LMFDB - Embedded newform 1296.3.e.i.161.7

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:	$8$
Coefficient field:	8.0.19269881856.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144 $ x^8 - 2x^7 + 15x^6 - 2x^5 + 133x^4 - 84x^3 + 276x^2 + 144*x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.7
Root			$0.831167 + 1.43962i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.3.e.i.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.97562i q^{5} -3.60938 q^{7} +O(q^{10})$	$q+3.97562i q^{5} -3.60938 q^{7} +13.6631i q^{11} +17.9287 q^{13} -21.6750i q^{17} +23.4245 q^{19} -15.0875i q^{23} +9.19442 q^{25} +23.6554i q^{29} -47.0033 q^{31} -14.3495i q^{35} +54.6126 q^{37} +28.4586i q^{41} -47.7505 q^{43} +35.2556i q^{47} -35.9724 q^{49} +65.0193i q^{53} -54.3192 q^{55} +87.7752i q^{59} +12.9370 q^{61} +71.2778i q^{65} +3.11190 q^{67} +49.5089i q^{71} +102.151 q^{73} -49.3152i q^{77} -37.4441 q^{79} +16.9740i q^{83} +86.1718 q^{85} -14.1990i q^{89} -64.7115 q^{91} +93.1271i q^{95} +48.2968 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{7}+O(q^{10}) $ 8 * q + 12 * q^7	$ 8 q + 12 q^{7} - 28 q^{13} - 4 q^{19} - 20 q^{25} - 100 q^{31} + 120 q^{37} - 56 q^{43} - 188 q^{49} + 244 q^{55} - 28 q^{61} - 40 q^{67} - 76 q^{73} + 52 q^{79} + 448 q^{85} - 108 q^{91} + 472 q^{97}+O(q^{100}) $ 8 * q + 12 * q^7 - 28 * q^13 - 4 * q^19 - 20 * q^25 - 100 * q^31 + 120 * q^37 - 56 * q^43 - 188 * q^49 + 244 * q^55 - 28 * q^61 - 40 * q^67 - 76 * q^73 + 52 * q^79 + 448 * q^85 - 108 * q^91 + 472 * 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
$5$	3.97562i	0.795125i	0.917575	+	0.397562i	$0.130144\pi$
$5$	3.97562i	0.795125i	−0.917575	+	0.397562i	$0.869856\pi$
$6$	0	0
$7$	−3.60938	−0.515626	−0.257813	−	0.966195i	$-0.583002\pi$
$7$	−3.60938	−0.515626	−0.257813	+	0.966195i	$0.583002\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	13.6631i	1.24210i	0.783773	+	0.621048i	$0.213293\pi$
$11$	13.6631i	1.24210i	−0.783773	+	0.621048i	$0.786707\pi$
$12$	0	0
$13$	17.9287	1.37913	0.689565	−	0.724223i	$-0.257802\pi$
$13$	17.9287	1.37913	0.689565	+	0.724223i	$0.257802\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 21.6750i	− 1.27500i	−0.770449	−	0.637501i	$-0.779968\pi$
$17$	− 21.6750i	− 1.27500i	0.770449	−	0.637501i	$-0.220032\pi$
$18$	0	0
$19$	23.4245	1.23287	0.616435	−	0.787406i	$-0.288576\pi$
$19$	23.4245	1.23287	0.616435	+	0.787406i	$0.288576\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 15.0875i	− 0.655979i	−0.944681	−	0.327990i	$-0.893629\pi$
$23$	− 15.0875i	− 0.655979i	0.944681	−	0.327990i	$-0.106371\pi$
$24$	0	0
$25$	9.19442	0.367777
$26$	0	0
$27$	0	0
$28$	0	0
$29$	23.6554i	0.815703i	0.913048	+	0.407852i	$0.133722\pi$
$29$	23.6554i	0.815703i	−0.913048	+	0.407852i	$0.866278\pi$
$30$	0	0
$31$	−47.0033	−1.51623	−0.758117	−	0.652118i	$-0.773880\pi$
$31$	−47.0033	−1.51623	−0.758117	+	0.652118i	$0.773880\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 14.3495i	− 0.409987i
$36$	0	0
$37$	54.6126	1.47602	0.738009	−	0.674791i	$-0.235766\pi$
$37$	54.6126	1.47602	0.738009	+	0.674791i	$0.235766\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	28.4586i	0.694112i	0.937844	+	0.347056i	$0.112819\pi$
$41$	28.4586i	0.694112i	−0.937844	+	0.347056i	$0.887181\pi$
$42$	0	0
$43$	−47.7505	−1.11048	−0.555239	−	0.831691i	$-0.687373\pi$
$43$	−47.7505	−1.11048	−0.555239	+	0.831691i	$0.687373\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	35.2556i	0.750120i	0.927001	+	0.375060i	$0.122378\pi$
$47$	35.2556i	0.750120i	−0.927001	+	0.375060i	$0.877622\pi$
$48$	0	0
$49$	−35.9724	−0.734130
$50$	0	0
$51$	0	0
$52$	0	0
$53$	65.0193i	1.22678i	0.789780	+	0.613390i	$0.210195\pi$
$53$	65.0193i	1.22678i	−0.789780	+	0.613390i	$0.789805\pi$
$54$	0	0
$55$	−54.3192	−0.987621
$56$	0	0
$57$	0	0
$58$	0	0
$59$	87.7752i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$59$	87.7752i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$60$	0	0
$61$	12.9370	0.212083	0.106041	−	0.994362i	$-0.466182\pi$
$61$	12.9370	0.212083	0.106041	+	0.994362i	$0.466182\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	71.2778i	1.09658i
$66$	0	0
$67$	3.11190	0.0464463	0.0232232	−	0.999730i	$-0.492607\pi$
$67$	3.11190	0.0464463	0.0232232	+	0.999730i	$0.492607\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	49.5089i	0.697308i	0.937251	+	0.348654i	$0.113361\pi$
$71$	49.5089i	0.697308i	−0.937251	+	0.348654i	$0.886639\pi$
$72$	0	0
$73$	102.151	1.39932	0.699662	−	0.714474i	$-0.253334\pi$
$73$	102.151	1.39932	0.699662	+	0.714474i	$0.253334\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 49.3152i	− 0.640457i
$78$	0	0
$79$	−37.4441	−0.473976	−0.236988	−	0.971513i	$-0.576160\pi$
