LMFDB - Embedded newform 875.2.b.c.624.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [875,2,Mod(624,875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(875, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("875.624");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 875 = 5^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	875.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.98691017686$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 14x^{10} + 71x^{8} + 156x^{6} + 135x^{4} + 27x^{2} + 1 $ x^12 + 14x^10 + 71x^8 + 156x^6 + 135x^4 + 27*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			624.12
Root			$-1.72131i$ of defining polynomial
Character	$\chi$	$=$	875.624
Dual form			875.2.b.c.624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.55803i q^{2} +0.353843i q^{3} -4.54354 q^{4} -0.905143 q^{6} -1.00000i q^{7} -6.50646i q^{8} +2.87480 q^{9} +5.69412 q^{11} -1.60770i q^{12} -6.47971i q^{13} +2.55803 q^{14} +7.55666 q^{16} -2.19206i q^{17} +7.35382i q^{18} +5.09571 q^{19} +0.353843 q^{21} +14.5657i q^{22} -6.78323i q^{23} +2.30226 q^{24} +16.5753 q^{26} +2.07876i q^{27} +4.54354i q^{28} -1.62794 q^{29} -8.51029 q^{31} +6.31728i q^{32} +2.01482i q^{33} +5.60737 q^{34} -13.0617 q^{36} -1.22831i q^{37} +13.0350i q^{38} +2.29280 q^{39} -1.53221 q^{41} +0.905143i q^{42} +6.00854i q^{43} -25.8714 q^{44} +17.3517 q^{46} -2.06257i q^{47} +2.67387i q^{48} -1.00000 q^{49} +0.775646 q^{51} +29.4408i q^{52} +6.26731i q^{53} -5.31753 q^{54} -6.50646 q^{56} +1.80308i q^{57} -4.16432i q^{58} +6.61824 q^{59} -7.69499 q^{61} -21.7696i q^{62} -2.87480i q^{63} -1.04650 q^{64} -5.15399 q^{66} +6.29996i q^{67} +9.95971i q^{68} +2.40020 q^{69} +0.839088 q^{71} -18.7047i q^{72} +4.62741i q^{73} +3.14205 q^{74} -23.1525 q^{76} -5.69412i q^{77} +5.86506i q^{78} +8.90370 q^{79} +7.88883 q^{81} -3.91945i q^{82} -3.59494i q^{83} -1.60770 q^{84} -15.3701 q^{86} -0.576035i q^{87} -37.0485i q^{88} +0.716374 q^{89} -6.47971 q^{91} +30.8199i q^{92} -3.01131i q^{93} +5.27613 q^{94} -2.23533 q^{96} +6.57786i q^{97} -2.55803i q^{98} +16.3694 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{4} - 4 q^{6} - 14 q^{9} + 2 q^{11} + 4 q^{14} + 16 q^{16} + 34 q^{19} - 10 q^{21} - 14 q^{24} + 32 q^{26} - 14 q^{29} - 22 q^{31} + 8 q^{34} + 20 q^{36} - 30 q^{39} + 6 q^{41} - 52 q^{44} - 16 q^{46}+ \cdots - 24 q^{99}+O(q^{100}) $ 12 * q - 8 * q^4 - 4 * q^6 - 14 * q^9 + 2 * q^11 + 4 * q^14 + 16 * q^16 + 34 * q^19 - 10 * q^21 - 14 * q^24 + 32 * q^26 - 14 * q^29 - 22 * q^31 + 8 * q^34 + 20 * q^36 - 30 * q^39 + 6 * q^41 - 52 * q^44 - 16 * q^46 - 12 * q^49 + 2 * q^51 + 52 * q^54 - 24 * q^56 + 42 * q^59 - 38 * q^61 + 44 * q^64 - 70 * q^66 + 4 * q^69 + 2 * q^71 + 58 * q^74 - 8 * q^76 - 44 * q^79 + 28 * q^81 + 22 * q^84 + 46 * q^86 + 16 * q^89 - 44 * q^91 + 52 * q^94 - 8 * q^96 - 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/875\mathbb{Z}\right)^\times$.

$n$	$127$	$626$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.55803i	1.80880i	0.426682	+	0.904402i	$0.359682\pi$
$2$	2.55803i	1.80880i	−0.426682	+	0.904402i	$0.640318\pi$
$3$	0.353843i	0.204291i	0.994769	+	0.102146i	$0.0325708\pi$
$3$	0.353843i	0.204291i	−0.994769	+	0.102146i	$0.967429\pi$
$4$	−4.54354	−2.27177
$5$	0	0
$5$	0	0
$6$	−0.905143	−0.369523
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	− 6.50646i	− 2.30038i
$9$	2.87480	0.958265
$10$	0	0
$11$	5.69412	1.71684	0.858420	−	0.512947i	$-0.171446\pi$
$11$	5.69412	1.71684	0.858420	+	0.512947i	$0.171446\pi$
$12$	− 1.60770i	− 0.464103i
$13$	− 6.47971i	− 1.79715i	−0.438822	−	0.898574i	$-0.644604\pi$
$13$	− 6.47971i	− 1.79715i	0.438822	−	0.898574i	$-0.355396\pi$
$14$	2.55803	0.683663
$15$	0	0
$16$	7.55666	1.88916
$17$	− 2.19206i	− 0.531653i	−0.964021	−	0.265826i	$-0.914355\pi$
$17$	− 2.19206i	− 0.531653i	0.964021	−	0.265826i	$-0.0856448\pi$
$18$	7.35382i	1.73331i
$19$	5.09571	1.16904	0.584518	−	0.811381i	$-0.301284\pi$
$19$	5.09571	1.16904	0.584518	+	0.811381i	$0.301284\pi$
$20$	0	0
$21$	0.353843	0.0772149
$22$	14.5657i	3.10543i
$23$	− 6.78323i	− 1.41440i	−0.707013	−	0.707201i	$-0.749958\pi$
$23$	− 6.78323i	− 1.41440i	0.707013	−	0.707201i	$-0.250042\pi$
$24$	2.30226	0.469948
$25$	0	0
$26$	16.5753	3.25069
$27$	2.07876i	0.400057i
$28$	4.54354i	0.858648i
$29$	−1.62794	−0.302301	−0.151150	−	0.988511i	$-0.548298\pi$
$29$	−1.62794	−0.302301	−0.151150	+	0.988511i	$0.548298\pi$
$30$	0	0
$31$	−8.51029	−1.52849	−0.764246	−	0.644925i	$-0.776889\pi$
$31$	−8.51029	−1.52849	−0.764246	+	0.644925i	$0.776889\pi$
$32$	6.31728i	1.11675i
$33$	2.01482i	0.350736i
$34$	5.60737	0.961656
$35$	0	0
$36$	−13.0617	−2.17696
$37$	− 1.22831i	− 0.201932i	−0.994890	−	0.100966i	$-0.967807\pi$
$37$	− 1.22831i	− 0.201932i	0.994890	−	0.100966i	$-0.0321934\pi$
$38$	13.0350i	2.11455i
$39$	2.29280	0.367142
$40$	0	0
$41$	−1.53221	−0.239291	−0.119646	−	0.992817i	$-0.538176\pi$
$41$	−1.53221	−0.239291	−0.119646	+	0.992817i	$0.538176\pi$
$42$	0.905143i	0.139667i
$43$	6.00854i	0.916294i	0.888876	+	0.458147i	$0.151487\pi$
$43$	6.00854i	0.916294i	−0.888876	+	0.458147i	$0.848513\pi$
$44$	−25.8714	−3.90026
$45$	0	0
$46$	17.3517	2.55837
$47$	− 2.06257i	− 0.300857i	−0.988621	−	0.150429i	$-0.951935\pi$
$47$	− 2.06257i	− 0.300857i	0.988621	−	0.150429i	$-0.0480653\pi$
$48$	2.67387i	0.385940i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0.775646	0.108612
$52$	29.4408i	4.08270i
$53$	6.26731i	0.860881i	0.902619	+	0.430440i	$0.141642\pi$
$53$	6.26731i	0.860881i	−0.902619	+	0.430440i	$0.858358\pi$
$54$	−5.31753	−0.723624
$55$	0	0
$56$	−6.50646	−0.869462
