LMFDB - Embedded newform 2268.4.f.a.1133.13

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			1133.13
Character	$\chi$	$=$	2268.1133
Dual form			2268.4.f.a.1133.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.3297 q^{5} +(2.73026 + 18.3179i) q^{7} +O(q^{10})$	$q-10.3297 q^{5} +(2.73026 + 18.3179i) q^{7} -31.3278i q^{11} -45.0393i q^{13} +62.4900 q^{17} +132.928i q^{19} -67.8975i q^{23} -18.2972 q^{25} +134.714i q^{29} -30.0123i q^{31} +(-28.2028 - 189.219i) q^{35} +40.4778 q^{37} -79.5029 q^{41} -322.904 q^{43} +342.535 q^{47} +(-328.091 + 100.025i) q^{49} -64.9125i q^{53} +323.607i q^{55} -158.760 q^{59} -570.006i q^{61} +465.243i q^{65} -301.667 q^{67} -719.100i q^{71} -558.706i q^{73} +(573.860 - 85.5331i) q^{77} -913.352 q^{79} +704.697 q^{83} -645.503 q^{85} +700.133 q^{89} +(825.026 - 122.969i) q^{91} -1373.11i q^{95} +234.158i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−10.3297			−0.923917			−0.461958	−	0.886902i	$-0.652853\pi$
$5$	−10.3297			−0.923917			−0.461958	+	0.886902i	$0.652853\pi$
$6$		0			0
$7$	2.73026	+	18.3179i	0.147420	+	0.989074i
$7$	2.73026	+	18.3179i	0.147420	+	0.989074i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	31.3278i		−	0.858700i	−0.903138	−	0.429350i	$-0.858743\pi$
$11$		−	31.3278i		−	0.858700i	0.903138	−	0.429350i	$-0.141257\pi$
$12$		0			0
$13$		−	45.0393i		−	0.960896i	−0.877023	−	0.480448i	$-0.840474\pi$
$13$		−	45.0393i		−	0.960896i	0.877023	−	0.480448i	$-0.159526\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	62.4900			0.891532			0.445766	−	0.895150i	$-0.352931\pi$
$17$	62.4900			0.891532			0.445766	+	0.895150i	$0.352931\pi$
$18$		0			0
$19$			132.928i			1.60504i	0.596624	+	0.802521i	$0.296508\pi$
$19$			132.928i			1.60504i	−0.596624	+	0.802521i	$0.703492\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	67.8975i		−	0.615548i	−0.951459	−	0.307774i	$-0.900416\pi$
$23$		−	67.8975i		−	0.615548i	0.951459	−	0.307774i	$-0.0995841\pi$
$24$		0			0
$25$	−18.2972			−0.146378
$26$		0			0
$27$		0			0
$28$		0			0
$29$			134.714i			0.862610i	0.902206	+	0.431305i	$0.141947\pi$
$29$			134.714i			0.862610i	−0.902206	+	0.431305i	$0.858053\pi$
$30$		0			0
$31$		−	30.0123i		−	0.173883i	−0.996213	−	0.0869413i	$-0.972291\pi$
$31$		−	30.0123i		−	0.173883i	0.996213	−	0.0869413i	$-0.0277093\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−28.2028	−	189.219i	−0.136204	−	0.913822i
$36$		0			0
$37$	40.4778			0.179852			0.0899259	−	0.995948i	$-0.471337\pi$
$37$	40.4778			0.179852			0.0899259	+	0.995948i	$0.471337\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−79.5029			−0.302836			−0.151418	−	0.988470i	$-0.548384\pi$
$41$	−79.5029			−0.302836			−0.151418	+	0.988470i	$0.548384\pi$
$42$		0			0
$43$	−322.904			−1.14517			−0.572586	−	0.819844i	$-0.694060\pi$
$43$	−322.904			−1.14517			−0.572586	+	0.819844i	$0.694060\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	342.535			1.06306			0.531531	−	0.847039i	$-0.321617\pi$
$47$	342.535			1.06306			0.531531	+	0.847039i	$0.321617\pi$
$48$		0			0
$49$	−328.091	+	100.025i	−0.956535	+	0.291619i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	64.9125i		−	0.168234i	−0.996456	−	0.0841172i	$-0.973193\pi$
$53$		−	64.9125i		−	0.168234i	0.996456	−	0.0841172i	$-0.0268070\pi$
$54$		0			0
$55$			323.607i			0.793367i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−158.760			−0.350318			−0.175159	−	0.984540i	$-0.556044\pi$
$59$	−158.760			−0.350318			−0.175159	+	0.984540i	$0.556044\pi$
$60$		0			0
$61$		−	570.006i		−	1.19642i	−0.801338	−	0.598212i	$-0.795878\pi$
$61$		−	570.006i		−	1.19642i	0.801338	−	0.598212i	$-0.204122\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			465.243i			0.887788i
$66$		0			0
$67$	−301.667			−0.550067			−0.275033	−	0.961435i	$-0.588689\pi$
$67$	−301.667			−0.550067			−0.275033	+	0.961435i	$0.588689\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	719.100i		−	1.20199i	−0.799252	−	0.600996i	$-0.794771\pi$
$71$		−	719.100i		−	1.20199i	0.799252	−	0.600996i	$-0.205229\pi$
$72$		0			0
$73$		−	558.706i		−	0.895775i	−0.894090	−	0.447888i	$-0.852176\pi$
$73$		−	558.706i		−	0.895775i	0.894090	−	0.447888i	$-0.147824\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	573.860	−	85.5331i	0.849318	−	0.126590i
$78$		0			0
$79$	−913.352			−1.30076			−0.650381	−	0.759608i	$-0.725391\pi$
$79$	−913.352			−1.30076			−0.650381	+	0.759608i	$0.725391\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	704.697			0.931934			0.465967	−	0.884802i	$-0.345707\pi$
$83$	704.697			0.931934			0.465967	+	0.884802i	$0.345707\pi$
$84$		0			0
$85$	−645.503			−0.823701
$86$		0			0
$87$		0			0
$88$		0			0
$89$	700.133			0.833865			0.416932	−	0.908938i	$-0.363105\pi$
$89$	700.133			0.833865			0.416932	+	0.908938i	$0.363105\pi$
$90$		0			0
$91$	825.026	−	122.969i	0.950398	−	0.141656i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	1373.11i		−	1.48293i
$96$		0			0
