LMFDB - Embedded newform 1792.3.d.j.1023.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,3,Mod(1023,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, 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("1792.1023");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1792.d (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:	$48.8284633734$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 5x^{14} + 48x^{12} + 180x^{10} + 1056x^{8} + 2880x^{6} + 12288x^{4} + 20480x^{2} + 65536 $ x^16 + 5x^14 + 48x^12 + 180x^10 + 1056x^8 + 2880x^6 + 12288x^4 + 20480*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{38} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1023.12
Root			$-1.69931 + 1.05468i$ of defining polynomial
Character	$\chi$	$=$	1792.1023
Dual form			1792.3.d.j.1023.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.44128i q^{3} +4.88287 q^{5} +2.64575i q^{7} -2.84239 q^{9} +O(q^{10})$	$q+3.44128i q^{3} +4.88287 q^{5} +2.64575i q^{7} -2.84239 q^{9} -21.4776i q^{11} -13.0760 q^{13} +16.8033i q^{15} -0.234889 q^{17} -4.55872i q^{19} -9.10476 q^{21} -10.9523i q^{23} -1.15761 q^{25} +21.1900i q^{27} +34.6435 q^{29} -34.1079i q^{31} +73.9103 q^{33} +12.9189i q^{35} +54.2370 q^{37} -44.9982i q^{39} +37.8300 q^{41} -4.84714i q^{43} -13.8790 q^{45} -72.3368i q^{47} -7.00000 q^{49} -0.808319i q^{51} -21.6707 q^{53} -104.872i q^{55} +15.6878 q^{57} +34.9007i q^{59} +63.6012 q^{61} -7.52026i q^{63} -63.8485 q^{65} -18.4344i q^{67} +37.6899 q^{69} +47.5244i q^{71} -55.9103 q^{73} -3.98365i q^{75} +56.8243 q^{77} -95.0135i q^{79} -98.5023 q^{81} -71.5156i q^{83} -1.14693 q^{85} +119.218i q^{87} +159.756 q^{89} -34.5959i q^{91} +117.375 q^{93} -22.2596i q^{95} -90.4794 q^{97} +61.0477i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 96 q^{9}+O(q^{10}) $ 16 * q - 96 * q^9	$ 16 q - 96 q^{9} - 160 q^{17} + 32 q^{25} + 64 q^{33} - 256 q^{41} - 112 q^{49} - 112 q^{57} - 144 q^{65} + 224 q^{73} + 96 q^{81} + 1024 q^{89} + 128 q^{97}+O(q^{100}) $ 16 * q - 96 * q^9 - 160 * q^17 + 32 * q^25 + 64 * q^33 - 256 * q^41 - 112 * q^49 - 112 * q^57 - 144 * q^65 + 224 * q^73 + 96 * q^81 + 1024 * q^89 + 128 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	3.44128i	1.14709i	0.819173	+	0.573546i	$0.194433\pi$
$3$	3.44128i	1.14709i	−0.819173	+	0.573546i	$0.805567\pi$
$4$	0	0
$5$	4.88287	0.976573	0.488287	−	0.872683i	$-0.337622\pi$
$5$	4.88287	0.976573	0.488287	+	0.872683i	$0.337622\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	−2.84239	−0.315821
$10$	0	0
$11$	− 21.4776i	− 1.95251i	−0.216632	−	0.976253i	$-0.569507\pi$
$11$	− 21.4776i	− 1.95251i	0.216632	−	0.976253i	$-0.430493\pi$
$12$	0	0
$13$	−13.0760	−1.00585	−0.502924	−	0.864331i	$-0.667742\pi$
$13$	−13.0760	−1.00585	−0.502924	+	0.864331i	$0.667742\pi$
$14$	0	0
$15$	16.8033i	1.12022i
$16$	0	0
$17$	−0.234889	−0.0138170	−0.00690851	−	0.999976i	$-0.502199\pi$
$17$	−0.234889	−0.0138170	−0.00690851	+	0.999976i	$0.502199\pi$
$18$	0	0
$19$	− 4.55872i	− 0.239933i	−0.992778	−	0.119966i	$-0.961721\pi$
$19$	− 4.55872i	− 0.239933i	0.992778	−	0.119966i	$-0.0382787\pi$
$20$	0	0
$21$	−9.10476	−0.433560
$22$	0	0
$23$	− 10.9523i	− 0.476187i	−0.971242	−	0.238094i	$-0.923478\pi$
$23$	− 10.9523i	− 0.476187i	0.971242	−	0.238094i	$-0.0765225\pi$
$24$	0	0
$25$	−1.15761	−0.0463043
$26$	0	0
$27$	21.1900i	0.784816i
$28$	0	0
$29$	34.6435	1.19460	0.597302	−	0.802016i	$-0.296239\pi$
$29$	34.6435	1.19460	0.597302	+	0.802016i	$0.296239\pi$
$30$	0	0
$31$	− 34.1079i	− 1.10025i	−0.835081	−	0.550127i	$-0.814579\pi$
$31$	− 34.1079i	− 1.10025i	0.835081	−	0.550127i	$-0.185421\pi$
$32$	0	0
$33$	73.9103	2.23971
$34$	0	0
$35$	12.9189i	0.369110i
$36$	0	0
$37$	54.2370	1.46586	0.732932	−	0.680302i	$-0.238151\pi$
$37$	54.2370	1.46586	0.732932	+	0.680302i	$0.238151\pi$
$38$	0	0
$39$	− 44.9982i	− 1.15380i
$40$	0	0
$41$	37.8300	0.922682	0.461341	−	0.887223i	$-0.347368\pi$
$41$	37.8300	0.922682	0.461341	+	0.887223i	$0.347368\pi$
$42$	0	0
$43$	− 4.84714i	− 0.112724i	−0.998410	−	0.0563621i	$-0.982050\pi$
$43$	− 4.84714i	− 0.112724i	0.998410	−	0.0563621i	$-0.0179501\pi$
$44$	0	0
$45$	−13.8790	−0.308423
$46$	0	0
$47$	− 72.3368i	− 1.53908i	−0.638598	−	0.769541i	$-0.720485\pi$
$47$	− 72.3368i	− 1.53908i	0.638598	−	0.769541i	$-0.279515\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 0.808319i	− 0.0158494i
$52$	0	0
$53$	−21.6707	−0.408881	−0.204440	−	0.978879i	$-0.565537\pi$
$53$	−21.6707	−0.408881	−0.204440	+	0.978879i	$0.565537\pi$
$54$	0	0
$55$	− 104.872i	− 1.90677i
$56$	0	0
$57$	15.6878	0.275225
$58$	0	0
$59$	34.9007i	0.591537i	0.955260	+	0.295768i	$0.0955757\pi$
$59$	34.9007i	0.591537i	−0.955260	+	0.295768i	$0.904424\pi$
$60$	0	0
$61$	63.6012	1.04264	0.521321	−	0.853360i	$-0.325439\pi$
$61$	63.6012	1.04264	0.521321	+	0.853360i	$0.325439\pi$
$62$	0	0
$63$	− 7.52026i	− 0.119369i
$64$	0	0
$65$	−63.8485	−0.982284
$66$	0	0
$67$	− 18.4344i	− 0.275140i	−0.990492	−	0.137570i	$-0.956071\pi$
