LMFDB - Embedded newform 1728.4.c.i.1727.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(1727,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1728.1727");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.c (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:	$101.955300490$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 12x^{10} + 112x^{8} - 368x^{6} + 928x^{4} - 256x^{2} + 64 $ x^12 - 12x^10 + 112x^8 - 368x^6 + 928x^4 - 256*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{36}\cdot 3^{12} $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1727.1
Root			$2.48442 - 1.43438i$ of defining polynomial
Character	$\chi$	$=$	1728.1727
Dual form			1728.4.c.i.1727.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-14.9230i q^{5} -30.0528i q^{7} +O(q^{10})$	$q-14.9230i q^{5} -30.0528i q^{7} -55.9380 q^{11} -57.4627 q^{13} -29.2840i q^{17} -0.709738i q^{19} +48.0368 q^{23} -97.6960 q^{25} -172.964i q^{29} -45.2268i q^{31} -448.477 q^{35} -248.625 q^{37} -51.3323i q^{41} +19.9660i q^{43} -10.8215 q^{47} -560.168 q^{49} -37.0817i q^{53} +834.763i q^{55} -411.262 q^{59} +308.855 q^{61} +857.516i q^{65} +113.616i q^{67} -1134.56 q^{71} +728.560 q^{73} +1681.09i q^{77} -487.025i q^{79} +1165.71 q^{83} -437.005 q^{85} -1198.68i q^{89} +1726.91i q^{91} -10.5914 q^{95} +624.472 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 36 q^{13} - 132 q^{25} - 516 q^{37} - 720 q^{49} + 972 q^{61} + 660 q^{73} - 1056 q^{85} + 2532 q^{97}+O(q^{100}) $ 12 * q - 36 * q^13 - 132 * q^25 - 516 * q^37 - 720 * q^49 + 972 * q^61 + 660 * q^73 - 1056 * q^85 + 2532 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	− 14.9230i	− 1.33475i	−0.744720	−	0.667377i	$-0.767417\pi$
$5$	− 14.9230i	− 1.33475i	0.744720	−	0.667377i	$-0.232583\pi$
$6$	0	0
$7$	− 30.0528i	− 1.62270i	−0.584563	−	0.811348i	$-0.698734\pi$
$7$	− 30.0528i	− 1.62270i	0.584563	−	0.811348i	$-0.301266\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−55.9380	−1.53327	−0.766634	−	0.642085i	$-0.778070\pi$
$11$	−55.9380	−1.53327	−0.766634	+	0.642085i	$0.778070\pi$
$12$	0	0
$13$	−57.4627	−1.22594	−0.612972	−	0.790105i	$-0.710026\pi$
$13$	−57.4627	−1.22594	−0.612972	+	0.790105i	$0.710026\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 29.2840i	− 0.417789i	−0.977938	−	0.208895i	$-0.933013\pi$
$17$	− 29.2840i	− 0.417789i	0.977938	−	0.208895i	$-0.0669865\pi$
$18$	0	0
$19$	− 0.709738i	− 0.00856974i	−0.999991	−	0.00428487i	$-0.998636\pi$
$19$	− 0.709738i	− 0.00856974i	0.999991	−	0.00428487i	$-0.00136392\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	48.0368	0.435494	0.217747	−	0.976005i	$-0.430129\pi$
$23$	48.0368	0.435494	0.217747	+	0.976005i	$0.430129\pi$
$24$	0	0
$25$	−97.6960	−0.781568
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 172.964i	− 1.10754i	−0.832670	−	0.553770i	$-0.813189\pi$
$29$	− 172.964i	− 1.10754i	0.832670	−	0.553770i	$-0.186811\pi$
$30$	0	0
$31$	− 45.2268i	− 0.262031i	−0.991380	−	0.131016i	$-0.958176\pi$
$31$	− 45.2268i	− 0.262031i	0.991380	−	0.131016i	$-0.0418238\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−448.477	−2.16590
$36$	0	0
$37$	−248.625	−1.10470	−0.552348	−	0.833613i	$-0.686268\pi$
$37$	−248.625	−1.10470	−0.552348	+	0.833613i	$0.686268\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 51.3323i	− 0.195531i	−0.995209	−	0.0977653i	$-0.968831\pi$
$41$	− 51.3323i	− 0.195531i	0.995209	−	0.0977653i	$-0.0311695\pi$
$42$	0	0
$43$	19.9660i	0.0708090i	0.999373	+	0.0354045i	$0.0112720\pi$
$43$	19.9660i	0.0708090i	−0.999373	+	0.0354045i	$0.988728\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.8215	−0.0335845	−0.0167923	−	0.999859i	$-0.505345\pi$
$47$	−10.8215	−0.0335845	−0.0167923	+	0.999859i	$0.505345\pi$
$48$	0	0
$49$	−560.168	−1.63314
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 37.0817i	− 0.0961051i	−0.998845	−	0.0480525i	$-0.984699\pi$
$53$	− 37.0817i	− 0.0961051i	0.998845	−	0.0480525i	$-0.0153015\pi$
$54$	0	0
$55$	834.763i	2.04653i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−411.262	−0.907487	−0.453744	−	0.891132i	$-0.649912\pi$
$59$	−411.262	−0.907487	−0.453744	+	0.891132i	$0.649912\pi$
$60$	0	0
$61$	308.855	0.648275	0.324138	−	0.946010i	$-0.394926\pi$
$61$	308.855	0.648275	0.324138	+	0.946010i	$0.394926\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	857.516i	1.63633i
$66$	0	0
$67$	113.616i	0.207170i	0.994621	+	0.103585i	$0.0330314\pi$
$67$	113.616i	0.207170i	−0.994621	+	0.103585i	$0.966969\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1134.56	−1.89645	−0.948224	−	0.317603i	$-0.897122\pi$
$71$	−1134.56	−1.89645	−0.948224	+	0.317603i	$0.897122\pi$
$72$	0	0
