LMFDB - Embedded newform 2304.3.g.v.1279.1

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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1279.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2304.1279
Dual form			2304.3.g.v.1279.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.46410 q^{5} -2.00000i q^{7} +O(q^{10})$	$q-3.46410 q^{5} -2.00000i q^{7} +13.8564i q^{11} -20.7846 q^{13} +18.0000 q^{17} +20.7846i q^{19} -36.0000i q^{23} -13.0000 q^{25} +31.1769 q^{29} +22.0000i q^{31} +6.92820i q^{35} -41.5692 q^{37} -54.0000 q^{41} +20.7846i q^{43} +36.0000i q^{47} +45.0000 q^{49} +100.459 q^{53} -48.0000i q^{55} -62.3538i q^{59} +72.0000 q^{65} -62.3538i q^{67} -108.000i q^{71} -10.0000 q^{73} +27.7128 q^{77} -50.0000i q^{79} +13.8564i q^{83} -62.3538 q^{85} -18.0000 q^{89} +41.5692i q^{91} -72.0000i q^{95} -34.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 72 q^{17} - 52 q^{25} - 216 q^{41} + 180 q^{49} + 288 q^{65} - 40 q^{73} - 72 q^{89} - 136 q^{97}+O(q^{100}) $ 4 * q + 72 * q^17 - 52 * q^25 - 216 * q^41 + 180 * q^49 + 288 * q^65 - 40 * q^73 - 72 * q^89 - 136 * 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
$5$	−3.46410	−0.692820	−0.346410	−	0.938083i	$-0.612599\pi$
$5$	−3.46410	−0.692820	−0.346410	+	0.938083i	$0.612599\pi$
$6$	0	0
$7$	− 2.00000i	− 0.285714i	−0.989743	−	0.142857i	$-0.954371\pi$
$7$	− 2.00000i	− 0.285714i	0.989743	−	0.142857i	$-0.0456289\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	13.8564i	1.25967i	0.776728	+	0.629837i	$0.216878\pi$
$11$	13.8564i	1.25967i	−0.776728	+	0.629837i	$0.783122\pi$
$12$	0	0
$13$	−20.7846	−1.59882	−0.799408	−	0.600788i	$-0.794853\pi$
$13$	−20.7846	−1.59882	−0.799408	+	0.600788i	$0.794853\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18.0000	1.05882	0.529412	−	0.848365i	$-0.322413\pi$
$17$	18.0000	1.05882	0.529412	+	0.848365i	$0.322413\pi$
$18$	0	0
$19$	20.7846i	1.09393i	0.837157	+	0.546963i	$0.184216\pi$
$19$	20.7846i	1.09393i	−0.837157	+	0.546963i	$0.815784\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 36.0000i	− 1.56522i	−0.622514	−	0.782609i	$-0.713889\pi$
$23$	− 36.0000i	− 1.56522i	0.622514	−	0.782609i	$-0.286111\pi$
$24$	0	0
$25$	−13.0000	−0.520000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	31.1769	1.07507	0.537533	−	0.843243i	$-0.319356\pi$
$29$	31.1769	1.07507	0.537533	+	0.843243i	$0.319356\pi$
$30$	0	0
$31$	22.0000i	0.709677i	0.934928	+	0.354839i	$0.115464\pi$
$31$	22.0000i	0.709677i	−0.934928	+	0.354839i	$0.884536\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.92820i	0.197949i
$36$	0	0
$37$	−41.5692	−1.12349	−0.561746	−	0.827310i	$-0.689870\pi$
$37$	−41.5692	−1.12349	−0.561746	+	0.827310i	$0.689870\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−54.0000	−1.31707	−0.658537	−	0.752549i	$-0.728824\pi$
$41$	−54.0000	−1.31707	−0.658537	+	0.752549i	$0.728824\pi$
$42$	0	0
$43$	20.7846i	0.483363i	0.970356	+	0.241682i	$0.0776989\pi$
$43$	20.7846i	0.483363i	−0.970356	+	0.241682i	$0.922301\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	36.0000i	0.765957i	0.923757	+	0.382979i	$0.125102\pi$
$47$	36.0000i	0.765957i	−0.923757	+	0.382979i	$0.874898\pi$
$48$	0	0
$49$	45.0000	0.918367
$50$	0	0
$51$	0	0
$52$	0	0
$53$	100.459	1.89545	0.947726	−	0.319086i	$-0.103376\pi$
$53$	100.459	1.89545	0.947726	+	0.319086i	$0.103376\pi$
$54$	0	0
$55$	− 48.0000i	− 0.872727i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 62.3538i	− 1.05684i	−0.848982	−	0.528422i	$-0.822784\pi$
$59$	− 62.3538i	− 1.05684i	0.848982	−	0.528422i	$-0.177216\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	72.0000	1.10769
$66$	0	0
$67$	− 62.3538i	− 0.930654i	−0.885139	−	0.465327i	$-0.845937\pi$
$67$	− 62.3538i	− 0.930654i	0.885139	−	0.465327i	$-0.154063\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 108.000i	− 1.52113i	−0.649264	−	0.760563i	$-0.724923\pi$
$71$	− 108.000i	− 1.52113i	0.649264	−	0.760563i	$-0.275077\pi$
$72$	0	0
$73$	−10.0000	−0.136986	−0.0684932	−	0.997652i	$-0.521819\pi$
$73$	−10.0000	−0.136986	−0.0684932	+	0.997652i	$0.521819\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	27.7128	0.359907
$78$	0	0
$79$	− 50.0000i	− 0.632911i	−0.948607	−	0.316456i	$-0.897507\pi$
$79$	− 50.0000i	− 0.632911i	0.948607	−	0.316456i	$-0.102493\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.8564i	0.166945i	0.996510	+	0.0834723i	$0.0266010\pi$
$83$	13.8564i	0.166945i	−0.996510	+	0.0834723i	$0.973399\pi$
$84$	0	0
$85$	−62.3538	−0.733574
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−18.0000	−0.202247	−0.101124	−	0.994874i	$-0.532244\pi$
