LMFDB - Embedded newform 1728.4.d.i.865.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(865,1728)]

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([0, 1, 0]))

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

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

chi := DirichletCharacter("1728.865");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.d (of order $2$, degree $1$, 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:	$16$
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} + 170x^{12} + 7609x^{8} + 59868x^{4} + 104976 $ x^16 + 170x^12 + 7609x^8 + 59868*x^4 + 104976
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			865.1
Root			$-1.14496 + 1.14496i$ of defining polynomial
Character	$\chi$	$=$	1728.865
Dual form			1728.4.d.i.865.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-18.8216i q^{5} -3.15603 q^{7} +O(q^{10})$	$q-18.8216i q^{5} -3.15603 q^{7} -12.5286i q^{11} -17.1201i q^{13} -41.2579 q^{17} -94.3335i q^{19} -74.9318 q^{23} -229.253 q^{25} +140.281i q^{29} -209.947 q^{31} +59.4015i q^{35} +350.829i q^{37} +348.713 q^{41} -323.894i q^{43} +81.2769 q^{47} -333.039 q^{49} -58.1320i q^{53} -235.807 q^{55} -280.970i q^{59} +38.3030i q^{61} -322.227 q^{65} -845.246i q^{67} -206.158 q^{71} -472.124 q^{73} +39.5405i q^{77} -146.441 q^{79} +1318.38i q^{83} +776.539i q^{85} +106.832 q^{89} +54.0315i q^{91} -1775.51 q^{95} -9.50431 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 240 q^{17} - 304 q^{25} + 1008 q^{41} + 1616 q^{49} + 2736 q^{65} + 128 q^{73} + 5856 q^{89} - 2576 q^{97}+O(q^{100}) $ 16 * q - 240 * q^17 - 304 * q^25 + 1008 * q^41 + 1616 * q^49 + 2736 * q^65 + 128 * q^73 + 5856 * q^89 - 2576 * 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
$5$	− 18.8216i	− 1.68346i	−0.539902	−	0.841728i	$-0.681539\pi$
$5$	− 18.8216i	− 1.68346i	0.539902	−	0.841728i	$-0.318461\pi$
$6$	0	0
$7$	−3.15603	−0.170410	−0.0852048	−	0.996363i	$-0.527154\pi$
$7$	−3.15603	−0.170410	−0.0852048	+	0.996363i	$0.527154\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 12.5286i	− 0.343409i	−0.985148	−	0.171705i	$-0.945073\pi$
$11$	− 12.5286i	− 0.343409i	0.985148	−	0.171705i	$-0.0549275\pi$
$12$	0	0
$13$	− 17.1201i	− 0.365251i	−0.983183	−	0.182625i	$-0.941540\pi$
$13$	− 17.1201i	− 0.365251i	0.983183	−	0.182625i	$-0.0584595\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−41.2579	−0.588618	−0.294309	−	0.955710i	$-0.595089\pi$
$17$	−41.2579	−0.588618	−0.294309	+	0.955710i	$0.595089\pi$
$18$	0	0
$19$	− 94.3335i	− 1.13903i	−0.821981	−	0.569516i	$-0.807131\pi$
$19$	− 94.3335i	− 1.13903i	0.821981	−	0.569516i	$-0.192869\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−74.9318	−0.679320	−0.339660	−	0.940548i	$-0.610312\pi$
$23$	−74.9318	−0.679320	−0.339660	+	0.940548i	$0.610312\pi$
$24$	0	0
$25$	−229.253	−1.83402
$26$	0	0
$27$	0	0
$28$	0	0
$29$	140.281i	0.898259i	0.893467	+	0.449129i	$0.148266\pi$
$29$	140.281i	0.898259i	−0.893467	+	0.449129i	$0.851734\pi$
$30$	0	0
$31$	−209.947	−1.21637	−0.608186	−	0.793795i	$-0.708103\pi$
$31$	−209.947	−1.21637	−0.608186	+	0.793795i	$0.708103\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	59.4015i	0.286877i
$36$	0	0
$37$	350.829i	1.55881i	0.626521	+	0.779405i	$0.284478\pi$
$37$	350.829i	1.55881i	−0.626521	+	0.779405i	$0.715522\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	348.713	1.32829	0.664145	−	0.747604i	$-0.268796\pi$
$41$	348.713	1.32829	0.664145	+	0.747604i	$0.268796\pi$
$42$	0	0
$43$	− 323.894i	− 1.14868i	−0.818616	−	0.574341i	$-0.805258\pi$
$43$	− 323.894i	− 1.14868i	0.818616	−	0.574341i	$-0.194742\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	81.2769	0.252244	0.126122	−	0.992015i	$-0.459747\pi$
$47$	81.2769	0.252244	0.126122	+	0.992015i	$0.459747\pi$
$48$	0	0
$49$	−333.039	−0.970961
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 58.1320i	− 0.150661i	−0.997159	−	0.0753306i	$-0.975999\pi$
$53$	− 58.1320i	− 0.150661i	0.997159	−	0.0753306i	$-0.0240012\pi$
$54$	0	0
$55$	−235.807	−0.578114
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 280.970i	− 0.619985i	−0.950739	−	0.309993i	$-0.899673\pi$
$59$	− 280.970i	− 0.619985i	0.950739	−	0.309993i	$-0.100327\pi$
$60$	0	0
$61$	38.3030i	0.0803966i	0.999192	+	0.0401983i	$0.0127990\pi$
$61$	38.3030i	0.0803966i	−0.999192	+	0.0401983i	$0.987201\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−322.227	−0.614883
$66$	0	0
$67$	− 845.246i	− 1.54124i	−0.637293	−	0.770621i	$-0.719946\pi$
$67$	− 845.246i	− 1.54124i	0.637293	−	0.770621i	$-0.280054\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−206.158	−0.344598	−0.172299	−	0.985045i	$-0.555120\pi$
