LMFDB - Embedded newform 1792.3.d.g.1023.4

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:	$8$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1023.4
Root			$-1.28897 + 0.581861i$ of defining polynomial
Character	$\chi$	$=$	1792.1023
Dual form			1792.3.d.g.1023.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.585786i q^{3} +9.03316 q^{5} +2.64575i q^{7} +8.65685 q^{9} +O(q^{10})$	$q-0.585786i q^{3} +9.03316 q^{5} +2.64575i q^{7} +8.65685 q^{9} +12.4853i q^{11} +9.03316 q^{13} -5.29150i q^{15} +12.3431 q^{17} -28.8701i q^{19} +1.54985 q^{21} -24.6418i q^{23} +56.5980 q^{25} -10.3431i q^{27} -22.4499 q^{29} +16.7824i q^{31} +7.31371 q^{33} +23.8995i q^{35} +16.2506 q^{37} -5.29150i q^{39} -6.97056 q^{41} -22.8284i q^{43} +78.1987 q^{45} +6.19938i q^{47} -7.00000 q^{49} -7.23045i q^{51} +8.01514 q^{53} +112.782i q^{55} -16.9117 q^{57} +30.4437i q^{59} +15.2325 q^{61} +22.9039i q^{63} +81.5980 q^{65} +78.6274i q^{67} -14.4348 q^{69} -17.5345i q^{71} -46.6863 q^{73} -33.1543i q^{75} -33.0329 q^{77} +81.0325i q^{79} +71.8528 q^{81} -40.3848i q^{83} +111.498 q^{85} +13.1509i q^{87} -111.941 q^{89} +23.8995i q^{91} +9.83089 q^{93} -260.788i q^{95} -164.108 q^{97} +108.083i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} + 144 q^{17} + 136 q^{25} - 32 q^{33} + 80 q^{41} - 56 q^{49} + 272 q^{57} + 336 q^{65} - 464 q^{73} - 104 q^{81} - 624 q^{89} - 272 q^{97}+O(q^{100}) $ 8 * q + 24 * q^9 + 144 * q^17 + 136 * q^25 - 32 * q^33 + 80 * q^41 - 56 * q^49 + 272 * q^57 + 336 * q^65 - 464 * q^73 - 104 * q^81 - 624 * q^89 - 272 * 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
$3$	− 0.585786i	− 0.195262i	−0.995223	−	0.0976311i	$-0.968873\pi$
$3$	− 0.585786i	− 0.195262i	0.995223	−	0.0976311i	$-0.0311265\pi$
$4$	0	0
$5$	9.03316	1.80663	0.903316	−	0.428976i	$-0.141125\pi$
$5$	9.03316	1.80663	0.903316	+	0.428976i	$0.141125\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	8.65685	0.961873
$10$	0	0
$11$	12.4853i	1.13503i	0.823365	+	0.567513i	$0.192094\pi$
$11$	12.4853i	1.13503i	−0.823365	+	0.567513i	$0.807906\pi$
$12$	0	0
$13$	9.03316	0.694858	0.347429	−	0.937706i	$-0.387055\pi$
$13$	9.03316	0.694858	0.347429	+	0.937706i	$0.387055\pi$
$14$	0	0
$15$	− 5.29150i	− 0.352767i
$16$	0	0
$17$	12.3431	0.726067	0.363034	−	0.931776i	$-0.381741\pi$
$17$	12.3431	0.726067	0.363034	+	0.931776i	$0.381741\pi$
$18$	0	0
$19$	− 28.8701i	− 1.51948i	−0.650229	−	0.759738i	$-0.725327\pi$
$19$	− 28.8701i	− 1.51948i	0.650229	−	0.759738i	$-0.274673\pi$
$20$	0	0
$21$	1.54985	0.0738022
$22$	0	0
$23$	− 24.6418i	− 1.07138i	−0.844414	−	0.535690i	$-0.820051\pi$
$23$	− 24.6418i	− 1.07138i	0.844414	−	0.535690i	$-0.179949\pi$
$24$	0	0
$25$	56.5980	2.26392
$26$	0	0
$27$	− 10.3431i	− 0.383079i
$28$	0	0
$29$	−22.4499	−0.774136	−0.387068	−	0.922051i	$-0.626512\pi$
$29$	−22.4499	−0.774136	−0.387068	+	0.922051i	$0.626512\pi$
$30$	0	0
$31$	16.7824i	0.541367i	0.962668	+	0.270684i	$0.0872497\pi$
$31$	16.7824i	0.541367i	−0.962668	+	0.270684i	$0.912750\pi$
$32$	0	0
$33$	7.31371	0.221628
$34$	0	0
$35$	23.8995i	0.682843i
$36$	0	0
$37$	16.2506	0.439204	0.219602	−	0.975589i	$-0.429524\pi$
$37$	16.2506	0.439204	0.219602	+	0.975589i	$0.429524\pi$
$38$	0	0
$39$	− 5.29150i	− 0.135680i
$40$	0	0
$41$	−6.97056	−0.170014	−0.0850069	−	0.996380i	$-0.527091\pi$
$41$	−6.97056	−0.170014	−0.0850069	+	0.996380i	$0.527091\pi$
$42$	0	0
$43$	− 22.8284i	− 0.530894i	−0.964126	−	0.265447i	$-0.914481\pi$
$43$	− 22.8284i	− 0.530894i	0.964126	−	0.265447i	$-0.0855195\pi$
$44$	0	0
$45$	78.1987	1.73775
$46$	0	0
$47$	6.19938i	0.131902i	0.997823	+	0.0659509i	$0.0210081\pi$
$47$	6.19938i	0.131902i	−0.997823	+	0.0659509i	$0.978992\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 7.23045i	− 0.141773i
$52$	0	0
$53$	8.01514	0.151229	0.0756145	−	0.997137i	$-0.475908\pi$
$53$	8.01514	0.151229	0.0756145	+	0.997137i	$0.475908\pi$
$54$	0	0
$55$	112.782i	2.05057i
$56$	0	0
$57$	−16.9117	−0.296696
$58$	0	0
$59$	30.4437i	0.515994i	0.966146	+	0.257997i	$0.0830625\pi$
$59$	30.4437i	0.515994i	−0.966146	+	0.257997i	$0.916938\pi$
$60$	0	0
$61$	15.2325	0.249714	0.124857	−	0.992175i	$-0.460153\pi$
$61$	15.2325	0.249714	0.124857	+	0.992175i	$0.460153\pi$
$62$	0	0
$63$	22.9039i	0.363554i
$64$	0	0
$65$	81.5980	1.25535
$66$	0	0
$67$	78.6274i	1.17354i	0.809752	+	0.586772i	$0.199601\pi$
$67$	78.6274i	1.17354i	−0.809752	+	0.586772i	$0.800399\pi$
$68$	0	0
$69$	−14.4348	−0.209200
$70$	0	0
$71$	− 17.5345i	− 0.246965i	−0.992347	−	0.123482i	$-0.960594\pi$
$71$	− 17.5345i	− 0.246965i	0.992347	−	0.123482i	$-0.0394062\pi$
$72$	0	0
$73$	−46.6863	−0.639538	−0.319769	−	0.947495i	$-0.603605\pi$
$73$	−46.6863	−0.639538	−0.319769	+	0.947495i	$0.603605\pi$
$74$	0	0
$75$	− 33.1543i	− 0.442058i
$76$	0	0
$77$	−33.0329	−0.428999
$78$	0	0
$79$	81.0325i	1.02573i	0.858470	+	0.512864i	$0.171416\pi$
$79$	81.0325i	1.02573i	−0.858470	+	0.512864i	$0.828584\pi$