$79$	−37.4441	−0.473976	−0.236988	+	0.971513i	$0.576160\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.9740i	0.204506i	0.994758	+	0.102253i	$0.0326052\pi$
$83$	16.9740i	0.204506i	−0.994758	+	0.102253i	$0.967395\pi$
$84$	0	0
$85$	86.1718	1.01379
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 14.1990i	− 0.159540i	−0.996813	−	0.0797698i	$-0.974581\pi$
$89$	− 14.1990i	− 0.159540i	0.996813	−	0.0797698i	$-0.0254185\pi$
$90$	0	0
$91$	−64.7115	−0.711116
$92$	0	0
$93$	0	0
$94$	0	0
$95$	93.1271i	0.980286i
$96$	0	0
$97$	48.2968	0.497905	0.248952	−	0.968516i	$-0.419914\pi$
$97$	48.2968	0.497905	0.248952	+	0.968516i	$0.419914\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	39.9503i	0.395548i	0.980248	+	0.197774i	$0.0633712\pi$
$101$	39.9503i	0.395548i	−0.980248	+	0.197774i	$0.936629\pi$
$102$	0	0
$103$	21.9498	0.213105	0.106552	−	0.994307i	$-0.466019\pi$
$103$	21.9498	0.213105	0.106552	+	0.994307i	$0.466019\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.521299i	− 0.00487195i	−0.999997	−	0.00243598i	$-0.999225\pi$
$107$	− 0.521299i	− 0.00487195i	0.999997	−	0.00243598i	$-0.000775396\pi$
$108$	0	0
$109$	−198.784	−1.82371	−0.911855	−	0.410513i	$-0.865350\pi$
$109$	−198.784	−1.82371	−0.911855	+	0.410513i	$0.865350\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	133.911i	1.18506i	0.805550	+	0.592528i	$0.201870\pi$
$113$	133.911i	1.18506i	−0.805550	+	0.592528i	$0.798130\pi$
$114$	0	0
$115$	59.9823	0.521585
$116$	0	0
$117$	0	0
$118$	0	0
$119$	78.2335i	0.657425i
$120$	0	0
$121$	−65.6790	−0.542802
$122$	0	0
$123$	0	0
$124$	0	0
$125$	135.944i	1.08755i
$126$	0	0
$127$	−45.8769	−0.361235	−0.180618	−	0.983553i	$-0.557810\pi$
$127$	−45.8769	−0.361235	−0.180618	+	0.983553i	$0.557810\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 193.891i	− 1.48008i	−0.672563	−	0.740040i	$-0.734806\pi$
$131$	− 193.891i	− 1.48008i	0.672563	−	0.740040i	$-0.265194\pi$
$132$	0	0
$133$	−84.5481	−0.635700
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 172.972i	− 1.26257i	−0.775552	−	0.631284i	$-0.782528\pi$
$137$	− 172.972i	− 1.26257i	0.775552	−	0.631284i	$-0.217472\pi$
$138$	0	0
$139$	1.60726	0.0115630	0.00578151	−	0.999983i	$-0.498160\pi$
$139$	1.60726	0.0115630	0.00578151	+	0.999983i	$0.498160\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	244.961i	1.71301i
$144$	0	0
$145$	−94.0450	−0.648586
$146$	0	0
$147$	0	0
$148$	0	0
$149$	87.8881i	0.589853i	0.955520	+	0.294927i	$0.0952952\pi$
$149$	87.8881i	0.589853i	−0.955520	+	0.294927i	$0.904705\pi$
$150$	0	0
$151$	13.2824	0.0879626	0.0439813	−	0.999032i	$-0.485996\pi$
$151$	13.2824	0.0879626	0.0439813	+	0.999032i	$0.485996\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 186.867i	− 1.20560i
$156$	0	0
$157$	187.785	1.19608	0.598042	−	0.801465i	$-0.295945\pi$
$157$	187.785	1.19608	0.598042	+	0.801465i	$0.295945\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	54.4566i	0.338240i
$162$	0	0
$163$	131.235	0.805123	0.402562	−	0.915393i	$-0.368120\pi$
$163$	131.235	0.805123	0.402562	+	0.915393i	$0.368120\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	183.177i	1.09687i	0.836193	+	0.548436i	$0.184776\pi$
$167$	183.177i	1.09687i	−0.836193	+	0.548436i	$0.815224\pi$
$168$	0	0
$169$	152.438	0.902002
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 26.1805i	− 0.151332i	−0.997133	−	0.0756662i	$-0.975892\pi$
$173$	− 26.1805i	− 0.151332i	0.997133	−	0.0756662i	$-0.0241083\pi$
$174$	0	0
$175$	−33.1862	−0.189635
$176$	0	0
$177$	0	0
$178$	0	0
$179$	175.092i	0.978165i	0.872238	+	0.489083i	$0.162668\pi$
$179$	175.092i	0.978165i	−0.872238	+	0.489083i	$0.837332\pi$
$180$	0	0
$181$	−172.814	−0.954772	−0.477386	−	0.878694i	$-0.658416\pi$
$181$	−172.814	−0.954772	−0.477386	+	0.878694i	$0.658416\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	217.119i	1.17362i
$186$	0	0
$187$	296.147	1.58368
$188$	0	0
$189$	0	0
$190$	0	0
$191$	87.8840i	0.460126i	0.973176	+	0.230063i	$0.0738932\pi$
$191$	87.8840i	0.460126i	−0.973176	+	0.230063i	$0.926107\pi$
$192$	0	0
$193$	−137.275	−0.711272	−0.355636	−	0.934625i	$-0.615736\pi$
$193$	−137.275	−0.711272	−0.355636	+	0.934625i	$0.615736\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	239.073i	1.21357i	0.794867	+	0.606784i	$0.207541\pi$
$197$	239.073i	1.21357i	−0.794867	+	0.606784i	$0.792459\pi$
$198$	0	0
$199$	250.574	1.25916	0.629582	−	0.776934i	$-0.283226\pi$
$199$	250.574	1.25916	0.629582	+	0.776934i	$0.283226\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 85.3814i	− 0.420598i
$204$	0	0
$205$	−113.141	−0.551905
$206$	0	0
$207$	0	0
$208$	0	0
$209$	320.051i	1.53134i
$210$	0	0
$211$	236.086	1.11889	0.559445	−	0.828867i	$-0.311014\pi$