$71$	−206.158	−0.344598	−0.172299	+	0.985045i	$0.555120\pi$
$72$	0	0
$73$	−472.124	−0.756958	−0.378479	−	0.925610i	$-0.623553\pi$
$73$	−472.124	−0.756958	−0.378479	+	0.925610i	$0.623553\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	39.5405i	0.0585202i
$78$	0	0
$79$	−146.441	−0.208556	−0.104278	−	0.994548i	$-0.533253\pi$
$79$	−146.441	−0.208556	−0.104278	+	0.994548i	$0.533253\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1318.38i	1.74351i	0.489942	+	0.871755i	$0.337018\pi$
$83$	1318.38i	1.74351i	−0.489942	+	0.871755i	$0.662982\pi$
$84$	0	0
$85$	776.539i	0.990912i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	106.832	0.127238	0.0636189	−	0.997974i	$-0.479736\pi$
$89$	106.832	0.127238	0.0636189	+	0.997974i	$0.479736\pi$
$90$	0	0
$91$	54.0315i	0.0622422i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1775.51	−1.91751
$96$	0	0
$97$	−9.50431	−0.00994863	−0.00497431	−	0.999988i	$-0.501583\pi$
$97$	−9.50431	−0.00994863	−0.00497431	+	0.999988i	$0.501583\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1511.75i	1.48935i	0.667427	+	0.744675i	$0.267395\pi$
$101$	1511.75i	1.48935i	−0.667427	+	0.744675i	$0.732605\pi$
$102$	0	0
$103$	506.324	0.484365	0.242183	−	0.970231i	$-0.422137\pi$
$103$	506.324	0.484365	0.242183	+	0.970231i	$0.422137\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	510.562i	0.461289i	0.973038	+	0.230644i	$0.0740834\pi$
$107$	510.562i	0.461289i	−0.973038	+	0.230644i	$0.925917\pi$
$108$	0	0
$109$	− 1653.76i	− 1.45322i	−0.687050	−	0.726610i	$-0.741095\pi$
$109$	− 1653.76i	− 1.45322i	0.687050	−	0.726610i	$-0.258905\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2134.26	1.77676	0.888382	−	0.459106i	$-0.151830\pi$
$113$	2134.26	1.77676	0.888382	+	0.459106i	$0.151830\pi$
$114$	0	0
$115$	1410.34i	1.14360i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	130.211	0.100306
$120$	0	0
$121$	1174.04	0.882070
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1962.20i	1.40404i
$126$	0	0
$127$	1203.23	0.840703	0.420351	−	0.907361i	$-0.361907\pi$
$127$	1203.23	0.840703	0.420351	+	0.907361i	$0.361907\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2460.49i	− 1.64103i	−0.571628	−	0.820513i	$-0.693688\pi$
$131$	− 2460.49i	− 1.64103i	0.571628	−	0.820513i	$-0.306312\pi$
$132$	0	0
$133$	297.719i	0.194102i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1679.34	−1.04727	−0.523634	−	0.851943i	$-0.675424\pi$
$137$	−1679.34	−1.04727	−0.523634	+	0.851943i	$0.675424\pi$
$138$	0	0
$139$	2413.78i	1.47291i	0.676487	+	0.736455i	$0.263502\pi$
$139$	2413.78i	1.47291i	−0.676487	+	0.736455i	$0.736498\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−214.490	−0.125430
$144$	0	0
$145$	2640.31	1.51218
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 330.871i	− 0.181919i	−0.995855	−	0.0909597i	$-0.971007\pi$
$149$	− 330.871i	− 0.181919i	0.995855	−	0.0909597i	$-0.0289934\pi$
$150$	0	0
$151$	−1111.82	−0.599197	−0.299598	−	0.954065i	$-0.596853\pi$
$151$	−1111.82	−0.599197	−0.299598	+	0.954065i	$0.596853\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3951.53i	2.04771i
$156$	0	0
$157$	− 3209.72i	− 1.63162i	−0.578322	−	0.815809i	$-0.696292\pi$
$157$	− 3209.72i	− 1.63162i	0.578322	−	0.815809i	$-0.303708\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	236.487	0.115763
$162$	0	0
$163$	965.607i	0.464001i	0.972716	+	0.232001i	$0.0745271\pi$
$163$	965.607i	0.464001i	−0.972716	+	0.232001i	$0.925473\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3186.94	−1.47672	−0.738362	−	0.674404i	$-0.764401\pi$
$167$	−3186.94	−1.47672	−0.738362	+	0.674404i	$0.764401\pi$
$168$	0	0
$169$	1903.90	0.866592
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2654.27i	1.16648i	0.812301	+	0.583238i	$0.198214\pi$
$173$	2654.27i	1.16648i	−0.812301	+	0.583238i	$0.801786\pi$
$174$	0	0
$175$	723.528	0.312535
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3168.30i	1.32296i	0.749962	+	0.661481i	$0.230072\pi$
$179$	3168.30i	1.32296i	−0.749962	+	0.661481i	$0.769928\pi$
$180$	0	0
$181$	3713.76i	1.52509i	0.646934	+	0.762546i	$0.276051\pi$
$181$	3713.76i	1.52509i	−0.646934	+	0.762546i	$0.723949\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6603.17	2.62419
$186$	0	0
$187$	516.901i	0.202137i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3166.76	−1.19968	−0.599840	−	0.800120i	$-0.704769\pi$
$191$	−3166.76	−1.19968	−0.599840	+	0.800120i	$0.704769\pi$
$192$	0	0
$193$	−748.459	−0.279146	−0.139573	−	0.990212i	$-0.544573\pi$
$193$	−748.459	−0.279146	−0.139573	+	0.990212i	$0.544573\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3400.72i	1.22990i	0.788565	+	0.614952i	$0.210824\pi$
$197$	3400.72i	1.22990i	−0.788565	+	0.614952i	$0.789176\pi$