$80$	0	0
$81$	71.8528	0.887072
$82$	0	0
$83$	− 40.3848i	− 0.486564i	−0.969956	−	0.243282i	$-0.921776\pi$
$83$	− 40.3848i	− 0.486564i	0.969956	−	0.243282i	$-0.0782240\pi$
$84$	0	0
$85$	111.498	1.31174
$86$	0	0
$87$	13.1509i	0.151159i
$88$	0	0
$89$	−111.941	−1.25777	−0.628883	−	0.777500i	$-0.716487\pi$
$89$	−111.941	−1.25777	−0.628883	+	0.777500i	$0.716487\pi$
$90$	0	0
$91$	23.8995i	0.262632i
$92$	0	0
$93$	9.83089	0.105709
$94$	0	0
$95$	− 260.788i	− 2.74514i
$96$	0	0
$97$	−164.108	−1.69183	−0.845916	−	0.533317i	$-0.820945\pi$
$97$	−164.108	−1.69183	−0.845916	+	0.533317i	$0.820945\pi$
$98$	0	0
$99$	108.083i	1.09175i
$100$	0	0
$101$	12.1329	0.120127	0.0600636	−	0.998195i	$-0.480870\pi$
$101$	12.1329	0.120127	0.0600636	+	0.998195i	$0.480870\pi$
$102$	0	0
$103$	− 106.582i	− 1.03478i	−0.855750	−	0.517389i	$-0.826904\pi$
$103$	− 106.582i	− 1.03478i	0.855750	−	0.517389i	$-0.173096\pi$
$104$	0	0
$105$	14.0000	0.133333
$106$	0	0
$107$	− 63.5980i	− 0.594374i	−0.954819	−	0.297187i	$-0.903952\pi$
$107$	− 63.5980i	− 0.594374i	0.954819	−	0.297187i	$-0.0960484\pi$
$108$	0	0
$109$	130.848	1.20044	0.600220	−	0.799835i	$-0.295080\pi$
$109$	130.848	1.20044	0.600220	+	0.799835i	$0.295080\pi$
$110$	0	0
$111$	− 9.51936i	− 0.0857600i
$112$	0	0
$113$	−138.225	−1.22323	−0.611617	−	0.791154i	$-0.709481\pi$
$113$	−138.225	−1.22323	−0.611617	+	0.791154i	$0.709481\pi$
$114$	0	0
$115$	− 222.593i	− 1.93559i
$116$	0	0
$117$	78.1987	0.668365
$118$	0	0
$119$	32.6569i	0.274428i
$120$	0	0
$121$	−34.8823	−0.288283
$122$	0	0
$123$	4.08326i	0.0331972i
$124$	0	0
$125$	285.430	2.28344
$126$	0	0
$127$	114.442i	0.901114i	0.892748	+	0.450557i	$0.148775\pi$
$127$	114.442i	0.901114i	−0.892748	+	0.450557i	$0.851225\pi$
$128$	0	0
$129$	−13.3726	−0.103663
$130$	0	0
$131$	168.350i	1.28512i	0.766237	+	0.642558i	$0.222127\pi$
$131$	168.350i	1.28512i	−0.766237	+	0.642558i	$0.777873\pi$
$132$	0	0
$133$	76.3830	0.574308
$134$	0	0
$135$	− 93.4313i	− 0.692084i
$136$	0	0
$137$	−34.6863	−0.253185	−0.126592	−	0.991955i	$-0.540404\pi$
$137$	−34.6863	−0.253185	−0.126592	+	0.991955i	$0.540404\pi$
$138$	0	0
$139$	107.664i	0.774561i	0.921962	+	0.387281i	$0.126586\pi$
$139$	107.664i	0.774561i	−0.921962	+	0.387281i	$0.873414\pi$
$140$	0	0
$141$	3.63151	0.0257554
$142$	0	0
$143$	112.782i	0.788682i
$144$	0	0
$145$	−202.794	−1.39858
$146$	0	0
$147$	4.10051i	0.0278946i
$148$	0	0
$149$	252.176	1.69246	0.846229	−	0.532819i	$-0.178867\pi$
$149$	252.176	1.69246	0.846229	+	0.532819i	$0.178867\pi$
$150$	0	0
$151$	− 234.486i	− 1.55289i	−0.630186	−	0.776444i	$-0.717021\pi$
$151$	− 234.486i	− 1.55289i	0.630186	−	0.776444i	$-0.282979\pi$
$152$	0	0
$153$	106.853	0.698384
$154$	0	0
$155$	151.598i	0.978051i
$156$	0	0
$157$	10.0968	0.0643109	0.0321554	−	0.999483i	$-0.489763\pi$
$157$	10.0968	0.0643109	0.0321554	+	0.999483i	$0.489763\pi$
$158$	0	0
$159$	− 4.69516i	− 0.0295293i
$160$	0	0
$161$	65.1960	0.404944
$162$	0	0
$163$	− 104.534i	− 0.641313i	−0.947196	−	0.320657i	$-0.896096\pi$
$163$	− 104.534i	− 0.641313i	0.947196	−	0.320657i	$-0.103904\pi$
$164$	0	0
$165$	66.0659	0.400399
$166$	0	0
$167$	296.765i	1.77703i	0.458843	+	0.888517i	$0.348264\pi$
$167$	296.765i	1.77703i	−0.458843	+	0.888517i	$0.651736\pi$
$168$	0	0
$169$	−87.4020	−0.517172
$170$	0	0
$171$	− 249.924i	− 1.46154i
$172$	0	0
$173$	40.0301	0.231388	0.115694	−	0.993285i	$-0.463091\pi$
$173$	40.0301	0.231388	0.115694	+	0.993285i	$0.463091\pi$
$174$	0	0
$175$	149.744i	0.855681i
$176$	0	0
$177$	17.8335	0.100754
$178$	0	0
$179$	− 294.794i	− 1.64689i	−0.567394	−	0.823447i	$-0.692048\pi$
$179$	− 294.794i	− 1.64689i	0.567394	−	0.823447i	$-0.307952\pi$
$180$	0	0
$181$	−40.4706	−0.223595	−0.111797	−	0.993731i	$-0.535661\pi$
$181$	−40.4706	−0.223595	−0.111797	+	0.993731i	$0.535661\pi$
$182$	0	0
$183$	− 8.92302i	− 0.0487596i
$184$	0	0
$185$	146.794	0.793481
$186$	0	0
$187$	154.108i	0.824105i
$188$	0	0
$189$	27.3654	0.144790
$190$	0	0
$191$	− 156.929i	− 0.821619i	−0.911721	−	0.410810i	$-0.865246\pi$
$191$	− 156.929i	− 0.821619i	0.911721	−	0.410810i	$-0.134754\pi$
$192$	0	0
$193$	−261.304	−1.35390	−0.676952	−	0.736027i	$-0.736700\pi$
$193$	−261.304	−1.35390	−0.676952	+	0.736027i	$0.736700\pi$
$194$	0	0
$195$	− 47.7990i	− 0.245123i
$196$	0	0
$197$	145.283	0.737475	0.368738	−	0.929533i	$-0.379790\pi$
$197$	145.283	0.737475	0.368738	+	0.929533i	$0.379790\pi$
$198$	0	0
$199$	390.508i	1.96235i	0.193122	+	0.981175i	$0.438139\pi$
$199$	390.508i	1.96235i	−0.193122	+	0.981175i	$0.561861\pi$
$200$	0	0
$201$	46.0589	0.229149
$202$	0	0
$203$	− 59.3970i	− 0.292596i
$204$	0	0
$205$	−62.9662	−0.307152
$206$	0	0
$207$	− 213.320i	− 1.03053i
$208$	0	0
$209$	360.451	1.72464
$210$	0	0
$211$	164.049i	0.777482i	0.921347	+	0.388741i	$0.127090\pi$
$211$	164.049i	0.777482i	−0.921347	+	0.388741i	$0.872910\pi$
$212$	0	0
$213$	−10.2715	−0.0482229
$214$	0	0
