LMFDB - Embedded newform 875.2.b.e.624.14

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			624.14
Root			$1.26874i$ of defining polynomial
Character	$\chi$	$=$	875.624
Dual form			875.2.b.e.624.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.64338i q^{2} -0.833911i q^{3} -4.98745 q^{4} +2.20434 q^{6} +1.00000i q^{7} -7.89697i q^{8} +2.30459 q^{9} -4.54519 q^{11} +4.15909i q^{12} -3.85791i q^{13} -2.64338 q^{14} +10.8998 q^{16} -4.62525i q^{17} +6.09191i q^{18} -6.66244 q^{19} +0.833911 q^{21} -12.0147i q^{22} -4.15079i q^{23} -6.58537 q^{24} +10.1979 q^{26} -4.42356i q^{27} -4.98745i q^{28} +8.32707 q^{29} -0.269795 q^{31} +13.0183i q^{32} +3.79028i q^{33} +12.2263 q^{34} -11.4940 q^{36} -3.02342i q^{37} -17.6114i q^{38} -3.21716 q^{39} -2.70751 q^{41} +2.20434i q^{42} +0.570645i q^{43} +22.6689 q^{44} +10.9721 q^{46} -6.18001i q^{47} -9.08944i q^{48} -1.00000 q^{49} -3.85704 q^{51} +19.2412i q^{52} +12.2587i q^{53} +11.6931 q^{54} +7.89697 q^{56} +5.55588i q^{57} +22.0116i q^{58} +0.419019 q^{59} +11.5307 q^{61} -0.713171i q^{62} +2.30459i q^{63} -12.6127 q^{64} -10.0192 q^{66} -0.289042i q^{67} +23.0682i q^{68} -3.46139 q^{69} -7.27856 q^{71} -18.1993i q^{72} -9.34497i q^{73} +7.99204 q^{74} +33.2286 q^{76} -4.54519i q^{77} -8.50417i q^{78} -4.37430 q^{79} +3.22492 q^{81} -7.15697i q^{82} -11.3312i q^{83} -4.15909 q^{84} -1.50843 q^{86} -6.94404i q^{87} +35.8932i q^{88} -14.6783 q^{89} +3.85791 q^{91} +20.7018i q^{92} +0.224985i q^{93} +16.3361 q^{94} +10.8561 q^{96} -9.86723i q^{97} -2.64338i q^{98} -10.4748 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.64338i	1.86915i	0.355765	+	0.934576i	$0.384220\pi$
$2$	2.64338i	1.86915i	−0.355765	+	0.934576i	$0.615780\pi$
$3$	− 0.833911i	− 0.481459i	−0.970592	−	0.240729i	$-0.922613\pi$
$3$	− 0.833911i	− 0.481459i	0.970592	−	0.240729i	$-0.0773866\pi$
$4$	−4.98745	−2.49373
$5$	0	0
$5$	0	0
$6$	2.20434	0.899919
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	− 7.89697i	− 2.79200i
$9$	2.30459	0.768197
$10$	0	0
$11$	−4.54519	−1.37043	−0.685213	−	0.728343i	$-0.740291\pi$
$11$	−4.54519	−1.37043	−0.685213	+	0.728343i	$0.740291\pi$
$12$	4.15909i	1.20063i
$13$	− 3.85791i	− 1.06999i	−0.844854	−	0.534996i	$-0.820313\pi$
$13$	− 3.85791i	− 1.06999i	0.844854	−	0.534996i	$-0.179687\pi$
$14$	−2.64338	−0.706473
$15$	0	0
$16$	10.8998	2.72494
$17$	− 4.62525i	− 1.12179i	−0.827888	−	0.560893i	$-0.810458\pi$
$17$	− 4.62525i	− 1.12179i	0.827888	−	0.560893i	$-0.189542\pi$
$18$	6.09191i	1.43588i
$19$	−6.66244	−1.52847	−0.764235	−	0.644938i	$-0.776883\pi$
$19$	−6.66244	−1.52847	−0.764235	+	0.644938i	$0.776883\pi$
$20$	0	0
$21$	0.833911	0.181974
$22$	− 12.0147i	− 2.56153i
$23$	− 4.15079i	− 0.865499i	−0.901514	−	0.432749i	$-0.857544\pi$
$23$	− 4.15079i	− 0.865499i	0.901514	−	0.432749i	$-0.142456\pi$
$24$	−6.58537	−1.34423
$25$	0	0
$26$	10.1979	1.99998
$27$	− 4.42356i	− 0.851314i
$28$	− 4.98745i	− 0.942540i
$29$	8.32707	1.54630	0.773149	−	0.634224i	$-0.218680\pi$
$29$	8.32707	1.54630	0.773149	+	0.634224i	$0.218680\pi$
$30$	0	0
$31$	−0.269795	−0.0484567	−0.0242283	−	0.999706i	$-0.507713\pi$
$31$	−0.269795	−0.0484567	−0.0242283	+	0.999706i	$0.507713\pi$
$32$	13.0183i	2.30133i
$33$	3.79028i	0.659803i
$34$	12.2263	2.09679
$35$	0	0
$36$	−11.4940	−1.91567
$37$	− 3.02342i	− 0.497047i	−0.968626	−	0.248524i	$-0.920055\pi$
$37$	− 3.02342i	− 0.497047i	0.968626	−	0.248524i	$-0.0799453\pi$
$38$	− 17.6114i	− 2.85694i
$39$	−3.21716	−0.515158
$40$	0	0
$41$	−2.70751	−0.422842	−0.211421	−	0.977395i	$-0.567809\pi$
$41$	−2.70751	−0.422842	−0.211421	+	0.977395i	$0.567809\pi$
$42$	2.20434i	0.340138i
$43$	0.570645i	0.0870226i	0.999053	+	0.0435113i	$0.0138544\pi$
$43$	0.570645i	0.0870226i	−0.999053	+	0.0435113i	$0.986146\pi$
$44$	22.6689	3.41747
$45$	0	0
$46$	10.9721	1.61775
$47$	− 6.18001i	− 0.901448i	−0.892663	−	0.450724i	$-0.851166\pi$
$47$	− 6.18001i	− 0.901448i	0.892663	−	0.450724i	$-0.148834\pi$
$48$	− 9.08944i	− 1.31195i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−3.85704	−0.540094
$52$	19.2412i	2.66827i
$53$	12.2587i	1.68387i	0.539581	+	0.841934i	$0.318583\pi$
$53$	12.2587i	1.68387i	−0.539581	+	0.841934i	$0.681417\pi$
$54$	11.6931	1.59123
$55$	0	0
$56$	7.89697	1.05528
$57$	5.55588i	0.735895i
$58$	22.0116i	2.89026i
$59$	0.419019	0.0545516	0.0272758	−	0.999628i	$-0.491317\pi$
$59$	0.419019	0.0545516	0.0272758	+	0.999628i	$0.491317\pi$
$60$	0	0
$61$	11.5307	1.47636	0.738180	−	0.674604i	$-0.235686\pi$
$61$	11.5307	1.47636	0.738180	+	0.674604i	$0.235686\pi$
$62$	− 0.713171i	− 0.0905728i
$63$	2.30459i	0.290351i
$64$	−12.6127	−1.57659
$65$	0	0
$66$	−10.0192	−1.23327
$67$	− 0.289042i	− 0.0353121i	−0.999844	−	0.0176560i	$-0.994380\pi$
$67$	− 0.289042i	− 0.0353121i	0.999844	−	0.0176560i	$-0.00562039\pi$
$68$	23.0682i	2.79743i
$69$	−3.46139	−0.416702
$70$	0	0
$71$	−7.27856	−0.863806	−0.431903	−	0.901920i	$-0.642158\pi$
$71$	−7.27856	−0.863806	−0.431903	+	0.901920i	$0.642158\pi$
$72$	− 18.1993i	− 2.14481i
$73$	− 9.34497i	− 1.09375i	−0.837215	−	0.546873i	$-0.815818\pi$
$73$	− 9.34497i	− 1.09375i	0.837215	−	0.546873i	$-0.184182\pi$
$74$	7.99204	0.929056
$75$	0	0
$76$	33.2286	3.81158
$77$	− 4.54519i	− 0.517972i
$78$	− 8.50417i	− 0.962907i
$79$	−4.37430	−0.492148	−0.246074	−	0.969251i	$-0.579141\pi$
$79$	−4.37430	−0.492148	−0.246074	+	0.969251i	$0.579141\pi$
$80$	0	0
$81$	3.22492	0.358325