$97$			234.158i			0.245104i	0.992462	+	0.122552i	$0.0391079\pi$
$97$			234.158i			0.245104i	−0.992462	+	0.122552i	$0.960892\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1876.20			−1.84840			−0.924200	−	0.381908i	$-0.875267\pi$
$101$	−1876.20			−1.84840			−0.924200	+	0.381908i	$0.875267\pi$
$102$		0			0
$103$		−	605.648i		−	0.579381i	−0.957120	−	0.289691i	$-0.906448\pi$
$103$		−	605.648i		−	0.579381i	0.957120	−	0.289691i	$-0.0935524\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1299.63i		−	1.17420i	−0.809513	−	0.587102i	$-0.800269\pi$
$107$		−	1299.63i		−	1.17420i	0.809513	−	0.587102i	$-0.199731\pi$
$108$		0			0
$109$	2079.24			1.82711			0.913554	−	0.406717i	$-0.133326\pi$
$109$	2079.24			1.82711			0.913554	+	0.406717i	$0.133326\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1259.23i			1.04831i	0.851624	+	0.524153i	$0.175618\pi$
$113$			1259.23i			1.04831i	−0.851624	+	0.524153i	$0.824382\pi$
$114$		0			0
$115$			701.361i			0.568715i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	170.614	+	1144.69i	0.131430	+	0.881791i
$120$		0			0
$121$	349.567			0.262635
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1480.22			1.05916
$126$		0			0
$127$	−1015.62			−0.709623			−0.354811	−	0.934938i	$-0.615455\pi$
$127$	−1015.62			−0.709623			−0.354811	+	0.934938i	$0.615455\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2707.60			1.80583			0.902915	−	0.429818i	$-0.141422\pi$
$131$	2707.60			1.80583			0.902915	+	0.429818i	$0.141422\pi$
$132$		0			0
$133$	−2434.96	+	362.928i	−1.58751	+	0.236616i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			3133.76i			1.95427i	0.212618	+	0.977135i	$0.431801\pi$
$137$			3133.76i			1.95427i	−0.212618	+	0.977135i	$0.568199\pi$
$138$		0			0
$139$			99.9821i			0.0610099i	0.999535	+	0.0305049i	$0.00971153\pi$
$139$			99.9821i			0.0610099i	−0.999535	+	0.0305049i	$0.990288\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1410.98			−0.825121
$144$		0			0
$145$		−	1391.55i		−	0.796980i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			2632.94i			1.44765i	0.689986	+	0.723823i	$0.257617\pi$
$149$			2632.94i			1.44765i	−0.689986	+	0.723823i	$0.742383\pi$
$150$		0			0
$151$	1888.86			1.01797			0.508985	−	0.860775i	$-0.330021\pi$
$151$	1888.86			1.01797			0.508985	+	0.860775i	$0.330021\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			310.018i			0.160653i
$156$		0			0
$157$			1282.19i			0.651785i	0.945407	+	0.325892i	$0.105665\pi$
$157$			1282.19i			0.651785i	−0.945407	+	0.325892i	$0.894335\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1243.74	−	185.378i	0.608822	−	0.0907442i
$162$		0			0
$163$	3558.63			1.71002			0.855011	−	0.518610i	$-0.173550\pi$
$163$	3558.63			1.71002			0.855011	+	0.518610i	$0.173550\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	943.207			0.437051			0.218526	−	0.975831i	$-0.429875\pi$
$167$	943.207			0.437051			0.218526	+	0.975831i	$0.429875\pi$
$168$		0			0
$169$	168.462			0.0766781
$170$		0			0
$171$		0			0
$172$		0			0
$173$	979.587			0.430501			0.215250	−	0.976559i	$-0.430943\pi$
$173$	979.587			0.430501			0.215250	+	0.976559i	$0.430943\pi$
$174$		0			0
$175$	−49.9562	−	335.167i	−0.0215791	−	0.144779i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			3739.96i			1.56166i	0.624742	+	0.780831i	$0.285204\pi$
$179$			3739.96i			1.56166i	−0.624742	+	0.780831i	$0.714796\pi$
$180$		0			0
$181$			1540.52i			0.632631i	0.948654	+	0.316316i	$0.102446\pi$
$181$			1540.52i			0.632631i	−0.948654	+	0.316316i	$0.897554\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−418.124			−0.166168
$186$		0			0
$187$		−	1957.68i		−	0.765558i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			750.177i			0.284193i	0.989853	+	0.142097i	$0.0453844\pi$
$191$			750.177i			0.284193i	−0.989853	+	0.142097i	$0.954616\pi$
$192$		0			0
$193$	−4052.06			−1.51126			−0.755632	−	0.654997i	$-0.772670\pi$
$193$	−4052.06			−1.51126			−0.755632	+	0.654997i	$0.772670\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	807.631i		−	0.292088i	−0.989278	−	0.146044i	$-0.953346\pi$
$197$		−	807.631i		−	0.292088i	0.989278	−	0.146044i	$-0.0466541\pi$
$198$		0			0
$199$			4386.10i			1.56243i	0.624265	+	0.781213i	$0.285399\pi$
$199$			4386.10i			1.56243i	−0.624265	+	0.781213i	$0.714601\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2467.67	+	367.803i	−0.853186	+	0.127166i
$204$		0			0
$205$	821.241			0.279795
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4164.35			1.37825
$210$		0			0
$211$	4265.25			1.39162			0.695811	−	0.718225i	$-0.255045\pi$
$211$	4265.25			1.39162			0.695811	+	0.718225i	$0.255045\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3335.50			1.05804
$216$		0			0
$217$	549.762	−	81.9413i	0.171983	−	0.0256338i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2814.50i		−	0.856670i
$222$		0			0
$223$			2046.06i			0.614413i	0.951643	+	0.307206i	$0.0993942\pi$