$211$	236.086	1.11889	0.559445	+	0.828867i	$0.311014\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 189.838i	− 0.882968i
$216$	0	0
$217$	169.653	0.781810
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 388.605i	− 1.75840i
$222$	0	0
$223$	85.7484	0.384522	0.192261	−	0.981344i	$-0.438418\pi$
$223$	85.7484	0.384522	0.192261	+	0.981344i	$0.438418\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	184.968i	0.814838i	0.913241	+	0.407419i	$0.133571\pi$
$227$	184.968i	0.814838i	−0.913241	+	0.407419i	$0.866429\pi$
$228$	0	0
$229$	−256.398	−1.11964	−0.559822	−	0.828613i	$-0.689130\pi$
$229$	−256.398	−1.11964	−0.559822	+	0.828613i	$0.689130\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 403.712i	− 1.73267i	−0.499464	−	0.866335i	$-0.666470\pi$
$233$	− 403.712i	− 1.73267i	0.499464	−	0.866335i	$-0.333530\pi$
$234$	0	0
$235$	−140.163	−0.596439
$236$	0	0
$237$	0	0
$238$	0	0
$239$	249.102i	1.04227i	0.853475	+	0.521133i	$0.174491\pi$
$239$	249.102i	1.04227i	−0.853475	+	0.521133i	$0.825509\pi$
$240$	0	0
$241$	−171.405	−0.711222	−0.355611	−	0.934634i	$-0.615727\pi$
$241$	−171.405	−0.711222	−0.355611	+	0.934634i	$0.615727\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 143.013i	− 0.583725i
$246$	0	0
$247$	419.971	1.70029
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 287.980i	− 1.14733i	−0.819089	−	0.573666i	$-0.805521\pi$
$251$	− 287.980i	− 1.14733i	0.819089	−	0.573666i	$-0.194479\pi$
$252$	0	0
$253$	206.142	0.814789
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 69.4285i	− 0.270150i	−0.990835	−	0.135075i	$-0.956873\pi$
$257$	− 69.4285i	− 0.270150i	0.990835	−	0.135075i	$-0.0431275\pi$
$258$	0	0
$259$	−197.118	−0.761073
$260$	0	0
$261$	0	0
$262$	0	0
$263$	79.3691i	0.301784i	0.988550	+	0.150892i	$0.0482145\pi$
$263$	79.3691i	0.301784i	−0.988550	+	0.150892i	$0.951785\pi$
$264$	0	0
$265$	−258.492	−0.975443
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 41.6217i	− 0.154728i	−0.997003	−	0.0773638i	$-0.975350\pi$
$269$	− 41.6217i	− 0.154728i	0.997003	−	0.0773638i	$-0.0246503\pi$
$270$	0	0
$271$	−98.0065	−0.361648	−0.180824	−	0.983516i	$-0.557876\pi$
$271$	−98.0065	−0.361648	−0.180824	+	0.983516i	$0.557876\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	125.624i	0.456814i
$276$	0	0
$277$	−306.800	−1.10758	−0.553790	−	0.832656i	$-0.686819\pi$
$277$	−306.800	−1.10758	−0.553790	+	0.832656i	$0.686819\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	63.3787i	0.225547i	0.993621	+	0.112774i	$0.0359735\pi$
$281$	63.3787i	0.225547i	−0.993621	+	0.112774i	$0.964027\pi$
$282$	0	0
$283$	−326.597	−1.15405	−0.577027	−	0.816725i	$-0.695787\pi$
$283$	−326.597	−1.15405	−0.577027	+	0.816725i	$0.695787\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 102.718i	− 0.357902i
$288$	0	0
$289$	−180.808	−0.625632
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 71.9517i	− 0.245569i	−0.992433	−	0.122784i	$-0.960818\pi$
$293$	− 71.9517i	− 0.245569i	0.992433	−	0.122784i	$-0.0391824\pi$
$294$	0	0
$295$	−348.961	−1.18292
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 270.500i	− 0.904681i
$300$	0	0
$301$	172.350	0.572591
$302$	0	0
$303$	0	0
$304$	0	0
$305$	51.4328i	0.168632i
$306$	0	0
$307$	161.023	0.524504	0.262252	−	0.964999i	$-0.415535\pi$
$307$	161.023	0.524504	0.262252	+	0.964999i	$0.415535\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 407.094i	− 1.30898i	−0.756069	−	0.654491i	$-0.772883\pi$
$311$	− 407.094i	− 1.30898i	0.756069	−	0.654491i	$-0.227117\pi$
$312$	0	0
$313$	−159.510	−0.509616	−0.254808	−	0.966992i	$-0.582012\pi$
$313$	−159.510	−0.509616	−0.254808	+	0.966992i	$0.582012\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	421.324i	1.32910i	0.747245	+	0.664549i	$0.231376\pi$
$317$	421.324i	1.32910i	−0.747245	+	0.664549i	$0.768624\pi$
$318$	0	0
$319$	−323.205	−1.01318
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 507.728i	− 1.57191i
$324$	0	0
$325$	164.844	0.507212
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 127.251i	− 0.386782i
$330$	0	0
$331$	−129.943	−0.392577	−0.196289	−	0.980546i	$-0.562889\pi$
$331$	−129.943	−0.392577	−0.196289	+	0.980546i	$0.562889\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.3718i	0.0369306i
$336$	0	0
$337$	−107.201	−0.318104	−0.159052	−	0.987270i	$-0.550844\pi$
$337$	−107.201	−0.318104	−0.159052	+	0.987270i	$0.550844\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 642.208i	− 1.88331i
$342$	0	0
$343$	306.698	0.894163
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 401.642i	− 1.15747i	−0.815516	−	0.578735i	$-0.803546\pi$
$347$	− 401.642i	− 1.15747i	0.815516	−	0.578735i	$-0.196454\pi$
$348$	0	0
$349$	437.457	1.25346	0.626730	−	0.779237i	$-0.284393\pi$
