LMFDB - Embedded newform 288.4.c.b.287.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,4,Mod(287,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("288.287");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	288.c (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:	$16.9925500817$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.77720518656.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 161x^{4} + 4096 $ x^8 + 161*x^4 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.4
Root			$-2.38456 + 2.38456i$ of defining polynomial
Character	$\chi$	$=$	288.287
Dual form			288.4.c.b.287.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-14.8339i q^{5} +30.9783i q^{7} +O(q^{10})$	$q-14.8339i q^{5} +30.9783i q^{7} -9.86874 q^{11} -54.9783 q^{13} +79.1652i q^{17} -43.9130i q^{19} +108.802 q^{23} -95.0435 q^{25} +231.116i q^{29} +318.978i q^{31} +459.527 q^{35} -38.1740 q^{37} -110.093i q^{41} +401.696i q^{43} -546.901 q^{47} -616.652 q^{49} +50.4660i q^{53} +146.391i q^{55} +500.693 q^{59} -865.478 q^{61} +815.540i q^{65} -306.043i q^{67} -194.024 q^{71} +867.043 q^{73} -305.716i q^{77} +816.674i q^{79} +1146.53 q^{83} +1174.33 q^{85} -468.043i q^{89} -1703.13i q^{91} -651.399 q^{95} +238.652 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 256 q^{13} - 1128 q^{25} - 1776 q^{37} - 1992 q^{49} - 2512 q^{61} - 1152 q^{73} + 3696 q^{85} + 7424 q^{97}+O(q^{100}) $ 8 * q - 256 * q^13 - 1128 * q^25 - 1776 * q^37 - 1992 * q^49 - 2512 * q^61 - 1152 * q^73 + 3696 * q^85 + 7424 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	− 14.8339i	− 1.32678i	−0.748273	−	0.663391i	$-0.769117\pi$
$5$	− 14.8339i	− 1.32678i	0.748273	−	0.663391i	$-0.230883\pi$
$6$	0	0
$7$	30.9783i	1.67267i	0.548220	+	0.836334i	$0.315306\pi$
$7$	30.9783i	1.67267i	−0.548220	+	0.836334i	$0.684694\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−9.86874	−0.270503	−0.135252	−	0.990811i	$-0.543184\pi$
$11$	−9.86874	−0.270503	−0.135252	+	0.990811i	$0.543184\pi$
$12$	0	0
$13$	−54.9783	−1.17294	−0.586470	−	0.809971i	$-0.699483\pi$
$13$	−54.9783	−1.17294	−0.586470	+	0.809971i	$0.699483\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	79.1652i	1.12943i	0.825285	+	0.564717i	$0.191015\pi$
$17$	79.1652i	1.12943i	−0.825285	+	0.564717i	$0.808985\pi$
$18$	0	0
$19$	− 43.9130i	− 0.530228i	−0.964217	−	0.265114i	$-0.914590\pi$
$19$	− 43.9130i	− 0.530228i	0.964217	−	0.265114i	$-0.0854096\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	108.802	0.986384	0.493192	−	0.869921i	$-0.335830\pi$
$23$	108.802	0.986384	0.493192	+	0.869921i	$0.335830\pi$
$24$	0	0
$25$	−95.0435	−0.760348
$26$	0	0
$27$	0	0
$28$	0	0
$29$	231.116i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$29$	231.116i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$30$	0	0
$31$	318.978i	1.84807i	0.382307	+	0.924035i	$0.375130\pi$
$31$	318.978i	1.84807i	−0.382307	+	0.924035i	$0.624870\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	459.527	2.21926
$36$	0	0
$37$	−38.1740	−0.169615	−0.0848077	−	0.996397i	$-0.527028\pi$
$37$	−38.1740	−0.169615	−0.0848077	+	0.996397i	$0.527028\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 110.093i	− 0.419358i	−0.977770	−	0.209679i	$-0.932758\pi$
$41$	− 110.093i	− 0.419358i	0.977770	−	0.209679i	$-0.0672420\pi$
$42$	0	0
$43$	401.696i	1.42460i	0.701873	+	0.712302i	$0.252347\pi$
$43$	401.696i	1.42460i	−0.701873	+	0.712302i	$0.747653\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−546.901	−1.69731	−0.848656	−	0.528945i	$-0.822588\pi$
$47$	−546.901	−1.69731	−0.848656	+	0.528945i	$0.822588\pi$
$48$	0	0
$49$	−616.652	−1.79782
$50$	0	0
$51$	0	0
$52$	0	0
$53$	50.4660i	0.130793i	0.997859	+	0.0653966i	$0.0208313\pi$
$53$	50.4660i	0.130793i	−0.997859	+	0.0653966i	$0.979169\pi$
$54$	0	0
$55$	146.391i	0.358899i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	500.693	1.10483	0.552413	−	0.833571i	$-0.313707\pi$
$59$	500.693	1.10483	0.552413	+	0.833571i	$0.313707\pi$
$60$	0	0
$61$	−865.478	−1.81661	−0.908304	−	0.418310i	$-0.862622\pi$
$61$	−865.478	−1.81661	−0.908304	+	0.418310i	$0.862622\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	815.540i	1.55623i
$66$	0	0
$67$	− 306.043i	− 0.558047i	−0.960284	−	0.279024i	$-0.909989\pi$
$67$	− 306.043i	− 0.558047i	0.960284	−	0.279024i	$-0.0900108\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−194.024	−0.324316	−0.162158	−	0.986765i	$-0.551845\pi$
$71$	−194.024	−0.324316	−0.162158	+	0.986765i	$0.551845\pi$
$72$	0	0
$73$	867.043	1.39013	0.695067	−	0.718945i	$-0.255375\pi$
$73$	867.043	1.39013	0.695067	+	0.718945i	$0.255375\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 305.716i	− 0.452462i
$78$	0	0
