LMFDB - Embedded newform 875.2.b.e.624.13

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 29x^{14} + 338x^{12} + 2040x^{10} + 6871x^{8} + 13035x^{6} + 13327x^{4} + 6338x^{2} + 841 $ x^16 + 29x^14 + 338x^12 + 2040x^10 + 6871x^8 + 13035x^6 + 13327x^4 + 6338*x^2 + 841
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			624.13
Root			$2.44932i$ of defining polynomial
Character	$\chi$	$=$	875.624
Dual form			875.2.b.e.624.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.88967i q^{2} -3.34505i q^{3} -1.57086 q^{4} +6.32104 q^{6} +1.00000i q^{7} +0.810940i q^{8} -8.18935 q^{9} -4.79106 q^{11} +5.25459i q^{12} +2.71167i q^{13} -1.88967 q^{14} -4.67412 q^{16} +0.520174i q^{17} -15.4752i q^{18} -0.405581 q^{19} +3.34505 q^{21} -9.05354i q^{22} -5.82371i q^{23} +2.71264 q^{24} -5.12416 q^{26} +17.3586i q^{27} -1.57086i q^{28} -0.163565 q^{29} -9.18965 q^{31} -7.21067i q^{32} +16.0263i q^{33} -0.982958 q^{34} +12.8643 q^{36} +4.98799i q^{37} -0.766414i q^{38} +9.07067 q^{39} -5.25682 q^{41} +6.32104i q^{42} +9.37859i q^{43} +7.52607 q^{44} +11.0049 q^{46} +1.98155i q^{47} +15.6352i q^{48} -1.00000 q^{49} +1.74001 q^{51} -4.25964i q^{52} +1.42426i q^{53} -32.8021 q^{54} -0.810940 q^{56} +1.35669i q^{57} -0.309083i q^{58} -2.21497 q^{59} +8.65223 q^{61} -17.3654i q^{62} -8.18935i q^{63} +4.27755 q^{64} -30.2845 q^{66} -7.72412i q^{67} -0.817119i q^{68} -19.4806 q^{69} +2.12012 q^{71} -6.64107i q^{72} -2.38846i q^{73} -9.42566 q^{74} +0.637109 q^{76} -4.79106i q^{77} +17.1406i q^{78} -11.7405 q^{79} +33.4974 q^{81} -9.93365i q^{82} -5.80296i q^{83} -5.25459 q^{84} -17.7225 q^{86} +0.547132i q^{87} -3.88527i q^{88} +14.5987 q^{89} -2.71167 q^{91} +9.14821i q^{92} +30.7398i q^{93} -3.74447 q^{94} -24.1201 q^{96} +16.4698i q^{97} -1.88967i q^{98} +39.2357 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 26 q^{4} + 4 q^{6} - 36 q^{9} - 10 q^{11} + 2 q^{14} + 70 q^{16} - 26 q^{19} + 16 q^{21} + 6 q^{24} - 10 q^{26} - 44 q^{29} + 6 q^{31} - 38 q^{34} + 54 q^{36} - 14 q^{39} + 24 q^{41} + 82 q^{44}+ \cdots + 62 q^{99}+O(q^{100}) $ 16 * q - 26 * q^4 + 4 * q^6 - 36 * q^9 - 10 * q^11 + 2 * q^14 + 70 * q^16 - 26 * q^19 + 16 * q^21 + 6 * q^24 - 10 * q^26 - 44 * q^29 + 6 * q^31 - 38 * q^34 + 54 * q^36 - 14 * q^39 + 24 * q^41 + 82 * q^44 + 22 * q^46 - 16 * q^49 + 20 * q^51 + 62 * q^54 - 24 * q^56 + 22 * q^59 + 90 * q^61 - 164 * q^64 - 84 * q^66 - 50 * q^69 - 30 * q^71 - 48 * q^74 + 52 * q^76 - 24 * q^79 + 72 * q^81 - 40 * q^84 - 96 * q^86 - 24 * q^89 + 12 * q^91 + 76 * q^94 - 60 * q^96 + 62 * 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 $

$2$	1.88967i	1.33620i	0.744072	+	0.668100i	$0.232892\pi$
$2$	1.88967i	1.33620i	−0.744072	+	0.668100i	$0.767108\pi$
$3$	− 3.34505i	− 1.93126i	−0.259912	−	0.965632i	$-0.583694\pi$
$3$	− 3.34505i	− 1.93126i	0.259912	−	0.965632i	$-0.416306\pi$
$4$	−1.57086	−0.785428
$5$	0	0
$5$	0	0
$6$	6.32104	2.58055
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0.810940i	0.286711i
$9$	−8.18935	−2.72978
$10$	0	0
$11$	−4.79106	−1.44456	−0.722280	−	0.691601i	$-0.756906\pi$
$11$	−4.79106	−1.44456	−0.722280	+	0.691601i	$0.756906\pi$
$12$	5.25459i	1.51687i
$13$	2.71167i	0.752082i	0.926603	+	0.376041i	$0.122715\pi$
$13$	2.71167i	0.752082i	−0.926603	+	0.376041i	$0.877285\pi$
$14$	−1.88967	−0.505036
$15$	0	0
$16$	−4.67412	−1.16853
$17$	0.520174i	0.126161i	0.998008	+	0.0630804i	$0.0200925\pi$
$17$	0.520174i	0.126161i	−0.998008	+	0.0630804i	$0.979908\pi$
$18$	− 15.4752i	− 3.64753i
$19$	−0.405581	−0.0930466	−0.0465233	−	0.998917i	$-0.514814\pi$
$19$	−0.405581	−0.0930466	−0.0465233	+	0.998917i	$0.514814\pi$
$20$	0	0
$21$	3.34505	0.729949
$22$	− 9.05354i	− 1.93022i
$23$	− 5.82371i	− 1.21433i	−0.794577	−	0.607164i	$-0.792307\pi$
$23$	− 5.82371i	− 1.21433i	0.794577	−	0.607164i	$-0.207693\pi$
$24$	2.71264	0.553714
$25$	0	0
$26$	−5.12416	−1.00493
$27$	17.3586i	3.34067i
$28$	− 1.57086i	− 0.296864i
$29$	−0.163565	−0.0303732	−0.0151866	−	0.999885i	$-0.504834\pi$
$29$	−0.163565	−0.0303732	−0.0151866	+	0.999885i	$0.504834\pi$
$30$	0	0
$31$	−9.18965	−1.65051	−0.825255	−	0.564760i	$-0.808969\pi$
$31$	−9.18965	−1.65051	−0.825255	+	0.564760i	$0.808969\pi$
$32$	− 7.21067i	− 1.27468i
$33$	16.0263i	2.78983i
$34$	−0.982958	−0.168576
$35$	0	0
$36$	12.8643	2.14405
$37$	4.98799i	0.820020i	0.912081	+	0.410010i	$0.134475\pi$
$37$	4.98799i	0.820020i	−0.912081	+	0.410010i	$0.865525\pi$
$38$	− 0.766414i	− 0.124329i
$39$	9.07067	1.45247
$40$	0	0
$41$	−5.25682	−0.820977	−0.410488	−	0.911866i	$-0.634642\pi$
$41$	−5.25682	−0.820977	−0.410488	+	0.911866i	$0.634642\pi$
$42$	6.32104i	0.975358i
$43$	9.37859i	1.43022i	0.699011	+	0.715111i	$0.253624\pi$
$43$	9.37859i	1.43022i	−0.699011	+	0.715111i	$0.746376\pi$
$44$	7.52607	1.13460
$45$	0	0
$46$	11.0049	1.62258
$47$	1.98155i	0.289038i	0.989502	+	0.144519i	$0.0461635\pi$
$47$	1.98155i	0.289038i	−0.989502	+	0.144519i	$0.953836\pi$
$48$	15.6352i	2.25674i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	1.74001	0.243650
$52$	− 4.25964i	− 0.590706i
$53$	1.42426i	0.195637i	0.995204	+	0.0978185i	$0.0311865\pi$
$53$	1.42426i	0.195637i	−0.995204	+	0.0978185i	$0.968814\pi$
$54$	−32.8021	−4.46380
$55$	0	0
$56$	−0.810940	−0.108366
$57$	1.35669i	0.179698i
$58$	− 0.309083i	− 0.0405846i
$59$	−2.21497	−0.288365	−0.144183	−	0.989551i	$-0.546055\pi$
$59$	−2.21497	−0.288365	−0.144183	+	0.989551i	$0.546055\pi$
$60$	0	0
$61$	8.65223	1.10780	0.553902	−	0.832582i	$-0.313138\pi$
$61$	8.65223	1.10780	0.553902	+	0.832582i	$0.313138\pi$
$62$	− 17.3654i	− 2.20541i
