LMFDB - Embedded newform 2304.4.f.e.1151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,4,Mod(1151,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2304.1151");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2304.f (of order $2$, degree $1$, not 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:	$135.940400653$
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^{16} $
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.1
Root			$1.67746 + 1.67746i$ of defining polynomial
Character	$\chi$	$=$	2304.1151
Dual form			2304.4.f.e.1151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-17.6623 q^{5} -14.9783i q^{7} +O(q^{10})$	$q-17.6623 q^{5} -14.9783i q^{7} -55.1236i q^{11} +9.02175i q^{13} +14.1729i q^{17} -139.913 q^{19} -86.1748 q^{23} +186.957 q^{25} +3.64319 q^{29} -273.022i q^{31} +264.550i q^{35} -405.826i q^{37} -344.853i q^{41} -241.696 q^{43} -297.999 q^{47} +118.652 q^{49} -602.901 q^{53} +973.609i q^{55} -630.678i q^{59} -237.478i q^{61} -159.345i q^{65} +397.957 q^{67} -778.955 q^{71} +1155.04 q^{73} -825.655 q^{77} -127.326i q^{79} +301.627i q^{83} -250.326i q^{85} +338.059i q^{89} +135.130 q^{91} +2471.18 q^{95} +1617.35 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 384 q^{19} + 1128 q^{25} + 640 q^{43} - 1992 q^{49} + 2816 q^{67} + 1152 q^{73} - 6272 q^{91} + 7424 q^{97}+O(q^{100}) $ 8 * q - 384 * q^19 + 1128 * q^25 + 640 * q^43 - 1992 * q^49 + 2816 * q^67 + 1152 * q^73 - 6272 * q^91 + 7424 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$-1$	$-1$

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
$5$	−17.6623	−1.57976	−0.789882	−	0.613259i	$-0.789858\pi$
$5$	−17.6623	−1.57976	−0.789882	+	0.613259i	$0.789858\pi$
$6$	0	0
$7$	− 14.9783i	− 0.808750i	−0.914593	−	0.404375i	$-0.867489\pi$
$7$	− 14.9783i	− 0.808750i	0.914593	−	0.404375i	$-0.132511\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 55.1236i	− 1.51094i	−0.655181	−	0.755472i	$-0.727408\pi$
$11$	− 55.1236i	− 1.51094i	0.655181	−	0.755472i	$-0.272592\pi$
$12$	0	0
$13$	9.02175i	0.192476i	0.995358	+	0.0962378i	$0.0306809\pi$
$13$	9.02175i	0.192476i	−0.995358	+	0.0962378i	$0.969319\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	14.1729i	0.202202i	0.994876	+	0.101101i	$0.0322365\pi$
$17$	14.1729i	0.202202i	−0.994876	+	0.101101i	$0.967763\pi$
$18$	0	0
$19$	−139.913	−1.68938	−0.844691	−	0.535255i	$-0.820216\pi$
$19$	−139.913	−1.68938	−0.844691	+	0.535255i	$0.820216\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−86.1748	−0.781247	−0.390623	−	0.920551i	$-0.627741\pi$
$23$	−86.1748	−0.781247	−0.390623	+	0.920551i	$0.627741\pi$
$24$	0	0
$25$	186.957	1.49565
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.64319	0.0233284	0.0116642	−	0.999932i	$-0.496287\pi$
$29$	3.64319	0.0233284	0.0116642	+	0.999932i	$0.496287\pi$
$30$	0	0
$31$	− 273.022i	− 1.58181i	−0.611938	−	0.790906i	$-0.709610\pi$
$31$	− 273.022i	− 1.58181i	0.611938	−	0.790906i	$-0.290390\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	264.550i	1.27763i
$36$	0	0
$37$	− 405.826i	− 1.80317i	−0.432599	−	0.901586i	$-0.642404\pi$
$37$	− 405.826i	− 1.80317i	0.432599	−	0.901586i	$-0.357596\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 344.853i	− 1.31358i	−0.754072	−	0.656792i	$-0.771913\pi$
$41$	− 344.853i	− 1.31358i	0.754072	−	0.656792i	$-0.228087\pi$
$42$	0	0
$43$	−241.696	−0.857168	−0.428584	−	0.903502i	$-0.640987\pi$
$43$	−241.696	−0.857168	−0.428584	+	0.903502i	$0.640987\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−297.999	−0.924844	−0.462422	−	0.886660i	$-0.653019\pi$
$47$	−297.999	−0.924844	−0.462422	+	0.886660i	$0.653019\pi$
$48$	0	0
$49$	118.652	0.345924
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−602.901	−1.56254	−0.781271	−	0.624191i	$-0.785429\pi$
$53$	−602.901	−1.56254	−0.781271	+	0.624191i	$0.785429\pi$
$54$	0	0
$55$	973.609i	2.38693i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 630.678i	− 1.39165i	−0.718212	−	0.695824i	$-0.755039\pi$
$59$	− 630.678i	− 1.39165i	0.718212	−	0.695824i	$-0.244961\pi$
$60$	0	0
$61$	− 237.478i	− 0.498458i	−0.968445	−	0.249229i	$-0.919823\pi$
$61$	− 237.478i	− 0.498458i	0.968445	−	0.249229i	$-0.0801772\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 159.345i	− 0.304066i
$66$	0	0
$67$	397.957	0.725644	0.362822	−	0.931859i	$-0.381813\pi$
$67$	397.957	0.725644	0.362822	+	0.931859i	$0.381813\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−778.955	−1.30204	−0.651021	−	0.759060i	$-0.725659\pi$
$71$	−778.955	−1.30204	−0.651021	+	0.759060i	$0.725659\pi$
$72$	0	0
$73$	1155.04	1.85188	0.925942	−	0.377665i	$-0.123273\pi$
$73$	1155.04	1.85188	0.925942	+	0.377665i	$0.123273\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−825.655	−1.22198