$223$			2046.06i			0.614413i	−0.951643	+	0.307206i	$0.900606\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5594.12			1.63566			0.817830	−	0.575459i	$-0.195177\pi$
$227$	5594.12			1.63566			0.817830	+	0.575459i	$0.195177\pi$
$228$		0			0
$229$		−	368.142i		−	0.106234i	−0.998588	−	0.0531168i	$-0.983084\pi$
$229$		−	368.142i		−	0.106234i	0.998588	−	0.0531168i	$-0.0169156\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	3900.42i		−	1.09667i	−0.836258	−	0.548336i	$-0.815261\pi$
$233$		−	3900.42i		−	1.09667i	0.836258	−	0.548336i	$-0.184739\pi$
$234$		0			0
$235$	−3538.29			−0.982180
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	2687.64i		−	0.727402i	−0.931516	−	0.363701i	$-0.881513\pi$
$239$		−	2687.64i		−	0.727402i	0.931516	−	0.363701i	$-0.118487\pi$
$240$		0			0
$241$		−	2454.70i		−	0.656105i	−0.944660	−	0.328052i	$-0.893608\pi$
$241$		−	2454.70i		−	0.656105i	0.944660	−	0.328052i	$-0.106392\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3389.09	−	1033.23i	0.883758	−	0.269432i
$246$		0			0
$247$	5986.99			1.54228
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3096.36			0.778648			0.389324	−	0.921101i	$-0.372709\pi$
$251$	3096.36			0.778648			0.389324	+	0.921101i	$0.372709\pi$
$252$		0			0
$253$	−2127.08			−0.528571
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2622.20			0.636453			0.318227	−	0.948015i	$-0.396913\pi$
$257$	2622.20			0.636453			0.318227	+	0.948015i	$0.396913\pi$
$258$		0			0
$259$	110.515	+	741.469i	0.0265138	+	0.177887i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	7135.59i		−	1.67300i	−0.547967	−	0.836500i	$-0.684598\pi$
$263$		−	7135.59i		−	1.67300i	0.547967	−	0.836500i	$-0.315402\pi$
$264$		0			0
$265$			670.527i			0.155435i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3846.62			0.871869			0.435934	−	0.899978i	$-0.356418\pi$
$269$	3846.62			0.871869			0.435934	+	0.899978i	$0.356418\pi$
$270$		0			0
$271$			446.364i			0.100054i	0.998748	+	0.0500271i	$0.0159308\pi$
$271$			446.364i			0.100054i	−0.998748	+	0.0500271i	$0.984069\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			573.213i			0.125695i
$276$		0			0
$277$	6810.12			1.47719			0.738593	−	0.674151i	$-0.235490\pi$
$277$	6810.12			1.47719			0.738593	+	0.674151i	$0.235490\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	3430.57i		−	0.728295i	−0.931341	−	0.364147i	$-0.881360\pi$
$281$		−	3430.57i		−	0.728295i	0.931341	−	0.364147i	$-0.118640\pi$
$282$		0			0
$283$			317.740i			0.0667409i	0.999443	+	0.0333705i	$0.0106241\pi$
$283$			317.740i			0.0667409i	−0.999443	+	0.0333705i	$0.989376\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−217.064	−	1456.33i	−0.0446441	−	0.299527i
$288$		0			0
$289$	−1008.00			−0.205171
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7403.58			1.47618			0.738091	−	0.674701i	$-0.235727\pi$
$293$	7403.58			1.47618			0.738091	+	0.674701i	$0.235727\pi$
$294$		0			0
$295$	1639.94			0.323665
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3058.05			−0.591478
$300$		0			0
$301$	−881.613	−	5914.93i	−0.168822	−	1.13266i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			5887.99i			1.10540i
$306$		0			0
$307$		−	9692.73i		−	1.80193i	−0.433888	−	0.900967i	$-0.642859\pi$
$307$		−	9692.73i		−	1.80193i	0.433888	−	0.900967i	$-0.357141\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	7914.39			1.44304			0.721518	−	0.692396i	$-0.243445\pi$
$311$	7914.39			1.44304			0.721518	+	0.692396i	$0.243445\pi$
$312$		0			0
$313$		−	5363.45i		−	0.968562i	−0.874913	−	0.484281i	$-0.839081\pi$
$313$		−	5363.45i		−	0.968562i	0.874913	−	0.484281i	$-0.160919\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	4320.07i		−	0.765423i	−0.923868	−	0.382712i	$-0.874990\pi$
$317$		−	4320.07i		−	0.765423i	0.923868	−	0.382712i	$-0.125010\pi$
$318$		0			0
$319$	4220.29			0.740723
$320$		0			0
$321$		0			0
$322$		0			0
$323$			8306.67i			1.43095i
$324$		0			0
$325$			824.095i			0.140654i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	935.210	+	6274.53i	0.156717	+	1.05145i
$330$		0			0
$331$	−743.429			−0.123452			−0.0617259	−	0.998093i	$-0.519660\pi$
$331$	−743.429			−0.123452			−0.0617259	+	0.998093i	$0.519660\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3116.13			0.508216
$336$		0			0
$337$	−9748.78			−1.57582			−0.787908	−	0.615793i	$-0.788836\pi$
$337$	−9748.78			−1.57582			−0.787908	+	0.615793i	$0.788836\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−940.219			−0.149313
$342$		0			0
$343$	−2728.03	−	5736.85i	−0.429445	−	0.903093i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	3410.01i		−	0.527548i	−0.964585	−	0.263774i	$-0.915033\pi$
$347$		−	3410.01i		−	0.527548i	0.964585	−	0.263774i	$-0.0849673\pi$
$348$		0			0
$349$			9419.34i			1.44471i	0.691520	+	0.722357i	$0.256941\pi$
$349$			9419.34i			1.44471i	−0.691520	+	0.722357i	$0.743059\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2726.28			0.411064			0.205532	−	0.978650i	$-0.434108\pi$
