LMFDB - Embedded newform 775.2.b.e.249.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [775,2,Mod(249,775)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("775.249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(775, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	775.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-18,0,16,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$6.18840615665$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.26394551296.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{5} + 28x^{4} - 12x^{3} + 2x^{2} + 8x + 16 $ x^8 - 2x^5 + 28x^4 - 12x^3 + 2x^2 + 8*x + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 155)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			249.8
Root			$-0.520627 - 0.520627i$ of defining polynomial
Character	$\chi$	$=$	775.249
Dual form			775.2.b.e.249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.80027i q^{2} -0.342376i q^{3} -5.84153 q^{4} +0.958747 q^{6} -1.04125i q^{7} -10.7573i q^{8} +2.88278 q^{9} +4.64180 q^{11} +2.00000i q^{12} +2.95875i q^{13} +2.91579 q^{14} +18.4404 q^{16} -6.29942i q^{17} +8.07256i q^{18} +1.11552 q^{19} -0.356500 q^{21} +12.9983i q^{22} +3.87454i q^{23} -3.68305 q^{24} -8.28530 q^{26} -2.01412i q^{27} +6.08251i q^{28} -2.35650 q^{29} +1.00000 q^{31} +30.1234i q^{32} -1.58924i q^{33} +17.6401 q^{34} -16.8398 q^{36} -1.30112i q^{37} +3.12376i q^{38} +1.01300 q^{39} +1.92573 q^{41} -0.998298i q^{42} -4.29942i q^{43} -27.1152 q^{44} -10.8498 q^{46} +3.31525i q^{47} -6.31355i q^{48} +5.91579 q^{49} -2.15677 q^{51} -17.2836i q^{52} +5.10964i q^{53} +5.64010 q^{54} -11.2011 q^{56} -0.381928i q^{57} -6.59884i q^{58} +5.07256 q^{59} +5.31525 q^{61} +2.80027i q^{62} -3.00170i q^{63} -47.4730 q^{64} +4.45031 q^{66} +8.64180i q^{67} +36.7982i q^{68} +1.32655 q^{69} -3.88278 q^{71} -31.0110i q^{72} +12.1740i q^{73} +3.64350 q^{74} -6.51634 q^{76} -4.83329i q^{77} +2.83669i q^{78} +7.12376 q^{79} +7.95875 q^{81} +5.39258i q^{82} -7.46614i q^{83} +2.08251 q^{84} +12.0396 q^{86} +0.806810i q^{87} -49.9333i q^{88} +12.1598 q^{89} +3.08080 q^{91} -22.6332i q^{92} -0.342376i q^{93} -9.28360 q^{94} +10.3135 q^{96} +0.915792i q^{97} +16.5658i q^{98} +13.3813 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 18 q^{4} + 16 q^{6} - 14 q^{9} - 12 q^{11} - 16 q^{14} + 22 q^{16} - 10 q^{19} + 4 q^{21} + 28 q^{24} - 24 q^{26} - 12 q^{29} + 8 q^{31} + 36 q^{34} - 50 q^{36} + 12 q^{39} + 26 q^{41} - 40 q^{44}+ \cdots + 104 q^{99}+O(q^{100}) $ 8 * q - 18 * q^4 + 16 * q^6 - 14 * q^9 - 12 * q^11 - 16 * q^14 + 22 * q^16 - 10 * q^19 + 4 * q^21 + 28 * q^24 - 24 * q^26 - 12 * q^29 + 8 * q^31 + 36 * q^34 - 50 * q^36 + 12 * q^39 + 26 * q^41 - 40 * q^44 - 52 * q^46 + 8 * q^49 + 10 * q^51 - 60 * q^54 - 8 * q^56 - 26 * q^59 + 44 * q^61 - 94 * q^64 - 40 * q^66 - 40 * q^69 + 6 * q^71 + 36 * q^74 + 28 * q^76 + 32 * q^79 + 72 * q^81 + 32 * q^86 + 24 * q^89 - 48 * q^91 + 24 * q^94 + 28 * q^96 + 104 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$.

$n$	$251$	$652$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.80027i	1.98009i	0.140746	+	0.990046i	$0.455050\pi$
$2$	2.80027i	1.98009i	−0.140746	+	0.990046i	$0.544950\pi$
$3$	− 0.342376i	− 0.197671i	−0.995104	−	0.0988355i	$-0.968488\pi$
$3$	− 0.342376i	− 0.197671i	0.995104	−	0.0988355i	$-0.0315118\pi$
$4$	−5.84153	−2.92076
$5$	0	0
$5$	0	0
$6$	0.958747	0.391407
$7$	− 1.04125i	− 0.393557i	−0.980448	−	0.196778i	$-0.936952\pi$
$7$	− 1.04125i	− 0.393557i	0.980448	−	0.196778i	$-0.0630479\pi$
$8$	− 10.7573i	− 3.80329i
$9$	2.88278	0.960926
$10$	0	0
$11$	4.64180	1.39955	0.699777	−	0.714361i	$-0.253283\pi$
$11$	4.64180	1.39955	0.699777	+	0.714361i	$0.253283\pi$
$12$	2.00000i	0.577350i
$13$	2.95875i	0.820609i	0.911949	+	0.410304i	$0.134578\pi$
$13$	2.95875i	0.820609i	−0.911949	+	0.410304i	$0.865422\pi$
$14$	2.91579	0.779278
$15$	0	0
$16$	18.4404	4.61009
$17$	− 6.29942i	− 1.52783i	−0.645314	−	0.763917i	$-0.723274\pi$
$17$	− 6.29942i	− 1.52783i	0.645314	−	0.763917i	$-0.276726\pi$
$18$	8.07256i	1.90272i
$19$	1.11552	0.255918	0.127959	−	0.991779i	$-0.459157\pi$
$19$	1.11552	0.255918	0.127959	+	0.991779i	$0.459157\pi$
$20$	0	0
$21$	−0.356500	−0.0777948
$22$	12.9983i	2.77125i
$23$	3.87454i	0.807897i	0.914782	+	0.403949i	$0.132363\pi$
$23$	3.87454i	0.807897i	−0.914782	+	0.403949i	$0.867637\pi$
$24$	−3.68305	−0.751800
$25$	0	0
$26$	−8.28530	−1.62488
$27$	− 2.01412i	− 0.387618i
$28$	6.08251i	1.14949i
$29$	−2.35650	−0.437591	−0.218796	−	0.975771i	$-0.570213\pi$
$29$	−2.35650	−0.437591	−0.218796	+	0.975771i	$0.570213\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	30.1234i	5.32512i
$33$	− 1.58924i	− 0.276651i
$34$	17.6401	3.02525
$35$	0	0
$36$	−16.8398	−2.80664
$37$	− 1.30112i	− 0.213903i	−0.994264	−	0.106952i	$-0.965891\pi$
$37$	− 1.30112i	− 0.213903i	0.994264	−	0.106952i	$-0.0341090\pi$
$38$	3.12376i	0.506741i
$39$	1.01300	0.162211
$40$	0	0
$41$	1.92573	0.300749	0.150375	−	0.988629i	$-0.451952\pi$
$41$	1.92573	0.300749	0.150375	+	0.988629i	$0.451952\pi$
$42$	− 0.998298i	− 0.154041i
$43$	− 4.29942i	− 0.655656i	−0.944737	−	0.327828i	$-0.893683\pi$
$43$	− 4.29942i	− 0.655656i	0.944737	−	0.327828i	$-0.106317\pi$
$44$	−27.1152	−4.08777
$45$	0	0
$46$	−10.8498	−1.59971
$47$	3.31525i	0.483579i	0.970329	+	0.241789i	$0.0777343\pi$
$47$	3.31525i	0.483579i	−0.970329	+	0.241789i	$0.922266\pi$
$48$	− 6.31355i	− 0.911282i
$49$	5.91579	0.845113
$50$	0	0
$51$	−2.15677	−0.302009
$52$	− 17.2836i	− 2.39680i
$53$	5.10964i	0.701862i	0.936401	+	0.350931i	$0.114135\pi$
$53$	5.10964i	0.701862i	−0.936401	+	0.350931i	$0.885865\pi$