$67$	− 18.4344i	− 0.275140i	0.990492	−	0.137570i	$-0.0439292\pi$
$68$	0	0
$69$	37.6899	0.546231
$70$	0	0
$71$	47.5244i	0.669358i	0.942332	+	0.334679i	$0.108628\pi$
$71$	47.5244i	0.669358i	−0.942332	+	0.334679i	$0.891372\pi$
$72$	0	0
$73$	−55.9103	−0.765894	−0.382947	−	0.923770i	$-0.625091\pi$
$73$	−55.9103	−0.765894	−0.382947	+	0.923770i	$0.625091\pi$
$74$	0	0
$75$	− 3.98365i	− 0.0531153i
$76$	0	0
$77$	56.8243	0.737978
$78$	0	0
$79$	− 95.0135i	− 1.20270i	−0.798985	−	0.601351i	$-0.794629\pi$
$79$	− 95.0135i	− 1.20270i	0.798985	−	0.601351i	$-0.205371\pi$
$80$	0	0
$81$	−98.5023	−1.21608
$82$	0	0
$83$	− 71.5156i	− 0.861634i	−0.902439	−	0.430817i	$-0.858225\pi$
$83$	− 71.5156i	− 0.861634i	0.902439	−	0.430817i	$-0.141775\pi$
$84$	0	0
$85$	−1.14693	−0.0134933
$86$	0	0
$87$	119.218i	1.37032i
$88$	0	0
$89$	159.756	1.79501	0.897504	−	0.441006i	$-0.145378\pi$
$89$	159.756	1.79501	0.897504	+	0.441006i	$0.145378\pi$
$90$	0	0
$91$	− 34.5959i	− 0.380175i
$92$	0	0
$93$	117.375	1.26209
$94$	0	0
$95$	− 22.2596i	− 0.234312i
$96$	0	0
$97$	−90.4794	−0.932777	−0.466389	−	0.884580i	$-0.654445\pi$
$97$	−90.4794	−0.932777	−0.466389	+	0.884580i	$0.654445\pi$
$98$	0	0
$99$	61.0477i	0.616643i
$100$	0	0
$101$	181.147	1.79353	0.896767	−	0.442503i	$-0.145909\pi$
$101$	181.147	1.79353	0.896767	+	0.442503i	$0.145909\pi$
$102$	0	0
$103$	− 39.3003i	− 0.381556i	−0.981633	−	0.190778i	$-0.938899\pi$
$103$	− 39.3003i	− 0.381556i	0.981633	−	0.190778i	$-0.0611010\pi$
$104$	0	0
$105$	−44.4574	−0.423403
$106$	0	0
$107$	38.4498i	0.359344i	0.983727	+	0.179672i	$0.0575037\pi$
$107$	38.4498i	0.359344i	−0.983727	+	0.179672i	$0.942496\pi$
$108$	0	0
$109$	27.8786	0.255767	0.127883	−	0.991789i	$-0.459182\pi$
$109$	27.8786	0.255767	0.127883	+	0.991789i	$0.459182\pi$
$110$	0	0
$111$	186.644i	1.68148i
$112$	0	0
$113$	82.4419	0.729574	0.364787	−	0.931091i	$-0.381142\pi$
$113$	82.4419	0.729574	0.364787	+	0.931091i	$0.381142\pi$
$114$	0	0
$115$	− 53.4786i	− 0.465032i
$116$	0	0
$117$	37.1672	0.317668
$118$	0	0
$119$	− 0.621458i	− 0.00522234i
$120$	0	0
$121$	−340.286	−2.81228
$122$	0	0
$123$	130.183i	1.05840i
$124$	0	0
$125$	−127.724	−1.02179
$126$	0	0
$127$	25.1408i	0.197959i	0.995089	+	0.0989796i	$0.0315579\pi$
$127$	25.1408i	0.197959i	−0.995089	+	0.0989796i	$0.968442\pi$
$128$	0	0
$129$	16.6804	0.129305
$130$	0	0
$131$	126.398i	0.964872i	0.875931	+	0.482436i	$0.160248\pi$
$131$	126.398i	0.964872i	−0.875931	+	0.482436i	$0.839752\pi$
$132$	0	0
$133$	12.0612	0.0906861
$134$	0	0
$135$	103.468i	0.766431i
$136$	0	0
$137$	−34.9456	−0.255078	−0.127539	−	0.991834i	$-0.540708\pi$
$137$	−34.9456	−0.255078	−0.127539	+	0.991834i	$0.540708\pi$
$138$	0	0
$139$	− 119.148i	− 0.857177i	−0.903500	−	0.428589i	$-0.859011\pi$
$139$	− 119.148i	− 0.857177i	0.903500	−	0.428589i	$-0.140989\pi$
$140$	0	0
$141$	248.931	1.76547
$142$	0	0
$143$	280.841i	1.96392i
$144$	0	0
$145$	169.160	1.16662
$146$	0	0
$147$	− 24.0889i	− 0.163870i
$148$	0	0
$149$	−121.932	−0.818334	−0.409167	−	0.912460i	$-0.634181\pi$
$149$	−121.932	−0.818334	−0.409167	+	0.912460i	$0.634181\pi$
$150$	0	0
$151$	− 220.404i	− 1.45963i	−0.683645	−	0.729815i	$-0.739606\pi$
$151$	− 220.404i	− 1.45963i	0.683645	−	0.729815i	$-0.260394\pi$
$152$	0	0
$153$	0.667647	0.00436371
$154$	0	0
$155$	− 166.544i	− 1.07448i
$156$	0	0
$157$	6.77014	0.0431219	0.0215610	−	0.999768i	$-0.493136\pi$
$157$	6.77014	0.0431219	0.0215610	+	0.999768i	$0.493136\pi$
$158$	0	0
$159$	− 74.5748i	− 0.469024i
$160$	0	0
$161$	28.9771	0.179982
$162$	0	0
$163$	207.243i	1.27143i	0.771924	+	0.635715i	$0.219294\pi$
$163$	207.243i	1.27143i	−0.771924	+	0.635715i	$0.780706\pi$
$164$	0	0
$165$	360.894	2.18724
$166$	0	0
$167$	165.529i	0.991193i	0.868553	+	0.495596i	$0.165050\pi$
$167$	165.529i	0.991193i	−0.868553	+	0.495596i	$0.834950\pi$
$168$	0	0
$169$	1.98237	0.0117300
$170$	0	0
$171$	12.9577i	0.0757759i
$172$	0	0
$173$	88.8530	0.513601	0.256800	−	0.966464i	$-0.417332\pi$
$173$	88.8530	0.513601	0.256800	+	0.966464i	$0.417332\pi$
$174$	0	0
$175$	− 3.06274i	− 0.0175014i
$176$	0	0
$177$	−120.103	−0.678548
$178$	0	0
$179$	− 80.3791i	− 0.449045i	−0.974469	−	0.224523i	$-0.927918\pi$
$179$	− 80.3791i	− 0.449045i	0.974469	−	0.224523i	$-0.0720823\pi$
$180$	0	0
$181$	276.353	1.52681	0.763406	−	0.645919i	$-0.223526\pi$
$181$	276.353	1.52681	0.763406	+	0.645919i	$0.223526\pi$
$182$	0	0
$183$	218.869i	1.19601i
$184$	0	0
$185$	264.832	1.43152
$186$	0	0
$187$	5.04485i	0.0269778i
$188$	0	0
$189$	−56.0636	−0.296633
$190$	0	0
$191$	− 203.015i	− 1.06290i	−0.847088	−	0.531452i	$-0.821647\pi$
$191$	− 203.015i	− 1.06290i	0.847088	−	0.531452i	$-0.178353\pi$
$192$	0	0
$193$	87.3328	0.452502	0.226251	−	0.974069i	$-0.427353\pi$
$193$	87.3328	0.452502	0.226251	+	0.974069i	$0.427353\pi$
$194$	0	0
$195$	− 219.720i	− 1.12677i
$196$	0	0
$197$	21.6639	0.109969	0.0549845	−	0.998487i	$-0.482489\pi$
$197$	21.6639	0.109969	0.0549845	+	0.998487i	$0.482489\pi$
$198$	0	0
$199$	− 181.933i	− 0.914235i	−0.889406	−	0.457118i	$-0.848882\pi$
