LMFDB - Embedded newform 2304.3.b.i.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(127,2304)]

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, 0]))

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

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

chi := DirichletCharacter("2304.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.b (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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2304.127
Dual form			2304.3.b.i.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{5} -4.00000i q^{7} +O(q^{10})$	$q-4.00000i q^{5} -4.00000i q^{7} +16.0000 q^{11} -2.00000i q^{13} +24.0000 q^{17} +24.0000 q^{19} -32.0000i q^{23} +9.00000 q^{25} +44.0000i q^{29} +52.0000i q^{31} -16.0000 q^{35} -18.0000i q^{37} -8.00000 q^{41} -56.0000 q^{43} +32.0000i q^{47} +33.0000 q^{49} +36.0000i q^{53} -64.0000i q^{55} -32.0000 q^{59} -62.0000i q^{61} -8.00000 q^{65} +80.0000 q^{67} +128.000i q^{71} +66.0000 q^{73} -64.0000i q^{77} +20.0000i q^{79} +16.0000 q^{83} -96.0000i q^{85} +144.000 q^{89} -8.00000 q^{91} -96.0000i q^{95} +94.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 32 q^{11} + 48 q^{17} + 48 q^{19} + 18 q^{25} - 32 q^{35} - 16 q^{41} - 112 q^{43} + 66 q^{49} - 64 q^{59} - 16 q^{65} + 160 q^{67} + 132 q^{73} + 32 q^{83} + 288 q^{89} - 16 q^{91} + 188 q^{97}+O(q^{100}) $ 2 * q + 32 * q^11 + 48 * q^17 + 48 * q^19 + 18 * q^25 - 32 * q^35 - 16 * q^41 - 112 * q^43 + 66 * q^49 - 64 * q^59 - 16 * q^65 + 160 * q^67 + 132 * q^73 + 32 * q^83 + 288 * q^89 - 16 * q^91 + 188 * 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
$5$	− 4.00000i	− 0.800000i	−0.916515	−	0.400000i	$-0.869010\pi$
$5$	− 4.00000i	− 0.800000i	0.916515	−	0.400000i	$-0.130990\pi$
$6$	0	0
$7$	− 4.00000i	− 0.571429i	−0.958315	−	0.285714i	$-0.907769\pi$
$7$	− 4.00000i	− 0.571429i	0.958315	−	0.285714i	$-0.0922308\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	16.0000	1.45455	0.727273	−	0.686349i	$-0.240788\pi$
$11$	16.0000	1.45455	0.727273	+	0.686349i	$0.240788\pi$
$12$	0	0
$13$	− 2.00000i	− 0.153846i	−0.997037	−	0.0769231i	$-0.975490\pi$
$13$	− 2.00000i	− 0.153846i	0.997037	−	0.0769231i	$-0.0245096\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	24.0000	1.41176	0.705882	−	0.708329i	$-0.250551\pi$
$17$	24.0000	1.41176	0.705882	+	0.708329i	$0.250551\pi$
$18$	0	0
$19$	24.0000	1.26316	0.631579	−	0.775312i	$-0.282407\pi$
$19$	24.0000	1.26316	0.631579	+	0.775312i	$0.282407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 32.0000i	− 1.39130i	−0.718379	−	0.695652i	$-0.755116\pi$
$23$	− 32.0000i	− 1.39130i	0.718379	−	0.695652i	$-0.244884\pi$
$24$	0	0
$25$	9.00000	0.360000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	44.0000i	1.51724i	0.651533	+	0.758621i	$0.274126\pi$
$29$	44.0000i	1.51724i	−0.651533	+	0.758621i	$0.725874\pi$
$30$	0	0
$31$	52.0000i	1.67742i	0.544579	+	0.838710i	$0.316690\pi$
$31$	52.0000i	1.67742i	−0.544579	+	0.838710i	$0.683310\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−16.0000	−0.457143
$36$	0	0
$37$	− 18.0000i	− 0.486486i	−0.969965	−	0.243243i	$-0.921789\pi$
$37$	− 18.0000i	− 0.486486i	0.969965	−	0.243243i	$-0.0782113\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.00000	−0.195122	−0.0975610	−	0.995230i	$-0.531104\pi$
$41$	−8.00000	−0.195122	−0.0975610	+	0.995230i	$0.531104\pi$
$42$	0	0
$43$	−56.0000	−1.30233	−0.651163	−	0.758938i	$-0.725718\pi$
$43$	−56.0000	−1.30233	−0.651163	+	0.758938i	$0.725718\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	32.0000i	0.680851i	0.940271	+	0.340426i	$0.110571\pi$
$47$	32.0000i	0.680851i	−0.940271	+	0.340426i	$0.889429\pi$
$48$	0	0
$49$	33.0000	0.673469
$50$	0	0
$51$	0	0
$52$	0	0
$53$	36.0000i	0.679245i	0.940562	+	0.339623i	$0.110299\pi$
$53$	36.0000i	0.679245i	−0.940562	+	0.339623i	$0.889701\pi$
$54$	0	0
$55$	− 64.0000i	− 1.16364i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−32.0000	−0.542373	−0.271186	−	0.962527i	$-0.587416\pi$
$59$	−32.0000	−0.542373	−0.271186	+	0.962527i	$0.587416\pi$
$60$	0	0
$61$	− 62.0000i	− 1.01639i	−0.861241	−	0.508197i	$-0.830312\pi$
$61$	− 62.0000i	− 1.01639i	0.861241	−	0.508197i	$-0.169688\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.00000	−0.123077
$66$	0	0
$67$	80.0000	1.19403	0.597015	−	0.802230i	$-0.296353\pi$
$67$	80.0000	1.19403	0.597015	+	0.802230i	$0.296353\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	128.000i	1.80282i	0.432970	+	0.901408i	$0.357466\pi$
$71$	128.000i	1.80282i	−0.432970	+	0.901408i	$0.642534\pi$
$72$	0	0
$73$	66.0000	0.904110	0.452055	−	0.891990i	$-0.350691\pi$
$73$	66.0000	0.904110	0.452055	+	0.891990i	$0.350691\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 64.0000i	− 0.831169i