$79$	816.674i	1.16308i	0.813519	+	0.581538i	$0.197549\pi$
$79$	816.674i	1.16308i	−0.813519	+	0.581538i	$0.802451\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1146.53	1.51624	0.758119	−	0.652116i	$-0.226119\pi$
$83$	1146.53	1.51624	0.758119	+	0.652116i	$0.226119\pi$
$84$	0	0
$85$	1174.33	1.49851
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 468.043i	− 0.557444i	−0.960372	−	0.278722i	$-0.910089\pi$
$89$	− 468.043i	− 0.557444i	0.960372	−	0.278722i	$-0.0899107\pi$
$90$	0	0
$91$	− 1703.13i	− 1.96194i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−651.399	−0.703497
$96$	0	0
$97$	238.652	0.249809	0.124905	−	0.992169i	$-0.460138\pi$
$97$	238.652	0.249809	0.124905	+	0.992169i	$0.460138\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	94.2144i	0.0928186i	0.998923	+	0.0464093i	$0.0147778\pi$
$101$	94.2144i	0.0928186i	−0.998923	+	0.0464093i	$0.985222\pi$
$102$	0	0
$103$	792.152i	0.757796i	0.925438	+	0.378898i	$0.123697\pi$
$103$	792.152i	0.757796i	−0.925438	+	0.378898i	$0.876303\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−496.819	−0.448872	−0.224436	−	0.974489i	$-0.572054\pi$
$107$	−496.819	−0.448872	−0.224436	+	0.974489i	$0.572054\pi$
$108$	0	0
$109$	131.239	0.115325	0.0576626	−	0.998336i	$-0.481635\pi$
$109$	131.239	0.115325	0.0576626	+	0.998336i	$0.481635\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1253.85i	− 1.04383i	−0.852998	−	0.521914i	$-0.825218\pi$
$113$	− 1253.85i	− 1.04383i	0.852998	−	0.521914i	$-0.174782\pi$
$114$	0	0
$115$	− 1613.96i	− 1.30871i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2452.40	−1.88917
$120$	0	0
$121$	−1233.61	−0.926828
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 444.371i	− 0.317966i
$126$	0	0
$127$	− 1428.76i	− 0.998284i	−0.866520	−	0.499142i	$-0.833649\pi$
$127$	− 1428.76i	− 0.998284i	0.866520	−	0.499142i	$-0.166351\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2626.19	−1.75154	−0.875770	−	0.482729i	$-0.839645\pi$
$131$	−2626.19	−1.75154	−0.875770	+	0.482729i	$0.839645\pi$
$132$	0	0
$133$	1360.35	0.886896
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2199.81i	− 1.37184i	−0.727676	−	0.685921i	$-0.759400\pi$
$137$	− 2199.81i	− 1.37184i	0.727676	−	0.685921i	$-0.240600\pi$
$138$	0	0
$139$	− 686.652i	− 0.419001i	−0.977809	−	0.209500i	$-0.932816\pi$
$139$	− 686.652i	− 0.419001i	0.977809	−	0.209500i	$-0.0671837\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	542.566	0.317284
$144$	0	0
$145$	3428.35	1.96351
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 440.850i	− 0.242388i	−0.992629	−	0.121194i	$-0.961328\pi$
$149$	− 440.850i	− 0.242388i	0.992629	−	0.121194i	$-0.0386724\pi$
$150$	0	0
$151$	− 860.239i	− 0.463611i	−0.972762	−	0.231805i	$-0.925537\pi$
$151$	− 860.239i	− 0.463611i	0.972762	−	0.231805i	$-0.0744633\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4731.68	2.45198
$156$	0	0
$157$	1408.78	0.716134	0.358067	−	0.933696i	$-0.383436\pi$
$157$	1408.78	0.716134	0.358067	+	0.933696i	$0.383436\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3370.50i	1.64989i
$162$	0	0
$163$	2608.30i	1.25336i	0.779276	+	0.626681i	$0.215587\pi$
$163$	2608.30i	1.25336i	−0.779276	+	0.626681i	$0.784413\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1473.67	−0.682851	−0.341426	−	0.939909i	$-0.610910\pi$
$167$	−1473.67	−0.682851	−0.341426	+	0.939909i	$0.610910\pi$
$168$	0	0
$169$	825.608	0.375789
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3158.69i	− 1.38816i	−0.719900	−	0.694078i	$-0.755812\pi$
$173$	− 3158.69i	− 1.38816i	0.719900	−	0.694078i	$-0.244188\pi$
$174$	0	0
$175$	− 2944.28i	− 1.27181i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−62.1024	−0.0259316	−0.0129658	−	0.999916i	$-0.504127\pi$
$179$	−62.1024	−0.0259316	−0.0129658	+	0.999916i	$0.504127\pi$
$180$	0	0
$181$	−2450.50	−1.00632	−0.503161	−	0.864193i	$-0.667830\pi$
$181$	−2450.50	−1.00632	−0.503161	+	0.864193i	$0.667830\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	566.268i	0.225042i
$186$	0	0
$187$	− 781.261i	− 0.305516i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1418.83	0.537501	0.268750	−	0.963210i	$-0.413389\pi$
$191$	1418.83	0.537501	0.268750	+	0.963210i	$0.413389\pi$
$192$	0	0
$193$	941.652	0.351200	0.175600	−	0.984462i	$-0.443813\pi$
$193$	941.652	0.351200	0.175600	+	0.984462i	$0.443813\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5158.88i	1.86576i	0.360185	+	0.932881i	$0.382714\pi$
$197$	5158.88i	1.86576i	−0.360185	+	0.932881i	$0.617286\pi$
$198$	0	0
$199$	− 44.6312i	− 0.0158986i	−0.999968	−	0.00794930i	$-0.997470\pi$
$199$	− 44.6312i	− 0.0158986i	0.999968	−	0.00794930i	$-0.00253037\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7159.58	−2.47539
$204$	0	0