$215$	− 206.213i	− 0.959129i
$216$	0	0
$217$	−44.4020	−0.204618
$218$	0	0
$219$	27.3482i	0.124878i
$220$	0	0
$221$	111.498	0.504514
$222$	0	0
$223$	− 10.5830i	− 0.0474574i	−0.999718	−	0.0237287i	$-0.992446\pi$
$223$	− 10.5830i	− 0.0474574i	0.999718	−	0.0237287i	$-0.00755379\pi$
$224$	0	0
$225$	489.960	2.17760
$226$	0	0
$227$	213.806i	0.941877i	0.882166	+	0.470939i	$0.156085\pi$
$227$	213.806i	0.941877i	−0.882166	+	0.470939i	$0.843915\pi$
$228$	0	0
$229$	232.028	1.01322	0.506612	−	0.862174i	$-0.330898\pi$
$229$	232.028	1.01322	0.506612	+	0.862174i	$0.330898\pi$
$230$	0	0
$231$	19.3503i	0.0837673i
$232$	0	0
$233$	192.863	0.827738	0.413869	−	0.910336i	$-0.364177\pi$
$233$	192.863	0.827738	0.413869	+	0.910336i	$0.364177\pi$
$234$	0	0
$235$	56.0000i	0.238298i
$236$	0	0
$237$	47.4678	0.200286
$238$	0	0
$239$	327.917i	1.37204i	0.727583	+	0.686020i	$0.240644\pi$
$239$	327.917i	1.37204i	−0.727583	+	0.686020i	$0.759356\pi$
$240$	0	0
$241$	71.8721	0.298225	0.149112	−	0.988820i	$-0.452358\pi$
$241$	71.8721	0.298225	0.149112	+	0.988820i	$0.452358\pi$
$242$	0	0
$243$	− 135.179i	− 0.556291i
$244$	0	0
$245$	−63.2321	−0.258090
$246$	0	0
$247$	− 260.788i	− 1.05582i
$248$	0	0
$249$	−23.6569	−0.0950074
$250$	0	0
$251$	− 256.919i	− 1.02358i	−0.859110	−	0.511790i	$-0.828982\pi$
$251$	− 256.919i	− 1.02358i	0.859110	−	0.511790i	$-0.171018\pi$
$252$	0	0
$253$	307.659	1.21604
$254$	0	0
$255$	− 65.3138i	− 0.256133i
$256$	0	0
$257$	319.352	1.24262	0.621308	−	0.783566i	$-0.286602\pi$
$257$	319.352	1.24262	0.621308	+	0.783566i	$0.286602\pi$
$258$	0	0
$259$	42.9949i	0.166004i
$260$	0	0
$261$	−194.346	−0.744620
$262$	0	0
$263$	− 377.357i	− 1.43482i	−0.696653	−	0.717408i	$-0.745328\pi$
$263$	− 377.357i	− 1.43482i	0.696653	−	0.717408i	$-0.254672\pi$
$264$	0	0
$265$	72.4020	0.273215
$266$	0	0
$267$	65.5736i	0.245594i
$268$	0	0
$269$	−28.1631	−0.104696	−0.0523478	−	0.998629i	$-0.516670\pi$
$269$	−28.1631	−0.104696	−0.0523478	+	0.998629i	$0.516670\pi$
$270$	0	0
$271$	− 399.715i	− 1.47496i	−0.675367	−	0.737482i	$-0.736015\pi$
$271$	− 399.715i	− 1.47496i	0.675367	−	0.737482i	$-0.263985\pi$
$272$	0	0
$273$	14.0000	0.0512821
$274$	0	0
$275$	706.642i	2.56961i
$276$	0	0
$277$	−102.951	−0.371663	−0.185831	−	0.982582i	$-0.559498\pi$
$277$	−102.951	−0.371663	−0.185831	+	0.982582i	$0.559498\pi$
$278$	0	0
$279$	145.283i	0.520726i
$280$	0	0
$281$	150.235	0.534646	0.267323	−	0.963607i	$-0.413861\pi$
$281$	150.235	0.534646	0.267323	+	0.963607i	$0.413861\pi$
$282$	0	0
$283$	− 178.561i	− 0.630959i	−0.948932	−	0.315480i	$-0.897835\pi$
$283$	− 178.561i	− 0.630959i	0.948932	−	0.315480i	$-0.102165\pi$
$284$	0	0
$285$	−152.766	−0.536021
$286$	0	0
$287$	− 18.4424i	− 0.0642591i
$288$	0	0
$289$	−136.647	−0.472826
$290$	0	0
$291$	96.1320i	0.330351i
$292$	0	0
$293$	−219.189	−0.748085	−0.374043	−	0.927411i	$-0.622029\pi$
$293$	−219.189	−0.748085	−0.374043	+	0.927411i	$0.622029\pi$
$294$	0	0
$295$	275.002i	0.932211i
$296$	0	0
$297$	129.137	0.434805
$298$	0	0
$299$	− 222.593i	− 0.744458i
$300$	0	0
$301$	60.3983	0.200659
$302$	0	0
$303$	− 7.10726i	− 0.0234563i
$304$	0	0
$305$	137.598	0.451141
$306$	0	0
$307$	− 316.669i	− 1.03150i	−0.856741	−	0.515748i	$-0.827514\pi$
$307$	− 316.669i	− 1.03150i	0.856741	−	0.515748i	$-0.172486\pi$
$308$	0	0
$309$	−62.4344	−0.202053
$310$	0	0
$311$	72.2653i	0.232364i	0.993228	+	0.116182i	$0.0370656\pi$
$311$	72.2653i	0.232364i	−0.993228	+	0.116182i	$0.962934\pi$
$312$	0	0
$313$	−81.9512	−0.261825	−0.130913	−	0.991394i	$-0.541791\pi$
$313$	−81.9512	−0.261825	−0.130913	+	0.991394i	$0.541791\pi$
$314$	0	0
$315$	206.894i	0.656808i
$316$	0	0
$317$	109.150	0.344322	0.172161	−	0.985069i	$-0.444925\pi$
$317$	109.150	0.344322	0.172161	+	0.985069i	$0.444925\pi$
$318$	0	0
$319$	− 280.294i	− 0.878664i
$320$	0	0
$321$	−37.2548	−0.116059
$322$	0	0
$323$	− 356.347i	− 1.10324i
$324$	0	0
$325$	511.259	1.57310
$326$	0	0
$327$	− 76.6489i	− 0.234400i
$328$	0	0
$329$	−16.4020	−0.0498542
$330$	0	0
$331$	321.740i	0.972025i	0.873952	+	0.486012i	$0.161549\pi$
$331$	321.740i	0.972025i	−0.873952	+	0.486012i	$0.838451\pi$
$332$	0	0
$333$	140.679	0.422459
$334$	0	0
$335$	710.254i	2.12016i
$336$	0	0
$337$	−164.049	−0.486792	−0.243396	−	0.969927i	$-0.578261\pi$
$337$	−164.049	−0.486792	−0.243396	+	0.969927i	$0.578261\pi$
$338$	0	0
$339$	80.9706i	0.238851i
$340$	0	0
$341$	−209.533	−0.614466
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	−130.392	−0.377948
$346$	0	0
$347$	− 330.309i	− 0.951898i	−0.879473	−	0.475949i	$-0.842105\pi$
$347$	− 330.309i	− 0.951898i	0.879473	−	0.475949i	$-0.157895\pi$
$348$	0	0
$349$	−262.402	−0.751869	−0.375934	−	0.926646i	$-0.622678\pi$
$349$	−262.402	−0.751869	−0.375934	+	0.926646i	$0.622678\pi$
$350$	0	0
$351$	− 93.4313i	− 0.266186i
$352$	0	0
$353$	−578.098	−1.63767	−0.818835	−	0.574029i	$-0.805380\pi$
$353$	−578.098	−1.63767	−0.818835	+	0.574029i	$0.805380\pi$
