LMFDB - Embedded newform 675.4.b.q.649.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(649,675)] 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("675.649"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-38,0,0,0,0,0,0,104,0,0,-276,0,-10,0,0,92] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 25x^{6} + 186x^{4} + 441x^{2} + 324 $ x^8 + 25x^6 + 186x^4 + 441*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$-1.24486i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.q.649.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.33012i q^{2} +6.23078 q^{4} +10.6979i q^{7} -18.9286i q^{8} -11.2588 q^{11} +2.74029i q^{13} +14.2294 q^{14} +24.6689 q^{16} -29.5692i q^{17} +31.1126 q^{19} +14.9755i q^{22} -116.944i q^{23} +3.64491 q^{26} +66.6560i q^{28} +108.384 q^{29} +70.7730 q^{31} -184.242i q^{32} -39.3305 q^{34} -282.289i q^{37} -41.3834i q^{38} +425.545 q^{41} +312.868i q^{43} -70.1509 q^{44} -155.549 q^{46} -193.619i q^{47} +228.556 q^{49} +17.0741i q^{52} -103.349i q^{53} +202.496 q^{56} -144.164i q^{58} -494.531 q^{59} +424.769 q^{61} -94.1365i q^{62} -47.7120 q^{64} +586.687i q^{67} -184.239i q^{68} +1139.86 q^{71} +302.564i q^{73} -375.478 q^{74} +193.856 q^{76} -120.445i q^{77} +525.354 q^{79} -566.026i q^{82} -1009.07i q^{83} +416.151 q^{86} +213.113i q^{88} -1424.57 q^{89} -29.3152 q^{91} -728.652i q^{92} -257.537 q^{94} -25.6808i q^{97} -304.007i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 38 q^{4} + 104 q^{11} - 276 q^{14} - 10 q^{16} + 92 q^{19} + 938 q^{26} - 940 q^{29} - 524 q^{31} + 84 q^{34} + 1396 q^{41} - 838 q^{44} + 1074 q^{46} - 1560 q^{49} + 4068 q^{56} - 200 q^{59} + 148 q^{61}+ \cdots - 5694 q^{94}+O(q^{100}) $ 8 * q - 38 * q^4 + 104 * q^11 - 276 * q^14 - 10 * q^16 + 92 * q^19 + 938 * q^26 - 940 * q^29 - 524 * q^31 + 84 * q^34 + 1396 * q^41 - 838 * q^44 + 1074 * q^46 - 1560 * q^49 + 4068 * q^56 - 200 * q^59 + 148 * q^61 + 1930 * q^64 + 3244 * q^71 + 506 * q^74 - 4136 * q^76 + 1016 * q^79 + 7880 * q^86 - 1512 * q^89 + 2240 * q^91 - 5694 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 1.33012i	− 0.470268i	−0.971963	−	0.235134i	$-0.924447\pi$
$2$	− 1.33012i	− 0.470268i	0.971963	−	0.235134i	$-0.0755529\pi$
$3$	0	0
$3$	0	0
$4$	6.23078	0.778848
$5$	0	0
$5$	0	0
$6$	0	0
$7$	10.6979i	0.577630i	0.957385	+	0.288815i	$0.0932612\pi$
$7$	10.6979i	0.577630i	−0.957385	+	0.288815i	$0.906739\pi$
$8$	− 18.9286i	− 0.836535i
$9$	0	0
$10$	0	0
$11$	−11.2588	−0.308604	−0.154302	−	0.988024i	$-0.549313\pi$
$11$	−11.2588	−0.308604	−0.154302	+	0.988024i	$0.549313\pi$
$12$	0	0
$13$	2.74029i	0.0584630i	0.999573	+	0.0292315i	$0.00930600\pi$
$13$	2.74029i	0.0584630i	−0.999573	+	0.0292315i	$0.990694\pi$
$14$	14.2294	0.271641
$15$	0	0
$16$	24.6689	0.385452
$17$	− 29.5692i	− 0.421858i	−0.977501	−	0.210929i	$-0.932351\pi$
$17$	− 29.5692i	− 0.421858i	0.977501	−	0.210929i	$-0.0676488\pi$
$18$	0	0
$19$	31.1126	0.375669	0.187835	−	0.982201i	$-0.439853\pi$
$19$	31.1126	0.375669	0.187835	+	0.982201i	$0.439853\pi$
$20$	0	0
$21$	0	0
$22$	14.9755i	0.145127i
$23$	− 116.944i	− 1.06020i	−0.847936	−	0.530098i	$-0.822155\pi$
$23$	− 116.944i	− 1.06020i	0.847936	−	0.530098i	$-0.177845\pi$
$24$	0	0
$25$	0	0
$26$	3.64491	0.0274933
$27$	0	0
$28$	66.6560i	0.449886i
$29$	108.384	0.694016	0.347008	−	0.937862i	$-0.387198\pi$
$29$	108.384	0.694016	0.347008	+	0.937862i	$0.387198\pi$
$30$	0	0
$31$	70.7730	0.410039	0.205019	−	0.978758i	$-0.434274\pi$
$31$	70.7730	0.410039	0.205019	+	0.978758i	$0.434274\pi$
$32$	− 184.242i	− 1.01780i
$33$	0	0
$34$	−39.3305	−0.198386
$35$	0	0
$36$	0	0
$37$	− 282.289i	− 1.25427i	−0.778910	−	0.627136i	$-0.784227\pi$
$37$	− 282.289i	− 1.25427i	0.778910	−	0.627136i	$-0.215773\pi$
$38$	− 41.3834i	− 0.176665i
$39$	0	0
$40$	0	0
$41$	425.545	1.62095	0.810475	−	0.585773i	$-0.199209\pi$
$41$	425.545	1.62095	0.810475	+	0.585773i	$0.199209\pi$
$42$	0	0
$43$	312.868i	1.10958i	0.831991	+	0.554789i	$0.187201\pi$
$43$	312.868i	1.10958i	−0.831991	+	0.554789i	$0.812799\pi$
$44$	−70.1509	−0.240355
$45$	0	0
$46$	−155.549	−0.498576
$47$	− 193.619i	− 0.600900i	−0.953798	−	0.300450i	$-0.902863\pi$
$47$	− 193.619i	− 0.600900i	0.953798	−	0.300450i	$-0.0971368\pi$
$48$	0	0
$49$	228.556	0.666344
$50$	0	0
$51$	0	0
$52$	17.0741i	0.0455338i
$53$	− 103.349i	− 0.267850i	−0.990991	−	0.133925i	$-0.957242\pi$
$53$	− 103.349i	− 0.267850i	0.990991	−	0.133925i	$-0.0427581\pi$
$54$	0	0
$55$	0	0
$56$	202.496	0.483208
$57$	0	0
$58$	− 144.164i	− 0.326374i
$59$	−494.531	−1.09123	−0.545614	−	0.838036i	$-0.683704\pi$
$59$	−494.531	−1.09123	−0.545614	+	0.838036i	$0.683704\pi$
$60$	0	0
$61$	424.769	0.891575	0.445787	−	0.895139i	$-0.352924\pi$
$61$	424.769	0.891575	0.445787	+	0.895139i	$0.352924\pi$
$62$	− 94.1365i	− 0.192828i
$63$	0	0
$64$	−47.7120	−0.0931875
$65$	0	0
$66$	0	0
$67$	586.687i	1.06978i	0.844922	+	0.534889i	$0.179647\pi$