$73$	728.560	1.16810	0.584051	−	0.811717i	$-0.301467\pi$
$73$	728.560	1.16810	0.584051	+	0.811717i	$0.301467\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1681.09i	2.48803i
$78$	0	0
$79$	− 487.025i	− 0.693602i	−0.937939	−	0.346801i	$-0.887268\pi$
$79$	− 487.025i	− 0.693602i	0.937939	−	0.346801i	$-0.112732\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1165.71	1.54161	0.770803	−	0.637074i	$-0.219855\pi$
$83$	1165.71	1.54161	0.770803	+	0.637074i	$0.219855\pi$
$84$	0	0
$85$	−437.005	−0.557646
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1198.68i	− 1.42764i	−0.700332	−	0.713818i	$-0.746965\pi$
$89$	− 1198.68i	− 1.42764i	0.700332	−	0.713818i	$-0.253035\pi$
$90$	0	0
$91$	1726.91i	1.98934i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−10.5914	−0.0114385
$96$	0	0
$97$	624.472	0.653665	0.326833	−	0.945082i	$-0.394019\pi$
$97$	624.472	0.653665	0.326833	+	0.945082i	$0.394019\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1757.10i	− 1.73107i	−0.500849	−	0.865535i	$-0.666979\pi$
$101$	− 1757.10i	− 1.73107i	0.500849	−	0.865535i	$-0.333021\pi$
$102$	0	0
$103$	− 3.90175i	− 0.00373254i	−0.999998	−	0.00186627i	$-0.999406\pi$
$103$	− 3.90175i	− 0.00373254i	0.999998	−	0.00186627i	$-0.000594052\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	77.3509	0.0698859	0.0349429	−	0.999389i	$-0.488875\pi$
$107$	77.3509	0.0698859	0.0349429	+	0.999389i	$0.488875\pi$
$108$	0	0
$109$	1660.75	1.45937	0.729683	−	0.683785i	$-0.239667\pi$
$109$	1660.75	1.45937	0.729683	+	0.683785i	$0.239667\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	253.390i	0.210946i	0.994422	+	0.105473i	$0.0336357\pi$
$113$	253.390i	0.210946i	−0.994422	+	0.105473i	$0.966364\pi$
$114$	0	0
$115$	− 716.853i	− 0.581277i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−880.065	−0.677945
$120$	0	0
$121$	1798.06	1.35091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 407.457i	− 0.291553i
$126$	0	0
$127$	1429.82i	0.999022i	0.866307	+	0.499511i	$0.166487\pi$
$127$	1429.82i	0.999022i	−0.866307	+	0.499511i	$0.833513\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2037.82	1.35912	0.679560	−	0.733620i	$-0.262171\pi$
$131$	2037.82	1.35912	0.679560	+	0.733620i	$0.262171\pi$
$132$	0	0
$133$	−21.3296	−0.0139061
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 804.869i	− 0.501931i	−0.967996	−	0.250966i	$-0.919252\pi$
$137$	− 804.869i	− 0.501931i	0.967996	−	0.250966i	$-0.0807481\pi$
$138$	0	0
$139$	− 413.813i	− 0.252512i	−0.991998	−	0.126256i	$-0.959704\pi$
$139$	− 413.813i	− 0.252512i	0.991998	−	0.126256i	$-0.0402961\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3214.35	1.87970
$144$	0	0
$145$	−2581.15	−1.47829
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1459.08i	0.802233i	0.916027	+	0.401116i	$0.131378\pi$
$149$	1459.08i	0.802233i	−0.916027	+	0.401116i	$0.868622\pi$
$150$	0	0
$151$	1668.38i	0.899144i	0.893244	+	0.449572i	$0.148424\pi$
$151$	1668.38i	0.899144i	−0.893244	+	0.449572i	$0.851576\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−674.920	−0.349747
$156$	0	0
$157$	1773.81	0.901691	0.450846	−	0.892602i	$-0.351123\pi$
$157$	1773.81	0.901691	0.450846	+	0.892602i	$0.351123\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1443.64i	− 0.706674i
$162$	0	0
$163$	3549.13i	1.70546i	0.522356	+	0.852728i	$0.325053\pi$
$163$	3549.13i	1.70546i	−0.522356	+	0.852728i	$0.674947\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1888.67	0.875146	0.437573	−	0.899183i	$-0.355838\pi$
$167$	1888.67	0.875146	0.437573	+	0.899183i	$0.355838\pi$
$168$	0	0
$169$	1104.96	0.502939
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2104.10i	− 0.924694i	−0.886699	−	0.462347i	$-0.847008\pi$
$173$	− 2104.10i	− 0.924694i	0.886699	−	0.462347i	$-0.152992\pi$
$174$	0	0
$175$	2936.03i	1.26825i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1830.10	0.764178	0.382089	−	0.924126i	$-0.375205\pi$
$179$	1830.10	0.764178	0.382089	+	0.924126i	$0.375205\pi$
$180$	0	0
$181$	−3333.54	−1.36895	−0.684475	−	0.729036i	$-0.739968\pi$
$181$	−3333.54	−1.36895	−0.684475	+	0.729036i	$0.739968\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3710.24i	1.47450i
$186$	0	0
$187$	1638.09i	0.640582i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3622.23	1.37223	0.686114	−	0.727494i	$-0.259315\pi$
$191$	3622.23	1.37223	0.686114	+	0.727494i	$0.259315\pi$
$192$	0	0
$193$	−2588.68	−0.965479	−0.482740	−	0.875764i	$-0.660358\pi$
$193$	−2588.68	−0.965479	−0.482740	+	0.875764i	$0.660358\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1752.24i	− 0.633717i	−0.948473	−	0.316858i	$-0.897372\pi$
$197$	− 1752.24i	− 0.633717i	0.948473	−	0.316858i	$-0.102628\pi$
$198$	0	0
