LMFDB - Embedded newform 875.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [875,2,Mod(1,875)]

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([0, 0]))

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

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

chi := DirichletCharacter("875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 875 = 5^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	875.a (trivial)

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:	yes
Analytic conductor:	$6.98691017686$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 10x^{6} + 30x^{5} + 29x^{4} - 79x^{3} - 43x^{2} + 62x + 29 $ x^8 - 3x^7 - 10x^6 + 30x^5 + 29x^4 - 79x^3 - 43x^2 + 62*x + 29
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44932$ of defining polynomial
Character	$\chi$	$=$	875.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.88967 q^{2} -3.34505 q^{3} +1.57086 q^{4} +6.32104 q^{6} -1.00000 q^{7} +0.810940 q^{8} +8.18935 q^{9} -4.79106 q^{11} -5.25459 q^{12} +2.71167 q^{13} +1.88967 q^{14} -4.67412 q^{16} -0.520174 q^{17} -15.4752 q^{18} +0.405581 q^{19} +3.34505 q^{21} +9.05354 q^{22} -5.82371 q^{23} -2.71264 q^{24} -5.12416 q^{26} -17.3586 q^{27} -1.57086 q^{28} +0.163565 q^{29} -9.18965 q^{31} +7.21067 q^{32} +16.0263 q^{33} +0.982958 q^{34} +12.8643 q^{36} -4.98799 q^{37} -0.766414 q^{38} -9.07067 q^{39} -5.25682 q^{41} -6.32104 q^{42} +9.37859 q^{43} -7.52607 q^{44} +11.0049 q^{46} -1.98155 q^{47} +15.6352 q^{48} +1.00000 q^{49} +1.74001 q^{51} +4.25964 q^{52} +1.42426 q^{53} +32.8021 q^{54} -0.810940 q^{56} -1.35669 q^{57} -0.309083 q^{58} +2.21497 q^{59} +8.65223 q^{61} +17.3654 q^{62} -8.18935 q^{63} -4.27755 q^{64} -30.2845 q^{66} +7.72412 q^{67} -0.817119 q^{68} +19.4806 q^{69} +2.12012 q^{71} +6.64107 q^{72} -2.38846 q^{73} +9.42566 q^{74} +0.637109 q^{76} +4.79106 q^{77} +17.1406 q^{78} +11.7405 q^{79} +33.4974 q^{81} +9.93365 q^{82} -5.80296 q^{83} +5.25459 q^{84} -17.7225 q^{86} -0.547132 q^{87} -3.88527 q^{88} -14.5987 q^{89} -2.71167 q^{91} -9.14821 q^{92} +30.7398 q^{93} +3.74447 q^{94} -24.1201 q^{96} -16.4698 q^{97} -1.88967 q^{98} -39.2357 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} - 8 q^{3} + 13 q^{4} + 2 q^{6} - 8 q^{7} + 12 q^{8} + 18 q^{9} - 5 q^{11} - 20 q^{12} - 6 q^{13} - q^{14} + 35 q^{16} + 13 q^{17} + 3 q^{18} + 13 q^{19} + 8 q^{21} + 22 q^{22} - 5 q^{23} - 3 q^{24}+ \cdots - 31 q^{99}+O(q^{100}) $ 8 * q + q^2 - 8 * q^3 + 13 * q^4 + 2 * q^6 - 8 * q^7 + 12 * q^8 + 18 * q^9 - 5 * q^11 - 20 * q^12 - 6 * q^13 - q^14 + 35 * q^16 + 13 * q^17 + 3 * q^18 + 13 * q^19 + 8 * q^21 + 22 * q^22 - 5 * q^23 - 3 * q^24 - 5 * q^26 - 29 * q^27 - 13 * q^28 + 22 * q^29 + 3 * q^31 + 35 * q^32 - 10 * q^33 + 19 * q^34 + 27 * q^36 + 7 * q^37 - 5 * q^38 + 7 * q^39 + 12 * q^41 - 2 * q^42 - q^43 - 41 * q^44 + 11 * q^46 - 2 * q^47 + 12 * q^48 + 8 * q^49 + 10 * q^51 - 49 * q^52 + 10 * q^53 - 31 * q^54 - 12 * q^56 + 17 * q^57 + 58 * q^58 - 11 * q^59 + 45 * q^61 + 26 * q^62 - 18 * q^63 + 82 * q^64 - 42 * q^66 + 6 * q^67 + 58 * q^68 + 25 * q^69 - 15 * q^71 + 85 * q^72 - 32 * q^73 + 24 * q^74 + 26 * q^76 + 5 * q^77 - 32 * q^78 + 12 * q^79 + 36 * q^81 - 15 * q^83 + 20 * q^84 - 48 * q^86 - 35 * q^87 - 2 * q^88 + 12 * q^89 + 6 * q^91 - 78 * q^92 + 33 * q^93 - 38 * q^94 - 30 * q^96 + 20 * q^97 + q^98 - 31 * q^99

