LMFDB - Embedded newform 1296.3.e.c.161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,3,Mod(161,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1296.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.e (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:	$35.3134422611$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.3
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.3.e.c.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.92480i q^{5} -12.7980 q^{7} +O(q^{10})$	$q+3.92480i q^{5} -12.7980 q^{7} +16.5099i q^{11} -2.79796 q^{13} -2.54270i q^{17} -21.5959 q^{19} +3.00340i q^{23} +9.59592 q^{25} -15.2385i q^{29} +27.2020 q^{31} -50.2295i q^{35} -10.4041 q^{37} +39.8372i q^{41} -34.1918 q^{43} -67.2000i q^{47} +114.788 q^{49} -100.680i q^{53} -64.7980 q^{55} +6.11756i q^{59} +40.7980 q^{61} -10.9814i q^{65} -108.980 q^{67} -52.8829i q^{71} +68.7878 q^{73} -211.293i q^{77} +25.5857 q^{79} -60.1434i q^{83} +9.97959 q^{85} -7.62809i q^{89} +35.8082 q^{91} -84.7597i q^{95} -101.404 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{7}+O(q^{10}) $ 4 * q - 12 * q^7	$ 4 q - 12 q^{7} + 28 q^{13} - 8 q^{19} - 40 q^{25} + 148 q^{31} - 120 q^{37} + 20 q^{43} + 224 q^{49} - 220 q^{55} + 124 q^{61} - 44 q^{67} + 40 q^{73} - 172 q^{79} - 352 q^{85} + 300 q^{91} - 484 q^{97}+O(q^{100}) $ 4 * q - 12 * q^7 + 28 * q^13 - 8 * q^19 - 40 * q^25 + 148 * q^31 - 120 * q^37 + 20 * q^43 + 224 * q^49 - 220 * q^55 + 124 * q^61 - 44 * q^67 + 40 * q^73 - 172 * q^79 - 352 * q^85 + 300 * q^91 - 484 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	3.92480i	0.784961i	0.919760	+	0.392480i	$0.128383\pi$
$5$	3.92480i	0.784961i	−0.919760	+	0.392480i	$0.871617\pi$
$6$	0	0
$7$	−12.7980	−1.82828	−0.914140	−	0.405399i	$-0.867133\pi$
$7$	−12.7980	−1.82828	−0.914140	+	0.405399i	$0.867133\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	16.5099i	1.50090i	0.660929	+	0.750448i	$0.270162\pi$
$11$	16.5099i	1.50090i	−0.660929	+	0.750448i	$0.729838\pi$
$12$	0	0
$13$	−2.79796	−0.215228	−0.107614	−	0.994193i	$-0.534321\pi$
$13$	−2.79796	−0.215228	−0.107614	+	0.994193i	$0.534321\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.54270i	− 0.149570i	−0.997200	−	0.0747852i	$-0.976173\pi$
$17$	− 2.54270i	− 0.149570i	0.997200	−	0.0747852i	$-0.0238271\pi$
$18$	0	0
$19$	−21.5959	−1.13663	−0.568314	−	0.822812i	$-0.692404\pi$
$19$	−21.5959	−1.13663	−0.568314	+	0.822812i	$0.692404\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.00340i	0.130583i	0.997866	+	0.0652913i	$0.0207977\pi$
$23$	3.00340i	0.130583i	−0.997866	+	0.0652913i	$0.979202\pi$
$24$	0	0
$25$	9.59592	0.383837
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 15.2385i	− 0.525466i	−0.964869	−	0.262733i	$-0.915376\pi$
$29$	− 15.2385i	− 0.525466i	0.964869	−	0.262733i	$-0.0846238\pi$
$30$	0	0
$31$	27.2020	0.877485	0.438743	−	0.898613i	$-0.355424\pi$
$31$	27.2020	0.877485	0.438743	+	0.898613i	$0.355424\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 50.2295i	− 1.43513i
$36$	0	0
$37$	−10.4041	−0.281191	−0.140596	−	0.990067i	$-0.544902\pi$
$37$	−10.4041	−0.281191	−0.140596	+	0.990067i	$0.544902\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	39.8372i	0.971638i	0.874059	+	0.485819i	$0.161479\pi$
$41$	39.8372i	0.971638i	−0.874059	+	0.485819i	$0.838521\pi$
$42$	0	0
$43$	−34.1918	−0.795159	−0.397579	−	0.917568i	$-0.630150\pi$
$43$	−34.1918	−0.795159	−0.397579	+	0.917568i	$0.630150\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 67.2000i	− 1.42979i	−0.699233	−	0.714894i	$-0.746475\pi$
$47$	− 67.2000i	− 1.42979i	0.699233	−	0.714894i	$-0.253525\pi$
$48$	0	0
$49$	114.788	2.34261
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 100.680i	− 1.89963i	−0.312810	−	0.949816i	$-0.601270\pi$
$53$	− 100.680i	− 1.89963i	0.312810	−	0.949816i	$-0.398730\pi$
$54$	0	0
$55$	−64.7980	−1.17814
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.11756i	0.103687i	0.998655	+	0.0518437i	$0.0165098\pi$
$59$	6.11756i	0.103687i	−0.998655	+	0.0518437i	$0.983490\pi$
$60$	0	0
$61$	40.7980	0.668819	0.334409	−	0.942428i	$-0.391463\pi$
$61$	40.7980	0.668819	0.334409	+	0.942428i	$0.391463\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 10.9814i	− 0.168945i
$66$	0	0
$67$	−108.980	−1.62656	−0.813281	−	0.581872i	$-0.802321\pi$
$67$	−108.980	−1.62656	−0.813281	+	0.581872i	$0.802321\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 52.8829i	− 0.744830i	−0.928066	−	0.372415i	$-0.878530\pi$
$71$	− 52.8829i	− 0.744830i	0.928066	−	0.372415i	$-0.121470\pi$
$72$	0	0
$73$	68.7878	0.942298	0.471149	−	0.882054i	$-0.343839\pi$
$73$	68.7878	0.942298	0.471149	+	0.882054i	$0.343839\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 211.293i	− 2.74406i