$199$	3316.58i	1.18144i	0.806877	+	0.590719i	$0.201156\pi$
$199$	3316.58i	1.18144i	−0.806877	+	0.590719i	$0.798844\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5198.05	−1.79720
$204$	0	0
$205$	−766.032	−0.260985
$206$	0	0
$207$	0	0
$208$	0	0
$209$	39.7013i	0.0131397i
$210$	0	0
$211$	5960.10i	1.94460i	0.233738	+	0.972300i	$0.424904\pi$
$211$	5960.10i	1.94460i	−0.233738	+	0.972300i	$0.575096\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	297.953	0.0945125
$216$	0	0
$217$	−1359.19	−0.425197
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1682.74i	0.512186i
$222$	0	0
$223$	1995.23i	0.599150i	0.954073	+	0.299575i	$0.0968449\pi$
$223$	1995.23i	0.599150i	−0.954073	+	0.299575i	$0.903155\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2280.12	−0.666682	−0.333341	−	0.942806i	$-0.608176\pi$
$227$	−2280.12	−0.666682	−0.333341	+	0.942806i	$0.608176\pi$
$228$	0	0
$229$	−4647.94	−1.34124	−0.670621	−	0.741800i	$-0.733972\pi$
$229$	−4647.94	−1.34124	−0.670621	+	0.741800i	$0.733972\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2824.81i	0.794247i	0.917765	+	0.397124i	$0.129992\pi$
$233$	2824.81i	0.794247i	−0.917765	+	0.397124i	$0.870008\pi$
$234$	0	0
$235$	161.489i	0.0448271i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2405.85	−0.651136	−0.325568	−	0.945519i	$-0.605555\pi$
$239$	−2405.85	−0.651136	−0.325568	+	0.945519i	$0.605555\pi$
$240$	0	0
$241$	−226.108	−0.0604352	−0.0302176	−	0.999543i	$-0.509620\pi$
$241$	−226.108	−0.0604352	−0.0302176	+	0.999543i	$0.509620\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	8359.39i	2.17984i
$246$	0	0
$247$	40.7834i	0.0105060i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1071.12	0.269357	0.134679	−	0.990889i	$-0.457000\pi$
$251$	1071.12	0.269357	0.134679	+	0.990889i	$0.457000\pi$
$252$	0	0
$253$	−2687.08	−0.667729
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1844.28i	0.447638i	0.974631	+	0.223819i	$0.0718525\pi$
$257$	1844.28i	0.447638i	−0.974631	+	0.223819i	$0.928148\pi$
$258$	0	0
$259$	7471.88i	1.79259i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7502.45	−1.75901	−0.879507	−	0.475886i	$-0.842128\pi$
$263$	−7502.45	−1.75901	−0.879507	+	0.475886i	$0.842128\pi$
$264$	0	0
$265$	−553.371	−0.128277
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 3465.21i	− 0.785418i	−0.919663	−	0.392709i	$-0.871538\pi$
$269$	− 3465.21i	− 0.785418i	0.919663	−	0.392709i	$-0.128462\pi$
$270$	0	0
$271$	− 2922.27i	− 0.655038i	−0.944845	−	0.327519i	$-0.893788\pi$
$271$	− 2922.27i	− 0.655038i	0.944845	−	0.327519i	$-0.106212\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5464.92	1.19835
$276$	0	0
$277$	6644.62	1.44129	0.720643	−	0.693306i	$-0.243847\pi$
$277$	6644.62	1.44129	0.720643	+	0.693306i	$0.243847\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 6612.13i	− 1.40373i	−0.712312	−	0.701863i	$-0.752352\pi$
$281$	− 6612.13i	− 1.40373i	0.712312	−	0.701863i	$-0.247648\pi$
$282$	0	0
$283$	− 2658.33i	− 0.558379i	−0.960236	−	0.279190i	$-0.909934\pi$
$283$	− 2658.33i	− 0.558379i	0.960236	−	0.279190i	$-0.0900658\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1542.68	−0.317287
$288$	0	0
$289$	4055.45	0.825452
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 5553.60i	− 1.10732i	−0.832743	−	0.553660i	$-0.813231\pi$
$293$	− 5553.60i	− 1.10732i	0.832743	−	0.553660i	$-0.186769\pi$
$294$	0	0
$295$	6137.26i	1.21127i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2760.32	−0.533891
$300$	0	0
$301$	600.033	0.114901
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 4609.04i	− 0.865288i
$306$	0	0
$307$	− 8694.95i	− 1.61644i	−0.588880	−	0.808220i	$-0.700431\pi$
$307$	− 8694.95i	− 1.61644i	0.588880	−	0.808220i	$-0.299569\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6717.09	−1.22473	−0.612365	−	0.790575i	$-0.709782\pi$
$311$	−6717.09	−1.22473	−0.612365	+	0.790575i	$0.709782\pi$
$312$	0	0
$313$	−1389.06	−0.250844	−0.125422	−	0.992103i	$-0.540028\pi$
$313$	−1389.06	−0.250844	−0.125422	+	0.992103i	$0.540028\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1457.63i	0.258260i	0.991628	+	0.129130i	$0.0412185\pi$
$317$	1457.63i	0.258260i	−0.991628	+	0.129130i	$0.958782\pi$
$318$	0	0
$319$	9675.28i	1.69816i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−20.7840	−0.00358034
$324$	0	0
$325$	5613.87	0.958159
$326$	0	0
$327$	0	0
$328$	0	0
$329$	325.215i	0.0544975i
$330$	0	0
$331$	4966.82i	0.824776i	0.911008	+	0.412388i	$0.135305\pi$
$331$	4966.82i	0.824776i	−0.911008	+	0.412388i	$0.864695\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1695.49	0.276521
$336$	0	0
$337$	−4892.90	−0.790899	−0.395450	−	0.918488i	$-0.629411\pi$