Coefficient data

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

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

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

$2$	−1.88967	−1.33620	−0.668100	−	0.744072i	$-0.732892\pi$
$2$	−1.88967	−1.33620	−0.668100	+	0.744072i	$0.732892\pi$
$3$	−3.34505	−1.93126	−0.965632	−	0.259912i	$-0.916306\pi$
$3$	−3.34505	−1.93126	−0.965632	+	0.259912i	$0.916306\pi$
$4$	1.57086	0.785428
$5$	0	0
$5$	0	0
$6$	6.32104	2.58055
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.810940	0.286711
$9$	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	$0.756906\pi$
$12$	−5.25459	−1.51687
$13$	2.71167	0.752082	0.376041	−	0.926603i	$-0.377285\pi$
$13$	2.71167	0.752082	0.376041	+	0.926603i	$0.377285\pi$
$14$	1.88967	0.505036
$15$	0	0
$16$	−4.67412	−1.16853
$17$	−0.520174	−0.126161	−0.0630804	−	0.998008i	$-0.520092\pi$
$17$	−0.520174	−0.126161	−0.0630804	+	0.998008i	$0.520092\pi$
$18$	−15.4752	−3.64753
$19$	0.405581	0.0930466	0.0465233	−	0.998917i	$-0.485186\pi$
$19$	0.405581	0.0930466	0.0465233	+	0.998917i	$0.485186\pi$
$20$	0	0
$21$	3.34505	0.729949
$22$	9.05354	1.93022
$23$	−5.82371	−1.21433	−0.607164	−	0.794577i	$-0.707693\pi$
$23$	−5.82371	−1.21433	−0.607164	+	0.794577i	$0.707693\pi$
$24$	−2.71264	−0.553714
$25$	0	0
$26$	−5.12416	−1.00493
$27$	−17.3586	−3.34067
$28$	−1.57086	−0.296864
$29$	0.163565	0.0303732	0.0151866	−	0.999885i	$-0.495166\pi$
$29$	0.163565	0.0303732	0.0151866	+	0.999885i	$0.495166\pi$
$30$	0	0
$31$	−9.18965	−1.65051	−0.825255	−	0.564760i	$-0.808969\pi$
$31$	−9.18965	−1.65051	−0.825255	+	0.564760i	$0.808969\pi$
$32$	7.21067	1.27468
$33$	16.0263	2.78983
$34$	0.982958	0.168576
$35$	0	0
$36$	12.8643	2.14405
$37$	−4.98799	−0.820020	−0.410010	−	0.912081i	$-0.634475\pi$
$37$	−4.98799	−0.820020	−0.410010	+	0.912081i	$0.634475\pi$
$38$	−0.766414	−0.124329
$39$	−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	$0.634642\pi$
$42$	−6.32104	−0.975358
$43$	9.37859	1.43022	0.715111	−	0.699011i	$-0.246376\pi$
$43$	9.37859	1.43022	0.715111	+	0.699011i	$0.246376\pi$
$44$	−7.52607	−1.13460
$45$	0	0
$46$	11.0049	1.62258
$47$	−1.98155	−0.289038	−0.144519	−	0.989502i	$-0.546164\pi$
$47$	−1.98155	−0.289038	−0.144519	+	0.989502i	$0.546164\pi$
$48$	15.6352	2.25674
$49$	1.00000	0.142857
$50$	0	0
$51$	1.74001	0.243650
$52$	4.25964	0.590706
$53$	1.42426	0.195637	0.0978185	−	0.995204i	$-0.468814\pi$
$53$	1.42426	0.195637	0.0978185	+	0.995204i	$0.468814\pi$
$54$	32.8021	4.46380
$55$	0	0
$56$	−0.810940	−0.108366
$57$	−1.35669	−0.179698
$58$	−0.309083	−0.0405846
$59$	2.21497	0.288365	0.144183	−	0.989551i	$-0.453945\pi$
$59$	2.21497	0.288365	0.144183	+	0.989551i	$0.453945\pi$
$60$	0	0
$61$	8.65223	1.10780	0.553902	−	0.832582i	$-0.313138\pi$
$61$	8.65223	1.10780	0.553902	+	0.832582i	$0.313138\pi$
$62$	17.3654	2.20541
$63$	−8.18935	−1.03176
$64$	−4.27755	−0.534694
$65$	0	0
$66$	−30.2845	−3.72777
$67$	7.72412	0.943652	0.471826	−	0.881692i	$-0.343595\pi$
$67$	7.72412	0.943652	0.471826	+	0.881692i	$0.343595\pi$
$68$	−0.817119	−0.0990902
$69$	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	$0.459848\pi$
$72$	6.64107	0.782658
$73$	−2.38846	−0.279549	−0.139774	−	0.990183i	$-0.544638\pi$
$73$	−2.38846	−0.279549	−0.139774	+	0.990183i	$0.544638\pi$
$74$	9.42566	1.09571
$75$	0	0
$76$	0.637109	0.0730814
$77$	4.79106	0.545992
$78$	17.1406	1.94079
$79$	11.7405	1.32091	0.660454	−	0.750867i	$-0.270364\pi$
$79$	11.7405	1.32091	0.660454	+	0.750867i	$0.270364\pi$
$80$	0	0
$81$	33.4974	3.72193
$82$	9.93365	1.09699
$83$	−5.80296	−0.636957	−0.318479	−	0.947930i	$-0.603172\pi$
$83$	−5.80296	−0.636957	−0.318479	+	0.947930i	$0.603172\pi$
$84$	5.25459	0.573323
$85$	0	0
$86$	−17.7225	−1.91106
$87$	−0.547132	−0.0586587
$88$	−3.88527	−0.414171
$89$	−14.5987	−1.54746	−0.773728	−	0.633517i	$-0.781610\pi$
$89$	−14.5987	−1.54746	−0.773728	+	0.633517i	$0.781610\pi$
$90$	0	0
$91$	−2.71167	−0.284260
$92$	−9.14821	−0.953767
$93$	30.7398	3.18757
$94$	3.74447	0.386213
$95$	0	0
$96$	−24.1201	−2.46174
$97$	−16.4698	−1.67225	−0.836127	−	0.548535i	$-0.815186\pi$
$97$	−16.4698	−1.67225	−0.836127	+	0.548535i	$0.815186\pi$
$98$	−1.88967	−0.190886
$99$	−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	$0.571144\pi$
$102$	−3.28804	−0.325565
$103$	2.13097	0.209971	0.104985	−	0.994474i	$-0.466520\pi$
$103$	2.13097	0.209971	0.104985	+	0.994474i	$0.466520\pi$
$104$	2.19900	0.215630
$105$	0	0
$106$	−2.69138	−0.261410
$107$	18.4519	1.78381	0.891904	−	0.452225i	$-0.149369\pi$
$107$	18.4519	1.78381	0.891904	+	0.452225i	$0.149369\pi$
$108$	−27.2679	−2.62385
$109$	15.8405	1.51724	0.758622	−	0.651530i	$-0.225873\pi$
$109$	15.8405	1.51724	0.758622	+	0.651530i	$0.225873\pi$
$110$	0	0
$111$	16.6851	1.58368
$112$	4.67412	0.441663
$113$	−7.17257	−0.674739	−0.337369	−	0.941372i	$-0.609537\pi$
$113$	−7.17257	−0.674739	−0.337369	+	0.941372i	$0.609537\pi$
$114$	2.56369	0.240112
$115$	0	0
$116$	0.256937	0.0238560
$117$	22.2068	2.05302
$118$	−4.18557	−0.385313
$119$	0.520174	0.0476843
$120$	0	0
$121$	11.9543	1.08675
$122$	−16.3499	−1.48025
$123$	17.5843	1.58552
$124$	−14.4356	−1.29636
$125$	0	0
$126$	15.4752	1.37864
$127$	9.53313	0.845929	0.422964	−	0.906146i	$-0.360990\pi$
$127$	9.53313	0.845929	0.422964	+	0.906146i	$0.360990\pi$
$128$	−6.33818	−0.560221
$129$	−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	$0.386780\pi$
$132$	25.1751	2.19121
$133$	−0.405581	−0.0351683
$134$	−14.5960	−1.26091
$135$	0	0
$136$	−0.421830	−0.0361717
$137$	3.58775	0.306522	0.153261	−	0.988186i	$-0.451023\pi$
$137$	3.58775	0.306522	0.153261	+	0.988186i	$0.451023\pi$
$138$	−36.8119	−3.13364
$139$	10.6898	0.906698	0.453349	−	0.891333i	$-0.350229\pi$
$139$	10.6898	0.906698	0.453349	+	0.891333i	$0.350229\pi$
$140$	0	0
$141$	6.62837	0.558210
$142$	−4.00632	−0.336203
$143$	−12.9918	−1.08643
$144$	−38.2780	−3.18984
$145$	0	0