$82$	− 7.15697i	− 0.790355i
$83$	− 11.3312i	− 1.24376i	−0.783114	−	0.621879i	$-0.786370\pi$
$83$	− 11.3312i	− 1.24376i	0.783114	−	0.621879i	$-0.213630\pi$
$84$	−4.15909	−0.453794
$85$	0	0
$86$	−1.50843	−0.162658
$87$	− 6.94404i	− 0.744479i
$88$	35.8932i	3.82623i
$89$	−14.6783	−1.55589	−0.777947	−	0.628330i	$-0.783739\pi$
$89$	−14.6783	−1.55589	−0.777947	+	0.628330i	$0.783739\pi$
$90$	0	0
$91$	3.85791	0.404419
$92$	20.7018i	2.15832i
$93$	0.224985i	0.0233299i
$94$	16.3361	1.68494
$95$	0	0
$96$	10.8561	1.10800
$97$	− 9.86723i	− 1.00187i	−0.865486	−	0.500933i	$-0.832990\pi$
$97$	− 9.86723i	− 1.00187i	0.865486	−	0.500933i	$-0.167010\pi$
$98$	− 2.64338i	− 0.267022i
$99$	−10.4748	−1.05276
$100$	0	0
$101$	6.27767	0.624651	0.312326	−	0.949975i	$-0.398892\pi$
$101$	6.27767	0.624651	0.312326	+	0.949975i	$0.398892\pi$
$102$	− 10.1956i	− 1.00952i
$103$	− 2.61593i	− 0.257756i	−0.991660	−	0.128878i	$-0.958862\pi$
$103$	− 2.61593i	− 0.257756i	0.991660	−	0.128878i	$-0.0411375\pi$
$104$	−30.4658	−2.98742
$105$	0	0
$106$	−32.4045	−3.14740
$107$	− 0.176723i	− 0.0170845i	−0.999964	−	0.00854225i	$-0.997281\pi$
$107$	− 0.176723i	− 0.0170845i	0.999964	−	0.00854225i	$-0.00271911\pi$
$108$	22.0623i	2.12294i
$109$	−10.7849	−1.03301	−0.516503	−	0.856285i	$-0.672767\pi$
$109$	−10.7849	−1.03301	−0.516503	+	0.856285i	$0.672767\pi$
$110$	0	0
$111$	−2.52126	−0.239308
$112$	10.8998i	1.02993i
$113$	8.12932i	0.764742i	0.924009	+	0.382371i	$0.124892\pi$
$113$	8.12932i	0.764742i	−0.924009	+	0.382371i	$0.875108\pi$
$114$	−14.6863	−1.37550
$115$	0	0
$116$	−41.5309	−3.85604
$117$	− 8.89092i	− 0.821966i
$118$	1.10762i	0.101965i
$119$	4.62525	0.423996
$120$	0	0
$121$	9.65873	0.878066
$122$	30.4801i	2.75954i
$123$	2.25782i	0.203581i
$124$	1.34559	0.120838
$125$	0	0
$126$	−6.09191	−0.542710
$127$	− 19.1636i	− 1.70050i	−0.526380	−	0.850249i	$-0.676451\pi$
$127$	− 19.1636i	− 1.70050i	0.526380	−	0.850249i	$-0.323549\pi$
$128$	− 7.30364i	− 0.645557i
$129$	0.475867	0.0418978
$130$	0	0
$131$	−18.0366	−1.57587	−0.787934	−	0.615760i	$-0.788849\pi$
$131$	−18.0366	−1.57587	−0.787934	+	0.615760i	$0.788849\pi$
$132$	− 18.9038i	− 1.64537i
$133$	− 6.66244i	− 0.577707i
$134$	0.764047	0.0660036
$135$	0	0
$136$	−36.5254	−3.13203
$137$	5.79298i	0.494927i	0.968897	+	0.247464i	$0.0795971\pi$
$137$	5.79298i	0.494927i	−0.968897	+	0.247464i	$0.920403\pi$
$138$	− 9.14976i	− 0.778879i
$139$	−4.68119	−0.397053	−0.198527	−	0.980095i	$-0.563616\pi$
$139$	−4.68119	−0.397053	−0.198527	+	0.980095i	$0.563616\pi$
$140$	0	0
$141$	−5.15358	−0.434010
$142$	− 19.2400i	− 1.61458i
$143$	17.5349i	1.46635i
$144$	25.1195	2.09329
$145$	0	0
$146$	24.7023	2.04438
$147$	0.833911i	0.0687798i
$148$	15.0792i	1.23950i
$149$	−19.6161	−1.60701	−0.803506	−	0.595297i	$-0.797035\pi$
$149$	−19.6161	−1.60701	−0.803506	+	0.595297i	$0.797035\pi$
$150$	0	0
$151$	16.5712	1.34855	0.674274	−	0.738481i	$-0.264457\pi$
$151$	16.5712	1.34855	0.674274	+	0.738481i	$0.264457\pi$
$152$	52.6131i	4.26748i
$153$	− 10.6593i	− 0.861754i
$154$	12.0147	0.968168
$155$	0	0
$156$	16.0454	1.28466
$157$	− 10.4246i	− 0.831972i	−0.909371	−	0.415986i	$-0.863436\pi$
$157$	− 10.4246i	− 0.831972i	0.909371	−	0.415986i	$-0.136564\pi$
$158$	− 11.5629i	− 0.919899i
$159$	10.2227	0.810713
$160$	0	0
$161$	4.15079	0.327128
$162$	8.52469i	0.669763i
$163$	4.02969i	0.315630i	0.987469	+	0.157815i	$0.0504450\pi$
$163$	4.02969i	0.315630i	−0.987469	+	0.157815i	$0.949555\pi$
$164$	13.5036	1.05445
$165$	0	0
$166$	29.9526	2.32477
$167$	22.2680i	1.72315i	0.507633	+	0.861573i	$0.330521\pi$
$167$	22.2680i	1.72315i	−0.507633	+	0.861573i	$0.669479\pi$
$168$	− 6.58537i	− 0.508072i
$169$	−1.88350	−0.144885
$170$	0	0
$171$	−15.3542	−1.17417
$172$	− 2.84606i	− 0.217010i
$173$	17.1363i	1.30285i	0.758712	+	0.651426i	$0.225829\pi$
$173$	17.1363i	1.30285i	−0.758712	+	0.651426i	$0.774171\pi$
$174$	18.3557	1.39154
$175$	0	0
$176$	−49.5415	−3.73433
$177$	− 0.349424i	− 0.0262643i
$178$	− 38.8003i	− 2.90820i
$179$	−15.3534	−1.14757	−0.573783	−	0.819007i	$-0.694525\pi$
$179$	−15.3534	−1.14757	−0.573783	+	0.819007i	$0.694525\pi$
$180$	0	0
$181$	−6.20953	−0.461551	−0.230775	−	0.973007i	$-0.574126\pi$
$181$	−6.20953	−0.461551	−0.230775	+	0.973007i	$0.574126\pi$
$182$	10.1979i	0.755921i
$183$	− 9.61561i	− 0.710806i
$184$	−32.7786	−2.41647
$185$	0	0
$186$	−0.594721	−0.0436071
$187$	21.0226i	1.53733i
$188$	30.8225i	2.24796i
$189$	4.42356	0.321767
$190$	0	0
$191$	7.32070	0.529707	0.264854	−	0.964289i	$-0.414676\pi$
$191$	7.32070	0.529707	0.264854	+	0.964289i	$0.414676\pi$
$192$	10.5179i	0.759064i
$193$	− 23.6227i	− 1.70040i	−0.526459	−	0.850200i	$-0.676481\pi$
$193$	− 23.6227i	− 1.70040i	0.526459	−	0.850200i	$-0.323519\pi$
$194$	26.0828	1.87264
$195$	0	0
$196$	4.98745	0.356247
$197$	− 8.09174i	− 0.576513i	−0.957553	−	0.288256i	$-0.906924\pi$
$197$	− 8.09174i	− 0.576513i	0.957553	−	0.288256i	$-0.0930755\pi$
$198$	− 27.6889i	− 1.96776i
$199$	24.6744	1.74912	0.874561	−	0.484915i	$-0.161149\pi$
$199$	24.6744	1.74912	0.874561	+	0.484915i	$0.161149\pi$
$200$	0	0
$201$	−0.241035	−0.0170013
$202$	16.5943i	1.16757i
$203$	8.32707i	0.584446i
$204$	19.2368	1.34685
$205$	0	0
$206$	6.91491	0.481784
$207$	− 9.56587i	− 0.664874i
$208$	− 42.0504i	− 2.91567i
$209$	30.2820	2.09465
$210$	0	0
$211$	−5.08112	−0.349798	−0.174899	−	0.984586i	$-0.555960\pi$