$353$	2726.28			0.411064			0.205532	+	0.978650i	$0.434108\pi$
$354$		0			0
$355$			7428.09i			1.11054i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			8978.19i			1.31992i	0.751302	+	0.659959i	$0.229426\pi$
$359$			8978.19i			1.31992i	−0.751302	+	0.659959i	$0.770574\pi$
$360$		0			0
$361$	−10810.9			−1.57616
$362$		0			0
$363$		0			0
$364$		0			0
$365$			5771.27i			0.827622i
$366$		0			0
$367$			7623.04i			1.08425i	0.840298	+	0.542124i	$0.182380\pi$
$367$			7623.04i			1.08425i	−0.840298	+	0.542124i	$0.817620\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1189.06	−	177.228i	0.166396	−	0.0248011i
$372$		0			0
$373$	2510.65			0.348516			0.174258	−	0.984700i	$-0.444247\pi$
$373$	2510.65			0.348516			0.174258	+	0.984700i	$0.444247\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6067.41			0.828879
$378$		0			0
$379$	−7234.68			−0.980529			−0.490265	−	0.871574i	$-0.663100\pi$
$379$	−7234.68			−0.980529			−0.490265	+	0.871574i	$0.663100\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12438.1			1.65942			0.829710	−	0.558195i	$-0.188506\pi$
$383$	12438.1			1.65942			0.829710	+	0.558195i	$0.188506\pi$
$384$		0			0
$385$	−5927.81	+	883.532i	−0.784699	+	0.116958i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			2994.19i			0.390261i	0.980777	+	0.195131i	$0.0625131\pi$
$389$			2994.19i			0.390261i	−0.980777	+	0.195131i	$0.937487\pi$
$390$		0			0
$391$		−	4242.91i		−	0.548781i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9434.66			1.20180
$396$		0			0
$397$			3404.67i			0.430417i	0.976568	+	0.215208i	$0.0690430\pi$
$397$			3404.67i			0.430417i	−0.976568	+	0.215208i	$0.930957\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			3088.04i			0.384561i	0.981340	+	0.192281i	$0.0615884\pi$
$401$			3088.04i			0.384561i	−0.981340	+	0.192281i	$0.938412\pi$
$402$		0			0
$403$	−1351.73			−0.167083
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1268.08i		−	0.154439i
$408$		0			0
$409$			10305.3i			1.24588i	0.782270	+	0.622940i	$0.214062\pi$
$409$			10305.3i			1.24588i	−0.782270	+	0.622940i	$0.785938\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−433.456	−	2908.15i	−0.0516440	−	0.346491i
$414$		0			0
$415$	−7279.31			−0.861029
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11468.7			−1.33719			−0.668596	−	0.743626i	$-0.733104\pi$
$419$	−11468.7			−1.33719			−0.668596	+	0.743626i	$0.733104\pi$
$420$		0			0
$421$	6004.53			0.695113			0.347557	−	0.937659i	$-0.387011\pi$
$421$	6004.53			0.695113			0.347557	+	0.937659i	$0.387011\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1143.39			−0.130501
$426$		0			0
$427$	10441.3	−	1556.27i	1.18335	−	0.176377i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			8312.17i			0.928963i	0.885583	+	0.464481i	$0.153759\pi$
$431$			8312.17i			0.928963i	−0.885583	+	0.464481i	$0.846241\pi$
$432$		0			0
$433$			4419.65i			0.490519i	0.969458	+	0.245259i	$0.0788731\pi$
$433$			4419.65i			0.490519i	−0.969458	+	0.245259i	$0.921127\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9025.48			0.987980
$438$		0			0
$439$		−	16181.4i		−	1.75922i	−0.475695	−	0.879610i	$-0.657803\pi$
$439$		−	16181.4i		−	1.75922i	0.475695	−	0.879610i	$-0.342197\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	3445.02i		−	0.369476i	−0.982788	−	0.184738i	$-0.940856\pi$
$443$		−	3445.02i		−	0.369476i	0.982788	−	0.184738i	$-0.0591436\pi$
$444$		0			0
$445$	−7232.17			−0.770421
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	3345.89i		−	0.351675i	−0.984419	−	0.175838i	$-0.943737\pi$
$449$		−	3345.89i		−	0.351675i	0.984419	−	0.175838i	$-0.0562634\pi$
$450$		0			0
$451$			2490.65i			0.260045i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8522.27	+	1270.23i	−0.878088	+	0.130878i
$456$		0			0
$457$	−5634.45			−0.576737			−0.288368	−	0.957520i	$-0.593113\pi$
$457$	−5634.45			−0.576737			−0.288368	+	0.957520i	$0.593113\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17621.1			1.78025			0.890124	−	0.455719i	$-0.150618\pi$
$461$	17621.1			1.78025			0.890124	+	0.455719i	$0.150618\pi$
$462$		0			0
$463$	−2101.73			−0.210962			−0.105481	−	0.994421i	$-0.533638\pi$
$463$	−2101.73			−0.210962			−0.105481	+	0.994421i	$0.533638\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4439.19			−0.439874			−0.219937	−	0.975514i	$-0.570585\pi$
$467$	−4439.19			−0.439874			−0.219937	+	0.975514i	$0.570585\pi$
$468$		0			0
$469$	−823.629	−	5525.91i	−0.0810910	−	0.544057i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			10115.9i			0.983359i
$474$		0			0
$475$		−	2432.22i		−	0.234943i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2385.54			0.227553			0.113777	−	0.993506i	$-0.463705\pi$
$479$	2385.54			0.227553			0.113777	+	0.993506i	$0.463705\pi$
$480$		0			0
$481$		−	1823.09i		−	0.172819i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	2418.78i		−	0.226456i
$486$		0			0
$487$	−1032.17			−0.0960409			−0.0480205	−	0.998846i	$-0.515291\pi$
