LMFDB - Embedded newform 675.4.b.q.649.8

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			649.8
Root			$-3.67875i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.q.649.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.72651i q^{2} -14.3399 q^{4} +17.6256i q^{7} -29.9655i q^{8} +34.2390 q^{11} -53.8784i q^{13} -83.3075 q^{14} +26.9130 q^{16} -74.7605i q^{17} +89.5599 q^{19} +161.831i q^{22} -176.169i q^{23} +254.657 q^{26} -252.749i q^{28} -194.211 q^{29} +107.939 q^{31} -112.519i q^{32} +353.356 q^{34} -430.818i q^{37} +423.305i q^{38} -108.894 q^{41} -409.261i q^{43} -490.982 q^{44} +832.666 q^{46} +409.188i q^{47} +32.3387 q^{49} +772.610i q^{52} +24.7760i q^{53} +528.159 q^{56} -917.940i q^{58} +295.748 q^{59} +305.325 q^{61} +510.176i q^{62} +747.127 q^{64} +915.415i q^{67} +1072.06i q^{68} +228.340 q^{71} +158.720i q^{73} +2036.27 q^{74} -1284.28 q^{76} +603.482i q^{77} +319.140 q^{79} -514.686i q^{82} +936.446i q^{83} +1934.38 q^{86} -1025.99i q^{88} -920.893 q^{89} +949.639 q^{91} +2526.25i q^{92} -1934.03 q^{94} -914.533i q^{97} +152.849i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.72651i	1.67107i	0.549435	+	0.835536i	$0.314843\pi$
$2$	4.72651i	1.67107i	−0.549435	+	0.835536i	$0.685157\pi$
$3$	0	0
$3$	0	0
$4$	−14.3399	−1.79248
$5$	0	0
$5$	0	0
$6$	0	0
$7$	17.6256i	0.951692i	0.879529	+	0.475846i	$0.157858\pi$
$7$	17.6256i	0.951692i	−0.879529	+	0.475846i	$0.842142\pi$
$8$	− 29.9655i	− 1.32430i
$9$	0	0
$10$	0	0
$11$	34.2390	0.938494	0.469247	−	0.883067i	$-0.344525\pi$
$11$	34.2390	0.938494	0.469247	+	0.883067i	$0.344525\pi$
$12$	0	0
$13$	− 53.8784i	− 1.14948i	−0.818337	−	0.574738i	$-0.805104\pi$
$13$	− 53.8784i	− 1.14948i	0.818337	−	0.574738i	$-0.194896\pi$
$14$	−83.3075	−1.59035
$15$	0	0
$16$	26.9130	0.420515
$17$	− 74.7605i	− 1.06659i	−0.845928	−	0.533296i	$-0.820953\pi$
$17$	− 74.7605i	− 1.06659i	0.845928	−	0.533296i	$-0.179047\pi$
$18$	0	0
$19$	89.5599	1.08139	0.540696	−	0.841218i	$-0.318161\pi$
$19$	89.5599	1.08139	0.540696	+	0.841218i	$0.318161\pi$
$20$	0	0
$21$	0	0
$22$	161.831i	1.56829i
$23$	− 176.169i	− 1.59712i	−0.601913	−	0.798562i	$-0.705595\pi$
$23$	− 176.169i	− 1.59712i	0.601913	−	0.798562i	$-0.294405\pi$
$24$	0	0
$25$	0	0
$26$	254.657	1.92086
$27$	0	0
$28$	− 252.749i	− 1.70589i
$29$	−194.211	−1.24359	−0.621795	−	0.783180i	$-0.713596\pi$
$29$	−194.211	−1.24359	−0.621795	+	0.783180i	$0.713596\pi$
$30$	0	0
$31$	107.939	0.625370	0.312685	−	0.949857i	$-0.398772\pi$
$31$	107.939	0.625370	0.312685	+	0.949857i	$0.398772\pi$
$32$	− 112.519i	− 0.621587i
$33$	0	0
$34$	353.356	1.78235
$35$	0	0
$36$	0	0
$37$	− 430.818i	− 1.91422i	−0.289726	−	0.957110i	$-0.593564\pi$
$37$	− 430.818i	− 1.91422i	0.289726	−	0.957110i	$-0.406436\pi$
$38$	423.305i	1.80708i
$39$	0	0
$40$	0	0
$41$	−108.894	−0.414788	−0.207394	−	0.978257i	$-0.566498\pi$
$41$	−108.894	−0.414788	−0.207394	+	0.978257i	$0.566498\pi$
$42$	0	0
$43$	− 409.261i	− 1.45144i	−0.687992	−	0.725718i	$-0.741508\pi$
$43$	− 409.261i	− 1.45144i	0.687992	−	0.725718i	$-0.258492\pi$
$44$	−490.982	−1.68224
$45$	0	0
$46$	832.666	2.66891
$47$	409.188i	1.26992i	0.772546	+	0.634959i	$0.218983\pi$
$47$	409.188i	1.26992i	−0.772546	+	0.634959i	$0.781017\pi$
$48$	0	0
$49$	32.3387	0.0942820
$50$	0	0
$51$	0	0
$52$	772.610i	2.06042i
$53$	24.7760i	0.0642121i	0.999484	+	0.0321061i	$0.0102214\pi$
$53$	24.7760i	0.0642121i	−0.999484	+	0.0321061i	$0.989779\pi$
$54$	0	0
$55$	0	0
$56$	528.159	1.26032
$57$	0	0
$58$	− 917.940i	− 2.07813i
$59$	295.748	0.652596	0.326298	−	0.945267i	$-0.394199\pi$
$59$	295.748	0.652596	0.326298	+	0.945267i	$0.394199\pi$
$60$	0	0
$61$	305.325	0.640868	0.320434	−	0.947271i	$-0.396171\pi$
$61$	305.325	0.640868	0.320434	+	0.947271i	$0.396171\pi$
$62$	510.176i	1.04504i
$63$	0	0
$64$	747.127	1.45923
$65$	0	0
$66$	0	0
$67$	915.415i	1.66919i	0.550864	+	0.834595i	$0.314298\pi$
$67$	915.415i	1.66919i	−0.550864	+	0.834595i	$0.685702\pi$
$68$	1072.06i	1.91185i
$69$	0	0
$70$	0	0
$71$	228.340	0.381675	0.190838	−	0.981622i	$-0.438880\pi$
$71$	228.340	0.381675	0.190838	+	0.981622i	$0.438880\pi$
$72$	0	0
$73$	158.720i	0.254476i	0.991872	+	0.127238i	$0.0406113\pi$
$73$	158.720i	0.254476i	−0.991872	+	0.127238i	$0.959389\pi$
$74$	2036.27	3.19880
$75$	0	0
$76$	−1284.28	−1.93838
$77$	603.482i	0.893157i
$78$	0	0
$79$	319.140	0.454507	0.227254	−	0.973836i	$-0.427025\pi$
$79$	319.140	0.454507	0.227254	+	0.973836i	$0.427025\pi$
$80$	0	0
$81$	0	0
$82$	− 514.686i	− 0.693141i
$83$	936.446i	1.23841i	0.785228	+	0.619207i	$0.212546\pi$
$83$	936.446i	1.23841i	−0.785228	+	0.619207i	$0.787454\pi$
$84$	0	0
$85$	0	0
$86$	1934.38	2.42545
$87$	0	0
$88$	− 1025.99i	− 1.24285i