$199$	− 181.933i	− 0.914235i	0.889406	−	0.457118i	$-0.151118\pi$
$200$	0	0
$201$	63.4378	0.315611
$202$	0	0
$203$	91.6582i	0.451518i
$204$	0	0
$205$	184.719	0.901067
$206$	0	0
$207$	31.1307i	0.150390i
$208$	0	0
$209$	−97.9103	−0.468470
$210$	0	0
$211$	21.4204i	0.101519i	0.998711	+	0.0507594i	$0.0161641\pi$
$211$	21.4204i	0.101519i	−0.998711	+	0.0507594i	$0.983836\pi$
$212$	0	0
$213$	−163.545	−0.767815
$214$	0	0
$215$	− 23.6680i	− 0.110084i
$216$	0	0
$217$	90.2410	0.415857
$218$	0	0
$219$	− 192.403i	− 0.878552i
$220$	0	0
$221$	3.07142	0.0138978
$222$	0	0
$223$	195.958i	0.878735i	0.898307	+	0.439367i	$0.144797\pi$
$223$	195.958i	0.878735i	−0.898307	+	0.439367i	$0.855203\pi$
$224$	0	0
$225$	3.29038	0.0146239
$226$	0	0
$227$	27.2652i	0.120111i	0.998195	+	0.0600554i	$0.0191277\pi$
$227$	27.2652i	0.120111i	−0.998195	+	0.0600554i	$0.980872\pi$
$228$	0	0
$229$	176.347	0.770076	0.385038	−	0.922901i	$-0.374188\pi$
$229$	176.347	0.770076	0.385038	+	0.922901i	$0.374188\pi$
$230$	0	0
$231$	195.548i	0.846529i
$232$	0	0
$233$	−71.8366	−0.308312	−0.154156	−	0.988047i	$-0.549266\pi$
$233$	−71.8366	−0.308312	−0.154156	+	0.988047i	$0.549266\pi$
$234$	0	0
$235$	− 353.211i	− 1.50303i
$236$	0	0
$237$	326.968	1.37961
$238$	0	0
$239$	− 71.0926i	− 0.297459i	−0.988878	−	0.148729i	$-0.952482\pi$
$239$	− 71.0926i	− 0.297459i	0.988878	−	0.148729i	$-0.0475183\pi$
$240$	0	0
$241$	56.1113	0.232827	0.116413	−	0.993201i	$-0.462860\pi$
$241$	56.1113	0.232827	0.116413	+	0.993201i	$0.462860\pi$
$242$	0	0
$243$	− 148.264i	− 0.610138i
$244$	0	0
$245$	−34.1801	−0.139510
$246$	0	0
$247$	59.6100i	0.241336i
$248$	0	0
$249$	246.105	0.988374
$250$	0	0
$251$	368.953i	1.46993i	0.678104	+	0.734966i	$0.262802\pi$
$251$	368.953i	1.46993i	−0.678104	+	0.734966i	$0.737198\pi$
$252$	0	0
$253$	−235.229	−0.929759
$254$	0	0
$255$	− 3.94691i	− 0.0154781i
$256$	0	0
$257$	23.7428	0.0923845	0.0461923	−	0.998933i	$-0.485291\pi$
$257$	23.7428	0.0923845	0.0461923	+	0.998933i	$0.485291\pi$
$258$	0	0
$259$	143.498i	0.554044i
$260$	0	0
$261$	−98.4705	−0.377282
$262$	0	0
$263$	73.9707i	0.281257i	0.990062	+	0.140629i	$0.0449124\pi$
$263$	73.9707i	0.281257i	−0.990062	+	0.140629i	$0.955088\pi$
$264$	0	0
$265$	−105.815	−0.399302
$266$	0	0
$267$	549.764i	2.05904i
$268$	0	0
$269$	335.593	1.24756	0.623779	−	0.781601i	$-0.285597\pi$
$269$	335.593	1.24756	0.623779	+	0.781601i	$0.285597\pi$
$270$	0	0
$271$	187.276i	0.691054i	0.938409	+	0.345527i	$0.112300\pi$
$271$	187.276i	0.691054i	−0.938409	+	0.345527i	$0.887700\pi$
$272$	0	0
$273$	119.054	0.436096
$274$	0	0
$275$	24.8626i	0.0904095i
$276$	0	0
$277$	−132.592	−0.478670	−0.239335	−	0.970937i	$-0.576929\pi$
$277$	−132.592	−0.478670	−0.239335	+	0.970937i	$0.576929\pi$
$278$	0	0
$279$	96.9480i	0.347484i
$280$	0	0
$281$	−331.520	−1.17979	−0.589894	−	0.807481i	$-0.700830\pi$
$281$	−331.520	−1.17979	−0.589894	+	0.807481i	$0.700830\pi$
$282$	0	0
$283$	− 66.7158i	− 0.235745i	−0.993029	−	0.117873i	$-0.962393\pi$
$283$	− 66.7158i	− 0.235745i	0.993029	−	0.117873i	$-0.0376074\pi$
$284$	0	0
$285$	76.6016	0.268778
$286$	0	0
$287$	100.089i	0.348741i
$288$	0	0
$289$	−288.945	−0.999809
$290$	0	0
$291$	− 311.365i	− 1.06998i
$292$	0	0
$293$	289.215	0.987082	0.493541	−	0.869723i	$-0.335702\pi$
$293$	289.215	0.987082	0.493541	+	0.869723i	$0.335702\pi$
$294$	0	0
$295$	170.415i	0.577679i
$296$	0	0
$297$	455.111	1.53236
$298$	0	0
$299$	143.213i	0.478972i
$300$	0	0
$301$	12.8243	0.0426058
$302$	0	0
$303$	623.377i	2.05735i
$304$	0	0
$305$	310.556	1.01822
$306$	0	0
$307$	− 0.693177i	− 0.00225790i	−0.999999	−	0.00112895i	$-0.999641\pi$
$307$	− 0.693177i	− 0.00225790i	0.999999	−	0.00112895i	$-0.000359357\pi$
$308$	0	0
$309$	135.243	0.437680
$310$	0	0
$311$	62.1583i	0.199866i	0.994994	+	0.0999330i	$0.0318628\pi$
$311$	62.1583i	0.199866i	−0.994994	+	0.0999330i	$0.968137\pi$
$312$	0	0
$313$	−213.594	−0.682408	−0.341204	−	0.939989i	$-0.610835\pi$
$313$	−213.594	−0.682408	−0.341204	+	0.939989i	$0.610835\pi$
$314$	0	0
$315$	− 36.7204i	− 0.116573i
$316$	0	0
$317$	−23.4577	−0.0739990	−0.0369995	−	0.999315i	$-0.511780\pi$
$317$	−23.4577	−0.0739990	−0.0369995	+	0.999315i	$0.511780\pi$
$318$	0	0
$319$	− 744.059i	− 2.33247i
$320$	0	0
$321$	−132.317	−0.412201
$322$	0	0
$323$	1.07079i	0.00331515i
$324$	0	0
$325$	15.1369	0.0465751
$326$	0	0
$327$	95.9380i	0.293388i
$328$	0	0
$329$	191.385	0.581718
$330$	0	0
$331$	507.406i	1.53295i	0.642275	+	0.766474i	$0.277991\pi$
$331$	507.406i	1.53295i	−0.642275	+	0.766474i	$0.722009\pi$
$332$	0	0
$333$	−154.163	−0.462951
$334$	0	0
$335$	− 90.0126i	− 0.268694i
$336$	0	0
$337$	−342.726	−1.01699	−0.508495	−	0.861065i	$-0.669798\pi$
$337$	−342.726	−1.01699	−0.508495	+	0.861065i	$0.669798\pi$
$338$	0	0
$339$	283.705i	0.836889i
$340$	0	0
$341$	−732.555	−2.14825
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	184.035	0.533434
$346$	0	0
$347$	136.745i	0.394079i	0.980396	+	0.197039i	$0.0631327\pi$