$354$	0	0
$355$	− 158.392i	− 0.446174i
$356$	0	0
$357$	19.1300	0.0535853
$358$	0	0
$359$	365.114i	1.01703i	0.861053	+	0.508515i	$0.169805\pi$
$359$	365.114i	1.01703i	−0.861053	+	0.508515i	$0.830195\pi$
$360$	0	0
$361$	−472.480	−1.30881
$362$	0	0
$363$	20.4335i	0.0562908i
$364$	0	0
$365$	−421.725	−1.15541
$366$	0	0
$367$	520.071i	1.41709i	0.705666	+	0.708544i	$0.250648\pi$
$367$	520.071i	1.41709i	−0.705666	+	0.708544i	$0.749352\pi$
$368$	0	0
$369$	−60.3431	−0.163532
$370$	0	0
$371$	21.2061i	0.0571592i
$372$	0	0
$373$	526.711	1.41210	0.706048	−	0.708164i	$-0.250476\pi$
$373$	526.711	1.41210	0.706048	+	0.708164i	$0.250476\pi$
$374$	0	0
$375$	− 167.201i	− 0.445869i
$376$	0	0
$377$	−202.794	−0.537915
$378$	0	0
$379$	121.976i	0.321835i	0.986968	+	0.160918i	$0.0514454\pi$
$379$	121.976i	0.321835i	−0.986968	+	0.160918i	$0.948555\pi$
$380$	0	0
$381$	67.0383	0.175954
$382$	0	0
$383$	− 316.427i	− 0.826179i	−0.910690	−	0.413089i	$-0.864450\pi$
$383$	− 316.427i	− 0.826179i	0.910690	−	0.413089i	$-0.135550\pi$
$384$	0	0
$385$	−298.392	−0.775044
$386$	0	0
$387$	− 197.622i	− 0.510652i
$388$	0	0
$389$	−92.1474	−0.236883	−0.118441	−	0.992961i	$-0.537790\pi$
$389$	−92.1474	−0.236883	−0.118441	+	0.992961i	$0.537790\pi$
$390$	0	0
$391$	− 304.157i	− 0.777895i
$392$	0	0
$393$	98.6173	0.250935
$394$	0	0
$395$	731.980i	1.85311i
$396$	0	0
$397$	562.267	1.41629	0.708144	−	0.706068i	$-0.249533\pi$
$397$	562.267	1.41629	0.708144	+	0.706068i	$0.249533\pi$
$398$	0	0
$399$	− 44.7441i	− 0.112141i
$400$	0	0
$401$	81.2061	0.202509	0.101254	−	0.994861i	$-0.467714\pi$
$401$	81.2061	0.202509	0.101254	+	0.994861i	$0.467714\pi$
$402$	0	0
$403$	151.598i	0.376174i
$404$	0	0
$405$	649.058	1.60261
$406$	0	0
$407$	202.893i	0.498508i
$408$	0	0
$409$	−450.735	−1.10204	−0.551021	−	0.834491i	$-0.685762\pi$
$409$	−450.735	−1.10204	−0.551021	+	0.834491i	$0.685762\pi$
$410$	0	0
$411$	20.3188i	0.0494374i
$412$	0	0
$413$	−80.5463	−0.195027
$414$	0	0
$415$	− 364.802i	− 0.879041i
$416$	0	0
$417$	63.0681	0.151242
$418$	0	0
$419$	− 624.988i	− 1.49162i	−0.666160	−	0.745809i	$-0.732063\pi$
$419$	− 624.988i	− 1.49162i	0.666160	−	0.745809i	$-0.267937\pi$
$420$	0	0
$421$	−566.476	−1.34555	−0.672774	−	0.739848i	$-0.734897\pi$
$421$	−566.476	−1.34555	−0.672774	+	0.739848i	$0.734897\pi$
$422$	0	0
$423$	53.6671i	0.126873i
$424$	0	0
$425$	698.597	1.64376
$426$	0	0
$427$	40.3015i	0.0943829i
$428$	0	0
$429$	66.0659	0.154000
$430$	0	0
$431$	289.528i	0.671760i	0.941905	+	0.335880i	$0.109034\pi$
$431$	289.528i	0.671760i	−0.941905	+	0.335880i	$0.890966\pi$
$432$	0	0
$433$	−597.696	−1.38036	−0.690180	−	0.723638i	$-0.742468\pi$
$433$	−597.696	−1.38036	−0.690180	+	0.723638i	$0.742468\pi$
$434$	0	0
$435$	118.794i	0.273090i
$436$	0	0
$437$	−711.409	−1.62794
$438$	0	0
$439$	38.3890i	0.0874464i	0.999044	+	0.0437232i	$0.0139220\pi$
$439$	38.3890i	0.0874464i	−0.999044	+	0.0437232i	$0.986078\pi$
$440$	0	0
$441$	−60.5980	−0.137410
$442$	0	0
$443$	599.058i	1.35228i	0.736775	+	0.676138i	$0.236348\pi$
$443$	599.058i	1.35228i	−0.736775	+	0.676138i	$0.763652\pi$
$444$	0	0
$445$	−1011.18	−2.27232
$446$	0	0
$447$	− 147.721i	− 0.330473i
$448$	0	0
$449$	−460.039	−1.02459	−0.512293	−	0.858811i	$-0.671204\pi$
$449$	−460.039	−1.02459	−0.512293	+	0.858811i	$0.671204\pi$
$450$	0	0
$451$	− 87.0294i	− 0.192970i
$452$	0	0
$453$	−137.359	−0.303220
$454$	0	0
$455$	215.888i	0.474479i
$456$	0	0
$457$	−266.323	−0.582764	−0.291382	−	0.956607i	$-0.594115\pi$
$457$	−266.323	−0.582764	−0.291382	+	0.956607i	$0.594115\pi$
$458$	0	0
$459$	− 127.667i	− 0.278142i
$460$	0	0
$461$	−763.123	−1.65537	−0.827683	−	0.561196i	$-0.810341\pi$
$461$	−763.123	−1.65537	−0.827683	+	0.561196i	$0.810341\pi$
$462$	0	0
$463$	123.988i	0.267792i	0.990995	+	0.133896i	$0.0427488\pi$
$463$	123.988i	0.267792i	−0.990995	+	0.133896i	$0.957251\pi$
$464$	0	0
$465$	88.8040	0.190976
$466$	0	0
$467$	768.718i	1.64608i	0.567986	+	0.823038i	$0.307723\pi$
$467$	768.718i	1.64608i	−0.567986	+	0.823038i	$0.692277\pi$
$468$	0	0
$469$	−208.029	−0.443558
$470$	0	0
$471$	− 5.91457i	− 0.0125575i
$472$	0	0
$473$	285.019	0.602578
$474$	0	0
$475$	− 1633.99i	− 3.43997i
$476$	0	0
$477$	69.3859	0.145463
$478$	0	0
$479$	− 118.981i	− 0.248394i	−0.992258	−	0.124197i	$-0.960364\pi$
$479$	− 118.981i	− 0.248394i	0.992258	−	0.124197i	$-0.0396355\pi$
$480$	0	0
$481$	146.794	0.305185
$482$	0	0
$483$	− 38.1909i	− 0.0790702i
$484$	0	0
$485$	−1482.41	−3.05652
$486$	0	0
$487$	282.577i	0.580240i	0.956990	+	0.290120i	$0.0936952\pi$
$487$	282.577i	0.580240i	−0.956990	+	0.290120i	$0.906305\pi$
$488$	0	0
$489$	−61.2346	−0.125224
$490$	0	0
$491$	− 388.049i	− 0.790323i	−0.918612	−	0.395162i	$-0.870689\pi$
$491$	− 388.049i	− 0.790323i	0.918612	−	0.395162i	$-0.129311\pi$
$492$	0	0
$493$	−277.103	−0.562075
$494$	0	0
$495$	976.333i	1.97239i
$496$	0	0
$497$	46.3919	0.0933439
$498$	0	0