$89$	−920.893	−1.09679	−0.548396	−	0.836219i	$-0.684761\pi$
$89$	−920.893	−1.09679	−0.548396	+	0.836219i	$0.684761\pi$
$90$	0	0
$91$	949.639	1.09395
$92$	2526.25i	2.86282i
$93$	0	0
$94$	−1934.03	−2.12213
$95$	0	0
$96$	0	0
$97$	− 914.533i	− 0.957286i	−0.878010	−	0.478643i	$-0.841129\pi$
$97$	− 914.533i	− 0.957286i	0.878010	−	0.478643i	$-0.158871\pi$
$98$	152.849i	0.157552i
$99$	0	0
$100$	0	0
$101$	−942.162	−0.928205	−0.464102	−	0.885782i	$-0.653623\pi$
$101$	−942.162	−0.928205	−0.464102	+	0.885782i	$0.653623\pi$
$102$	0	0
$103$	1204.14i	1.15192i	0.817478	+	0.575960i	$0.195372\pi$
$103$	1204.14i	1.15192i	−0.817478	+	0.575960i	$0.804628\pi$
$104$	−1614.49	−1.52225
$105$	0	0
$106$	−117.104	−0.107303
$107$	− 1416.82i	− 1.28008i	−0.768340	−	0.640042i	$-0.778917\pi$
$107$	− 1416.82i	− 1.28008i	0.768340	−	0.640042i	$-0.221083\pi$
$108$	0	0
$109$	1432.65	1.25893	0.629464	−	0.777029i	$-0.283274\pi$
$109$	1432.65	1.25893	0.629464	+	0.777029i	$0.283274\pi$
$110$	0	0
$111$	0	0
$112$	474.357i	0.400201i
$113$	− 713.619i	− 0.594085i	−0.954864	−	0.297043i	$-0.904000\pi$
$113$	− 713.619i	− 0.594085i	0.954864	−	0.297043i	$-0.0960003\pi$
$114$	0	0
$115$	0	0
$116$	2784.96	2.22911
$117$	0	0
$118$	1397.86i	1.09054i
$119$	1317.70	1.01507
$120$	0	0
$121$	−158.694	−0.119229
$122$	1443.12i	1.07094i
$123$	0	0
$124$	−1547.84	−1.12097
$125$	0	0
$126$	0	0
$127$	− 2507.42i	− 1.75195i	−0.482359	−	0.875974i	$-0.660220\pi$
$127$	− 2507.42i	− 1.75195i	0.482359	−	0.875974i	$-0.339780\pi$
$128$	2631.15i	1.81690i
$129$	0	0
$130$	0	0
$131$	1003.74	0.669444	0.334722	−	0.942317i	$-0.391358\pi$
$131$	1003.74	0.669444	0.334722	+	0.942317i	$0.391358\pi$
$132$	0	0
$133$	1578.55i	1.02915i
$134$	−4326.72	−2.78934
$135$	0	0
$136$	−2240.23	−1.41249
$137$	− 821.572i	− 0.512347i	−0.966631	−	0.256174i	$-0.917538\pi$
$137$	− 821.572i	− 0.512347i	0.966631	−	0.256174i	$-0.0824619\pi$
$138$	0	0
$139$	434.511	0.265142	0.132571	−	0.991174i	$-0.457677\pi$
$139$	434.511	0.265142	0.132571	+	0.991174i	$0.457677\pi$
$140$	0	0
$141$	0	0
$142$	1079.25i	0.637807i
$143$	− 1844.74i	− 1.07878i
$144$	0	0
$145$	0	0
$146$	−750.191	−0.425248
$147$	0	0
$148$	6177.88i	3.43121i
$149$	−3379.68	−1.85822	−0.929109	−	0.369807i	$-0.879424\pi$
$149$	−3379.68	−1.85822	−0.929109	+	0.369807i	$0.879424\pi$
$150$	0	0
$151$	1003.54	0.540838	0.270419	−	0.962743i	$-0.412838\pi$
$151$	1003.54	0.540838	0.270419	+	0.962743i	$0.412838\pi$
$152$	− 2683.70i	− 1.43209i
$153$	0	0
$154$	−2852.36	−1.49253
$155$	0	0
$156$	0	0
$157$	− 3652.95i	− 1.85693i	−0.371426	−	0.928463i	$-0.621131\pi$
$157$	− 3652.95i	− 1.85693i	0.371426	−	0.928463i	$-0.378869\pi$
$158$	1508.42i	0.759514i
$159$	0	0
$160$	0	0
$161$	3105.09	1.51997
$162$	0	0
$163$	705.127i	0.338833i	0.985545	+	0.169416i	$0.0541883\pi$
$163$	705.127i	0.338833i	−0.985545	+	0.169416i	$0.945812\pi$
$164$	1561.52	0.743501
$165$	0	0
$166$	−4426.12	−2.06948
$167$	− 2471.73i	− 1.14532i	−0.819794	−	0.572658i	$-0.805912\pi$
$167$	− 2471.73i	− 1.14532i	0.819794	−	0.572658i	$-0.194088\pi$
$168$	0	0
$169$	−705.886	−0.321296
$170$	0	0
$171$	0	0
$172$	5868.75i	2.60168i
$173$	384.262i	0.168872i	0.996429	+	0.0844360i	$0.0269089\pi$
$173$	384.262i	0.168872i	−0.996429	+	0.0844360i	$0.973091\pi$
$174$	0	0
$175$	0	0
$176$	921.472	0.394651
$177$	0	0
$178$	− 4352.61i	− 1.83282i
$179$	−2775.75	−1.15904	−0.579522	−	0.814956i	$-0.696761\pi$
$179$	−2775.75	−1.15904	−0.579522	+	0.814956i	$0.696761\pi$
$180$	0	0
$181$	−3653.40	−1.50030	−0.750151	−	0.661266i	$-0.770019\pi$
$181$	−3653.40	−1.50030	−0.750151	+	0.661266i	$0.770019\pi$
$182$	4488.48i	1.82807i
$183$	0	0
$184$	−5278.99	−2.11507
$185$	0	0
$186$	0	0
$187$	− 2559.72i	− 1.00099i
$188$	− 5867.70i	− 2.27631i
$189$	0	0
$190$	0	0
$191$	1045.38	0.396028	0.198014	−	0.980199i	$-0.436551\pi$
$191$	1045.38	0.396028	0.198014	+	0.980199i	$0.436551\pi$
$192$	0	0
$193$	1442.74i	0.538087i	0.963128	+	0.269043i	$0.0867075\pi$
$193$	1442.74i	0.538087i	−0.963128	+	0.269043i	$0.913292\pi$
$194$	4322.55	1.59969
$195$	0	0
$196$	−463.733	−0.168999
$197$	1363.99i	0.493300i	0.969105	+	0.246650i	$0.0793298\pi$
$197$	1363.99i	0.493300i	−0.969105	+	0.246650i	$0.920670\pi$
$198$	0	0
$199$	−921.087	−0.328111	−0.164056	−	0.986451i	$-0.552458\pi$
$199$	−921.087	−0.328111	−0.164056	+	0.986451i	$0.552458\pi$
$200$	0	0
$201$	0	0
$202$	− 4453.14i	− 1.55110i
$203$	− 3423.08i	− 1.18351i
$204$	0	0
$205$	0	0
$206$	−5691.39	−1.92494
$207$	0	0
$208$	− 1450.03i	− 0.483372i
$209$	3066.44	1.01488
$210$	0	0
$211$	4461.60	1.45568	0.727842	−	0.685745i	$-0.240523\pi$
$211$	4461.60	1.45568	0.727842	+	0.685745i	$0.240523\pi$
$212$	− 355.284i	− 0.115099i
$213$	0	0
$214$	6696.60	2.13911
$215$	0	0
$216$	0	0
$217$	1902.49i	0.595160i
