LMFDB - Embedded newform 2304.3.g.u.1279.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(1279,2304)]

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, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2304.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.g (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:	$62.7794529086$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 64)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1279.4
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2304.1279
Dual form			2304.3.g.u.1279.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.92820 q^{5} +8.00000i q^{7} +O(q^{10})$	$q+6.92820 q^{5} +8.00000i q^{7} -3.46410i q^{11} -6.92820 q^{13} +6.00000 q^{17} +24.2487i q^{19} -24.0000i q^{23} +23.0000 q^{25} -20.7846 q^{29} +32.0000i q^{31} +55.4256i q^{35} -6.92820 q^{37} +66.0000 q^{41} +31.1769i q^{43} +48.0000i q^{47} -15.0000 q^{49} +90.0666 q^{53} -24.0000i q^{55} -31.1769i q^{59} -90.0666 q^{61} -48.0000 q^{65} +79.6743i q^{67} +120.000i q^{71} -58.0000 q^{73} +27.7128 q^{77} -16.0000i q^{79} +58.8897i q^{83} +41.5692 q^{85} -102.000 q^{89} -55.4256i q^{91} +168.000i q^{95} +26.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 24 q^{17} + 92 q^{25} + 264 q^{41} - 60 q^{49} - 192 q^{65} - 232 q^{73} - 408 q^{89} + 104 q^{97}+O(q^{100}) $ 4 * q + 24 * q^17 + 92 * q^25 + 264 * q^41 - 60 * q^49 - 192 * q^65 - 232 * q^73 - 408 * q^89 + 104 * 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
$5$	6.92820	1.38564	0.692820	−	0.721110i	$-0.256368\pi$
$5$	6.92820	1.38564	0.692820	+	0.721110i	$0.256368\pi$
$6$	0	0
$7$	8.00000i	1.14286i	0.820652	+	0.571429i	$0.193611\pi$
$7$	8.00000i	1.14286i	−0.820652	+	0.571429i	$0.806389\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.46410i	− 0.314918i	−0.987525	−	0.157459i	$-0.949670\pi$
$11$	− 3.46410i	− 0.314918i	0.987525	−	0.157459i	$-0.0503303\pi$
$12$	0	0
$13$	−6.92820	−0.532939	−0.266469	−	0.963843i	$-0.585857\pi$
$13$	−6.92820	−0.532939	−0.266469	+	0.963843i	$0.585857\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000	0.352941	0.176471	−	0.984306i	$-0.443532\pi$
$17$	6.00000	0.352941	0.176471	+	0.984306i	$0.443532\pi$
$18$	0	0
$19$	24.2487i	1.27625i	0.769934	+	0.638124i	$0.220289\pi$
$19$	24.2487i	1.27625i	−0.769934	+	0.638124i	$0.779711\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 24.0000i	− 1.04348i	−0.853105	−	0.521739i	$-0.825283\pi$
$23$	− 24.0000i	− 1.04348i	0.853105	−	0.521739i	$-0.174717\pi$
$24$	0	0
$25$	23.0000	0.920000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−20.7846	−0.716711	−0.358355	−	0.933585i	$-0.616662\pi$
$29$	−20.7846	−0.716711	−0.358355	+	0.933585i	$0.616662\pi$
$30$	0	0
$31$	32.0000i	1.03226i	0.856511	+	0.516129i	$0.172628\pi$
$31$	32.0000i	1.03226i	−0.856511	+	0.516129i	$0.827372\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	55.4256i	1.58359i
$36$	0	0
$37$	−6.92820	−0.187249	−0.0936244	−	0.995608i	$-0.529845\pi$
$37$	−6.92820	−0.187249	−0.0936244	+	0.995608i	$0.529845\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	66.0000	1.60976	0.804878	−	0.593440i	$-0.202231\pi$
$41$	66.0000	1.60976	0.804878	+	0.593440i	$0.202231\pi$
$42$	0	0
$43$	31.1769i	0.725045i	0.931975	+	0.362522i	$0.118084\pi$
$43$	31.1769i	0.725045i	−0.931975	+	0.362522i	$0.881916\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	48.0000i	1.02128i	0.859796	+	0.510638i	$0.170591\pi$
$47$	48.0000i	1.02128i	−0.859796	+	0.510638i	$0.829409\pi$
$48$	0	0
$49$	−15.0000	−0.306122
$50$	0	0
$51$	0	0
$52$	0	0
$53$	90.0666	1.69937	0.849685	−	0.527290i	$-0.176792\pi$
$53$	90.0666	1.69937	0.849685	+	0.527290i	$0.176792\pi$
$54$	0	0
$55$	− 24.0000i	− 0.436364i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 31.1769i	− 0.528422i	−0.964465	−	0.264211i	$-0.914888\pi$
$59$	− 31.1769i	− 0.528422i	0.964465	−	0.264211i	$-0.0851116\pi$
$60$	0	0
$61$	−90.0666	−1.47650	−0.738251	−	0.674526i	$-0.764348\pi$
$61$	−90.0666	−1.47650	−0.738251	+	0.674526i	$0.764348\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−48.0000	−0.738462
$66$	0	0
$67$	79.6743i	1.18917i	0.804033	+	0.594585i	$0.202683\pi$
$67$	79.6743i	1.18917i	−0.804033	+	0.594585i	$0.797317\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	120.000i	1.69014i	0.534655	+	0.845070i	$0.320442\pi$
$71$	120.000i	1.69014i	−0.534655	+	0.845070i	$0.679558\pi$
$72$	0	0
$73$	−58.0000	−0.794521	−0.397260	−	0.917706i	$-0.630039\pi$
$73$	−58.0000	−0.794521	−0.397260	+	0.917706i	$0.630039\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	27.7128	0.359907
$78$	0	0
$79$	− 16.0000i	− 0.202532i	−0.994859	−	0.101266i	$-0.967711\pi$
$79$	− 16.0000i	− 0.202532i	0.994859	−	0.101266i	$-0.0322893\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	58.8897i	0.709515i	0.934958	+	0.354757i	$0.115437\pi$