$499$	27.7157i	0.0555425i	0.999614	+	0.0277713i	$0.00884101\pi$
$499$	27.7157i	0.0555425i	−0.999614	+	0.0277713i	$0.991159\pi$
$500$	0	0
$501$	173.841	0.346988
$502$	0	0
$503$	− 727.477i	− 1.44628i	−0.690703	−	0.723138i	$-0.742699\pi$
$503$	− 727.477i	− 1.44628i	0.690703	−	0.723138i	$-0.257301\pi$
$504$	0	0
$505$	109.598	0.217026
$506$	0	0
$507$	51.1989i	0.100984i
$508$	0	0
$509$	−634.183	−1.24594	−0.622969	−	0.782246i	$-0.714074\pi$
$509$	−634.183	−1.24594	−0.622969	+	0.782246i	$0.714074\pi$
$510$	0	0
$511$	− 123.520i	− 0.241723i
$512$	0	0
$513$	−298.607	−0.582080
$514$	0	0
$515$	− 962.774i	− 1.86946i
$516$	0	0
$517$	−77.4010	−0.149712
$518$	0	0
$519$	− 23.4491i	− 0.0451813i
$520$	0	0
$521$	−833.127	−1.59909	−0.799546	−	0.600605i	$-0.794927\pi$
$521$	−833.127	−1.59909	−0.799546	+	0.600605i	$0.794927\pi$
$522$	0	0
$523$	876.434i	1.67578i	0.545838	+	0.837891i	$0.316211\pi$
$523$	876.434i	1.67578i	−0.545838	+	0.837891i	$0.683789\pi$
$524$	0	0
$525$	87.7181	0.167082
$526$	0	0
$527$	207.147i	0.393069i
$528$	0	0
$529$	−78.2162	−0.147857
$530$	0	0
$531$	263.546i	0.496321i
$532$	0	0
$533$	−62.9662	−0.118135
$534$	0	0
$535$	− 574.491i	− 1.07381i
$536$	0	0
$537$	−172.686	−0.321576
$538$	0	0
$539$	− 87.3970i	− 0.162147i
$540$	0	0
$541$	405.915	0.750305	0.375152	−	0.926963i	$-0.377590\pi$
$541$	405.915	0.750305	0.375152	+	0.926963i	$0.377590\pi$
$542$	0	0
$543$	23.7072i	0.0436596i
$544$	0	0
$545$	1181.97	2.16875
$546$	0	0
$547$	606.024i	1.10791i	0.832548	+	0.553953i	$0.186881\pi$
$547$	606.024i	1.10791i	−0.832548	+	0.553953i	$0.813119\pi$
$548$	0	0
$549$	131.866	0.240193
$550$	0	0
$551$	648.131i	1.17628i
$552$	0	0
$553$	−214.392	−0.387689
$554$	0	0
$555$	− 85.9899i	− 0.154937i
$556$	0	0
$557$	−36.4442	−0.0654294	−0.0327147	−	0.999465i	$-0.510415\pi$
$557$	−36.4442	−0.0654294	−0.0327147	+	0.999465i	$0.510415\pi$
$558$	0	0
$559$	− 206.213i	− 0.368896i
$560$	0	0
$561$	90.2742	0.160917
$562$	0	0
$563$	186.389i	0.331064i	0.986204	+	0.165532i	$0.0529341\pi$
$563$	186.389i	0.331064i	−0.986204	+	0.165532i	$0.947066\pi$
$564$	0	0
$565$	−1248.61	−2.20993
$566$	0	0
$567$	190.105i	0.335282i
$568$	0	0
$569$	670.891	1.17907	0.589536	−	0.807742i	$-0.299311\pi$
$569$	670.891	1.17907	0.589536	+	0.807742i	$0.299311\pi$
$570$	0	0
$571$	− 677.082i	− 1.18578i	−0.805282	−	0.592892i	$-0.797986\pi$
$571$	− 677.082i	− 1.18578i	0.805282	−	0.592892i	$-0.202014\pi$
$572$	0	0
$573$	−91.9271	−0.160431
$574$	0	0
$575$	− 1394.67i	− 2.42552i
$576$	0	0
$577$	927.901	1.60815	0.804073	−	0.594530i	$-0.202662\pi$
$577$	927.901	1.60815	0.804073	+	0.594530i	$0.202662\pi$
$578$	0	0
$579$	153.068i	0.264366i
$580$	0	0
$581$	106.848	0.183904
$582$	0	0
$583$	100.071i	0.171649i
$584$	0	0
$585$	706.382	1.20749
$586$	0	0
$587$	− 321.120i	− 0.547053i	−0.961865	−	0.273526i	$-0.911810\pi$
$587$	− 321.120i	− 0.547053i	0.961865	−	0.273526i	$-0.0881900\pi$
$588$	0	0
$589$	484.508	0.822595
$590$	0	0
$591$	− 85.1046i	− 0.144001i
$592$	0	0
$593$	−219.255	−0.369738	−0.184869	−	0.982763i	$-0.559186\pi$
$593$	−219.255	−0.369738	−0.184869	+	0.982763i	$0.559186\pi$
$594$	0	0
$595$	294.995i	0.495790i
$596$	0	0
$597$	228.754	0.383173
$598$	0	0
$599$	154.802i	0.258434i	0.991616	+	0.129217i	$0.0412464\pi$
$599$	154.802i	0.258434i	−0.991616	+	0.129217i	$0.958754\pi$
$600$	0	0
$601$	−205.862	−0.342533	−0.171266	−	0.985225i	$-0.554786\pi$
$601$	−205.862	−0.342533	−0.171266	+	0.985225i	$0.554786\pi$
$602$	0	0
$603$	680.666i	1.12880i
$604$	0	0
$605$	−315.097	−0.520821
$606$	0	0
$607$	− 790.663i	− 1.30258i	−0.758831	−	0.651288i	$-0.774229\pi$
$607$	− 790.663i	− 1.30258i	0.758831	−	0.651288i	$-0.225771\pi$
$608$	0	0
$609$	−34.7939	−0.0571329
$610$	0	0
$611$	56.0000i	0.0916530i
$612$	0	0
$613$	741.471	1.20958	0.604789	−	0.796386i	$-0.293257\pi$
$613$	741.471	1.20958	0.604789	+	0.796386i	$0.293257\pi$
$614$	0	0
$615$	36.8848i	0.0599752i
$616$	0	0
$617$	−171.578	−0.278084	−0.139042	−	0.990286i	$-0.544402\pi$
$617$	−171.578	−0.278084	−0.139042	+	0.990286i	$0.544402\pi$
$618$	0	0
$619$	− 540.198i	− 0.872695i	−0.899778	−	0.436347i	$-0.856272\pi$
$619$	− 540.198i	− 0.872695i	0.899778	−	0.436347i	$-0.143728\pi$
$620$	0	0
$621$	−254.873	−0.410424
$622$	0	0
$623$	− 296.168i	− 0.475391i
$624$	0	0
$625$	1163.38	1.86141
$626$	0	0
$627$	− 211.147i	− 0.336758i
$628$	0	0
$629$	200.583	0.318892
$630$	0	0
$631$	269.399i	0.426940i	0.976950	+	0.213470i	$0.0684766\pi$
$631$	269.399i	0.426940i	−0.976950	+	0.213470i	$0.931523\pi$
$632$	0	0
$633$	96.0975	0.151813
$634$	0	0
$635$	1033.77i	1.62798i
$636$	0	0
$637$	−63.2321	−0.0992655
$638$	0	0
$639$	− 151.794i	− 0.237549i
$640$	0	0
$641$	36.1867	0.0564535	0.0282268	−	0.999602i	$-0.491014\pi$
$641$	36.1867	0.0564535	0.0282268	+	0.999602i	$0.491014\pi$
$642$	0	0
$643$	− 266.297i	− 0.414148i	−0.978325	−	0.207074i	$-0.933606\pi$
$643$	− 266.297i	− 0.414148i	0.978325	−	0.207074i	$-0.0663941\pi$