$57$	1.80308i	0.238824i
$58$	− 4.16432i	− 0.546802i
$59$	6.61824	0.861621	0.430811	−	0.902442i	$-0.358228\pi$
$59$	6.61824	0.861621	0.430811	+	0.902442i	$0.358228\pi$
$60$	0	0
$61$	−7.69499	−0.985242	−0.492621	−	0.870244i	$-0.663961\pi$
$61$	−7.69499	−0.985242	−0.492621	+	0.870244i	$0.663961\pi$
$62$	− 21.7696i	− 2.76474i
$63$	− 2.87480i	− 0.362190i
$64$	−1.04650	−0.130812
$65$	0	0
$66$	−5.15399	−0.634412
$67$	6.29996i	0.769663i	0.922987	+	0.384832i	$0.125740\pi$
$67$	6.29996i	0.769663i	−0.922987	+	0.384832i	$0.874260\pi$
$68$	9.95971i	1.20779i
$69$	2.40020	0.288950
$70$	0	0
$71$	0.839088	0.0995814	0.0497907	−	0.998760i	$-0.484145\pi$
$71$	0.839088	0.0995814	0.0497907	+	0.998760i	$0.484145\pi$
$72$	− 18.7047i	− 2.20437i
$73$	4.62741i	0.541598i	0.962636	+	0.270799i	$0.0872878\pi$
$73$	4.62741i	0.541598i	−0.962636	+	0.270799i	$0.912712\pi$
$74$	3.14205	0.365256
$75$	0	0
$76$	−23.1525	−2.65578
$77$	− 5.69412i	− 0.648905i
$78$	5.86506i	0.664087i
$79$	8.90370	1.00174	0.500872	−	0.865521i	$-0.333013\pi$
$79$	8.90370	1.00174	0.500872	+	0.865521i	$0.333013\pi$
$80$	0	0
$81$	7.88883	0.876537
$82$	− 3.91945i	− 0.432831i
$83$	− 3.59494i	− 0.394596i	−0.980344	−	0.197298i	$-0.936783\pi$
$83$	− 3.59494i	− 0.394596i	0.980344	−	0.197298i	$-0.0632167\pi$
$84$	−1.60770	−0.175414
$85$	0	0
$86$	−15.3701	−1.65740
$87$	− 0.576035i	− 0.0617574i
$88$	− 37.0485i	− 3.94938i
$89$	0.716374	0.0759355	0.0379677	−	0.999279i	$-0.487912\pi$
$89$	0.716374	0.0759355	0.0379677	+	0.999279i	$0.487912\pi$
$90$	0	0
$91$	−6.47971	−0.679258
$92$	30.8199i	3.21319i
$93$	− 3.01131i	− 0.312258i
$94$	5.27613	0.544191
$95$	0	0
$96$	−2.23533	−0.228142
$97$	6.57786i	0.667880i	0.942594	+	0.333940i	$0.108378\pi$
$97$	6.57786i	0.667880i	−0.942594	+	0.333940i	$0.891622\pi$
$98$	− 2.55803i	− 0.258400i
$99$	16.3694	1.64519
$100$	0	0
$101$	−10.1795	−1.01290	−0.506449	−	0.862270i	$-0.669042\pi$
$101$	−10.1795	−1.01290	−0.506449	+	0.862270i	$0.669042\pi$
$102$	1.98413i	0.196458i
$103$	− 12.4409i	− 1.22584i	−0.790145	−	0.612920i	$-0.789995\pi$
$103$	− 12.4409i	− 1.22584i	0.790145	−	0.612920i	$-0.210005\pi$
$104$	−42.1599	−4.13412
$105$	0	0
$106$	−16.0320	−1.55716
$107$	1.70563i	0.164890i	0.996596	+	0.0824450i	$0.0262729\pi$
$107$	1.70563i	0.164890i	−0.996596	+	0.0824450i	$0.973727\pi$
$108$	− 9.44491i	− 0.908837i
$109$	12.1453	1.16331	0.581653	−	0.813437i	$-0.302406\pi$
$109$	12.1453	1.16331	0.581653	+	0.813437i	$0.302406\pi$
$110$	0	0
$111$	0.434628	0.0412530
$112$	− 7.55666i	− 0.714037i
$113$	− 12.7733i	− 1.20161i	−0.799395	−	0.600806i	$-0.794846\pi$
$113$	− 12.7733i	− 1.20161i	0.799395	−	0.600806i	$-0.205154\pi$
$114$	−4.61234	−0.431985
$115$	0	0
$116$	7.39660	0.686757
$117$	− 18.6278i	− 1.72214i
$118$	16.9297i	1.55850i
$119$	−2.19206	−0.200946
$120$	0	0
$121$	21.4229	1.94754
$122$	− 19.6840i	− 1.78211i
$123$	− 0.542163i	− 0.0488851i
$124$	38.6668	3.47238
$125$	0	0
$126$	7.35382	0.655131
$127$	3.87656i	0.343989i	0.985098	+	0.171994i	$0.0550211\pi$
$127$	3.87656i	0.343989i	−0.985098	+	0.171994i	$0.944979\pi$
$128$	9.95759i	0.880135i
$129$	−2.12608	−0.187191
$130$	0	0
$131$	1.55620	0.135966	0.0679831	−	0.997686i	$-0.478344\pi$
$131$	1.55620	0.135966	0.0679831	+	0.997686i	$0.478344\pi$
$132$	− 9.15443i	− 0.796791i
$133$	− 5.09571i	− 0.441854i
$134$	−16.1155	−1.39217
$135$	0	0
$136$	−14.2626	−1.22300
$137$	11.6597i	0.996159i	0.867131	+	0.498080i	$0.165961\pi$
$137$	11.6597i	0.996159i	−0.867131	+	0.498080i	$0.834039\pi$
$138$	6.13979i	0.522654i
$139$	−9.16488	−0.777355	−0.388677	−	0.921374i	$-0.627068\pi$
$139$	−9.16488	−0.777355	−0.388677	+	0.921374i	$0.627068\pi$
$140$	0	0
$141$	0.729827	0.0614625
$142$	2.14642i	0.180123i
$143$	− 36.8962i	− 3.08542i
$144$	21.7238	1.81032
$145$	0	0
$146$	−11.8371	−0.979644
$147$	− 0.353843i	− 0.0291845i
$148$	5.58085i	0.458743i
$149$	−4.36230	−0.357373	−0.178687	−	0.983906i	$-0.557185\pi$
$149$	−4.36230	−0.357373	−0.178687	+	0.983906i	$0.557185\pi$
$150$	0	0
$151$	−10.3209	−0.839902	−0.419951	−	0.907547i	$-0.637953\pi$
$151$	−10.3209	−0.839902	−0.419951	+	0.907547i	$0.637953\pi$
$152$	− 33.1550i	− 2.68922i
$153$	− 6.30173i	− 0.509464i
$154$	14.5657	1.17374
$155$	0	0
$156$	−10.4174	−0.834061
$157$	23.8740i	1.90535i	0.303985	+	0.952677i	$0.401683\pi$
$157$	23.8740i	1.90535i	−0.303985	+	0.952677i	$0.598317\pi$
$158$	22.7760i	1.81196i
$159$	−2.21764	−0.175871
$160$	0	0
$161$	−6.78323	−0.534593
$162$	20.1799i	1.58548i
$163$	20.2290i	1.58446i	0.610222	+	0.792231i	$0.291080\pi$
$163$	20.2290i	1.58446i	−0.610222	+	0.792231i	$0.708920\pi$
$164$	6.96166	0.543614
$165$	0	0
$166$	9.19599	0.713747
$167$	− 8.90008i	− 0.688709i	−0.938840	−	0.344354i	$-0.888098\pi$
$167$	− 8.90008i	− 0.688709i	0.938840	−	0.344354i	$-0.111902\pi$
$168$	− 2.30226i	− 0.177624i
$169$	−28.9866	−2.22974
$170$	0	0
$171$	14.6491	1.12025
$172$	− 27.3000i	− 2.08161i
$173$	15.0386i	1.14337i	0.820474	+	0.571683i	$0.193709\pi$
$173$	15.0386i	1.14337i	−0.820474	+	0.571683i	$0.806291\pi$
$174$	1.47352	0.111707
$175$	0	0
$176$	43.0285	3.24339
$177$	2.34182i	0.176022i
$178$	1.83251i	0.137352i
$179$	−1.62454	−0.121424	−0.0607120	−	0.998155i	$-0.519337\pi$
$179$	−1.62454	−0.121424	−0.0607120	+	0.998155i	$0.519337\pi$
$180$	0	0
$181$	7.78605	0.578733	0.289366	−	0.957218i	$-0.406555\pi$
$181$	7.78605	0.578733	0.289366	+	0.957218i	$0.406555\pi$
$182$	− 16.5753i	− 1.22864i
$183$	− 2.72282i	− 0.201276i
$184$	−44.1348	−3.25366
$185$	0	0
$186$	7.70302	0.564813
$187$	− 12.4819i	− 0.912763i
$188$	9.37137i	0.683478i
$189$	2.07876	0.151207
$190$	0	0
$191$	−16.6452	−1.20440	−0.602201	−	0.798344i	$-0.705709\pi$