$78$	0	0
$79$	20.0000i	0.253165i	0.991956	+	0.126582i	$0.0404008\pi$
$79$	20.0000i	0.253165i	−0.991956	+	0.126582i	$0.959599\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.0000	0.192771	0.0963855	−	0.995344i	$-0.469272\pi$
$83$	16.0000	0.192771	0.0963855	+	0.995344i	$0.469272\pi$
$84$	0	0
$85$	− 96.0000i	− 1.12941i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	144.000	1.61798	0.808989	−	0.587824i	$-0.200015\pi$
$89$	144.000	1.61798	0.808989	+	0.587824i	$0.200015\pi$
$90$	0	0
$91$	−8.00000	−0.0879121
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 96.0000i	− 1.01053i
$96$	0	0
$97$	94.0000	0.969072	0.484536	−	0.874771i	$-0.338988\pi$
$97$	94.0000	0.969072	0.484536	+	0.874771i	$0.338988\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	20.0000i	0.198020i	0.995086	+	0.0990099i	$0.0315676\pi$
$101$	20.0000i	0.198020i	−0.995086	+	0.0990099i	$0.968432\pi$
$102$	0	0
$103$	− 44.0000i	− 0.427184i	−0.976923	−	0.213592i	$-0.931484\pi$
$103$	− 44.0000i	− 0.427184i	0.976923	−	0.213592i	$-0.0685164\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−96.0000	−0.897196	−0.448598	−	0.893734i	$-0.648077\pi$
$107$	−96.0000	−0.897196	−0.448598	+	0.893734i	$0.648077\pi$
$108$	0	0
$109$	− 66.0000i	− 0.605505i	−0.953069	−	0.302752i	$-0.902095\pi$
$109$	− 66.0000i	− 0.605505i	0.953069	−	0.302752i	$-0.0979055\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−80.0000	−0.707965	−0.353982	−	0.935252i	$-0.615173\pi$
$113$	−80.0000	−0.707965	−0.353982	+	0.935252i	$0.615173\pi$
$114$	0	0
$115$	−128.000	−1.11304
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 96.0000i	− 0.806723i
$120$	0	0
$121$	135.000	1.11570
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 136.000i	− 1.08800i
$126$	0	0
$127$	− 132.000i	− 1.03937i	−0.854358	−	0.519685i	$-0.826049\pi$
$127$	− 132.000i	− 1.03937i	0.854358	−	0.519685i	$-0.173951\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	128.000	0.977099	0.488550	−	0.872536i	$-0.337526\pi$
$131$	128.000	0.977099	0.488550	+	0.872536i	$0.337526\pi$
$132$	0	0
$133$	− 96.0000i	− 0.721805i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−248.000	−1.81022	−0.905109	−	0.425179i	$-0.860211\pi$
$137$	−248.000	−1.81022	−0.905109	+	0.425179i	$0.860211\pi$
$138$	0	0
$139$	−16.0000	−0.115108	−0.0575540	−	0.998342i	$-0.518330\pi$
$139$	−16.0000	−0.115108	−0.0575540	+	0.998342i	$0.518330\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 32.0000i	− 0.223776i
$144$	0	0
$145$	176.000	1.21379
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 116.000i	− 0.778523i	−0.921127	−	0.389262i	$-0.872730\pi$
$149$	− 116.000i	− 0.778523i	0.921127	−	0.389262i	$-0.127270\pi$
$150$	0	0
$151$	220.000i	1.45695i	0.685070	+	0.728477i	$0.259771\pi$
$151$	220.000i	1.45695i	−0.685070	+	0.728477i	$0.740229\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	208.000	1.34194
$156$	0	0
$157$	− 286.000i	− 1.82166i	−0.412786	−	0.910828i	$-0.635444\pi$
$157$	− 286.000i	− 1.82166i	0.412786	−	0.910828i	$-0.364556\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−128.000	−0.795031
$162$	0	0
$163$	−168.000	−1.03067	−0.515337	−	0.856987i	$-0.672333\pi$
$163$	−168.000	−1.03067	−0.515337	+	0.856987i	$0.672333\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 160.000i	− 0.958084i	−0.877792	−	0.479042i	$-0.840984\pi$
$167$	− 160.000i	− 0.958084i	0.877792	−	0.479042i	$-0.159016\pi$
$168$	0	0
$169$	165.000	0.976331
$170$	0	0
$171$	0	0
$172$	0	0
$173$	244.000i	1.41040i	0.709006	+	0.705202i	$0.249144\pi$
$173$	244.000i	1.41040i	−0.709006	+	0.705202i	$0.750856\pi$
$174$	0	0
$175$	− 36.0000i	− 0.205714i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−96.0000	−0.536313	−0.268156	−	0.963375i	$-0.586414\pi$
$179$	−96.0000	−0.536313	−0.268156	+	0.963375i	$0.586414\pi$
$180$	0	0
$181$	− 270.000i	− 1.49171i	−0.666107	−	0.745856i	$-0.732041\pi$
$181$	− 270.000i	− 1.49171i	0.666107	−	0.745856i	$-0.267959\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−72.0000	−0.389189
$186$	0	0
$187$	384.000	2.05348
$188$	0	0
$189$	0	0
$190$	0	0
$191$	224.000i	1.17277i	0.810031	+	0.586387i	$0.199450\pi$
$191$	224.000i	1.17277i	−0.810031	+	0.586387i	$0.800550\pi$
$192$	0	0
$193$	178.000	0.922280	0.461140	−	0.887327i	$-0.347441\pi$
$193$	178.000	0.922280	0.461140	+	0.887327i	$0.347441\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 92.0000i	− 0.467005i	−0.972356	−	0.233503i	$-0.924981\pi$
$197$	− 92.0000i	− 0.467005i	0.972356	−	0.233503i	$-0.0750187\pi$
$198$	0	0
$199$	164.000i	0.824121i	0.911157	+	0.412060i	$0.135191\pi$
$199$	164.000i	0.824121i	−0.911157	+	0.412060i	$0.864809\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	176.000	0.866995