$198$	0	0
$199$	3695.21	1.31631	0.658157	−	0.752881i	$-0.271336\pi$
$199$	3695.21	1.31631	0.658157	+	0.752881i	$0.271336\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 442.730i	− 0.153072i
$204$	0	0
$205$	− 6563.34i	− 2.23612i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1181.86	−0.391154
$210$	0	0
$211$	469.899i	0.153314i	0.997058	+	0.0766568i	$0.0244246\pi$
$211$	469.899i	0.153314i	−0.997058	+	0.0766568i	$0.975575\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6096.20	−1.93376
$216$	0	0
$217$	662.598	0.207281
$218$	0	0
$219$	0	0
$220$	0	0
$221$	706.338i	0.214993i
$222$	0	0
$223$	446.278	0.134013	0.0670066	−	0.997753i	$-0.478655\pi$
$223$	446.278	0.134013	0.0670066	+	0.997753i	$0.478655\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1569.17i	0.458809i	0.973331	+	0.229404i	$0.0736778\pi$
$227$	1569.17i	0.458809i	−0.973331	+	0.229404i	$0.926322\pi$
$228$	0	0
$229$	5393.37i	1.55635i	0.628049	+	0.778174i	$0.283854\pi$
$229$	5393.37i	1.55635i	−0.628049	+	0.778174i	$0.716146\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3190.18	0.896978	0.448489	−	0.893788i	$-0.351962\pi$
$233$	3190.18	0.896978	0.448489	+	0.893788i	$0.351962\pi$
$234$	0	0
$235$	− 1529.76i	− 0.424641i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3188.93	−0.863074	−0.431537	−	0.902095i	$-0.642029\pi$
$239$	−3188.93	−0.863074	−0.431537	+	0.902095i	$0.642029\pi$
$240$	0	0
$241$	−3377.43	−0.902736	−0.451368	−	0.892338i	$-0.649064\pi$
$241$	−3377.43	−0.902736	−0.451368	+	0.892338i	$0.649064\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6268.34i	1.63457i
$246$	0	0
$247$	−1615.00	−0.416032
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 96.7247i	− 0.0243235i	−0.999926	−	0.0121618i	$-0.996129\pi$
$251$	− 96.7247i	− 0.0243235i	0.999926	−	0.0121618i	$-0.00387131\pi$
$252$	0	0
$253$	938.787i	0.233285i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1586.02	−0.384954	−0.192477	−	0.981301i	$-0.561652\pi$
$257$	−1586.02	−0.384954	−0.192477	+	0.981301i	$0.561652\pi$
$258$	0	0
$259$	− 1107.23i	− 0.265636i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3605.14	0.845255	0.422628	−	0.906303i	$-0.361108\pi$
$263$	3605.14	0.845255	0.422628	+	0.906303i	$0.361108\pi$
$264$	0	0
$265$	−1094.14	−0.253632
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2264.95i	0.513369i	0.966495	+	0.256685i	$0.0826302\pi$
$269$	2264.95i	0.513369i	−0.966495	+	0.256685i	$0.917370\pi$
$270$	0	0
$271$	−6075.27	−1.36179	−0.680897	−	0.732379i	$-0.738410\pi$
$271$	−6075.27	−1.36179	−0.680897	+	0.732379i	$0.738410\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2872.20i	0.629820i
$276$	0	0
$277$	3539.55i	0.767765i	0.923382	+	0.383883i	$0.125413\pi$
$277$	3539.55i	0.767765i	−0.923382	+	0.383883i	$0.874587\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5606.52	1.19024	0.595120	−	0.803637i	$-0.297105\pi$
$281$	5606.52	1.19024	0.595120	+	0.803637i	$0.297105\pi$
$282$	0	0
$283$	− 8588.35i	− 1.80397i	−0.431764	−	0.901987i	$-0.642109\pi$
$283$	− 8588.35i	− 1.80397i	0.431764	−	0.901987i	$-0.357891\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1100.55	−0.226353
$288$	0	0
$289$	−3210.79	−0.653529
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5460.19i	1.08869i	0.838860	+	0.544347i	$0.183223\pi$
$293$	5460.19i	1.08869i	−0.838860	+	0.544347i	$0.816777\pi$
$294$	0	0
$295$	−5288.30	−1.04372
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1282.84i	0.248122i
$300$	0	0
$301$	1022.22i	0.195747i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	720.923	0.135344
$306$	0	0
$307$	− 7108.38i	− 1.32149i	−0.750612	−	0.660743i	$-0.770241\pi$
$307$	− 7108.38i	− 1.32149i	0.750612	−	0.660743i	$-0.229759\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6063.78	−1.10561	−0.552806	−	0.833310i	$-0.686443\pi$
$311$	−6063.78	−1.10561	−0.552806	+	0.833310i	$0.686443\pi$
$312$	0	0
$313$	−10247.5	−1.85056	−0.925279	−	0.379286i	$-0.876170\pi$
$313$	−10247.5	−1.85056	−0.925279	+	0.379286i	$0.876170\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2923.57i	0.517994i	0.965878	+	0.258997i	$0.0833919\pi$
$317$	2923.57i	0.517994i	−0.965878	+	0.258997i	$0.916608\pi$
$318$	0	0
$319$	1757.52	0.308470
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3892.00i	0.670454i
$324$	0	0
$325$	3924.83i	0.669877i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−256.512	−0.0429848
$330$	0	0
$331$	2186.04i	0.363009i	0.983390	+	0.181504i	$0.0580966\pi$
$331$	2186.04i	0.363009i	−0.983390	+	0.181504i	$0.941903\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15908.9	−2.59461
$336$	0	0
$337$	1569.52	0.253701	0.126850	−	0.991922i	$-0.459513\pi$