$191$	−16.6452	−1.20440	−0.602201	+	0.798344i	$0.705709\pi$
$192$	− 0.370295i	− 0.0267238i
$193$	3.76432i	0.270962i	0.990780	+	0.135481i	$0.0432579\pi$
$193$	3.76432i	0.270962i	−0.990780	+	0.135481i	$0.956742\pi$
$194$	−16.8264	−1.20806
$195$	0	0
$196$	4.54354	0.324538
$197$	4.86416i	0.346557i	0.984873	+	0.173279i	$0.0554361\pi$
$197$	4.86416i	0.346557i	−0.984873	+	0.173279i	$0.944564\pi$
$198$	41.8735i	2.97582i
$199$	−7.04266	−0.499241	−0.249621	−	0.968344i	$-0.580306\pi$
$199$	−7.04266	−0.499241	−0.249621	+	0.968344i	$0.580306\pi$
$200$	0	0
$201$	−2.22920	−0.157236
$202$	− 26.0395i	− 1.83213i
$203$	1.62794i	0.114259i
$204$	−3.52418	−0.246742
$205$	0	0
$206$	31.8243	2.21730
$207$	− 19.5004i	− 1.35537i
$208$	− 48.9649i	− 3.39511i
$209$	29.0155	2.00705
$210$	0	0
$211$	11.5218	0.793191	0.396595	−	0.917994i	$-0.370192\pi$
$211$	11.5218	0.793191	0.396595	+	0.917994i	$0.370192\pi$
$212$	− 28.4757i	− 1.95572i
$213$	0.296906i	0.0203436i
$214$	−4.36307	−0.298253
$215$	0	0
$216$	13.5253	0.920282
$217$	8.51029i	0.577716i
$218$	31.0680i	2.10419i
$219$	−1.63738	−0.110644
$220$	0	0
$221$	−14.2039	−0.955459
$222$	1.11179i	0.0746186i
$223$	0.639877i	0.0428493i	0.999770	+	0.0214247i	$0.00682020\pi$
$223$	0.639877i	0.0428493i	−0.999770	+	0.0214247i	$0.993180\pi$
$224$	6.31728	0.422091
$225$	0	0
$226$	32.6746	2.17348
$227$	− 10.7993i	− 0.716774i	−0.933573	−	0.358387i	$-0.883327\pi$
$227$	− 10.7993i	− 0.716774i	0.933573	−	0.358387i	$-0.116673\pi$
$228$	− 8.19237i	− 0.542553i
$229$	20.2209	1.33623	0.668116	−	0.744057i	$-0.267101\pi$
$229$	20.2209	1.33623	0.668116	+	0.744057i	$0.267101\pi$
$230$	0	0
$231$	2.01482	0.132566
$232$	10.5921i	0.695406i
$233$	7.15167i	0.468521i	0.972174	+	0.234261i	$0.0752669\pi$
$233$	7.15167i	0.468521i	−0.972174	+	0.234261i	$0.924733\pi$
$234$	47.6506	3.11502
$235$	0	0
$236$	−30.0702	−1.95740
$237$	3.15051i	0.204648i
$238$	− 5.60737i	− 0.363472i
$239$	12.7774	0.826500	0.413250	−	0.910618i	$-0.364394\pi$
$239$	12.7774	0.826500	0.413250	+	0.910618i	$0.364394\pi$
$240$	0	0
$241$	26.3455	1.69706	0.848530	−	0.529147i	$-0.177488\pi$
$241$	26.3455	1.69706	0.848530	+	0.529147i	$0.177488\pi$
$242$	54.8006i	3.52272i
$243$	9.02768i	0.579126i
$244$	34.9625	2.23824
$245$	0	0
$246$	1.38687	0.0884236
$247$	− 33.0187i	− 2.10093i
$248$	55.3718i	3.51611i
$249$	1.27205	0.0806126
$250$	0	0
$251$	−7.27060	−0.458916	−0.229458	−	0.973319i	$-0.573695\pi$
$251$	−7.27060	−0.458916	−0.229458	+	0.973319i	$0.573695\pi$
$252$	13.0617i	0.822812i
$253$	− 38.6245i	− 2.42830i
$254$	−9.91637	−0.622208
$255$	0	0
$256$	−27.5648	−1.72280
$257$	29.7439i	1.85537i	0.373362	+	0.927686i	$0.378205\pi$
$257$	29.7439i	1.85537i	−0.373362	+	0.927686i	$0.621795\pi$
$258$	− 5.43859i	− 0.338592i
$259$	−1.22831	−0.0763232
$260$	0	0
$261$	−4.67999	−0.289684
$262$	3.98082i	0.245936i
$263$	22.4784i	1.38608i	0.720899	+	0.693040i	$0.243729\pi$
$263$	22.4784i	1.38608i	−0.720899	+	0.693040i	$0.756271\pi$
$264$	13.1094	0.806825
$265$	0	0
$266$	13.0350	0.799227
$267$	0.253484i	0.0155130i
$268$	− 28.6241i	− 1.74850i
$269$	−18.3297	−1.11758	−0.558789	−	0.829310i	$-0.688734\pi$
$269$	−18.3297	−1.11758	−0.558789	+	0.829310i	$0.688734\pi$
$270$	0	0
$271$	16.1329	0.980005	0.490002	−	0.871721i	$-0.336996\pi$
$271$	16.1329	0.980005	0.490002	+	0.871721i	$0.336996\pi$
$272$	− 16.5647i	− 1.00438i
$273$	− 2.29280i	− 0.138767i
$274$	−29.8260	−1.80186
$275$	0	0
$276$	−10.9054	−0.656428
$277$	3.26856i	0.196389i	0.995167	+	0.0981945i	$0.0313067\pi$
$277$	3.26856i	0.196389i	−0.995167	+	0.0981945i	$0.968693\pi$
$278$	− 23.4441i	− 1.40608i
$279$	−24.4653	−1.46470
$280$	0	0
$281$	−3.85635	−0.230050	−0.115025	−	0.993363i	$-0.536695\pi$
$281$	−3.85635	−0.230050	−0.115025	+	0.993363i	$0.536695\pi$
$282$	1.86692i	0.111174i
$283$	− 22.0923i	− 1.31325i	−0.754217	−	0.656626i	$-0.771983\pi$
$283$	− 22.0923i	− 1.31325i	0.754217	−	0.656626i	$-0.228017\pi$
$284$	−3.81243	−0.226226
$285$	0	0
$286$	94.3817	5.58091
$287$	1.53221i	0.0904436i
$288$	18.1609i	1.07014i
$289$	12.1949	0.717345
$290$	0	0
$291$	−2.32753	−0.136442
$292$	− 21.0248i	− 1.23038i
$293$	− 19.8463i	− 1.15943i	−0.814819	−	0.579715i	$-0.803164\pi$
$293$	− 19.8463i	− 1.15943i	0.814819	−	0.579715i	$-0.196836\pi$
$294$	0.905143	0.0527890
$295$	0	0
$296$	−7.99192	−0.464521
$297$	11.8367i	0.686834i
$298$	− 11.1589i	− 0.646418i
$299$	−43.9533	−2.54189
$300$	0	0
$301$	6.00854	0.346327
$302$	− 26.4012i	− 1.51922i
$303$	− 3.60195i	− 0.206926i
$304$	38.5065	2.20850
$305$	0	0
$306$	16.1200	0.921521
$307$	− 28.9241i	− 1.65079i	−0.564559	−	0.825393i	$-0.690954\pi$
$307$	− 28.9241i	− 1.65079i	0.564559	−	0.825393i	$-0.309046\pi$
$308$	25.8714i	1.47416i
$309$	4.40213	0.250429
$310$	0	0
$311$	25.8008	1.46303	0.731515	−	0.681825i	$-0.238813\pi$
$311$	25.8008	1.46303	0.731515	+	0.681825i	$0.238813\pi$
$312$	− 14.9180i	− 0.844566i
$313$	4.34404i	0.245540i	0.992435	+	0.122770i	$0.0391777\pi$
$313$	4.34404i	0.245540i	−0.992435	+	0.122770i	$0.960822\pi$
$314$	−61.0705	−3.44641
$315$	0	0
$316$	−40.4543	−2.27573
$317$	11.9946i	0.673685i	0.941561	+	0.336842i	$0.109359\pi$
$317$	11.9946i	0.673685i	−0.941561	+	0.336842i	$0.890641\pi$
$318$	− 5.67281i	− 0.318115i
$319$	−9.26967	−0.519002
$320$	0	0
$321$	−0.603527	−0.0336856
$322$	− 17.3517i	− 0.966974i
$323$	− 11.1701i	− 0.621521i
$324$	−35.8432	−1.99129
$325$	0	0
$326$	−51.7466	−2.86598
$327$	4.29752i	0.237653i
$328$	9.96927i	0.550461i
$329$	−2.06257	−0.113713
$330$	0	0
$331$	−25.4301	−1.39776	−0.698882	−	0.715237i	$-0.746319\pi$
$331$	−25.4301	−1.39776	−0.698882	+	0.715237i	$0.746319\pi$