$83$	58.8897i	0.709515i	−0.934958	+	0.354757i	$0.884563\pi$
$84$	0	0
$85$	41.5692	0.489050
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−102.000	−1.14607	−0.573034	−	0.819532i	$-0.694234\pi$
$89$	−102.000	−1.14607	−0.573034	+	0.819532i	$0.694234\pi$
$90$	0	0
$91$	− 55.4256i	− 0.609073i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	168.000i	1.76842i
$96$	0	0
$97$	26.0000	0.268041	0.134021	−	0.990979i	$-0.457211\pi$
$97$	26.0000	0.268041	0.134021	+	0.990979i	$0.457211\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.92820	0.0685961	0.0342980	−	0.999412i	$-0.489080\pi$
$101$	6.92820	0.0685961	0.0342980	+	0.999412i	$0.489080\pi$
$102$	0	0
$103$	40.0000i	0.388350i	0.980967	+	0.194175i	$0.0622029\pi$
$103$	40.0000i	0.388350i	−0.980967	+	0.194175i	$0.937797\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 58.8897i	− 0.550371i	−0.961391	−	0.275186i	$-0.911261\pi$
$107$	− 58.8897i	− 0.550371i	0.961391	−	0.275186i	$-0.0887393\pi$
$108$	0	0
$109$	103.923	0.953422	0.476711	−	0.879060i	$-0.341829\pi$
$109$	103.923	0.953422	0.476711	+	0.879060i	$0.341829\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−66.0000	−0.584071	−0.292035	−	0.956408i	$-0.594333\pi$
$113$	−66.0000	−0.584071	−0.292035	+	0.956408i	$0.594333\pi$
$114$	0	0
$115$	− 166.277i	− 1.44589i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	48.0000i	0.403361i
$120$	0	0
$121$	109.000	0.900826
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−13.8564	−0.110851
$126$	0	0
$127$	64.0000i	0.503937i	0.967735	+	0.251969i	$0.0810779\pi$
$127$	64.0000i	0.503937i	−0.967735	+	0.251969i	$0.918922\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 245.951i	− 1.87749i	−0.344612	−	0.938745i	$-0.611990\pi$
$131$	− 245.951i	− 1.87749i	0.344612	−	0.938745i	$-0.388010\pi$
$132$	0	0
$133$	−193.990	−1.45857
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−30.0000	−0.218978	−0.109489	−	0.993988i	$-0.534921\pi$
$137$	−30.0000	−0.218978	−0.109489	+	0.993988i	$0.534921\pi$
$138$	0	0
$139$	− 79.6743i	− 0.573197i	−0.958051	−	0.286598i	$-0.907475\pi$
$139$	− 79.6743i	− 0.573197i	0.958051	−	0.286598i	$-0.0925245\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	24.0000i	0.167832i
$144$	0	0
$145$	−144.000	−0.993103
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−131.636	−0.883462	−0.441731	−	0.897148i	$-0.645635\pi$
$149$	−131.636	−0.883462	−0.441731	+	0.897148i	$0.645635\pi$
$150$	0	0
$151$	152.000i	1.00662i	0.864105	+	0.503311i	$0.167885\pi$
$151$	152.000i	1.00662i	−0.864105	+	0.503311i	$0.832115\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	221.703i	1.43034i
$156$	0	0
$157$	−145.492	−0.926702	−0.463351	−	0.886175i	$-0.653353\pi$
$157$	−145.492	−0.926702	−0.463351	+	0.886175i	$0.653353\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	192.000	1.19255
$162$	0	0
$163$	− 169.741i	− 1.04136i	−0.853753	−	0.520678i	$-0.825679\pi$
$163$	− 169.741i	− 1.04136i	0.853753	−	0.520678i	$-0.174321\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	216.000i	1.29341i	0.762739	+	0.646707i	$0.223854\pi$
$167$	216.000i	1.29341i	−0.762739	+	0.646707i	$0.776146\pi$
$168$	0	0
$169$	−121.000	−0.715976
$170$	0	0
$171$	0	0
$172$	0	0
$173$	173.205	1.00119	0.500593	−	0.865683i	$-0.333115\pi$
$173$	173.205	1.00119	0.500593	+	0.865683i	$0.333115\pi$
$174$	0	0
$175$	184.000i	1.05143i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 107.387i	− 0.599928i	−0.953950	−	0.299964i	$-0.903025\pi$
$179$	− 107.387i	− 0.599928i	0.953950	−	0.299964i	$-0.0969747\pi$
$180$	0	0
$181$	76.2102	0.421051	0.210526	−	0.977588i	$-0.432482\pi$
$181$	76.2102	0.421051	0.210526	+	0.977588i	$0.432482\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−48.0000	−0.259459
$186$	0	0
$187$	− 20.7846i	− 0.111148i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	218.000	1.12953	0.564767	−	0.825251i	$-0.308966\pi$
$193$	218.000	1.12953	0.564767	+	0.825251i	$0.308966\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−214.774	−1.09022	−0.545112	−	0.838363i	$-0.683513\pi$
$197$	−214.774	−1.09022	−0.545112	+	0.838363i	$0.683513\pi$
$198$	0	0
$199$	328.000i	1.64824i	0.566414	+	0.824121i	$0.308330\pi$
$199$	328.000i	1.64824i	−0.566414	+	0.824121i	$0.691670\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 166.277i	− 0.819098i
$204$	0	0
$205$	457.261	2.23054
$206$	0	0
$207$	0	0
$208$	0	0
$209$	84.0000	0.401914
$210$	0	0
$211$	107.387i	0.508944i	0.967080	+	0.254472i	$0.0819016\pi$
$211$	107.387i	0.508944i	−0.967080	+	0.254472i	$0.918098\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	216.000i	1.00465i
$216$	0	0
$217$	−256.000	−1.17972
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−41.5692	−0.188096