$337$	1569.52	0.253701	0.126850	+	0.991922i	$0.459513\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2630.33i	0.417713i
$342$	0	0
$343$	2133.60	0.335871
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4527.86i	0.700485i	0.936659	+	0.350243i	$0.113901\pi$
$347$	4527.86i	0.700485i	−0.936659	+	0.350243i	$0.886099\pi$
$348$	0	0
$349$	− 3236.15i	− 0.496353i	−0.968715	−	0.248177i	$-0.920169\pi$
$349$	− 3236.15i	− 0.496353i	0.968715	−	0.248177i	$-0.0798313\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4801.65	0.723984	0.361992	−	0.932181i	$-0.382097\pi$
$353$	4801.65	0.723984	0.361992	+	0.932181i	$0.382097\pi$
$354$	0	0
$355$	3880.23i	0.580115i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7526.17	−1.10645	−0.553226	−	0.833031i	$-0.686603\pi$
$359$	−7526.17	−1.10645	−0.553226	+	0.833031i	$0.686603\pi$
$360$	0	0
$361$	−2039.81	−0.297392
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8886.13i	1.27431i
$366$	0	0
$367$	−11202.9	−1.59343	−0.796714	−	0.604356i	$-0.793430\pi$
$367$	−11202.9	−1.59343	−0.796714	+	0.604356i	$0.793430\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	183.466i	0.0256741i
$372$	0	0
$373$	− 2705.50i	− 0.375564i	−0.982211	−	0.187782i	$-0.939870\pi$
$373$	− 2705.50i	− 0.375564i	0.982211	−	0.187782i	$-0.0601298\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2401.62	0.328089
$378$	0	0
$379$	− 2273.84i	− 0.308178i	−0.988057	−	0.154089i	$-0.950756\pi$
$379$	− 2273.84i	− 0.308178i	0.988057	−	0.154089i	$-0.0492442\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7959.30	1.06188	0.530942	−	0.847408i	$-0.321838\pi$
$383$	7959.30	1.06188	0.530942	+	0.847408i	$0.321838\pi$
$384$	0	0
$385$	744.215	0.0985162
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 9937.77i	− 1.29528i	−0.761945	−	0.647641i	$-0.775755\pi$
$389$	− 9937.77i	− 1.29528i	0.761945	−	0.647641i	$-0.224245\pi$
$390$	0	0
$391$	3091.52	0.399860
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2756.25i	0.351094i
$396$	0	0
$397$	9273.20i	1.17231i	0.810197	+	0.586157i	$0.199360\pi$
$397$	9273.20i	1.17231i	−0.810197	+	0.586157i	$0.800640\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7192.80	−0.895739	−0.447869	−	0.894099i	$-0.647817\pi$
$401$	−7192.80	−0.895739	−0.447869	+	0.894099i	$0.647817\pi$
$402$	0	0
$403$	3594.30i	0.444281i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4395.38	0.535310
$408$	0	0
$409$	1158.41	0.140048	0.0700240	−	0.997545i	$-0.477692\pi$
$409$	1158.41	0.140048	0.0700240	+	0.997545i	$0.477692\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	886.748i	0.105651i
$414$	0	0
$415$	24814.1	2.93512
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12712.7i	1.48224i	0.671374	+	0.741119i	$0.265704\pi$
$419$	12712.7i	1.48224i	−0.671374	+	0.741119i	$0.734296\pi$
$420$	0	0
$421$	− 14247.9i	− 1.64941i	−0.565564	−	0.824704i	$-0.691342\pi$
$421$	− 14247.9i	− 1.64941i	0.565564	−	0.824704i	$-0.308658\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9458.48	1.07954
$426$	0	0
$427$	− 120.885i	− 0.0137003i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14635.2	1.63563	0.817813	−	0.575484i	$-0.195186\pi$
$431$	14635.2	1.63563	0.817813	+	0.575484i	$0.195186\pi$
$432$	0	0
$433$	8629.39	0.957742	0.478871	−	0.877885i	$-0.341046\pi$
$433$	8629.39	0.957742	0.478871	+	0.877885i	$0.341046\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7068.58i	0.773766i
$438$	0	0
$439$	−9944.65	−1.08117	−0.540583	−	0.841290i	$-0.681796\pi$
$439$	−9944.65	−1.08117	−0.540583	+	0.841290i	$0.681796\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 15717.1i	− 1.68564i	−0.538193	−	0.842822i	$-0.680893\pi$
$443$	− 15717.1i	− 1.68564i	0.538193	−	0.842822i	$-0.319107\pi$
$444$	0	0
$445$	− 2010.75i	− 0.214199i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1950.61	−0.205022	−0.102511	−	0.994732i	$-0.532688\pi$
$449$	−1950.61	−0.205022	−0.102511	+	0.994732i	$0.532688\pi$
$450$	0	0
$451$	− 4368.87i	− 0.456147i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1016.96	0.104782
$456$	0	0
$457$	7176.54	0.734583	0.367291	−	0.930106i	$-0.380285\pi$
$457$	7176.54	0.734583	0.367291	+	0.930106i	$0.380285\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1712.19i	0.172982i	0.996253	+	0.0864910i	$0.0275654\pi$
$461$	1712.19i	0.172982i	−0.996253	+	0.0864910i	$0.972435\pi$
$462$	0	0
$463$	−4660.16	−0.467767	−0.233883	−	0.972265i	$-0.575143\pi$
$463$	−4660.16	−0.467767	−0.233883	+	0.972265i	$0.575143\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 17095.0i	− 1.69392i	−0.531657	−	0.846960i	$-0.678431\pi$
$467$	− 17095.0i	− 1.69392i	0.531657	−	0.846960i	$-0.321569\pi$
$468$	0	0
$469$	2667.62i	0.262642i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4057.92	−0.394468