$349$	437.457	1.25346	0.626730	+	0.779237i	$0.284393\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 350.612i	− 0.993236i	−0.867969	−	0.496618i	$-0.834575\pi$
$353$	− 350.612i	− 0.993236i	0.867969	−	0.496618i	$-0.165425\pi$
$354$	0	0
$355$	−196.829	−0.554447
$356$	0	0
$357$	0	0
$358$	0	0
$359$	529.816i	1.47581i	0.674904	+	0.737906i	$0.264185\pi$
$359$	529.816i	1.47581i	−0.674904	+	0.737906i	$0.735815\pi$
$360$	0	0
$361$	187.709	0.519969
$362$	0	0
$363$	0	0
$364$	0	0
$365$	406.113i	1.11264i
$366$	0	0
$367$	723.943	1.97260	0.986298	−	0.164973i	$-0.0527537\pi$
$367$	723.943	1.97260	0.986298	+	0.164973i	$0.0527537\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 234.680i	− 0.632560i
$372$	0	0
$373$	59.2010	0.158716	0.0793580	−	0.996846i	$-0.474713\pi$
$373$	59.2010	0.158716	0.0793580	+	0.996846i	$0.474713\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	424.111i	1.12496i
$378$	0	0
$379$	−32.5366	−0.0858485	−0.0429243	−	0.999078i	$-0.513667\pi$
$379$	−32.5366	−0.0858485	−0.0429243	+	0.999078i	$0.513667\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	217.389i	0.567595i	0.958884	+	0.283797i	$0.0915943\pi$
$383$	217.389i	0.567595i	−0.958884	+	0.283797i	$0.908406\pi$
$384$	0	0
$385$	196.059	0.509243
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 669.394i	− 1.72081i	−0.509614	−	0.860403i	$-0.670212\pi$
$389$	− 669.394i	− 1.72081i	0.509614	−	0.860403i	$-0.329788\pi$
$390$	0	0
$391$	−327.023	−0.836375
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 148.864i	− 0.376870i
$396$	0	0
$397$	22.3088	0.0561935	0.0280967	−	0.999605i	$-0.491055\pi$
$397$	22.3088	0.0561935	0.0280967	+	0.999605i	$0.491055\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 89.7972i	− 0.223933i	−0.993712	−	0.111967i	$-0.964285\pi$
$401$	− 89.7972i	− 0.223933i	0.993712	−	0.111967i	$-0.0357149\pi$
$402$	0	0
$403$	−842.707	−2.09109
$404$	0	0
$405$	0	0
$406$	0	0
$407$	746.175i	1.83335i
$408$	0	0
$409$	616.928	1.50838	0.754191	−	0.656655i	$-0.228029\pi$
$409$	616.928	1.50838	0.754191	+	0.656655i	$0.228029\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 316.814i	− 0.767105i
$414$	0	0
$415$	−67.4824	−0.162608
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 654.138i	− 1.56119i	−0.625037	−	0.780595i	$-0.714916\pi$
$419$	− 654.138i	− 1.56119i	0.625037	−	0.780595i	$-0.285084\pi$
$420$	0	0
$421$	418.573	0.994236	0.497118	−	0.867683i	$-0.334392\pi$
$421$	418.573	0.994236	0.497118	+	0.867683i	$0.334392\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 199.289i	− 0.468916i
$426$	0	0
$427$	−46.6947	−0.109355
$428$	0	0
$429$	0	0
$430$	0	0
$431$	111.970i	0.259792i	0.991528	+	0.129896i	$0.0414644\pi$
$431$	111.970i	0.259792i	−0.991528	+	0.129896i	$0.958536\pi$
$432$	0	0
$433$	−158.770	−0.366673	−0.183337	−	0.983050i	$-0.558690\pi$
$433$	−158.770	−0.366673	−0.183337	+	0.983050i	$0.558690\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 353.418i	− 0.808737i
$438$	0	0
$439$	123.642	0.281645	0.140823	−	0.990035i	$-0.455025\pi$
$439$	123.642	0.281645	0.140823	+	0.990035i	$0.455025\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 785.654i	− 1.77349i	−0.462263	−	0.886743i	$-0.652962\pi$
$443$	− 785.654i	− 1.77349i	0.462263	−	0.886743i	$-0.347038\pi$
$444$	0	0
$445$	56.4500	0.126854
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 387.277i	− 0.862533i	−0.902225	−	0.431266i	$-0.858067\pi$
$449$	− 387.277i	− 0.862533i	0.902225	−	0.431266i	$-0.141933\pi$
$450$	0	0
$451$	−388.831	−0.862153
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 257.269i	− 0.565426i
$456$	0	0
$457$	−321.335	−0.703141	−0.351570	−	0.936161i	$-0.614352\pi$
$457$	−321.335	−0.703141	−0.351570	+	0.936161i	$0.614352\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 70.5019i	− 0.152933i	−0.997072	−	0.0764663i	$-0.975636\pi$
$461$	− 70.5019i	− 0.152933i	0.997072	−	0.0764663i	$-0.0243638\pi$
$462$	0	0
$463$	362.341	0.782594	0.391297	−	0.920265i	$-0.372027\pi$
$463$	362.341	0.782594	0.391297	+	0.920265i	$0.372027\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 478.151i	− 1.02388i	−0.859022	−	0.511939i	$-0.828927\pi$
$467$	− 478.151i	− 1.02388i	0.859022	−	0.511939i	$-0.171073\pi$
$468$	0	0
$469$	−11.2320	−0.0239489
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 652.418i	− 1.37932i
$474$	0	0
$475$	215.375	0.453421
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 913.364i	− 1.90681i	−0.301688	−	0.953407i	$-0.597550\pi$
$479$	− 913.364i	− 1.90681i	0.301688	−	0.953407i	$-0.402450\pi$
$480$	0	0
$481$	979.134	2.03562
$482$	0	0
$483$	0	0
$484$	0	0
$485$	192.010i	0.395896i
$486$	0	0
$487$	−700.870	−1.43916	−0.719579	−	0.694411i	$-0.755665\pi$
