LMFDB - Embedded newform 1152.2.l.a.863.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(287,1152)]

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([2, 1, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1152.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.l (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:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 $ x^16 - 4x^14 + 6x^12 - 12x^10 + 33x^8 - 48x^6 + 96x^4 - 256*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			863.7
Root			$-0.944649 - 1.05244i$ of defining polynomial
Character	$\chi$	$=$	1152.863
Dual form			1152.2.l.a.287.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.10489 + 2.10489i) q^{5} -4.40731 q^{7} +O(q^{10})$	$q+(2.10489 + 2.10489i) q^{5} -4.40731 q^{7} +(-0.215589 + 0.215589i) q^{11} +(2.73544 + 2.73544i) q^{13} -2.36438i q^{17} +(-0.758681 + 0.758681i) q^{19} +1.75549i q^{23} +3.86110i q^{25} +(-5.54221 + 5.54221i) q^{29} +9.01709i q^{31} +(-9.27690 - 9.27690i) q^{35} +(-3.10242 + 3.10242i) q^{37} -10.1014 q^{41} +(3.54621 + 3.54621i) q^{43} -3.90136 q^{47} +12.4244 q^{49} +(-2.71378 - 2.71378i) q^{53} -0.907583 q^{55} +(3.40445 - 3.40445i) q^{59} +(1.75868 + 1.75868i) q^{61} +11.5156i q^{65} +(-9.11951 + 9.11951i) q^{67} -11.8897i q^{71} -0.482639i q^{73} +(0.950169 - 0.950169i) q^{77} -6.88995i q^{79} +(4.79951 + 4.79951i) q^{83} +(4.97676 - 4.97676i) q^{85} +7.00534 q^{89} +(-12.0559 - 12.0559i) q^{91} -3.19387 q^{95} -3.34374 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{19} + 32 q^{43} + 16 q^{49} - 64 q^{55} + 32 q^{61} + 16 q^{67} + 32 q^{85} - 48 q^{91}+O(q^{100}) $ 16 * q - 16 * q^19 + 32 * q^43 + 16 * q^49 - 64 * q^55 + 32 * q^61 + 16 * q^67 + 32 * q^85 - 48 * 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
$5$	2.10489	+	2.10489i	0.941334	+	0.941334i	0.998372	−	0.0570377i	$-0.0181655\pi$
$5$	2.10489	+	2.10489i	0.941334	+	0.941334i	−0.0570377	+	0.998372i	$0.518166\pi$
$6$		0			0
$7$	−4.40731			−1.66581			−0.832904	−	0.553418i	$-0.813323\pi$
$7$	−4.40731			−1.66581			−0.832904	+	0.553418i	$0.813323\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.215589	+	0.215589i	−0.0650026	+	0.0650026i	−0.738861	−	0.673858i	$-0.764636\pi$
$11$	−0.215589	+	0.215589i	−0.0650026	+	0.0650026i	0.673858	+	0.738861i	$0.264636\pi$
$12$		0			0
$13$	2.73544	+	2.73544i	0.758675	+	0.758675i	0.976081	−	0.217406i	$-0.0697597\pi$
$13$	2.73544	+	2.73544i	0.758675	+	0.758675i	−0.217406	+	0.976081i	$0.569760\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.36438i		−	0.573447i	−0.958013	−	0.286724i	$-0.907434\pi$
$17$		−	2.36438i		−	0.573447i	0.958013	−	0.286724i	$-0.0925661\pi$
$18$		0			0
$19$	−0.758681	+	0.758681i	−0.174053	+	0.174053i	−0.788758	−	0.614704i	$-0.789275\pi$
$19$	−0.758681	+	0.758681i	−0.174053	+	0.174053i	0.614704	+	0.788758i	$0.289275\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.75549i			0.366045i	0.983109	+	0.183023i	$0.0585881\pi$
$23$			1.75549i			0.366045i	−0.983109	+	0.183023i	$0.941412\pi$
$24$		0			0
$25$			3.86110i			0.772221i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.54221	+	5.54221i	−1.02916	+	1.02916i	−0.0296002	+	0.999562i	$0.509423\pi$
$29$	−5.54221	+	5.54221i	−1.02916	+	1.02916i	−0.999562	+	0.0296002i	$0.990577\pi$
$30$		0			0
$31$			9.01709i			1.61952i	0.586763	+	0.809759i	$0.300402\pi$
$31$			9.01709i			1.61952i	−0.586763	+	0.809759i	$0.699598\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.27690	−	9.27690i	−1.56808	−	1.56808i
$36$		0			0
$37$	−3.10242	+	3.10242i	−0.510035	+	0.510035i	−0.914537	−	0.404502i	$-0.867445\pi$
$37$	−3.10242	+	3.10242i	−0.510035	+	0.510035i	0.404502	+	0.914537i	$0.367445\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10.1014			−1.57757			−0.788785	−	0.614669i	$-0.789290\pi$
$41$	−10.1014			−1.57757			−0.788785	+	0.614669i	$0.789290\pi$
$42$		0			0
$43$	3.54621	+	3.54621i	0.540792	+	0.540792i	0.923761	−	0.382969i	$-0.125099\pi$
$43$	3.54621	+	3.54621i	0.540792	+	0.540792i	−0.382969	+	0.923761i	$0.625099\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.90136			−0.569072			−0.284536	−	0.958665i	$-0.591840\pi$
$47$	−3.90136			−0.569072			−0.284536	+	0.958665i	$0.591840\pi$
$48$		0			0
$49$	12.4244			1.77491
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.71378	−	2.71378i	−0.372766	−	0.372766i	0.495717	−	0.868484i	$-0.334905\pi$
$53$	−2.71378	−	2.71378i	−0.372766	−	0.372766i	−0.868484	+	0.495717i	$0.834905\pi$
$54$		0			0
$55$	−0.907583			−0.122378
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.40445	−	3.40445i	0.443222	−	0.443222i	−0.449871	−	0.893093i	$-0.648530\pi$
$59$	3.40445	−	3.40445i	0.443222	−	0.443222i	0.893093	+	0.449871i	$0.148530\pi$
$60$		0			0
$61$	1.75868	+	1.75868i	0.225176	+	0.225176i	0.810674	−	0.585498i	$-0.199101\pi$
$61$	1.75868	+	1.75868i	0.225176	+	0.225176i	−0.585498	+	0.810674i	$0.699101\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			11.5156i			1.42833i
$66$		0			0
$67$	−9.11951	+	9.11951i	−1.11413	+	1.11413i	−0.121539	+	0.992587i	$0.538783\pi$
$67$	−9.11951	+	9.11951i	−1.11413	+	1.11413i	−0.992587	+	0.121539i	$0.961217\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.8897i		−	1.41105i	−0.708684	−	0.705526i	$-0.750711\pi$
$71$		−	11.8897i		−	1.41105i	0.708684	−	0.705526i	$-0.249289\pi$
$72$		0			0
$73$		−	0.482639i		−	0.0564886i	−0.999601	−	0.0282443i	$-0.991008\pi$
$73$		−	0.482639i		−	0.0564886i	0.999601	−	0.0282443i	$-0.00899163\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.950169	−	0.950169i	0.108282	−	0.108282i
$78$		0			0
$79$		−	6.88995i		−	0.775180i	−0.921832	−	0.387590i	$-0.873308\pi$