$474$	0	0
$475$	21626.2i	2.08901i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17881.4	−1.70568	−0.852840	−	0.522173i	$-0.825122\pi$
$479$	−17881.4	−1.70568	−0.852840	+	0.522173i	$0.825122\pi$
$480$	0	0
$481$	6006.22	0.569356
$482$	0	0
$483$	0	0
$484$	0	0
$485$	178.886i	0.0167481i
$486$	0	0
$487$	10510.2	0.977948	0.488974	−	0.872298i	$-0.337371\pi$
$487$	10510.2	0.977948	0.488974	+	0.872298i	$0.337371\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17293.2i	1.58947i	0.606955	+	0.794737i	$0.292391\pi$
$491$	17293.2i	1.58947i	−0.606955	+	0.794737i	$0.707609\pi$
$492$	0	0
$493$	− 5787.69i	− 0.528731i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	650.641	0.0587228
$498$	0	0
$499$	9946.00i	0.892273i	0.894965	+	0.446136i	$0.147200\pi$
$499$	9946.00i	0.892273i	−0.894965	+	0.446136i	$0.852800\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18362.3	−1.62770	−0.813850	−	0.581075i	$-0.802632\pi$
$503$	−18362.3	−1.62770	−0.813850	+	0.581075i	$0.802632\pi$
$504$	0	0
$505$	28453.5	2.50725
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 1113.89i	− 0.0969989i	−0.998823	−	0.0484994i	$-0.984556\pi$
$509$	− 1113.89i	− 0.0969989i	0.998823	−	0.0484994i	$-0.0154439\pi$
$510$	0	0
$511$	1490.04	0.128993
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 9529.83i	− 0.815407i
$516$	0	0
$517$	− 1018.28i	− 0.0866228i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16996.3	1.42922	0.714608	−	0.699526i	$-0.246605\pi$
$521$	16996.3	1.42922	0.714608	+	0.699526i	$0.246605\pi$
$522$	0	0
$523$	6149.03i	0.514108i	0.966397	+	0.257054i	$0.0827518\pi$
$523$	6149.03i	0.514108i	−0.966397	+	0.257054i	$0.917248\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8661.95	0.715978
$528$	0	0
$529$	−6552.23	−0.538525
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 5970.00i	− 0.485158i
$534$	0	0
$535$	9609.60	0.776559
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4172.50i	0.333437i
$540$	0	0
$541$	12755.4i	1.01368i	0.862041	+	0.506838i	$0.169186\pi$
$541$	12755.4i	1.01368i	−0.862041	+	0.506838i	$0.830814\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−31126.3	−2.44643
$546$	0	0
$547$	10290.6i	0.804380i	0.915556	+	0.402190i	$0.131751\pi$
$547$	10290.6i	0.804380i	−0.915556	+	0.402190i	$0.868249\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13233.2	1.02314
$552$	0	0
$553$	462.172	0.0355399
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 13981.0i	− 1.06354i	−0.846888	−	0.531772i	$-0.821526\pi$
$557$	− 13981.0i	− 1.06354i	0.846888	−	0.531772i	$-0.178474\pi$
$558$	0	0
$559$	−5545.09	−0.419557
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 617.101i	− 0.0461949i	−0.999733	−	0.0230974i	$-0.992647\pi$
$563$	− 617.101i	− 0.0461949i	0.999733	−	0.0230974i	$-0.00735280\pi$
$564$	0	0
$565$	− 40170.2i	− 2.99110i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12885.6	0.949370	0.474685	−	0.880156i	$-0.342562\pi$
$569$	12885.6	0.949370	0.474685	+	0.880156i	$0.342562\pi$
$570$	0	0
$571$	− 878.774i	− 0.0644055i	−0.999481	−	0.0322028i	$-0.989748\pi$
$571$	− 878.774i	− 0.0644055i	0.999481	−	0.0322028i	$-0.0102522\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	17178.3	1.24589
$576$	0	0
$577$	−4191.89	−0.302444	−0.151222	−	0.988500i	$-0.548321\pi$
$577$	−4191.89	−0.302444	−0.151222	+	0.988500i	$0.548321\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4160.86i	− 0.297111i
$582$	0	0
$583$	−728.310	−0.0517385
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1447.61i	− 0.101788i	−0.998704	−	0.0508939i	$-0.983793\pi$
$587$	− 1447.61i	− 0.101788i	0.998704	−	0.0508939i	$-0.0162070\pi$
$588$	0	0
$589$	19805.0i	1.38549i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26573.8	−1.84023	−0.920114	−	0.391650i	$-0.871905\pi$
$593$	−26573.8	−1.84023	−0.920114	+	0.391650i	$0.871905\pi$
$594$	0	0
$595$	− 2450.78i	− 0.168861i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21840.9	1.48981	0.744905	−	0.667171i	$-0.232495\pi$
$599$	21840.9	1.48981	0.744905	+	0.667171i	$0.232495\pi$
$600$	0	0
$601$	−22895.8	−1.55398	−0.776988	−	0.629515i	$-0.783254\pi$
$601$	−22895.8	−1.55398	−0.776988	+	0.629515i	$0.783254\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 22097.2i	− 1.48493i
$606$	0	0
$607$	15291.8	1.02253	0.511266	−	0.859423i	$-0.329177\pi$
$607$	15291.8	1.02253	0.511266	+	0.859423i	$0.329177\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1391.47i	− 0.0921322i
$612$	0	0
$613$	− 195.623i	− 0.0128893i	−0.999979	−	0.00644466i	$-0.997949\pi$
$613$	− 195.623i	− 0.0128893i	0.999979	−	0.00644466i	$-0.00205141\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11987.0	0.782137	0.391068	−	0.920362i	$-0.372106\pi$