$487$	−700.870	−1.43916	−0.719579	+	0.694411i	$0.755665\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	553.848i	1.12800i	0.825775	+	0.564000i	$0.190738\pi$
$491$	553.848i	1.12800i	−0.825775	+	0.564000i	$0.809262\pi$
$492$	0	0
$493$	512.732	1.04002
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 178.696i	− 0.359550i
$498$	0	0
$499$	325.749	0.652803	0.326401	−	0.945231i	$-0.394164\pi$
$499$	325.749	0.652803	0.326401	+	0.945231i	$0.394164\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	610.492i	1.21370i	0.794816	+	0.606851i	$0.207567\pi$
$503$	610.492i	1.21370i	−0.794816	+	0.606851i	$0.792433\pi$
$504$	0	0
$505$	−158.827	−0.314510
$506$	0	0
$507$	0	0
$508$	0	0
$509$	894.466i	1.75730i	0.477466	+	0.878650i	$0.341555\pi$
$509$	894.466i	1.75730i	−0.477466	+	0.878650i	$0.658445\pi$
$510$	0	0
$511$	−368.701	−0.721528
$512$	0	0
$513$	0	0
$514$	0	0
$515$	87.2641i	0.169445i
$516$	0	0
$517$	−481.700	−0.931721
$518$	0	0
$519$	0	0
$520$	0	0
$521$	244.209i	0.468732i	0.972148	+	0.234366i	$0.0753014\pi$
$521$	244.209i	0.468732i	−0.972148	+	0.234366i	$0.924699\pi$
$522$	0	0
$523$	4.79146	0.00916150	0.00458075	−	0.999990i	$-0.498542\pi$
$523$	4.79146	0.00916150	0.00458075	+	0.999990i	$0.498542\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1018.80i	1.93320i
$528$	0	0
$529$	301.367	0.569691
$530$	0	0
$531$	0	0
$532$	0	0
$533$	510.225i	0.957271i
$534$	0	0
$535$	2.07249	0.00387381
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 491.492i	− 0.911859i
$540$	0	0
$541$	406.633	0.751633	0.375816	−	0.926694i	$-0.377362\pi$
$541$	406.633	0.751633	0.375816	+	0.926694i	$0.377362\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 790.292i	− 1.45008i
$546$	0	0
$547$	500.566	0.915111	0.457556	−	0.889181i	$-0.348725\pi$
$547$	500.566	0.915111	0.457556	+	0.889181i	$0.348725\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	554.117i	1.00566i
$552$	0	0
$553$	135.150	0.244394
$554$	0	0
$555$	0	0
$556$	0	0
$557$	549.645i	0.986796i	0.869804	+	0.493398i	$0.164245\pi$
$557$	549.645i	0.986796i	−0.869804	+	0.493398i	$0.835755\pi$
$558$	0	0
$559$	−856.105	−1.53149
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 922.634i	− 1.63878i	−0.573235	−	0.819391i	$-0.694312\pi$
$563$	− 922.634i	− 1.63878i	0.573235	−	0.819391i	$-0.305688\pi$
$564$	0	0
$565$	−532.381	−0.942267
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 325.417i	− 0.571911i	−0.958243	−	0.285955i	$-0.907689\pi$
$569$	− 325.417i	− 0.571911i	0.958243	−	0.285955i	$-0.0923109\pi$
$570$	0	0
$571$	129.697	0.227141	0.113570	−	0.993530i	$-0.463771\pi$
$571$	129.697	0.227141	0.113570	+	0.993530i	$0.463771\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 138.721i	− 0.241254i
$576$	0	0
$577$	401.374	0.695621	0.347811	−	0.937565i	$-0.386925\pi$
$577$	401.374	0.695621	0.347811	+	0.937565i	$0.386925\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 61.2658i	− 0.105449i
$582$	0	0
$583$	−888.363	−1.52378
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 750.646i	− 1.27878i	−0.768881	−	0.639392i	$-0.779186\pi$
$587$	− 750.646i	− 1.27878i	0.768881	−	0.639392i	$-0.220814\pi$
$588$	0	0
$589$	−1101.03	−1.86932
$590$	0	0
$591$	0	0
$592$	0	0
$593$	969.193i	1.63439i	0.576362	+	0.817195i	$0.304472\pi$
$593$	969.193i	1.63439i	−0.576362	+	0.817195i	$0.695528\pi$
$594$	0	0
$595$	−311.027	−0.522735
$596$	0	0
$597$	0	0
$598$	0	0
$599$	11.2802i	0.0188318i	0.999956	+	0.00941589i	$0.00299722\pi$
$599$	11.2802i	0.0188318i	−0.999956	+	0.00941589i	$0.997003\pi$
$600$	0	0
$601$	850.417	1.41500	0.707502	−	0.706711i	$-0.249822\pi$
$601$	850.417	1.41500	0.707502	+	0.706711i	$0.249822\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 261.115i	− 0.431595i
$606$	0	0
$607$	−182.417	−0.300522	−0.150261	−	0.988646i	$-0.548011\pi$
$607$	−182.417	−0.300522	−0.150261	+	0.988646i	$0.548011\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	632.088i	1.03451i
$612$	0	0
$613$	−334.345	−0.545425	−0.272712	−	0.962096i	$-0.587921\pi$
$613$	−334.345	−0.545425	−0.272712	+	0.962096i	$0.587921\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 177.986i	− 0.288470i	−0.989543	−	0.144235i	$-0.953928\pi$
$617$	− 177.986i	− 0.288470i	0.989543	−	0.144235i	$-0.0460721\pi$
$618$	0	0
$619$	1059.45	1.71155	0.855775	−	0.517349i	$-0.173081\pi$
$619$	1059.45	1.71155	0.855775	+	0.517349i	$0.173081\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	51.2497i	0.0822628i
$624$	0	0
$625$	−310.602	−0.496964
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1183.73i	− 1.88193i
$630$	0	0
$631$	950.275	1.50598	0.752991	−	0.658031i	$-0.228610\pi$
$631$	950.275	1.50598	0.752991	+	0.658031i	$0.228610\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 182.389i	− 0.287227i