$337$	−4892.90	−0.790899	−0.395450	+	0.918488i	$0.629411\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2529.90i	0.401764i
$342$	0	0
$343$	6526.50i	1.02740i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−253.981	−0.0392923	−0.0196461	−	0.999807i	$-0.506254\pi$
$347$	−253.981	−0.0392923	−0.0196461	+	0.999807i	$0.506254\pi$
$348$	0	0
$349$	6464.21	0.991465	0.495732	−	0.868475i	$-0.334900\pi$
$349$	6464.21	0.991465	0.495732	+	0.868475i	$0.334900\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 4033.67i	− 0.608188i	−0.952642	−	0.304094i	$-0.901646\pi$
$353$	− 4033.67i	− 0.608188i	0.952642	−	0.304094i	$-0.0983537\pi$
$354$	0	0
$355$	16931.1i	2.53129i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10670.4	−1.56870	−0.784350	−	0.620319i	$-0.787003\pi$
$359$	−10670.4	−1.56870	−0.784350	+	0.620319i	$0.787003\pi$
$360$	0	0
$361$	6858.50	0.999927
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 10872.3i	− 1.55913i
$366$	0	0
$367$	2559.84i	0.364094i	0.983290	+	0.182047i	$0.0582724\pi$
$367$	2559.84i	0.364094i	−0.983290	+	0.182047i	$0.941728\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1114.41	−0.155949
$372$	0	0
$373$	−1935.92	−0.268735	−0.134368	−	0.990932i	$-0.542900\pi$
$373$	−1935.92	−0.268735	−0.134368	+	0.990932i	$0.542900\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9938.99i	1.35778i
$378$	0	0
$379$	5944.90i	0.805722i	0.915261	+	0.402861i	$0.131984\pi$
$379$	5944.90i	0.805722i	−0.915261	+	0.402861i	$0.868016\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1696.80	−0.226378	−0.113189	−	0.993573i	$-0.536107\pi$
$383$	−1696.80	−0.226378	−0.113189	+	0.993573i	$0.536107\pi$
$384$	0	0
$385$	25086.9	3.32090
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3913.71i	0.510111i	0.966926	+	0.255055i	$0.0820937\pi$
$389$	3913.71i	0.510111i	−0.966926	+	0.255055i	$0.917906\pi$
$390$	0	0
$391$	− 1406.71i	− 0.181945i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7267.87	−0.925788
$396$	0	0
$397$	−5079.98	−0.642208	−0.321104	−	0.947044i	$-0.604054\pi$
$397$	−5079.98	−0.642208	−0.321104	+	0.947044i	$0.604054\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 9374.44i	− 1.16742i	−0.811961	−	0.583712i	$-0.801600\pi$
$401$	− 9374.44i	− 1.16742i	0.811961	−	0.583712i	$-0.198400\pi$
$402$	0	0
$403$	2598.85i	0.321236i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	13907.6	1.69379
$408$	0	0
$409$	−12392.7	−1.49824	−0.749120	−	0.662434i	$-0.769523\pi$
$409$	−12392.7	−1.49824	−0.749120	+	0.662434i	$0.769523\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	12359.6i	1.47258i
$414$	0	0
$415$	− 17395.9i	− 2.05766i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	266.866	0.0311151	0.0155576	−	0.999879i	$-0.495048\pi$
$419$	266.866	0.0311151	0.0155576	+	0.999879i	$0.495048\pi$
$420$	0	0
$421$	9952.62	1.15216	0.576082	−	0.817392i	$-0.304581\pi$
$421$	9952.62	1.15216	0.576082	+	0.817392i	$0.304581\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2860.93i	0.326531i
$426$	0	0
$427$	− 9281.94i	− 1.05195i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6583.33	0.735749	0.367875	−	0.929875i	$-0.380086\pi$
$431$	6583.33	0.735749	0.367875	+	0.929875i	$0.380086\pi$
$432$	0	0
$433$	−13747.0	−1.52572	−0.762860	−	0.646564i	$-0.776205\pi$
$433$	−13747.0	−1.52572	−0.762860	+	0.646564i	$0.776205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 34.0935i	− 0.00373207i
$438$	0	0
$439$	13380.0i	1.45466i	0.686290	+	0.727328i	$0.259238\pi$
$439$	13380.0i	1.45466i	−0.686290	+	0.727328i	$0.740762\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16108.1	1.72758	0.863791	−	0.503851i	$-0.168084\pi$
$443$	16108.1	1.72758	0.863791	+	0.503851i	$0.168084\pi$
$444$	0	0
$445$	−17887.9	−1.90554
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11283.5i	1.18597i	0.805214	+	0.592984i	$0.202050\pi$
$449$	11283.5i	1.18597i	−0.805214	+	0.592984i	$0.797950\pi$
$450$	0	0
$451$	2871.42i	0.299801i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	25770.7	2.65527
$456$	0	0
$457$	1984.72	0.203154	0.101577	−	0.994828i	$-0.467611\pi$
$457$	1984.72	0.203154	0.101577	+	0.994828i	$0.467611\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 9634.41i	− 0.973360i	−0.873580	−	0.486680i	$-0.838208\pi$
$461$	− 9634.41i	− 0.973360i	0.873580	−	0.486680i	$-0.161792\pi$
$462$	0	0
$463$	1392.38i	0.139762i	0.997555	+	0.0698808i	$0.0222619\pi$
$463$	1392.38i	0.139762i	−0.997555	+	0.0698808i	$0.977738\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15161.4	−1.50232	−0.751161	−	0.660119i	$-0.770506\pi$
$467$	−15161.4	−1.50232	−0.751161	+	0.660119i	$0.770506\pi$
$468$	0	0
$469$	3414.47	0.336174
