LMFDB - Embedded newform 2268.4.f.a.1133.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,4,Mod(1133,2268)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2268.1133");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2268.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$133.816331893$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1133.15
Character	$\chi$	$=$	2268.1133
Dual form			2268.4.f.a.1133.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.06893 q^{5} +(-18.4258 - 1.86777i) q^{7} +O(q^{10})$	$q-7.06893 q^{5} +(-18.4258 - 1.86777i) q^{7} -8.54562i q^{11} -52.3503i q^{13} +38.9547 q^{17} +66.4008i q^{19} +200.801i q^{23} -75.0302 q^{25} -61.1555i q^{29} +134.408i q^{31} +(130.251 + 13.2032i) q^{35} +298.967 q^{37} +442.556 q^{41} -52.2742 q^{43} -275.364 q^{47} +(336.023 + 68.8306i) q^{49} -136.637i q^{53} +60.4084i q^{55} -382.537 q^{59} -302.201i q^{61} +370.061i q^{65} -637.880 q^{67} +228.249i q^{71} +1241.68i q^{73} +(-15.9613 + 157.460i) q^{77} +201.387 q^{79} -646.904 q^{83} -275.368 q^{85} +826.042 q^{89} +(-97.7786 + 964.598i) q^{91} -469.383i q^{95} +19.7107i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 12 q^{7}+O(q^{10}) $ 48 * q - 12 * q^7	$ 48 q - 12 q^{7} + 1200 q^{25} + 336 q^{37} - 168 q^{43} - 636 q^{49} + 1176 q^{67} - 408 q^{79} + 720 q^{85} - 1080 q^{91}+O(q^{100}) $ 48 * q - 12 * q^7 + 1200 * q^25 + 336 * q^37 - 168 * q^43 - 636 * q^49 + 1176 * q^67 - 408 * q^79 + 720 * q^85 - 1080 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−7.06893			−0.632265			−0.316132	−	0.948715i	$-0.602384\pi$
$5$	−7.06893			−0.632265			−0.316132	+	0.948715i	$0.602384\pi$
$6$		0			0
$7$	−18.4258	−	1.86777i	−0.994902	−	0.100850i
$7$	−18.4258	−	1.86777i	−0.994902	−	0.100850i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	8.54562i		−	0.234236i	−0.993118	−	0.117118i	$-0.962634\pi$
$11$		−	8.54562i		−	0.234236i	0.993118	−	0.117118i	$-0.0373656\pi$
$12$		0			0
$13$		−	52.3503i		−	1.11687i	−0.829547	−	0.558437i	$-0.811401\pi$
$13$		−	52.3503i		−	1.11687i	0.829547	−	0.558437i	$-0.188599\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	38.9547			0.555759			0.277879	−	0.960616i	$-0.410368\pi$
$17$	38.9547			0.555759			0.277879	+	0.960616i	$0.410368\pi$
$18$		0			0
$19$			66.4008i			0.801757i	0.916131	+	0.400879i	$0.131295\pi$
$19$			66.4008i			0.801757i	−0.916131	+	0.400879i	$0.868705\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			200.801i			1.82043i	0.414133	+	0.910217i	$0.364085\pi$
$23$			200.801i			1.82043i	−0.414133	+	0.910217i	$0.635915\pi$
$24$		0			0
$25$	−75.0302			−0.600242
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	61.1555i		−	0.391596i	−0.980644	−	0.195798i	$-0.937270\pi$
$29$		−	61.1555i		−	0.391596i	0.980644	−	0.195798i	$-0.0627297\pi$
$30$		0			0
$31$			134.408i			0.778724i	0.921085	+	0.389362i	$0.127304\pi$
$31$			134.408i			0.778724i	−0.921085	+	0.389362i	$0.872696\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	130.251	+	13.2032i	0.629041	+	0.0637641i
$36$		0			0
$37$	298.967			1.32838			0.664188	−	0.747565i	$-0.268777\pi$
$37$	298.967			1.32838			0.664188	+	0.747565i	$0.268777\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	442.556			1.68575			0.842874	−	0.538110i	$-0.180862\pi$
$41$	442.556			1.68575			0.842874	+	0.538110i	$0.180862\pi$
$42$		0			0
$43$	−52.2742			−0.185389			−0.0926947	−	0.995695i	$-0.529548\pi$
$43$	−52.2742			−0.185389			−0.0926947	+	0.995695i	$0.529548\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−275.364			−0.854595			−0.427298	−	0.904111i	$-0.640534\pi$
$47$	−275.364			−0.854595			−0.427298	+	0.904111i	$0.640534\pi$
$48$		0			0
$49$	336.023	+	68.8306i	0.979658	+	0.200672i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	136.637i		−	0.354123i	−0.984200	−	0.177061i	$-0.943341\pi$
$53$		−	136.637i		−	0.354123i	0.984200	−	0.177061i	$-0.0566591\pi$
$54$		0			0
$55$			60.4084i			0.148099i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−382.537			−0.844102			−0.422051	−	0.906572i	$-0.638690\pi$
$59$	−382.537			−0.844102			−0.422051	+	0.906572i	$0.638690\pi$
$60$		0			0
$61$		−	302.201i		−	0.634309i	−0.948374	−	0.317155i	$-0.897273\pi$
$61$		−	302.201i		−	0.634309i	0.948374	−	0.317155i	$-0.102727\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			370.061i			0.706160i
$66$		0			0
$67$	−637.880			−1.16313			−0.581563	−	0.813502i	$-0.697558\pi$
$67$	−637.880			−1.16313			−0.581563	+	0.813502i	$0.697558\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			228.249i			0.381524i	0.981636	+	0.190762i	$0.0610959\pi$
$71$			228.249i			0.381524i	−0.981636	+	0.190762i	$0.938904\pi$
$72$		0			0
$73$			1241.68i			1.99079i	0.0958607	+	0.995395i	$0.469440\pi$
$73$			1241.68i			1.99079i	−0.0958607	+	0.995395i	$0.530560\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−15.9613	+	157.460i	−0.0236228	+	0.233042i
$78$		0			0
$79$	201.387			0.286808			0.143404	−	0.989664i	$-0.454195\pi$
$79$	201.387			0.286808			0.143404	+	0.989664i	$0.454195\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−646.904			−0.855505			−0.427752	−	0.903896i	$-0.640694\pi$
