LMFDB - Embedded newform 1728.3.h.j.161.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(161,1728)]

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, 1]))

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

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

chi := DirichletCharacter("1728.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.h (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:	$47.0845896815$
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} - 13x^{10} + 129x^{8} - 512x^{6} + 1548x^{4} - 160x^{2} + 16 $ x^12 - 13x^10 + 129x^8 - 512x^6 + 1548x^4 - 160*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.11
Root			$-0.278546 + 0.160819i$ of defining polynomial
Character	$\chi$	$=$	1728.161
Dual form			1728.3.h.j.161.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.03816 q^{5} +7.92398 q^{7} +O(q^{10})$	$q+9.03816 q^{5} +7.92398 q^{7} +1.07018 q^{11} -15.7264i q^{13} +23.3793i q^{17} +31.2389i q^{19} -4.34147i q^{23} +56.6884 q^{25} -0.634271 q^{29} +18.1947 q^{31} +71.6182 q^{35} +8.79821i q^{37} -39.2389i q^{41} +62.8989i q^{43} +71.0368i q^{47} +13.7895 q^{49} +90.1026 q^{53} +9.67244 q^{55} -28.7586 q^{59} -108.701i q^{61} -142.138i q^{65} -70.8989i q^{67} -14.2930i q^{71} -73.6884 q^{73} +8.48006 q^{77} -1.35100 q^{79} -119.065 q^{83} +211.306i q^{85} +100.815i q^{89} -124.616i q^{91} +282.343i q^{95} -83.0568 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 12 q^{11} + 72 q^{25} + 252 q^{35} + 168 q^{49} + 264 q^{59} - 276 q^{73} + 396 q^{83} - 396 q^{97}+O(q^{100}) $ 12 * q + 12 * q^11 + 72 * q^25 + 252 * q^35 + 168 * q^49 + 264 * q^59 - 276 * q^73 + 396 * q^83 - 396 * 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
$5$	9.03816	1.80763	0.903816	−	0.427920i	$-0.140754\pi$
$5$	9.03816	1.80763	0.903816	+	0.427920i	$0.140754\pi$
$6$	0	0
$7$	7.92398	1.13200	0.565999	−	0.824406i	$-0.308491\pi$
$7$	7.92398	1.13200	0.565999	+	0.824406i	$0.308491\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.07018	0.0972888	0.0486444	−	0.998816i	$-0.484510\pi$
$11$	1.07018	0.0972888	0.0486444	+	0.998816i	$0.484510\pi$
$12$	0	0
$13$	− 15.7264i	− 1.20972i	−0.796330	−	0.604862i	$-0.793228\pi$
$13$	− 15.7264i	− 1.20972i	0.796330	−	0.604862i	$-0.206772\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	23.3793i	1.37525i	0.726065	+	0.687626i	$0.241347\pi$
$17$	23.3793i	1.37525i	−0.726065	+	0.687626i	$0.758653\pi$
$18$	0	0
$19$	31.2389i	1.64415i	0.569376	+	0.822077i	$0.307185\pi$
$19$	31.2389i	1.64415i	−0.569376	+	0.822077i	$0.692815\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.34147i	− 0.188760i	−0.995536	−	0.0943798i	$-0.969913\pi$
$23$	− 4.34147i	− 0.188760i	0.995536	−	0.0943798i	$-0.0300868\pi$
$24$	0	0
$25$	56.6884	2.26754
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.634271	−0.0218714	−0.0109357	−	0.999940i	$-0.503481\pi$
$29$	−0.634271	−0.0218714	−0.0109357	+	0.999940i	$0.503481\pi$
$30$	0	0
$31$	18.1947	0.586927	0.293463	−	0.955970i	$-0.405192\pi$
$31$	18.1947	0.586927	0.293463	+	0.955970i	$0.405192\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	71.6182	2.04624
$36$	0	0
$37$	8.79821i	0.237789i	0.992907	+	0.118895i	$0.0379351\pi$
$37$	8.79821i	0.237789i	−0.992907	+	0.118895i	$0.962065\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 39.2389i	− 0.957047i	−0.878075	−	0.478524i	$-0.841172\pi$
$41$	− 39.2389i	− 0.957047i	0.878075	−	0.478524i	$-0.158828\pi$
$42$	0	0
$43$	62.8989i	1.46277i	0.681967	+	0.731383i	$0.261125\pi$
$43$	62.8989i	1.46277i	−0.681967	+	0.731383i	$0.738875\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	71.0368i	1.51142i	0.654906	+	0.755710i	$0.272708\pi$
$47$	71.0368i	1.51142i	−0.654906	+	0.755710i	$0.727292\pi$
$48$	0	0
$49$	13.7895	0.281418
$50$	0	0
$51$	0	0
$52$	0	0
$53$	90.1026	1.70005	0.850024	−	0.526743i	$-0.176587\pi$
$53$	90.1026	1.70005	0.850024	+	0.526743i	$0.176587\pi$
$54$	0	0
$55$	9.67244	0.175862
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−28.7586	−0.487434	−0.243717	−	0.969846i	$-0.578367\pi$
$59$	−28.7586	−0.487434	−0.243717	+	0.969846i	$0.578367\pi$
$60$	0	0
$61$	− 108.701i	− 1.78198i	−0.454018	−	0.890992i	$-0.650010\pi$
$61$	− 108.701i	− 1.78198i	0.454018	−	0.890992i	$-0.349990\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 142.138i	− 2.18674i
$66$	0	0
$67$	− 70.8989i	− 1.05819i	−0.848562	−	0.529097i	$-0.822531\pi$
$67$	− 70.8989i	− 1.05819i	0.848562	−	0.529097i	$-0.177469\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 14.2930i	− 0.201309i	−0.994921	−	0.100655i	$-0.967906\pi$
$71$	− 14.2930i	− 0.201309i	0.994921	−	0.100655i	$-0.0320937\pi$
$72$	0	0