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1116.86i	− 0.108569i
$474$	0	0
$475$	69.3385i	0.00669783i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−273.224	−0.0260625	−0.0130312	−	0.999915i	$-0.504148\pi$
$479$	−273.224	−0.0260625	−0.0130312	+	0.999915i	$0.504148\pi$
$480$	0	0
$481$	14286.7	1.35430
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 9319.00i	− 0.872482i
$486$	0	0
$487$	8706.60i	0.810131i	0.914288	+	0.405065i	$0.132751\pi$
$487$	8706.60i	0.810131i	−0.914288	+	0.405065i	$0.867249\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4442.47	−0.408321	−0.204161	−	0.978937i	$-0.565446\pi$
$491$	−4442.47	−0.408321	−0.204161	+	0.978937i	$0.565446\pi$
$492$	0	0
$493$	−5065.09	−0.462718
$494$	0	0
$495$	0	0
$496$	0	0
$497$	34096.7i	3.07736i
$498$	0	0
$499$	− 4920.29i	− 0.441408i	−0.975341	−	0.220704i	$-0.929165\pi$
$499$	− 4920.29i	− 0.441408i	0.975341	−	0.220704i	$-0.0708355\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5575.54	0.494237	0.247118	−	0.968985i	$-0.420516\pi$
$503$	5575.54	0.494237	0.247118	+	0.968985i	$0.420516\pi$
$504$	0	0
$505$	−26221.2	−2.31055
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17508.5i	1.52466i	0.647190	+	0.762328i	$0.275944\pi$
$509$	17508.5i	1.52466i	−0.647190	+	0.762328i	$0.724056\pi$
$510$	0	0
$511$	− 21895.2i	− 1.89548i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−58.2259	−0.00498202
$516$	0	0
$517$	605.331	0.0514941
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 12662.1i	− 1.06475i	−0.846507	−	0.532377i	$-0.821299\pi$
$521$	− 12662.1i	− 1.06475i	0.846507	−	0.532377i	$-0.178701\pi$
$522$	0	0
$523$	− 2988.40i	− 0.249854i	−0.992166	−	0.124927i	$-0.960130\pi$
$523$	− 2988.40i	− 0.249854i	0.992166	−	0.124927i	$-0.0398697\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1324.42	−0.109474
$528$	0	0
$529$	−9859.47	−0.810345
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2949.69i	0.239710i
$534$	0	0
$535$	− 1154.31i	− 0.0932805i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	31334.7	2.50404
$540$	0	0
$541$	−9079.39	−0.721541	−0.360770	−	0.932655i	$-0.617486\pi$
$541$	−9079.39	−0.721541	−0.360770	+	0.932655i	$0.617486\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 24783.4i	− 1.94790i
$546$	0	0
$547$	− 21690.2i	− 1.69544i	−0.530444	−	0.847720i	$-0.677975\pi$
$547$	− 21690.2i	− 1.69544i	0.530444	−	0.847720i	$-0.322025\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−122.759	−0.00949133
$552$	0	0
$553$	−14636.4	−1.12551
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18129.0i	1.37908i	0.724247	+	0.689541i	$0.242188\pi$
$557$	18129.0i	1.37908i	−0.724247	+	0.689541i	$0.757812\pi$
$558$	0	0
$559$	− 1147.30i	− 0.0868078i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4922.54	0.368491	0.184245	−	0.982880i	$-0.441016\pi$
$563$	4922.54	0.368491	0.184245	+	0.982880i	$0.441016\pi$
$564$	0	0
$565$	3781.34	0.281562
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 17731.7i	− 1.30641i	−0.757179	−	0.653207i	$-0.773423\pi$
$569$	− 17731.7i	− 1.30641i	0.757179	−	0.653207i	$-0.226577\pi$
$570$	0	0
$571$	− 17237.0i	− 1.26330i	−0.775254	−	0.631650i	$-0.782378\pi$
$571$	− 17237.0i	− 1.26330i	0.775254	−	0.631650i	$-0.217622\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4693.00	−0.340368
$576$	0	0
$577$	6166.73	0.444929	0.222465	−	0.974941i	$-0.428590\pi$
$577$	6166.73	0.444929	0.222465	+	0.974941i	$0.428590\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 35032.8i	− 2.50156i
$582$	0	0
$583$	2074.28i	0.147355i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11268.2	−0.792311	−0.396155	−	0.918183i	$-0.629656\pi$
$587$	−11268.2	−0.792311	−0.396155	+	0.918183i	$0.629656\pi$
$588$	0	0
$589$	−32.0992	−0.00224554
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 4766.20i	− 0.330058i	−0.986289	−	0.165029i	$-0.947228\pi$
$593$	− 4766.20i	− 0.330058i	0.986289	−	0.165029i	$-0.0527718\pi$
$594$	0	0
$595$	13133.2i	0.904889i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−11994.7	−0.818177	−0.409089	−	0.912495i	$-0.634153\pi$
$599$	−11994.7	−0.818177	−0.409089	+	0.912495i	$0.634153\pi$
$600$	0	0
$601$	23975.3	1.62724	0.813622	−	0.581394i	$-0.197493\pi$
$601$	23975.3	1.62724	0.813622	+	0.581394i	$0.197493\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 26832.4i	− 1.80313i
$606$	0	0
$607$	− 18629.0i	− 1.24568i	−0.782349	−	0.622840i	$-0.785979\pi$
$607$	− 18629.0i	− 1.24568i	0.782349	−	0.622840i	$-0.214021\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	621.830	0.0411728
$612$	0	0
$613$	−18552.5	−1.22240	−0.611198	−	0.791478i	$-0.709312\pi$