$63$	− 8.18935i	− 1.03176i
$64$	4.27755	0.534694
$65$	0	0
$66$	−30.2845	−3.72777
$67$	− 7.72412i	− 0.943652i	−0.881692	−	0.471826i	$-0.843595\pi$
$67$	− 7.72412i	− 0.943652i	0.881692	−	0.471826i	$-0.156405\pi$
$68$	− 0.817119i	− 0.0990902i
$69$	−19.4806	−2.34519
$70$	0	0
$71$	2.12012	0.251612	0.125806	−	0.992055i	$-0.459848\pi$
$71$	2.12012	0.251612	0.125806	+	0.992055i	$0.459848\pi$
$72$	− 6.64107i	− 0.782658i
$73$	− 2.38846i	− 0.279549i	−0.990183	−	0.139774i	$-0.955362\pi$
$73$	− 2.38846i	− 0.279549i	0.990183	−	0.139774i	$-0.0446377\pi$
$74$	−9.42566	−1.09571
$75$	0	0
$76$	0.637109	0.0730814
$77$	− 4.79106i	− 0.545992i
$78$	17.1406i	1.94079i
$79$	−11.7405	−1.32091	−0.660454	−	0.750867i	$-0.729636\pi$
$79$	−11.7405	−1.32091	−0.660454	+	0.750867i	$0.729636\pi$
$80$	0	0
$81$	33.4974	3.72193
$82$	− 9.93365i	− 1.09699i
$83$	− 5.80296i	− 0.636957i	−0.947930	−	0.318479i	$-0.896828\pi$
$83$	− 5.80296i	− 0.636957i	0.947930	−	0.318479i	$-0.103172\pi$
$84$	−5.25459	−0.573323
$85$	0	0
$86$	−17.7225	−1.91106
$87$	0.547132i	0.0586587i
$88$	− 3.88527i	− 0.414171i
$89$	14.5987	1.54746	0.773728	−	0.633517i	$-0.218390\pi$
$89$	14.5987	1.54746	0.773728	+	0.633517i	$0.218390\pi$
$90$	0	0
$91$	−2.71167	−0.284260
$92$	9.14821i	0.953767i
$93$	30.7398i	3.18757i
$94$	−3.74447	−0.386213
$95$	0	0
$96$	−24.1201	−2.46174
$97$	16.4698i	1.67225i	0.548535	+	0.836127i	$0.315186\pi$
$97$	16.4698i	1.67225i	−0.548535	+	0.836127i	$0.684814\pi$
$98$	− 1.88967i	− 0.190886i
$99$	39.2357	3.94334
$100$	0	0
$101$	−4.45512	−0.443301	−0.221651	−	0.975126i	$-0.571144\pi$
$101$	−4.45512	−0.443301	−0.221651	+	0.975126i	$0.571144\pi$
$102$	3.28804i	0.325565i
$103$	2.13097i	0.209971i	0.994474	+	0.104985i	$0.0334796\pi$
$103$	2.13097i	0.209971i	−0.994474	+	0.104985i	$0.966520\pi$
$104$	−2.19900	−0.215630
$105$	0	0
$106$	−2.69138	−0.261410
$107$	− 18.4519i	− 1.78381i	−0.452225	−	0.891904i	$-0.649369\pi$
$107$	− 18.4519i	− 1.78381i	0.452225	−	0.891904i	$-0.350631\pi$
$108$	− 27.2679i	− 2.62385i
$109$	−15.8405	−1.51724	−0.758622	−	0.651530i	$-0.774127\pi$
$109$	−15.8405	−1.51724	−0.758622	+	0.651530i	$0.774127\pi$
$110$	0	0
$111$	16.6851	1.58368
$112$	− 4.67412i	− 0.441663i
$113$	− 7.17257i	− 0.674739i	−0.941372	−	0.337369i	$-0.890463\pi$
$113$	− 7.17257i	− 0.674739i	0.941372	−	0.337369i	$-0.109537\pi$
$114$	−2.56369	−0.240112
$115$	0	0
$116$	0.256937	0.0238560
$117$	− 22.2068i	− 2.05302i
$118$	− 4.18557i	− 0.385313i
$119$	−0.520174	−0.0476843
$120$	0	0
$121$	11.9543	1.08675
$122$	16.3499i	1.48025i
$123$	17.5843i	1.58552i
$124$	14.4356	1.29636
$125$	0	0
$126$	15.4752	1.37864
$127$	− 9.53313i	− 0.845929i	−0.906146	−	0.422964i	$-0.860990\pi$
$127$	− 9.53313i	− 0.845929i	0.906146	−	0.422964i	$-0.139010\pi$
$128$	− 6.33818i	− 0.560221i
$129$	31.3718	2.76214
$130$	0	0
$131$	7.97152	0.696475	0.348237	−	0.937406i	$-0.386780\pi$
$131$	7.97152	0.696475	0.348237	+	0.937406i	$0.386780\pi$
$132$	− 25.1751i	− 2.19121i
$133$	− 0.405581i	− 0.0351683i
$134$	14.5960	1.26091
$135$	0	0
$136$	−0.421830	−0.0361717
$137$	− 3.58775i	− 0.306522i	−0.988186	−	0.153261i	$-0.951023\pi$
$137$	− 3.58775i	− 0.306522i	0.988186	−	0.153261i	$-0.0489775\pi$
$138$	− 36.8119i	− 3.13364i
$139$	−10.6898	−0.906698	−0.453349	−	0.891333i	$-0.649771\pi$
$139$	−10.6898	−0.906698	−0.453349	+	0.891333i	$0.649771\pi$
$140$	0	0
$141$	6.62837	0.558210
$142$	4.00632i	0.336203i
$143$	− 12.9918i	− 1.08643i
$144$	38.2780	3.18984
$145$	0	0
$146$	4.51341	0.373533
$147$	3.34505i	0.275895i
$148$	− 7.83541i	− 0.644067i
$149$	−1.22331	−0.100217	−0.0501087	−	0.998744i	$-0.515957\pi$
$149$	−1.22331	−0.100217	−0.0501087	+	0.998744i	$0.515957\pi$
$150$	0	0
$151$	−1.65614	−0.134774	−0.0673872	−	0.997727i	$-0.521466\pi$
$151$	−1.65614	−0.134774	−0.0673872	+	0.997727i	$0.521466\pi$
$152$	− 0.328902i	− 0.0266775i
$153$	− 4.25989i	− 0.344392i
$154$	9.05354	0.729555
$155$	0	0
$156$	−14.2487	−1.14081
$157$	11.1522i	0.890042i	0.895520	+	0.445021i	$0.146804\pi$
$157$	11.1522i	0.890042i	−0.895520	+	0.445021i	$0.853196\pi$
$158$	− 22.1856i	− 1.76500i
$159$	4.76421	0.377827
$160$	0	0
$161$	5.82371	0.458973
$162$	63.2990i	4.97324i
$163$	1.03739i	0.0812546i	0.999174	+	0.0406273i	$0.0129356\pi$
$163$	1.03739i	0.0812546i	−0.999174	+	0.0406273i	$0.987064\pi$
$164$	8.25770	0.644818
$165$	0	0
$166$	10.9657	0.851102
$167$	− 9.27166i	− 0.717463i	−0.933441	−	0.358731i	$-0.883209\pi$
$167$	− 9.27166i	− 0.717463i	0.933441	−	0.358731i	$-0.116791\pi$
$168$	2.71264i	0.209284i
$169$	5.64685	0.434373
$170$	0	0
$171$	3.32144	0.253997
$172$	− 14.7324i	− 1.12334i
$173$	− 10.2811i	− 0.781660i	−0.920463	−	0.390830i	$-0.872188\pi$
$173$	− 10.2811i	− 0.781660i	0.920463	−	0.390830i	$-0.127812\pi$
$174$	−1.03390	−0.0783797
$175$	0	0
$176$	22.3940	1.68801
$177$	7.40920i	0.556909i
$178$	27.5867i	2.06771i
$179$	−4.36578	−0.326314	−0.163157	−	0.986600i	$-0.552168\pi$
$179$	−4.36578	−0.326314	−0.163157	+	0.986600i	$0.552168\pi$
$180$	0	0
$181$	4.26928	0.317333	0.158666	−	0.987332i	$-0.449281\pi$
$181$	4.26928	0.317333	0.158666	+	0.987332i	$0.449281\pi$
$182$	− 5.12416i	− 0.379828i
$183$	− 28.9421i	− 2.13946i
$184$	4.72268	0.348161
$185$	0	0
$186$	−58.0882	−4.25923
$187$	− 2.49219i	− 0.182247i
$188$	− 3.11273i	− 0.227019i
$189$	−17.3586	−1.26265
$190$	0	0
$191$	−17.9357	−1.29778	−0.648891	−	0.760882i	$-0.724767\pi$
$191$	−17.9357	−1.29778	−0.648891	+	0.760882i	$0.724767\pi$
$192$	− 14.3086i	− 1.03264i
$193$	27.2695i	1.96290i	0.191707	+	0.981452i	$0.438598\pi$
$193$	27.2695i	1.96290i	−0.191707	+	0.981452i	$0.561402\pi$