$67$	586.687i	1.06978i	−0.844922	+	0.534889i	$0.820353\pi$
$68$	− 184.239i	− 0.328563i
$69$	0	0
$70$	0	0
$71$	1139.86	1.90531	0.952653	−	0.304060i	$-0.0983424\pi$
$71$	1139.86	1.90531	0.952653	+	0.304060i	$0.0983424\pi$
$72$	0	0
$73$	302.564i	0.485102i	0.970139	+	0.242551i	$0.0779842\pi$
$73$	302.564i	0.485102i	−0.970139	+	0.242551i	$0.922016\pi$
$74$	−375.478	−0.589844
$75$	0	0
$76$	193.856	0.292589
$77$	− 120.445i	− 0.178259i
$78$	0	0
$79$	525.354	0.748189	0.374095	−	0.927391i	$-0.377954\pi$
$79$	525.354	0.748189	0.374095	+	0.927391i	$0.377954\pi$
$80$	0	0
$81$	0	0
$82$	− 566.026i	− 0.762281i
$83$	− 1009.07i	− 1.33446i	−0.744854	−	0.667228i	$-0.767481\pi$
$83$	− 1009.07i	− 1.33446i	0.744854	−	0.667228i	$-0.232519\pi$
$84$	0	0
$85$	0	0
$86$	416.151	0.521799
$87$	0	0
$88$	213.113i	0.258158i
$89$	−1424.57	−1.69668	−0.848340	−	0.529451i	$-0.822398\pi$
$89$	−1424.57	−1.69668	−0.848340	+	0.529451i	$0.822398\pi$
$90$	0	0
$91$	−29.3152	−0.0337700
$92$	− 728.652i	− 0.825731i
$93$	0	0
$94$	−257.537	−0.282584
$95$	0	0
$96$	0	0
$97$	− 25.6808i	− 0.0268814i	−0.999910	−	0.0134407i	$-0.995722\pi$
$97$	− 25.6808i	− 0.0268814i	0.999910	−	0.0134407i	$-0.00427843\pi$
$98$	− 304.007i	− 0.313360i
$99$	0	0
$100$	0	0
$101$	1523.35	1.50078	0.750392	−	0.660993i	$-0.229865\pi$
$101$	1523.35	1.50078	0.750392	+	0.660993i	$0.229865\pi$
$102$	0	0
$103$	− 1237.75i	− 1.18407i	−0.805913	−	0.592034i	$-0.798325\pi$
$103$	− 1237.75i	− 1.18407i	0.805913	−	0.592034i	$-0.201675\pi$
$104$	51.8699	0.0489064
$105$	0	0
$106$	−137.466	−0.125961
$107$	− 465.820i	− 0.420865i	−0.977608	−	0.210432i	$-0.932513\pi$
$107$	− 465.820i	− 0.420865i	0.977608	−	0.210432i	$-0.0674871\pi$
$108$	0	0
$109$	−748.032	−0.657325	−0.328663	−	0.944447i	$-0.606598\pi$
$109$	−748.032	−0.657325	−0.328663	+	0.944447i	$0.606598\pi$
$110$	0	0
$111$	0	0
$112$	263.905i	0.222649i
$113$	− 324.843i	− 0.270431i	−0.990816	−	0.135215i	$-0.956827\pi$
$113$	− 324.843i	− 0.270431i	0.990816	−	0.135215i	$-0.0431726\pi$
$114$	0	0
$115$	0	0
$116$	675.319	0.540533
$117$	0	0
$118$	657.786i	0.513170i
$119$	316.327	0.243678
$120$	0	0
$121$	−1204.24	−0.904764
$122$	− 564.993i	− 0.419279i
$123$	0	0
$124$	440.971	0.319358
$125$	0	0
$126$	0	0
$127$	29.9720i	0.0209416i	0.999945	+	0.0104708i	$0.00333302\pi$
$127$	29.9720i	0.0209416i	−0.999945	+	0.0104708i	$0.996667\pi$
$128$	− 1410.47i	− 0.973978i
$129$	0	0
$130$	0	0
$131$	906.495	0.604587	0.302293	−	0.953215i	$-0.402248\pi$
$131$	906.495	0.604587	0.302293	+	0.953215i	$0.402248\pi$
$132$	0	0
$133$	332.838i	0.216998i
$134$	780.363	0.503083
$135$	0	0
$136$	−559.704	−0.352899
$137$	− 2359.95i	− 1.47171i	−0.677139	−	0.735855i	$-0.736780\pi$
$137$	− 2359.95i	− 1.47171i	0.677139	−	0.735855i	$-0.263220\pi$
$138$	0	0
$139$	−1709.46	−1.04313	−0.521563	−	0.853213i	$-0.674651\pi$
$139$	−1709.46	−1.04313	−0.521563	+	0.853213i	$0.674651\pi$
$140$	0	0
$141$	0	0
$142$	− 1516.15i	− 0.896005i
$143$	− 30.8522i	− 0.0180419i
$144$	0	0
$145$	0	0
$146$	402.446	0.228128
$147$	0	0
$148$	− 1758.88i	− 0.976886i
$149$	−119.170	−0.0655221	−0.0327611	−	0.999463i	$-0.510430\pi$
$149$	−119.170	−0.0655221	−0.0327611	+	0.999463i	$0.510430\pi$
$150$	0	0
$151$	768.157	0.413985	0.206992	−	0.978343i	$-0.433632\pi$
$151$	768.157	0.413985	0.206992	+	0.978343i	$0.433632\pi$
$152$	− 588.919i	− 0.314261i
$153$	0	0
$154$	−160.206	−0.0838294
$155$	0	0
$156$	0	0
$157$	1999.27i	1.01630i	0.861268	+	0.508151i	$0.169671\pi$
$157$	1999.27i	1.01630i	−0.861268	+	0.508151i	$0.830329\pi$
$158$	− 698.783i	− 0.351850i
$159$	0	0
$160$	0	0
$161$	1251.05	0.612401
$162$	0	0
$163$	− 1206.73i	− 0.579867i	−0.957047	−	0.289934i	$-0.906367\pi$
$163$	− 1206.73i	− 0.579867i	0.957047	−	0.289934i	$-0.0936332\pi$
$164$	2651.48	1.26247
$165$	0	0
$166$	−1342.18	−0.627552
$167$	102.994i	0.0477239i	0.999715	+	0.0238619i	$0.00759621\pi$
$167$	102.994i	0.0477239i	−0.999715	+	0.0238619i	$0.992404\pi$
$168$	0	0
$169$	2189.49	0.996582
$170$	0	0
$171$	0	0
$172$	1949.41i	0.864193i
$173$	− 308.318i	− 0.135497i	−0.997702	−	0.0677485i	$-0.978418\pi$
$173$	− 308.318i	− 0.135497i	0.997702	−	0.0677485i	$-0.0215815\pi$
$174$	0	0
$175$	0	0
$176$	−277.741	−0.118952
$177$	0	0
$178$	1894.85i	0.797895i
$179$	−3145.61	−1.31349	−0.656743	−	0.754115i	$-0.728066\pi$
$179$	−3145.61	−1.31349	−0.656743	+	0.754115i	$0.728066\pi$
$180$	0	0
$181$	2321.35	0.953284	0.476642	−	0.879098i	$-0.341854\pi$
$181$	2321.35	0.953284	0.476642	+	0.879098i	$0.341854\pi$
$182$	38.9927i	0.0158809i
$183$	0	0
$184$	−2213.59	−0.886891
$185$	0	0
$186$	0	0
$187$	332.912i	0.130187i
$188$	− 1206.40i	− 0.468009i
$189$	0	0
$190$	0	0
$191$	−2261.68	−0.856804	−0.428402	−	0.903588i	$-0.640923\pi$
$191$	−2261.68	−0.856804	−0.428402	+	0.903588i	$0.640923\pi$
$192$	0	0
$193$	− 4792.77i	− 1.78752i	−0.448545	−	0.893760i	$-0.648058\pi$
$193$	− 4792.77i	− 1.78752i	0.448545	−	0.893760i	$-0.351942\pi$
$194$	−34.1585	−0.0126414
$195$	0	0
$196$	1424.08	0.518981