$54$	5.64010	0.767520
$55$	0	0
$56$	−11.2011	−1.49681
$57$	− 0.381928i	− 0.0505875i
$58$	− 6.59884i	− 0.866471i
$59$	5.07256	0.660392	0.330196	−	0.943912i	$-0.392885\pi$
$59$	5.07256	0.660392	0.330196	+	0.943912i	$0.392885\pi$
$60$	0	0
$61$	5.31525	0.680548	0.340274	−	0.940326i	$-0.389480\pi$
$61$	5.31525	0.680548	0.340274	+	0.940326i	$0.389480\pi$
$62$	2.80027i	0.355635i
$63$	− 3.00170i	− 0.378179i
$64$	−47.4730	−5.93413
$65$	0	0
$66$	4.45031	0.547795
$67$	8.64180i	1.05576i	0.849318	+	0.527882i	$0.177014\pi$
$67$	8.64180i	1.05576i	−0.849318	+	0.527882i	$0.822986\pi$
$68$	36.7982i	4.46244i
$69$	1.32655	0.159698
$70$	0	0
$71$	−3.88278	−0.460801	−0.230401	−	0.973096i	$-0.574004\pi$
$71$	−3.88278	−0.460801	−0.230401	+	0.973096i	$0.574004\pi$
$72$	− 31.0110i	− 3.65468i
$73$	12.1740i	1.42485i	0.701746	+	0.712427i	$0.252404\pi$
$73$	12.1740i	1.42485i	−0.701746	+	0.712427i	$0.747596\pi$
$74$	3.64350	0.423548
$75$	0	0
$76$	−6.51634	−0.747475
$77$	− 4.83329i	− 0.550804i
$78$	2.83669i	0.321192i
$79$	7.12376	0.801486	0.400743	−	0.916191i	$-0.368752\pi$
$79$	7.12376	0.801486	0.400743	+	0.916191i	$0.368752\pi$
$80$	0	0
$81$	7.95875	0.884305
$82$	5.39258i	0.595511i
$83$	− 7.46614i	− 0.819515i	−0.912195	−	0.409757i	$-0.865613\pi$
$83$	− 7.46614i	− 0.819515i	0.912195	−	0.409757i	$-0.134387\pi$
$84$	2.08251	0.227220
$85$	0	0
$86$	12.0396	1.29826
$87$	0.806810i	0.0864991i
$88$	− 49.9333i	− 5.32291i
$89$	12.1598	1.28894	0.644470	−	0.764630i	$-0.277078\pi$
$89$	12.1598	1.28894	0.644470	+	0.764630i	$0.277078\pi$
$90$	0	0
$91$	3.08080	0.322956
$92$	− 22.6332i	− 2.35968i
$93$	− 0.342376i	− 0.0355028i
$94$	−9.28360	−0.957530
$95$	0	0
$96$	10.3135	1.05262
$97$	0.915792i	0.0929846i	0.998919	+	0.0464923i	$0.0148043\pi$
$97$	0.915792i	0.0929846i	−0.998919	+	0.0464923i	$0.985196\pi$
$98$	16.5658i	1.67340i
$99$	13.3813	1.34487
$100$	0	0
$101$	−4.11382	−0.409340	−0.204670	−	0.978831i	$-0.565612\pi$
$101$	−4.11382	−0.409340	−0.204670	+	0.978831i	$0.565612\pi$
$102$	− 6.03955i	− 0.598005i
$103$	− 16.9983i	− 1.67489i	−0.546520	−	0.837446i	$-0.684048\pi$
$103$	− 16.9983i	− 1.67489i	0.546520	−	0.837446i	$-0.315952\pi$
$104$	31.8282	3.12101
$105$	0	0
$106$	−14.3084	−1.38975
$107$	− 4.84976i	− 0.468844i	−0.972135	−	0.234422i	$-0.924680\pi$
$107$	− 4.84976i	− 0.468844i	0.972135	−	0.234422i	$-0.0753198\pi$
$108$	11.7656i	1.13214i
$109$	−9.19803	−0.881011	−0.440506	−	0.897750i	$-0.645201\pi$
$109$	−9.19803	−0.881011	−0.440506	+	0.897750i	$0.645201\pi$
$110$	0	0
$111$	−0.445474	−0.0422825
$112$	− 19.2011i	− 1.81433i
$113$	3.47508i	0.326908i	0.986551	+	0.163454i	$0.0522636\pi$
$113$	3.47508i	0.326908i	−0.986551	+	0.163454i	$0.947736\pi$
$114$	1.06950	0.100168
$115$	0	0
$116$	13.7656	1.27810
$117$	8.52941i	0.788544i
$118$	14.2046i	1.30764i
$119$	−6.55929	−0.601289
$120$	0	0
$121$	10.5463	0.958753
$122$	14.8841i	1.34755i
$123$	− 0.659325i	− 0.0594494i
$124$	−5.84153	−0.524584
$125$	0	0
$126$	8.40558	0.748829
$127$	− 12.8728i	− 1.14228i	−0.820853	−	0.571140i	$-0.806501\pi$
$127$	− 12.8728i	− 1.14228i	0.820853	−	0.571140i	$-0.193499\pi$
$128$	− 72.6906i	− 6.42500i
$129$	−1.47202	−0.129604
$130$	0	0
$131$	15.4668	1.35134	0.675672	−	0.737202i	$-0.263853\pi$
$131$	15.4668	1.35134	0.675672	+	0.737202i	$0.263853\pi$
$132$	9.28360i	0.808033i
$133$	− 1.16154i	− 0.100718i
$134$	−24.1994	−2.09051
$135$	0	0
$136$	−67.7649	−5.81079
$137$	16.5734i	1.41596i	0.706231	+	0.707981i	$0.250394\pi$
$137$	16.5734i	1.41596i	−0.706231	+	0.707981i	$0.749606\pi$
$138$	3.71470i	0.316216i
$139$	−21.5259	−1.82581	−0.912903	−	0.408176i	$-0.866165\pi$
$139$	−21.5259	−1.82581	−0.912903	+	0.408176i	$0.866165\pi$
$140$	0	0
$141$	1.13506	0.0955895
$142$	− 10.8728i	− 0.912428i
$143$	13.7339i	1.14849i
$144$	53.1595	4.42996
$145$	0	0
$146$	−34.0904	−2.82134
$147$	− 2.02543i	− 0.167054i
$148$	7.60054i	0.624761i
$149$	−20.7556	−1.70037	−0.850183	−	0.526487i	$-0.823509\pi$
$149$	−20.7556	−1.70037	−0.850183	+	0.526487i	$0.823509\pi$
$150$	0	0
$151$	−18.4451	−1.50104	−0.750522	−	0.660846i	$-0.770198\pi$
$151$	−18.4451	−1.50104	−0.750522	+	0.660846i	$0.770198\pi$
$152$	− 12.0000i	− 0.973329i
$153$	− 18.1598i	− 1.46814i
$154$	13.5345	1.09064
$155$	0	0
$156$	−5.91749	−0.473779
$157$	− 5.64350i	− 0.450400i	−0.974313	−	0.225200i	$-0.927696\pi$
$157$	− 5.64350i	− 0.450400i	0.974313	−	0.225200i	$-0.0723036\pi$
$158$	19.9485i	1.58701i
$159$	1.74942	0.138738
$160$	0	0
$161$	4.03438	0.317953
$162$	22.2867i	1.75101i
$163$	− 16.1994i	− 1.26883i	−0.772991	−	0.634417i	$-0.781240\pi$
$163$	− 16.1994i	− 1.26883i	0.772991	−	0.634417i	$-0.218760\pi$
$164$	−11.2492	−0.878417
$165$	0	0
$166$	20.9072	1.62271
$167$	− 5.14919i	− 0.398456i	−0.979953	−	0.199228i	$-0.936157\pi$
$167$	− 5.14919i	− 0.398456i	0.979953	−	0.199228i	$-0.0638434\pi$
$168$	3.83499i	0.295876i
$169$	4.24582	0.326601
$170$	0	0
$171$	3.21580	0.245918
$172$	25.1152i	1.91501i
$173$	10.7621i	0.818226i	0.912484	+	0.409113i	$0.134162\pi$
$173$	10.7621i	0.818226i	−0.912484	+	0.409113i	$0.865838\pi$
$174$	−2.25929	−0.171276
$175$	0	0
$176$	85.5965	6.45208
$177$	− 1.73673i	− 0.130540i
$178$	34.0509i	2.55222i
$179$	10.2740	0.767914	0.383957	−	0.923351i	$-0.374561\pi$
$179$	10.2740	0.767914	0.383957	+	0.923351i	$0.374561\pi$
$180$	0	0
$181$	2.53622	0.188516	0.0942578	−	0.995548i	$-0.469952\pi$
$181$	2.53622	0.188516	0.0942578	+	0.995548i	$0.469952\pi$
$182$	8.62709i	0.639483i
$183$	− 1.81981i	− 0.134525i
$184$	41.6796	3.07266
$185$	0	0
$186$	0.958747	0.0702987
$187$	− 29.2406i	− 2.13829i
$188$	− 19.3661i	− 1.41242i
$189$	−2.09721	−0.152550
$190$	0	0