$194$	−31.1225	−2.23447
$195$	0	0
$196$	1.57086	0.112204
$197$	− 9.05598i	− 0.645212i	−0.946533	−	0.322606i	$-0.895441\pi$
$197$	− 9.05598i	− 0.645212i	0.946533	−	0.322606i	$-0.104559\pi$
$198$	74.1426i	5.26908i
$199$	−16.9716	−1.20308	−0.601541	−	0.798842i	$-0.705446\pi$
$199$	−16.9716	−1.20308	−0.601541	+	0.798842i	$0.705446\pi$
$200$	0	0
$201$	−25.8376	−1.82244
$202$	− 8.41872i	− 0.592339i
$203$	− 0.163565i	− 0.0114800i
$204$	−2.73330	−0.191369
$205$	0	0
$206$	−4.02684	−0.280563
$207$	47.6924i	3.31485i
$208$	− 12.6747i	− 0.878831i
$209$	1.94316	0.134411
$210$	0	0
$211$	14.5470	1.00146	0.500729	−	0.865604i	$-0.333065\pi$
$211$	14.5470	1.00146	0.500729	+	0.865604i	$0.333065\pi$
$212$	− 2.23731i	− 0.153659i
$213$	− 7.09190i	− 0.485929i
$214$	34.8679	2.38352
$215$	0	0
$216$	−14.0768	−0.957806
$217$	− 9.18965i	− 0.623834i
$218$	− 29.9333i	− 2.02734i
$219$	−7.98953	−0.539882
$220$	0	0
$221$	−1.41054	−0.0948833
$222$	31.5293i	2.11611i
$223$	1.21243i	0.0811900i	0.999176	+	0.0405950i	$0.0129253\pi$
$223$	1.21243i	0.0811900i	−0.999176	+	0.0405950i	$0.987075\pi$
$224$	7.21067	0.481783
$225$	0	0
$226$	13.5538	0.901586
$227$	16.8192i	1.11633i	0.829729	+	0.558166i	$0.188495\pi$
$227$	16.8192i	1.11633i	−0.829729	+	0.558166i	$0.811505\pi$
$228$	− 2.13116i	− 0.141140i
$229$	−23.0409	−1.52259	−0.761294	−	0.648407i	$-0.775435\pi$
$229$	−23.0409	−1.52259	−0.761294	+	0.648407i	$0.775435\pi$
$230$	0	0
$231$	−16.0263	−1.05446
$232$	− 0.132641i	− 0.00870832i
$233$	15.8533i	1.03859i	0.854596	+	0.519293i	$0.173805\pi$
$233$	15.8533i	1.03859i	−0.854596	+	0.519293i	$0.826195\pi$
$234$	41.9636	2.74324
$235$	0	0
$236$	3.47941	0.226490
$237$	39.2725i	2.55102i
$238$	− 0.982958i	− 0.0637157i
$239$	4.29785	0.278005	0.139002	−	0.990292i	$-0.455610\pi$
$239$	4.29785	0.278005	0.139002	+	0.990292i	$0.455610\pi$
$240$	0	0
$241$	−2.38839	−0.153850	−0.0769248	−	0.997037i	$-0.524510\pi$
$241$	−2.38839	−0.153850	−0.0769248	+	0.997037i	$0.524510\pi$
$242$	22.5897i	1.45212i
$243$	− 59.9745i	− 3.84737i
$244$	−13.5914	−0.870101
$245$	0	0
$246$	−33.2285	−2.11857
$247$	− 1.09980i	− 0.0699787i
$248$	− 7.45226i	− 0.473219i
$249$	−19.4112	−1.23013
$250$	0	0
$251$	4.75420	0.300083	0.150041	−	0.988680i	$-0.452059\pi$
$251$	4.75420	0.300083	0.150041	+	0.988680i	$0.452059\pi$
$252$	12.8643i	0.810374i
$253$	27.9018i	1.75417i
$254$	18.0145	1.13033
$255$	0	0
$256$	20.5322	1.28326
$257$	− 15.4336i	− 0.962724i	−0.876522	−	0.481362i	$-0.840142\pi$
$257$	− 15.4336i	− 0.962724i	0.876522	−	0.481362i	$-0.159858\pi$
$258$	59.2825i	3.69077i
$259$	−4.98799	−0.309939
$260$	0	0
$261$	1.33949	0.0829122
$262$	15.0635i	0.930629i
$263$	− 0.273283i	− 0.0168513i	−0.999965	−	0.00842567i	$-0.997318\pi$
$263$	− 0.273283i	− 0.0168513i	0.999965	−	0.00842567i	$-0.00268201\pi$
$264$	−12.9964	−0.799874
$265$	0	0
$266$	0.766414	0.0469919
$267$	− 48.8333i	− 2.98855i
$268$	12.1335i	0.741170i
$269$	−14.3844	−0.877033	−0.438516	−	0.898723i	$-0.644496\pi$
$269$	−14.3844	−0.877033	−0.438516	+	0.898723i	$0.644496\pi$
$270$	0	0
$271$	15.6313	0.949532	0.474766	−	0.880112i	$-0.342533\pi$
$271$	15.6313	0.949532	0.474766	+	0.880112i	$0.342533\pi$
$272$	− 2.43136i	− 0.147423i
$273$	9.07067i	0.548982i
$274$	6.77966	0.409574
$275$	0	0
$276$	30.6012	1.84198
$277$	− 21.3893i	− 1.28516i	−0.766220	−	0.642579i	$-0.777865\pi$
$277$	− 21.3893i	− 1.28516i	0.766220	−	0.642579i	$-0.222135\pi$
$278$	− 20.2002i	− 1.21153i
$279$	75.2573	4.50554
$280$	0	0
$281$	−17.5303	−1.04577	−0.522886	−	0.852402i	$-0.675145\pi$
$281$	−17.5303	−1.04577	−0.522886	+	0.852402i	$0.675145\pi$
$282$	12.5254i	0.745879i
$283$	− 1.87403i	− 0.111400i	−0.998448	−	0.0556999i	$-0.982261\pi$
$283$	− 1.87403i	− 0.111400i	0.998448	−	0.0556999i	$-0.0177390\pi$
$284$	−3.33040	−0.197623
$285$	0	0
$286$	24.5502	1.45168
$287$	− 5.25682i	− 0.310300i
$288$	59.0507i	3.47960i
$289$	16.7294	0.984083
$290$	0	0
$291$	55.0923	3.22957
$292$	3.75193i	0.219565i
$293$	18.2486i	1.06609i	0.846086	+	0.533047i	$0.178953\pi$
$293$	18.2486i	1.06609i	−0.846086	+	0.533047i	$0.821047\pi$
$294$	−6.32104	−0.368651
$295$	0	0
$296$	−4.04496	−0.235109
$297$	− 83.1663i	− 4.82580i
$298$	− 2.31165i	− 0.133910i
$299$	15.7920	0.913274
$300$	0	0
$301$	−9.37859	−0.540573
$302$	− 3.12955i	− 0.180086i
$303$	14.9026i	0.856132i
$304$	1.89573	0.108728
$305$	0	0
$306$	8.04979	0.460176
$307$	− 12.9856i	− 0.741125i	−0.928808	−	0.370562i	$-0.879165\pi$
$307$	− 12.9856i	− 0.741125i	0.928808	−	0.370562i	$-0.120835\pi$
$308$	7.52607i	0.428838i
$309$	7.12821	0.405509
$310$	0	0
$311$	18.1000	1.02635	0.513177	−	0.858283i	$-0.328468\pi$
$311$	18.1000	1.02635	0.513177	+	0.858283i	$0.328468\pi$
$312$	7.35577i	0.416439i
$313$	2.59623i	0.146747i	0.997305	+	0.0733737i	$0.0233766\pi$
$313$	2.59623i	0.146747i	−0.997305	+	0.0733737i	$0.976623\pi$
$314$	−21.0740	−1.18927
$315$	0	0
$316$	18.4426	1.03748
$317$	23.8818i	1.34134i	0.741757	+	0.670669i	$0.233993\pi$
$317$	23.8818i	1.34134i	−0.741757	+	0.670669i	$0.766007\pi$
$318$	9.00280i	0.504852i
$319$	0.783649	0.0438759
$320$	0	0
$321$	−61.7223	−3.44501
$322$	11.0049i	0.613279i
$323$	− 0.210973i	− 0.0117388i
$324$	−52.6196	−2.92331
$325$	0	0
$326$	−1.96032	−0.108572
$327$	52.9872i	2.93020i
$328$	− 4.26296i	− 0.235383i
$329$	−1.98155	−0.109246
$330$	0	0
$331$	18.4514	1.01418	0.507092	−	0.861892i	$-0.330721\pi$
$331$	18.4514	1.01418	0.507092	+	0.861892i	$0.330721\pi$
$332$	9.11561i	0.500284i
$333$	− 40.8484i	− 2.23848i
$334$	17.5204	0.958673
$335$	0	0
$336$	−15.6352	−0.852968
$337$	− 10.1586i	− 0.553377i	−0.960960	−	0.276688i	$-0.910763\pi$
