LMFDB - Embedded newform 1152.3.j.b.161.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(161,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("1152.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.j (of order $4$, degree $2$, 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:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			161.5
Character	$\chi$	$=$	1152.161
Dual form			1152.3.j.b.737.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.92848 - 1.92848i) q^{5} +2.43162i q^{7} +O(q^{10})$	$q+(-1.92848 - 1.92848i) q^{5} +2.43162i q^{7} +(2.79147 + 2.79147i) q^{11} +(1.54064 + 1.54064i) q^{13} +20.3746i q^{17} +(-25.0179 - 25.0179i) q^{19} -3.55894 q^{23} -17.5619i q^{25} +(23.4539 - 23.4539i) q^{29} +13.3241 q^{31} +(4.68934 - 4.68934i) q^{35} +(25.4406 - 25.4406i) q^{37} -64.0726 q^{41} +(-24.6791 + 24.6791i) q^{43} +79.5718i q^{47} +43.0872 q^{49} +(-39.8061 - 39.8061i) q^{53} -10.7666i q^{55} +(-19.2371 - 19.2371i) q^{59} +(-63.5441 - 63.5441i) q^{61} -5.94218i q^{65} +(-65.1837 - 65.1837i) q^{67} -84.6981 q^{71} -39.7473i q^{73} +(-6.78781 + 6.78781i) q^{77} -109.386 q^{79} +(-14.7433 + 14.7433i) q^{83} +(39.2920 - 39.2920i) q^{85} -32.3085 q^{89} +(-3.74625 + 3.74625i) q^{91} +96.4929i q^{95} +123.467 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 64 q^{19} + 128 q^{43} - 224 q^{49} - 64 q^{61} - 64 q^{67} + 512 q^{79} - 320 q^{85} - 192 q^{91}+O(q^{100}) $ 32 * q + 64 * q^19 + 128 * q^43 - 224 * q^49 - 64 * q^61 - 64 * q^67 + 512 * q^79 - 320 * q^85 - 192 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$5$	−1.92848	−	1.92848i	−0.385696	−	0.385696i	0.487453	−	0.873149i	$-0.337926\pi$
$5$	−1.92848	−	1.92848i	−0.385696	−	0.385696i	−0.873149	+	0.487453i	$0.837926\pi$
$6$		0			0
$7$			2.43162i			0.347375i	0.984801	+	0.173687i	$0.0555682\pi$
$7$			2.43162i			0.347375i	−0.984801	+	0.173687i	$0.944432\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.79147	+	2.79147i	0.253770	+	0.253770i	0.822514	−	0.568744i	$-0.192571\pi$
$11$	2.79147	+	2.79147i	0.253770	+	0.253770i	−0.568744	+	0.822514i	$0.692571\pi$
$12$		0			0
$13$	1.54064	+	1.54064i	0.118511	+	0.118511i	0.763875	−	0.645364i	$-0.223294\pi$
$13$	1.54064	+	1.54064i	0.118511	+	0.118511i	−0.645364	+	0.763875i	$0.723294\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			20.3746i			1.19850i	0.800561	+	0.599252i	$0.204535\pi$
$17$			20.3746i			1.19850i	−0.800561	+	0.599252i	$0.795465\pi$
$18$		0			0
$19$	−25.0179	−	25.0179i	−1.31673	−	1.31673i	−0.916349	−	0.400380i	$-0.868878\pi$
$19$	−25.0179	−	25.0179i	−1.31673	−	1.31673i	−0.400380	−	0.916349i	$-0.631122\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.55894			−0.154737			−0.0773683	−	0.997003i	$-0.524652\pi$
$23$	−3.55894			−0.154737			−0.0773683	+	0.997003i	$0.524652\pi$
$24$		0			0
$25$		−	17.5619i		−	0.702477i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	23.4539	−	23.4539i	0.808756	−	0.808756i	−0.175690	−	0.984446i	$-0.556216\pi$
$29$	23.4539	−	23.4539i	0.808756	−	0.808756i	0.984446	+	0.175690i	$0.0562156\pi$
$30$		0			0
$31$	13.3241			0.429808			0.214904	−	0.976635i	$-0.431056\pi$
$31$	13.3241			0.429808			0.214904	+	0.976635i	$0.431056\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.68934	−	4.68934i	0.133981	−	0.133981i
$36$		0			0
$37$	25.4406	−	25.4406i	0.687584	−	0.687584i	−0.274113	−	0.961697i	$-0.588384\pi$
$37$	25.4406	−	25.4406i	0.687584	−	0.687584i	0.961697	+	0.274113i	$0.0883844\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−64.0726			−1.56275			−0.781373	−	0.624064i	$-0.785480\pi$
$41$	−64.0726			−1.56275			−0.781373	+	0.624064i	$0.785480\pi$
$42$		0			0
$43$	−24.6791	+	24.6791i	−0.573932	+	0.573932i	−0.933225	−	0.359293i	$-0.883018\pi$
$43$	−24.6791	+	24.6791i	−0.573932	+	0.573932i	0.359293	+	0.933225i	$0.383018\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			79.5718i			1.69302i	0.532375	+	0.846508i	$0.321299\pi$
$47$			79.5718i			1.69302i	−0.532375	+	0.846508i	$0.678701\pi$
$48$		0			0
$49$	43.0872			0.879331
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−39.8061	−	39.8061i	−0.751058	−	0.751058i	0.223619	−	0.974677i	$-0.428213\pi$
$53$	−39.8061	−	39.8061i	−0.751058	−	0.751058i	−0.974677	+	0.223619i	$0.928213\pi$
$54$		0			0
$55$		−	10.7666i		−	0.195757i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−19.2371	−	19.2371i	−0.326053	−	0.326053i	0.525030	−	0.851083i	$-0.324054\pi$
$59$	−19.2371	−	19.2371i	−0.326053	−	0.326053i	−0.851083	+	0.525030i	$0.824054\pi$
$60$		0			0
$61$	−63.5441	−	63.5441i	−1.04171	−	1.04171i	−0.999092	−	0.0426158i	$-0.986431\pi$
$61$	−63.5441	−	63.5441i	−1.04171	−	1.04171i	−0.0426158	−	0.999092i	$-0.513569\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	5.94218i		−	0.0914182i
$66$		0			0
$67$	−65.1837	−	65.1837i	−0.972891	−	0.972891i	0.0267514	−	0.999642i	$-0.491484\pi$
$67$	−65.1837	−	65.1837i	−0.972891	−	0.972891i	−0.999642	+	0.0267514i	$0.991484\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−84.6981			−1.19293			−0.596466	−	0.802639i	$-0.703429\pi$
$71$	−84.6981			−1.19293			−0.596466	+	0.802639i	$0.703429\pi$
$72$		0			0
$73$		−	39.7473i		−	0.544483i	−0.962229	−	0.272242i	$-0.912235\pi$
$73$		−	39.7473i		−	0.544483i	0.962229	−	0.272242i	$-0.0877650\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.78781	+	6.78781i	−0.0881533	+	0.0881533i
$78$		0			0
$79$	−109.386			−1.38463			−0.692317	−	0.721593i	$-0.743410\pi$
$79$	−109.386			−1.38463			−0.692317	+	0.721593i	$0.743410\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−14.7433	+	14.7433i	−0.177630	+	0.177630i	−0.790322	−	0.612692i	$-0.790087\pi$
