LMFDB - Embedded newform 875.2.a.h.1.6

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:	$6$
Coefficient field:	6.6.1241125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{4} - 2x^{3} + 11x^{2} + 7x + 1 $ x^6 - 7x^4 - 2x^3 + 11x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.72131$ 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)

$f(q)$	$=$	\(q+2.55803 q^{2} -0.353843 q^{3} +4.54354 q^{4} -0.905143 q^{6} -1.00000 q^{7} +6.50646 q^{8} -2.87480 q^{9} +5.69412 q^{11} -1.60770 q^{12} +6.47971 q^{13} -2.55803 q^{14} +7.55666 q^{16} -2.19206 q^{17} -7.35382 q^{18} -5.09571 q^{19} +0.353843 q^{21} +14.5657 q^{22} +6.78323 q^{23} -2.30226 q^{24} +16.5753 q^{26} +2.07876 q^{27} -4.54354 q^{28} +1.62794 q^{29} -8.51029 q^{31} +6.31728 q^{32} -2.01482 q^{33} -5.60737 q^{34} -13.0617 q^{36} -1.22831 q^{37} -13.0350 q^{38} -2.29280 q^{39} -1.53221 q^{41} +0.905143 q^{42} -6.00854 q^{43} +25.8714 q^{44} +17.3517 q^{46} -2.06257 q^{47} -2.67387 q^{48} +1.00000 q^{49} +0.775646 q^{51} +29.4408 q^{52} -6.26731 q^{53} +5.31753 q^{54} -6.50646 q^{56} +1.80308 q^{57} +4.16432 q^{58} -6.61824 q^{59} -7.69499 q^{61} -21.7696 q^{62} +2.87480 q^{63} +1.04650 q^{64} -5.15399 q^{66} +6.29996 q^{67} -9.95971 q^{68} -2.40020 q^{69} +0.839088 q^{71} -18.7047 q^{72} -4.62741 q^{73} -3.14205 q^{74} -23.1525 q^{76} -5.69412 q^{77} -5.86506 q^{78} -8.90370 q^{79} +7.88883 q^{81} -3.91945 q^{82} +3.59494 q^{83} +1.60770 q^{84} -15.3701 q^{86} -0.576035 q^{87} +37.0485 q^{88} -0.716374 q^{89} -6.47971 q^{91} +30.8199 q^{92} +3.01131 q^{93} -5.27613 q^{94} -2.23533 q^{96} +6.57786 q^{97} +2.55803 q^{98} -16.3694 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 5 q^{3} + 4 q^{4} - 2 q^{6} - 6 q^{7} + 12 q^{8} + 7 q^{9} + q^{11} + 11 q^{12} + 22 q^{13} - 2 q^{14} + 8 q^{16} + 4 q^{17} - 14 q^{18} - 17 q^{19} - 5 q^{21} + 9 q^{22} + 3 q^{23} + 7 q^{24}+ \cdots + 12 q^{99}+O(q^{100}) $ 6 * q + 2 * q^2 + 5 * q^3 + 4 * q^4 - 2 * q^6 - 6 * q^7 + 12 * q^8 + 7 * q^9 + q^11 + 11 * q^12 + 22 * q^13 - 2 * q^14 + 8 * q^16 + 4 * q^17 - 14 * q^18 - 17 * q^19 - 5 * q^21 + 9 * q^22 + 3 * q^23 + 7 * q^24 + 16 * q^26 + 26 * q^27 - 4 * q^28 + 7 * q^29 - 11 * q^31 + q^32 + 21 * q^33 - 4 * q^34 + 10 * q^36 + 33 * q^37 + 15 * q^39 + 3 * q^41 + 2 * q^42 + 2 * q^43 + 26 * q^44 - 8 * q^46 - 6 * q^47 - 20 * q^48 + 6 * q^49 + q^51 + 47 * q^52 + 33 * q^53 - 26 * q^54 - 12 * q^56 - 3 * q^57 + 43 * q^58 - 21 * q^59 - 19 * q^61 - 37 * q^62 - 7 * q^63 - 22 * q^64 - 35 * q^66 + 2 * q^67 - 30 * q^68 - 2 * q^69 + q^71 - 10 * q^72 + 16 * q^73 - 29 * q^74 - 4 * q^76 - q^77 - q^78 + 22 * q^79 + 14 * q^81 - 13 * q^82 + 3 * q^83 - 11 * q^84 + 23 * q^86 + 28 * q^88 - 8 * q^89 - 22 * q^91 - 2 * q^92 - 16 * q^93 - 26 * q^94 - 4 * q^96 + 16 * q^97 + 2 * q^98 + 12 * 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 $

$2$	2.55803	1.80880	0.904402	−	0.426682i	$-0.140318\pi$
$2$	2.55803	1.80880	0.904402	+	0.426682i	$0.140318\pi$
$3$	−0.353843	−0.204291	−0.102146	−	0.994769i	$-0.532571\pi$
$3$	−0.353843	−0.204291	−0.102146	+	0.994769i	$0.532571\pi$
$4$	4.54354	2.27177
$5$	0	0
$5$	0	0
$6$	−0.905143	−0.369523
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	6.50646	2.30038
$9$	−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	$0.171446\pi$
$12$	−1.60770	−0.464103
$13$	6.47971	1.79715	0.898574	−	0.438822i	$-0.144604\pi$
$13$	6.47971	1.79715	0.898574	+	0.438822i	$0.144604\pi$
$14$	−2.55803	−0.683663
$15$	0	0
$16$	7.55666	1.88916
$17$	−2.19206	−0.531653	−0.265826	−	0.964021i	$-0.585645\pi$
$17$	−2.19206	−0.531653	−0.265826	+	0.964021i	$0.585645\pi$
$18$	−7.35382	−1.73331
$19$	−5.09571	−1.16904	−0.584518	−	0.811381i	$-0.698716\pi$
$19$	−5.09571	−1.16904	−0.584518	+	0.811381i	$0.698716\pi$
$20$	0	0
$21$	0.353843	0.0772149
$22$	14.5657	3.10543
$23$	6.78323	1.41440	0.707201	−	0.707013i	$-0.249958\pi$
$23$	6.78323	1.41440	0.707201	+	0.707013i	$0.249958\pi$
$24$	−2.30226	−0.469948
$25$	0	0
$26$	16.5753	3.25069
$27$	2.07876	0.400057
$28$	−4.54354	−0.858648
$29$	1.62794	0.302301	0.151150	−	0.988511i	$-0.451702\pi$
$29$	1.62794	0.302301	0.151150	+	0.988511i	$0.451702\pi$
$30$	0	0
$31$	−8.51029	−1.52849	−0.764246	−	0.644925i	$-0.776889\pi$
$31$	−8.51029	−1.52849	−0.764246	+	0.644925i	$0.776889\pi$
$32$	6.31728	1.11675
$33$	−2.01482	−0.350736
$34$	−5.60737	−0.961656
$35$	0	0
$36$	−13.0617	−2.17696
$37$	−1.22831	−0.201932	−0.100966	−	0.994890i	$-0.532193\pi$
$37$	−1.22831	−0.201932	−0.100966	+	0.994890i	$0.532193\pi$
$38$	−13.0350	−2.11455
$39$	−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	$0.538176\pi$
$42$	0.905143	0.139667
$43$	−6.00854	−0.916294	−0.458147	−	0.888876i	$-0.651487\pi$
$43$	−6.00854	−0.916294	−0.458147	+	0.888876i	$0.651487\pi$
$44$	25.8714	3.90026
$45$	0	0
$46$	17.3517	2.55837
$47$	−2.06257	−0.300857	−0.150429	−	0.988621i	$-0.548065\pi$
$47$	−2.06257	−0.300857	−0.150429	+	0.988621i	$0.548065\pi$
$48$	−2.67387	−0.385940
$49$	1.00000	0.142857
$50$	0	0
$51$	0.775646	0.108612
$52$	29.4408	4.08270
$53$	−6.26731	−0.860881	−0.430440	−	0.902619i	$-0.641642\pi$