$211$	−5.08112	−0.349798	−0.174899	+	0.984586i	$0.555960\pi$
$212$	− 61.1399i	− 4.19910i
$213$	6.06967i	0.415887i
$214$	0.467147	0.0319335
$215$	0	0
$216$	−34.9327	−2.37687
$217$	− 0.269795i	− 0.0183149i
$218$	− 28.5086i	− 1.93085i
$219$	−7.79288	−0.526594
$220$	0	0
$221$	−17.8438	−1.20030
$222$	− 6.66465i	− 0.447302i
$223$	1.25572i	0.0840896i	0.999116	+	0.0420448i	$0.0133872\pi$
$223$	1.25572i	0.0840896i	−0.999116	+	0.0420448i	$0.986613\pi$
$224$	−13.0183	−0.869821
$225$	0	0
$226$	−21.4889	−1.42942
$227$	6.42534i	0.426465i	0.977002	+	0.213232i	$0.0683991\pi$
$227$	6.42534i	0.426465i	−0.977002	+	0.213232i	$0.931601\pi$
$228$	− 27.7097i	− 1.83512i
$229$	0.848045	0.0560404	0.0280202	−	0.999607i	$-0.491080\pi$
$229$	0.848045	0.0560404	0.0280202	+	0.999607i	$0.491080\pi$
$230$	0	0
$231$	−3.79028	−0.249382
$232$	− 65.7586i	− 4.31726i
$233$	− 1.13822i	− 0.0745672i	−0.999305	−	0.0372836i	$-0.988130\pi$
$233$	− 1.13822i	− 0.0745672i	0.999305	−	0.0372836i	$-0.0118705\pi$
$234$	23.5021	1.53638
$235$	0	0
$236$	−2.08983	−0.136037
$237$	3.64778i	0.236949i
$238$	12.2263i	0.792512i
$239$	−6.74663	−0.436403	−0.218202	−	0.975904i	$-0.570019\pi$
$239$	−6.74663	−0.436403	−0.218202	+	0.975904i	$0.570019\pi$
$240$	0	0
$241$	6.87736	0.443010	0.221505	−	0.975159i	$-0.428903\pi$
$241$	6.87736	0.443010	0.221505	+	0.975159i	$0.428903\pi$
$242$	25.5317i	1.64124i
$243$	− 15.9600i	− 1.02383i
$244$	−57.5090	−3.68164
$245$	0	0
$246$	−5.96827	−0.380523
$247$	25.7031i	1.63545i
$248$	2.13056i	0.135291i
$249$	−9.44919	−0.598818
$250$	0	0
$251$	27.9049	1.76134	0.880671	−	0.473728i	$-0.157092\pi$
$251$	27.9049	1.76134	0.880671	+	0.473728i	$0.157092\pi$
$252$	− 11.4940i	− 0.724057i
$253$	18.8661i	1.18610i
$254$	50.6568	3.17849
$255$	0	0
$256$	−5.91916	−0.369948
$257$	− 1.48316i	− 0.0925169i	−0.998930	−	0.0462585i	$-0.985270\pi$
$257$	− 1.48316i	− 0.0925169i	0.998930	−	0.0462585i	$-0.0147298\pi$
$258$	1.25790i	0.0783133i
$259$	3.02342	0.187866
$260$	0	0
$261$	19.1905	1.18786
$262$	− 47.6776i	− 2.94553i
$263$	− 0.390033i	− 0.0240504i	−0.999928	−	0.0120252i	$-0.996172\pi$
$263$	− 0.390033i	− 0.0240504i	0.999928	−	0.0120252i	$-0.00382784\pi$
$264$	29.9317	1.84217
$265$	0	0
$266$	17.6114	1.07982
$267$	12.2404i	0.749099i
$268$	1.44158i	0.0880587i
$269$	−29.5173	−1.79970	−0.899851	−	0.436197i	$-0.856325\pi$
$269$	−29.5173	−1.79970	−0.899851	+	0.436197i	$0.856325\pi$
$270$	0	0
$271$	1.82944	0.111131	0.0555654	−	0.998455i	$-0.482304\pi$
$271$	1.82944	0.111131	0.0555654	+	0.998455i	$0.482304\pi$
$272$	− 50.4141i	− 3.05680i
$273$	− 3.21716i	− 0.194711i
$274$	−15.3130	−0.925094
$275$	0	0
$276$	17.2635	1.03914
$277$	− 1.49518i	− 0.0898364i	−0.998991	−	0.0449182i	$-0.985697\pi$
$277$	− 1.49518i	− 0.0898364i	0.998991	−	0.0449182i	$-0.0143027\pi$
$278$	− 12.3742i	− 0.742153i
$279$	−0.621768	−0.0372243
$280$	0	0
$281$	16.0659	0.958412	0.479206	−	0.877703i	$-0.340925\pi$
$281$	16.0659	0.958412	0.479206	+	0.877703i	$0.340925\pi$
$282$	− 13.6229i	− 0.811230i
$283$	17.1301i	1.01828i	0.860684	+	0.509139i	$0.170036\pi$
$283$	17.1301i	1.01828i	−0.860684	+	0.509139i	$0.829964\pi$
$284$	36.3015	2.15410
$285$	0	0
$286$	−46.3515	−2.74082
$287$	− 2.70751i	− 0.159819i
$288$	30.0018i	1.76788i
$289$	−4.39290	−0.258406
$290$	0	0
$291$	−8.22840	−0.482357
$292$	46.6076i	2.72750i
$293$	− 5.47224i	− 0.319691i	−0.987142	−	0.159846i	$-0.948900\pi$
$293$	− 5.47224i	− 0.319691i	0.987142	−	0.159846i	$-0.0510997\pi$
$294$	−2.20434	−0.128560
$295$	0	0
$296$	−23.8758	−1.38776
$297$	20.1059i	1.16666i
$298$	− 51.8527i	− 3.00375i
$299$	−16.0134	−0.926077
$300$	0	0
$301$	−0.570645	−0.0328914
$302$	43.8041i	2.52064i
$303$	− 5.23502i	− 0.300744i
$304$	−72.6191	−4.16499
$305$	0	0
$306$	28.1766	1.61075
$307$	− 0.329124i	− 0.0187841i	−0.999956	−	0.00939205i	$-0.997010\pi$
$307$	− 0.329124i	− 0.0187841i	0.999956	−	0.00939205i	$-0.00298963\pi$
$308$	22.6689i	1.29168i
$309$	−2.18146	−0.124099
$310$	0	0
$311$	−2.26993	−0.128716	−0.0643579	−	0.997927i	$-0.520500\pi$
$311$	−2.26993	−0.128716	−0.0643579	+	0.997927i	$0.520500\pi$
$312$	25.4058i	1.43832i
$313$	− 13.2388i	− 0.748301i	−0.927368	−	0.374150i	$-0.877934\pi$
$313$	− 13.2388i	− 0.748301i	0.927368	−	0.374150i	$-0.122066\pi$
$314$	27.5561	1.55508
$315$	0	0
$316$	21.8166	1.22728
$317$	1.18652i	0.0666417i	0.999445	+	0.0333209i	$0.0106083\pi$
$317$	1.18652i	0.0666417i	−0.999445	+	0.0333209i	$0.989392\pi$
$318$	27.0225i	1.51534i
$319$	−37.8481	−2.11909
$320$	0	0
$321$	−0.147372	−0.00822548
$322$	10.9721i	0.611451i
$323$	30.8154i	1.71462i
$324$	−16.0841	−0.893563
$325$	0	0
$326$	−10.6520	−0.589960
$327$	8.99366i	0.497350i
$328$	21.3811i	1.18057i
$329$	6.18001	0.340715
$330$	0	0
$331$	−24.5431	−1.34901	−0.674504	−	0.738271i	$-0.735643\pi$
$331$	−24.5431	−1.34901	−0.674504	+	0.738271i	$0.735643\pi$
$332$	56.5136i	3.10159i
$333$	− 6.96775i	− 0.381830i
$334$	−58.8627	−3.22082
$335$	0	0
$336$	9.08944	0.495870
$337$	21.3287i	1.16185i	0.813957	+	0.580925i	$0.197309\pi$
$337$	21.3287i	1.16185i	−0.813957	+	0.580925i	$0.802691\pi$
$338$	− 4.97881i	− 0.270812i
$339$	6.77913	0.368192
$340$	0	0
$341$	1.22627	0.0664063
$342$	− 40.5870i	− 2.19469i
$343$	− 1.00000i	− 0.0539949i
$344$	4.50637	0.242967
$345$	0	0
$346$	−45.2979	−2.43523
$347$	9.25146i	0.496644i	0.968678	+	0.248322i	$0.0798791\pi$
$347$	9.25146i	0.496644i	−0.968678	+	0.248322i	$0.920121\pi$
$348$	34.6330i	1.85653i
