LMFDB - Embedded newform 2268.4.f.a.1133.18

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			1133.18
Character	$\chi$	$=$	2268.1133
Dual form			2268.4.f.a.1133.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.99994 q^{5} +(-15.8480 + 9.58340i) q^{7} +O(q^{10})$	$q-5.99994 q^{5} +(-15.8480 + 9.58340i) q^{7} +45.4733i q^{11} -26.2841i q^{13} -19.7249 q^{17} -27.9162i q^{19} -69.6822i q^{23} -89.0007 q^{25} +137.445i q^{29} +159.582i q^{31} +(95.0869 - 57.4998i) q^{35} -287.829 q^{37} +40.8576 q^{41} -110.832 q^{43} -219.333 q^{47} +(159.317 - 303.755i) q^{49} +209.770i q^{53} -272.837i q^{55} -827.759 q^{59} +686.133i q^{61} +157.703i q^{65} +342.897 q^{67} -387.476i q^{71} +220.721i q^{73} +(-435.789 - 720.660i) q^{77} -484.602 q^{79} +708.646 q^{83} +118.348 q^{85} -140.929 q^{89} +(251.891 + 416.550i) q^{91} +167.495i q^{95} +1506.08i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−5.99994			−0.536651			−0.268325	−	0.963328i	$-0.586470\pi$
$5$	−5.99994			−0.536651			−0.268325	+	0.963328i	$0.586470\pi$
$6$		0			0
$7$	−15.8480	+	9.58340i	−0.855710	+	0.517455i
$7$	−15.8480	+	9.58340i	−0.855710	+	0.517455i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			45.4733i			1.24643i	0.782051	+	0.623214i	$0.214174\pi$
$11$			45.4733i			1.24643i	−0.782051	+	0.623214i	$0.785826\pi$
$12$		0			0
$13$		−	26.2841i		−	0.560762i	−0.959889	−	0.280381i	$-0.909539\pi$
$13$		−	26.2841i		−	0.560762i	0.959889	−	0.280381i	$-0.0904608\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−19.7249			−0.281411			−0.140706	−	0.990051i	$-0.544937\pi$
$17$	−19.7249			−0.281411			−0.140706	+	0.990051i	$0.544937\pi$
$18$		0			0
$19$		−	27.9162i		−	0.337074i	−0.985695	−	0.168537i	$-0.946096\pi$
$19$		−	27.9162i		−	0.337074i	0.985695	−	0.168537i	$-0.0539043\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	69.6822i		−	0.631728i	−0.948804	−	0.315864i	$-0.897706\pi$
$23$		−	69.6822i		−	0.631728i	0.948804	−	0.315864i	$-0.102294\pi$
$24$		0			0
$25$	−89.0007			−0.712006
$26$		0			0
$27$		0			0
$28$		0			0
$29$			137.445i			0.880103i	0.897973	+	0.440051i	$0.145040\pi$
$29$			137.445i			0.880103i	−0.897973	+	0.440051i	$0.854960\pi$
$30$		0			0
$31$			159.582i			0.924571i	0.886731	+	0.462286i	$0.152970\pi$
$31$			159.582i			0.924571i	−0.886731	+	0.462286i	$0.847030\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	95.0869	−	57.4998i	0.459218	−	0.277693i
$36$		0			0
$37$	−287.829			−1.27889			−0.639443	−	0.768839i	$-0.720835\pi$
$37$	−287.829			−1.27889			−0.639443	+	0.768839i	$0.720835\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	40.8576			0.155631			0.0778157	−	0.996968i	$-0.475205\pi$
$41$	40.8576			0.155631			0.0778157	+	0.996968i	$0.475205\pi$
$42$		0			0
$43$	−110.832			−0.393062			−0.196531	−	0.980498i	$-0.562968\pi$
$43$	−110.832			−0.393062			−0.196531	+	0.980498i	$0.562968\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−219.333			−0.680701			−0.340351	−	0.940299i	$-0.610546\pi$
$47$	−219.333			−0.680701			−0.340351	+	0.940299i	$0.610546\pi$
$48$		0			0
$49$	159.317	−	303.755i	0.464481	−	0.885583i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			209.770i			0.543664i	0.962345	+	0.271832i	$0.0876295\pi$
$53$			209.770i			0.543664i	−0.962345	+	0.271832i	$0.912371\pi$
$54$		0			0
$55$		−	272.837i		−	0.668897i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−827.759			−1.82653			−0.913263	−	0.407370i	$-0.866446\pi$
$59$	−827.759			−1.82653			−0.913263	+	0.407370i	$0.866446\pi$
$60$		0			0
$61$			686.133i			1.44017i	0.693886	+	0.720085i	$0.255897\pi$
$61$			686.133i			1.44017i	−0.693886	+	0.720085i	$0.744103\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			157.703i			0.300933i
$66$		0			0
$67$	342.897			0.625247			0.312624	−	0.949877i	$-0.398792\pi$
$67$	342.897			0.625247			0.312624	+	0.949877i	$0.398792\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	387.476i		−	0.647675i	−0.946113	−	0.323838i	$-0.895027\pi$
$71$		−	387.476i		−	0.647675i	0.946113	−	0.323838i	$-0.104973\pi$
$72$		0			0
$73$			220.721i			0.353882i	0.984221	+	0.176941i	$0.0566202\pi$
$73$			220.721i			0.353882i	−0.984221	+	0.176941i	$0.943380\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−435.789	−	720.660i	−0.644971	−	1.06658i
$78$		0			0
$79$	−484.602			−0.690151			−0.345076	−	0.938575i	$-0.612147\pi$
$79$	−484.602			−0.690151			−0.345076	+	0.938575i	$0.612147\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	708.646			0.937157			0.468579	−	0.883422i	$-0.344766\pi$
$83$	708.646			0.937157			0.468579	+	0.883422i	$0.344766\pi$
$84$		0			0
$85$	118.348			0.151020
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−140.929			−0.167848			−0.0839240	−	0.996472i	$-0.526745\pi$
$89$	−140.929			−0.167848			−0.0839240	+	0.996472i	$0.526745\pi$
$90$		0			0
$91$	251.891	+	416.550i	0.290169	+	0.479850i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			167.495i			0.180891i
$96$		0			0