$146$	4.51341	0.373533
$147$	−3.34505	−0.275895
$148$	−7.83541	−0.644067
$149$	1.22331	0.100217	0.0501087	−	0.998744i	$-0.484043\pi$
$149$	1.22331	0.100217	0.0501087	+	0.998744i	$0.484043\pi$
$150$	0	0
$151$	−1.65614	−0.134774	−0.0673872	−	0.997727i	$-0.521466\pi$
$151$	−1.65614	−0.134774	−0.0673872	+	0.997727i	$0.521466\pi$
$152$	0.328902	0.0266775
$153$	−4.25989	−0.344392
$154$	−9.05354	−0.729555
$155$	0	0
$156$	−14.2487	−1.14081
$157$	−11.1522	−0.890042	−0.445021	−	0.895520i	$-0.646804\pi$
$157$	−11.1522	−0.890042	−0.445021	+	0.895520i	$0.646804\pi$
$158$	−22.1856	−1.76500
$159$	−4.76421	−0.377827
$160$	0	0
$161$	5.82371	0.458973
$162$	−63.2990	−4.97324
$163$	1.03739	0.0812546	0.0406273	−	0.999174i	$-0.487064\pi$
$163$	1.03739	0.0812546	0.0406273	+	0.999174i	$0.487064\pi$
$164$	−8.25770	−0.644818
$165$	0	0
$166$	10.9657	0.851102
$167$	9.27166	0.717463	0.358731	−	0.933441i	$-0.383209\pi$
$167$	9.27166	0.717463	0.358731	+	0.933441i	$0.383209\pi$
$168$	2.71264	0.209284
$169$	−5.64685	−0.434373
$170$	0	0
$171$	3.32144	0.253997
$172$	14.7324	1.12334
$173$	−10.2811	−0.781660	−0.390830	−	0.920463i	$-0.627812\pi$
$173$	−10.2811	−0.781660	−0.390830	+	0.920463i	$0.627812\pi$
$174$	1.03390	0.0783797
$175$	0	0
$176$	22.3940	1.68801
$177$	−7.40920	−0.556909
$178$	27.5867	2.06771
$179$	4.36578	0.326314	0.163157	−	0.986600i	$-0.447832\pi$
$179$	4.36578	0.326314	0.163157	+	0.986600i	$0.447832\pi$
$180$	0	0
$181$	4.26928	0.317333	0.158666	−	0.987332i	$-0.449281\pi$
$181$	4.26928	0.317333	0.158666	+	0.987332i	$0.449281\pi$
$182$	5.12416	0.379828
$183$	−28.9421	−2.13946
$184$	−4.72268	−0.348161
$185$	0	0
$186$	−58.0882	−4.25923
$187$	2.49219	0.182247
$188$	−3.11273	−0.227019
$189$	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	$0.724767\pi$
$192$	14.3086	1.03264
$193$	27.2695	1.96290	0.981452	−	0.191707i	$-0.0614023\pi$
$193$	27.2695	1.96290	0.981452	+	0.191707i	$0.0614023\pi$
$194$	31.1225	2.23447
$195$	0	0
$196$	1.57086	0.112204
$197$	9.05598	0.645212	0.322606	−	0.946533i	$-0.395441\pi$
$197$	9.05598	0.645212	0.322606	+	0.946533i	$0.395441\pi$
$198$	74.1426	5.26908
$199$	16.9716	1.20308	0.601541	−	0.798842i	$-0.294554\pi$
$199$	16.9716	1.20308	0.601541	+	0.798842i	$0.294554\pi$
$200$	0	0
$201$	−25.8376	−1.82244
$202$	8.41872	0.592339
$203$	−0.163565	−0.0114800
$204$	2.73330	0.191369
$205$	0	0
$206$	−4.02684	−0.280563
$207$	−47.6924	−3.31485
$208$	−12.6747	−0.878831
$209$	−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	$0.333065\pi$
$212$	2.23731	0.153659
$213$	−7.09190	−0.485929
$214$	−34.8679	−2.38352
$215$	0	0
$216$	−14.0768	−0.957806
$217$	9.18965	0.623834
$218$	−29.9333	−2.02734
$219$	7.98953	0.539882
$220$	0	0
$221$	−1.41054	−0.0948833
$222$	−31.5293	−2.11611
$223$	1.21243	0.0811900	0.0405950	−	0.999176i	$-0.487075\pi$
$223$	1.21243	0.0811900	0.0405950	+	0.999176i	$0.487075\pi$
$224$	−7.21067	−0.481783
$225$	0	0
$226$	13.5538	0.901586
$227$	−16.8192	−1.11633	−0.558166	−	0.829729i	$-0.688495\pi$
$227$	−16.8192	−1.11633	−0.558166	+	0.829729i	$0.688495\pi$
$228$	−2.13116	−0.141140
$229$	23.0409	1.52259	0.761294	−	0.648407i	$-0.224565\pi$
$229$	23.0409	1.52259	0.761294	+	0.648407i	$0.224565\pi$
$230$	0	0
$231$	−16.0263	−1.05446
$232$	0.132641	0.00870832
$233$	15.8533	1.03859	0.519293	−	0.854596i	$-0.326195\pi$
$233$	15.8533	1.03859	0.519293	+	0.854596i	$0.326195\pi$
$234$	−41.9636	−2.74324
$235$	0	0
$236$	3.47941	0.226490
$237$	−39.2725	−2.55102
$238$	−0.982958	−0.0637157
$239$	−4.29785	−0.278005	−0.139002	−	0.990292i	$-0.544390\pi$
$239$	−4.29785	−0.278005	−0.139002	+	0.990292i	$0.544390\pi$
$240$	0	0
$241$	−2.38839	−0.153850	−0.0769248	−	0.997037i	$-0.524510\pi$
$241$	−2.38839	−0.153850	−0.0769248	+	0.997037i	$0.524510\pi$
$242$	−22.5897	−1.45212
$243$	−59.9745	−3.84737
$244$	13.5914	0.870101
$245$	0	0
$246$	−33.2285	−2.11857
$247$	1.09980	0.0699787
$248$	−7.45226	−0.473219
$249$	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	$0.452059\pi$
$252$	−12.8643	−0.810374
$253$	27.9018	1.75417
$254$	−18.0145	−1.13033
$255$	0	0
$256$	20.5322	1.28326
$257$	15.4336	0.962724	0.481362	−	0.876522i	$-0.340142\pi$
$257$	15.4336	0.962724	0.481362	+	0.876522i	$0.340142\pi$
$258$	59.2825	3.69077
$259$	4.98799	0.309939
$260$	0	0
$261$	1.33949	0.0829122
$262$	−15.0635	−0.930629
$263$	−0.273283	−0.0168513	−0.00842567	−	0.999965i	$-0.502682\pi$
$263$	−0.273283	−0.0168513	−0.00842567	+	0.999965i	$0.502682\pi$
$264$	12.9964	0.799874
$265$	0	0
$266$	0.766414	0.0469919
$267$	48.8333	2.98855
$268$	12.1335	0.741170
$269$	14.3844	0.877033	0.438516	−	0.898723i	$-0.355504\pi$
$269$	14.3844	0.877033	0.438516	+	0.898723i	$0.355504\pi$
$270$	0	0
$271$	15.6313	0.949532	0.474766	−	0.880112i	$-0.342533\pi$
$271$	15.6313	0.949532	0.474766	+	0.880112i	$0.342533\pi$
$272$	2.43136	0.147423
$273$	9.07067	0.548982
$274$	−6.77966	−0.409574
$275$	0	0
$276$	30.6012	1.84198
$277$	21.3893	1.28516	0.642579	−	0.766220i	$-0.277865\pi$
$277$	21.3893	1.28516	0.642579	+	0.766220i	$0.277865\pi$
$278$	−20.2002	−1.21153
$279$	−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	$0.675145\pi$
$282$	−12.5254	−0.745879
$283$	−1.87403	−0.111400	−0.0556999	−	0.998448i	$-0.517739\pi$
$283$	−1.87403	−0.111400	−0.0556999	+	0.998448i	$0.517739\pi$
$284$	3.33040	0.197623
$285$	0	0
$286$	24.5502	1.45168
$287$	5.25682	0.310300
$288$	59.0507	3.47960
$289$	−16.7294	−0.984083
$290$	0	0
$291$	55.0923	3.22957
$292$	−3.75193	−0.219565
$293$	18.2486	1.06609	0.533047	−	0.846086i	$-0.321047\pi$
$293$	18.2486	1.06609	0.533047	+	0.846086i	$0.321047\pi$
$294$	6.32104	0.368651
$295$	0	0
$296$	−4.04496	−0.235109
$297$	83.1663	4.82580
$298$	−2.31165	−0.133910
$299$	−15.7920	−0.913274
$300$	0	0
$301$	−9.37859	−0.540573
$302$	3.12955	0.180086
$303$	14.9026	0.856132
$304$	−1.89573	−0.108728
$305$	0	0
$306$	8.04979	0.460176
$307$	12.9856	0.741125	0.370562	−	0.928808i	$-0.379165\pi$