$636$	0	0
$637$	−644.938	−1.01246
$638$	0	0
$639$	0	0
$640$	0	0
$641$	85.6396i	0.133603i	0.997766	+	0.0668016i	$0.0212794\pi$
$641$	85.6396i	0.133603i	−0.997766	+	0.0668016i	$0.978721\pi$
$642$	0	0
$643$	278.894	0.433739	0.216869	−	0.976201i	$-0.430415\pi$
$643$	278.894	0.433739	0.216869	+	0.976201i	$0.430415\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 654.758i	− 1.01199i	−0.862536	−	0.505995i	$-0.831125\pi$
$647$	− 654.758i	− 1.01199i	0.862536	−	0.505995i	$-0.168875\pi$
$648$	0	0
$649$	−1199.28	−1.84789
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 184.401i	− 0.282390i	−0.989982	−	0.141195i	$-0.954906\pi$
$653$	− 184.401i	− 0.282390i	0.989982	−	0.141195i	$-0.0450945\pi$
$654$	0	0
$655$	770.836	1.17685
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 191.266i	− 0.290236i	−0.989414	−	0.145118i	$-0.953644\pi$
$659$	− 191.266i	− 0.290236i	0.989414	−	0.145118i	$-0.0463562\pi$
$660$	0	0
$661$	−133.202	−0.201516	−0.100758	−	0.994911i	$-0.532127\pi$
$661$	−133.202	−0.201516	−0.100758	+	0.994911i	$0.532127\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 336.131i	− 0.505461i
$666$	0	0
$667$	356.901	0.535085
$668$	0	0
$669$	0	0
$670$	0	0
$671$	176.759i	0.263427i
$672$	0	0
$673$	−1323.04	−1.96589	−0.982943	−	0.183909i	$-0.941125\pi$
$673$	−1323.04	−1.96589	−0.982943	+	0.183909i	$0.941125\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 332.799i	− 0.491579i	−0.969323	−	0.245790i	$-0.920953\pi$
$677$	− 332.799i	− 0.491579i	0.969323	−	0.245790i	$-0.0790473\pi$
$678$	0	0
$679$	−174.321	−0.256733
$680$	0	0
$681$	0	0
$682$	0	0
$683$	190.785i	0.279333i	0.990199	+	0.139667i	$0.0446031\pi$
$683$	190.785i	0.279333i	−0.990199	+	0.139667i	$0.955397\pi$
$684$	0	0
$685$	687.671	1.00390
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1165.71i	1.69189i
$690$	0	0
$691$	−720.269	−1.04236	−0.521179	−	0.853447i	$-0.674508\pi$
$691$	−720.269	−1.04236	−0.521179	+	0.853447i	$0.674508\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.38986i	0.00919405i
$696$	0	0
$697$	616.841	0.884994
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 145.736i	− 0.207897i	−0.994583	−	0.103949i	$-0.966852\pi$
$701$	− 145.736i	− 0.207897i	0.994583	−	0.103949i	$-0.0331477\pi$
$702$	0	0
$703$	1279.28	1.81974
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 144.196i	− 0.203955i
$708$	0	0
$709$	788.860	1.11264	0.556319	−	0.830969i	$-0.312213\pi$
$709$	788.860	1.11264	0.556319	+	0.830969i	$0.312213\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	709.163i	0.994618i
$714$	0	0
$715$	−973.872	−1.36206
$716$	0	0
$717$	0	0
$718$	0	0
$719$	721.671i	1.00372i	0.864950	+	0.501858i	$0.167350\pi$
$719$	721.671i	1.00372i	−0.864950	+	0.501858i	$0.832650\pi$
$720$	0	0
$721$	−79.2252	−0.109882
$722$	0	0
$723$	0	0
$724$	0	0
$725$	217.498i	0.299997i
$726$	0	0
$727$	−1086.87	−1.49500	−0.747500	−	0.664262i	$-0.768746\pi$
$727$	−1086.87	−1.49500	−0.747500	+	0.664262i	$0.768746\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1035.00i	1.41586i
$732$	0	0
$733$	705.505	0.962490	0.481245	−	0.876586i	$-0.340185\pi$
$733$	705.505	0.962490	0.481245	+	0.876586i	$0.340185\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	42.5181i	0.0576908i
$738$	0	0
$739$	1183.74	1.60182	0.800909	−	0.598786i	$-0.204350\pi$
$739$	1183.74	1.60182	0.800909	+	0.598786i	$0.204350\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 216.769i	− 0.291748i	−0.989303	−	0.145874i	$-0.953401\pi$
$743$	− 216.769i	− 0.291748i	0.989303	−	0.145874i	$-0.0465994\pi$
$744$	0	0
$745$	−349.410	−0.469007
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.88157i	0.00251211i
$750$	0	0
$751$	153.409	0.204273	0.102137	−	0.994770i	$-0.467432\pi$
$751$	153.409	0.204273	0.102137	+	0.994770i	$0.467432\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	52.8056i	0.0699412i
$756$	0	0
$757$	−585.893	−0.773967	−0.386983	−	0.922087i	$-0.626483\pi$
$757$	−585.893	−0.773967	−0.386983	+	0.922087i	$0.626483\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 335.513i	− 0.440884i	−0.975400	−	0.220442i	$-0.929250\pi$
$761$	− 335.513i	− 0.440884i	0.975400	−	0.220442i	$-0.0707500\pi$
$762$	0	0
$763$	717.489	0.940352
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1573.70i	2.05175i
$768$	0	0
$769$	−1227.59	−1.59635	−0.798175	−	0.602425i	$-0.794201\pi$
$769$	−1227.59	−1.59635	−0.798175	+	0.602425i	$0.794201\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 512.944i	− 0.663576i	−0.943354	−	0.331788i	$-0.892348\pi$
$773$	− 512.944i	− 0.663576i	0.943354	−	0.331788i	$-0.107652\pi$
$774$	0	0
$775$	−432.168	−0.557636
$776$	0	0
$777$	0	0
$778$	0	0
$779$	666.629i	0.855750i