$204$	0	0
$205$	32.0000i	0.156098i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	384.000	1.83732
$210$	0	0
$211$	80.0000	0.379147	0.189573	−	0.981867i	$-0.439289\pi$
$211$	80.0000	0.379147	0.189573	+	0.981867i	$0.439289\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	224.000i	1.04186i
$216$	0	0
$217$	208.000	0.958525
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 48.0000i	− 0.217195i
$222$	0	0
$223$	− 212.000i	− 0.950673i	−0.879804	−	0.475336i	$-0.842326\pi$
$223$	− 212.000i	− 0.950673i	0.879804	−	0.475336i	$-0.157674\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−400.000	−1.76211	−0.881057	−	0.473010i	$-0.843168\pi$
$227$	−400.000	−1.76211	−0.881057	+	0.473010i	$0.843168\pi$
$228$	0	0
$229$	− 110.000i	− 0.480349i	−0.970730	−	0.240175i	$-0.922795\pi$
$229$	− 110.000i	− 0.480349i	0.970730	−	0.240175i	$-0.0772047\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.0000	−0.0686695	−0.0343348	−	0.999410i	$-0.510931\pi$
$233$	−16.0000	−0.0686695	−0.0343348	+	0.999410i	$0.510931\pi$
$234$	0	0
$235$	128.000	0.544681
$236$	0	0
$237$	0	0
$238$	0	0
$239$	32.0000i	0.133891i	0.997757	+	0.0669456i	$0.0213254\pi$
$239$	32.0000i	0.133891i	−0.997757	+	0.0669456i	$0.978675\pi$
$240$	0	0
$241$	110.000	0.456432	0.228216	−	0.973611i	$-0.426711\pi$
$241$	110.000	0.456432	0.228216	+	0.973611i	$0.426711\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 132.000i	− 0.538776i
$246$	0	0
$247$	− 48.0000i	− 0.194332i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−432.000	−1.72112	−0.860558	−	0.509353i	$-0.829885\pi$
$251$	−432.000	−1.72112	−0.860558	+	0.509353i	$0.829885\pi$
$252$	0	0
$253$	− 512.000i	− 2.02372i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−32.0000	−0.124514	−0.0622568	−	0.998060i	$-0.519830\pi$
$257$	−32.0000	−0.124514	−0.0622568	+	0.998060i	$0.519830\pi$
$258$	0	0
$259$	−72.0000	−0.277992
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 256.000i	− 0.973384i	−0.873574	−	0.486692i	$-0.838203\pi$
$263$	− 256.000i	− 0.973384i	0.873574	−	0.486692i	$-0.161797\pi$
$264$	0	0
$265$	144.000	0.543396
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 132.000i	− 0.490706i	−0.969434	−	0.245353i	$-0.921096\pi$
$269$	− 132.000i	− 0.490706i	0.969434	−	0.245353i	$-0.0789039\pi$
$270$	0	0
$271$	12.0000i	0.0442804i	0.999755	+	0.0221402i	$0.00704803\pi$
$271$	12.0000i	0.0442804i	−0.999755	+	0.0221402i	$0.992952\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	144.000	0.523636
$276$	0	0
$277$	− 206.000i	− 0.743682i	−0.928296	−	0.371841i	$-0.878727\pi$
$277$	− 206.000i	− 0.743682i	0.928296	−	0.371841i	$-0.121273\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	352.000	1.25267	0.626335	−	0.779554i	$-0.284554\pi$
$281$	352.000	1.25267	0.626335	+	0.779554i	$0.284554\pi$
$282$	0	0
$283$	−352.000	−1.24382	−0.621908	−	0.783090i	$-0.713642\pi$
$283$	−352.000	−1.24382	−0.621908	+	0.783090i	$0.713642\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	32.0000i	0.111498i
$288$	0	0
$289$	287.000	0.993080
$290$	0	0
$291$	0	0
$292$	0	0
$293$	260.000i	0.887372i	0.896182	+	0.443686i	$0.146329\pi$
$293$	260.000i	0.887372i	−0.896182	+	0.443686i	$0.853671\pi$
$294$	0	0
$295$	128.000i	0.433898i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−64.0000	−0.214047
$300$	0	0
$301$	224.000i	0.744186i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−248.000	−0.813115
$306$	0	0
$307$	−48.0000	−0.156352	−0.0781759	−	0.996940i	$-0.524910\pi$
$307$	−48.0000	−0.156352	−0.0781759	+	0.996940i	$0.524910\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	448.000i	1.44051i	0.693707	+	0.720257i	$0.255976\pi$
$311$	448.000i	1.44051i	−0.693707	+	0.720257i	$0.744024\pi$
$312$	0	0
$313$	94.0000	0.300319	0.150160	−	0.988662i	$-0.452021\pi$
$313$	94.0000	0.300319	0.150160	+	0.988662i	$0.452021\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	332.000i	1.04732i	0.851928	+	0.523659i	$0.175434\pi$
$317$	332.000i	1.04732i	−0.851928	+	0.523659i	$0.824566\pi$
$318$	0	0
$319$	704.000i	2.20690i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	576.000	1.78328
$324$	0	0
$325$	− 18.0000i	− 0.0553846i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	128.000	0.389058
$330$	0	0
$331$	32.0000	0.0966767	0.0483384	−	0.998831i	$-0.484607\pi$
$331$	32.0000	0.0966767	0.0483384	+	0.998831i	$0.484607\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 320.000i	− 0.955224i
$336$	0	0
$337$	110.000	0.326409	0.163205	−	0.986592i	$-0.447817\pi$
$337$	110.000	0.326409	0.163205	+	0.986592i	$0.447817\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	832.000i	2.43988i
$342$	0	0