$97$			1506.08i			1.57649i	0.615361	+	0.788245i	$0.289010\pi$
$97$			1506.08i			1.57649i	−0.615361	+	0.788245i	$0.710990\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1847.03			1.81967			0.909834	−	0.414971i	$-0.136208\pi$
$101$	1847.03			1.81967			0.909834	+	0.414971i	$0.136208\pi$
$102$		0			0
$103$		−	804.227i		−	0.769348i	−0.923052	−	0.384674i	$-0.874314\pi$
$103$		−	804.227i		−	0.769348i	0.923052	−	0.384674i	$-0.125686\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			282.997i			0.255685i	0.991794	+	0.127843i	$0.0408053\pi$
$107$			282.997i			0.255685i	−0.991794	+	0.127843i	$0.959195\pi$
$108$		0			0
$109$	1826.43			1.60496			0.802478	−	0.596681i	$-0.203514\pi$
$109$	1826.43			1.60496			0.802478	+	0.596681i	$0.203514\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	942.017i		−	0.784226i	−0.919917	−	0.392113i	$-0.871744\pi$
$113$		−	942.017i		−	0.784226i	0.919917	−	0.392113i	$-0.128256\pi$
$114$		0			0
$115$			418.089i			0.339017i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	312.600	−	189.032i	0.240807	−	0.145618i
$120$		0			0
$121$	−736.821			−0.553584
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1283.99			0.918749
$126$		0			0
$127$	320.551			0.223971			0.111985	−	0.993710i	$-0.464279\pi$
$127$	320.551			0.223971			0.111985	+	0.993710i	$0.464279\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−613.813			−0.409382			−0.204691	−	0.978827i	$-0.565619\pi$
$131$	−613.813			−0.409382			−0.204691	+	0.978827i	$0.565619\pi$
$132$		0			0
$133$	267.532	+	442.415i	0.174421	+	0.288438i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1750.36i		−	1.09156i	−0.837929	−	0.545780i	$-0.816234\pi$
$137$		−	1750.36i		−	1.09156i	0.837929	−	0.545780i	$-0.183766\pi$
$138$		0			0
$139$		−	1294.10i		−	0.789668i	−0.918752	−	0.394834i	$-0.870802\pi$
$139$		−	1294.10i		−	0.789668i	0.918752	−	0.394834i	$-0.129198\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1195.23			0.698950
$144$		0			0
$145$		−	824.664i		−	0.472308i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	2885.86i		−	1.58670i	−0.608763	−	0.793352i	$-0.708334\pi$
$149$		−	2885.86i		−	1.58670i	0.608763	−	0.793352i	$-0.291666\pi$
$150$		0			0
$151$	−3081.26			−1.66059			−0.830297	−	0.557321i	$-0.811829\pi$
$151$	−3081.26			−1.66059			−0.830297	+	0.557321i	$0.811829\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	957.480i		−	0.496172i
$156$		0			0
$157$		−	1258.78i		−	0.639881i	−0.947438	−	0.319941i	$-0.896337\pi$
$157$		−	1258.78i		−	0.639881i	0.947438	−	0.319941i	$-0.103663\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	667.793	+	1104.32i	0.326891	+	0.540576i
$162$		0			0
$163$	−2906.94			−1.39687			−0.698434	−	0.715675i	$-0.746119\pi$
$163$	−2906.94			−1.39687			−0.698434	+	0.715675i	$0.746119\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	573.550			0.265764			0.132882	−	0.991132i	$-0.457577\pi$
$167$	573.550			0.265764			0.132882	+	0.991132i	$0.457577\pi$
$168$		0			0
$169$	1506.14			0.685546
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3862.27			1.69736			0.848679	−	0.528909i	$-0.177399\pi$
$173$	3862.27			1.69736			0.848679	+	0.528909i	$0.177399\pi$
$174$		0			0
$175$	1410.48	−	852.930i	0.609271	−	0.368431i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	2248.26i		−	0.938787i	−0.882989	−	0.469394i	$-0.844473\pi$
$179$		−	2248.26i		−	0.938787i	0.882989	−	0.469394i	$-0.155527\pi$
$180$		0			0
$181$		−	166.407i		−	0.0683368i	−0.999416	−	0.0341684i	$-0.989122\pi$
$181$		−	166.407i		−	0.0683368i	0.999416	−	0.0341684i	$-0.0108783\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1726.96			0.686315
$186$		0			0
$187$		−	896.956i		−	0.350759i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			2030.98i			0.769405i	0.923041	+	0.384703i	$0.125696\pi$
$191$			2030.98i			0.769405i	−0.923041	+	0.384703i	$0.874304\pi$
$192$		0			0
$193$	−502.236			−0.187315			−0.0936573	−	0.995604i	$-0.529856\pi$
$193$	−502.236			−0.187315			−0.0936573	+	0.995604i	$0.529856\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3949.11i		−	1.42823i	−0.700026	−	0.714117i	$-0.746828\pi$
$197$		−	3949.11i		−	1.42823i	0.700026	−	0.714117i	$-0.253172\pi$
$198$		0			0
$199$		−	629.709i		−	0.224316i	−0.993690	−	0.112158i	$-0.964224\pi$
$199$		−	629.709i		−	0.224316i	0.993690	−	0.112158i	$-0.0357763\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1317.19	−	2178.23i	−0.455413	−	0.753113i
$204$		0			0
$205$	−245.143			−0.0835197
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1269.44			0.420139
$210$		0			0
$211$	2989.90			0.975514			0.487757	−	0.872979i	$-0.337815\pi$
$211$	2989.90			0.975514			0.487757	+	0.872979i	$0.337815\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	664.983			0.210937
$216$		0			0
$217$	−1529.33	−	2529.05i	−0.478424	−	0.791165i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			518.452i			0.157805i
$222$		0			0
$223$		−	3267.48i		−	0.981195i	−0.871386	−	0.490598i	$-0.836778\pi$