$487$	−1032.17			−0.0960409			−0.0480205	+	0.998846i	$0.515291\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	21157.0i		−	1.94461i	−0.233716	−	0.972305i	$-0.575089\pi$
$491$		−	21157.0i		−	1.94461i	0.233716	−	0.972305i	$-0.424911\pi$
$492$		0			0
$493$			8418.25i			0.769045i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	13172.4	−	1963.33i	1.18886	−	0.177198i
$498$		0			0
$499$	−10604.6			−0.951358			−0.475679	−	0.879619i	$-0.657798\pi$
$499$	−10604.6			−0.951358			−0.475679	+	0.879619i	$0.657798\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4721.95			−0.418572			−0.209286	−	0.977855i	$-0.567114\pi$
$503$	−4721.95			−0.418572			−0.209286	+	0.977855i	$0.567114\pi$
$504$		0			0
$505$	19380.5			1.70777
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12591.9			1.09652			0.548259	−	0.836308i	$-0.315291\pi$
$509$	12591.9			1.09652			0.548259	+	0.836308i	$0.315291\pi$
$510$		0			0
$511$	10234.3	−	1525.41i	0.885988	−	0.132055i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			6256.16i			0.535300i
$516$		0			0
$517$		−	10730.9i		−	0.912850i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11584.6			0.974145			0.487072	−	0.873362i	$-0.338065\pi$
$521$	11584.6			0.974145			0.487072	+	0.873362i	$0.338065\pi$
$522$		0			0
$523$			3526.37i			0.294833i	0.989075	+	0.147416i	$0.0470957\pi$
$523$			3526.37i			0.294833i	−0.989075	+	0.147416i	$0.952904\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	1875.47i		−	0.155022i
$528$		0			0
$529$	7556.93			0.621101
$530$		0			0
$531$		0			0
$532$		0			0
$533$			3580.75i			0.290994i
$534$		0			0
$535$			13424.8i			1.08487i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3133.58	+	10278.4i	0.250413	+	0.821376i
$540$		0			0
$541$	7849.84			0.623828			0.311914	−	0.950110i	$-0.399030\pi$
$541$	7849.84			0.623828			0.311914	+	0.950110i	$0.399030\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−21477.9			−1.68810
$546$		0			0
$547$	−15684.5			−1.22600			−0.613001	−	0.790082i	$-0.710038\pi$
$547$	−15684.5			−1.22600			−0.613001	+	0.790082i	$0.710038\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−17907.2			−1.38453
$552$		0			0
$553$	−2493.69	−	16730.7i	−0.191759	−	1.28655i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	21036.4i		−	1.60025i	−0.599833	−	0.800126i	$-0.704766\pi$
$557$		−	21036.4i		−	1.60025i	0.599833	−	0.800126i	$-0.295234\pi$
$558$		0			0
$559$			14543.4i			1.10039i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12694.0			0.950248			0.475124	−	0.879919i	$-0.342403\pi$
$563$	12694.0			0.950248			0.475124	+	0.879919i	$0.342403\pi$
$564$		0			0
$565$		−	13007.5i		−	0.968548i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			18190.1i			1.34019i	0.742277	+	0.670094i	$0.233746\pi$
$569$			18190.1i			1.34019i	−0.742277	+	0.670094i	$0.766254\pi$
$570$		0			0
$571$	20291.9			1.48720			0.743599	−	0.668626i	$-0.233117\pi$
$571$	20291.9			1.48720			0.743599	+	0.668626i	$0.233117\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1242.34i			0.0901026i
$576$		0			0
$577$			12356.1i			0.891494i	0.895159	+	0.445747i	$0.147062\pi$
$577$			12356.1i			0.891494i	−0.895159	+	0.445747i	$0.852938\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1924.01	+	12908.6i	0.137386	+	0.921751i
$582$		0			0
$583$	−2033.57			−0.144463
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2868.85			0.201721			0.100860	−	0.994901i	$-0.467840\pi$
$587$	2868.85			0.201721			0.100860	+	0.994901i	$0.467840\pi$
$588$		0			0
$589$	3989.47			0.279089
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3259.99			0.225754			0.112877	−	0.993609i	$-0.463993\pi$
$593$	3259.99			0.225754			0.112877	+	0.993609i	$0.463993\pi$
$594$		0			0
$595$	−1762.39	−	11824.3i	−0.121430	−	0.814702i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			23060.8i			1.57302i	0.617578	+	0.786509i	$0.288114\pi$
$599$			23060.8i			1.57302i	−0.617578	+	0.786509i	$0.711886\pi$
$600$		0			0
$601$			12502.8i			0.848586i	0.905525	+	0.424293i	$0.139477\pi$
$601$			12502.8i			0.848586i	−0.905525	+	0.424293i	$0.860523\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3610.92			−0.242653
$606$		0			0
$607$			7665.84i			0.512598i	0.966598	+	0.256299i	$0.0825031\pi$
$607$			7665.84i			0.512598i	−0.966598	+	0.256299i	$0.917497\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	15427.5i		−	1.02149i
$612$		0			0
$613$	20585.4			1.35634			0.678170	−	0.734905i	$-0.262773\pi$
$613$	20585.4			1.35634			0.678170	+	0.734905i	$0.262773\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	16367.0i		−	1.06793i	−0.845508	−	0.533963i	$-0.820702\pi$
$617$		−	16367.0i		−	1.06793i	0.845508	−	0.533963i	$-0.179298\pi$
$618$		0			0
$619$		−	6993.79i		−	0.454126i	−0.973880	−	0.227063i	$-0.927088\pi$
$619$		−	6993.79i		−	0.454126i	0.973880	−	0.227063i	$-0.0729124\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1911.55	+	12825.0i	0.122928	+	0.824754i
$624$		0			0
$625$	−13003.1			−0.832196