$191$	−4.45031	−0.322013	−0.161007	−	0.986953i	$-0.551474\pi$
$191$	−4.45031	−0.322013	−0.161007	+	0.986953i	$0.551474\pi$
$192$	16.2536i	1.17301i
$193$	− 2.71300i	− 0.195286i	−0.995222	−	0.0976430i	$-0.968870\pi$
$193$	− 2.71300i	− 0.195286i	0.995222	−	0.0976430i	$-0.0311303\pi$
$194$	−2.56447	−0.184118
$195$	0	0
$196$	−34.5573	−2.46838
$197$	5.04125i	0.359174i	0.983742	+	0.179587i	$0.0574762\pi$
$197$	5.04125i	0.359174i	−0.983742	+	0.179587i	$0.942524\pi$
$198$	37.4712i	2.66296i
$199$	14.3531	1.01746	0.508732	−	0.860925i	$-0.330114\pi$
$199$	14.3531	1.01746	0.508732	+	0.860925i	$0.330114\pi$
$200$	0	0
$201$	2.95875	0.208694
$202$	− 11.5198i	− 0.810531i
$203$	2.45371i	0.172217i
$204$	12.5988	0.882095
$205$	0	0
$206$	47.5999	3.31644
$207$	11.1694i	0.776330i
$208$	54.5604i	3.78308i
$209$	5.17802	0.358171
$210$	0	0
$211$	−11.6288	−0.800559	−0.400280	−	0.916393i	$-0.631087\pi$
$211$	−11.6288	−0.800559	−0.400280	+	0.916393i	$0.631087\pi$
$212$	− 29.8481i	− 2.04997i
$213$	1.32937i	0.0910870i
$214$	13.5807	0.928355
$215$	0	0
$216$	−21.6666	−1.47422
$217$	− 1.04125i	− 0.0706849i
$218$	− 25.7570i	− 1.74448i
$219$	4.16808	0.281652
$220$	0	0
$221$	18.6384	1.25375
$222$	− 1.24745i	− 0.0837232i
$223$	5.45483i	0.365283i	0.983180	+	0.182641i	$0.0584647\pi$
$223$	5.45483i	0.365283i	−0.983180	+	0.182641i	$0.941535\pi$
$224$	31.3661	2.09574
$225$	0	0
$226$	−9.73118	−0.647309
$227$	7.97352i	0.529221i	0.964355	+	0.264611i	$0.0852434\pi$
$227$	7.97352i	0.529221i	−0.964355	+	0.264611i	$0.914757\pi$
$228$	2.23104i	0.147754i
$229$	−22.0197	−1.45510	−0.727550	−	0.686054i	$-0.759341\pi$
$229$	−22.0197	−1.45510	−0.727550	+	0.686054i	$0.759341\pi$
$230$	0	0
$231$	−1.65480	−0.108878
$232$	25.3496i	1.66428i
$233$	− 17.2723i	− 1.13155i	−0.824561	−	0.565773i	$-0.808578\pi$
$233$	− 17.2723i	− 1.13155i	0.824561	−	0.565773i	$-0.191422\pi$
$234$	−23.8847	−1.56139
$235$	0	0
$236$	−29.6315	−1.92885
$237$	− 2.43901i	− 0.158430i
$238$	− 18.3678i	− 1.19061i
$239$	21.3462	1.38077	0.690386	−	0.723441i	$-0.257441\pi$
$239$	21.3462	1.38077	0.690386	+	0.723441i	$0.257441\pi$
$240$	0	0
$241$	−4.15984	−0.267959	−0.133979	−	0.990984i	$-0.542776\pi$
$241$	−4.15984	−0.267959	−0.133979	+	0.990984i	$0.542776\pi$
$242$	29.5325i	1.89842i
$243$	− 8.76726i	− 0.562420i
$244$	−31.0492	−1.98772
$245$	0	0
$246$	1.84629	0.117715
$247$	3.30054i	0.210008i
$248$	− 10.7573i	− 0.683090i
$249$	−2.55623	−0.161994
$250$	0	0
$251$	5.76896	0.364134	0.182067	−	0.983286i	$-0.441721\pi$
$251$	5.76896	0.364134	0.182067	+	0.983286i	$0.441721\pi$
$252$	17.5345i	1.10457i
$253$	17.9848i	1.13070i
$254$	36.0475	2.26182
$255$	0	0
$256$	108.607	6.78796
$257$	22.4417i	1.39988i	0.714203	+	0.699938i	$0.246789\pi$
$257$	22.4417i	1.39988i	−0.714203	+	0.699938i	$0.753211\pi$
$258$	− 4.12206i	− 0.256628i
$259$	−1.35480	−0.0841831
$260$	0	0
$261$	−6.79327	−0.420493
$262$	43.3114i	2.67579i
$263$	10.1904i	0.628369i	0.949362	+	0.314185i	$0.101731\pi$
$263$	10.1904i	0.628369i	−0.949362	+	0.314185i	$0.898269\pi$
$264$	−17.0960	−1.05218
$265$	0	0
$266$	3.25262	0.199431
$267$	− 4.16324i	− 0.254786i
$268$	− 50.4813i	− 3.08364i
$269$	−28.0904	−1.71270	−0.856351	−	0.516394i	$-0.827274\pi$
$269$	−28.0904	−1.71270	−0.856351	+	0.516394i	$0.827274\pi$
$270$	0	0
$271$	0.0429548	0.00260932	0.00130466	−	0.999999i	$-0.499585\pi$
$271$	0.0429548	0.00260932	0.00130466	+	0.999999i	$0.499585\pi$
$272$	− 116.164i	− 7.04346i
$273$	− 1.05479i	− 0.0638391i
$274$	−46.4101	−2.80374
$275$	0	0
$276$	−7.74908	−0.466440
$277$	21.8288i	1.31156i	0.754951	+	0.655782i	$0.227661\pi$
$277$	21.8288i	1.31156i	−0.754951	+	0.655782i	$0.772339\pi$
$278$	− 60.2785i	− 3.61526i
$279$	2.88278	0.172587
$280$	0	0
$281$	−9.20415	−0.549074	−0.274537	−	0.961577i	$-0.588525\pi$
$281$	−9.20415	−0.549074	−0.274537	+	0.961577i	$0.588525\pi$
$282$	3.17848i	0.189276i
$283$	− 6.85317i	− 0.407379i	−0.979036	−	0.203689i	$-0.934707\pi$
$283$	− 6.85317i	− 0.407379i	0.979036	−	0.203689i	$-0.0652932\pi$
$284$	22.6813	1.34589
$285$	0	0
$286$	−38.4587	−2.27411
$287$	− 2.00518i	− 0.118362i
$288$	86.8391i	5.11705i
$289$	−22.6827	−1.33428
$290$	0	0
$291$	0.313546	0.0183804
$292$	− 71.1145i	− 4.16166i
$293$	− 16.6101i	− 0.970375i	−0.874410	−	0.485188i	$-0.838751\pi$
$293$	− 16.6101i	− 0.970375i	0.874410	−	0.485188i	$-0.161249\pi$
$294$	5.67175	0.330783
$295$	0	0
$296$	−13.9966	−0.813536
$297$	− 9.34916i	− 0.542493i
$298$	− 58.1214i	− 3.36688i
$299$	−11.4638	−0.662968
$300$	0	0
$301$	−4.47679	−0.258038
$302$	− 51.6514i	− 2.97220i
$303$	1.40847i	0.0809147i
$304$	20.5706	1.17980
$305$	0	0
$306$	50.8525	2.90704
$307$	− 16.9553i	− 0.967693i	−0.875153	−	0.483846i	$-0.839239\pi$
$307$	− 16.9553i	− 0.967693i	0.875153	−	0.483846i	$-0.160761\pi$
$308$	28.2338i	1.60877i
$309$	−5.81981	−0.331078
$310$	0	0
$311$	−0.638734	−0.0362193	−0.0181096	−	0.999836i	$-0.505765\pi$
$311$	−0.638734	−0.0362193	−0.0181096	+	0.999836i	$0.505765\pi$
$312$	− 10.8972i	− 0.616933i
$313$	− 18.1141i	− 1.02387i	−0.859025	−	0.511934i	$-0.828929\pi$
$313$	− 18.1141i	− 1.02387i	0.859025	−	0.511934i	$-0.171071\pi$
$314$	15.8033	0.891834
$315$	0	0
$316$	−41.6136	−2.34095
$317$	5.06433i	0.284441i	0.989835	+	0.142220i	$0.0454242\pi$
$317$	5.06433i	0.284441i	−0.989835	+	0.142220i	$0.954576\pi$
$318$	4.89885i	0.274714i
$319$	−10.9384	−0.612433
$320$	0	0
$321$	−1.66044	−0.0926770
$322$	11.2974i	0.629577i
$323$	− 7.02713i	− 0.391000i
$324$	−46.4912	−2.58285
$325$	0	0
$326$	45.3627	2.51241
$327$	3.14919i	0.174150i
$328$	− 20.7157i	− 1.14383i
$329$	3.45201	0.190316
$330$	0	0
$331$	−20.0344	−1.10119	−0.550594	−	0.834773i	$-0.685599\pi$
$331$	−20.0344	−1.10119	−0.550594	+	0.834773i	$0.685599\pi$