$223$		−	3267.48i		−	0.981195i	0.871386	−	0.490598i	$-0.163222\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5673.72			−1.65894			−0.829468	−	0.558555i	$-0.811356\pi$
$227$	−5673.72			−1.65894			−0.829468	+	0.558555i	$0.811356\pi$
$228$		0			0
$229$			5229.04i			1.50893i	0.656340	+	0.754465i	$0.272103\pi$
$229$			5229.04i			1.50893i	−0.656340	+	0.754465i	$0.727897\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	3433.84i		−	0.965487i	−0.875762	−	0.482743i	$-0.839640\pi$
$233$		−	3433.84i		−	0.965487i	0.875762	−	0.482743i	$-0.160360\pi$
$234$		0			0
$235$	1315.98			0.365299
$236$		0			0
$237$		0			0
$238$		0			0
$239$			542.336i			0.146782i	0.997303	+	0.0733908i	$0.0233820\pi$
$239$			542.336i			0.146782i	−0.997303	+	0.0733908i	$0.976618\pi$
$240$		0			0
$241$			3591.49i			0.959952i	0.877282	+	0.479976i	$0.159355\pi$
$241$			3591.49i			0.959952i	−0.877282	+	0.479976i	$0.840645\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−955.892	+	1822.51i	−0.249264	+	0.475249i
$246$		0			0
$247$	−733.752			−0.189018
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4334.39			1.08998			0.544988	−	0.838444i	$-0.316534\pi$
$251$	4334.39			1.08998			0.544988	+	0.838444i	$0.316534\pi$
$252$		0			0
$253$	3168.68			0.787404
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6565.77			1.59362			0.796812	−	0.604227i	$-0.206518\pi$
$257$	6565.77			1.59362			0.796812	+	0.604227i	$0.206518\pi$
$258$		0			0
$259$	4561.51	−	2758.38i	1.09436	−	0.661766i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	7245.51i		−	1.69877i	−0.527772	−	0.849386i	$-0.676973\pi$
$263$		−	7245.51i		−	1.69877i	0.527772	−	0.849386i	$-0.323027\pi$
$264$		0			0
$265$		−	1258.61i		−	0.291758i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3455.87			−0.783302			−0.391651	−	0.920114i	$-0.628096\pi$
$269$	−3455.87			−0.783302			−0.391651	+	0.920114i	$0.628096\pi$
$270$		0			0
$271$		−	2471.30i		−	0.553951i	−0.960877	−	0.276975i	$-0.910668\pi$
$271$		−	2471.30i		−	0.553951i	0.960877	−	0.276975i	$-0.0893321\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	4047.16i		−	0.887465i
$276$		0			0
$277$	−4513.69			−0.979067			−0.489533	−	0.871985i	$-0.662833\pi$
$277$	−4513.69			−0.979067			−0.489533	+	0.871985i	$0.662833\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	2413.88i		−	0.512455i	−0.966616	−	0.256228i	$-0.917520\pi$
$281$		−	2413.88i		−	0.512455i	0.966616	−	0.256228i	$-0.0824796\pi$
$282$		0			0
$283$			4495.97i			0.944373i	0.881499	+	0.472187i	$0.156535\pi$
$283$			4495.97i			0.944373i	−0.881499	+	0.472187i	$0.843465\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−647.511	+	391.555i	−0.133175	+	0.0805322i
$288$		0			0
$289$	−4523.93			−0.920808
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3139.27			−0.625933			−0.312966	−	0.949764i	$-0.601323\pi$
$293$	−3139.27			−0.625933			−0.312966	+	0.949764i	$0.601323\pi$
$294$		0			0
$295$	4966.50			0.980207
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1831.54			−0.354249
$300$		0			0
$301$	1756.46	−	1062.14i	0.336347	−	0.203392i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	4116.75i		−	0.772868i
$306$		0			0
$307$			10294.0i			1.91371i	0.290560	+	0.956857i	$0.406158\pi$
$307$			10294.0i			1.91371i	−0.290560	+	0.956857i	$0.593842\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	7881.17			1.43698			0.718489	−	0.695538i	$-0.244834\pi$
$311$	7881.17			1.43698			0.718489	+	0.695538i	$0.244834\pi$
$312$		0			0
$313$		−	8194.72i		−	1.47985i	−0.672689	−	0.739925i	$-0.734861\pi$
$313$		−	8194.72i		−	1.47985i	0.672689	−	0.739925i	$-0.265139\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	8765.10i		−	1.55299i	−0.630125	−	0.776494i	$-0.716996\pi$
$317$		−	8765.10i		−	1.55299i	0.630125	−	0.776494i	$-0.283004\pi$
$318$		0			0
$319$	−6250.10			−1.09699
$320$		0			0
$321$		0			0
$322$		0			0
$323$			550.644i			0.0948564i
$324$		0			0
$325$			2339.31i			0.399266i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3475.98	−	2101.95i	0.582483	−	0.352232i
$330$		0			0
$331$	6863.52			1.13974			0.569869	−	0.821736i	$-0.306994\pi$
$331$	6863.52			1.13974			0.569869	+	0.821736i	$0.306994\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2057.36			−0.335539
$336$		0			0
$337$	−4647.63			−0.751253			−0.375627	−	0.926771i	$-0.622573\pi$
$337$	−4647.63			−0.751253			−0.375627	+	0.926771i	$0.622573\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7256.70			−1.15241
$342$		0			0
$343$	386.154	+	6340.70i	0.0607882	+	0.998151i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			9578.29i			1.48181i	0.671608	+	0.740907i	$0.265604\pi$
$347$			9578.29i			1.48181i	−0.671608	+	0.740907i	$0.734396\pi$
$348$		0			0
$349$			871.993i			0.133744i	0.997762	+	0.0668721i	$0.0213019\pi$
$349$			871.993i			0.133744i	−0.997762	+	0.0668721i	$0.978698\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8297.27			1.25105			0.625523	−	0.780206i	$-0.284886\pi$
$353$	8297.27			1.25105			0.625523	+	0.780206i	$0.284886\pi$