$79$		−	6.88995i		−	0.775180i	0.921832	−	0.387590i	$-0.126692\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.79951	+	4.79951i	0.526814	+	0.526814i	0.919621	−	0.392807i	$-0.128496\pi$
$83$	4.79951	+	4.79951i	0.526814	+	0.526814i	−0.392807	+	0.919621i	$0.628496\pi$
$84$		0			0
$85$	4.97676	−	4.97676i	0.539805	−	0.539805i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.00534			0.742564			0.371282	−	0.928520i	$-0.378918\pi$
$89$	7.00534			0.742564			0.371282	+	0.928520i	$0.378918\pi$
$90$		0			0
$91$	−12.0559	−	12.0559i	−1.26381	−	1.26381i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.19387			−0.327685
$96$		0			0
$97$	−3.34374			−0.339506			−0.169753	−	0.985487i	$-0.554297\pi$
$97$	−3.34374			−0.339506			−0.169753	+	0.985487i	$0.554297\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.76361	+	1.76361i	0.175486	+	0.175486i	0.789385	−	0.613899i	$-0.210400\pi$
$101$	1.76361	+	1.76361i	0.175486	+	0.175486i	−0.613899	+	0.789385i	$0.710400\pi$
$102$		0			0
$103$	−1.01709			−0.100217			−0.0501085	−	0.998744i	$-0.515957\pi$
$103$	−1.01709			−0.100217			−0.0501085	+	0.998744i	$0.515957\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.33152	+	2.33152i	−0.225396	+	0.225396i	−0.810766	−	0.585370i	$-0.800949\pi$
$107$	−2.33152	+	2.33152i	−0.225396	+	0.225396i	0.585370	+	0.810766i	$0.300949\pi$
$108$		0			0
$109$	8.07918	+	8.07918i	0.773845	+	0.773845i	0.978776	−	0.204931i	$-0.0656970\pi$
$109$	8.07918	+	8.07918i	0.773845	+	0.773845i	−0.204931	+	0.978776i	$0.565697\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			2.09677i			0.197247i	0.995125	+	0.0986237i	$0.0314440\pi$
$113$			2.09677i			0.197247i	−0.995125	+	0.0986237i	$0.968556\pi$
$114$		0			0
$115$	−3.69511	+	3.69511i	−0.344571	+	0.344571i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			10.4206i			0.955253i
$120$		0			0
$121$			10.9070i			0.991549i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.39725	−	2.39725i	0.214416	−	0.214416i
$126$		0			0
$127$			10.2802i			0.912218i	0.889924	+	0.456109i	$0.150757\pi$
$127$			10.2802i			0.912218i	−0.889924	+	0.456109i	$0.849243\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.80891	+	6.80891i	0.594897	+	0.594897i	0.938950	−	0.344053i	$-0.111800\pi$
$131$	6.80891	+	6.80891i	0.594897	+	0.594897i	−0.344053	+	0.938950i	$0.611800\pi$
$132$		0			0
$133$	3.34374	−	3.34374i	0.289939	−	0.289939i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.7922			1.17834			0.589172	−	0.808008i	$-0.299454\pi$
$137$	13.7922			1.17834			0.589172	+	0.808008i	$0.299454\pi$
$138$		0			0
$139$	13.1195	+	13.1195i	1.11278	+	1.11278i	0.992773	+	0.120010i	$0.0382927\pi$
$139$	13.1195	+	13.1195i	1.11278	+	1.11278i	0.120010	+	0.992773i	$0.461707\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.17946			−0.0986317
$144$		0			0
$145$	−23.3314			−1.93757
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.76174	−	7.76174i	−0.635867	−	0.635867i	0.313667	−	0.949533i	$-0.398443\pi$
$149$	−7.76174	−	7.76174i	−0.635867	−	0.635867i	−0.949533	+	0.313667i	$0.898443\pi$
$150$		0			0
$151$	−0.202466			−0.0164765			−0.00823823	−	0.999966i	$-0.502622\pi$
$151$	−0.202466			−0.0164765			−0.00823823	+	0.999966i	$0.502622\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−18.9800	+	18.9800i	−1.52451	+	1.52451i
$156$		0			0
$157$	−3.75868	−	3.75868i	−0.299975	−	0.299975i	0.541029	−	0.841004i	$-0.318035\pi$
$157$	−3.75868	−	3.75868i	−0.299975	−	0.299975i	−0.841004	+	0.541029i	$0.818035\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	7.73700i		−	0.609761i
$162$		0			0
$163$	13.2684	−	13.2684i	1.03926	−	1.03926i	0.0400655	−	0.999197i	$-0.487243\pi$
$163$	13.2684	−	13.2684i	1.03926	−	1.03926i	0.999197	−	0.0400655i	$-0.0127567\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	20.1644i		−	1.56037i	−0.625548	−	0.780186i	$-0.715125\pi$
$167$		−	20.1644i		−	1.56037i	0.625548	−	0.780186i	$-0.284875\pi$
$168$		0			0
$169$			1.96528i			0.151175i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.98315	−	3.98315i	0.302833	−	0.302833i	−0.539288	−	0.842121i	$-0.681307\pi$
$173$	3.98315	−	3.98315i	0.302833	−	0.302833i	0.842121	+	0.539288i	$0.181307\pi$
$174$		0			0
$175$		−	17.0171i		−	1.28637i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	14.7182	+	14.7182i	1.10009	+	1.10009i	0.994399	+	0.105688i	$0.0337044\pi$
$179$	14.7182	+	14.7182i	1.10009	+	1.10009i	0.105688	+	0.994399i	$0.466296\pi$
$180$		0			0
$181$	−3.26456	+	3.26456i	−0.242653	+	0.242653i	−0.817947	−	0.575294i	$-0.804888\pi$
$181$	−3.26456	+	3.26456i	−0.242653	+	0.242653i	0.575294	+	0.817947i	$0.304888\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−13.0605			−0.960227
$186$		0			0
$187$	0.509736	+	0.509736i	0.0372756	+	0.0372756i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.69868			0.557057			0.278528	−	0.960428i	$-0.410153\pi$
$191$	7.69868			0.557057			0.278528	+	0.960428i	$0.410153\pi$
$192$		0			0
$193$	−3.51736			−0.253185			−0.126593	−	0.991955i	$-0.540404\pi$
$193$	−3.51736			−0.253185			−0.126593	+	0.991955i	$0.540404\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.40456	−	6.40456i	−0.456306	−	0.456306i	0.441135	−	0.897441i	$-0.354576\pi$
$197$	−6.40456	−	6.40456i	−0.456306	−	0.456306i	−0.897441	+	0.441135i	$0.854576\pi$
$198$		0			0
$199$	23.3491			1.65517			0.827586	−	0.561339i	$-0.189714\pi$
$199$	23.3491			1.65517			0.827586	+	0.561339i	$0.189714\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	24.4262	−	24.4262i	1.71439	−	1.71439i
$204$		0			0
$205$	−21.2623	−	21.2623i	−1.48502	−	1.48502i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	0.327127i		−	0.0226278i