$307$	12.9856	0.741125	0.370562	+	0.928808i	$0.379165\pi$
$308$	7.52607	0.428838
$309$	−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	$0.328468\pi$
$312$	−7.35577	−0.416439
$313$	2.59623	0.146747	0.0733737	−	0.997305i	$-0.476623\pi$
$313$	2.59623	0.146747	0.0733737	+	0.997305i	$0.476623\pi$
$314$	21.0740	1.18927
$315$	0	0
$316$	18.4426	1.03748
$317$	−23.8818	−1.34134	−0.670669	−	0.741757i	$-0.733993\pi$
$317$	−23.8818	−1.34134	−0.670669	+	0.741757i	$0.733993\pi$
$318$	9.00280	0.504852
$319$	−0.783649	−0.0438759
$320$	0	0
$321$	−61.7223	−3.44501
$322$	−11.0049	−0.613279
$323$	−0.210973	−0.0117388
$324$	52.6196	2.92331
$325$	0	0
$326$	−1.96032	−0.108572
$327$	−52.9872	−2.93020
$328$	−4.26296	−0.235383
$329$	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	$0.330721\pi$
$332$	−9.11561	−0.500284
$333$	−40.8484	−2.23848
$334$	−17.5204	−0.958673
$335$	0	0
$336$	−15.6352	−0.852968
$337$	10.1586	0.553377	0.276688	−	0.960960i	$-0.410763\pi$
$337$	10.1586	0.553377	0.276688	+	0.960960i	$0.410763\pi$
$338$	10.6707	0.580409
$339$	23.9926	1.30310
$340$	0	0
$341$	44.0282	2.38426
$342$	−6.27643	−0.339391
$343$	−1.00000	−0.0539949
$344$	7.60548	0.410060
$345$	0	0
$346$	19.4279	1.04445
$347$	17.5771	0.943590	0.471795	−	0.881708i	$-0.343606\pi$
$347$	17.5771	0.943590	0.471795	+	0.881708i	$0.343606\pi$
$348$	−0.859465	−0.0460722
$349$	−9.74642	−0.521714	−0.260857	−	0.965377i	$-0.584005\pi$
$349$	−9.74642	−0.521714	−0.260857	+	0.965377i	$0.584005\pi$
$350$	0	0
$351$	−47.0709	−2.51246
$352$	−34.5468	−1.84135
$353$	12.9348	0.688449	0.344224	−	0.938887i	$-0.388142\pi$
$353$	12.9348	0.688449	0.344224	+	0.938887i	$0.388142\pi$
$354$	14.0009	0.744142
$355$	0	0
$356$	−22.9324	−1.21542
$357$	−1.74001	−0.0920910
$358$	−8.24990	−0.436021
$359$	−0.176146	−0.00929664	−0.00464832	−	0.999989i	$-0.501480\pi$
$359$	−0.176146	−0.00929664	−0.00464832	+	0.999989i	$0.501480\pi$
$360$	0	0
$361$	−18.8355	−0.991342
$362$	−8.06753	−0.424020
$363$	−39.9877	−2.09881
$364$	−4.25964	−0.223266
$365$	0	0
$366$	54.6911	2.85875
$367$	−27.4851	−1.43471	−0.717355	−	0.696708i	$-0.754647\pi$
$367$	−27.4851	−1.43471	−0.717355	+	0.696708i	$0.754647\pi$
$368$	27.2207	1.41898
$369$	−43.0499	−2.24109
$370$	0	0
$371$	−1.42426	−0.0739438
$372$	48.2879	2.50361
$373$	−18.6763	−0.967020	−0.483510	−	0.875339i	$-0.660638\pi$
$373$	−18.6763	−0.967020	−0.483510	+	0.875339i	$0.660638\pi$
$374$	−4.70942	−0.243518
$375$	0	0
$376$	−1.60692	−0.0828704
$377$	0.443533	0.0228431
$378$	−32.8021	−1.68716
$379$	−19.5390	−1.00365	−0.501825	−	0.864969i	$-0.667338\pi$
$379$	−19.5390	−1.00365	−0.501825	+	0.864969i	$0.667338\pi$
$380$	0	0
$381$	−31.8888	−1.63371
$382$	33.8926	1.73409
$383$	26.7503	1.36688	0.683439	−	0.730008i	$-0.260484\pi$
$383$	26.7503	1.36688	0.683439	+	0.730008i	$0.260484\pi$
$384$	21.2015	1.08194
$385$	0	0
$386$	−51.5305	−2.62283
$387$	76.8046	3.90420
$388$	−25.8717	−1.31344
$389$	13.9701	0.708314	0.354157	−	0.935186i	$-0.384768\pi$
$389$	13.9701	0.708314	0.354157	+	0.935186i	$0.384768\pi$
$390$	0	0
$391$	3.02934	0.153201
$392$	0.810940	0.0409587
$393$	−26.6651	−1.34508
$394$	−17.1128	−0.862132
$395$	0	0
$396$	−61.6336	−3.09721
$397$	−28.1378	−1.41219	−0.706097	−	0.708115i	$-0.749546\pi$
$397$	−28.1378	−1.41219	−0.706097	+	0.708115i	$0.749546\pi$
$398$	−32.0707	−1.60756
$399$	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	$0.530578\pi$
$402$	48.8245	2.43514
$403$	−24.9193	−1.24132
$404$	−6.99836	−0.348181
$405$	0	0
$406$	0.309083	0.0153395
$407$	23.8978	1.18457
$408$	1.41104	0.0698570
$409$	29.9871	1.48277	0.741384	−	0.671081i	$-0.234170\pi$
$409$	29.9871	1.48277	0.741384	+	0.671081i	$0.234170\pi$
$410$	0	0
$411$	−12.0012	−0.591975
$412$	3.34745	0.164917
$413$	−2.21497	−0.108992
$414$	90.1229	4.42930
$415$	0	0
$416$	19.5530	0.958663
$417$	−35.7579	−1.75107
$418$	3.67194	0.179600
$419$	33.5276	1.63793	0.818964	−	0.573845i	$-0.194549\pi$
$419$	33.5276	1.63793	0.818964	+	0.573845i	$0.194549\pi$
$420$	0	0
$421$	39.6034	1.93015	0.965075	−	0.261975i	$-0.0843738\pi$
$421$	39.6034	1.93015	0.965075	+	0.261975i	$0.0843738\pi$
$422$	−27.4891	−1.33815
$423$	−16.2276	−0.789012
$424$	1.15499	0.0560912
$425$	0	0
$426$	13.4013	0.649297
$427$	−8.65223	−0.418711
$428$	28.9852	1.40105
$429$	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	$0.534787\pi$
$432$	81.1363	3.90367
$433$	17.4520	0.838689	0.419344	−	0.907827i	$-0.362260\pi$
$433$	17.4520	0.838689	0.419344	+	0.907827i	$0.362260\pi$
$434$	−17.3654	−0.833567
$435$	0	0
$436$	24.8832	1.19169
$437$	−2.36198	−0.112989
$438$	−15.0976	−0.721390
$439$	−0.254860	−0.0121638	−0.00608189	−	0.999982i	$-0.501936\pi$
$439$	−0.254860	−0.0121638	−0.00608189	+	0.999982i	$0.501936\pi$
$440$	0	0
$441$	8.18935	0.389969
$442$	2.66546	0.126783
$443$	−18.4511	−0.876637	−0.438319	−	0.898820i	$-0.644426\pi$
$443$	−18.4511	−0.876637	−0.438319	+	0.898820i	$0.644426\pi$
$444$	26.2098	1.24386
$445$	0	0
$446$	−2.29108	−0.108486
$447$	−4.09203	−0.193546
$448$	4.27755	0.202095
$449$	9.88811	0.466649	0.233324	−	0.972399i	$-0.425040\pi$
$449$	9.88811	0.466649	0.233324	+	0.972399i	$0.425040\pi$
$450$	0	0
$451$	25.1857	1.18595
$452$	−11.2671	−0.529959
$453$	5.53986	0.260285
$454$	31.7828	1.49164
$455$	0	0
$456$	−1.10019	−0.0515212
$457$	−24.5336	−1.14763	−0.573817	−	0.818984i	$-0.694538\pi$
$457$	−24.5336	−1.14763	−0.573817	+	0.818984i	$0.694538\pi$
$458$	−43.5398	−2.03448
$459$	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	$0.822446\pi$
$462$	30.2845	1.40896
$463$	34.6332	1.60954	0.804770	−	0.593587i	$-0.202289\pi$
$463$	34.6332	1.60954	0.804770	+	0.593587i	$0.202289\pi$
$464$	−0.764521	−0.0354920
$465$	0	0
$466$	−29.9576	−1.38776
$467$	−7.61216	−0.352249	−0.176124	−	0.984368i	$-0.556356\pi$
$467$	−7.61216	−0.352249	−0.176124	+	0.984368i	$0.556356\pi$
$468$	34.8837	1.61250
$469$	−7.72412	−0.356667