$354$		0			0
$355$			2324.83i			0.347575i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			976.696i			0.143588i	0.997419	+	0.0717939i	$0.0228724\pi$
$359$			976.696i			0.143588i	−0.997419	+	0.0717939i	$0.977128\pi$
$360$		0			0
$361$	6079.69			0.886381
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	1324.31i		−	0.189911i
$366$		0			0
$367$		−	6929.94i		−	0.985668i	−0.870123	−	0.492834i	$-0.835961\pi$
$367$		−	6929.94i		−	0.985668i	0.870123	−	0.492834i	$-0.164039\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2010.31	−	3324.44i	−0.281322	−	0.465219i
$372$		0			0
$373$	−4519.37			−0.627357			−0.313678	−	0.949529i	$-0.601561\pi$
$373$	−4519.37			−0.627357			−0.313678	+	0.949529i	$0.601561\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3612.63			0.493528
$378$		0			0
$379$	2781.32			0.376958			0.188479	−	0.982077i	$-0.439644\pi$
$379$	2781.32			0.376958			0.188479	+	0.982077i	$0.439644\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10898.9			1.45407			0.727033	−	0.686603i	$-0.240899\pi$
$383$	10898.9			1.45407			0.727033	+	0.686603i	$0.240899\pi$
$384$		0			0
$385$	2614.71	+	4323.91i	0.346124	+	0.572382i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	2219.44i		−	0.289280i	−0.989484	−	0.144640i	$-0.953798\pi$
$389$		−	2219.44i		−	0.289280i	0.989484	−	0.144640i	$-0.0462024\pi$
$390$		0			0
$391$			1374.47i			0.177775i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2907.58			0.370370
$396$		0			0
$397$		−	10873.7i		−	1.37464i	−0.726354	−	0.687321i	$-0.758786\pi$
$397$		−	10873.7i		−	1.37464i	0.726354	−	0.687321i	$-0.241214\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	6121.06i		−	0.762273i	−0.924519	−	0.381136i	$-0.875533\pi$
$401$		−	6121.06i		−	0.762273i	0.924519	−	0.381136i	$-0.124467\pi$
$402$		0			0
$403$	4194.46			0.518464
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	13088.5i		−	1.59404i
$408$		0			0
$409$			14035.4i			1.69683i	0.529329	+	0.848416i	$0.322444\pi$
$409$			14035.4i			1.69683i	−0.529329	+	0.848416i	$0.677556\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	13118.3	−	7932.75i	1.56298	−	0.945145i
$414$		0			0
$415$	−4251.83			−0.502926
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8288.37			0.966380			0.483190	−	0.875515i	$-0.339478\pi$
$419$	8288.37			0.966380			0.483190	+	0.875515i	$0.339478\pi$
$420$		0			0
$421$	−15534.6			−1.79836			−0.899181	−	0.437578i	$-0.855836\pi$
$421$	−15534.6			−1.79836			−0.899181	+	0.437578i	$0.855836\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1755.53			0.200366
$426$		0			0
$427$	−6575.49	−	10873.8i	−0.745223	−	1.23237i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	3167.89i		−	0.354041i	−0.984207	−	0.177021i	$-0.943354\pi$
$431$		−	3167.89i		−	0.354041i	0.984207	−	0.177021i	$-0.0566459\pi$
$432$		0			0
$433$		−	2187.70i		−	0.242804i	−0.992603	−	0.121402i	$-0.961261\pi$
$433$		−	2187.70i		−	0.242804i	0.992603	−	0.121402i	$-0.0387391\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1945.26			−0.212939
$438$		0			0
$439$		−	1967.37i		−	0.213889i	−0.994265	−	0.106945i	$-0.965893\pi$
$439$		−	1967.37i		−	0.213889i	0.994265	−	0.106945i	$-0.0341068\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	2574.07i		−	0.276067i	−0.990428	−	0.138034i	$-0.955922\pi$
$443$		−	2574.07i		−	0.276067i	0.990428	−	0.138034i	$-0.0440782\pi$
$444$		0			0
$445$	845.567			0.0900757
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	9168.11i		−	0.963630i	−0.876273	−	0.481815i	$-0.839978\pi$
$449$		−	9168.11i		−	0.963630i	0.876273	−	0.481815i	$-0.160022\pi$
$450$		0			0
$451$			1857.93i			0.193983i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1511.33	−	2499.28i	−0.155719	−	0.257512i
$456$		0			0
$457$	−898.836			−0.0920039			−0.0460019	−	0.998941i	$-0.514648\pi$
$457$	−898.836			−0.0920039			−0.0460019	+	0.998941i	$0.514648\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1501.91			0.151737			0.0758686	−	0.997118i	$-0.475827\pi$
$461$	1501.91			0.151737			0.0758686	+	0.997118i	$0.475827\pi$
$462$		0			0
$463$	−1031.76			−0.103564			−0.0517820	−	0.998658i	$-0.516490\pi$
$463$	−1031.76			−0.103564			−0.0517820	+	0.998658i	$0.516490\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8619.68			−0.854114			−0.427057	−	0.904225i	$-0.640450\pi$
$467$	−8619.68			−0.854114			−0.427057	+	0.904225i	$0.640450\pi$
$468$		0			0
$469$	−5434.23	+	3286.12i	−0.535031	+	0.323537i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	5039.88i		−	0.489924i
$474$		0			0
$475$			2484.56i			0.239999i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4075.02			0.388711			0.194355	−	0.980931i	$-0.437738\pi$
$479$	4075.02			0.388711			0.194355	+	0.980931i	$0.437738\pi$
$480$		0			0
$481$			7565.33i			0.717150i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	9036.40i		−	0.846025i
$486$		0			0
$487$	−14817.8			−1.37876			−0.689381	−	0.724399i	$-0.742117\pi$