$53$	−6.26731	−0.860881	−0.430440	+	0.902619i	$0.641642\pi$
$54$	5.31753	0.723624
$55$	0	0
$56$	−6.50646	−0.869462
$57$	1.80308	0.238824
$58$	4.16432	0.546802
$59$	−6.61824	−0.861621	−0.430811	−	0.902442i	$-0.641772\pi$
$59$	−6.61824	−0.861621	−0.430811	+	0.902442i	$0.641772\pi$
$60$	0	0
$61$	−7.69499	−0.985242	−0.492621	−	0.870244i	$-0.663961\pi$
$61$	−7.69499	−0.985242	−0.492621	+	0.870244i	$0.663961\pi$
$62$	−21.7696	−2.76474
$63$	2.87480	0.362190
$64$	1.04650	0.130812
$65$	0	0
$66$	−5.15399	−0.634412
$67$	6.29996	0.769663	0.384832	−	0.922987i	$-0.374260\pi$
$67$	6.29996	0.769663	0.384832	+	0.922987i	$0.374260\pi$
$68$	−9.95971	−1.20779
$69$	−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	$0.484145\pi$
$72$	−18.7047	−2.20437
$73$	−4.62741	−0.541598	−0.270799	−	0.962636i	$-0.587288\pi$
$73$	−4.62741	−0.541598	−0.270799	+	0.962636i	$0.587288\pi$
$74$	−3.14205	−0.365256
$75$	0	0
$76$	−23.1525	−2.65578
$77$	−5.69412	−0.648905
$78$	−5.86506	−0.664087
$79$	−8.90370	−1.00174	−0.500872	−	0.865521i	$-0.666987\pi$
$79$	−8.90370	−1.00174	−0.500872	+	0.865521i	$0.666987\pi$
$80$	0	0
$81$	7.88883	0.876537
$82$	−3.91945	−0.432831
$83$	3.59494	0.394596	0.197298	−	0.980344i	$-0.436783\pi$
$83$	3.59494	0.394596	0.197298	+	0.980344i	$0.436783\pi$
$84$	1.60770	0.175414
$85$	0	0
$86$	−15.3701	−1.65740
$87$	−0.576035	−0.0617574
$88$	37.0485	3.94938
$89$	−0.716374	−0.0759355	−0.0379677	−	0.999279i	$-0.512088\pi$
$89$	−0.716374	−0.0759355	−0.0379677	+	0.999279i	$0.512088\pi$
$90$	0	0
$91$	−6.47971	−0.679258
$92$	30.8199	3.21319
$93$	3.01131	0.312258
$94$	−5.27613	−0.544191
$95$	0	0
$96$	−2.23533	−0.228142
$97$	6.57786	0.667880	0.333940	−	0.942594i	$-0.391622\pi$
$97$	6.57786	0.667880	0.333940	+	0.942594i	$0.391622\pi$
$98$	2.55803	0.258400
$99$	−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	$0.669042\pi$
$102$	1.98413	0.196458
$103$	12.4409	1.22584	0.612920	−	0.790145i	$-0.289995\pi$
$103$	12.4409	1.22584	0.612920	+	0.790145i	$0.289995\pi$
$104$	42.1599	4.13412
$105$	0	0
$106$	−16.0320	−1.55716
$107$	1.70563	0.164890	0.0824450	−	0.996596i	$-0.473727\pi$
$107$	1.70563	0.164890	0.0824450	+	0.996596i	$0.473727\pi$
$108$	9.44491	0.908837
$109$	−12.1453	−1.16331	−0.581653	−	0.813437i	$-0.697594\pi$
$109$	−12.1453	−1.16331	−0.581653	+	0.813437i	$0.697594\pi$
$110$	0	0
$111$	0.434628	0.0412530
$112$	−7.55666	−0.714037
$113$	12.7733	1.20161	0.600806	−	0.799395i	$-0.294846\pi$
$113$	12.7733	1.20161	0.600806	+	0.799395i	$0.294846\pi$
$114$	4.61234	0.431985
$115$	0	0
$116$	7.39660	0.686757
$117$	−18.6278	−1.72214
$118$	−16.9297	−1.55850
$119$	2.19206	0.200946
$120$	0	0
$121$	21.4229	1.94754
$122$	−19.6840	−1.78211
$123$	0.542163	0.0488851
$124$	−38.6668	−3.47238
$125$	0	0
$126$	7.35382	0.655131
$127$	3.87656	0.343989	0.171994	−	0.985098i	$-0.444979\pi$
$127$	3.87656	0.343989	0.171994	+	0.985098i	$0.444979\pi$
$128$	−9.95759	−0.880135
$129$	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	$0.478344\pi$
$132$	−9.15443	−0.796791
$133$	5.09571	0.441854
$134$	16.1155	1.39217
$135$	0	0
$136$	−14.2626	−1.22300
$137$	11.6597	0.996159	0.498080	−	0.867131i	$-0.334039\pi$
$137$	11.6597	0.996159	0.498080	+	0.867131i	$0.334039\pi$
$138$	−6.13979	−0.522654
$139$	9.16488	0.777355	0.388677	−	0.921374i	$-0.372932\pi$
$139$	9.16488	0.777355	0.388677	+	0.921374i	$0.372932\pi$
$140$	0	0
$141$	0.729827	0.0614625
$142$	2.14642	0.180123
$143$	36.8962	3.08542
$144$	−21.7238	−1.81032
$145$	0	0
$146$	−11.8371	−0.979644
$147$	−0.353843	−0.0291845
$148$	−5.58085	−0.458743
$149$	4.36230	0.357373	0.178687	−	0.983906i	$-0.442815\pi$
$149$	4.36230	0.357373	0.178687	+	0.983906i	$0.442815\pi$
$150$	0	0
$151$	−10.3209	−0.839902	−0.419951	−	0.907547i	$-0.637953\pi$
$151$	−10.3209	−0.839902	−0.419951	+	0.907547i	$0.637953\pi$
$152$	−33.1550	−2.68922
$153$	6.30173	0.509464
$154$	−14.5657	−1.17374
$155$	0	0
$156$	−10.4174	−0.834061
$157$	23.8740	1.90535	0.952677	−	0.303985i	$-0.0983173\pi$
$157$	23.8740	1.90535	0.952677	+	0.303985i	$0.0983173\pi$
$158$	−22.7760	−1.81196
$159$	2.21764	0.175871
$160$	0	0
$161$	−6.78323	−0.534593
$162$	20.1799	1.58548
$163$	−20.2290	−1.58446	−0.792231	−	0.610222i	$-0.791080\pi$
$163$	−20.2290	−1.58446	−0.792231	+	0.610222i	$0.791080\pi$
$164$	−6.96166	−0.543614
$165$	0	0
$166$	9.19599	0.713747
$167$	−8.90008	−0.688709	−0.344354	−	0.938840i	$-0.611902\pi$
$167$	−8.90008	−0.688709	−0.344354	+	0.938840i	$0.611902\pi$
$168$	2.30226	0.177624
$169$	28.9866	2.22974
$170$	0	0
$171$	14.6491	1.12025
$172$	−27.3000	−2.08161
$173$	−15.0386	−1.14337	−0.571683	−	0.820474i	$-0.693709\pi$
$173$	−15.0386	−1.14337	−0.571683	+	0.820474i	$0.693709\pi$
$174$	−1.47352	−0.111707
$175$	0	0
$176$	43.0285	3.24339
$177$	2.34182	0.176022
$178$	−1.83251	−0.137352
$179$	1.62454	0.121424	0.0607120	−	0.998155i	$-0.480663\pi$
$179$	1.62454	0.121424	0.0607120	+	0.998155i	$0.480663\pi$
$180$	0	0
$181$	7.78605	0.578733	0.289366	−	0.957218i	$-0.406555\pi$
$181$	7.78605	0.578733	0.289366	+	0.957218i	$0.406555\pi$
$182$	−16.5753	−1.22864
$183$	2.72282	0.201276
$184$	44.1348	3.25366
$185$	0	0