$332$	43.6136i	2.39361i
$333$	− 3.75085i	− 0.205545i
$334$	14.4191	0.788979
$335$	0	0
$336$	−6.57400	−0.358641
$337$	4.86729i	0.265138i	0.991174	+	0.132569i	$0.0423227\pi$
$337$	4.86729i	0.265138i	−0.991174	+	0.132569i	$0.957677\pi$
$338$	11.8894i	0.646700i
$339$	1.18979	0.0646203
$340$	0	0
$341$	4.64180	0.251367
$342$	9.00511i	0.486940i
$343$	− 13.4486i	− 0.726157i
$344$	−46.2502	−2.49365
$345$	0	0
$346$	−30.1368	−1.62016
$347$	10.1221i	0.543381i	0.962385	+	0.271690i	$0.0875826\pi$
$347$	10.1221i	0.543381i	−0.962385	+	0.271690i	$0.912417\pi$
$348$	− 4.71300i	− 0.252643i
$349$	18.3245	0.980888	0.490444	−	0.871473i	$-0.336835\pi$
$349$	18.3245	0.980888	0.490444	+	0.871473i	$0.336835\pi$
$350$	0	0
$351$	5.95928	0.318083
$352$	139.827i	7.45279i
$353$	− 14.3248i	− 0.762435i	−0.924485	−	0.381217i	$-0.875505\pi$
$353$	− 14.3248i	− 0.762435i	0.924485	−	0.381217i	$-0.124495\pi$
$354$	4.86331	0.258482
$355$	0	0
$356$	−71.0320	−3.76469
$357$	2.24575i	0.118857i
$358$	28.7700i	1.52054i
$359$	−12.3197	−0.650207	−0.325104	−	0.945678i	$-0.605399\pi$
$359$	−12.3197	−0.650207	−0.325104	+	0.945678i	$0.605399\pi$
$360$	0	0
$361$	−17.7556	−0.934506
$362$	7.10210i	0.373278i
$363$	− 3.61080i	− 0.189518i
$364$	−17.9966	−0.943278
$365$	0	0
$366$	5.09598	0.266371
$367$	− 24.4706i	− 1.27735i	−0.769475	−	0.638676i	$-0.779482\pi$
$367$	− 24.4706i	− 1.27735i	0.769475	−	0.638676i	$-0.220518\pi$
$368$	71.4479i	3.72448i
$369$	5.55146	0.288998
$370$	0	0
$371$	5.32042	0.276223
$372$	2.00000i	0.103695i
$373$	26.5276i	1.37355i	0.726870	+	0.686775i	$0.240974\pi$
$373$	26.5276i	1.37355i	−0.726870	+	0.686775i	$0.759026\pi$
$374$	81.8818	4.23400
$375$	0	0
$376$	35.6632	1.83919
$377$	− 6.97229i	− 0.359091i
$378$	− 5.87277i	− 0.302063i
$379$	−12.5528	−0.644795	−0.322398	−	0.946604i	$-0.604489\pi$
$379$	−12.5528	−0.644795	−0.322398	+	0.946604i	$0.604489\pi$
$380$	0	0
$381$	−4.40735	−0.225796
$382$	− 12.4621i	− 0.637615i
$383$	11.5882i	0.592129i	0.955168	+	0.296064i	$0.0956743\pi$
$383$	11.5882i	0.592129i	−0.955168	+	0.296064i	$0.904326\pi$
$384$	−24.8875	−1.27004
$385$	0	0
$386$	7.59714	0.386684
$387$	− 12.3943i	− 0.630037i
$388$	− 5.34962i	− 0.271586i
$389$	−32.5790	−1.65182	−0.825909	−	0.563803i	$-0.809338\pi$
$389$	−32.5790	−1.65182	−0.825909	+	0.563803i	$0.809338\pi$
$390$	0	0
$391$	24.4074	1.23433
$392$	− 63.6381i	− 3.21421i
$393$	− 5.29548i	− 0.267122i
$394$	−14.1169	−0.711198
$395$	0	0
$396$	−78.1671	−3.92804
$397$	− 29.8508i	− 1.49817i	−0.662475	−	0.749084i	$-0.730494\pi$
$397$	− 29.8508i	− 1.49817i	0.662475	−	0.749084i	$-0.269506\pi$
$398$	40.1926i	2.01467i
$399$	−0.397683	−0.0199091
$400$	0	0
$401$	−24.9176	−1.24432	−0.622162	−	0.782889i	$-0.713745\pi$
$401$	−24.9176	−1.24432	−0.622162	+	0.782889i	$0.713745\pi$
$402$	8.28530i	0.413233i
$403$	2.95875i	0.147386i
$404$	24.0310	1.19559
$405$	0	0
$406$	−6.87107	−0.341005
$407$	− 6.03955i	− 0.299369i
$408$	23.2011i	1.14863i
$409$	−13.8085	−0.682787	−0.341393	−	0.939920i	$-0.610899\pi$
$409$	−13.8085	−0.682787	−0.341393	+	0.939920i	$0.610899\pi$
$410$	0	0
$411$	5.67435	0.279895
$412$	99.2960i	4.89196i
$413$	− 5.28182i	− 0.259902i
$414$	−31.2775	−1.53720
$415$	0	0
$416$	−89.1276	−4.36984
$417$	7.36997i	0.360909i
$418$	14.4999i	0.709211i
$419$	21.7799	1.06402	0.532009	−	0.846738i	$-0.321437\pi$
$419$	21.7799	1.06402	0.532009	+	0.846738i	$0.321437\pi$
$420$	0	0
$421$	3.23758	0.157790	0.0788949	−	0.996883i	$-0.474861\pi$
$421$	3.23758	0.157790	0.0788949	+	0.996883i	$0.474861\pi$
$422$	− 32.5638i	− 1.58518i
$423$	9.55712i	0.464683i
$424$	54.9660	2.66938
$425$	0	0
$426$	−3.72260	−0.180361
$427$	− 5.53452i	− 0.267834i
$428$	28.3300i	1.36938i
$429$	4.70216	0.227023
$430$	0	0
$431$	−12.6466	−0.609167	−0.304583	−	0.952486i	$-0.598517\pi$
$431$	−12.6466	−0.609167	−0.304583	+	0.952486i	$0.598517\pi$
$432$	− 37.1412i	− 1.78696i
$433$	31.9914i	1.53741i	0.639604	+	0.768705i	$0.279098\pi$
$433$	31.9914i	1.53741i	−0.639604	+	0.768705i	$0.720902\pi$
$434$	2.91579	0.139963
$435$	0	0
$436$	53.7305	2.57322
$437$	4.32212i	0.206755i
$438$	11.6717i	0.557698i
$439$	13.8432	0.660701	0.330351	−	0.943858i	$-0.392833\pi$
$439$	13.8432	0.660701	0.330351	+	0.943858i	$0.392833\pi$
$440$	0	0
$441$	17.0539	0.812091
$442$	52.1926i	2.48255i
$443$	− 20.3479i	− 0.966759i	−0.875411	−	0.483379i	$-0.839409\pi$
$443$	− 20.3479i	− 0.966759i	0.875411	−	0.483379i	$-0.160591\pi$
$444$	2.60225	0.123497
$445$	0	0
$446$	−15.2750	−0.723293
$447$	7.10623i	0.336113i
$448$	49.4314i	2.33542i
$449$	−8.71640	−0.411353	−0.205676	−	0.978620i	$-0.565939\pi$
$449$	−8.71640	−0.411353	−0.205676	+	0.978620i	$0.565939\pi$
$450$	0	0
$451$	8.93886	0.420915
$452$	− 20.2998i	− 0.954822i
$453$	6.31518i	0.296713i
$454$	−22.3280	−1.04791
$455$	0	0
$456$	−4.10852	−0.192399
$457$	0.0311855i	0.00145879i	1.00000		0.000729397i	$0.000232174\pi$
$457$	0.0311855i	0.00145879i	−1.00000		0.000729397i	$0.999768\pi$
$458$	− 61.6611i	− 2.88123i
$459$	−12.6878	−0.592217
$460$	0	0
$461$	−29.4882	−1.37340	−0.686700	−	0.726941i	$-0.740941\pi$
$461$	−29.4882	−1.37340	−0.686700	+	0.726941i	$0.740941\pi$
$462$	− 4.63390i	− 0.215588i
$463$	− 19.1547i	− 0.890196i	−0.895482	−	0.445098i	$-0.853169\pi$
$463$	− 19.1547i	− 0.890196i	0.895482	−	0.445098i	$-0.146831\pi$
$464$	−43.4547	−2.01734
$465$	0	0
$466$	48.3671	2.24056
$467$	23.9367i	1.10766i	0.832630	+	0.553829i	$0.186834\pi$
$467$	23.9367i	1.10766i	−0.832630	+	0.553829i	$0.813166\pi$
$468$	− 49.8248i	− 2.30315i
$469$	8.99830	0.415503
$470$	0	0
$471$	−1.93220	−0.0890311
$472$	− 54.5672i	− 2.51166i
$473$	− 19.9570i	− 0.917626i