$78$	0	0
$79$	25.5857	0.323870	0.161935	−	0.986801i	$-0.448227\pi$
$79$	25.5857	0.323870	0.161935	+	0.986801i	$0.448227\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 60.1434i	− 0.724619i	−0.932058	−	0.362310i	$-0.881988\pi$
$83$	− 60.1434i	− 0.724619i	0.932058	−	0.362310i	$-0.118012\pi$
$84$	0	0
$85$	9.97959	0.117407
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 7.62809i	− 0.0857089i	−0.999081	−	0.0428545i	$-0.986355\pi$
$89$	− 7.62809i	− 0.0857089i	0.999081	−	0.0428545i	$-0.0136452\pi$
$90$	0	0
$91$	35.8082	0.393496
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 84.7597i	− 0.892208i
$96$	0	0
$97$	−101.404	−1.04540	−0.522701	−	0.852516i	$-0.675076\pi$
$97$	−101.404	−1.04540	−0.522701	+	0.852516i	$0.675076\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.38211i	− 0.0136842i	−0.999977	−	0.00684211i	$-0.997822\pi$
$101$	− 1.38211i	− 0.0136842i	0.999977	−	0.00684211i	$-0.00217793\pi$
$102$	0	0
$103$	1.58571	0.0153953	0.00769763	−	0.999970i	$-0.497550\pi$
$103$	1.58571	0.0153953	0.00769763	+	0.999970i	$0.497550\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	103.223i	0.964702i	0.875978	+	0.482351i	$0.160217\pi$
$107$	103.223i	0.964702i	−0.875978	+	0.482351i	$0.839783\pi$
$108$	0	0
$109$	14.4041	0.132148	0.0660738	−	0.997815i	$-0.478953\pi$
$109$	14.4041	0.132148	0.0660738	+	0.997815i	$0.478953\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	106.320i	0.940882i	0.882432	+	0.470441i	$0.155905\pi$
$113$	106.320i	0.940882i	−0.882432	+	0.470441i	$0.844095\pi$
$114$	0	0
$115$	−11.7878	−0.102502
$116$	0	0
$117$	0	0
$118$	0	0
$119$	32.5413i	0.273457i
$120$	0	0
$121$	−151.576	−1.25269
$122$	0	0
$123$	0	0
$124$	0	0
$125$	135.782i	1.08626i
$126$	0	0
$127$	183.980	1.44866	0.724329	−	0.689454i	$-0.242150\pi$
$127$	183.980	1.44866	0.724329	+	0.689454i	$0.242150\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 105.877i	− 0.808218i	−0.914711	−	0.404109i	$-0.867582\pi$
$131$	− 105.877i	− 0.808218i	0.914711	−	0.404109i	$-0.132418\pi$
$132$	0	0
$133$	276.384	2.07807
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 190.858i	− 1.39312i	−0.717497	−	0.696562i	$-0.754712\pi$
$137$	− 190.858i	− 1.39312i	0.717497	−	0.696562i	$-0.245288\pi$
$138$	0	0
$139$	94.9796	0.683306	0.341653	−	0.939826i	$-0.389013\pi$
$139$	94.9796	0.683306	0.341653	+	0.939826i	$0.389013\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 46.1939i	− 0.323034i
$144$	0	0
$145$	59.8082	0.412470
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 91.9271i	− 0.616961i	−0.951231	−	0.308480i	$-0.900180\pi$
$149$	− 91.9271i	− 0.616961i	0.951231	−	0.308480i	$-0.0998204\pi$
$150$	0	0
$151$	16.4143	0.108704	0.0543519	−	0.998522i	$-0.482691\pi$
$151$	16.4143	0.108704	0.0543519	+	0.998522i	$0.482691\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	106.763i	0.688791i
$156$	0	0
$157$	−211.969	−1.35012	−0.675062	−	0.737761i	$-0.735883\pi$
$157$	−211.969	−1.35012	−0.675062	+	0.737761i	$0.735883\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 38.4374i	− 0.238742i
$162$	0	0
$163$	−172.788	−1.06005	−0.530024	−	0.847983i	$-0.677817\pi$
$163$	−172.788	−1.06005	−0.530024	+	0.847983i	$0.677817\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 188.444i	− 1.12840i	−0.825637	−	0.564202i	$-0.809184\pi$
$167$	− 188.444i	− 1.12840i	0.825637	−	0.564202i	$-0.190816\pi$
$168$	0	0
$169$	−161.171	−0.953677
$170$	0	0
$171$	0	0
$172$	0	0
$173$	60.2718i	0.348392i	0.984711	+	0.174196i	$0.0557326\pi$
$173$	60.2718i	0.348392i	−0.984711	+	0.174196i	$0.944267\pi$
$174$	0	0
$175$	−122.808	−0.701761
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 72.7108i	− 0.406205i	−0.979157	−	0.203103i	$-0.934897\pi$
$179$	− 72.7108i	− 0.406205i	0.979157	−	0.203103i	$-0.0651025\pi$
$180$	0	0
$181$	−97.5959	−0.539204	−0.269602	−	0.962972i	$-0.586892\pi$
$181$	−97.5959	−0.539204	−0.269602	+	0.962972i	$0.586892\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 40.8340i	− 0.220724i
$186$	0	0
$187$	41.9796	0.224490
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 326.272i	− 1.70823i	−0.520082	−	0.854116i	$-0.674099\pi$
$191$	− 326.272i	− 1.70823i	0.520082	−	0.854116i	$-0.325901\pi$
$192$	0	0
$193$	21.0204	0.108914	0.0544570	−	0.998516i	$-0.482657\pi$
$193$	21.0204	0.108914	0.0544570	+	0.998516i	$0.482657\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	32.5413i	0.165184i	0.996583	+	0.0825922i	$0.0263199\pi$
$197$	32.5413i	0.165184i	−0.996583	+	0.0825922i	$0.973680\pi$
$198$	0	0
$199$	62.0000	0.311558	0.155779	−	0.987792i	$-0.450211\pi$
$199$	62.0000	0.311558	0.155779	+	0.987792i	$0.450211\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	195.022i	0.960699i
$204$	0	0
$205$	−156.353	−0.762698