$186$	7.70302	0.564813
$187$	−12.4819	−0.912763
$188$	−9.37137	−0.683478
$189$	−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	$0.705709\pi$
$192$	−0.370295	−0.0267238
$193$	−3.76432	−0.270962	−0.135481	−	0.990780i	$-0.543258\pi$
$193$	−3.76432	−0.270962	−0.135481	+	0.990780i	$0.543258\pi$
$194$	16.8264	1.20806
$195$	0	0
$196$	4.54354	0.324538
$197$	4.86416	0.346557	0.173279	−	0.984873i	$-0.444564\pi$
$197$	4.86416	0.346557	0.173279	+	0.984873i	$0.444564\pi$
$198$	−41.8735	−2.97582
$199$	7.04266	0.499241	0.249621	−	0.968344i	$-0.419694\pi$
$199$	7.04266	0.499241	0.249621	+	0.968344i	$0.419694\pi$
$200$	0	0
$201$	−2.22920	−0.157236
$202$	−26.0395	−1.83213
$203$	−1.62794	−0.114259
$204$	3.52418	0.246742
$205$	0	0
$206$	31.8243	2.21730
$207$	−19.5004	−1.35537
$208$	48.9649	3.39511
$209$	−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	$0.370192\pi$
$212$	−28.4757	−1.95572
$213$	−0.296906	−0.0203436
$214$	4.36307	0.298253
$215$	0	0
$216$	13.5253	0.920282
$217$	8.51029	0.577716
$218$	−31.0680	−2.10419
$219$	1.63738	0.110644
$220$	0	0
$221$	−14.2039	−0.955459
$222$	1.11179	0.0746186
$223$	−0.639877	−0.0428493	−0.0214247	−	0.999770i	$-0.506820\pi$
$223$	−0.639877	−0.0428493	−0.0214247	+	0.999770i	$0.506820\pi$
$224$	−6.31728	−0.422091
$225$	0	0
$226$	32.6746	2.17348
$227$	−10.7993	−0.716774	−0.358387	−	0.933573i	$-0.616673\pi$
$227$	−10.7993	−0.716774	−0.358387	+	0.933573i	$0.616673\pi$
$228$	8.19237	0.542553
$229$	−20.2209	−1.33623	−0.668116	−	0.744057i	$-0.732899\pi$
$229$	−20.2209	−1.33623	−0.668116	+	0.744057i	$0.732899\pi$
$230$	0	0
$231$	2.01482	0.132566
$232$	10.5921	0.695406
$233$	−7.15167	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$233$	−7.15167	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$234$	−47.6506	−3.11502
$235$	0	0
$236$	−30.0702	−1.95740
$237$	3.15051	0.204648
$238$	5.60737	0.363472
$239$	−12.7774	−0.826500	−0.413250	−	0.910618i	$-0.635606\pi$
$239$	−12.7774	−0.826500	−0.413250	+	0.910618i	$0.635606\pi$
$240$	0	0
$241$	26.3455	1.69706	0.848530	−	0.529147i	$-0.177488\pi$
$241$	26.3455	1.69706	0.848530	+	0.529147i	$0.177488\pi$
$242$	54.8006	3.52272
$243$	−9.02768	−0.579126
$244$	−34.9625	−2.23824
$245$	0	0
$246$	1.38687	0.0884236
$247$	−33.0187	−2.10093
$248$	−55.3718	−3.51611
$249$	−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	$0.573695\pi$
$252$	13.0617	0.822812
$253$	38.6245	2.42830
$254$	9.91637	0.622208
$255$	0	0
$256$	−27.5648	−1.72280
$257$	29.7439	1.85537	0.927686	−	0.373362i	$-0.121795\pi$
$257$	29.7439	1.85537	0.927686	+	0.373362i	$0.121795\pi$
$258$	5.43859	0.338592
$259$	1.22831	0.0763232
$260$	0	0
$261$	−4.67999	−0.289684
$262$	3.98082	0.245936
$263$	−22.4784	−1.38608	−0.693040	−	0.720899i	$-0.743729\pi$
$263$	−22.4784	−1.38608	−0.693040	+	0.720899i	$0.743729\pi$
$264$	−13.1094	−0.806825
$265$	0	0
$266$	13.0350	0.799227
$267$	0.253484	0.0155130
$268$	28.6241	1.74850
$269$	18.3297	1.11758	0.558789	−	0.829310i	$-0.311266\pi$
$269$	18.3297	1.11758	0.558789	+	0.829310i	$0.311266\pi$
$270$	0	0
$271$	16.1329	0.980005	0.490002	−	0.871721i	$-0.336996\pi$
$271$	16.1329	0.980005	0.490002	+	0.871721i	$0.336996\pi$
$272$	−16.5647	−1.00438
$273$	2.29280	0.138767
$274$	29.8260	1.80186
$275$	0	0
$276$	−10.9054	−0.656428
$277$	3.26856	0.196389	0.0981945	−	0.995167i	$-0.468693\pi$
$277$	3.26856	0.196389	0.0981945	+	0.995167i	$0.468693\pi$
$278$	23.4441	1.40608
$279$	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	$0.536695\pi$
$282$	1.86692	0.111174
$283$	22.0923	1.31325	0.656626	−	0.754217i	$-0.271983\pi$
$283$	22.0923	1.31325	0.656626	+	0.754217i	$0.271983\pi$
$284$	3.81243	0.226226
$285$	0	0
$286$	94.3817	5.58091
$287$	1.53221	0.0904436
$288$	−18.1609	−1.07014
$289$	−12.1949	−0.717345
$290$	0	0
$291$	−2.32753	−0.136442
$292$	−21.0248	−1.23038
$293$	19.8463	1.15943	0.579715	−	0.814819i	$-0.303164\pi$
$293$	19.8463	1.15943	0.579715	+	0.814819i	$0.303164\pi$
$294$	−0.905143	−0.0527890
$295$	0	0
$296$	−7.99192	−0.464521
$297$	11.8367	0.686834
$298$	11.1589	0.646418
$299$	43.9533	2.54189
$300$	0	0
$301$	6.00854	0.346327
$302$	−26.4012	−1.51922
$303$	3.60195	0.206926
$304$	−38.5065	−2.20850
$305$	0	0
$306$	16.1200	0.921521
$307$	−28.9241	−1.65079	−0.825393	−	0.564559i	$-0.809046\pi$
$307$	−28.9241	−1.65079	−0.825393	+	0.564559i	$0.809046\pi$
$308$	−25.8714	−1.47416
$309$	−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	$0.238813\pi$
$312$	−14.9180	−0.844566
$313$	−4.34404	−0.245540	−0.122770	−	0.992435i	$-0.539178\pi$
$313$	−4.34404	−0.245540	−0.122770	+	0.992435i	$0.539178\pi$
$314$	61.0705	3.44641
$315$	0	0
$316$	−40.4543	−2.27573
$317$	11.9946	0.673685	0.336842	−	0.941561i	$-0.390641\pi$
$317$	11.9946	0.673685	0.336842	+	0.941561i	$0.390641\pi$
$318$	5.67281	0.318115
$319$	9.26967	0.519002
$320$	0	0
$321$	−0.603527	−0.0336856
$322$	−17.3517	−0.966974
$323$	11.1701	0.621521
$324$	35.8432	1.99129
$325$	0	0
$326$	−51.7466	−2.86598
$327$	4.29752	0.237653
$328$	−9.96927	−0.550461
$329$	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	$0.746319\pi$
$332$	16.3338	0.896431