$210$		0			0
$211$	6.63688	−	6.63688i	0.456901	−	0.456901i	−0.440736	−	0.897637i	$-0.645282\pi$
$211$	6.63688	−	6.63688i	0.456901	−	0.456901i	0.897637	+	0.440736i	$0.145282\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			14.9287i			1.01813i
$216$		0			0
$217$		−	39.7411i		−	2.69780i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.46763	−	6.46763i	0.435060	−	0.435060i
$222$		0			0
$223$		−	13.2219i		−	0.885406i	−0.896668	−	0.442703i	$-0.854020\pi$
$223$		−	13.2219i		−	0.885406i	0.896668	−	0.442703i	$-0.145980\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−16.4029	−	16.4029i	−1.08870	−	1.08870i	−0.995663	−	0.0930369i	$-0.970343\pi$
$227$	−16.4029	−	16.4029i	−1.08870	−	1.08870i	−0.0930369	−	0.995663i	$-0.529657\pi$
$228$		0			0
$229$	4.98677	−	4.98677i	0.329535	−	0.329535i	−0.522875	−	0.852410i	$-0.675140\pi$
$229$	4.98677	−	4.98677i	0.329535	−	0.329535i	0.852410	+	0.522875i	$0.175140\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.05879			−0.0693634			−0.0346817	−	0.999398i	$-0.511042\pi$
$233$	−1.05879			−0.0693634			−0.0346817	+	0.999398i	$0.511042\pi$
$234$		0			0
$235$	−8.21193	−	8.21193i	−0.535687	−	0.535687i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.317107			−0.0205119			−0.0102560	−	0.999947i	$-0.503265\pi$
$239$	−0.317107			−0.0205119			−0.0102560	+	0.999947i	$0.503265\pi$
$240$		0			0
$241$	11.7334			0.755816			0.377908	−	0.925843i	$-0.376644\pi$
$241$	11.7334			0.755816			0.377908	+	0.925843i	$0.376644\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	26.1520	+	26.1520i	1.67079	+	1.67079i
$246$		0			0
$247$	−4.15065			−0.264100
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.02450	−	7.02450i	0.443382	−	0.443382i	−0.449765	−	0.893147i	$-0.648492\pi$
$251$	7.02450	−	7.02450i	0.443382	−	0.443382i	0.893147	+	0.449765i	$0.148492\pi$
$252$		0			0
$253$	−0.378465	−	0.378465i	−0.0237939	−	0.0237939i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	23.5830i		−	1.47107i	−0.677487	−	0.735535i	$-0.736931\pi$
$257$		−	23.5830i		−	1.47107i	0.677487	−	0.735535i	$-0.263069\pi$
$258$		0			0
$259$	13.6733	−	13.6733i	0.849621	−	0.849621i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			20.1370i			1.24170i	0.783928	+	0.620851i	$0.213213\pi$
$263$			20.1370i			1.24170i	−0.783928	+	0.620851i	$0.786787\pi$
$264$		0			0
$265$		−	11.4244i		−	0.701796i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.62201	+	8.62201i	−0.525693	+	0.525693i	−0.919285	−	0.393592i	$-0.871232\pi$
$269$	−8.62201	+	8.62201i	−0.525693	+	0.525693i	0.393592	+	0.919285i	$0.371232\pi$
$270$		0			0
$271$			2.18722i			0.132864i	0.997791	+	0.0664319i	$0.0211615\pi$
$271$			2.18722i			0.132864i	−0.997791	+	0.0664319i	$0.978838\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.832413	−	0.832413i	−0.0501964	−	0.0501964i
$276$		0			0
$277$	−8.64248	+	8.64248i	−0.519277	+	0.519277i	−0.917352	−	0.398076i	$-0.869678\pi$
$277$	−8.64248	+	8.64248i	−0.519277	+	0.519277i	0.398076	+	0.917352i	$0.369678\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.8081			0.883375			0.441688	−	0.897169i	$-0.354380\pi$
$281$	14.8081			0.883375			0.441688	+	0.897169i	$0.354380\pi$
$282$		0			0
$283$	−6.12714	−	6.12714i	−0.364221	−	0.364221i	0.501144	−	0.865364i	$-0.332913\pi$
$283$	−6.12714	−	6.12714i	−0.364221	−	0.364221i	−0.865364	+	0.501144i	$0.832913\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	44.5199			2.62793
$288$		0			0
$289$	11.4097			0.671158
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−11.8979	−	11.8979i	−0.695080	−	0.695080i	0.268265	−	0.963345i	$-0.413550\pi$
$293$	−11.8979	−	11.8979i	−0.695080	−	0.695080i	−0.963345	+	0.268265i	$0.913550\pi$
$294$		0			0
$295$	14.3320			0.834441
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.80204	+	4.80204i	−0.277709	+	0.277709i
$300$		0			0
$301$	−15.6293	−	15.6293i	−0.900855	−	0.900855i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			7.40365i			0.423932i
$306$		0			0
$307$	−4.91467	+	4.91467i	−0.280495	+	0.280495i	−0.833306	−	0.552811i	$-0.813555\pi$
$307$	−4.91467	+	4.91467i	−0.280495	+	0.280495i	0.552811	+	0.833306i	$0.313555\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			20.7712i			1.17783i	0.808195	+	0.588915i	$0.200445\pi$
$311$			20.7712i			1.17783i	−0.808195	+	0.588915i	$0.799555\pi$
$312$		0			0
$313$			16.3897i			0.926400i	0.886254	+	0.463200i	$0.153299\pi$
$313$			16.3897i			0.926400i	−0.886254	+	0.463200i	$0.846701\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	22.3592	−	22.3592i	1.25582	−	1.25582i	0.302749	−	0.953070i	$-0.402096\pi$
$317$	22.3592	−	22.3592i	1.25582	−	1.25582i	0.953070	−	0.302749i	$-0.0979044\pi$
$318$		0			0
$319$		−	2.38968i		−	0.133796i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.79381	+	1.79381i	0.0998103	+	0.0998103i
$324$		0			0
$325$	−10.5618	+	10.5618i	−0.585865	+	0.585865i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	17.1945			0.947965
$330$		0			0
$331$	7.08533	+	7.08533i	0.389445	+	0.389445i	0.874490	−	0.485044i	$-0.161197\pi$
$331$	7.08533	+	7.08533i	0.389445	+	0.389445i	−0.485044	+	0.874490i	$0.661197\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−38.3911			−2.09753
$336$		0			0
$337$	−23.3314			−1.27094			−0.635472	−	0.772124i	$-0.719195\pi$
$337$	−23.3314			−1.27094			−0.635472	+	0.772124i	$0.719195\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.94399	−	1.94399i	−0.105273	−	0.105273i
$342$		0			0
$343$	−23.9070			−1.29086
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−21.2074	+	21.2074i	−1.13847	+	1.13847i	−0.149751	+	0.988724i	$0.547847\pi$