$347$	136.745i	0.394079i	−0.980396	+	0.197039i	$0.936867\pi$
$348$	0	0
$349$	−82.0565	−0.235119	−0.117559	−	0.993066i	$-0.537507\pi$
$349$	−82.0565	−0.235119	−0.117559	+	0.993066i	$0.537507\pi$
$350$	0	0
$351$	− 277.081i	− 0.789406i
$352$	0	0
$353$	507.367	1.43730	0.718651	−	0.695371i	$-0.244760\pi$
$353$	507.367	1.43730	0.718651	+	0.695371i	$0.244760\pi$
$354$	0	0
$355$	232.055i	0.653677i
$356$	0	0
$357$	2.13861	0.00599051
$358$	0	0
$359$	− 560.809i	− 1.56214i	−0.624442	−	0.781071i	$-0.714674\pi$
$359$	− 560.809i	− 1.56214i	0.624442	−	0.781071i	$-0.285326\pi$
$360$	0	0
$361$	340.218	0.942432
$362$	0	0
$363$	− 1171.02i	− 3.22595i
$364$	0	0
$365$	−273.003	−0.747952
$366$	0	0
$367$	26.9431i	0.0734145i	0.999326	+	0.0367072i	$0.0116869\pi$
$367$	26.9431i	0.0734145i	−0.999326	+	0.0367072i	$0.988313\pi$
$368$	0	0
$369$	−107.528	−0.291403
$370$	0	0
$371$	− 57.3352i	− 0.154542i
$372$	0	0
$373$	538.034	1.44245	0.721225	−	0.692701i	$-0.243579\pi$
$373$	538.034	1.44245	0.721225	+	0.692701i	$0.243579\pi$
$374$	0	0
$375$	− 439.534i	− 1.17209i
$376$	0	0
$377$	−453.000	−1.20159
$378$	0	0
$379$	− 182.132i	− 0.480560i	−0.970704	−	0.240280i	$-0.922761\pi$
$379$	− 182.132i	− 0.480560i	0.970704	−	0.240280i	$-0.0772393\pi$
$380$	0	0
$381$	−86.5166	−0.227078
$382$	0	0
$383$	− 333.271i	− 0.870160i	−0.900392	−	0.435080i	$-0.856720\pi$
$383$	− 333.271i	− 0.870160i	0.900392	−	0.435080i	$-0.143280\pi$
$384$	0	0
$385$	277.466	0.720690
$386$	0	0
$387$	13.7775i	0.0356007i
$388$	0	0
$389$	−109.639	−0.281847	−0.140924	−	0.990020i	$-0.545007\pi$
$389$	−109.639	−0.281847	−0.140924	+	0.990020i	$0.545007\pi$
$390$	0	0
$391$	2.57258i	0.00657948i
$392$	0	0
$393$	−434.971	−1.10680
$394$	0	0
$395$	− 463.938i	− 1.17453i
$396$	0	0
$397$	−310.938	−0.783219	−0.391610	−	0.920131i	$-0.628082\pi$
$397$	−310.938	−0.783219	−0.391610	+	0.920131i	$0.628082\pi$
$398$	0	0
$399$	41.5061i	0.104025i
$400$	0	0
$401$	−423.903	−1.05711	−0.528557	−	0.848898i	$-0.677267\pi$
$401$	−423.903	−1.05711	−0.528557	+	0.848898i	$0.677267\pi$
$402$	0	0
$403$	445.995i	1.10669i
$404$	0	0
$405$	−480.974	−1.18759
$406$	0	0
$407$	− 1164.88i	− 2.86211i
$408$	0	0
$409$	−444.543	−1.08690	−0.543451	−	0.839441i	$-0.682883\pi$
$409$	−444.543	−1.08690	−0.543451	+	0.839441i	$0.682883\pi$
$410$	0	0
$411$	− 120.258i	− 0.292598i
$412$	0	0
$413$	−92.3385	−0.223580
$414$	0	0
$415$	− 349.201i	− 0.841449i
$416$	0	0
$417$	410.020	0.983261
$418$	0	0
$419$	− 457.129i	− 1.09100i	−0.838111	−	0.545500i	$-0.816340\pi$
$419$	− 457.129i	− 1.09100i	0.838111	−	0.545500i	$-0.183660\pi$
$420$	0	0
$421$	−25.4812	−0.0605255	−0.0302628	−	0.999542i	$-0.509634\pi$
$421$	−25.4812	−0.0605255	−0.0302628	+	0.999542i	$0.509634\pi$
$422$	0	0
$423$	205.610i	0.486075i
$424$	0	0
$425$	0.271910	0.000639787 0
$426$	0	0
$427$	168.273i	0.394082i
$428$	0	0
$429$	−966.453	−2.25280
$430$	0	0
$431$	− 124.595i	− 0.289084i	−0.989499	−	0.144542i	$-0.953829\pi$
$431$	− 124.595i	− 0.289084i	0.989499	−	0.144542i	$-0.0461709\pi$
$432$	0	0
$433$	−272.271	−0.628802	−0.314401	−	0.949290i	$-0.601804\pi$
$433$	−272.271	−0.628802	−0.314401	+	0.949290i	$0.601804\pi$
$434$	0	0
$435$	582.126i	1.33822i
$436$	0	0
$437$	−49.9285	−0.114253
$438$	0	0
$439$	− 255.069i	− 0.581023i	−0.956871	−	0.290512i	$-0.906174\pi$
$439$	− 255.069i	− 0.581023i	0.956871	−	0.290512i	$-0.0938255\pi$
$440$	0	0
$441$	19.8967	0.0451173
$442$	0	0
$443$	131.274i	0.296330i	0.988963	+	0.148165i	$0.0473366\pi$
$443$	131.274i	0.296330i	−0.988963	+	0.148165i	$0.952663\pi$
$444$	0	0
$445$	780.066	1.75296
$446$	0	0
$447$	− 419.601i	− 0.938704i
$448$	0	0
$449$	−642.824	−1.43168	−0.715839	−	0.698265i	$-0.753956\pi$
$449$	−642.824	−1.43168	−0.715839	+	0.698265i	$0.753956\pi$
$450$	0	0
$451$	− 812.496i	− 1.80154i
$452$	0	0
$453$	758.472	1.67433
$454$	0	0
$455$	− 168.927i	− 0.371269i
$456$	0	0
$457$	−693.088	−1.51660	−0.758302	−	0.651903i	$-0.773971\pi$
$457$	−693.088	−1.51660	−0.758302	+	0.651903i	$0.773971\pi$
$458$	0	0
$459$	− 4.97731i	− 0.0108438i
$460$	0	0
$461$	−258.699	−0.561170	−0.280585	−	0.959829i	$-0.590528\pi$
$461$	−258.699	−0.561170	−0.280585	+	0.959829i	$0.590528\pi$
$462$	0	0
$463$	637.226i	1.37630i	0.725569	+	0.688150i	$0.241577\pi$
$463$	637.226i	1.37630i	−0.725569	+	0.688150i	$0.758423\pi$
$464$	0	0
$465$	573.125	1.23253
$466$	0	0
$467$	199.483i	0.427159i	0.976926	+	0.213580i	$0.0685123\pi$
$467$	199.483i	0.427159i	−0.976926	+	0.213580i	$0.931488\pi$
$468$	0	0
$469$	48.7727	0.103993
$470$	0	0
$471$	23.2979i	0.0494648i
$472$	0	0
$473$	−104.105	−0.220095
$474$	0	0
$475$	5.27721i	0.0111099i
$476$	0	0
$477$	61.5966	0.129133
$478$	0	0
$479$	− 674.160i	− 1.40743i	−0.710481	−	0.703716i	$-0.751523\pi$
$479$	− 674.160i	− 1.40743i	0.710481	−	0.703716i	$-0.248477\pi$
$480$	0	0
$481$	−709.204	−1.47444
$482$	0	0
$483$	99.7182i	0.206456i
$484$	0	0
$485$	−441.799	−0.910926
$486$	0	0
$487$	401.718i	0.824883i	0.910984	+	0.412442i	$0.135324\pi$
$487$	401.718i	0.824883i	−0.910984	+	0.412442i	$0.864676\pi$
$488$	0	0