$83$	−14.7433	+	14.7433i	−0.177630	+	0.177630i	0.612692	+	0.790322i	$0.290087\pi$
$84$		0			0
$85$	39.2920	−	39.2920i	0.462258	−	0.462258i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−32.3085			−0.363017			−0.181508	−	0.983389i	$-0.558098\pi$
$89$	−32.3085			−0.363017			−0.181508	+	0.983389i	$0.558098\pi$
$90$		0			0
$91$	−3.74625	+	3.74625i	−0.0411676	+	0.0411676i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			96.4929i			1.01572i
$96$		0			0
$97$	123.467			1.27286			0.636428	−	0.771336i	$-0.280411\pi$
$97$	123.467			1.27286			0.636428	+	0.771336i	$0.280411\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−83.5696	−	83.5696i	−0.827422	−	0.827422i	0.159737	−	0.987160i	$-0.448935\pi$
$101$	−83.5696	−	83.5696i	−0.827422	−	0.827422i	−0.987160	+	0.159737i	$0.948935\pi$
$102$		0			0
$103$		−	38.0120i		−	0.369049i	−0.982828	−	0.184524i	$-0.940926\pi$
$103$		−	38.0120i		−	0.369049i	0.982828	−	0.184524i	$-0.0590744\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	69.3864	+	69.3864i	0.648471	+	0.648471i	0.952623	−	0.304153i	$-0.0983733\pi$
$107$	69.3864	+	69.3864i	0.648471	+	0.648471i	−0.304153	+	0.952623i	$0.598373\pi$
$108$		0			0
$109$	−46.2338	−	46.2338i	−0.424163	−	0.424163i	0.462471	−	0.886634i	$-0.346963\pi$
$109$	−46.2338	−	46.2338i	−0.424163	−	0.424163i	−0.886634	+	0.462471i	$0.846963\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	97.1603i		−	0.859826i	−0.902870	−	0.429913i	$-0.858544\pi$
$113$		−	97.1603i		−	0.859826i	0.902870	−	0.429913i	$-0.141456\pi$
$114$		0			0
$115$	6.86335	+	6.86335i	0.0596813	+	0.0596813i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−49.5432			−0.416330
$120$		0			0
$121$		−	105.415i		−	0.871201i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−82.0799	+	82.0799i	−0.656639	+	0.656639i
$126$		0			0
$127$	−187.433			−1.47585			−0.737926	−	0.674881i	$-0.764195\pi$
$127$	−187.433			−1.47585			−0.737926	+	0.674881i	$0.764195\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−79.2297	+	79.2297i	−0.604807	+	0.604807i	−0.941584	−	0.336778i	$-0.890663\pi$
$131$	−79.2297	+	79.2297i	−0.604807	+	0.604807i	0.336778	+	0.941584i	$0.390663\pi$
$132$		0			0
$133$	60.8340	−	60.8340i	0.457398	−	0.457398i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.46186			−0.0252691			−0.0126345	−	0.999920i	$-0.504022\pi$
$137$	−3.46186			−0.0252691			−0.0126345	+	0.999920i	$0.504022\pi$
$138$		0			0
$139$	−14.1054	+	14.1054i	−0.101478	+	0.101478i	−0.756023	−	0.654545i	$-0.772860\pi$
$139$	−14.1054	+	14.1054i	−0.101478	+	0.101478i	0.654545	+	0.756023i	$0.272860\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			8.60130i			0.0601489i
$144$		0			0
$145$	−90.4609			−0.623868
$146$		0			0
$147$		0			0
$148$		0			0
$149$	85.7760	+	85.7760i	0.575678	+	0.575678i	0.933709	−	0.358032i	$-0.116552\pi$
$149$	85.7760	+	85.7760i	0.575678	+	0.575678i	−0.358032	+	0.933709i	$0.616552\pi$
$150$		0			0
$151$		−	221.163i		−	1.46465i	−0.680953	−	0.732327i	$-0.738434\pi$
$151$		−	221.163i		−	1.46465i	0.680953	−	0.732327i	$-0.261566\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−25.6952	−	25.6952i	−0.165775	−	0.165775i
$156$		0			0
$157$	114.407	+	114.407i	0.728709	+	0.728709i	0.970363	−	0.241654i	$-0.0776898\pi$
$157$	114.407	+	114.407i	0.728709	+	0.728709i	−0.241654	+	0.970363i	$0.577690\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	8.65400i		−	0.0537516i
$162$		0			0
$163$	−124.266	−	124.266i	−0.762366	−	0.762366i	0.214383	−	0.976750i	$-0.431226\pi$
$163$	−124.266	−	124.266i	−0.762366	−	0.762366i	−0.976750	+	0.214383i	$0.931226\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	230.474			1.38008			0.690041	−	0.723770i	$-0.257592\pi$
$167$	230.474			1.38008			0.690041	+	0.723770i	$0.257592\pi$
$168$		0			0
$169$		−	164.253i		−	0.971910i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−43.4882	+	43.4882i	−0.251377	+	0.251377i	−0.821535	−	0.570158i	$-0.806882\pi$
$173$	−43.4882	+	43.4882i	−0.251377	+	0.251377i	0.570158	+	0.821535i	$0.306882\pi$
$174$		0			0
$175$	42.7039			0.244023
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−145.904	+	145.904i	−0.815104	+	0.815104i	−0.985394	−	0.170290i	$-0.945530\pi$
$179$	−145.904	+	145.904i	−0.815104	+	0.815104i	0.170290	+	0.985394i	$0.445530\pi$
$180$		0			0
$181$	184.775	−	184.775i	1.02085	−	1.02085i	0.0210758	−	0.999778i	$-0.493291\pi$
$181$	184.775	−	184.775i	1.02085	−	1.02085i	0.999778	−	0.0210758i	$-0.00670914\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−98.1235			−0.530397
$186$		0			0
$187$	−56.8750	+	56.8750i	−0.304145	+	0.304145i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	352.205i		−	1.84401i	−0.387182	−	0.922003i	$-0.626551\pi$
$191$		−	352.205i		−	1.84401i	0.387182	−	0.922003i	$-0.373449\pi$
$192$		0			0
$193$	−145.660			−0.754715			−0.377357	−	0.926068i	$-0.623167\pi$
$193$	−145.660			−0.754715			−0.377357	+	0.926068i	$0.623167\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.7746	−	18.7746i	−0.0953027	−	0.0953027i	0.657848	−	0.753151i	$-0.271467\pi$
$197$	−18.7746	−	18.7746i	−0.0953027	−	0.0953027i	−0.753151	+	0.657848i	$0.771467\pi$
$198$		0			0
$199$			260.124i			1.30716i	0.756858	+	0.653579i	$0.226733\pi$
$199$			260.124i			1.30716i	−0.756858	+	0.653579i	$0.773267\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	57.0310	+	57.0310i	0.280941	+	0.280941i
$204$		0			0
$205$	123.563	+	123.563i	0.602745	+	0.602745i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	139.673i		−	0.668293i
$210$		0			0
$211$	127.020	+	127.020i	0.601991	+	0.601991i	0.940840	−	0.338850i	$-0.110038\pi$