$617$	11987.0	0.782137	0.391068	+	0.920362i	$0.372106\pi$
$618$	0	0
$619$	− 16401.0i	− 1.06496i	−0.846442	−	0.532481i	$-0.821260\pi$
$619$	− 16401.0i	− 1.06496i	0.846442	−	0.532481i	$-0.178740\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−337.165	−0.0216825
$624$	0	0
$625$	8275.21	0.529613
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 14474.5i	− 0.917543i
$630$	0	0
$631$	−4917.53	−0.310244	−0.155122	−	0.987895i	$-0.549577\pi$
$631$	−4917.53	−0.310244	−0.155122	+	0.987895i	$0.549577\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 22646.7i	− 1.41529i
$636$	0	0
$637$	5701.66i	0.354644i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−20177.3	−1.24330	−0.621652	−	0.783294i	$-0.713538\pi$
$641$	−20177.3	−1.24330	−0.621652	+	0.783294i	$0.713538\pi$
$642$	0	0
$643$	− 19879.8i	− 1.21926i	−0.792688	−	0.609628i	$-0.791319\pi$
$643$	− 19879.8i	− 1.21926i	0.792688	−	0.609628i	$-0.208681\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−937.184	−0.0569467	−0.0284733	−	0.999595i	$-0.509065\pi$
$647$	−937.184	−0.0569467	−0.0284733	+	0.999595i	$0.509065\pi$
$648$	0	0
$649$	−3520.14	−0.212909
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2760.79i	− 0.165449i	−0.996572	−	0.0827244i	$-0.973638\pi$
$653$	− 2760.79i	− 0.165449i	0.996572	−	0.0827244i	$-0.0263621\pi$
$654$	0	0
$655$	−46310.4	−2.76259
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 11657.9i	− 0.689115i	−0.938765	−	0.344557i	$-0.888029\pi$
$659$	− 11657.9i	− 0.689115i	0.938765	−	0.344557i	$-0.111971\pi$
$660$	0	0
$661$	− 6460.30i	− 0.380146i	−0.981770	−	0.190073i	$-0.939128\pi$
$661$	− 6460.30i	− 0.380146i	0.981770	−	0.190073i	$-0.0608725\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5603.55	0.326762
$666$	0	0
$667$	− 10511.5i	− 0.610205i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	479.881	0.0276089
$672$	0	0
$673$	−13654.2	−0.782066	−0.391033	−	0.920377i	$-0.627882\pi$
$673$	−13654.2	−0.782066	−0.391033	+	0.920377i	$0.627882\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3909.82i	0.221960i	0.993823	+	0.110980i	$0.0353989\pi$
$677$	3909.82i	0.221960i	−0.993823	+	0.110980i	$0.964601\pi$
$678$	0	0
$679$	29.9959	0.00169534
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 6975.27i	− 0.390778i	−0.980726	−	0.195389i	$-0.937403\pi$
$683$	− 6975.27i	− 0.390778i	0.980726	−	0.195389i	$-0.0625969\pi$
$684$	0	0
$685$	31607.9i	1.76303i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−995.225	−0.0550291
$690$	0	0
$691$	1797.15i	0.0989388i	0.998776	+	0.0494694i	$0.0157530\pi$
$691$	1797.15i	0.0989388i	−0.998776	+	0.0494694i	$0.984247\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	45431.3	2.47958
$696$	0	0
$697$	−14387.2	−0.781855
$698$	0	0
$699$	0	0
$700$	0	0
$701$	12103.2i	0.652111i	0.945351	+	0.326056i	$0.105720\pi$
$701$	12103.2i	0.652111i	−0.945351	+	0.326056i	$0.894280\pi$
$702$	0	0
$703$	33094.9	1.77553
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4771.11i	− 0.253799i
$708$	0	0
$709$	− 21036.5i	− 1.11431i	−0.830410	−	0.557153i	$-0.811894\pi$
$709$	− 21036.5i	− 1.11431i	0.830410	−	0.557153i	$-0.188106\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	15731.7	0.826305
$714$	0	0
$715$	4037.04i	0.211156i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24103.3	1.25021	0.625105	−	0.780541i	$-0.285056\pi$
$719$	24103.3	1.25021	0.625105	+	0.780541i	$0.285056\pi$
$720$	0	0
$721$	−1597.97	−0.0825404
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 32159.8i	− 1.64743i
$726$	0	0
$727$	−35807.7	−1.82673	−0.913366	−	0.407139i	$-0.866526\pi$
$727$	−35807.7	−1.82673	−0.913366	+	0.407139i	$0.866526\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	13363.2i	0.676135i
$732$	0	0
$733$	22396.2i	1.12854i	0.825589	+	0.564272i	$0.190843\pi$
$733$	22396.2i	1.12854i	−0.825589	+	0.564272i	$0.809157\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10589.7	−0.529277
$738$	0	0
$739$	− 33482.3i	− 1.66667i	−0.552771	−	0.833333i	$-0.686430\pi$
$739$	− 33482.3i	− 1.66667i	0.552771	−	0.833333i	$-0.313570\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7694.74	−0.379936	−0.189968	−	0.981790i	$-0.560838\pi$
$743$	−7694.74	−0.379936	−0.189968	+	0.981790i	$0.560838\pi$
$744$	0	0
$745$	−6227.52	−0.306253
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1611.35i	− 0.0786080i
$750$	0	0
$751$	−22791.3	−1.10741	−0.553707	−	0.832712i	$-0.686787\pi$
$751$	−22791.3	−1.10741	−0.553707	+	0.832712i	$0.686787\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	20926.2i	1.00872i
$756$	0	0
$757$	22563.9i	1.08335i	0.840587	+	0.541677i	$0.182210\pi$