$222$	0	0
$223$	− 32.0000i	− 0.143498i	−0.997423	−	0.0717489i	$-0.977142\pi$
$223$	− 32.0000i	− 0.143498i	0.997423	−	0.0717489i	$-0.0228580\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	114.315i	0.503592i	0.967780	+	0.251796i	$0.0810212\pi$
$227$	114.315i	0.503592i	−0.967780	+	0.251796i	$0.918979\pi$
$228$	0	0
$229$	270.200	1.17991	0.589956	−	0.807435i	$-0.299145\pi$
$229$	270.200	1.17991	0.589956	+	0.807435i	$0.299145\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	186.000	0.798283	0.399142	−	0.916889i	$-0.369308\pi$
$233$	186.000	0.798283	0.399142	+	0.916889i	$0.369308\pi$
$234$	0	0
$235$	332.554i	1.41512i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	48.0000i	0.200837i	0.994945	+	0.100418i	$0.0320181\pi$
$239$	48.0000i	0.200837i	−0.994945	+	0.100418i	$0.967982\pi$
$240$	0	0
$241$	250.000	1.03734	0.518672	−	0.854973i	$-0.326427\pi$
$241$	250.000	1.03734	0.518672	+	0.854973i	$0.326427\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−103.923	−0.424176
$246$	0	0
$247$	− 168.000i	− 0.680162i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	301.377i	1.20070i	0.799736	+	0.600352i	$0.204973\pi$
$251$	301.377i	1.20070i	−0.799736	+	0.600352i	$0.795027\pi$
$252$	0	0
$253$	−83.1384	−0.328610
$254$	0	0
$255$	0	0
$256$	0	0
$257$	222.000	0.863813	0.431907	−	0.901918i	$-0.357841\pi$
$257$	222.000	0.863813	0.431907	+	0.901918i	$0.357841\pi$
$258$	0	0
$259$	− 55.4256i	− 0.213999i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	312.000i	1.18631i	0.805088	+	0.593156i	$0.202118\pi$
$263$	312.000i	1.18631i	−0.805088	+	0.593156i	$0.797882\pi$
$264$	0	0
$265$	624.000	2.35472
$266$	0	0
$267$	0	0
$268$	0	0
$269$	62.3538	0.231799	0.115899	−	0.993261i	$-0.463025\pi$
$269$	62.3538	0.231799	0.115899	+	0.993261i	$0.463025\pi$
$270$	0	0
$271$	112.000i	0.413284i	0.978417	+	0.206642i	$0.0662536\pi$
$271$	112.000i	0.413284i	−0.978417	+	0.206642i	$0.933746\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 79.6743i	− 0.289725i
$276$	0	0
$277$	−145.492	−0.525243	−0.262621	−	0.964899i	$-0.584587\pi$
$277$	−145.492	−0.525243	−0.262621	+	0.964899i	$0.584587\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	90.0000	0.320285	0.160142	−	0.987094i	$-0.448805\pi$
$281$	90.0000	0.320285	0.160142	+	0.987094i	$0.448805\pi$
$282$	0	0
$283$	− 218.238i	− 0.771160i	−0.922674	−	0.385580i	$-0.874001\pi$
$283$	− 218.238i	− 0.771160i	0.922674	−	0.385580i	$-0.125999\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	528.000i	1.83972i
$288$	0	0
$289$	−253.000	−0.875433
$290$	0	0
$291$	0	0
$292$	0	0
$293$	62.3538	0.212812	0.106406	−	0.994323i	$-0.466066\pi$
$293$	62.3538	0.212812	0.106406	+	0.994323i	$0.466066\pi$
$294$	0	0
$295$	− 216.000i	− 0.732203i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	166.277i	0.556110i
$300$	0	0
$301$	−249.415	−0.828622
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−624.000	−2.04590
$306$	0	0
$307$	135.100i	0.440065i	0.975493	+	0.220033i	$0.0706163\pi$
$307$	135.100i	0.440065i	−0.975493	+	0.220033i	$0.929384\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	264.000i	0.848875i	0.905457	+	0.424437i	$0.139528\pi$
$311$	264.000i	0.848875i	−0.905457	+	0.424437i	$0.860472\pi$
$312$	0	0
$313$	−226.000	−0.722045	−0.361022	−	0.932557i	$-0.617572\pi$
$313$	−226.000	−0.722045	−0.361022	+	0.932557i	$0.617572\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−408.764	−1.28948	−0.644738	−	0.764404i	$-0.723034\pi$
$317$	−408.764	−1.28948	−0.644738	+	0.764404i	$0.723034\pi$
$318$	0	0
$319$	72.0000i	0.225705i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	145.492i	0.450440i
$324$	0	0
$325$	−159.349	−0.490304
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−384.000	−1.16717
$330$	0	0
$331$	502.295i	1.51751i	0.651378	+	0.758753i	$0.274191\pi$
$331$	502.295i	1.51751i	−0.651378	+	0.758753i	$0.725809\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	552.000i	1.64776i
$336$	0	0
$337$	−382.000	−1.13353	−0.566766	−	0.823879i	$-0.691805\pi$
$337$	−382.000	−1.13353	−0.566766	+	0.823879i	$0.691805\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	110.851	0.325077
$342$	0	0
$343$	272.000i	0.793003i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 640.859i	− 1.84686i	−0.383773	−	0.923428i	$-0.625375\pi$
$347$	− 640.859i	− 1.84686i	0.383773	−	0.923428i	$-0.374625\pi$
$348$	0	0
$349$	464.190	1.33006	0.665028	−	0.746818i	$-0.268420\pi$
$349$	464.190	1.33006	0.665028	+	0.746818i	$0.268420\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	318.000	0.900850	0.450425	−	0.892814i	$-0.351272\pi$
$353$	318.000	0.900850	0.450425	+	0.892814i	$0.351272\pi$
$354$	0	0
$355$	831.384i	2.34193i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 552.000i	− 1.53760i	−0.639487	−	0.768802i	$-0.720853\pi$