$489$	−713.181	−1.45845
$490$	0	0
$491$	428.880i	0.873482i	0.899587	+	0.436741i	$0.143867\pi$
$491$	428.880i	0.873482i	−0.899587	+	0.436741i	$0.856133\pi$
$492$	0	0
$493$	−8.13739	−0.0165059
$494$	0	0
$495$	298.088i	0.602198i
$496$	0	0
$497$	−125.738	−0.252993
$498$	0	0
$499$	− 182.619i	− 0.365970i	−0.983116	−	0.182985i	$-0.941424\pi$
$499$	− 182.619i	− 0.365970i	0.983116	−	0.182985i	$-0.0585760\pi$
$500$	0	0
$501$	−569.632	−1.13699
$502$	0	0
$503$	380.158i	0.755781i	0.925850	+	0.377891i	$0.123350\pi$
$503$	380.158i	0.755781i	−0.925850	+	0.377891i	$0.876650\pi$
$504$	0	0
$505$	884.516	1.75152
$506$	0	0
$507$	6.82188i	0.0134554i
$508$	0	0
$509$	−289.538	−0.568836	−0.284418	−	0.958700i	$-0.591800\pi$
$509$	−289.538	−0.568836	−0.284418	+	0.958700i	$0.591800\pi$
$510$	0	0
$511$	− 147.925i	− 0.289481i
$512$	0	0
$513$	96.5995	0.188303
$514$	0	0
$515$	− 191.898i	− 0.372617i
$516$	0	0
$517$	−1553.62	−3.00507
$518$	0	0
$519$	305.768i	0.589148i
$520$	0	0
$521$	738.899	1.41823	0.709116	−	0.705092i	$-0.249094\pi$
$521$	738.899	1.41823	0.709116	+	0.705092i	$0.249094\pi$
$522$	0	0
$523$	647.126i	1.23734i	0.785653	+	0.618668i	$0.212327\pi$
$523$	647.126i	1.23734i	−0.785653	+	0.618668i	$0.787673\pi$
$524$	0	0
$525$	10.5397	0.0200757
$526$	0	0
$527$	8.01157i	0.0152022i
$528$	0	0
$529$	409.047	0.773246
$530$	0	0
$531$	− 99.2014i	− 0.186820i
$532$	0	0
$533$	−494.666	−0.928078
$534$	0	0
$535$	187.745i	0.350926i
$536$	0	0
$537$	276.607	0.515097
$538$	0	0
$539$	150.343i	0.278930i
$540$	0	0
$541$	178.722	0.330355	0.165178	−	0.986264i	$-0.447180\pi$
$541$	178.722	0.330355	0.165178	+	0.986264i	$0.447180\pi$
$542$	0	0
$543$	951.008i	1.75140i
$544$	0	0
$545$	136.127	0.249775
$546$	0	0
$547$	− 452.236i	− 0.826758i	−0.910559	−	0.413379i	$-0.864349\pi$
$547$	− 452.236i	− 0.826758i	0.910559	−	0.413379i	$-0.135651\pi$
$548$	0	0
$549$	−180.780	−0.329289
$550$	0	0
$551$	− 157.930i	− 0.286625i
$552$	0	0
$553$	251.382	0.454579
$554$	0	0
$555$	911.360i	1.64209i
$556$	0	0
$557$	−854.108	−1.53341	−0.766704	−	0.642001i	$-0.778105\pi$
$557$	−854.108	−1.53341	−0.766704	+	0.642001i	$0.778105\pi$
$558$	0	0
$559$	63.3814i	0.113383i
$560$	0	0
$561$	−17.3607	−0.0309460
$562$	0	0
$563$	− 249.654i	− 0.443436i	−0.975111	−	0.221718i	$-0.928834\pi$
$563$	− 249.654i	− 0.443436i	0.975111	−	0.221718i	$-0.0711664\pi$
$564$	0	0
$565$	402.553	0.712483
$566$	0	0
$567$	− 260.613i	− 0.459634i
$568$	0	0
$569$	−104.353	−0.183396	−0.0916982	−	0.995787i	$-0.529229\pi$
$569$	−104.353	−0.183396	−0.0916982	+	0.995787i	$0.529229\pi$
$570$	0	0
$571$	− 649.705i	− 1.13784i	−0.822394	−	0.568919i	$-0.807362\pi$
$571$	− 649.705i	− 1.13784i	0.822394	−	0.568919i	$-0.192638\pi$
$572$	0	0
$573$	698.630	1.21925
$574$	0	0
$575$	12.6785i	0.0220495i
$576$	0	0
$577$	−346.022	−0.599692	−0.299846	−	0.953988i	$-0.596935\pi$
$577$	−346.022	−0.599692	−0.299846	+	0.953988i	$0.596935\pi$
$578$	0	0
$579$	300.537i	0.519061i
$580$	0	0
$581$	189.213	0.325667
$582$	0	0
$583$	465.434i	0.798342i
$584$	0	0
$585$	181.482	0.310226
$586$	0	0
$587$	− 1153.54i	− 1.96514i	−0.185885	−	0.982572i	$-0.559515\pi$
$587$	− 1153.54i	− 1.96514i	0.185885	−	0.982572i	$-0.440485\pi$
$588$	0	0
$589$	−155.488	−0.263987
$590$	0	0
$591$	74.5515i	0.126145i
$592$	0	0
$593$	880.135	1.48421	0.742104	−	0.670285i	$-0.233828\pi$
$593$	880.135	1.48421	0.742104	+	0.670285i	$0.233828\pi$
$594$	0	0
$595$	− 3.03450i	− 0.00510000i
$596$	0	0
$597$	626.081	1.04871
$598$	0	0
$599$	554.939i	0.926442i	0.886243	+	0.463221i	$0.153306\pi$
$599$	554.939i	0.926442i	−0.886243	+	0.463221i	$0.846694\pi$
$600$	0	0
$601$	666.057	1.10825	0.554124	−	0.832434i	$-0.313054\pi$
$601$	666.057	1.10825	0.554124	+	0.832434i	$0.313054\pi$
$602$	0	0
$603$	52.3977i	0.0868950i
$604$	0	0
$605$	−1661.57	−2.74640
$606$	0	0
$607$	− 192.927i	− 0.317836i	−0.987292	−	0.158918i	$-0.949199\pi$
$607$	− 192.927i	− 0.317836i	0.987292	−	0.158918i	$-0.0508006\pi$
$608$	0	0
$609$	−315.421	−0.517933
$610$	0	0
$611$	945.878i	1.54808i
$612$	0	0
$613$	−608.234	−0.992226	−0.496113	−	0.868258i	$-0.665240\pi$
$613$	−608.234	−0.992226	−0.496113	+	0.868258i	$0.665240\pi$
$614$	0	0
$615$	635.668i	1.03361i
$616$	0	0
$617$	−0.884056	−0.00143283	−0.000716415	−	1.00000i	$-0.500228\pi$
$617$	−0.884056	−0.00143283	−0.000716415		1.00000i	$0.500228\pi$
$618$	0	0
$619$	358.525i	0.579200i	0.957148	+	0.289600i	$0.0935222\pi$
$619$	358.525i	0.579200i	−0.957148	+	0.289600i	$0.906478\pi$
$620$	0	0
$621$	232.080	0.373719
$622$	0	0
$623$	422.674i	0.678449i
$624$	0	0
$625$	−594.720	−0.951552
$626$	0	0
$627$	− 336.937i	− 0.537379i
$628$	0	0
$629$	−12.7397	−0.0202539
$630$	0	0
$631$	390.515i	0.618883i	0.950918	+	0.309442i	$0.100142\pi$
$631$	390.515i	0.618883i	−0.950918	+	0.309442i	$0.899858\pi$
$632$	0	0
$633$	−73.7137	−0.116451
$634$	0	0
$635$	122.759i	0.193322i
$636$	0	0
$637$	91.5322	0.143693
$638$	0	0
$639$	− 135.083i	− 0.211397i
$640$	0	0
$641$	−431.936	−0.673848	−0.336924	−	0.941532i	$-0.609386\pi$