$218$	6771.44i	2.10376i
$219$	0	0
$220$	0	0
$221$	−4027.98	−1.22602
$222$	0	0
$223$	2623.53i	0.787822i	0.919149	+	0.393911i	$0.128878\pi$
$223$	2623.53i	0.787822i	−0.919149	+	0.393911i	$0.871122\pi$
$224$	1983.22	0.591559
$225$	0	0
$226$	3372.92	0.992759
$227$	2571.02i	0.751739i	0.926673	+	0.375870i	$0.122656\pi$
$227$	2571.02i	0.751739i	−0.926673	+	0.375870i	$0.877344\pi$
$228$	0	0
$229$	3814.54	1.10075	0.550375	−	0.834918i	$-0.314485\pi$
$229$	3814.54	1.10075	0.550375	+	0.834918i	$0.314485\pi$
$230$	0	0
$231$	0	0
$232$	5819.62i	1.64688i
$233$	4827.35i	1.35730i	0.734463	+	0.678649i	$0.237434\pi$
$233$	4827.35i	1.35730i	−0.734463	+	0.678649i	$0.762566\pi$
$234$	0	0
$235$	0	0
$236$	−4240.99	−1.16977
$237$	0	0
$238$	6228.11i	1.69625i
$239$	4165.97	1.12751	0.563753	−	0.825943i	$-0.309357\pi$
$239$	4165.97	1.12751	0.563753	+	0.825943i	$0.309357\pi$
$240$	0	0
$241$	5232.03	1.39844	0.699221	−	0.714906i	$-0.253530\pi$
$241$	5232.03	1.39844	0.699221	+	0.714906i	$0.253530\pi$
$242$	− 750.069i	− 0.199241i
$243$	0	0
$244$	−4378.33	−1.14874
$245$	0	0
$246$	0	0
$247$	− 4825.35i	− 1.24303i
$248$	− 3234.45i	− 0.828177i
$249$	0	0
$250$	0	0
$251$	4222.11	1.06174	0.530871	−	0.847453i	$-0.321865\pi$
$251$	4222.11	1.06174	0.530871	+	0.847453i	$0.321865\pi$
$252$	0	0
$253$	− 6031.85i	− 1.49889i
$254$	11851.3	2.92763
$255$	0	0
$256$	−6459.12	−1.57693
$257$	− 6104.54i	− 1.48168i	−0.671684	−	0.740838i	$-0.734429\pi$
$257$	− 6104.54i	− 1.48168i	0.671684	−	0.740838i	$-0.265571\pi$
$258$	0	0
$259$	7593.43	1.82175
$260$	0	0
$261$	0	0
$262$	4744.18i	1.11869i
$263$	− 4328.06i	− 1.01475i	−0.861725	−	0.507376i	$-0.830616\pi$
$263$	− 4328.06i	− 1.01475i	0.861725	−	0.507376i	$-0.169384\pi$
$264$	0	0
$265$	0	0
$266$	−7461.01	−1.71979
$267$	0	0
$268$	− 13126.9i	− 2.99200i
$269$	−4131.70	−0.936484	−0.468242	−	0.883600i	$-0.655112\pi$
$269$	−4131.70	−0.936484	−0.468242	+	0.883600i	$0.655112\pi$
$270$	0	0
$271$	3274.65	0.734024	0.367012	−	0.930216i	$-0.380381\pi$
$271$	3274.65	0.734024	0.367012	+	0.930216i	$0.380381\pi$
$272$	− 2012.03i	− 0.448519i
$273$	0	0
$274$	3883.16	0.856170
$275$	0	0
$276$	0	0
$277$	− 2779.74i	− 0.602955i	−0.953473	−	0.301478i	$-0.902520\pi$
$277$	− 2779.74i	− 0.602955i	0.953473	−	0.301478i	$-0.0974799\pi$
$278$	2053.72i	0.443072i
$279$	0	0
$280$	0	0
$281$	−1386.05	−0.294252	−0.147126	−	0.989118i	$-0.547002\pi$
$281$	−1386.05	−0.294252	−0.147126	+	0.989118i	$0.547002\pi$
$282$	0	0
$283$	− 743.439i	− 0.156159i	−0.996947	−	0.0780793i	$-0.975121\pi$
$283$	− 743.439i	− 0.156159i	0.996947	−	0.0780793i	$-0.0248787\pi$
$284$	−3274.36	−0.684147
$285$	0	0
$286$	8719.18	1.80271
$287$	− 1919.31i	− 0.394751i
$288$	0	0
$289$	−676.129	−0.137620
$290$	0	0
$291$	0	0
$292$	− 2276.02i	− 0.456145i
$293$	4366.87i	0.870701i	0.900261	+	0.435350i	$0.143376\pi$
$293$	4366.87i	0.870701i	−0.900261	+	0.435350i	$0.856624\pi$
$294$	0	0
$295$	0	0
$296$	−12909.7	−2.53500
$297$	0	0
$298$	− 15974.1i	− 3.10522i
$299$	−9491.73	−1.83586
$300$	0	0
$301$	7213.47	1.38132
$302$	4743.22i	0.903780i
$303$	0	0
$304$	2410.32	0.454742
$305$	0	0
$306$	0	0
$307$	2887.01i	0.536712i	0.963320	+	0.268356i	$0.0864803\pi$
$307$	2887.01i	0.536712i	−0.963320	+	0.268356i	$0.913520\pi$
$308$	− 8653.85i	− 1.60097i
$309$	0	0
$310$	0	0
$311$	−4549.32	−0.829481	−0.414740	−	0.909940i	$-0.636128\pi$
$311$	−4549.32	−0.829481	−0.414740	+	0.909940i	$0.636128\pi$
$312$	0	0
$313$	− 3385.58i	− 0.611387i	−0.952130	−	0.305694i	$-0.901112\pi$
$313$	− 3385.58i	− 0.611387i	0.952130	−	0.305694i	$-0.0988884\pi$
$314$	17265.7	3.10306
$315$	0	0
$316$	−4576.43	−0.814697
$317$	− 5150.39i	− 0.912539i	−0.889842	−	0.456270i	$-0.849185\pi$
$317$	− 5150.39i	− 0.912539i	0.889842	−	0.456270i	$-0.150815\pi$
$318$	0	0
$319$	−6649.58	−1.16710
$320$	0	0
$321$	0	0
$322$	14676.2i	2.53998i
$323$	− 6695.54i	− 1.15340i
$324$	0	0
$325$	0	0
$326$	−3332.79	−0.566215
$327$	0	0
$328$	3263.05i	0.549303i
$329$	−7212.17	−1.20857
$330$	0	0
$331$	−5835.19	−0.968976	−0.484488	−	0.874798i	$-0.660994\pi$
$331$	−5835.19	−0.968976	−0.484488	+	0.874798i	$0.660994\pi$
$332$	− 13428.5i	− 2.21984i
$333$	0	0
$334$	11682.6	1.91391
$335$	0	0
$336$	0	0
$337$	3879.29i	0.627057i	0.949579	+	0.313529i	$0.101511\pi$
$337$	3879.29i	0.627057i	−0.949579	+	0.313529i	$0.898489\pi$
$338$	− 3336.38i	− 0.536908i
$339$	0	0
$340$	0	0
$341$	3695.73	0.586906
$342$	0	0
$343$	6615.56i	1.04142i
$344$	−12263.7	−1.92213
$345$	0	0
$346$	−1816.22	−0.282198
$347$	2674.45i	0.413752i	0.978367	+	0.206876i	$0.0663297\pi$
$347$	2674.45i	0.413752i	−0.978367	+	0.206876i	$0.933670\pi$
$348$	0	0
$349$	4887.15	0.749579	0.374790	−	0.927110i	$-0.377715\pi$
$349$	4887.15	0.749579	0.374790	+	0.927110i	$0.377715\pi$