$757$	22563.9i	1.08335i	−0.840587	+	0.541677i	$0.817790\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8054.95	−0.383695	−0.191847	−	0.981425i	$-0.561448\pi$
$761$	−8054.95	−0.383695	−0.191847	+	0.981425i	$0.561448\pi$
$762$	0	0
$763$	5219.30i	0.247643i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4810.22	−0.226450
$768$	0	0
$769$	−21302.4	−0.998940	−0.499470	−	0.866331i	$-0.666472\pi$
$769$	−21302.4	−0.998940	−0.499470	+	0.866331i	$0.666472\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 12887.0i	− 0.599628i	−0.953998	−	0.299814i	$-0.903075\pi$
$773$	− 12887.0i	− 0.599628i	0.953998	−	0.299814i	$-0.0969245\pi$
$774$	0	0
$775$	48130.8	2.23085
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 32895.3i	− 1.51296i
$780$	0	0
$781$	2582.86i	0.118338i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−60412.2	−2.74675
$786$	0	0
$787$	− 18751.2i	− 0.849310i	−0.905355	−	0.424655i	$-0.860395\pi$
$787$	− 18751.2i	− 0.849310i	0.905355	−	0.424655i	$-0.139605\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6735.79	−0.302777
$792$	0	0
$793$	655.750	0.0293649
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 43467.7i	− 1.93187i	−0.258778	−	0.965937i	$-0.583320\pi$
$797$	− 43467.7i	− 1.93187i	0.258778	−	0.965937i	$-0.416680\pi$
$798$	0	0
$799$	−3353.31	−0.148475
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5915.03i	0.259946i
$804$	0	0
$805$	− 4451.06i	− 0.194881i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−17404.3	−0.756369	−0.378184	−	0.925730i	$-0.623451\pi$
$809$	−17404.3	−0.756369	−0.378184	+	0.925730i	$0.623451\pi$
$810$	0	0
$811$	− 28843.2i	− 1.24885i	−0.781083	−	0.624427i	$-0.785333\pi$
$811$	− 28843.2i	− 1.24885i	0.781083	−	0.624427i	$-0.214667\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18174.3	0.781125
$816$	0	0
$817$	−30554.0	−1.30839
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 10223.6i	− 0.434601i	−0.976105	−	0.217301i	$-0.930275\pi$
$821$	− 10223.6i	− 0.434601i	0.976105	−	0.217301i	$-0.0697252\pi$
$822$	0	0
$823$	−35728.6	−1.51327	−0.756635	−	0.653837i	$-0.773158\pi$
$823$	−35728.6	−1.51327	−0.756635	+	0.653837i	$0.773158\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 33604.6i	− 1.41299i	−0.707717	−	0.706496i	$-0.750275\pi$
$827$	− 33604.6i	− 1.41299i	0.707717	−	0.706496i	$-0.249725\pi$
$828$	0	0
$829$	22432.8i	0.939836i	0.882710	+	0.469918i	$0.155716\pi$
$829$	22432.8i	0.939836i	−0.882710	+	0.469918i	$0.844284\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	13740.5	0.571525
$834$	0	0
$835$	59983.4i	2.48600i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	32899.6	1.35378	0.676890	−	0.736084i	$-0.263327\pi$
$839$	32899.6	1.35378	0.676890	+	0.736084i	$0.263327\pi$
$840$	0	0
$841$	4710.28	0.193132
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 35834.5i	− 1.45887i
$846$	0	0
$847$	−3705.29	−0.150313
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 26288.2i	− 1.05893i
$852$	0	0
$853$	− 3786.95i	− 0.152008i	−0.997108	−	0.0760039i	$-0.975784\pi$
$853$	− 3786.95i	− 0.152008i	0.997108	−	0.0760039i	$-0.0242161\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27304.5	−1.08833	−0.544167	−	0.838977i	$-0.683154\pi$
$857$	−27304.5	−1.08833	−0.544167	+	0.838977i	$0.683154\pi$
$858$	0	0
$859$	− 21708.0i	− 0.862244i	−0.902294	−	0.431122i	$-0.858118\pi$
$859$	− 21708.0i	− 0.862244i	0.902294	−	0.431122i	$-0.141882\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	42670.2	1.68309	0.841546	−	0.540185i	$-0.181646\pi$
$863$	42670.2	1.68309	0.841546	+	0.540185i	$0.181646\pi$
$864$	0	0
$865$	49957.6	1.96371
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1834.69i	0.0716200i
$870$	0	0
$871$	−14470.7	−0.562940
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 6192.77i	− 0.239261i
$876$	0	0
$877$	− 27440.7i	− 1.05656i	−0.849069	−	0.528282i	$-0.822836\pi$
$877$	− 27440.7i	− 1.05656i	0.849069	−	0.528282i	$-0.177164\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−16408.8	−0.627498	−0.313749	−	0.949506i	$-0.601585\pi$
$881$	−16408.8	−0.627498	−0.313749	+	0.949506i	$0.601585\pi$
$882$	0	0
$883$	− 13786.1i	− 0.525413i	−0.964876	−	0.262707i	$-0.915385\pi$
$883$	− 13786.1i	− 0.525413i	0.964876	−	0.262707i	$-0.0846151\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7999.50	−0.302815	−0.151408	−	0.988471i	$-0.548381\pi$
$887$	−7999.50	−0.302815	−0.151408	+	0.988471i	$0.548381\pi$
$888$	0	0
$889$	−3797.42	−0.143264
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 7667.14i	− 0.287314i
$894$	0	0
$895$	59632.5	2.22715
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 29451.5i	− 1.09262i
$900$	0	0
$901$	2398.40i	0.0886819i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	69899.0	2.56743
$906$	0	0