$347$	−21.2074	+	21.2074i	−1.13847	+	1.13847i	−0.988724	+	0.149751i	$0.952153\pi$
$348$		0			0
$349$	−5.38022	−	5.38022i	−0.287996	−	0.287996i	0.548291	−	0.836288i	$-0.315279\pi$
$349$	−5.38022	−	5.38022i	−0.287996	−	0.287996i	−0.836288	+	0.548291i	$0.815279\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	10.5163i		−	0.559727i	−0.960040	−	0.279864i	$-0.909711\pi$
$353$		−	10.5163i		−	0.559727i	0.960040	−	0.279864i	$-0.0902892\pi$
$354$		0			0
$355$	25.0266	−	25.0266i	1.32827	−	1.32827i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			11.7583i			0.620578i	0.950642	+	0.310289i	$0.100426\pi$
$359$			11.7583i			0.620578i	−0.950642	+	0.310289i	$0.899574\pi$
$360$		0			0
$361$			17.8488i			0.939411i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.01590	−	1.01590i	0.0531747	−	0.0531747i
$366$		0			0
$367$			3.49973i			0.182684i	0.995820	+	0.0913422i	$0.0291157\pi$
$367$			3.49973i			0.182684i	−0.995820	+	0.0913422i	$0.970884\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	11.9605	+	11.9605i	0.620957	+	0.620957i
$372$		0			0
$373$	−19.1831	+	19.1831i	−0.993262	+	0.993262i	−0.999977	−	0.00671500i	$-0.997863\pi$
$373$	−19.1831	+	19.1831i	−0.993262	+	0.993262i	0.00671500	+	0.999977i	$0.497863\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−30.3208			−1.56160
$378$		0			0
$379$	14.9977	+	14.9977i	0.770381	+	0.770381i	0.978173	−	0.207792i	$-0.0666278\pi$
$379$	14.9977	+	14.9977i	0.770381	+	0.770381i	−0.207792	+	0.978173i	$0.566628\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	30.4027			1.55351			0.776754	−	0.629805i	$-0.216865\pi$
$383$	30.4027			1.55351			0.776754	+	0.629805i	$0.216865\pi$
$384$		0			0
$385$	4.00000			0.203859
$386$		0			0
$387$		0			0
$388$		0			0
$389$	19.4571	+	19.4571i	0.986516	+	0.986516i	0.999910	−	0.0133943i	$-0.00426365\pi$
$389$	19.4571	+	19.4571i	0.986516	+	0.986516i	−0.0133943	+	0.999910i	$0.504264\pi$
$390$		0			0
$391$	4.15065			0.209908
$392$		0			0
$393$		0			0
$394$		0			0
$395$	14.5026	−	14.5026i	0.729704	−	0.729704i
$396$		0			0
$397$	20.4339	+	20.4339i	1.02555	+	1.02555i	0.999665	+	0.0258815i	$0.00823925\pi$
$397$	20.4339	+	20.4339i	1.02555	+	1.02555i	0.0258815	+	0.999665i	$0.491761\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	22.5671i		−	1.12695i	−0.826133	−	0.563475i	$-0.809464\pi$
$401$		−	22.5671i		−	1.12695i	0.826133	−	0.563475i	$-0.190536\pi$
$402$		0			0
$403$	−24.6657	+	24.6657i	−1.22869	+	1.22869i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1.33770i		−	0.0663073i
$408$		0			0
$409$		−	9.78286i		−	0.483731i	−0.970310	−	0.241866i	$-0.922241\pi$
$409$		−	9.78286i		−	0.483731i	0.970310	−	0.241866i	$-0.0777593\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−15.0045	+	15.0045i	−0.738323	+	0.738323i
$414$		0			0
$415$			20.2048i			0.991817i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9.09125	−	9.09125i	−0.444137	−	0.444137i	0.449263	−	0.893400i	$-0.351687\pi$
$419$	−9.09125	−	9.09125i	−0.444137	−	0.444137i	−0.893400	+	0.449263i	$0.851687\pi$
$420$		0			0
$421$	17.9862	−	17.9862i	0.876595	−	0.876595i	−0.116586	−	0.993181i	$-0.537195\pi$
$421$	17.9862	−	17.9862i	0.876595	−	0.876595i	0.993181	+	0.116586i	$0.0371949\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	9.12913			0.442828
$426$		0			0
$427$	−7.75106	−	7.75106i	−0.375100	−	0.375100i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−30.7707			−1.48217			−0.741085	−	0.671411i	$-0.765689\pi$
$431$	−30.7707			−1.48217			−0.741085	+	0.671411i	$0.765689\pi$
$432$		0			0
$433$	3.49735			0.168072			0.0840360	−	0.996463i	$-0.473219\pi$
$433$	3.49735			0.168072			0.0840360	+	0.996463i	$0.473219\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.33186	−	1.33186i	−0.0637113	−	0.0637113i
$438$		0			0
$439$	−21.1290			−1.00843			−0.504216	−	0.863578i	$-0.668218\pi$
$439$	−21.1290			−1.00843			−0.504216	+	0.863578i	$0.668218\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.6022	+	12.6022i	−0.598750	+	0.598750i	−0.939980	−	0.341230i	$-0.889157\pi$
$443$	−12.6022	+	12.6022i	−0.598750	+	0.598750i	0.341230	+	0.939980i	$0.389157\pi$
$444$		0			0
$445$	14.7454	+	14.7454i	0.699001	+	0.699001i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			9.98731i			0.471330i	0.971834	+	0.235665i	$0.0757268\pi$
$449$			9.98731i			0.471330i	−0.971834	+	0.235665i	$0.924273\pi$
$450$		0			0
$451$	2.17775	−	2.17775i	0.102546	−	0.102546i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	50.7528i		−	2.37933i
$456$		0			0
$457$		−	15.2508i		−	0.713402i	−0.934219	−	0.356701i	$-0.883902\pi$
$457$		−	15.2508i		−	0.713402i	0.934219	−	0.356701i	$-0.116098\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14.3850	+	14.3850i	−0.669976	+	0.669976i	−0.957710	−	0.287734i	$-0.907098\pi$
$461$	−14.3850	+	14.3850i	−0.669976	+	0.669976i	0.287734	+	0.957710i	$0.407098\pi$
$462$		0			0
$463$		−	22.1295i		−	1.02845i	−0.857657	−	0.514223i	$-0.828080\pi$
$463$		−	22.1295i		−	1.02845i	0.857657	−	0.514223i	$-0.171920\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.79951	+	4.79951i	0.222095	+	0.222095i	0.809380	−	0.587285i	$-0.199803\pi$
$467$	4.79951	+	4.79951i	0.222095	+	0.222095i	−0.587285	+	0.809380i	$0.699803\pi$
$468$		0			0
$469$	40.1926	−	40.1926i	1.85592	−	1.85592i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.52905			−0.0703058
$474$		0			0
$475$	−2.92934	−	2.92934i	−0.134408	−	0.134408i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−23.2801			−1.06369			−0.531847	−	0.846840i	$-0.678502\pi$
$479$	−23.2801			−1.06369			−0.531847	+	0.846840i	$0.678502\pi$
$480$		0			0
$481$	−16.9730			−0.773902