$474$	6.82988	0.313707
$475$	0	0
$476$	38.3163	1.75622
$477$	14.7299i	0.674438i
$478$	59.7752i	2.73406i
$479$	−21.4039	−0.977968	−0.488984	−	0.872293i	$-0.662632\pi$
$479$	−21.4039	−0.977968	−0.488984	+	0.872293i	$0.662632\pi$
$480$	0	0
$481$	3.84969	0.175531
$482$	− 11.6487i	− 0.530583i
$483$	− 1.38127i	− 0.0628502i
$484$	−61.6064	−2.80029
$485$	0	0
$486$	24.5507	1.11364
$487$	− 26.0763i	− 1.18163i	−0.806807	−	0.590815i	$-0.798806\pi$
$487$	− 26.0763i	− 1.18163i	0.806807	−	0.590815i	$-0.201194\pi$
$488$	− 57.1778i	− 2.58832i
$489$	−5.54629	−0.250812
$490$	0	0
$491$	−11.9367	−0.538696	−0.269348	−	0.963043i	$-0.586808\pi$
$491$	−11.9367	−0.538696	−0.269348	+	0.963043i	$0.586808\pi$
$492$	3.85147i	0.173638i
$493$	14.8446i	0.668567i
$494$	−9.24241	−0.415836
$495$	0	0
$496$	18.4404	0.827997
$497$	4.04295i	0.181351i
$498$	− 7.15813i	− 0.320764i
$499$	−16.5394	−0.740406	−0.370203	−	0.928951i	$-0.620712\pi$
$499$	−16.5394	−0.740406	−0.370203	+	0.928951i	$0.620712\pi$
$500$	0	0
$501$	−1.76296	−0.0787632
$502$	16.1547i	0.721018i
$503$	16.6418i	0.742021i	0.928629	+	0.371011i	$0.120989\pi$
$503$	16.6418i	0.742021i	−0.928629	+	0.371011i	$0.879011\pi$
$504$	−32.2903	−1.43832
$505$	0	0
$506$	−50.3624	−2.23888
$507$	− 1.45367i	− 0.0645596i
$508$	75.1970i	3.33633i
$509$	43.0388	1.90766	0.953831	−	0.300345i	$-0.0971018\pi$
$509$	43.0388	1.90766	0.953831	+	0.300345i	$0.0971018\pi$
$510$	0	0
$511$	12.6762	0.560761
$512$	158.749i	7.01579i
$513$	− 2.24680i	− 0.0991984i
$514$	−62.8430	−2.77188
$515$	0	0
$516$	8.59884	0.378543
$517$	15.3887i	0.676795i
$518$	− 3.79381i	− 0.166690i
$519$	3.68468	0.161740
$520$	0	0
$521$	37.2138	1.63037	0.815184	−	0.579203i	$-0.196636\pi$
$521$	37.2138	1.63037	0.815184	+	0.579203i	$0.196636\pi$
$522$	− 19.0230i	− 0.832614i
$523$	− 23.4085i	− 1.02358i	−0.859110	−	0.511791i	$-0.828982\pi$
$523$	− 23.4085i	− 1.02358i	0.859110	−	0.511791i	$-0.171018\pi$
$524$	−90.3500	−3.94696
$525$	0	0
$526$	−28.5360	−1.24423
$527$	− 6.29942i	− 0.274407i
$528$	− 29.3062i	− 1.27539i
$529$	7.98795	0.347302
$530$	0	0
$531$	14.6231	0.634588
$532$	6.78516i	0.294174i
$533$	5.69776i	0.246797i
$534$	11.6582	0.504500
$535$	0	0
$536$	92.9626	4.01537
$537$	− 3.51757i	− 0.151794i
$538$	− 78.6608i	− 3.39131i
$539$	27.4599	1.18278
$540$	0	0
$541$	2.38298	0.102452	0.0512261	−	0.998687i	$-0.483687\pi$
$541$	2.38298	0.102452	0.0512261	+	0.998687i	$0.483687\pi$
$542$	0.120285i	0.00516669i
$543$	− 0.868341i	− 0.0372641i
$544$	189.760	8.13590
$545$	0	0
$546$	2.95371	0.126407
$547$	21.4400i	0.916709i	0.888769	+	0.458355i	$0.151561\pi$
$547$	21.4400i	0.916709i	−0.888769	+	0.458355i	$0.848439\pi$
$548$	− 96.8140i	− 4.13569i
$549$	15.3227	0.653956
$550$	0	0
$551$	−2.62872	−0.111987
$552$	− 14.2701i	− 0.607377i
$553$	− 7.41764i	− 0.315430i
$554$	−61.1265	−2.59702
$555$	0	0
$556$	125.744	5.33275
$557$	12.5276i	0.530813i	0.964137	+	0.265407i	$0.0855062\pi$
$557$	12.5276i	0.530813i	−0.964137	+	0.265407i	$0.914494\pi$
$558$	8.07256i	0.341739i
$559$	12.7209	0.538037
$560$	0	0
$561$	−10.0113	−0.422678
$562$	− 25.7741i	− 1.08722i
$563$	10.7361i	0.452472i	0.974073	+	0.226236i	$0.0726420\pi$
$563$	10.7361i	0.452472i	−0.974073	+	0.226236i	$0.927358\pi$
$564$	−6.63049	−0.279194
$565$	0	0
$566$	19.1907	0.806647
$567$	− 8.28707i	− 0.348024i
$568$	41.7683i	1.75256i
$569$	−1.00783	−0.0422504	−0.0211252	−	0.999777i	$-0.506725\pi$
$569$	−1.00783	−0.0422504	−0.0211252	+	0.999777i	$0.506725\pi$
$570$	0	0
$571$	−9.88631	−0.413729	−0.206865	−	0.978370i	$-0.566326\pi$
$571$	−9.88631	−0.413729	−0.206865	+	0.978370i	$0.566326\pi$
$572$	− 80.2270i	− 3.35446i
$573$	1.52368i	0.0636527i
$574$	5.61504	0.234367
$575$	0	0
$576$	−136.854	−5.70226
$577$	− 22.1564i	− 0.922384i	−0.887300	−	0.461192i	$-0.847422\pi$
$577$	− 22.1564i	− 0.922384i	0.887300	−	0.461192i	$-0.152578\pi$
$578$	− 63.5178i	− 2.64199i
$579$	−0.928867	−0.0386024
$580$	0	0
$581$	−7.77414	−0.322526
$582$	0.878013i	0.0363948i
$583$	23.7179i	0.982295i
$584$	130.959	5.41913
$585$	0	0
$586$	46.5129	1.92143
$587$	− 16.7526i	− 0.691452i	−0.938336	−	0.345726i	$-0.887633\pi$
$587$	− 16.7526i	− 0.691452i	0.938336	−	0.345726i	$-0.112367\pi$
$588$	11.8316i	0.487926i
$589$	1.11552	0.0459642
$590$	0	0
$591$	1.72601	0.0709984
$592$	− 23.9932i	− 0.986114i
$593$	− 21.0526i	− 0.864525i	−0.901748	−	0.432262i	$-0.857715\pi$
$593$	− 21.0526i	− 0.864525i	0.901748	−	0.432262i	$-0.142285\pi$
$594$	26.1802	1.07419
$595$	0	0
$596$	121.244	4.96637
$597$	− 4.91416i	− 0.201123i
$598$	− 32.1017i	− 1.31274i
$599$	24.1994	0.988760	0.494380	−	0.869246i	$-0.335395\pi$
$599$	24.1994	0.988760	0.494380	+	0.869246i	$0.335395\pi$
$600$	0	0
$601$	−37.0275	−1.51038	−0.755192	−	0.655504i	$-0.772456\pi$
$601$	−37.0275	−1.51038	−0.755192	+	0.655504i	$0.772456\pi$
$602$	− 12.5362i	− 0.510938i
$603$	24.9124i	1.01451i
$604$	107.748	4.38419
$605$	0	0
$606$	−3.94411	−0.160219
$607$	10.9011i	0.442461i	0.975222	+	0.221231i	$0.0710074\pi$
$607$	10.9011i	0.442461i	−0.975222	+	0.221231i	$0.928993\pi$
$608$	33.6033i	1.36279i
$609$	0.840093	0.0340423
$610$	0	0
$611$	−9.80898	−0.396829
$612$	106.081i	4.28808i
$613$	17.8882i	0.722497i	0.932469	+	0.361249i	$0.117649\pi$
$613$	17.8882i	0.722497i	−0.932469	+	0.361249i	$0.882351\pi$
$614$	47.4796	1.91612
$615$	0	0
$616$	−51.9932	−2.09487
$617$	39.3548i	1.58436i	0.610285	+	0.792182i	$0.291055\pi$
$617$	39.3548i	1.58436i	−0.610285	+	0.792182i	$0.708945\pi$
$618$	− 16.2971i	− 0.655564i
$619$	−19.5429	−0.785495	−0.392747	−	0.919646i	$-0.628475\pi$
$619$	−19.5429	−0.785495	−0.392747	+	0.919646i	$0.628475\pi$
$620$	0	0
$621$	7.80380	0.313156