$205$	−1633.11	−0.556397
$206$	0	0
$207$	0	0
$208$	0	0
$209$	433.366i	0.143428i
$210$	0	0
$211$	1591.52i	0.519265i	0.965707	+	0.259633i	$0.0836014\pi$
$211$	1591.52i	0.519265i	−0.965707	+	0.259633i	$0.916399\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5958.70	1.89014
$216$	0	0
$217$	−9881.39	−3.09121
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 4352.36i	− 1.32476i
$222$	0	0
$223$	3940.37i	1.18326i	0.806210	+	0.591629i	$0.201515\pi$
$223$	3940.37i	1.18326i	−0.806210	+	0.591629i	$0.798485\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1123.41	−0.328472	−0.164236	−	0.986421i	$-0.552516\pi$
$227$	−1123.41	−0.328472	−0.164236	+	0.986421i	$0.552516\pi$
$228$	0	0
$229$	−221.760	−0.0639927	−0.0319964	−	0.999488i	$-0.510186\pi$
$229$	−221.760	−0.0639927	−0.0319964	+	0.999488i	$0.510186\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3109.18i	− 0.874202i	−0.899412	−	0.437101i	$-0.856005\pi$
$233$	− 3109.18i	− 0.874202i	0.899412	−	0.437101i	$-0.143995\pi$
$234$	0	0
$235$	8112.65i	2.25196i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2871.71	0.777221	0.388610	−	0.921402i	$-0.372955\pi$
$239$	2871.71	0.777221	0.388610	+	0.921402i	$0.372955\pi$
$240$	0	0
$241$	1802.82	0.481868	0.240934	−	0.970542i	$-0.422546\pi$
$241$	1802.82	0.481868	0.240934	+	0.970542i	$0.422546\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9147.33i	2.38531i
$246$	0	0
$247$	2414.26i	0.621926i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	707.015	0.177794	0.0888972	−	0.996041i	$-0.471666\pi$
$251$	707.015	0.177794	0.0888972	+	0.996041i	$0.471666\pi$
$252$	0	0
$253$	−1073.74	−0.266820
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1564.18i	− 0.379653i	−0.981818	−	0.189827i	$-0.939207\pi$
$257$	− 1564.18i	− 0.379653i	0.981818	−	0.189827i	$-0.0607926\pi$
$258$	0	0
$259$	− 1182.56i	− 0.283710i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8146.79	1.91009	0.955043	−	0.296467i	$-0.0958084\pi$
$263$	8146.79	1.91009	0.955043	+	0.296467i	$0.0958084\pi$
$264$	0	0
$265$	748.606	0.173534
$266$	0	0
$267$	0	0
$268$	0	0
$269$	47.3601i	0.0107346i	0.999986	+	0.00536728i	$0.00170847\pi$
$269$	47.3601i	0.0107346i	−0.999986	+	0.00536728i	$0.998292\pi$
$270$	0	0
$271$	1112.11i	0.249283i	0.992202	+	0.124642i	$0.0397781\pi$
$271$	1112.11i	0.249283i	−0.992202	+	0.124642i	$0.960222\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	937.959	0.205677
$276$	0	0
$277$	2773.59	0.601620	0.300810	−	0.953684i	$-0.402743\pi$
$277$	2773.59	0.601620	0.300810	+	0.953684i	$0.402743\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7526.50i	1.59784i	0.601436	+	0.798921i	$0.294595\pi$
$281$	7526.50i	1.59784i	−0.601436	+	0.798921i	$0.705405\pi$
$282$	0	0
$283$	166.175i	0.0349049i	0.999848	+	0.0174524i	$0.00555556\pi$
$283$	166.175i	0.0349049i	−0.999848	+	0.0174524i	$0.994444\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3410.50	0.701447
$288$	0	0
$289$	−1354.13	−0.275622
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 740.755i	− 0.147698i	−0.997269	−	0.0738488i	$-0.976472\pi$
$293$	− 740.755i	− 0.147698i	0.997269	−	0.0738488i	$-0.0235282\pi$
$294$	0	0
$295$	− 7427.21i	− 1.46586i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5981.75	−1.15697
$300$	0	0
$301$	−12443.8	−2.38289
$302$	0	0
$303$	0	0
$304$	0	0
$305$	12838.4i	2.41024i
$306$	0	0
$307$	4096.30i	0.761526i	0.924673	+	0.380763i	$0.124339\pi$
$307$	4096.30i	0.761526i	−0.924673	+	0.380763i	$0.875661\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1427.93	−0.260354	−0.130177	−	0.991491i	$-0.541555\pi$
$311$	−1427.93	−0.260354	−0.130177	+	0.991491i	$0.541555\pi$
$312$	0	0
$313$	−5362.17	−0.968332	−0.484166	−	0.874976i	$-0.660877\pi$
$313$	−5362.17	−0.968332	−0.484166	+	0.874976i	$0.660877\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2520.79i	− 0.446630i	−0.974746	−	0.223315i	$-0.928312\pi$
$317$	− 2520.79i	− 0.446630i	0.974746	−	0.223315i	$-0.0716878\pi$
$318$	0	0
$319$	− 2280.83i	− 0.400319i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3476.38	0.598858
$324$	0	0
$325$	5225.33	0.891843
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 16942.0i	− 2.83904i
$330$	0	0
$331$	7083.08i	1.17620i	0.808789	+	0.588099i	$0.200123\pi$
$331$	7083.08i	1.17620i	−0.808789	+	0.588099i	$0.799877\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4539.81	−0.740406
$336$	0	0
$337$	9814.78	1.58648	0.793242	−	0.608906i	$-0.208391\pi$
$337$	9814.78	1.58648	0.793242	+	0.608906i	$0.208391\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3147.91i	− 0.499909i
$342$	0	0
$343$	− 8477.26i	− 1.33449i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5362.98	−0.829682	−0.414841	−	0.909894i	$-0.636163\pi$