$197$	2262.41i	0.818223i	0.912484	+	0.409111i	$0.134161\pi$
$197$	2262.41i	0.818223i	−0.912484	+	0.409111i	$0.865839\pi$
$198$	0	0
$199$	2188.42	0.779562	0.389781	−	0.920908i	$-0.372551\pi$
$199$	2188.42	0.779562	0.389781	+	0.920908i	$0.372551\pi$
$200$	0	0
$201$	0	0
$202$	− 2026.24i	− 0.705771i
$203$	1159.48i	0.400884i
$204$	0	0
$205$	0	0
$206$	−1646.35	−0.556830
$207$	0	0
$208$	67.6000i	0.0225347i
$209$	−350.289	−0.115933
$210$	0	0
$211$	−4907.42	−1.60114	−0.800570	−	0.599240i	$-0.795470\pi$
$211$	−4907.42	−1.60114	−0.800570	+	0.599240i	$0.795470\pi$
$212$	− 643.943i	− 0.208614i
$213$	0	0
$214$	−619.596	−0.197919
$215$	0	0
$216$	0	0
$217$	757.119i	0.236851i
$218$	994.971i	0.309119i
$219$	0	0
$220$	0	0
$221$	81.0281	0.0246631
$222$	0	0
$223$	5787.33i	1.73789i	0.494912	+	0.868943i	$0.335200\pi$
$223$	5787.33i	1.73789i	−0.494912	+	0.868943i	$0.664800\pi$
$224$	1970.99	0.587912
$225$	0	0
$226$	−432.080	−0.127175
$227$	4358.48i	1.27437i	0.770710	+	0.637186i	$0.219902\pi$
$227$	4358.48i	1.27437i	−0.770710	+	0.637186i	$0.780098\pi$
$228$	0	0
$229$	−3944.00	−1.13811	−0.569054	−	0.822300i	$-0.692690\pi$
$229$	−3944.00	−1.13811	−0.569054	+	0.822300i	$0.692690\pi$
$230$	0	0
$231$	0	0
$232$	− 2051.57i	− 0.580569i
$233$	5530.41i	1.55497i	0.628899	+	0.777487i	$0.283506\pi$
$233$	5530.41i	1.55497i	−0.628899	+	0.777487i	$0.716494\pi$
$234$	0	0
$235$	0	0
$236$	−3081.32	−0.849901
$237$	0	0
$238$	− 420.752i	− 0.114594i
$239$	−3823.19	−1.03473	−0.517367	−	0.855763i	$-0.673088\pi$
$239$	−3823.19	−1.03473	−0.517367	+	0.855763i	$0.673088\pi$
$240$	0	0
$241$	3976.26	1.06280	0.531398	−	0.847122i	$-0.321667\pi$
$241$	3976.26	1.06280	0.531398	+	0.847122i	$0.321667\pi$
$242$	1601.78i	0.425481i
$243$	0	0
$244$	2646.64	0.694401
$245$	0	0
$246$	0	0
$247$	85.2574i	0.0219628i
$248$	− 1339.64i	− 0.343012i
$249$	0	0
$250$	0	0
$251$	−1758.83	−0.442297	−0.221148	−	0.975240i	$-0.570981\pi$
$251$	−1758.83	−0.442297	−0.221148	+	0.975240i	$0.570981\pi$
$252$	0	0
$253$	1316.64i	0.327181i
$254$	39.8663	0.00984817
$255$	0	0
$256$	−2257.79	−0.551218
$257$	3396.20i	0.824316i	0.911112	+	0.412158i	$0.135225\pi$
$257$	3396.20i	0.824316i	−0.911112	+	0.412158i	$0.864775\pi$
$258$	0	0
$259$	3019.89	0.724504
$260$	0	0
$261$	0	0
$262$	− 1205.75i	− 0.284318i
$263$	4664.71i	1.09368i	0.837236	+	0.546841i	$0.184170\pi$
$263$	4664.71i	1.09368i	−0.837236	+	0.546841i	$0.815830\pi$
$264$	0	0
$265$	0	0
$266$	442.714	0.102047
$267$	0	0
$268$	3655.52i	0.833195i
$269$	214.904	0.0487099	0.0243549	−	0.999703i	$-0.492247\pi$
$269$	214.904	0.0487099	0.0243549	+	0.999703i	$0.492247\pi$
$270$	0	0
$271$	3260.69	0.730896	0.365448	−	0.930832i	$-0.380916\pi$
$271$	3260.69	0.730896	0.365448	+	0.930832i	$0.380916\pi$
$272$	− 729.440i	− 0.162606i
$273$	0	0
$274$	−3139.02	−0.692098
$275$	0	0
$276$	0	0
$277$	− 1992.86i	− 0.432273i	−0.976363	−	0.216136i	$-0.930654\pi$
$277$	− 1992.86i	− 0.432273i	0.976363	−	0.216136i	$-0.0693456\pi$
$278$	2273.78i	0.490549i
$279$	0	0
$280$	0	0
$281$	−3029.98	−0.643251	−0.321626	−	0.946867i	$-0.604229\pi$
$281$	−3029.98	−0.643251	−0.321626	+	0.946867i	$0.604229\pi$
$282$	0	0
$283$	− 4950.46i	− 1.03984i	−0.854215	−	0.519920i	$-0.825962\pi$
$283$	− 4950.46i	− 1.03984i	0.854215	−	0.519920i	$-0.174038\pi$
$284$	7102.23	1.48394
$285$	0	0
$286$	−41.0372	−0.00848454
$287$	4552.42i	0.936309i
$288$	0	0
$289$	4038.66	0.822036
$290$	0	0
$291$	0	0
$292$	1885.21i	0.377821i
$293$	8087.47i	1.61254i	0.591546	+	0.806272i	$0.298518\pi$
$293$	8087.47i	1.61254i	−0.591546	+	0.806272i	$0.701482\pi$
$294$	0	0
$295$	0	0
$296$	−5343.35	−1.04924
$297$	0	0
$298$	158.510i	0.0308130i
$299$	320.460	0.0619822
$300$	0	0
$301$	−3347.01	−0.640926
$302$	− 1021.74i	− 0.194684i
$303$	0	0
$304$	767.514	0.144802
$305$	0	0
$306$	0	0
$307$	− 272.579i	− 0.0506740i	−0.999679	−	0.0253370i	$-0.991934\pi$
$307$	− 272.579i	− 0.0506740i	0.999679	−	0.0253370i	$-0.00806588\pi$
$308$	− 750.464i	− 0.138836i
$309$	0	0
$310$	0	0
$311$	−9093.68	−1.65806	−0.829028	−	0.559208i	$-0.811105\pi$
$311$	−9093.68	−1.65806	−0.829028	+	0.559208i	$0.811105\pi$
$312$	0	0
$313$	8213.36i	1.48322i	0.670834	+	0.741608i	$0.265936\pi$
$313$	8213.36i	1.48322i	−0.670834	+	0.741608i	$0.734064\pi$
$314$	2659.27	0.477935
$315$	0	0
$316$	3273.37	0.582726
$317$	4859.27i	0.860959i	0.902600	+	0.430480i	$0.141656\pi$
$317$	4859.27i	0.860959i	−0.902600	+	0.430480i	$0.858344\pi$
$318$	0	0
$319$	−1220.27	−0.214176
$320$	0	0
$321$	0	0
$322$	− 1664.04i	− 0.287992i
$323$	− 919.973i	− 0.158479i
$324$	0	0
$325$	0	0
$326$	−1605.09	−0.272693
$327$	0	0
$328$	− 8054.99i	− 1.35598i
$329$	2071.31	0.347098
$330$	0	0
$331$	10784.9	1.79091	0.895454	−	0.445155i	$-0.146851\pi$
$331$	10784.9	1.79091	0.895454	+	0.445155i	$0.146851\pi$
$332$	− 6287.30i	− 1.03934i
$333$	0	0
$334$	136.994	0.0224430
$335$	0	0
$336$	0	0
$337$	4752.67i	0.768232i	0.923285	+	0.384116i	$0.125494\pi$