$89$	−18.0000	−0.202247	−0.101124	+	0.994874i	$0.532244\pi$
$90$	0	0
$91$	41.5692i	0.456805i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 72.0000i	− 0.757895i
$96$	0	0
$97$	−34.0000	−0.350515	−0.175258	−	0.984523i	$-0.556076\pi$
$97$	−34.0000	−0.350515	−0.175258	+	0.984523i	$0.556076\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.3205	0.171490	0.0857451	−	0.996317i	$-0.472673\pi$
$101$	17.3205	0.171490	0.0857451	+	0.996317i	$0.472673\pi$
$102$	0	0
$103$	− 118.000i	− 1.14563i	−0.819684	−	0.572816i	$-0.805851\pi$
$103$	− 118.000i	− 1.14563i	0.819684	−	0.572816i	$-0.194149\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	131.636i	1.23024i	0.788433	+	0.615121i	$0.210893\pi$
$107$	131.636i	1.23024i	−0.788433	+	0.615121i	$0.789107\pi$
$108$	0	0
$109$	−145.492	−1.33479	−0.667396	−	0.744703i	$-0.732591\pi$
$109$	−145.492	−1.33479	−0.667396	+	0.744703i	$0.732591\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−90.0000	−0.796460	−0.398230	−	0.917286i	$-0.630375\pi$
$113$	−90.0000	−0.796460	−0.398230	+	0.917286i	$0.630375\pi$
$114$	0	0
$115$	124.708i	1.08441i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 36.0000i	− 0.302521i
$120$	0	0
$121$	−71.0000	−0.586777
$122$	0	0
$123$	0	0
$124$	0	0
$125$	131.636	1.05309
$126$	0	0
$127$	− 118.000i	− 0.929134i	−0.885538	−	0.464567i	$-0.846210\pi$
$127$	− 118.000i	− 0.929134i	0.885538	−	0.464567i	$-0.153790\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 34.6410i	− 0.264435i	−0.991221	−	0.132218i	$-0.957790\pi$
$131$	− 34.6410i	− 0.264435i	0.991221	−	0.132218i	$-0.0422098\pi$
$132$	0	0
$133$	41.5692	0.312551
$134$	0	0
$135$	0	0
$136$	0	0
$137$	90.0000	0.656934	0.328467	−	0.944515i	$-0.393468\pi$
$137$	90.0000	0.656934	0.328467	+	0.944515i	$0.393468\pi$
$138$	0	0
$139$	− 270.200i	− 1.94388i	−0.235220	−	0.971942i	$-0.575581\pi$
$139$	− 270.200i	− 1.94388i	0.235220	−	0.971942i	$-0.424419\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 288.000i	− 2.01399i
$144$	0	0
$145$	−108.000	−0.744828
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−183.597	−1.23220	−0.616099	−	0.787669i	$-0.711288\pi$
$149$	−183.597	−1.23220	−0.616099	+	0.787669i	$0.711288\pi$
$150$	0	0
$151$	142.000i	0.940397i	0.882561	+	0.470199i	$0.155818\pi$
$151$	142.000i	0.940397i	−0.882561	+	0.470199i	$0.844182\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 76.2102i	− 0.491679i
$156$	0	0
$157$	41.5692	0.264772	0.132386	−	0.991198i	$-0.457736\pi$
$157$	41.5692	0.264772	0.132386	+	0.991198i	$0.457736\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−72.0000	−0.447205
$162$	0	0
$163$	− 103.923i	− 0.637565i	−0.947828	−	0.318782i	$-0.896726\pi$
$163$	− 103.923i	− 0.637565i	0.947828	−	0.318782i	$-0.103274\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 72.0000i	− 0.431138i	−0.976489	−	0.215569i	$-0.930839\pi$
$167$	− 72.0000i	− 0.431138i	0.976489	−	0.215569i	$-0.0691606\pi$
$168$	0	0
$169$	263.000	1.55621
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−169.741	−0.981162	−0.490581	−	0.871396i	$-0.663215\pi$
$173$	−169.741	−0.981162	−0.490581	+	0.871396i	$0.663215\pi$
$174$	0	0
$175$	26.0000i	0.148571i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 131.636i	− 0.735396i	−0.929945	−	0.367698i	$-0.880146\pi$
$179$	− 131.636i	− 0.735396i	0.929945	−	0.367698i	$-0.119854\pi$
$180$	0	0
$181$	20.7846	0.114832	0.0574160	−	0.998350i	$-0.481714\pi$
$181$	20.7846	0.114832	0.0574160	+	0.998350i	$0.481714\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	144.000	0.778378
$186$	0	0
$187$	249.415i	1.33377i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 360.000i	− 1.88482i	−0.334464	−	0.942408i	$-0.608555\pi$
$191$	− 360.000i	− 1.88482i	0.334464	−	0.942408i	$-0.391445\pi$
$192$	0	0
$193$	194.000	1.00518	0.502591	−	0.864525i	$-0.332380\pi$
$193$	194.000	1.00518	0.502591	+	0.864525i	$0.332380\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	86.6025	0.439607	0.219803	−	0.975544i	$-0.429458\pi$
$197$	86.6025	0.439607	0.219803	+	0.975544i	$0.429458\pi$
$198$	0	0
$199$	26.0000i	0.130653i	0.997864	+	0.0653266i	$0.0208089\pi$
$199$	26.0000i	0.130653i	−0.997864	+	0.0653266i	$0.979191\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 62.3538i	− 0.307162i
$204$	0	0
$205$	187.061	0.912495
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−288.000	−1.37799
$210$	0	0
$211$	− 228.631i	− 1.08356i	−0.840521	−	0.541779i	$-0.817751\pi$
$211$	− 228.631i	− 1.08356i	0.840521	−	0.541779i	$-0.182249\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 72.0000i	− 0.334884i