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 356.546i	− 1.70596i
$210$	0	0
$211$	168.596	0.799033	0.399516	−	0.916726i	$-0.369178\pi$
$211$	168.596	0.799033	0.399516	+	0.916726i	$0.369178\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 134.196i	− 0.624169i
$216$	0	0
$217$	−348.131	−1.60429
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.11436i	0.0321917i
$222$	0	0
$223$	156.373	0.701226	0.350613	−	0.936520i	$-0.385973\pi$
$223$	156.373	0.701226	0.350613	+	0.936520i	$0.385973\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 100.056i	− 0.440775i	−0.975412	−	0.220388i	$-0.929268\pi$
$227$	− 100.056i	− 0.440775i	0.975412	−	0.220388i	$-0.0707322\pi$
$228$	0	0
$229$	−209.990	−0.916986	−0.458493	−	0.888698i	$-0.651611\pi$
$229$	−209.990	−0.916986	−0.458493	+	0.888698i	$0.651611\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 307.127i	− 1.31814i	−0.752081	−	0.659070i	$-0.770950\pi$
$233$	− 307.127i	− 1.31814i	0.752081	−	0.659070i	$-0.229050\pi$
$234$	0	0
$235$	263.747	1.12233
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 81.2780i	− 0.340075i	−0.985438	−	0.170038i	$-0.945611\pi$
$239$	− 81.2780i	− 0.340075i	0.985438	−	0.170038i	$-0.0543889\pi$
$240$	0	0
$241$	−360.192	−1.49457	−0.747286	−	0.664503i	$-0.768643\pi$
$241$	−360.192	−1.49457	−0.747286	+	0.664503i	$0.768643\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	450.519i	1.83885i
$246$	0	0
$247$	60.4245	0.244634
$248$	0	0
$249$	0	0
$250$	0	0
$251$	400.179i	1.59434i	0.603755	+	0.797170i	$0.293670\pi$
$251$	400.179i	1.59434i	−0.603755	+	0.797170i	$0.706330\pi$
$252$	0	0
$253$	−49.5857	−0.195991
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 363.771i	− 1.41545i	−0.706488	−	0.707725i	$-0.749722\pi$
$257$	− 363.771i	− 1.41545i	0.706488	−	0.707725i	$-0.250278\pi$
$258$	0	0
$259$	133.151	0.514097
$260$	0	0
$261$	0	0
$262$	0	0
$263$	377.091i	1.43381i	0.697173	+	0.716903i	$0.254441\pi$
$263$	377.091i	1.43381i	−0.697173	+	0.716903i	$0.745559\pi$
$264$	0	0
$265$	395.151	1.49114
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 170.849i	− 0.635125i	−0.948237	−	0.317562i	$-0.897136\pi$
$269$	− 170.849i	− 0.635125i	0.948237	−	0.317562i	$-0.102864\pi$
$270$	0	0
$271$	50.4041	0.185993	0.0929965	−	0.995666i	$-0.470355\pi$
$271$	50.4041	0.185993	0.0929965	+	0.995666i	$0.470355\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	158.427i	0.576099i
$276$	0	0
$277$	25.6061	0.0924409	0.0462204	−	0.998931i	$-0.485282\pi$
$277$	25.6061	0.0924409	0.0462204	+	0.998931i	$0.485282\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	335.172i	1.19278i	0.802694	+	0.596391i	$0.203399\pi$
$281$	335.172i	1.19278i	−0.802694	+	0.596391i	$0.796601\pi$
$282$	0	0
$283$	14.9796	0.0529314	0.0264657	−	0.999650i	$-0.491575\pi$
$283$	14.9796	0.0529314	0.0264657	+	0.999650i	$0.491575\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 509.834i	− 1.77643i
$288$	0	0
$289$	282.535	0.977629
$290$	0	0
$291$	0	0
$292$	0	0
$293$	301.395i	1.02865i	0.857595	+	0.514325i	$0.171958\pi$
$293$	301.395i	1.02865i	−0.857595	+	0.514325i	$0.828042\pi$
$294$	0	0
$295$	−24.0102	−0.0813905
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 8.40339i	− 0.0281050i
$300$	0	0
$301$	437.586	1.45377
$302$	0	0
$303$	0	0
$304$	0	0
$305$	160.124i	0.524997i
$306$	0	0
$307$	44.7469	0.145755	0.0728777	−	0.997341i	$-0.476782\pi$
$307$	44.7469	0.145755	0.0728777	+	0.997341i	$0.476782\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	324.873i	1.04461i	0.852760	+	0.522303i	$0.174927\pi$
$311$	324.873i	1.04461i	−0.852760	+	0.522303i	$0.825073\pi$
$312$	0	0
$313$	313.788	1.00252	0.501258	−	0.865298i	$-0.332871\pi$
$313$	313.788	1.00252	0.501258	+	0.865298i	$0.332871\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 55.4080i	− 0.174788i	−0.996174	−	0.0873942i	$-0.972146\pi$
$317$	− 55.4080i	− 0.174788i	0.996174	−	0.0873942i	$-0.0278540\pi$
$318$	0	0
$319$	251.586	0.788670
$320$	0	0
$321$	0	0
$322$	0	0
$323$	54.9119i	0.170006i
$324$	0	0
$325$	−26.8490	−0.0826123
$326$	0	0
$327$	0	0
$328$	0	0
$329$	860.023i	2.61405i
$330$	0	0
$331$	−15.4041	−0.0465380	−0.0232690	−	0.999729i	$-0.507407\pi$
$331$	−15.4041	−0.0465380	−0.0232690	+	0.999729i	$0.507407\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 427.723i	− 1.27679i
$336$	0	0
$337$	75.7673	0.224829	0.112414	−	0.993661i	$-0.464142\pi$
$337$	75.7673	0.224829	0.112414	+	0.993661i	$0.464142\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	449.102i	1.31701i
$342$	0	0
$343$	−841.949	−2.45466
$344$	0	0
$345$	0	0
$346$	0	0
$347$	668.169i	1.92556i	0.270290	+	0.962779i	$0.412880\pi$
$347$	668.169i	1.92556i	−0.270290	+	0.962779i	$0.587120\pi$
$348$	0	0
$349$	−246.031	−0.704959	−0.352479	−	0.935820i	$-0.614661\pi$