$78$	0	0
$79$	− 127.326i	− 0.181333i	−0.995881	−	0.0906666i	$-0.971100\pi$
$79$	− 127.326i	− 0.181333i	0.995881	−	0.0906666i	$-0.0288998\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	301.627i	0.398890i	0.979909	+	0.199445i	$0.0639140\pi$
$83$	301.627i	0.398890i	−0.979909	+	0.199445i	$0.936086\pi$
$84$	0	0
$85$	− 250.326i	− 0.319431i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	338.059i	0.402631i	0.979526	+	0.201315i	$0.0645216\pi$
$89$	338.059i	0.402631i	−0.979526	+	0.201315i	$0.935478\pi$
$90$	0	0
$91$	135.130	0.155665
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2471.18	2.66882
$96$	0	0
$97$	1617.35	1.69296	0.846478	−	0.532423i	$-0.178719\pi$
$97$	1617.35	1.69296	0.846478	+	0.532423i	$0.178719\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−451.672	−0.444981	−0.222490	−	0.974935i	$-0.571419\pi$
$101$	−451.672	−0.444981	−0.222490	+	0.974935i	$0.571419\pi$
$102$	0	0
$103$	− 1000.15i	− 0.956776i	−0.878149	−	0.478388i	$-0.841221\pi$
$103$	− 1000.15i	− 0.956776i	0.878149	−	0.478388i	$-0.158779\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1582.93i	− 1.43017i	−0.699038	−	0.715085i	$-0.746388\pi$
$107$	− 1582.93i	− 1.43017i	0.699038	−	0.715085i	$-0.253612\pi$
$108$	0	0
$109$	− 636.761i	− 0.559547i	−0.960066	−	0.279773i	$-0.909741\pi$
$109$	− 636.761i	− 0.559547i	0.960066	−	0.279773i	$-0.0902594\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 83.9922i	− 0.0699232i	−0.999389	−	0.0349616i	$-0.988869\pi$
$113$	− 83.9922i	− 0.0699232i	0.999389	−	0.0349616i	$-0.0111309\pi$
$114$	0	0
$115$	1522.04	1.23419
$116$	0	0
$117$	0	0
$118$	0	0
$119$	212.285	0.163531
$120$	0	0
$121$	−1707.61	−1.28295
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1094.29	−0.783013
$126$	0	0
$127$	923.239i	0.645073i	0.946557	+	0.322536i	$0.104535\pi$
$127$	923.239i	0.645073i	−0.946557	+	0.322536i	$0.895465\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	363.452i	0.242404i	0.992628	+	0.121202i	$0.0386749\pi$
$131$	363.452i	0.242404i	−0.992628	+	0.121202i	$0.961325\pi$
$132$	0	0
$133$	2095.65i	1.36629i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	704.986i	0.439642i	0.975540	+	0.219821i	$0.0705474\pi$
$137$	704.986i	0.439642i	−0.975540	+	0.219821i	$0.929453\pi$
$138$	0	0
$139$	−2065.35	−1.26029	−0.630146	−	0.776477i	$-0.717005\pi$
$139$	−2065.35	−1.26029	−0.630146	+	0.776477i	$0.717005\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	497.311	0.290820
$144$	0	0
$145$	−64.3470	−0.0368533
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1123.27	0.617596	0.308798	−	0.951128i	$-0.400073\pi$
$149$	1123.27	0.617596	0.308798	+	0.951128i	$0.400073\pi$
$150$	0	0
$151$	748.239i	0.403250i	0.979463	+	0.201625i	$0.0646223\pi$
$151$	748.239i	0.403250i	−0.979463	+	0.201625i	$0.935378\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4822.19i	2.49889i
$156$	0	0
$157$	1164.78i	0.592100i	0.955172	+	0.296050i	$0.0956695\pi$
$157$	1164.78i	0.592100i	−0.955172	+	0.296050i	$0.904330\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1290.75i	0.631833i
$162$	0	0
$163$	976.304	0.469141	0.234571	−	0.972099i	$-0.424632\pi$
$163$	976.304	0.469141	0.234571	+	0.972099i	$0.424632\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1603.66	−0.743082	−0.371541	−	0.928417i	$-0.621170\pi$
$167$	−1603.66	−0.743082	−0.371541	+	0.928417i	$0.621170\pi$
$168$	0	0
$169$	2115.61	0.962953
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−266.534	−0.117134	−0.0585670	−	0.998283i	$-0.518653\pi$
$173$	−266.534	−0.117134	−0.0585670	+	0.998283i	$0.518653\pi$
$174$	0	0
$175$	− 2800.28i	− 1.20961i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	197.867i	0.0826216i	0.999146	+	0.0413108i	$0.0131534\pi$
$179$	197.867i	0.0826216i	−0.999146	+	0.0413108i	$0.986847\pi$
$180$	0	0
$181$	2834.50i	1.16401i	0.813184	+	0.582007i	$0.197732\pi$
$181$	2834.50i	1.16401i	−0.813184	+	0.582007i	$0.802268\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7167.82i	2.84859i
$186$	0	0
$187$	781.261	0.305516
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5188.38	−1.96554	−0.982770	−	0.184835i	$-0.940825\pi$
$191$	−5188.38	−1.96554	−0.982770	+	0.184835i	$0.940825\pi$
$192$	0	0
$193$	206.348	0.0769599	0.0384799	−	0.999259i	$-0.487748\pi$
$193$	206.348	0.0769599	0.0384799	+	0.999259i	$0.487748\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3306.60	−1.19587	−0.597933	−	0.801546i	$-0.704011\pi$
$197$	−3306.60	−1.19587	−0.597933	+	0.801546i	$0.704011\pi$
$198$	0	0
$199$	− 3491.37i	− 1.24370i	−0.783136	−	0.621851i	$-0.786381\pi$
$199$	− 3491.37i	− 1.24370i	0.783136	−	0.621851i	$-0.213619\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 54.5686i	− 0.0188668i
$204$	0	0