$216$	0	0
$217$	44.0000	0.202765
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−374.123	−1.69286
$222$	0	0
$223$	− 310.000i	− 1.39013i	−0.718945	−	0.695067i	$-0.755375\pi$
$223$	− 310.000i	− 1.39013i	0.718945	−	0.695067i	$-0.244625\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	249.415i	1.09875i	0.835577	+	0.549373i	$0.185133\pi$
$227$	249.415i	1.09875i	−0.835577	+	0.549373i	$0.814867\pi$
$228$	0	0
$229$	228.631	0.998387	0.499194	−	0.866490i	$-0.333630\pi$
$229$	228.631	0.998387	0.499194	+	0.866490i	$0.333630\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	198.000	0.849785	0.424893	−	0.905244i	$-0.360312\pi$
$233$	198.000	0.849785	0.424893	+	0.905244i	$0.360312\pi$
$234$	0	0
$235$	− 124.708i	− 0.530671i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 288.000i	− 1.20502i	−0.798111	−	0.602510i	$-0.794167\pi$
$239$	− 288.000i	− 1.20502i	0.798111	−	0.602510i	$-0.205833\pi$
$240$	0	0
$241$	−374.000	−1.55187	−0.775934	−	0.630815i	$-0.782721\pi$
$241$	−374.000	−1.55187	−0.775934	+	0.630815i	$0.782721\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−155.885	−0.636264
$246$	0	0
$247$	− 432.000i	− 1.74899i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 374.123i	− 1.49053i	−0.666769	−	0.745265i	$-0.732323\pi$
$251$	− 374.123i	− 1.49053i	0.666769	−	0.745265i	$-0.267677\pi$
$252$	0	0
$253$	498.831	1.97166
$254$	0	0
$255$	0	0
$256$	0	0
$257$	270.000	1.05058	0.525292	−	0.850922i	$-0.323956\pi$
$257$	270.000	1.05058	0.525292	+	0.850922i	$0.323956\pi$
$258$	0	0
$259$	83.1384i	0.320998i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 144.000i	− 0.547529i	−0.961797	−	0.273764i	$-0.911731\pi$
$263$	− 144.000i	− 0.547529i	0.961797	−	0.273764i	$-0.0882688\pi$
$264$	0	0
$265$	−348.000	−1.31321
$266$	0	0
$267$	0	0
$268$	0	0
$269$	31.1769	0.115899	0.0579497	−	0.998320i	$-0.481544\pi$
$269$	31.1769	0.115899	0.0579497	+	0.998320i	$0.481544\pi$
$270$	0	0
$271$	− 262.000i	− 0.966790i	−0.875402	−	0.483395i	$-0.839404\pi$
$271$	− 262.000i	− 0.966790i	0.875402	−	0.483395i	$-0.160596\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 180.133i	− 0.655030i
$276$	0	0
$277$	103.923	0.375173	0.187587	−	0.982248i	$-0.439933\pi$
$277$	103.923	0.375173	0.187587	+	0.982248i	$0.439933\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−234.000	−0.832740	−0.416370	−	0.909195i	$-0.636698\pi$
$281$	−234.000	−0.832740	−0.416370	+	0.909195i	$0.636698\pi$
$282$	0	0
$283$	353.338i	1.24855i	0.781206	+	0.624273i	$0.214605\pi$
$283$	353.338i	1.24855i	−0.781206	+	0.624273i	$0.785395\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	108.000i	0.376307i
$288$	0	0
$289$	35.0000	0.121107
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−93.5307	−0.319218	−0.159609	−	0.987180i	$-0.551023\pi$
$293$	−93.5307	−0.319218	−0.159609	+	0.987180i	$0.551023\pi$
$294$	0	0
$295$	216.000i	0.732203i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	748.246i	2.50249i
$300$	0	0
$301$	41.5692	0.138104
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	436.477i	1.42175i	0.703319	+	0.710874i	$0.251701\pi$
$307$	436.477i	1.42175i	−0.703319	+	0.710874i	$0.748299\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 216.000i	− 0.694534i	−0.937766	−	0.347267i	$-0.887110\pi$
$311$	− 216.000i	− 0.694534i	0.937766	−	0.347267i	$-0.112890\pi$
$312$	0	0
$313$	290.000	0.926518	0.463259	−	0.886223i	$-0.346680\pi$
$313$	290.000	0.926518	0.463259	+	0.886223i	$0.346680\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	142.028	0.448038	0.224019	−	0.974585i	$-0.428082\pi$
$317$	142.028	0.448038	0.224019	+	0.974585i	$0.428082\pi$
$318$	0	0
$319$	432.000i	1.35423i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	374.123i	1.15828i
$324$	0	0
$325$	270.200	0.831384
$326$	0	0
$327$	0	0
$328$	0	0
$329$	72.0000	0.218845
$330$	0	0
$331$	− 353.338i	− 1.06749i	−0.845646	−	0.533744i	$-0.820785\pi$
$331$	− 353.338i	− 1.06749i	0.845646	−	0.533744i	$-0.179215\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	216.000i	0.644776i
$336$	0	0
$337$	566.000	1.67953	0.839763	−	0.542954i	$-0.182694\pi$
$337$	566.000	1.67953	0.839763	+	0.542954i	$0.182694\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−304.841	−0.893962
$342$	0	0
$343$	− 188.000i	− 0.548105i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	27.7128i	0.0798640i	0.999202	+	0.0399320i	$0.0127141\pi$
$347$	27.7128i	0.0798640i	−0.999202	+	0.0399320i	$0.987286\pi$
$348$	0	0
$349$	−581.969	−1.66753	−0.833767	−	0.552117i	$-0.813820\pi$