$333$	3.53113	0.193505
$334$	−22.7667	−1.24574
$335$	0	0
$336$	2.67387	0.145872
$337$	6.65856	0.362715	0.181358	−	0.983417i	$-0.441951\pi$
$337$	6.65856	0.362715	0.181358	+	0.983417i	$0.441951\pi$
$338$	74.1487	4.03316
$339$	−4.51975	−0.245479
$340$	0	0
$341$	−48.4585	−2.62418
$342$	37.4729	2.02630
$343$	−1.00000	−0.0539949
$344$	−39.0943	−2.10782
$345$	0	0
$346$	−38.4693	−2.06812
$347$	−1.27571	−0.0684837	−0.0342419	−	0.999414i	$-0.510902\pi$
$347$	−1.27571	−0.0684837	−0.0342419	+	0.999414i	$0.510902\pi$
$348$	−2.61724	−0.140299
$349$	23.4771	1.25670	0.628351	−	0.777930i	$-0.283730\pi$
$349$	23.4771	1.25670	0.628351	+	0.777930i	$0.283730\pi$
$350$	0	0
$351$	13.4697	0.718961
$352$	35.9713	1.91728
$353$	−17.6711	−0.940537	−0.470268	−	0.882523i	$-0.655843\pi$
$353$	−17.6711	−0.940537	−0.470268	+	0.882523i	$0.655843\pi$
$354$	5.99045	0.318389
$355$	0	0
$356$	−3.25487	−0.172508
$357$	−0.775646	−0.0410515
$358$	4.15564	0.219632
$359$	32.7103	1.72638	0.863191	−	0.504878i	$-0.168463\pi$
$359$	32.7103	1.72638	0.863191	+	0.504878i	$0.168463\pi$
$360$	0	0
$361$	6.96623	0.366644
$362$	19.9170	1.04681
$363$	−7.58036	−0.397866
$364$	−29.4408	−1.54312
$365$	0	0
$366$	6.96506	0.364070
$367$	2.73145	0.142580	0.0712902	−	0.997456i	$-0.477288\pi$
$367$	2.73145	0.142580	0.0712902	+	0.997456i	$0.477288\pi$
$368$	51.2586	2.67204
$369$	4.40479	0.229304
$370$	0	0
$371$	6.26731	0.325382
$372$	13.6820	0.709378
$373$	8.37626	0.433706	0.216853	−	0.976204i	$-0.430421\pi$
$373$	8.37626	0.433706	0.216853	+	0.976204i	$0.430421\pi$
$374$	−31.9290	−1.65101
$375$	0	0
$376$	−13.4200	−0.692085
$377$	10.5486	0.543279
$378$	−5.31753	−0.273504
$379$	−12.6454	−0.649548	−0.324774	−	0.945792i	$-0.605288\pi$
$379$	−12.6454	−0.649548	−0.324774	+	0.945792i	$0.605288\pi$
$380$	0	0
$381$	−1.37169	−0.0702740
$382$	−42.5789	−2.17853
$383$	−20.3777	−1.04125	−0.520625	−	0.853785i	$-0.674301\pi$
$383$	−20.3777	−1.04125	−0.520625	+	0.853785i	$0.674301\pi$
$384$	3.52342	0.179804
$385$	0	0
$386$	−9.62926	−0.490116
$387$	17.2733	0.878053
$388$	29.8867	1.51727
$389$	17.4785	0.886196	0.443098	−	0.896473i	$-0.353879\pi$
$389$	17.4785	0.886196	0.443098	+	0.896473i	$0.353879\pi$
$390$	0	0
$391$	−14.8693	−0.751971
$392$	6.50646	0.328626
$393$	−0.550652	−0.0277767
$394$	12.4427	0.626854
$395$	0	0
$396$	−74.3750	−3.73749
$397$	22.0398	1.10615	0.553073	−	0.833133i	$-0.313455\pi$
$397$	22.0398	1.10615	0.553073	+	0.833133i	$0.313455\pi$
$398$	18.0154	0.903029
$399$	−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	$0.564411\pi$
$402$	−5.70237	−0.284408
$403$	−55.1442	−2.74693
$404$	−46.2510	−2.30107
$405$	0	0
$406$	−4.16432	−0.206672
$407$	−6.99412	−0.346685
$408$	5.04671	0.249849
$409$	23.3486	1.15451	0.577256	−	0.816563i	$-0.304123\pi$
$409$	23.3486	1.15451	0.577256	+	0.816563i	$0.304123\pi$
$410$	0	0
$411$	−4.12572	−0.203507
$412$	56.5258	2.78483
$413$	6.61824	0.325662
$414$	−49.8827	−2.45160
$415$	0	0
$416$	40.9341	2.00696
$417$	−3.24293	−0.158807
$418$	−74.2227	−3.63035
$419$	−13.0497	−0.637518	−0.318759	−	0.947836i	$-0.603266\pi$
$419$	−13.0497	−0.637518	−0.318759	+	0.947836i	$0.603266\pi$
$420$	0	0
$421$	38.2475	1.86407	0.932034	−	0.362372i	$-0.118033\pi$
$421$	38.2475	1.86407	0.932034	+	0.362372i	$0.118033\pi$
$422$	29.4731	1.43473
$423$	5.92947	0.288301
$424$	−40.7779	−1.98035
$425$	0	0
$426$	−0.759494	−0.0367976
$427$	7.69499	0.372386
$428$	7.74961	0.374592
$429$	−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	$0.444715\pi$
$432$	15.7085	0.755773
$433$	33.9429	1.63119	0.815595	−	0.578623i	$-0.196410\pi$
$433$	33.9429	1.63119	0.815595	+	0.578623i	$0.196410\pi$
$434$	21.7696	1.04497
$435$	0	0
$436$	−55.1825	−2.64276
$437$	−34.5654	−1.65349
$438$	4.18847	0.200133
$439$	−1.11980	−0.0534451	−0.0267226	−	0.999643i	$-0.508507\pi$
$439$	−1.11980	−0.0534451	−0.0267226	+	0.999643i	$0.508507\pi$
$440$	0	0
$441$	−2.87480	−0.136895
$442$	−36.3341	−1.72824
$443$	21.9887	1.04472	0.522358	−	0.852727i	$-0.325053\pi$
$443$	21.9887	1.04472	0.522358	+	0.852727i	$0.325053\pi$
$444$	1.97475	0.0937173
$445$	0	0
$446$	−1.63683	−0.0775060
$447$	−1.54357	−0.0730083
$448$	−1.04650	−0.0494423
$449$	17.5039	0.826058	0.413029	−	0.910718i	$-0.364471\pi$
$449$	17.5039	0.826058	0.413029	+	0.910718i	$0.364471\pi$
$450$	0	0
$451$	−8.72459	−0.410825
$452$	58.0360	2.72978
$453$	3.65197	0.171585
$454$	−27.6249	−1.29650
$455$	0	0
$456$	11.7317	0.549386
$457$	5.33726	0.249667	0.124833	−	0.992178i	$-0.460160\pi$
$457$	5.33726	0.249667	0.124833	+	0.992178i	$0.460160\pi$
$458$	−51.7256	−2.41698
$459$	−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	$0.446899\pi$
$462$	5.15399	0.239785
$463$	−15.6224	−0.726036	−0.363018	−	0.931782i	$-0.618254\pi$
$463$	−15.6224	−0.726036	−0.363018	+	0.931782i	$0.618254\pi$
$464$	12.3018	0.571096
$465$	0	0
$466$	−18.2942	−0.847463
$467$	−28.1640	−1.30328	−0.651638	−	0.758530i	$-0.725918\pi$
$467$	−28.1640	−1.30328	−0.651638	+	0.758530i	$0.725918\pi$
$468$	−84.6363	−3.91231
$469$	−6.29996	−0.290905
$470$	0	0
$471$	−8.44766	−0.389247
$472$	−43.0613	−1.98206
$473$	−34.2133	−1.57313