$337$	− 10.1586i	− 0.553377i	0.960960	−	0.276688i	$-0.0892370\pi$
$338$	10.6707i	0.580409i
$339$	−23.9926	−1.30310
$340$	0	0
$341$	44.0282	2.38426
$342$	6.27643i	0.339391i
$343$	− 1.00000i	− 0.0539949i
$344$	−7.60548	−0.410060
$345$	0	0
$346$	19.4279	1.04445
$347$	− 17.5771i	− 0.943590i	−0.881708	−	0.471795i	$-0.843606\pi$
$347$	− 17.5771i	− 0.943590i	0.881708	−	0.471795i	$-0.156394\pi$
$348$	− 0.859465i	− 0.0460722i
$349$	9.74642	0.521714	0.260857	−	0.965377i	$-0.415995\pi$
$349$	9.74642	0.521714	0.260857	+	0.965377i	$0.415995\pi$
$350$	0	0
$351$	−47.0709	−2.51246
$352$	34.5468i	1.84135i
$353$	12.9348i	0.688449i	0.938887	+	0.344224i	$0.111858\pi$
$353$	12.9348i	0.688449i	−0.938887	+	0.344224i	$0.888142\pi$
$354$	−14.0009	−0.744142
$355$	0	0
$356$	−22.9324	−1.21542
$357$	1.74001i	0.0920910i
$358$	− 8.24990i	− 0.436021i
$359$	0.176146	0.00929664	0.00464832	−	0.999989i	$-0.498520\pi$
$359$	0.176146	0.00929664	0.00464832	+	0.999989i	$0.498520\pi$
$360$	0	0
$361$	−18.8355	−0.991342
$362$	8.06753i	0.424020i
$363$	− 39.9877i	− 2.09881i
$364$	4.25964	0.223266
$365$	0	0
$366$	54.6911	2.85875
$367$	27.4851i	1.43471i	0.696708	+	0.717355i	$0.254647\pi$
$367$	27.4851i	1.43471i	−0.696708	+	0.717355i	$0.745353\pi$
$368$	27.2207i	1.41898i
$369$	43.0499	2.24109
$370$	0	0
$371$	−1.42426	−0.0739438
$372$	− 48.2879i	− 2.50361i
$373$	− 18.6763i	− 0.967020i	−0.875339	−	0.483510i	$-0.839362\pi$
$373$	− 18.6763i	− 0.967020i	0.875339	−	0.483510i	$-0.160638\pi$
$374$	4.70942	0.243518
$375$	0	0
$376$	−1.60692	−0.0828704
$377$	− 0.443533i	− 0.0228431i
$378$	− 32.8021i	− 1.68716i
$379$	19.5390	1.00365	0.501825	−	0.864969i	$-0.332662\pi$
$379$	19.5390	1.00365	0.501825	+	0.864969i	$0.332662\pi$
$380$	0	0
$381$	−31.8888	−1.63371
$382$	− 33.8926i	− 1.73409i
$383$	26.7503i	1.36688i	0.730008	+	0.683439i	$0.239516\pi$
$383$	26.7503i	1.36688i	−0.730008	+	0.683439i	$0.760484\pi$
$384$	−21.2015	−1.08194
$385$	0	0
$386$	−51.5305	−2.62283
$387$	− 76.8046i	− 3.90420i
$388$	− 25.8717i	− 1.31344i
$389$	−13.9701	−0.708314	−0.354157	−	0.935186i	$-0.615232\pi$
$389$	−13.9701	−0.708314	−0.354157	+	0.935186i	$0.615232\pi$
$390$	0	0
$391$	3.02934	0.153201
$392$	− 0.810940i	− 0.0409587i
$393$	− 26.6651i	− 1.34508i
$394$	17.1128	0.862132
$395$	0	0
$396$	−61.6336	−3.09721
$397$	28.1378i	1.41219i	0.708115	+	0.706097i	$0.249546\pi$
$397$	28.1378i	1.41219i	−0.708115	+	0.706097i	$0.750454\pi$
$398$	− 32.0707i	− 1.60756i
$399$	−1.35669	−0.0679193
$400$	0	0
$401$	−3.84138	−0.191830	−0.0959148	−	0.995390i	$-0.530578\pi$
$401$	−3.84138	−0.191830	−0.0959148	+	0.995390i	$0.530578\pi$
$402$	− 48.8245i	− 2.43514i
$403$	− 24.9193i	− 1.24132i
$404$	6.99836	0.348181
$405$	0	0
$406$	0.309083	0.0153395
$407$	− 23.8978i	− 1.18457i
$408$	1.41104i	0.0698570i
$409$	−29.9871	−1.48277	−0.741384	−	0.671081i	$-0.765830\pi$
$409$	−29.9871	−1.48277	−0.741384	+	0.671081i	$0.765830\pi$
$410$	0	0
$411$	−12.0012	−0.591975
$412$	− 3.34745i	− 0.164917i
$413$	− 2.21497i	− 0.108992i
$414$	−90.1229	−4.42930
$415$	0	0
$416$	19.5530	0.958663
$417$	35.7579i	1.75107i
$418$	3.67194i	0.179600i
$419$	−33.5276	−1.63793	−0.818964	−	0.573845i	$-0.805451\pi$
$419$	−33.5276	−1.63793	−0.818964	+	0.573845i	$0.805451\pi$
$420$	0	0
$421$	39.6034	1.93015	0.965075	−	0.261975i	$-0.0843738\pi$
$421$	39.6034	1.93015	0.965075	+	0.261975i	$0.0843738\pi$
$422$	27.4891i	1.33815i
$423$	− 16.2276i	− 0.789012i
$424$	−1.15499	−0.0560912
$425$	0	0
$426$	13.4013	0.649297
$427$	8.65223i	0.418711i
$428$	28.9852i	1.40105i
$429$	−43.4582	−2.09818
$430$	0	0
$431$	−4.52867	−0.218138	−0.109069	−	0.994034i	$-0.534787\pi$
$431$	−4.52867	−0.218138	−0.109069	+	0.994034i	$0.534787\pi$
$432$	− 81.1363i	− 3.90367i
$433$	17.4520i	0.838689i	0.907827	+	0.419344i	$0.137740\pi$
$433$	17.4520i	0.838689i	−0.907827	+	0.419344i	$0.862260\pi$
$434$	17.3654	0.833567
$435$	0	0
$436$	24.8832	1.19169
$437$	2.36198i	0.112989i
$438$	− 15.0976i	− 0.721390i
$439$	0.254860	0.0121638	0.00608189	−	0.999982i	$-0.498064\pi$
$439$	0.254860	0.0121638	0.00608189	+	0.999982i	$0.498064\pi$
$440$	0	0
$441$	8.18935	0.389969
$442$	− 2.66546i	− 0.126783i
$443$	− 18.4511i	− 0.876637i	−0.898820	−	0.438319i	$-0.855574\pi$
$443$	− 18.4511i	− 0.876637i	0.898820	−	0.438319i	$-0.144426\pi$
$444$	−26.2098	−1.24386
$445$	0	0
$446$	−2.29108	−0.108486
$447$	4.09203i	0.193546i
$448$	4.27755i	0.202095i
$449$	−9.88811	−0.466649	−0.233324	−	0.972399i	$-0.574960\pi$
$449$	−9.88811	−0.466649	−0.233324	+	0.972399i	$0.574960\pi$
$450$	0	0
$451$	25.1857	1.18595
$452$	11.2671i	0.529959i
$453$	5.53986i	0.260285i
$454$	−31.7828	−1.49164
$455$	0	0
$456$	−1.10019	−0.0515212
$457$	24.5336i	1.14763i	0.818984	+	0.573817i	$0.194538\pi$
$457$	24.5336i	1.14763i	−0.818984	+	0.573817i	$0.805462\pi$
$458$	− 43.5398i	− 2.03448i
$459$	−9.02951	−0.421461
$460$	0	0
$461$	−36.4327	−1.69684	−0.848421	−	0.529322i	$-0.822446\pi$
$461$	−36.4327	−1.69684	−0.848421	+	0.529322i	$0.822446\pi$
$462$	− 30.2845i	− 1.40896i
$463$	34.6332i	1.60954i	0.593587	+	0.804770i	$0.297711\pi$
$463$	34.6332i	1.60954i	−0.593587	+	0.804770i	$0.702289\pi$
$464$	0.764521	0.0354920
$465$	0	0
$466$	−29.9576	−1.38776
$467$	7.61216i	0.352249i	0.984368	+	0.176124i	$0.0563561\pi$
$467$	7.61216i	0.352249i	−0.984368	+	0.176124i	$0.943644\pi$
$468$	34.8837i	1.61250i
$469$	7.72412	0.356667
$470$	0	0
$471$	37.3046	1.71891
$472$	− 1.79621i	− 0.0826774i
$473$	− 44.9334i	− 2.06604i
$474$	−74.2121	−3.40867
$475$	0	0
$476$	0.817119	0.0374526
$477$	− 11.6638i	− 0.534046i
$478$	8.12152i	0.371470i