$359$	− 552.000i	− 1.53760i	0.639487	−	0.768802i	$-0.279147\pi$
$360$	0	0
$361$	−227.000	−0.628809
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−401.836	−1.10092
$366$	0	0
$367$	− 496.000i	− 1.35150i	−0.737132	−	0.675749i	$-0.763820\pi$
$367$	− 496.000i	− 1.35150i	0.737132	−	0.675749i	$-0.236180\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	720.533i	1.94214i
$372$	0	0
$373$	575.041	1.54166	0.770832	−	0.637038i	$-0.219841\pi$
$373$	575.041	1.54166	0.770832	+	0.637038i	$0.219841\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	144.000	0.381963
$378$	0	0
$379$	− 245.951i	− 0.648948i	−0.945895	−	0.324474i	$-0.894813\pi$
$379$	− 245.951i	− 0.648948i	0.945895	−	0.324474i	$-0.105187\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 384.000i	− 1.00261i	−0.865270	−	0.501305i	$-0.832853\pi$
$383$	− 384.000i	− 1.00261i	0.865270	−	0.501305i	$-0.167147\pi$
$384$	0	0
$385$	192.000	0.498701
$386$	0	0
$387$	0	0
$388$	0	0
$389$	284.056	0.730222	0.365111	−	0.930964i	$-0.381031\pi$
$389$	284.056	0.730222	0.365111	+	0.930964i	$0.381031\pi$
$390$	0	0
$391$	− 144.000i	− 0.368286i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 110.851i	− 0.280636i
$396$	0	0
$397$	−394.908	−0.994729	−0.497365	−	0.867542i	$-0.665699\pi$
$397$	−394.908	−0.994729	−0.497365	+	0.867542i	$0.665699\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−378.000	−0.942643	−0.471322	−	0.881961i	$-0.656223\pi$
$401$	−378.000	−0.942643	−0.471322	+	0.881961i	$0.656223\pi$
$402$	0	0
$403$	− 221.703i	− 0.550130i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.0000i	0.0589681i
$408$	0	0
$409$	−34.0000	−0.0831296	−0.0415648	−	0.999136i	$-0.513234\pi$
$409$	−34.0000	−0.0831296	−0.0415648	+	0.999136i	$0.513234\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	249.415	0.603911
$414$	0	0
$415$	408.000i	0.983133i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 301.377i	− 0.719276i	−0.933092	−	0.359638i	$-0.882900\pi$
$419$	− 301.377i	− 0.719276i	0.933092	−	0.359638i	$-0.117100\pi$
$420$	0	0
$421$	48.4974	0.115196	0.0575979	−	0.998340i	$-0.481656\pi$
$421$	48.4974	0.115196	0.0575979	+	0.998340i	$0.481656\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	138.000	0.324706
$426$	0	0
$427$	− 720.533i	− 1.68743i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 144.000i	− 0.334107i	−0.985948	−	0.167053i	$-0.946575\pi$
$431$	− 144.000i	− 0.334107i	0.985948	−	0.167053i	$-0.0534252\pi$
$432$	0	0
$433$	−134.000	−0.309469	−0.154734	−	0.987956i	$-0.549452\pi$
$433$	−134.000	−0.309469	−0.154734	+	0.987956i	$0.549452\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	581.969	1.33174
$438$	0	0
$439$	376.000i	0.856492i	0.903662	+	0.428246i	$0.140868\pi$
$439$	376.000i	0.856492i	−0.903662	+	0.428246i	$0.859132\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	384.515i	0.867980i	0.900918	+	0.433990i	$0.142895\pi$
$443$	384.515i	0.867980i	−0.900918	+	0.433990i	$0.857105\pi$
$444$	0	0
$445$	−706.677	−1.58804
$446$	0	0
$447$	0	0
$448$	0	0
$449$	102.000	0.227171	0.113586	−	0.993528i	$-0.463766\pi$
$449$	102.000	0.227171	0.113586	+	0.993528i	$0.463766\pi$
$450$	0	0
$451$	− 228.631i	− 0.506942i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 384.000i	− 0.843956i
$456$	0	0
$457$	254.000	0.555799	0.277899	−	0.960610i	$-0.410362\pi$
$457$	254.000	0.555799	0.277899	+	0.960610i	$0.410362\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−48.4974	−0.105200	−0.0526002	−	0.998616i	$-0.516751\pi$
$461$	−48.4974	−0.105200	−0.0526002	+	0.998616i	$0.516751\pi$
$462$	0	0
$463$	880.000i	1.90065i	0.311263	+	0.950324i	$0.399248\pi$
$463$	880.000i	1.90065i	−0.311263	+	0.950324i	$0.600752\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	142.028i	0.304129i	0.988371	+	0.152064i	$0.0485921\pi$
$467$	142.028i	0.304129i	−0.988371	+	0.152064i	$0.951408\pi$
$468$	0	0
$469$	−637.395	−1.35905
$470$	0	0
$471$	0	0
$472$	0	0
$473$	108.000	0.228330
$474$	0	0
$475$	557.720i	1.17415i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 480.000i	− 1.00209i	−0.865422	−	0.501044i	$-0.832950\pi$
$479$	− 480.000i	− 1.00209i	0.865422	−	0.501044i	$-0.167050\pi$
$480$	0	0
$481$	48.0000	0.0997921
$482$	0	0
$483$	0	0
$484$	0	0
$485$	180.133	0.371409
$486$	0	0
$487$	− 280.000i	− 0.574949i	−0.957788	−	0.287474i	$-0.907184\pi$
$487$	− 280.000i	− 0.574949i	0.957788	−	0.287474i	$-0.0928156\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 751.710i	− 1.53098i	−0.643449	−	0.765489i	$-0.722497\pi$
$491$	− 751.710i	− 1.53098i	0.643449	−	0.765489i	$-0.277503\pi$
$492$	0	0
$493$	−124.708	−0.252957
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−960.000	−1.93159
$498$	0	0
$499$	− 280.592i	− 0.562309i	−0.959663	−	0.281155i	$-0.909283\pi$