$83$	−646.904			−0.855505			−0.427752	+	0.903896i	$0.640694\pi$
$84$		0			0
$85$	−275.368			−0.351386
$86$		0			0
$87$		0			0
$88$		0			0
$89$	826.042			0.983824			0.491912	−	0.870645i	$-0.336298\pi$
$89$	826.042			0.983824			0.491912	+	0.870645i	$0.336298\pi$
$90$		0			0
$91$	−97.7786	+	964.598i	−0.112637	+	1.11118i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	469.383i		−	0.506923i
$96$		0			0
$97$			19.7107i			0.0206321i	0.999947	+	0.0103161i	$0.00328376\pi$
$97$			19.7107i			0.0206321i	−0.999947	+	0.0103161i	$0.996716\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1628.58			−1.60445			−0.802226	−	0.597020i	$-0.796351\pi$
$101$	−1628.58			−1.60445			−0.802226	+	0.597020i	$0.796351\pi$
$102$		0			0
$103$		−	743.754i		−	0.711498i	−0.934582	−	0.355749i	$-0.884226\pi$
$103$		−	743.754i		−	0.711498i	0.934582	−	0.355749i	$-0.115774\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	836.505i		−	0.755775i	−0.925851	−	0.377888i	$-0.876651\pi$
$107$		−	836.505i		−	0.755775i	0.925851	−	0.377888i	$-0.123349\pi$
$108$		0			0
$109$	−1564.06			−1.37440			−0.687202	−	0.726467i	$-0.741161\pi$
$109$	−1564.06			−1.37440			−0.687202	+	0.726467i	$0.741161\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2080.54i		−	1.73204i	−0.500007	−	0.866021i	$-0.666669\pi$
$113$		−	2080.54i		−	1.73204i	0.500007	−	0.866021i	$-0.333331\pi$
$114$		0			0
$115$		−	1419.45i		−	1.15100i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−717.772	−	72.7585i	−0.552925	−	0.0560484i
$120$		0			0
$121$	1257.97			0.945133
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1414.00			1.01178
$126$		0			0
$127$	774.362			0.541051			0.270526	−	0.962713i	$-0.412803\pi$
$127$	774.362			0.541051			0.270526	+	0.962713i	$0.412803\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	805.668			0.537340			0.268670	−	0.963232i	$-0.413416\pi$
$131$	805.668			0.537340			0.268670	+	0.963232i	$0.413416\pi$
$132$		0			0
$133$	124.022	−	1223.49i	0.0808575	−	0.797669i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1832.97i		−	1.14307i	−0.820577	−	0.571537i	$-0.806348\pi$
$137$		−	1832.97i		−	1.14307i	0.820577	−	0.571537i	$-0.193652\pi$
$138$		0			0
$139$			883.909i			0.539368i	0.962949	+	0.269684i	$0.0869193\pi$
$139$			883.909i			0.539368i	−0.962949	+	0.269684i	$0.913081\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−447.366			−0.261613
$144$		0			0
$145$			432.304i			0.247592i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	1612.05i		−	0.886337i	−0.896438	−	0.443169i	$-0.853854\pi$
$149$		−	1612.05i		−	0.886337i	0.896438	−	0.443169i	$-0.146146\pi$
$150$		0			0
$151$	1150.14			0.619849			0.309925	−	0.950761i	$-0.399696\pi$
$151$	1150.14			0.619849			0.309925	+	0.950761i	$0.399696\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	950.123i		−	0.492359i
$156$		0			0
$157$		−	1254.62i		−	0.637767i	−0.947794	−	0.318883i	$-0.896692\pi$
$157$		−	1254.62i		−	0.637767i	0.947794	−	0.318883i	$-0.103308\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	375.051	−	3699.93i	0.183591	−	1.81115i
$162$		0			0
$163$	1005.42			0.483131			0.241565	−	0.970385i	$-0.422339\pi$
$163$	1005.42			0.483131			0.241565	+	0.970385i	$0.422339\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2396.98			1.11068			0.555340	−	0.831623i	$-0.312588\pi$
$167$	2396.98			1.11068			0.555340	+	0.831623i	$0.312588\pi$
$168$		0			0
$169$	−543.555			−0.247408
$170$		0			0
$171$		0			0
$172$		0			0
$173$	31.7845			0.0139684			0.00698419	−	0.999976i	$-0.497777\pi$
$173$	31.7845			0.0139684			0.00698419	+	0.999976i	$0.497777\pi$
$174$		0			0
$175$	1382.49	+	140.139i	0.597181	+	0.0605346i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			1553.05i			0.648494i	0.945972	+	0.324247i	$0.105111\pi$
$179$			1553.05i			0.648494i	−0.945972	+	0.324247i	$0.894889\pi$
$180$		0			0
$181$			3516.58i			1.44412i	0.691831	+	0.722059i	$0.256804\pi$
$181$			3516.58i			1.44412i	−0.691831	+	0.722059i	$0.743196\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2113.38			−0.839886
$186$		0			0
$187$		−	332.892i		−	0.130179i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	2275.94i		−	0.862205i	−0.902303	−	0.431103i	$-0.858125\pi$
$191$		−	2275.94i		−	0.862205i	0.902303	−	0.431103i	$-0.141875\pi$
$192$		0			0
$193$	−1590.63			−0.593242			−0.296621	−	0.954995i	$-0.595860\pi$
$193$	−1590.63			−0.593242			−0.296621	+	0.954995i	$0.595860\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1715.67i		−	0.620490i	−0.950657	−	0.310245i	$-0.899589\pi$
$197$		−	1715.67i		−	0.620490i	0.950657	−	0.310245i	$-0.100411\pi$
$198$		0			0
$199$		−	2534.03i		−	0.902676i	−0.892353	−	0.451338i	$-0.850947\pi$
$199$		−	2534.03i		−	0.902676i	0.892353	−	0.451338i	$-0.149053\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−114.225	+	1126.84i	−0.0394926	+	0.389600i
$204$		0			0
$205$	−3128.40			−1.06584
$206$		0			0
$207$		0			0
$208$		0			0
$209$	567.436			0.187801
$210$		0			0
$211$	−4299.58			−1.40282			−0.701411	−	0.712757i	$-0.747446\pi$