$73$	−73.6884	−1.00943	−0.504715	−	0.863286i	$-0.668402\pi$
$73$	−73.6884	−1.00943	−0.504715	+	0.863286i	$0.668402\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.48006	0.110131
$78$	0	0
$79$	−1.35100	−0.0171012	−0.00855061	−	0.999963i	$-0.502722\pi$
$79$	−1.35100	−0.0171012	−0.00855061	+	0.999963i	$0.502722\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−119.065	−1.43452	−0.717261	−	0.696805i	$-0.754604\pi$
$83$	−119.065	−1.43452	−0.717261	+	0.696805i	$0.754604\pi$
$84$	0	0
$85$	211.306i	2.48595i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	100.815i	1.13276i	0.824145	+	0.566379i	$0.191656\pi$
$89$	100.815i	1.13276i	−0.824145	+	0.566379i	$0.808344\pi$
$90$	0	0
$91$	− 124.616i	− 1.36940i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	282.343i	2.97203i
$96$	0	0
$97$	−83.0568	−0.856256	−0.428128	−	0.903718i	$-0.640827\pi$
$97$	−83.0568	−0.856256	−0.428128	+	0.903718i	$0.640827\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	43.9984	0.435628	0.217814	−	0.975990i	$-0.430107\pi$
$101$	43.9984	0.435628	0.217814	+	0.975990i	$0.430107\pi$
$102$	0	0
$103$	−75.9629	−0.737504	−0.368752	−	0.929528i	$-0.620215\pi$
$103$	−75.9629	−0.737504	−0.368752	+	0.929528i	$0.620215\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	204.096	1.90744	0.953720	−	0.300695i	$-0.0972188\pi$
$107$	204.096	1.90744	0.953720	+	0.300695i	$0.0972188\pi$
$108$	0	0
$109$	− 77.5940i	− 0.711872i	−0.934510	−	0.355936i	$-0.884162\pi$
$109$	− 77.5940i	− 0.711872i	0.934510	−	0.355936i	$-0.115838\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 151.520i	− 1.34088i	−0.741963	−	0.670441i	$-0.766105\pi$
$113$	− 151.520i	− 1.34088i	0.741963	−	0.670441i	$-0.233895\pi$
$114$	0	0
$115$	− 39.2389i	− 0.341208i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	185.257i	1.55678i
$120$	0	0
$121$	−119.855	−0.990535
$122$	0	0
$123$	0	0
$124$	0	0
$125$	286.405	2.29124
$126$	0	0
$127$	−225.793	−1.77790	−0.888951	−	0.458003i	$-0.848565\pi$
$127$	−225.793	−1.77790	−0.888951	+	0.458003i	$0.848565\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	114.039	0.870529	0.435265	−	0.900303i	$-0.356655\pi$
$131$	114.039	0.870529	0.435265	+	0.900303i	$0.356655\pi$
$132$	0	0
$133$	247.537i	1.86118i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 132.535i	− 0.967407i	−0.875232	−	0.483703i	$-0.839291\pi$
$137$	− 132.535i	− 0.967407i	0.875232	−	0.483703i	$-0.160709\pi$
$138$	0	0
$139$	− 104.000i	− 0.748201i	−0.927388	−	0.374101i	$-0.877951\pi$
$139$	− 104.000i	− 0.748201i	0.927388	−	0.374101i	$-0.122049\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 16.8300i	− 0.117693i
$144$	0	0
$145$	−5.73264	−0.0395355
$146$	0	0
$147$	0	0
$148$	0	0
$149$	122.450	0.821810	0.410905	−	0.911678i	$-0.365213\pi$
$149$	122.450	0.821810	0.410905	+	0.911678i	$0.365213\pi$
$150$	0	0
$151$	11.8183	0.0782672	0.0391336	−	0.999234i	$-0.487540\pi$
$151$	11.8183	0.0782672	0.0391336	+	0.999234i	$0.487540\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	164.447	1.06095
$156$	0	0
$157$	− 82.5781i	− 0.525975i	−0.964799	−	0.262988i	$-0.915292\pi$
$157$	− 82.5781i	− 0.525975i	0.964799	−	0.262988i	$-0.0847078\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 34.4017i	− 0.213675i
$162$	0	0
$163$	49.1821i	0.301731i	0.988554	+	0.150865i	$0.0482060\pi$
$163$	49.1821i	0.301731i	−0.988554	+	0.150865i	$0.951794\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 174.465i	− 1.04470i	−0.852731	−	0.522351i	$-0.825055\pi$
$167$	− 174.465i	− 1.04470i	0.852731	−	0.522351i	$-0.174945\pi$
$168$	0	0
$169$	−78.3200	−0.463432
$170$	0	0
$171$	0	0
$172$	0	0
$173$	153.801	0.889024	0.444512	−	0.895773i	$-0.353377\pi$
$173$	153.801	0.889024	0.444512	+	0.895773i	$0.353377\pi$
$174$	0	0
$175$	449.198	2.56685
$176$	0	0
$177$	0	0
$178$	0	0
$179$	212.658	1.18803	0.594015	−	0.804454i	$-0.297542\pi$
$179$	212.658	1.18803	0.594015	+	0.804454i	$0.297542\pi$
$180$	0	0
$181$	− 161.359i	− 0.891487i	−0.895161	−	0.445743i	$-0.852939\pi$
$181$	− 161.359i	− 0.891487i	0.895161	−	0.445743i	$-0.147061\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	79.5197i	0.429836i
$186$	0	0
$187$	25.0200i	0.133797i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	40.6465i	0.212809i	0.994323	+	0.106404i	$0.0339338\pi$
$191$	40.6465i	0.212809i	−0.994323	+	0.106404i	$0.966066\pi$
$192$	0	0
$193$	301.486	1.56211	0.781053	−	0.624465i	$-0.214683\pi$
$193$	301.486	1.56211	0.781053	+	0.624465i	$0.214683\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.0193	0.121925	0.0609627	−	0.998140i	$-0.480583\pi$
$197$	24.0193	0.121925	0.0609627	+	0.998140i	$0.480583\pi$
$198$	0	0