$349$	−246.031	−0.704959	−0.352479	+	0.935820i	$0.614661\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	180.722i	0.511962i	0.966682	+	0.255981i	$0.0823984\pi$
$353$	180.722i	0.511962i	−0.966682	+	0.255981i	$0.917602\pi$
$354$	0	0
$355$	207.555	0.584662
$356$	0	0
$357$	0	0
$358$	0	0
$359$	256.273i	0.713852i	0.934133	+	0.356926i	$0.116175\pi$
$359$	256.273i	0.713852i	−0.934133	+	0.356926i	$0.883825\pi$
$360$	0	0
$361$	105.384	0.291922
$362$	0	0
$363$	0	0
$364$	0	0
$365$	269.978i	0.739667i
$366$	0	0
$367$	−540.716	−1.47334	−0.736671	−	0.676252i	$-0.763603\pi$
$367$	−540.716	−1.47334	−0.736671	+	0.676252i	$0.763603\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1288.50i	3.47306i
$372$	0	0
$373$	−171.969	−0.461044	−0.230522	−	0.973067i	$-0.574043\pi$
$373$	−171.969	−0.461044	−0.230522	+	0.973067i	$0.574043\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	42.6367i	0.113095i
$378$	0	0
$379$	170.849	0.450789	0.225394	−	0.974268i	$-0.427633\pi$
$379$	170.849	0.450789	0.225394	+	0.974268i	$0.427633\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 583.910i	− 1.52457i	−0.647243	−	0.762284i	$-0.724078\pi$
$383$	− 583.910i	− 1.52457i	0.647243	−	0.762284i	$-0.275922\pi$
$384$	0	0
$385$	829.282	2.15398
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 444.273i	− 1.14209i	−0.820918	−	0.571045i	$-0.806538\pi$
$389$	− 444.273i	− 1.14209i	0.820918	−	0.571045i	$-0.193462\pi$
$390$	0	0
$391$	7.63674	0.0195313
$392$	0	0
$393$	0	0
$394$	0	0
$395$	100.419i	0.254225i
$396$	0	0
$397$	−575.090	−1.44859	−0.724294	−	0.689491i	$-0.757834\pi$
$397$	−575.090	−1.44859	−0.724294	+	0.689491i	$0.757834\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 163.180i	− 0.406934i	−0.979082	−	0.203467i	$-0.934779\pi$
$401$	− 163.180i	− 0.406934i	0.979082	−	0.203467i	$-0.0652209\pi$
$402$	0	0
$403$	−76.1102	−0.188859
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 171.770i	− 0.422039i
$408$	0	0
$409$	−601.282	−1.47013	−0.735063	−	0.677999i	$-0.762847\pi$
$409$	−601.282	−1.47013	−0.735063	+	0.677999i	$0.762847\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 78.2922i	− 0.189570i
$414$	0	0
$415$	236.051	0.568798
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 183.159i	− 0.437134i	−0.975822	−	0.218567i	$-0.929862\pi$
$419$	− 183.159i	− 0.437134i	0.975822	−	0.218567i	$-0.0701382\pi$
$420$	0	0
$421$	167.545	0.397969	0.198984	−	0.980003i	$-0.436236\pi$
$421$	167.545	0.397969	0.198984	+	0.980003i	$0.436236\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 24.3995i	− 0.0574106i
$426$	0	0
$427$	−522.131	−1.22279
$428$	0	0
$429$	0	0
$430$	0	0
$431$	644.766i	1.49598i	0.663712	+	0.747988i	$0.268980\pi$
$431$	644.766i	1.49598i	−0.663712	+	0.747988i	$0.731020\pi$
$432$	0	0
$433$	133.514	0.308347	0.154174	−	0.988044i	$-0.450729\pi$
$433$	133.514	0.308347	0.154174	+	0.988044i	$0.450729\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 64.8612i	− 0.148424i
$438$	0	0
$439$	92.4551	0.210604	0.105302	−	0.994440i	$-0.466419\pi$
$439$	92.4551	0.210604	0.105302	+	0.994440i	$0.466419\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	173.387i	0.391392i	0.980665	+	0.195696i	$0.0626965\pi$
$443$	173.387i	0.391392i	−0.980665	+	0.195696i	$0.937303\pi$
$444$	0	0
$445$	29.9388	0.0672781
$446$	0	0
$447$	0	0
$448$	0	0
$449$	430.692i	0.959224i	0.877481	+	0.479612i	$0.159223\pi$
$449$	430.692i	0.959224i	−0.877481	+	0.479612i	$0.840777\pi$
$450$	0	0
$451$	−657.706	−1.45833
$452$	0	0
$453$	0	0
$454$	0	0
$455$	140.540i	0.308879i
$456$	0	0
$457$	−525.686	−1.15030	−0.575148	−	0.818049i	$-0.695056\pi$
$457$	−525.686	−1.15030	−0.575148	+	0.818049i	$0.695056\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.78978i	0.00388239i	0.999998	+	0.00194119i	$0.000617901\pi$
$461$	1.78978i	0.00388239i	−0.999998	+	0.00194119i	$0.999382\pi$
$462$	0	0
$463$	−526.333	−1.13679	−0.568394	−	0.822757i	$-0.692435\pi$
$463$	−526.333	−1.13679	−0.568394	+	0.822757i	$0.692435\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 409.322i	− 0.876494i	−0.898855	−	0.438247i	$-0.855600\pi$
$467$	− 409.322i	− 0.876494i	0.898855	−	0.438247i	$-0.144400\pi$
$468$	0	0
$469$	1394.72	2.97381
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 564.502i	− 1.19345i
$474$	0	0
$475$	−207.233	−0.436279
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 606.457i	− 1.26609i	−0.774115	−	0.633045i	$-0.781805\pi$
$479$	− 606.457i	− 1.26609i	0.774115	−	0.633045i	$-0.218195\pi$
$480$	0	0
$481$	29.1102	0.0605202
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 397.991i	− 0.820600i
$486$	0	0
$487$	275.131	0.564950	0.282475	−	0.959275i	$-0.408845\pi$
$487$	275.131	0.564950	0.282475	+	0.959275i	$0.408845\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 147.189i	− 0.299774i	−0.988703	−	0.149887i	$-0.952109\pi$