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2529.46			0.160344
$630$		0			0
$631$	8776.78			0.553721			0.276861	−	0.960910i	$-0.410706\pi$
$631$	8776.78			0.553721			0.276861	+	0.960910i	$0.410706\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10491.1			0.655633
$636$		0			0
$637$	4505.07	+	14777.0i	0.280216	+	0.919131i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			2958.45i			0.182296i	0.995837	+	0.0911481i	$0.0290537\pi$
$641$			2958.45i			0.182296i	−0.995837	+	0.0911481i	$0.970946\pi$
$642$		0			0
$643$		−	13383.7i		−	0.820842i	−0.911896	−	0.410421i	$-0.865382\pi$
$643$		−	13383.7i		−	0.820842i	0.911896	−	0.410421i	$-0.134618\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−20353.0			−1.23672			−0.618362	−	0.785893i	$-0.712203\pi$
$647$	−20353.0			−1.23672			−0.618362	+	0.785893i	$0.712203\pi$
$648$		0			0
$649$			4973.61i			0.300818i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	15618.2i		−	0.935970i	−0.883736	−	0.467985i	$-0.844980\pi$
$653$		−	15618.2i		−	0.935970i	0.883736	−	0.467985i	$-0.155020\pi$
$654$		0			0
$655$	−27968.7			−1.66844
$656$		0			0
$657$		0			0
$658$		0			0
$659$			6752.47i			0.399149i	0.979883	+	0.199574i	$0.0639559\pi$
$659$			6752.47i			0.399149i	−0.979883	+	0.199574i	$0.936044\pi$
$660$		0			0
$661$		−	9998.14i		−	0.588325i	−0.955755	−	0.294162i	$-0.904959\pi$
$661$		−	9998.14i		−	0.588325i	0.955755	−	0.294162i	$-0.0950407\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	25152.5	−	3748.94i	1.46672	−	0.218613i
$666$		0			0
$667$	9146.72			0.530978
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−17857.1			−1.02737
$672$		0			0
$673$	63.4176			0.00363235			0.00181617	−	0.999998i	$-0.499422\pi$
$673$	63.4176			0.00363235			0.00181617	+	0.999998i	$0.499422\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−17748.7			−1.00759			−0.503793	−	0.863824i	$-0.668063\pi$
$677$	−17748.7			−1.00759			−0.503793	+	0.863824i	$0.668063\pi$
$678$		0			0
$679$	−4289.28	+	639.312i	−0.242426	+	0.0361333i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	8356.42i		−	0.468155i	−0.972218	−	0.234077i	$-0.924793\pi$
$683$		−	8356.42i		−	0.468155i	0.972218	−	0.234077i	$-0.0752069\pi$
$684$		0			0
$685$		−	32370.8i		−	1.80558i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2923.61			−0.161656
$690$		0			0
$691$		−	29142.1i		−	1.60437i	−0.597076	−	0.802185i	$-0.703671\pi$
$691$		−	29142.1i		−	1.60437i	0.597076	−	0.802185i	$-0.296329\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	1032.79i		−	0.0563681i
$696$		0			0
$697$	−4968.13			−0.269988
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	30359.4i		−	1.63574i	−0.575400	−	0.817872i	$-0.695154\pi$
$701$		−	30359.4i		−	1.63574i	0.575400	−	0.817872i	$-0.304846\pi$
$702$		0			0
$703$			5380.64i			0.288670i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5122.50	−	34368.0i	−0.272492	−	1.82820i
$708$		0			0
$709$	12127.7			0.642403			0.321202	−	0.947011i	$-0.395913\pi$
$709$	12127.7			0.642403			0.321202	+	0.947011i	$0.395913\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2037.76			−0.107033
$714$		0			0
$715$	14575.0			0.762344
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9928.20			0.514964			0.257482	−	0.966283i	$-0.417107\pi$
$719$	9928.20			0.514964			0.257482	+	0.966283i	$0.417107\pi$
$720$		0			0
$721$	11094.2	−	1653.58i	0.573051	−	0.0854125i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	2464.89i		−	0.126267i
$726$		0			0
$727$			30312.8i			1.54641i	0.634157	+	0.773204i	$0.281347\pi$
$727$			30312.8i			1.54641i	−0.634157	+	0.773204i	$0.718653\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−20178.3			−1.02096
$732$		0			0
$733$			18873.3i			0.951027i	0.879709	+	0.475513i	$0.157738\pi$
$733$			18873.3i			0.951027i	−0.879709	+	0.475513i	$0.842262\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			9450.57i			0.472342i
$738$		0			0
$739$	24825.4			1.23575			0.617873	−	0.786278i	$-0.287995\pi$
$739$	24825.4			1.23575			0.617873	+	0.786278i	$0.287995\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			26095.2i			1.28848i	0.764823	+	0.644241i	$0.222826\pi$
$743$			26095.2i			1.28848i	−0.764823	+	0.644241i	$0.777174\pi$
$744$		0			0
$745$		−	27197.5i		−	1.33750i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	23806.5	−	3548.33i	1.16138	−	0.173102i
$750$		0			0
$751$	−26422.4			−1.28384			−0.641922	−	0.766770i	$-0.721863\pi$
$751$	−26422.4			−1.28384			−0.641922	+	0.766770i	$0.721863\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−19511.4			−0.940520
$756$		0			0
$757$	28263.9			1.35703			0.678513	−	0.734588i	$-0.262625\pi$
$757$	28263.9			1.35703			0.678513	+	0.734588i	$0.262625\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	38390.9			1.82874			0.914369	−	0.404883i	$-0.132688\pi$
$761$	38390.9			1.82874			0.914369	+	0.404883i	$0.132688\pi$
$762$		0			0