$499$	− 280.592i	− 0.562309i	0.959663	−	0.281155i	$-0.0907174\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 120.000i	− 0.238569i	−0.992860	−	0.119284i	$-0.961940\pi$
$503$	− 120.000i	− 0.238569i	0.992860	−	0.119284i	$-0.0380600\pi$
$504$	0	0
$505$	48.0000	0.0950495
$506$	0	0
$507$	0	0
$508$	0	0
$509$	533.472	1.04808	0.524039	−	0.851694i	$-0.324425\pi$
$509$	533.472	1.04808	0.524039	+	0.851694i	$0.324425\pi$
$510$	0	0
$511$	− 464.000i	− 0.908023i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	277.128i	0.538113i
$516$	0	0
$517$	166.277	0.321619
$518$	0	0
$519$	0	0
$520$	0	0
$521$	354.000	0.679463	0.339731	−	0.940523i	$-0.389664\pi$
$521$	354.000	0.679463	0.339731	+	0.940523i	$0.389664\pi$
$522$	0	0
$523$	− 162.813i	− 0.311305i	−0.987812	−	0.155653i	$-0.950252\pi$
$523$	− 162.813i	− 0.311305i	0.987812	−	0.155653i	$-0.0497481\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	192.000i	0.364326i
$528$	0	0
$529$	−47.0000	−0.0888469
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−457.261	−0.857901
$534$	0	0
$535$	− 408.000i	− 0.762617i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	51.9615i	0.0964036i
$540$	0	0
$541$	1073.87	1.98498	0.992488	−	0.122346i	$-0.0390417\pi$
$541$	1073.87	1.98498	0.992488	+	0.122346i	$0.0390417\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	720.000	1.32110
$546$	0	0
$547$	− 917.987i	− 1.67822i	−0.543961	−	0.839111i	$-0.683076\pi$
$547$	− 917.987i	− 1.67822i	0.543961	−	0.839111i	$-0.316924\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 504.000i	− 0.914701i
$552$	0	0
$553$	128.000	0.231465
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−381.051	−0.684113	−0.342057	−	0.939679i	$-0.611123\pi$
$557$	−381.051	−0.684113	−0.342057	+	0.939679i	$0.611123\pi$
$558$	0	0
$559$	− 216.000i	− 0.386404i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 911.059i	− 1.61822i	−0.587656	−	0.809111i	$-0.699949\pi$
$563$	− 911.059i	− 1.61822i	0.587656	−	0.809111i	$-0.300051\pi$
$564$	0	0
$565$	−457.261	−0.809312
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−702.000	−1.23374	−0.616872	−	0.787064i	$-0.711600\pi$
$569$	−702.000	−1.23374	−0.616872	+	0.787064i	$0.711600\pi$
$570$	0	0
$571$	− 938.772i	− 1.64408i	−0.569427	−	0.822042i	$-0.692835\pi$
$571$	− 938.772i	− 1.64408i	0.569427	−	0.822042i	$-0.307165\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 552.000i	− 0.960000i
$576$	0	0
$577$	802.000	1.38995	0.694974	−	0.719035i	$-0.255416\pi$
$577$	802.000	1.38995	0.694974	+	0.719035i	$0.255416\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−471.118	−0.810874
$582$	0	0
$583$	− 312.000i	− 0.535163i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	107.387i	0.182942i	0.995808	+	0.0914712i	$0.0291569\pi$
$587$	107.387i	0.182942i	−0.995808	+	0.0914712i	$0.970843\pi$
$588$	0	0
$589$	−775.959	−1.31742
$590$	0	0
$591$	0	0
$592$	0	0
$593$	798.000	1.34570	0.672850	−	0.739779i	$-0.265070\pi$
$593$	798.000	1.34570	0.672850	+	0.739779i	$0.265070\pi$
$594$	0	0
$595$	332.554i	0.558914i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 24.0000i	− 0.0400668i	−0.999799	−	0.0200334i	$-0.993623\pi$
$599$	− 24.0000i	− 0.0400668i	0.999799	−	0.0200334i	$-0.00637725\pi$
$600$	0	0
$601$	−218.000	−0.362729	−0.181364	−	0.983416i	$-0.558051\pi$
$601$	−218.000	−0.362729	−0.181364	+	0.983416i	$0.558051\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	755.174	1.24822
$606$	0	0
$607$	− 608.000i	− 1.00165i	−0.865549	−	0.500824i	$-0.833030\pi$
$607$	− 608.000i	− 1.00165i	0.865549	−	0.500824i	$-0.166970\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 332.554i	− 0.544278i
$612$	0	0
$613$	381.051	0.621617	0.310808	−	0.950473i	$-0.399400\pi$
$613$	381.051	0.621617	0.310808	+	0.950473i	$0.399400\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	954.000	1.54619	0.773096	−	0.634289i	$-0.218707\pi$
$617$	954.000	1.54619	0.773096	+	0.634289i	$0.218707\pi$
$618$	0	0
$619$	225.167i	0.363759i	0.983321	+	0.181879i	$0.0582180\pi$
$619$	225.167i	0.363759i	−0.983321	+	0.181879i	$0.941782\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 816.000i	− 1.30979i
$624$	0	0
$625$	−671.000	−1.07360
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−41.5692	−0.0660878
$630$	0	0
$631$	248.000i	0.393027i	0.980501	+	0.196513i	$0.0629619\pi$
$631$	248.000i	0.393027i	−0.980501	+	0.196513i	$0.937038\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	443.405i	0.698276i
$636$	0	0
$637$	103.923	0.163145
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1062.00	1.65679	0.828393	−	0.560147i	$-0.189255\pi$
$641$	1062.00	1.65679	0.828393	+	0.560147i	$0.189255\pi$
$642$	0	0
$643$	301.377i	0.468704i	0.972152	+	0.234352i	$0.0752969\pi$