$211$	−4299.58			−1.40282			−0.701411	+	0.712757i	$0.747446\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	369.523			0.117215
$216$		0			0
$217$	251.044	−	2476.58i	0.0785345	−	0.774753i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2039.29i		−	0.620712i
$222$		0			0
$223$		−	2920.33i		−	0.876949i	−0.898744	−	0.438474i	$-0.855519\pi$
$223$		−	2920.33i		−	0.876949i	0.898744	−	0.438474i	$-0.144481\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−780.237			−0.228133			−0.114066	−	0.993473i	$-0.536388\pi$
$227$	−780.237			−0.228133			−0.114066	+	0.993473i	$0.536388\pi$
$228$		0			0
$229$		−	3283.21i		−	0.947426i	−0.880679	−	0.473713i	$-0.842913\pi$
$229$		−	3283.21i		−	0.947426i	0.880679	−	0.473713i	$-0.157087\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	5223.64i		−	1.46872i	−0.678759	−	0.734361i	$-0.737482\pi$
$233$		−	5223.64i		−	1.46872i	0.678759	−	0.734361i	$-0.262518\pi$
$234$		0			0
$235$	1946.53			0.540330
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	1035.19i		−	0.280171i	−0.990139	−	0.140085i	$-0.955262\pi$
$239$		−	1035.19i		−	0.280171i	0.990139	−	0.140085i	$-0.0447377\pi$
$240$		0			0
$241$			5555.88i			1.48500i	0.669845	+	0.742501i	$0.266361\pi$
$241$			5555.88i			1.48500i	−0.669845	+	0.742501i	$0.733639\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2375.32	−	486.559i	−0.619403	−	0.126878i
$246$		0			0
$247$	3476.10			0.895462
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−974.838			−0.245144			−0.122572	−	0.992460i	$-0.539114\pi$
$251$	−974.838			−0.245144			−0.122572	+	0.992460i	$0.539114\pi$
$252$		0			0
$253$	1715.97			0.426412
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4812.66			−1.16811			−0.584057	−	0.811713i	$-0.698536\pi$
$257$	−4812.66			−1.16811			−0.584057	+	0.811713i	$0.698536\pi$
$258$		0			0
$259$	−5508.73	−	558.404i	−1.32160	−	0.133967i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	2462.36i		−	0.577323i	−0.957431	−	0.288661i	$-0.906790\pi$
$263$		−	2462.36i		−	0.577323i	0.957431	−	0.288661i	$-0.0932101\pi$
$264$		0			0
$265$			965.876i			0.223899i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2092.97			−0.474389			−0.237194	−	0.971462i	$-0.576228\pi$
$269$	−2092.97			−0.474389			−0.237194	+	0.971462i	$0.576228\pi$
$270$		0			0
$271$		−	5006.83i		−	1.12230i	−0.827714	−	0.561150i	$-0.810359\pi$
$271$		−	5006.83i		−	1.12230i	0.827714	−	0.561150i	$-0.189641\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			641.179i			0.140598i
$276$		0			0
$277$	8317.87			1.80423			0.902117	−	0.431493i	$-0.142013\pi$
$277$	8317.87			1.80423			0.902117	+	0.431493i	$0.142013\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	4647.40i		−	0.986622i	−0.869853	−	0.493311i	$-0.835786\pi$
$281$		−	4647.40i		−	0.986622i	0.869853	−	0.493311i	$-0.164214\pi$
$282$		0			0
$283$			4584.33i			0.962934i	0.876464	+	0.481467i	$0.159896\pi$
$283$			4584.33i			0.962934i	−0.876464	+	0.481467i	$0.840104\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8154.47	−	826.595i	−1.67715	−	0.170008i
$288$		0			0
$289$	−3395.53			−0.691132
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3922.26			−0.782050			−0.391025	−	0.920380i	$-0.627879\pi$
$293$	−3922.26			−0.782050			−0.391025	+	0.920380i	$0.627879\pi$
$294$		0			0
$295$	2704.13			0.533696
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10512.0			2.03319
$300$		0			0
$301$	963.196	+	97.6365i	0.184444	+	0.0186966i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			2136.24i			0.401051i
$306$		0			0
$307$		−	8034.65i		−	1.49369i	−0.665000	−	0.746843i	$-0.731569\pi$
$307$		−	8034.65i		−	1.49369i	0.665000	−	0.746843i	$-0.268431\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4430.93			0.807894			0.403947	−	0.914782i	$-0.367638\pi$
$311$	4430.93			0.807894			0.403947	+	0.914782i	$0.367638\pi$
$312$		0			0
$313$			5919.25i			1.06893i	0.845190	+	0.534467i	$0.179487\pi$
$313$			5919.25i			1.06893i	−0.845190	+	0.534467i	$0.820513\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	528.900i		−	0.0937097i	−0.998902	−	0.0468548i	$-0.985080\pi$
$317$		−	528.900i		−	0.0937097i	0.998902	−	0.0468548i	$-0.0149198\pi$
$318$		0			0
$319$	−522.611			−0.0917261
$320$		0			0
$321$		0			0
$322$		0			0
$323$			2586.62i			0.445583i
$324$		0			0
$325$			3927.85i			0.670394i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5073.81	+	514.318i	0.850238	+	0.0861862i
$330$		0			0
$331$	−3649.76			−0.606069			−0.303035	−	0.952980i	$-0.598000\pi$
$331$	−3649.76			−0.606069			−0.303035	+	0.952980i	$0.598000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4509.13			0.735403
$336$		0			0
$337$	6569.39			1.06189			0.530946	−	0.847406i	$-0.321837\pi$
$337$	6569.39			1.06189			0.530946	+	0.847406i	$0.321837\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1148.60			0.182405
$342$		0			0
$343$	−6062.94	−	1895.88i	−0.954426	−	0.298448i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	11858.8i		−	1.83461i	−0.398180	−	0.917307i	$-0.630358\pi$
$347$		−	11858.8i		−	1.83461i	0.398180	−	0.917307i	$-0.369642\pi$