$349$	28.6781	1.53510	0.767552	−	0.640987i	$-0.221475\pi$
$349$	28.6781	1.53510	0.767552	+	0.640987i	$0.221475\pi$
$350$	0	0
$351$	−17.0657	−0.910900
$352$	− 59.1706i	− 3.15380i
$353$	8.63065i	0.459363i	0.973266	+	0.229682i	$0.0737685\pi$
$353$	8.63065i	0.459363i	−0.973266	+	0.229682i	$0.926231\pi$
$354$	0.923661	0.0490920
$355$	0	0
$356$	73.2072	3.87997
$357$	− 3.85704i	− 0.204136i
$358$	− 40.5848i	− 2.14497i
$359$	20.5429	1.08421	0.542106	−	0.840310i	$-0.317627\pi$
$359$	20.5429	1.08421	0.542106	+	0.840310i	$0.317627\pi$
$360$	0	0
$361$	25.3881	1.33622
$362$	− 16.4141i	− 0.862708i
$363$	− 8.05452i	− 0.422753i
$364$	−19.2412	−1.00851
$365$	0	0
$366$	25.4177	1.32860
$367$	2.41855i	0.126247i	0.998006	+	0.0631236i	$0.0201062\pi$
$367$	2.41855i	0.126247i	−0.998006	+	0.0631236i	$0.979894\pi$
$368$	− 45.2426i	− 2.35843i
$369$	−6.23970	−0.324826
$370$	0	0
$371$	−12.2587	−0.636442
$372$	− 1.12210i	− 0.0581783i
$373$	− 5.97422i	− 0.309333i	−0.987967	−	0.154667i	$-0.950570\pi$
$373$	− 5.97422i	− 0.309333i	0.987967	−	0.154667i	$-0.0494303\pi$
$374$	−55.5707	−2.87349
$375$	0	0
$376$	−48.8034	−2.51684
$377$	− 32.1251i	− 1.65453i
$378$	11.6931i	0.601430i
$379$	31.8336	1.63518	0.817591	−	0.575799i	$-0.195309\pi$
$379$	31.8336	1.63518	0.817591	+	0.575799i	$0.195309\pi$
$380$	0	0
$381$	−15.9808	−0.818720
$382$	19.3514i	0.990103i
$383$	− 21.3560i	− 1.09124i	−0.838033	−	0.545619i	$-0.816295\pi$
$383$	− 21.3560i	− 1.09124i	0.838033	−	0.545619i	$-0.183705\pi$
$384$	−6.09059	−0.310809
$385$	0	0
$386$	62.4438	3.17831
$387$	1.31510i	0.0668505i
$388$	49.2124i	2.49838i
$389$	−30.6084	−1.55191	−0.775954	−	0.630789i	$-0.782731\pi$
$389$	−30.6084	−1.55191	−0.775954	+	0.630789i	$0.782731\pi$
$390$	0	0
$391$	−19.1984	−0.970905
$392$	7.89697i	0.398857i
$393$	15.0409i	0.758715i
$394$	21.3895	1.07759
$395$	0	0
$396$	52.2426	2.62529
$397$	− 15.0292i	− 0.754294i	−0.926153	−	0.377147i	$-0.876905\pi$
$397$	− 15.0292i	− 0.754294i	0.926153	−	0.377147i	$-0.123095\pi$
$398$	65.2238i	3.26937i
$399$	−5.55588	−0.278142
$400$	0	0
$401$	12.0828	0.603387	0.301693	−	0.953405i	$-0.402448\pi$
$401$	12.0828	0.603387	0.301693	+	0.953405i	$0.402448\pi$
$402$	− 0.637147i	− 0.0317780i
$403$	1.04085i	0.0518483i
$404$	−31.3096	−1.55771
$405$	0	0
$406$	−22.0116	−1.09242
$407$	13.7420i	0.681166i
$408$	30.4589i	1.50794i
$409$	10.8652	0.537248	0.268624	−	0.963245i	$-0.413431\pi$
$409$	10.8652	0.537248	0.268624	+	0.963245i	$0.413431\pi$
$410$	0	0
$411$	4.83083	0.238287
$412$	13.0468i	0.642772i
$413$	0.419019i	0.0206186i
$414$	25.2862	1.24275
$415$	0	0
$416$	50.2234	2.46241
$417$	3.90370i	0.191165i
$418$	80.0469i	3.91522i
$419$	9.71688	0.474700	0.237350	−	0.971424i	$-0.423721\pi$
$419$	9.71688	0.474700	0.237350	+	0.971424i	$0.423721\pi$
$420$	0	0
$421$	−6.31066	−0.307563	−0.153781	−	0.988105i	$-0.549145\pi$
$421$	−6.31066	−0.307563	−0.153781	+	0.988105i	$0.549145\pi$
$422$	− 13.4313i	− 0.653826i
$423$	− 14.2424i	− 0.692490i
$424$	96.8069	4.70136
$425$	0	0
$426$	−16.0444	−0.777356
$427$	11.5307i	0.558011i
$428$	0.881399i	0.0426040i
$429$	14.6226	0.705985
$430$	0	0
$431$	−4.92653	−0.237303	−0.118651	−	0.992936i	$-0.537857\pi$
$431$	−4.92653	−0.237303	−0.118651	+	0.992936i	$0.537857\pi$
$432$	− 48.2158i	− 2.31978i
$433$	37.1049i	1.78315i	0.452874	+	0.891575i	$0.350399\pi$
$433$	37.1049i	1.78315i	−0.452874	+	0.891575i	$0.649601\pi$
$434$	0.713171	0.0342333
$435$	0	0
$436$	53.7892	2.57604
$437$	27.6544i	1.32289i
$438$	− 20.5995i	− 0.984284i
$439$	9.50092	0.453454	0.226727	−	0.973958i	$-0.427197\pi$
$439$	9.50092	0.453454	0.226727	+	0.973958i	$0.427197\pi$
$440$	0	0
$441$	−2.30459	−0.109742
$442$	− 47.1679i	− 2.24355i
$443$	13.2415i	0.629123i	0.949237	+	0.314561i	$0.101857\pi$
$443$	13.2415i	0.629123i	−0.949237	+	0.314561i	$0.898143\pi$
$444$	12.5747	0.596768
$445$	0	0
$446$	−3.31936	−0.157176
$447$	16.3581i	0.773710i
$448$	− 12.6127i	− 0.595895i
$449$	20.2167	0.954085	0.477043	−	0.878880i	$-0.341709\pi$
$449$	20.2167	0.954085	0.477043	+	0.878880i	$0.341709\pi$
$450$	0	0
$451$	12.3061	0.579473
$452$	− 40.5446i	− 1.90706i
$453$	− 13.8189i	− 0.649271i
$454$	−16.9846	−0.797127
$455$	0	0
$456$	43.8746	2.05462
$457$	23.1239i	1.08169i	0.841122	+	0.540845i	$0.181895\pi$
$457$	23.1239i	1.08169i	−0.841122	+	0.540845i	$0.818105\pi$
$458$	2.24170i	0.104748i
$459$	−20.4600	−0.954993
$460$	0	0
$461$	7.94071	0.369836	0.184918	−	0.982754i	$-0.440798\pi$
$461$	7.94071	0.369836	0.184918	+	0.982754i	$0.440798\pi$
$462$	− 10.0192i	− 0.466133i
$463$	15.2021i	0.706503i	0.935528	+	0.353252i	$0.114924\pi$
$463$	15.2021i	0.706503i	−0.935528	+	0.353252i	$0.885076\pi$
$464$	90.7631	4.21357
$465$	0	0
$466$	3.00874	0.139377
$467$	− 3.86560i	− 0.178879i	−0.995992	−	0.0894393i	$-0.971493\pi$
$467$	− 3.86560i	− 0.178879i	0.995992	−	0.0894393i	$-0.0285075\pi$
$468$	44.3430i	2.04976i
$469$	0.289042	0.0133467
$470$	0	0
$471$	−8.69318	−0.400560
$472$	− 3.30898i	− 0.152308i
$473$	− 2.59369i	− 0.119258i
$474$	−9.64247	−0.442893
$475$	0	0
$476$	−23.0682	−1.05733
$477$	28.2514i	1.29354i
$478$	− 17.8339i	− 0.815703i
$479$	36.4902	1.66728	0.833641	−	0.552307i	$-0.186252\pi$
$479$	36.4902	1.66728	0.833641	+	0.552307i	$0.186252\pi$
$480$	0	0
$481$	−11.6641	−0.531837
$482$	18.1795i	0.828052i
$483$	− 3.46139i	− 0.157499i
$484$	−48.1724	−2.18966
$485$	0	0
$486$	42.1883	1.91370
$487$	− 28.0516i	− 1.27114i	−0.772044	−	0.635570i	$-0.780765\pi$