$474$	8.05912	0.370168
$475$	0	0
$476$	9.95971	0.456503
$477$	18.0172	0.824952
$478$	−32.6850	−1.49498
$479$	−26.4738	−1.20962	−0.604809	−	0.796371i	$-0.706750\pi$
$479$	−26.4738	−1.20962	−0.604809	+	0.796371i	$0.706750\pi$
$480$	0	0
$481$	−7.95906	−0.362902
$482$	67.3926	3.06965
$483$	2.40020	0.109213
$484$	97.3360	4.42436
$485$	0	0
$486$	−23.0931	−1.04752
$487$	−10.4405	−0.473106	−0.236553	−	0.971619i	$-0.576018\pi$
$487$	−10.4405	−0.473106	−0.236553	+	0.971619i	$0.576018\pi$
$488$	−50.0671	−2.26643
$489$	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	$0.532618\pi$
$492$	2.46334	0.111056
$493$	−3.56854	−0.160719
$494$	−84.4629	−3.80017
$495$	0	0
$496$	−64.3093	−2.88757
$497$	−0.839088	−0.0376382
$498$	−3.25394	−0.145812
$499$	−1.36104	−0.0609283	−0.0304642	−	0.999536i	$-0.509699\pi$
$499$	−1.36104	−0.0609283	−0.0304642	+	0.999536i	$0.509699\pi$
$500$	0	0
$501$	3.14923	0.140697
$502$	−18.5984	−0.830089
$503$	27.9936	1.24817	0.624086	−	0.781356i	$-0.285472\pi$
$503$	27.9936	1.24817	0.624086	+	0.781356i	$0.285472\pi$
$504$	18.7047	0.833175
$505$	0	0
$506$	98.8028	4.39232
$507$	−10.2567	−0.455517
$508$	17.6133	0.781463
$509$	1.40214	0.0621488	0.0310744	−	0.999517i	$-0.490107\pi$
$509$	1.40214	0.0621488	0.0310744	+	0.999517i	$0.490107\pi$
$510$	0	0
$511$	4.62741	0.204705
$512$	−50.5966	−2.23608
$513$	−10.5927	−0.467681
$514$	76.0858	3.35600
$515$	0	0
$516$	9.65993	0.425255
$517$	−11.7445	−0.516524
$518$	3.14205	0.138054
$519$	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	$0.670154\pi$
$522$	−11.9716	−0.523981
$523$	−3.80015	−0.166169	−0.0830845	−	0.996543i	$-0.526477\pi$
$523$	−3.80015	−0.166169	−0.0830845	+	0.996543i	$0.526477\pi$
$524$	7.07067	0.308884
$525$	0	0
$526$	−57.5006	−2.50715
$527$	18.6551	0.812628
$528$	−15.2253	−0.662598
$529$	23.0122	1.00053
$530$	0	0
$531$	19.0261	0.825662
$532$	23.1525	1.00379
$533$	−9.92828	−0.430042
$534$	0.648420	0.0280599
$535$	0	0
$536$	40.9904	1.77052
$537$	−0.574833	−0.0248059
$538$	46.8879	2.02148
$539$	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	$0.611884\pi$
$542$	41.2685	1.77264
$543$	−2.75504	−0.118230
$544$	−13.8479	−0.593722
$545$	0	0
$546$	5.86506	0.251001
$547$	−30.2390	−1.29293	−0.646464	−	0.762944i	$-0.723753\pi$
$547$	−30.2390	−1.29293	−0.646464	+	0.762944i	$0.723753\pi$
$548$	52.9765	2.26304
$549$	22.1215	0.944123
$550$	0	0
$551$	−8.29550	−0.353400
$552$	−15.6168	−0.664695
$553$	8.90370	0.378624
$554$	8.36110	0.355229
$555$	0	0
$556$	41.6410	1.76597
$557$	24.2216	1.02630	0.513152	−	0.858298i	$-0.328478\pi$
$557$	24.2216	1.02630	0.513152	+	0.858298i	$0.328478\pi$
$558$	62.5831	2.64936
$559$	−38.9336	−1.64672
$560$	0	0
$561$	4.41662	0.186470
$562$	−9.86467	−0.416116
$563$	11.9124	0.502048	0.251024	−	0.967981i	$-0.419233\pi$
$563$	11.9124	0.502048	0.251024	+	0.967981i	$0.419233\pi$
$564$	3.31600	0.139629
$565$	0	0
$566$	56.5129	2.37541
$567$	−7.88883	−0.331300
$568$	5.45949	0.229075
$569$	22.2725	0.933712	0.466856	−	0.884333i	$-0.345387\pi$
$569$	22.2725	0.933712	0.466856	+	0.884333i	$0.345387\pi$
$570$	0	0
$571$	19.6713	0.823216	0.411608	−	0.911361i	$-0.364967\pi$
$571$	19.6713	0.823216	0.411608	+	0.911361i	$0.364967\pi$
$572$	167.639	7.00935
$573$	5.88978	0.246049
$574$	3.91945	0.163595
$575$	0	0
$576$	−3.00846	−0.125353
$577$	22.5166	0.937378	0.468689	−	0.883363i	$-0.344726\pi$
$577$	22.5166	0.937378	0.468689	+	0.883363i	$0.344726\pi$
$578$	−31.1949	−1.29754
$579$	1.33198	0.0553551
$580$	0	0
$581$	−3.59494	−0.149143
$582$	−5.95390	−0.246797
$583$	−35.6868	−1.47799
$584$	−30.1081	−1.24588
$585$	0	0
$586$	50.7674	2.09718
$587$	−28.2290	−1.16514	−0.582568	−	0.812782i	$-0.697952\pi$
$587$	−28.2290	−1.16514	−0.582568	+	0.812782i	$0.697952\pi$
$588$	−1.60770	−0.0663004
$589$	43.3659	1.78686
$590$	0	0
$591$	−1.72115	−0.0707987
$592$	−9.28189	−0.381483
$593$	−27.2560	−1.11927	−0.559635	−	0.828739i	$-0.689059\pi$
$593$	−27.2560	−1.11927	−0.559635	+	0.828739i	$0.689059\pi$
$594$	30.2786	1.24235
$595$	0	0
$596$	19.8203	0.811870
$597$	−2.49200	−0.101991
$598$	112.434	4.59777
$599$	19.7263	0.805997	0.402998	−	0.915201i	$-0.367968\pi$
$599$	19.7263	0.805997	0.402998	+	0.915201i	$0.367968\pi$
$600$	0	0
$601$	−39.6721	−1.61826	−0.809130	−	0.587629i	$-0.800061\pi$
$601$	−39.6721	−1.61826	−0.809130	+	0.587629i	$0.800061\pi$
$602$	15.3701	0.626437
$603$	−18.1111	−0.737541
$604$	−46.8933	−1.90806
$605$	0	0
$606$	9.21390	0.374289
$607$	23.8572	0.968336	0.484168	−	0.874975i	$-0.339122\pi$
$607$	23.8572	0.968336	0.484168	+	0.874975i	$0.339122\pi$
$608$	−32.1910	−1.30552
$609$	0.576035	0.0233421
$610$	0	0
$611$	−13.3649	−0.540685
$612$	28.6321	1.15739
$613$	−20.6564	−0.834302	−0.417151	−	0.908837i	$-0.636971\pi$
$613$	−20.6564	−0.834302	−0.417151	+	0.908837i	$0.636971\pi$
$614$	−73.9889	−2.98595
$615$	0	0
$616$	−37.0485	−1.49273
$617$	−48.1688	−1.93920	−0.969602	−	0.244688i	$-0.921314\pi$
$617$	−48.1688	−1.93920	−0.969602	+	0.244688i	$0.921314\pi$
$618$	−11.2608	−0.452976
$619$	−19.1752	−0.770716	−0.385358	−	0.922767i	$-0.625922\pi$
$619$	−19.1752	−0.770716	−0.385358	+	0.922767i	$0.625922\pi$