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.03820	−	7.03820i	−0.319588	−	0.319588i
$486$		0			0
$487$	21.0748			0.954990			0.477495	−	0.878635i	$-0.341545\pi$
$487$	21.0748			0.954990			0.477495	+	0.878635i	$0.341545\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	26.1459	−	26.1459i	1.17995	−	1.17995i	0.200194	−	0.979756i	$-0.435843\pi$
$491$	26.1459	−	26.1459i	1.17995	−	1.17995i	0.979756	−	0.200194i	$-0.0641571\pi$
$492$		0			0
$493$	13.1039	+	13.1039i	0.590170	+	0.590170i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			52.4018i			2.35054i
$498$		0			0
$499$	−21.0195	+	21.0195i	−0.940961	+	0.940961i	−0.998352	−	0.0573910i	$-0.981722\pi$
$499$	−21.0195	+	21.0195i	−0.940961	+	0.940961i	0.0573910	+	0.998352i	$0.481722\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	44.4925i		−	1.98382i	−0.126929	−	0.991912i	$-0.540512\pi$
$503$		−	44.4925i		−	1.98382i	0.126929	−	0.991912i	$-0.459488\pi$
$504$		0			0
$505$			7.42440i			0.330382i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11.7664	+	11.7664i	−0.521536	+	0.521536i	−0.918035	−	0.396499i	$-0.870225\pi$
$509$	−11.7664	+	11.7664i	−0.521536	+	0.521536i	0.396499	+	0.918035i	$0.370225\pi$
$510$		0			0
$511$			2.12714i			0.0940991i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.14086	−	2.14086i	−0.0943377	−	0.0943377i
$516$		0			0
$517$	0.841092	−	0.841092i	0.0369912	−	0.0369912i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−29.6216			−1.29775			−0.648873	−	0.760897i	$-0.724759\pi$
$521$	−29.6216			−1.29775			−0.648873	+	0.760897i	$0.724759\pi$
$522$		0			0
$523$	−4.96353	−	4.96353i	−0.217040	−	0.217040i	0.590210	−	0.807250i	$-0.299045\pi$
$523$	−4.96353	−	4.96353i	−0.217040	−	0.217040i	−0.807250	+	0.590210i	$0.799045\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	21.3199			0.928708
$528$		0			0
$529$	19.9183			0.866011
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−27.6317	−	27.6317i	−1.19686	−	1.19686i
$534$		0			0
$535$	−9.81517			−0.424347
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.67857	+	2.67857i	−0.115374	+	0.115374i
$540$		0			0
$541$	−10.0792	−	10.0792i	−0.433338	−	0.433338i	0.456424	−	0.889762i	$-0.349130\pi$
$541$	−10.0792	−	10.0792i	−0.433338	−	0.433338i	−0.889762	+	0.456424i	$0.849130\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			34.0115i			1.45689i
$546$		0			0
$547$	−2.48797	+	2.48797i	−0.106378	+	0.106378i	−0.758293	−	0.651914i	$-0.773966\pi$
$547$	−2.48797	+	2.48797i	−0.106378	+	0.106378i	0.651914	+	0.758293i	$0.273966\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	8.40953i		−	0.358258i
$552$		0			0
$553$			30.3662i			1.29130i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24.2575	−	24.2575i	1.02782	−	1.02782i	0.0282205	−	0.999602i	$-0.491016\pi$
$557$	24.2575	−	24.2575i	1.02782	−	1.02782i	0.999602	−	0.0282205i	$-0.00898407\pi$
$558$		0			0
$559$			19.4009i			0.820570i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−14.4061	−	14.4061i	−0.607143	−	0.607143i	0.335055	−	0.942198i	$-0.391245\pi$
$563$	−14.4061	−	14.4061i	−0.607143	−	0.607143i	−0.942198	+	0.335055i	$0.891245\pi$
$564$		0			0
$565$	−4.41346	+	4.41346i	−0.185676	+	0.185676i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	33.5254			1.40546			0.702729	−	0.711457i	$-0.251964\pi$
$569$	33.5254			1.40546			0.702729	+	0.711457i	$0.251964\pi$
$570$		0			0
$571$	−14.5368	−	14.5368i	−0.608348	−	0.608348i	0.334167	−	0.942514i	$-0.391545\pi$
$571$	−14.5368	−	14.5368i	−0.608348	−	0.608348i	−0.942514	+	0.334167i	$0.891545\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.77813			−0.282668
$576$		0			0
$577$	20.3662			0.847855			0.423927	−	0.905696i	$-0.360651\pi$
$577$	20.3662			0.847855			0.423927	+	0.905696i	$0.360651\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−21.1529	−	21.1529i	−0.877571	−	0.877571i
$582$		0			0
$583$	1.17012			0.0484616
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−17.4809	+	17.4809i	−0.721512	+	0.721512i	−0.968913	−	0.247401i	$-0.920424\pi$
$587$	−17.4809	+	17.4809i	−0.721512	+	0.721512i	0.247401	+	0.968913i	$0.420424\pi$
$588$		0			0
$589$	−6.84109	−	6.84109i	−0.281882	−	0.281882i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	25.6632i		−	1.05386i	−0.849908	−	0.526930i	$-0.823343\pi$
$593$		−	25.6632i		−	1.05386i	0.849908	−	0.526930i	$-0.176657\pi$
$594$		0			0
$595$	−21.9341	+	21.9341i	−0.899212	+	0.899212i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			22.4652i			0.917902i	0.888461	+	0.458951i	$0.151775\pi$
$599$			22.4652i			0.917902i	−0.888461	+	0.458951i	$0.848225\pi$
$600$		0			0
$601$			31.8106i			1.29758i	0.760967	+	0.648790i	$0.224725\pi$
$601$			31.8106i			1.29758i	−0.760967	+	0.648790i	$0.775275\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−22.9581	+	22.9581i	−0.933379	+	0.933379i
$606$		0			0
$607$			14.9829i			0.608138i	0.952650	+	0.304069i	$0.0983453\pi$
$607$			14.9829i			0.608138i	−0.952650	+	0.304069i	$0.901655\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10.6720	−	10.6720i	−0.431741	−	0.431741i
$612$		0			0
$613$	−25.6734	+	25.6734i	−1.03694	+	1.03694i	−0.0376494	+	0.999291i	$0.511987\pi$
$613$	−25.6734	+	25.6734i	−1.03694	+	1.03694i	−0.999291	+	0.0376494i	$0.988013\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−27.1598			−1.09341			−0.546705	−	0.837325i	$-0.684118\pi$
$617$	−27.1598			−1.09341			−0.546705	+	0.837325i	$0.684118\pi$
$618$		0			0
$619$	29.4244	+	29.4244i	1.18267	+	1.18267i	0.979051	+	0.203616i	$0.0652693\pi$
$619$	29.4244	+	29.4244i	1.18267	+	1.18267i	0.203616	+	0.979051i	$0.434731\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−30.8747			−1.23697