$487$	−14817.8			−1.37876			−0.689381	+	0.724399i	$0.742117\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	7179.19i		−	0.659862i	−0.944005	−	0.329931i	$-0.892975\pi$
$491$		−	7179.19i		−	0.659862i	0.944005	−	0.329931i	$-0.107025\pi$
$492$		0			0
$493$		−	2711.10i		−	0.247671i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3713.34	+	6140.71i	0.335143	+	0.554223i
$498$		0			0
$499$	10761.2			0.965405			0.482702	−	0.875784i	$-0.339655\pi$
$499$	10761.2			0.965405			0.482702	+	0.875784i	$0.339655\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	11285.3			1.00037			0.500186	−	0.865918i	$-0.333265\pi$
$503$	11285.3			1.00037			0.500186	+	0.865918i	$0.333265\pi$
$504$		0			0
$505$	−11082.1			−0.976527
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10809.0			0.941262			0.470631	−	0.882330i	$-0.344026\pi$
$509$	10809.0			0.941262			0.470631	+	0.882330i	$0.344026\pi$
$510$		0			0
$511$	−2115.25	−	3497.98i	−0.183118	−	0.302821i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			4825.31i			0.412871i
$516$		0			0
$517$		−	9973.78i		−	0.848445i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3085.93			−0.259495			−0.129747	−	0.991547i	$-0.541417\pi$
$521$	−3085.93			−0.259495			−0.129747	+	0.991547i	$0.541417\pi$
$522$		0			0
$523$		−	2155.94i		−	0.180254i	−0.995930	−	0.0901268i	$-0.971273\pi$
$523$		−	2155.94i		−	0.180254i	0.995930	−	0.0901268i	$-0.0287272\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	3147.73i		−	0.260185i
$528$		0			0
$529$	7311.39			0.600920
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	1073.91i		−	0.0872722i
$534$		0			0
$535$		−	1697.96i		−	0.137214i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	13812.7	+	7244.67i	1.10382	+	0.578942i
$540$		0			0
$541$	19410.4			1.54255			0.771273	−	0.636504i	$-0.219620\pi$
$541$	19410.4			1.54255			0.771273	+	0.636504i	$0.219620\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−10958.5			−0.861301
$546$		0			0
$547$	4558.31			0.356306			0.178153	−	0.984003i	$-0.442988\pi$
$547$	4558.31			0.356306			0.178153	+	0.984003i	$0.442988\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3836.95			0.296660
$552$		0			0
$553$	7679.95	−	4644.13i	0.590569	−	0.357122i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	4710.38i		−	0.358322i	−0.983820	−	0.179161i	$-0.942662\pi$
$557$		−	4710.38i		−	0.358322i	0.983820	−	0.179161i	$-0.0573383\pi$
$558$		0			0
$559$			2913.11i			0.220414i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4632.51			−0.346780			−0.173390	−	0.984853i	$-0.555472\pi$
$563$	−4632.51			−0.346780			−0.173390	+	0.984853i	$0.555472\pi$
$564$		0			0
$565$			5652.04i			0.420855i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			7698.40i			0.567194i	0.958943	+	0.283597i	$0.0915279\pi$
$569$			7698.40i			0.567194i	−0.958943	+	0.283597i	$0.908472\pi$
$570$		0			0
$571$	17112.7			1.25419			0.627096	−	0.778942i	$-0.284243\pi$
$571$	17112.7			1.25419			0.627096	+	0.778942i	$0.284243\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			6201.77i			0.449794i
$576$		0			0
$577$		−	2389.01i		−	0.172367i	−0.996279	−	0.0861834i	$-0.972533\pi$
$577$		−	2389.01i		−	0.172367i	0.996279	−	0.0861834i	$-0.0274671\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−11230.6	+	6791.24i	−0.801935	+	0.484937i
$582$		0			0
$583$	−9538.95			−0.677638
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15754.4			−1.10776			−0.553880	−	0.832597i	$-0.686853\pi$
$587$	−15754.4			−1.10776			−0.553880	+	0.832597i	$0.686853\pi$
$588$		0			0
$589$	4454.91			0.311649
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−14439.8			−0.999953			−0.499976	−	0.866039i	$-0.666658\pi$
$593$	−14439.8			−0.999953			−0.499976	+	0.866039i	$0.666658\pi$
$594$		0			0
$595$	−1875.58	+	1134.18i	−0.129229	+	0.0781458i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	15091.5i		−	1.02942i	−0.857364	−	0.514710i	$-0.827900\pi$
$599$		−	15091.5i		−	1.02942i	0.857364	−	0.514710i	$-0.172100\pi$
$600$		0			0
$601$			25271.1i			1.71519i	0.514326	+	0.857595i	$0.328042\pi$
$601$			25271.1i			1.71519i	−0.514326	+	0.857595i	$0.671958\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4420.88			0.297081
$606$		0			0
$607$			19539.9i			1.30659i	0.757105	+	0.653293i	$0.226613\pi$
$607$			19539.9i			1.30659i	−0.757105	+	0.653293i	$0.773387\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			5764.97i			0.381711i
$612$		0			0
$613$	22968.2			1.51334			0.756671	−	0.653796i	$-0.226825\pi$
$613$	22968.2			1.51334			0.756671	+	0.653796i	$0.226825\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	28335.8i		−	1.84887i	−0.381334	−	0.924437i	$-0.624535\pi$
$617$		−	28335.8i		−	1.84887i	0.381334	−	0.924437i	$-0.375465\pi$
$618$		0			0
$619$			20030.9i			1.30066i	0.759650	+	0.650332i	$0.225370\pi$
$619$			20030.9i			1.30066i	−0.759650	+	0.650332i	$0.774630\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2233.44	−	1350.58i	0.143629	−	0.0868537i
$624$		0			0
$625$	3421.23			0.218959