$350$	0	0
$351$	0	0
$352$	− 3852.54i	− 0.583356i
$353$	− 11808.5i	− 1.78046i	−0.455508	−	0.890232i	$-0.650542\pi$
$353$	− 11808.5i	− 1.78046i	0.455508	−	0.890232i	$-0.349458\pi$
$354$	0	0
$355$	0	0
$356$	13205.5	1.96598
$357$	0	0
$358$	− 13119.6i	− 1.93685i
$359$	−8464.47	−1.24440	−0.622198	−	0.782860i	$-0.713760\pi$
$359$	−8464.47	−1.24440	−0.622198	+	0.782860i	$0.713760\pi$
$360$	0	0
$361$	1161.97	0.169408
$362$	− 17267.8i	− 2.50712i
$363$	0	0
$364$	−13617.7	−1.96088
$365$	0	0
$366$	0	0
$367$	− 692.882i	− 0.0985508i	−0.998785	−	0.0492754i	$-0.984309\pi$
$367$	− 692.882i	− 0.0985508i	0.998785	−	0.0492754i	$-0.0156912\pi$
$368$	− 4741.24i	− 0.671615i
$369$	0	0
$370$	0	0
$371$	−436.691	−0.0611102
$372$	0	0
$373$	5892.36i	0.817948i	0.912546	+	0.408974i	$0.134113\pi$
$373$	5892.36i	0.817948i	−0.912546	+	0.408974i	$0.865887\pi$
$374$	12098.5	1.67273
$375$	0	0
$376$	12261.5	1.68175
$377$	10463.8i	1.42948i
$378$	0	0
$379$	9962.11	1.35018	0.675091	−	0.737734i	$-0.264104\pi$
$379$	9962.11	1.35018	0.675091	+	0.737734i	$0.264104\pi$
$380$	0	0
$381$	0	0
$382$	4941.01i	0.661791i
$383$	− 9038.77i	− 1.20590i	−0.797779	−	0.602950i	$-0.793992\pi$
$383$	− 9038.77i	− 1.20590i	0.797779	−	0.602950i	$-0.206008\pi$
$384$	0	0
$385$	0	0
$386$	−6819.13	−0.899182
$387$	0	0
$388$	13114.3i	1.71592i
$389$	−9387.09	−1.22351	−0.611754	−	0.791048i	$-0.709536\pi$
$389$	−9387.09	−1.22351	−0.611754	+	0.791048i	$0.709536\pi$
$390$	0	0
$391$	−13170.5	−1.70348
$392$	− 969.045i	− 0.124858i
$393$	0	0
$394$	−6446.90	−0.824340
$395$	0	0
$396$	0	0
$397$	− 8786.21i	− 1.11075i	−0.831600	−	0.555374i	$-0.812575\pi$
$397$	− 8786.21i	− 1.11075i	0.831600	−	0.555374i	$-0.187425\pi$
$398$	− 4353.53i	− 0.548298i
$399$	0	0
$400$	0	0
$401$	4867.79	0.606199	0.303099	−	0.952959i	$-0.401979\pi$
$401$	4867.79	0.606199	0.303099	+	0.952959i	$0.401979\pi$
$402$	0	0
$403$	− 5815.60i	− 0.718848i
$404$	13510.5	1.66379
$405$	0	0
$406$	16179.2	1.97774
$407$	− 14750.8i	− 1.79648i
$408$	0	0
$409$	2001.50	0.241975	0.120988	−	0.992654i	$-0.461394\pi$
$409$	2001.50	0.241975	0.120988	+	0.992654i	$0.461394\pi$
$410$	0	0
$411$	0	0
$412$	− 17267.3i	− 2.06480i
$413$	5212.74i	0.621070i
$414$	0	0
$415$	0	0
$416$	−6062.36	−0.714499
$417$	0	0
$418$	14493.5i	1.69594i
$419$	2270.88	0.264772	0.132386	−	0.991198i	$-0.457736\pi$
$419$	2270.88	0.264772	0.132386	+	0.991198i	$0.457736\pi$
$420$	0	0
$421$	7383.89	0.854795	0.427397	−	0.904064i	$-0.359431\pi$
$421$	7383.89	0.854795	0.427397	+	0.904064i	$0.359431\pi$
$422$	21087.8i	2.43255i
$423$	0	0
$424$	742.424	0.0850360
$425$	0	0
$426$	0	0
$427$	5381.54i	0.609909i
$428$	20317.0i	2.29453i
$429$	0	0
$430$	0	0
$431$	962.054	0.107519	0.0537593	−	0.998554i	$-0.482880\pi$
$431$	962.054	0.107519	0.0537593	+	0.998554i	$0.482880\pi$
$432$	0	0
$433$	2416.32i	0.268178i	0.990969	+	0.134089i	$0.0428108\pi$
$433$	2416.32i	0.268178i	−0.990969	+	0.134089i	$0.957189\pi$
$434$	−8992.15	−0.994555
$435$	0	0
$436$	−20544.1	−2.25661
$437$	− 15777.7i	− 1.72712i
$438$	0	0
$439$	−6535.97	−0.710580	−0.355290	−	0.934756i	$-0.615618\pi$
$439$	−6535.97	−0.710580	−0.355290	+	0.934756i	$0.615618\pi$
$440$	0	0
$441$	0	0
$442$	− 19038.3i	− 2.04877i
$443$	5684.43i	0.609651i	0.952408	+	0.304826i	$0.0985982\pi$
$443$	5684.43i	0.609651i	−0.952408	+	0.304826i	$0.901402\pi$
$444$	0	0
$445$	0	0
$446$	−12400.1	−1.31651
$447$	0	0
$448$	13168.6i	1.38874i
$449$	7486.60	0.786892	0.393446	−	0.919348i	$-0.371283\pi$
$449$	7486.60	0.786892	0.393446	+	0.919348i	$0.371283\pi$
$450$	0	0
$451$	−3728.40	−0.389276
$452$	10233.2i	1.06489i
$453$	0	0
$454$	−12152.0	−1.25621
$455$	0	0
$456$	0	0
$457$	2773.87i	0.283931i	0.989872	+	0.141965i	$0.0453421\pi$
$457$	2773.87i	0.283931i	−0.989872	+	0.141965i	$0.954658\pi$
$458$	18029.4i	1.83943i
$459$	0	0
$460$	0	0
$461$	17868.7	1.80527	0.902633	−	0.430411i	$-0.141631\pi$
$461$	17868.7	1.80527	0.902633	+	0.430411i	$0.141631\pi$
$462$	0	0
$463$	− 18349.1i	− 1.84180i	−0.389799	−	0.920900i	$-0.627456\pi$
$463$	− 18349.1i	− 1.84180i	0.389799	−	0.920900i	$-0.372544\pi$
$464$	−5226.80	−0.522948
$465$	0	0
$466$	−22816.5	−2.26814
$467$	15896.5i	1.57517i	0.616207	+	0.787584i	$0.288668\pi$
$467$	15896.5i	1.57517i	−0.616207	+	0.787584i	$0.711332\pi$
$468$	0	0
$469$	−16134.7	−1.58856
$470$	0	0
$471$	0	0
$472$	− 8862.24i	− 0.864232i
$473$	− 14012.7i	− 1.36216i
$474$	0	0
$475$	0	0
$476$	−18895.6	−1.81949
$477$	0	0
$478$	19690.5i	1.88415i
$479$	331.824	0.0316522	0.0158261	−	0.999875i	$-0.494962\pi$
$479$	331.824	0.0316522	0.0158261	+	0.999875i	$0.494962\pi$
$480$	0	0
$481$	−23211.8	−2.20035
$482$	24729.2i	2.33690i
$483$	0	0
$484$	2275.65	0.213717
$485$	0	0
$486$	0	0
$487$	9442.70i	0.878623i	0.898335	+	0.439312i	$0.144778\pi$
