	−2.06146
$760$	0	0
$761$	−9.58422	−0.347428	−0.173714	−	0.984796i	$-0.555577\pi$
$761$	−9.58422	−0.347428	−0.173714	+	0.984796i	$0.555577\pi$
$762$	− 41.1866i	− 1.49203i
$763$	22.5589i	0.816689i
$764$	−5.12599	−0.185452
$765$	0	0
$766$	−22.0646	−0.797225
$767$	− 33.1293i	− 1.19623i
$768$	51.7598i	1.86772i
$769$	30.2404	1.09050	0.545248	−	0.838275i	$-0.316436\pi$
$769$	30.2404	1.09050	0.545248	+	0.838275i	$0.316436\pi$
$770$	0	0
$771$	−70.7842	−2.54923
$772$	− 72.5656i	− 2.61169i
$773$	− 8.88586i	− 0.319602i	−0.987149	−	0.159801i	$-0.948915\pi$
$773$	− 8.88586i	− 0.319602i	0.987149	−	0.159801i	$-0.0510853\pi$
$774$	30.2619	1.08774
$775$	0	0
$776$	49.3846	1.77280
$777$	− 54.7364i	− 1.96366i
$778$	− 5.22797i	− 0.187432i
$779$	4.96073	0.177737
$780$	0	0
$781$	14.8347	0.530826
$782$	− 89.8248i	− 3.21213i
$783$	17.6584i	0.631061i
$784$	21.6400	0.772858
$785$	0	0
$786$	−37.0211	−1.32050
$787$	32.3032i	1.15148i	0.817632	+	0.575742i	$0.195287\pi$
$787$	32.3032i	1.15148i	−0.817632	+	0.575742i	$0.804713\pi$
$788$	− 57.2547i	− 2.03962i
$789$	−46.2496	−1.64653
$790$	0	0
$791$	−1.59559	−0.0567326
$792$	− 65.7465i	− 2.33620i
$793$	12.0292i	0.427169i
$794$	21.7496	0.771864
$795$	0	0
$796$	−30.3778	−1.07671
$797$	− 24.1925i	− 0.856942i	−0.903556	−	0.428471i	$-0.859052\pi$
$797$	− 24.1925i	− 0.856942i	0.903556	−	0.428471i	$-0.140948\pi$
$798$	9.37309i	0.331804i
$799$	56.0813	1.98401
$800$	0	0
$801$	−0.426227	−0.0150600
$802$	24.1079i	0.851278i
$803$	63.2539i	2.23218i
$804$	−81.9903	−2.89157
$805$	0	0
$806$	9.13865	0.321895
$807$	− 48.9323i	− 1.72250i
$808$	59.5329i	2.09436i
$809$	1.57516	0.0553796	0.0276898	−	0.999617i	$-0.491185\pi$
$809$	1.57516	0.0553796	0.0276898	+	0.999617i	$0.491185\pi$
$810$	0	0
$811$	−22.5306	−0.791157	−0.395579	−	0.918432i	$-0.629456\pi$
$811$	−22.5306	−0.791157	−0.395579	+	0.918432i	$0.629456\pi$
$812$	− 79.3559i	− 2.78485i
$813$	− 20.3506i	− 0.713728i
$814$	140.737	4.93282
$815$	0	0
$816$	123.222	4.31364
$817$	− 4.44737i	− 0.155594i
$818$	− 57.4979i	− 2.01037i
$819$	−15.0794	−0.526917
$820$	0	0
$821$	−24.5336	−0.856228	−0.428114	−	0.903725i	$-0.640822\pi$
$821$	−24.5336	−0.856228	−0.428114	+	0.903725i	$0.640822\pi$
$822$	78.8814i	2.75130i
$823$	− 9.88629i	− 0.344614i	−0.985043	−	0.172307i	$-0.944878\pi$
$823$	− 9.88629i	− 0.344614i	0.985043	−	0.172307i	$-0.0551222\pi$
$824$	−10.8955	−0.379562
$825$	0	0
$826$	−51.1193	−1.77867
$827$	38.5489i	1.34047i	0.742147	+	0.670237i	$0.233808\pi$
$827$	38.5489i	1.34047i	−0.742147	+	0.670237i	$0.766192\pi$
$828$	50.8455i	1.76700i
$829$	19.0794	0.662655	0.331327	−	0.943516i	$-0.392504\pi$
$829$	19.0794	0.662655	0.331327	+	0.943516i	$0.392504\pi$
$830$	0	0
$831$	−50.3863	−1.74788
$832$	− 14.5016i	− 0.502754i
$833$	16.3963i	0.568098i
$834$	5.28186	0.182896
$835$	0	0
$836$	−16.8811	−0.583844
$837$	2.19793i	0.0759714i
$838$	17.7590i	0.613473i