$337$	4752.67i	0.768232i	−0.923285	+	0.384116i	$0.874506\pi$
$338$	− 2912.28i	− 0.468661i
$339$	0	0
$340$	0	0
$341$	−796.816	−0.126540
$342$	0	0
$343$	6114.42i	0.962530i
$344$	5922.16	0.928202
$345$	0	0
$346$	−410.099	−0.0637199
$347$	10303.2i	1.59397i	0.604000	+	0.796984i	$0.293573\pi$
$347$	10303.2i	1.59397i	−0.604000	+	0.796984i	$0.706427\pi$
$348$	0	0
$349$	9058.09	1.38931	0.694654	−	0.719344i	$-0.255558\pi$
$349$	9058.09	1.38931	0.694654	+	0.719344i	$0.255558\pi$
$350$	0	0
$351$	0	0
$352$	2074.33i	0.314097i
$353$	1275.21i	0.192273i	0.995368	+	0.0961366i	$0.0306486\pi$
$353$	1275.21i	0.192273i	−0.995368	+	0.0961366i	$0.969351\pi$
$354$	0	0
$355$	0	0
$356$	−8876.21	−1.32146
$357$	0	0
$358$	4184.03i	0.617690i
$359$	2556.89	0.375899	0.187949	−	0.982179i	$-0.439816\pi$
$359$	2556.89	0.375899	0.187949	+	0.982179i	$0.439816\pi$
$360$	0	0
$361$	−5891.01	−0.858873
$362$	− 3087.67i	− 0.448299i
$363$	0	0
$364$	−182.657	−0.0263017
$365$	0	0
$366$	0	0
$367$	− 754.620i	− 0.107332i	−0.998559	−	0.0536660i	$-0.982909\pi$
$367$	− 754.620i	− 0.107332i	0.998559	−	0.0536660i	$-0.0170906\pi$
$368$	− 2884.88i	− 0.408655i
$369$	0	0
$370$	0	0
$371$	1105.61	0.154718
$372$	0	0
$373$	− 6177.49i	− 0.857529i	−0.903416	−	0.428765i	$-0.858949\pi$
$373$	− 6177.49i	− 0.857529i	0.903416	−	0.428765i	$-0.141051\pi$
$374$	442.813	0.0612227
$375$	0	0
$376$	−3664.95	−0.502674
$377$	297.004i	0.0405743i
$378$	0	0
$379$	−8146.93	−1.10417	−0.552084	−	0.833789i	$-0.686167\pi$
$379$	−8146.93	−1.10417	−0.552084	+	0.833789i	$0.686167\pi$
$380$	0	0
$381$	0	0
$382$	3008.30i	0.402927i
$383$	4710.88i	0.628498i	0.949341	+	0.314249i	$0.101753\pi$
$383$	4710.88i	0.628498i	−0.949341	+	0.314249i	$0.898247\pi$
$384$	0	0
$385$	0	0
$386$	−6374.96	−0.840614
$387$	0	0
$388$	− 160.012i	− 0.0209365i
$389$	−9508.93	−1.23939	−0.619694	−	0.784844i	$-0.712743\pi$
$389$	−9508.93	−1.23939	−0.619694	+	0.784844i	$0.712743\pi$
$390$	0	0
$391$	−3457.94	−0.447252
$392$	− 4326.25i	− 0.557420i
$393$	0	0
$394$	3009.27	0.384784
$395$	0	0
$396$	0	0
$397$	− 9615.34i	− 1.21557i	−0.794103	−	0.607783i	$-0.792059\pi$
$397$	− 9615.34i	− 1.21557i	0.794103	−	0.607783i	$-0.207941\pi$
$398$	− 2910.86i	− 0.366603i
$399$	0	0
$400$	0	0
$401$	481.837	0.0600045	0.0300022	−	0.999550i	$-0.490449\pi$
$401$	481.837	0.0600045	0.0300022	+	0.999550i	$0.490449\pi$
$402$	0	0
$403$	193.938i	0.0239721i
$404$	9491.68	1.16888
$405$	0	0
$406$	1542.25	0.188523
$407$	3178.22i	0.387073i
$408$	0	0
$409$	−12331.6	−1.49085	−0.745423	−	0.666592i	$-0.767752\pi$
$409$	−12331.6	−1.49085	−0.745423	+	0.666592i	$0.767752\pi$
$410$	0	0
$411$	0	0
$412$	− 7712.15i	− 0.922209i
$413$	− 5290.42i	− 0.630326i
$414$	0	0
$415$	0	0
$416$	504.875	0.0595037
$417$	0	0
$418$	465.926i	0.0545196i
$419$	553.776	0.0645674	0.0322837	−	0.999479i	$-0.489722\pi$
$419$	553.776	0.0645674	0.0322837	+	0.999479i	$0.489722\pi$
$420$	0	0
$421$	−522.671	−0.0605070	−0.0302535	−	0.999542i	$-0.509631\pi$
$421$	−522.671	−0.0605070	−0.0302535	+	0.999542i	$0.509631\pi$
$422$	6527.45i	0.752965i
$423$	0	0
$424$	−1956.25	−0.224066
$425$	0	0
$426$	0	0
$427$	4544.11i	0.515000i
$428$	− 2902.42i	− 0.327789i
$429$	0	0
$430$	0	0
$431$	1294.01	0.144618	0.0723089	−	0.997382i	$-0.476963\pi$
$431$	1294.01	0.144618	0.0723089	+	0.997382i	$0.476963\pi$
$432$	0	0
$433$	− 179.997i	− 0.0199771i	−0.999950	−	0.00998856i	$-0.996820\pi$
$433$	− 179.997i	− 0.0199771i	0.999950	−	0.00998856i	$-0.00317951\pi$
$434$	1007.06	0.111383
$435$	0	0
$436$	−4660.82	−0.511956
$437$	− 3638.43i	− 0.398283i
$438$	0	0
$439$	6068.09	0.659714	0.329857	−	0.944031i	$-0.393000\pi$
$439$	6068.09	0.659714	0.329857	+	0.944031i	$0.393000\pi$
$440$	0	0
$441$	0	0
$442$	− 107.777i	− 0.0115983i
$443$	15649.3i	1.67838i	0.543838	+	0.839190i	$0.316971\pi$
$443$	15649.3i	1.67838i	−0.543838	+	0.839190i	$0.683029\pi$
$444$	0	0
$445$	0	0
$446$	7697.84	0.817272
$447$	0	0
$448$	− 510.416i	− 0.0538279i
$449$	−7274.81	−0.764631	−0.382316	−	0.924032i	$-0.624873\pi$
$449$	−7274.81	−0.764631	−0.382316	+	0.924032i	$0.624873\pi$
$450$	0	0
$451$	−4791.11	−0.500232
$452$	− 2024.03i	− 0.210624i
$453$	0	0
$454$	5797.29	0.599296
$455$	0	0
$456$	0	0
$457$	4356.56i	0.445933i	0.974826	+	0.222967i	$0.0715741\pi$
$457$	4356.56i	0.445933i	−0.974826	+	0.222967i	$0.928426\pi$
$458$	5245.98i	0.535216i
$459$	0	0
$460$	0	0
$461$	9318.22	0.941416	0.470708	−	0.882289i	$-0.343999\pi$
$461$	9318.22	0.941416	0.470708	+	0.882289i	$0.343999\pi$
$462$	0	0
$463$	− 18699.8i	− 1.87701i	−0.345270	−	0.938503i	$-0.612213\pi$
$463$	− 18699.8i	− 1.87701i	0.345270	−	0.938503i	$-0.387787\pi$
$464$	2673.72	0.267510
$465$	0	0
$466$	7356.10	0.731255
$467$	− 5345.39i	− 0.529669i	−0.964294	−	0.264834i	$-0.914683\pi$
$467$	− 5345.39i	− 0.529669i	0.964294	−	0.264834i	$-0.0853173\pi$
$468$	0	0
$469$	−6276.29	−0.617936
$470$	0	0
$471$	0	0
$472$	9360.81i	0.912852i
$473$	− 3522.50i	− 0.342420i
$474$	0	0
$475$	0	0