$470$	0	0
$471$	37.3046	1.71891
$472$	1.79621	0.0826774
$473$	−44.9334	−2.06604
$474$	74.2121	3.40867
$475$	0	0
$476$	0.817119	0.0374526
$477$	11.6638	0.534046
$478$	8.12152	0.371470
$479$	−21.4501	−0.980080	−0.490040	−	0.871700i	$-0.663018\pi$
$479$	−21.4501	−0.980080	−0.490040	+	0.871700i	$0.663018\pi$
$480$	0	0
$481$	−13.5258	−0.616722
$482$	4.51326	0.205574
$483$	−19.4806	−0.886398
$484$	18.7785	0.853568
$485$	0	0
$486$	113.332	5.14085
$487$	−3.34199	−0.151440	−0.0757199	−	0.997129i	$-0.524125\pi$
$487$	−3.34199	−0.151440	−0.0757199	+	0.997129i	$0.524125\pi$
$488$	7.01644	0.317619
$489$	−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	$0.771753\pi$
$492$	27.6224	1.24531
$493$	−0.0850821	−0.00383191
$494$	−2.07826	−0.0935054
$495$	0	0
$496$	42.9536	1.92867
$497$	−2.12012	−0.0951003
$498$	−36.6807	−1.64370
$499$	−36.9475	−1.65400	−0.826999	−	0.562204i	$-0.809954\pi$
$499$	−36.9475	−1.65400	−0.826999	+	0.562204i	$0.809954\pi$
$500$	0	0
$501$	−31.0142	−1.38561
$502$	−8.98388	−0.400970
$503$	30.7679	1.37187	0.685936	−	0.727662i	$-0.259393\pi$
$503$	30.7679	1.37187	0.685936	+	0.727662i	$0.259393\pi$
$504$	−6.64107	−0.295817
$505$	0	0
$506$	−52.7252	−2.34392
$507$	18.8890	0.838889
$508$	14.9752	0.664416
$509$	10.9758	0.486494	0.243247	−	0.969964i	$-0.421787\pi$
$509$	10.9758	0.486494	0.243247	+	0.969964i	$0.421787\pi$
$510$	0	0
$511$	2.38846	0.105659
$512$	−26.1227	−1.15447
$513$	−7.04032	−0.310838
$514$	−29.1645	−1.28639
$515$	0	0
$516$	−49.2807	−2.16946
$517$	9.49372	0.417533
$518$	−9.42566	−0.414140
$519$	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	$0.348079\pi$
$522$	−2.53119	−0.110787
$523$	−6.71426	−0.293594	−0.146797	−	0.989167i	$-0.546896\pi$
$523$	−6.71426	−0.293594	−0.146797	+	0.989167i	$0.546896\pi$
$524$	12.5221	0.547031
$525$	0	0
$526$	0.516415	0.0225168
$527$	4.78022	0.208230
$528$	−74.9091	−3.26000
$529$	10.9156	0.474591
$530$	0	0
$531$	18.1392	0.787174
$532$	−0.637109	−0.0276222
$533$	−14.2548	−0.617442
$534$	−92.2788	−3.99330
$535$	0	0
$536$	6.26380	0.270555
$537$	−14.6038	−0.630199
$538$	−27.1818	−1.17189
$539$	−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	$0.475995\pi$
$542$	−29.5379	−1.26876
$543$	−14.2809	−0.612854
$544$	−3.75081	−0.160815
$545$	0	0
$546$	−17.1406	−0.733549
$547$	−25.9535	−1.10969	−0.554847	−	0.831953i	$-0.687223\pi$
$547$	−25.9535	−1.10969	−0.554847	+	0.831953i	$0.687223\pi$
$548$	5.63583	0.240751
$549$	70.8561	3.02406
$550$	0	0
$551$	0.0663387	0.00282612
$552$	15.7976	0.672390
$553$	−11.7405	−0.499256
$554$	−40.4187	−1.71723
$555$	0	0
$556$	16.7922	0.712146
$557$	3.45603	0.146437	0.0732184	−	0.997316i	$-0.476673\pi$
$557$	3.45603	0.146437	0.0732184	+	0.997316i	$0.476673\pi$
$558$	142.211	6.02029
$559$	25.4316	1.07564
$560$	0	0
$561$	−8.33649	−0.351967
$562$	33.1266	1.39736
$563$	27.6750	1.16636	0.583181	−	0.812342i	$-0.301808\pi$
$563$	27.6750	1.16636	0.583181	+	0.812342i	$0.301808\pi$
$564$	10.4122	0.438434
$565$	0	0
$566$	3.54131	0.148852
$567$	−33.4974	−1.40676
$568$	1.71929	0.0721398
$569$	18.0239	0.755602	0.377801	−	0.925887i	$-0.376680\pi$
$569$	18.0239	0.755602	0.377801	+	0.925887i	$0.376680\pi$
$570$	0	0
$571$	−10.4491	−0.437281	−0.218641	−	0.975805i	$-0.570162\pi$
$571$	−10.4491	−0.437281	−0.218641	+	0.975805i	$0.570162\pi$
$572$	−20.4082	−0.853311
$573$	59.9958	2.50636
$574$	−9.93365	−0.414623
$575$	0	0
$576$	−35.0304	−1.45960
$577$	−25.4222	−1.05834	−0.529169	−	0.848516i	$-0.677496\pi$
$577$	−25.4222	−1.05834	−0.529169	+	0.848516i	$0.677496\pi$
$578$	31.6131	1.31493
$579$	−91.2179	−3.79089
$580$	0	0
$581$	5.80296	0.240747
$582$	−104.106	−4.31534
$583$	−6.82372	−0.282609
$584$	−1.93690	−0.0801496
$585$	0	0
$586$	−34.4838	−1.42451
$587$	38.7065	1.59759	0.798793	−	0.601606i	$-0.205472\pi$
$587$	38.7065	1.59759	0.798793	+	0.601606i	$0.205472\pi$
$588$	−5.25459	−0.216696
$589$	−3.72715	−0.153574
$590$	0	0
$591$	−30.2927	−1.24608
$592$	23.3145	0.958219
$593$	31.0697	1.27588	0.637939	−	0.770087i	$-0.279787\pi$
$593$	31.0697	1.27588	0.637939	+	0.770087i	$0.279787\pi$
$594$	−157.157	−6.44823
$595$	0	0
$596$	1.92164	0.0787136
$597$	−56.7707	−2.32347
$598$	29.8416	1.22032
$599$	−0.389923	−0.0159318	−0.00796591	−	0.999968i	$-0.502536\pi$
$599$	−0.389923	−0.0159318	−0.00796591	+	0.999968i	$0.502536\pi$
$600$	0	0
$601$	−3.35540	−0.136870	−0.0684349	−	0.997656i	$-0.521801\pi$
$601$	−3.35540	−0.136870	−0.0684349	+	0.997656i	$0.521801\pi$
$602$	17.7225	0.722313
$603$	63.2555	2.57596
$604$	−2.60155	−0.105856
$605$	0	0
$606$	−28.1610	−1.14396
$607$	6.93329	0.281413	0.140707	−	0.990051i	$-0.455063\pi$
$607$	6.93329	0.281413	0.140707	+	0.990051i	$0.455063\pi$
$608$	2.92451	0.118605
$609$	0.547132	0.0221709
$610$	0	0
$611$	−5.37330	−0.217381
$612$	−6.69167	−0.270495
$613$	−10.5240	−0.425060	−0.212530	−	0.977155i	$-0.568170\pi$
$613$	−10.5240	−0.425060	−0.212530	+	0.977155i	$0.568170\pi$
$614$	−24.5384	−0.990290
$615$	0	0
$616$	3.88527	0.156542
$617$	−5.97280	−0.240456	−0.120228	−	0.992746i	$-0.538363\pi$
$617$	−5.97280	−0.240456	−0.120228	+	0.992746i	$0.538363\pi$
$618$	13.4700	0.541841
$619$	−38.0234	−1.52829	−0.764145	−	0.645045i	$-0.776839\pi$
$619$	−38.0234	−1.52829	−0.764145	+	0.645045i	$0.776839\pi$
$620$	0	0
$621$	101.092	4.05667
$622$	−34.2030	−1.37141
$623$	14.5987	0.584884
$624$	42.3974	1.69725
$625$	0	0
$626$	−4.90602	−0.196084
$627$	6.49998	0.259584
$628$	−17.5185	−0.699064
$629$	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	$0.667526\pi$
$632$	9.52083	0.378718
$633$	−48.6605	−1.93408
$634$	45.1288	1.79229
$635$	0	0
$636$	−7.48390	−0.296756
$637$	2.71167	0.107440
$638$	1.48084	0.0586269
$639$	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	$0.343109\pi$
$642$	116.635	4.60321
$643$	−38.5274	−1.51937	−0.759686	−	0.650290i	$-0.774648\pi$
$643$	−38.5274	−1.51937	−0.759686	+	0.650290i	$0.774648\pi$
$644$	9.14821	0.360490