$613$	−18552.5	−1.22240	−0.611198	+	0.791478i	$0.709312\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7836.95i	− 0.511351i	−0.966763	−	0.255676i	$-0.917702\pi$
$617$	− 7836.95i	− 0.511351i	0.966763	−	0.255676i	$-0.0822979\pi$
$618$	0	0
$619$	− 16023.6i	− 1.04045i	−0.854028	−	0.520227i	$-0.825847\pi$
$619$	− 16023.6i	− 1.04045i	0.854028	−	0.520227i	$-0.174153\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−36023.6	−2.31662
$624$	0	0
$625$	−18292.5	−1.17072
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7280.75i	0.461530i
$630$	0	0
$631$	18451.1i	1.16407i	0.813164	+	0.582035i	$0.197743\pi$
$631$	18451.1i	1.16407i	−0.813164	+	0.582035i	$0.802257\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	21337.2	1.33345
$636$	0	0
$637$	32188.7	2.00214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20544.4i	1.26592i	0.774185	+	0.632960i	$0.218160\pi$
$641$	20544.4i	1.26592i	−0.774185	+	0.632960i	$0.781840\pi$
$642$	0	0
$643$	27413.8i	1.68133i	0.541555	+	0.840665i	$0.317836\pi$
$643$	27413.8i	1.68133i	−0.541555	+	0.840665i	$0.682164\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23051.0	1.40066	0.700332	−	0.713817i	$-0.253035\pi$
$647$	23051.0	1.40066	0.700332	+	0.713817i	$0.253035\pi$
$648$	0	0
$649$	23005.2	1.39142
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27762.7i	1.66376i	0.554953	+	0.831882i	$0.312736\pi$
$653$	27762.7i	1.66376i	−0.554953	+	0.831882i	$0.687264\pi$
$654$	0	0
$655$	− 30410.3i	− 1.81409i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9857.22	0.582675	0.291338	−	0.956620i	$-0.405900\pi$
$659$	9857.22	0.582675	0.291338	+	0.956620i	$0.405900\pi$
$660$	0	0
$661$	−3542.53	−0.208454	−0.104227	−	0.994554i	$-0.533237\pi$
$661$	−3542.53	−0.208454	−0.104227	+	0.994554i	$0.533237\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	318.301i	0.0185612i
$666$	0	0
$667$	− 8308.65i	− 0.482327i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−17276.7	−0.993979
$672$	0	0
$673$	19216.6	1.10066	0.550330	−	0.834947i	$-0.314502\pi$
$673$	19216.6	1.10066	0.550330	+	0.834947i	$0.314502\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15731.9i	0.893093i	0.894760	+	0.446547i	$0.147346\pi$
$677$	15731.9i	0.893093i	−0.894760	+	0.446547i	$0.852654\pi$
$678$	0	0
$679$	− 18767.1i	− 1.06070i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16870.5	−0.945142	−0.472571	−	0.881293i	$-0.656674\pi$
$683$	−16870.5	−0.945142	−0.472571	+	0.881293i	$0.656674\pi$
$684$	0	0
$685$	−12011.1	−0.669955
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2130.82i	0.117819i
$690$	0	0
$691$	29234.5i	1.60945i	0.593645	+	0.804727i	$0.297688\pi$
$691$	29234.5i	1.60945i	−0.593645	+	0.804727i	$0.702312\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6175.33	−0.337041
$696$	0	0
$697$	−1503.21	−0.0816905
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 14904.4i	− 0.803041i	−0.915850	−	0.401520i	$-0.868482\pi$
$701$	− 14904.4i	− 0.803041i	0.915850	−	0.401520i	$-0.131518\pi$
$702$	0	0
$703$	176.459i	0.00946696i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−52805.7	−2.80900
$708$	0	0
$709$	17930.1	0.949758	0.474879	−	0.880051i	$-0.342492\pi$
$709$	17930.1	0.949758	0.474879	+	0.880051i	$0.342492\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2172.55i	− 0.114113i
$714$	0	0
$715$	− 47967.7i	− 2.50894i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−19033.4	−0.987238	−0.493619	−	0.869678i	$-0.664326\pi$
$719$	−19033.4	−0.987238	−0.493619	+	0.869678i	$0.664326\pi$
$720$	0	0
$721$	−117.258	−0.00605677
$722$	0	0
$723$	0	0
$724$	0	0
$725$	16897.9i	0.865618i
$726$	0	0
$727$	− 19643.0i	− 1.00209i	−0.865422	−	0.501043i	$-0.832950\pi$
$727$	− 19643.0i	− 1.00209i	0.865422	−	0.501043i	$-0.167050\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	584.684	0.0295832
$732$	0	0
$733$	−8335.12	−0.420006	−0.210003	−	0.977701i	$-0.567347\pi$
$733$	−8335.12	−0.420006	−0.210003	+	0.977701i	$0.567347\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 6355.44i	− 0.317647i
$738$	0	0
$739$	− 12308.0i	− 0.612661i	−0.951925	−	0.306331i	$-0.900899\pi$
$739$	− 12308.0i	− 0.612661i	0.951925	−	0.306331i	$-0.0991013\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26227.7	−1.29502	−0.647511	−	0.762056i	$-0.724190\pi$
$743$	−26227.7	−1.29502	−0.647511	+	0.762056i	$0.724190\pi$
$744$	0	0
$745$	21773.9	1.07078
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 2324.61i	− 0.113404i
$750$	0	0
$751$	− 2741.63i	− 0.133214i	−0.997779	−	0.0666068i	$-0.978783\pi$
$751$	− 2741.63i	− 0.133214i	0.997779	−	0.0666068i	$-0.0212173\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	24897.2	1.20014
$756$	0	0
$757$	17816.8	0.855434	0.427717	−	0.903913i	$-0.359318\pi$