$641$	−431.936	−0.673848	−0.336924	+	0.941532i	$0.609386\pi$
$642$	0	0
$643$	− 49.9370i	− 0.0776625i	−0.999246	−	0.0388313i	$-0.987637\pi$
$643$	− 49.9370i	− 0.0776625i	0.999246	−	0.0388313i	$-0.0123635\pi$
$644$	0	0
$645$	81.4480	0.126276
$646$	0	0
$647$	224.141i	0.346431i	0.984884	+	0.173216i	$0.0554157\pi$
$647$	224.141i	0.346431i	−0.984884	+	0.173216i	$0.944584\pi$
$648$	0	0
$649$	749.582	1.15498
$650$	0	0
$651$	310.544i	0.477027i
$652$	0	0
$653$	80.7637	0.123681	0.0618405	−	0.998086i	$-0.480303\pi$
$653$	80.7637	0.123681	0.0618405	+	0.998086i	$0.480303\pi$
$654$	0	0
$655$	617.186i	0.942268i
$656$	0	0
$657$	158.919	0.241886
$658$	0	0
$659$	940.466i	1.42711i	0.700599	+	0.713555i	$0.252916\pi$
$659$	940.466i	1.42711i	−0.700599	+	0.713555i	$0.747084\pi$
$660$	0	0
$661$	−119.930	−0.181437	−0.0907184	−	0.995877i	$-0.528916\pi$
$661$	−119.930	−0.181437	−0.0907184	+	0.995877i	$0.528916\pi$
$662$	0	0
$663$	10.5696i	0.0159421i
$664$	0	0
$665$	58.8935	0.0885616
$666$	0	0
$667$	− 379.426i	− 0.568855i
$668$	0	0
$669$	−674.346	−1.00799
$670$	0	0
$671$	− 1366.00i	− 2.03577i
$672$	0	0
$673$	1085.06	1.61227	0.806136	−	0.591731i	$-0.201555\pi$
$673$	1085.06	1.61227	0.806136	+	0.591731i	$0.201555\pi$
$674$	0	0
$675$	− 24.5298i	− 0.0363404i
$676$	0	0
$677$	−949.901	−1.40310	−0.701552	−	0.712618i	$-0.747509\pi$
$677$	−949.901	−1.40310	−0.701552	+	0.712618i	$0.747509\pi$
$678$	0	0
$679$	− 239.386i	− 0.352557i
$680$	0	0
$681$	−93.8270	−0.137778
$682$	0	0
$683$	− 893.785i	− 1.30862i	−0.756228	−	0.654308i	$-0.772960\pi$
$683$	− 893.785i	− 1.30862i	0.756228	−	0.654308i	$-0.227040\pi$
$684$	0	0
$685$	−170.635	−0.249102
$686$	0	0
$687$	606.861i	0.883349i
$688$	0	0
$689$	283.366	0.411272
$690$	0	0
$691$	− 1208.56i	− 1.74901i	−0.485019	−	0.874504i	$-0.661187\pi$
$691$	− 1208.56i	− 1.74901i	0.485019	−	0.874504i	$-0.338813\pi$
$692$	0	0
$693$	−161.517	−0.233069
$694$	0	0
$695$	− 581.782i	− 0.837096i
$696$	0	0
$697$	−8.88585	−0.0127487
$698$	0	0
$699$	− 247.210i	− 0.353662i
$700$	0	0
$701$	219.477	0.313091	0.156546	−	0.987671i	$-0.449964\pi$
$701$	219.477	0.313091	0.156546	+	0.987671i	$0.449964\pi$
$702$	0	0
$703$	− 247.251i	− 0.351709i
$704$	0	0
$705$	1215.50	1.72411
$706$	0	0
$707$	479.270i	0.677892i
$708$	0	0
$709$	−1265.13	−1.78439	−0.892195	−	0.451651i	$-0.850835\pi$
$709$	−1265.13	−1.78439	−0.892195	+	0.451651i	$0.850835\pi$
$710$	0	0
$711$	270.066i	0.379839i
$712$	0	0
$713$	−373.560	−0.523927
$714$	0	0
$715$	1371.31i	1.91792i
$716$	0	0
$717$	244.650	0.341213
$718$	0	0
$719$	1163.47i	1.61818i	0.587687	+	0.809089i	$0.300039\pi$
$719$	1163.47i	1.61818i	−0.587687	+	0.809089i	$0.699961\pi$
$720$	0	0
$721$	103.979	0.144215
$722$	0	0
$723$	193.094i	0.267074i
$724$	0	0
$725$	−40.1036	−0.0553153
$726$	0	0
$727$	− 1303.68i	− 1.79324i	−0.442803	−	0.896619i	$-0.646016\pi$
$727$	− 1303.68i	− 1.79324i	0.442803	−	0.896619i	$-0.353984\pi$
$728$	0	0
$729$	−376.305	−0.516193
$730$	0	0
$731$	1.13854i	0.00155751i
$732$	0	0
$733$	−1256.12	−1.71367	−0.856836	−	0.515589i	$-0.827573\pi$
$733$	−1256.12	−1.71367	−0.856836	+	0.515589i	$0.827573\pi$
$734$	0	0
$735$	− 117.623i	− 0.160031i
$736$	0	0
$737$	−395.925	−0.537212
$738$	0	0
$739$	− 687.168i	− 0.929862i	−0.885347	−	0.464931i	$-0.846079\pi$
$739$	− 687.168i	− 0.929862i	0.885347	−	0.464931i	$-0.153921\pi$
$740$	0	0
$741$	−205.134	−0.276835
$742$	0	0
$743$	362.628i	0.488059i	0.969768	+	0.244030i	$0.0784694\pi$
$743$	362.628i	0.488059i	−0.969768	+	0.244030i	$0.921531\pi$
$744$	0	0
$745$	−595.376	−0.799163
$746$	0	0
$747$	203.275i	0.272122i
$748$	0	0
$749$	−101.729	−0.135819
$750$	0	0
$751$	261.366i	0.348024i	0.984744	+	0.174012i	$0.0556732\pi$
$751$	261.366i	0.348024i	−0.984744	+	0.174012i	$0.944327\pi$
$752$	0	0
$753$	−1269.67	−1.68615
$754$	0	0
$755$	− 1076.20i	− 1.42544i
$756$	0	0
$757$	1395.34	1.84325	0.921625	−	0.388081i	$-0.126862\pi$
$757$	1395.34	1.84325	0.921625	+	0.388081i	$0.126862\pi$
$758$	0	0
$759$	− 809.488i	− 1.06652i
$760$	0	0
$761$	319.500	0.419843	0.209921	−	0.977718i	$-0.432679\pi$
$761$	319.500	0.419843	0.209921	+	0.977718i	$0.432679\pi$
$762$	0	0
$763$	73.7598i	0.0966708i
$764$	0	0
$765$	3.26003	0.00426148
$766$	0	0
$767$	− 456.362i	− 0.594996i
$768$	0	0
$769$	634.936	0.825664	0.412832	−	0.910807i	$-0.364540\pi$
$769$	634.936	0.825664	0.412832	+	0.910807i	$0.364540\pi$
$770$	0	0
$771$	81.7056i	0.105974i
$772$	0	0
$773$	−96.1663	−0.124407	−0.0622033	−	0.998063i	$-0.519813\pi$
$773$	−96.1663	−0.124407	−0.0622033	+	0.998063i	$0.519813\pi$
$774$	0	0
$775$	39.4836i	0.0509465i
$776$	0	0
$777$	−493.815	−0.635540
$778$	0	0
$779$	− 172.456i	− 0.221382i
$780$	0	0
$781$	1020.71	1.30693
$782$	0	0
$783$	734.098i	0.937545i
$784$	0	0
$785$	33.0577	0.0421117
$786$	0	0
$787$	1319.25i	1.67630i	0.545442	+	0.838148i	$0.316362\pi$
$787$	1319.25i	1.67630i	−0.545442	+	0.838148i	$0.683638\pi$
$788$	0	0
$789$	−254.554	−0.322628
$790$	0	0
$791$	218.121i	0.275753i
$792$	0	0
$793$	−831.651	−1.04874
$794$	0	0