$643$	301.377i	0.468704i	−0.972152	+	0.234352i	$0.924703\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	696.000i	1.07573i	0.843030	+	0.537867i	$0.180770\pi$
$647$	696.000i	1.07573i	−0.843030	+	0.537867i	$0.819230\pi$
$648$	0	0
$649$	−108.000	−0.166410
$650$	0	0
$651$	0	0
$652$	0	0
$653$	505.759	0.774516	0.387258	−	0.921971i	$-0.373422\pi$
$653$	505.759	0.774516	0.387258	+	0.921971i	$0.373422\pi$
$654$	0	0
$655$	− 1704.00i	− 2.60153i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 938.772i	− 1.42454i	−0.701906	−	0.712270i	$-0.747667\pi$
$659$	− 938.772i	− 1.42454i	0.701906	−	0.712270i	$-0.252333\pi$
$660$	0	0
$661$	−200.918	−0.303961	−0.151980	−	0.988384i	$-0.548565\pi$
$661$	−200.918	−0.303961	−0.151980	+	0.988384i	$0.548565\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1344.00	−2.02105
$666$	0	0
$667$	498.831i	0.747872i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	312.000i	0.464978i
$672$	0	0
$673$	730.000	1.08470	0.542348	−	0.840154i	$-0.317536\pi$
$673$	730.000	1.08470	0.542348	+	0.840154i	$0.317536\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	893.738	1.32015	0.660073	−	0.751202i	$-0.270526\pi$
$677$	893.738	1.32015	0.660073	+	0.751202i	$0.270526\pi$
$678$	0	0
$679$	208.000i	0.306333i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	467.654i	0.684705i	0.939572	+	0.342353i	$0.111224\pi$
$683$	467.654i	0.684705i	−0.939572	+	0.342353i	$0.888776\pi$
$684$	0	0
$685$	−207.846	−0.303425
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−624.000	−0.905660
$690$	0	0
$691$	− 446.869i	− 0.646699i	−0.946280	−	0.323350i	$-0.895191\pi$
$691$	− 446.869i	− 0.646699i	0.946280	−	0.323350i	$-0.104809\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 552.000i	− 0.794245i
$696$	0	0
$697$	396.000	0.568149
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−131.636	−0.187783	−0.0938915	−	0.995582i	$-0.529931\pi$
$701$	−131.636	−0.187783	−0.0938915	+	0.995582i	$0.529931\pi$
$702$	0	0
$703$	− 168.000i	− 0.238976i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	55.4256i	0.0783955i
$708$	0	0
$709$	−1281.72	−1.80778	−0.903891	−	0.427763i	$-0.859302\pi$
$709$	−1281.72	−1.80778	−0.903891	+	0.427763i	$0.859302\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	768.000	1.07714
$714$	0	0
$715$	166.277i	0.232555i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1008.00i	− 1.40195i	−0.713187	−	0.700974i	$-0.752749\pi$
$719$	− 1008.00i	− 1.40195i	0.713187	−	0.700974i	$-0.247251\pi$
$720$	0	0
$721$	−320.000	−0.443828
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−478.046	−0.659374
$726$	0	0
$727$	− 1384.00i	− 1.90371i	−0.306543	−	0.951857i	$-0.599172\pi$
$727$	− 1384.00i	− 1.90371i	0.306543	−	0.951857i	$-0.400828\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	187.061i	0.255898i
$732$	0	0
$733$	1018.45	1.38942	0.694711	−	0.719289i	$-0.255532\pi$
$733$	1018.45	1.38942	0.694711	+	0.719289i	$0.255532\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	276.000	0.374491
$738$	0	0
$739$	− 446.869i	− 0.604694i	−0.953198	−	0.302347i	$-0.902230\pi$
$739$	− 446.869i	− 0.604694i	0.953198	−	0.302347i	$-0.0977702\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 744.000i	− 1.00135i	−0.865637	−	0.500673i	$-0.833086\pi$
$743$	− 744.000i	− 1.00135i	0.865637	−	0.500673i	$-0.166914\pi$
$744$	0	0
$745$	−912.000	−1.22416
$746$	0	0
$747$	0	0
$748$	0	0
$749$	471.118	0.628996
$750$	0	0
$751$	848.000i	1.12916i	0.825378	+	0.564581i	$0.190962\pi$
$751$	848.000i	1.12916i	−0.825378	+	0.564581i	$0.809038\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1053.09i	1.39482i
$756$	0	0
$757$	297.913	0.393544	0.196772	−	0.980449i	$-0.436954\pi$
$757$	297.913	0.393544	0.196772	+	0.980449i	$0.436954\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−414.000	−0.544021	−0.272011	−	0.962294i	$-0.587689\pi$
$761$	−414.000	−0.544021	−0.272011	+	0.962294i	$0.587689\pi$
$762$	0	0
$763$	831.384i	1.08963i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	216.000i	0.281617i
$768$	0	0
$769$	−358.000	−0.465540	−0.232770	−	0.972532i	$-0.574779\pi$
$769$	−358.000	−0.465540	−0.232770	+	0.972532i	$0.574779\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1267.86	−1.64018	−0.820091	−	0.572233i	$-0.806077\pi$
$773$	−1267.86	−1.64018	−0.820091	+	0.572233i	$0.806077\pi$
$774$	0	0
$775$	736.000i	0.949677i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1600.41i	2.05445i
$780$	0	0
$781$	415.692	0.532256
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1008.00	−1.28408
$786$	0	0
$787$	772.495i	0.981569i	0.871281	+	0.490784i	$0.163290\pi$
$787$	772.495i	0.981569i	−0.871281	+	0.490784i	$0.836710\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 528.000i	− 0.667509i
$792$	0	0
$793$	624.000	0.786885