$347$	−5362.98	−0.829682	−0.414841	+	0.909894i	$0.636163\pi$
$348$	0	0
$349$	752.958	0.115487	0.0577434	−	0.998331i	$-0.481609\pi$
$349$	752.958	0.115487	0.0577434	+	0.998331i	$0.481609\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5317.93i	0.801827i	0.916116	+	0.400914i	$0.131307\pi$
$353$	5317.93i	0.801827i	−0.916116	+	0.400914i	$0.868693\pi$
$354$	0	0
$355$	2878.13i	0.430296i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1064.38	−0.156478	−0.0782392	−	0.996935i	$-0.524930\pi$
$359$	−1064.38	−0.156478	−0.0782392	+	0.996935i	$0.524930\pi$
$360$	0	0
$361$	4930.65	0.718858
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 12861.6i	− 1.84440i
$366$	0	0
$367$	3455.59i	0.491499i	0.969333	+	0.245749i	$0.0790340\pi$
$367$	3455.59i	0.491499i	−0.969333	+	0.245749i	$0.920966\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1563.35	−0.218774
$372$	0	0
$373$	7013.13	0.973528	0.486764	−	0.873533i	$-0.338177\pi$
$373$	7013.13	0.973528	0.486764	+	0.873533i	$0.338177\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12706.4i	− 1.73584i
$378$	0	0
$379$	− 4644.35i	− 0.629457i	−0.949182	−	0.314728i	$-0.898087\pi$
$379$	− 4644.35i	− 0.629457i	0.949182	−	0.314728i	$-0.101913\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12025.7	−1.60440	−0.802201	−	0.597053i	$-0.796338\pi$
$383$	−12025.7	−1.60440	−0.802201	+	0.597053i	$0.796338\pi$
$384$	0	0
$385$	−4534.95	−0.600318
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9478.32i	1.23540i	0.786415	+	0.617699i	$0.211935\pi$
$389$	9478.32i	1.23540i	−0.786415	+	0.617699i	$0.788065\pi$
$390$	0	0
$391$	8613.35i	1.11406i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12114.4	1.54315
$396$	0	0
$397$	6747.39	0.853002	0.426501	−	0.904487i	$-0.359746\pi$
$397$	6747.39	0.853002	0.426501	+	0.904487i	$0.359746\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 524.089i	− 0.0652663i	−0.999467	−	0.0326331i	$-0.989611\pi$
$401$	− 524.089i	− 0.0652663i	0.999467	−	0.0326331i	$-0.0103893\pi$
$402$	0	0
$403$	− 17536.9i	− 2.16768i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	376.729	0.0458815
$408$	0	0
$409$	1225.87	0.148204	0.0741019	−	0.997251i	$-0.476391\pi$
$409$	1225.87	0.148204	0.0741019	+	0.997251i	$0.476391\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	15510.6i	1.84801i
$414$	0	0
$415$	− 17007.4i	− 2.01172i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9769.48	−1.13907	−0.569535	−	0.821967i	$-0.692877\pi$
$419$	−9769.48	−1.13907	−0.569535	+	0.821967i	$0.692877\pi$
$420$	0	0
$421$	10097.4	1.16893	0.584463	−	0.811421i	$-0.301305\pi$
$421$	10097.4	1.16893	0.584463	+	0.811421i	$0.301305\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 7524.14i	− 0.858763i
$426$	0	0
$427$	− 26811.0i	− 3.03858i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9338.69	1.04369	0.521843	−	0.853041i	$-0.325245\pi$
$431$	9338.69	1.04369	0.521843	+	0.853041i	$0.325245\pi$
$432$	0	0
$433$	3872.09	0.429748	0.214874	−	0.976642i	$-0.431066\pi$
$433$	3872.09	0.429748	0.214874	+	0.976642i	$0.431066\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4777.83i	− 0.523008i
$438$	0	0
$439$	10236.7i	1.11292i	0.830875	+	0.556460i	$0.187841\pi$
$439$	10236.7i	1.11292i	−0.830875	+	0.556460i	$0.812159\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2296.53	−0.246301	−0.123150	−	0.992388i	$-0.539300\pi$
$443$	−2296.53	−0.246301	−0.123150	+	0.992388i	$0.539300\pi$
$444$	0	0
$445$	−6942.89	−0.739606
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14250.3i	1.49780i	0.662684	+	0.748899i	$0.269417\pi$
$449$	14250.3i	1.49780i	−0.662684	+	0.748899i	$0.730583\pi$
$450$	0	0
$451$	1086.48i	0.113438i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−25264.0	−2.60306
$456$	0	0
$457$	7673.43	0.785444	0.392722	−	0.919657i	$-0.371533\pi$
$457$	7673.43	0.785444	0.392722	+	0.919657i	$0.371533\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 17198.5i	− 1.73756i	−0.495197	−	0.868781i	$-0.664904\pi$
$461$	− 17198.5i	− 1.73756i	0.495197	−	0.868781i	$-0.335096\pi$
$462$	0	0
$463$	3819.89i	0.383424i	0.981451	+	0.191712i	$0.0614040\pi$
$463$	3819.89i	0.383424i	−0.981451	+	0.191712i	$0.938596\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1280.20	−0.126853	−0.0634266	−	0.997987i	$-0.520203\pi$
$467$	−1280.20	−0.126853	−0.0634266	+	0.997987i	$0.520203\pi$
$468$	0	0
$469$	9480.69	0.933428
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 3964.23i	− 0.385360i
$474$	0	0
$475$	4173.65i	0.403158i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13847.2	1.32087	0.660433	−	0.750885i	$-0.270373\pi$
$479$	13847.2	1.32087	0.660433	+	0.750885i	$0.270373\pi$
$480$	0	0
$481$	2098.74	0.198949
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 3540.14i	− 0.331442i