$491$	− 147.189i	− 0.299774i	0.988703	−	0.149887i	$-0.0478910\pi$
$492$	0	0
$493$	−38.7469	−0.0785942
$494$	0	0
$495$	0	0
$496$	0	0
$497$	676.794i	1.36176i
$498$	0	0
$499$	−452.616	−0.907047	−0.453523	−	0.891244i	$-0.649833\pi$
$499$	−452.616	−0.907047	−0.453523	+	0.891244i	$0.649833\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 571.541i	− 1.13627i	−0.822937	−	0.568133i	$-0.807666\pi$
$503$	− 571.541i	− 1.13627i	0.822937	−	0.568133i	$-0.192334\pi$
$504$	0	0
$505$	5.42449	0.0107416
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 455.109i	− 0.894123i	−0.894503	−	0.447062i	$-0.852470\pi$
$509$	− 455.109i	− 0.894123i	0.894503	−	0.447062i	$-0.147530\pi$
$510$	0	0
$511$	−880.343	−1.72278
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.22361i	0.0120847i
$516$	0	0
$517$	1109.46	2.14596
$518$	0	0
$519$	0	0
$520$	0	0
$521$	846.127i	1.62404i	0.583627	+	0.812022i	$0.301633\pi$
$521$	846.127i	1.62404i	−0.583627	+	0.812022i	$0.698367\pi$
$522$	0	0
$523$	487.616	0.932345	0.466172	−	0.884694i	$-0.345633\pi$
$523$	487.616	0.932345	0.466172	+	0.884694i	$0.345633\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 69.1666i	− 0.131246i
$528$	0	0
$529$	519.980	0.982948
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 111.463i	− 0.209123i
$534$	0	0
$535$	−405.131	−0.757253
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1895.13i	3.51601i
$540$	0	0
$541$	608.302	1.12440	0.562202	−	0.827000i	$-0.309955\pi$
$541$	608.302	1.12440	0.562202	+	0.827000i	$0.309955\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	56.5332i	0.103731i
$546$	0	0
$547$	863.343	1.57832	0.789162	−	0.614186i	$-0.210515\pi$
$547$	863.343	1.57832	0.789162	+	0.614186i	$0.210515\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	329.090i	0.597259i
$552$	0	0
$553$	−327.445	−0.592125
$554$	0	0
$555$	0	0
$556$	0	0
$557$	331.526i	0.595200i	0.954691	+	0.297600i	$0.0961861\pi$
$557$	331.526i	0.595200i	−0.954691	+	0.297600i	$0.903814\pi$
$558$	0	0
$559$	95.6674	0.171140
$560$	0	0
$561$	0	0
$562$	0	0
$563$	593.544i	1.05425i	0.849787	+	0.527126i	$0.176731\pi$
$563$	593.544i	1.05425i	−0.849787	+	0.527126i	$0.823269\pi$
$564$	0	0
$565$	−417.284	−0.738555
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1026.71i	1.80441i	0.431310	+	0.902204i	$0.358051\pi$
$569$	1026.71i	1.80441i	−0.431310	+	0.902204i	$0.641949\pi$
$570$	0	0
$571$	−383.686	−0.671954	−0.335977	−	0.941870i	$-0.609066\pi$
$571$	−383.686	−0.671954	−0.335977	+	0.941870i	$0.609066\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	28.8204i	0.0501224i
$576$	0	0
$577$	−777.433	−1.34737	−0.673685	−	0.739019i	$-0.735290\pi$
$577$	−777.433	−1.34737	−0.673685	+	0.739019i	$0.735290\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	769.713i	1.32481i
$582$	0	0
$583$	1662.22	2.85115
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 671.482i	− 1.14392i	−0.820281	−	0.571961i	$-0.806183\pi$
$587$	− 671.482i	− 1.14392i	0.820281	−	0.571961i	$-0.193817\pi$
$588$	0	0
$589$	−587.453	−0.997374
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 536.457i	− 0.904650i	−0.891853	−	0.452325i	$-0.850595\pi$
$593$	− 536.457i	− 0.904650i	0.891853	−	0.452325i	$-0.149405\pi$
$594$	0	0
$595$	−127.718	−0.214653
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 849.210i	− 1.41771i	−0.705352	−	0.708857i	$-0.749211\pi$
$599$	− 849.210i	− 1.41771i	0.705352	−	0.708857i	$-0.250789\pi$
$600$	0	0
$601$	661.384	1.10047	0.550236	−	0.835009i	$-0.314538\pi$
$601$	661.384	1.10047	0.550236	+	0.835009i	$0.314538\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 594.904i	− 0.983313i
$606$	0	0
$607$	21.9082	0.0360925	0.0180463	−	0.999837i	$-0.494255\pi$
$607$	21.9082	0.0360925	0.0180463	+	0.999837i	$0.494255\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	188.023i	0.307730i
$612$	0	0
$613$	−1164.75	−1.90008	−0.950038	−	0.312133i	$-0.898956\pi$
$613$	−1164.75	−1.90008	−0.950038	+	0.312133i	$0.898956\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 644.584i	− 1.04471i	−0.852729	−	0.522354i	$-0.825054\pi$
$617$	− 644.584i	− 1.04471i	0.852729	−	0.522354i	$-0.174946\pi$
$618$	0	0
$619$	−1128.94	−1.82381	−0.911905	−	0.410401i	$-0.865389\pi$
$619$	−1128.94	−1.82381	−0.911905	+	0.410401i	$0.865389\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	97.6240i	0.156700i
$624$	0	0
$625$	−293.020	−0.468833
$626$	0	0
$627$	0	0
$628$	0	0
$629$	26.4544i	0.0420579i
$630$	0	0
$631$	−143.212	−0.226961	−0.113480	−	0.993540i	$-0.536200\pi$
$631$	−143.212	−0.226961	−0.113480	+	0.993540i	$0.536200\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	722.084i	1.13714i
$636$	0	0
$637$	−321.171	−0.504194