$479$	21.4501	0.980080	0.490040	−	0.871700i	$-0.336982\pi$
$479$	21.4501	0.980080	0.490040	+	0.871700i	$0.336982\pi$
$480$	0	0
$481$	−13.5258	−0.616722
$482$	− 4.51326i	− 0.205574i
$483$	− 19.4806i	− 0.886398i
$484$	−18.7785	−0.853568
$485$	0	0
$486$	113.332	5.14085
$487$	3.34199i	0.151440i	0.997129	+	0.0757199i	$0.0241255\pi$
$487$	3.34199i	0.151440i	−0.997129	+	0.0757199i	$0.975875\pi$
$488$	7.01644i	0.317619i
$489$	3.47012	0.156924
$490$	0	0
$491$	−33.4036	−1.50748	−0.753741	−	0.657171i	$-0.771753\pi$
$491$	−33.4036	−1.50748	−0.753741	+	0.657171i	$0.771753\pi$
$492$	− 27.6224i	− 1.24531i
$493$	− 0.0850821i	− 0.00383191i
$494$	2.07826	0.0935054
$495$	0	0
$496$	42.9536	1.92867
$497$	2.12012i	0.0951003i
$498$	− 36.6807i	− 1.64370i
$499$	36.9475	1.65400	0.826999	−	0.562204i	$-0.190046\pi$
$499$	36.9475	1.65400	0.826999	+	0.562204i	$0.190046\pi$
$500$	0	0
$501$	−31.0142	−1.38561
$502$	8.98388i	0.400970i
$503$	30.7679i	1.37187i	0.727662	+	0.685936i	$0.240607\pi$
$503$	30.7679i	1.37187i	−0.727662	+	0.685936i	$0.759393\pi$
$504$	6.64107	0.295817
$505$	0	0
$506$	−52.7252	−2.34392
$507$	− 18.8890i	− 0.838889i
$508$	14.9752i	0.664416i
$509$	−10.9758	−0.486494	−0.243247	−	0.969964i	$-0.578213\pi$
$509$	−10.9758	−0.486494	−0.243247	+	0.969964i	$0.578213\pi$
$510$	0	0
$511$	2.38846	0.105659
$512$	26.1227i	1.15447i
$513$	− 7.04032i	− 0.310838i
$514$	29.1645	1.28639
$515$	0	0
$516$	−49.2807	−2.16946
$517$	− 9.49372i	− 0.417533i
$518$	− 9.42566i	− 0.414140i
$519$	−34.3909	−1.50959
$520$	0	0
$521$	20.9701	0.918719	0.459359	−	0.888250i	$-0.348079\pi$
$521$	20.9701	0.918719	0.459359	+	0.888250i	$0.348079\pi$
$522$	2.53119i	0.110787i
$523$	− 6.71426i	− 0.293594i	−0.989167	−	0.146797i	$-0.953104\pi$
$523$	− 6.71426i	− 0.293594i	0.989167	−	0.146797i	$-0.0468964\pi$
$524$	−12.5221	−0.547031
$525$	0	0
$526$	0.516415	0.0225168
$527$	− 4.78022i	− 0.208230i
$528$	− 74.9091i	− 3.26000i
$529$	−10.9156	−0.474591
$530$	0	0
$531$	18.1392	0.787174
$532$	0.637109i	0.0276222i
$533$	− 14.2548i	− 0.617442i
$534$	92.2788	3.99330
$535$	0	0
$536$	6.26380	0.270555
$537$	14.6038i	0.630199i
$538$	− 27.1818i	− 1.17189i
$539$	4.79106	0.206366
$540$	0	0
$541$	3.50487	0.150686	0.0753431	−	0.997158i	$-0.475995\pi$
$541$	3.50487	0.150686	0.0753431	+	0.997158i	$0.475995\pi$
$542$	29.5379i	1.26876i
$543$	− 14.2809i	− 0.612854i
$544$	3.75081	0.160815
$545$	0	0
$546$	−17.1406	−0.733549
$547$	25.9535i	1.10969i	0.831953	+	0.554847i	$0.187223\pi$
$547$	25.9535i	1.10969i	−0.831953	+	0.554847i	$0.812777\pi$
$548$	5.63583i	0.240751i
$549$	−70.8561	−3.02406
$550$	0	0
$551$	0.0663387	0.00282612
$552$	− 15.7976i	− 0.672390i
$553$	− 11.7405i	− 0.499256i
$554$	40.4187	1.71723
$555$	0	0
$556$	16.7922	0.712146
$557$	− 3.45603i	− 0.146437i	−0.997316	−	0.0732184i	$-0.976673\pi$
$557$	− 3.45603i	− 0.146437i	0.997316	−	0.0732184i	$-0.0233270\pi$
$558$	142.211i	6.02029i
$559$	−25.4316	−1.07564
$560$	0	0
$561$	−8.33649	−0.351967
$562$	− 33.1266i	− 1.39736i
$563$	27.6750i	1.16636i	0.812342	+	0.583181i	$0.198192\pi$
$563$	27.6750i	1.16636i	−0.812342	+	0.583181i	$0.801808\pi$
$564$	−10.4122	−0.438434
$565$	0	0
$566$	3.54131	0.148852
$567$	33.4974i	1.40676i
$568$	1.71929i	0.0721398i
$569$	−18.0239	−0.755602	−0.377801	−	0.925887i	$-0.623320\pi$
$569$	−18.0239	−0.755602	−0.377801	+	0.925887i	$0.623320\pi$
$570$	0	0
$571$	−10.4491	−0.437281	−0.218641	−	0.975805i	$-0.570162\pi$
$571$	−10.4491	−0.437281	−0.218641	+	0.975805i	$0.570162\pi$
$572$	20.4082i	0.853311i
$573$	59.9958i	2.50636i
$574$	9.93365	0.414623
$575$	0	0
$576$	−35.0304	−1.45960
$577$	25.4222i	1.05834i	0.848516	+	0.529169i	$0.177496\pi$
$577$	25.4222i	1.05834i	−0.848516	+	0.529169i	$0.822504\pi$
$578$	31.6131i	1.31493i
$579$	91.2179	3.79089
$580$	0	0
$581$	5.80296	0.240747
$582$	104.106i	4.31534i
$583$	− 6.82372i	− 0.282609i
$584$	1.93690	0.0801496
$585$	0	0
$586$	−34.4838	−1.42451
$587$	− 38.7065i	− 1.59759i	−0.601606	−	0.798793i	$-0.705472\pi$
$587$	− 38.7065i	− 1.59759i	0.601606	−	0.798793i	$-0.294528\pi$
$588$	− 5.25459i	− 0.216696i
$589$	3.72715	0.153574
$590$	0	0
$591$	−30.2927	−1.24608
$592$	− 23.3145i	− 0.958219i
$593$	31.0697i	1.27588i	0.770087	+	0.637939i	$0.220213\pi$
$593$	31.0697i	1.27588i	−0.770087	+	0.637939i	$0.779787\pi$
$594$	157.157	6.44823
$595$	0	0
$596$	1.92164	0.0787136
$597$	56.7707i	2.32347i
$598$	29.8416i	1.22032i
$599$	0.389923	0.0159318	0.00796591	−	0.999968i	$-0.497464\pi$
$599$	0.389923	0.0159318	0.00796591	+	0.999968i	$0.497464\pi$
$600$	0	0
$601$	−3.35540	−0.136870	−0.0684349	−	0.997656i	$-0.521801\pi$
$601$	−3.35540	−0.136870	−0.0684349	+	0.997656i	$0.521801\pi$
$602$	− 17.7225i	− 0.722313i
$603$	63.2555i	2.57596i
$604$	2.60155	0.105856
$605$	0	0
$606$	−28.1610	−1.14396
$607$	− 6.93329i	− 0.281413i	−0.990051	−	0.140707i	$-0.955063\pi$
$607$	− 6.93329i	− 0.281413i	0.990051	−	0.140707i	$-0.0449375\pi$
$608$	2.92451i	0.118605i
$609$	−0.547132	−0.0221709
$610$	0	0
$611$	−5.37330	−0.217381
$612$	6.69167i	0.270495i
$613$	− 10.5240i	− 0.425060i	−0.977155	−	0.212530i	$-0.931830\pi$
$613$	− 10.5240i	− 0.425060i	0.977155	−	0.212530i	$-0.0681703\pi$
$614$	24.5384	0.990290
$615$	0	0
$616$	3.88527	0.156542
$617$	5.97280i	0.240456i	0.992746	+	0.120228i	$0.0383625\pi$
$617$	5.97280i	0.240456i	−0.992746	+	0.120228i	$0.961637\pi$
$618$	13.4700i	0.541841i
$619$	38.0234	1.52829	0.764145	−	0.645045i	$-0.223161\pi$
$619$	38.0234	1.52829	0.764145	+	0.645045i	$0.223161\pi$
$620$	0	0
$621$	101.092	4.05667
$622$	34.2030i	1.37141i
$623$	14.5987i	0.584884i
$624$	−42.3974	−1.69725