$349$	−581.969	−1.66753	−0.833767	+	0.552117i	$0.813820\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	342.000	0.968839	0.484419	−	0.874836i	$-0.339031\pi$
$353$	342.000	0.968839	0.484419	+	0.874836i	$0.339031\pi$
$354$	0	0
$355$	374.123i	1.05387i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 108.000i	− 0.300836i	−0.988623	−	0.150418i	$-0.951938\pi$
$359$	− 108.000i	− 0.300836i	0.988623	−	0.150418i	$-0.0480619\pi$
$360$	0	0
$361$	−71.0000	−0.196676
$362$	0	0
$363$	0	0
$364$	0	0
$365$	34.6410	0.0949069
$366$	0	0
$367$	− 74.0000i	− 0.201635i	−0.994905	−	0.100817i	$-0.967854\pi$
$367$	− 74.0000i	− 0.201635i	0.994905	−	0.100817i	$-0.0321458\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 200.918i	− 0.541558i
$372$	0	0
$373$	−83.1384	−0.222891	−0.111446	−	0.993771i	$-0.535548\pi$
$373$	−83.1384	−0.222891	−0.111446	+	0.993771i	$0.535548\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−648.000	−1.71883
$378$	0	0
$379$	228.631i	0.603247i	0.953427	+	0.301624i	$0.0975286\pi$
$379$	228.631i	0.603247i	−0.953427	+	0.301624i	$0.902471\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	216.000i	0.563969i	0.959419	+	0.281984i	$0.0909926\pi$
$383$	216.000i	0.563969i	−0.959419	+	0.281984i	$0.909007\pi$
$384$	0	0
$385$	−96.0000	−0.249351
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−142.028	−0.365111	−0.182555	−	0.983196i	$-0.558437\pi$
$389$	−142.028	−0.365111	−0.182555	+	0.983196i	$0.558437\pi$
$390$	0	0
$391$	− 648.000i	− 1.65729i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	173.205i	0.438494i
$396$	0	0
$397$	−332.554	−0.837667	−0.418833	−	0.908063i	$-0.637561\pi$
$397$	−332.554	−0.837667	−0.418833	+	0.908063i	$0.637561\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−126.000	−0.314214	−0.157107	−	0.987582i	$-0.550217\pi$
$401$	−126.000	−0.314214	−0.157107	+	0.987582i	$0.550217\pi$
$402$	0	0
$403$	− 457.261i	− 1.13464i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 576.000i	− 1.41523i
$408$	0	0
$409$	−562.000	−1.37408	−0.687042	−	0.726618i	$-0.741091\pi$
$409$	−562.000	−1.37408	−0.687042	+	0.726618i	$0.741091\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−124.708	−0.301956
$414$	0	0
$415$	− 48.0000i	− 0.115663i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 498.831i	− 1.19053i	−0.803531	−	0.595263i	$-0.797048\pi$
$419$	− 498.831i	− 1.19053i	0.803531	−	0.595263i	$-0.202952\pi$
$420$	0	0
$421$	−644.323	−1.53046	−0.765229	−	0.643758i	$-0.777374\pi$
$421$	−644.323	−1.53046	−0.765229	+	0.643758i	$0.777374\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−234.000	−0.550588
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	252.000i	0.584687i	0.956313	+	0.292343i	$0.0944350\pi$
$431$	252.000i	0.584687i	−0.956313	+	0.292343i	$0.905565\pi$
$432$	0	0
$433$	−386.000	−0.891455	−0.445727	−	0.895169i	$-0.647055\pi$
$433$	−386.000	−0.891455	−0.445727	+	0.895169i	$0.647055\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	748.246	1.71223
$438$	0	0
$439$	866.000i	1.97267i	0.164767	+	0.986333i	$0.447313\pi$
$439$	866.000i	1.97267i	−0.164767	+	0.986333i	$0.552687\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 748.246i	− 1.68904i	−0.535522	−	0.844521i	$-0.679885\pi$
$443$	− 748.246i	− 1.68904i	0.535522	−	0.844521i	$-0.320115\pi$
$444$	0	0
$445$	62.3538	0.140121
$446$	0	0
$447$	0	0
$448$	0	0
$449$	450.000	1.00223	0.501114	−	0.865382i	$-0.332924\pi$
$449$	450.000	1.00223	0.501114	+	0.865382i	$0.332924\pi$
$450$	0	0
$451$	− 748.246i	− 1.65908i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 144.000i	− 0.316484i
$456$	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
$461$	−17.3205	−0.0375716	−0.0187858	−	0.999824i	$-0.505980\pi$
$461$	−17.3205	−0.0375716	−0.0187858	+	0.999824i	$0.505980\pi$
$462$	0	0
$463$	626.000i	1.35205i	0.736878	+	0.676026i	$0.236299\pi$
$463$	626.000i	1.35205i	−0.736878	+	0.676026i	$0.763701\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	304.841i	0.652764i	0.945238	+	0.326382i	$0.105830\pi$
$467$	304.841i	0.652764i	−0.945238	+	0.326382i	$0.894170\pi$
$468$	0	0
$469$	−124.708	−0.265901
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−288.000	−0.608879
$474$	0	0
$475$	− 270.200i	− 0.568842i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 252.000i	− 0.526096i	−0.964783	−	0.263048i	$-0.915272\pi$
$479$	− 252.000i	− 0.526096i	0.964783	−	0.263048i	$-0.0847278\pi$
$480$	0	0
$481$	864.000	1.79626
$482$	0	0
$483$	0	0
$484$	0	0
$485$	117.779	0.242844
$486$	0	0
$487$	− 218.000i	− 0.447639i	−0.974631	−	0.223819i	$-0.928147\pi$