$211$	127.020	+	127.020i	0.601991	+	0.601991i	−0.338850	+	0.940840i	$0.610038\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	95.1863			0.442727
$216$		0			0
$217$			32.3991i			0.149304i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−31.3898	+	31.3898i	−0.142035	+	0.142035i
$222$		0			0
$223$	70.2119			0.314852			0.157426	−	0.987531i	$-0.449680\pi$
$223$	70.2119			0.314852			0.157426	+	0.987531i	$0.449680\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−57.6438	+	57.6438i	−0.253938	+	0.253938i	−0.822583	−	0.568645i	$-0.807468\pi$
$227$	−57.6438	+	57.6438i	−0.253938	+	0.253938i	0.568645	+	0.822583i	$0.307468\pi$
$228$		0			0
$229$	−136.406	+	136.406i	−0.595660	+	0.595660i	−0.939155	−	0.343495i	$-0.888389\pi$
$229$	−136.406	+	136.406i	−0.595660	+	0.595660i	0.343495	+	0.939155i	$0.388389\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	171.152			0.734556			0.367278	−	0.930111i	$-0.380290\pi$
$233$	171.152			0.734556			0.367278	+	0.930111i	$0.380290\pi$
$234$		0			0
$235$	153.453	−	153.453i	0.652990	−	0.652990i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			79.7138i			0.333531i	0.985997	+	0.166765i	$0.0533322\pi$
$239$			79.7138i			0.333531i	−0.985997	+	0.166765i	$0.946668\pi$
$240$		0			0
$241$	−68.5687			−0.284517			−0.142259	−	0.989830i	$-0.545436\pi$
$241$	−68.5687			−0.284517			−0.142259	+	0.989830i	$0.545436\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−83.0929	−	83.0929i	−0.339155	−	0.339155i
$246$		0			0
$247$		−	77.0869i		−	0.312093i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	11.6232	+	11.6232i	0.0463074	+	0.0463074i	0.729881	−	0.683574i	$-0.239575\pi$
$251$	11.6232	+	11.6232i	0.0463074	+	0.0463074i	−0.683574	+	0.729881i	$0.739575\pi$
$252$		0			0
$253$	−9.93469	−	9.93469i	−0.0392675	−	0.0392675i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			44.8750i			0.174611i	0.996182	+	0.0873055i	$0.0278256\pi$
$257$			44.8750i			0.174611i	−0.996182	+	0.0873055i	$0.972174\pi$
$258$		0			0
$259$	61.8619	+	61.8619i	0.238849	+	0.238849i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−384.441			−1.46175			−0.730876	−	0.682510i	$-0.760888\pi$
$263$	−384.441			−1.46175			−0.730876	+	0.682510i	$0.760888\pi$
$264$		0			0
$265$			153.531i			0.579360i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.844060	+	0.844060i	−0.00313777	+	0.00313777i	−0.708674	−	0.705536i	$-0.750706\pi$
$269$	−0.844060	+	0.844060i	−0.00313777	+	0.00313777i	0.705536	+	0.708674i	$0.250706\pi$
$270$		0			0
$271$	37.3816			0.137939			0.0689697	−	0.997619i	$-0.478029\pi$
$271$	37.3816			0.137939			0.0689697	+	0.997619i	$0.478029\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	49.0236	−	49.0236i	0.178268	−	0.178268i
$276$		0			0
$277$	−342.799	+	342.799i	−1.23754	+	1.23754i	−0.276541	+	0.961002i	$0.589188\pi$
$277$	−342.799	+	342.799i	−1.23754	+	1.23754i	−0.961002	+	0.276541i	$0.910812\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	303.846			1.08130			0.540651	−	0.841247i	$-0.318178\pi$
$281$	303.846			1.08130			0.540651	+	0.841247i	$0.318178\pi$
$282$		0			0
$283$	−10.3395	+	10.3395i	−0.0365354	+	0.0365354i	−0.725138	−	0.688603i	$-0.758224\pi$
$283$	−10.3395	+	10.3395i	−0.0365354	+	0.0365354i	0.688603	+	0.725138i	$0.258224\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	155.800i		−	0.542858i
$288$		0			0
$289$	−126.123			−0.436411
$290$		0			0
$291$		0			0
$292$		0			0
$293$	230.634	+	230.634i	0.787148	+	0.787148i	0.981026	−	0.193877i	$-0.0621064\pi$
$293$	230.634	+	230.634i	0.787148	+	0.787148i	−0.193877	+	0.981026i	$0.562106\pi$
$294$		0			0
$295$			74.1969i			0.251515i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5.48304	−	5.48304i	−0.0183379	−	0.0183379i
$300$		0			0
$301$	−60.0102	−	60.0102i	−0.199369	−	0.199369i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			245.087i			0.803565i
$306$		0			0
$307$	255.002	+	255.002i	0.830625	+	0.830625i	0.987602	−	0.156978i	$-0.0501750\pi$
$307$	255.002	+	255.002i	0.830625	+	0.830625i	−0.156978	+	0.987602i	$0.550175\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	333.987			1.07391			0.536957	−	0.843610i	$-0.319574\pi$
$311$	333.987			1.07391			0.536957	+	0.843610i	$0.319574\pi$
$312$		0			0
$313$			159.757i			0.510405i	0.966888	+	0.255203i	$0.0821421\pi$
$313$			159.757i			0.510405i	−0.966888	+	0.255203i	$0.917858\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	265.214	−	265.214i	0.836637	−	0.836637i	−0.151778	−	0.988415i	$-0.548500\pi$
$317$	265.214	−	265.214i	0.836637	−	0.836637i	0.988415	+	0.151778i	$0.0484998\pi$
$318$		0			0
$319$	130.942			0.410476
$320$		0			0
$321$		0			0
$322$		0			0
$323$	509.728	−	509.728i	1.57810	−	1.57810i
$324$		0			0
$325$	27.0566	−	27.0566i	0.0832509	−	0.0832509i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−193.488			−0.588111
$330$		0			0
$331$	−83.6950	+	83.6950i	−0.252855	+	0.252855i	−0.822140	−	0.569285i	$-0.807220\pi$
$331$	−83.6950	+	83.6950i	−0.252855	+	0.252855i	0.569285	+	0.822140i	$0.307220\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			251.411i			0.750481i
$336$		0			0
$337$	346.530			1.02828			0.514140	−	0.857706i	$-0.328111\pi$
$337$	346.530			1.02828			0.514140	+	0.857706i	$0.328111\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	37.1937	+	37.1937i	0.109073	+	0.109073i
$342$		0			0
$343$			223.921i			0.652832i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−394.994	−	394.994i	−1.13831	−	1.13831i	−0.988753	−	0.149559i	$-0.952214\pi$
$347$	−394.994	−	394.994i	−1.13831	−	1.13831i	−0.149559	−	0.988753i	$-0.547786\pi$
$348$		0			0
$349$	−100.763	−	100.763i	−0.288719	−	0.288719i	0.547854	−	0.836574i	$-0.315445\pi$