$625$	0	0
$626$	−4.90602	−0.196084
$627$	− 6.49998i	− 0.259584i
$628$	− 17.5185i	− 0.699064i
$629$	−2.59462	−0.103454
$630$	0	0
$631$	−25.2371	−1.00467	−0.502337	−	0.864672i	$-0.667526\pi$
$631$	−25.2371	−1.00467	−0.502337	+	0.864672i	$0.667526\pi$
$632$	− 9.52083i	− 0.378718i
$633$	− 48.6605i	− 1.93408i
$634$	−45.1288	−1.79229
$635$	0	0
$636$	−7.48390	−0.296756
$637$	− 2.71167i	− 0.107440i
$638$	1.48084i	0.0586269i
$639$	−17.3624	−0.686845
$640$	0	0
$641$	23.9595	0.946341	0.473171	−	0.880971i	$-0.343109\pi$
$641$	23.9595	0.946341	0.473171	+	0.880971i	$0.343109\pi$
$642$	− 116.635i	− 4.60321i
$643$	− 38.5274i	− 1.51937i	−0.650290	−	0.759686i	$-0.725352\pi$
$643$	− 38.5274i	− 1.51937i	0.650290	−	0.759686i	$-0.274648\pi$
$644$	−9.14821	−0.360490
$645$	0	0
$646$	0.398669	0.0156854
$647$	33.6150i	1.32154i	0.750588	+	0.660771i	$0.229771\pi$
$647$	33.6150i	1.32154i	−0.750588	+	0.660771i	$0.770229\pi$
$648$	27.1644i	1.06712i
$649$	10.6121	0.416561
$650$	0	0
$651$	−30.7398	−1.20479
$652$	− 1.62959i	− 0.0638196i
$653$	− 22.6590i	− 0.886716i	−0.896345	−	0.443358i	$-0.853787\pi$
$653$	− 22.6590i	− 0.886716i	0.896345	−	0.443358i	$-0.146213\pi$
$654$	−100.128	−3.91533
$655$	0	0
$656$	24.5710	0.959337
$657$	19.5600i	0.763107i
$658$	− 3.74447i	− 0.145975i
$659$	−45.0061	−1.75319	−0.876595	−	0.481229i	$-0.840190\pi$
$659$	−45.0061	−1.75319	−0.876595	+	0.481229i	$0.840190\pi$
$660$	0	0
$661$	−13.7843	−0.536148	−0.268074	−	0.963398i	$-0.586387\pi$
$661$	−13.7843	−0.536148	−0.268074	+	0.963398i	$0.586387\pi$
$662$	34.8672i	1.35515i
$663$	4.71833i	0.183245i
$664$	4.70585	0.182622
$665$	0	0
$666$	77.1900	2.99105
$667$	0.952553i	0.0368830i
$668$	14.5644i	0.563515i
$669$	4.05562	0.156799
$670$	0	0
$671$	−41.4534	−1.60029
$672$	− 24.1201i	− 0.930451i
$673$	9.76275i	0.376326i	0.982138	+	0.188163i	$0.0602534\pi$
$673$	9.76275i	0.376326i	−0.982138	+	0.188163i	$0.939747\pi$
$674$	19.1965	0.739422
$675$	0	0
$676$	−8.87038	−0.341169
$677$	− 17.3365i	− 0.666295i	−0.942875	−	0.333147i	$-0.891889\pi$
$677$	− 17.3365i	− 0.666295i	0.942875	−	0.333147i	$-0.108111\pi$
$678$	− 45.3381i	− 1.74120i
$679$	−16.4698	−0.632053
$680$	0	0
$681$	56.2612	2.15593
$682$	83.1989i	3.18585i
$683$	− 10.2687i	− 0.392921i	−0.980512	−	0.196460i	$-0.937055\pi$
$683$	− 10.2687i	− 0.392921i	0.980512	−	0.196460i	$-0.0629447\pi$
$684$	−5.21751	−0.199496
$685$	0	0
$686$	1.88967	0.0721480
$687$	77.0730i	2.94052i
$688$	− 43.8367i	− 1.67126i
$689$	−3.86212	−0.147135
$690$	0	0
$691$	−27.0508	−1.02906	−0.514531	−	0.857472i	$-0.672034\pi$
$691$	−27.0508	−1.02906	−0.514531	+	0.857472i	$0.672034\pi$
$692$	16.1502i	0.613937i
$693$	39.2357i	1.49044i
$694$	33.2150	1.26082
$695$	0	0
$696$	−0.443691	−0.0168181
$697$	− 2.73446i	− 0.103575i
$698$	18.4175i	0.697114i
$699$	53.0301	2.00578
$700$	0	0
$701$	19.3032	0.729073	0.364536	−	0.931189i	$-0.381227\pi$
$701$	19.3032	0.729073	0.364536	+	0.931189i	$0.381227\pi$
$702$	− 88.9484i	− 3.35714i
$703$	− 2.02303i	− 0.0763001i
$704$	−20.4940	−0.772398
$705$	0	0
$706$	−24.4425	−0.919904
$707$	− 4.45512i	− 0.167552i
$708$	− 11.6388i	− 0.437412i
$709$	−27.9701	−1.05044	−0.525219	−	0.850967i	$-0.676017\pi$
$709$	−27.9701	−1.05044	−0.525219	+	0.850967i	$0.676017\pi$
$710$	0	0
$711$	96.1469	3.60579
$712$	11.8387i	0.443672i
$713$	53.5179i	2.00426i
$714$	−3.28804	−0.123052
$715$	0	0
$716$	6.85802	0.256296
$717$	− 14.3765i	− 0.536901i
$718$	0.332858i	0.0124222i
$719$	39.2359	1.46325	0.731626	−	0.681706i	$-0.238762\pi$
$719$	39.2359	1.46325	0.731626	+	0.681706i	$0.238762\pi$
$720$	0	0
$721$	−2.13097	−0.0793616
$722$	− 35.5929i	− 1.32463i
$723$	7.98927i	0.297124i
$724$	−6.70642	−0.249242
$725$	0	0
$726$	75.5636	2.80443
$727$	33.0645i	1.22629i	0.789969	+	0.613147i	$0.210097\pi$
$727$	33.0645i	1.22629i	−0.789969	+	0.613147i	$0.789903\pi$
$728$	− 2.19900i	− 0.0815005i
$729$	−100.125	−3.70835
$730$	0	0
$731$	−4.87850	−0.180438
$732$	45.4639i	1.68039i
$733$	− 13.8421i	− 0.511268i	−0.966774	−	0.255634i	$-0.917716\pi$
$733$	− 13.8421i	− 0.511268i	0.966774	−	0.255634i	$-0.0822843\pi$
$734$	−51.9378	−1.91706
$735$	0	0
$736$	−41.9929	−1.54788
$737$	37.0068i	1.36316i
$738$	81.3501i	2.99454i
$739$	−40.3699	−1.48503	−0.742515	−	0.669829i	$-0.766367\pi$
$739$	−40.3699	−1.48503	−0.742515	+	0.669829i	$0.766367\pi$
$740$	0	0
$741$	−3.67889	−0.135147
$742$	− 2.69138i	− 0.0988037i
$743$	5.77743i	0.211953i	0.994369	+	0.105977i	$0.0337969\pi$
$743$	5.77743i	0.211953i	−0.994369	+	0.105977i	$0.966203\pi$
$744$	−24.9282	−0.913911
$745$	0	0
$746$	35.2920	1.29213
$747$	47.5224i	1.73875i
$748$	3.91487i	0.143142i
$749$	18.4519	0.674216
$750$	0	0
$751$	−17.2304	−0.628745	−0.314372	−	0.949300i	$-0.601794\pi$
$751$	−17.2304	−0.628745	−0.314372	+	0.949300i	$0.601794\pi$
$752$	− 9.26200i	− 0.337750i
$753$	− 15.9030i	− 0.579539i
$754$	0.838132	0.0305230
$755$	0	0
$756$	27.2679	0.991724
$757$	3.11588i	0.113248i	0.998396	+	0.0566242i	$0.0180337\pi$
$757$	3.11588i	0.113248i	−0.998396	+	0.0566242i	$0.981966\pi$
$758$	36.9222i	1.34108i
$759$	93.3328	3.38776
$760$	0	0
$761$	−10.5773	−0.383425	−0.191713	−	0.981451i	$-0.561404\pi$
$761$	−10.5773	−0.383425	−0.191713	+	0.981451i	$0.561404\pi$
$762$	− 60.2593i	− 2.18296i
$763$	− 15.8405i	− 0.573465i
$764$	28.1744	1.01931
$765$	0	0
$766$	−50.5493	−1.82642
$767$	− 6.00628i	− 0.216874i
$768$	− 68.6811i	− 2.47832i
$769$	6.97652	0.251580	0.125790	−	0.992057i	$-0.459854\pi$
$769$	6.97652	0.251580	0.125790	+	0.992057i	$0.459854\pi$
$770$	0	0
$771$	−51.6263	−1.85927
$772$	− 42.8365i	− 1.54172i