$645$	0	0
$646$	0.398669	0.0156854
$647$	−33.6150	−1.32154	−0.660771	−	0.750588i	$-0.729771\pi$
$647$	−33.6150	−1.32154	−0.660771	+	0.750588i	$0.729771\pi$
$648$	27.1644	1.06712
$649$	−10.6121	−0.416561
$650$	0	0
$651$	−30.7398	−1.20479
$652$	1.62959	0.0638196
$653$	−22.6590	−0.886716	−0.443358	−	0.896345i	$-0.646213\pi$
$653$	−22.6590	−0.886716	−0.443358	+	0.896345i	$0.646213\pi$
$654$	100.128	3.91533
$655$	0	0
$656$	24.5710	0.959337
$657$	−19.5600	−0.763107
$658$	−3.74447	−0.145975
$659$	45.0061	1.75319	0.876595	−	0.481229i	$-0.159810\pi$
$659$	45.0061	1.75319	0.876595	+	0.481229i	$0.159810\pi$
$660$	0	0
$661$	−13.7843	−0.536148	−0.268074	−	0.963398i	$-0.586387\pi$
$661$	−13.7843	−0.536148	−0.268074	+	0.963398i	$0.586387\pi$
$662$	−34.8672	−1.35515
$663$	4.71833	0.183245
$664$	−4.70585	−0.182622
$665$	0	0
$666$	77.1900	2.99105
$667$	−0.952553	−0.0368830
$668$	14.5644	0.563515
$669$	−4.05562	−0.156799
$670$	0	0
$671$	−41.4534	−1.60029
$672$	24.1201	0.930451
$673$	9.76275	0.376326	0.188163	−	0.982138i	$-0.439747\pi$
$673$	9.76275	0.376326	0.188163	+	0.982138i	$0.439747\pi$
$674$	−19.1965	−0.739422
$675$	0	0
$676$	−8.87038	−0.341169
$677$	17.3365	0.666295	0.333147	−	0.942875i	$-0.391889\pi$
$677$	17.3365	0.666295	0.333147	+	0.942875i	$0.391889\pi$
$678$	−45.3381	−1.74120
$679$	16.4698	0.632053
$680$	0	0
$681$	56.2612	2.15593
$682$	−83.1989	−3.18585
$683$	−10.2687	−0.392921	−0.196460	−	0.980512i	$-0.562945\pi$
$683$	−10.2687	−0.392921	−0.196460	+	0.980512i	$0.562945\pi$
$684$	5.21751	0.199496
$685$	0	0
$686$	1.88967	0.0721480
$687$	−77.0730	−2.94052
$688$	−43.8367	−1.67126
$689$	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	$0.672034\pi$
$692$	−16.1502	−0.613937
$693$	39.2357	1.49044
$694$	−33.2150	−1.26082
$695$	0	0
$696$	−0.443691	−0.0168181
$697$	2.73446	0.103575
$698$	18.4175	0.697114
$699$	−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	$0.381227\pi$
$702$	88.9484	3.35714
$703$	−2.02303	−0.0763001
$704$	20.4940	0.772398
$705$	0	0
$706$	−24.4425	−0.919904
$707$	4.45512	0.167552
$708$	−11.6388	−0.437412
$709$	27.9701	1.05044	0.525219	−	0.850967i	$-0.323983\pi$
$709$	27.9701	1.05044	0.525219	+	0.850967i	$0.323983\pi$
$710$	0	0
$711$	96.1469	3.60579
$712$	−11.8387	−0.443672
$713$	53.5179	2.00426
$714$	3.28804	0.123052
$715$	0	0
$716$	6.85802	0.256296
$717$	14.3765	0.536901
$718$	0.332858	0.0124222
$719$	−39.2359	−1.46325	−0.731626	−	0.681706i	$-0.761238\pi$
$719$	−39.2359	−1.46325	−0.731626	+	0.681706i	$0.761238\pi$
$720$	0	0
$721$	−2.13097	−0.0793616
$722$	35.5929	1.32463
$723$	7.98927	0.297124
$724$	6.70642	0.249242
$725$	0	0
$726$	75.5636	2.80443
$727$	−33.0645	−1.22629	−0.613147	−	0.789969i	$-0.710097\pi$
$727$	−33.0645	−1.22629	−0.613147	+	0.789969i	$0.710097\pi$
$728$	−2.19900	−0.0815005
$729$	100.125	3.70835
$730$	0	0
$731$	−4.87850	−0.180438
$732$	−45.4639	−1.68039
$733$	−13.8421	−0.511268	−0.255634	−	0.966774i	$-0.582284\pi$
$733$	−13.8421	−0.511268	−0.255634	+	0.966774i	$0.582284\pi$
$734$	51.9378	1.91706
$735$	0	0
$736$	−41.9929	−1.54788
$737$	−37.0068	−1.36316
$738$	81.3501	2.99454
$739$	40.3699	1.48503	0.742515	−	0.669829i	$-0.233633\pi$
$739$	40.3699	1.48503	0.742515	+	0.669829i	$0.233633\pi$
$740$	0	0
$741$	−3.67889	−0.135147
$742$	2.69138	0.0988037
$743$	5.77743	0.211953	0.105977	−	0.994369i	$-0.466203\pi$
$743$	5.77743	0.211953	0.105977	+	0.994369i	$0.466203\pi$
$744$	24.9282	0.913911
$745$	0	0
$746$	35.2920	1.29213
$747$	−47.5224	−1.73875
$748$	3.91487	0.143142
$749$	−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	$0.601794\pi$
$752$	9.26200	0.337750
$753$	−15.9030	−0.579539
$754$	−0.838132	−0.0305230
$755$	0	0
$756$	27.2679	0.991724
$757$	−3.11588	−0.113248	−0.0566242	−	0.998396i	$-0.518034\pi$
$757$	−3.11588	−0.113248	−0.0566242	+	0.998396i	$0.518034\pi$
$758$	36.9222	1.34108
$759$	−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	$0.561404\pi$
$762$	60.2593	2.18296
$763$	−15.8405	−0.573465
$764$	−28.1744	−1.01931
$765$	0	0
$766$	−50.5493	−1.82642
$767$	6.00628	0.216874
$768$	−68.6811	−2.47832
$769$	−6.97652	−0.251580	−0.125790	−	0.992057i	$-0.540146\pi$
$769$	−6.97652	−0.251580	−0.125790	+	0.992057i	$0.540146\pi$
$770$	0	0
$771$	−51.6263	−1.85927
$772$	42.8365	1.54172
$773$	27.4955	0.988945	0.494473	−	0.869193i	$-0.335361\pi$
$773$	27.4955	0.988945	0.494473	+	0.869193i	$0.335361\pi$
$774$	−145.135	−5.21678
$775$	0	0
$776$	−13.3560	−0.479453
$777$	−16.6851	−0.598573
$778$	−26.3989	−0.946448
$779$	−2.13206	−0.0763891
$780$	0	0
$781$	−10.1576	−0.363468
$782$	−5.72446	−0.204706
$783$	−2.83926	−0.101467
$784$	−4.67412	−0.166933
$785$	0	0
$786$	50.3883	1.79729
$787$	−1.16804	−0.0416361	−0.0208181	−	0.999783i	$-0.506627\pi$
$787$	−1.16804	−0.0416361	−0.0208181	+	0.999783i	$0.506627\pi$
$788$	14.2256	0.506768
$789$	0.914145	0.0325444
$790$	0	0
$791$	7.17257	0.255027
$792$	−31.8178	−1.13060
$793$	23.4620	0.833159
$794$	53.1711	1.88697
$795$	0	0
$796$	26.6599	0.944934
$797$	−5.25703	−0.186214	−0.0931068	−	0.995656i	$-0.529680\pi$
$797$	−5.25703	−0.186214	−0.0931068	+	0.995656i	$0.529680\pi$
$798$	−2.56369	−0.0907537
$799$	1.03075	0.0364653
$800$	0	0
$801$	−119.554	−4.22422
$802$	7.25895	0.256323
$803$	11.4433	0.403825
$804$	−40.5871	−1.43140
$805$	0	0
$806$	47.0893	1.65865
$807$	−48.1165	−1.69378
$808$	−3.61284	−0.127099
$809$	−32.2856	−1.13510	−0.567551	−	0.823338i	$-0.692109\pi$
$809$	−32.2856	−1.13510	−0.567551	+	0.823338i	$0.692109\pi$
$810$	0	0
$811$	0.362439	0.0127270	0.00636348	−	0.999980i	$-0.497974\pi$
$811$	0.362439	0.0127270	0.00636348	+	0.999980i	$0.497974\pi$
$812$	−0.256937	−0.00901670
$813$	−52.2873	−1.83380
$814$	−45.1589	−1.58282
$815$	0	0
$816$	−8.13301	−0.284712
$817$	3.80378	0.133077
$818$	−56.6658	−1.98127
$819$	−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	$0.734133\pi$
$822$	22.6783	0.790996