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 229.114i	− 0.357432i	−0.983901	−	0.178716i	$-0.942806\pi$
$641$	− 229.114i	− 0.357432i	0.983901	−	0.178716i	$-0.0571943\pi$
$642$	0	0
$643$	1109.36	1.72529	0.862646	−	0.505807i	$-0.168805\pi$
$643$	1109.36	1.72529	0.862646	+	0.505807i	$0.168805\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1052.57i	− 1.62685i	−0.581669	−	0.813426i	$-0.697600\pi$
$647$	− 1052.57i	− 1.62685i	0.581669	−	0.813426i	$-0.302400\pi$
$648$	0	0
$649$	−101.000	−0.155624
$650$	0	0
$651$	0	0
$652$	0	0
$653$	918.563i	1.40668i	0.710853	+	0.703341i	$0.248309\pi$
$653$	918.563i	1.40668i	−0.710853	+	0.703341i	$0.751691\pi$
$654$	0	0
$655$	415.545	0.634420
$656$	0	0
$657$	0	0
$658$	0	0
$659$	239.355i	0.363210i	0.983372	+	0.181605i	$0.0581292\pi$
$659$	239.355i	0.363210i	−0.983372	+	0.181605i	$0.941871\pi$
$660$	0	0
$661$	355.586	0.537951	0.268976	−	0.963147i	$-0.413315\pi$
$661$	355.586	0.537951	0.268976	+	0.963147i	$0.413315\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1084.75i	1.63121i
$666$	0	0
$667$	45.7673	0.0686167
$668$	0	0
$669$	0	0
$670$	0	0
$671$	673.569i	1.00383i
$672$	0	0
$673$	−582.857	−0.866058	−0.433029	−	0.901380i	$-0.642555\pi$
$673$	−582.857	−0.866058	−0.433029	+	0.901380i	$0.642555\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 178.344i	− 0.263432i	−0.991287	−	0.131716i	$-0.957951\pi$
$677$	− 178.344i	− 0.263432i	0.991287	−	0.131716i	$-0.0420487\pi$
$678$	0	0
$679$	1297.77	1.91129
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 660.510i	− 0.967072i	−0.875325	−	0.483536i	$-0.839352\pi$
$683$	− 660.510i	− 0.967072i	0.875325	−	0.483536i	$-0.160648\pi$
$684$	0	0
$685$	749.080	1.09355
$686$	0	0
$687$	0	0
$688$	0	0
$689$	281.700i	0.408853i
$690$	0	0
$691$	−86.5143	−0.125202	−0.0626008	−	0.998039i	$-0.519939\pi$
$691$	−86.5143	−0.125202	−0.0626008	+	0.998039i	$0.519939\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	372.776i	0.536369i
$696$	0	0
$697$	101.294	0.145328
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1204.11i	− 1.71770i	−0.512227	−	0.858850i	$-0.671179\pi$
$701$	− 1204.11i	− 1.71770i	0.512227	−	0.858850i	$-0.328821\pi$
$702$	0	0
$703$	224.686	0.319610
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.6881i	0.0250186i
$708$	0	0
$709$	−803.888	−1.13383	−0.566917	−	0.823775i	$-0.691864\pi$
$709$	−803.888	−1.13383	−0.566917	+	0.823775i	$0.691864\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	81.6986i	0.114584i
$714$	0	0
$715$	181.302	0.253569
$716$	0	0
$717$	0	0
$718$	0	0
$719$	66.1101i	0.0919474i	0.998943	+	0.0459737i	$0.0146390\pi$
$719$	66.1101i	0.0919474i	−0.998943	+	0.0459737i	$0.985361\pi$
$720$	0	0
$721$	−20.2939	−0.0281469
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 146.228i	− 0.201693i
$726$	0	0
$727$	−519.667	−0.714811	−0.357405	−	0.933949i	$-0.616339\pi$
$727$	−519.667	−0.714811	−0.357405	+	0.933949i	$0.616339\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	86.9395i	0.118932i
$732$	0	0
$733$	−160.292	−0.218679	−0.109340	−	0.994004i	$-0.534874\pi$
$733$	−160.292	−0.218679	−0.109340	+	0.994004i	$0.534874\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1799.24i	− 2.44130i
$738$	0	0
$739$	122.849	0.166237	0.0831184	−	0.996540i	$-0.473512\pi$
$739$	122.849	0.166237	0.0831184	+	0.996540i	$0.473512\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 676.590i	− 0.910619i	−0.890333	−	0.455309i	$-0.849529\pi$
$743$	− 676.590i	− 0.910619i	0.890333	−	0.455309i	$-0.150471\pi$
$744$	0	0
$745$	360.796	0.484290
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1321.05i	− 1.76375i
$750$	0	0
$751$	−342.859	−0.456537	−0.228268	−	0.973598i	$-0.573306\pi$
$751$	−342.859	−0.456537	−0.228268	+	0.973598i	$0.573306\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	64.4229i	0.0853283i
$756$	0	0
$757$	−452.220	−0.597385	−0.298692	−	0.954349i	$-0.596550\pi$
$757$	−452.220	−0.597385	−0.298692	+	0.954349i	$0.596550\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 235.005i	− 0.308811i	−0.988008	−	0.154405i	$-0.950654\pi$
$761$	− 235.005i	− 0.308811i	0.988008	−	0.154405i	$-0.0493462\pi$
$762$	0	0
$763$	−184.343	−0.241603
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 17.1167i	− 0.0223164i
$768$	0	0
$769$	−636.878	−0.828189	−0.414095	−	0.910234i	$-0.635902\pi$
$769$	−636.878	−0.828189	−0.414095	+	0.910234i	$0.635902\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	592.885i	0.766992i	0.923543	+	0.383496i	$0.125280\pi$
$773$	592.885i	0.766992i	−0.923543	+	0.383496i	$0.874720\pi$
$774$	0	0
$775$	261.029	0.336811
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 860.320i	− 1.10439i
$780$	0	0
$781$	873.090	1.11791
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 831.938i	− 1.05979i