$476$	1970.96	0.189788
$477$	0	0
$478$	5085.30i	0.486603i
$479$	−16930.5	−1.61498	−0.807489	−	0.589883i	$-0.799174\pi$
$479$	−16930.5	−1.61498	−0.807489	+	0.589883i	$0.799174\pi$
$480$	0	0
$481$	773.553	0.0733285
$482$	− 5288.90i	− 0.499799i
$483$	0	0
$484$	−7503.36	−0.704673
$485$	0	0
$486$	0	0
$487$	− 3052.84i	− 0.284060i	−0.989862	−	0.142030i	$-0.954637\pi$
$487$	− 3052.84i	− 0.284060i	0.989862	−	0.142030i	$-0.0453630\pi$
$488$	− 8040.29i	− 0.745834i
$489$	0	0
$490$	0	0
$491$	7591.68	0.697775	0.348888	−	0.937165i	$-0.386560\pi$
$491$	7591.68	0.697775	0.348888	+	0.937165i	$0.386560\pi$
$492$	0	0
$493$	− 3204.84i	− 0.292776i
$494$	113.403	0.0103284
$495$	0	0
$496$	1745.89	0.158050
$497$	12194.1i	1.10056i
$498$	0	0
$499$	3688.97	0.330944	0.165472	−	0.986215i	$-0.447085\pi$
$499$	3688.97	0.330944	0.165472	+	0.986215i	$0.447085\pi$
$500$	0	0
$501$	0	0
$502$	2339.46i	0.207998i
$503$	2212.28i	0.196105i	0.995181	+	0.0980523i	$0.0312612\pi$
$503$	2212.28i	0.196105i	−0.995181	+	0.0980523i	$0.968739\pi$
$504$	0	0
$505$	0	0
$506$	1751.29	0.153863
$507$	0	0
$508$	186.749i	0.0163103i
$509$	−13842.8	−1.20544	−0.602722	−	0.797952i	$-0.705917\pi$
$509$	−13842.8	−1.20544	−0.602722	+	0.797952i	$0.705917\pi$
$510$	0	0
$511$	−3236.79	−0.280209
$512$	− 8280.64i	− 0.714758i
$513$	0	0
$514$	4517.35	0.387649
$515$	0	0
$516$	0	0
$517$	2179.91i	0.185440i
$518$	− 4016.81i	− 0.340711i
$519$	0	0
$520$	0	0
$521$	21683.4	1.82335	0.911677	−	0.410908i	$-0.134788\pi$
$521$	21683.4	1.82335	0.911677	+	0.410908i	$0.134788\pi$
$522$	0	0
$523$	2021.57i	0.169020i	0.996423	+	0.0845098i	$0.0269324\pi$
$523$	2021.57i	0.169020i	−0.996423	+	0.0845098i	$0.973068\pi$
$524$	5648.17	0.470881
$525$	0	0
$526$	6204.62	0.514324
$527$	− 2092.70i	− 0.172978i
$528$	0	0
$529$	−1508.89	−0.124015
$530$	0	0
$531$	0	0
$532$	2073.84i	0.169008i
$533$	1166.12i	0.0947657i
$534$	0	0
$535$	0	0
$536$	11105.2	0.894908
$537$	0	0
$538$	− 285.848i	− 0.0229067i
$539$	−2573.26	−0.205636
$540$	0	0
$541$	−9538.79	−0.758049	−0.379025	−	0.925387i	$-0.623740\pi$
$541$	−9538.79	−0.758049	−0.379025	+	0.925387i	$0.623740\pi$
$542$	− 4337.11i	− 0.343717i
$543$	0	0
$544$	−5447.88	−0.429367
$545$	0	0
$546$	0	0
$547$	7163.31i	0.559929i	0.960010	+	0.279964i	$0.0903226\pi$
$547$	7163.31i	0.559929i	−0.960010	+	0.279964i	$0.909677\pi$
$548$	− 14704.3i	− 1.14624i
$549$	0	0
$550$	0	0
$551$	3372.12	0.260720
$552$	0	0
$553$	5620.16i	0.432176i
$554$	−2650.75	−0.203284
$555$	0	0
$556$	−10651.3	−0.812436
$557$	− 14886.5i	− 1.13243i	−0.824258	−	0.566214i	$-0.808408\pi$
$557$	− 14886.5i	− 1.13243i	0.824258	−	0.566214i	$-0.191592\pi$
$558$	0	0
$559$	−857.348	−0.0648693
$560$	0	0
$561$	0	0
$562$	4030.24i	0.302501i
$563$	− 8134.04i	− 0.608897i	−0.952529	−	0.304448i	$-0.901528\pi$
$563$	− 8134.04i	− 0.608897i	0.952529	−	0.304448i	$-0.0984721\pi$
$564$	0	0
$565$	0	0
$566$	−6584.71	−0.489003
$567$	0	0
$568$	− 21576.0i	− 1.59386i
$569$	−10031.8	−0.739115	−0.369557	−	0.929208i	$-0.620491\pi$
$569$	−10031.8	−0.739115	−0.369557	+	0.929208i	$0.620491\pi$
$570$	0	0
$571$	−2507.05	−0.183742	−0.0918712	−	0.995771i	$-0.529285\pi$
$571$	−2507.05	−0.183742	−0.0918712	+	0.995771i	$0.529285\pi$
$572$	− 192.234i	− 0.0140519i
$573$	0	0
$574$	6055.26	0.440316
$575$	0	0
$576$	0	0
$577$	− 5378.18i	− 0.388036i	−0.980998	−	0.194018i	$-0.937848\pi$
$577$	− 5378.18i	− 0.388036i	0.980998	−	0.194018i	$-0.0621520\pi$
$578$	− 5371.90i	− 0.386577i
$579$	0	0
$580$	0	0
$581$	10794.9	0.770821
$582$	0	0
$583$	1163.58i	0.0826594i
$584$	5727.13	0.405805
$585$	0	0
$586$	10757.3	0.758328
$587$	− 6153.77i	− 0.432697i	−0.976316	−	0.216349i	$-0.930585\pi$
$587$	− 6153.77i	− 0.432697i	0.976316	−	0.216349i	$-0.0694148\pi$
$588$	0	0
$589$	2201.93	0.154039
$590$	0	0
$591$	0	0
$592$	− 6963.77i	− 0.483461i
$593$	− 14278.1i	− 0.988753i	−0.869248	−	0.494376i	$-0.835396\pi$
$593$	− 14278.1i	− 0.988753i	0.869248	−	0.494376i	$-0.164604\pi$
$594$	0	0
$595$	0	0
$596$	−742.523	−0.0510318
$597$	0	0
$598$	− 426.250i	− 0.0291483i
$599$	−2071.22	−0.141282	−0.0706410	−	0.997502i	$-0.522504\pi$
$599$	−2071.22	−0.141282	−0.0706410	+	0.997502i	$0.522504\pi$
$600$	0	0
$601$	12307.3	0.835319	0.417660	−	0.908604i	$-0.362850\pi$
$601$	12307.3	0.835319	0.417660	+	0.908604i	$0.362850\pi$
$602$	4451.93i	0.301407i
$603$	0	0
$604$	4786.22	0.322431
$605$	0	0
$606$	0	0
$607$	− 22963.7i	− 1.53553i	−0.640732	−	0.767765i	$-0.721369\pi$
$607$	− 22963.7i	− 1.53553i	0.640732	−	0.767765i	$-0.278631\pi$
$608$	− 5732.23i	− 0.382356i
$609$	0	0
$610$	0	0
$611$	530.573	0.0351304
$612$	0	0
$613$	12746.7i	0.839862i	0.907556	+	0.419931i	$0.137946\pi$
$613$	12746.7i	0.839862i	−0.907556	+	0.419931i	$0.862054\pi$
$614$	−362.563	−0.0238304
$615$	0	0
$616$	−2279.85	−0.149120
$617$	− 13959.6i	− 0.910844i	−0.890276	−	0.455422i	$-0.849488\pi$
$617$	− 13959.6i	− 0.910844i	0.890276	−	0.455422i	$-0.150512\pi$
$618$	0	0