$486$	0	0
$487$	1663.54i	0.154789i	0.997001	+	0.0773945i	$0.0246601\pi$
$487$	1663.54i	0.154789i	−0.997001	+	0.0773945i	$0.975340\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18521.0	1.70233	0.851163	−	0.524901i	$-0.175898\pi$
$491$	18521.0	1.70233	0.851163	+	0.524901i	$0.175898\pi$
$492$	0	0
$493$	−18296.4	−1.67145
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6010.53i	− 0.542473i
$498$	0	0
$499$	8732.34i	0.783393i	0.920094	+	0.391697i	$0.128112\pi$
$499$	8732.34i	0.783393i	−0.920094	+	0.391697i	$0.871888\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14898.7	−1.32068	−0.660338	−	0.750968i	$-0.729587\pi$
$503$	−14898.7	−1.32068	−0.660338	+	0.750968i	$0.729587\pi$
$504$	0	0
$505$	1397.56	0.123150
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 11287.0i	− 0.982886i	−0.870910	−	0.491443i	$-0.836470\pi$
$509$	− 11287.0i	− 0.982886i	0.870910	−	0.491443i	$-0.163530\pi$
$510$	0	0
$511$	26859.5i	2.32523i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11750.7	1.00543
$516$	0	0
$517$	5397.22	0.459129
$518$	0	0
$519$	0	0
$520$	0	0
$521$	11544.2i	0.970749i	0.874306	+	0.485375i	$0.161317\pi$
$521$	11544.2i	0.970749i	−0.874306	+	0.485375i	$0.838683\pi$
$522$	0	0
$523$	− 1393.21i	− 0.116484i	−0.998303	−	0.0582419i	$-0.981451\pi$
$523$	− 1393.21i	− 0.116484i	0.998303	−	0.0582419i	$-0.0185495\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25252.0	−2.08727
$528$	0	0
$529$	−329.088	−0.0270476
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6052.74i	0.491882i
$534$	0	0
$535$	7369.74i	0.595555i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6085.58	0.486316
$540$	0	0
$541$	−9100.41	−0.723211	−0.361606	−	0.932331i	$-0.617771\pi$
$541$	−9100.41	−0.723211	−0.361606	+	0.932331i	$0.617771\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1946.78i	− 0.153011i
$546$	0	0
$547$	− 18886.8i	− 1.47631i	−0.674630	−	0.738156i	$-0.735697\pi$
$547$	− 18886.8i	− 1.47631i	0.674630	−	0.738156i	$-0.264303\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10149.0	0.784687
$552$	0	0
$553$	−25299.1	−1.94544
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4816.02i	0.366358i	0.983080	+	0.183179i	$0.0586388\pi$
$557$	4816.02i	0.366358i	−0.983080	+	0.183179i	$0.941361\pi$
$558$	0	0
$559$	− 22084.5i	− 1.67098i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4690.73	0.351138	0.175569	−	0.984467i	$-0.443824\pi$
$563$	4690.73	0.351138	0.175569	+	0.984467i	$0.443824\pi$
$564$	0	0
$565$	−18599.5	−1.38493
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6547.96i	0.482434i	0.970471	+	0.241217i	$0.0775465\pi$
$569$	6547.96i	0.482434i	−0.970471	+	0.241217i	$0.922453\pi$
$570$	0	0
$571$	13043.0i	0.955927i	0.878380	+	0.477964i	$0.158625\pi$
$571$	13043.0i	0.955927i	−0.878380	+	0.477964i	$0.841375\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−10340.9	−0.749995
$576$	0	0
$577$	8853.82	0.638803	0.319402	−	0.947619i	$-0.396518\pi$
$577$	8853.82	0.638803	0.319402	+	0.947619i	$0.396518\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	35517.4i	2.53616i
$582$	0	0
$583$	− 498.036i	− 0.0353800i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18849.9	1.32542	0.662708	−	0.748878i	$-0.269407\pi$
$587$	18849.9	1.32542	0.662708	+	0.748878i	$0.269407\pi$
$588$	0	0
$589$	14007.3	0.979899
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4673.42i	0.323633i	0.986821	+	0.161817i	$0.0517353\pi$
$593$	4673.42i	0.323633i	−0.986821	+	0.161817i	$0.948265\pi$
$594$	0	0
$595$	36378.6i	2.50651i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12367.0	0.843573	0.421786	−	0.906695i	$-0.361403\pi$
$599$	12367.0	0.843573	0.421786	+	0.906695i	$0.361403\pi$
$600$	0	0
$601$	−14617.0	−0.992077	−0.496038	−	0.868301i	$-0.665212\pi$
$601$	−14617.0	−0.992077	−0.496038	+	0.868301i	$0.665212\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	18299.2i	1.22970i
$606$	0	0
$607$	4724.02i	0.315885i	0.987448	+	0.157942i	$0.0504860\pi$
$607$	4724.02i	0.315885i	−0.987448	+	0.157942i	$0.949514\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30067.6	1.99085
$612$	0	0
$613$	−11318.3	−0.745748	−0.372874	−	0.927882i	$-0.621628\pi$
$613$	−11318.3	−0.745748	−0.372874	+	0.927882i	$0.621628\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6420.71i	0.418943i	0.977815	+	0.209472i	$0.0671744\pi$
$617$	6420.71i	0.418943i	−0.977815	+	0.209472i	$0.932826\pi$
$618$	0	0
$619$	− 9684.86i	− 0.628865i	−0.949280	−	0.314433i	$-0.898186\pi$
$619$	− 9684.86i	− 0.628865i	0.949280	−	0.314433i	$-0.101814\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14499.2	0.932418
$624$	0	0
$625$	−18472.2	−1.18222
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3022.05i	− 0.191569i