$487$	− 28.0516i	− 1.27114i	0.772044	−	0.635570i	$-0.219235\pi$
$488$	− 91.0578i	− 4.12199i
$489$	3.36041	0.151963
$490$	0	0
$491$	1.94688	0.0878616	0.0439308	−	0.999035i	$-0.486012\pi$
$491$	1.94688	0.0878616	0.0439308	+	0.999035i	$0.486012\pi$
$492$	− 11.2608i	− 0.507675i
$493$	− 38.5147i	− 1.73462i
$494$	−67.9431	−3.05691
$495$	0	0
$496$	−2.94071	−0.132042
$497$	− 7.27856i	− 0.326488i
$498$	− 24.9778i	− 1.11928i
$499$	−9.04201	−0.404776	−0.202388	−	0.979305i	$-0.564870\pi$
$499$	−9.04201	−0.404776	−0.202388	+	0.979305i	$0.564870\pi$
$500$	0	0
$501$	18.5695	0.829624
$502$	73.7632i	3.29221i
$503$	− 8.34548i	− 0.372107i	−0.982540	−	0.186053i	$-0.940430\pi$
$503$	− 8.34548i	− 0.372107i	0.982540	−	0.186053i	$-0.0595697\pi$
$504$	18.1993	0.810661
$505$	0	0
$506$	−49.8702	−2.21700
$507$	1.57067i	0.0697561i
$508$	95.5778i	4.24058i
$509$	−29.1093	−1.29025	−0.645123	−	0.764079i	$-0.723194\pi$
$509$	−29.1093	−1.29025	−0.645123	+	0.764079i	$0.723194\pi$
$510$	0	0
$511$	9.34497	0.413397
$512$	− 30.2539i	− 1.33705i
$513$	29.4717i	1.30121i
$514$	3.92055	0.172928
$515$	0	0
$516$	−2.37337	−0.104482
$517$	28.0893i	1.23537i
$518$	7.99204i	0.351150i
$519$	14.2902	0.627270
$520$	0	0
$521$	42.4571	1.86008	0.930039	−	0.367461i	$-0.119773\pi$
$521$	42.4571	1.86008	0.930039	+	0.367461i	$0.119773\pi$
$522$	50.7278i	2.22029i
$523$	− 2.39751i	− 0.104836i	−0.998625	−	0.0524179i	$-0.983307\pi$
$523$	− 2.39751i	− 0.104836i	0.998625	−	0.0524179i	$-0.0166928\pi$
$524$	89.9568	3.92978
$525$	0	0
$526$	1.03100	0.0449539
$527$	1.24787i	0.0543581i
$528$	41.3132i	1.79793i
$529$	5.77098	0.250912
$530$	0	0
$531$	0.965667	0.0419064
$532$	33.2286i	1.44064i
$533$	10.4453i	0.452438i
$534$	−32.3560	−1.40018
$535$	0	0
$536$	−2.28255	−0.0985913
$537$	12.8034i	0.552506i
$538$	− 78.0254i	− 3.36392i
$539$	4.54519	0.195775
$540$	0	0
$541$	−10.2880	−0.442314	−0.221157	−	0.975238i	$-0.570983\pi$
$541$	−10.2880	−0.442314	−0.221157	+	0.975238i	$0.570983\pi$
$542$	4.83591i	0.207720i
$543$	5.17819i	0.222218i
$544$	60.2128	2.58160
$545$	0	0
$546$	8.50417	0.363945
$547$	− 36.1810i	− 1.54699i	−0.633804	−	0.773494i	$-0.718507\pi$
$547$	− 36.1810i	− 1.54699i	0.633804	−	0.773494i	$-0.281493\pi$
$548$	− 28.8922i	− 1.23421i
$549$	26.5736	1.13414
$550$	0	0
$551$	−55.4786	−2.36347
$552$	27.3345i	1.16343i
$553$	− 4.37430i	− 0.186014i
$554$	3.95232	0.167918
$555$	0	0
$556$	23.3472	0.990142
$557$	37.5288i	1.59015i	0.606514	+	0.795073i	$0.292568\pi$
$557$	37.5288i	1.59015i	−0.606514	+	0.795073i	$0.707432\pi$
$558$	− 1.64357i	− 0.0695778i
$559$	2.20150	0.0931135
$560$	0	0
$561$	17.5310	0.740159
$562$	42.4683i	1.79142i
$563$	5.50239i	0.231898i	0.993255	+	0.115949i	$0.0369909\pi$
$563$	5.50239i	0.231898i	−0.993255	+	0.115949i	$0.963009\pi$
$564$	25.7032	1.08230
$565$	0	0
$566$	−45.2814	−1.90332
$567$	3.22492i	0.135434i
$568$	57.4785i	2.41175i
$569$	−12.1804	−0.510629	−0.255314	−	0.966858i	$-0.582179\pi$
$569$	−12.1804	−0.510629	−0.255314	+	0.966858i	$0.582179\pi$
$570$	0	0
$571$	−10.3473	−0.433019	−0.216510	−	0.976281i	$-0.569467\pi$
$571$	−10.3473	−0.433019	−0.216510	+	0.976281i	$0.569467\pi$
$572$	− 87.4547i	− 3.65666i
$573$	− 6.10481i	− 0.255032i
$574$	7.15697	0.298726
$575$	0	0
$576$	−29.0672	−1.21113
$577$	− 29.3627i	− 1.22238i	−0.791482	−	0.611192i	$-0.790690\pi$
$577$	− 29.3627i	− 1.22238i	0.791482	−	0.611192i	$-0.209310\pi$
$578$	− 11.6121i	− 0.482999i
$579$	−19.6992	−0.818673
$580$	0	0
$581$	11.3312	0.470096
$582$	− 21.7508i	− 0.901598i
$583$	− 55.7183i	− 2.30762i
$584$	−73.7969	−3.05374
$585$	0	0
$586$	14.4652	0.597552
$587$	18.1347i	0.748498i	0.927328	+	0.374249i	$0.122099\pi$
$587$	18.1347i	0.748498i	−0.927328	+	0.374249i	$0.877901\pi$
$588$	− 4.15909i	− 0.171518i
$589$	1.79750	0.0740645
$590$	0	0
$591$	−6.74779	−0.277567
$592$	− 32.9546i	− 1.35442i
$593$	− 23.7551i	− 0.975505i	−0.872982	−	0.487752i	$-0.837817\pi$
$593$	− 23.7551i	− 0.975505i	0.872982	−	0.487752i	$-0.162183\pi$
$594$	−53.1475	−2.18067
$595$	0	0
$596$	97.8343	4.00745
$597$	− 20.5763i	− 0.842131i
$598$	− 42.3294i	− 1.73098i
$599$	−9.08142	−0.371057	−0.185528	−	0.982639i	$-0.559400\pi$
$599$	−9.08142	−0.371057	−0.185528	+	0.982639i	$0.559400\pi$
$600$	0	0
$601$	27.1808	1.10873	0.554364	−	0.832274i	$-0.312962\pi$
$601$	27.1808	1.10873	0.554364	+	0.832274i	$0.312962\pi$
$602$	− 1.50843i	− 0.0614791i
$603$	− 0.666124i	− 0.0271267i
$604$	−82.6483	−3.36291
$605$	0	0
$606$	13.8381	0.562136
$607$	− 8.67473i	− 0.352096i	−0.984382	−	0.176048i	$-0.943669\pi$
$607$	− 8.67473i	− 0.352096i	0.984382	−	0.176048i	$-0.0563314\pi$
$608$	− 86.7336i	− 3.51751i
$609$	6.94404	0.281387
$610$	0	0
$611$	−23.8420	−0.964543
$612$	53.1628i	2.14898i
$613$	10.2136i	0.412522i	0.978497	+	0.206261i	$0.0661297\pi$
$613$	10.2136i	0.412522i	−0.978497	+	0.206261i	$0.933870\pi$
$614$	0.869999	0.0351103
$615$	0	0
$616$	−35.8932	−1.44618
$617$	− 8.10309i	− 0.326218i	−0.986608	−	0.163109i	$-0.947848\pi$
$617$	− 8.10309i	− 0.326218i	0.986608	−	0.163109i	$-0.0521522\pi$
$618$	− 5.76642i	− 0.231959i
$619$	−9.49309	−0.381559	−0.190780	−	0.981633i	$-0.561102\pi$
$619$	−9.49309	−0.381559	−0.190780	+	0.981633i	$0.561102\pi$
$620$	0	0
$621$	−18.3612	−0.736811
$622$	− 6.00028i	− 0.240589i
$623$	− 14.6783i	− 0.588073i
$624$	−35.0663	−1.40377
$625$	0	0
$626$	34.9951	1.39869
$627$	− 25.2525i	− 1.00849i