$644$	0	0
$645$	−120.797	−0.187282
$646$	0	0
$647$	− 1086.24i	− 1.67888i	−0.543452	−	0.839440i	$-0.682883\pi$
$647$	− 1086.24i	− 1.67888i	0.543452	−	0.839440i	$-0.317117\pi$
$648$	0	0
$649$	−380.098	−0.585666
$650$	0	0
$651$	26.0101i	0.0399541i
$652$	0	0
$653$	−1195.35	−1.83055	−0.915274	−	0.402832i	$-0.868026\pi$
$653$	−1195.35	−1.83055	−0.915274	+	0.402832i	$0.868026\pi$
$654$	0	0
$655$	1520.74i	2.32173i
$656$	0	0
$657$	−404.156	−0.615154
$658$	0	0
$659$	685.220i	1.03979i	0.854231	+	0.519894i	$0.174029\pi$
$659$	685.220i	1.03979i	−0.854231	+	0.519894i	$0.825971\pi$
$660$	0	0
$661$	−993.382	−1.50285	−0.751423	−	0.659820i	$-0.770632\pi$
$661$	−993.382	−1.50285	−0.751423	+	0.659820i	$0.770632\pi$
$662$	0	0
$663$	− 65.3138i	− 0.0985125i
$664$	0	0
$665$	689.980	1.03756
$666$	0	0
$667$	553.206i	0.829394i
$668$	0	0
$669$	−6.19938	−0.00926664
$670$	0	0
$671$	190.183i	0.283432i
$672$	0	0
$673$	106.569	0.158349	0.0791743	−	0.996861i	$-0.474772\pi$
$673$	106.569	0.158349	0.0791743	+	0.996861i	$0.474772\pi$
$674$	0	0
$675$	− 585.401i	− 0.867261i
$676$	0	0
$677$	1004.18	1.48329	0.741643	−	0.670795i	$-0.234047\pi$
$677$	1004.18	1.48329	0.741643	+	0.670795i	$0.234047\pi$
$678$	0	0
$679$	− 434.188i	− 0.639452i
$680$	0	0
$681$	125.245	0.183913
$682$	0	0
$683$	− 678.225i	− 0.993009i	−0.868034	−	0.496505i	$-0.834617\pi$
$683$	− 678.225i	− 0.993009i	0.868034	−	0.496505i	$-0.165383\pi$
$684$	0	0
$685$	−313.327	−0.457411
$686$	0	0
$687$	− 135.919i	− 0.197844i
$688$	0	0
$689$	72.4020	0.105083
$690$	0	0
$691$	365.175i	0.528473i	0.964458	+	0.264236i	$0.0851199\pi$
$691$	365.175i	0.528473i	−0.964458	+	0.264236i	$0.914880\pi$
$692$	0	0
$693$	−285.961	−0.412643
$694$	0	0
$695$	972.546i	1.39935i
$696$	0	0
$697$	−86.0387	−0.123441
$698$	0	0
$699$	− 112.976i	− 0.161626i
$700$	0	0
$701$	−940.292	−1.34136	−0.670679	−	0.741748i	$-0.733997\pi$
$701$	−940.292	−1.34136	−0.670679	+	0.741748i	$0.733997\pi$
$702$	0	0
$703$	− 469.155i	− 0.667361i
$704$	0	0
$705$	32.8040	0.0465306
$706$	0	0
$707$	32.1005i	0.0454038i
$708$	0	0
$709$	−1057.46	−1.49148	−0.745738	−	0.666239i	$-0.767903\pi$
$709$	−1057.46	−1.49148	−0.745738	+	0.666239i	$0.767903\pi$
$710$	0	0
$711$	701.487i	0.986620i
$712$	0	0
$713$	413.547	0.580010
$714$	0	0
$715$	1018.77i	1.42486i
$716$	0	0
$717$	192.090	0.267907
$718$	0	0
$719$	1034.82i	1.43926i	0.694360	+	0.719628i	$0.255688\pi$
$719$	1034.82i	1.43926i	−0.694360	+	0.719628i	$0.744312\pi$
$720$	0	0
$721$	281.990	0.391109
$722$	0	0
$723$	− 42.1017i	− 0.0582320i
$724$	0	0
$725$	−1270.62	−1.75258
$726$	0	0
$727$	− 495.145i	− 0.681080i	−0.940230	−	0.340540i	$-0.889390\pi$
$727$	− 495.145i	− 0.681080i	0.940230	−	0.340540i	$-0.110610\pi$
$728$	0	0
$729$	567.489	0.778449
$730$	0	0
$731$	− 281.775i	− 0.385465i
$732$	0	0
$733$	567.494	0.774207	0.387103	−	0.922036i	$-0.373476\pi$
$733$	567.494	0.774207	0.387103	+	0.922036i	$0.373476\pi$
$734$	0	0
$735$	37.0405i	0.0503953i
$736$	0	0
$737$	−981.685	−1.33200
$738$	0	0
$739$	544.701i	0.737078i	0.929612	+	0.368539i	$0.120142\pi$
$739$	544.701i	0.737078i	−0.929612	+	0.368539i	$0.879858\pi$
$740$	0	0
$741$	−152.766	−0.206162
$742$	0	0
$743$	− 731.264i	− 0.984205i	−0.870537	−	0.492102i	$-0.836229\pi$
$743$	− 731.264i	− 0.984205i	0.870537	−	0.492102i	$-0.163771\pi$
$744$	0	0
$745$	2277.95	3.05765
$746$	0	0
$747$	− 349.605i	− 0.468012i
$748$	0	0
$749$	168.264	0.224652
$750$	0	0
$751$	666.262i	0.887166i	0.896233	+	0.443583i	$0.146293\pi$
$751$	666.262i	0.887166i	−0.896233	+	0.443583i	$0.853707\pi$
$752$	0	0
$753$	−150.500	−0.199867
$754$	0	0
$755$	− 2118.15i	− 2.80550i
$756$	0	0
$757$	238.623	0.315222	0.157611	−	0.987501i	$-0.449621\pi$
$757$	238.623	0.315222	0.157611	+	0.987501i	$0.449621\pi$
$758$	0	0
$759$	− 180.223i	− 0.237447i
$760$	0	0
$761$	−614.930	−0.808055	−0.404028	−	0.914747i	$-0.632390\pi$
$761$	−614.930	−0.808055	−0.404028	+	0.914747i	$0.632390\pi$
$762$	0	0
$763$	346.191i	0.453723i
$764$	0	0
$765$	965.219	1.26172
$766$	0	0
$767$	275.002i	0.358543i
$768$	0	0
$769$	178.950	0.232705	0.116353	−	0.993208i	$-0.462880\pi$
$769$	178.950	0.232705	0.116353	+	0.993208i	$0.462880\pi$
$770$	0	0
$771$	− 187.072i	− 0.242636i
$772$	0	0
$773$	631.615	0.817095	0.408548	−	0.912737i	$-0.366035\pi$
$773$	631.615	0.817095	0.408548	+	0.912737i	$0.366035\pi$
$774$	0	0
$775$	949.849i	1.22561i
$776$	0	0
$777$	25.1859	0.0324142
$778$	0	0
$779$	201.241i	0.258332i
$780$	0	0
$781$	218.923	0.280311
$782$	0	0
$783$	232.203i	0.296556i
$784$	0	0
$785$	91.2061	0.116186
$786$	0	0
$787$	− 456.655i	− 0.580247i	−0.956989	−	0.290124i	$-0.906304\pi$
$787$	− 456.655i	− 0.580247i	0.956989	−	0.290124i	$-0.0936965\pi$
$788$	0	0
$789$	−221.050	−0.280165
$790$	0	0
$791$	− 365.710i	− 0.462339i
$792$	0	0
$793$	137.598	0.173516
$794$	0	0
$795$	− 42.4121i	− 0.0533486i
$796$	0	0
$797$	218.566	0.274236	0.137118	−	0.990555i	$-0.456216\pi$