$839$	−47.1714	−1.62854	−0.814269	−	0.580488i	$-0.802862\pi$
$839$	−47.1714	−1.62854	−0.814269	+	0.580488i	$0.802862\pi$
$840$	0	0
$841$	35.5474	1.22577
$842$	54.1681i	1.86676i
$843$	22.1464i	0.762764i
$844$	−39.3159	−1.35331
$845$	0	0
$846$	−45.3199	−1.55813
$847$	− 23.5766i	− 0.810100i
$848$	− 34.5120i	− 1.18515i
$849$	10.2167	0.350637
$850$	0	0
$851$	−62.2970	−2.13551
$852$	− 33.0162i	− 1.13112i
$853$	− 24.5249i	− 0.839717i	−0.907590	−	0.419858i	$-0.862080\pi$
$853$	− 24.5249i	− 0.839717i	0.907590	−	0.419858i	$-0.137920\pi$
$854$	18.5613	0.635154
$855$	0	0
$856$	8.64523	0.295488
$857$	42.9761i	1.46803i	0.679131	+	0.734017i	$0.262357\pi$
$857$	42.9761i	1.46803i	−0.679131	+	0.734017i	$0.737643\pi$
$858$	− 96.3846i	− 3.29052i
$859$	48.9415	1.66986	0.834932	−	0.550353i	$-0.185507\pi$
$859$	48.9415	1.66986	0.834932	+	0.550353i	$0.185507\pi$
$860$	0	0
$861$	30.6133	1.04330
$862$	86.0277i	2.93011i
$863$	− 32.7812i	− 1.11589i	−0.829880	−	0.557943i	$-0.811591\pi$
$863$	− 32.7812i	− 1.11589i	0.829880	−	0.557943i	$-0.188409\pi$
$864$	17.9822	0.611767
$865$	0	0
$866$	−1.80096	−0.0611989
$867$	55.2786i	1.87736i
$868$	− 9.87733i	− 0.335258i
$869$	−54.9929	−1.86551
$870$	0	0
$871$	−27.6748	−0.937725
$872$	73.8889i	2.50219i
$873$	− 14.4137i	− 0.487828i
$874$	10.6678	0.360843
$875$	0	0
$876$	140.779	4.75647
$877$	1.30919i	0.0442083i	0.999756	+	0.0221041i	$0.00703654\pi$
$877$	1.30919i	0.0442083i	−0.999756	+	0.0221041i	$0.992963\pi$
$878$	45.6421i	1.54035i
$879$	73.8894	2.49223
$880$	0	0
$881$	32.1645	1.08365	0.541825	−	0.840491i	$-0.317733\pi$
$881$	32.1645	1.08365	0.541825	+	0.840491i	$0.317733\pi$
$882$	− 13.2500i	− 0.446152i
$883$	57.3931i	1.93143i	0.259604	+	0.965715i	$0.416408\pi$
$883$	57.3931i	1.93143i	−0.259604	+	0.965715i	$0.583592\pi$
$884$	106.781	3.59143
$885$	0	0
$886$	−78.1992	−2.62715
$887$	16.6057i	0.557566i	0.960354	+	0.278783i	$0.0899311\pi$
$887$	16.6057i	0.557566i	−0.960354	+	0.278783i	$0.910069\pi$
$888$	− 179.282i	− 6.01631i
$889$	−15.0257	−0.503947
$890$	0	0
$891$	51.6954	1.73186
$892$	47.3696i	1.58605i
$893$	6.66033i	0.222879i
$894$	39.8545	1.33293
$895$	0	0
$896$	12.1805	0.406922
$897$	42.6645i	1.42453i
$898$	− 2.34541i	− 0.0782675i
$899$	8.03414	0.267954
$900$	0	0
$901$	26.1492	0.871156
$902$	78.7122i	2.62083i
$903$	− 27.4453i	− 0.913324i
$904$	−5.22614	−0.173819
$905$	0	0
$906$	−89.4274	−2.97103
$907$	− 3.74422i	− 0.124325i	−0.998066	−	0.0621624i	$-0.980200\pi$
$907$	− 3.74422i	− 0.124325i	0.998066	−	0.0621624i	$-0.0197997\pi$
$908$	17.4825i	0.580178i
$909$	17.3756	0.576312
$910$	0	0
$911$	−25.6118	−0.848558	−0.424279	−	0.905531i	$-0.639472\pi$
$911$	−25.6118	−0.848558	−0.424279	+	0.905531i	$0.639472\pi$
$912$	14.6341i	0.484584i
$913$	46.1120i	1.52608i
$914$	−81.7527	−2.70414
$915$	0	0
$916$	−92.4314	−3.05402
$917$	13.5061i	0.446010i
$918$	36.6640i	1.21009i
$919$	11.9450	0.394030	0.197015	−	0.980400i	$-0.436875\pi$