$199$	−56.8231	−0.285543	−0.142772	−	0.989756i	$-0.545601\pi$
$199$	−56.8231	−0.285543	−0.142772	+	0.989756i	$0.545601\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.02595	−0.0247584
$204$	0	0
$205$	− 354.648i	− 1.72999i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	33.4312i	0.159958i
$210$	0	0
$211$	116.357i	0.551454i	0.961236	+	0.275727i	$0.0889186\pi$
$211$	116.357i	0.551454i	−0.961236	+	0.275727i	$0.911081\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	568.491i	2.64414i
$216$	0	0
$217$	144.175	0.664400
$218$	0	0
$219$	0	0
$220$	0	0
$221$	367.672	1.66368
$222$	0	0
$223$	−341.311	−1.53054	−0.765270	−	0.643709i	$-0.777395\pi$
$223$	−341.311	−1.53054	−0.765270	+	0.643709i	$0.777395\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	174.956	0.770730	0.385365	−	0.922764i	$-0.374075\pi$
$227$	174.956	0.770730	0.385365	+	0.922764i	$0.374075\pi$
$228$	0	0
$229$	211.520i	0.923670i	0.886966	+	0.461835i	$0.152809\pi$
$229$	211.520i	0.923670i	−0.886966	+	0.461835i	$0.847191\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 60.9776i	− 0.261706i	−0.991402	−	0.130853i	$-0.958228\pi$
$233$	− 60.9776i	− 0.261706i	0.991402	−	0.130853i	$-0.0417716\pi$
$234$	0	0
$235$	642.042i	2.73209i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	331.136i	1.38551i	0.721174	+	0.692754i	$0.243603\pi$
$239$	331.136i	1.38551i	−0.721174	+	0.692754i	$0.756397\pi$
$240$	0	0
$241$	−23.6842	−0.0982749	−0.0491374	−	0.998792i	$-0.515647\pi$
$241$	−23.6842	−0.0982749	−0.0491374	+	0.998792i	$0.515647\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	124.631	0.508700
$246$	0	0
$247$	491.276	1.98897
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−389.429	−1.55151	−0.775754	−	0.631035i	$-0.782630\pi$
$251$	−389.429	−1.55151	−0.775754	+	0.631035i	$0.782630\pi$
$252$	0	0
$253$	− 4.64614i	− 0.0183642i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 326.419i	− 1.27011i	−0.772466	−	0.635056i	$-0.780977\pi$
$257$	− 326.419i	− 1.27011i	0.772466	−	0.635056i	$-0.219023\pi$
$258$	0	0
$259$	69.7168i	0.269177i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	178.610i	0.679124i	0.940584	+	0.339562i	$0.110279\pi$
$263$	178.610i	0.679124i	−0.940584	+	0.339562i	$0.889721\pi$
$264$	0	0
$265$	814.362	3.07306
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−403.038	−1.49828	−0.749142	−	0.662409i	$-0.769534\pi$
$269$	−403.038	−1.49828	−0.749142	+	0.662409i	$0.769534\pi$
$270$	0	0
$271$	178.330	0.658044	0.329022	−	0.944322i	$-0.393281\pi$
$271$	178.330	0.658044	0.329022	+	0.944322i	$0.393281\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	60.6666	0.220606
$276$	0	0
$277$	− 140.359i	− 0.506712i	−0.967373	−	0.253356i	$-0.918466\pi$
$277$	− 140.359i	− 0.506712i	0.967373	−	0.253356i	$-0.0815344\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	396.335i	1.41045i	0.708986	+	0.705223i	$0.249153\pi$
$281$	396.335i	1.41045i	−0.708986	+	0.705223i	$0.750847\pi$
$282$	0	0
$283$	− 178.283i	− 0.629976i	−0.949096	−	0.314988i	$-0.898000\pi$
$283$	− 178.283i	− 0.629976i	0.949096	−	0.314988i	$-0.102000\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 310.929i	− 1.08338i
$288$	0	0
$289$	−257.592	−0.891320
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−47.6578	−0.162655	−0.0813273	−	0.996687i	$-0.525916\pi$
$293$	−47.6578	−0.162655	−0.0813273	+	0.996687i	$0.525916\pi$
$294$	0	0
$295$	−259.925	−0.881101
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−68.2758	−0.228347
$300$	0	0
$301$	498.410i	1.65585i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 982.458i	− 3.22117i
$306$	0	0
$307$	74.5421i	0.242808i	0.992603	+	0.121404i	$0.0387397\pi$
$307$	74.5421i	0.242808i	−0.992603	+	0.121404i	$0.961260\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 322.453i	− 1.03683i	−0.855130	−	0.518414i	$-0.826523\pi$
$311$	− 322.453i	− 1.03683i	0.855130	−	0.518414i	$-0.173477\pi$
$312$	0	0
$313$	−239.324	−0.764614	−0.382307	−	0.924035i	$-0.624870\pi$
$313$	−239.324	−0.764614	−0.382307	+	0.924035i	$0.624870\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	93.5590	0.295139	0.147569	−	0.989052i	$-0.452855\pi$
$317$	93.5590	0.295139	0.147569	+	0.989052i	$0.452855\pi$
$318$	0	0
$319$	−0.678782	−0.00212784
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−730.345	−2.26113
$324$	0	0
$325$	− 891.505i	− 2.74309i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	562.894i	1.71092i
$330$	0	0
$331$	− 338.042i	− 1.02128i	−0.859796	−	0.510638i	$-0.829409\pi$
$331$	− 338.042i	− 1.02128i	0.859796	−	0.510638i	$-0.170591\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 640.796i	− 1.91282i
$336$	0	0
$337$	−235.507	−0.698835	−0.349417	−	0.936967i	$-0.613620\pi$