$622$	− 1.78863i	− 0.0717175i
$623$	− 12.6615i	− 0.507271i
$624$	18.6802	0.747806
$625$	0	0
$626$	50.7243	2.02735
$627$	− 1.77283i	− 0.0708000i
$628$	32.9666i	1.31551i
$629$	−8.19632	−0.326809
$630$	0	0
$631$	−33.4486	−1.33157	−0.665784	−	0.746145i	$-0.731903\pi$
$631$	−33.4486	−1.33157	−0.665784	+	0.746145i	$0.731903\pi$
$632$	− 76.6325i	− 3.04828i
$633$	3.98142i	0.158247i
$634$	−14.1815	−0.563219
$635$	0	0
$636$	−10.2193	−0.405220
$637$	17.5033i	0.693507i
$638$	− 30.6305i	− 1.21267i
$639$	−11.1932	−0.442796
$640$	0	0
$641$	7.13506	0.281818	0.140909	−	0.990023i	$-0.454998\pi$
$641$	7.13506	0.281818	0.140909	+	0.990023i	$0.454998\pi$
$642$	− 4.64970i	− 0.183509i
$643$	21.1141i	0.832660i	0.909214	+	0.416330i	$0.136684\pi$
$643$	21.1141i	0.832660i	−0.909214	+	0.416330i	$0.863316\pi$
$644$	−23.5669	−0.928666
$645$	0	0
$646$	19.6779	0.774216
$647$	1.08185i	0.0425320i	0.999774	+	0.0212660i	$0.00676969\pi$
$647$	1.08185i	0.0425320i	−0.999774	+	0.0212660i	$0.993230\pi$
$648$	− 85.6148i	− 3.36327i
$649$	23.5458	0.924254
$650$	0	0
$651$	−0.356500	−0.0139724
$652$	94.6291i	3.70596i
$653$	47.5112i	1.85926i	0.368497	+	0.929629i	$0.379873\pi$
$653$	47.5112i	1.85926i	−0.368497	+	0.929629i	$0.620127\pi$
$654$	−8.81858	−0.344834
$655$	0	0
$656$	35.5112	1.38648
$657$	35.0948i	1.36918i
$658$	9.66657i	0.376842i
$659$	24.3048	0.946782	0.473391	−	0.880852i	$-0.343030\pi$
$659$	24.3048	0.946782	0.473391	+	0.880852i	$0.343030\pi$
$660$	0	0
$661$	1.23308	0.0479613	0.0239806	−	0.999712i	$-0.492366\pi$
$661$	1.23308	0.0479613	0.0239806	+	0.999712i	$0.492366\pi$
$662$	− 56.1017i	− 2.18045i
$663$	− 6.38134i	− 0.247831i
$664$	−80.3156	−3.11685
$665$	0	0
$666$	10.5034	0.406999
$667$	− 9.13035i	− 0.353529i
$668$	30.0791i	1.16380i
$669$	1.86761	0.0722058
$670$	0	0
$671$	24.6723	0.952464
$672$	− 10.7390i	− 0.414266i
$673$	− 11.0242i	− 0.424951i	−0.977166	−	0.212476i	$-0.931847\pi$
$673$	− 11.0242i	− 0.424951i	0.977166	−	0.212476i	$-0.0681526\pi$
$674$	−13.6297	−0.524998
$675$	0	0
$676$	−24.8020	−0.953925
$677$	39.3994i	1.51424i	0.653274	+	0.757122i	$0.273395\pi$
$677$	39.3994i	1.51424i	−0.653274	+	0.757122i	$0.726605\pi$
$678$	3.33173i	0.127954i
$679$	0.953571	0.0365947
$680$	0	0
$681$	2.72995	0.104612
$682$	12.9983i	0.497731i
$683$	42.8213i	1.63851i	0.573428	+	0.819256i	$0.305613\pi$
$683$	42.8213i	1.63851i	−0.573428	+	0.819256i	$0.694387\pi$
$684$	−18.7852	−0.718268
$685$	0	0
$686$	37.6598	1.43786
$687$	7.53901i	0.287631i
$688$	− 79.2829i	− 3.02263i
$689$	−15.1181	−0.575954
$690$	0	0
$691$	−34.1911	−1.30069	−0.650346	−	0.759638i	$-0.725376\pi$
$691$	−34.1911	−1.30069	−0.650346	+	0.759638i	$0.725376\pi$
$692$	− 62.8670i	− 2.38984i
$693$	− 13.9333i	− 0.529282i
$694$	−28.3445	−1.07594
$695$	0	0
$696$	8.67911	0.328981
$697$	− 12.1310i	− 0.459495i
$698$	51.3136i	1.94225i
$699$	−5.91362	−0.223674
$700$	0	0
$701$	−31.6598	−1.19577	−0.597886	−	0.801581i	$-0.703993\pi$
$701$	−31.6598	−1.19577	−0.597886	+	0.801581i	$0.703993\pi$
$702$	16.6876i	0.629834i
$703$	− 1.45143i	− 0.0547417i
$704$	−220.360	−8.30514
$705$	0	0
$706$	40.1135	1.50969
$707$	4.28353i	0.161099i
$708$	10.1451i	0.381277i
$709$	19.8198	0.744349	0.372174	−	0.928163i	$-0.378612\pi$
$709$	19.8198	0.744349	0.372174	+	0.928163i	$0.378612\pi$
$710$	0	0
$711$	20.5362	0.770168
$712$	− 130.807i	− 4.90221i
$713$	3.87454i	0.145103i
$714$	−6.28870	−0.235349
$715$	0	0
$716$	−60.0158	−2.24290
$717$	− 7.30844i	− 0.272939i
$718$	− 34.4984i	− 1.28747i
$719$	−25.7339	−0.959713	−0.479856	−	0.877347i	$-0.659311\pi$
$719$	−25.7339	−0.959713	−0.479856	+	0.877347i	$0.659311\pi$
$720$	0	0
$721$	−17.6995	−0.659165
$722$	− 49.7206i	− 1.85041i
$723$	1.42423i	0.0529677i
$724$	−14.8154	−0.550610
$725$	0	0
$726$	10.1112	0.375263
$727$	− 24.6909i	− 0.915734i	−0.889021	−	0.457867i	$-0.848614\pi$
$727$	− 24.6909i	− 0.915734i	0.889021	−	0.457867i	$-0.151386\pi$
$728$	− 33.1412i	− 1.22829i
$729$	20.8745	0.773131
$730$	0	0
$731$	−27.0839	−1.00173
$732$	10.6305i	0.392914i
$733$	23.0265i	0.850505i	0.905075	+	0.425252i	$0.139815\pi$
$733$	23.0265i	0.850505i	−0.905075	+	0.425252i	$0.860185\pi$
$734$	68.5242	2.52928
$735$	0	0
$736$	−116.714	−4.30215
$737$	40.1135i	1.47760i
$738$	15.5456i	0.572242i
$739$	27.2176	1.00121	0.500607	−	0.865675i	$-0.333110\pi$
$739$	27.2176	1.00121	0.500607	+	0.865675i	$0.333110\pi$
$740$	0	0
$741$	1.13003	0.0415126
$742$	14.8986i	0.546946i
$743$	− 23.0434i	− 0.845380i	−0.906274	−	0.422690i	$-0.861086\pi$
$743$	− 23.0434i	− 0.845380i	0.906274	−	0.422690i	$-0.138914\pi$
$744$	−3.68305	−0.135027
$745$	0	0
$746$	−74.2846	−2.71975
$747$	− 21.5232i	− 0.787493i
$748$	170.810i	6.24543i
$749$	−5.04983	−0.184517
$750$	0	0
$751$	−12.5480	−0.457883	−0.228941	−	0.973440i	$-0.573526\pi$
$751$	−12.5480	−0.457883	−0.228941	+	0.973440i	$0.573526\pi$
$752$	61.1344i	2.22934i
$753$	− 1.97516i	− 0.0719787i
$754$	19.5243	0.711033
$755$	0	0
$756$	12.2509	0.445562
$757$	0.441361i	0.0160415i	0.999968	+	0.00802077i	$0.00255312\pi$
$757$	0.441361i	0.0160415i	−0.999968	+	0.00802077i	$0.997447\pi$
$758$	− 35.1513i	− 1.27675i
$759$	6.15758	0.223506
$760$	0	0
$761$	−23.0903	−0.837024	−0.418512	−	0.908211i	$-0.637448\pi$
$761$	−23.0903	−0.837024	−0.418512	+	0.908211i	$0.637448\pi$
$762$	− 12.3418i	− 0.447096i
$763$	9.57747i	0.346728i
$764$	25.9966	0.940524
$765$	0	0
$766$	−32.4501	−1.17247
$767$	15.0084i	0.541923i
$768$	− 37.1846i	− 1.34178i
$769$	16.5906	0.598272	0.299136	−	0.954210i	$-0.403301\pi$
$769$	16.5906	0.598272	0.299136	+	0.954210i	$0.403301\pi$
$770$	0	0
$771$	7.68352	0.276715
$772$	15.8481i	0.570384i
$773$	50.6033i	1.82008i	0.414525	+	0.910038i	$0.363948\pi$