$332$	16.3338i	0.896431i
$333$	− 3.53113i	− 0.193505i
$334$	22.7667	1.24574
$335$	0	0
$336$	2.67387	0.145872
$337$	6.65856i	0.362715i	0.983417	+	0.181358i	$0.0580491\pi$
$337$	6.65856i	0.362715i	−0.983417	+	0.181358i	$0.941951\pi$
$338$	− 74.1487i	− 4.03316i
$339$	4.51975	0.245479
$340$	0	0
$341$	−48.4585	−2.62418
$342$	37.4729i	2.02630i
$343$	1.00000i	0.0539949i
$344$	39.0943	2.10782
$345$	0	0
$346$	−38.4693	−2.06812
$347$	− 1.27571i	− 0.0684837i	−0.999414	−	0.0342419i	$-0.989098\pi$
$347$	− 1.27571i	− 0.0684837i	0.999414	−	0.0342419i	$-0.0109017\pi$
$348$	2.61724i	0.140299i
$349$	−23.4771	−1.25670	−0.628351	−	0.777930i	$-0.716270\pi$
$349$	−23.4771	−1.25670	−0.628351	+	0.777930i	$0.716270\pi$
$350$	0	0
$351$	13.4697	0.718961
$352$	35.9713i	1.91728i
$353$	17.6711i	0.940537i	0.882523	+	0.470268i	$0.155843\pi$
$353$	17.6711i	0.940537i	−0.882523	+	0.470268i	$0.844157\pi$
$354$	−5.99045	−0.318389
$355$	0	0
$356$	−3.25487	−0.172508
$357$	− 0.775646i	− 0.0410515i
$358$	− 4.15564i	− 0.219632i
$359$	−32.7103	−1.72638	−0.863191	−	0.504878i	$-0.831537\pi$
$359$	−32.7103	−1.72638	−0.863191	+	0.504878i	$0.831537\pi$
$360$	0	0
$361$	6.96623	0.366644
$362$	19.9170i	1.04681i
$363$	7.58036i	0.397866i
$364$	29.4408	1.54312
$365$	0	0
$366$	6.96506	0.364070
$367$	2.73145i	0.142580i	0.997456	+	0.0712902i	$0.0227116\pi$
$367$	2.73145i	0.142580i	−0.997456	+	0.0712902i	$0.977288\pi$
$368$	− 51.2586i	− 2.67204i
$369$	−4.40479	−0.229304
$370$	0	0
$371$	6.26731	0.325382
$372$	13.6820i	0.709378i
$373$	− 8.37626i	− 0.433706i	−0.976204	−	0.216853i	$-0.930421\pi$
$373$	− 8.37626i	− 0.433706i	0.976204	−	0.216853i	$-0.0695793\pi$
$374$	31.9290	1.65101
$375$	0	0
$376$	−13.4200	−0.692085
$377$	10.5486i	0.543279i
$378$	5.31753i	0.273504i
$379$	12.6454	0.649548	0.324774	−	0.945792i	$-0.394712\pi$
$379$	12.6454	0.649548	0.324774	+	0.945792i	$0.394712\pi$
$380$	0	0
$381$	−1.37169	−0.0702740
$382$	− 42.5789i	− 2.17853i
$383$	20.3777i	1.04125i	0.853785	+	0.520625i	$0.174301\pi$
$383$	20.3777i	1.04125i	−0.853785	+	0.520625i	$0.825699\pi$
$384$	−3.52342	−0.179804
$385$	0	0
$386$	−9.62926	−0.490116
$387$	17.2733i	0.878053i
$388$	− 29.8867i	− 1.51727i
$389$	−17.4785	−0.886196	−0.443098	−	0.896473i	$-0.646121\pi$
$389$	−17.4785	−0.886196	−0.443098	+	0.896473i	$0.646121\pi$
$390$	0	0
$391$	−14.8693	−0.751971
$392$	6.50646i	0.328626i
$393$	0.550652i	0.0277767i
$394$	−12.4427	−0.626854
$395$	0	0
$396$	−74.3750	−3.73749
$397$	22.0398i	1.10615i	0.833133	+	0.553073i	$0.186545\pi$
$397$	22.0398i	1.10615i	−0.833133	+	0.553073i	$0.813455\pi$
$398$	− 18.0154i	− 0.903029i
$399$	1.80308	0.0902670
$400$	0	0
$401$	−8.04900	−0.401948	−0.200974	−	0.979597i	$-0.564411\pi$
$401$	−8.04900	−0.401948	−0.200974	+	0.979597i	$0.564411\pi$
$402$	− 5.70237i	− 0.284408i
$403$	55.1442i	2.74693i
$404$	46.2510	2.30107
$405$	0	0
$406$	−4.16432	−0.206672
$407$	− 6.99412i	− 0.346685i
$408$	− 5.04671i	− 0.249849i
$409$	−23.3486	−1.15451	−0.577256	−	0.816563i	$-0.695877\pi$
$409$	−23.3486	−1.15451	−0.577256	+	0.816563i	$0.695877\pi$
$410$	0	0
$411$	−4.12572	−0.203507
$412$	56.5258i	2.78483i
$413$	− 6.61824i	− 0.325662i
$414$	49.8827	2.45160
$415$	0	0
$416$	40.9341	2.00696
$417$	− 3.24293i	− 0.158807i
$418$	74.2227i	3.63035i
$419$	13.0497	0.637518	0.318759	−	0.947836i	$-0.396734\pi$
$419$	13.0497	0.637518	0.318759	+	0.947836i	$0.396734\pi$
$420$	0	0
$421$	38.2475	1.86407	0.932034	−	0.362372i	$-0.118033\pi$
$421$	38.2475	1.86407	0.932034	+	0.362372i	$0.118033\pi$
$422$	29.4731i	1.43473i
$423$	− 5.92947i	− 0.288301i
$424$	40.7779	1.98035
$425$	0	0
$426$	−0.759494	−0.0367976
$427$	7.69499i	0.372386i
$428$	− 7.74961i	− 0.374592i
$429$	13.0555	0.630324
$430$	0	0
$431$	7.17527	0.345621	0.172810	−	0.984955i	$-0.444715\pi$
$431$	7.17527	0.345621	0.172810	+	0.984955i	$0.444715\pi$
$432$	15.7085i	0.755773i
$433$	− 33.9429i	− 1.63119i	−0.578623	−	0.815595i	$-0.696410\pi$
$433$	− 33.9429i	− 1.63119i	0.578623	−	0.815595i	$-0.303590\pi$
$434$	−21.7696	−1.04497
$435$	0	0
$436$	−55.1825	−2.64276
$437$	− 34.5654i	− 1.65349i
$438$	− 4.18847i	− 0.200133i
$439$	1.11980	0.0534451	0.0267226	−	0.999643i	$-0.491493\pi$
$439$	1.11980	0.0534451	0.0267226	+	0.999643i	$0.491493\pi$
$440$	0	0
$441$	−2.87480	−0.136895
$442$	− 36.3341i	− 1.72824i
$443$	− 21.9887i	− 1.04472i	−0.852727	−	0.522358i	$-0.825053\pi$
$443$	− 21.9887i	− 1.04472i	0.852727	−	0.522358i	$-0.174947\pi$
$444$	−1.97475	−0.0937173
$445$	0	0
$446$	−1.63683	−0.0775060
$447$	− 1.54357i	− 0.0730083i
$448$	1.04650i	0.0494423i
$449$	−17.5039	−0.826058	−0.413029	−	0.910718i	$-0.635529\pi$
$449$	−17.5039	−0.826058	−0.413029	+	0.910718i	$0.635529\pi$
$450$	0	0
$451$	−8.72459	−0.410825
$452$	58.0360i	2.72978i
$453$	− 3.65197i	− 0.171585i
$454$	27.6249	1.29650
$455$	0	0
$456$	11.7317	0.549386
$457$	5.33726i	0.249667i	0.992178	+	0.124833i	$0.0398396\pi$
$457$	5.33726i	0.249667i	−0.992178	+	0.124833i	$0.960160\pi$
$458$	51.7256i	2.41698i
$459$	4.55676	0.212691
$460$	0	0
$461$	7.13045	0.332098	0.166049	−	0.986117i	$-0.446899\pi$
$461$	7.13045	0.332098	0.166049	+	0.986117i	$0.446899\pi$
$462$	5.15399i	0.239785i
$463$	15.6224i	0.726036i	0.931782	+	0.363018i	$0.118254\pi$
$463$	15.6224i	0.726036i	−0.931782	+	0.363018i	$0.881746\pi$
$464$	−12.3018	−0.571096
$465$	0	0
$466$	−18.2942	−0.847463
$467$	− 28.1640i	− 1.30328i	−0.758530	−	0.651638i	$-0.774082\pi$
$467$	− 28.1640i	− 1.30328i	0.758530	−	0.651638i	$-0.225918\pi$
$468$	84.6363i	3.91231i
$469$	6.29996	0.290905
$470$	0	0
$471$	−8.44766	−0.389247
$472$	− 43.0613i	− 1.98206i
$473$	34.2133i	1.57313i