$343$	− 328.000i	− 0.956268i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	176.000	0.507205	0.253602	−	0.967309i	$-0.418385\pi$
$347$	176.000	0.507205	0.253602	+	0.967309i	$0.418385\pi$
$348$	0	0
$349$	− 382.000i	− 1.09456i	−0.836951	−	0.547278i	$-0.815664\pi$
$349$	− 382.000i	− 1.09456i	0.836951	−	0.547278i	$-0.184336\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	176.000	0.498584	0.249292	−	0.968428i	$-0.419802\pi$
$353$	176.000	0.498584	0.249292	+	0.968428i	$0.419802\pi$
$354$	0	0
$355$	512.000	1.44225
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	215.000	0.595568
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 264.000i	− 0.723288i
$366$	0	0
$367$	− 364.000i	− 0.991826i	−0.868372	−	0.495913i	$-0.834834\pi$
$367$	− 364.000i	− 0.991826i	0.868372	−	0.495913i	$-0.165166\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	144.000	0.388140
$372$	0	0
$373$	− 274.000i	− 0.734584i	−0.930106	−	0.367292i	$-0.880285\pi$
$373$	− 274.000i	− 0.734584i	0.930106	−	0.367292i	$-0.119715\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	88.0000	0.233422
$378$	0	0
$379$	−632.000	−1.66755	−0.833773	−	0.552107i	$-0.813824\pi$
$379$	−632.000	−1.66755	−0.833773	+	0.552107i	$0.813824\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 416.000i	− 1.08616i	−0.839680	−	0.543081i	$-0.817258\pi$
$383$	− 416.000i	− 1.08616i	0.839680	−	0.543081i	$-0.182742\pi$
$384$	0	0
$385$	−256.000	−0.664935
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 148.000i	− 0.380463i	−0.981739	−	0.190231i	$-0.939076\pi$
$389$	− 148.000i	− 0.380463i	0.981739	−	0.190231i	$-0.0609238\pi$
$390$	0	0
$391$	− 768.000i	− 1.96419i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	80.0000	0.202532
$396$	0	0
$397$	450.000i	1.13350i	0.823889	+	0.566751i	$0.191800\pi$
$397$	450.000i	1.13350i	−0.823889	+	0.566751i	$0.808200\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−376.000	−0.937656	−0.468828	−	0.883290i	$-0.655324\pi$
$401$	−376.000	−0.937656	−0.468828	+	0.883290i	$0.655324\pi$
$402$	0	0
$403$	104.000	0.258065
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 288.000i	− 0.707617i
$408$	0	0
$409$	−110.000	−0.268949	−0.134474	−	0.990917i	$-0.542935\pi$
$409$	−110.000	−0.268949	−0.134474	+	0.990917i	$0.542935\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	128.000i	0.309927i
$414$	0	0
$415$	− 64.0000i	− 0.154217i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−272.000	−0.649165	−0.324582	−	0.945857i	$-0.605224\pi$
$419$	−272.000	−0.649165	−0.324582	+	0.945857i	$0.605224\pi$
$420$	0	0
$421$	530.000i	1.25891i	0.777038	+	0.629454i	$0.216721\pi$
$421$	530.000i	1.25891i	−0.777038	+	0.629454i	$0.783279\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	216.000	0.508235
$426$	0	0
$427$	−248.000	−0.580796
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 96.0000i	− 0.222738i	−0.993779	−	0.111369i	$-0.964476\pi$
$431$	− 96.0000i	− 0.222738i	0.993779	−	0.111369i	$-0.0355235\pi$
$432$	0	0
$433$	66.0000	0.152425	0.0762125	−	0.997092i	$-0.475717\pi$
$433$	66.0000	0.152425	0.0762125	+	0.997092i	$0.475717\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 768.000i	− 1.75744i
$438$	0	0
$439$	− 188.000i	− 0.428246i	−0.976807	−	0.214123i	$-0.931311\pi$
$439$	− 188.000i	− 0.428246i	0.976807	−	0.214123i	$-0.0686893\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	176.000	0.397291	0.198646	−	0.980071i	$-0.436346\pi$
$443$	176.000	0.397291	0.198646	+	0.980071i	$0.436346\pi$
$444$	0	0
$445$	− 576.000i	− 1.29438i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	312.000	0.694878	0.347439	−	0.937703i	$-0.387051\pi$
$449$	312.000	0.694878	0.347439	+	0.937703i	$0.387051\pi$
$450$	0	0
$451$	−128.000	−0.283814
$452$	0	0
$453$	0	0
$454$	0	0
$455$	32.0000i	0.0703297i
$456$	0	0
$457$	−526.000	−1.15098	−0.575492	−	0.817807i	$-0.695190\pi$
$457$	−526.000	−1.15098	−0.575492	+	0.817807i	$0.695190\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 316.000i	− 0.685466i	−0.939433	−	0.342733i	$-0.888647\pi$
$461$	− 316.000i	− 0.685466i	0.939433	−	0.342733i	$-0.111353\pi$
$462$	0	0
$463$	796.000i	1.71922i	0.510949	+	0.859611i	$0.329294\pi$
$463$	796.000i	1.71922i	−0.510949	+	0.859611i	$0.670706\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−336.000	−0.719486	−0.359743	−	0.933051i	$-0.617136\pi$
$467$	−336.000	−0.719486	−0.359743	+	0.933051i	$0.617136\pi$
$468$	0	0
$469$	− 320.000i	− 0.682303i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−896.000	−1.89429
$474$	0	0
$475$	216.000	0.454737
$476$	0	0
$477$	0	0
$478$	0	0
$479$	640.000i	1.33612i	0.744109	+	0.668058i	$0.232874\pi$
$479$	640.000i	1.33612i	−0.744109	+	0.668058i	$0.767126\pi$
$480$	0	0
$481$	−36.0000	−0.0748441