$773$	27.4955i	0.988945i	0.869193	+	0.494473i	$0.164639\pi$
$773$	27.4955i	0.988945i	−0.869193	+	0.494473i	$0.835361\pi$
$774$	145.135	5.21678
$775$	0	0
$776$	−13.3560	−0.479453
$777$	16.6851i	0.598573i
$778$	− 26.3989i	− 0.946448i
$779$	2.13206	0.0763891
$780$	0	0
$781$	−10.1576	−0.363468
$782$	5.72446i	0.204706i
$783$	− 2.83926i	− 0.101467i
$784$	4.67412	0.166933
$785$	0	0
$786$	50.3883	1.79729
$787$	1.16804i	0.0416361i	0.999783	+	0.0208181i	$0.00662707\pi$
$787$	1.16804i	0.0416361i	−0.999783	+	0.0208181i	$0.993373\pi$
$788$	14.2256i	0.506768i
$789$	−0.914145	−0.0325444
$790$	0	0
$791$	7.17257	0.255027
$792$	31.8178i	1.13060i
$793$	23.4620i	0.833159i
$794$	−53.1711	−1.88697
$795$	0	0
$796$	26.6599	0.944934
$797$	5.25703i	0.186214i	0.995656	+	0.0931068i	$0.0296798\pi$
$797$	5.25703i	0.186214i	−0.995656	+	0.0931068i	$0.970320\pi$
$798$	− 2.56369i	− 0.0907537i
$799$	−1.03075	−0.0364653
$800$	0	0
$801$	−119.554	−4.22422
$802$	− 7.25895i	− 0.256323i
$803$	11.4433i	0.403825i
$804$	40.5871	1.43140
$805$	0	0
$806$	47.0893	1.65865
$807$	48.1165i	1.69378i
$808$	− 3.61284i	− 0.127099i
$809$	32.2856	1.13510	0.567551	−	0.823338i	$-0.307891\pi$
$809$	32.2856	1.13510	0.567551	+	0.823338i	$0.307891\pi$
$810$	0	0
$811$	0.362439	0.0127270	0.00636348	−	0.999980i	$-0.497974\pi$
$811$	0.362439	0.0127270	0.00636348	+	0.999980i	$0.497974\pi$
$812$	0.256937i	0.00901670i
$813$	− 52.2873i	− 1.83380i
$814$	45.1589	1.58282
$815$	0	0
$816$	−8.13301	−0.284712
$817$	− 3.80378i	− 0.133077i
$818$	− 56.6658i	− 1.98127i
$819$	22.2068	0.775969
$820$	0	0
$821$	−38.4523	−1.34199	−0.670996	−	0.741461i	$-0.734133\pi$
$821$	−38.4523	−1.34199	−0.670996	+	0.741461i	$0.734133\pi$
$822$	− 22.6783i	− 0.790996i
$823$	5.90272i	0.205756i	0.994694	+	0.102878i	$0.0328051\pi$
$823$	5.90272i	0.205756i	−0.994694	+	0.102878i	$0.967195\pi$
$824$	−1.72809	−0.0602009
$825$	0	0
$826$	4.18557	0.145635
$827$	− 41.8778i	− 1.45624i	−0.685452	−	0.728118i	$-0.740395\pi$
$827$	− 41.8778i	− 1.45624i	0.685452	−	0.728118i	$-0.259605\pi$
$828$	− 74.9179i	− 2.60358i
$829$	−19.9050	−0.691331	−0.345665	−	0.938358i	$-0.612347\pi$
$829$	−19.9050	−0.691331	−0.345665	+	0.938358i	$0.612347\pi$
$830$	0	0
$831$	−71.5482	−2.48198
$832$	11.5993i	0.402134i
$833$	− 0.520174i	− 0.0180230i
$834$	−67.5708	−2.33978
$835$	0	0
$836$	−3.05243	−0.105571
$837$	− 159.520i	− 5.51381i
$838$	− 63.3560i	− 2.18860i
$839$	37.3674	1.29007	0.645033	−	0.764155i	$-0.276844\pi$
$839$	37.3674	1.29007	0.645033	+	0.764155i	$0.276844\pi$
$840$	0	0
$841$	−28.9732	−0.999077
$842$	74.8373i	2.57906i
$843$	58.6398i	2.01966i
$844$	−22.8513	−0.786573
$845$	0	0
$846$	30.6648	1.05428
$847$	11.9543i	0.410755i
$848$	− 6.65716i	− 0.228608i
$849$	−6.26873	−0.215142
$850$	0	0
$851$	29.0486	0.995773
$852$	11.1403i	0.381662i
$853$	− 47.5502i	− 1.62809i	−0.580804	−	0.814044i	$-0.697262\pi$
$853$	− 47.5502i	− 1.62809i	0.580804	−	0.814044i	$-0.302738\pi$
$854$	−16.3499	−0.559481
$855$	0	0
$856$	14.9634	0.511437
$857$	− 30.6116i	− 1.04567i	−0.852433	−	0.522836i	$-0.824874\pi$
$857$	− 30.6116i	− 1.04567i	0.852433	−	0.522836i	$-0.175126\pi$
$858$	− 82.1216i	− 2.80359i
$859$	38.0855	1.29946	0.649730	−	0.760165i	$-0.274882\pi$
$859$	38.0855	1.29946	0.649730	+	0.760165i	$0.274882\pi$
$860$	0	0
$861$	−17.5843	−0.599271
$862$	− 8.55769i	− 0.291476i
$863$	38.9217i	1.32491i	0.749101	+	0.662455i	$0.230486\pi$
$863$	38.9217i	1.32491i	−0.749101	+	0.662455i	$0.769514\pi$
$864$	125.167	4.25828
$865$	0	0
$866$	−32.9785	−1.12066
$867$	− 55.9607i	− 1.90053i
$868$	14.4356i	0.489977i
$869$	56.2494	1.90813
$870$	0	0
$871$	20.9453	0.709703
$872$	− 12.8457i	− 0.435010i
$873$	− 134.877i	− 4.56489i
$874$	−4.46337	−0.150976
$875$	0	0
$876$	12.5504	0.424039
$877$	− 27.5668i	− 0.930863i	−0.885084	−	0.465432i	$-0.845899\pi$
$877$	− 27.5668i	− 0.930863i	0.885084	−	0.465432i	$-0.154101\pi$
$878$	0.481601i	0.0162532i
$879$	61.0424	2.05891
$880$	0	0
$881$	6.81222	0.229509	0.114755	−	0.993394i	$-0.463392\pi$
$881$	6.81222	0.229509	0.114755	+	0.993394i	$0.463392\pi$
$882$	15.4752i	0.521076i
$883$	41.1531i	1.38491i	0.721459	+	0.692457i	$0.243472\pi$
$883$	41.1531i	1.38491i	−0.721459	+	0.692457i	$0.756528\pi$
$884$	2.21576	0.0745240
$885$	0	0
$886$	34.8665	1.17136
$887$	10.3979i	0.349128i	0.984646	+	0.174564i	$0.0558515\pi$
$887$	10.3979i	0.349128i	−0.984646	+	0.174564i	$0.944148\pi$
$888$	13.5306i	0.454057i
$889$	9.53313	0.319731
$890$	0	0
$891$	−160.488	−5.37655
$892$	− 1.90455i	− 0.0637689i
$893$	− 0.803678i	− 0.0268940i
$894$	−7.73259	−0.258616
$895$	0	0
$896$	6.33818	0.211744
$897$	− 52.8249i	− 1.76377i
$898$	− 18.6853i	− 0.623536i
$899$	1.50310	0.0501313
$900$	0	0
$901$	−0.740863	−0.0246817
$902$	47.5928i	1.58467i
$903$	31.3718i	1.04399i
$904$	5.81653	0.193455
$905$	0	0
$906$	−10.4685	−0.347793
$907$	14.3126i	0.475242i	0.971358	+	0.237621i	$0.0763677\pi$
$907$	14.3126i	0.475242i	−0.971358	+	0.237621i	$0.923632\pi$
$908$	− 26.4206i	− 0.876798i
$909$	36.4846	1.21012
$910$	0	0
$911$	−48.4129	−1.60399	−0.801996	−	0.597329i	$-0.796229\pi$
$911$	−48.4129	−1.60399	−0.801996	+	0.597329i	$0.796229\pi$
$912$	− 6.34132i	− 0.209982i
$913$	27.8023i	0.920123i
$914$	−46.3604	−1.53347
$915$	0	0
$916$	36.1940	1.19588
$917$	7.97152i	0.263243i
$918$	− 17.0628i	− 0.563156i
$919$	0.526360	0.0173630	0.00868151	−	0.999962i	$-0.497237\pi$
$919$	0.526360	0.0173630	0.00868151	+	0.999962i	$0.497237\pi$
$920$	0	0
$921$	−43.4373	−1.43131
$922$	− 68.8459i	− 2.26732i
$923$	5.74906i	0.189233i
$924$	25.1751	0.828199
$925$	0	0
$926$	−65.4453	−2.15067
$927$	− 17.4513i	− 0.573175i