$919$	11.9450	0.394030	0.197015	+	0.980400i	$0.436875\pi$
$920$	0	0
$921$	−25.2742	−0.832813
$922$	− 10.6498i	− 0.350731i
$923$	− 11.1442i	− 0.366816i
$924$	−104.175	−3.42712
$925$	0	0
$926$	−4.01214	−0.131847
$927$	3.18001i	0.104445i
$928$	− 65.7309i	− 2.15772i
$929$	−13.3211	−0.437051	−0.218525	−	0.975831i	$-0.570125\pi$
$929$	−13.3211	−0.437051	−0.218525	+	0.975831i	$0.570125\pi$
$930$	0	0
$931$	−1.94726	−0.0638188
$932$	124.352i	4.07328i
$933$	27.4235i	0.897806i
$934$	1.28638	0.0420917
$935$	0	0
$936$	−49.3906	−1.61438
$937$	− 17.3862i	− 0.567981i	−0.958827	−	0.283991i	$-0.908342\pi$
$937$	− 17.3862i	− 0.567981i	0.958827	−	0.283991i	$-0.0916584\pi$
$938$	42.7028i	1.39430i
$939$	22.4622	0.733025
$940$	0	0
$941$	26.0346	0.848702	0.424351	−	0.905498i	$-0.360502\pi$
$941$	26.0346	0.848702	0.424351	+	0.905498i	$0.360502\pi$
$942$	15.8390i	0.516063i
$943$	− 34.8419i	− 1.13461i
$944$	−79.8121	−2.59766
$945$	0	0
$946$	70.5668	2.29432
$947$	1.30792i	0.0425018i	0.999774	+	0.0212509i	$0.00676489\pi$
$947$	1.30792i	0.0425018i	−0.999774	+	0.0212509i	$0.993235\pi$
$948$	122.393i	3.97514i
$949$	47.5181	1.54250
$950$	0	0
$951$	5.17345	0.167761
$952$	− 94.3073i	− 3.05652i
$953$	− 40.9822i	− 1.32755i	−0.747935	−	0.663773i	$-0.768954\pi$
$953$	− 40.9822i	− 1.32755i	0.747935	−	0.663773i	$-0.231046\pi$
$954$	−21.1315	−0.684156
$955$	0	0
$956$	−77.7428	−2.51438
$957$	− 84.7354i	− 2.73911i
$958$	− 43.0824i	− 1.39193i
$959$	28.7776	0.929276
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 105.725i	− 3.40872i
$963$	− 2.52324i	− 0.0813104i
$964$	0.422515	0.0136083
$965$	0	0
$966$	65.8323	2.11812
$967$	− 38.8300i	− 1.24869i	−0.781150	−	0.624344i	$-0.785367\pi$
$967$	− 38.8300i	− 1.24869i	0.781150	−	0.624344i	$-0.214633\pi$
$968$	− 77.2220i	− 2.48201i
$969$	−11.0880	−0.356199
$970$	0	0
$971$	−20.8747	−0.669901	−0.334950	−	0.942236i	$-0.608720\pi$
$971$	−20.8747	−0.669901	−0.334950	+	0.942236i	$0.608720\pi$
$972$	− 84.2150i	− 2.70120i
$973$	− 1.92693i	− 0.0617746i
$974$	−4.09299	−0.131148
$975$	0	0
$976$	28.9796	0.927613
$977$	− 10.9170i	− 0.349265i	−0.984634	−	0.174633i	$-0.944126\pi$
$977$	− 10.9170i	− 0.349265i	0.984634	−	0.174633i	$-0.0558738\pi$
$978$	− 35.7504i	− 1.14317i
$979$	−0.993905	−0.0317653
$980$	0	0
$981$	21.5656	0.688537
$982$	54.7090i	1.74583i
$983$	− 23.2316i	− 0.740973i	−0.928838	−	0.370486i	$-0.879191\pi$
$983$	− 23.2316i	− 0.740973i	0.928838	−	0.370486i	$-0.120809\pi$
$984$	100.270	3.19649
$985$	0	0
$986$	134.019	4.26803
$987$	41.1018i	1.30829i
$988$	12.6815i	0.403453i
$989$	−31.2363	−0.993257
$990$	0	0
$991$	18.8123	0.597594	0.298797	−	0.954317i	$-0.403415\pi$
$991$	18.8123	0.597594	0.298797	+	0.954317i	$0.403415\pi$
$992$	− 8.18144i	− 0.259761i
$993$	− 20.2715i	− 0.643296i
$994$	−17.1958	−0.545416
$995$	0	0
$996$	102.627	3.25187
$997$	− 22.3280i	− 0.707134i	−0.935409	−	0.353567i	$-0.884969\pi$