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5677.39			0.359893
$630$		0			0
$631$	7219.42			0.455469			0.227734	−	0.973723i	$-0.426868\pi$
$631$	7219.42			0.455469			0.227734	+	0.973723i	$0.426868\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1923.29			−0.120194
$636$		0			0
$637$	−7983.93	−	4187.51i	−0.496601	−	0.260463i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			17258.6i			1.06345i	0.846916	+	0.531726i	$0.178456\pi$
$641$			17258.6i			1.06345i	−0.846916	+	0.531726i	$0.821544\pi$
$642$		0			0
$643$		−	25347.2i		−	1.55458i	−0.629141	−	0.777291i	$-0.716593\pi$
$643$		−	25347.2i		−	1.55458i	0.629141	−	0.777291i	$-0.283407\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6799.46			0.413160			0.206580	−	0.978430i	$-0.433767\pi$
$647$	6799.46			0.413160			0.206580	+	0.978430i	$0.433767\pi$
$648$		0			0
$649$		−	37640.9i		−	2.27663i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			22462.2i			1.34612i	0.739590	+	0.673058i	$0.235019\pi$
$653$			22462.2i			1.34612i	−0.739590	+	0.673058i	$0.764981\pi$
$654$		0			0
$655$	3682.84			0.219695
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	3737.15i		−	0.220908i	−0.993881	−	0.110454i	$-0.964769\pi$
$659$		−	3737.15i		−	0.220908i	0.993881	−	0.110454i	$-0.0352305\pi$
$660$		0			0
$661$		−	11784.7i		−	0.693450i	−0.937967	−	0.346725i	$-0.887294\pi$
$661$		−	11784.7i		−	0.693450i	0.937967	−	0.346725i	$-0.112706\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1605.17	−	2654.46i	−0.0936030	−	0.154790i
$666$		0			0
$667$	9577.50			0.555986
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−31200.7			−1.79507
$672$		0			0
$673$	−11424.7			−0.654368			−0.327184	−	0.944961i	$-0.606100\pi$
$673$	−11424.7			−0.654368			−0.327184	+	0.944961i	$0.606100\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	30027.0			1.70462			0.852312	−	0.523034i	$-0.175200\pi$
$677$	30027.0			1.70462			0.852312	+	0.523034i	$0.175200\pi$
$678$		0			0
$679$	−14433.4	−	23868.4i	−0.815763	−	1.34902i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			18482.4i			1.03545i	0.855548	+	0.517723i	$0.173220\pi$
$683$			18482.4i			1.03545i	−0.855548	+	0.517723i	$0.826780\pi$
$684$		0			0
$685$			10502.1i			0.585786i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5513.63			0.304866
$690$		0			0
$691$		−	18946.8i		−	1.04308i	−0.853227	−	0.521540i	$-0.825358\pi$
$691$		−	18946.8i		−	1.04308i	0.853227	−	0.521540i	$-0.174642\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			7764.50i			0.423776i
$696$		0			0
$697$	−805.912			−0.0437964
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	9045.09i		−	0.487344i	−0.969858	−	0.243672i	$-0.921648\pi$
$701$		−	9045.09i		−	0.487344i	0.969858	−	0.243672i	$-0.0783521\pi$
$702$		0			0
$703$			8035.08i			0.431079i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−29271.7	+	17700.8i	−1.55711	+	0.941597i
$708$		0			0
$709$	3733.78			0.197779			0.0988893	−	0.995098i	$-0.468471\pi$
$709$	3733.78			0.197779			0.0988893	+	0.995098i	$0.468471\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	11120.0			0.584078
$714$		0			0
$715$	−7171.28			−0.375092
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−35410.2			−1.83669			−0.918343	−	0.395785i	$-0.870473\pi$
$719$	−35410.2			−1.83669			−0.918343	+	0.395785i	$0.870473\pi$
$720$		0			0
$721$	7707.23	+	12745.4i	0.398103	+	0.658339i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	12232.7i		−	0.626638i
$726$		0			0
$727$		−	2689.21i		−	0.137190i	−0.997645	−	0.0685951i	$-0.978148\pi$
$727$		−	2689.21i		−	0.137190i	0.997645	−	0.0685951i	$-0.0218517\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2186.14			0.110612
$732$		0			0
$733$			12147.9i			0.612133i	0.952010	+	0.306066i	$0.0990130\pi$
$733$			12147.9i			0.612133i	−0.952010	+	0.306066i	$0.900987\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			15592.7i			0.779326i
$738$		0			0
$739$	−33517.3			−1.66841			−0.834204	−	0.551456i	$-0.814072\pi$
$739$	−33517.3			−1.66841			−0.834204	+	0.551456i	$0.814072\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			24689.2i			1.21906i	0.792764	+	0.609529i	$0.208641\pi$
$743$			24689.2i			1.21906i	−0.792764	+	0.609529i	$0.791359\pi$
$744$		0			0
$745$			17315.0i			0.851506i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2712.07	−	4484.93i	−0.132306	−	0.218793i
$750$		0			0
$751$	9589.49			0.465946			0.232973	−	0.972483i	$-0.425155\pi$
$751$	9589.49			0.465946			0.232973	+	0.972483i	$0.425155\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	18487.4			0.891159
$756$		0			0
$757$	−13621.8			−0.654022			−0.327011	−	0.945021i	$-0.606041\pi$
$757$	−13621.8			−0.654022			−0.327011	+	0.945021i	$0.606041\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1679.81			−0.0800173			−0.0400086	−	0.999199i	$-0.512739\pi$
$761$	−1679.81			−0.0800173			−0.0400086	+	0.999199i	$0.512739\pi$
$762$		0			0