$624$		0			0
$625$	29.3974			1.17590
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.33532	+	7.33532i	0.292478	+	0.292478i
$630$		0			0
$631$	−7.84697			−0.312383			−0.156191	−	0.987727i	$-0.549922\pi$
$631$	−7.84697			−0.312383			−0.156191	+	0.987727i	$0.549922\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−21.6386	+	21.6386i	−0.858702	+	0.858702i
$636$		0			0
$637$	33.9862	+	33.9862i	1.34658	+	1.34658i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			37.7956i			1.49284i	0.665477	+	0.746418i	$0.268228\pi$
$641$			37.7956i			1.49284i	−0.665477	+	0.746418i	$0.731772\pi$
$642$		0			0
$643$	−27.3026	+	27.3026i	−1.07671	+	1.07671i	−0.0799071	+	0.996802i	$0.525462\pi$
$643$	−27.3026	+	27.3026i	−1.07671	+	1.07671i	−0.996802	+	0.0799071i	$0.974538\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	0.100686i		−	0.00395836i	−0.999998	−	0.00197918i	$-0.999370\pi$
$647$		−	0.100686i		−	0.00395836i	0.999998	−	0.00197918i	$-0.000629993\pi$
$648$		0			0
$649$			1.46793i			0.0576212i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.10477	+	9.10477i	−0.356297	+	0.356297i	−0.862446	−	0.506149i	$-0.831069\pi$
$653$	−9.10477	+	9.10477i	−0.356297	+	0.356297i	0.506149	+	0.862446i	$0.331069\pi$
$654$		0			0
$655$			28.6640i			1.11999i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14.6141	+	14.6141i	0.569285	+	0.569285i	0.931928	−	0.362643i	$-0.118126\pi$
$659$	14.6141	+	14.6141i	0.569285	+	0.569285i	−0.362643	+	0.931928i	$0.618126\pi$
$660$		0			0
$661$	−5.87057	+	5.87057i	−0.228339	+	0.228339i	−0.811998	−	0.583660i	$-0.801620\pi$
$661$	−5.87057	+	5.87057i	−0.228339	+	0.228339i	0.583660	+	0.811998i	$0.301620\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	14.0764			0.545860
$666$		0			0
$667$	−9.72929	−	9.72929i	−0.376720	−	0.376720i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.758305			−0.0292741
$672$		0			0
$673$	−27.3515			−1.05432			−0.527161	−	0.849766i	$-0.676743\pi$
$673$	−27.3515			−1.05432			−0.527161	+	0.849766i	$0.676743\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	31.2462	+	31.2462i	1.20089	+	1.20089i	0.973897	+	0.226992i	$0.0728891\pi$
$677$	31.2462	+	31.2462i	1.20089	+	1.20089i	0.226992	+	0.973897i	$0.427111\pi$
$678$		0			0
$679$	14.7369			0.565551
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.0378	−	15.0378i	0.575406	−	0.575406i	−0.358228	−	0.933634i	$-0.616619\pi$
$683$	15.0378	−	15.0378i	0.575406	−	0.575406i	0.933634	+	0.358228i	$0.116619\pi$
$684$		0			0
$685$	29.0310	+	29.0310i	1.10922	+	1.10922i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	14.8468i		−	0.565617i
$690$		0			0
$691$	−4.24894	+	4.24894i	−0.161637	+	0.161637i	−0.783292	−	0.621654i	$-0.786461\pi$
$691$	−4.24894	+	4.24894i	−0.161637	+	0.161637i	0.621654	+	0.783292i	$0.286461\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			55.2302i			2.09500i
$696$		0			0
$697$			23.8835i			0.904653i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	23.4629	−	23.4629i	0.886183	−	0.886183i	−0.107971	−	0.994154i	$-0.534435\pi$
$701$	23.4629	−	23.4629i	0.886183	−	0.886183i	0.994154	+	0.107971i	$0.0344354\pi$
$702$		0			0
$703$		−	4.70750i		−	0.177547i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.77278	−	7.77278i	−0.292326	−	0.292326i
$708$		0			0
$709$	−2.79314	+	2.79314i	−0.104898	+	0.104898i	−0.757608	−	0.652710i	$-0.773632\pi$
$709$	−2.79314	+	2.79314i	−0.104898	+	0.104898i	0.652710	+	0.757608i	$0.273632\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−15.8294			−0.592816
$714$		0			0
$715$	−2.48264	−	2.48264i	−0.0928454	−	0.0928454i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−33.8130			−1.26101			−0.630507	−	0.776184i	$-0.717153\pi$
$719$	−33.8130			−1.26101			−0.630507	+	0.776184i	$0.717153\pi$
$720$		0			0
$721$	4.48264			0.166942
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−21.3990	−	21.3990i	−0.794740	−	0.794740i
$726$		0			0
$727$	−2.42732			−0.0900245			−0.0450122	−	0.998986i	$-0.514333\pi$
$727$	−2.42732			−0.0900245			−0.0450122	+	0.998986i	$0.514333\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.38460	−	8.38460i	0.310115	−	0.310115i
$732$		0			0
$733$	−16.8596	−	16.8596i	−0.622725	−	0.622725i	0.323503	−	0.946227i	$-0.395140\pi$
$733$	−16.8596	−	16.8596i	−0.622725	−	0.622725i	−0.946227	+	0.323503i	$0.895140\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	3.93214i		−	0.144842i
$738$		0			0
$739$	−25.2243	+	25.2243i	−0.927892	+	0.927892i	−0.997570	−	0.0696780i	$-0.977803\pi$
$739$	−25.2243	+	25.2243i	−0.927892	+	0.927892i	0.0696780	+	0.997570i	$0.477803\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	33.4926i		−	1.22872i	−0.789025	−	0.614361i	$-0.789414\pi$
$743$		−	33.4926i		−	1.22872i	0.789025	−	0.614361i	$-0.210586\pi$
$744$		0			0
$745$		−	32.6752i		−	1.19713i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.2757	−	10.2757i	0.375467	−	0.375467i
$750$		0			0
$751$		−	17.9247i		−	0.654081i	−0.945010	−	0.327040i	$-0.893949\pi$
$751$		−	17.9247i		−	0.654081i	0.945010	−	0.327040i	$-0.106051\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.426168	−	0.426168i	−0.0155099	−	0.0155099i
$756$		0			0
$757$	−3.10619	+	3.10619i	−0.112896	+	0.112896i	−0.761298	−	0.648402i	$-0.775438\pi$
$757$	−3.10619	+	3.10619i	−0.112896	+	0.112896i	0.648402	+	0.761298i	$0.275438\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.10917			0.0402073			0.0201037	−	0.999798i	$-0.493600\pi$
$761$	1.10917			0.0402073			0.0201037	+	0.999798i	$0.493600\pi$
$762$		0			0
$763$	−35.6075	−	35.6075i	−1.28908	−	1.28908i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	18.6254			0.672523