$630$	0	0
$631$	3642.85i	0.229825i	0.993376	+	0.114912i	$0.0366587\pi$
$631$	3642.85i	0.229825i	−0.993376	+	0.114912i	$0.963341\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−21194.0	−1.32450
$636$	0	0
$637$	33902.4	2.10873
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 2768.44i	− 0.170588i	−0.996356	−	0.0852939i	$-0.972817\pi$
$641$	− 2768.44i	− 0.170588i	0.996356	−	0.0852939i	$-0.0271829\pi$
$642$	0	0
$643$	− 8558.82i	− 0.524925i	−0.964942	−	0.262463i	$-0.915465\pi$
$643$	− 8558.82i	− 0.524925i	0.964942	−	0.262463i	$-0.0845347\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12937.7	0.786144	0.393072	−	0.919508i	$-0.371412\pi$
$647$	12937.7	0.786144	0.393072	+	0.919508i	$0.371412\pi$
$648$	0	0
$649$	−4941.21	−0.298859
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 25595.6i	− 1.53389i	−0.641712	−	0.766946i	$-0.721775\pi$
$653$	− 25595.6i	− 1.53389i	0.641712	−	0.766946i	$-0.278225\pi$
$654$	0	0
$655$	38956.6i	2.32391i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12541.3	−0.741335	−0.370668	−	0.928766i	$-0.620871\pi$
$659$	−12541.3	−0.741335	−0.370668	+	0.928766i	$0.620871\pi$
$660$	0	0
$661$	−16135.2	−0.949451	−0.474726	−	0.880134i	$-0.657453\pi$
$661$	−16135.2	−0.949451	−0.474726	+	0.880134i	$0.657453\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 20179.2i	− 1.17672i
$666$	0	0
$667$	25146.0i	1.45975i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8541.17	0.491398
$672$	0	0
$673$	25243.7	1.44588	0.722938	−	0.690913i	$-0.242791\pi$
$673$	25243.7	1.44588	0.722938	+	0.690913i	$0.242791\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 11027.7i	− 0.626039i	−0.949747	−	0.313019i	$-0.898659\pi$
$677$	− 11027.7i	− 0.626039i	0.949747	−	0.313019i	$-0.101341\pi$
$678$	0	0
$679$	7393.04i	0.417848i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−15409.7	−0.863304	−0.431652	−	0.902040i	$-0.642069\pi$
$683$	−15409.7	−0.863304	−0.431652	+	0.902040i	$0.642069\pi$
$684$	0	0
$685$	−32631.7	−1.82013
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 2774.53i	− 0.153413i
$690$	0	0
$691$	− 23759.2i	− 1.30802i	−0.756486	−	0.654010i	$-0.773086\pi$
$691$	− 23759.2i	− 1.30802i	0.756486	−	0.654010i	$-0.226914\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10185.7	−0.555922
$696$	0	0
$697$	8715.56	0.473638
$698$	0	0
$699$	0	0
$700$	0	0
$701$	24574.3i	1.32405i	0.749481	+	0.662026i	$0.230303\pi$
$701$	24574.3i	1.32405i	−0.749481	+	0.662026i	$0.769697\pi$
$702$	0	0
$703$	1676.33i	0.0899348i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2918.60	−0.155255
$708$	0	0
$709$	−20614.2	−1.09194	−0.545968	−	0.837806i	$-0.683838\pi$
$709$	−20614.2	−1.09194	−0.545968	+	0.837806i	$0.683838\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	34705.5i	1.82291i
$714$	0	0
$715$	− 8048.35i	− 0.420967i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3239.50	−0.168029	−0.0840146	−	0.996465i	$-0.526774\pi$
$719$	−3239.50	−0.168029	−0.0840146	+	0.996465i	$0.526774\pi$
$720$	0	0
$721$	−24539.5	−1.26754
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 21966.1i	− 1.12524i
$726$	0	0
$727$	− 27590.8i	− 1.40755i	−0.710425	−	0.703773i	$-0.751497\pi$
$727$	− 27590.8i	− 1.40755i	0.710425	−	0.703773i	$-0.248503\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−31800.3	−1.60900
$732$	0	0
$733$	32932.1	1.65945	0.829725	−	0.558173i	$-0.188497\pi$
$733$	32932.1	1.65945	0.829725	+	0.558173i	$0.188497\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3020.26i	0.150954i
$738$	0	0
$739$	7766.70i	0.386607i	0.981139	+	0.193304i	$0.0619202\pi$
$739$	7766.70i	0.386607i	−0.981139	+	0.193304i	$0.938080\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32172.3	1.58854	0.794271	−	0.607563i	$-0.207853\pi$
$743$	32172.3	1.58854	0.794271	+	0.607563i	$0.207853\pi$
$744$	0	0
$745$	−6539.52	−0.321596
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 15390.6i	− 0.750814i
$750$	0	0
$751$	− 12175.5i	− 0.591600i	−0.955250	−	0.295800i	$-0.904414\pi$
$751$	− 12175.5i	− 0.591600i	0.955250	−	0.295800i	$-0.0955862\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12760.7	−0.615110
$756$	0	0
$757$	−14541.7	−0.698185	−0.349092	−	0.937088i	$-0.613510\pi$
$757$	−14541.7	−0.698185	−0.349092	+	0.937088i	$0.613510\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8669.16i	0.412952i	0.978452	+	0.206476i	$0.0661996\pi$
$761$	8669.16i	0.412952i	−0.978452	+	0.206476i	$0.933800\pi$
$762$	0	0
$763$	4065.56i	0.192901i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−27527.2	−1.29589
$768$	0	0
$769$	28389.1	1.33126	0.665629	−	0.746282i	$-0.268163\pi$
$769$	28389.1	1.33126	0.665629	+	0.746282i	$0.268163\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 15782.6i	− 0.734362i	−0.930149	−	0.367181i	$-0.880323\pi$