$763$	5676.86	+	38087.3i	0.269353	+	1.80715i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			7150.44i			0.336620i
$768$		0			0
$769$			30969.4i			1.45226i	0.687558	+	0.726129i	$0.258683\pi$
$769$			30969.4i			1.45226i	−0.687558	+	0.726129i	$0.741317\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−28691.6			−1.33501			−0.667507	−	0.744604i	$-0.732638\pi$
$773$	−28691.6			−1.33501			−0.667507	+	0.744604i	$0.732638\pi$
$774$		0			0
$775$			549.142i			0.0254526i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	10568.2i		−	0.486064i
$780$		0			0
$781$	−22527.8			−1.03215
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	13244.7i		−	0.602195i
$786$		0			0
$787$			22699.3i			1.02814i	0.857749	+	0.514068i	$0.171862\pi$
$787$			22699.3i			1.02814i	−0.857749	+	0.514068i	$0.828138\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−23066.5	+	3438.03i	−1.03685	+	0.154542i
$792$		0			0
$793$	−25672.7			−1.14964
$794$		0			0
$795$		0			0
$796$		0			0
$797$	9891.20			0.439604			0.219802	−	0.975545i	$-0.429459\pi$
$797$	9891.20			0.439604			0.219802	+	0.975545i	$0.429459\pi$
$798$		0			0
$799$	21405.0			0.947753
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−17503.1			−0.769202
$804$		0			0
$805$	−12847.5	+	1914.90i	−0.562501	+	0.0838401i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	15328.4i		−	0.666155i	−0.942900	−	0.333077i	$-0.891913\pi$
$809$		−	15328.4i		−	0.666155i	0.942900	−	0.333077i	$-0.108087\pi$
$810$		0			0
$811$		−	4463.96i		−	0.193281i	−0.995319	−	0.0966406i	$-0.969190\pi$
$811$		−	4463.96i		−	0.193281i	0.995319	−	0.0966406i	$-0.0308097\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−36759.6			−1.57992
$816$		0			0
$817$		−	42923.0i		−	1.83805i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	15773.9i		−	0.670539i	−0.942122	−	0.335269i	$-0.891173\pi$
$821$		−	15773.9i		−	0.670539i	0.942122	−	0.335269i	$-0.108827\pi$
$822$		0			0
$823$	1921.62			0.0813894			0.0406947	−	0.999172i	$-0.487043\pi$
$823$	1921.62			0.0813894			0.0406947	+	0.999172i	$0.487043\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	28140.7i		−	1.18325i	−0.806214	−	0.591624i	$-0.798487\pi$
$827$		−	28140.7i		−	1.18325i	0.806214	−	0.591624i	$-0.201513\pi$
$828$		0			0
$829$			13305.9i			0.557460i	0.960370	+	0.278730i	$0.0899134\pi$
$829$			13305.9i			0.557460i	−0.960370	+	0.278730i	$0.910087\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−20502.4	+	6250.58i	−0.852781	+	0.259988i
$834$		0			0
$835$	−9743.05			−0.403799
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−17826.4			−0.733535			−0.366767	−	0.930313i	$-0.619536\pi$
$839$	−17826.4			−0.733535			−0.366767	+	0.930313i	$0.619536\pi$
$840$		0			0
$841$	6241.22			0.255903
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1740.16			−0.0708442
$846$		0			0
$847$	954.409	+	6403.33i	0.0387177	+	0.259765i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	2748.34i		−	0.110707i
$852$		0			0
$853$			40919.9i			1.64252i	0.570553	+	0.821261i	$0.306729\pi$
$853$			40919.9i			1.64252i	−0.570553	+	0.821261i	$0.693271\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−28162.9			−1.12255			−0.561275	−	0.827629i	$-0.689689\pi$
$857$	−28162.9			−1.12255			−0.561275	+	0.827629i	$0.689689\pi$
$858$		0			0
$859$		−	3591.66i		−	0.142661i	−0.997453	−	0.0713304i	$-0.977276\pi$
$859$		−	3591.66i		−	0.142661i	0.997453	−	0.0713304i	$-0.0227245\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			18091.2i			0.713593i	0.934182	+	0.356796i	$0.116131\pi$
$863$			18091.2i			0.713593i	−0.934182	+	0.356796i	$0.883869\pi$
$864$		0			0
$865$	−10118.8			−0.397747
$866$		0			0
$867$		0			0
$868$		0			0
$869$			28613.3i			1.11696i
$870$		0			0
$871$			13586.9i			0.528557i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4041.38	+	27114.5i	0.156141	+	1.04759i
$876$		0			0
$877$	46430.6			1.78774			0.893871	−	0.448323i	$-0.147979\pi$
$877$	46430.6			1.78774			0.893871	+	0.448323i	$0.147979\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	5007.33			0.191488			0.0957441	−	0.995406i	$-0.469477\pi$
$881$	5007.33			0.191488			0.0957441	+	0.995406i	$0.469477\pi$
$882$		0			0
$883$	−49999.0			−1.90555			−0.952775	−	0.303677i	$-0.901786\pi$
$883$	−49999.0			−1.90555			−0.952775	+	0.303677i	$0.901786\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3460.82			−0.131007			−0.0655034	−	0.997852i	$-0.520865\pi$
$887$	−3460.82			−0.131007			−0.0655034	+	0.997852i	$0.520865\pi$
$888$		0			0
$889$	−2772.92	−	18604.1i	−0.104613	−	0.701870i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			45532.5i			1.70626i
$894$		0			0
$895$		−	38632.7i		−	1.44285i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4043.06			0.149993
$900$		0			0
$901$		−	4056.38i		−	0.149986i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	15913.2i		−	0.584498i
$906$		0			0
$907$	−29093.9			−1.06510			−0.532550	−	0.846398i	$-0.678766\pi$