$205$	6090.89i	2.07515i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7712.50i	2.55256i
$210$	0	0
$211$	−4808.48	−1.56886	−0.784429	−	0.620218i	$-0.787044\pi$
$211$	−4808.48	−1.56886	−0.784429	+	0.620218i	$0.787044\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4268.90	1.35412
$216$	0	0
$217$	−4089.39	−1.27929
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−127.864	−0.0389189
$222$	0	0
$223$	1620.37i	0.486582i	0.969953	+	0.243291i	$0.0782271\pi$
$223$	1620.37i	0.486582i	−0.969953	+	0.243291i	$0.921773\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	761.369i	0.222616i	0.993786	+	0.111308i	$0.0355040\pi$
$227$	761.369i	0.222616i	−0.993786	+	0.111308i	$0.964496\pi$
$228$	0	0
$229$	2397.76i	0.691915i	0.938250	+	0.345957i	$0.112446\pi$
$229$	2397.76i	0.691915i	−0.938250	+	0.345957i	$0.887554\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 4299.95i	− 1.20901i	−0.796602	−	0.604504i	$-0.793371\pi$
$233$	− 4299.95i	− 1.20901i	0.796602	−	0.604504i	$-0.206629\pi$
$234$	0	0
$235$	5263.35	1.46103
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1287.79	0.348538	0.174269	−	0.984698i	$-0.444244\pi$
$239$	1287.79	0.348538	0.174269	+	0.984698i	$0.444244\pi$
$240$	0	0
$241$	−4906.82	−1.31152	−0.655760	−	0.754969i	$-0.727652\pi$
$241$	−4906.82	−1.31152	−0.655760	+	0.754969i	$0.727652\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2095.67	−0.546478
$246$	0	0
$247$	− 1262.26i	− 0.325165i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 3501.68i	− 0.880575i	−0.897857	−	0.440288i	$-0.854876\pi$
$251$	− 3501.68i	− 0.880575i	0.897857	−	0.440288i	$-0.145124\pi$
$252$	0	0
$253$	4750.26i	1.18042i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1295.48i	0.314435i	0.987564	+	0.157218i	$0.0502524\pi$
$257$	1295.48i	0.314435i	−0.987564	+	0.157218i	$0.949748\pi$
$258$	0	0
$259$	−6078.56	−1.45831
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4117.27	0.965329	0.482665	−	0.875805i	$-0.339669\pi$
$263$	4117.27	0.965329	0.482665	+	0.875805i	$0.339669\pi$
$264$	0	0
$265$	10648.6	2.46845
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2974.78	−0.674259	−0.337129	−	0.941458i	$-0.609456\pi$
$269$	−2974.78	−0.674259	−0.337129	+	0.941458i	$0.609456\pi$
$270$	0	0
$271$	5000.11i	1.12079i	0.828224	+	0.560396i	$0.189351\pi$
$271$	5000.11i	1.12079i	−0.828224	+	0.560396i	$0.810649\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 10305.7i	− 2.25985i
$276$	0	0
$277$	− 213.586i	− 0.0463291i	−0.999732	−	0.0231645i	$-0.992626\pi$
$277$	− 213.586i	− 0.0463291i	0.999732	−	0.0231645i	$-0.00737416\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7291.74i	1.54800i	0.633184	+	0.774002i	$0.281748\pi$
$281$	7291.74i	1.54800i	−0.633184	+	0.774002i	$0.718252\pi$
$282$	0	0
$283$	4761.83	1.00022	0.500108	−	0.865963i	$-0.333294\pi$
$283$	4761.83	1.00022	0.500108	+	0.865963i	$0.333294\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5165.29	−1.06236
$288$	0	0
$289$	4712.13	0.959114
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1371.49	−0.273459	−0.136730	−	0.990608i	$-0.543659\pi$
$293$	−1371.49	−0.273459	−0.136730	+	0.990608i	$0.543659\pi$
$294$	0	0
$295$	11139.2i	2.19847i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 777.447i	− 0.150371i
$300$	0	0
$301$	3620.18i	0.693234i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4194.41i	0.787446i
$306$	0	0
$307$	−511.696	−0.0951272	−0.0475636	−	0.998868i	$-0.515146\pi$
$307$	−511.696	−0.0951272	−0.0475636	+	0.998868i	$0.515146\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−518.033	−0.0944532	−0.0472266	−	0.998884i	$-0.515038\pi$
$311$	−518.033	−0.0944532	−0.0472266	+	0.998884i	$0.515038\pi$
$312$	0	0
$313$	−2726.17	−0.492308	−0.246154	−	0.969231i	$-0.579167\pi$
$313$	−2726.17	−0.492308	−0.246154	+	0.969231i	$0.579167\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	566.346	0.100344	0.0501722	−	0.998741i	$-0.484023\pi$
$317$	566.346	0.100344	0.0501722	+	0.998741i	$0.484023\pi$
$318$	0	0
$319$	− 200.826i	− 0.0352479i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1982.97i	− 0.341596i
$324$	0	0
$325$	1686.67i	0.287877i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4463.51i	0.747967i
$330$	0	0
$331$	−7531.08	−1.25059	−0.625296	−	0.780388i	$-0.715022\pi$
$331$	−7531.08	−1.25059	−0.625296	+	0.780388i	$0.715022\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7028.82	−1.14635
$336$	0	0
$337$	7241.22	1.17049	0.585244	−	0.810857i	$-0.300999\pi$
$337$	7241.22	1.17049	0.585244	+	0.810857i	$0.300999\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−15049.9	−2.39003
$342$	0	0
$343$	− 6914.74i	− 1.08852i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8937.55i	1.38269i	0.722525	+	0.691344i	$0.242981\pi$