$474$	−8.05912	−0.370168
$475$	0	0
$476$	9.95971	0.456503
$477$	18.0172i	0.824952i
$478$	32.6850i	1.49498i
$479$	26.4738	1.20962	0.604809	−	0.796371i	$-0.293250\pi$
$479$	26.4738	1.20962	0.604809	+	0.796371i	$0.293250\pi$
$480$	0	0
$481$	−7.95906	−0.362902
$482$	67.3926i	3.06965i
$483$	− 2.40020i	− 0.109213i
$484$	−97.3360	−4.42436
$485$	0	0
$486$	−23.0931	−1.04752
$487$	− 10.4405i	− 0.473106i	−0.971619	−	0.236553i	$-0.923982\pi$
$487$	− 10.4405i	− 0.473106i	0.971619	−	0.236553i	$-0.0760177\pi$
$488$	50.0671i	2.26643i
$489$	−7.15791	−0.323692
$490$	0	0
$491$	−4.53334	−0.204587	−0.102293	−	0.994754i	$-0.532618\pi$
$491$	−4.53334	−0.204587	−0.102293	+	0.994754i	$0.532618\pi$
$492$	2.46334i	0.111056i
$493$	3.56854i	0.160719i
$494$	84.4629	3.80017
$495$	0	0
$496$	−64.3093	−2.88757
$497$	− 0.839088i	− 0.0376382i
$498$	3.25394i	0.145812i
$499$	1.36104	0.0609283	0.0304642	−	0.999536i	$-0.490301\pi$
$499$	1.36104	0.0609283	0.0304642	+	0.999536i	$0.490301\pi$
$500$	0	0
$501$	3.14923	0.140697
$502$	− 18.5984i	− 0.830089i
$503$	− 27.9936i	− 1.24817i	−0.781356	−	0.624086i	$-0.785472\pi$
$503$	− 27.9936i	− 1.24817i	0.781356	−	0.624086i	$-0.214528\pi$
$504$	−18.7047	−0.833175
$505$	0	0
$506$	98.8028	4.39232
$507$	− 10.2567i	− 0.455517i
$508$	− 17.6133i	− 0.781463i
$509$	−1.40214	−0.0621488	−0.0310744	−	0.999517i	$-0.509893\pi$
$509$	−1.40214	−0.0621488	−0.0310744	+	0.999517i	$0.509893\pi$
$510$	0	0
$511$	4.62741	0.204705
$512$	− 50.5966i	− 2.23608i
$513$	10.5927i	0.467681i
$514$	−76.0858	−3.35600
$515$	0	0
$516$	9.65993	0.425255
$517$	− 11.7445i	− 0.516524i
$518$	− 3.14205i	− 0.138054i
$519$	−5.32132	−0.233580
$520$	0	0
$521$	−23.2572	−1.01892	−0.509458	−	0.860495i	$-0.670154\pi$
$521$	−23.2572	−1.01892	−0.509458	+	0.860495i	$0.670154\pi$
$522$	− 11.9716i	− 0.523981i
$523$	3.80015i	0.166169i	0.996543	+	0.0830845i	$0.0264771\pi$
$523$	3.80015i	0.166169i	−0.996543	+	0.0830845i	$0.973523\pi$
$524$	−7.07067	−0.308884
$525$	0	0
$526$	−57.5006	−2.50715
$527$	18.6551i	0.812628i
$528$	15.2253i	0.662598i
$529$	−23.0122	−1.00053
$530$	0	0
$531$	19.0261	0.825662
$532$	23.1525i	1.00379i
$533$	9.92828i	0.430042i
$534$	−0.648420	−0.0280599
$535$	0	0
$536$	40.9904	1.77052
$537$	− 0.574833i	− 0.0248059i
$538$	− 46.8879i	− 2.02148i
$539$	−5.69412	−0.245263
$540$	0	0
$541$	−16.0165	−0.688602	−0.344301	−	0.938859i	$-0.611884\pi$
$541$	−16.0165	−0.688602	−0.344301	+	0.938859i	$0.611884\pi$
$542$	41.2685i	1.77264i
$543$	2.75504i	0.118230i
$544$	13.8479	0.593722
$545$	0	0
$546$	5.86506	0.251001
$547$	− 30.2390i	− 1.29293i	−0.762944	−	0.646464i	$-0.776247\pi$
$547$	− 30.2390i	− 1.29293i	0.762944	−	0.646464i	$-0.223753\pi$
$548$	− 52.9765i	− 2.26304i
$549$	−22.1215	−0.944123
$550$	0	0
$551$	−8.29550	−0.353400
$552$	− 15.6168i	− 0.664695i
$553$	− 8.90370i	− 0.378624i
$554$	−8.36110	−0.355229
$555$	0	0
$556$	41.6410	1.76597
$557$	24.2216i	1.02630i	0.858298	+	0.513152i	$0.171522\pi$
$557$	24.2216i	1.02630i	−0.858298	+	0.513152i	$0.828478\pi$
$558$	− 62.5831i	− 2.64936i
$559$	38.9336	1.64672
$560$	0	0
$561$	4.41662	0.186470
$562$	− 9.86467i	− 0.416116i
$563$	− 11.9124i	− 0.502048i	−0.967981	−	0.251024i	$-0.919233\pi$
$563$	− 11.9124i	− 0.502048i	0.967981	−	0.251024i	$-0.0807672\pi$
$564$	−3.31600	−0.139629
$565$	0	0
$566$	56.5129	2.37541
$567$	− 7.88883i	− 0.331300i
$568$	− 5.45949i	− 0.229075i
$569$	−22.2725	−0.933712	−0.466856	−	0.884333i	$-0.654613\pi$
$569$	−22.2725	−0.933712	−0.466856	+	0.884333i	$0.654613\pi$
$570$	0	0
$571$	19.6713	0.823216	0.411608	−	0.911361i	$-0.364967\pi$
$571$	19.6713	0.823216	0.411608	+	0.911361i	$0.364967\pi$
$572$	167.639i	7.00935i
$573$	− 5.88978i	− 0.246049i
$574$	−3.91945	−0.163595
$575$	0	0
$576$	−3.00846	−0.125353
$577$	22.5166i	0.937378i	0.883363	+	0.468689i	$0.155274\pi$
$577$	22.5166i	0.937378i	−0.883363	+	0.468689i	$0.844726\pi$
$578$	31.1949i	1.29754i
$579$	−1.33198	−0.0553551
$580$	0	0
$581$	−3.59494	−0.149143
$582$	− 5.95390i	− 0.246797i
$583$	35.6868i	1.47799i
$584$	30.1081	1.24588
$585$	0	0
$586$	50.7674	2.09718
$587$	− 28.2290i	− 1.16514i	−0.812782	−	0.582568i	$-0.802048\pi$
$587$	− 28.2290i	− 1.16514i	0.812782	−	0.582568i	$-0.197952\pi$
$588$	1.60770i	0.0663004i
$589$	−43.3659	−1.78686
$590$	0	0
$591$	−1.72115	−0.0707987
$592$	− 9.28189i	− 0.381483i
$593$	27.2560i	1.11927i	0.828739	+	0.559635i	$0.189059\pi$
$593$	27.2560i	1.11927i	−0.828739	+	0.559635i	$0.810941\pi$
$594$	−30.2786	−1.24235
$595$	0	0
$596$	19.8203	0.811870
$597$	− 2.49200i	− 0.101991i
$598$	− 112.434i	− 4.59777i
$599$	−19.7263	−0.805997	−0.402998	−	0.915201i	$-0.632032\pi$
$599$	−19.7263	−0.805997	−0.402998	+	0.915201i	$0.632032\pi$
$600$	0	0
$601$	−39.6721	−1.61826	−0.809130	−	0.587629i	$-0.800061\pi$
$601$	−39.6721	−1.61826	−0.809130	+	0.587629i	$0.800061\pi$
$602$	15.3701i	0.626437i
$603$	18.1111i	0.737541i
$604$	46.8933	1.90806
$605$	0	0
$606$	9.21390	0.374289
$607$	23.8572i	0.968336i	0.874975	+	0.484168i	$0.160878\pi$
$607$	23.8572i	0.968336i	−0.874975	+	0.484168i	$0.839122\pi$
$608$	32.1910i	1.30552i
$609$	−0.576035	−0.0233421
$610$	0	0
$611$	−13.3649	−0.540685
$612$	28.6321i	1.15739i
$613$	20.6564i	0.834302i	0.908837	+	0.417151i	$0.136971\pi$
$613$	20.6564i	0.834302i	−0.908837	+	0.417151i	$0.863029\pi$
$614$	73.9889	2.98595
$615$	0	0
$616$	−37.0485	−1.49273
$617$	− 48.1688i	− 1.93920i	−0.244688	−	0.969602i	$-0.578686\pi$
$617$	− 48.1688i	− 1.93920i	0.244688	−	0.969602i	$-0.421314\pi$
$618$	11.2608i	0.452976i
$619$	19.1752	0.770716	0.385358	−	0.922767i	$-0.374078\pi$
$619$	19.1752	0.770716	0.385358	+	0.922767i	$0.374078\pi$
$620$	0	0