$786$	0	0
$787$	1333.81	1.69480	0.847400	−	0.530954i	$-0.178166\pi$
$787$	1333.81	1.69480	0.847400	+	0.530954i	$0.178166\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1360.67i	− 1.72020i
$792$	0	0
$793$	−114.151	−0.143948
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 737.322i	− 0.925122i	−0.886587	−	0.462561i	$-0.846931\pi$
$797$	− 737.322i	− 0.925122i	0.886587	−	0.462561i	$-0.153069\pi$
$798$	0	0
$799$	−170.869	−0.213854
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1135.68i	1.41429i
$804$	0	0
$805$	150.859	0.187403
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 71.6833i	− 0.0886073i	−0.999018	−	0.0443037i	$-0.985893\pi$
$809$	− 71.6833i	− 0.0886073i	0.999018	−	0.0443037i	$-0.0141069\pi$
$810$	0	0
$811$	−1086.24	−1.33938	−0.669692	−	0.742639i	$-0.733574\pi$
$811$	−1086.24	−1.33938	−0.669692	+	0.742639i	$0.733574\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 678.158i	− 0.832096i
$816$	0	0
$817$	738.404	0.903799
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1423.66i	1.73405i	0.498262	+	0.867026i	$0.333972\pi$
$821$	1423.66i	1.73405i	−0.498262	+	0.867026i	$0.666028\pi$
$822$	0	0
$823$	434.757	0.528259	0.264129	−	0.964487i	$-0.414915\pi$
$823$	434.757	0.528259	0.264129	+	0.964487i	$0.414915\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 677.821i	− 0.819614i	−0.912172	−	0.409807i	$-0.865596\pi$
$827$	− 677.821i	− 0.819614i	0.912172	−	0.409807i	$-0.134404\pi$
$828$	0	0
$829$	−995.775	−1.20118	−0.600588	−	0.799558i	$-0.705067\pi$
$829$	−995.775	−1.20118	−0.600588	+	0.799558i	$0.705067\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 291.871i	− 0.350385i
$834$	0	0
$835$	739.604	0.885753
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1327.36i	− 1.58208i	−0.611767	−	0.791038i	$-0.709541\pi$
$839$	− 1327.36i	− 1.58208i	0.611767	−	0.791038i	$-0.290459\pi$
$840$	0	0
$841$	608.788	0.723886
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 632.566i	− 0.748599i
$846$	0	0
$847$	1939.86	2.29027
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 31.2476i	− 0.0367187i
$852$	0	0
$853$	−1370.63	−1.60684	−0.803420	−	0.595413i	$-0.796989\pi$
$853$	−1370.63	−1.60684	−0.803420	+	0.595413i	$0.796989\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1199.43i	− 1.39957i	−0.714352	−	0.699787i	$-0.753278\pi$
$857$	− 1199.43i	− 1.39957i	0.714352	−	0.699787i	$-0.246722\pi$
$858$	0	0
$859$	−62.2327	−0.0724478	−0.0362239	−	0.999344i	$-0.511533\pi$
$859$	−62.2327	−0.0724478	−0.0362239	+	0.999344i	$0.511533\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	154.565i	0.179102i	0.995982	+	0.0895509i	$0.0285432\pi$
$863$	154.565i	0.179102i	−0.995982	+	0.0895509i	$0.971457\pi$
$864$	0	0
$865$	−236.555	−0.273474
$866$	0	0
$867$	0	0
$868$	0	0
$869$	422.417i	0.486095i
$870$	0	0
$871$	304.920	0.350081
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1737.73i	− 1.98598i
$876$	0	0
$877$	−1096.37	−1.25014	−0.625070	−	0.780568i	$-0.714930\pi$
$877$	−1096.37	−1.25014	−0.625070	+	0.780568i	$0.714930\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1241.22i	− 1.40888i	−0.709765	−	0.704438i	$-0.751199\pi$
$881$	− 1241.22i	− 1.40888i	0.709765	−	0.704438i	$-0.248801\pi$
$882$	0	0
$883$	−553.555	−0.626903	−0.313451	−	0.949604i	$-0.601485\pi$
$883$	−553.555	−0.626903	−0.313451	+	0.949604i	$0.601485\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 406.345i	− 0.458112i	−0.973413	−	0.229056i	$-0.926436\pi$
$887$	− 406.345i	− 0.458112i	0.973413	−	0.229056i	$-0.0735638\pi$
$888$	0	0
$889$	−2354.56	−2.64855
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1451.25i	1.62514i
$894$	0	0
$895$	285.376	0.318855
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 414.519i	− 0.461089i
$900$	0	0
$901$	−256.000	−0.284129
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 383.045i	− 0.423254i
$906$	0	0
$907$	463.424	0.510942	0.255471	−	0.966817i	$-0.417769\pi$
$907$	463.424	0.510942	0.255471	+	0.966817i	$0.417769\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1151.80i	− 1.26433i	−0.774836	−	0.632163i	$-0.782167\pi$
$911$	− 1151.80i	− 1.26433i	0.774836	−	0.632163i	$-0.217833\pi$
$912$	0	0
$913$	992.959	1.08758
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1355.00i	1.47765i
$918$	0	0
$919$	−944.665	−1.02793	−0.513964	−	0.857812i	$-0.671823\pi$
$919$	−944.665	−1.02793	−0.513964	+	0.857812i	$0.671823\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	147.964i	0.160308i
$924$	0	0
$925$	−99.8367	−0.107932
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1053.90i	− 1.13444i	−0.823565	−	0.567221i	$-0.808018\pi$
$929$	− 1053.90i	− 1.13444i	0.823565	−	0.567221i	$-0.191982\pi$
$930$	0	0
$931$	−2478.95	−2.66267
$932$	0	0
$933$	0	0
$934$	0	0