$619$	−15542.0	−1.00918	−0.504591	−	0.863358i	$-0.668357\pi$
$619$	−15542.0	−1.00918	−0.504591	+	0.863358i	$0.668357\pi$
$620$	0	0
$621$	0	0
$622$	12095.7i	0.779730i
$623$	− 15239.9i	− 0.980053i
$624$	0	0
$625$	0	0
$626$	10924.7	0.697509
$627$	0	0
$628$	12457.0i	0.791545i
$629$	−8347.06	−0.529124
$630$	0	0
$631$	−12297.5	−0.775844	−0.387922	−	0.921692i	$-0.626807\pi$
$631$	−12297.5	−0.775844	−0.387922	+	0.921692i	$0.626807\pi$
$632$	− 9944.24i	− 0.625887i
$633$	0	0
$634$	6463.41	0.404882
$635$	0	0
$636$	0	0
$637$	626.309i	0.0389565i
$638$	1623.11i	0.100720i
$639$	0	0
$640$	0	0
$641$	16523.4	1.01815	0.509075	−	0.860722i	$-0.329987\pi$
$641$	16523.4	1.01815	0.509075	+	0.860722i	$0.329987\pi$
$642$	0	0
$643$	25293.3i	1.55128i	0.631177	+	0.775639i	$0.282572\pi$
$643$	25293.3i	1.55128i	−0.631177	+	0.775639i	$0.717428\pi$
$644$	7795.02	0.476967
$645$	0	0
$646$	−1223.67	−0.0745276
$647$	− 1442.91i	− 0.0876763i	−0.999039	−	0.0438381i	$-0.986041\pi$
$647$	− 1442.91i	− 0.0876763i	0.999039	−	0.0438381i	$-0.0139586\pi$
$648$	0	0
$649$	5567.81	0.336757
$650$	0	0
$651$	0	0
$652$	− 7518.87i	− 0.451628i
$653$	− 9042.22i	− 0.541883i	−0.962596	−	0.270941i	$-0.912665\pi$
$653$	− 9042.22i	− 0.541883i	0.962596	−	0.270941i	$-0.0873350\pi$
$654$	0	0
$655$	0	0
$656$	10497.7	0.624799
$657$	0	0
$658$	− 2755.09i	− 0.163229i
$659$	10084.1	0.596085	0.298042	−	0.954553i	$-0.403666\pi$
$659$	10084.1	0.596085	0.298042	+	0.954553i	$0.403666\pi$
$660$	0	0
$661$	4181.32	0.246043	0.123022	−	0.992404i	$-0.460742\pi$
$661$	4181.32	0.246043	0.123022	+	0.992404i	$0.460742\pi$
$662$	− 14345.2i	− 0.842207i
$663$	0	0
$664$	−19100.3	−1.11632
$665$	0	0
$666$	0	0
$667$	− 12674.9i	− 0.735793i
$668$	641.731i	0.0371696i
$669$	0	0
$670$	0	0
$671$	−4782.37	−0.275143
$672$	0	0
$673$	− 21108.5i	− 1.20902i	−0.796596	−	0.604512i	$-0.793368\pi$
$673$	− 21108.5i	− 1.20902i	0.796596	−	0.604512i	$-0.206632\pi$
$674$	6321.61	0.361275
$675$	0	0
$676$	13642.2	0.776186
$677$	− 19793.4i	− 1.12367i	−0.827250	−	0.561834i	$-0.810096\pi$
$677$	− 19793.4i	− 1.12367i	0.827250	−	0.561834i	$-0.189904\pi$
$678$	0	0
$679$	274.730	0.0155275
$680$	0	0
$681$	0	0
$682$	1059.86i	0.0595075i
$683$	7226.49i	0.404852i	0.979298	+	0.202426i	$0.0648826\pi$
$683$	7226.49i	0.404852i	−0.979298	+	0.202426i	$0.935117\pi$
$684$	0	0
$685$	0	0
$686$	8132.91	0.452647
$687$	0	0
$688$	7718.11i	0.427689i
$689$	283.205	0.0156593
$690$	0	0
$691$	−1152.43	−0.0634451	−0.0317225	−	0.999497i	$-0.510099\pi$
$691$	−1152.43	−0.0634451	−0.0317225	+	0.999497i	$0.510099\pi$
$692$	− 1921.06i	− 0.105531i
$693$	0	0
$694$	13704.5	0.749593
$695$	0	0
$696$	0	0
$697$	− 12583.0i	− 0.683810i
$698$	− 12048.3i	− 0.653347i
$699$	0	0
$700$	0	0
$701$	13364.5	0.720073	0.360036	−	0.932938i	$-0.382764\pi$
$701$	13364.5	0.720073	0.360036	+	0.932938i	$0.382764\pi$
$702$	0	0
$703$	− 8782.74i	− 0.471191i
$704$	537.178	0.0287580
$705$	0	0
$706$	1696.18	0.0904200
$707$	16296.6i	0.866898i
$708$	0	0
$709$	15588.0	0.825696	0.412848	−	0.910800i	$-0.364534\pi$
$709$	15588.0	0.825696	0.412848	+	0.910800i	$0.364534\pi$
$710$	0	0
$711$	0	0
$712$	26965.2i	1.41933i
$713$	− 8276.47i	− 0.434721i
$714$	0	0
$715$	0	0
$716$	−19599.6	−1.02301
$717$	0	0
$718$	− 3400.97i	− 0.176773i
$719$	−1086.42	−0.0563513	−0.0281757	−	0.999603i	$-0.508970\pi$
$719$	−1086.42	−0.0563513	−0.0281757	+	0.999603i	$0.508970\pi$
$720$	0	0
$721$	13241.3	0.683953
$722$	7835.74i	0.403900i
$723$	0	0
$724$	14463.8	0.742463
$725$	0	0
$726$	0	0
$727$	27592.3i	1.40762i	0.710387	+	0.703811i	$0.248520\pi$
$727$	27592.3i	1.40762i	−0.710387	+	0.703811i	$0.751480\pi$
$728$	554.897i	0.0282498i
$729$	0	0
$730$	0	0
$731$	9251.24	0.468084
$732$	0	0
$733$	28759.0i	1.44916i	0.689189	+	0.724582i	$0.257967\pi$
$733$	28759.0i	1.44916i	−0.689189	+	0.724582i	$0.742033\pi$
$734$	−1003.73	−0.0504748
$735$	0	0
$736$	−21546.0	−1.07907
$737$	− 6605.36i	− 0.330138i
$738$	0	0
$739$	−113.264	−0.00563800	−0.00281900	−	0.999996i	$-0.500897\pi$
$739$	−113.264	−0.00563800	−0.00281900	+	0.999996i	$0.500897\pi$
$740$	0	0
$741$	0	0
$742$	− 1470.59i	− 0.0727589i
$743$	37767.7i	1.86482i	0.361399	+	0.932411i	$0.382299\pi$
$743$	37767.7i	1.86482i	−0.361399	+	0.932411i	$0.617701\pi$
$744$	0	0
$745$	0	0
$746$	−8216.80	−0.403269
$747$	0	0
$748$	2074.30i	0.101396i
$749$	4983.27	0.243104
$750$	0	0
$751$	26372.9	1.28144	0.640720	−	0.767774i	$-0.278636\pi$
$751$	26372.9	1.28144	0.640720	+	0.767774i	$0.278636\pi$
$752$	− 4776.38i	− 0.231618i
$753$	0	0
$754$	395.051	0.0190808
$755$	0	0
$756$	0	0
$757$	13224.5i	0.634943i	0.948268	+	0.317472i	$0.102834\pi$
$757$	13224.5i	0.634943i	−0.948268	+	0.317472i	$0.897166\pi$
$758$	10836.4i	0.519255i
$759$	0	0
$760$	0	0
$761$	1709.91	0.0814511	0.0407256	−	0.999170i	$-0.487033\pi$
$761$	1709.91	0.0814511	0.0407256	+	0.999170i	$0.487033\pi$
$762$	0	0
$763$	− 8002.33i	− 0.379691i
$764$	−14092.0	−0.667320
$765$	0	0
$766$	6266.03	0.295563