$349$	−100.763	−	100.763i	−0.288719	−	0.288719i	−0.836574	+	0.547854i	$0.815445\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	211.250i		−	0.598442i	−0.954184	−	0.299221i	$-0.903273\pi$
$353$		−	211.250i		−	0.598442i	0.954184	−	0.299221i	$-0.0967268\pi$
$354$		0			0
$355$	163.339	+	163.339i	0.460109	+	0.460109i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−180.058			−0.501555			−0.250777	−	0.968045i	$-0.580686\pi$
$359$	−180.058			−0.501555			−0.250777	+	0.968045i	$0.580686\pi$
$360$		0			0
$361$			890.786i			2.46755i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−76.6519	+	76.6519i	−0.210005	+	0.210005i
$366$		0			0
$367$	611.746			1.66688			0.833442	−	0.552608i	$-0.186367\pi$
$367$	611.746			1.66688			0.833442	+	0.552608i	$0.186367\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	96.7933	−	96.7933i	0.260898	−	0.260898i
$372$		0			0
$373$	−250.082	+	250.082i	−0.670462	+	0.670462i	−0.957823	−	0.287360i	$-0.907222\pi$
$373$	−250.082	+	250.082i	−0.670462	+	0.670462i	0.287360	+	0.957823i	$0.407222\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	72.2680			0.191692
$378$		0			0
$379$	196.949	−	196.949i	0.519655	−	0.519655i	−0.397812	−	0.917467i	$-0.630230\pi$
$379$	196.949	−	196.949i	0.519655	−	0.519655i	0.917467	+	0.397812i	$0.130230\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			27.7580i			0.0724752i	0.999343	+	0.0362376i	$0.0115373\pi$
$383$			27.7580i			0.0724752i	−0.999343	+	0.0362376i	$0.988463\pi$
$384$		0			0
$385$	26.1803			0.0680008
$386$		0			0
$387$		0			0
$388$		0			0
$389$	250.685	+	250.685i	0.644434	+	0.644434i	0.951642	−	0.307208i	$-0.0993947\pi$
$389$	250.685	+	250.685i	0.644434	+	0.644434i	−0.307208	+	0.951642i	$0.599395\pi$
$390$		0			0
$391$		−	72.5119i		−	0.185452i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	210.949	+	210.949i	0.534048	+	0.534048i
$396$		0			0
$397$	−211.973	−	211.973i	−0.533936	−	0.533936i	0.387805	−	0.921741i	$-0.373233\pi$
$397$	−211.973	−	211.973i	−0.533936	−	0.533936i	−0.921741	+	0.387805i	$0.873233\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			603.861i			1.50589i	0.658085	+	0.752944i	$0.271367\pi$
$401$			603.861i			1.50589i	−0.658085	+	0.752944i	$0.728633\pi$
$402$		0			0
$403$	20.5275	+	20.5275i	0.0509368	+	0.0509368i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	142.034			0.348977
$408$		0			0
$409$			229.079i			0.560094i	0.959986	+	0.280047i	$0.0903501\pi$
$409$			229.079i			0.560094i	−0.959986	+	0.280047i	$0.909650\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	46.7774	−	46.7774i	0.113262	−	0.113262i
$414$		0			0
$415$	56.8642			0.137022
$416$		0			0
$417$		0			0
$418$		0			0
$419$	581.441	−	581.441i	1.38769	−	1.38769i	0.557531	−	0.830156i	$-0.311749\pi$
$419$	581.441	−	581.441i	1.38769	−	1.38769i	0.830156	−	0.557531i	$-0.188251\pi$
$420$		0			0
$421$	−212.037	+	212.037i	−0.503650	+	0.503650i	−0.912570	−	0.408920i	$-0.865905\pi$
$421$	−212.037	+	212.037i	−0.503650	+	0.503650i	0.408920	+	0.912570i	$0.365905\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	357.816			0.841921
$426$		0			0
$427$	154.515	−	154.515i	0.361863	−	0.361863i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			71.0623i			0.164878i	0.996596	+	0.0824388i	$0.0262709\pi$
$431$			71.0623i			0.164878i	−0.996596	+	0.0824388i	$0.973729\pi$
$432$		0			0
$433$	−71.8456			−0.165925			−0.0829626	−	0.996553i	$-0.526438\pi$
$433$	−71.8456			−0.165925			−0.0829626	+	0.996553i	$0.526438\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	89.0371	+	89.0371i	0.203746	+	0.203746i
$438$		0			0
$439$			75.0323i			0.170916i	0.996342	+	0.0854582i	$0.0272354\pi$
$439$			75.0323i			0.170916i	−0.996342	+	0.0854582i	$0.972765\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−407.393	−	407.393i	−0.919623	−	0.919623i	0.0773783	−	0.997002i	$-0.475345\pi$
$443$	−407.393	−	407.393i	−0.919623	−	0.919623i	−0.997002	+	0.0773783i	$0.975345\pi$
$444$		0			0
$445$	62.3064	+	62.3064i	0.140014	+	0.140014i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			537.780i			1.19773i	0.800851	+	0.598864i	$0.204381\pi$
$449$			537.780i			1.19773i	−0.800851	+	0.598864i	$0.795619\pi$
$450$		0			0
$451$	−178.857	−	178.857i	−0.396578	−	0.396578i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	14.4491			0.0317564
$456$		0			0
$457$		−	670.548i		−	1.46728i	−0.679537	−	0.733641i	$-0.737820\pi$
$457$		−	670.548i		−	1.46728i	0.679537	−	0.733641i	$-0.262180\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	624.560	−	624.560i	1.35479	−	1.35479i	0.474584	−	0.880210i	$-0.342599\pi$
$461$	624.560	−	624.560i	1.35479	−	1.35479i	0.880210	−	0.474584i	$-0.157401\pi$
$462$		0			0
$463$	556.663			1.20230			0.601148	−	0.799138i	$-0.294710\pi$
$463$	556.663			1.20230			0.601148	+	0.799138i	$0.294710\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−135.903	+	135.903i	−0.291013	+	0.291013i	−0.837480	−	0.546468i	$-0.815972\pi$
$467$	−135.903	+	135.903i	−0.291013	+	0.291013i	0.546468	+	0.837480i	$0.315972\pi$
$468$		0			0
$469$	158.502	−	158.502i	0.337957	−	0.337957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−137.782			−0.291294
$474$		0			0
$475$	−439.362	+	439.362i	−0.924972	+	0.924972i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	754.590i		−	1.57534i	−0.616095	−	0.787672i	$-0.711286\pi$
$479$		−	754.590i		−	1.57534i	0.616095	−	0.787672i	$-0.288714\pi$
$480$		0			0
$481$	78.3895			0.162972
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−238.104	−	238.104i	−0.490936	−	0.490936i
$486$		0			0
$487$			752.536i			1.54525i	0.634863	+	0.772624i	$0.281056\pi$