$763$	−28945.2	+	17503.4i	−1.37338	+	0.830493i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			21756.9i			1.02425i
$768$		0			0
$769$		−	35443.6i		−	1.66207i	−0.556221	−	0.831034i	$-0.687749\pi$
$769$		−	35443.6i		−	1.66207i	0.556221	−	0.831034i	$-0.312251\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−13901.4			−0.646829			−0.323415	−	0.946257i	$-0.604831\pi$
$773$	−13901.4			−0.646829			−0.323415	+	0.946257i	$0.604831\pi$
$774$		0			0
$775$		−	14202.9i		−	0.658300i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1140.59i		−	0.0524593i
$780$		0			0
$781$	17619.8			0.807281
$782$		0			0
$783$		0			0
$784$		0			0
$785$			7552.58i			0.343393i
$786$		0			0
$787$			4510.99i			0.204319i	0.994768	+	0.102160i	$0.0325753\pi$
$787$			4510.99i			0.204319i	−0.994768	+	0.102160i	$0.967425\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9027.72	+	14929.1i	0.405801	+	0.671070i
$792$		0			0
$793$	18034.4			0.807592
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−31241.2			−1.38848			−0.694240	−	0.719743i	$-0.744259\pi$
$797$	−31241.2			−1.38848			−0.694240	+	0.719743i	$0.744259\pi$
$798$		0			0
$799$	4326.31			0.191557
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10036.9			−0.441089
$804$		0			0
$805$	−4006.71	−	6625.87i	−0.175426	−	0.290101i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			42276.5i			1.83728i	0.395091	+	0.918642i	$0.370713\pi$
$809$			42276.5i			1.83728i	−0.395091	+	0.918642i	$0.629287\pi$
$810$		0			0
$811$			29402.1i			1.27305i	0.771254	+	0.636527i	$0.219630\pi$
$811$			29402.1i			1.27305i	−0.771254	+	0.636527i	$0.780370\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	17441.5			0.749630
$816$		0			0
$817$			3093.99i			0.132491i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	41689.7i		−	1.77221i	−0.463489	−	0.886103i	$-0.653403\pi$
$821$		−	41689.7i		−	1.77221i	0.463489	−	0.886103i	$-0.346597\pi$
$822$		0			0
$823$	−8324.18			−0.352567			−0.176284	−	0.984339i	$-0.556408\pi$
$823$	−8324.18			−0.352567			−0.176284	+	0.984339i	$0.556408\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			7260.76i			0.305298i	0.988280	+	0.152649i	$0.0487804\pi$
$827$			7260.76i			0.305298i	−0.988280	+	0.152649i	$0.951220\pi$
$828$		0			0
$829$		−	26509.9i		−	1.11065i	−0.831634	−	0.555324i	$-0.812594\pi$
$829$		−	26509.9i		−	1.11065i	0.831634	−	0.555324i	$-0.187406\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3142.51	+	5991.54i	−0.130710	+	0.249213i
$834$		0			0
$835$	−3441.26			−0.142623
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−8876.79			−0.365269			−0.182635	−	0.983181i	$-0.558463\pi$
$839$	−8876.79			−0.365269			−0.182635	+	0.983181i	$0.558463\pi$
$840$		0			0
$841$	5497.75			0.225419
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9036.78			−0.367899
$846$		0			0
$847$	11677.1	−	7061.25i	0.473708	−	0.286455i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			20056.6i			0.807908i
$852$		0			0
$853$		−	15206.0i		−	0.610370i	−0.952293	−	0.305185i	$-0.901282\pi$
$853$		−	15206.0i		−	0.610370i	0.952293	−	0.305185i	$-0.0987182\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−23653.8			−0.942820			−0.471410	−	0.881914i	$-0.656255\pi$
$857$	−23653.8			−0.942820			−0.471410	+	0.881914i	$0.656255\pi$
$858$		0			0
$859$			29497.3i			1.17164i	0.810442	+	0.585818i	$0.199227\pi$
$859$			29497.3i			1.17164i	−0.810442	+	0.585818i	$0.800773\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	36862.0i		−	1.45399i	−0.686640	−	0.726997i	$-0.740915\pi$
$863$		−	36862.0i		−	1.45399i	0.686640	−	0.726997i	$-0.259085\pi$
$864$		0			0
$865$	−23173.4			−0.910888
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	22036.4i		−	0.860224i
$870$		0			0
$871$		−	9012.75i		−	0.350615i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−20348.7	+	12305.0i	−0.786183	+	0.475411i
$876$		0			0
$877$	−8707.85			−0.335283			−0.167642	−	0.985848i	$-0.553615\pi$
$877$	−8707.85			−0.335283			−0.167642	+	0.985848i	$0.553615\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18950.2			0.724685			0.362342	−	0.932045i	$-0.381977\pi$
$881$	18950.2			0.724685			0.362342	+	0.932045i	$0.381977\pi$
$882$		0			0
$883$	5510.89			0.210030			0.105015	−	0.994471i	$-0.466511\pi$
$883$	5510.89			0.210030			0.105015	+	0.994471i	$0.466511\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	25305.6			0.957924			0.478962	−	0.877836i	$-0.341013\pi$
$887$	25305.6			0.957924			0.478962	+	0.877836i	$0.341013\pi$
$888$		0			0
$889$	−5080.08	+	3071.97i	−0.191654	+	0.115895i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			6122.93i			0.229447i
$894$		0			0
$895$			13489.4i			0.503801i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−21933.8			−0.813717
$900$		0			0
$901$		−	4137.70i		−	0.152993i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			998.434i			0.0366730i
$906$		0			0
$907$	−18512.8			−0.677738			−0.338869	−	0.940834i	$-0.610044\pi$
$907$	−18512.8			−0.677738			−0.338869	+	0.940834i	$0.610044\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			31568.2i			1.14808i	0.818827	+	0.574040i	$0.194625\pi$