$780$	0	0
$781$	−676.442	−0.866123
$782$	0	0
$783$	0	0
$784$	0	0
$785$	746.564i	0.951036i
$786$	0	0
$787$	568.774	0.722711	0.361356	−	0.932428i	$-0.382314\pi$
$787$	568.774	0.722711	0.361356	+	0.932428i	$0.382314\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 483.337i	− 0.611046i
$792$	0	0
$793$	231.944	0.292490
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1185.64i	1.48763i	0.668388	+	0.743813i	$0.266985\pi$
$797$	1185.64i	1.48763i	−0.668388	+	0.743813i	$0.733015\pi$
$798$	0	0
$799$	764.168	0.956405
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1395.69i	1.73809i
$804$	0	0
$805$	−216.499	−0.268943
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 185.707i	− 0.229552i	−0.993391	−	0.114776i	$-0.963385\pi$
$809$	− 185.707i	− 0.229552i	0.993391	−	0.114776i	$-0.0366150\pi$
$810$	0	0
$811$	−451.817	−0.557111	−0.278555	−	0.960420i	$-0.589856\pi$
$811$	−451.817	−0.557111	−0.278555	+	0.960420i	$0.589856\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	521.741i	0.640173i
$816$	0	0
$817$	−1118.53	−1.36907
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1188.96i	1.44818i	0.689705	+	0.724090i	$0.257740\pi$
$821$	1188.96i	1.44818i	−0.689705	+	0.724090i	$0.742260\pi$
$822$	0	0
$823$	659.664	0.801536	0.400768	−	0.916179i	$-0.368743\pi$
$823$	659.664	0.801536	0.400768	+	0.916179i	$0.368743\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	291.986i	0.353067i	0.984295	+	0.176533i	$0.0564883\pi$
$827$	291.986i	0.353067i	−0.984295	+	0.176533i	$0.943512\pi$
$828$	0	0
$829$	−452.809	−0.546212	−0.273106	−	0.961984i	$-0.588051\pi$
$829$	−452.809	−0.546212	−0.273106	+	0.961984i	$0.588051\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	779.702i	0.936017i
$834$	0	0
$835$	−728.245	−0.872149
$836$	0	0
$837$	0	0
$838$	0	0
$839$	72.4111i	0.0863064i	0.999068	+	0.0431532i	$0.0137404\pi$
$839$	72.4111i	0.0863064i	−0.999068	+	0.0431532i	$0.986260\pi$
$840$	0	0
$841$	281.422	0.334628
$842$	0	0
$843$	0	0
$844$	0	0
$845$	606.037i	0.717204i
$846$	0	0
$847$	237.061	0.279883
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 823.969i	− 0.968237i
$852$	0	0
$853$	331.617	0.388766	0.194383	−	0.980926i	$-0.437730\pi$
$853$	331.617	0.388766	0.194383	+	0.980926i	$0.437730\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	492.225i	0.574358i	0.957877	+	0.287179i	$0.0927174\pi$
$857$	492.225i	0.574358i	−0.957877	+	0.287179i	$0.907283\pi$
$858$	0	0
$859$	759.770	0.884482	0.442241	−	0.896896i	$-0.354184\pi$
$859$	759.770	0.884482	0.442241	+	0.896896i	$0.354184\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 263.506i	− 0.305337i	−0.988277	−	0.152668i	$-0.951213\pi$
$863$	− 263.506i	− 0.305337i	0.988277	−	0.152668i	$-0.0487867\pi$
$864$	0	0
$865$	104.084	0.120328
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 511.601i	− 0.588723i
$870$	0	0
$871$	55.7924	0.0640555
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 490.674i	− 0.560771i
$876$	0	0
$877$	−1415.89	−1.61447	−0.807236	−	0.590228i	$-0.799038\pi$
$877$	−1415.89	−1.61447	−0.807236	+	0.590228i	$0.799038\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	992.699i	1.12679i	0.826189	+	0.563393i	$0.190504\pi$
$881$	992.699i	1.12679i	−0.826189	+	0.563393i	$0.809496\pi$
$882$	0	0
$883$	800.265	0.906303	0.453151	−	0.891434i	$-0.350300\pi$
$883$	800.265	0.906303	0.453151	+	0.891434i	$0.350300\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	603.879i	0.680811i	0.940279	+	0.340405i	$0.110564\pi$
$887$	603.879i	0.680811i	−0.940279	+	0.340405i	$0.889436\pi$
$888$	0	0
$889$	165.587	0.186262
$890$	0	0
$891$	0	0
$892$	0	0
$893$	825.847i	0.924801i
$894$	0	0
$895$	−696.098	−0.777763
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1111.88i	− 1.23680i
$900$	0	0
$901$	1409.30	1.56415
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 687.042i	− 0.759163i
$906$	0	0
$907$	654.989	0.722149	0.361074	−	0.932537i	$-0.382410\pi$
$907$	654.989	0.722149	0.361074	+	0.932537i	$0.382410\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1703.81i	− 1.87027i	−0.354294	−	0.935134i	$-0.615279\pi$
$911$	− 1703.81i	− 1.87027i	0.354294	−	0.935134i	$-0.384721\pi$
$912$	0	0
$913$	−231.917	−0.254017
$914$	0	0
$915$	0	0
$916$	0	0
$917$	699.825i	0.763168i
$918$	0	0
$919$	412.015	0.448330	0.224165	−	0.974551i	$-0.428035\pi$
$919$	412.015	0.448330	0.224165	+	0.974551i	$0.428035\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	887.630i	0.961679i
$924$	0	0
$925$	502.131	0.542845
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 745.542i	− 0.802521i	−0.915964	−	0.401260i	$-0.868572\pi$
$929$	− 745.542i	− 0.802521i	0.915964	−	0.401260i	$-0.131428\pi$
$930$	0	0
$931$	−842.636	−0.905087