$487$			752.536i			1.54525i	−0.634863	+	0.772624i	$0.718944\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	577.864	+	577.864i	1.17691	+	1.17691i	0.980527	+	0.196387i	$0.0629208\pi$
$491$	577.864	+	577.864i	1.17691	+	1.17691i	0.196387	+	0.980527i	$0.437079\pi$
$492$		0			0
$493$	477.863	+	477.863i	0.969297	+	0.969297i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	205.954i		−	0.414394i
$498$		0			0
$499$	46.1997	+	46.1997i	0.0925846	+	0.0925846i	0.751882	−	0.659298i	$-0.229146\pi$
$499$	46.1997	+	46.1997i	0.0925846	+	0.0925846i	−0.659298	+	0.751882i	$0.729146\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	442.969			0.880655			0.440327	−	0.897837i	$-0.354862\pi$
$503$	442.969			0.880655			0.440327	+	0.897837i	$0.354862\pi$
$504$		0			0
$505$			322.325i			0.638267i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−166.024	+	166.024i	−0.326178	+	0.326178i	−0.851131	−	0.524953i	$-0.824083\pi$
$509$	−166.024	+	166.024i	−0.326178	+	0.326178i	0.524953	+	0.851131i	$0.324083\pi$
$510$		0			0
$511$	96.6504			0.189140
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−73.3054	+	73.3054i	−0.142341	+	0.142341i
$516$		0			0
$517$	−222.122	+	222.122i	−0.429637	+	0.429637i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7.42117			−0.0142441			−0.00712204	−	0.999975i	$-0.502267\pi$
$521$	−7.42117			−0.0142441			−0.00712204	+	0.999975i	$0.502267\pi$
$522$		0			0
$523$	653.700	−	653.700i	1.24990	−	1.24990i	0.294143	−	0.955761i	$-0.404966\pi$
$523$	653.700	−	653.700i	1.24990	−	1.24990i	0.955761	−	0.294143i	$-0.0950343\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			271.472i			0.515127i
$528$		0			0
$529$	−516.334			−0.976057
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−98.7126	−	98.7126i	−0.185202	−	0.185202i
$534$		0			0
$535$		−	267.621i		−	0.500226i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	120.277	+	120.277i	0.223148	+	0.223148i
$540$		0			0
$541$	192.630	+	192.630i	0.356063	+	0.356063i	0.862360	−	0.506296i	$-0.168986\pi$
$541$	192.630	+	192.630i	0.356063	+	0.356063i	−0.506296	+	0.862360i	$0.668986\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			178.322i			0.327196i
$546$		0			0
$547$	−131.598	−	131.598i	−0.240582	−	0.240582i	0.576509	−	0.817091i	$-0.304415\pi$
$547$	−131.598	−	131.598i	−0.240582	−	0.240582i	−0.817091	+	0.576509i	$0.804415\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1173.53			−2.12982
$552$		0			0
$553$		−	265.986i		−	0.480987i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−109.222	+	109.222i	−0.196090	+	0.196090i	−0.798322	−	0.602231i	$-0.794278\pi$
$557$	−109.222	+	109.222i	−0.196090	+	0.196090i	0.602231	+	0.798322i	$0.294278\pi$
$558$		0			0
$559$	−76.0430			−0.136034
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−640.646	+	640.646i	−1.13791	+	1.13791i	−0.149090	+	0.988824i	$0.547635\pi$
$563$	−640.646	+	640.646i	−1.13791	+	1.13791i	−0.988824	+	0.149090i	$0.952365\pi$
$564$		0			0
$565$	−187.372	+	187.372i	−0.331632	+	0.331632i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−273.069			−0.479911			−0.239955	−	0.970784i	$-0.577133\pi$
$569$	−273.069			−0.479911			−0.239955	+	0.970784i	$0.577133\pi$
$570$		0			0
$571$	−663.916	+	663.916i	−1.16273	+	1.16273i	−0.178849	+	0.983877i	$0.557237\pi$
$571$	−663.916	+	663.916i	−1.16273	+	1.16273i	−0.983877	+	0.178849i	$0.942763\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			62.5018i			0.108699i
$576$		0			0
$577$	−403.711			−0.699672			−0.349836	−	0.936811i	$-0.613763\pi$
$577$	−403.711			−0.699672			−0.349836	+	0.936811i	$0.613763\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−35.8501	−	35.8501i	−0.0617041	−	0.0617041i
$582$		0			0
$583$		−	222.235i		−	0.381192i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−703.189	−	703.189i	−1.19794	−	1.19794i	−0.974783	−	0.223154i	$-0.928365\pi$
$587$	−703.189	−	703.189i	−1.19794	−	1.19794i	−0.223154	−	0.974783i	$-0.571635\pi$
$588$		0			0
$589$	−333.339	−	333.339i	−0.565941	−	0.565941i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	668.046i		−	1.12655i	−0.826268	−	0.563277i	$-0.809540\pi$
$593$		−	668.046i		−	1.12655i	0.826268	−	0.563277i	$-0.190460\pi$
$594$		0			0
$595$	95.5432	+	95.5432i	0.160577	+	0.160577i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	677.684			1.13136			0.565679	−	0.824625i	$-0.308614\pi$
$599$	677.684			1.13136			0.565679	+	0.824625i	$0.308614\pi$
$600$		0			0
$601$			170.941i			0.284427i	0.989836	+	0.142214i	$0.0454220\pi$
$601$			170.941i			0.284427i	−0.989836	+	0.142214i	$0.954578\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−203.292	+	203.292i	−0.336019	+	0.336019i
$606$		0			0
$607$	−1112.69			−1.83310			−0.916548	−	0.399925i	$-0.869036\pi$
$607$	−1112.69			−1.83310			−0.916548	+	0.399925i	$0.869036\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−122.591	+	122.591i	−0.200640	+	0.200640i
$612$		0			0
$613$	455.272	−	455.272i	0.742696	−	0.742696i	−0.230400	−	0.973096i	$-0.574004\pi$
$613$	455.272	−	455.272i	0.742696	−	0.742696i	0.973096	+	0.230400i	$0.0740036\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	229.336			0.371695			0.185847	−	0.982579i	$-0.440497\pi$
$617$	229.336			0.371695			0.185847	+	0.982579i	$0.440497\pi$
$618$		0			0
$619$	163.423	−	163.423i	0.264011	−	0.264011i	−0.562670	−	0.826681i	$-0.690226\pi$
$619$	163.423	−	163.423i	0.264011	−	0.264011i	0.826681	+	0.562670i	$0.190226\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	78.5621i		−	0.126103i
$624$		0			0
$625$	−122.469			−0.195950
$626$		0			0
$627$		0			0
$628$		0			0