$767$	− 1355.16i	− 0.0637965i
$768$	0	0
$769$	4705.78	0.220669	0.110335	−	0.993894i	$-0.464808\pi$
$769$	4705.78	0.220669	0.110335	+	0.993894i	$0.464808\pi$
$770$	0	0
$771$	0	0
$772$	− 29862.7i	− 1.39221i
$773$	− 19109.6i	− 0.889164i	−0.895738	−	0.444582i	$-0.853352\pi$
$773$	− 19109.6i	− 0.889164i	0.895738	−	0.444582i	$-0.146648\pi$
$774$	0	0
$775$	0	0
$776$	−486.103	−0.0224872
$777$	0	0
$778$	12648.0i	0.582844i
$779$	13239.8	0.608941
$780$	0	0
$781$	−12833.4	−0.587985
$782$	4599.47i	0.210328i
$783$	0	0
$784$	5638.23	0.256844
$785$	0	0
$786$	0	0
$787$	− 17209.4i	− 0.779480i	−0.920925	−	0.389740i	$-0.872565\pi$
$787$	− 17209.4i	− 0.779480i	0.920925	−	0.389740i	$-0.127435\pi$
$788$	14096.6i	0.637271i
$789$	0	0
$790$	0	0
$791$	3475.12	0.156209
$792$	0	0
$793$	1163.99i	0.0521242i
$794$	−12789.5	−0.571642
$795$	0	0
$796$	13635.6	0.607160
$797$	34775.4i	1.54556i	0.634676	+	0.772779i	$0.281134\pi$
$797$	34775.4i	1.54556i	−0.634676	+	0.772779i	$0.718866\pi$
$798$	0	0
$799$	−5725.17	−0.253494
$800$	0	0
$801$	0	0
$802$	− 640.901i	− 0.0282182i
$803$	− 3406.50i	− 0.149704i
$804$	0	0
$805$	0	0
$806$	257.961	0.0112733
$807$	0	0
$808$	− 28835.0i	− 1.25546i
$809$	−22233.7	−0.966250	−0.483125	−	0.875551i	$-0.660498\pi$
$809$	−22233.7	−0.966250	−0.483125	+	0.875551i	$0.660498\pi$
$810$	0	0
$811$	−3576.54	−0.154857	−0.0774287	−	0.996998i	$-0.524671\pi$
$811$	−3576.54	−0.154857	−0.0774287	+	0.996998i	$0.524671\pi$
$812$	7224.47i	0.312228i
$813$	0	0
$814$	4227.42	0.182028
$815$	0	0
$816$	0	0
$817$	9734.12i	0.416834i
$818$	16402.4i	0.701097i
$819$	0	0
$820$	0	0
$821$	−29910.6	−1.27148	−0.635741	−	0.771902i	$-0.719306\pi$
$821$	−29910.6	−1.27148	−0.635741	+	0.771902i	$0.719306\pi$
$822$	0	0
$823$	34421.1i	1.45789i	0.684572	+	0.728945i	$0.259989\pi$
$823$	34421.1i	1.45789i	−0.684572	+	0.728945i	$0.740011\pi$
$824$	−23428.9	−0.990515
$825$	0	0
$826$	−7036.89	−0.296422
$827$	30778.8i	1.29417i	0.762416	+	0.647087i	$0.224013\pi$
$827$	30778.8i	1.29417i	−0.762416	+	0.647087i	$0.775987\pi$
$828$	0	0
$829$	−21156.8	−0.886375	−0.443188	−	0.896429i	$-0.646152\pi$
$829$	−21156.8	−0.886375	−0.443188	+	0.896429i	$0.646152\pi$
$830$	0	0
$831$	0	0
$832$	− 130.745i	− 0.00544802i
$833$	− 6758.21i	− 0.281102i
$834$	0	0
$835$	0	0
$836$	−2182.57	−0.0902941
$837$	0	0
$838$	− 736.588i	− 0.0303640i
$839$	20004.9	0.823178	0.411589	−	0.911370i	$-0.364974\pi$
$839$	20004.9	0.823178	0.411589	+	0.911370i	$0.364974\pi$
$840$	0	0
$841$	−12641.8	−0.518342
$842$	695.215i	0.0284545i
$843$	0	0
$844$	−30577.0	−1.24704
$845$	0	0
$846$	0	0
$847$	− 12882.8i	− 0.522618i
$848$	− 2549.50i	− 0.103243i
$849$	0	0
$850$	0	0
$851$	−33012.0	−1.32977
$852$	0	0
$853$	− 32918.6i	− 1.32135i	−0.750672	−	0.660675i	$-0.770270\pi$
$853$	− 32918.6i	− 1.32135i	0.750672	−	0.660675i	$-0.229730\pi$
$854$	6044.21	0.242188
$855$	0	0
$856$	−8817.33	−0.352068
$857$	− 8865.73i	− 0.353381i	−0.984266	−	0.176691i	$-0.943461\pi$
$857$	− 8865.73i	− 0.353381i	0.984266	−	0.176691i	$-0.0565392\pi$
$858$	0	0
$859$	32088.1	1.27454	0.637271	−	0.770640i	$-0.280063\pi$
$859$	32088.1	1.27454	0.637271	+	0.770640i	$0.280063\pi$
$860$	0	0
$861$	0	0
$862$	− 1721.19i	− 0.0680091i
$863$	40687.9i	1.60491i	0.596715	+	0.802453i	$0.296472\pi$
$863$	40687.9i	1.60491i	−0.596715	+	0.802453i	$0.703528\pi$
$864$	0	0
$865$	0	0
$866$	−239.417	−0.00939460
$867$	0	0
$868$	4717.44i	0.184471i
$869$	−5914.83	−0.230894
$870$	0	0
$871$	−1607.69	−0.0625425
$872$	14159.2i	0.549876i
$873$	0	0
$874$	−4839.54	−0.187300
$875$	0	0
$876$	0	0
$877$	− 18440.9i	− 0.710041i	−0.934858	−	0.355021i	$-0.884474\pi$
$877$	− 18440.9i	− 0.710041i	0.934858	−	0.355021i	$-0.115526\pi$
$878$	− 8071.29i	− 0.310242i
$879$	0	0
$880$	0	0
$881$	−13603.8	−0.520230	−0.260115	−	0.965578i	$-0.583760\pi$
$881$	−13603.8	−0.520230	−0.260115	+	0.965578i	$0.583760\pi$
$882$	0	0
$883$	1727.71i	0.0658461i	0.999458	+	0.0329231i	$0.0104816\pi$
$883$	1727.71i	0.0658461i	−0.999458	+	0.0329231i	$0.989518\pi$
$884$	504.868	0.0192088
$885$	0	0
$886$	20815.5	0.789289
$887$	− 7445.07i	− 0.281827i	−0.990022	−	0.140914i	$-0.954996\pi$
$887$	− 7445.07i	− 0.281827i	0.990022	−	0.140914i	$-0.0450040\pi$
$888$	0	0
$889$	−320.636	−0.0120965
$890$	0	0
$891$	0	0
$892$	36059.6i	1.35355i
$893$	− 6024.00i	− 0.225739i
$894$	0	0
$895$	0	0
$896$	15089.0	0.562599
$897$	0	0
$898$	9676.36i	0.359582i
$899$	7670.68	0.284574
$900$	0	0
$901$	−3055.94	−0.112994
$902$	6372.74i	0.235243i
$903$	0	0
$904$	−6148.84	−0.226225
$905$	0	0
$906$	0	0
$907$	16050.6i	0.587598i	0.955867	+	0.293799i	$0.0949196\pi$
$907$	16050.6i	0.587598i	−0.955867	+	0.293799i	$0.905080\pi$
$908$	27156.7i	0.992541i
$909$	0	0
$910$	0	0
$911$	−52398.4	−1.90564	−0.952818	−	0.303541i	$-0.901831\pi$
$911$	−52398.4	−1.90564	−0.952818	+	0.303541i	$0.901831\pi$
$912$	0	0
$913$	11360.9i	0.411818i
$914$	5794.75	0.209708
$915$	0	0
$916$	−24574.2	−0.886413
$917$	9697.55i	0.349227i
$918$	0	0