$628$	51.9921i	2.07471i
$629$	−13.9841	−0.557581
$630$	0	0
$631$	−19.0891	−0.759925	−0.379962	−	0.925002i	$-0.624063\pi$
$631$	−19.0891	−0.759925	−0.379962	+	0.925002i	$0.624063\pi$
$632$	34.5437i	1.37408i
$633$	4.23720i	0.168414i
$634$	−3.13643	−0.124563
$635$	0	0
$636$	−50.9852	−2.02170
$637$	3.85791i	0.152856i
$638$	− 100.047i	− 3.96089i
$639$	−16.7741	−0.663574
$640$	0	0
$641$	−0.0611719	−0.00241614	−0.00120807	−	0.999999i	$-0.500385\pi$
$641$	−0.0611719	−0.00241614	−0.00120807	+	0.999999i	$0.500385\pi$
$642$	− 0.389559i	− 0.0153747i
$643$	− 8.28259i	− 0.326634i	−0.986574	−	0.163317i	$-0.947781\pi$
$643$	− 8.28259i	− 0.326634i	0.986574	−	0.163317i	$-0.0522193\pi$
$644$	−20.7018	−0.815767
$645$	0	0
$646$	−81.4569	−3.20488
$647$	− 43.9925i	− 1.72952i	−0.502182	−	0.864762i	$-0.667469\pi$
$647$	− 43.9925i	− 1.72952i	0.502182	−	0.864762i	$-0.332531\pi$
$648$	− 25.4671i	− 1.00044i
$649$	−1.90452	−0.0747589
$650$	0	0
$651$	−0.224985	−0.00881787
$652$	− 20.0979i	− 0.787095i
$653$	18.6070i	0.728147i	0.931370	+	0.364073i	$0.118614\pi$
$653$	18.6070i	0.728147i	−0.931370	+	0.364073i	$0.881386\pi$
$654$	−23.7736	−0.929623
$655$	0	0
$656$	−29.5112	−1.15222
$657$	− 21.5364i	− 0.840213i
$658$	16.3361i	0.636848i
$659$	42.8021	1.66733	0.833667	−	0.552268i	$-0.186237\pi$
$659$	42.8021	1.66733	0.833667	+	0.552268i	$0.186237\pi$
$660$	0	0
$661$	17.0003	0.661234	0.330617	−	0.943765i	$-0.392743\pi$
$661$	17.0003	0.661234	0.330617	+	0.943765i	$0.392743\pi$
$662$	− 64.8766i	− 2.52150i
$663$	14.8801i	0.577897i
$664$	−89.4818	−3.47257
$665$	0	0
$666$	18.4184	0.713698
$667$	− 34.5639i	− 1.33832i
$668$	− 111.060i	− 4.29706i
$669$	1.04716	0.0404857
$670$	0	0
$671$	−52.4093	−2.02324
$672$	10.8561i	0.418783i
$673$	− 5.43715i	− 0.209587i	−0.994494	−	0.104793i	$-0.966582\pi$
$673$	− 5.43715i	− 0.209587i	0.994494	−	0.104793i	$-0.0334181\pi$
$674$	−56.3799	−2.17167
$675$	0	0
$676$	9.39388	0.361303
$677$	− 5.00554i	− 0.192378i	−0.995363	−	0.0961892i	$-0.969335\pi$
$677$	− 5.00554i	− 0.192378i	0.995363	−	0.0961892i	$-0.0306654\pi$
$678$	17.9198i	0.688206i
$679$	9.86723	0.378670
$680$	0	0
$681$	5.35816	0.205325
$682$	3.24150i	0.124123i
$683$	39.6432i	1.51691i	0.651728	+	0.758453i	$0.274044\pi$
$683$	39.6432i	1.51691i	−0.651728	+	0.758453i	$0.725956\pi$
$684$	76.5784	2.92805
$685$	0	0
$686$	2.64338	0.100925
$687$	− 0.707194i	− 0.0269811i
$688$	6.21990i	0.237131i
$689$	47.2932	1.80173
$690$	0	0
$691$	19.5149	0.742383	0.371191	−	0.928556i	$-0.378949\pi$
$691$	19.5149	0.742383	0.371191	+	0.928556i	$0.378949\pi$
$692$	− 85.4667i	− 3.24896i
$693$	− 10.4748i	− 0.397905i
$694$	−24.4551	−0.928303
$695$	0	0
$696$	−54.8368	−2.07858
$697$	12.5229i	0.474338i
$698$	75.8071i	2.86934i
$699$	−0.949173	−0.0359010
$700$	0	0
$701$	21.8539	0.825412	0.412706	−	0.910864i	$-0.364584\pi$
$701$	21.8539	0.825412	0.412706	+	0.910864i	$0.364584\pi$
$702$	− 45.1111i	− 1.70261i
$703$	20.1434i	0.759721i
$704$	57.3272	2.16060
$705$	0	0
$706$	−22.8141	−0.858620
$707$	6.27767i	0.236096i
$708$	1.74274i	0.0654961i
$709$	−40.7664	−1.53101	−0.765507	−	0.643427i	$-0.777512\pi$
$709$	−40.7664	−1.53101	−0.765507	+	0.643427i	$0.777512\pi$
$710$	0	0
$711$	−10.0810	−0.378067
$712$	115.914i	4.34406i
$713$	1.11986i	0.0419392i
$714$	10.1956	0.381562
$715$	0	0
$716$	76.5743	2.86172
$717$	5.62609i	0.210110i
$718$	54.3026i	2.02656i
$719$	−20.4371	−0.762176	−0.381088	−	0.924539i	$-0.624451\pi$
$719$	−20.4371	−0.762176	−0.381088	+	0.924539i	$0.624451\pi$
$720$	0	0
$721$	2.61593	0.0974225
$722$	67.1105i	2.49759i
$723$	− 5.73511i	− 0.213291i
$724$	30.9697	1.15098
$725$	0	0
$726$	21.2911	0.790189
$727$	32.0766i	1.18966i	0.803853	+	0.594828i	$0.202780\pi$
$727$	32.0766i	1.18966i	−0.803853	+	0.594828i	$0.797220\pi$
$728$	− 30.4658i	− 1.12914i
$729$	−3.63443	−0.134609
$730$	0	0
$731$	2.63937	0.0976208
$732$	47.9574i	1.77256i
$733$	− 44.5082i	− 1.64395i	−0.569525	−	0.821974i	$-0.692873\pi$
$733$	− 44.5082i	− 1.64395i	0.569525	−	0.821974i	$-0.307127\pi$
$734$	−6.39314	−0.235975
$735$	0	0
$736$	54.0361	1.99180
$737$	1.31375i	0.0483926i
$738$	− 16.4939i	− 0.607149i
$739$	−0.0772344	−0.00284111	−0.00142056	−	0.999999i	$-0.500452\pi$
$739$	−0.0772344	−0.00284111	−0.00142056	+	0.999999i	$0.500452\pi$
$740$	0	0
$741$	21.4341	0.787402
$742$	− 32.4045i	− 1.18961i
$743$	14.8925i	0.546353i	0.961964	+	0.273177i	$0.0880743\pi$
$743$	14.8925i	0.546353i	−0.961964	+	0.273177i	$0.911926\pi$
$744$	1.77670	0.0651370
$745$	0	0
$746$	15.7921	0.578191
$747$	− 26.1137i	− 0.955451i
$748$	− 104.849i	− 3.83367i
$749$	0.176723	0.00645733
$750$	0	0
$751$	16.8102	0.613414	0.306707	−	0.951804i	$-0.400773\pi$
$751$	16.8102	0.613414	0.306707	+	0.951804i	$0.400773\pi$
$752$	− 67.3607i	− 2.45639i
$753$	− 23.2702i	− 0.848014i
$754$	84.9189	3.09256
$755$	0	0
$756$	−22.0623	−0.802398
$757$	− 12.0534i	− 0.438090i	−0.975715	−	0.219045i	$-0.929706\pi$
$757$	− 12.0534i	− 0.438090i	0.975715	−	0.219045i	$-0.0702941\pi$
$758$	84.1483i	3.05640i
$759$	15.7327	0.571059
$760$	0	0
$761$	13.6602	0.495182	0.247591	−	0.968865i	$-0.420361\pi$
$761$	13.6602	0.495182	0.247591	+	0.968865i	$0.420361\pi$
$762$	− 42.2433i	− 1.53031i
$763$	− 10.7849i	− 0.390440i
$764$	−36.5116	−1.32094
$765$	0	0
$766$	56.4519	2.03969
$767$	− 1.61654i	− 0.0583698i
$768$	4.93606i	0.178115i
$769$	37.1200	1.33858	0.669290	−	0.743001i	$-0.266598\pi$