$347$	8937.55i	1.38269i	−0.722525	+	0.691344i	$0.757019\pi$
$348$	0	0
$349$	− 7003.04i	− 1.07411i	−0.843547	−	0.537055i	$-0.819537\pi$
$349$	− 7003.04i	− 1.07411i	0.843547	−	0.537055i	$-0.180463\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3368.17i	0.507845i	0.967225	+	0.253923i	$0.0817208\pi$
$353$	3368.17i	0.507845i	−0.967225	+	0.253923i	$0.918279\pi$
$354$	0	0
$355$	13758.1	2.05692
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6019.78	0.884992	0.442496	−	0.896771i	$-0.354093\pi$
$359$	6019.78	0.884992	0.442496	+	0.896771i	$0.354093\pi$
$360$	0	0
$361$	12716.6	1.85401
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−20400.7	−2.92554
$366$	0	0
$367$	3759.59i	0.534738i	0.963594	+	0.267369i	$0.0861542\pi$
$367$	3759.59i	0.534738i	−0.963594	+	0.267369i	$0.913846\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9030.40i	1.26371i
$372$	0	0
$373$	5174.87i	0.718350i	0.933270	+	0.359175i	$0.116942\pi$
$373$	5174.87i	0.718350i	−0.933270	+	0.359175i	$0.883058\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.8679i	0.00449014i
$378$	0	0
$379$	3076.35	0.416943	0.208471	−	0.978028i	$-0.433151\pi$
$379$	3076.35	0.416943	0.208471	+	0.978028i	$0.433151\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10465.9	1.39630	0.698151	−	0.715951i	$-0.254007\pi$
$383$	10465.9	1.39630	0.698151	+	0.715951i	$0.254007\pi$
$384$	0	0
$385$	14583.0	1.93043
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2556.64	−0.333231	−0.166615	−	0.986022i	$-0.553284\pi$
$389$	−2556.64	−0.333231	−0.166615	+	0.986022i	$0.553284\pi$
$390$	0	0
$391$	− 1221.35i	− 0.157970i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2248.87i	0.286464i
$396$	0	0
$397$	− 1232.61i	− 0.155826i	−0.996960	−	0.0779130i	$-0.975174\pi$
$397$	− 1232.61i	− 0.155826i	0.996960	−	0.0779130i	$-0.0248256\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 2278.88i	− 0.283795i	−0.989881	−	0.141898i	$-0.954680\pi$
$401$	− 2278.88i	− 0.283795i	0.989881	−	0.141898i	$-0.0453204\pi$
$402$	0	0
$403$	2463.13	0.304460
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−22370.6	−2.72449
$408$	0	0
$409$	−950.130	−0.114868	−0.0574339	−	0.998349i	$-0.518292\pi$
$409$	−950.130	−0.114868	−0.0574339	+	0.998349i	$0.518292\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9446.45	−1.12549
$414$	0	0
$415$	− 5327.43i	− 0.630152i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 6974.81i	− 0.813226i	−0.913600	−	0.406613i	$-0.866710\pi$
$419$	− 6974.81i	− 0.813226i	0.913600	−	0.406613i	$-0.133290\pi$
$420$	0	0
$421$	− 1713.41i	− 0.198353i	−0.995070	−	0.0991764i	$-0.968379\pi$
$421$	− 1713.41i	− 0.198353i	0.995070	−	0.0991764i	$-0.0316208\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2649.71i	0.302424i
$426$	0	0
$427$	−3557.01	−0.403128
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7234.34	0.808506	0.404253	−	0.914647i	$-0.367532\pi$
$431$	7234.34	0.808506	0.404253	+	0.914647i	$0.367532\pi$
$432$	0	0
$433$	8283.91	0.919398	0.459699	−	0.888075i	$-0.347957\pi$
$433$	8283.91	0.919398	0.459699	+	0.888075i	$0.347957\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12057.0	1.31982
$438$	0	0
$439$	− 11500.7i	− 1.25034i	−0.780489	−	0.625170i	$-0.785030\pi$
$439$	− 11500.7i	− 1.25034i	0.780489	−	0.625170i	$-0.214970\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 12976.7i	− 1.39174i	−0.718169	−	0.695869i	$-0.755019\pi$
$443$	− 12976.7i	− 1.39174i	0.718169	−	0.695869i	$-0.244981\pi$
$444$	0	0
$445$	− 5970.89i	− 0.636061i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 8302.07i	− 0.872604i	−0.899800	−	0.436302i	$-0.856288\pi$
$449$	− 8302.07i	− 0.872604i	0.899800	−	0.436302i	$-0.143712\pi$
$450$	0	0
$451$	−19009.5	−1.98475
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2386.71	−0.245913
$456$	0	0
$457$	1977.43	0.202408	0.101204	−	0.994866i	$-0.467731\pi$
$457$	1977.43	0.202408	0.101204	+	0.994866i	$0.467731\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1616.73	0.163338	0.0816690	−	0.996660i	$-0.473975\pi$
$461$	1616.73	0.163338	0.0816690	+	0.996660i	$0.473975\pi$
$462$	0	0
$463$	− 1476.11i	− 0.148165i	−0.997252	−	0.0740827i	$-0.976397\pi$
$463$	− 1476.11i	− 0.148165i	0.997252	−	0.0740827i	$-0.0236029\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15942.8i	1.57975i	0.613268	+	0.789875i	$0.289855\pi$
$467$	15942.8i	1.57975i	−0.613268	+	0.789875i	$0.710145\pi$
$468$	0	0
$469$	− 5960.69i	− 0.586864i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13323.1i	1.29513i
$474$	0	0
$475$	−26157.6	−2.52673
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14684.4	1.40073	0.700363	−	0.713786i	$-0.253021\pi$
$479$	14684.4	1.40073	0.700363	+	0.713786i	$0.253021\pi$
$480$	0	0
$481$	3661.26	0.347067