$757$	17816.8	0.855434	0.427717	+	0.903913i	$0.359318\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32228.2i	1.53518i	0.640940	+	0.767591i	$0.278545\pi$
$761$	32228.2i	1.53518i	−0.640940	+	0.767591i	$0.721455\pi$
$762$	0	0
$763$	− 49910.1i	− 2.36811i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	23632.2	1.11253
$768$	0	0
$769$	−4236.60	−0.198668	−0.0993340	−	0.995054i	$-0.531671\pi$
$769$	−4236.60	−0.198668	−0.0993340	+	0.995054i	$0.531671\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1754.64i	0.0816431i	0.999166	+	0.0408215i	$0.0129975\pi$
$773$	1754.64i	0.0816431i	−0.999166	+	0.0408215i	$0.987002\pi$
$774$	0	0
$775$	4418.48i	0.204795i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−36.4324	−0.00167565
$780$	0	0
$781$	63465.1	2.90776
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 26470.6i	− 1.20354i
$786$	0	0
$787$	− 11896.9i	− 0.538857i	−0.963020	−	0.269428i	$-0.913165\pi$
$787$	− 11896.9i	− 0.538857i	0.963020	−	0.269428i	$-0.0868347\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7615.07	0.342302
$792$	0	0
$793$	−17747.6	−0.794749
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 16762.4i	− 0.744986i	−0.928035	−	0.372493i	$-0.878503\pi$
$797$	− 16762.4i	− 0.744986i	0.928035	−	0.372493i	$-0.121497\pi$
$798$	0	0
$799$	316.896i	0.0140313i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−40754.2	−1.79101
$804$	0	0
$805$	−21543.4	−0.943236
$806$	0	0
$807$	0	0
$808$	0	0
$809$	28039.3i	1.21855i	0.792957	+	0.609277i	$0.208540\pi$
$809$	28039.3i	1.21855i	−0.792957	+	0.609277i	$0.791460\pi$
$810$	0	0
$811$	− 14131.8i	− 0.611882i	−0.952051	−	0.305941i	$-0.901029\pi$
$811$	− 14131.8i	− 0.611882i	0.952051	−	0.305941i	$-0.0989710\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	52963.7	2.27636
$816$	0	0
$817$	14.1706	0.000606814 0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8233.70i	0.350010i	0.984568	+	0.175005i	$0.0559942\pi$
$821$	8233.70i	0.350010i	−0.984568	+	0.175005i	$0.944006\pi$
$822$	0	0
$823$	− 5824.72i	− 0.246704i	−0.992363	−	0.123352i	$-0.960636\pi$
$823$	− 5824.72i	− 0.246704i	0.992363	−	0.123352i	$-0.0393644\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20581.6	−0.865406	−0.432703	−	0.901536i	$-0.642440\pi$
$827$	−20581.6	−0.865406	−0.432703	+	0.901536i	$0.642440\pi$
$828$	0	0
$829$	−2844.67	−0.119179	−0.0595896	−	0.998223i	$-0.518979\pi$
$829$	−2844.67	−0.119179	−0.0595896	+	0.998223i	$0.518979\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	16404.0i	0.682309i
$834$	0	0
$835$	− 28184.6i	− 1.16810i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12881.0	0.530037	0.265019	−	0.964243i	$-0.414622\pi$
$839$	12881.0	0.530037	0.265019	+	0.964243i	$0.414622\pi$
$840$	0	0
$841$	−5527.65	−0.226645
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 16489.3i	− 0.671300i
$846$	0	0
$847$	− 54036.6i	− 2.19211i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−11943.2	−0.481089
$852$	0	0
$853$	−43159.3	−1.73241	−0.866205	−	0.499689i	$-0.833448\pi$
$853$	−43159.3	−1.73241	−0.866205	+	0.499689i	$0.833448\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 236.518i	− 0.00942741i	−0.999989	−	0.00471371i	$-0.998500\pi$
$857$	− 236.518i	− 0.00942741i	0.999989	−	0.00471371i	$-0.00150042\pi$
$858$	0	0
$859$	1103.24i	0.0438210i	0.999760	+	0.0219105i	$0.00697488\pi$
$859$	1103.24i	0.0438210i	−0.999760	+	0.0219105i	$0.993025\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22297.9	−0.879522	−0.439761	−	0.898115i	$-0.644937\pi$
$863$	−22297.9	−0.879522	−0.439761	+	0.898115i	$0.644937\pi$
$864$	0	0
$865$	−31399.5	−1.23424
$866$	0	0
$867$	0	0
$868$	0	0
$869$	27243.2i	1.06348i
$870$	0	0
$871$	− 6528.67i	− 0.253979i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12245.2	−0.473101
$876$	0	0
$877$	−29308.0	−1.12846	−0.564231	−	0.825617i	$-0.690827\pi$
$877$	−29308.0	−1.12846	−0.564231	+	0.825617i	$0.690827\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 25857.5i	− 0.988831i	−0.869226	−	0.494415i	$-0.835382\pi$
$881$	− 25857.5i	− 0.988831i	0.869226	−	0.494415i	$-0.164618\pi$
$882$	0	0
$883$	15147.3i	0.577290i	0.957436	+	0.288645i	$0.0932047\pi$
$883$	15147.3i	0.577290i	−0.957436	+	0.288645i	$0.906795\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18719.8	0.708624	0.354312	−	0.935127i	$-0.384715\pi$
$887$	18719.8	0.708624	0.354312	+	0.935127i	$0.384715\pi$
$888$	0	0
$889$	42970.0	1.62111
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.68040i	0 0.000287811i
$894$	0	0
$895$	− 27310.5i	− 1.01999i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7822.62	−0.290210
$900$	0	0
$901$	−1085.90	−0.0401516
$902$	0	0
$903$	0	0
$904$	0	0
$905$	49746.4i	1.82721i
$906$	0	0