$823$	5.90272	0.205756	0.102878	−	0.994694i	$-0.467195\pi$
$823$	5.90272	0.205756	0.102878	+	0.994694i	$0.467195\pi$
$824$	1.72809	0.0602009
$825$	0	0
$826$	4.18557	0.145635
$827$	41.8778	1.45624	0.728118	−	0.685452i	$-0.240395\pi$
$827$	41.8778	1.45624	0.728118	+	0.685452i	$0.240395\pi$
$828$	−74.9179	−2.60358
$829$	19.9050	0.691331	0.345665	−	0.938358i	$-0.387653\pi$
$829$	19.9050	0.691331	0.345665	+	0.938358i	$0.387653\pi$
$830$	0	0
$831$	−71.5482	−2.48198
$832$	−11.5993	−0.402134
$833$	−0.520174	−0.0180230
$834$	67.5708	2.33978
$835$	0	0
$836$	−3.05243	−0.105571
$837$	159.520	5.51381
$838$	−63.3560	−2.18860
$839$	−37.3674	−1.29007	−0.645033	−	0.764155i	$-0.723156\pi$
$839$	−37.3674	−1.29007	−0.645033	+	0.764155i	$0.723156\pi$
$840$	0	0
$841$	−28.9732	−0.999077
$842$	−74.8373	−2.57906
$843$	58.6398	2.01966
$844$	22.8513	0.786573
$845$	0	0
$846$	30.6648	1.05428
$847$	−11.9543	−0.410755
$848$	−6.65716	−0.228608
$849$	6.26873	0.215142
$850$	0	0
$851$	29.0486	0.995773
$852$	−11.1403	−0.381662
$853$	−47.5502	−1.62809	−0.814044	−	0.580804i	$-0.802738\pi$
$853$	−47.5502	−1.62809	−0.814044	+	0.580804i	$0.802738\pi$
$854$	16.3499	0.559481
$855$	0	0
$856$	14.9634	0.511437
$857$	30.6116	1.04567	0.522836	−	0.852433i	$-0.324874\pi$
$857$	30.6116	1.04567	0.522836	+	0.852433i	$0.324874\pi$
$858$	−82.1216	−2.80359
$859$	−38.0855	−1.29946	−0.649730	−	0.760165i	$-0.725118\pi$
$859$	−38.0855	−1.29946	−0.649730	+	0.760165i	$0.725118\pi$
$860$	0	0
$861$	−17.5843	−0.599271
$862$	8.55769	0.291476
$863$	38.9217	1.32491	0.662455	−	0.749101i	$-0.269514\pi$
$863$	38.9217	1.32491	0.662455	+	0.749101i	$0.269514\pi$
$864$	−125.167	−4.25828
$865$	0	0
$866$	−32.9785	−1.12066
$867$	55.9607	1.90053
$868$	14.4356	0.489977
$869$	−56.2494	−1.90813
$870$	0	0
$871$	20.9453	0.709703
$872$	12.8457	0.435010
$873$	−134.877	−4.56489
$874$	4.46337	0.150976
$875$	0	0
$876$	12.5504	0.424039
$877$	27.5668	0.930863	0.465432	−	0.885084i	$-0.345899\pi$
$877$	27.5668	0.930863	0.465432	+	0.885084i	$0.345899\pi$
$878$	0.481601	0.0162532
$879$	−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	$0.463392\pi$
$882$	−15.4752	−0.521076
$883$	41.1531	1.38491	0.692457	−	0.721459i	$-0.256528\pi$
$883$	41.1531	1.38491	0.692457	+	0.721459i	$0.256528\pi$
$884$	−2.21576	−0.0745240
$885$	0	0
$886$	34.8665	1.17136
$887$	−10.3979	−0.349128	−0.174564	−	0.984646i	$-0.555852\pi$
$887$	−10.3979	−0.349128	−0.174564	+	0.984646i	$0.555852\pi$
$888$	13.5306	0.454057
$889$	−9.53313	−0.319731
$890$	0	0
$891$	−160.488	−5.37655
$892$	1.90455	0.0637689
$893$	−0.803678	−0.0268940
$894$	7.73259	0.258616
$895$	0	0
$896$	6.33818	0.211744
$897$	52.8249	1.76377
$898$	−18.6853	−0.623536
$899$	−1.50310	−0.0501313
$900$	0	0
$901$	−0.740863	−0.0246817
$902$	−47.5928	−1.58467
$903$	31.3718	1.04399
$904$	−5.81653	−0.193455
$905$	0	0
$906$	−10.4685	−0.347793
$907$	−14.3126	−0.475242	−0.237621	−	0.971358i	$-0.576368\pi$
$907$	−14.3126	−0.475242	−0.237621	+	0.971358i	$0.576368\pi$
$908$	−26.4206	−0.876798
$909$	−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	$0.796229\pi$
$912$	6.34132	0.209982
$913$	27.8023	0.920123
$914$	46.3604	1.53347
$915$	0	0
$916$	36.1940	1.19588
$917$	−7.97152	−0.263243
$918$	−17.0628	−0.563156
$919$	−0.526360	−0.0173630	−0.00868151	−	0.999962i	$-0.502763\pi$
$919$	−0.526360	−0.0173630	−0.00868151	+	0.999962i	$0.502763\pi$
$920$	0	0
$921$	−43.4373	−1.43131
$922$	68.8459	2.26732
$923$	5.74906	0.189233
$924$	−25.1751	−0.828199
$925$	0	0
$926$	−65.4453	−2.15067
$927$	17.4513	0.573175
$928$	1.17941	0.0387161
$929$	−1.36557	−0.0448030	−0.0224015	−	0.999749i	$-0.507131\pi$
$929$	−1.36557	−0.0448030	−0.0224015	+	0.999749i	$0.507131\pi$
$930$	0	0
$931$	0.405581	0.0132924
$932$	24.9033	0.815734
$933$	−60.5453	−1.98216
$934$	14.3845	0.470674
$935$	0	0
$936$	18.0084	0.588623
$937$	−0.859041	−0.0280636	−0.0140318	−	0.999902i	$-0.504467\pi$
$937$	−0.859041	−0.0280636	−0.0140318	+	0.999902i	$0.504467\pi$
$938$	14.5960	0.476578
$939$	−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	$0.258263\pi$
$942$	−70.4935	−2.29680
$943$	30.6142	0.996935
$944$	−10.3531	−0.336964
$945$	0	0
$946$	84.9094	2.76064
$947$	40.7967	1.32571	0.662857	−	0.748746i	$-0.269344\pi$
$947$	40.7967	1.32571	0.662857	+	0.748746i	$0.269344\pi$
$948$	−61.6914	−2.00364
$949$	−6.47673	−0.210243
$950$	0	0
$951$	79.8859	2.59048
$952$	0.421830	0.0136716
$953$	16.0122	0.518688	0.259344	−	0.965785i	$-0.416494\pi$
$953$	16.0122	0.518688	0.259344	+	0.965785i	$0.416494\pi$
$954$	−22.0407	−0.713592
$955$	0	0
$956$	−6.75130	−0.218353
$957$	2.62134	0.0847360
$958$	40.5336	1.30958
$959$	−3.58775	−0.115854
$960$	0	0
$961$	53.4497	1.72418
$962$	25.5593	0.824064
$963$	151.109	4.86941
$964$	−3.75181	−0.120838
$965$	0	0
$966$	36.8119	1.18440
$967$	44.6342	1.43534	0.717669	−	0.696384i	$-0.245209\pi$
$967$	44.6342	1.43534	0.717669	+	0.696384i	$0.245209\pi$
$968$	9.69423	0.311584
$969$	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	$0.688825\pi$
$972$	−94.2113	−3.02183
$973$	−10.6898	−0.342700
$974$	6.31525	0.202354
$975$	0	0
$976$	−40.4416	−1.29450
$977$	21.1094	0.675348	0.337674	−	0.941263i	$-0.390360\pi$
$977$	21.1094	0.675348	0.337674	+	0.941263i	$0.390360\pi$
$978$	6.55738	0.209682
$979$	69.9432	2.23539
$980$	0	0
$981$	129.723	4.14175
$982$	63.1218	2.01430
$983$	20.7396	0.661491	0.330745	−	0.943720i	$-0.392700\pi$
$983$	20.7396	0.661491	0.330745	+	0.943720i	$0.392700\pi$
$984$	14.2598	0.454587
$985$	0	0
$986$	0.160777	0.00512019
$987$	−6.62837	−0.210983
$988$	1.72763	0.0549632
$989$	−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	$0.597358\pi$
$992$	−66.2636	−2.10387
$993$	−61.7210	−1.95866
$994$	4.00632	0.127073
$995$	0	0
$996$	30.4922	0.966181
$997$	−20.9224	−0.662618	−0.331309	−	0.943522i	$-0.607490\pi$
$997$	−20.9224	−0.662618	−0.331309	+	0.943522i	$0.607490\pi$
$998$	69.8186	2.21007
$999$	86.5846	2.73942