$797$	218.566	0.274236	0.137118	+	0.990555i	$0.456216\pi$
$798$	0	0
$799$	76.5199i	0.0957695i
$800$	0	0
$801$	−969.058	−1.20981
$802$	0	0
$803$	− 582.891i	− 0.725892i
$804$	0	0
$805$	588.926	0.731585
$806$	0	0
$807$	16.4976i	0.0204431i
$808$	0	0
$809$	−1347.46	−1.66559	−0.832794	−	0.553584i	$-0.813260\pi$
$809$	−1347.46	−1.66559	−0.832794	+	0.553584i	$0.813260\pi$
$810$	0	0
$811$	− 672.620i	− 0.829371i	−0.909965	−	0.414686i	$-0.863892\pi$
$811$	− 672.620i	− 0.829371i	0.909965	−	0.414686i	$-0.136108\pi$
$812$	0	0
$813$	−234.148	−0.288005
$814$	0	0
$815$	− 944.273i	− 1.15862i
$816$	0	0
$817$	−659.058	−0.806681
$818$	0	0
$819$	206.894i	0.252618i
$820$	0	0
$821$	1162.57	1.41604	0.708022	−	0.706190i	$-0.249588\pi$
$821$	1162.57	1.41604	0.708022	+	0.706190i	$0.249588\pi$
$822$	0	0
$823$	1041.65i	1.26567i	0.774286	+	0.632835i	$0.218109\pi$
$823$	1041.65i	1.26567i	−0.774286	+	0.632835i	$0.781891\pi$
$824$	0	0
$825$	413.941	0.501747
$826$	0	0
$827$	278.432i	0.336678i	0.985729	+	0.168339i	$0.0538403\pi$
$827$	278.432i	0.336678i	−0.985729	+	0.168339i	$0.946160\pi$
$828$	0	0
$829$	−1065.74	−1.28557	−0.642785	−	0.766046i	$-0.722221\pi$
$829$	−1065.74	−1.28557	−0.642785	+	0.766046i	$0.722221\pi$
$830$	0	0
$831$	60.3071i	0.0725717i
$832$	0	0
$833$	−86.4020	−0.103724
$834$	0	0
$835$	2680.72i	3.21045i
$836$	0	0
$837$	173.583	0.207387
$838$	0	0
$839$	− 305.844i	− 0.364533i	−0.983249	−	0.182267i	$-0.941657\pi$
$839$	− 305.844i	− 0.364533i	0.983249	−	0.182267i	$-0.0583434\pi$
$840$	0	0
$841$	−337.000	−0.400713
$842$	0	0
$843$	− 88.0059i	− 0.104396i
$844$	0	0
$845$	−789.516	−0.934339
$846$	0	0
$847$	− 92.2898i	− 0.108961i
$848$	0	0
$849$	−104.599	−0.123202
$850$	0	0
$851$	− 400.442i	− 0.470555i
$852$	0	0
$853$	164.018	0.192283	0.0961417	−	0.995368i	$-0.469350\pi$
$853$	164.018	0.192283	0.0961417	+	0.995368i	$0.469350\pi$
$854$	0	0
$855$	− 2257.60i	− 2.64047i
$856$	0	0
$857$	−851.068	−0.993078	−0.496539	−	0.868014i	$-0.665396\pi$
$857$	−851.068	−0.993078	−0.496539	+	0.868014i	$0.665396\pi$
$858$	0	0
$859$	− 1179.69i	− 1.37333i	−0.726973	−	0.686666i	$-0.759073\pi$
$859$	− 1179.69i	− 1.37333i	0.726973	−	0.686666i	$-0.240927\pi$
$860$	0	0
$861$	−10.8033	−0.0125474
$862$	0	0
$863$	279.048i	0.323346i	0.986844	+	0.161673i	$0.0516890\pi$
$863$	279.048i	0.323346i	−0.986844	+	0.161673i	$0.948311\pi$
$864$	0	0
$865$	361.598	0.418032
$866$	0	0
$867$	80.0458i	0.0923250i
$868$	0	0
$869$	−1011.71	−1.16423
$870$	0	0
$871$	710.254i	0.815447i
$872$	0	0
$873$	−1420.66	−1.62733
$874$	0	0
$875$	755.176i	0.863058i
$876$	0	0
$877$	−674.159	−0.768711	−0.384355	−	0.923185i	$-0.625576\pi$
$877$	−674.159	−0.768711	−0.384355	+	0.923185i	$0.625576\pi$
$878$	0	0
$879$	128.398i	0.146073i
$880$	0	0
$881$	−1001.29	−1.13654	−0.568271	−	0.822841i	$-0.692387\pi$
$881$	−1001.29	−1.13654	−0.568271	+	0.822841i	$0.692387\pi$
$882$	0	0
$883$	− 882.010i	− 0.998879i	−0.866349	−	0.499439i	$-0.833539\pi$
$883$	− 882.010i	− 0.998879i	0.866349	−	0.499439i	$-0.166461\pi$
$884$	0	0
$885$	161.093	0.182026
$886$	0	0
$887$	− 7.08053i	− 0.00798256i	−0.999992	−	0.00399128i	$-0.998730\pi$
$887$	− 7.08053i	− 0.00798256i	0.999992	−	0.00399128i	$-0.00127047\pi$
$888$	0	0
$889$	−302.784	−0.340589
$890$	0	0
$891$	897.103i	1.00685i
$892$	0	0
$893$	178.976	0.200422
$894$	0	0
$895$	− 2662.92i	− 2.97533i
$896$	0	0
$897$	−130.392	−0.145364
$898$	0	0
$899$	− 376.764i	− 0.419092i
$900$	0	0
$901$	98.9320	0.109802
$902$	0	0
$903$	− 35.3805i	− 0.0391811i
$904$	0	0
$905$	−365.578	−0.403953
$906$	0	0
$907$	450.372i	0.496551i	0.968689	+	0.248275i	$0.0798638\pi$
$907$	450.372i	0.496551i	−0.968689	+	0.248275i	$0.920136\pi$
$908$	0	0
$909$	105.032	0.115547
$910$	0	0
$911$	202.426i	0.222201i	0.993809	+	0.111101i	$0.0354376\pi$
$911$	202.426i	0.222201i	−0.993809	+	0.111101i	$0.964562\pi$
$912$	0	0
$913$	504.215	0.552262
$914$	0	0
$915$	− 80.6030i	− 0.0880907i
$916$	0	0
$917$	−445.413	−0.485728
$918$	0	0
$919$	− 1593.73i	− 1.73420i	−0.498138	−	0.867098i	$-0.665983\pi$
$919$	− 1593.73i	− 1.73420i	0.498138	−	0.867098i	$-0.334017\pi$
$920$	0	0
$921$	−185.500	−0.201412
$922$	0	0
$923$	− 158.392i	− 0.171606i
$924$	0	0
$925$	919.749	0.994323
$926$	0	0
$927$	− 922.666i	− 0.995325i
$928$	0	0
$929$	−1039.40	−1.11884	−0.559419	−	0.828885i	$-0.688976\pi$
$929$	−1039.40	−1.11884	−0.559419	+	0.828885i	$0.688976\pi$
$930$	0	0
$931$	202.090i	0.217068i
$932$	0	0
$933$	42.3320	0.0453719
$934$	0	0
$935$	1392.08i	1.48885i
$936$	0	0
$937$	−881.765	−0.941051	−0.470525	−	0.882386i	$-0.655936\pi$
$937$	−881.765	−0.941051	−0.470525	+	0.882386i	$0.655936\pi$
$938$	0	0
$939$	48.0059i	0.0511245i
$940$	0	0
$941$	953.344	1.01312	0.506559	−	0.862205i	$-0.330917\pi$
$941$	953.344	1.01312	0.506559	+	0.862205i	$0.330917\pi$
$942$	0	0
$943$	171.767i	0.182149i
$944$	0	0
$945$	247.196	0.261583
$946$	0	0