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 376.000i	− 0.775258i
$486$	0	0
$487$	300.000i	0.616016i	0.951384	+	0.308008i	$0.0996624\pi$
$487$	300.000i	0.616016i	−0.951384	+	0.308008i	$0.900338\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−512.000	−1.04277	−0.521385	−	0.853322i	$-0.674584\pi$
$491$	−512.000	−1.04277	−0.521385	+	0.853322i	$0.674584\pi$
$492$	0	0
$493$	1056.00i	2.14199i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	512.000	1.03018
$498$	0	0
$499$	736.000	1.47495	0.737475	−	0.675374i	$-0.236018\pi$
$499$	736.000	1.47495	0.737475	+	0.675374i	$0.236018\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 704.000i	− 1.39960i	−0.714338	−	0.699801i	$-0.753272\pi$
$503$	− 704.000i	− 1.39960i	0.714338	−	0.699801i	$-0.246728\pi$
$504$	0	0
$505$	80.0000	0.158416
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 100.000i	− 0.196464i	−0.995164	−	0.0982318i	$-0.968681\pi$
$509$	− 100.000i	− 0.196464i	0.995164	−	0.0982318i	$-0.0313187\pi$
$510$	0	0
$511$	− 264.000i	− 0.516634i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−176.000	−0.341748
$516$	0	0
$517$	512.000i	0.990329i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	88.0000	0.168906	0.0844530	−	0.996427i	$-0.473086\pi$
$521$	88.0000	0.168906	0.0844530	+	0.996427i	$0.473086\pi$
$522$	0	0
$523$	200.000	0.382409	0.191205	−	0.981550i	$-0.438761\pi$
$523$	200.000	0.382409	0.191205	+	0.981550i	$0.438761\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1248.00i	2.36812i
$528$	0	0
$529$	−495.000	−0.935728
$530$	0	0
$531$	0	0
$532$	0	0
$533$	16.0000i	0.0300188i
$534$	0	0
$535$	384.000i	0.717757i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	528.000	0.979592
$540$	0	0
$541$	− 578.000i	− 1.06839i	−0.845361	−	0.534196i	$-0.820614\pi$
$541$	− 578.000i	− 1.06839i	0.845361	−	0.534196i	$-0.179386\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−264.000	−0.484404
$546$	0	0
$547$	−824.000	−1.50640	−0.753199	−	0.657792i	$-0.771490\pi$
$547$	−824.000	−1.50640	−0.753199	+	0.657792i	$0.771490\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1056.00i	1.91652i
$552$	0	0
$553$	80.0000	0.144665
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1044.00i	1.87433i	0.348891	+	0.937163i	$0.386558\pi$
$557$	1044.00i	1.87433i	−0.348891	+	0.937163i	$0.613442\pi$
$558$	0	0
$559$	112.000i	0.200358i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−464.000	−0.824156	−0.412078	−	0.911149i	$-0.635197\pi$
$563$	−464.000	−0.824156	−0.412078	+	0.911149i	$0.635197\pi$
$564$	0	0
$565$	320.000i	0.566372i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−616.000	−1.08260	−0.541301	−	0.840829i	$-0.682068\pi$
$569$	−616.000	−1.08260	−0.541301	+	0.840829i	$0.682068\pi$
$570$	0	0
$571$	−272.000	−0.476357	−0.238179	−	0.971221i	$-0.576550\pi$
$571$	−272.000	−0.476357	−0.238179	+	0.971221i	$0.576550\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 288.000i	− 0.500870i
$576$	0	0
$577$	−462.000	−0.800693	−0.400347	−	0.916364i	$-0.631110\pi$
$577$	−462.000	−0.800693	−0.400347	+	0.916364i	$0.631110\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 64.0000i	− 0.110155i
$582$	0	0
$583$	576.000i	0.987993i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	128.000	0.218058	0.109029	−	0.994039i	$-0.465226\pi$
$587$	128.000	0.218058	0.109029	+	0.994039i	$0.465226\pi$
$588$	0	0
$589$	1248.00i	2.11885i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−432.000	−0.728499	−0.364250	−	0.931301i	$-0.618674\pi$
$593$	−432.000	−0.728499	−0.364250	+	0.931301i	$0.618674\pi$
$594$	0	0
$595$	−384.000	−0.645378
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 352.000i	− 0.587646i	−0.955860	−	0.293823i	$-0.905072\pi$
$599$	− 352.000i	− 0.587646i	0.955860	−	0.293823i	$-0.0949276\pi$
$600$	0	0
$601$	−994.000	−1.65391	−0.826955	−	0.562268i	$-0.809929\pi$
$601$	−994.000	−1.65391	−0.826955	+	0.562268i	$0.809929\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 540.000i	− 0.892562i
$606$	0	0
$607$	− 580.000i	− 0.955519i	−0.878491	−	0.477759i	$-0.841449\pi$
$607$	− 580.000i	− 0.955519i	0.878491	−	0.477759i	$-0.158551\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	64.0000	0.104746
$612$	0	0
$613$	462.000i	0.753670i	0.926280	+	0.376835i	$0.122988\pi$
$613$	462.000i	0.753670i	−0.926280	+	0.376835i	$0.877012\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−32.0000	−0.0518639	−0.0259319	−	0.999664i	$-0.508255\pi$
$617$	−32.0000	−0.0518639	−0.0259319	+	0.999664i	$0.508255\pi$
$618$	0	0
$619$	800.000	1.29241	0.646204	−	0.763165i	$-0.276356\pi$
$619$	800.000	1.29241	0.646204	+	0.763165i	$0.276356\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 576.000i	− 0.924559i
$624$	0	0