$337$	−235.507	−0.698835	−0.349417	+	0.936967i	$0.613620\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.4716	0.0571014
$342$	0	0
$343$	−279.008	−0.813433
$344$	0	0
$345$	0	0
$346$	0	0
$347$	449.450	1.29524	0.647622	−	0.761962i	$-0.275764\pi$
$347$	449.450	1.29524	0.647622	+	0.761962i	$0.275764\pi$
$348$	0	0
$349$	208.735i	0.598095i	0.954238	+	0.299048i	$0.0966689\pi$
$349$	208.735i	0.598095i	−0.954238	+	0.299048i	$0.903331\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 404.076i	− 1.14469i	−0.820012	−	0.572346i	$-0.806034\pi$
$353$	− 404.076i	− 1.14469i	0.820012	−	0.572346i	$-0.193966\pi$
$354$	0	0
$355$	− 129.182i	− 0.363893i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	605.760i	1.68735i	0.536852	+	0.843677i	$0.319614\pi$
$359$	605.760i	1.68735i	−0.536852	+	0.843677i	$0.680386\pi$
$360$	0	0
$361$	−614.872	−1.70325
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−666.008	−1.82468
$366$	0	0
$367$	346.991	0.945481	0.472740	−	0.881202i	$-0.343265\pi$
$367$	346.991	0.945481	0.472740	+	0.881202i	$0.343265\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	713.971	1.92445
$372$	0	0
$373$	413.336i	1.10814i	0.832470	+	0.554070i	$0.186926\pi$
$373$	413.336i	1.10814i	−0.832470	+	0.554070i	$0.813074\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.97480i	0.0264584i
$378$	0	0
$379$	− 348.503i	− 0.919533i	−0.888040	−	0.459767i	$-0.847933\pi$
$379$	− 348.503i	− 0.919533i	0.888040	−	0.459767i	$-0.152067\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	236.283i	0.616927i	0.951236	+	0.308464i	$0.0998148\pi$
$383$	236.283i	0.616927i	−0.951236	+	0.308464i	$0.900185\pi$
$384$	0	0
$385$	76.6442	0.199076
$386$	0	0
$387$	0	0
$388$	0	0
$389$	138.591	0.356276	0.178138	−	0.984005i	$-0.442993\pi$
$389$	138.591	0.356276	0.178138	+	0.984005i	$0.442993\pi$
$390$	0	0
$391$	101.501	0.259592
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−12.2105	−0.0309127
$396$	0	0
$397$	512.390i	1.29065i	0.763906	+	0.645327i	$0.223279\pi$
$397$	512.390i	1.29065i	−0.763906	+	0.645327i	$0.776721\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	442.345i	1.10310i	0.834141	+	0.551552i	$0.185964\pi$
$401$	442.345i	1.10310i	−0.834141	+	0.551552i	$0.814036\pi$
$402$	0	0
$403$	− 286.138i	− 0.710020i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.41564i	0.0231342i
$408$	0	0
$409$	−260.305	−0.636443	−0.318222	−	0.948016i	$-0.603086\pi$
$409$	−260.305	−0.636443	−0.318222	+	0.948016i	$0.603086\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−227.883	−0.551774
$414$	0	0
$415$	−1076.13	−2.59309
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−460.986	−1.10020	−0.550102	−	0.835097i	$-0.685411\pi$
$419$	−460.986	−1.10020	−0.550102	+	0.835097i	$0.685411\pi$
$420$	0	0
$421$	− 215.681i	− 0.512307i	−0.966636	−	0.256153i	$-0.917545\pi$
$421$	− 215.681i	− 0.512307i	0.966636	−	0.256153i	$-0.0824552\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1325.34i	3.11844i
$426$	0	0
$427$	− 861.345i	− 2.01720i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 621.965i	− 1.44307i	−0.692376	−	0.721537i	$-0.743436\pi$
$431$	− 621.965i	− 1.44307i	0.692376	−	0.721537i	$-0.256564\pi$
$432$	0	0
$433$	−117.566	−0.271516	−0.135758	−	0.990742i	$-0.543347\pi$
$433$	−117.566	−0.271516	−0.135758	+	0.990742i	$0.543347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	135.623	0.310350
$438$	0	0
$439$	−235.424	−0.536273	−0.268136	−	0.963381i	$-0.586408\pi$
$439$	−235.424	−0.536273	−0.268136	+	0.963381i	$0.586408\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−325.320	−0.734357	−0.367178	−	0.930151i	$-0.619676\pi$
$443$	−325.320	−0.734357	−0.367178	+	0.930151i	$0.619676\pi$
$444$	0	0
$445$	911.186i	2.04761i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	430.658i	0.959149i	0.877501	+	0.479574i	$0.159209\pi$
$449$	430.658i	0.959149i	−0.877501	+	0.479574i	$0.840791\pi$
$450$	0	0
$451$	− 41.9926i	− 0.0931100i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 1126.30i	− 2.47538i
$456$	0	0
$457$	−544.402	−1.19125	−0.595626	−	0.803262i	$-0.703096\pi$
$457$	−544.402	−1.19125	−0.595626	+	0.803262i	$0.703096\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−304.995	−0.661594	−0.330797	−	0.943702i	$-0.607318\pi$
$461$	−304.995	−0.661594	−0.330797	+	0.943702i	$0.607318\pi$
$462$	0	0
$463$	591.711	1.27799	0.638997	−	0.769209i	$-0.279350\pi$
$463$	591.711	1.27799	0.638997	+	0.769209i	$0.279350\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−327.438	−0.701152	−0.350576	−	0.936534i	$-0.614014\pi$
$467$	−327.438	−0.701152	−0.350576	+	0.936534i	$0.614014\pi$
$468$	0	0