$348$		0			0
$349$			35.5137i			0.00544700i	0.999996	+	0.00272350i	$0.000866919\pi$
$349$			35.5137i			0.00544700i	−0.999996	+	0.00272350i	$0.999133\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9936.13			1.49815			0.749075	−	0.662485i	$-0.230498\pi$
$353$	9936.13			1.49815			0.749075	+	0.662485i	$0.230498\pi$
$354$		0			0
$355$		−	1613.48i		−	0.241224i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	10087.1i		−	1.48295i	−0.670983	−	0.741473i	$-0.734128\pi$
$359$		−	10087.1i		−	1.48295i	0.670983	−	0.741473i	$-0.265872\pi$
$360$		0			0
$361$	2449.94			0.357186
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	8777.35i		−	1.25871i
$366$		0			0
$367$			9784.64i			1.39170i	0.718187	+	0.695850i	$0.244972\pi$
$367$			9784.64i			1.39170i	−0.718187	+	0.695850i	$0.755028\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−255.207	+	2517.65i	−0.0357134	+	0.352317i
$372$		0			0
$373$	−7768.62			−1.07840			−0.539201	−	0.842177i	$-0.681274\pi$
$373$	−7768.62			−1.07840			−0.539201	+	0.842177i	$0.681274\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3201.51			−0.437364
$378$		0			0
$379$	−10683.1			−1.44789			−0.723947	−	0.689856i	$-0.757674\pi$
$379$	−10683.1			−1.44789			−0.723947	+	0.689856i	$0.757674\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12913.8			−1.72288			−0.861440	−	0.507859i	$-0.830437\pi$
$383$	−12913.8			−1.72288			−0.861440	+	0.507859i	$0.830437\pi$
$384$		0			0
$385$	112.829	−	1113.08i	0.0149359	−	0.147344i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	11270.5i		−	1.46899i	−0.678613	−	0.734496i	$-0.737418\pi$
$389$		−	11270.5i		−	1.46899i	0.678613	−	0.734496i	$-0.262582\pi$
$390$		0			0
$391$			7822.15i			1.01172i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1423.59			−0.181339
$396$		0			0
$397$		−	4908.58i		−	0.620540i	−0.950648	−	0.310270i	$-0.899581\pi$
$397$		−	4908.58i		−	0.620540i	0.950648	−	0.310270i	$-0.100419\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	4842.40i		−	0.603038i	−0.953460	−	0.301519i	$-0.902506\pi$
$401$		−	4842.40i		−	0.603038i	0.953460	−	0.301519i	$-0.0974936\pi$
$402$		0			0
$403$	7036.31			0.869736
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2554.86i		−	0.311154i
$408$		0			0
$409$			486.806i			0.0588533i	0.999567	+	0.0294266i	$0.00936814\pi$
$409$			486.806i			0.0588533i	−0.999567	+	0.0294266i	$0.990632\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7048.56	+	714.492i	0.839799	+	0.0851280i
$414$		0			0
$415$	4572.92			0.540905
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13161.8			1.53459			0.767296	−	0.641293i	$-0.221602\pi$
$419$	13161.8			1.53459			0.767296	+	0.641293i	$0.221602\pi$
$420$		0			0
$421$	1696.14			0.196353			0.0981767	−	0.995169i	$-0.468699\pi$
$421$	1696.14			0.196353			0.0981767	+	0.995169i	$0.468699\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2922.78			−0.333589
$426$		0			0
$427$	−564.443	+	5568.30i	−0.0639703	+	0.631075i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	3662.97i		−	0.409372i	−0.978828	−	0.204686i	$-0.934383\pi$
$431$		−	3662.97i		−	0.409372i	0.978828	−	0.204686i	$-0.0656173\pi$
$432$		0			0
$433$			5695.06i			0.632071i	0.948747	+	0.316036i	$0.102352\pi$
$433$			5695.06i			0.632071i	−0.948747	+	0.316036i	$0.897648\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−13333.4			−1.45955
$438$		0			0
$439$			14671.1i			1.59502i	0.603308	+	0.797508i	$0.293849\pi$
$439$			14671.1i			1.59502i	−0.603308	+	0.797508i	$0.706151\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	4084.40i		−	0.438049i	−0.975719	−	0.219025i	$-0.929712\pi$
$443$		−	4084.40i		−	0.438049i	0.975719	−	0.219025i	$-0.0702875\pi$
$444$		0			0
$445$	−5839.24			−0.622037
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	17057.8i		−	1.79289i	−0.443160	−	0.896443i	$-0.646143\pi$
$449$		−	17057.8i		−	1.79289i	0.443160	−	0.896443i	$-0.353857\pi$
$450$		0			0
$451$		−	3781.92i		−	0.394864i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	691.190	−	6818.68i	0.0712165	−	0.702560i
$456$		0			0
$457$	3700.39			0.378768			0.189384	−	0.981903i	$-0.439351\pi$
$457$	3700.39			0.378768			0.189384	+	0.981903i	$0.439351\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−13473.3			−1.36120			−0.680601	−	0.732654i	$-0.738281\pi$
$461$	−13473.3			−1.36120			−0.680601	+	0.732654i	$0.738281\pi$
$462$		0			0
$463$	3101.92			0.311357			0.155678	−	0.987808i	$-0.450244\pi$
$463$	3101.92			0.311357			0.155678	+	0.987808i	$0.450244\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12591.3			−1.24766			−0.623828	−	0.781561i	$-0.714424\pi$
$467$	−12591.3			−1.24766			−0.623828	+	0.781561i	$0.714424\pi$
$468$		0			0
$469$	11753.5	+	1191.41i	1.15720	+	0.117302i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			446.716i			0.0434250i
$474$		0			0
$475$		−	4982.06i		−	0.481248i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5281.09			0.503756			0.251878	−	0.967759i	$-0.418952\pi$
$479$	5281.09			0.503756			0.251878	+	0.967759i	$0.418952\pi$
$480$		0			0