$795$	− 364.139i	− 0.458036i
$796$	0	0
$797$	−818.575	−1.02707	−0.513535	−	0.858068i	$-0.671664\pi$
$797$	−818.575	−1.02707	−0.513535	+	0.858068i	$0.671664\pi$
$798$	0	0
$799$	16.9911i	0.0212655i
$800$	0	0
$801$	−454.088	−0.566902
$802$	0	0
$803$	1200.82i	1.49541i
$804$	0	0
$805$	141.491	0.175765
$806$	0	0
$807$	1154.87i	1.43106i
$808$	0	0
$809$	−1232.72	−1.52376	−0.761881	−	0.647717i	$-0.775724\pi$
$809$	−1232.72	−1.52376	−0.761881	+	0.647717i	$0.775724\pi$
$810$	0	0
$811$	− 1009.05i	− 1.24421i	−0.782935	−	0.622103i	$-0.786278\pi$
$811$	− 1009.05i	− 1.24421i	0.782935	−	0.622103i	$-0.213722\pi$
$812$	0	0
$813$	−644.468	−0.792703
$814$	0	0
$815$	1011.94i	1.24164i
$816$	0	0
$817$	−22.0968	−0.0270462
$818$	0	0
$819$	98.3351i	0.120067i
$820$	0	0
$821$	939.093	1.14384	0.571920	−	0.820309i	$-0.306199\pi$
$821$	939.093	1.14384	0.571920	+	0.820309i	$0.306199\pi$
$822$	0	0
$823$	911.100i	1.10705i	0.832833	+	0.553524i	$0.186717\pi$
$823$	911.100i	1.10705i	−0.832833	+	0.553524i	$0.813283\pi$
$824$	0	0
$825$	−85.5591	−0.103708
$826$	0	0
$827$	− 65.6564i	− 0.0793910i	−0.999212	−	0.0396955i	$-0.987361\pi$
$827$	− 65.6564i	− 0.0793910i	0.999212	−	0.0396955i	$-0.0126388\pi$
$828$	0	0
$829$	1515.94	1.82864	0.914318	−	0.404997i	$-0.132728\pi$
$829$	1515.94	1.82864	0.914318	+	0.404997i	$0.132728\pi$
$830$	0	0
$831$	− 456.285i	− 0.549079i
$832$	0	0
$833$	1.64422	0.00197386
$834$	0	0
$835$	808.257i	0.967972i
$836$	0	0
$837$	722.747	0.863497
$838$	0	0
$839$	− 869.972i	− 1.03692i	−0.855103	−	0.518458i	$-0.826506\pi$
$839$	− 869.972i	− 1.03692i	0.855103	−	0.518458i	$-0.173494\pi$
$840$	0	0
$841$	359.174	0.427080
$842$	0	0
$843$	− 1140.85i	− 1.35333i
$844$	0	0
$845$	9.67964	0.0114552
$846$	0	0
$847$	− 900.313i	− 1.06294i
$848$	0	0
$849$	229.588	0.270421
$850$	0	0
$851$	− 594.020i	− 0.698026i
$852$	0	0
$853$	1643.91	1.92721	0.963607	−	0.267322i	$-0.0861387\pi$
$853$	1643.91	1.92721	0.963607	+	0.267322i	$0.0861387\pi$
$854$	0	0
$855$	63.2706i	0.0740007i
$856$	0	0
$857$	−286.059	−0.333791	−0.166895	−	0.985975i	$-0.553374\pi$
$857$	−286.059	−0.333791	−0.166895	+	0.985975i	$0.553374\pi$
$858$	0	0
$859$	719.782i	0.837930i	0.908002	+	0.418965i	$0.137607\pi$
$859$	719.782i	0.837930i	−0.908002	+	0.418965i	$0.862393\pi$
$860$	0	0
$861$	−344.433	−0.400038
$862$	0	0
$863$	1120.47i	1.29835i	0.760641	+	0.649173i	$0.224885\pi$
$863$	1120.47i	1.29835i	−0.760641	+	0.649173i	$0.775115\pi$
$864$	0	0
$865$	433.857	0.501569
$866$	0	0
$867$	− 994.339i	− 1.14687i
$868$	0	0
$869$	−2040.66	−2.34828
$870$	0	0
$871$	241.048i	0.276749i
$872$	0	0
$873$	257.178	0.294591
$874$	0	0
$875$	− 337.926i	− 0.386201i
$876$	0	0
$877$	145.400	0.165792	0.0828960	−	0.996558i	$-0.473583\pi$
$877$	145.400	0.165792	0.0828960	+	0.996558i	$0.473583\pi$
$878$	0	0
$879$	995.269i	1.13227i
$880$	0	0
$881$	476.080	0.540386	0.270193	−	0.962806i	$-0.412913\pi$
$881$	476.080	0.540386	0.270193	+	0.962806i	$0.412913\pi$
$882$	0	0
$883$	1101.22i	1.24714i	0.781769	+	0.623568i	$0.214318\pi$
$883$	1101.22i	1.24714i	−0.781769	+	0.623568i	$0.785682\pi$
$884$	0	0
$885$	−586.447	−0.662652
$886$	0	0
$887$	1491.49i	1.68150i	0.541427	+	0.840748i	$0.317884\pi$
$887$	1491.49i	1.68150i	−0.541427	+	0.840748i	$0.682116\pi$
$888$	0	0
$889$	−66.5164	−0.0748216
$890$	0	0
$891$	2115.59i	2.37440i
$892$	0	0
$893$	−329.764	−0.369276
$894$	0	0
$895$	− 392.481i	− 0.438526i
$896$	0	0
$897$	−492.834	−0.549425
$898$	0	0
$899$	− 1181.62i	− 1.31437i
$900$	0	0
$901$	5.09021	0.00564951
$902$	0	0
$903$	44.1321i	0.0488728i
$904$	0	0
$905$	1349.40	1.49104
$906$	0	0
$907$	1155.46i	1.27394i	0.770889	+	0.636969i	$0.219812\pi$
$907$	1155.46i	1.27394i	−0.770889	+	0.636969i	$0.780188\pi$
$908$	0	0
$909$	−514.891	−0.566436
$910$	0	0
$911$	− 944.690i	− 1.03698i	−0.855083	−	0.518491i	$-0.826494\pi$
$911$	− 944.690i	− 1.03698i	0.855083	−	0.518491i	$-0.173506\pi$
$912$	0	0
$913$	−1535.98	−1.68235
$914$	0	0
$915$	1068.71i	1.16799i
$916$	0	0
$917$	−334.418	−0.364687
$918$	0	0
$919$	− 149.150i	− 0.162296i	−0.996702	−	0.0811478i	$-0.974141\pi$
$919$	− 149.150i	− 0.162296i	0.996702	−	0.0811478i	$-0.0258586\pi$
$920$	0	0
$921$	2.38541	0.00259003
$922$	0	0
$923$	− 621.430i	− 0.673272i
$924$	0	0
$925$	−62.7851	−0.0678758
$926$	0	0
$927$	111.707i	0.120503i
$928$	0	0
$929$	−58.8399	−0.0633368	−0.0316684	−	0.999498i	$-0.510082\pi$
$929$	−58.8399	−0.0633368	−0.0316684	+	0.999498i	$0.510082\pi$
$930$	0	0
$931$	31.9111i	0.0342761i
$932$	0	0
$933$	−213.904	−0.229265
$934$	0	0
$935$	24.6333i	0.0263458i
$936$	0	0
$937$	1700.18	1.81449	0.907246	−	0.420601i	$-0.138181\pi$
$937$	1700.18	1.81449	0.907246	+	0.420601i	$0.138181\pi$
$938$	0	0
$939$	− 735.036i	− 0.782786i
$940$	0	0
$941$	−56.6116	−0.0601611	−0.0300805	−	0.999547i	$-0.509576\pi$
$941$	−56.6116	−0.0601611	−0.0300805	+	0.999547i	$0.509576\pi$
$942$	0	0
$943$	− 414.325i	− 0.439369i
$944$	0	0
$945$	−273.751	−0.289684
$946$	0	0
$947$	− 242.533i	− 0.256107i	−0.991767	−	0.128053i	$-0.959127\pi$