$629$	518.341	+	518.341i	0.824072	+	0.824072i
$630$		0			0
$631$		−	555.849i		−	0.880902i	−0.897777	−	0.440451i	$-0.854819\pi$
$631$		−	555.849i		−	0.880902i	0.897777	−	0.440451i	$-0.145181\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	361.462	+	361.462i	0.569231	+	0.569231i
$636$		0			0
$637$	66.3818	+	66.3818i	0.104210	+	0.104210i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	982.994i		−	1.53353i	−0.641927	−	0.766766i	$-0.721865\pi$
$641$		−	982.994i		−	1.53353i	0.641927	−	0.766766i	$-0.278135\pi$
$642$		0			0
$643$	−163.762	−	163.762i	−0.254684	−	0.254684i	0.568204	−	0.822888i	$-0.307639\pi$
$643$	−163.762	−	163.762i	−0.254684	−	0.254684i	−0.822888	+	0.568204i	$0.807639\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	655.404			1.01299			0.506495	−	0.862243i	$-0.330941\pi$
$647$	655.404			1.01299			0.506495	+	0.862243i	$0.330941\pi$
$648$		0			0
$649$		−	107.400i		−	0.165485i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−469.752	+	469.752i	−0.719376	+	0.719376i	−0.968477	−	0.249102i	$-0.919865\pi$
$653$	−469.752	+	469.752i	−0.719376	+	0.719376i	0.249102	+	0.968477i	$0.419865\pi$
$654$		0			0
$655$	305.586			0.466543
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−576.834	+	576.834i	−0.875317	+	0.875317i	−0.993046	−	0.117728i	$-0.962439\pi$
$659$	−576.834	+	576.834i	−0.875317	+	0.875317i	0.117728	+	0.993046i	$0.462439\pi$
$660$		0			0
$661$	679.417	−	679.417i	1.02786	−	1.02786i	0.0282616	−	0.999601i	$-0.491003\pi$
$661$	679.417	−	679.417i	1.02786	−	1.02786i	0.999601	−	0.0282616i	$-0.00899714\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−234.634			−0.352834
$666$		0			0
$667$	−83.4711	+	83.4711i	−0.125144	+	0.125144i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	354.764i		−	0.528709i
$672$		0			0
$673$	929.909			1.38174			0.690868	−	0.722981i	$-0.257228\pi$
$673$	929.909			1.38174			0.690868	+	0.722981i	$0.257228\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−597.623	−	597.623i	−0.882751	−	0.882751i	0.111062	−	0.993813i	$-0.464575\pi$
$677$	−597.623	−	597.623i	−0.882751	−	0.882751i	−0.993813	+	0.111062i	$0.964575\pi$
$678$		0			0
$679$			300.225i			0.442158i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	286.314	+	286.314i	0.419200	+	0.419200i	0.884928	−	0.465728i	$-0.154207\pi$
$683$	286.314	+	286.314i	0.419200	+	0.419200i	−0.465728	+	0.884928i	$0.654207\pi$
$684$		0			0
$685$	6.67614	+	6.67614i	0.00974619	+	0.00974619i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	122.653i		−	0.178017i
$690$		0			0
$691$	483.066	+	483.066i	0.699083	+	0.699083i	0.964213	−	0.265130i	$-0.0854149\pi$
$691$	483.066	+	483.066i	0.699083	+	0.699083i	−0.265130	+	0.964213i	$0.585415\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	54.4039			0.0782790
$696$		0			0
$697$		−	1305.45i		−	1.87296i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−349.679	+	349.679i	−0.498829	+	0.498829i	−0.911073	−	0.412244i	$-0.864745\pi$
$701$	−349.679	+	349.679i	−0.498829	+	0.498829i	0.412244	+	0.911073i	$0.364745\pi$
$702$		0			0
$703$	−1272.94			−1.81072
$704$		0			0
$705$		0			0
$706$		0			0
$707$	203.210	−	203.210i	0.287425	−	0.287425i
$708$		0			0
$709$	374.404	−	374.404i	0.528074	−	0.528074i	−0.391924	−	0.919998i	$-0.628190\pi$
$709$	374.404	−	374.404i	0.528074	−	0.528074i	0.919998	+	0.391924i	$0.128190\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−47.4195			−0.0665071
$714$		0			0
$715$	16.5874	−	16.5874i	0.0231992	−	0.0231992i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			819.536i			1.13983i	0.821704	+	0.569914i	$0.193024\pi$
$719$			819.536i			1.13983i	−0.821704	+	0.569914i	$0.806976\pi$
$720$		0			0
$721$	92.4308			0.128198
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−411.896	−	411.896i	−0.568132	−	0.568132i
$726$		0			0
$727$		−	1089.99i		−	1.49930i	−0.661836	−	0.749649i	$-0.730222\pi$
$727$		−	1089.99i		−	1.49930i	0.661836	−	0.749649i	$-0.269778\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−502.825	−	502.825i	−0.687860	−	0.687860i
$732$		0			0
$733$	−165.917	−	165.917i	−0.226354	−	0.226354i	0.584814	−	0.811168i	$-0.301168\pi$
$733$	−165.917	−	165.917i	−0.226354	−	0.226354i	−0.811168	+	0.584814i	$0.801168\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	363.917i		−	0.493781i
$738$		0			0
$739$	−316.451	−	316.451i	−0.428215	−	0.428215i	0.459805	−	0.888020i	$-0.347919\pi$
$739$	−316.451	−	316.451i	−0.428215	−	0.428215i	−0.888020	+	0.459805i	$0.847919\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1074.58			−1.44627			−0.723134	−	0.690708i	$-0.757299\pi$
$743$	−1074.58			−1.44627			−0.723134	+	0.690708i	$0.757299\pi$
$744$		0			0
$745$		−	330.835i		−	0.444073i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−168.721	+	168.721i	−0.225262	+	0.225262i
$750$		0			0
$751$	−425.307			−0.566320			−0.283160	−	0.959073i	$-0.591383\pi$
$751$	−425.307			−0.566320			−0.283160	+	0.959073i	$0.591383\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−426.508	+	426.508i	−0.564912	+	0.564912i
$756$		0			0
$757$	−93.5146	+	93.5146i	−0.123533	+	0.123533i	−0.766170	−	0.642637i	$-0.777840\pi$
$757$	−93.5146	+	93.5146i	−0.123533	+	0.123533i	0.642637	+	0.766170i	$0.277840\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	430.074			0.565144			0.282572	−	0.959246i	$-0.408812\pi$
$761$	430.074			0.565144			0.282572	+	0.959246i	$0.408812\pi$
$762$		0			0
$763$	112.423	−	112.423i	0.147343	−	0.147343i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	59.2749i		−	0.0772814i
$768$		0			0
$769$	−307.931			−0.400430			−0.200215	−	0.979752i	$-0.564164\pi$