$625$	−319.000	−0.510400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 432.000i	− 0.686804i
$630$	0	0
$631$	204.000i	0.323296i	0.986848	+	0.161648i	$0.0516810\pi$
$631$	204.000i	0.323296i	−0.986848	+	0.161648i	$0.948319\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−528.000	−0.831496
$636$	0	0
$637$	− 66.0000i	− 0.103611i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−680.000	−1.06084	−0.530421	−	0.847734i	$-0.677966\pi$
$641$	−680.000	−1.06084	−0.530421	+	0.847734i	$0.677966\pi$
$642$	0	0
$643$	200.000	0.311042	0.155521	−	0.987833i	$-0.450294\pi$
$643$	200.000	0.311042	0.155521	+	0.987833i	$0.450294\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 96.0000i	− 0.148377i	−0.997244	−	0.0741886i	$-0.976363\pi$
$647$	− 96.0000i	− 0.148377i	0.997244	−	0.0741886i	$-0.0236367\pi$
$648$	0	0
$649$	−512.000	−0.788906
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 380.000i	− 0.581930i	−0.956734	−	0.290965i	$-0.906024\pi$
$653$	− 380.000i	− 0.581930i	0.956734	−	0.290965i	$-0.0939762\pi$
$654$	0	0
$655$	− 512.000i	− 0.781679i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	704.000	1.06829	0.534143	−	0.845394i	$-0.320635\pi$
$659$	704.000	1.06829	0.534143	+	0.845394i	$0.320635\pi$
$660$	0	0
$661$	238.000i	0.360061i	0.983661	+	0.180030i	$0.0576196\pi$
$661$	238.000i	0.360061i	−0.983661	+	0.180030i	$0.942380\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−384.000	−0.577444
$666$	0	0
$667$	1408.00	2.11094
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 992.000i	− 1.47839i
$672$	0	0
$673$	−350.000	−0.520059	−0.260030	−	0.965601i	$-0.583732\pi$
$673$	−350.000	−0.520059	−0.260030	+	0.965601i	$0.583732\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1100.00i	1.62482i	0.583090	+	0.812408i	$0.301844\pi$
$677$	1100.00i	1.62482i	−0.583090	+	0.812408i	$0.698156\pi$
$678$	0	0
$679$	− 376.000i	− 0.553756i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	912.000	1.33529	0.667643	−	0.744482i	$-0.267303\pi$
$683$	912.000	1.33529	0.667643	+	0.744482i	$0.267303\pi$
$684$	0	0
$685$	992.000i	1.44818i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	72.0000	0.104499
$690$	0	0
$691$	−616.000	−0.891462	−0.445731	−	0.895167i	$-0.647056\pi$
$691$	−616.000	−0.891462	−0.445731	+	0.895167i	$0.647056\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	64.0000i	0.0920863i
$696$	0	0
$697$	−192.000	−0.275466
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 900.000i	− 1.28388i	−0.766755	−	0.641940i	$-0.778130\pi$
$701$	− 900.000i	− 1.28388i	0.766755	−	0.641940i	$-0.221870\pi$
$702$	0	0
$703$	− 432.000i	− 0.614509i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	80.0000	0.113154
$708$	0	0
$709$	1234.00i	1.74048i	0.492628	+	0.870240i	$0.336036\pi$
$709$	1234.00i	1.74048i	−0.492628	+	0.870240i	$0.663964\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1664.00	2.33380
$714$	0	0
$715$	−128.000	−0.179021
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 576.000i	− 0.801113i	−0.916272	−	0.400556i	$-0.868817\pi$
$719$	− 576.000i	− 0.801113i	0.916272	−	0.400556i	$-0.131183\pi$
$720$	0	0
$721$	−176.000	−0.244105
$722$	0	0
$723$	0	0
$724$	0	0
$725$	396.000i	0.546207i
$726$	0	0
$727$	1228.00i	1.68913i	0.535450	+	0.844567i	$0.320142\pi$
$727$	1228.00i	1.68913i	−0.535450	+	0.844567i	$0.679858\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1344.00	−1.83858
$732$	0	0
$733$	− 770.000i	− 1.05048i	−0.850955	−	0.525239i	$-0.823976\pi$
$733$	− 770.000i	− 1.05048i	0.850955	−	0.525239i	$-0.176024\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1280.00	1.73677
$738$	0	0
$739$	592.000	0.801083	0.400541	−	0.916279i	$-0.368822\pi$
$739$	592.000	0.801083	0.400541	+	0.916279i	$0.368822\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 160.000i	− 0.215343i	−0.994187	−	0.107672i	$-0.965660\pi$
$743$	− 160.000i	− 0.215343i	0.994187	−	0.107672i	$-0.0343395\pi$
$744$	0	0
$745$	−464.000	−0.622819
$746$	0	0
$747$	0	0
$748$	0	0
$749$	384.000i	0.512684i
$750$	0	0
$751$	− 508.000i	− 0.676431i	−0.941069	−	0.338216i	$-0.890177\pi$
$751$	− 508.000i	− 0.676431i	0.941069	−	0.338216i	$-0.109823\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	880.000	1.16556
$756$	0	0
$757$	− 398.000i	− 0.525760i	−0.964829	−	0.262880i	$-0.915328\pi$
$757$	− 398.000i	− 0.525760i	0.964829	−	0.262880i	$-0.0846723\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1384.00	1.81866	0.909330	−	0.416076i	$-0.136595\pi$
$761$	1384.00	1.81866	0.909330	+	0.416076i	$0.136595\pi$
$762$	0	0
$763$	−264.000	−0.346003
$764$	0	0
$765$	0	0
$766$	0	0
$767$	64.0000i	0.0834420i
$768$	0	0
$769$	−910.000	−1.18336	−0.591678	−	0.806175i	$-0.701534\pi$