$907$	9526.43i	0.348754i	0.984679	+	0.174377i	$0.0557912\pi$
$907$	9526.43i	0.348754i	−0.984679	+	0.174377i	$0.944209\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6532.37	−0.237571	−0.118785	−	0.992920i	$-0.537900\pi$
$911$	−6532.37	−0.237571	−0.118785	+	0.992920i	$0.537900\pi$
$912$	0	0
$913$	16517.4	0.598737
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7765.39i	0.279646i
$918$	0	0
$919$	9808.79	0.352081	0.176040	−	0.984383i	$-0.443671\pi$
$919$	9808.79	0.352081	0.176040	+	0.984383i	$0.443671\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3529.44i	0.125865i
$924$	0	0
$925$	− 80428.5i	− 2.85889i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−17001.0	−0.600414	−0.300207	−	0.953874i	$-0.597056\pi$
$929$	−17001.0	−0.600414	−0.300207	+	0.953874i	$0.597056\pi$
$930$	0	0
$931$	31416.8i	1.10595i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9728.91	0.340288
$936$	0	0
$937$	−9299.04	−0.324212	−0.162106	−	0.986773i	$-0.551829\pi$
$937$	−9299.04	−0.324212	−0.162106	+	0.986773i	$0.551829\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 3082.33i	− 0.106781i	−0.998574	−	0.0533905i	$-0.982997\pi$
$941$	− 3082.33i	− 0.106781i	0.998574	−	0.0533905i	$-0.0170028\pi$
$942$	0	0
$943$	−26129.7	−0.902333
$944$	0	0
$945$	0	0
$946$	0	0
$947$	50937.6i	1.74789i	0.486028	+	0.873944i	$0.338445\pi$
$947$	50937.6i	1.74789i	−0.486028	+	0.873944i	$0.661555\pi$
$948$	0	0
$949$	8082.81i	0.276479i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	50040.6	1.70092	0.850459	−	0.526042i	$-0.176325\pi$
$953$	50040.6	1.70092	0.850459	+	0.526042i	$0.176325\pi$
$954$	0	0
$955$	59603.5i	2.01961i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5300.05	0.178465
$960$	0	0
$961$	14286.6	0.479561
$962$	0	0
$963$	0	0
$964$	0	0
$965$	14087.2i	0.469930i
$966$	0	0
$967$	47168.3	1.56859	0.784297	−	0.620385i	$-0.213024\pi$
$967$	47168.3	1.56859	0.784297	+	0.620385i	$0.213024\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36701.9i	1.21300i	0.795085	+	0.606499i	$0.207426\pi$
$971$	36701.9i	1.21300i	−0.795085	+	0.606499i	$0.792574\pi$
$972$	0	0
$973$	− 7617.97i	− 0.250998i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−29687.6	−0.972151	−0.486076	−	0.873917i	$-0.661572\pi$
$977$	−29687.6	−0.972151	−0.486076	+	0.873917i	$0.661572\pi$
$978$	0	0
$979$	− 1338.45i	− 0.0436946i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	21936.6	0.711769	0.355884	−	0.934530i	$-0.384180\pi$
$983$	21936.6	0.711769	0.355884	+	0.934530i	$0.384180\pi$
$984$	0	0
$985$	64006.9	2.07049
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24269.9i	0.780323i
$990$	0	0
$991$	19571.5	0.627355	0.313677	−	0.949530i	$-0.398439\pi$
$991$	19571.5	0.627355	0.313677	+	0.949530i	$0.398439\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 69549.8i	− 2.21596i
$996$	0	0
$997$	1821.90i	0.0578738i	0.999581	+	0.0289369i	$0.00921218\pi$
$997$	1821.90i	0.0578738i	−0.999581	+	0.0289369i	$0.990788\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.d.i.865.1	✓	16
3.2	odd	2		1728.4.d.k.865.15	yes	16
4.3	odd	2	inner	1728.4.d.i.865.2	yes	16
8.3	odd	2	inner	1728.4.d.i.865.16	yes	16
8.5	even	2	inner	1728.4.d.i.865.15	yes	16
12.11	even	2		1728.4.d.k.865.16	yes	16
24.5	odd	2		1728.4.d.k.865.1	yes	16
24.11	even	2		1728.4.d.k.865.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.d.i.865.1	✓	16	1.1	even	1	trivial
1728.4.d.i.865.2	yes	16	4.3	odd	2	inner
1728.4.d.i.865.15	yes	16	8.5	even	2	inner
1728.4.d.i.865.16	yes	16	8.3	odd	2	inner
1728.4.d.k.865.1	yes	16	24.5	odd	2
1728.4.d.k.865.2	yes	16	24.11	even	2
1728.4.d.k.865.15	yes	16	3.2	odd	2
1728.4.d.k.865.16	yes	16	12.11	even	2

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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-18.8216i q^{5} -3.15603 q^{7} +O(q^{10})\)	\(q-18.8216i q^{5} -3.15603 q^{7} -12.5286i q^{11} -17.1201i q^{13} -41.2579 q^{17} -94.3335i q^{19} -74.9318 q^{23} -229.253 q^{25} +140.281i q^{29} -209.947 q^{31} +59.4015i q^{35} +350.829i q^{37} +348.713 q^{41} -323.894i q^{43} +81.2769 q^{47} -333.039 q^{49} -58.1320i q^{53} -235.807 q^{55} -280.970i q^{59} +38.3030i q^{61} -322.227 q^{65} -845.246i q^{67} -206.158 q^{71} -472.124 q^{73} +39.5405i q^{77} -146.441 q^{79} +1318.38i q^{83} +776.539i q^{85} +106.832 q^{89} +54.0315i q^{91} -1775.51 q^{95} -9.50431 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 240 q^{17} - 304 q^{25} + 1008 q^{41} + 1616 q^{49} + 2736 q^{65} + 128 q^{73} + 5856 q^{89} - 2576 q^{97}+O(q^{100}) \) 16 * q - 240 * q^17 - 304 * q^25 + 1008 * q^41 + 1616 * q^49 + 2736 * q^65 + 128 * q^73 + 5856 * q^89 - 2576 * q^97

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