$481$		−	15651.0i		−	1.48363i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	139.333i		−	0.0130450i
$486$		0			0
$487$	945.042			0.0879341			0.0439671	−	0.999033i	$-0.486000\pi$
$487$	945.042			0.0879341			0.0439671	+	0.999033i	$0.486000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	1531.66i		−	0.140779i	−0.997520	−	0.0703897i	$-0.977576\pi$
$491$		−	1531.66i		−	0.140779i	0.997520	−	0.0703897i	$-0.0224243\pi$
$492$		0			0
$493$		−	2382.29i		−	0.217633i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	426.318	−	4205.69i	0.0384768	−	0.379579i
$498$		0			0
$499$	13707.4			1.22972			0.614859	−	0.788637i	$-0.289213\pi$
$499$	13707.4			1.22972			0.614859	+	0.788637i	$0.289213\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−17261.6			−1.53013			−0.765065	−	0.643953i	$-0.777293\pi$
$503$	−17261.6			−1.53013			−0.765065	+	0.643953i	$0.777293\pi$
$504$		0			0
$505$	11512.3			1.01444
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−13084.6			−1.13942			−0.569711	−	0.821845i	$-0.692945\pi$
$509$	−13084.6			−1.13942			−0.569711	+	0.821845i	$0.692945\pi$
$510$		0			0
$511$	2319.18	−	22879.0i	0.200772	−	1.98064i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			5257.55i			0.449855i
$516$		0			0
$517$			2353.16i			0.200177i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12720.6			1.06967			0.534835	−	0.844956i	$-0.320374\pi$
$521$	12720.6			1.06967			0.534835	+	0.844956i	$0.320374\pi$
$522$		0			0
$523$		−	1352.20i		−	0.113055i	−0.998401	−	0.0565274i	$-0.981997\pi$
$523$		−	1352.20i		−	0.113055i	0.998401	−	0.0565274i	$-0.0180028\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			5235.83i			0.432782i
$528$		0			0
$529$	−28154.2			−2.31398
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	23168.0i		−	1.88277i
$534$		0			0
$535$			5913.20i			0.477850i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	588.200	−	2871.52i	0.0470048	−	0.229472i
$540$		0			0
$541$	−8023.06			−0.637594			−0.318797	−	0.947823i	$-0.603279\pi$
$541$	−8023.06			−0.637594			−0.318797	+	0.947823i	$0.603279\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	11056.3			0.868987
$546$		0			0
$547$	−1932.83			−0.151082			−0.0755411	−	0.997143i	$-0.524068\pi$
$547$	−1932.83			−0.151082			−0.0755411	+	0.997143i	$0.524068\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4060.77			0.313965
$552$		0			0
$553$	−3710.73	−	376.146i	−0.285346	−	0.0289247i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			3185.91i			0.242354i	0.992631	+	0.121177i	$0.0386669\pi$
$557$			3185.91i			0.242354i	−0.992631	+	0.121177i	$0.961333\pi$
$558$		0			0
$559$			2736.57i			0.207057i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	11088.3			0.830050			0.415025	−	0.909810i	$-0.363773\pi$
$563$	11088.3			0.830050			0.415025	+	0.909810i	$0.363773\pi$
$564$		0			0
$565$			14707.2i			1.09511i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	4811.38i		−	0.354488i	−0.984167	−	0.177244i	$-0.943282\pi$
$569$		−	4811.38i		−	0.354488i	0.984167	−	0.177244i	$-0.0567181\pi$
$570$		0			0
$571$	−3422.61			−0.250844			−0.125422	−	0.992104i	$-0.540028\pi$
$571$	−3422.61			−0.250844			−0.125422	+	0.992104i	$0.540028\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	15066.2i		−	1.09270i
$576$		0			0
$577$			15586.3i			1.12455i	0.826950	+	0.562275i	$0.190074\pi$
$577$			15586.3i			1.12455i	−0.826950	+	0.562275i	$0.809926\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	11919.7	+	1208.27i	0.851143	+	0.0862779i
$582$		0			0
$583$	−1167.65			−0.0829484
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−24617.1			−1.73093			−0.865465	−	0.500969i	$-0.832977\pi$
$587$	−24617.1			−1.73093			−0.865465	+	0.500969i	$0.832977\pi$
$588$		0			0
$589$	−8924.81			−0.624347
$590$		0			0
$591$		0			0
$592$		0			0
$593$	25080.0			1.73678			0.868392	−	0.495878i	$-0.165154\pi$
$593$	25080.0			1.73678			0.868392	+	0.495878i	$0.165154\pi$
$594$		0			0
$595$	5073.88	+	514.325i	0.349595	+	0.0354374i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			12941.8i			0.882787i	0.897314	+	0.441394i	$0.145516\pi$
$599$			12941.8i			0.882787i	−0.897314	+	0.441394i	$0.854484\pi$
$600$		0			0
$601$		−	13358.2i		−	0.906643i	−0.891347	−	0.453321i	$-0.850239\pi$
$601$		−	13358.2i		−	0.906643i	0.891347	−	0.453321i	$-0.149761\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8892.52			−0.597574
$606$		0			0
$607$		−	11029.7i		−	0.737528i	−0.929523	−	0.368764i	$-0.879781\pi$
$607$		−	11029.7i		−	0.737528i	0.929523	−	0.368764i	$-0.120219\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			14415.4i			0.954475i
$612$		0			0
$613$	−9716.71			−0.640219			−0.320109	−	0.947381i	$-0.603720\pi$
$613$	−9716.71			−0.640219			−0.320109	+	0.947381i	$0.603720\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			22363.7i			1.45920i	0.683872	+	0.729602i	$0.260295\pi$
$617$			22363.7i			1.45920i	−0.683872	+	0.729602i	$0.739705\pi$
$618$		0			0
$619$		−	5809.00i		−	0.377194i	−0.982055	−	0.188597i	$-0.939606\pi$
$619$		−	5809.00i		−	0.377194i	0.982055	−	0.188597i	$-0.0603940\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−15220.5	−	1542.86i	−0.978808	−	0.0992189i