$919$	−34880.1	−1.25200	−0.626001	−	0.779822i	$-0.715309\pi$
$919$	−34880.1	−1.25200	−0.626001	+	0.779822i	$0.715309\pi$
$920$	0	0
$921$	0	0
$922$	− 12394.3i	− 0.442718i
$923$	3123.55i	0.111390i
$924$	0	0
$925$	0	0
$926$	−24873.0	−0.882696
$927$	0	0
$928$	− 19968.9i	− 0.706370i
$929$	42172.6	1.48938	0.744692	−	0.667408i	$-0.232596\pi$
$929$	42172.6	1.48938	0.744692	+	0.667408i	$0.232596\pi$
$930$	0	0
$931$	7110.96	0.250325
$932$	34458.8i	1.21109i
$933$	0	0
$934$	−7110.01	−0.249086
$935$	0	0
$936$	0	0
$937$	− 33853.9i	− 1.18032i	−0.807287	−	0.590159i	$-0.799065\pi$
$937$	− 33853.9i	− 1.18032i	0.807287	−	0.590159i	$-0.200935\pi$
$938$	8348.21i	0.290596i
$939$	0	0
$940$	0	0
$941$	14665.4	0.508053	0.254027	−	0.967197i	$-0.418245\pi$
$941$	14665.4	0.508053	0.254027	+	0.967197i	$0.418245\pi$
$942$	0	0
$943$	− 49764.9i	− 1.71853i
$944$	−12199.6	−0.420616
$945$	0	0
$946$	−4685.35	−0.161029
$947$	43534.3i	1.49385i	0.664908	+	0.746925i	$0.268471\pi$
$947$	43534.3i	1.49385i	−0.664908	+	0.746925i	$0.731529\pi$
$948$	0	0
$949$	−829.113	−0.0283605
$950$	0	0
$951$	0	0
$952$	− 5987.63i	− 0.203845i
$953$	− 3831.21i	− 0.130226i	−0.997878	−	0.0651128i	$-0.979259\pi$
$953$	− 3831.21i	− 0.130226i	0.997878	−	0.0651128i	$-0.0207407\pi$
$954$	0	0
$955$	0	0
$956$	−23821.5	−0.805901
$957$	0	0
$958$	22519.6i	0.759473i
$959$	25246.4	0.850104
$960$	0	0
$961$	−24782.2	−0.831868
$962$	− 1028.92i	− 0.0344840i
$963$	0	0
$964$	24775.2	0.827756
$965$	0	0
$966$	0	0
$967$	27975.3i	0.930324i	0.885225	+	0.465162i	$0.154004\pi$
$967$	27975.3i	0.930324i	−0.885225	+	0.465162i	$0.845996\pi$
$968$	22794.6i	0.756867i
$969$	0	0
$970$	0	0
$971$	−39865.6	−1.31756	−0.658779	−	0.752337i	$-0.728927\pi$
$971$	−39865.6	−1.31756	−0.658779	+	0.752337i	$0.728927\pi$
$972$	0	0
$973$	− 18287.5i	− 0.602540i
$974$	−4060.64	−0.133584
$975$	0	0
$976$	10478.6	0.343659
$977$	− 38912.9i	− 1.27424i	−0.770764	−	0.637121i	$-0.780125\pi$
$977$	− 38912.9i	− 1.27424i	0.770764	−	0.637121i	$-0.219875\pi$
$978$	0	0
$979$	16038.9	0.523602
$980$	0	0
$981$	0	0
$982$	− 10097.8i	− 0.328141i
$983$	27331.9i	0.886829i	0.896317	+	0.443415i	$0.146233\pi$
$983$	27331.9i	0.886829i	−0.896317	+	0.443415i	$0.853767\pi$
$984$	0	0
$985$	0	0
$986$	−4262.81	−0.137683
$987$	0	0
$988$	531.221i	0.0171056i
$989$	36588.0	1.17637
$990$	0	0
$991$	−47040.3	−1.50786	−0.753928	−	0.656957i	$-0.771843\pi$
$991$	−47040.3	−1.50786	−0.753928	+	0.656957i	$0.771843\pi$
$992$	− 13039.3i	− 0.417338i
$993$	0	0
$994$	16219.6	0.517559
$995$	0	0
$996$	0	0
$997$	60011.9i	1.90632i	0.302473	+	0.953158i	$0.402188\pi$
$997$	60011.9i	1.90632i	−0.302473	+	0.953158i	$0.597812\pi$
$998$	− 4906.76i	− 0.155632i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.b.q.649.4		8
3.2	odd	2		675.4.b.p.649.5		8
5.2	odd	4		675.4.a.v.1.3	yes	4
5.3	odd	4		675.4.a.y.1.2	yes	4
5.4	even	2	inner	675.4.b.q.649.5		8
15.2	even	4		675.4.a.z.1.2	yes	4
15.8	even	4		675.4.a.u.1.3	✓	4
15.14	odd	2		675.4.b.p.649.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.4.a.u.1.3	✓	4	15.8	even	4
675.4.a.v.1.3	yes	4	5.2	odd	4
675.4.a.y.1.2	yes	4	5.3	odd	4
675.4.a.z.1.2	yes	4	15.2	even	4
675.4.b.p.649.4		8	15.14	odd	2
675.4.b.p.649.5		8	3.2	odd	2
675.4.b.q.649.4		8	1.1	even	1	trivial
675.4.b.q.649.5		8	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.33012i q^{2} +6.23078 q^{4} +10.6979i q^{7} -18.9286i q^{8} -11.2588 q^{11} +2.74029i q^{13} +14.2294 q^{14} +24.6689 q^{16} -29.5692i q^{17} +31.1126 q^{19} +14.9755i q^{22} -116.944i q^{23} +3.64491 q^{26} +66.6560i q^{28} +108.384 q^{29} +70.7730 q^{31} -184.242i q^{32} -39.3305 q^{34} -282.289i q^{37} -41.3834i q^{38} +425.545 q^{41} +312.868i q^{43} -70.1509 q^{44} -155.549 q^{46} -193.619i q^{47} +228.556 q^{49} +17.0741i q^{52} -103.349i q^{53} +202.496 q^{56} -144.164i q^{58} -494.531 q^{59} +424.769 q^{61} -94.1365i q^{62} -47.7120 q^{64} +586.687i q^{67} -184.239i q^{68} +1139.86 q^{71} +302.564i q^{73} -375.478 q^{74} +193.856 q^{76} -120.445i q^{77} +525.354 q^{79} -566.026i q^{82} -1009.07i q^{83} +416.151 q^{86} +213.113i q^{88} -1424.57 q^{89} -29.3152 q^{91} -728.652i q^{92} -257.537 q^{94} -25.6808i q^{97} -304.007i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 38 q^{4} + 104 q^{11} - 276 q^{14} - 10 q^{16} + 92 q^{19} + 938 q^{26} - 940 q^{29} - 524 q^{31} + 84 q^{34} + 1396 q^{41} - 838 q^{44} + 1074 q^{46} - 1560 q^{49} + 4068 q^{56} - 200 q^{59} + 148 q^{61}+ \cdots - 5694 q^{94}+O(q^{100}) \) 8 * q - 38 * q^4 + 104 * q^11 - 276 * q^14 - 10 * q^16 + 92 * q^19 + 938 * q^26 - 940 * q^29 - 524 * q^31 + 84 * q^34 + 1396 * q^41 - 838 * q^44 + 1074 * q^46 - 1560 * q^49 + 4068 * q^56 - 200 * q^59 + 148 * q^61 + 1930 * q^64 + 3244 * q^71 + 506 * q^74 - 4136 * q^76 + 1016 * q^79 + 7880 * q^86 - 1512 * q^89 + 2240 * q^91 - 5694 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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