$469$	− 561.802i	− 1.19787i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	67.3130i	0.142311i
$474$	0	0
$475$	1770.89i	3.72818i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 745.601i	− 1.55658i	−0.627906	−	0.778289i	$-0.716088\pi$
$479$	− 745.601i	− 1.55658i	0.627906	−	0.778289i	$-0.283912\pi$
$480$	0	0
$481$	138.364	0.287660
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−750.681	−1.54780
$486$	0	0
$487$	−804.238	−1.65141	−0.825707	−	0.564099i	$-0.809223\pi$
$487$	−804.238	−1.65141	−0.825707	+	0.564099i	$0.809223\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	152.679	0.310956	0.155478	−	0.987839i	$-0.450308\pi$
$491$	152.679	0.310956	0.155478	+	0.987839i	$0.450308\pi$
$492$	0	0
$493$	− 14.8288i	− 0.0300787i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 113.257i	− 0.227882i
$498$	0	0
$499$	− 753.686i	− 1.51039i	−0.655498	−	0.755197i	$-0.727541\pi$
$499$	− 753.686i	− 1.51039i	0.655498	−	0.755197i	$-0.272459\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 429.795i	− 0.854463i	−0.904142	−	0.427232i	$-0.859489\pi$
$503$	− 429.795i	− 0.854463i	0.904142	−	0.427232i	$-0.140511\pi$
$504$	0	0
$505$	397.665	0.787456
$506$	0	0
$507$	0	0
$508$	0	0
$509$	491.207	0.965042	0.482521	−	0.875884i	$-0.339721\pi$
$509$	491.207	0.965042	0.482521	+	0.875884i	$0.339721\pi$
$510$	0	0
$511$	−583.906	−1.14267
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−686.565	−1.33314
$516$	0	0
$517$	76.0219i	0.147044i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	666.985i	1.28020i	0.768291	+	0.640101i	$0.221107\pi$
$521$	666.985i	1.28020i	−0.768291	+	0.640101i	$0.778893\pi$
$522$	0	0
$523$	− 443.127i	− 0.847280i	−0.905831	−	0.423640i	$-0.860752\pi$
$523$	− 443.127i	− 0.847280i	0.905831	−	0.423640i	$-0.139248\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	425.380i	0.807173i
$528$	0	0
$529$	510.152	0.964370
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−617.088	−1.15776
$534$	0	0
$535$	1844.65	3.44795
$536$	0	0
$537$	0	0
$538$	0	0
$539$	14.7572	0.0273788
$540$	0	0
$541$	− 172.653i	− 0.319137i	−0.987187	−	0.159569i	$-0.948990\pi$
$541$	− 172.653i	− 0.319137i	0.987187	−	0.159569i	$-0.0510103\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 701.308i	− 1.28680i
$546$	0	0
$547$	− 55.1084i	− 0.100747i	−0.998730	−	0.0503733i	$-0.983959\pi$
$547$	− 55.1084i	− 0.100747i	0.998730	−	0.0503733i	$-0.0160411\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 19.8140i	− 0.0359600i
$552$	0	0
$553$	−10.7053	−0.0193585
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−49.5862	−0.0890237	−0.0445119	−	0.999009i	$-0.514173\pi$
$557$	−49.5862	−0.0890237	−0.0445119	+	0.999009i	$0.514173\pi$
$558$	0	0
$559$	989.175	1.76954
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−172.517	−0.306425	−0.153213	−	0.988193i	$-0.548962\pi$
$563$	−172.517	−0.306425	−0.153213	+	0.988193i	$0.548962\pi$
$564$	0	0
$565$	− 1369.46i	− 2.42382i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 572.646i	− 1.00641i	−0.864168	−	0.503203i	$-0.832155\pi$
$569$	− 572.646i	− 1.00641i	0.864168	−	0.503203i	$-0.167845\pi$
$570$	0	0
$571$	− 132.517i	− 0.232079i	−0.993245	−	0.116039i	$-0.962980\pi$
$571$	− 132.517i	− 0.232079i	0.993245	−	0.116039i	$-0.0370199\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 246.111i	− 0.428019i
$576$	0	0
$577$	−198.324	−0.343716	−0.171858	−	0.985122i	$-0.554977\pi$
$577$	−198.324	−0.343716	−0.171858	+	0.985122i	$0.554977\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−943.471	−1.62387
$582$	0	0
$583$	96.4257	0.165396
$584$	0	0
$585$	0	0
$586$	0	0
$587$	60.3600	0.102828	0.0514140	−	0.998677i	$-0.483627\pi$
$587$	60.3600	0.102828	0.0514140	+	0.998677i	$0.483627\pi$
$588$	0	0
$589$	568.384i	0.964999i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 462.144i	− 0.779332i	−0.920956	−	0.389666i	$-0.872590\pi$
$593$	− 462.144i	− 0.779332i	0.920956	−	0.389666i	$-0.127410\pi$
$594$	0	0
$595$	1674.38i	2.81409i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 965.054i	− 1.61111i	−0.592522	−	0.805554i	$-0.701868\pi$
$599$	− 965.054i	− 1.61111i	0.592522	−	0.805554i	$-0.298132\pi$
$600$	0	0
$601$	822.800	1.36905	0.684526	−	0.728989i	$-0.260009\pi$
$601$	822.800	1.36905	0.684526	+	0.728989i	$0.260009\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1083.27	−1.79052
$606$	0	0
$607$	−512.002	−0.843496	−0.421748	−	0.906713i	$-0.638583\pi$
$607$	−512.002	−0.843496	−0.421748	+	0.906713i	$0.638583\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1117.15	1.82840
$612$	0	0
$613$	− 430.232i	− 0.701846i	−0.936404	−	0.350923i	$-0.885868\pi$