$769$	37.1200	1.33858	0.669290	+	0.743001i	$0.266598\pi$
$770$	0	0
$771$	−1.23682	−0.0445431
$772$	117.817i	4.24033i
$773$	29.4332i	1.05864i	0.848423	+	0.529318i	$0.177552\pi$
$773$	29.4332i	1.05864i	−0.848423	+	0.529318i	$0.822448\pi$
$774$	−3.47632	−0.124954
$775$	0	0
$776$	−77.9212	−2.79721
$777$	− 2.52126i	− 0.0904498i
$778$	− 80.9096i	− 2.90075i
$779$	18.0386	0.646300
$780$	0	0
$781$	33.0824	1.18378
$782$	− 50.7487i	− 1.81477i
$783$	− 36.8353i	− 1.31639i
$784$	−10.8998	−0.389278
$785$	0	0
$786$	−39.7589	−1.41815
$787$	37.0603i	1.32106i	0.750801	+	0.660529i	$0.229668\pi$
$787$	37.0603i	1.32106i	−0.750801	+	0.660529i	$0.770332\pi$
$788$	40.3572i	1.43766i
$789$	−0.325253	−0.0115793
$790$	0	0
$791$	−8.12932	−0.289045
$792$	82.7192i	2.93930i
$793$	− 44.4846i	− 1.57969i
$794$	39.7279	1.40989
$795$	0	0
$796$	−123.062	−4.36183
$797$	4.72508i	0.167371i	0.996492	+	0.0836855i	$0.0266691\pi$
$797$	4.72508i	0.167371i	−0.996492	+	0.0836855i	$0.973331\pi$
$798$	− 14.6863i	− 0.519890i
$799$	−28.5841	−1.01123
$800$	0	0
$801$	−33.8274	−1.19523
$802$	31.9394i	1.12782i
$803$	42.4747i	1.49890i
$804$	1.20215	0.0423966
$805$	0	0
$806$	−2.75135	−0.0969123
$807$	24.6148i	0.866483i
$808$	− 49.5745i	− 1.74403i
$809$	27.4587	0.965397	0.482699	−	0.875786i	$-0.339657\pi$
$809$	27.4587	0.965397	0.482699	+	0.875786i	$0.339657\pi$
$810$	0	0
$811$	−22.2780	−0.782286	−0.391143	−	0.920330i	$-0.627920\pi$
$811$	−22.2780	−0.782286	−0.391143	+	0.920330i	$0.627920\pi$
$812$	− 41.5309i	− 1.45745i
$813$	− 1.52559i	− 0.0535049i
$814$	−36.3253	−1.27320
$815$	0	0
$816$	−42.0409	−1.47173
$817$	− 3.80189i	− 0.133011i
$818$	28.7208i	1.00420i
$819$	8.89092	0.310674
$820$	0	0
$821$	20.7446	0.723991	0.361996	−	0.932180i	$-0.382096\pi$
$821$	20.7446	0.723991	0.361996	+	0.932180i	$0.382096\pi$
$822$	12.7697i	0.445395i
$823$	35.7195i	1.24510i	0.782578	+	0.622552i	$0.213904\pi$
$823$	35.7195i	1.24510i	−0.782578	+	0.622552i	$0.786096\pi$
$824$	−20.6579	−0.719654
$825$	0	0
$826$	−1.10762	−0.0385392
$827$	25.4541i	0.885126i	0.896737	+	0.442563i	$0.145931\pi$
$827$	25.4541i	0.885126i	−0.896737	+	0.442563i	$0.854069\pi$
$828$	47.7093i	1.65801i
$829$	13.4046	0.465561	0.232780	−	0.972529i	$-0.425218\pi$
$829$	13.4046	0.465561	0.232780	+	0.972529i	$0.425218\pi$
$830$	0	0
$831$	−1.24684	−0.0432525
$832$	48.6588i	1.68694i
$833$	4.62525i	0.160255i
$834$	−10.3189	−0.357316
$835$	0	0
$836$	−151.030	−5.22349
$837$	1.19346i	0.0412519i
$838$	25.6854i	0.887287i
$839$	10.6811	0.368754	0.184377	−	0.982856i	$-0.440973\pi$
$839$	10.6811	0.368754	0.184377	+	0.982856i	$0.440973\pi$
$840$	0	0
$841$	40.3401	1.39104
$842$	− 16.6815i	− 0.574881i
$843$	− 13.3975i	− 0.461436i
$844$	25.3418	0.872301
$845$	0	0
$846$	37.6481	1.29437
$847$	9.65873i	0.331878i
$848$	133.617i	4.58844i
$849$	14.2850	0.490259
$850$	0	0
$851$	−12.5496	−0.430194
$852$	− 30.2722i	− 1.03711i
$853$	− 44.8085i	− 1.53422i	−0.641518	−	0.767108i	$-0.721695\pi$
$853$	− 44.8085i	− 1.53422i	0.641518	−	0.767108i	$-0.278305\pi$
$854$	−30.4801	−1.04301
$855$	0	0
$856$	−1.39558	−0.0476999
$857$	− 36.9801i	− 1.26322i	−0.775288	−	0.631608i	$-0.782395\pi$
$857$	− 36.9801i	− 1.26322i	0.775288	−	0.631608i	$-0.217605\pi$
$858$	38.6530i	1.31959i
$859$	46.8718	1.59924	0.799622	−	0.600504i	$-0.205033\pi$
$859$	46.8718	1.59924	0.799622	+	0.600504i	$0.205033\pi$
$860$	0	0
$861$	−2.25782	−0.0769463
$862$	− 13.0227i	− 0.443554i
$863$	15.3188i	0.521459i	0.965412	+	0.260730i	$0.0839631\pi$
$863$	15.3188i	0.521459i	−0.965412	+	0.260730i	$0.916037\pi$
$864$	57.5872	1.95916
$865$	0	0
$866$	−98.0824	−3.33298
$867$	3.66329i	0.124412i
$868$	1.34559i	0.0456723i
$869$	19.8820	0.674452
$870$	0	0
$871$	−1.11510	−0.0377837
$872$	85.1681i	2.88415i
$873$	− 22.7400i	− 0.769631i
$874$	−73.1010	−2.47268
$875$	0	0
$876$	38.8666	1.31318
$877$	− 43.4733i	− 1.46799i	−0.679156	−	0.733994i	$-0.737654\pi$
$877$	− 43.4733i	− 1.46799i	0.679156	−	0.733994i	$-0.262346\pi$
$878$	25.1145i	0.847575i
$879$	−4.56336	−0.153918
$880$	0	0
$881$	−44.7337	−1.50712	−0.753558	−	0.657381i	$-0.771664\pi$
$881$	−44.7337	−1.50712	−0.753558	+	0.657381i	$0.771664\pi$
$882$	− 6.09191i	− 0.205125i
$883$	− 32.1748i	− 1.08277i	−0.840776	−	0.541384i	$-0.817901\pi$
$883$	− 32.1748i	− 1.08277i	0.840776	−	0.541384i	$-0.182099\pi$
$884$	88.9951	2.99323
$885$	0	0
$886$	−35.0023	−1.17593
$887$	− 32.3009i	− 1.08456i	−0.840199	−	0.542279i	$-0.817562\pi$
$887$	− 32.3009i	− 1.08456i	0.840199	−	0.542279i	$-0.182438\pi$
$888$	19.9103i	0.668147i
$889$	19.1636	0.642728
$890$	0	0
$891$	−14.6579	−0.491057
$892$	− 6.26287i	− 0.209696i
$893$	41.1740i	1.37783i
$894$	−43.2406	−1.44618
$895$	0	0
$896$	7.30364	0.243998
$897$	13.3537i	0.445868i
$898$	53.4404i	1.78333i
$899$	−2.24660	−0.0749284
$900$	0	0
$901$	56.6997	1.88894
$902$	32.5298i	1.08312i
$903$	0.475867i	0.0158359i
$904$	64.1969	2.13516
$905$	0	0
$906$	36.5287	1.21358
$907$	− 32.6364i	− 1.08367i	−0.840484	−	0.541836i	$-0.817729\pi$
$907$	− 32.6364i	− 1.08367i	0.840484	−	0.541836i	$-0.182271\pi$
$908$	− 32.0461i	− 1.06349i
$909$	14.4675	0.479855
$910$	0	0
$911$	14.5262	0.481276	0.240638	−	0.970615i	$-0.422643\pi$
$911$	14.5262	0.481276	0.240638	+	0.970615i	$0.422643\pi$
$912$	60.5579i	2.00527i
$913$	51.5023i	1.70448i
$914$	−61.1252	−2.02184
$915$	0	0
$916$	−4.22958	−0.139749
$917$	− 18.0366i	− 0.595622i
$918$	− 54.0836i	− 1.78503i
$919$	28.2136	0.930681	0.465341	−	0.885132i	$-0.345932\pi$