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−28566.1	−2.67447
$486$	0	0
$487$	− 9871.54i	− 0.918526i	−0.888300	−	0.459263i	$-0.848114\pi$
$487$	− 9871.54i	− 0.918526i	0.888300	−	0.459263i	$-0.151886\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4356.26i	0.400398i	0.979755	+	0.200199i	$0.0641589\pi$
$491$	4356.26i	0.400398i	−0.979755	+	0.200199i	$0.935841\pi$
$492$	0	0
$493$	51.6345i	0.00471704i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11667.4i	1.05303i
$498$	0	0
$499$	15900.3	1.42645	0.713224	−	0.700937i	$-0.247234\pi$
$499$	15900.3	1.42645	0.713224	+	0.700937i	$0.247234\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9083.46	0.805192	0.402596	−	0.915378i	$-0.368108\pi$
$503$	9083.46	0.805192	0.402596	+	0.915378i	$0.368108\pi$
$504$	0	0
$505$	7977.56	0.702964
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4733.57	0.412204	0.206102	−	0.978531i	$-0.433922\pi$
$509$	4733.57	0.412204	0.206102	+	0.978531i	$0.433922\pi$
$510$	0	0
$511$	− 17300.5i	− 1.49771i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17665.0i	1.51148i
$516$	0	0
$517$	16426.8i	1.39739i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 14468.8i	− 1.21668i	−0.793675	−	0.608341i	$-0.791835\pi$
$521$	− 14468.8i	− 1.21668i	0.793675	−	0.608341i	$-0.208165\pi$
$522$	0	0
$523$	12945.2	1.08232	0.541161	−	0.840919i	$-0.317985\pi$
$523$	12945.2	1.08232	0.541161	+	0.840919i	$0.317985\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3869.51	0.319845
$528$	0	0
$529$	−4740.91	−0.389653
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3111.18	0.252833
$534$	0	0
$535$	27958.3i	2.25933i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 6540.52i	− 0.522672i
$540$	0	0
$541$	7859.59i	0.624603i	0.949983	+	0.312301i	$0.101100\pi$
$541$	7859.59i	0.624603i	−0.949983	+	0.312301i	$0.898900\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	11246.7i	0.883952i
$546$	0	0
$547$	20633.2	1.61282	0.806408	−	0.591359i	$-0.201408\pi$
$547$	20633.2	1.61282	0.806408	+	0.591359i	$0.201408\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−509.729	−0.0394105
$552$	0	0
$553$	−1907.12	−0.146653
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22253.3	−1.69282	−0.846411	−	0.532531i	$-0.821241\pi$
$557$	−22253.3	−1.69282	−0.846411	+	0.532531i	$0.821241\pi$
$558$	0	0
$559$	− 2180.52i	− 0.164984i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 7722.80i	− 0.578112i	−0.957312	−	0.289056i	$-0.906659\pi$
$563$	− 7722.80i	− 0.578112i	0.957312	−	0.289056i	$-0.0933415\pi$
$564$	0	0
$565$	1483.50i	0.110462i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5085.66i	0.374696i	0.982294	+	0.187348i	$0.0599892\pi$
$569$	5085.66i	0.374696i	−0.982294	+	0.187348i	$0.940011\pi$
$570$	0	0
$571$	11021.0	0.807728	0.403864	−	0.914819i	$-0.367667\pi$
$571$	11021.0	0.807728	0.403864	+	0.914819i	$0.367667\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−16110.9	−1.16847
$576$	0	0
$577$	−16881.8	−1.21802	−0.609012	−	0.793161i	$-0.708434\pi$
$577$	−16881.8	−1.21802	−0.609012	+	0.793161i	$0.708434\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4517.85	0.322602
$582$	0	0
$583$	33234.0i	2.36091i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3641.73i	− 0.256065i	−0.991770	−	0.128032i	$-0.959134\pi$
$587$	− 3641.73i	− 0.256065i	0.991770	−	0.128032i	$-0.0408662\pi$
$588$	0	0
$589$	38199.3i	2.67228i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 2475.73i	− 0.171444i	−0.996319	−	0.0857219i	$-0.972680\pi$
$593$	− 2475.73i	− 0.171444i	0.996319	−	0.0857219i	$-0.0273197\pi$
$594$	0	0
$595$	−3749.44	−0.258340
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−306.543	−0.0209099	−0.0104549	−	0.999945i	$-0.503328\pi$
$599$	−306.543	−0.0209099	−0.0104549	+	0.999945i	$0.503328\pi$
$600$	0	0
$601$	12411.0	0.842358	0.421179	−	0.906978i	$-0.361616\pi$
$601$	12411.0	0.842358	0.421179	+	0.906978i	$0.361616\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	30160.3	2.02676
$606$	0	0
$607$	1572.02i	0.105118i	0.998618	+	0.0525588i	$0.0167377\pi$
$607$	1572.02i	0.105118i	−0.998618	+	0.0525588i	$0.983262\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 2688.47i	− 0.178010i
$612$	0	0
$613$	− 3597.65i	− 0.237044i	−0.992951	−	0.118522i	$-0.962184\pi$
$613$	− 3597.65i	− 0.237044i	0.992951	−	0.118522i	$-0.0378156\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26075.4i	1.70139i	0.525659	+	0.850695i	$0.323819\pi$
$617$	26075.4i	1.70139i	−0.525659	+	0.850695i	$0.676181\pi$
$618$	0	0
$619$	13844.9	0.898986	0.449493	−	0.893284i	$-0.351605\pi$
$619$	13844.9	0.898986	0.449493	+	0.893284i	$0.351605\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5063.53	0.325627
$624$	0	0
$625$	−4041.83	−0.258677