$487$	9442.70i	0.878623i	−0.898335	+	0.439312i	$0.855222\pi$
$488$	− 9149.22i	− 0.848700i
$489$	0	0
$490$	0	0
$491$	−15333.4	−1.40934	−0.704671	−	0.709534i	$-0.748905\pi$
$491$	−15333.4	−1.40934	−0.704671	+	0.709534i	$0.748905\pi$
$492$	0	0
$493$	14519.3i	1.32640i
$494$	22807.0	2.07720
$495$	0	0
$496$	2904.97	0.262978
$497$	4024.62i	0.363238i
$498$	0	0
$499$	−12847.3	−1.15256	−0.576278	−	0.817254i	$-0.695495\pi$
$499$	−12847.3	−1.15256	−0.576278	+	0.817254i	$0.695495\pi$
$500$	0	0
$501$	0	0
$502$	19955.8i	1.77425i
$503$	19623.4i	1.73949i	0.493498	+	0.869747i	$0.335718\pi$
$503$	19623.4i	1.73949i	−0.493498	+	0.869747i	$0.664282\pi$
$504$	0	0
$505$	0	0
$506$	28509.6	2.50476
$507$	0	0
$508$	35956.1i	3.14034i
$509$	7398.97	0.644310	0.322155	−	0.946687i	$-0.395593\pi$
$509$	7398.97	0.644310	0.322155	+	0.946687i	$0.395593\pi$
$510$	0	0
$511$	−2797.53	−0.242183
$512$	− 9479.91i	− 0.818275i
$513$	0	0
$514$	28853.1	2.47599
$515$	0	0
$516$	0	0
$517$	14010.2i	1.19181i
$518$	35890.4i	3.04427i
$519$	0	0
$520$	0	0
$521$	19995.7	1.68144	0.840718	−	0.541473i	$-0.182133\pi$
$521$	19995.7	1.68144	0.840718	+	0.541473i	$0.182133\pi$
$522$	0	0
$523$	1340.93i	0.112113i	0.998428	+	0.0560564i	$0.0178526\pi$
$523$	1340.93i	0.112113i	−0.998428	+	0.0560564i	$0.982147\pi$
$524$	−14393.5	−1.19997
$525$	0	0
$526$	20456.6	1.69572
$527$	− 8069.59i	− 0.667015i
$528$	0	0
$529$	−18868.6	−1.55080
$530$	0	0
$531$	0	0
$532$	− 22636.1i	− 1.84474i
$533$	5867.01i	0.476789i
$534$	0	0
$535$	0	0
$536$	27430.8	2.21051
$537$	0	0
$538$	− 19528.5i	− 1.56493i
$539$	1107.24	0.0884831
$540$	0	0
$541$	−588.601	−0.0467762	−0.0233881	−	0.999726i	$-0.507445\pi$
$541$	−588.601	−0.0467762	−0.0233881	+	0.999726i	$0.507445\pi$
$542$	15477.6i	1.22661i
$543$	0	0
$544$	−8411.99	−0.662980
$545$	0	0
$546$	0	0
$547$	− 5606.58i	− 0.438245i	−0.975697	−	0.219123i	$-0.929681\pi$
$547$	− 5606.58i	− 0.438245i	0.975697	−	0.219123i	$-0.0703195\pi$
$548$	11781.2i	0.918375i
$549$	0	0
$550$	0	0
$551$	−17393.5	−1.34481
$552$	0	0
$553$	5625.03i	0.432551i
$554$	13138.5	1.00758
$555$	0	0
$556$	−6230.83	−0.475263
$557$	− 9216.20i	− 0.701083i	−0.936547	−	0.350541i	$-0.885998\pi$
$557$	− 9216.20i	− 0.701083i	0.936547	−	0.350541i	$-0.114002\pi$
$558$	0	0
$559$	−22050.4	−1.66839
$560$	0	0
$561$	0	0
$562$	− 6551.17i	− 0.491716i
$563$	− 26712.3i	− 1.99963i	−0.0192945	−	0.999814i	$-0.506142\pi$
$563$	− 26712.3i	− 1.99963i	0.0192945	−	0.999814i	$-0.493858\pi$
$564$	0	0
$565$	0	0
$566$	3513.87	0.260952
$567$	0	0
$568$	− 6842.31i	− 0.505452i
$569$	−19903.6	−1.46643	−0.733217	−	0.679994i	$-0.761982\pi$
$569$	−19903.6	−1.46643	−0.733217	+	0.679994i	$0.761982\pi$
$570$	0	0
$571$	9848.73	0.721815	0.360908	−	0.932602i	$-0.382467\pi$
$571$	9848.73	0.721815	0.360908	+	0.932602i	$0.382467\pi$
$572$	26453.4i	1.93369i
$573$	0	0
$574$	9071.65	0.659657
$575$	0	0
$576$	0	0
$577$	20534.4i	1.48155i	0.671751	+	0.740777i	$0.265543\pi$
$577$	20534.4i	1.48155i	−0.671751	+	0.740777i	$0.734457\pi$
$578$	− 3195.73i	− 0.229974i
$579$	0	0
$580$	0	0
$581$	−16505.4	−1.17859
$582$	0	0
$583$	848.304i	0.0602627i
$584$	4756.12	0.337003
$585$	0	0
$586$	−20640.1	−1.45500
$587$	− 17036.4i	− 1.19790i	−0.800786	−	0.598950i	$-0.795585\pi$
$587$	− 17036.4i	− 1.19790i	0.800786	−	0.598950i	$-0.204415\pi$
$588$	0	0
$589$	9667.03	0.676270
$590$	0	0
$591$	0	0
$592$	− 11594.6i	− 0.804959i
$593$	5045.69i	0.349413i	0.984621	+	0.174706i	$0.0558976\pi$
$593$	5045.69i	0.349413i	−0.984621	+	0.174706i	$0.944102\pi$
$594$	0	0
$595$	0	0
$596$	48464.2	3.33083
$597$	0	0
$598$	− 44862.7i	− 3.06785i
$599$	−1428.92	−0.0974693	−0.0487346	−	0.998812i	$-0.515519\pi$
$599$	−1428.92	−0.0974693	−0.0487346	+	0.998812i	$0.515519\pi$
$600$	0	0
$601$	4679.36	0.317596	0.158798	−	0.987311i	$-0.449238\pi$
$601$	4679.36	0.317596	0.158798	+	0.987311i	$0.449238\pi$
$602$	34094.5i	2.30829i
$603$	0	0
$604$	−14390.6	−0.969444
$605$	0	0
$606$	0	0
$607$	− 4793.69i	− 0.320544i	−0.987073	−	0.160272i	$-0.948763\pi$
$607$	− 4793.69i	− 0.320544i	0.987073	−	0.160272i	$-0.0512371\pi$
$608$	− 10077.2i	− 0.672179i
$609$	0	0
$610$	0	0
$611$	22046.4	1.45974
$612$	0	0
$613$	− 4528.68i	− 0.298387i	−0.988808	−	0.149194i	$-0.952332\pi$
$613$	− 4528.68i	− 0.298387i	0.988808	−	0.149194i	$-0.0476678\pi$
$614$	−13645.5	−0.896885
$615$	0	0
$616$	18083.6	1.18281
$617$	3749.88i	0.244675i	0.992489	+	0.122337i	$0.0390390\pi$
$617$	3749.88i	0.244675i	−0.992489	+	0.122337i	$0.960961\pi$
$618$	0	0
$619$	23973.1	1.55664	0.778320	−	0.627868i	$-0.216072\pi$
$619$	23973.1	1.55664	0.778320	+	0.627868i	$0.216072\pi$
$620$	0	0
$621$	0	0
$622$	− 21502.4i	− 1.38612i
$623$	− 16231.3i	− 1.04381i
$624$	0	0
$625$	0	0