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−76.2102	−0.0956214	−0.0478107	−	0.998856i	$-0.515224\pi$
$797$	−76.2102	−0.0956214	−0.0478107	+	0.998856i	$0.515224\pi$
$798$	0	0
$799$	288.000i	0.360451i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	200.918i	0.250209i
$804$	0	0
$805$	1330.22	1.65244
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−702.000	−0.867738	−0.433869	−	0.900976i	$-0.642852\pi$
$809$	−702.000	−0.867738	−0.433869	+	0.900976i	$0.642852\pi$
$810$	0	0
$811$	446.869i	0.551010i	0.961300	+	0.275505i	$0.0888451\pi$
$811$	446.869i	0.551010i	−0.961300	+	0.275505i	$0.911155\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1176.00i	− 1.44294i
$816$	0	0
$817$	−756.000	−0.925337
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1351.00	−1.64555	−0.822777	−	0.568365i	$-0.807576\pi$
$821$	−1351.00	−1.64555	−0.822777	+	0.568365i	$0.807576\pi$
$822$	0	0
$823$	− 520.000i	− 0.631835i	−0.948787	−	0.315917i	$-0.897688\pi$
$823$	− 520.000i	− 0.631835i	0.948787	−	0.315917i	$-0.102312\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1111.98i	− 1.34459i	−0.740283	−	0.672295i	$-0.765308\pi$
$827$	− 1111.98i	− 1.34459i	0.740283	−	0.672295i	$-0.234692\pi$
$828$	0	0
$829$	−200.918	−0.242362	−0.121181	−	0.992630i	$-0.538668\pi$
$829$	−200.918	−0.242362	−0.121181	+	0.992630i	$0.538668\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−90.0000	−0.108043
$834$	0	0
$835$	1496.49i	1.79221i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 72.0000i	− 0.0858164i	−0.999079	−	0.0429082i	$-0.986338\pi$
$839$	− 72.0000i	− 0.0858164i	0.999079	−	0.0429082i	$-0.0136623\pi$
$840$	0	0
$841$	−409.000	−0.486326
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−838.313	−0.992086
$846$	0	0
$847$	872.000i	1.02952i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	166.277i	0.195390i
$852$	0	0
$853$	−256.344	−0.300520	−0.150260	−	0.988647i	$-0.548011\pi$
$853$	−256.344	−0.300520	−0.150260	+	0.988647i	$0.548011\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−510.000	−0.595099	−0.297550	−	0.954706i	$-0.596169\pi$
$857$	−510.000	−0.595099	−0.297550	+	0.954706i	$0.596169\pi$
$858$	0	0
$859$	973.413i	1.13319i	0.823995	+	0.566596i	$0.191740\pi$
$859$	973.413i	1.13319i	−0.823995	+	0.566596i	$0.808260\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1248.00i	− 1.44612i	−0.690786	−	0.723059i	$-0.742735\pi$
$863$	− 1248.00i	− 1.44612i	0.690786	−	0.723059i	$-0.257265\pi$
$864$	0	0
$865$	1200.00	1.38728
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−55.4256	−0.0637809
$870$	0	0
$871$	− 552.000i	− 0.633754i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 110.851i	− 0.126687i
$876$	0	0
$877$	−450.333	−0.513493	−0.256746	−	0.966479i	$-0.582651\pi$
$877$	−450.333	−0.513493	−0.256746	+	0.966479i	$0.582651\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1218.00	−1.38252	−0.691260	−	0.722606i	$-0.742944\pi$
$881$	−1218.00	−1.38252	−0.691260	+	0.722606i	$0.742944\pi$
$882$	0	0
$883$	− 252.879i	− 0.286387i	−0.989695	−	0.143193i	$-0.954263\pi$
$883$	− 252.879i	− 0.286387i	0.989695	−	0.143193i	$-0.0457371\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1464.00i	− 1.65051i	−0.564762	−	0.825254i	$-0.691032\pi$
$887$	− 1464.00i	− 1.65051i	0.564762	−	0.825254i	$-0.308968\pi$
$888$	0	0
$889$	−512.000	−0.575928
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1163.94	−1.30340
$894$	0	0
$895$	− 744.000i	− 0.831285i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 665.108i	− 0.739830i
$900$	0	0
$901$	540.400	0.599778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	528.000	0.583425
$906$	0	0
$907$	− 1243.61i	− 1.37113i	−0.728013	−	0.685564i	$-0.759556\pi$
$907$	− 1243.61i	− 1.37113i	0.728013	−	0.685564i	$-0.240444\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 624.000i	− 0.684962i	−0.939525	−	0.342481i	$-0.888733\pi$
$911$	− 624.000i	− 0.684962i	0.939525	−	0.342481i	$-0.111267\pi$
$912$	0	0
$913$	204.000	0.223439
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1967.61	2.14570
$918$	0	0
$919$	1304.00i	1.41893i	0.704739	+	0.709467i	$0.251064\pi$
$919$	1304.00i	1.41893i	−0.704739	+	0.709467i	$0.748936\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 831.384i	− 0.900741i
$924$	0	0
$925$	−159.349	−0.172269
$926$	0	0
$927$	0	0
$928$	0	0
$929$	486.000	0.523143	0.261572	−	0.965184i	$-0.415759\pi$
$929$	486.000	0.523143	0.261572	+	0.965184i	$0.415759\pi$
$930$	0	0
$931$	− 363.731i	− 0.390688i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 144.000i	− 0.154011i
$936$	0	0
$937$	1094.00	1.16756	0.583778	−	0.811913i	$-0.301574\pi$
$937$	1094.00	1.16756	0.583778	+	0.811913i	$0.301574\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−214.774	−0.228240	−0.114120	−	0.993467i	$-0.536405\pi$