$769$	−307.931			−0.400430			−0.200215	+	0.979752i	$0.564164\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	531.375	+	531.375i	0.687419	+	0.687419i	0.961661	−	0.274241i	$-0.0884268\pi$
$773$	531.375	+	531.375i	0.687419	+	0.687419i	−0.274241	+	0.961661i	$0.588427\pi$
$774$		0			0
$775$		−	233.996i		−	0.301930i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1602.96	+	1602.96i	2.05771	+	2.05771i
$780$		0			0
$781$	−236.432	−	236.432i	−0.302730	−	0.302730i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	441.265i		−	0.562121i
$786$		0			0
$787$	−114.817	−	114.817i	−0.145892	−	0.145892i	0.630388	−	0.776280i	$-0.282896\pi$
$787$	−114.817	−	114.817i	−0.145892	−	0.145892i	−0.776280	+	0.630388i	$0.782896\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	236.257			0.298682
$792$		0			0
$793$		−	195.797i		−	0.246907i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−699.220	+	699.220i	−0.877315	+	0.877315i	−0.993256	−	0.115941i	$-0.963012\pi$
$797$	−699.220	+	699.220i	−0.877315	+	0.877315i	0.115941	+	0.993256i	$0.463012\pi$
$798$		0			0
$799$	−1621.24			−2.02909
$800$		0			0
$801$		0			0
$802$		0			0
$803$	110.953	−	110.953i	0.138174	−	0.138174i
$804$		0			0
$805$	−16.6891	+	16.6891i	−0.0207318	+	0.0207318i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1160.71			−1.43475			−0.717376	−	0.696686i	$-0.754657\pi$
$809$	−1160.71			−1.43475			−0.717376	+	0.696686i	$0.754657\pi$
$810$		0			0
$811$	297.004	−	297.004i	0.366220	−	0.366220i	−0.499877	−	0.866097i	$-0.666621\pi$
$811$	297.004	−	297.004i	0.366220	−	0.366220i	0.866097	+	0.499877i	$0.166621\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			479.288i			0.588084i
$816$		0			0
$817$	1234.84			1.51143
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−606.627	−	606.627i	−0.738888	−	0.738888i	0.233475	−	0.972363i	$-0.424990\pi$
$821$	−606.627	−	606.627i	−0.738888	−	0.738888i	−0.972363	+	0.233475i	$0.924990\pi$
$822$		0			0
$823$		−	320.341i		−	0.389236i	−0.980879	−	0.194618i	$-0.937653\pi$
$823$		−	320.341i		−	0.389236i	0.980879	−	0.194618i	$-0.0623467\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−686.333	−	686.333i	−0.829907	−	0.829907i	0.157597	−	0.987504i	$-0.449625\pi$
$827$	−686.333	−	686.333i	−0.829907	−	0.829907i	−0.987504	+	0.157597i	$0.949625\pi$
$828$		0			0
$829$	−317.721	−	317.721i	−0.383258	−	0.383258i	0.489017	−	0.872275i	$-0.337356\pi$
$829$	−317.721	−	317.721i	−0.383258	−	0.383258i	−0.872275	+	0.489017i	$0.837356\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			877.883i			1.05388i
$834$		0			0
$835$	−444.465	−	444.465i	−0.532293	−	0.532293i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−999.424			−1.19121			−0.595605	−	0.803278i	$-0.703087\pi$
$839$	−999.424			−1.19121			−0.595605	+	0.803278i	$0.703087\pi$
$840$		0			0
$841$		−	259.172i		−	0.308171i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−316.759	+	316.759i	−0.374862	+	0.374862i
$846$		0			0
$847$	256.330			0.302633
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−90.5416	+	90.5416i	−0.106394	+	0.106394i
$852$		0			0
$853$	437.602	−	437.602i	0.513015	−	0.513015i	−0.402434	−	0.915449i	$-0.631836\pi$
$853$	437.602	−	437.602i	0.513015	−	0.513015i	0.915449	+	0.402434i	$0.131836\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	734.356			0.856891			0.428445	−	0.903568i	$-0.359061\pi$
$857$	734.356			0.856891			0.428445	+	0.903568i	$0.359061\pi$
$858$		0			0
$859$	−473.257	+	473.257i	−0.550939	+	0.550939i	−0.926712	−	0.375773i	$-0.877377\pi$
$859$	−473.257	+	473.257i	−0.550939	+	0.550939i	0.375773	+	0.926712i	$0.377377\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	415.374i		−	0.481315i	−0.970610	−	0.240657i	$-0.922637\pi$
$863$		−	415.374i		−	0.481315i	0.970610	−	0.240657i	$-0.0773630\pi$
$864$		0			0
$865$	167.732			0.193910
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−305.348	−	305.348i	−0.351379	−	0.351379i
$870$		0			0
$871$		−	200.849i		−	0.230596i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−199.587	−	199.587i	−0.228100	−	0.228100i
$876$		0			0
$877$	639.234	+	639.234i	0.728887	+	0.728887i	0.970398	−	0.241511i	$-0.0776429\pi$
$877$	639.234	+	639.234i	0.728887	+	0.728887i	−0.241511	+	0.970398i	$0.577643\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			434.503i			0.493193i	0.969118	+	0.246596i	$0.0793122\pi$
$881$			434.503i			0.493193i	−0.969118	+	0.246596i	$0.920688\pi$
$882$		0			0
$883$	−597.662	−	597.662i	−0.676854	−	0.676854i	0.282433	−	0.959287i	$-0.408858\pi$
$883$	−597.662	−	597.662i	−0.676854	−	0.676854i	−0.959287	+	0.282433i	$0.908858\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1100.10			1.24025			0.620125	−	0.784503i	$-0.287082\pi$
$887$	1100.10			1.24025			0.620125	+	0.784503i	$0.287082\pi$
$888$		0			0
$889$		−	455.767i		−	0.512674i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1990.72	−	1990.72i	2.22924	−	2.22924i
$894$		0			0
$895$	562.745			0.628765
$896$		0			0
$897$		0			0
$898$		0			0
$899$	312.501	−	312.501i	0.347610	−	0.347610i
$900$		0			0
$901$	811.031	−	811.031i	0.900146	−	0.900146i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−712.669			−0.787479
$906$		0			0
$907$	−344.291	+	344.291i	−0.379593	+	0.379593i	−0.870955	−	0.491362i	$-0.836499\pi$
$907$	−344.291	+	344.291i	−0.379593	+	0.379593i	0.491362	+	0.870955i	$0.336499\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			803.034i			0.881486i	0.897633	+	0.440743i	$0.145285\pi$
$911$			803.034i			0.881486i	−0.897633	+	0.440743i	$0.854715\pi$
$912$		0			0
$913$	−82.3109			−0.0901543
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−192.657	−	192.657i	−0.210094	−	0.210094i