$626$	16002.0	1.02167
$627$	0	0
$628$	52382.9i	3.32851i
$629$	−32208.2	−2.04169
$630$	0	0
$631$	3635.10	0.229336	0.114668	−	0.993404i	$-0.463420\pi$
$631$	3635.10	0.229336	0.114668	+	0.993404i	$0.463420\pi$
$632$	− 9563.18i	− 0.601903i
$633$	0	0
$634$	24343.4	1.52492
$635$	0	0
$636$	0	0
$637$	− 1742.36i	− 0.108375i
$638$	− 31429.3i	− 1.95031i
$639$	0	0
$640$	0	0
$641$	6743.48	0.415525	0.207762	−	0.978179i	$-0.433382\pi$
$641$	6743.48	0.415525	0.207762	+	0.978179i	$0.433382\pi$
$642$	0	0
$643$	7401.83i	0.453965i	0.973899	+	0.226983i	$0.0728861\pi$
$643$	7401.83i	0.453965i	−0.973899	+	0.226983i	$0.927114\pi$
$644$	−44526.6	−2.72452
$645$	0	0
$646$	31646.5	1.92742
$647$	− 15410.5i	− 0.936395i	−0.883624	−	0.468197i	$-0.844904\pi$
$647$	− 15410.5i	− 0.936395i	0.883624	−	0.468197i	$-0.155096\pi$
$648$	0	0
$649$	10126.1	0.612457
$650$	0	0
$651$	0	0
$652$	− 10111.4i	− 0.607353i
$653$	12798.9i	0.767013i	0.923538	+	0.383507i	$0.125284\pi$
$653$	12798.9i	0.767013i	−0.923538	+	0.383507i	$0.874716\pi$
$654$	0	0
$655$	0	0
$656$	−2930.65	−0.174425
$657$	0	0
$658$	− 34088.4i	− 2.01961i
$659$	−2194.59	−0.129726	−0.0648628	−	0.997894i	$-0.520661\pi$
$659$	−2194.59	−0.129726	−0.0648628	+	0.997894i	$0.520661\pi$
$660$	0	0
$661$	−14915.1	−0.877653	−0.438826	−	0.898572i	$-0.644606\pi$
$661$	−14915.1	−0.877653	−0.438826	+	0.898572i	$0.644606\pi$
$662$	− 27580.1i	− 1.61923i
$663$	0	0
$664$	28061.0	1.64003
$665$	0	0
$666$	0	0
$667$	34214.0i	1.98617i
$668$	35444.2i	2.05296i
$669$	0	0
$670$	0	0
$671$	10454.0	0.601450
$672$	0	0
$673$	− 27650.0i	− 1.58370i	−0.610718	−	0.791848i	$-0.709119\pi$
$673$	− 27650.0i	− 1.58370i	0.610718	−	0.791848i	$-0.290881\pi$
$674$	−18335.5	−1.04786
$675$	0	0
$676$	10122.3	0.575917
$677$	17963.4i	1.01978i	0.860240	+	0.509889i	$0.170314\pi$
$677$	17963.4i	1.01978i	−0.860240	+	0.509889i	$0.829686\pi$
$678$	0	0
$679$	16119.2	0.911042
$680$	0	0
$681$	0	0
$682$	17467.9i	0.980763i
$683$	22747.4i	1.27438i	0.770705	+	0.637192i	$0.219904\pi$
$683$	22747.4i	1.27438i	−0.770705	+	0.637192i	$0.780096\pi$
$684$	0	0
$685$	0	0
$686$	−31268.5	−1.74029
$687$	0	0
$688$	− 11014.4i	− 0.610351i
$689$	1334.89	0.0738103
$690$	0	0
$691$	−16494.4	−0.908071	−0.454036	−	0.890984i	$-0.650016\pi$
$691$	−16494.4	−0.908071	−0.454036	+	0.890984i	$0.650016\pi$
$692$	− 5510.26i	− 0.302701i
$693$	0	0
$694$	−12640.8	−0.691410
$695$	0	0
$696$	0	0
$697$	8140.93i	0.442410i
$698$	23099.2i	1.25260i
$699$	0	0
$700$	0	0
$701$	23463.2	1.26418	0.632092	−	0.774894i	$-0.282197\pi$
$701$	23463.2	1.26418	0.632092	+	0.774894i	$0.282197\pi$
$702$	0	0
$703$	− 38584.0i	− 2.07002i
$704$	25580.8	1.36948
$705$	0	0
$706$	55813.0	2.97528
$707$	− 16606.2i	− 0.883365i
$708$	0	0
$709$	−16436.0	−0.870615	−0.435308	−	0.900282i	$-0.643360\pi$
$709$	−16436.0	−0.870615	−0.435308	+	0.900282i	$0.643360\pi$
$710$	0	0
$711$	0	0
$712$	27595.0i	1.45248i
$713$	− 19015.6i	− 0.998793i
$714$	0	0
$715$	0	0
$716$	39803.9	2.07757
$717$	0	0
$718$	− 40007.4i	− 2.07947i
$719$	−543.595	−0.0281957	−0.0140978	−	0.999901i	$-0.504488\pi$
$719$	−543.595	−0.0281957	−0.0140978	+	0.999901i	$0.504488\pi$
$720$	0	0
$721$	−21223.7	−1.09627
$722$	5492.06i	0.283093i
$723$	0	0
$724$	52389.2	2.68927
$725$	0	0
$726$	0	0
$727$	1097.53i	0.0559905i	0.999608	+	0.0279953i	$0.00891234\pi$
$727$	1097.53i	0.0559905i	−0.999608	+	0.0279953i	$0.991088\pi$
$728$	− 28456.4i	− 1.44871i
$729$	0	0
$730$	0	0
$731$	−30596.6	−1.54809
$732$	0	0
$733$	− 26094.8i	− 1.31492i	−0.753490	−	0.657459i	$-0.771631\pi$
$733$	− 26094.8i	− 1.31492i	0.753490	−	0.657459i	$-0.228369\pi$
$734$	3274.91	0.164685
$735$	0	0
$736$	−19822.4	−0.992751
$737$	31342.9i	1.56652i
$738$	0	0
$739$	−17477.5	−0.869985	−0.434992	−	0.900434i	$-0.643249\pi$
$739$	−17477.5	−0.869985	−0.434992	+	0.900434i	$0.643249\pi$
$740$	0	0
$741$	0	0
$742$	− 2064.02i	− 0.102120i
$743$	11534.6i	0.569533i	0.958597	+	0.284766i	$0.0919161\pi$
$743$	11534.6i	0.569533i	−0.958597	+	0.284766i	$0.908084\pi$
$744$	0	0
$745$	0	0
$746$	−27850.3	−1.36685
$747$	0	0
$748$	36706.1i	1.79426i
$749$	24972.2	1.21825
$750$	0	0
$751$	22616.9	1.09894	0.549470	−	0.835514i	$-0.314830\pi$
$751$	22616.9	1.09894	0.549470	+	0.835514i	$0.314830\pi$
$752$	11012.5i	0.534020i
$753$	0	0
$754$	−49457.2	−2.38876
$755$	0	0
$756$	0	0
$757$	39919.4i	1.91664i	0.285699	+	0.958320i	$0.407774\pi$
$757$	39919.4i	1.91664i	−0.285699	+	0.958320i	$0.592226\pi$
$758$	47086.0i	2.25625i
$759$	0	0
$760$	0	0
$761$	−5317.72	−0.253308	−0.126654	−	0.991947i	$-0.540424\pi$
$761$	−5317.72	−0.253308	−0.126654	+	0.991947i	$0.540424\pi$
$762$	0	0
$763$	25251.3i	1.19811i
$764$	−14990.7	−0.709873
$765$	0	0
$766$	42721.8	2.01515
$767$	− 15934.5i	− 0.750143i
$768$	0	0