$624$		0			0
$625$	−616.696			−0.0394686
$626$		0			0
$627$		0			0
$628$		0			0
$629$	11646.2			0.738257
$630$		0			0
$631$	−11462.0			−0.723128			−0.361564	−	0.932347i	$-0.617757\pi$
$631$	−11462.0			−0.723128			−0.361564	+	0.932347i	$0.617757\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5473.91			−0.342088
$636$		0			0
$637$	3603.30	−	17590.9i	0.224126	−	1.09416i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			25699.7i			1.58358i	0.610792	+	0.791791i	$0.290851\pi$
$641$			25699.7i			1.58358i	−0.610792	+	0.791791i	$0.709149\pi$
$642$		0			0
$643$			16807.7i			1.03084i	0.856937	+	0.515421i	$0.172364\pi$
$643$			16807.7i			1.03084i	−0.856937	+	0.515421i	$0.827636\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14607.3			0.887595			0.443798	−	0.896127i	$-0.353631\pi$
$647$	14607.3			0.887595			0.443798	+	0.896127i	$0.353631\pi$
$648$		0			0
$649$			3269.01i			0.197720i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	26932.7i		−	1.61403i	−0.590534	−	0.807013i	$-0.701083\pi$
$653$		−	26932.7i		−	1.61403i	0.590534	−	0.807013i	$-0.298917\pi$
$654$		0			0
$655$	−5695.21			−0.339741
$656$		0			0
$657$		0			0
$658$		0			0
$659$			24879.7i			1.47068i	0.677699	+	0.735339i	$0.262977\pi$
$659$			24879.7i			1.47068i	−0.677699	+	0.735339i	$0.737023\pi$
$660$		0			0
$661$		−	28717.0i		−	1.68980i	−0.534920	−	0.844902i	$-0.679658\pi$
$661$		−	28717.0i		−	1.68980i	0.534920	−	0.844902i	$-0.320342\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−876.701	+	8648.77i	−0.0511233	+	0.504338i
$666$		0			0
$667$	12280.1			0.712874
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2582.49			−0.148578
$672$		0			0
$673$	974.881			0.0558379			0.0279190	−	0.999610i	$-0.491112\pi$
$673$	974.881			0.0558379			0.0279190	+	0.999610i	$0.491112\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	21278.1			1.20795			0.603976	−	0.797003i	$-0.293582\pi$
$677$	21278.1			1.20795			0.603976	+	0.797003i	$0.293582\pi$
$678$		0			0
$679$	36.8151	−	363.186i	0.00208076	−	0.0205269i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			6554.78i			0.367221i	0.982999	+	0.183610i	$0.0587785\pi$
$683$			6554.78i			0.367221i	−0.982999	+	0.183610i	$0.941222\pi$
$684$		0			0
$685$			12957.1i			0.722725i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7152.97			−0.395510
$690$		0			0
$691$		−	8738.16i		−	0.481064i	−0.970641	−	0.240532i	$-0.922678\pi$
$691$		−	8738.16i		−	0.481064i	0.970641	−	0.240532i	$-0.0773219\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	6248.29i		−	0.341023i
$696$		0			0
$697$	17239.6			0.936869
$698$		0			0
$699$		0			0
$700$		0			0
$701$			9016.22i			0.485789i	0.970053	+	0.242894i	$0.0780969\pi$
$701$			9016.22i			0.485789i	−0.970053	+	0.242894i	$0.921903\pi$
$702$		0			0
$703$			19851.7i			1.06504i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	30007.9	+	3041.82i	1.59627	+	0.161810i
$708$		0			0
$709$	28263.4			1.49711			0.748557	−	0.663071i	$-0.230747\pi$
$709$	28263.4			1.49711			0.748557	+	0.663071i	$0.230747\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−26989.4			−1.41761
$714$		0			0
$715$	3162.40			0.165408
$716$		0			0
$717$		0			0
$718$		0			0
$719$	23753.1			1.23205			0.616023	−	0.787728i	$-0.288743\pi$
$719$	23753.1			1.23205			0.616023	+	0.787728i	$0.288743\pi$
$720$		0			0
$721$	−1389.16	+	13704.3i	−0.0717548	+	0.707870i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			4588.51i			0.235052i
$726$		0			0
$727$		−	26684.1i		−	1.36129i	−0.732613	−	0.680645i	$-0.761699\pi$
$727$		−	26684.1i		−	1.36129i	0.732613	−	0.680645i	$-0.238301\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2036.33			−0.103032
$732$		0			0
$733$		−	33684.6i		−	1.69737i	−0.528901	−	0.848684i	$-0.677396\pi$
$733$		−	33684.6i		−	1.69737i	0.528901	−	0.848684i	$-0.322604\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			5451.08i			0.272446i
$738$		0			0
$739$	1602.94			0.0797905			0.0398952	−	0.999204i	$-0.487298\pi$
$739$	1602.94			0.0797905			0.0398952	+	0.999204i	$0.487298\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	39150.9i		−	1.93312i	−0.256439	−	0.966560i	$-0.582549\pi$
$743$		−	39150.9i		−	1.93312i	0.256439	−	0.966560i	$-0.417451\pi$
$744$		0			0
$745$			11395.5i			0.560400i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1562.40	+	15413.3i	−0.0762202	+	0.751922i
$750$		0			0
$751$	−20835.8			−1.01240			−0.506199	−	0.862417i	$-0.668950\pi$
$751$	−20835.8			−1.01240			−0.506199	+	0.862417i	$0.668950\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8130.28			−0.391909
$756$		0			0
$757$	1105.00			0.0530538			0.0265269	−	0.999648i	$-0.491555\pi$
$757$	1105.00			0.0530538			0.0265269	+	0.999648i	$0.491555\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−27544.5			−1.31207			−0.656036	−	0.754730i	$-0.727768\pi$
$761$	−27544.5			−1.31207			−0.656036	+	0.754730i	$0.727768\pi$
$762$		0			0
$763$	28819.2	+	2921.32i	1.36740	+	0.138609i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			20025.9i			0.942756i
$768$		0			0