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1177.37i	1.25922i
$936$	0	0
$937$	227.643	0.242948	0.121474	−	0.992595i	$-0.461238\pi$
$937$	227.643	0.242948	0.121474	+	0.992595i	$0.461238\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 124.799i	− 0.132624i	−0.997799	−	0.0663119i	$-0.978877\pi$
$941$	− 124.799i	− 0.132624i	0.997799	−	0.0663119i	$-0.0211232\pi$
$942$	0	0
$943$	429.369	0.455323
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 587.532i	− 0.620414i	−0.950669	−	0.310207i	$-0.899602\pi$
$947$	− 587.532i	− 0.620414i	0.950669	−	0.310207i	$-0.100398\pi$
$948$	0	0
$949$	1831.43	1.92985
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 413.855i	− 0.434266i	−0.976142	−	0.217133i	$-0.930330\pi$
$953$	− 413.855i	− 0.434266i	0.976142	−	0.217133i	$-0.0696705\pi$
$954$	0	0
$955$	−349.394	−0.365857
$956$	0	0
$957$	0	0
$958$	0	0
$959$	624.322i	0.651013i
$960$	0	0
$961$	1248.31	1.29897
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 545.756i	− 0.565550i
$966$	0	0
$967$	1025.93	1.06094	0.530468	−	0.847705i	$-0.322016\pi$
$967$	1025.93	1.06094	0.530468	+	0.847705i	$0.322016\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	610.884i	0.629129i	0.949236	+	0.314564i	$0.101858\pi$
$971$	610.884i	0.629129i	−0.949236	+	0.314564i	$0.898142\pi$
$972$	0	0
$973$	−5.80122	−0.00596220
$974$	0	0
$975$	0	0
$976$	0	0
$977$	471.348i	0.482445i	0.970470	+	0.241222i	$0.0775483\pi$
$977$	471.348i	0.482445i	−0.970470	+	0.241222i	$0.922452\pi$
$978$	0	0
$979$	194.002	0.198163
$980$	0	0
$981$	0	0
$982$	0	0
$983$	956.760i	0.973306i	0.873595	+	0.486653i	$0.161782\pi$
$983$	956.760i	0.973306i	−0.873595	+	0.486653i	$0.838218\pi$
$984$	0	0
$985$	−950.464	−0.964938
$986$	0	0
$987$	0	0
$988$	0	0
$989$	720.437i	0.728450i
$990$	0	0
$991$	−1743.95	−1.75978	−0.879892	−	0.475174i	$-0.842385\pi$
$991$	−1743.95	−1.75978	−0.879892	+	0.475174i	$0.842385\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	996.187i	1.00119i
$996$	0	0
$997$	−163.729	−0.164222	−0.0821109	−	0.996623i	$-0.526166\pi$
$997$	−163.729	−0.164222	−0.0821109	+	0.996623i	$0.526166\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.e.i.161.7		8
3.2	odd	2	inner	1296.3.e.i.161.2		8
4.3	odd	2		648.3.e.c.161.7		8
9.2	odd	6		144.3.q.e.113.2		8
9.4	even	3		144.3.q.e.65.2		8
9.5	odd	6		432.3.q.e.305.1		8
9.7	even	3		432.3.q.e.17.1		8
12.11	even	2		648.3.e.c.161.2		8
36.7	odd	6		216.3.m.b.17.1		8
36.11	even	6		72.3.m.b.41.3	✓	8
36.23	even	6		216.3.m.b.89.1		8
36.31	odd	6		72.3.m.b.65.3	yes	8
72.5	odd	6		1728.3.q.i.1601.4		8
72.11	even	6		576.3.q.i.257.2		8
72.13	even	6		576.3.q.j.65.3		8
72.29	odd	6		576.3.q.j.257.3		8
72.43	odd	6		1728.3.q.j.449.4		8
72.59	even	6		1728.3.q.j.1601.4		8
72.61	even	6		1728.3.q.i.449.4		8
72.67	odd	6		576.3.q.i.65.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.m.b.41.3	✓	8	36.11	even	6
72.3.m.b.65.3	yes	8	36.31	odd	6
144.3.q.e.65.2		8	9.4	even	3
144.3.q.e.113.2		8	9.2	odd	6
216.3.m.b.17.1		8	36.7	odd	6
216.3.m.b.89.1		8	36.23	even	6
432.3.q.e.17.1		8	9.7	even	3
432.3.q.e.305.1		8	9.5	odd	6
576.3.q.i.65.2		8	72.67	odd	6
576.3.q.i.257.2		8	72.11	even	6
576.3.q.j.65.3		8	72.13	even	6
576.3.q.j.257.3		8	72.29	odd	6
648.3.e.c.161.2		8	12.11	even	2
648.3.e.c.161.7		8	4.3	odd	2
1296.3.e.i.161.2		8	3.2	odd	2	inner
1296.3.e.i.161.7		8	1.1	even	1	trivial
1728.3.q.i.449.4		8	72.61	even	6
1728.3.q.i.1601.4		8	72.5	odd	6
1728.3.q.j.449.4		8	72.43	odd	6
1728.3.q.j.1601.4		8	72.59	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)

\(f(q)\)	\(=\)	\(q+3.97562i q^{5} -3.60938 q^{7} +O(q^{10})\)	\(q+3.97562i q^{5} -3.60938 q^{7} +13.6631i q^{11} +17.9287 q^{13} -21.6750i q^{17} +23.4245 q^{19} -15.0875i q^{23} +9.19442 q^{25} +23.6554i q^{29} -47.0033 q^{31} -14.3495i q^{35} +54.6126 q^{37} +28.4586i q^{41} -47.7505 q^{43} +35.2556i q^{47} -35.9724 q^{49} +65.0193i q^{53} -54.3192 q^{55} +87.7752i q^{59} +12.9370 q^{61} +71.2778i q^{65} +3.11190 q^{67} +49.5089i q^{71} +102.151 q^{73} -49.3152i q^{77} -37.4441 q^{79} +16.9740i q^{83} +86.1718 q^{85} -14.1990i q^{89} -64.7115 q^{91} +93.1271i q^{95} +48.2968 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{7}+O(q^{10}) \) 8 * q + 12 * q^7	\( 8 q + 12 q^{7} - 28 q^{13} - 4 q^{19} - 20 q^{25} - 100 q^{31} + 120 q^{37} - 56 q^{43} - 188 q^{49} + 244 q^{55} - 28 q^{61} - 40 q^{67} - 76 q^{73} + 52 q^{79} + 448 q^{85} - 108 q^{91} + 472 q^{97}+O(q^{100}) \) 8 * q + 12 * q^7 - 28 * q^13 - 4 * q^19 - 20 * q^25 - 100 * q^31 + 120 * q^37 - 56 * q^43 - 188 * q^49 + 244 * q^55 - 28 * q^61 - 40 * q^67 - 76 * q^73 + 52 * q^79 + 448 * q^85 - 108 * q^91 + 472 * q^97

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