$621$	14.1007	0.565841
$622$	65.9994i	2.64633i
$623$	− 0.716374i	− 0.0287009i
$624$	17.3259	0.693591
$625$	0	0
$626$	−11.1122	−0.444133
$627$	10.2670i	0.410023i
$628$	− 108.472i	− 4.32852i
$629$	−2.69252	−0.107358
$630$	0	0
$631$	30.8692	1.22889	0.614443	−	0.788962i	$-0.289381\pi$
$631$	30.8692	1.22889	0.614443	+	0.788962i	$0.289381\pi$
$632$	− 57.9315i	− 2.30439i
$633$	4.07690i	0.162042i
$634$	−30.6826	−1.21856
$635$	0	0
$636$	10.0759	0.399537
$637$	6.47971i	0.256735i
$638$	− 23.7121i	− 0.938772i
$639$	2.41221	0.0954254
$640$	0	0
$641$	12.4835	0.493070	0.246535	−	0.969134i	$-0.420708\pi$
$641$	12.4835	0.493070	0.246535	+	0.969134i	$0.420708\pi$
$642$	− 1.54384i	− 0.0609306i
$643$	− 32.0891i	− 1.26547i	−0.774369	−	0.632735i	$-0.781932\pi$
$643$	− 32.0891i	− 1.26547i	0.774369	−	0.632735i	$-0.218068\pi$
$644$	30.8199	1.21447
$645$	0	0
$646$	28.5735	1.12421
$647$	− 28.7155i	− 1.12892i	−0.825459	−	0.564462i	$-0.809084\pi$
$647$	− 28.7155i	− 1.12892i	0.825459	−	0.564462i	$-0.190916\pi$
$648$	− 51.3283i	− 2.01637i
$649$	37.6850	1.47927
$650$	0	0
$651$	−3.01131	−0.118022
$652$	− 91.9114i	− 3.59953i
$653$	16.4333i	0.643086i	0.946895	+	0.321543i	$0.104201\pi$
$653$	16.4333i	0.643086i	−0.946895	+	0.321543i	$0.895799\pi$
$654$	−10.9932	−0.429868
$655$	0	0
$656$	−11.5784	−0.452061
$657$	13.3029i	0.518994i
$658$	− 5.27613i	− 0.205685i
$659$	36.3315	1.41528	0.707638	−	0.706575i	$-0.249761\pi$
$659$	36.3315	1.41528	0.707638	+	0.706575i	$0.249761\pi$
$660$	0	0
$661$	20.5035	0.797493	0.398746	−	0.917061i	$-0.369445\pi$
$661$	20.5035	0.797493	0.398746	+	0.917061i	$0.369445\pi$
$662$	− 65.0510i	− 2.52828i
$663$	− 5.02596i	− 0.195192i
$664$	−23.3903	−0.907721
$665$	0	0
$666$	9.03274	0.350012
$667$	11.0427i	0.427574i
$668$	40.4378i	1.56459i
$669$	−0.226416	−0.00875375
$670$	0	0
$671$	−43.8161	−1.69150
$672$	2.23533i	0.0862296i
$673$	25.6104i	0.987208i	0.869687	+	0.493604i	$0.164321\pi$
$673$	25.6104i	0.987208i	−0.869687	+	0.493604i	$0.835679\pi$
$674$	−17.0328	−0.656080
$675$	0	0
$676$	131.702	5.06545
$677$	− 33.7573i	− 1.29740i	−0.761045	−	0.648699i	$-0.775313\pi$
$677$	− 33.7573i	− 1.29740i	0.761045	−	0.648699i	$-0.224687\pi$
$678$	11.5617i	0.444023i
$679$	6.57786	0.252435
$680$	0	0
$681$	3.82125	0.146431
$682$	− 123.959i	− 4.74662i
$683$	2.25100i	0.0861320i	0.999072	+	0.0430660i	$0.0137126\pi$
$683$	2.25100i	0.0861320i	−0.999072	+	0.0430660i	$0.986287\pi$
$684$	−66.5588	−2.54494
$685$	0	0
$686$	−2.55803	−0.0976662
$687$	7.15501i	0.272981i
$688$	45.4045i	1.73103i
$689$	40.6103	1.54713
$690$	0	0
$691$	12.0924	0.460018	0.230009	−	0.973188i	$-0.426124\pi$
$691$	12.0924	0.460018	0.230009	+	0.973188i	$0.426124\pi$
$692$	− 68.3286i	− 2.59746i
$693$	− 16.3694i	− 0.621823i
$694$	3.26331	0.123874
$695$	0	0
$696$	−3.74795	−0.142066
$697$	3.35870i	0.127220i
$698$	− 60.0553i	− 2.27313i
$699$	−2.53057	−0.0957149
$700$	0	0
$701$	27.7239	1.04712	0.523559	−	0.851990i	$-0.324604\pi$
$701$	27.7239	1.04712	0.523559	+	0.851990i	$0.324604\pi$
$702$	34.4560i	1.30046i
$703$	− 6.25909i	− 0.236066i
$704$	−5.95887	−0.224583
$705$	0	0
$706$	−45.2032	−1.70125
$707$	10.1795i	0.382840i
$708$	− 10.6401i	− 0.399881i
$709$	39.7082	1.49127	0.745636	−	0.666354i	$-0.232146\pi$
$709$	39.7082	1.49127	0.745636	+	0.666354i	$0.232146\pi$
$710$	0	0
$711$	25.5963	0.959937
$712$	− 4.66105i	− 0.174680i
$713$	57.7272i	2.16190i
$714$	1.98413	0.0742541
$715$	0	0
$716$	7.38117	0.275847
$717$	4.52119i	0.168847i
$718$	− 83.6740i	− 3.12268i
$719$	−17.8189	−0.664531	−0.332266	−	0.943186i	$-0.607813\pi$
$719$	−17.8189	−0.664531	−0.332266	+	0.943186i	$0.607813\pi$
$720$	0	0
$721$	−12.4409	−0.463324
$722$	17.8199i	0.663186i
$723$	9.32216i	0.346695i
$724$	−35.3762	−1.31475
$725$	0	0
$726$	−19.3908	−0.719661
$727$	51.8897i	1.92448i	0.272197	+	0.962241i	$0.412250\pi$
$727$	51.8897i	1.92448i	−0.272197	+	0.962241i	$0.587750\pi$
$728$	42.1599i	1.56255i
$729$	20.4721	0.758226
$730$	0	0
$731$	13.1711	0.487151
$732$	12.3712i	0.457254i
$733$	− 20.0529i	− 0.740669i	−0.928898	−	0.370334i	$-0.879243\pi$
$733$	− 20.0529i	− 0.740669i	0.928898	−	0.370334i	$-0.120757\pi$
$734$	−6.98714	−0.257900
$735$	0	0
$736$	42.8516	1.57953
$737$	35.8727i	1.32139i
$738$	− 11.2676i	− 0.414767i
$739$	−6.39375	−0.235198	−0.117599	−	0.993061i	$-0.537520\pi$
$739$	−6.39375	−0.235198	−0.117599	+	0.993061i	$0.537520\pi$
$740$	0	0
$741$	11.6834	0.429202
$742$	16.0320i	0.588552i
$743$	3.15082i	0.115592i	0.998328	+	0.0577962i	$0.0184073\pi$
$743$	3.15082i	0.115592i	−0.998328	+	0.0577962i	$0.981593\pi$
$744$	−19.5929	−0.718312
$745$	0	0
$746$	21.4268	0.784490
$747$	− 10.3347i	− 0.378128i
$748$	56.7118i	2.07359i
$749$	1.70563	0.0623225
$750$	0	0
$751$	−21.1642	−0.772293	−0.386147	−	0.922437i	$-0.626194\pi$
$751$	−21.1642	−0.772293	−0.386147	+	0.922437i	$0.626194\pi$
$752$	− 15.5862i	− 0.568369i
$753$	− 2.57265i	− 0.0937526i
$754$	−26.9836	−0.982684
$755$	0	0
$756$	−9.44491	−0.343508
$757$	45.1346i	1.64044i	0.572045	+	0.820222i	$0.306150\pi$
$757$	45.1346i	1.64044i	−0.572045	+	0.820222i	$0.693850\pi$
$758$	32.3473i	1.17491i
$759$	13.6670	0.496081
$760$	0	0
$761$	25.4211	0.921513	0.460757	−	0.887526i	$-0.347578\pi$
$761$	25.4211	0.921513	0.460757	+	0.887526i	$0.347578\pi$
$762$	− 3.50884i	− 0.127112i
$763$	− 12.1453i	− 0.439688i
$764$	75.6279	2.73612
$765$	0	0
$766$	−52.1268	−1.88342
$767$	− 42.8843i	− 1.54846i
$768$	− 9.75363i	− 0.351954i
$769$	−8.36177	−0.301533	−0.150767	−	0.988569i	$-0.548174\pi$
$769$	−8.36177	−0.301533	−0.150767	+	0.988569i	$0.548174\pi$
$770$	0	0
$771$	−10.5247	−0.379037
$772$	− 17.1033i	− 0.615562i