$768$		0			0
$769$	22.4591			0.809897			0.404948	−	0.914340i	$-0.367289\pi$
$769$	22.4591			0.809897			0.404948	+	0.914340i	$0.367289\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	22.6327	+	22.6327i	0.814041	+	0.814041i	0.985237	−	0.171196i	$-0.0547631\pi$
$773$	22.6327	+	22.6327i	0.814041	+	0.814041i	−0.171196	+	0.985237i	$0.554763\pi$
$774$		0			0
$775$	−34.8159			−1.25062
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7.66372	−	7.66372i	0.274581	−	0.274581i
$780$		0			0
$781$	2.56330	+	2.56330i	0.0917221	+	0.0917221i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	15.8232i		−	0.564754i
$786$		0			0
$787$	38.6505	−	38.6505i	1.37774	−	1.37774i	0.529315	−	0.848425i	$-0.322449\pi$
$787$	38.6505	−	38.6505i	1.37774	−	1.37774i	0.848425	−	0.529315i	$-0.177551\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	9.24111i		−	0.328576i
$792$		0			0
$793$			9.62153i			0.341671i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−37.8969	+	37.8969i	−1.34238	+	1.34238i	−0.448686	+	0.893689i	$0.648108\pi$
$797$	−37.8969	+	37.8969i	−1.34238	+	1.34238i	−0.893689	+	0.448686i	$0.851892\pi$
$798$		0			0
$799$			9.22432i			0.326333i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.104052	+	0.104052i	0.00367191	+	0.00367191i
$804$		0			0
$805$	16.2855	−	16.2855i	0.573989	−	0.573989i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	27.0719			0.951799			0.475899	−	0.879500i	$-0.342123\pi$
$809$	27.0719			0.951799			0.475899	+	0.879500i	$0.342123\pi$
$810$		0			0
$811$	19.0554	+	19.0554i	0.669126	+	0.669126i	0.957514	−	0.288388i	$-0.0931193\pi$
$811$	19.0554	+	19.0554i	0.669126	+	0.669126i	−0.288388	+	0.957514i	$0.593119\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	55.8571			1.95659
$816$		0			0
$817$	−5.38088			−0.188253
$818$		0			0
$819$		0			0
$820$		0			0
$821$	37.5393	+	37.5393i	1.31013	+	1.31013i	0.921315	+	0.388817i	$0.127116\pi$
$821$	37.5393	+	37.5393i	1.31013	+	1.31013i	0.388817	+	0.921315i	$0.372884\pi$
$822$		0			0
$823$	−27.9589			−0.974584			−0.487292	−	0.873239i	$-0.662015\pi$
$823$	−27.9589			−0.974584			−0.487292	+	0.873239i	$0.662015\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.04182	−	2.04182i	0.0710011	−	0.0710011i	−0.670715	−	0.741716i	$-0.734012\pi$
$827$	2.04182	−	2.04182i	0.0710011	−	0.0710011i	0.741716	+	0.670715i	$0.234012\pi$
$828$		0			0
$829$	−34.5808	−	34.5808i	−1.20104	−	1.20104i	−0.973851	−	0.227188i	$-0.927047\pi$
$829$	−34.5808	−	34.5808i	−1.20104	−	1.20104i	−0.227188	−	0.973851i	$-0.572953\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	29.3760i		−	1.01782i
$834$		0			0
$835$	42.4439	−	42.4439i	1.46883	−	1.46883i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	0.865723i		−	0.0298881i	−0.999888	−	0.0149440i	$-0.995243\pi$
$839$		−	0.865723i		−	0.0298881i	0.999888	−	0.0149440i	$-0.00475701\pi$
$840$		0			0
$841$		−	32.4321i		−	1.11835i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.13669	+	4.13669i	−0.142306	+	0.142306i
$846$		0			0
$847$		−	48.0707i		−	1.65173i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5.44628	−	5.44628i	−0.186696	−	0.186696i
$852$		0			0
$853$	14.4262	−	14.4262i	0.493942	−	0.493942i	−0.415604	−	0.909546i	$-0.636430\pi$
$853$	14.4262	−	14.4262i	0.493942	−	0.493942i	0.909546	+	0.415604i	$0.136430\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−37.6532			−1.28621			−0.643105	−	0.765778i	$-0.722354\pi$
$857$	−37.6532			−1.28621			−0.643105	+	0.765778i	$0.722354\pi$
$858$		0			0
$859$	38.7470	+	38.7470i	1.32203	+	1.32203i	0.912130	+	0.409901i	$0.134437\pi$
$859$	38.7470	+	38.7470i	1.32203	+	1.32203i	0.409901	+	0.912130i	$0.365563\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14.8672			0.506085			0.253043	−	0.967455i	$-0.418569\pi$
$863$	14.8672			0.506085			0.253043	+	0.967455i	$0.418569\pi$
$864$		0			0
$865$	16.7681			0.570134
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.48540	+	1.48540i	0.0503887	+	0.0503887i
$870$		0			0
$871$	−49.8918			−1.69052
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10.5654	+	10.5654i	−0.357176	+	0.357176i
$876$		0			0
$877$	13.7699	+	13.7699i	0.464976	+	0.464976i	0.900283	−	0.435306i	$-0.143360\pi$
$877$	13.7699	+	13.7699i	0.464976	+	0.464976i	−0.435306	+	0.900283i	$0.643360\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	20.6232i		−	0.694812i	−0.937715	−	0.347406i	$-0.887063\pi$
$881$		−	20.6232i		−	0.694812i	0.937715	−	0.347406i	$-0.112937\pi$
$882$		0			0
$883$	41.5416	−	41.5416i	1.39799	−	1.39799i	0.592185	−	0.805802i	$-0.298265\pi$
$883$	41.5416	−	41.5416i	1.39799	−	1.39799i	0.805802	−	0.592185i	$-0.201735\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			7.44312i			0.249916i	0.992162	+	0.124958i	$0.0398795\pi$
$887$			7.44312i			0.249916i	−0.992162	+	0.124958i	$0.960120\pi$
$888$		0			0
$889$		−	45.3079i		−	1.51958i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.95989	−	2.95989i	0.0990489	−	0.0990489i
$894$		0			0
$895$			61.9602i			2.07110i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−49.9746	−	49.9746i	−1.66675	−	1.66675i
$900$		0			0
$901$	−6.41642	+	6.41642i	−0.213762	+	0.213762i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−13.7431			−0.456835
$906$		0			0
$907$	−3.75106	−	3.75106i	−0.124552	−	0.124552i	0.642083	−	0.766635i	$-0.278070\pi$
$907$	−3.75106	−	3.75106i	−0.124552	−	0.124552i	−0.766635	+	0.642083i	$0.778070\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	29.9816			0.993334			0.496667	−	0.867941i	$-0.334557\pi$
$911$	29.9816			0.993334			0.496667	+	0.867941i	$0.334557\pi$
$912$		0			0
$913$	−2.06944			−0.0684886