$773$	50.6033i	1.82008i	−0.414525	+	0.910038i	$0.636052\pi$
$774$	34.7074	1.24753
$775$	0	0
$776$	9.85147	0.353647
$777$	0.463851i	0.0166406i
$778$	− 91.2300i	− 3.27075i
$779$	2.14819	0.0769670
$780$	0	0
$781$	−18.0231	−0.644916
$782$	68.3472i	2.44409i
$783$	4.74628i	0.169618i
$784$	109.089	3.89605
$785$	0	0
$786$	14.8288	0.528925
$787$	− 42.9038i	− 1.52936i	−0.644413	−	0.764678i	$-0.722898\pi$
$787$	− 42.9038i	− 1.52936i	0.644413	−	0.764678i	$-0.277102\pi$
$788$	− 29.4486i	− 1.04906i
$789$	3.48897	0.124210
$790$	0	0
$791$	3.61844	0.128657
$792$	− 143.947i	− 5.11492i
$793$	15.7265i	0.558463i
$794$	83.5903	2.96651
$795$	0	0
$796$	−83.8440	−2.97177
$797$	33.5707i	1.18913i	0.804046	+	0.594567i	$0.202676\pi$
$797$	33.5707i	1.18913i	−0.804046	+	0.594567i	$0.797324\pi$
$798$	− 1.11362i	− 0.0394218i
$799$	20.8841	0.738828
$800$	0	0
$801$	35.0541	1.23858
$802$	− 69.7760i	− 2.46388i
$803$	56.5091i	1.99416i
$804$	−17.2836	−0.609545
$805$	0	0
$806$	−8.28530	−0.291837
$807$	9.61749i	0.338552i
$808$	44.2536i	1.55684i
$809$	−35.3774	−1.24380	−0.621902	−	0.783095i	$-0.713640\pi$
$809$	−35.3774	−1.24380	−0.621902	+	0.783095i	$0.713640\pi$
$810$	0	0
$811$	54.9833	1.93072	0.965362	−	0.260916i	$-0.0840245\pi$
$811$	54.9833	1.93072	0.965362	+	0.260916i	$0.0840245\pi$
$812$	− 14.3334i	− 0.503005i
$813$	− 0.0147067i	0 0.000515787i
$814$	16.9124	0.592779
$815$	0	0
$816$	−39.7717	−1.39229
$817$	− 4.79609i	− 0.167794i
$818$	− 38.6676i	− 1.35198i
$819$	8.88128	0.310337
$820$	0	0
$821$	47.1995	1.64727	0.823636	−	0.567118i	$-0.191942\pi$
$821$	47.1995	1.64727	0.823636	+	0.567118i	$0.191942\pi$
$822$	15.8897i	0.554217i
$823$	− 31.1661i	− 1.08638i	−0.839609	−	0.543192i	$-0.817216\pi$
$823$	− 31.1661i	− 1.08638i	0.839609	−	0.543192i	$-0.182784\pi$
$824$	−182.856	−6.37009
$825$	0	0
$826$	14.7905	0.514629
$827$	− 2.62427i	− 0.0912548i	−0.998959	−	0.0456274i	$-0.985471\pi$
$827$	− 2.62427i	− 0.0912548i	0.998959	−	0.0456274i	$-0.0145287\pi$
$828$	− 65.2466i	− 2.26747i
$829$	−30.1876	−1.04846	−0.524230	−	0.851577i	$-0.675647\pi$
$829$	−30.1876	−1.04846	−0.524230	+	0.851577i	$0.675647\pi$
$830$	0	0
$831$	7.47365	0.259258
$832$	− 140.461i	− 4.86960i
$833$	− 37.2661i	− 1.29119i
$834$	−20.6379	−0.714633
$835$	0	0
$836$	−30.2475	−1.04613
$837$	− 2.01412i	− 0.0696183i
$838$	60.9897i	2.10685i
$839$	9.75555	0.336799	0.168399	−	0.985719i	$-0.446140\pi$
$839$	9.75555	0.336799	0.168399	+	0.985719i	$0.446140\pi$
$840$	0	0
$841$	−23.4469	−0.808514
$842$	9.06610i	0.312438i
$843$	3.15128i	0.108536i
$844$	67.9299	2.33824
$845$	0	0
$846$	−26.7625	−0.920115
$847$	− 10.9814i	− 0.377324i
$848$	94.2236i	3.23565i
$849$	−2.34636	−0.0805270
$850$	0	0
$851$	5.04125	0.172812
$852$	− 7.76556i	− 0.266044i
$853$	39.4547i	1.35090i	0.737404	+	0.675452i	$0.236052\pi$
$853$	39.4547i	1.35090i	−0.737404	+	0.675452i	$0.763948\pi$
$854$	15.4982	0.530336
$855$	0	0
$856$	−52.1705	−1.78315
$857$	10.6006i	0.362110i	0.983473	+	0.181055i	$0.0579512\pi$
$857$	10.6006i	0.362110i	−0.983473	+	0.181055i	$0.942049\pi$
$858$	13.1673i	0.449526i
$859$	30.1660	1.02925	0.514625	−	0.857416i	$-0.327931\pi$
$859$	30.1660	1.02925	0.514625	+	0.857416i	$0.327931\pi$
$860$	0	0
$861$	−0.686525	−0.0233967
$862$	− 35.4140i	− 1.20621i
$863$	11.7962i	0.401546i	0.979638	+	0.200773i	$0.0643454\pi$
$863$	11.7962i	0.401546i	−0.979638	+	0.200773i	$0.935655\pi$
$864$	60.6723	2.06411
$865$	0	0
$866$	−89.5847	−3.04421
$867$	7.76602i	0.263748i
$868$	6.08251i	0.206454i
$869$	33.0670	1.12172
$870$	0	0
$871$	−25.5689	−0.866369
$872$	98.9461i	3.35074i
$873$	2.64003i	0.0893513i
$874$	−12.1031	−0.409394
$875$	0	0
$876$	−24.3479	−0.822640
$877$	− 12.3531i	− 0.417134i	−0.978008	−	0.208567i	$-0.933120\pi$
$877$	− 12.3531i	− 0.417134i	0.978008	−	0.208567i	$-0.0668800\pi$
$878$	38.7648i	1.30825i
$879$	−5.68692	−0.191815
$880$	0	0
$881$	28.2740	0.952575	0.476288	−	0.879290i	$-0.341982\pi$
$881$	28.2740	0.952575	0.476288	+	0.879290i	$0.341982\pi$
$882$	47.7556i	1.60802i
$883$	14.7328i	0.495798i	0.968786	+	0.247899i	$0.0797401\pi$
$883$	14.7328i	0.495798i	−0.968786	+	0.247899i	$0.920260\pi$
$884$	−108.877	−3.66192
$885$	0	0
$886$	56.9797	1.91427
$887$	− 26.9237i	− 0.904009i	−0.892016	−	0.452005i	$-0.850709\pi$
$887$	− 26.9237i	− 0.904009i	0.892016	−	0.452005i	$-0.149291\pi$
$888$	4.79210i	0.160812i
$889$	−13.4039	−0.449552
$890$	0	0
$891$	36.9429	1.23763
$892$	− 31.8645i	− 1.06690i
$893$	3.69822i	0.123756i
$894$	−19.8994	−0.665535
$895$	0	0
$896$	−75.6893	−2.52860
$897$	3.92493i	0.131050i
$898$	− 24.4083i	− 0.814516i
$899$	−2.35650	−0.0785937
$900$	0	0
$901$	32.1877	1.07233
$902$	25.0313i	0.833450i
$903$	1.53275i	0.0510066i
$904$	37.3826	1.24333
$905$	0	0
$906$	−17.6842	−0.587519
$907$	25.7869i	0.856239i	0.903722	+	0.428119i	$0.140824\pi$
$907$	25.7869i	0.856239i	−0.903722	+	0.428119i	$0.859176\pi$
$908$	− 46.5775i	− 1.54573i
$909$	−11.8592	−0.393346
$910$	0	0
$911$	24.4130	0.808839	0.404419	−	0.914574i	$-0.367474\pi$
$911$	24.4130	0.808839	0.404419	+	0.914574i	$0.367474\pi$
$912$	− 7.04288i	− 0.233213i
$913$	− 34.6563i	− 1.14696i
$914$	−0.0873278	−0.00288855
$915$	0	0
$916$	128.628	4.25000
$917$	− 16.1049i	− 0.531831i
$918$	− 35.5293i	− 1.17264i
$919$	6.86800	0.226554	0.113277	−	0.993563i	$-0.463865\pi$
$919$	6.86800	0.226554	0.113277	+	0.993563i	$0.463865\pi$
$920$	0	0
$921$	−5.80511	−0.191285
$922$	− 82.5749i	− 2.71946i
$923$	− 11.4882i	− 0.378137i
$924$	9.66657	0.318007
$925$	0	0
$926$	53.6385	1.76267
$927$	− 49.0023i	− 1.60945i
$928$	− 70.9858i	− 2.33022i
$929$	−12.6470	−0.414934	−0.207467	−	0.978242i	$-0.566522\pi$