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5751.73	0.364605
$630$	0	0
$631$	− 906.847i	− 0.0572124i	−0.999591	−	0.0286062i	$-0.990893\pi$
$631$	− 906.847i	− 0.0572124i	0.999591	−	0.0286062i	$-0.00910687\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 16306.5i	− 1.01906i
$636$	0	0
$637$	1070.45i	0.0665820i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	28362.9i	1.74769i	0.486209	+	0.873843i	$0.338379\pi$
$641$	28362.9i	1.74769i	−0.486209	+	0.873843i	$0.661621\pi$
$642$	0	0
$643$	−6606.82	−0.405206	−0.202603	−	0.979261i	$-0.564940\pi$
$643$	−6606.82	−0.405206	−0.202603	+	0.979261i	$0.564940\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18843.5	−1.14500	−0.572499	−	0.819905i	$-0.694026\pi$
$647$	−18843.5	−1.14500	−0.572499	+	0.819905i	$0.694026\pi$
$648$	0	0
$649$	−34765.2	−2.10270
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15382.1	0.921819	0.460910	−	0.887447i	$-0.347523\pi$
$653$	15382.1	0.921819	0.460910	+	0.887447i	$0.347523\pi$
$654$	0	0
$655$	− 6419.40i	− 0.382941i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 6692.00i	− 0.395574i	−0.980245	−	0.197787i	$-0.936625\pi$
$659$	− 6692.00i	− 0.395574i	0.980245	−	0.197787i	$-0.0633754\pi$
$660$	0	0
$661$	− 18708.8i	− 1.10089i	−0.834872	−	0.550444i	$-0.814458\pi$
$661$	− 18708.8i	− 1.10089i	0.834872	−	0.550444i	$-0.185542\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 37014.0i	− 2.15841i
$666$	0	0
$667$	−313.951	−0.0182252
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−13090.6	−0.753142
$672$	0	0
$673$	−21815.7	−1.24953	−0.624766	−	0.780812i	$-0.714805\pi$
$673$	−21815.7	−1.24953	−0.624766	+	0.780812i	$0.714805\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14764.7	0.838191	0.419095	−	0.907942i	$-0.362347\pi$
$677$	14764.7	0.838191	0.419095	+	0.907942i	$0.362347\pi$
$678$	0	0
$679$	− 24225.0i	− 1.36918i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 10522.2i	− 0.589489i	−0.955576	−	0.294744i	$-0.904765\pi$
$683$	− 10522.2i	− 0.589489i	0.955576	−	0.294744i	$-0.0952345\pi$
$684$	0	0
$685$	− 12451.7i	− 0.694531i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 5439.22i	− 0.300751i
$690$	0	0
$691$	−24495.2	−1.34854	−0.674269	−	0.738486i	$-0.735541\pi$
$691$	−24495.2	−1.34854	−0.674269	+	0.738486i	$0.735541\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	36478.8	1.99096
$696$	0	0
$697$	4887.56	0.265609
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22137.1	1.19274	0.596368	−	0.802711i	$-0.296610\pi$
$701$	22137.1	1.19274	0.596368	+	0.802711i	$0.296610\pi$
$702$	0	0
$703$	56780.3i	3.04625i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6765.26i	0.359878i
$708$	0	0
$709$	23366.2i	1.23771i	0.785506	+	0.618854i	$0.212403\pi$
$709$	23366.2i	1.23771i	−0.785506	+	0.618854i	$0.787597\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	23527.6i	1.23579i
$714$	0	0
$715$	−8783.65	−0.459427
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3714.68	−0.192676	−0.0963380	−	0.995349i	$-0.530713\pi$
$719$	−3714.68	−0.192676	−0.0963380	+	0.995349i	$0.530713\pi$
$720$	0	0
$721$	−14980.5	−0.773792
$722$	0	0
$723$	0	0
$724$	0	0
$725$	681.118	0.0348911
$726$	0	0
$727$	6646.80i	0.339087i	0.985523	+	0.169543i	$0.0542293\pi$
$727$	6646.80i	0.339087i	−0.985523	+	0.169543i	$0.945771\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 3425.52i	− 0.173321i
$732$	0	0
$733$	− 22683.9i	− 1.14304i	−0.820589	−	0.571519i	$-0.806354\pi$
$733$	− 22683.9i	− 1.14304i	0.820589	−	0.571519i	$-0.193646\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 21936.8i	− 1.09641i
$738$	0	0
$739$	−30377.3	−1.51211	−0.756053	−	0.654510i	$-0.772875\pi$
$739$	−30377.3	−1.51211	−0.756053	+	0.654510i	$0.772875\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17871.8	−0.882439	−0.441219	−	0.897399i	$-0.645454\pi$
$743$	−17871.8	−0.882439	−0.441219	+	0.897399i	$0.645454\pi$
$744$	0	0
$745$	−19839.5	−0.975656
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−23709.6	−1.15665
$750$	0	0
$751$	− 33183.5i	− 1.61236i	−0.591669	−	0.806181i	$-0.701531\pi$
$751$	− 33183.5i	− 1.61236i	0.591669	−	0.806181i	$-0.298469\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 13215.6i	− 0.637040i
$756$	0	0
$757$	30541.7i	1.46639i	0.680020	+	0.733194i	$0.261971\pi$
$757$	30541.7i	1.46639i	−0.680020	+	0.733194i	$0.738029\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 14713.4i	− 0.700870i	−0.936587	−	0.350435i	$-0.886034\pi$
$761$	− 14713.4i	− 0.700870i	0.936587	−	0.350435i	$-0.113966\pi$
$762$	0	0
$763$	−9537.56	−0.452533
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5689.82	0.267858
$768$	0	0
$769$	1182.88	0.0554689	0.0277345	−	0.999615i	$-0.491171\pi$