$487$	− 218.000i	− 0.447639i	0.974631	−	0.223819i	$-0.0718525\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 713.605i	− 1.45337i	−0.686971	−	0.726685i	$-0.741060\pi$
$491$	− 713.605i	− 1.45337i	0.686971	−	0.726685i	$-0.258940\pi$
$492$	0	0
$493$	561.184	1.13831
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−216.000	−0.434608
$498$	0	0
$499$	561.184i	1.12462i	0.826927	+	0.562309i	$0.190087\pi$
$499$	561.184i	1.12462i	−0.826927	+	0.562309i	$0.809913\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	324.000i	0.644135i	0.946717	+	0.322068i	$0.104378\pi$
$503$	324.000i	0.644135i	−0.946717	+	0.322068i	$0.895622\pi$
$504$	0	0
$505$	−60.0000	−0.118812
$506$	0	0
$507$	0	0
$508$	0	0
$509$	751.710	1.47684	0.738419	−	0.674343i	$-0.235573\pi$
$509$	751.710	1.47684	0.738419	+	0.674343i	$0.235573\pi$
$510$	0	0
$511$	20.0000i	0.0391389i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	408.764i	0.793716i
$516$	0	0
$517$	−498.831	−0.964856
$518$	0	0
$519$	0	0
$520$	0	0
$521$	738.000	1.41651	0.708253	−	0.705958i	$-0.249483\pi$
$521$	738.000	1.41651	0.708253	+	0.705958i	$0.249483\pi$
$522$	0	0
$523$	62.3538i	0.119223i	0.998222	+	0.0596117i	$0.0189862\pi$
$523$	62.3538i	0.119223i	−0.998222	+	0.0596117i	$0.981014\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	396.000i	0.751423i
$528$	0	0
$529$	−767.000	−1.44991
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1122.37	2.10576
$534$	0	0
$535$	− 456.000i	− 0.852336i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	623.538i	1.15684i
$540$	0	0
$541$	−228.631	−0.422608	−0.211304	−	0.977420i	$-0.567771\pi$
$541$	−228.631	−0.422608	−0.211304	+	0.977420i	$0.567771\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	504.000	0.924771
$546$	0	0
$547$	− 436.477i	− 0.797947i	−0.916963	−	0.398973i	$-0.869367\pi$
$547$	− 436.477i	− 0.797947i	0.916963	−	0.398973i	$-0.130633\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	648.000i	1.17604i
$552$	0	0
$553$	−100.000	−0.180832
$554$	0	0
$555$	0	0
$556$	0	0
$557$	273.664	0.491318	0.245659	−	0.969356i	$-0.420996\pi$
$557$	273.664	0.491318	0.245659	+	0.969356i	$0.420996\pi$
$558$	0	0
$559$	− 432.000i	− 0.772809i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	27.7128i	0.0492235i	0.999697	+	0.0246117i	$0.00783495\pi$
$563$	27.7128i	0.0492235i	−0.999697	+	0.0246117i	$0.992165\pi$
$564$	0	0
$565$	311.769	0.551804
$566$	0	0
$567$	0	0
$568$	0	0
$569$	666.000	1.17047	0.585237	−	0.810862i	$-0.301001\pi$
$569$	666.000	1.17047	0.585237	+	0.810862i	$0.301001\pi$
$570$	0	0
$571$	270.200i	0.473205i	0.971607	+	0.236602i	$0.0760339\pi$
$571$	270.200i	0.473205i	−0.971607	+	0.236602i	$0.923966\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	468.000i	0.813913i
$576$	0	0
$577$	142.000	0.246101	0.123050	−	0.992400i	$-0.460732\pi$
$577$	142.000	0.246101	0.123050	+	0.992400i	$0.460732\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	27.7128	0.0476985
$582$	0	0
$583$	1392.00i	2.38765i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	422.620i	0.719967i	0.932959	+	0.359983i	$0.117218\pi$
$587$	422.620i	0.719967i	−0.932959	+	0.359983i	$0.882782\pi$
$588$	0	0
$589$	−457.261	−0.776335
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−954.000	−1.60877	−0.804384	−	0.594109i	$-0.797505\pi$
$593$	−954.000	−1.60877	−0.804384	+	0.594109i	$0.797505\pi$
$594$	0	0
$595$	124.708i	0.209593i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1044.00i	1.74290i	0.490480	+	0.871452i	$0.336821\pi$
$599$	1044.00i	1.74290i	−0.490480	+	0.871452i	$0.663179\pi$
$600$	0	0
$601$	430.000	0.715474	0.357737	−	0.933822i	$-0.383548\pi$
$601$	430.000	0.715474	0.357737	+	0.933822i	$0.383548\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	245.951	0.406531
$606$	0	0
$607$	458.000i	0.754530i	0.926105	+	0.377265i	$0.123135\pi$
$607$	458.000i	0.754530i	−0.926105	+	0.377265i	$0.876865\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 748.246i	− 1.22463i
$612$	0	0
$613$	−249.415	−0.406877	−0.203438	−	0.979088i	$-0.565212\pi$
$613$	−249.415	−0.406877	−0.203438	+	0.979088i	$0.565212\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−234.000	−0.379254	−0.189627	−	0.981856i	$-0.560728\pi$
$617$	−234.000	−0.379254	−0.189627	+	0.981856i	$0.560728\pi$
$618$	0	0
$619$	− 311.769i	− 0.503666i	−0.967771	−	0.251833i	$-0.918967\pi$
$619$	− 311.769i	− 0.503666i	0.967771	−	0.251833i	$-0.0810333\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	36.0000i	0.0577849i
$624$	0	0
$625$	−131.000	−0.209600
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−748.246	−1.18958
$630$	0	0
$631$	− 26.0000i	− 0.0412044i	−0.999788	−	0.0206022i	$-0.993442\pi$