$620$	0	0
$621$	14.1007	0.565841
$622$	65.9994	2.64633
$623$	0.716374	0.0287009
$624$	−17.3259	−0.693591
$625$	0	0
$626$	−11.1122	−0.444133
$627$	10.2670	0.410023
$628$	108.472	4.32852
$629$	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	$0.289381\pi$
$632$	−57.9315	−2.30439
$633$	−4.07690	−0.162042
$634$	30.6826	1.21856
$635$	0	0
$636$	10.0759	0.399537
$637$	6.47971	0.256735
$638$	23.7121	0.938772
$639$	−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	$0.420708\pi$
$642$	−1.54384	−0.0609306
$643$	32.0891	1.26547	0.632735	−	0.774369i	$-0.281932\pi$
$643$	32.0891	1.26547	0.632735	+	0.774369i	$0.281932\pi$
$644$	−30.8199	−1.21447
$645$	0	0
$646$	28.5735	1.12421
$647$	−28.7155	−1.12892	−0.564462	−	0.825459i	$-0.690916\pi$
$647$	−28.7155	−1.12892	−0.564462	+	0.825459i	$0.690916\pi$
$648$	51.3283	2.01637
$649$	−37.6850	−1.47927
$650$	0	0
$651$	−3.01131	−0.118022
$652$	−91.9114	−3.59953
$653$	−16.4333	−0.643086	−0.321543	−	0.946895i	$-0.604201\pi$
$653$	−16.4333	−0.643086	−0.321543	+	0.946895i	$0.604201\pi$
$654$	10.9932	0.429868
$655$	0	0
$656$	−11.5784	−0.452061
$657$	13.3029	0.518994
$658$	5.27613	0.205685
$659$	−36.3315	−1.41528	−0.707638	−	0.706575i	$-0.750239\pi$
$659$	−36.3315	−1.41528	−0.707638	+	0.706575i	$0.750239\pi$
$660$	0	0
$661$	20.5035	0.797493	0.398746	−	0.917061i	$-0.369445\pi$
$661$	20.5035	0.797493	0.398746	+	0.917061i	$0.369445\pi$
$662$	−65.0510	−2.52828
$663$	5.02596	0.195192
$664$	23.3903	0.907721
$665$	0	0
$666$	9.03274	0.350012
$667$	11.0427	0.427574
$668$	−40.4378	−1.56459
$669$	0.226416	0.00875375
$670$	0	0
$671$	−43.8161	−1.69150
$672$	2.23533	0.0862296
$673$	−25.6104	−0.987208	−0.493604	−	0.869687i	$-0.664321\pi$
$673$	−25.6104	−0.987208	−0.493604	+	0.869687i	$0.664321\pi$
$674$	17.0328	0.656080
$675$	0	0
$676$	131.702	5.06545
$677$	−33.7573	−1.29740	−0.648699	−	0.761045i	$-0.724687\pi$
$677$	−33.7573	−1.29740	−0.648699	+	0.761045i	$0.724687\pi$
$678$	−11.5617	−0.444023
$679$	−6.57786	−0.252435
$680$	0	0
$681$	3.82125	0.146431
$682$	−123.959	−4.74662
$683$	−2.25100	−0.0861320	−0.0430660	−	0.999072i	$-0.513713\pi$
$683$	−2.25100	−0.0861320	−0.0430660	+	0.999072i	$0.513713\pi$
$684$	66.5588	2.54494
$685$	0	0
$686$	−2.55803	−0.0976662
$687$	7.15501	0.272981
$688$	−45.4045	−1.73103
$689$	−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	$0.426124\pi$
$692$	−68.3286	−2.59746
$693$	16.3694	0.621823
$694$	−3.26331	−0.123874
$695$	0	0
$696$	−3.74795	−0.142066
$697$	3.35870	0.127220
$698$	60.0553	2.27313
$699$	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	$0.324604\pi$
$702$	34.4560	1.30046
$703$	6.25909	0.236066
$704$	5.95887	0.224583
$705$	0	0
$706$	−45.2032	−1.70125
$707$	10.1795	0.382840
$708$	10.6401	0.399881
$709$	−39.7082	−1.49127	−0.745636	−	0.666354i	$-0.767854\pi$
$709$	−39.7082	−1.49127	−0.745636	+	0.666354i	$0.767854\pi$
$710$	0	0
$711$	25.5963	0.959937
$712$	−4.66105	−0.174680
$713$	−57.7272	−2.16190
$714$	−1.98413	−0.0742541
$715$	0	0
$716$	7.38117	0.275847
$717$	4.52119	0.168847
$718$	83.6740	3.12268
$719$	17.8189	0.664531	0.332266	−	0.943186i	$-0.392187\pi$
$719$	17.8189	0.664531	0.332266	+	0.943186i	$0.392187\pi$
$720$	0	0
$721$	−12.4409	−0.463324
$722$	17.8199	0.663186
$723$	−9.32216	−0.346695
$724$	35.3762	1.31475
$725$	0	0
$726$	−19.3908	−0.719661
$727$	51.8897	1.92448	0.962241	−	0.272197i	$-0.0877504\pi$
$727$	51.8897	1.92448	0.962241	+	0.272197i	$0.0877504\pi$
$728$	−42.1599	−1.56255
$729$	−20.4721	−0.758226
$730$	0	0
$731$	13.1711	0.487151
$732$	12.3712	0.457254
$733$	20.0529	0.740669	0.370334	−	0.928898i	$-0.379243\pi$
$733$	20.0529	0.740669	0.370334	+	0.928898i	$0.379243\pi$
$734$	6.98714	0.257900
$735$	0	0
$736$	42.8516	1.57953
$737$	35.8727	1.32139
$738$	11.2676	0.414767
$739$	6.39375	0.235198	0.117599	−	0.993061i	$-0.462480\pi$
$739$	6.39375	0.235198	0.117599	+	0.993061i	$0.462480\pi$
$740$	0	0
$741$	11.6834	0.429202
$742$	16.0320	0.588552
$743$	−3.15082	−0.115592	−0.0577962	−	0.998328i	$-0.518407\pi$
$743$	−3.15082	−0.115592	−0.0577962	+	0.998328i	$0.518407\pi$
$744$	19.5929	0.718312
$745$	0	0
$746$	21.4268	0.784490
$747$	−10.3347	−0.378128
$748$	−56.7118	−2.07359
$749$	−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	$0.626194\pi$
$752$	−15.5862	−0.568369
$753$	2.57265	0.0937526
$754$	26.9836	0.982684
$755$	0	0
$756$	−9.44491	−0.343508
$757$	45.1346	1.64044	0.820222	−	0.572045i	$-0.193850\pi$
$757$	45.1346	1.64044	0.820222	+	0.572045i	$0.193850\pi$
$758$	−32.3473	−1.17491
$759$	−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	$0.347578\pi$
$762$	−3.50884	−0.127112
$763$	12.1453	0.439688
$764$	−75.6279	−2.73612
$765$	0	0
$766$	−52.1268	−1.88342
$767$	−42.8843	−1.54846
$768$	9.75363	0.351954
$769$	8.36177	0.301533	0.150767	−	0.988569i	$-0.451826\pi$
$769$	8.36177	0.301533	0.150767	+	0.988569i	$0.451826\pi$
$770$	0	0
$771$	−10.5247	−0.379037
$772$	−17.1033	−0.615562
$773$	22.3094	0.802414	0.401207	−	0.915987i	$-0.368591\pi$
$773$	22.3094	0.802414	0.401207	+	0.915987i	$0.368591\pi$
$774$	44.1858	1.58822
$775$	0	0
$776$	42.7985	1.53638