$773$	− 15782.6i	− 0.734362i	0.930149	−	0.367181i	$-0.119677\pi$
$774$	0	0
$775$	− 30316.8i	− 1.40518i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4834.53	−0.222356
$780$	0	0
$781$	1914.77	0.0877285
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 20897.7i	− 0.950153i
$786$	0	0
$787$	− 25460.6i	− 1.15321i	−0.817025	−	0.576603i	$-0.804378\pi$
$787$	− 25460.6i	− 1.15321i	0.817025	−	0.576603i	$-0.195622\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	38842.2	1.74598
$792$	0	0
$793$	47582.5	2.13077
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19600.9i	0.871139i	0.900155	+	0.435570i	$0.143453\pi$
$797$	19600.9i	0.871139i	−0.900155	+	0.435570i	$0.856547\pi$
$798$	0	0
$799$	− 43295.5i	− 1.91700i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8556.62	−0.376036
$804$	0	0
$805$	49997.5	2.18905
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 3065.74i	− 0.133233i	−0.997779	−	0.0666166i	$-0.978780\pi$
$809$	− 3065.74i	− 0.133233i	0.997779	−	0.0666166i	$-0.0212204\pi$
$810$	0	0
$811$	44326.1i	1.91923i	0.281309	+	0.959617i	$0.409231\pi$
$811$	44326.1i	1.91923i	−0.281309	+	0.959617i	$0.590769\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	38691.2	1.66294
$816$	0	0
$817$	17639.7	0.755365
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21842.2i	0.928498i	0.885705	+	0.464249i	$0.153676\pi$
$821$	21842.2i	0.928498i	−0.885705	+	0.464249i	$0.846324\pi$
$822$	0	0
$823$	− 5363.37i	− 0.227163i	−0.993529	−	0.113582i	$-0.963768\pi$
$823$	− 5363.37i	− 0.227163i	0.993529	−	0.113582i	$-0.0362323\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11117.1	−0.467450	−0.233725	−	0.972303i	$-0.575091\pi$
$827$	−11117.1	−0.467450	−0.233725	+	0.972303i	$0.575091\pi$
$828$	0	0
$829$	41554.2	1.74094	0.870469	−	0.492224i	$-0.163816\pi$
$829$	41554.2	1.74094	0.870469	+	0.492224i	$0.163816\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 48817.4i	− 2.03052i
$834$	0	0
$835$	21860.2i	0.905994i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34881.7	−1.43534	−0.717669	−	0.696384i	$-0.754791\pi$
$839$	−34881.7	−1.43534	−0.717669	+	0.696384i	$0.754791\pi$
$840$	0	0
$841$	−29025.7	−1.19012
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 12247.0i	− 0.498589i
$846$	0	0
$847$	− 38215.0i	− 1.55028i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4153.41	−0.167306
$852$	0	0
$853$	26716.2	1.07239	0.536194	−	0.844095i	$-0.319862\pi$
$853$	26716.2	1.07239	0.536194	+	0.844095i	$0.319862\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 11875.9i	− 0.473362i	−0.971587	−	0.236681i	$-0.923940\pi$
$857$	− 11875.9i	− 0.473362i	0.971587	−	0.236681i	$-0.0760597\pi$
$858$	0	0
$859$	− 26907.2i	− 1.06876i	−0.845246	−	0.534378i	$-0.820546\pi$
$859$	− 26907.2i	− 1.06876i	0.845246	−	0.534378i	$-0.179454\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5839.41	0.230331	0.115166	−	0.993346i	$-0.463260\pi$
$863$	5839.41	0.230331	0.115166	+	0.993346i	$0.463260\pi$
$864$	0	0
$865$	−46855.6	−1.84178
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 8059.54i	− 0.314616i
$870$	0	0
$871$	16825.7i	0.654556i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	13765.8	0.531851
$876$	0	0
$877$	−43994.2	−1.69393	−0.846965	−	0.531648i	$-0.821573\pi$
$877$	−43994.2	−1.69393	−0.846965	+	0.531648i	$0.821573\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 12290.1i	− 0.469992i	−0.971996	−	0.234996i	$-0.924492\pi$
$881$	− 12290.1i	− 0.469992i	0.971996	−	0.234996i	$-0.0755077\pi$
$882$	0	0
$883$	33329.9i	1.27026i	0.772404	+	0.635131i	$0.219054\pi$
$883$	33329.9i	1.27026i	−0.772404	+	0.635131i	$0.780946\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39661.2	−1.50135	−0.750673	−	0.660674i	$-0.770271\pi$
$887$	−39661.2	−1.50135	−0.750673	+	0.660674i	$0.770271\pi$
$888$	0	0
$889$	44260.5	1.66980
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24016.1i	0.899963i
$894$	0	0
$895$	921.218i	0.0344055i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−73721.1	−2.73497
$900$	0	0
$901$	−3995.15	−0.147722
$902$	0	0
$903$	0	0
$904$	0	0
$905$	36350.4i	1.33517i
$906$	0	0
$907$	− 42455.0i	− 1.55424i	−0.629353	−	0.777120i	$-0.716680\pi$
$907$	− 42455.0i	− 1.55424i	0.629353	−	0.777120i	$-0.283320\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−22949.8	−0.834643	−0.417322	−	0.908759i	$-0.637031\pi$
$911$	−22949.8	−0.834643	−0.417322	+	0.908759i	$0.637031\pi$
$912$	0	0
$913$	−11314.8	−0.410147
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 81354.9i	− 2.92974i
$918$	0	0
$919$	− 44604.0i	− 1.60103i	−0.599311	−	0.800516i	$-0.704559\pi$