$935$	164.762i	0.176216i
$936$	0	0
$937$	561.392	0.599137	0.299569	−	0.954075i	$-0.403157\pi$
$937$	561.392	0.599137	0.299569	+	0.954075i	$0.403157\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	113.633i	0.120758i	0.998176	+	0.0603789i	$0.0192309\pi$
$941$	113.633i	0.120758i	−0.998176	+	0.0603789i	$0.980769\pi$
$942$	0	0
$943$	−119.647	−0.126879
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 72.3432i	− 0.0763919i	−0.999270	−	0.0381960i	$-0.987839\pi$
$947$	− 72.3432i	− 0.0763919i	0.999270	−	0.0381960i	$-0.0121611\pi$
$948$	0	0
$949$	−192.465	−0.202809
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 776.447i	− 0.814739i	−0.913263	−	0.407370i	$-0.866446\pi$
$953$	− 776.447i	− 0.814739i	0.913263	−	0.407370i	$-0.133554\pi$
$954$	0	0
$955$	1280.56	1.34090
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2442.59i	2.54702i
$960$	0	0
$961$	−221.049	−0.230020
$962$	0	0
$963$	0	0
$964$	0	0
$965$	82.5010i	0.0854932i
$966$	0	0
$967$	−22.7775	−0.0235549	−0.0117774	−	0.999931i	$-0.503749\pi$
$967$	−22.7775	−0.0235549	−0.0117774	+	0.999931i	$0.503749\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1016.98i	− 1.04735i	−0.851919	−	0.523674i	$-0.824561\pi$
$971$	− 1016.98i	− 1.04735i	0.851919	−	0.523674i	$-0.175439\pi$
$972$	0	0
$973$	−1215.54	−1.24928
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1126.76i	− 1.15328i	−0.816997	−	0.576642i	$-0.804363\pi$
$977$	− 1126.76i	− 1.15328i	0.816997	−	0.576642i	$-0.195637\pi$
$978$	0	0
$979$	125.939	0.128640
$980$	0	0
$981$	0	0
$982$	0	0
$983$	323.438i	0.329031i	0.986374	+	0.164516i	$0.0526061\pi$
$983$	323.438i	0.329031i	−0.986374	+	0.164516i	$0.947394\pi$
$984$	0	0
$985$	−127.718	−0.129663
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 102.692i	− 0.103834i
$990$	0	0
$991$	−1961.47	−1.97929	−0.989644	−	0.143547i	$-0.954149\pi$
$991$	−1961.47	−1.97929	−0.989644	+	0.143547i	$0.954149\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	243.338i	0.244561i
$996$	0	0
$997$	1119.91	1.12328	0.561639	−	0.827382i	$-0.310171\pi$
$997$	1119.91	1.12328	0.561639	+	0.827382i	$0.310171\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.e.c.161.3		4
3.2	odd	2	inner	1296.3.e.c.161.2		4
4.3	odd	2		648.3.e.b.161.3		4
9.2	odd	6		432.3.q.c.17.2		4
9.4	even	3		432.3.q.c.305.2		4
9.5	odd	6		144.3.q.d.65.1		4
9.7	even	3		144.3.q.d.113.1		4
12.11	even	2		648.3.e.b.161.2		4
36.7	odd	6		72.3.m.a.41.1	✓	4
36.11	even	6		216.3.m.a.17.2		4
36.23	even	6		72.3.m.a.65.1	yes	4
36.31	odd	6		216.3.m.a.89.2		4
72.5	odd	6		576.3.q.c.65.2		4
72.11	even	6		1728.3.q.e.449.1		4
72.13	even	6		1728.3.q.f.1601.1		4
72.29	odd	6		1728.3.q.f.449.1		4
72.43	odd	6		576.3.q.h.257.2		4
72.59	even	6		576.3.q.h.65.2		4
72.61	even	6		576.3.q.c.257.2		4
72.67	odd	6		1728.3.q.e.1601.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.m.a.41.1	✓	4	36.7	odd	6
72.3.m.a.65.1	yes	4	36.23	even	6
144.3.q.d.65.1		4	9.5	odd	6
144.3.q.d.113.1		4	9.7	even	3
216.3.m.a.17.2		4	36.11	even	6
216.3.m.a.89.2		4	36.31	odd	6
432.3.q.c.17.2		4	9.2	odd	6
432.3.q.c.305.2		4	9.4	even	3
576.3.q.c.65.2		4	72.5	odd	6
576.3.q.c.257.2		4	72.61	even	6
576.3.q.h.65.2		4	72.59	even	6
576.3.q.h.257.2		4	72.43	odd	6
648.3.e.b.161.2		4	12.11	even	2
648.3.e.b.161.3		4	4.3	odd	2
1296.3.e.c.161.2		4	3.2	odd	2	inner
1296.3.e.c.161.3		4	1.1	even	1	trivial
1728.3.q.e.449.1		4	72.11	even	6
1728.3.q.e.1601.1		4	72.67	odd	6
1728.3.q.f.449.1		4	72.29	odd	6
1728.3.q.f.1601.1		4	72.13	even	6

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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.92480i q^{5} -12.7980 q^{7} +O(q^{10})\)	\(q+3.92480i q^{5} -12.7980 q^{7} +16.5099i q^{11} -2.79796 q^{13} -2.54270i q^{17} -21.5959 q^{19} +3.00340i q^{23} +9.59592 q^{25} -15.2385i q^{29} +27.2020 q^{31} -50.2295i q^{35} -10.4041 q^{37} +39.8372i q^{41} -34.1918 q^{43} -67.2000i q^{47} +114.788 q^{49} -100.680i q^{53} -64.7980 q^{55} +6.11756i q^{59} +40.7980 q^{61} -10.9814i q^{65} -108.980 q^{67} -52.8829i q^{71} +68.7878 q^{73} -211.293i q^{77} +25.5857 q^{79} -60.1434i q^{83} +9.97959 q^{85} -7.62809i q^{89} +35.8082 q^{91} -84.7597i q^{95} -101.404 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{7}+O(q^{10}) \) 4 * q - 12 * q^7	\( 4 q - 12 q^{7} + 28 q^{13} - 8 q^{19} - 40 q^{25} + 148 q^{31} - 120 q^{37} + 20 q^{43} + 224 q^{49} - 220 q^{55} + 124 q^{61} - 44 q^{67} + 40 q^{73} - 172 q^{79} - 352 q^{85} + 300 q^{91} - 484 q^{97}+O(q^{100}) \) 4 * q - 12 * q^7 + 28 * q^13 - 8 * q^19 - 40 * q^25 + 148 * q^31 - 120 * q^37 + 20 * q^43 + 224 * q^49 - 220 * q^55 + 124 * q^61 - 44 * q^67 + 40 * q^73 - 172 * q^79 - 352 * q^85 + 300 * q^91 - 484 * q^97

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