$941$	−214.774	−0.228240	−0.114120	+	0.993467i	$0.536405\pi$
$942$	0	0
$943$	− 1584.00i	− 1.67975i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	640.859i	0.676725i	0.941016	+	0.338363i	$0.109873\pi$
$947$	640.859i	0.676725i	−0.941016	+	0.338363i	$0.890127\pi$
$948$	0	0
$949$	401.836	0.423431
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1218.00	1.27807	0.639035	−	0.769178i	$-0.279334\pi$
$953$	1218.00	1.27807	0.639035	+	0.769178i	$0.279334\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 240.000i	− 0.250261i
$960$	0	0
$961$	−63.0000	−0.0655567
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1510.35	1.56513
$966$	0	0
$967$	− 56.0000i	− 0.0579111i	−0.999581	−	0.0289555i	$-0.990782\pi$
$967$	− 56.0000i	− 0.0579111i	0.999581	−	0.0289555i	$-0.00921812\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 86.6025i	− 0.0891890i	−0.999005	−	0.0445945i	$-0.985800\pi$
$971$	− 86.6025i	− 0.0891890i	0.999005	−	0.0445945i	$-0.0141996\pi$
$972$	0	0
$973$	637.395	0.655082
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−570.000	−0.583419	−0.291709	−	0.956507i	$-0.594224\pi$
$977$	−570.000	−0.583419	−0.291709	+	0.956507i	$0.594224\pi$
$978$	0	0
$979$	353.338i	0.360918i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 792.000i	− 0.805697i	−0.915267	−	0.402848i	$-0.868020\pi$
$983$	− 792.000i	− 0.805697i	0.915267	−	0.402848i	$-0.131980\pi$
$984$	0	0
$985$	−1488.00	−1.51066
$986$	0	0
$987$	0	0
$988$	0	0
$989$	748.246	0.756568
$990$	0	0
$991$	1184.00i	1.19475i	0.801961	+	0.597376i	$0.203790\pi$
$991$	1184.00i	1.19475i	−0.801961	+	0.597376i	$0.796210\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2272.45i	2.28387i
$996$	0	0
$997$	769.031	0.771345	0.385672	−	0.922636i	$-0.373970\pi$
$997$	769.031	0.771345	0.385672	+	0.922636i	$0.373970\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.g.u.1279.4		4
3.2	odd	2		256.3.c.g.255.1		4
4.3	odd	2	inner	2304.3.g.u.1279.3		4
8.3	odd	2	inner	2304.3.g.u.1279.1		4
8.5	even	2	inner	2304.3.g.u.1279.2		4
12.11	even	2		256.3.c.g.255.3		4
16.3	odd	4		576.3.b.d.415.2		4
16.5	even	4		576.3.b.d.415.3		4
16.11	odd	4		576.3.b.d.415.4		4
16.13	even	4		576.3.b.d.415.1		4
24.5	odd	2		256.3.c.g.255.4		4
24.11	even	2		256.3.c.g.255.2		4
48.5	odd	4		64.3.d.a.31.1	✓	4
48.11	even	4		64.3.d.a.31.3	yes	4
48.29	odd	4		64.3.d.a.31.4	yes	4
48.35	even	4		64.3.d.a.31.2	yes	4
240.29	odd	4		1600.3.g.c.351.2		4
240.53	even	4		1600.3.e.a.799.1		4
240.59	even	4		1600.3.g.c.351.1		4
240.77	even	4		1600.3.e.f.799.2		4
240.83	odd	4		1600.3.e.f.799.1		4
240.107	odd	4		1600.3.e.a.799.2		4
240.149	odd	4		1600.3.g.c.351.4		4
240.173	even	4		1600.3.e.a.799.4		4
240.179	even	4		1600.3.g.c.351.3		4
240.197	even	4		1600.3.e.f.799.3		4
240.203	odd	4		1600.3.e.f.799.4		4
240.227	odd	4		1600.3.e.a.799.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.3.d.a.31.1	✓	4	48.5	odd	4
64.3.d.a.31.2	yes	4	48.35	even	4
64.3.d.a.31.3	yes	4	48.11	even	4
64.3.d.a.31.4	yes	4	48.29	odd	4
256.3.c.g.255.1		4	3.2	odd	2
256.3.c.g.255.2		4	24.11	even	2
256.3.c.g.255.3		4	12.11	even	2
256.3.c.g.255.4		4	24.5	odd	2
576.3.b.d.415.1		4	16.13	even	4
576.3.b.d.415.2		4	16.3	odd	4
576.3.b.d.415.3		4	16.5	even	4
576.3.b.d.415.4		4	16.11	odd	4
1600.3.e.a.799.1		4	240.53	even	4
1600.3.e.a.799.2		4	240.107	odd	4
1600.3.e.a.799.3		4	240.227	odd	4
1600.3.e.a.799.4		4	240.173	even	4
1600.3.e.f.799.1		4	240.83	odd	4
1600.3.e.f.799.2		4	240.77	even	4
1600.3.e.f.799.3		4	240.197	even	4
1600.3.e.f.799.4		4	240.203	odd	4
1600.3.g.c.351.1		4	240.59	even	4
1600.3.g.c.351.2		4	240.29	odd	4
1600.3.g.c.351.3		4	240.179	even	4
1600.3.g.c.351.4		4	240.149	odd	4
2304.3.g.u.1279.1		4	8.3	odd	2	inner
2304.3.g.u.1279.2		4	8.5	even	2	inner
2304.3.g.u.1279.3		4	4.3	odd	2	inner
2304.3.g.u.1279.4		4	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.92820 q^{5} +8.00000i q^{7} +O(q^{10})\)	\(q+6.92820 q^{5} +8.00000i q^{7} -3.46410i q^{11} -6.92820 q^{13} +6.00000 q^{17} +24.2487i q^{19} -24.0000i q^{23} +23.0000 q^{25} -20.7846 q^{29} +32.0000i q^{31} +55.4256i q^{35} -6.92820 q^{37} +66.0000 q^{41} +31.1769i q^{43} +48.0000i q^{47} -15.0000 q^{49} +90.0666 q^{53} -24.0000i q^{55} -31.1769i q^{59} -90.0666 q^{61} -48.0000 q^{65} +79.6743i q^{67} +120.000i q^{71} -58.0000 q^{73} +27.7128 q^{77} -16.0000i q^{79} +58.8897i q^{83} +41.5692 q^{85} -102.000 q^{89} -55.4256i q^{91} +168.000i q^{95} +26.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 24 q^{17} + 92 q^{25} + 264 q^{41} - 60 q^{49} - 192 q^{65} - 232 q^{73} - 408 q^{89} + 104 q^{97}+O(q^{100}) \) 4 * q + 24 * q^17 + 92 * q^25 + 264 * q^41 - 60 * q^49 - 192 * q^65 - 232 * q^73 - 408 * q^89 + 104 * q^97

Introduction

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