$773$	− 22.3094i	− 0.802414i	−0.915987	−	0.401207i	$-0.868591\pi$
$773$	− 22.3094i	− 0.802414i	0.915987	−	0.401207i	$-0.131409\pi$
$774$	−44.1858	−1.58822
$775$	0	0
$776$	42.7985	1.53638
$777$	− 0.434628i	− 0.0155922i
$778$	− 44.7106i	− 1.60295i
$779$	−7.80770	−0.279740
$780$	0	0
$781$	4.77786	0.170965
$782$	− 38.0361i	− 1.36017i
$783$	− 3.38409i	− 0.120937i
$784$	−7.55666	−0.269881
$785$	0	0
$786$	−1.40859	−0.0502426
$787$	19.4098i	0.691884i	0.938256	+	0.345942i	$0.112441\pi$
$787$	19.4098i	0.691884i	−0.938256	+	0.345942i	$0.887559\pi$
$788$	− 22.1005i	− 0.787298i
$789$	−7.95384	−0.283164
$790$	0	0
$791$	−12.7733	−0.454166
$792$	− 106.507i	− 3.78456i
$793$	49.8613i	1.77063i
$794$	−56.3786	−2.00080
$795$	0	0
$796$	31.9986	1.13416
$797$	30.7974i	1.09090i	0.838143	+	0.545450i	$0.183641\pi$
$797$	30.7974i	1.09090i	−0.838143	+	0.545450i	$0.816359\pi$
$798$	4.61234i	0.163275i
$799$	−4.52129	−0.159952
$800$	0	0
$801$	2.05943	0.0727663
$802$	− 20.5896i	− 0.727045i
$803$	26.3490i	0.929837i
$804$	10.1285	0.357203
$805$	0	0
$806$	−141.061	−4.96865
$807$	− 6.48582i	− 0.228312i
$808$	66.2325i	2.33005i
$809$	48.1014	1.69115	0.845577	−	0.533854i	$-0.179257\pi$
$809$	48.1014	1.69115	0.845577	+	0.533854i	$0.179257\pi$
$810$	0	0
$811$	−31.3020	−1.09916	−0.549580	−	0.835441i	$-0.685212\pi$
$811$	−31.3020	−1.09916	−0.549580	+	0.835441i	$0.685212\pi$
$812$	− 7.39660i	− 0.259570i
$813$	5.70852i	0.200207i
$814$	17.8912	0.627086
$815$	0	0
$816$	5.86129	0.205186
$817$	30.6178i	1.07118i
$818$	− 59.7264i	− 2.08829i
$819$	−18.6278	−0.650909
$820$	0	0
$821$	7.79834	0.272164	0.136082	−	0.990698i	$-0.456549\pi$
$821$	7.79834	0.272164	0.136082	+	0.990698i	$0.456549\pi$
$822$	− 10.5537i	− 0.368104i
$823$	53.0240i	1.84830i	0.382029	+	0.924150i	$0.375226\pi$
$823$	53.0240i	1.84830i	−0.382029	+	0.924150i	$0.624774\pi$
$824$	−80.9463	−2.81990
$825$	0	0
$826$	16.9297	0.589059
$827$	− 52.0766i	− 1.81088i	−0.424473	−	0.905441i	$-0.639541\pi$
$827$	− 52.0766i	− 1.81088i	0.424473	−	0.905441i	$-0.360459\pi$
$828$	88.6008i	3.07909i
$829$	8.49580	0.295071	0.147536	−	0.989057i	$-0.452866\pi$
$829$	8.49580	0.295071	0.147536	+	0.989057i	$0.452866\pi$
$830$	0	0
$831$	−1.15656	−0.0401206
$832$	6.78099i	0.235088i
$833$	2.19206i	0.0759504i
$834$	8.29552	0.287250
$835$	0	0
$836$	−131.833	−4.55955
$837$	− 17.6908i	− 0.611484i
$838$	33.3815i	1.15314i
$839$	31.4762	1.08668	0.543339	−	0.839513i	$-0.317160\pi$
$839$	31.4762	1.08668	0.543339	+	0.839513i	$0.317160\pi$
$840$	0	0
$841$	−26.3498	−0.908614
$842$	97.8383i	3.37173i
$843$	− 1.36454i	− 0.0469973i
$844$	−52.3496	−1.80195
$845$	0	0
$846$	15.1678	0.521479
$847$	− 21.4229i	− 0.736101i
$848$	47.3599i	1.62635i
$849$	7.81721	0.268286
$850$	0	0
$851$	−8.33188	−0.285613
$852$	− 1.34900i	− 0.0462160i
$853$	28.8426i	0.987552i	0.869589	+	0.493776i	$0.164384\pi$
$853$	28.8426i	0.987552i	−0.869589	+	0.493776i	$0.835616\pi$
$854$	−19.6840	−0.673574
$855$	0	0
$856$	11.0976	0.379309
$857$	− 26.8954i	− 0.918730i	−0.888248	−	0.459365i	$-0.848077\pi$
$857$	− 26.8954i	− 0.918730i	0.888248	−	0.459365i	$-0.151923\pi$
$858$	33.3963i	1.14013i
$859$	−4.29573	−0.146569	−0.0732843	−	0.997311i	$-0.523348\pi$
$859$	−4.29573	−0.146569	−0.0732843	+	0.997311i	$0.523348\pi$
$860$	0	0
$861$	−0.542163	−0.0184768
$862$	18.3546i	0.625160i
$863$	− 39.2114i	− 1.33477i	−0.744712	−	0.667386i	$-0.767413\pi$
$863$	− 39.2114i	− 1.33477i	0.744712	−	0.667386i	$-0.232587\pi$
$864$	−13.1321	−0.446762
$865$	0	0
$866$	86.8270	2.95050
$867$	4.31507i	0.146547i
$868$	− 38.6668i	− 1.31244i
$869$	50.6987	1.71984
$870$	0	0
$871$	40.8219	1.38320
$872$	− 79.0227i	− 2.67604i
$873$	18.9100i	0.640006i
$874$	88.4193	2.99083
$875$	0	0
$876$	7.43949	0.251357
$877$	11.1744i	0.377331i	0.982041	+	0.188666i	$0.0604162\pi$
$877$	11.1744i	0.377331i	−0.982041	+	0.188666i	$0.939584\pi$
$878$	2.86449i	0.0966717i
$879$	7.02246	0.236862
$880$	0	0
$881$	−55.5215	−1.87057	−0.935284	−	0.353899i	$-0.884856\pi$
$881$	−55.5215	−1.87057	−0.935284	+	0.353899i	$0.884856\pi$
$882$	− 7.35382i	− 0.247616i
$883$	− 45.3210i	− 1.52517i	−0.646886	−	0.762586i	$-0.723929\pi$
$883$	− 45.3210i	− 1.52517i	0.646886	−	0.762586i	$-0.276071\pi$
$884$	64.5360	2.17058
$885$	0	0
$886$	56.2479	1.88968
$887$	35.9580i	1.20735i	0.797229	+	0.603676i	$0.206298\pi$
$887$	35.9580i	1.20735i	−0.797229	+	0.603676i	$0.793702\pi$
$888$	− 2.82789i	− 0.0948976i
$889$	3.87656	0.130016
$890$	0	0
$891$	44.9199	1.50487
$892$	− 2.90730i	− 0.0973438i
$893$	− 10.5103i	− 0.351713i
$894$	3.94850	0.132058
$895$	0	0
$896$	9.95759	0.332660
$897$	− 15.5526i	− 0.519286i
$898$	− 44.7754i	− 1.49418i
$899$	13.8542	0.462064
$900$	0	0
$901$	13.7383	0.457690
$902$	− 22.3178i	− 0.743101i
$903$	2.12608i	0.0707516i
$904$	−83.1090	−2.76416
$905$	0	0
$906$	9.34187	0.310363
$907$	30.2781i	1.00537i	0.864470	+	0.502685i	$0.167654\pi$
$907$	30.2781i	1.00537i	−0.864470	+	0.502685i	$0.832346\pi$
$908$	49.0670i	1.62834i
$909$	−29.2640	−0.970625
$910$	0	0
$911$	−22.6499	−0.750424	−0.375212	−	0.926939i	$-0.622430\pi$
$911$	−22.6499	−0.750424	−0.375212	+	0.926939i	$0.622430\pi$
$912$	13.6253i	0.451178i
$913$	− 20.4700i	− 0.677459i
$914$	−13.6529	−0.451598
$915$	0	0
$916$	−91.8742	−3.03561
$917$	− 1.55620i	− 0.0513904i
$918$	11.6563i	0.384717i
$919$	−25.2435	−0.832707	−0.416354	−	0.909203i	$-0.636692\pi$
$919$	−25.2435	−0.832707	−0.416354	+	0.909203i	$0.636692\pi$
$920$	0	0
$921$	10.2346	0.337241
$922$	18.2399i	0.600700i
$923$	− 5.43705i	− 0.178963i
$924$	−9.15443	−0.301159
$925$	0	0
$926$	−39.9627	−1.31326
$927$	− 35.7651i	− 1.17468i
$928$	− 10.2841i	− 0.337593i