$777$	−0.434628	−0.0155922
$778$	44.7106	1.60295
$779$	7.80770	0.279740
$780$	0	0
$781$	4.77786	0.170965
$782$	−38.0361	−1.36017
$783$	3.38409	0.120937
$784$	7.55666	0.269881
$785$	0	0
$786$	−1.40859	−0.0502426
$787$	19.4098	0.691884	0.345942	−	0.938256i	$-0.387559\pi$
$787$	19.4098	0.691884	0.345942	+	0.938256i	$0.387559\pi$
$788$	22.1005	0.787298
$789$	7.95384	0.283164
$790$	0	0
$791$	−12.7733	−0.454166
$792$	−106.507	−3.78456
$793$	−49.8613	−1.77063
$794$	56.3786	2.00080
$795$	0	0
$796$	31.9986	1.13416
$797$	30.7974	1.09090	0.545450	−	0.838143i	$-0.316359\pi$
$797$	30.7974	1.09090	0.545450	+	0.838143i	$0.316359\pi$
$798$	−4.61234	−0.163275
$799$	4.52129	0.159952
$800$	0	0
$801$	2.05943	0.0727663
$802$	−20.5896	−0.727045
$803$	−26.3490	−0.929837
$804$	−10.1285	−0.357203
$805$	0	0
$806$	−141.061	−4.96865
$807$	−6.48582	−0.228312
$808$	−66.2325	−2.33005
$809$	−48.1014	−1.69115	−0.845577	−	0.533854i	$-0.820743\pi$
$809$	−48.1014	−1.69115	−0.845577	+	0.533854i	$0.820743\pi$
$810$	0	0
$811$	−31.3020	−1.09916	−0.549580	−	0.835441i	$-0.685212\pi$
$811$	−31.3020	−1.09916	−0.549580	+	0.835441i	$0.685212\pi$
$812$	−7.39660	−0.259570
$813$	−5.70852	−0.200207
$814$	−17.8912	−0.627086
$815$	0	0
$816$	5.86129	0.205186
$817$	30.6178	1.07118
$818$	59.7264	2.08829
$819$	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	$0.456549\pi$
$822$	−10.5537	−0.368104
$823$	−53.0240	−1.84830	−0.924150	−	0.382029i	$-0.875226\pi$
$823$	−53.0240	−1.84830	−0.924150	+	0.382029i	$0.875226\pi$
$824$	80.9463	2.81990
$825$	0	0
$826$	16.9297	0.589059
$827$	−52.0766	−1.81088	−0.905441	−	0.424473i	$-0.860459\pi$
$827$	−52.0766	−1.81088	−0.905441	+	0.424473i	$0.860459\pi$
$828$	−88.6008	−3.07909
$829$	−8.49580	−0.295071	−0.147536	−	0.989057i	$-0.547134\pi$
$829$	−8.49580	−0.295071	−0.147536	+	0.989057i	$0.547134\pi$
$830$	0	0
$831$	−1.15656	−0.0401206
$832$	6.78099	0.235088
$833$	−2.19206	−0.0759504
$834$	−8.29552	−0.287250
$835$	0	0
$836$	−131.833	−4.55955
$837$	−17.6908	−0.611484
$838$	−33.3815	−1.15314
$839$	−31.4762	−1.08668	−0.543339	−	0.839513i	$-0.682840\pi$
$839$	−31.4762	−1.08668	−0.543339	+	0.839513i	$0.682840\pi$
$840$	0	0
$841$	−26.3498	−0.908614
$842$	97.8383	3.37173
$843$	1.36454	0.0469973
$844$	52.3496	1.80195
$845$	0	0
$846$	15.1678	0.521479
$847$	−21.4229	−0.736101
$848$	−47.3599	−1.62635
$849$	−7.81721	−0.268286
$850$	0	0
$851$	−8.33188	−0.285613
$852$	−1.34900	−0.0462160
$853$	−28.8426	−0.987552	−0.493776	−	0.869589i	$-0.664384\pi$
$853$	−28.8426	−0.987552	−0.493776	+	0.869589i	$0.664384\pi$
$854$	19.6840	0.673574
$855$	0	0
$856$	11.0976	0.379309
$857$	−26.8954	−0.918730	−0.459365	−	0.888248i	$-0.651923\pi$
$857$	−26.8954	−0.918730	−0.459365	+	0.888248i	$0.651923\pi$
$858$	−33.3963	−1.14013
$859$	4.29573	0.146569	0.0732843	−	0.997311i	$-0.476652\pi$
$859$	4.29573	0.146569	0.0732843	+	0.997311i	$0.476652\pi$
$860$	0	0
$861$	−0.542163	−0.0184768
$862$	18.3546	0.625160
$863$	39.2114	1.33477	0.667386	−	0.744712i	$-0.267413\pi$
$863$	39.2114	1.33477	0.667386	+	0.744712i	$0.267413\pi$
$864$	13.1321	0.446762
$865$	0	0
$866$	86.8270	2.95050
$867$	4.31507	0.146547
$868$	38.6668	1.31244
$869$	−50.6987	−1.71984
$870$	0	0
$871$	40.8219	1.38320
$872$	−79.0227	−2.67604
$873$	−18.9100	−0.640006
$874$	−88.4193	−2.99083
$875$	0	0
$876$	7.43949	0.251357
$877$	11.1744	0.377331	0.188666	−	0.982041i	$-0.439584\pi$
$877$	11.1744	0.377331	0.188666	+	0.982041i	$0.439584\pi$
$878$	−2.86449	−0.0966717
$879$	−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	$0.884856\pi$
$882$	−7.35382	−0.247616
$883$	45.3210	1.52517	0.762586	−	0.646886i	$-0.223929\pi$
$883$	45.3210	1.52517	0.762586	+	0.646886i	$0.223929\pi$
$884$	−64.5360	−2.17058
$885$	0	0
$886$	56.2479	1.88968
$887$	35.9580	1.20735	0.603676	−	0.797229i	$-0.293702\pi$
$887$	35.9580	1.20735	0.603676	+	0.797229i	$0.293702\pi$
$888$	2.82789	0.0948976
$889$	−3.87656	−0.130016
$890$	0	0
$891$	44.9199	1.50487
$892$	−2.90730	−0.0973438
$893$	10.5103	0.351713
$894$	−3.94850	−0.132058
$895$	0	0
$896$	9.95759	0.332660
$897$	−15.5526	−0.519286
$898$	44.7754	1.49418
$899$	−13.8542	−0.462064
$900$	0	0
$901$	13.7383	0.457690
$902$	−22.3178	−0.743101
$903$	−2.12608	−0.0707516
$904$	83.1090	2.76416
$905$	0	0
$906$	9.34187	0.310363
$907$	30.2781	1.00537	0.502685	−	0.864470i	$-0.332346\pi$
$907$	30.2781	1.00537	0.502685	+	0.864470i	$0.332346\pi$
$908$	−49.0670	−1.62834
$909$	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	$0.622430\pi$
$912$	13.6253	0.451178
$913$	20.4700	0.677459
$914$	13.6529	0.451598
$915$	0	0
$916$	−91.8742	−3.03561
$917$	−1.55620	−0.0513904
$918$	−11.6563	−0.384717
$919$	25.2435	0.832707	0.416354	−	0.909203i	$-0.363308\pi$
$919$	25.2435	0.832707	0.416354	+	0.909203i	$0.363308\pi$
$920$	0	0
$921$	10.2346	0.337241
$922$	18.2399	0.600700
$923$	5.43705	0.178963
$924$	9.15443	0.301159
$925$	0	0
$926$	−39.9627	−1.31326
$927$	−35.7651	−1.17468
$928$	10.2841	0.337593
$929$	−3.46732	−0.113759	−0.0568795	−	0.998381i	$-0.518115\pi$
$929$	−3.46732	−0.113759	−0.0568795	+	0.998381i	$0.518115\pi$
$930$	0	0
$931$	−5.09571	−0.167005