$769$			12029.9i			0.564121i	0.959397	+	0.282060i	$0.0910179\pi$
$769$			12029.9i			0.564121i	−0.959397	+	0.282060i	$0.908982\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20325.9			−0.945758			−0.472879	−	0.881127i	$-0.656785\pi$
$773$	−20325.9			−0.945758			−0.472879	+	0.881127i	$0.656785\pi$
$774$		0			0
$775$		−	10084.7i		−	0.467422i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			29386.1i			1.35156i
$780$		0			0
$781$	1950.53			0.0893669
$782$		0			0
$783$		0			0
$784$		0			0
$785$			8868.81i			0.403237i
$786$		0			0
$787$			26442.6i			1.19768i	0.800867	+	0.598842i	$0.204372\pi$
$787$			26442.6i			1.19768i	−0.800867	+	0.598842i	$0.795628\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3885.98	+	38335.7i	−0.174677	+	1.72321i
$792$		0			0
$793$	−15820.3			−0.708443
$794$		0			0
$795$		0			0
$796$		0			0
$797$	26352.9			1.17122			0.585612	−	0.810591i	$-0.300854\pi$
$797$	26352.9			1.17122			0.585612	+	0.810591i	$0.300854\pi$
$798$		0			0
$799$	−10726.7			−0.474949
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10610.9			0.466316
$804$		0			0
$805$	−2651.21	+	26154.6i	−0.116078	+	1.14513i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	36209.0i		−	1.57360i	−0.617211	−	0.786798i	$-0.711737\pi$
$809$		−	36209.0i		−	1.57360i	0.617211	−	0.786798i	$-0.288263\pi$
$810$		0			0
$811$			12130.9i			0.525245i	0.964899	+	0.262622i	$0.0845874\pi$
$811$			12130.9i			0.525245i	−0.964899	+	0.262622i	$0.915413\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7107.22			−0.305466
$816$		0			0
$817$		−	3471.05i		−	0.148637i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			19239.0i			0.817841i	0.912570	+	0.408920i	$0.134095\pi$
$821$			19239.0i			0.817841i	−0.912570	+	0.408920i	$0.865905\pi$
$822$		0			0
$823$	1974.53			0.0836305			0.0418152	−	0.999125i	$-0.486686\pi$
$823$	1974.53			0.0836305			0.0418152	+	0.999125i	$0.486686\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			18657.5i			0.784506i	0.919857	+	0.392253i	$0.128304\pi$
$827$			18657.5i			0.784506i	−0.919857	+	0.392253i	$0.871696\pi$
$828$		0			0
$829$		−	39035.0i		−	1.63539i	−0.575650	−	0.817696i	$-0.695251\pi$
$829$		−	39035.0i		−	1.63539i	0.575650	−	0.817696i	$-0.304749\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	13089.7	+	2681.27i	0.544454	+	0.111525i
$834$		0			0
$835$	−16944.1			−0.702244
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15291.4			0.629221			0.314610	−	0.949221i	$-0.398126\pi$
$839$	15291.4			0.629221			0.314610	+	0.949221i	$0.398126\pi$
$840$		0			0
$841$	20649.0			0.846653
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3842.36			0.156427
$846$		0			0
$847$	−23179.2	−	2349.61i	−0.940315	−	0.0953170i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			60033.0i			2.41822i
$852$		0			0
$853$		−	27831.6i		−	1.11716i	−0.829452	−	0.558579i	$-0.811347\pi$
$853$		−	27831.6i		−	1.11716i	0.829452	−	0.558579i	$-0.188653\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−24381.1			−0.971811			−0.485905	−	0.874011i	$-0.661510\pi$
$857$	−24381.1			−0.971811			−0.485905	+	0.874011i	$0.661510\pi$
$858$		0			0
$859$		−	27582.0i		−	1.09556i	−0.836623	−	0.547779i	$-0.815473\pi$
$859$		−	27582.0i		−	1.09556i	0.836623	−	0.547779i	$-0.184527\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			26388.4i			1.04087i	0.853901	+	0.520436i	$0.174230\pi$
$863$			26388.4i			1.04087i	−0.853901	+	0.520436i	$0.825770\pi$
$864$		0			0
$865$	−224.682			−0.00883171
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1720.98i		−	0.0671809i
$870$		0			0
$871$			33393.2i			1.29906i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−26054.1	−	2641.03i	−1.00662	−	0.102038i
$876$		0			0
$877$	−31506.9			−1.21313			−0.606564	−	0.795035i	$-0.707453\pi$
$877$	−31506.9			−1.21313			−0.606564	+	0.795035i	$0.707453\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1153.83			−0.0441243			−0.0220621	−	0.999757i	$-0.507023\pi$
$881$	−1153.83			−0.0441243			−0.0220621	+	0.999757i	$0.507023\pi$
$882$		0			0
$883$	22903.1			0.872877			0.436439	−	0.899734i	$-0.356240\pi$
$883$	22903.1			0.872877			0.436439	+	0.899734i	$0.356240\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−29805.0			−1.12825			−0.564123	−	0.825691i	$-0.690786\pi$
$887$	−29805.0			−1.12825			−0.564123	+	0.825691i	$0.690786\pi$
$888$		0			0
$889$	−14268.3	−	1446.33i	−0.538293	−	0.0545652i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	18284.4i		−	0.685178i
$894$		0			0
$895$		−	10978.4i		−	0.410020i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8219.80			0.304945
$900$		0			0
$901$		−	5322.64i		−	0.196807i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	24858.5i		−	0.913065i
$906$		0			0
$907$	−20347.0			−0.744884			−0.372442	−	0.928055i	$-0.621479\pi$
$907$	−20347.0			−0.744884			−0.372442	+	0.928055i	$0.621479\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	31138.7i		−	1.13246i	−0.824247	−	0.566231i	$-0.808401\pi$
$911$		−	31138.7i		−	1.13246i	0.824247	−	0.566231i	$-0.191599\pi$
$912$		0			0