$919$	28.2136	0.930681	0.465341	+	0.885132i	$0.345932\pi$
$920$	0	0
$921$	−0.274460	−0.00904377
$922$	20.9903i	0.691279i
$923$	28.0801i	0.924266i
$924$	18.9038	0.621891
$925$	0	0
$926$	−40.1850	−1.32056
$927$	− 6.02866i	− 0.198007i
$928$	108.404i	3.55854i
$929$	2.45383	0.0805075	0.0402537	−	0.999189i	$-0.487183\pi$
$929$	2.45383	0.0805075	0.0402537	+	0.999189i	$0.487183\pi$
$930$	0	0
$931$	6.66244	0.218353
$932$	5.67681i	0.185950i
$933$	1.89292i	0.0619713i
$934$	10.2182	0.334351
$935$	0	0
$936$	−70.2113	−2.29493
$937$	− 36.6194i	− 1.19630i	−0.801383	−	0.598151i	$-0.795902\pi$
$937$	− 36.6194i	− 1.19630i	0.801383	−	0.598151i	$-0.204098\pi$
$938$	0.764047i	0.0249470i
$939$	−11.0400	−0.360276
$940$	0	0
$941$	14.7576	0.481085	0.240543	−	0.970639i	$-0.422675\pi$
$941$	14.7576	0.481085	0.240543	+	0.970639i	$0.422675\pi$
$942$	− 22.9794i	− 0.748708i
$943$	11.2383i	0.365969i
$944$	4.56721	0.148650
$945$	0	0
$946$	6.85610	0.222911
$947$	− 54.3926i	− 1.76752i	−0.467939	−	0.883761i	$-0.655003\pi$
$947$	− 54.3926i	− 1.76752i	0.467939	−	0.883761i	$-0.344997\pi$
$948$	− 18.1931i	− 0.590886i
$949$	−36.0521	−1.17030
$950$	0	0
$951$	0.989454	0.0320852
$952$	− 36.5254i	− 1.18380i
$953$	42.6983i	1.38313i	0.722313	+	0.691566i	$0.243079\pi$
$953$	42.6983i	1.38313i	−0.722313	+	0.691566i	$0.756921\pi$
$954$	−74.6792	−2.41783
$955$	0	0
$956$	33.6485	1.08827
$957$	31.5619i	1.02025i
$958$	96.4575i	3.11640i
$959$	−5.79298	−0.187065
$960$	0	0
$961$	−30.9272	−0.997652
$962$	− 30.8326i	− 0.994083i
$963$	− 0.407275i	− 0.0131243i
$964$	−34.3005	−1.10475
$965$	0	0
$966$	9.14976	0.294389
$967$	− 44.4649i	− 1.42989i	−0.699179	−	0.714947i	$-0.746451\pi$
$967$	− 44.4649i	− 1.42989i	0.699179	−	0.714947i	$-0.253549\pi$
$968$	− 76.2746i	− 2.45156i
$969$	25.6973	0.825517
$970$	0	0
$971$	26.5710	0.852704	0.426352	−	0.904557i	$-0.359799\pi$
$971$	26.5710	0.852704	0.426352	+	0.904557i	$0.359799\pi$
$972$	79.5996i	2.55316i
$973$	− 4.68119i	− 0.150072i
$974$	74.1510	2.37595
$975$	0	0
$976$	125.682	4.02299
$977$	− 33.1445i	− 1.06038i	−0.847877	−	0.530192i	$-0.822120\pi$
$977$	− 33.1445i	− 1.06038i	0.847877	−	0.530192i	$-0.177880\pi$
$978$	8.88283i	0.284042i
$979$	66.7155	2.13224
$980$	0	0
$981$	−24.8548	−0.793553
$982$	5.14635i	0.164227i
$983$	− 15.1403i	− 0.482900i	−0.970413	−	0.241450i	$-0.922377\pi$
$983$	− 15.1403i	− 0.482900i	0.970413	−	0.241450i	$-0.0776230\pi$
$984$	17.8299	0.568397
$985$	0	0
$986$	101.809	3.24226
$987$	− 5.15358i	− 0.164040i
$988$	− 128.193i	− 4.07837i
$989$	2.36863	0.0753179
$990$	0	0
$991$	3.95460	0.125622	0.0628109	−	0.998025i	$-0.479993\pi$
$991$	3.95460	0.125622	0.0628109	+	0.998025i	$0.479993\pi$
$992$	− 3.51227i	− 0.111515i
$993$	20.4667i	0.649492i
$994$	19.2400	0.610255
$995$	0	0
$996$	47.1274	1.49329
$997$	1.11672i	0.0353668i	0.999844	+	0.0176834i	$0.00562909\pi$
$997$	1.11672i	0.0353668i	−0.999844	+	0.0176834i	$0.994371\pi$
$998$	− 23.9015i	− 0.756587i
$999$	−13.3743	−0.423143

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	875.2.b.e.624.14		16
5.2	odd	4		875.2.a.j.1.1	yes	8
5.3	odd	4		875.2.a.i.1.8	✓	8
5.4	even	2	inner	875.2.b.e.624.3		16
15.2	even	4		7875.2.a.w.1.8		8
15.8	even	4		7875.2.a.bb.1.1		8
35.13	even	4		6125.2.a.v.1.8		8
35.27	even	4		6125.2.a.w.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
875.2.a.i.1.8	✓	8	5.3	odd	4
875.2.a.j.1.1	yes	8	5.2	odd	4
875.2.b.e.624.3		16	5.4	even	2	inner
875.2.b.e.624.14		16	1.1	even	1	trivial
6125.2.a.v.1.8		8	35.13	even	4
6125.2.a.w.1.1		8	35.27	even	4
7875.2.a.w.1.8		8	15.2	even	4
7875.2.a.bb.1.1		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)

\(f(q)\)	\(=\)	\(q+2.64338i q^{2} -0.833911i q^{3} -4.98745 q^{4} +2.20434 q^{6} +1.00000i q^{7} -7.89697i q^{8} +2.30459 q^{9} -4.54519 q^{11} +4.15909i q^{12} -3.85791i q^{13} -2.64338 q^{14} +10.8998 q^{16} -4.62525i q^{17} +6.09191i q^{18} -6.66244 q^{19} +0.833911 q^{21} -12.0147i q^{22} -4.15079i q^{23} -6.58537 q^{24} +10.1979 q^{26} -4.42356i q^{27} -4.98745i q^{28} +8.32707 q^{29} -0.269795 q^{31} +13.0183i q^{32} +3.79028i q^{33} +12.2263 q^{34} -11.4940 q^{36} -3.02342i q^{37} -17.6114i q^{38} -3.21716 q^{39} -2.70751 q^{41} +2.20434i q^{42} +0.570645i q^{43} +22.6689 q^{44} +10.9721 q^{46} -6.18001i q^{47} -9.08944i q^{48} -1.00000 q^{49} -3.85704 q^{51} +19.2412i q^{52} +12.2587i q^{53} +11.6931 q^{54} +7.89697 q^{56} +5.55588i q^{57} +22.0116i q^{58} +0.419019 q^{59} +11.5307 q^{61} -0.713171i q^{62} +2.30459i q^{63} -12.6127 q^{64} -10.0192 q^{66} -0.289042i q^{67} +23.0682i q^{68} -3.46139 q^{69} -7.27856 q^{71} -18.1993i q^{72} -9.34497i q^{73} +7.99204 q^{74} +33.2286 q^{76} -4.54519i q^{77} -8.50417i q^{78} -4.37430 q^{79} +3.22492 q^{81} -7.15697i q^{82} -11.3312i q^{83} -4.15909 q^{84} -1.50843 q^{86} -6.94404i q^{87} +35.8932i q^{88} -14.6783 q^{89} +3.85791 q^{91} +20.7018i q^{92} +0.224985i q^{93} +16.3361 q^{94} +10.8561 q^{96} -9.86723i q^{97} -2.64338i q^{98} -10.4748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 26 q^{4} + 4 q^{6} - 36 q^{9} - 10 q^{11} + 2 q^{14} + 70 q^{16} - 26 q^{19} + 16 q^{21} + 6 q^{24} - 10 q^{26} - 44 q^{29} + 6 q^{31} - 38 q^{34} + 54 q^{36} - 14 q^{39} + 24 q^{41} + 82 q^{44}+ \cdots + 62 q^{99}+O(q^{100}) \) 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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