$928$	1.17941i	0.0387161i
$929$	1.36557	0.0448030	0.0224015	−	0.999749i	$-0.492869\pi$
$929$	1.36557	0.0448030	0.0224015	+	0.999749i	$0.492869\pi$
$930$	0	0
$931$	0.405581	0.0132924
$932$	− 24.9033i	− 0.815734i
$933$	− 60.5453i	− 1.98216i
$934$	−14.3845	−0.470674
$935$	0	0
$936$	18.0084	0.588623
$937$	0.859041i	0.0280636i	0.999902	+	0.0140318i	$0.00446661\pi$
$937$	0.859041i	0.0280636i	−0.999902	+	0.0140318i	$0.995533\pi$
$938$	14.5960i	0.476578i
$939$	8.68451	0.283408
$940$	0	0
$941$	42.2414	1.37703	0.688516	−	0.725221i	$-0.258263\pi$
$941$	42.2414	1.37703	0.688516	+	0.725221i	$0.258263\pi$
$942$	70.4935i	2.29680i
$943$	30.6142i	0.996935i
$944$	10.3531	0.336964
$945$	0	0
$946$	84.9094	2.76064
$947$	− 40.7967i	− 1.32571i	−0.748746	−	0.662857i	$-0.769344\pi$
$947$	− 40.7967i	− 1.32571i	0.748746	−	0.662857i	$-0.230656\pi$
$948$	− 61.6914i	− 2.00364i
$949$	6.47673	0.210243
$950$	0	0
$951$	79.8859	2.59048
$952$	− 0.421830i	− 0.0136716i
$953$	16.0122i	0.518688i	0.965785	+	0.259344i	$0.0835063\pi$
$953$	16.0122i	0.518688i	−0.965785	+	0.259344i	$0.916494\pi$
$954$	22.0407	0.713592
$955$	0	0
$956$	−6.75130	−0.218353
$957$	− 2.62134i	− 0.0847360i
$958$	40.5336i	1.30958i
$959$	3.58775	0.115854
$960$	0	0
$961$	53.4497	1.72418
$962$	− 25.5593i	− 0.824064i
$963$	151.109i	4.86941i
$964$	3.75181	0.120838
$965$	0	0
$966$	36.8119	1.18440
$967$	− 44.6342i	− 1.43534i	−0.696384	−	0.717669i	$-0.745209\pi$
$967$	− 44.6342i	− 1.43534i	0.696384	−	0.717669i	$-0.254791\pi$
$968$	9.69423i	0.311584i
$969$	−0.705714	−0.0226708
$970$	0	0
$971$	−34.8396	−1.11805	−0.559027	−	0.829149i	$-0.688825\pi$
$971$	−34.8396	−1.11805	−0.559027	+	0.829149i	$0.688825\pi$
$972$	94.2113i	3.02183i
$973$	− 10.6898i	− 0.342700i
$974$	−6.31525	−0.202354
$975$	0	0
$976$	−40.4416	−1.29450
$977$	− 21.1094i	− 0.675348i	−0.941263	−	0.337674i	$-0.890360\pi$
$977$	− 21.1094i	− 0.675348i	0.941263	−	0.337674i	$-0.109640\pi$
$978$	6.55738i	0.209682i
$979$	−69.9432	−2.23539
$980$	0	0
$981$	129.723	4.14175
$982$	− 63.1218i	− 2.01430i
$983$	20.7396i	0.661491i	0.943720	+	0.330745i	$0.107300\pi$
$983$	20.7396i	0.661491i	−0.943720	+	0.330745i	$0.892700\pi$
$984$	−14.2598	−0.454587
$985$	0	0
$986$	0.160777	0.00512019
$987$	6.62837i	0.210983i
$988$	1.72763i	0.0549632i
$989$	54.6182	1.73676
$990$	0	0
$991$	−18.9582	−0.602227	−0.301113	−	0.953588i	$-0.597358\pi$
$991$	−18.9582	−0.602227	−0.301113	+	0.953588i	$0.597358\pi$
$992$	66.2636i	2.10387i
$993$	− 61.7210i	− 1.95866i
$994$	−4.00632	−0.127073
$995$	0	0
$996$	30.4922	0.966181
$997$	20.9224i	0.662618i	0.943522	+	0.331309i	$0.107490\pi$
$997$	20.9224i	0.662618i	−0.943522	+	0.331309i	$0.892510\pi$
$998$	69.8186i	2.21007i
$999$	−86.5846	−2.73942

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	875.2.b.e.624.13		16
5.2	odd	4		875.2.a.j.1.2	yes	8
5.3	odd	4		875.2.a.i.1.7	✓	8
5.4	even	2	inner	875.2.b.e.624.4		16
15.2	even	4		7875.2.a.w.1.7		8
15.8	even	4		7875.2.a.bb.1.2		8
35.13	even	4		6125.2.a.v.1.7		8
35.27	even	4		6125.2.a.w.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
875.2.a.i.1.7	✓	8	5.3	odd	4
875.2.a.j.1.2	yes	8	5.2	odd	4
875.2.b.e.624.4		16	5.4	even	2	inner
875.2.b.e.624.13		16	1.1	even	1	trivial
6125.2.a.v.1.7		8	35.13	even	4
6125.2.a.w.1.2		8	35.27	even	4
7875.2.a.w.1.7		8	15.2	even	4
7875.2.a.bb.1.2		8	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.88967i q^{2} -3.34505i q^{3} -1.57086 q^{4} +6.32104 q^{6} +1.00000i q^{7} +0.810940i q^{8} -8.18935 q^{9} -4.79106 q^{11} +5.25459i q^{12} +2.71167i q^{13} -1.88967 q^{14} -4.67412 q^{16} +0.520174i q^{17} -15.4752i q^{18} -0.405581 q^{19} +3.34505 q^{21} -9.05354i q^{22} -5.82371i q^{23} +2.71264 q^{24} -5.12416 q^{26} +17.3586i q^{27} -1.57086i q^{28} -0.163565 q^{29} -9.18965 q^{31} -7.21067i q^{32} +16.0263i q^{33} -0.982958 q^{34} +12.8643 q^{36} +4.98799i q^{37} -0.766414i q^{38} +9.07067 q^{39} -5.25682 q^{41} +6.32104i q^{42} +9.37859i q^{43} +7.52607 q^{44} +11.0049 q^{46} +1.98155i q^{47} +15.6352i q^{48} -1.00000 q^{49} +1.74001 q^{51} -4.25964i q^{52} +1.42426i q^{53} -32.8021 q^{54} -0.810940 q^{56} +1.35669i q^{57} -0.309083i q^{58} -2.21497 q^{59} +8.65223 q^{61} -17.3654i q^{62} -8.18935i q^{63} +4.27755 q^{64} -30.2845 q^{66} -7.72412i q^{67} -0.817119i q^{68} -19.4806 q^{69} +2.12012 q^{71} -6.64107i q^{72} -2.38846i q^{73} -9.42566 q^{74} +0.637109 q^{76} -4.79106i q^{77} +17.1406i q^{78} -11.7405 q^{79} +33.4974 q^{81} -9.93365i q^{82} -5.80296i q^{83} -5.25459 q^{84} -17.7225 q^{86} +0.547132i q^{87} -3.88527i q^{88} +14.5987 q^{89} -2.71167 q^{91} +9.14821i q^{92} +30.7398i q^{93} -3.74447 q^{94} -24.1201 q^{96} +16.4698i q^{97} -1.88967i q^{98} +39.2357 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 26 q^{4} + 4 q^{6} - 36 q^{9} - 10 q^{11} + 2 q^{14} + 70 q^{16} - 26 q^{19} + 16 q^{21} + 6 q^{24} - 10 q^{26} - 44 q^{29} + 6 q^{31} - 38 q^{34} + 54 q^{36} - 14 q^{39} + 24 q^{41} + 82 q^{44}+ \cdots + 62 q^{99}+O(q^{100}) \) 16 * q - 26 * q^4 + 4 * q^6 - 36 * q^9 - 10 * q^11 + 2 * q^14 + 70 * q^16 - 26 * q^19 + 16 * q^21 + 6 * q^24 - 10 * q^26 - 44 * q^29 + 6 * q^31 - 38 * q^34 + 54 * q^36 - 14 * q^39 + 24 * q^41 + 82 * q^44 + 22 * q^46 - 16 * q^49 + 20 * q^51 + 62 * q^54 - 24 * q^56 + 22 * q^59 + 90 * q^61 - 164 * q^64 - 84 * q^66 - 50 * q^69 - 30 * q^71 - 48 * q^74 + 52 * q^76 - 24 * q^79 + 72 * q^81 - 40 * q^84 - 96 * q^86 - 24 * q^89 + 12 * q^91 + 76 * q^94 - 60 * q^96 + 62 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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