$631$	− 26.0000i	− 0.0412044i	0.999788	−	0.0206022i	$-0.00655835\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	408.764i	0.643723i
$636$	0	0
$637$	−935.307	−1.46830
$638$	0	0
$639$	0	0
$640$	0	0
$641$	378.000	0.589704	0.294852	−	0.955543i	$-0.404730\pi$
$641$	378.000	0.589704	0.294852	+	0.955543i	$0.404730\pi$
$642$	0	0
$643$	62.3538i	0.0969733i	0.998824	+	0.0484866i	$0.0154398\pi$
$643$	62.3538i	0.0969733i	−0.998824	+	0.0484866i	$0.984560\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	684.000i	1.05719i	0.848875	+	0.528594i	$0.177280\pi$
$647$	684.000i	1.05719i	−0.848875	+	0.528594i	$0.822720\pi$
$648$	0	0
$649$	864.000	1.33128
$650$	0	0
$651$	0	0
$652$	0	0
$653$	495.367	0.758601	0.379301	−	0.925274i	$-0.376165\pi$
$653$	495.367	0.758601	0.379301	+	0.925274i	$0.376165\pi$
$654$	0	0
$655$	120.000i	0.183206i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	325.626i	0.494121i	0.969000	+	0.247060i	$0.0794646\pi$
$659$	325.626i	0.494121i	−0.969000	+	0.247060i	$0.920535\pi$
$660$	0	0
$661$	374.123	0.565995	0.282998	−	0.959121i	$-0.408671\pi$
$661$	374.123	0.565995	0.282998	+	0.959121i	$0.408671\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−144.000	−0.216541
$666$	0	0
$667$	− 1122.37i	− 1.68271i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−146.000	−0.216939	−0.108470	−	0.994100i	$-0.534595\pi$
$673$	−146.000	−0.216939	−0.108470	+	0.994100i	$0.534595\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−530.008	−0.782877	−0.391438	−	0.920204i	$-0.628022\pi$
$677$	−530.008	−0.782877	−0.391438	+	0.920204i	$0.628022\pi$
$678$	0	0
$679$	68.0000i	0.100147i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 124.708i	− 0.182588i	−0.995824	−	0.0912940i	$-0.970900\pi$
$683$	− 124.708i	− 0.182588i	0.995824	−	0.0912940i	$-0.0291003\pi$
$684$	0	0
$685$	−311.769	−0.455137
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2088.00	−3.03048
$690$	0	0
$691$	478.046i	0.691818i	0.938268	+	0.345909i	$0.112429\pi$
$691$	478.046i	0.691818i	−0.938268	+	0.345909i	$0.887571\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	936.000i	1.34676i
$696$	0	0
$697$	−972.000	−1.39455
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1285.18	−1.83335	−0.916677	−	0.399628i	$-0.869139\pi$
$701$	−1285.18	−1.83335	−0.916677	+	0.399628i	$0.869139\pi$
$702$	0	0
$703$	− 864.000i	− 1.22902i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 34.6410i	− 0.0489972i
$708$	0	0
$709$	394.908	0.556992	0.278496	−	0.960437i	$-0.410164\pi$
$709$	394.908	0.556992	0.278496	+	0.960437i	$0.410164\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	792.000	1.11080
$714$	0	0
$715$	997.661i	1.39533i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 612.000i	− 0.851182i	−0.904916	−	0.425591i	$-0.860066\pi$
$719$	− 612.000i	− 0.851182i	0.904916	−	0.425591i	$-0.139934\pi$
$720$	0	0
$721$	−236.000	−0.327323
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−405.300	−0.559034
$726$	0	0
$727$	502.000i	0.690509i	0.938509	+	0.345254i	$0.112207\pi$
$727$	502.000i	0.690509i	−0.938509	+	0.345254i	$0.887793\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	374.123i	0.511796i
$732$	0	0
$733$	−20.7846	−0.0283555	−0.0141778	−	0.999899i	$-0.504513\pi$
$733$	−20.7846	−0.0283555	−0.0141778	+	0.999899i	$0.504513\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	864.000	1.17232
$738$	0	0
$739$	644.323i	0.871885i	0.899975	+	0.435942i	$0.143585\pi$
$739$	644.323i	0.871885i	−0.899975	+	0.435942i	$0.856415\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	720.000i	0.969044i	0.874779	+	0.484522i	$0.161007\pi$
$743$	720.000i	0.969044i	−0.874779	+	0.484522i	$0.838993\pi$
$744$	0	0
$745$	636.000	0.853691
$746$	0	0
$747$	0	0
$748$	0	0
$749$	263.272	0.351498
$750$	0	0
$751$	310.000i	0.412783i	0.978469	+	0.206391i	$0.0661720\pi$
$751$	310.000i	0.412783i	−0.978469	+	0.206391i	$0.933828\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 491.902i	− 0.651526i
$756$	0	0
$757$	−1184.72	−1.56502	−0.782512	−	0.622636i	$-0.786062\pi$
$757$	−1184.72	−1.56502	−0.782512	+	0.622636i	$0.786062\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	450.000	0.591327	0.295664	−	0.955292i	$-0.404459\pi$
$761$	450.000	0.591327	0.295664	+	0.955292i	$0.404459\pi$
$762$	0	0
$763$	290.985i	0.381369i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1296.00i	1.68970i
$768$	0	0
$769$	50.0000	0.0650195	0.0325098	−	0.999471i	$-0.489650\pi$
$769$	50.0000	0.0650195	0.0325098	+	0.999471i	$0.489650\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	779.423	1.00831	0.504155	−	0.863613i	$-0.331804\pi$