$907$	10374.7i	0.379807i	0.981803	+	0.189904i	$0.0608175\pi$
$907$	10374.7i	0.379807i	−0.981803	+	0.189904i	$0.939182\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−10860.1	−0.394962	−0.197481	−	0.980307i	$-0.563276\pi$
$911$	−10860.1	−0.394962	−0.197481	+	0.980307i	$0.563276\pi$
$912$	0	0
$913$	−65207.4	−2.36369
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 61242.0i	− 2.20544i
$918$	0	0
$919$	− 5743.96i	− 0.206176i	−0.994672	−	0.103088i	$-0.967128\pi$
$919$	− 5743.96i	− 0.206176i	0.994672	−	0.103088i	$-0.0328723\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	65195.0	2.32494
$924$	0	0
$925$	24289.7	0.863396
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 28379.7i	− 1.00227i	−0.865369	−	0.501135i	$-0.832916\pi$
$929$	− 28379.7i	− 1.00227i	0.865369	−	0.501135i	$-0.167084\pi$
$930$	0	0
$931$	397.572i	0.0139956i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24445.2	0.855020
$936$	0	0
$937$	24896.7	0.868025	0.434013	−	0.900907i	$-0.357097\pi$
$937$	24896.7	0.868025	0.434013	+	0.900907i	$0.357097\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 601.120i	− 0.0208246i	−0.999946	−	0.0104123i	$-0.996686\pi$
$941$	− 601.120i	− 0.0208246i	0.999946	−	0.0104123i	$-0.00331440\pi$
$942$	0	0
$943$	− 2465.84i	− 0.0851524i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25774.5	0.884432	0.442216	−	0.896908i	$-0.354192\pi$
$947$	25774.5	0.884432	0.442216	+	0.896908i	$0.354192\pi$
$948$	0	0
$949$	−41865.0	−1.43203
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 40615.9i	− 1.38056i	−0.723541	−	0.690282i	$-0.757487\pi$
$953$	− 40615.9i	− 1.38056i	0.723541	−	0.690282i	$-0.242513\pi$
$954$	0	0
$955$	− 54054.6i	− 1.83159i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−24188.5	−0.814482
$960$	0	0
$961$	27745.5	0.931340
$962$	0	0
$963$	0	0
$964$	0	0
$965$	38630.9i	1.28868i
$966$	0	0
$967$	52148.0i	1.73420i	0.498138	+	0.867098i	$0.334017\pi$
$967$	52148.0i	1.73420i	−0.498138	+	0.867098i	$0.665983\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29877.8	−0.987461	−0.493731	−	0.869615i	$-0.664367\pi$
$971$	−29877.8	−0.987461	−0.493731	+	0.869615i	$0.664367\pi$
$972$	0	0
$973$	−12436.2	−0.409750
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10385.8i	0.340094i	0.985436	+	0.170047i	$0.0543920\pi$
$977$	10385.8i	0.340094i	−0.985436	+	0.170047i	$0.945608\pi$
$978$	0	0
$979$	67051.6i	2.18895i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	14316.7	0.464530	0.232265	−	0.972652i	$-0.425386\pi$
$983$	14316.7	0.464530	0.232265	+	0.972652i	$0.425386\pi$
$984$	0	0
$985$	−26148.7	−0.845856
$986$	0	0
$987$	0	0
$988$	0	0
$989$	959.102i	0.0308369i
$990$	0	0
$991$	− 30199.0i	− 0.968013i	−0.875064	−	0.484007i	$-0.839181\pi$
$991$	− 30199.0i	− 0.968013i	0.875064	−	0.484007i	$-0.160819\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	49493.3	1.57693
$996$	0	0
$997$	−17508.8	−0.556178	−0.278089	−	0.960555i	$-0.589701\pi$
$997$	−17508.8	−0.556178	−0.278089	+	0.960555i	$0.589701\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.c.i.1727.1		12
3.2	odd	2	inner	1728.4.c.i.1727.11		12
4.3	odd	2	inner	1728.4.c.i.1727.2		12
8.3	odd	2		108.4.b.a.107.7	yes	12
8.5	even	2		108.4.b.a.107.5	✓	12
12.11	even	2	inner	1728.4.c.i.1727.12		12
24.5	odd	2		108.4.b.a.107.8	yes	12
24.11	even	2		108.4.b.a.107.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.b.a.107.5	✓	12	8.5	even	2
108.4.b.a.107.6	yes	12	24.11	even	2
108.4.b.a.107.7	yes	12	8.3	odd	2
108.4.b.a.107.8	yes	12	24.5	odd	2
1728.4.c.i.1727.1		12	1.1	even	1	trivial
1728.4.c.i.1727.2		12	4.3	odd	2	inner
1728.4.c.i.1727.11		12	3.2	odd	2	inner
1728.4.c.i.1727.12		12	12.11	even	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-14.9230i q^{5} -30.0528i q^{7} +O(q^{10})\)	\(q-14.9230i q^{5} -30.0528i q^{7} -55.9380 q^{11} -57.4627 q^{13} -29.2840i q^{17} -0.709738i q^{19} +48.0368 q^{23} -97.6960 q^{25} -172.964i q^{29} -45.2268i q^{31} -448.477 q^{35} -248.625 q^{37} -51.3323i q^{41} +19.9660i q^{43} -10.8215 q^{47} -560.168 q^{49} -37.0817i q^{53} +834.763i q^{55} -411.262 q^{59} +308.855 q^{61} +857.516i q^{65} +113.616i q^{67} -1134.56 q^{71} +728.560 q^{73} +1681.09i q^{77} -487.025i q^{79} +1165.71 q^{83} -437.005 q^{85} -1198.68i q^{89} +1726.91i q^{91} -10.5914 q^{95} +624.472 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 36 q^{13} - 132 q^{25} - 516 q^{37} - 720 q^{49} + 972 q^{61} + 660 q^{73} - 1056 q^{85} + 2532 q^{97}+O(q^{100}) \) 12 * q - 36 * q^13 - 132 * q^25 - 516 * q^37 - 720 * q^49 + 972 * q^61 + 660 * q^73 - 1056 * q^85 + 2532 * q^97

Introduction

L-functions

Modular forms

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