$613$	− 430.232i	− 0.701846i	0.936404	−	0.350923i	$-0.114132\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	840.028i	1.36147i	0.732529	+	0.680736i	$0.238340\pi$
$617$	840.028i	1.36147i	−0.732529	+	0.680736i	$0.761660\pi$
$618$	0	0
$619$	662.138i	1.06969i	0.844950	+	0.534845i	$0.179630\pi$
$619$	662.138i	1.06969i	−0.844950	+	0.534845i	$0.820370\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	798.859i	1.28228i
$624$	0	0
$625$	1171.37	1.87419
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−205.696	−0.327021
$630$	0	0
$631$	−594.485	−0.942131	−0.471065	−	0.882098i	$-0.656130\pi$
$631$	−594.485	−0.942131	−0.471065	+	0.882098i	$0.656130\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2040.76	−3.21379
$636$	0	0
$637$	− 216.859i	− 0.340438i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 393.431i	− 0.613777i	−0.951745	−	0.306889i	$-0.900712\pi$
$641$	− 393.431i	− 0.613777i	0.951745	−	0.306889i	$-0.0992879\pi$
$642$	0	0
$643$	945.265i	1.47009i	0.678021	+	0.735043i	$0.262838\pi$
$643$	945.265i	1.47009i	−0.678021	+	0.735043i	$0.737162\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 460.952i	− 0.712446i	−0.934401	−	0.356223i	$-0.884064\pi$
$647$	− 460.952i	− 0.712446i	0.934401	−	0.356223i	$-0.115936\pi$
$648$	0	0
$649$	−30.7768	−0.0474219
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1163.60	−1.78192	−0.890961	−	0.454080i	$-0.849968\pi$
$653$	−1163.60	−1.78192	−0.890961	+	0.454080i	$0.849968\pi$
$654$	0	0
$655$	1030.71	1.57360
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−894.773	−1.35777	−0.678887	−	0.734243i	$-0.737537\pi$
$659$	−894.773	−1.35777	−0.678887	+	0.734243i	$0.737537\pi$
$660$	0	0
$661$	571.283i	0.864271i	0.901809	+	0.432136i	$0.142240\pi$
$661$	571.283i	0.864271i	−0.901809	+	0.432136i	$0.857760\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2237.28i	3.36433i
$666$	0	0
$667$	2.75367i	0.00412844i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 116.329i	− 0.173367i
$672$	0	0
$673$	−521.714	−0.775206	−0.387603	−	0.921826i	$-0.626697\pi$
$673$	−521.714	−0.775206	−0.387603	+	0.921826i	$0.626697\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−735.979	−1.08712	−0.543559	−	0.839371i	$-0.682924\pi$
$677$	−735.979	−1.08712	−0.543559	+	0.839371i	$0.682924\pi$
$678$	0	0
$679$	−658.141	−0.969279
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.32560	0.0121898	0.00609488	−	0.999981i	$-0.498060\pi$
$683$	8.32560	0.0121898	0.00609488	+	0.999981i	$0.498060\pi$
$684$	0	0
$685$	− 1197.87i	− 1.74872i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1416.99i	− 2.05659i
$690$	0	0
$691$	− 833.459i	− 1.20616i	−0.797679	−	0.603082i	$-0.793939\pi$
$691$	− 833.459i	− 1.20616i	0.797679	−	0.603082i	$-0.206061\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 939.969i	− 1.35247i
$696$	0	0
$697$	917.379	1.31618
$698$	0	0
$699$	0	0
$700$	0	0
$701$	748.081	1.06716	0.533581	−	0.845749i	$-0.320846\pi$
$701$	748.081	1.06716	0.533581	+	0.845749i	$0.320846\pi$
$702$	0	0
$703$	−274.847	−0.390963
$704$	0	0
$705$	0	0
$706$	0	0
$707$	348.643	0.493130
$708$	0	0
$709$	870.918i	1.22837i	0.789160	+	0.614187i	$0.210516\pi$
$709$	870.918i	1.22837i	−0.789160	+	0.614187i	$0.789484\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 78.9919i	− 0.110788i
$714$	0	0
$715$	− 152.113i	− 0.212745i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 734.578i	− 1.02167i	−0.859680	−	0.510833i	$-0.829337\pi$
$719$	− 734.578i	− 1.02167i	0.859680	−	0.510833i	$-0.170663\pi$
$720$	0	0
$721$	−601.928	−0.834852
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−35.9558	−0.0495942
$726$	0	0
$727$	20.8359	0.0286601	0.0143300	−	0.999897i	$-0.495438\pi$
$727$	20.8359	0.0286601	0.0143300	+	0.999897i	$0.495438\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1470.53	−2.01167
$732$	0	0
$733$	− 9.75355i	− 0.0133063i	−0.999978	−	0.00665317i	$-0.997882\pi$
$733$	− 9.75355i	− 0.0133063i	0.999978	−	0.00665317i	$-0.00211779\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 75.8744i	− 0.102950i
$738$	0	0
$739$	352.340i	0.476779i	0.971170	+	0.238390i	$0.0766196\pi$
$739$	352.340i	0.476779i	−0.971170	+	0.238390i	$0.923380\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1063.07i	1.43078i	0.698726	+	0.715390i	$0.253751\pi$
$743$	1063.07i	1.43078i	−0.698726	+	0.715390i	$0.746249\pi$
$744$	0	0
$745$	1106.72	1.48553
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1617.25	2.15922
$750$	0	0
$751$	421.237	0.560902	0.280451	−	0.959868i	$-0.409516\pi$
$751$	421.237	0.560902	0.280451	+	0.959868i	$0.409516\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	106.816	0.141478
$756$	0	0
$757$	1155.52i	1.52644i	0.646137	+	0.763221i	$0.276383\pi$