$773$	779.423	1.00831	0.504155	+	0.863613i	$0.331804\pi$
$774$	0	0
$775$	− 286.000i	− 0.369032i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1122.37i	− 1.44078i
$780$	0	0
$781$	1496.49	1.91612
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−144.000	−0.183439
$786$	0	0
$787$	− 1060.02i	− 1.34691i	−0.739230	−	0.673453i	$-0.764810\pi$
$787$	− 1060.02i	− 1.34691i	0.739230	−	0.673453i	$-0.235190\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	180.000i	0.227560i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−128.172	−0.160818	−0.0804089	−	0.996762i	$-0.525623\pi$
$797$	−128.172	−0.160818	−0.0804089	+	0.996762i	$0.525623\pi$
$798$	0	0
$799$	648.000i	0.811014i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 138.564i	− 0.172558i
$804$	0	0
$805$	249.415	0.309833
$806$	0	0
$807$	0	0
$808$	0	0
$809$	234.000	0.289246	0.144623	−	0.989487i	$-0.453803\pi$
$809$	234.000	0.289246	0.144623	+	0.989487i	$0.453803\pi$
$810$	0	0
$811$	644.323i	0.794480i	0.917715	+	0.397240i	$0.130032\pi$
$811$	644.323i	0.794480i	−0.917715	+	0.397240i	$0.869968\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	360.000i	0.441718i
$816$	0	0
$817$	−432.000	−0.528764
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−841.777	−1.02531	−0.512653	−	0.858596i	$-0.671337\pi$
$821$	−841.777	−1.02531	−0.512653	+	0.858596i	$0.671337\pi$
$822$	0	0
$823$	382.000i	0.464156i	0.972697	+	0.232078i	$0.0745524\pi$
$823$	382.000i	0.464156i	−0.972697	+	0.232078i	$0.925448\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 810.600i	− 0.980169i	−0.871675	−	0.490085i	$-0.836966\pi$
$827$	− 810.600i	− 0.980169i	0.871675	−	0.490085i	$-0.163034\pi$
$828$	0	0
$829$	1018.45	1.22852	0.614262	−	0.789102i	$-0.289454\pi$
$829$	1018.45	1.22852	0.614262	+	0.789102i	$0.289454\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	810.000	0.972389
$834$	0	0
$835$	249.415i	0.298701i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	396.000i	0.471990i	0.971754	+	0.235995i	$0.0758350\pi$
$839$	396.000i	0.471990i	−0.971754	+	0.235995i	$0.924165\pi$
$840$	0	0
$841$	131.000	0.155767
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−911.059	−1.07818
$846$	0	0
$847$	142.000i	0.167651i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1496.49i	1.75851i
$852$	0	0
$853$	623.538	0.730994	0.365497	−	0.930812i	$-0.380899\pi$
$853$	623.538	0.730994	0.365497	+	0.930812i	$0.380899\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1206.00	−1.40723	−0.703617	−	0.710579i	$-0.748433\pi$
$857$	−1206.00	−1.40723	−0.703617	+	0.710579i	$0.748433\pi$
$858$	0	0
$859$	− 1226.29i	− 1.42758i	−0.700359	−	0.713790i	$-0.746977\pi$
$859$	− 1226.29i	− 1.42758i	0.700359	−	0.713790i	$-0.253023\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 72.0000i	− 0.0834299i	−0.999130	−	0.0417149i	$-0.986718\pi$
$863$	− 72.0000i	− 0.0834299i	0.999130	−	0.0417149i	$-0.0132821\pi$
$864$	0	0
$865$	588.000	0.679769
$866$	0	0
$867$	0	0
$868$	0	0
$869$	692.820	0.797262
$870$	0	0
$871$	1296.00i	1.48794i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 263.272i	− 0.300882i
$876$	0	0
$877$	665.108	0.758389	0.379195	−	0.925317i	$-0.376201\pi$
$877$	665.108	0.758389	0.379195	+	0.925317i	$0.376201\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−594.000	−0.674234	−0.337117	−	0.941463i	$-0.609452\pi$
$881$	−594.000	−0.674234	−0.337117	+	0.941463i	$0.609452\pi$
$882$	0	0
$883$	− 769.031i	− 0.870929i	−0.900206	−	0.435465i	$-0.856584\pi$
$883$	− 769.031i	− 0.870929i	0.900206	−	0.435465i	$-0.143416\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 720.000i	− 0.811725i	−0.913934	−	0.405862i	$-0.866971\pi$
$887$	− 720.000i	− 0.811725i	0.913934	−	0.405862i	$-0.133029\pi$
$888$	0	0
$889$	−236.000	−0.265467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−748.246	−0.837901
$894$	0	0
$895$	456.000i	0.509497i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	685.892i	0.762950i
$900$	0	0
$901$	1808.26	2.00695
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−72.0000	−0.0795580
$906$	0	0
$907$	1517.28i	1.67285i	0.548080	+	0.836426i	$0.315359\pi$
$907$	1517.28i	1.67285i	−0.548080	+	0.836426i	$0.684641\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	720.000i	0.790340i	0.918608	+	0.395170i	$0.129314\pi$
$911$	720.000i	0.790340i	−0.918608	+	0.395170i	$0.870686\pi$
$912$	0	0
$913$	−192.000	−0.210296
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−69.2820	−0.0755529
$918$	0	0
$919$	46.0000i	0.0500544i	0.999687	+	0.0250272i	$0.00796724\pi$