$911$			31568.2i			1.14808i	−0.818827	+	0.574040i	$0.805375\pi$
$912$		0			0
$913$			32224.5i			1.16810i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9727.70	−	5882.42i	0.350313	−	0.211837i
$918$		0			0
$919$	2352.27			0.0844332			0.0422166	−	0.999108i	$-0.486558\pi$
$919$	2352.27			0.0844332			0.0422166	+	0.999108i	$0.486558\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10184.5			−0.363192
$924$		0			0
$925$	25617.0			0.910574
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−9456.80			−0.333980			−0.166990	−	0.985959i	$-0.553405\pi$
$929$	−9456.80			−0.333980			−0.166990	+	0.985959i	$0.553405\pi$
$930$		0			0
$931$	−8479.68	−	4447.52i	−0.298507	−	0.156564i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			5381.68i			0.188235i
$936$		0			0
$937$			34791.9i			1.21302i	0.795075	+	0.606511i	$0.207431\pi$
$937$			34791.9i			1.21302i	−0.795075	+	0.606511i	$0.792569\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11030.8			0.382139			0.191069	−	0.981577i	$-0.438804\pi$
$941$	11030.8			0.382139			0.191069	+	0.981577i	$0.438804\pi$
$942$		0			0
$943$		−	2847.05i		−	0.0983167i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			30487.0i			1.04614i	0.852289	+	0.523071i	$0.175214\pi$
$947$			30487.0i			1.04614i	−0.852289	+	0.523071i	$0.824786\pi$
$948$		0			0
$949$	5801.45			0.198444
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	21378.9i		−	0.726685i	−0.931656	−	0.363343i	$-0.881635\pi$
$953$		−	21378.9i		−	0.726685i	0.931656	−	0.363343i	$-0.118365\pi$
$954$		0			0
$955$		−	12185.7i		−	0.412902i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	16774.4	+	27739.7i	0.564833	+	0.934059i
$960$		0			0
$961$	4324.71			0.145168
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3013.38			0.100523
$966$		0			0
$967$	−3998.63			−0.132975			−0.0664877	−	0.997787i	$-0.521179\pi$
$967$	−3998.63			−0.132975			−0.0664877	+	0.997787i	$0.521179\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−48223.6			−1.59379			−0.796895	−	0.604118i	$-0.793525\pi$
$971$	−48223.6			−1.59379			−0.796895	+	0.604118i	$0.793525\pi$
$972$		0			0
$973$	12401.8	+	20508.8i	0.408618	+	0.675727i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	3646.48i		−	0.119408i	−0.998216	−	0.0597038i	$-0.980984\pi$
$977$		−	3646.48i		−	0.119408i	0.998216	−	0.0597038i	$-0.0190156\pi$
$978$		0			0
$979$		−	6408.52i		−	0.209210i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	20946.9			0.679656			0.339828	−	0.940488i	$-0.389631\pi$
$983$	20946.9			0.679656			0.339828	+	0.940488i	$0.389631\pi$
$984$		0			0
$985$			23694.4i			0.766463i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			7722.99i			0.248308i
$990$		0			0
$991$	6732.72			0.215814			0.107907	−	0.994161i	$-0.465585\pi$
$991$	6732.72			0.215814			0.107907	+	0.994161i	$0.465585\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			3778.22i			0.120379i
$996$		0			0
$997$		−	41524.1i		−	1.31904i	−0.751687	−	0.659520i	$-0.770760\pi$
$997$		−	41524.1i		−	1.31904i	0.751687	−	0.659520i	$-0.229240\pi$
$998$		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.1133.18		48
3.2	odd	2	inner	2268.4.f.a.1133.31		48
7.6	odd	2	inner	2268.4.f.a.1133.32		48
9.2	odd	6		252.4.x.a.41.15	yes	48
9.4	even	3		252.4.x.a.209.10	yes	48
9.5	odd	6		756.4.x.a.629.9		48
9.7	even	3		756.4.x.a.125.16		48
21.20	even	2	inner	2268.4.f.a.1133.17		48
63.13	odd	6		252.4.x.a.209.15	yes	48
63.20	even	6		252.4.x.a.41.10	✓	48
63.34	odd	6		756.4.x.a.125.9		48
63.41	even	6		756.4.x.a.629.16		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.x.a.41.10	✓	48	63.20	even	6
252.4.x.a.41.15	yes	48	9.2	odd	6
252.4.x.a.209.10	yes	48	9.4	even	3
252.4.x.a.209.15	yes	48	63.13	odd	6
756.4.x.a.125.9		48	63.34	odd	6
756.4.x.a.125.16		48	9.7	even	3
756.4.x.a.629.9		48	9.5	odd	6
756.4.x.a.629.16		48	63.41	even	6
2268.4.f.a.1133.17		48	21.20	even	2	inner
2268.4.f.a.1133.18		48	1.1	even	1	trivial
2268.4.f.a.1133.31		48	3.2	odd	2	inner
2268.4.f.a.1133.32		48	7.6	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)

\(f(q)\)	\(=\)	\(q-5.99994 q^{5} +(-15.8480 + 9.58340i) q^{7} +O(q^{10})\)	\(q-5.99994 q^{5} +(-15.8480 + 9.58340i) q^{7} +45.4733i q^{11} -26.2841i q^{13} -19.7249 q^{17} -27.9162i q^{19} -69.6822i q^{23} -89.0007 q^{25} +137.445i q^{29} +159.582i q^{31} +(95.0869 - 57.4998i) q^{35} -287.829 q^{37} +40.8576 q^{41} -110.832 q^{43} -219.333 q^{47} +(159.317 - 303.755i) q^{49} +209.770i q^{53} -272.837i q^{55} -827.759 q^{59} +686.133i q^{61} +157.703i q^{65} +342.897 q^{67} -387.476i q^{71} +220.721i q^{73} +(-435.789 - 720.660i) q^{77} -484.602 q^{79} +708.646 q^{83} +118.348 q^{85} -140.929 q^{89} +(251.891 + 416.550i) q^{91} +167.495i q^{95} +1506.08i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{7}+O(q^{10}) \) 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

Related objects

Downloads

Learn more