$918$		0			0
$919$			1119.49i			1.21816i	0.793107	+	0.609082i	$0.208462\pi$
$919$			1119.49i			1.21816i	−0.793107	+	0.609082i	$0.791538\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−130.489	−	130.489i	−0.141375	−	0.141375i
$924$		0			0
$925$	−446.786	−	446.786i	−0.483012	−	0.483012i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1069.69i		−	1.15144i	−0.817647	−	0.575721i	$-0.804722\pi$
$929$		−	1069.69i		−	1.15144i	0.817647	−	0.575721i	$-0.195278\pi$
$930$		0			0
$931$	−1077.95	−	1077.95i	−1.15784	−	1.15784i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	219.365			0.234615
$936$		0			0
$937$			61.7823i			0.0659363i	0.999456	+	0.0329682i	$0.0104960\pi$
$937$			61.7823i			0.0659363i	−0.999456	+	0.0329682i	$0.989504\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−455.261	+	455.261i	−0.483806	+	0.483806i	−0.906345	−	0.422539i	$-0.861139\pi$
$941$	−455.261	+	455.261i	−0.483806	+	0.483806i	0.422539	+	0.906345i	$0.361139\pi$
$942$		0			0
$943$	228.031			0.241814
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1145.09	+	1145.09i	−1.20918	+	1.20918i	−0.237883	+	0.971294i	$0.576453\pi$
$947$	−1145.09	+	1145.09i	−1.20918	+	1.20918i	−0.971294	+	0.237883i	$0.923547\pi$
$948$		0			0
$949$	61.2362	−	61.2362i	0.0645271	−	0.0645271i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1187.58			−1.24615			−0.623076	−	0.782161i	$-0.714117\pi$
$953$	−1187.58			−1.24615			−0.623076	+	0.782161i	$0.714117\pi$
$954$		0			0
$955$	−679.221	+	679.221i	−0.711227	+	0.711227i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	8.41794i		−	0.00877783i
$960$		0			0
$961$	−783.470			−0.815265
$962$		0			0
$963$		0			0
$964$		0			0
$965$	280.903	+	280.903i	0.291091	+	0.291091i
$966$		0			0
$967$			464.902i			0.480767i	0.970678	+	0.240384i	$0.0772732\pi$
$967$			464.902i			0.480767i	−0.970678	+	0.240384i	$0.922727\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	102.522	+	102.522i	0.105584	+	0.105584i	0.757925	−	0.652341i	$-0.226213\pi$
$971$	102.522	+	102.522i	0.105584	+	0.105584i	−0.652341	+	0.757925i	$0.726213\pi$
$972$		0			0
$973$	−34.2990	−	34.2990i	−0.0352507	−	0.0352507i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1273.15i			1.30312i	0.758597	+	0.651560i	$0.225885\pi$
$977$			1273.15i			1.30312i	−0.758597	+	0.651560i	$0.774115\pi$
$978$		0			0
$979$	−90.1883	−	90.1883i	−0.0921229	−	0.0921229i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−523.931			−0.532992			−0.266496	−	0.963836i	$-0.585866\pi$
$983$	−523.931			−0.532992			−0.266496	+	0.963836i	$0.585866\pi$
$984$		0			0
$985$			72.4130i			0.0735158i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	87.8314	−	87.8314i	0.0888083	−	0.0888083i
$990$		0			0
$991$	1675.46			1.69067			0.845336	−	0.534235i	$-0.179400\pi$
$991$	1675.46			1.69067			0.845336	+	0.534235i	$0.179400\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	501.645	−	501.645i	0.504166	−	0.504166i
$996$		0			0
$997$	−911.770	+	911.770i	−0.914513	+	0.914513i	−0.996623	−	0.0821099i	$-0.973834\pi$
$997$	−911.770	+	911.770i	−0.914513	+	0.914513i	0.0821099	+	0.996623i	$0.473834\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.j.b.161.5		32
3.2	odd	2	inner	1152.3.j.b.161.12		32
4.3	odd	2		1152.3.j.a.161.5		32
8.3	odd	2		144.3.j.a.125.16	yes	32
8.5	even	2		576.3.j.a.17.12		32
12.11	even	2		1152.3.j.a.161.12		32
16.3	odd	4		144.3.j.a.53.1	✓	32
16.5	even	4	inner	1152.3.j.b.737.12		32
16.11	odd	4		1152.3.j.a.737.12		32
16.13	even	4		576.3.j.a.305.5		32
24.5	odd	2		576.3.j.a.17.5		32
24.11	even	2		144.3.j.a.125.1	yes	32
48.5	odd	4	inner	1152.3.j.b.737.5		32
48.11	even	4		1152.3.j.a.737.5		32
48.29	odd	4		576.3.j.a.305.12		32
48.35	even	4		144.3.j.a.53.16	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.j.a.53.1	✓	32	16.3	odd	4
144.3.j.a.53.16	yes	32	48.35	even	4
144.3.j.a.125.1	yes	32	24.11	even	2
144.3.j.a.125.16	yes	32	8.3	odd	2
576.3.j.a.17.5		32	24.5	odd	2
576.3.j.a.17.12		32	8.5	even	2
576.3.j.a.305.5		32	16.13	even	4
576.3.j.a.305.12		32	48.29	odd	4
1152.3.j.a.161.5		32	4.3	odd	2
1152.3.j.a.161.12		32	12.11	even	2
1152.3.j.a.737.5		32	48.11	even	4
1152.3.j.a.737.12		32	16.11	odd	4
1152.3.j.b.161.5		32	1.1	even	1	trivial
1152.3.j.b.161.12		32	3.2	odd	2	inner
1152.3.j.b.737.5		32	48.5	odd	4	inner
1152.3.j.b.737.12		32	16.5	even	4	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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.92848 - 1.92848i) q^{5} +2.43162i q^{7} +O(q^{10})\)	\(q+(-1.92848 - 1.92848i) q^{5} +2.43162i q^{7} +(2.79147 + 2.79147i) q^{11} +(1.54064 + 1.54064i) q^{13} +20.3746i q^{17} +(-25.0179 - 25.0179i) q^{19} -3.55894 q^{23} -17.5619i q^{25} +(23.4539 - 23.4539i) q^{29} +13.3241 q^{31} +(4.68934 - 4.68934i) q^{35} +(25.4406 - 25.4406i) q^{37} -64.0726 q^{41} +(-24.6791 + 24.6791i) q^{43} +79.5718i q^{47} +43.0872 q^{49} +(-39.8061 - 39.8061i) q^{53} -10.7666i q^{55} +(-19.2371 - 19.2371i) q^{59} +(-63.5441 - 63.5441i) q^{61} -5.94218i q^{65} +(-65.1837 - 65.1837i) q^{67} -84.6981 q^{71} -39.7473i q^{73} +(-6.78781 + 6.78781i) q^{77} -109.386 q^{79} +(-14.7433 + 14.7433i) q^{83} +(39.2920 - 39.2920i) q^{85} -32.3085 q^{89} +(-3.74625 + 3.74625i) q^{91} +96.4929i q^{95} +123.467 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 64 q^{19} + 128 q^{43} - 224 q^{49} - 64 q^{61} - 64 q^{67} + 512 q^{79} - 320 q^{85} - 192 q^{91}+O(q^{100}) \) 32 * q + 64 * q^19 + 128 * q^43 - 224 * q^49 - 64 * q^61 - 64 * q^67 + 512 * q^79 - 320 * q^85 - 192 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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