$932$	−32.4939	−1.06437
$933$	−9.12945	−0.298885
$934$	−72.0445	−2.35737
$935$	0	0
$936$	−121.201	−3.96158
$937$	10.9233	0.356848	0.178424	−	0.983954i	$-0.442900\pi$
$937$	10.9233	0.356848	0.178424	+	0.983954i	$0.442900\pi$
$938$	−16.1155	−0.526190
$939$	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	$0.546709\pi$
$942$	−21.6094	−0.704072
$943$	−10.3933	−0.338454
$944$	−50.0118	−1.62774
$945$	0	0
$946$	−87.5189	−2.84548
$947$	−33.5680	−1.09081	−0.545406	−	0.838172i	$-0.683625\pi$
$947$	−33.5680	−1.09081	−0.545406	+	0.838172i	$0.683625\pi$
$948$	14.3145	0.464913
$949$	−29.9843	−0.973331
$950$	0	0
$951$	−4.24421	−0.137628
$952$	14.2626	0.462252
$953$	37.9396	1.22898	0.614492	−	0.788923i	$-0.289361\pi$
$953$	37.9396	1.22898	0.614492	+	0.788923i	$0.289361\pi$
$954$	46.0887	1.49218
$955$	0	0
$956$	−58.0545	−1.87762
$957$	−3.28001	−0.106028
$958$	−67.7208	−2.18796
$959$	−11.6597	−0.376513
$960$	0	0
$961$	41.4250	1.33629
$962$	−20.3596	−0.656418
$963$	−4.90335	−0.158008
$964$	119.702	3.85533
$965$	0	0
$966$	6.13979	0.197545
$967$	12.4715	0.401056	0.200528	−	0.979688i	$-0.435734\pi$
$967$	12.4715	0.401056	0.200528	+	0.979688i	$0.435734\pi$
$968$	139.387	4.48008
$969$	−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	$0.787558\pi$
$972$	−41.0176	−1.31564
$973$	−9.16488	−0.293812
$974$	−26.7073	−0.855756
$975$	0	0
$976$	−58.1484	−1.86128
$977$	12.0315	0.384921	0.192461	−	0.981305i	$-0.438353\pi$
$977$	12.0315	0.384921	0.192461	+	0.981305i	$0.438353\pi$
$978$	18.3102	0.585495
$979$	−4.07911	−0.130369
$980$	0	0
$981$	34.9152	1.11476
$982$	−11.5964	−0.370057
$983$	−23.4581	−0.748199	−0.374099	−	0.927389i	$-0.622048\pi$
$983$	−23.4581	−0.748199	−0.374099	+	0.927389i	$0.622048\pi$
$984$	3.52756	0.112454
$985$	0	0
$986$	−9.12845	−0.290709
$987$	−0.729827	−0.0232307
$988$	−150.022	−4.77283
$989$	−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	$0.490248\pi$
$992$	−53.7618	−1.70694
$993$	8.99826	0.285551
$994$	−2.14642	−0.0680802
$995$	0	0
$996$	−5.77959	−0.183133
$997$	33.7803	1.06983	0.534917	−	0.844905i	$-0.320343\pi$
$997$	33.7803	1.06983	0.534917	+	0.844905i	$0.320343\pi$
$998$	−3.48158	−0.110207
$999$	−2.55335	−0.0807844

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
875.2.a.e.1.1	✓	6	5.4	even	2
875.2.a.h.1.6	yes	6	1.1	even	1	trivial
875.2.b.c.624.1		12	5.3	odd	4
875.2.b.c.624.12		12	5.2	odd	4
6125.2.a.s.1.1		6	35.34	odd	2
6125.2.a.t.1.6		6	7.6	odd	2
7875.2.a.q.1.1		6	3.2	odd	2
7875.2.a.t.1.6		6	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)

\(f(q)\)	\(=\)	\(q+2.55803 q^{2} -0.353843 q^{3} +4.54354 q^{4} -0.905143 q^{6} -1.00000 q^{7} +6.50646 q^{8} -2.87480 q^{9} +5.69412 q^{11} -1.60770 q^{12} +6.47971 q^{13} -2.55803 q^{14} +7.55666 q^{16} -2.19206 q^{17} -7.35382 q^{18} -5.09571 q^{19} +0.353843 q^{21} +14.5657 q^{22} +6.78323 q^{23} -2.30226 q^{24} +16.5753 q^{26} +2.07876 q^{27} -4.54354 q^{28} +1.62794 q^{29} -8.51029 q^{31} +6.31728 q^{32} -2.01482 q^{33} -5.60737 q^{34} -13.0617 q^{36} -1.22831 q^{37} -13.0350 q^{38} -2.29280 q^{39} -1.53221 q^{41} +0.905143 q^{42} -6.00854 q^{43} +25.8714 q^{44} +17.3517 q^{46} -2.06257 q^{47} -2.67387 q^{48} +1.00000 q^{49} +0.775646 q^{51} +29.4408 q^{52} -6.26731 q^{53} +5.31753 q^{54} -6.50646 q^{56} +1.80308 q^{57} +4.16432 q^{58} -6.61824 q^{59} -7.69499 q^{61} -21.7696 q^{62} +2.87480 q^{63} +1.04650 q^{64} -5.15399 q^{66} +6.29996 q^{67} -9.95971 q^{68} -2.40020 q^{69} +0.839088 q^{71} -18.7047 q^{72} -4.62741 q^{73} -3.14205 q^{74} -23.1525 q^{76} -5.69412 q^{77} -5.86506 q^{78} -8.90370 q^{79} +7.88883 q^{81} -3.91945 q^{82} +3.59494 q^{83} +1.60770 q^{84} -15.3701 q^{86} -0.576035 q^{87} +37.0485 q^{88} -0.716374 q^{89} -6.47971 q^{91} +30.8199 q^{92} +3.01131 q^{93} -5.27613 q^{94} -2.23533 q^{96} +6.57786 q^{97} +2.55803 q^{98} -16.3694 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 5 q^{3} + 4 q^{4} - 2 q^{6} - 6 q^{7} + 12 q^{8} + 7 q^{9} + q^{11} + 11 q^{12} + 22 q^{13} - 2 q^{14} + 8 q^{16} + 4 q^{17} - 14 q^{18} - 17 q^{19} - 5 q^{21} + 9 q^{22} + 3 q^{23} + 7 q^{24}+ \cdots + 12 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 + 5 * q^3 + 4 * q^4 - 2 * q^6 - 6 * q^7 + 12 * q^8 + 7 * q^9 + q^11 + 11 * q^12 + 22 * q^13 - 2 * q^14 + 8 * q^16 + 4 * q^17 - 14 * q^18 - 17 * q^19 - 5 * q^21 + 9 * q^22 + 3 * q^23 + 7 * q^24 + 16 * q^26 + 26 * q^27 - 4 * q^28 + 7 * q^29 - 11 * q^31 + q^32 + 21 * q^33 - 4 * q^34 + 10 * q^36 + 33 * q^37 + 15 * q^39 + 3 * q^41 + 2 * q^42 + 2 * q^43 + 26 * q^44 - 8 * q^46 - 6 * q^47 - 20 * q^48 + 6 * q^49 + q^51 + 47 * q^52 + 33 * q^53 - 26 * q^54 - 12 * q^56 - 3 * q^57 + 43 * q^58 - 21 * q^59 - 19 * q^61 - 37 * q^62 - 7 * q^63 - 22 * q^64 - 35 * q^66 + 2 * q^67 - 30 * q^68 - 2 * q^69 + q^71 - 10 * q^72 + 16 * q^73 - 29 * q^74 - 4 * q^76 - q^77 - q^78 + 22 * q^79 + 14 * q^81 - 13 * q^82 + 3 * q^83 - 11 * q^84 + 23 * q^86 + 28 * q^88 - 8 * q^89 - 22 * q^91 - 2 * q^92 - 16 * q^93 - 26 * q^94 - 4 * q^96 + 16 * q^97 + 2 * q^98 + 12 * q^99

Introduction

L-functions

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