$997$	− 22.3280i	− 0.707134i	0.935409	−	0.353567i	$-0.115031\pi$
$998$	− 22.0319i	− 0.697408i
$999$	25.4279	0.804502

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.b.g.249.10		10
5.2	odd	4		775.2.a.h.1.1	✓	5
5.3	odd	4		775.2.a.k.1.5	yes	5
5.4	even	2	inner	775.2.b.g.249.1		10
15.2	even	4		6975.2.a.by.1.5		5
15.8	even	4		6975.2.a.bp.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
775.2.a.h.1.1	✓	5	5.2	odd	4
775.2.a.k.1.5	yes	5	5.3	odd	4
775.2.b.g.249.1		10	5.4	even	2	inner
775.2.b.g.249.10		10	1.1	even	1	trivial
6975.2.a.bp.1.1		5	15.8	even	4
6975.2.a.by.1.5		5	15.2	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:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.58398i q^{2} -2.24029i q^{3} -4.67698 q^{4} +5.78888 q^{6} -2.11190i q^{7} -6.91727i q^{8} -2.01891 q^{9} -4.70783 q^{11} +10.4778i q^{12} +3.53665i q^{13} +5.45713 q^{14} +8.52017 q^{16} +6.45560i q^{17} -5.21684i q^{18} -0.766680 q^{19} -4.73128 q^{21} -12.1650i q^{22} +5.38481i q^{23} -15.4967 q^{24} -9.13865 q^{26} -2.19793i q^{27} +9.87733i q^{28} -8.03414 q^{29} -1.00000 q^{31} +8.18144i q^{32} +10.5469i q^{33} -16.6812 q^{34} +9.44240 q^{36} +11.5690i q^{37} -1.98109i q^{38} +7.92313 q^{39} -6.47040 q^{41} -12.2256i q^{42} +5.80082i q^{43} +22.0184 q^{44} -13.9143 q^{46} -8.68724i q^{47} -19.0877i q^{48} +2.53986 q^{49} +14.4624 q^{51} -16.5408i q^{52} -4.05063i q^{53} +5.67941 q^{54} -14.6086 q^{56} +1.71759i q^{57} -20.7601i q^{58} -9.36743 q^{59} +3.40129 q^{61} -2.58398i q^{62} +4.26375i q^{63} -4.10039 q^{64} -27.2531 q^{66} +7.82514i q^{67} -30.1927i q^{68} +12.0635 q^{69} -3.15106 q^{71} +13.9654i q^{72} -13.4359i q^{73} -29.8942 q^{74} +3.58574 q^{76} +9.94249i q^{77} +20.4733i q^{78} +11.6812 q^{79} -10.9807 q^{81} -16.7194i q^{82} -9.79474i q^{83} +22.1281 q^{84} -14.9892 q^{86} +17.9988i q^{87} +32.5653i q^{88} +0.211117 q^{89} +7.46907 q^{91} -25.1846i q^{92} +2.24029i q^{93} +22.4477 q^{94} +18.3288 q^{96} +7.13932i q^{97} +6.56296i q^{98} +9.50469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 12 q^{4} - 2 q^{6} - 12 q^{9} - 4 q^{11} - 4 q^{14} + 8 q^{16} - 16 q^{19} - 30 q^{21} - 52 q^{24} - 32 q^{26} - 12 q^{29} - 10 q^{31} - 62 q^{34} - 26 q^{36} + 26 q^{39} - 4 q^{41} + 8 q^{44} - 22 q^{46}+ \cdots + 66 q^{99}+O(q^{100}) \) 10 * q - 12 * q^4 - 2 * q^6 - 12 * q^9 - 4 * q^11 - 4 * q^14 + 8 * q^16 - 16 * q^19 - 30 * q^21 - 52 * q^24 - 32 * q^26 - 12 * q^29 - 10 * q^31 - 62 * q^34 - 26 * q^36 + 26 * q^39 - 4 * q^41 + 8 * q^44 - 22 * q^46 + 18 * q^49 + 10 * q^51 - 20 * q^54 - 18 * q^56 + 8 * q^59 - 10 * q^61 + 10 * q^64 - 50 * q^66 - 20 * q^69 + 12 * q^71 - 8 * q^74 - 10 * q^76 + 12 * q^79 - 14 * q^81 + 30 * q^84 - 14 * q^86 + 62 * q^89 + 16 * q^91 + 28 * q^94 + 20 * q^96 + 66 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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