$769$	1182.88	0.0554689	0.0277345	+	0.999615i	$0.491171\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8341.02	0.388106	0.194053	−	0.980991i	$-0.437837\pi$
$773$	8341.02	0.388106	0.194053	+	0.980991i	$0.437837\pi$
$774$	0	0
$775$	− 51043.2i	− 2.36584i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	48249.4i	2.21914i
$780$	0	0
$781$	42938.8i	1.96731i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 20572.7i	− 0.935378i
$786$	0	0
$787$	9835.39	0.445482	0.222741	−	0.974878i	$-0.428500\pi$
$787$	9835.39	0.445482	0.222741	+	0.974878i	$0.428500\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1258.06	−0.0565504
$792$	0	0
$793$	2142.47	0.0959410
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20803.2	0.924577	0.462288	−	0.886730i	$-0.347028\pi$
$797$	20803.2	0.924577	0.462288	+	0.886730i	$0.347028\pi$
$798$	0	0
$799$	− 4223.51i	− 0.187005i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 63670.1i	− 2.79809i
$804$	0	0
$805$	− 22797.5i	− 0.998147i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	14439.4i	0.627518i	0.949503	+	0.313759i	$0.101588\pi$
$809$	14439.4i	0.627518i	−0.949503	+	0.313759i	$0.898412\pi$
$810$	0	0
$811$	6457.92	0.279616	0.139808	−	0.990179i	$-0.455352\pi$
$811$	6457.92	0.279616	0.139808	+	0.990179i	$0.455352\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−17243.8	−0.741132
$816$	0	0
$817$	33816.3	1.44808
$818$	0	0
$819$	0	0
$820$	0	0
$821$	37203.3	1.58149	0.790747	−	0.612144i	$-0.209693\pi$
$821$	37203.3	1.58149	0.790747	+	0.612144i	$0.209693\pi$
$822$	0	0
$823$	− 1916.63i	− 0.0811781i	−0.999176	−	0.0405890i	$-0.987077\pi$
$823$	− 1916.63i	− 0.0811781i	0.999176	−	0.0405890i	$-0.0129234\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44003.2i	1.85023i	0.379683	+	0.925117i	$0.376033\pi$
$827$	44003.2i	1.85023i	−0.379683	+	0.925117i	$0.623967\pi$
$828$	0	0
$829$	− 39853.8i	− 1.66970i	−0.550479	−	0.834849i	$-0.685555\pi$
$829$	− 39853.8i	− 1.66970i	0.550479	−	0.834849i	$-0.314445\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1681.64i	0.0699465i
$834$	0	0
$835$	28324.2	1.17389
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30695.6	1.26309	0.631543	−	0.775341i	$-0.282422\pi$
$839$	30695.6	1.26309	0.631543	+	0.775341i	$0.282422\pi$
$840$	0	0
$841$	−24375.7	−0.999456
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−37366.5	−1.52124
$846$	0	0
$847$	25577.0i	1.03759i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	34972.0i	1.40872i
$852$	0	0
$853$	− 36152.2i	− 1.45115i	−0.688144	−	0.725574i	$-0.741574\pi$
$853$	− 36152.2i	− 1.45115i	0.688144	−	0.725574i	$-0.258426\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 13536.1i	− 0.539540i	−0.962925	−	0.269770i	$-0.913052\pi$
$857$	− 13536.1i	− 0.539540i	0.962925	−	0.269770i	$-0.0869477\pi$
$858$	0	0
$859$	−20932.8	−0.831454	−0.415727	−	0.909489i	$-0.636473\pi$
$859$	−20932.8	−0.831454	−0.415727	+	0.909489i	$0.636473\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15068.3	−0.594359	−0.297179	−	0.954822i	$-0.596046\pi$
$863$	−15068.3	−0.594359	−0.297179	+	0.954822i	$0.596046\pi$
$864$	0	0
$865$	4707.60	0.185044
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7018.68	−0.273984
$870$	0	0
$871$	3590.26i	0.139669i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	16390.6i	0.633261i
$876$	0	0
$877$	− 40198.2i	− 1.54777i	−0.633325	−	0.773886i	$-0.718311\pi$
$877$	− 40198.2i	− 1.54777i	0.633325	−	0.773886i	$-0.281689\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 34387.5i	− 1.31503i	−0.753441	−	0.657516i	$-0.771607\pi$
$881$	− 34387.5i	− 1.31503i	0.753441	−	0.657516i	$-0.228393\pi$
$882$	0	0
$883$	9041.95	0.344604	0.172302	−	0.985044i	$-0.444879\pi$
$883$	9041.95	0.344604	0.172302	+	0.985044i	$0.444879\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11974.5	−0.453286	−0.226643	−	0.973978i	$-0.572775\pi$
$887$	−11974.5	−0.453286	−0.226643	+	0.973978i	$0.572775\pi$
$888$	0	0
$889$	13828.5	0.521702
$890$	0	0
$891$	0	0
$892$	0	0
$893$	41694.0	1.56241
$894$	0	0
$895$	− 3494.78i	− 0.130523i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 994.670i	− 0.0369011i
$900$	0	0
$901$	− 8544.85i	− 0.315949i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 50063.7i	− 1.83887i
$906$	0	0
$907$	−27657.0	−1.01250	−0.506249	−	0.862387i	$-0.668968\pi$
$907$	−27657.0	−1.01250	−0.506249	+	0.862387i	$0.668968\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−42952.4	−1.56210	−0.781052	−	0.624466i	$-0.785317\pi$
$911$	−42952.4	−1.56210	−0.781052	+	0.624466i	$0.785317\pi$
$912$	0	0
$913$	16626.8	0.602701
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5443.88	0.196044
$918$	0	0
$919$	− 17076.0i	− 0.612934i	−0.951881	−	0.306467i	$-0.900853\pi$
$919$	− 17076.0i	− 0.612934i	0.951881	−	0.306467i	$-0.0991469\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 7027.54i	− 0.250611i