$769$	−19615.7	−0.919846	−0.459923	−	0.887959i	$-0.652123\pi$
$769$	−19615.7	−0.919846	−0.459923	+	0.887959i	$0.652123\pi$
$770$	0	0
$771$	0	0
$772$	− 20688.7i	− 0.964512i
$773$	− 27548.0i	− 1.28180i	−0.767624	−	0.640901i	$-0.778561\pi$
$773$	− 27548.0i	− 1.28180i	0.767624	−	0.640901i	$-0.221439\pi$
$774$	0	0
$775$	0	0
$776$	−27404.4	−1.26773
$777$	0	0
$778$	− 44368.2i	− 2.04457i
$779$	−9752.49	−0.448549
$780$	0	0
$781$	7818.12	0.358200
$782$	− 62250.5i	− 2.84664i
$783$	0	0
$784$	870.331	0.0396470
$785$	0	0
$786$	0	0
$787$	362.900i	0.0164371i	0.999966	+	0.00821854i	$0.00261607\pi$
$787$	362.900i	0.0164371i	−0.999966	+	0.00821854i	$0.997384\pi$
$788$	− 19559.4i	− 0.884233i
$789$	0	0
$790$	0	0
$791$	12577.9	0.565386
$792$	0	0
$793$	− 16450.5i	− 0.736662i
$794$	41528.1	1.85614
$795$	0	0
$796$	13208.3	0.588134
$797$	35122.3i	1.56097i	0.625172	+	0.780487i	$0.285029\pi$
$797$	35122.3i	1.56097i	−0.625172	+	0.780487i	$0.714971\pi$
$798$	0	0
$799$	30591.1	1.35449
$800$	0	0
$801$	0	0
$802$	23007.6i	1.01300i
$803$	5434.41i	0.238824i
$804$	0	0
$805$	0	0
$806$	27487.5	1.20125
$807$	0	0
$808$	28232.3i	1.22922i
$809$	20667.7	0.898194	0.449097	−	0.893483i	$-0.351746\pi$
$809$	20667.7	0.898194	0.449097	+	0.893483i	$0.351746\pi$
$810$	0	0
$811$	−45901.0	−1.98743	−0.993713	−	0.111958i	$-0.964288\pi$
$811$	−45901.0	−1.98743	−0.993713	+	0.111958i	$0.964288\pi$
$812$	49086.6i	2.12143i
$813$	0	0
$814$	69719.6	3.00205
$815$	0	0
$816$	0	0
$817$	− 36653.4i	− 1.56957i
$818$	9460.11i	0.404358i
$819$	0	0
$820$	0	0
$821$	265.370	0.0112807	0.00564037	−	0.999984i	$-0.498205\pi$
$821$	265.370	0.0112807	0.00564037	+	0.999984i	$0.498205\pi$
$822$	0	0
$823$	960.254i	0.0406711i	0.999793	+	0.0203356i	$0.00647346\pi$
$823$	960.254i	0.0406711i	−0.999793	+	0.0203356i	$0.993527\pi$
$824$	36082.7	1.52549
$825$	0	0
$826$	−24638.0	−1.03785
$827$	8116.35i	0.341273i	0.985334	+	0.170637i	$0.0545825\pi$
$827$	8116.35i	0.341273i	−0.985334	+	0.170637i	$0.945418\pi$
$828$	0	0
$829$	−2257.09	−0.0945620	−0.0472810	−	0.998882i	$-0.515056\pi$
$829$	−2257.09	−0.0945620	−0.0472810	+	0.998882i	$0.515056\pi$
$830$	0	0
$831$	0	0
$832$	− 40254.0i	− 1.67735i
$833$	− 2417.66i	− 0.100560i
$834$	0	0
$835$	0	0
$836$	−43972.3	−1.81916
$837$	0	0
$838$	10733.3i	0.442454i
$839$	33981.9	1.39831	0.699156	−	0.714969i	$-0.253559\pi$
$839$	33981.9	1.39831	0.699156	+	0.714969i	$0.253559\pi$
$840$	0	0
$841$	13328.9	0.546514
$842$	34900.0i	1.42842i
$843$	0	0
$844$	−63978.8	−2.60929
$845$	0	0
$846$	0	0
$847$	− 2797.08i	− 0.113470i
$848$	666.796i	0.0270022i
$849$	0	0
$850$	0	0
$851$	−75897.0	−3.05724
$852$	0	0
$853$	8331.29i	0.334417i	0.985922	+	0.167209i	$0.0534753\pi$
$853$	8331.29i	0.334417i	−0.985922	+	0.167209i	$0.946525\pi$
$854$	−25435.9	−1.01920
$855$	0	0
$856$	−42455.6	−1.69521
$857$	25427.2i	1.01351i	0.862091	+	0.506753i	$0.169155\pi$
$857$	25427.2i	1.01351i	−0.862091	+	0.506753i	$0.830845\pi$
$858$	0	0
$859$	−22758.3	−0.903961	−0.451981	−	0.892028i	$-0.649282\pi$
$859$	−22758.3	−0.903961	−0.451981	+	0.892028i	$0.649282\pi$
$860$	0	0
$861$	0	0
$862$	4547.16i	0.179671i
$863$	− 4016.10i	− 0.158412i	−0.996858	−	0.0792060i	$-0.974762\pi$
$863$	− 4016.10i	− 0.158412i	0.996858	−	0.0792060i	$-0.0252385\pi$
$864$	0	0
$865$	0	0
$866$	−11420.8	−0.448145
$867$	0	0
$868$	− 27281.5i	− 1.06681i
$869$	10927.0	0.426552
$870$	0	0
$871$	49321.1	1.91869
$872$	− 42930.1i	− 1.66720i
$873$	0	0
$874$	74573.4	2.88614
$875$	0	0
$876$	0	0
$877$	− 2913.80i	− 0.112192i	−0.998425	−	0.0560958i	$-0.982135\pi$
$877$	− 2913.80i	− 0.112192i	0.998425	−	0.0560958i	$-0.0178652\pi$
$878$	− 30892.3i	− 1.18743i
$879$	0	0
$880$	0	0
$881$	15832.1	0.605446	0.302723	−	0.953079i	$-0.402104\pi$
$881$	15832.1	0.605446	0.302723	+	0.953079i	$0.402104\pi$
$882$	0	0
$883$	41471.6i	1.58056i	0.612748	+	0.790278i	$0.290064\pi$
$883$	41471.6i	1.58056i	−0.612748	+	0.790278i	$0.709936\pi$
$884$	57760.7	2.19763
$885$	0	0
$886$	−26867.5	−1.01877
$887$	− 41362.0i	− 1.56573i	−0.622193	−	0.782864i	$-0.713758\pi$
$887$	− 41362.0i	− 1.56573i	0.622193	−	0.782864i	$-0.286242\pi$
$888$	0	0
$889$	44194.7	1.66732
$890$	0	0
$891$	0	0
$892$	− 37621.0i	− 1.41216i
$893$	36646.8i	1.37328i
$894$	0	0
$895$	0	0
$896$	−46375.5	−1.72913
$897$	0	0
$898$	35385.5i	1.31495i
$899$	−20963.0	−0.777703
$900$	0	0
$901$	1852.26	0.0684882
$902$	− 17622.3i	− 0.650509i
$903$	0	0
$904$	−21383.9	−0.786746
$905$	0	0
$906$	0	0
$907$	6539.32i	0.239399i	0.992810	+	0.119699i	$0.0381931\pi$
$907$	6539.32i	0.239399i	−0.992810	+	0.119699i	$0.961807\pi$
$908$	− 36868.1i	− 1.34748i
$909$	0	0
$910$	0	0
$911$	7793.75	0.283445	0.141723	−	0.989906i	$-0.454736\pi$
$911$	7793.75	0.283445	0.141723	+	0.989906i	$0.454736\pi$