$913$			5528.19i			0.200390i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−14845.1	−	1504.81i	−0.534600	−	0.0541909i
$918$		0			0
$919$	18807.6			0.675089			0.337545	−	0.941310i	$-0.390404\pi$
$919$	18807.6			0.675089			0.337545	+	0.941310i	$0.390404\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	11948.9			0.426115
$924$		0			0
$925$	−22431.6			−0.797347
$926$		0			0
$927$		0			0
$928$		0			0
$929$	27397.5			0.967583			0.483791	−	0.875183i	$-0.339259\pi$
$929$	27397.5			0.967583			0.483791	+	0.875183i	$0.339259\pi$
$930$		0			0
$931$	−4570.41	+	22312.2i	−0.160890	+	0.785448i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			2353.19i			0.0823075i
$936$		0			0
$937$			27022.7i			0.942149i	0.882093	+	0.471075i	$0.156134\pi$
$937$			27022.7i			0.942149i	−0.882093	+	0.471075i	$0.843866\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−16048.4			−0.555964			−0.277982	−	0.960586i	$-0.589666\pi$
$941$	−16048.4			−0.555964			−0.277982	+	0.960586i	$0.589666\pi$
$942$		0			0
$943$			88865.9i			3.06879i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			3526.00i			0.120992i	0.998168	+	0.0604962i	$0.0192683\pi$
$947$			3526.00i			0.120992i	−0.998168	+	0.0604962i	$0.980732\pi$
$948$		0			0
$949$	65002.3			2.22346
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	31452.5i		−	1.06909i	−0.845139	−	0.534547i	$-0.820482\pi$
$953$		−	31452.5i		−	1.06909i	0.845139	−	0.534547i	$-0.179518\pi$
$954$		0			0
$955$			16088.5i			0.545142i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3423.57	+	33774.0i	−0.115279	+	1.13725i
$960$		0			0
$961$	11725.4			0.393589
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11244.0			0.375086
$966$		0			0
$967$	−9870.97			−0.328262			−0.164131	−	0.986439i	$-0.552482\pi$
$967$	−9870.97			−0.328262			−0.164131	+	0.986439i	$0.552482\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−9155.66			−0.302594			−0.151297	−	0.988488i	$-0.548345\pi$
$971$	−9155.66			−0.302594			−0.151297	+	0.988488i	$0.548345\pi$
$972$		0			0
$973$	1650.94	−	16286.8i	0.0543954	−	0.536618i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			13641.8i			0.446713i	0.974737	+	0.223356i	$0.0717014\pi$
$977$			13641.8i			0.446713i	−0.974737	+	0.223356i	$0.928299\pi$
$978$		0			0
$979$		−	7059.04i		−	0.230447i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	57673.7			1.87132			0.935659	−	0.352907i	$-0.114807\pi$
$983$	57673.7			1.87132			0.935659	+	0.352907i	$0.114807\pi$
$984$		0			0
$985$			12128.0i			0.392314i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	10496.7i		−	0.337489i
$990$		0			0
$991$	28668.4			0.918953			0.459476	−	0.888190i	$-0.348037\pi$
$991$	28668.4			0.918953			0.459476	+	0.888190i	$0.348037\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			17912.9i			0.570730i
$996$		0			0
$997$			4536.02i			0.144090i	0.997401	+	0.0720448i	$0.0229525\pi$
$997$			4536.02i			0.144090i	−0.997401	+	0.0720448i	$0.977048\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.4.f.a.1133.15		48
3.2	odd	2	inner	2268.4.f.a.1133.34		48
7.6	odd	2	inner	2268.4.f.a.1133.33		48
9.2	odd	6		252.4.x.a.41.19	yes	48
9.4	even	3		252.4.x.a.209.6	yes	48
9.5	odd	6		756.4.x.a.629.8		48
9.7	even	3		756.4.x.a.125.17		48
21.20	even	2	inner	2268.4.f.a.1133.16		48
63.13	odd	6		252.4.x.a.209.19	yes	48
63.20	even	6		252.4.x.a.41.6	✓	48
63.34	odd	6		756.4.x.a.125.8		48
63.41	even	6		756.4.x.a.629.17		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.x.a.41.6	✓	48	63.20	even	6
252.4.x.a.41.19	yes	48	9.2	odd	6
252.4.x.a.209.6	yes	48	9.4	even	3
252.4.x.a.209.19	yes	48	63.13	odd	6
756.4.x.a.125.8		48	63.34	odd	6
756.4.x.a.125.17		48	9.7	even	3
756.4.x.a.629.8		48	9.5	odd	6
756.4.x.a.629.17		48	63.41	even	6
2268.4.f.a.1133.15		48	1.1	even	1	trivial
2268.4.f.a.1133.16		48	21.20	even	2	inner
2268.4.f.a.1133.33		48	7.6	odd	2	inner
2268.4.f.a.1133.34		48	3.2	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2268.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.06893 q^{5} +(-18.4258 - 1.86777i) q^{7} +O(q^{10})\)	\(q-7.06893 q^{5} +(-18.4258 - 1.86777i) q^{7} -8.54562i q^{11} -52.3503i q^{13} +38.9547 q^{17} +66.4008i q^{19} +200.801i q^{23} -75.0302 q^{25} -61.1555i q^{29} +134.408i q^{31} +(130.251 + 13.2032i) q^{35} +298.967 q^{37} +442.556 q^{41} -52.2742 q^{43} -275.364 q^{47} +(336.023 + 68.8306i) q^{49} -136.637i q^{53} +60.4084i q^{55} -382.537 q^{59} -302.201i q^{61} +370.061i q^{65} -637.880 q^{67} +228.249i q^{71} +1241.68i q^{73} +(-15.9613 + 157.460i) q^{77} +201.387 q^{79} -646.904 q^{83} -275.368 q^{85} +826.042 q^{89} +(-97.7786 + 964.598i) q^{91} -469.383i q^{95} +19.7107i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{7}+O(q^{10}) \) 48 * q - 12 * q^7	\( 48 q - 12 q^{7} + 1200 q^{25} + 336 q^{37} - 168 q^{43} - 636 q^{49} + 1176 q^{67} - 408 q^{79} + 720 q^{85} - 1080 q^{91}+O(q^{100}) \) 48 * q - 12 * q^7 + 1200 * q^25 + 336 * q^37 - 168 * q^43 - 636 * q^49 + 1176 * q^67 - 408 * q^79 + 720 * q^85 - 1080 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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