$919$	46.0000i	0.0500544i	−0.999687	+	0.0250272i	$0.992033\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2244.74i	2.43200i
$924$	0	0
$925$	540.400	0.584216
$926$	0	0
$927$	0	0
$928$	0	0
$929$	18.0000	0.0193757	0.00968784	−	0.999953i	$-0.496916\pi$
$929$	18.0000	0.0193757	0.00968784	+	0.999953i	$0.496916\pi$
$930$	0	0
$931$	935.307i	1.00463i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 864.000i	− 0.924064i
$936$	0	0
$937$	−1198.00	−1.27855	−0.639274	−	0.768979i	$-0.720765\pi$
$937$	−1198.00	−1.27855	−0.639274	+	0.768979i	$0.720765\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−682.428	−0.725216	−0.362608	−	0.931942i	$-0.618114\pi$
$941$	−682.428	−0.725216	−0.362608	+	0.931942i	$0.618114\pi$
$942$	0	0
$943$	1944.00i	2.06151i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1046.16i	− 1.10471i	−0.833610	−	0.552354i	$-0.813730\pi$
$947$	− 1046.16i	− 1.10471i	0.833610	−	0.552354i	$-0.186270\pi$
$948$	0	0
$949$	207.846	0.219016
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−414.000	−0.434418	−0.217209	−	0.976125i	$-0.569695\pi$
$953$	−414.000	−0.434418	−0.217209	+	0.976125i	$0.569695\pi$
$954$	0	0
$955$	1247.08i	1.30584i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 180.000i	− 0.187696i
$960$	0	0
$961$	477.000	0.496358
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−672.036	−0.696410
$966$	0	0
$967$	− 1174.00i	− 1.21406i	−0.794677	−	0.607032i	$-0.792360\pi$
$967$	− 1174.00i	− 1.21406i	0.794677	−	0.607032i	$-0.207640\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	180.133i	0.185513i	0.995689	+	0.0927566i	$0.0295678\pi$
$971$	180.133i	0.185513i	−0.995689	+	0.0927566i	$0.970432\pi$
$972$	0	0
$973$	−540.400	−0.555396
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1746.00	1.78710	0.893552	−	0.448960i	$-0.148206\pi$
$977$	1746.00	1.78710	0.893552	+	0.448960i	$0.148206\pi$
$978$	0	0
$979$	− 249.415i	− 0.254765i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1008.00i	1.02543i	0.858558	+	0.512716i	$0.171361\pi$
$983$	1008.00i	1.02543i	−0.858558	+	0.512716i	$0.828639\pi$
$984$	0	0
$985$	−300.000	−0.304569
$986$	0	0
$987$	0	0
$988$	0	0
$989$	748.246	0.756568
$990$	0	0
$991$	− 314.000i	− 0.316852i	−0.987371	−	0.158426i	$-0.949358\pi$
$991$	− 314.000i	− 0.316852i	0.987371	−	0.158426i	$-0.0506419\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 90.0666i	− 0.0905192i
$996$	0	0
$997$	−249.415	−0.250166	−0.125083	−	0.992146i	$-0.539920\pi$
$997$	−249.415	−0.250166	−0.125083	+	0.992146i	$0.539920\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.g.v.1279.1		4
3.2	odd	2		768.3.g.d.511.4		4
4.3	odd	2	inner	2304.3.g.v.1279.2		4
8.3	odd	2	inner	2304.3.g.v.1279.4		4
8.5	even	2	inner	2304.3.g.v.1279.3		4
12.11	even	2		768.3.g.d.511.2		4
16.3	odd	4		576.3.b.e.415.3		4
16.5	even	4		576.3.b.e.415.2		4
16.11	odd	4		576.3.b.e.415.1		4
16.13	even	4		576.3.b.e.415.4		4
24.5	odd	2		768.3.g.d.511.1		4
24.11	even	2		768.3.g.d.511.3		4
48.5	odd	4		192.3.b.a.31.4	yes	4
48.11	even	4		192.3.b.a.31.2	yes	4
48.29	odd	4		192.3.b.a.31.1	✓	4
48.35	even	4		192.3.b.a.31.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.b.a.31.1	✓	4	48.29	odd	4
192.3.b.a.31.2	yes	4	48.11	even	4
192.3.b.a.31.3	yes	4	48.35	even	4
192.3.b.a.31.4	yes	4	48.5	odd	4
576.3.b.e.415.1		4	16.11	odd	4
576.3.b.e.415.2		4	16.5	even	4
576.3.b.e.415.3		4	16.3	odd	4
576.3.b.e.415.4		4	16.13	even	4
768.3.g.d.511.1		4	24.5	odd	2
768.3.g.d.511.2		4	12.11	even	2
768.3.g.d.511.3		4	24.11	even	2
768.3.g.d.511.4		4	3.2	odd	2
2304.3.g.v.1279.1		4	1.1	even	1	trivial
2304.3.g.v.1279.2		4	4.3	odd	2	inner
2304.3.g.v.1279.3		4	8.5	even	2	inner
2304.3.g.v.1279.4		4	8.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.46410 q^{5} -2.00000i q^{7} +O(q^{10})\)	\(q-3.46410 q^{5} -2.00000i q^{7} +13.8564i q^{11} -20.7846 q^{13} +18.0000 q^{17} +20.7846i q^{19} -36.0000i q^{23} -13.0000 q^{25} +31.1769 q^{29} +22.0000i q^{31} +6.92820i q^{35} -41.5692 q^{37} -54.0000 q^{41} +20.7846i q^{43} +36.0000i q^{47} +45.0000 q^{49} +100.459 q^{53} -48.0000i q^{55} -62.3538i q^{59} +72.0000 q^{65} -62.3538i q^{67} -108.000i q^{71} -10.0000 q^{73} +27.7128 q^{77} -50.0000i q^{79} +13.8564i q^{83} -62.3538 q^{85} -18.0000 q^{89} +41.5692i q^{91} -72.0000i q^{95} -34.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 72 q^{17} - 52 q^{25} - 216 q^{41} + 180 q^{49} + 288 q^{65} - 40 q^{73} - 72 q^{89} - 136 q^{97}+O(q^{100}) \) 4 * q + 72 * q^17 - 52 * q^25 - 216 * q^41 + 180 * q^49 + 288 * q^65 - 40 * q^73 - 72 * q^89 - 136 * q^97

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