$947$	− 242.533i	− 0.256107i	0.991767	−	0.128053i	$-0.0408729\pi$
$948$	0	0
$949$	731.084	0.770373
$950$	0	0
$951$	− 80.7244i	− 0.0848837i
$952$	0	0
$953$	364.070	0.382025	0.191013	−	0.981588i	$-0.438823\pi$
$953$	364.070	0.382025	0.191013	+	0.981588i	$0.438823\pi$
$954$	0	0
$955$	− 991.294i	− 1.03800i
$956$	0	0
$957$	2560.51	2.67556
$958$	0	0
$959$	− 92.4575i	− 0.0964103i
$960$	0	0
$961$	−202.348	−0.210560
$962$	0	0
$963$	− 109.290i	− 0.113489i
$964$	0	0
$965$	426.435	0.441901
$966$	0	0
$967$	− 1221.99i	− 1.26369i	−0.775093	−	0.631847i	$-0.782297\pi$
$967$	− 1221.99i	− 1.26369i	0.775093	−	0.631847i	$-0.217703\pi$
$968$	0	0
$969$	−3.68490	−0.00380279
$970$	0	0
$971$	− 1088.53i	− 1.12104i	−0.828140	−	0.560521i	$-0.810601\pi$
$971$	− 1088.53i	− 1.12104i	0.828140	−	0.560521i	$-0.189399\pi$
$972$	0	0
$973$	315.235	0.323982
$974$	0	0
$975$	52.0903i	0.0534259i
$976$	0	0
$977$	−1061.51	−1.08650	−0.543252	−	0.839570i	$-0.682807\pi$
$977$	−1061.51	−1.08650	−0.543252	+	0.839570i	$0.682807\pi$
$978$	0	0
$979$	− 3431.17i	− 3.50477i
$980$	0	0
$981$	−79.2419	−0.0807766
$982$	0	0
$983$	− 1322.61i	− 1.34549i	−0.739877	−	0.672743i	$-0.765116\pi$
$983$	− 1322.61i	− 1.34549i	0.739877	−	0.672743i	$-0.234884\pi$
$984$	0	0
$985$	105.782	0.107393
$986$	0	0
$987$	658.610i	0.667285i
$988$	0	0
$989$	−53.0874	−0.0536778
$990$	0	0
$991$	− 675.806i	− 0.681944i	−0.940073	−	0.340972i	$-0.889244\pi$
$991$	− 675.806i	− 0.681944i	0.940073	−	0.340972i	$-0.110756\pi$
$992$	0	0
$993$	−1746.12	−1.75843
$994$	0	0
$995$	− 888.354i	− 0.892818i
$996$	0	0
$997$	409.220	0.410452	0.205226	−	0.978715i	$-0.434207\pi$
$997$	409.220	0.410452	0.205226	+	0.978715i	$0.434207\pi$
$998$	0	0
$999$	1149.28i	1.15043i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.3.d.j.1023.12		16
4.3	odd	2	inner	1792.3.d.j.1023.6		16
8.3	odd	2	inner	1792.3.d.j.1023.11		16
8.5	even	2	inner	1792.3.d.j.1023.5		16
16.3	odd	4		224.3.g.b.15.5		8
16.5	even	4		224.3.g.b.15.6		8
16.11	odd	4		56.3.g.b.43.3	✓	8
16.13	even	4		56.3.g.b.43.4	yes	8
48.5	odd	4		2016.3.g.b.1135.3		8
48.11	even	4		504.3.g.b.379.6		8
48.29	odd	4		504.3.g.b.379.5		8
48.35	even	4		2016.3.g.b.1135.6		8
112.11	odd	12		392.3.k.o.275.8		16
112.13	odd	4		392.3.g.m.99.4		8
112.27	even	4		392.3.g.m.99.3		8
112.45	odd	12		392.3.k.n.275.3		16
112.59	even	12		392.3.k.n.275.8		16
112.61	odd	12		392.3.k.n.67.8		16
112.69	odd	4		1568.3.g.m.687.3		8
112.75	even	12		392.3.k.n.67.3		16
112.83	even	4		1568.3.g.m.687.4		8
112.93	even	12		392.3.k.o.67.8		16
112.107	odd	12		392.3.k.o.67.3		16
112.109	even	12		392.3.k.o.275.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.g.b.43.3	✓	8	16.11	odd	4
56.3.g.b.43.4	yes	8	16.13	even	4
224.3.g.b.15.5		8	16.3	odd	4
224.3.g.b.15.6		8	16.5	even	4
392.3.g.m.99.3		8	112.27	even	4
392.3.g.m.99.4		8	112.13	odd	4
392.3.k.n.67.3		16	112.75	even	12
392.3.k.n.67.8		16	112.61	odd	12
392.3.k.n.275.3		16	112.45	odd	12
392.3.k.n.275.8		16	112.59	even	12
392.3.k.o.67.3		16	112.107	odd	12
392.3.k.o.67.8		16	112.93	even	12
392.3.k.o.275.3		16	112.109	even	12
392.3.k.o.275.8		16	112.11	odd	12
504.3.g.b.379.5		8	48.29	odd	4
504.3.g.b.379.6		8	48.11	even	4
1568.3.g.m.687.3		8	112.69	odd	4
1568.3.g.m.687.4		8	112.83	even	4
1792.3.d.j.1023.5		16	8.5	even	2	inner
1792.3.d.j.1023.6		16	4.3	odd	2	inner
1792.3.d.j.1023.11		16	8.3	odd	2	inner
1792.3.d.j.1023.12		16	1.1	even	1	trivial
2016.3.g.b.1135.3		8	48.5	odd	4
2016.3.g.b.1135.6		8	48.35	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1792.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.44128i q^{3} +4.88287 q^{5} +2.64575i q^{7} -2.84239 q^{9} +O(q^{10})\)	\(q+3.44128i q^{3} +4.88287 q^{5} +2.64575i q^{7} -2.84239 q^{9} -21.4776i q^{11} -13.0760 q^{13} +16.8033i q^{15} -0.234889 q^{17} -4.55872i q^{19} -9.10476 q^{21} -10.9523i q^{23} -1.15761 q^{25} +21.1900i q^{27} +34.6435 q^{29} -34.1079i q^{31} +73.9103 q^{33} +12.9189i q^{35} +54.2370 q^{37} -44.9982i q^{39} +37.8300 q^{41} -4.84714i q^{43} -13.8790 q^{45} -72.3368i q^{47} -7.00000 q^{49} -0.808319i q^{51} -21.6707 q^{53} -104.872i q^{55} +15.6878 q^{57} +34.9007i q^{59} +63.6012 q^{61} -7.52026i q^{63} -63.8485 q^{65} -18.4344i q^{67} +37.6899 q^{69} +47.5244i q^{71} -55.9103 q^{73} -3.98365i q^{75} +56.8243 q^{77} -95.0135i q^{79} -98.5023 q^{81} -71.5156i q^{83} -1.14693 q^{85} +119.218i q^{87} +159.756 q^{89} -34.5959i q^{91} +117.375 q^{93} -22.2596i q^{95} -90.4794 q^{97} +61.0477i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 96 q^{9}+O(q^{10}) \) 16 * q - 96 * q^9	\( 16 q - 96 q^{9} - 160 q^{17} + 32 q^{25} + 64 q^{33} - 256 q^{41} - 112 q^{49} - 112 q^{57} - 144 q^{65} + 224 q^{73} + 96 q^{81} + 1024 q^{89} + 128 q^{97}+O(q^{100}) \) 16 * q - 96 * q^9 - 160 * q^17 + 32 * q^25 + 64 * q^33 - 256 * q^41 - 112 * q^49 - 112 * q^57 - 144 * q^65 + 224 * q^73 + 96 * q^81 + 1024 * q^89 + 128 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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