$769$	−910.000	−1.18336	−0.591678	+	0.806175i	$0.701534\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 492.000i	− 0.636481i	−0.948010	−	0.318241i	$-0.896908\pi$
$773$	− 492.000i	− 0.636481i	0.948010	−	0.318241i	$-0.103092\pi$
$774$	0	0
$775$	468.000i	0.603871i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−192.000	−0.246470
$780$	0	0
$781$	2048.00i	2.62228i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1144.00	−1.45732
$786$	0	0
$787$	−1496.00	−1.90089	−0.950445	−	0.310894i	$-0.899372\pi$
$787$	−1496.00	−1.90089	−0.950445	+	0.310894i	$0.899372\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	320.000i	0.404551i
$792$	0	0
$793$	−124.000	−0.156368
$794$	0	0
$795$	0	0
$796$	0	0
$797$	836.000i	1.04893i	0.851431	+	0.524467i	$0.175735\pi$
$797$	836.000i	1.04893i	−0.851431	+	0.524467i	$0.824265\pi$
$798$	0	0
$799$	768.000i	0.961202i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1056.00	1.31507
$804$	0	0
$805$	512.000i	0.636025i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−312.000	−0.385661	−0.192831	−	0.981232i	$-0.561767\pi$
$809$	−312.000	−0.385661	−0.192831	+	0.981232i	$0.561767\pi$
$810$	0	0
$811$	−648.000	−0.799014	−0.399507	−	0.916730i	$-0.630819\pi$
$811$	−648.000	−0.799014	−0.399507	+	0.916730i	$0.630819\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	672.000i	0.824540i
$816$	0	0
$817$	−1344.00	−1.64504
$818$	0	0
$819$	0	0
$820$	0	0
$821$	964.000i	1.17418i	0.809522	+	0.587089i	$0.199726\pi$
$821$	964.000i	1.17418i	−0.809522	+	0.587089i	$0.800274\pi$
$822$	0	0
$823$	− 1300.00i	− 1.57959i	−0.613373	−	0.789793i	$-0.710188\pi$
$823$	− 1300.00i	− 1.57959i	0.613373	−	0.789793i	$-0.289812\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1056.00	1.27690	0.638452	−	0.769661i	$-0.279575\pi$
$827$	1056.00	1.27690	0.638452	+	0.769661i	$0.279575\pi$
$828$	0	0
$829$	− 930.000i	− 1.12183i	−0.827872	−	0.560917i	$-0.810449\pi$
$829$	− 930.000i	− 1.12183i	0.827872	−	0.560917i	$-0.189551\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	792.000	0.950780
$834$	0	0
$835$	−640.000	−0.766467
$836$	0	0
$837$	0	0
$838$	0	0
$839$	256.000i	0.305125i	0.988294	+	0.152563i	$0.0487526\pi$
$839$	256.000i	0.305125i	−0.988294	+	0.152563i	$0.951247\pi$
$840$	0	0
$841$	−1095.00	−1.30202
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 660.000i	− 0.781065i
$846$	0	0
$847$	− 540.000i	− 0.637544i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−576.000	−0.676851
$852$	0	0
$853$	302.000i	0.354045i	0.984207	+	0.177022i	$0.0566465\pi$
$853$	302.000i	0.354045i	−0.984207	+	0.177022i	$0.943354\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−584.000	−0.681447	−0.340723	−	0.940164i	$-0.610672\pi$
$857$	−584.000	−0.681447	−0.340723	+	0.940164i	$0.610672\pi$
$858$	0	0
$859$	−584.000	−0.679860	−0.339930	−	0.940451i	$-0.610403\pi$
$859$	−584.000	−0.679860	−0.339930	+	0.940451i	$0.610403\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1056.00i	1.22364i	0.790998	+	0.611819i	$0.209562\pi$
$863$	1056.00i	1.22364i	−0.790998	+	0.611819i	$0.790438\pi$
$864$	0	0
$865$	976.000	1.12832
$866$	0	0
$867$	0	0
$868$	0	0
$869$	320.000i	0.368239i
$870$	0	0
$871$	− 160.000i	− 0.183697i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−544.000	−0.621714
$876$	0	0
$877$	1378.00i	1.57127i	0.618693	+	0.785633i	$0.287662\pi$
$877$	1378.00i	1.57127i	−0.618693	+	0.785633i	$0.712338\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	480.000	0.544835	0.272418	−	0.962179i	$-0.412177\pi$
$881$	480.000	0.544835	0.272418	+	0.962179i	$0.412177\pi$
$882$	0	0
$883$	792.000	0.896942	0.448471	−	0.893797i	$-0.351969\pi$
$883$	792.000	0.896942	0.448471	+	0.893797i	$0.351969\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	832.000i	0.937993i	0.883200	+	0.468997i	$0.155384\pi$
$887$	832.000i	0.937993i	−0.883200	+	0.468997i	$0.844616\pi$
$888$	0	0
$889$	−528.000	−0.593926
$890$	0	0
$891$	0	0
$892$	0	0
$893$	768.000i	0.860022i
$894$	0	0
$895$	384.000i	0.429050i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2288.00	−2.54505
$900$	0	0
$901$	864.000i	0.958935i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1080.00	−1.19337
$906$	0	0
$907$	−488.000	−0.538037	−0.269019	−	0.963135i	$-0.586699\pi$
$907$	−488.000	−0.538037	−0.269019	+	0.963135i	$0.586699\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 992.000i	− 1.08891i	−0.838789	−	0.544457i	$-0.816736\pi$
$911$	− 992.000i	− 1.08891i	0.838789	−	0.544457i	$-0.183264\pi$
$912$	0	0
$913$	256.000	0.280394
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 512.000i	− 0.558342i
$918$	0	0
$919$	− 852.000i	− 0.927095i	−0.886072	−	0.463547i	$-0.846576\pi$
$919$	− 852.000i	− 0.927095i	0.886072	−	0.463547i	$-0.153424\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	256.000	0.277356