$757$	1155.52i	1.52644i	−0.646137	+	0.763221i	$0.723617\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1281.10i	1.68345i	0.539909	+	0.841723i	$0.318459\pi$
$761$	1281.10i	1.68345i	−0.539909	+	0.841723i	$0.681541\pi$
$762$	0	0
$763$	− 614.854i	− 0.805837i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	452.269i	0.589660i
$768$	0	0
$769$	−156.377	−0.203351	−0.101676	−	0.994818i	$-0.532420\pi$
$769$	−156.377	−0.203351	−0.101676	+	0.994818i	$0.532420\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1063.74	1.37612	0.688058	−	0.725656i	$-0.258463\pi$
$773$	1063.74	1.37612	0.688058	+	0.725656i	$0.258463\pi$
$774$	0	0
$775$	1031.43	1.33088
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1225.78	1.57353
$780$	0	0
$781$	− 15.2960i	− 0.0195851i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 746.354i	− 0.950770i
$786$	0	0
$787$	− 133.863i	− 0.170093i	−0.996377	−	0.0850464i	$-0.972896\pi$
$787$	− 133.863i	− 0.170093i	0.996377	−	0.0850464i	$-0.0271039\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1200.64i	− 1.51787i
$792$	0	0
$793$	−1709.48	−2.15571
$794$	0	0
$795$	0	0
$796$	0	0
$797$	287.344	0.360532	0.180266	−	0.983618i	$-0.442304\pi$
$797$	287.344	0.360532	0.180266	+	0.983618i	$0.442304\pi$
$798$	0	0
$799$	−1660.79	−2.07859
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−78.8596	−0.0982063
$804$	0	0
$805$	− 310.929i	− 0.386247i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	427.228i	0.528094i	0.964510	+	0.264047i	$0.0850574\pi$
$809$	427.228i	0.528094i	−0.964510	+	0.264047i	$0.914943\pi$
$810$	0	0
$811$	257.702i	0.317758i	0.987298	+	0.158879i	$0.0507880\pi$
$811$	257.702i	0.317758i	−0.987298	+	0.158879i	$0.949212\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	444.516i	0.545418i
$816$	0	0
$817$	−1964.90	−2.40501
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−514.687	−0.626903	−0.313452	−	0.949604i	$-0.601485\pi$
$821$	−514.687	−0.626903	−0.313452	+	0.949604i	$0.601485\pi$
$822$	0	0
$823$	271.043	0.329335	0.164667	−	0.986349i	$-0.447345\pi$
$823$	271.043	0.329335	0.164667	+	0.986349i	$0.447345\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1266.64	1.53161	0.765804	−	0.643073i	$-0.222341\pi$
$827$	1266.64	1.53161	0.765804	+	0.643073i	$0.222341\pi$
$828$	0	0
$829$	− 1076.89i	− 1.29902i	−0.760354	−	0.649509i	$-0.774975\pi$
$829$	− 1076.89i	− 1.29902i	0.760354	−	0.649509i	$-0.225025\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	322.388i	0.387021i
$834$	0	0
$835$	− 1576.84i	− 1.88844i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	311.036i	0.370723i	0.982670	+	0.185361i	$0.0593456\pi$
$839$	311.036i	0.370723i	−0.982670	+	0.185361i	$0.940654\pi$
$840$	0	0
$841$	−840.598	−0.999522
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−707.869	−0.837715
$846$	0	0
$847$	−949.726	−1.12128
$848$	0	0
$849$	0	0
$850$	0	0
$851$	38.1972	0.0448851
$852$	0	0
$853$	97.6779i	0.114511i	0.998360	+	0.0572555i	$0.0182350\pi$
$853$	97.6779i	0.114511i	−0.998360	+	0.0572555i	$0.981765\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	738.242i	0.861426i	0.902489	+	0.430713i	$0.141738\pi$
$857$	738.242i	0.861426i	−0.902489	+	0.430713i	$0.858262\pi$
$858$	0	0
$859$	− 1325.05i	− 1.54255i	−0.636499	−	0.771277i	$-0.719618\pi$
$859$	− 1325.05i	− 1.54255i	0.636499	−	0.771277i	$-0.280382\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	111.255i	0.128917i	0.997920	+	0.0644584i	$0.0205320\pi$
$863$	111.255i	0.128917i	−0.997920	+	0.0644584i	$0.979468\pi$
$864$	0	0
$865$	1390.08	1.60703
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.44581	−0.00166376
$870$	0	0
$871$	−1114.99	−1.28012
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2269.47	2.59368
$876$	0	0
$877$	− 15.1739i	− 0.0173021i	−0.999963	−	0.00865105i	$-0.997246\pi$
$877$	− 15.1739i	− 0.0173021i	0.999963	−	0.00865105i	$-0.00275375\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	965.029i	1.09538i	0.836682	+	0.547690i	$0.184493\pi$
$881$	965.029i	1.09538i	−0.836682	+	0.547690i	$0.815507\pi$
$882$	0	0
$883$	− 1588.94i	− 1.79948i	−0.436423	−	0.899742i	$-0.643755\pi$
$883$	− 1588.94i	− 1.79948i	0.436423	−	0.899742i	$-0.356245\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	99.4993i	0.112175i	0.998426	+	0.0560876i	$0.0178626\pi$
$887$	99.4993i	0.112175i	−0.998426	+	0.0560876i	$0.982137\pi$
$888$	0	0
$889$	−1789.18	−2.01258
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2219.11	−2.48501
$894$	0	0
$895$	1922.03	2.14752
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11.5404	−0.0128369
$900$	0	0
$901$	2106.54i	2.33800i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1458.39i	− 1.61148i