$924$	0	0
$925$	− 75871.8i	− 2.69692i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 26818.3i	− 0.947126i	−0.880760	−	0.473563i	$-0.842968\pi$
$929$	− 26818.3i	− 0.947126i	0.880760	−	0.473563i	$-0.157032\pi$
$930$	0	0
$931$	−16601.0	−0.584398
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−13798.9	−0.482642
$936$	0	0
$937$	−38163.9	−1.33059	−0.665294	−	0.746582i	$-0.731694\pi$
$937$	−38163.9	−1.33059	−0.665294	+	0.746582i	$0.731694\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−10728.0	−0.371650	−0.185825	−	0.982583i	$-0.559496\pi$
$941$	−10728.0	−0.371650	−0.185825	+	0.982583i	$0.559496\pi$
$942$	0	0
$943$	29717.6i	1.02623i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 28971.4i	− 0.994133i	−0.867713	−	0.497066i	$-0.834411\pi$
$947$	− 28971.4i	− 0.994133i	0.867713	−	0.497066i	$-0.165589\pi$
$948$	0	0
$949$	10420.5i	0.356443i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 36897.2i	− 1.25416i	−0.778954	−	0.627081i	$-0.784250\pi$
$953$	− 36897.2i	− 1.25416i	0.778954	−	0.627081i	$-0.215750\pi$
$954$	0	0
$955$	91638.7	3.10509
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10559.5	0.355561
$960$	0	0
$961$	−44749.9	−1.50213
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3644.58	−0.121578
$966$	0	0
$967$	− 2016.50i	− 0.0670591i	−0.999438	−	0.0335295i	$-0.989325\pi$
$967$	− 2016.50i	− 0.0670591i	0.999438	−	0.0335295i	$-0.0106748\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 18482.0i	− 0.610830i	−0.952219	−	0.305415i	$-0.901205\pi$
$971$	− 18482.0i	− 0.610830i	0.952219	−	0.305415i	$-0.0987952\pi$
$972$	0	0
$973$	30935.3i	1.01926i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 22871.4i	− 0.748945i	−0.927238	−	0.374473i	$-0.877824\pi$
$977$	− 22871.4i	− 0.748945i	0.927238	−	0.374473i	$-0.122176\pi$
$978$	0	0
$979$	18635.0	0.608352
$980$	0	0
$981$	0	0
$982$	0	0
$983$	16630.8	0.539615	0.269808	−	0.962914i	$-0.413040\pi$
$983$	16630.8	0.539615	0.269808	+	0.962914i	$0.413040\pi$
$984$	0	0
$985$	58402.1	1.88918
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20828.1	0.669660
$990$	0	0
$991$	− 21036.2i	− 0.674307i	−0.941450	−	0.337153i	$-0.890536\pi$
$991$	− 21036.2i	− 0.674307i	0.941450	−	0.337153i	$-0.109464\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	61665.6i	1.96475i
$996$	0	0
$997$	− 31241.8i	− 0.992416i	−0.868204	−	0.496208i	$-0.834725\pi$
$997$	− 31241.8i	− 0.992416i	0.868204	−	0.496208i	$-0.165275\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.4.f.e.1151.1		8
3.2	odd	2	inner	2304.4.f.e.1151.7		8
4.3	odd	2		2304.4.f.h.1151.2		8
8.3	odd	2	inner	2304.4.f.e.1151.8		8
8.5	even	2		2304.4.f.h.1151.7		8
12.11	even	2		2304.4.f.h.1151.8		8
16.3	odd	4		288.4.c.b.287.7	yes	8
16.5	even	4		576.4.c.f.575.2		8
16.11	odd	4		576.4.c.f.575.1		8
16.13	even	4		288.4.c.b.287.8	yes	8
24.5	odd	2		2304.4.f.h.1151.1		8
24.11	even	2	inner	2304.4.f.e.1151.2		8
48.5	odd	4		576.4.c.f.575.8		8
48.11	even	4		576.4.c.f.575.7		8
48.29	odd	4		288.4.c.b.287.2	yes	8
48.35	even	4		288.4.c.b.287.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.4.c.b.287.1	✓	8	48.35	even	4
288.4.c.b.287.2	yes	8	48.29	odd	4
288.4.c.b.287.7	yes	8	16.3	odd	4
288.4.c.b.287.8	yes	8	16.13	even	4
576.4.c.f.575.1		8	16.11	odd	4
576.4.c.f.575.2		8	16.5	even	4
576.4.c.f.575.7		8	48.11	even	4
576.4.c.f.575.8		8	48.5	odd	4
2304.4.f.e.1151.1		8	1.1	even	1	trivial
2304.4.f.e.1151.2		8	24.11	even	2	inner
2304.4.f.e.1151.7		8	3.2	odd	2	inner
2304.4.f.e.1151.8		8	8.3	odd	2	inner
2304.4.f.h.1151.1		8	24.5	odd	2
2304.4.f.h.1151.2		8	4.3	odd	2
2304.4.f.h.1151.7		8	8.5	even	2
2304.4.f.h.1151.8		8	12.11	even	2

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2304.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-17.6623 q^{5} -14.9783i q^{7} +O(q^{10})\)	\(q-17.6623 q^{5} -14.9783i q^{7} -55.1236i q^{11} +9.02175i q^{13} +14.1729i q^{17} -139.913 q^{19} -86.1748 q^{23} +186.957 q^{25} +3.64319 q^{29} -273.022i q^{31} +264.550i q^{35} -405.826i q^{37} -344.853i q^{41} -241.696 q^{43} -297.999 q^{47} +118.652 q^{49} -602.901 q^{53} +973.609i q^{55} -630.678i q^{59} -237.478i q^{61} -159.345i q^{65} +397.957 q^{67} -778.955 q^{71} +1155.04 q^{73} -825.655 q^{77} -127.326i q^{79} +301.627i q^{83} -250.326i q^{85} +338.059i q^{89} +135.130 q^{91} +2471.18 q^{95} +1617.35 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 384 q^{19} + 1128 q^{25} + 640 q^{43} - 1992 q^{49} + 2816 q^{67} + 1152 q^{73} - 6272 q^{91} + 7424 q^{97}+O(q^{100}) \) 8 * q - 384 * q^19 + 1128 * q^25 + 640 * q^43 - 1992 * q^49 + 2816 * q^67 + 1152 * q^73 - 6272 * q^91 + 7424 * q^97

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