$929$	−12.6470	−0.414934	−0.207467	+	0.978242i	$0.566522\pi$
$930$	0	0
$931$	6.59918	0.216279
$932$	100.897i	3.30498i
$933$	0.218687i	0.00715950i
$934$	−67.0293	−2.19326
$935$	0	0
$936$	91.7536	2.99906
$937$	− 42.8873i	− 1.40107i	−0.713619	−	0.700534i	$-0.752945\pi$
$937$	− 42.8873i	− 1.40107i	0.713619	−	0.700534i	$-0.247055\pi$
$938$	25.1977i	0.822734i
$939$	−6.20183	−0.202389
$940$	0	0
$941$	24.2254	0.789725	0.394863	−	0.918740i	$-0.370792\pi$
$941$	24.2254	0.789725	0.394863	+	0.918740i	$0.370792\pi$
$942$	− 5.41069i	− 0.176290i
$943$	7.46133i	0.242974i
$944$	93.5400	3.04447
$945$	0	0
$946$	55.8852	1.81698
$947$	16.2682i	0.528647i	0.964434	+	0.264323i	$0.0851486\pi$
$947$	16.2682i	0.528647i	−0.964434	+	0.264323i	$0.914851\pi$
$948$	14.2475i	0.462738i
$949$	−36.0197	−1.16925
$950$	0	0
$951$	1.73391	0.0562257
$952$	70.5604i	2.28688i
$953$	19.2891i	0.624837i	0.949945	+	0.312418i	$0.101139\pi$
$953$	19.2891i	0.624837i	−0.949945	+	0.312418i	$0.898861\pi$
$954$	−41.2479	−1.33545
$955$	0	0
$956$	−124.694	−4.03291
$957$	3.74505i	0.121060i
$958$	− 59.9367i	− 1.93647i
$959$	17.2571	0.557261
$960$	0	0
$961$	1.00000	0.0322581
$962$	10.7802i	0.347567i
$963$	− 13.9808i	− 0.450525i
$964$	24.2998	0.782644
$965$	0	0
$966$	3.86795	0.124449
$967$	− 20.9745i	− 0.674496i	−0.941416	−	0.337248i	$-0.890504\pi$
$967$	− 20.9745i	− 0.674496i	0.941416	−	0.337248i	$-0.109496\pi$
$968$	− 113.450i	− 3.64641i
$969$	−2.40592	−0.0772894
$970$	0	0
$971$	−16.2572	−0.521718	−0.260859	−	0.965377i	$-0.584006\pi$
$971$	−16.2572	−0.521718	−0.260859	+	0.965377i	$0.584006\pi$
$972$	51.2142i	1.64270i
$973$	22.4139i	0.718558i
$974$	73.0207	2.33973
$975$	0	0
$976$	98.0151	3.13739
$977$	− 12.3649i	− 0.395587i	−0.980244	−	0.197794i	$-0.936622\pi$
$977$	− 12.3649i	− 0.395587i	0.980244	−	0.197794i	$-0.0633776\pi$
$978$	− 15.5311i	− 0.496630i
$979$	56.4435	1.80394
$980$	0	0
$981$	−26.5159	−0.846587
$982$	− 33.4260i	− 1.06667i
$983$	39.4586i	1.25853i	0.777189	+	0.629267i	$0.216645\pi$
$983$	39.4586i	1.25853i	−0.777189	+	0.629267i	$0.783355\pi$
$984$	−7.09257	−0.226103
$985$	0	0
$986$	−41.5689	−1.32382
$987$	− 1.18189i	− 0.0376199i
$988$	− 19.2802i	− 0.613385i
$989$	16.6583	0.529702
$990$	0	0
$991$	−55.5390	−1.76425	−0.882127	−	0.471011i	$-0.843889\pi$
$991$	−55.5390	−1.76425	−0.882127	+	0.471011i	$0.843889\pi$
$992$	30.1234i	0.956420i
$993$	6.85930i	0.217673i
$994$	−11.3214	−0.359092
$995$	0	0
$996$	14.9323	0.473147
$997$	− 13.4120i	− 0.424762i	−0.977187	−	0.212381i	$-0.931878\pi$
$997$	− 13.4120i	− 0.424762i	0.977187	−	0.212381i	$-0.0681218\pi$
$998$	− 46.3149i	− 1.46607i
$999$	−2.62062	−0.0829129

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.b.e.249.8		8
5.2	odd	4		155.2.a.d.1.1	✓	4
5.3	odd	4		775.2.a.g.1.4		4
5.4	even	2	inner	775.2.b.e.249.1		8
15.2	even	4		1395.2.a.m.1.4		4
15.8	even	4		6975.2.a.bj.1.1		4
20.7	even	4		2480.2.a.z.1.2		4
35.27	even	4		7595.2.a.q.1.1		4
40.27	even	4		9920.2.a.cd.1.3		4
40.37	odd	4		9920.2.a.ch.1.2		4
155.92	even	4		4805.2.a.j.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.a.d.1.1	✓	4	5.2	odd	4
775.2.a.g.1.4		4	5.3	odd	4
775.2.b.e.249.1		8	5.4	even	2	inner
775.2.b.e.249.8		8	1.1	even	1	trivial
1395.2.a.m.1.4		4	15.2	even	4
2480.2.a.z.1.2		4	20.7	even	4
4805.2.a.j.1.1		4	155.92	even	4
6975.2.a.bj.1.1		4	15.8	even	4
7595.2.a.q.1.1		4	35.27	even	4
9920.2.a.cd.1.3		4	40.27	even	4
9920.2.a.ch.1.2		4	40.37	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.80027i q^{2} -0.342376i q^{3} -5.84153 q^{4} +0.958747 q^{6} -1.04125i q^{7} -10.7573i q^{8} +2.88278 q^{9} +4.64180 q^{11} +2.00000i q^{12} +2.95875i q^{13} +2.91579 q^{14} +18.4404 q^{16} -6.29942i q^{17} +8.07256i q^{18} +1.11552 q^{19} -0.356500 q^{21} +12.9983i q^{22} +3.87454i q^{23} -3.68305 q^{24} -8.28530 q^{26} -2.01412i q^{27} +6.08251i q^{28} -2.35650 q^{29} +1.00000 q^{31} +30.1234i q^{32} -1.58924i q^{33} +17.6401 q^{34} -16.8398 q^{36} -1.30112i q^{37} +3.12376i q^{38} +1.01300 q^{39} +1.92573 q^{41} -0.998298i q^{42} -4.29942i q^{43} -27.1152 q^{44} -10.8498 q^{46} +3.31525i q^{47} -6.31355i q^{48} +5.91579 q^{49} -2.15677 q^{51} -17.2836i q^{52} +5.10964i q^{53} +5.64010 q^{54} -11.2011 q^{56} -0.381928i q^{57} -6.59884i q^{58} +5.07256 q^{59} +5.31525 q^{61} +2.80027i q^{62} -3.00170i q^{63} -47.4730 q^{64} +4.45031 q^{66} +8.64180i q^{67} +36.7982i q^{68} +1.32655 q^{69} -3.88278 q^{71} -31.0110i q^{72} +12.1740i q^{73} +3.64350 q^{74} -6.51634 q^{76} -4.83329i q^{77} +2.83669i q^{78} +7.12376 q^{79} +7.95875 q^{81} +5.39258i q^{82} -7.46614i q^{83} +2.08251 q^{84} +12.0396 q^{86} +0.806810i q^{87} -49.9333i q^{88} +12.1598 q^{89} +3.08080 q^{91} -22.6332i q^{92} -0.342376i q^{93} -9.28360 q^{94} +10.3135 q^{96} +0.915792i q^{97} +16.5658i q^{98} +13.3813 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 18 q^{4} + 16 q^{6} - 14 q^{9} - 12 q^{11} - 16 q^{14} + 22 q^{16} - 10 q^{19} + 4 q^{21} + 28 q^{24} - 24 q^{26} - 12 q^{29} + 8 q^{31} + 36 q^{34} - 50 q^{36} + 12 q^{39} + 26 q^{41} - 40 q^{44}+ \cdots + 104 q^{99}+O(q^{100}) \) 8 * q - 18 * q^4 + 16 * q^6 - 14 * q^9 - 12 * q^11 - 16 * q^14 + 22 * q^16 - 10 * q^19 + 4 * q^21 + 28 * q^24 - 24 * q^26 - 12 * q^29 + 8 * q^31 + 36 * q^34 - 50 * q^36 + 12 * q^39 + 26 * q^41 - 40 * q^44 - 52 * q^46 + 8 * q^49 + 10 * q^51 - 60 * q^54 - 8 * q^56 - 26 * q^59 + 44 * q^61 - 94 * q^64 - 40 * q^66 - 40 * q^69 + 6 * q^71 + 36 * q^74 + 28 * q^76 + 32 * q^79 + 72 * q^81 + 32 * q^86 + 24 * q^89 - 48 * q^91 + 24 * q^94 + 28 * q^96 + 104 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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