$929$	3.46732	0.113759	0.0568795	−	0.998381i	$-0.481885\pi$
$929$	3.46732	0.113759	0.0568795	+	0.998381i	$0.481885\pi$
$930$	0	0
$931$	−5.09571	−0.167005
$932$	− 32.4939i	− 1.06437i
$933$	9.12945i	0.298885i
$934$	72.0445	2.35737
$935$	0	0
$936$	−121.201	−3.96158
$937$	10.9233i	0.356848i	0.983954	+	0.178424i	$0.0570999\pi$
$937$	10.9233i	0.356848i	−0.983954	+	0.178424i	$0.942900\pi$
$938$	16.1155i	0.526190i
$939$	−1.53711	−0.0501617
$940$	0	0
$941$	−8.97045	−0.292428	−0.146214	−	0.989253i	$-0.546709\pi$
$941$	−8.97045	−0.292428	−0.146214	+	0.989253i	$0.546709\pi$
$942$	− 21.6094i	− 0.704072i
$943$	10.3933i	0.338454i
$944$	50.0118	1.62774
$945$	0	0
$946$	−87.5189	−2.84548
$947$	− 33.5680i	− 1.09081i	−0.838172	−	0.545406i	$-0.816375\pi$
$947$	− 33.5680i	− 1.09081i	0.838172	−	0.545406i	$-0.183625\pi$
$948$	− 14.3145i	− 0.464913i
$949$	29.9843	0.973331
$950$	0	0
$951$	−4.24421	−0.137628
$952$	14.2626i	0.462252i
$953$	− 37.9396i	− 1.22898i	−0.788923	−	0.614492i	$-0.789361\pi$
$953$	− 37.9396i	− 1.22898i	0.788923	−	0.614492i	$-0.210639\pi$
$954$	−46.0887	−1.49218
$955$	0	0
$956$	−58.0545	−1.87762
$957$	− 3.28001i	− 0.106028i
$958$	67.7208i	2.18796i
$959$	11.6597	0.376513
$960$	0	0
$961$	41.4250	1.33629
$962$	− 20.3596i	− 0.656418i
$963$	4.90335i	0.158008i
$964$	−119.702	−3.85533
$965$	0	0
$966$	6.13979	0.197545
$967$	12.4715i	0.401056i	0.979688	+	0.200528i	$0.0642657\pi$
$967$	12.4715i	0.401056i	−0.979688	+	0.200528i	$0.935734\pi$
$968$	− 139.387i	− 4.48008i
$969$	3.95246	0.126971
$970$	0	0
$971$	−48.9493	−1.57086	−0.785429	−	0.618951i	$-0.787558\pi$
$971$	−48.9493	−1.57086	−0.785429	+	0.618951i	$0.787558\pi$
$972$	− 41.0176i	− 1.31564i
$973$	9.16488i	0.293812i
$974$	26.7073	0.855756
$975$	0	0
$976$	−58.1484	−1.86128
$977$	12.0315i	0.384921i	0.981305	+	0.192461i	$0.0616467\pi$
$977$	12.0315i	0.384921i	−0.981305	+	0.192461i	$0.938353\pi$
$978$	− 18.3102i	− 0.585495i
$979$	4.07911	0.130369
$980$	0	0
$981$	34.9152	1.11476
$982$	− 11.5964i	− 0.370057i
$983$	23.4581i	0.748199i	0.927389	+	0.374099i	$0.122048\pi$
$983$	23.4581i	0.748199i	−0.927389	+	0.374099i	$0.877952\pi$
$984$	−3.52756	−0.112454
$985$	0	0
$986$	−9.12845	−0.290709
$987$	− 0.729827i	− 0.0232307i
$988$	150.022i	4.77283i
$989$	40.7573	1.29601
$990$	0	0
$991$	1.92852	0.0612615	0.0306308	−	0.999531i	$-0.490248\pi$
$991$	1.92852	0.0612615	0.0306308	+	0.999531i	$0.490248\pi$
$992$	− 53.7618i	− 1.70694i
$993$	− 8.99826i	− 0.285551i
$994$	2.14642	0.0680802
$995$	0	0
$996$	−5.77959	−0.183133
$997$	33.7803i	1.06983i	0.844905	+	0.534917i	$0.179657\pi$
$997$	33.7803i	1.06983i	−0.844905	+	0.534917i	$0.820343\pi$
$998$	3.48158i	0.110207i
$999$	2.55335	0.0807844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	875.2.b.c.624.12		12
5.2	odd	4		875.2.a.e.1.1	✓	6
5.3	odd	4		875.2.a.h.1.6	yes	6
5.4	even	2	inner	875.2.b.c.624.1		12
15.2	even	4		7875.2.a.t.1.6		6
15.8	even	4		7875.2.a.q.1.1		6
35.13	even	4		6125.2.a.t.1.6		6
35.27	even	4		6125.2.a.s.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
875.2.a.e.1.1	✓	6	5.2	odd	4
875.2.a.h.1.6	yes	6	5.3	odd	4
875.2.b.c.624.1		12	5.4	even	2	inner
875.2.b.c.624.12		12	1.1	even	1	trivial
6125.2.a.s.1.1		6	35.27	even	4
6125.2.a.t.1.6		6	35.13	even	4
7875.2.a.q.1.1		6	15.8	even	4
7875.2.a.t.1.6		6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 875 = 5^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	875.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.55803i q^{2} +0.353843i q^{3} -4.54354 q^{4} -0.905143 q^{6} -1.00000i q^{7} -6.50646i q^{8} +2.87480 q^{9} +5.69412 q^{11} -1.60770i q^{12} -6.47971i q^{13} +2.55803 q^{14} +7.55666 q^{16} -2.19206i q^{17} +7.35382i q^{18} +5.09571 q^{19} +0.353843 q^{21} +14.5657i q^{22} -6.78323i q^{23} +2.30226 q^{24} +16.5753 q^{26} +2.07876i q^{27} +4.54354i q^{28} -1.62794 q^{29} -8.51029 q^{31} +6.31728i q^{32} +2.01482i q^{33} +5.60737 q^{34} -13.0617 q^{36} -1.22831i q^{37} +13.0350i q^{38} +2.29280 q^{39} -1.53221 q^{41} +0.905143i q^{42} +6.00854i q^{43} -25.8714 q^{44} +17.3517 q^{46} -2.06257i q^{47} +2.67387i q^{48} -1.00000 q^{49} +0.775646 q^{51} +29.4408i q^{52} +6.26731i q^{53} -5.31753 q^{54} -6.50646 q^{56} +1.80308i q^{57} -4.16432i q^{58} +6.61824 q^{59} -7.69499 q^{61} -21.7696i q^{62} -2.87480i q^{63} -1.04650 q^{64} -5.15399 q^{66} +6.29996i q^{67} +9.95971i q^{68} +2.40020 q^{69} +0.839088 q^{71} -18.7047i q^{72} +4.62741i q^{73} +3.14205 q^{74} -23.1525 q^{76} -5.69412i q^{77} +5.86506i q^{78} +8.90370 q^{79} +7.88883 q^{81} -3.91945i q^{82} -3.59494i q^{83} -1.60770 q^{84} -15.3701 q^{86} -0.576035i q^{87} -37.0485i q^{88} +0.716374 q^{89} -6.47971 q^{91} +30.8199i q^{92} -3.01131i q^{93} +5.27613 q^{94} -2.23533 q^{96} +6.57786i q^{97} -2.55803i q^{98} +16.3694 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{4} - 4 q^{6} - 14 q^{9} + 2 q^{11} + 4 q^{14} + 16 q^{16} + 34 q^{19} - 10 q^{21} - 14 q^{24} + 32 q^{26} - 14 q^{29} - 22 q^{31} + 8 q^{34} + 20 q^{36} - 30 q^{39} + 6 q^{41} - 52 q^{44} - 16 q^{46}+ \cdots - 24 q^{99}+O(q^{100}) \) 12 * q - 8 * q^4 - 4 * q^6 - 14 * q^9 + 2 * q^11 + 4 * q^14 + 16 * q^16 + 34 * q^19 - 10 * q^21 - 14 * q^24 + 32 * q^26 - 14 * q^29 - 22 * q^31 + 8 * q^34 + 20 * q^36 - 30 * q^39 + 6 * q^41 - 52 * q^44 - 16 * q^46 - 12 * q^49 + 2 * q^51 + 52 * q^54 - 24 * q^56 + 42 * q^59 - 38 * q^61 + 44 * q^64 - 70 * q^66 + 4 * q^69 + 2 * q^71 + 58 * q^74 - 8 * q^76 - 44 * q^79 + 28 * q^81 + 22 * q^84 + 46 * q^86 + 16 * q^89 - 44 * q^91 + 52 * q^94 - 8 * q^96 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more