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−30.0090	−	30.0090i	−0.990984	−	0.990984i
$918$		0			0
$919$	38.1402			1.25813			0.629064	−	0.777353i	$-0.283438\pi$
$919$	38.1402			1.25813			0.629064	+	0.777353i	$0.283438\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	32.5237	−	32.5237i	1.07053	−	1.07053i
$924$		0			0
$925$	−11.9788	−	11.9788i	−0.393860	−	0.393860i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	13.8738i		−	0.455183i	−0.973757	−	0.227592i	$-0.926915\pi$
$929$		−	13.8738i		−	0.455183i	0.973757	−	0.227592i	$-0.0730851\pi$
$930$		0			0
$931$	−9.42615	+	9.42615i	−0.308930	+	0.308930i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			2.14587i			0.0701775i
$936$		0			0
$937$			26.4097i			0.862767i	0.902169	+	0.431384i	$0.141974\pi$
$937$			26.4097i			0.862767i	−0.902169	+	0.431384i	$0.858026\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.91123	−	4.91123i	0.160102	−	0.160102i	−0.622510	−	0.782612i	$-0.713887\pi$
$941$	4.91123	−	4.91123i	0.160102	−	0.160102i	0.782612	+	0.622510i	$0.213887\pi$
$942$		0			0
$943$		−	17.7329i		−	0.577462i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.84386	+	6.84386i	0.222396	+	0.222396i	0.809507	−	0.587111i	$-0.199735\pi$
$947$	6.84386	+	6.84386i	0.222396	+	0.222396i	−0.587111	+	0.809507i	$0.699735\pi$
$948$		0			0
$949$	1.32023	−	1.32023i	0.0428565	−	0.0428565i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4.33045			0.140277			0.0701386	−	0.997537i	$-0.477656\pi$
$953$	4.33045			0.140277			0.0701386	+	0.997537i	$0.477656\pi$
$954$		0			0
$955$	16.2048	+	16.2048i	0.524377	+	0.524377i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−60.7864			−1.96289
$960$		0			0
$961$	−50.3079			−1.62284
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.40365	−	7.40365i	−0.238332	−	0.238332i
$966$		0			0
$967$	40.6664			1.30774			0.653871	−	0.756606i	$-0.273144\pi$
$967$	40.6664			1.30774			0.653871	+	0.756606i	$0.273144\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24.0742	+	24.0742i	−0.772577	+	0.772577i	−0.978556	−	0.205979i	$-0.933962\pi$
$971$	−24.0742	+	24.0742i	−0.772577	+	0.772577i	0.205979	+	0.978556i	$0.433962\pi$
$972$		0			0
$973$	−57.8218	−	57.8218i	−1.85368	−	1.85368i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			21.0381i			0.673068i	0.941671	+	0.336534i	$0.109255\pi$
$977$			21.0381i			0.673068i	−0.941671	+	0.336534i	$0.890745\pi$
$978$		0			0
$979$	−1.51028	+	1.51028i	−0.0482686	+	0.0482686i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			14.1206i			0.450376i	0.974315	+	0.225188i	$0.0722997\pi$
$983$			14.1206i			0.450376i	−0.974315	+	0.225188i	$0.927700\pi$
$984$		0			0
$985$		−	26.9618i		−	0.859074i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6.22534	+	6.22534i	−0.197954	+	0.197954i
$990$		0			0
$991$		−	33.3338i		−	1.05888i	−0.848346	−	0.529442i	$-0.822401\pi$
$991$		−	33.3338i		−	1.05888i	0.848346	−	0.529442i	$-0.177599\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	49.1472	+	49.1472i	1.55807	+	1.55807i
$996$		0			0
$997$	−24.8358	+	24.8358i	−0.786559	+	0.786559i	−0.980928	−	0.194369i	$-0.937734\pi$
$997$	−24.8358	+	24.8358i	−0.786559	+	0.786559i	0.194369	+	0.980928i	$0.437734\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.2.l.a.863.7		16
3.2	odd	2	inner	1152.2.l.a.863.2		16
4.3	odd	2		1152.2.l.b.863.7		16
8.3	odd	2		576.2.l.a.431.2		16
8.5	even	2		144.2.l.a.35.1	✓	16
12.11	even	2		1152.2.l.b.863.2		16
16.3	odd	4		144.2.l.a.107.8	yes	16
16.5	even	4		1152.2.l.b.287.2		16
16.11	odd	4	inner	1152.2.l.a.287.2		16
16.13	even	4		576.2.l.a.143.7		16
24.5	odd	2		144.2.l.a.35.8	yes	16
24.11	even	2		576.2.l.a.431.7		16
48.5	odd	4		1152.2.l.b.287.7		16
48.11	even	4	inner	1152.2.l.a.287.7		16
48.29	odd	4		576.2.l.a.143.2		16
48.35	even	4		144.2.l.a.107.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.l.a.35.1	✓	16	8.5	even	2
144.2.l.a.35.8	yes	16	24.5	odd	2
144.2.l.a.107.1	yes	16	48.35	even	4
144.2.l.a.107.8	yes	16	16.3	odd	4
576.2.l.a.143.2		16	48.29	odd	4
576.2.l.a.143.7		16	16.13	even	4
576.2.l.a.431.2		16	8.3	odd	2
576.2.l.a.431.7		16	24.11	even	2
1152.2.l.a.287.2		16	16.11	odd	4	inner
1152.2.l.a.287.7		16	48.11	even	4	inner
1152.2.l.a.863.2		16	3.2	odd	2	inner
1152.2.l.a.863.7		16	1.1	even	1	trivial
1152.2.l.b.287.2		16	16.5	even	4
1152.2.l.b.287.7		16	48.5	odd	4
1152.2.l.b.863.2		16	12.11	even	2
1152.2.l.b.863.7		16	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.10489 + 2.10489i) q^{5} -4.40731 q^{7} +O(q^{10})\)	\(q+(2.10489 + 2.10489i) q^{5} -4.40731 q^{7} +(-0.215589 + 0.215589i) q^{11} +(2.73544 + 2.73544i) q^{13} -2.36438i q^{17} +(-0.758681 + 0.758681i) q^{19} +1.75549i q^{23} +3.86110i q^{25} +(-5.54221 + 5.54221i) q^{29} +9.01709i q^{31} +(-9.27690 - 9.27690i) q^{35} +(-3.10242 + 3.10242i) q^{37} -10.1014 q^{41} +(3.54621 + 3.54621i) q^{43} -3.90136 q^{47} +12.4244 q^{49} +(-2.71378 - 2.71378i) q^{53} -0.907583 q^{55} +(3.40445 - 3.40445i) q^{59} +(1.75868 + 1.75868i) q^{61} +11.5156i q^{65} +(-9.11951 + 9.11951i) q^{67} -11.8897i q^{71} -0.482639i q^{73} +(0.950169 - 0.950169i) q^{77} -6.88995i q^{79} +(4.79951 + 4.79951i) q^{83} +(4.97676 - 4.97676i) q^{85} +7.00534 q^{89} +(-12.0559 - 12.0559i) q^{91} -3.19387 q^{95} -3.34374 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{19} + 32 q^{43} + 16 q^{49} - 64 q^{55} + 32 q^{61} + 16 q^{67} + 32 q^{85} - 48 q^{91}+O(q^{100}) \) 16 * q - 16 * q^19 + 32 * q^43 + 16 * q^49 - 64 * q^55 + 32 * q^61 + 16 * q^67 + 32 * q^85 - 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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