$912$	0	0
$913$	32062.9i	1.16224i
$914$	−13110.7	−0.474469
$915$	0	0
$916$	−54700.0	−1.97308
$917$	17691.5i	0.637104i
$918$	0	0
$919$	−38128.9	−1.36862	−0.684308	−	0.729193i	$-0.739896\pi$
$919$	−38128.9	−1.36862	−0.684308	+	0.729193i	$0.739896\pi$
$920$	0	0
$921$	0	0
$922$	84456.5i	3.01673i
$923$	− 12302.6i	− 0.438727i
$924$	0	0
$925$	0	0
$926$	86727.0	3.07778
$927$	0	0
$928$	21852.5i	0.772999i
$929$	35726.2	1.26172	0.630860	−	0.775897i	$-0.282702\pi$
$929$	35726.2	1.26172	0.630860	+	0.775897i	$0.282702\pi$
$930$	0	0
$931$	2896.25	0.101956
$932$	− 69223.6i	− 2.43293i
$933$	0	0
$934$	−75135.1	−2.63222
$935$	0	0
$936$	0	0
$937$	− 49634.6i	− 1.73052i	−0.501327	−	0.865258i	$-0.667155\pi$
$937$	− 49634.6i	− 1.73052i	0.501327	−	0.865258i	$-0.332845\pi$
$938$	− 76260.9i	− 2.65459i
$939$	0	0
$940$	0	0
$941$	7852.19	0.272024	0.136012	−	0.990707i	$-0.456571\pi$
$941$	7852.19	0.272024	0.136012	+	0.990707i	$0.456571\pi$
$942$	0	0
$943$	19183.7i	0.662468i
$944$	7959.47	0.274427
$945$	0	0
$946$	66231.0	2.27627
$947$	45715.9i	1.56871i	0.620313	+	0.784355i	$0.287006\pi$
$947$	45715.9i	1.56871i	−0.620313	+	0.784355i	$0.712994\pi$
$948$	0	0
$949$	8551.59	0.292515
$950$	0	0
$951$	0	0
$952$	− 39485.4i	− 1.34425i
$953$	41726.6i	1.41832i	0.705048	+	0.709160i	$0.250925\pi$
$953$	41726.6i	1.41832i	−0.705048	+	0.709160i	$0.749075\pi$
$954$	0	0
$955$	0	0
$956$	−59739.4	−2.02104
$957$	0	0
$958$	1568.37i	0.0528932i
$959$	14480.7	0.487597
$960$	0	0
$961$	−18140.1	−0.608912
$962$	− 109711.i	− 3.67694i
$963$	0	0
$964$	−75026.6	−2.50669
$965$	0	0
$966$	0	0
$967$	4802.46i	0.159707i	0.996807	+	0.0798535i	$0.0254452\pi$
$967$	4802.46i	0.159707i	−0.996807	+	0.0798535i	$0.974555\pi$
$968$	4755.34i	0.157895i
$969$	0	0
$970$	0	0
$971$	−50899.2	−1.68222	−0.841109	−	0.540865i	$-0.818097\pi$
$971$	−50899.2	−1.68222	−0.841109	+	0.540865i	$0.818097\pi$
$972$	0	0
$973$	7658.51i	0.252334i
$974$	−44631.0	−1.46824
$975$	0	0
$976$	8217.22	0.269495
$977$	− 18227.1i	− 0.596865i	−0.954431	−	0.298433i	$-0.903536\pi$
$977$	− 18227.1i	− 0.596865i	0.954431	−	0.298433i	$-0.0964638\pi$
$978$	0	0
$979$	−31530.4	−1.02933
$980$	0	0
$981$	0	0
$982$	− 72473.4i	− 2.35511i
$983$	− 9327.86i	− 0.302658i	−0.988483	−	0.151329i	$-0.951645\pi$
$983$	− 9327.86i	− 0.302658i	0.988483	−	0.151329i	$-0.0483553\pi$
$984$	0	0
$985$	0	0
$986$	−68625.6	−2.21652
$987$	0	0
$988$	69194.9i	2.22812i
$989$	−72099.2	−2.31812
$990$	0	0
$991$	−20920.4	−0.670593	−0.335296	−	0.942113i	$-0.608836\pi$
$991$	−20920.4	−0.670593	−0.335296	+	0.942113i	$0.608836\pi$
$992$	− 12145.3i	− 0.388722i
$993$	0	0
$994$	−19022.4	−0.606996
$995$	0	0
$996$	0	0
$997$	− 39491.7i	− 1.25448i	−0.778827	−	0.627239i	$-0.784185\pi$
$997$	− 39491.7i	− 1.25448i	0.778827	−	0.627239i	$-0.215815\pi$
$998$	− 60723.0i	− 1.92600i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.b.q.649.8		8
3.2	odd	2		675.4.b.p.649.1		8
5.2	odd	4		675.4.a.y.1.1	yes	4
5.3	odd	4		675.4.a.v.1.4	yes	4
5.4	even	2	inner	675.4.b.q.649.1		8
15.2	even	4		675.4.a.u.1.4	✓	4
15.8	even	4		675.4.a.z.1.1	yes	4
15.14	odd	2		675.4.b.p.649.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.4.a.u.1.4	✓	4	15.2	even	4
675.4.a.v.1.4	yes	4	5.3	odd	4
675.4.a.y.1.1	yes	4	5.2	odd	4
675.4.a.z.1.1	yes	4	15.8	even	4
675.4.b.p.649.1		8	3.2	odd	2
675.4.b.p.649.8		8	15.14	odd	2
675.4.b.q.649.1		8	5.4	even	2	inner
675.4.b.q.649.8		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+4.72651i q^{2} -14.3399 q^{4} +17.6256i q^{7} -29.9655i q^{8} +34.2390 q^{11} -53.8784i q^{13} -83.3075 q^{14} +26.9130 q^{16} -74.7605i q^{17} +89.5599 q^{19} +161.831i q^{22} -176.169i q^{23} +254.657 q^{26} -252.749i q^{28} -194.211 q^{29} +107.939 q^{31} -112.519i q^{32} +353.356 q^{34} -430.818i q^{37} +423.305i q^{38} -108.894 q^{41} -409.261i q^{43} -490.982 q^{44} +832.666 q^{46} +409.188i q^{47} +32.3387 q^{49} +772.610i q^{52} +24.7760i q^{53} +528.159 q^{56} -917.940i q^{58} +295.748 q^{59} +305.325 q^{61} +510.176i q^{62} +747.127 q^{64} +915.415i q^{67} +1072.06i q^{68} +228.340 q^{71} +158.720i q^{73} +2036.27 q^{74} -1284.28 q^{76} +603.482i q^{77} +319.140 q^{79} -514.686i q^{82} +936.446i q^{83} +1934.38 q^{86} -1025.99i q^{88} -920.893 q^{89} +949.639 q^{91} +2526.25i q^{92} -1934.03 q^{94} -914.533i q^{97} +152.849i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 38 q^{4} + 104 q^{11} - 276 q^{14} - 10 q^{16} + 92 q^{19} + 938 q^{26} - 940 q^{29} - 524 q^{31} + 84 q^{34} + 1396 q^{41} - 838 q^{44} + 1074 q^{46} - 1560 q^{49} + 4068 q^{56} - 200 q^{59} + 148 q^{61}+ \cdots - 5694 q^{94}+O(q^{100}) \) 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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