$906$	0	0
$907$	763.643i	0.841944i	0.907074	+	0.420972i	$0.138311\pi$
$907$	763.643i	0.841944i	−0.907074	+	0.420972i	$0.861689\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 484.661i	− 0.532010i	−0.963972	−	0.266005i	$-0.914296\pi$
$911$	− 484.661i	− 0.532010i	0.963972	−	0.266005i	$-0.0857038\pi$
$912$	0	0
$913$	−127.421	−0.139563
$914$	0	0
$915$	0	0
$916$	0	0
$917$	903.645	0.985436
$918$	0	0
$919$	−518.271	−0.563951	−0.281975	−	0.959422i	$-0.590990\pi$
$919$	−518.271	−0.563951	−0.281975	+	0.959422i	$0.590990\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−224.777	−0.243529
$924$	0	0
$925$	498.756i	0.539196i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 709.355i	− 0.763568i	−0.924251	−	0.381784i	$-0.875310\pi$
$929$	− 709.355i	− 0.763568i	0.924251	−	0.381784i	$-0.124690\pi$
$930$	0	0
$931$	430.768i	0.462694i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	226.135i	0.241855i
$936$	0	0
$937$	1308.30	1.39626	0.698132	−	0.715969i	$-0.254015\pi$
$937$	1308.30	1.39626	0.698132	+	0.715969i	$0.254015\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	256.150	0.272211	0.136105	−	0.990694i	$-0.456541\pi$
$941$	256.150	0.272211	0.136105	+	0.990694i	$0.456541\pi$
$942$	0	0
$943$	−170.355	−0.180652
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−542.855	−0.573237	−0.286618	−	0.958045i	$-0.592531\pi$
$947$	−542.855	−0.573237	−0.286618	+	0.958045i	$0.592531\pi$
$948$	0	0
$949$	1158.85i	1.22113i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1161.82i	− 1.21912i	−0.792742	−	0.609558i	$-0.791347\pi$
$953$	− 1161.82i	− 1.21912i	0.792742	−	0.609558i	$-0.208653\pi$
$954$	0	0
$955$	367.369i	0.384680i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1050.20i	− 1.09510i
$960$	0	0
$961$	−629.952	−0.655517
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2724.88	2.82371
$966$	0	0
$967$	−682.976	−0.706283	−0.353142	−	0.935570i	$-0.614887\pi$
$967$	−682.976	−0.706283	−0.353142	+	0.935570i	$0.614887\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−252.827	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$971$	−252.827	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$972$	0	0
$973$	− 824.094i	− 0.846962i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	208.186i	0.213087i	0.994308	+	0.106543i	$0.0339783\pi$
$977$	208.186i	0.213087i	−0.994308	+	0.106543i	$0.966022\pi$
$978$	0	0
$979$	107.890i	0.110205i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1154.92i	1.17489i	0.809264	+	0.587446i	$0.199866\pi$
$983$	1154.92i	1.17489i	−0.809264	+	0.587446i	$0.800134\pi$
$984$	0	0
$985$	217.090	0.220396
$986$	0	0
$987$	0	0
$988$	0	0
$989$	273.074	0.276111
$990$	0	0
$991$	−668.987	−0.675062	−0.337531	−	0.941314i	$-0.609592\pi$
$991$	−668.987	−0.675062	−0.337531	+	0.941314i	$0.609592\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−513.577	−0.516158
$996$	0	0
$997$	846.458i	0.849005i	0.905427	+	0.424502i	$0.139551\pi$
$997$	846.458i	0.849005i	−0.905427	+	0.424502i	$0.860449\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.h.j.161.11	yes	12
3.2	odd	2		1728.3.h.i.161.1	✓	12
4.3	odd	2		1728.3.h.i.161.11	yes	12
8.3	odd	2	inner	1728.3.h.j.161.2	yes	12
8.5	even	2		1728.3.h.i.161.2	yes	12
12.11	even	2	inner	1728.3.h.j.161.1	yes	12
24.5	odd	2	inner	1728.3.h.j.161.12	yes	12
24.11	even	2		1728.3.h.i.161.12	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.h.i.161.1	✓	12	3.2	odd	2
1728.3.h.i.161.2	yes	12	8.5	even	2
1728.3.h.i.161.11	yes	12	4.3	odd	2
1728.3.h.i.161.12	yes	12	24.11	even	2
1728.3.h.j.161.1	yes	12	12.11	even	2	inner
1728.3.h.j.161.2	yes	12	8.3	odd	2	inner
1728.3.h.j.161.11	yes	12	1.1	even	1	trivial
1728.3.h.j.161.12	yes	12	24.5	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.03816 q^{5} +7.92398 q^{7} +O(q^{10})\)	\(q+9.03816 q^{5} +7.92398 q^{7} +1.07018 q^{11} -15.7264i q^{13} +23.3793i q^{17} +31.2389i q^{19} -4.34147i q^{23} +56.6884 q^{25} -0.634271 q^{29} +18.1947 q^{31} +71.6182 q^{35} +8.79821i q^{37} -39.2389i q^{41} +62.8989i q^{43} +71.0368i q^{47} +13.7895 q^{49} +90.1026 q^{53} +9.67244 q^{55} -28.7586 q^{59} -108.701i q^{61} -142.138i q^{65} -70.8989i q^{67} -14.2930i q^{71} -73.6884 q^{73} +8.48006 q^{77} -1.35100 q^{79} -119.065 q^{83} +211.306i q^{85} +100.815i q^{89} -124.616i q^{91} +282.343i q^{95} -83.0568 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 12 q^{11} + 72 q^{25} + 252 q^{35} + 168 q^{49} + 264 q^{59} - 276 q^{73} + 396 q^{83} - 396 q^{97}+O(q^{100}) \) 12 * q + 12 * q^11 + 72 * q^25 + 252 * q^35 + 168 * q^49 + 264 * q^59 - 276 * q^73 + 396 * q^83 - 396 * q^97

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