$919$	− 44604.0i	− 1.60103i	0.599311	−	0.800516i	$-0.295441\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10667.1	0.380403
$924$	0	0
$925$	3628.19	0.128967
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 25063.5i	− 0.885153i	−0.896731	−	0.442577i	$-0.854064\pi$
$929$	− 25063.5i	− 0.885153i	0.896731	−	0.442577i	$-0.145936\pi$
$930$	0	0
$931$	27079.0i	0.953254i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−11589.1	−0.405352
$936$	0	0
$937$	33752.1	1.17677	0.588385	−	0.808581i	$-0.299764\pi$
$937$	33752.1	1.17677	0.588385	+	0.808581i	$0.299764\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	39088.6i	1.35415i	0.735915	+	0.677074i	$0.236752\pi$
$941$	39088.6i	1.35415i	−0.735915	+	0.677074i	$0.763248\pi$
$942$	0	0
$943$	− 11978.4i	− 0.413648i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7294.32	0.250299	0.125150	−	0.992138i	$-0.460059\pi$
$947$	7294.32	0.250299	0.125150	+	0.992138i	$0.460059\pi$
$948$	0	0
$949$	−47668.5	−1.63054
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 16201.6i	− 0.550703i	−0.961344	−	0.275351i	$-0.911206\pi$
$953$	− 16201.6i	− 0.550703i	0.961344	−	0.275351i	$-0.0887942\pi$
$954$	0	0
$955$	− 21046.7i	− 0.713146i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	68146.2	2.29464
$960$	0	0
$961$	−71956.1	−2.41536
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 13968.3i	− 0.465965i
$966$	0	0
$967$	15952.5i	0.530504i	0.964179	+	0.265252i	$0.0854552\pi$
$967$	15952.5i	0.530504i	−0.964179	+	0.265252i	$0.914545\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−30067.2	−0.993722	−0.496861	−	0.867830i	$-0.665514\pi$
$971$	−30067.2	−0.993722	−0.496861	+	0.867830i	$0.665514\pi$
$972$	0	0
$973$	21271.3	0.700849
$974$	0	0
$975$	0	0
$976$	0	0
$977$	59343.9i	1.94328i	0.236474	+	0.971638i	$0.424008\pi$
$977$	59343.9i	1.94328i	−0.236474	+	0.971638i	$0.575992\pi$
$978$	0	0
$979$	4618.99i	0.150790i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33918.8	1.10055	0.550276	−	0.834983i	$-0.314523\pi$
$983$	33918.8	1.10055	0.550276	+	0.834983i	$0.314523\pi$
$984$	0	0
$985$	76526.1	2.47546
$986$	0	0
$987$	0	0
$988$	0	0
$989$	43705.3i	1.40521i
$990$	0	0
$991$	19427.8i	0.622748i	0.950287	+	0.311374i	$0.100789\pi$
$991$	19427.8i	0.622748i	−0.950287	+	0.311374i	$0.899211\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−662.053	−0.0210940
$996$	0	0
$997$	2949.82	0.0937028	0.0468514	−	0.998902i	$-0.485081\pi$
$997$	2949.82	0.0937028	0.0468514	+	0.998902i	$0.485081\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.4.c.b.287.4	yes	8
3.2	odd	2	inner	288.4.c.b.287.6	yes	8
4.3	odd	2	inner	288.4.c.b.287.3	✓	8
8.3	odd	2		576.4.c.f.575.5		8
8.5	even	2		576.4.c.f.575.6		8
12.11	even	2	inner	288.4.c.b.287.5	yes	8
16.3	odd	4		2304.4.f.h.1151.4		8
16.5	even	4		2304.4.f.h.1151.5		8
16.11	odd	4		2304.4.f.e.1151.6		8
16.13	even	4		2304.4.f.e.1151.3		8
24.5	odd	2		576.4.c.f.575.4		8
24.11	even	2		576.4.c.f.575.3		8
48.5	odd	4		2304.4.f.h.1151.3		8
48.11	even	4		2304.4.f.e.1151.4		8
48.29	odd	4		2304.4.f.e.1151.5		8
48.35	even	4		2304.4.f.h.1151.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.4.c.b.287.3	✓	8	4.3	odd	2	inner
288.4.c.b.287.4	yes	8	1.1	even	1	trivial
288.4.c.b.287.5	yes	8	12.11	even	2	inner
288.4.c.b.287.6	yes	8	3.2	odd	2	inner
576.4.c.f.575.3		8	24.11	even	2
576.4.c.f.575.4		8	24.5	odd	2
576.4.c.f.575.5		8	8.3	odd	2
576.4.c.f.575.6		8	8.5	even	2
2304.4.f.e.1151.3		8	16.13	even	4
2304.4.f.e.1151.4		8	48.11	even	4
2304.4.f.e.1151.5		8	48.29	odd	4
2304.4.f.e.1151.6		8	16.11	odd	4
2304.4.f.h.1151.3		8	48.5	odd	4
2304.4.f.h.1151.4		8	16.3	odd	4
2304.4.f.h.1151.5		8	16.5	even	4
2304.4.f.h.1151.6		8	48.35	even	4

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:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	288.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-14.8339i q^{5} +30.9783i q^{7} +O(q^{10})\)	\(q-14.8339i q^{5} +30.9783i q^{7} -9.86874 q^{11} -54.9783 q^{13} +79.1652i q^{17} -43.9130i q^{19} +108.802 q^{23} -95.0435 q^{25} +231.116i q^{29} +318.978i q^{31} +459.527 q^{35} -38.1740 q^{37} -110.093i q^{41} +401.696i q^{43} -546.901 q^{47} -616.652 q^{49} +50.4660i q^{53} +146.391i q^{55} +500.693 q^{59} -865.478 q^{61} +815.540i q^{65} -306.043i q^{67} -194.024 q^{71} +867.043 q^{73} -305.716i q^{77} +816.674i q^{79} +1146.53 q^{83} +1174.33 q^{85} -468.043i q^{89} -1703.13i q^{91} -651.399 q^{95} +238.652 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 256 q^{13} - 1128 q^{25} - 1776 q^{37} - 1992 q^{49} - 2512 q^{61} - 1152 q^{73} + 3696 q^{85} + 7424 q^{97}+O(q^{100}) \) 8 * q - 256 * q^13 - 1128 * q^25 - 1776 * q^37 - 1992 * q^49 - 2512 * q^61 - 1152 * q^73 + 3696 * q^85 + 7424 * q^97

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