$947$	− 16.8957i	− 0.0178413i	−0.999960	−	0.00892063i	$-0.997160\pi$
$947$	− 16.8957i	− 0.0178413i	0.999960	−	0.00892063i	$-0.00283956\pi$
$948$	0	0
$949$	−421.725	−0.444389
$950$	0	0
$951$	− 63.9386i	− 0.0672330i
$952$	0	0
$953$	−1526.31	−1.60159	−0.800794	−	0.598940i	$-0.795589\pi$
$953$	−1526.31	−1.60159	−0.800794	+	0.598940i	$0.795589\pi$
$954$	0	0
$955$	− 1417.57i	− 1.48436i
$956$	0	0
$957$	−164.192	−0.171570
$958$	0	0
$959$	− 91.7713i	− 0.0956948i
$960$	0	0
$961$	679.352	0.706921
$962$	0	0
$963$	− 550.558i	− 0.571712i
$964$	0	0
$965$	−2360.40	−2.44601
$966$	0	0
$967$	1410.39i	1.45852i	0.684235	+	0.729262i	$0.260136\pi$
$967$	1410.39i	1.45852i	−0.684235	+	0.729262i	$0.739864\pi$
$968$	0	0
$969$	−208.743	−0.215421
$970$	0	0
$971$	− 596.497i	− 0.614312i	−0.951659	−	0.307156i	$-0.900623\pi$
$971$	− 596.497i	− 0.614312i	0.951659	−	0.307156i	$-0.0993773\pi$
$972$	0	0
$973$	−284.852	−0.292757
$974$	0	0
$975$	− 299.488i	− 0.307168i
$976$	0	0
$977$	146.686	0.150140	0.0750698	−	0.997178i	$-0.476082\pi$
$977$	146.686	0.150140	0.0750698	+	0.997178i	$0.476082\pi$
$978$	0	0
$979$	− 1397.62i	− 1.42760i
$980$	0	0
$981$	1132.73	1.15467
$982$	0	0
$983$	169.457i	0.172388i	0.996278	+	0.0861939i	$0.0274704\pi$
$983$	169.457i	0.172388i	−0.996278	+	0.0861939i	$0.972530\pi$
$984$	0	0
$985$	1312.36	1.33235
$986$	0	0
$987$	9.60808i	0.00973463i
$988$	0	0
$989$	−562.533	−0.568789
$990$	0	0
$991$	1686.90i	1.70222i	0.524988	+	0.851109i	$0.324070\pi$
$991$	1686.90i	1.70222i	−0.524988	+	0.851109i	$0.675930\pi$
$992$	0	0
$993$	188.471	0.189800
$994$	0	0
$995$	3527.52i	3.54524i
$996$	0	0
$997$	1736.14	1.74136	0.870680	−	0.491849i	$-0.163679\pi$
$997$	1736.14	1.74136	0.870680	+	0.491849i	$0.163679\pi$
$998$	0	0
$999$	− 168.082i	− 0.168250i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.3.d.g.1023.4		8
4.3	odd	2	inner	1792.3.d.g.1023.6		8
8.3	odd	2	inner	1792.3.d.g.1023.3		8
8.5	even	2	inner	1792.3.d.g.1023.5		8
16.3	odd	4		224.3.g.a.15.3		4
16.5	even	4		224.3.g.a.15.4		4
16.11	odd	4		56.3.g.a.43.2	yes	4
16.13	even	4		56.3.g.a.43.1	✓	4
48.5	odd	4		2016.3.g.a.1135.1		4
48.11	even	4		504.3.g.a.379.3		4
48.29	odd	4		504.3.g.a.379.4		4
48.35	even	4		2016.3.g.a.1135.4		4
112.11	odd	12		392.3.k.i.275.2		8
112.13	odd	4		392.3.g.h.99.1		4
112.27	even	4		392.3.g.h.99.2		4
112.45	odd	12		392.3.k.j.275.4		8
112.59	even	12		392.3.k.j.275.2		8
112.61	odd	12		392.3.k.j.67.2		8
112.69	odd	4		1568.3.g.h.687.1		4
112.75	even	12		392.3.k.j.67.4		8
112.83	even	4		1568.3.g.h.687.2		4
112.93	even	12		392.3.k.i.67.2		8
112.107	odd	12		392.3.k.i.67.4		8
112.109	even	12		392.3.k.i.275.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.g.a.43.1	✓	4	16.13	even	4
56.3.g.a.43.2	yes	4	16.11	odd	4
224.3.g.a.15.3		4	16.3	odd	4
224.3.g.a.15.4		4	16.5	even	4
392.3.g.h.99.1		4	112.13	odd	4
392.3.g.h.99.2		4	112.27	even	4
392.3.k.i.67.2		8	112.93	even	12
392.3.k.i.67.4		8	112.107	odd	12
392.3.k.i.275.2		8	112.11	odd	12
392.3.k.i.275.4		8	112.109	even	12
392.3.k.j.67.2		8	112.61	odd	12
392.3.k.j.67.4		8	112.75	even	12
392.3.k.j.275.2		8	112.59	even	12
392.3.k.j.275.4		8	112.45	odd	12
504.3.g.a.379.3		4	48.11	even	4
504.3.g.a.379.4		4	48.29	odd	4
1568.3.g.h.687.1		4	112.69	odd	4
1568.3.g.h.687.2		4	112.83	even	4
1792.3.d.g.1023.3		8	8.3	odd	2	inner
1792.3.d.g.1023.4		8	1.1	even	1	trivial
1792.3.d.g.1023.5		8	8.5	even	2	inner
1792.3.d.g.1023.6		8	4.3	odd	2	inner
2016.3.g.a.1135.1		4	48.5	odd	4
2016.3.g.a.1135.4		4	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)

\(f(q)\)	\(=\)	\(q-0.585786i q^{3} +9.03316 q^{5} +2.64575i q^{7} +8.65685 q^{9} +O(q^{10})\)	\(q-0.585786i q^{3} +9.03316 q^{5} +2.64575i q^{7} +8.65685 q^{9} +12.4853i q^{11} +9.03316 q^{13} -5.29150i q^{15} +12.3431 q^{17} -28.8701i q^{19} +1.54985 q^{21} -24.6418i q^{23} +56.5980 q^{25} -10.3431i q^{27} -22.4499 q^{29} +16.7824i q^{31} +7.31371 q^{33} +23.8995i q^{35} +16.2506 q^{37} -5.29150i q^{39} -6.97056 q^{41} -22.8284i q^{43} +78.1987 q^{45} +6.19938i q^{47} -7.00000 q^{49} -7.23045i q^{51} +8.01514 q^{53} +112.782i q^{55} -16.9117 q^{57} +30.4437i q^{59} +15.2325 q^{61} +22.9039i q^{63} +81.5980 q^{65} +78.6274i q^{67} -14.4348 q^{69} -17.5345i q^{71} -46.6863 q^{73} -33.1543i q^{75} -33.0329 q^{77} +81.0325i q^{79} +71.8528 q^{81} -40.3848i q^{83} +111.498 q^{85} +13.1509i q^{87} -111.941 q^{89} +23.8995i q^{91} +9.83089 q^{93} -260.788i q^{95} -164.108 q^{97} +108.083i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} + 144 q^{17} + 136 q^{25} - 32 q^{33} + 80 q^{41} - 56 q^{49} + 272 q^{57} + 336 q^{65} - 464 q^{73} - 104 q^{81} - 624 q^{89} - 272 q^{97}+O(q^{100}) \) 8 * q + 24 * q^9 + 144 * q^17 + 136 * q^25 - 32 * q^33 + 80 * q^41 - 56 * q^49 + 272 * q^57 + 336 * q^65 - 464 * q^73 - 104 * q^81 - 624 * q^89 - 272 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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