$924$	0	0
$925$	− 162.000i	− 0.175135i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−952.000	−1.02476	−0.512379	−	0.858759i	$-0.671236\pi$
$929$	−952.000	−1.02476	−0.512379	+	0.858759i	$0.671236\pi$
$930$	0	0
$931$	792.000	0.850698
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1536.00i	− 1.64278i
$936$	0	0
$937$	750.000	0.800427	0.400213	−	0.916422i	$-0.368936\pi$
$937$	750.000	0.800427	0.400213	+	0.916422i	$0.368936\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	772.000i	0.820404i	0.911995	+	0.410202i	$0.134542\pi$
$941$	772.000i	0.820404i	−0.911995	+	0.410202i	$0.865458\pi$
$942$	0	0
$943$	256.000i	0.271474i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1312.00	−1.38543	−0.692714	−	0.721213i	$-0.743585\pi$
$947$	−1312.00	−1.38543	−0.692714	+	0.721213i	$0.743585\pi$
$948$	0	0
$949$	− 132.000i	− 0.139094i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−264.000	−0.277020	−0.138510	−	0.990361i	$-0.544231\pi$
$953$	−264.000	−0.277020	−0.138510	+	0.990361i	$0.544231\pi$
$954$	0	0
$955$	896.000	0.938220
$956$	0	0
$957$	0	0
$958$	0	0
$959$	992.000i	1.03441i
$960$	0	0
$961$	−1743.00	−1.81374
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 712.000i	− 0.737824i
$966$	0	0
$967$	212.000i	0.219235i	0.993974	+	0.109617i	$0.0349626\pi$
$967$	212.000i	0.219235i	−0.993974	+	0.109617i	$0.965037\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−880.000	−0.906282	−0.453141	−	0.891439i	$-0.649697\pi$
$971$	−880.000	−0.906282	−0.453141	+	0.891439i	$0.649697\pi$
$972$	0	0
$973$	64.0000i	0.0657760i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1384.00	1.41658	0.708291	−	0.705921i	$-0.249467\pi$
$977$	1384.00	1.41658	0.708291	+	0.705921i	$0.249467\pi$
$978$	0	0
$979$	2304.00	2.35342
$980$	0	0
$981$	0	0
$982$	0	0
$983$	704.000i	0.716175i	0.933688	+	0.358087i	$0.116571\pi$
$983$	704.000i	0.716175i	−0.933688	+	0.358087i	$0.883429\pi$
$984$	0	0
$985$	−368.000	−0.373604
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1792.00i	1.81193i
$990$	0	0
$991$	1092.00i	1.10192i	0.834533	+	0.550959i	$0.185738\pi$
$991$	1092.00i	1.10192i	−0.834533	+	0.550959i	$0.814262\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	656.000	0.659296
$996$	0	0
$997$	1198.00i	1.20160i	0.799398	+	0.600802i	$0.205152\pi$
$997$	1198.00i	1.20160i	−0.799398	+	0.600802i	$0.794848\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.b.i.127.1		2
3.2	odd	2		2304.3.b.a.127.2		2
4.3	odd	2		2304.3.b.b.127.1		2
8.3	odd	2	inner	2304.3.b.i.127.2		2
8.5	even	2		2304.3.b.b.127.2		2
12.11	even	2		2304.3.b.h.127.2		2
16.3	odd	4		576.3.g.d.127.1		2
16.5	even	4		288.3.g.c.127.2	yes	2
16.11	odd	4		288.3.g.c.127.1	yes	2
16.13	even	4		576.3.g.d.127.2		2
24.5	odd	2		2304.3.b.h.127.1		2
24.11	even	2		2304.3.b.a.127.1		2
48.5	odd	4		288.3.g.a.127.2	yes	2
48.11	even	4		288.3.g.a.127.1	✓	2
48.29	odd	4		576.3.g.h.127.2		2
48.35	even	4		576.3.g.h.127.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.g.a.127.1	✓	2	48.11	even	4
288.3.g.a.127.2	yes	2	48.5	odd	4
288.3.g.c.127.1	yes	2	16.11	odd	4
288.3.g.c.127.2	yes	2	16.5	even	4
576.3.g.d.127.1		2	16.3	odd	4
576.3.g.d.127.2		2	16.13	even	4
576.3.g.h.127.1		2	48.35	even	4
576.3.g.h.127.2		2	48.29	odd	4
2304.3.b.a.127.1		2	24.11	even	2
2304.3.b.a.127.2		2	3.2	odd	2
2304.3.b.b.127.1		2	4.3	odd	2
2304.3.b.b.127.2		2	8.5	even	2
2304.3.b.h.127.1		2	24.5	odd	2
2304.3.b.h.127.2		2	12.11	even	2
2304.3.b.i.127.1		2	1.1	even	1	trivial
2304.3.b.i.127.2		2	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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{5} -4.00000i q^{7} +O(q^{10})\)	\(q-4.00000i q^{5} -4.00000i q^{7} +16.0000 q^{11} -2.00000i q^{13} +24.0000 q^{17} +24.0000 q^{19} -32.0000i q^{23} +9.00000 q^{25} +44.0000i q^{29} +52.0000i q^{31} -16.0000 q^{35} -18.0000i q^{37} -8.00000 q^{41} -56.0000 q^{43} +32.0000i q^{47} +33.0000 q^{49} +36.0000i q^{53} -64.0000i q^{55} -32.0000 q^{59} -62.0000i q^{61} -8.00000 q^{65} +80.0000 q^{67} +128.000i q^{71} +66.0000 q^{73} -64.0000i q^{77} +20.0000i q^{79} +16.0000 q^{83} -96.0000i q^{85} +144.000 q^{89} -8.00000 q^{91} -96.0000i q^{95} +94.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 32 q^{11} + 48 q^{17} + 48 q^{19} + 18 q^{25} - 32 q^{35} - 16 q^{41} - 112 q^{43} + 66 q^{49} - 64 q^{59} - 16 q^{65} + 160 q^{67} + 132 q^{73} + 32 q^{83} + 288 q^{89} - 16 q^{91} + 188 q^{97}+O(q^{100}) \) 2 * q + 32 * q^11 + 48 * q^17 + 48 * q^19 + 18 * q^25 - 32 * q^35 - 16 * q^41 - 112 * q^43 + 66 * q^49 - 64 * q^59 - 16 * q^65 + 160 * q^67 + 132 * q^73 + 32 * q^83 + 288 * q^89 - 16 * q^91 + 188 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more