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
875.2.a.i.1.7	✓	8	5.4	even	2
875.2.a.j.1.2	yes	8	1.1	even	1	trivial
875.2.b.e.624.4		16	5.2	odd	4
875.2.b.e.624.13		16	5.3	odd	4
6125.2.a.v.1.7		8	35.34	odd	2
6125.2.a.w.1.2		8	7.6	odd	2
7875.2.a.w.1.7		8	3.2	odd	2
7875.2.a.bb.1.2		8	15.14	odd	2

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.a (trivial)

\(f(q)\)	\(=\)	\(q-1.88967 q^{2} -3.34505 q^{3} +1.57086 q^{4} +6.32104 q^{6} -1.00000 q^{7} +0.810940 q^{8} +8.18935 q^{9} -4.79106 q^{11} -5.25459 q^{12} +2.71167 q^{13} +1.88967 q^{14} -4.67412 q^{16} -0.520174 q^{17} -15.4752 q^{18} +0.405581 q^{19} +3.34505 q^{21} +9.05354 q^{22} -5.82371 q^{23} -2.71264 q^{24} -5.12416 q^{26} -17.3586 q^{27} -1.57086 q^{28} +0.163565 q^{29} -9.18965 q^{31} +7.21067 q^{32} +16.0263 q^{33} +0.982958 q^{34} +12.8643 q^{36} -4.98799 q^{37} -0.766414 q^{38} -9.07067 q^{39} -5.25682 q^{41} -6.32104 q^{42} +9.37859 q^{43} -7.52607 q^{44} +11.0049 q^{46} -1.98155 q^{47} +15.6352 q^{48} +1.00000 q^{49} +1.74001 q^{51} +4.25964 q^{52} +1.42426 q^{53} +32.8021 q^{54} -0.810940 q^{56} -1.35669 q^{57} -0.309083 q^{58} +2.21497 q^{59} +8.65223 q^{61} +17.3654 q^{62} -8.18935 q^{63} -4.27755 q^{64} -30.2845 q^{66} +7.72412 q^{67} -0.817119 q^{68} +19.4806 q^{69} +2.12012 q^{71} +6.64107 q^{72} -2.38846 q^{73} +9.42566 q^{74} +0.637109 q^{76} +4.79106 q^{77} +17.1406 q^{78} +11.7405 q^{79} +33.4974 q^{81} +9.93365 q^{82} -5.80296 q^{83} +5.25459 q^{84} -17.7225 q^{86} -0.547132 q^{87} -3.88527 q^{88} -14.5987 q^{89} -2.71167 q^{91} -9.14821 q^{92} +30.7398 q^{93} +3.74447 q^{94} -24.1201 q^{96} -16.4698 q^{97} -1.88967 q^{98} -39.2357 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} - 8 q^{3} + 13 q^{4} + 2 q^{6} - 8 q^{7} + 12 q^{8} + 18 q^{9} - 5 q^{11} - 20 q^{12} - 6 q^{13} - q^{14} + 35 q^{16} + 13 q^{17} + 3 q^{18} + 13 q^{19} + 8 q^{21} + 22 q^{22} - 5 q^{23} - 3 q^{24}+ \cdots - 31 q^{99}+O(q^{100}) \) 8 * q + q^2 - 8 * q^3 + 13 * q^4 + 2 * q^6 - 8 * q^7 + 12 * q^8 + 18 * q^9 - 5 * q^11 - 20 * q^12 - 6 * q^13 - q^14 + 35 * q^16 + 13 * q^17 + 3 * q^18 + 13 * q^19 + 8 * q^21 + 22 * q^22 - 5 * q^23 - 3 * q^24 - 5 * q^26 - 29 * q^27 - 13 * q^28 + 22 * q^29 + 3 * q^31 + 35 * q^32 - 10 * q^33 + 19 * q^34 + 27 * q^36 + 7 * q^37 - 5 * q^38 + 7 * q^39 + 12 * q^41 - 2 * q^42 - q^43 - 41 * q^44 + 11 * q^46 - 2 * q^47 + 12 * q^48 + 8 * q^49 + 10 * q^51 - 49 * q^52 + 10 * q^53 - 31 * q^54 - 12 * q^56 + 17 * q^57 + 58 * q^58 - 11 * q^59 + 45 * q^61 + 26 * q^62 - 18 * q^63 + 82 * q^64 - 42 * q^66 + 6 * q^67 + 58 * q^68 + 25 * q^69 - 15 * q^71 + 85 * q^72 - 32 * q^73 + 24 * q^74 + 26 * q^76 + 5 * q^77 - 32 * q^78 + 12 * q^79 + 36 * q^81 - 15 * q^83 + 20 * q^84 - 48 * q^86 - 35 * q^87 - 2 * q^88 + 12 * q^89 + 6 * q^91 - 78 * q^92 + 33 * q^93 - 38 * q^94 - 30 * q^96 + 20 * q^97 + q^98 - 31 * q^99

Introduction

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