$907$	−29093.9			−1.06510			−0.532550	+	0.846398i	$0.678766\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	33227.9i		−	1.20844i	−0.796818	−	0.604220i	$-0.793485\pi$
$911$		−	33227.9i		−	1.20844i	0.796818	−	0.604220i	$-0.206515\pi$
$912$		0			0
$913$		−	22076.6i		−	0.800251i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7392.44	+	49597.5i	0.266216	+	1.78610i
$918$		0			0
$919$	−11642.5			−0.417902			−0.208951	−	0.977926i	$-0.567005\pi$
$919$	−11642.5			−0.417902			−0.208951	+	0.977926i	$0.567005\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−32387.7			−1.15499
$924$		0			0
$925$	−740.633			−0.0263263
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−38113.9			−1.34605			−0.673024	−	0.739621i	$-0.735005\pi$
$929$	−38113.9			−1.34605			−0.673024	+	0.739621i	$0.735005\pi$
$930$		0			0
$931$	−13296.2	−	43612.6i	−0.468061	−	1.53528i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			20222.2i			0.707312i
$936$		0			0
$937$			22454.2i			0.782869i	0.920206	+	0.391435i	$0.128021\pi$
$937$			22454.2i			0.782869i	−0.920206	+	0.391435i	$0.871979\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12319.8			−0.426796			−0.213398	−	0.976965i	$-0.568453\pi$
$941$	−12319.8			−0.426796			−0.213398	+	0.976965i	$0.568453\pi$
$942$		0			0
$943$			5398.05i			0.186410i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	15468.9i		−	0.530805i	−0.964138	−	0.265403i	$-0.914495\pi$
$947$		−	15468.9i		−	0.530805i	0.964138	−	0.265403i	$-0.0855049\pi$
$948$		0			0
$949$	−25163.7			−0.860747
$950$		0			0
$951$		0			0
$952$		0			0
$953$			5267.67i			0.179052i	0.995984	+	0.0895260i	$0.0285352\pi$
$953$			5267.67i			0.179052i	−0.995984	+	0.0895260i	$0.971465\pi$
$954$		0			0
$955$		−	7749.10i		−	0.262571i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−57403.9	+	8555.98i	−1.93292	+	0.288099i
$960$		0			0
$961$	28890.3			0.969765
$962$		0			0
$963$		0			0
$964$		0			0
$965$	41856.6			1.39628
$966$		0			0
$967$	−12474.2			−0.414832			−0.207416	−	0.978253i	$-0.566505\pi$
$967$	−12474.2			−0.414832			−0.207416	+	0.978253i	$0.566505\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−57914.6			−1.91407			−0.957037	−	0.289964i	$-0.906357\pi$
$971$	−57914.6			−1.91407			−0.957037	+	0.289964i	$0.906357\pi$
$972$		0			0
$973$	−1831.46	+	272.977i	−0.0603433	+	0.00899409i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			18423.0i			0.603278i	0.953422	+	0.301639i	$0.0975337\pi$
$977$			18423.0i			0.603278i	−0.953422	+	0.301639i	$0.902466\pi$
$978$		0			0
$979$		−	21933.6i		−	0.716039i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	8587.39			0.278632			0.139316	−	0.990248i	$-0.455510\pi$
$983$	8587.39			0.278632			0.139316	+	0.990248i	$0.455510\pi$
$984$		0			0
$985$			8342.59i			0.269865i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			21924.4i			0.704909i
$990$		0			0
$991$	−17639.5			−0.565426			−0.282713	−	0.959205i	$-0.591234\pi$
$991$	−17639.5			−0.565426			−0.282713	+	0.959205i	$0.591234\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	45307.2i		−	1.44355i
$996$		0			0
$997$		−	4827.63i		−	0.153353i	−0.997056	−	0.0766764i	$-0.975569\pi$
$997$		−	4827.63i		−	0.153353i	0.997056	−	0.0766764i	$-0.0244308\pi$
$998$		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.1133.13		48
3.2	odd	2	inner	2268.4.f.a.1133.36		48
7.6	odd	2	inner	2268.4.f.a.1133.35		48
9.2	odd	6		252.4.x.a.41.4	✓	48
9.4	even	3		252.4.x.a.209.21	yes	48
9.5	odd	6		756.4.x.a.629.7		48
9.7	even	3		756.4.x.a.125.18		48
21.20	even	2	inner	2268.4.f.a.1133.14		48
63.13	odd	6		252.4.x.a.209.4	yes	48
63.20	even	6		252.4.x.a.41.21	yes	48
63.34	odd	6		756.4.x.a.125.7		48
63.41	even	6		756.4.x.a.629.18		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.x.a.41.4	✓	48	9.2	odd	6
252.4.x.a.41.21	yes	48	63.20	even	6
252.4.x.a.209.4	yes	48	63.13	odd	6
252.4.x.a.209.21	yes	48	9.4	even	3
756.4.x.a.125.7		48	63.34	odd	6
756.4.x.a.125.18		48	9.7	even	3
756.4.x.a.629.7		48	9.5	odd	6
756.4.x.a.629.18		48	63.41	even	6
2268.4.f.a.1133.13		48	1.1	even	1	trivial
2268.4.f.a.1133.14		48	21.20	even	2	inner
2268.4.f.a.1133.35		48	7.6	odd	2	inner
2268.4.f.a.1133.36		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)

\(f(q)\)	\(=\)	\(q-10.3297 q^{5} +(2.73026 + 18.3179i) q^{7} +O(q^{10})\)	\(q-10.3297 q^{5} +(2.73026 + 18.3179i) q^{7} -31.3278i q^{11} -45.0393i q^{13} +62.4900 q^{17} +132.928i q^{19} -67.8975i q^{23} -18.2972 q^{25} +134.714i q^{29} -30.0123i q^{31} +(-28.2028 - 189.219i) q^{35} +40.4778 q^{37} -79.5029 q^{41} -322.904 q^{43} +342.535 q^{47} +(-328.091 + 100.025i) q^{49} -64.9125i q^{53} +323.607i q^{55} -158.760 q^{59} -570.006i q^{61} +465.243i q^{65} -301.667 q^{67} -719.100i q^{71} -558.706i q^{73} +(573.860 - 85.5331i) q^{77} -913.352 q^{79} +704.697 q^{83} -645.503 q^{85} +700.133 q^{89} +(825.026 - 122.969i) q^{91} -1373.11i q^{95} +234.158i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{7}+O(q^{10}) \) 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