LMFDB - Embedded newform 1152.3.j.a.161.4

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			161.4
Character	$\chi$	$=$	1152.161
Dual form			1152.3.j.a.737.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.90343 - 3.90343i) q^{5} -0.778757i q^{7} +O(q^{10})$	$q+(-3.90343 - 3.90343i) q^{5} -0.778757i q^{7} +(-4.29098 - 4.29098i) q^{11} +(-6.44007 - 6.44007i) q^{13} +31.3383i q^{17} +(-1.11906 - 1.11906i) q^{19} -34.2197 q^{23} +5.47346i q^{25} +(-8.77471 + 8.77471i) q^{29} +50.8507 q^{31} +(-3.03982 + 3.03982i) q^{35} +(29.3064 - 29.3064i) q^{37} +31.4271 q^{41} +(-55.9022 + 55.9022i) q^{43} -26.5249i q^{47} +48.3935 q^{49} +(-9.76389 - 9.76389i) q^{53} +33.4991i q^{55} +(54.3706 + 54.3706i) q^{59} +(47.1104 + 47.1104i) q^{61} +50.2766i q^{65} +(66.1339 + 66.1339i) q^{67} -75.9403 q^{71} +24.1992i q^{73} +(-3.34163 + 3.34163i) q^{77} -80.5454 q^{79} +(-82.6319 + 82.6319i) q^{83} +(122.327 - 122.327i) q^{85} +82.6445 q^{89} +(-5.01524 + 5.01524i) q^{91} +8.73634i q^{95} +48.9478 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.90343	−	3.90343i	−0.780685	−	0.780685i	0.199261	−	0.979946i	$-0.436146\pi$
$5$	−3.90343	−	3.90343i	−0.780685	−	0.780685i	−0.979946	+	0.199261i	$0.936146\pi$
$6$		0			0
$7$		−	0.778757i		−	0.111251i	−0.998452	−	0.0556255i	$-0.982285\pi$
$7$		−	0.778757i		−	0.111251i	0.998452	−	0.0556255i	$-0.0177153\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.29098	−	4.29098i	−0.390089	−	0.390089i	0.484630	−	0.874719i	$-0.338954\pi$
$11$	−4.29098	−	4.29098i	−0.390089	−	0.390089i	−0.874719	+	0.484630i	$0.838954\pi$
$12$		0			0
$13$	−6.44007	−	6.44007i	−0.495390	−	0.495390i	0.414610	−	0.909999i	$-0.363918\pi$
$13$	−6.44007	−	6.44007i	−0.495390	−	0.495390i	−0.909999	+	0.414610i	$0.863918\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			31.3383i			1.84343i	0.387872	+	0.921713i	$0.373210\pi$
$17$			31.3383i			1.84343i	−0.387872	+	0.921713i	$0.626790\pi$
$18$		0			0
$19$	−1.11906	−	1.11906i	−0.0588979	−	0.0588979i	0.677044	−	0.735942i	$-0.263261\pi$
$19$	−1.11906	−	1.11906i	−0.0588979	−	0.0588979i	−0.735942	+	0.677044i	$0.763261\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−34.2197			−1.48781			−0.743906	−	0.668285i	$-0.767029\pi$
$23$	−34.2197			−1.48781			−0.743906	+	0.668285i	$0.767029\pi$
$24$		0			0
$25$			5.47346i			0.218939i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.77471	+	8.77471i	−0.302576	+	0.302576i	−0.842021	−	0.539445i	$-0.818634\pi$
$29$	−8.77471	+	8.77471i	−0.302576	+	0.302576i	0.539445	+	0.842021i	$0.318634\pi$
$30$		0			0
$31$	50.8507			1.64035			0.820173	−	0.572116i	$-0.193877\pi$
$31$	50.8507			1.64035			0.820173	+	0.572116i	$0.193877\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.03982	+	3.03982i	−0.0868520	+	0.0868520i
$36$		0			0
$37$	29.3064	−	29.3064i	0.792065	−	0.792065i	−0.189765	−	0.981830i	$-0.560773\pi$
$37$	29.3064	−	29.3064i	0.792065	−	0.792065i	0.981830	+	0.189765i	$0.0607725\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	31.4271			0.766513			0.383257	−	0.923642i	$-0.374802\pi$
$41$	31.4271			0.766513			0.383257	+	0.923642i	$0.374802\pi$
$42$		0			0
$43$	−55.9022	+	55.9022i	−1.30005	+	1.30005i	−0.371696	+	0.928354i	$0.621224\pi$
$43$	−55.9022	+	55.9022i	−1.30005	+	1.30005i	−0.928354	+	0.371696i	$0.878776\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	26.5249i		−	0.564359i	−0.959362	−	0.282180i	$-0.908943\pi$
$47$		−	26.5249i		−	0.564359i	0.959362	−	0.282180i	$-0.0910574\pi$
$48$		0			0
$49$	48.3935			0.987623
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.76389	−	9.76389i	−0.184224	−	0.184224i	0.608969	−	0.793194i	$-0.291583\pi$
$53$	−9.76389	−	9.76389i	−0.184224	−	0.184224i	−0.793194	+	0.608969i	$0.791583\pi$
$54$		0			0
$55$			33.4991i			0.609074i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	54.3706	+	54.3706i	0.921535	+	0.921535i	0.997138	−	0.0756026i	$-0.0240881\pi$
$59$	54.3706	+	54.3706i	0.921535	+	0.921535i	−0.0756026	+	0.997138i	$0.524088\pi$
$60$		0			0
$61$	47.1104	+	47.1104i	0.772302	+	0.772302i	0.978508	−	0.206207i	$-0.0661119\pi$
$61$	47.1104	+	47.1104i	0.772302	+	0.772302i	−0.206207	+	0.978508i	$0.566112\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			50.2766i			0.773487i
$66$		0			0
$67$	66.1339	+	66.1339i	0.987073	+	0.987073i	0.999918	−	0.0128446i	$-0.00408867\pi$
$67$	66.1339	+	66.1339i	0.987073	+	0.987073i	−0.0128446	+	0.999918i	$0.504089\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−75.9403			−1.06958			−0.534791	−	0.844985i	$-0.679609\pi$
$71$	−75.9403			−1.06958			−0.534791	+	0.844985i	$0.679609\pi$
$72$		0			0
$73$			24.1992i			0.331497i	0.986168	+	0.165748i	$0.0530039\pi$
$73$			24.1992i			0.331497i	−0.986168	+	0.165748i	$0.946996\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.34163	+	3.34163i	−0.0433978	+	0.0433978i
$78$		0			0
$79$	−80.5454			−1.01956			−0.509781	−	0.860304i	$-0.670274\pi$
$79$	−80.5454			−1.01956			−0.509781	+	0.860304i	$0.670274\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−82.6319	+	82.6319i	−0.995565	+	0.995565i	−0.999990	−	0.00442530i	$-0.998591\pi$
$83$	−82.6319	+	82.6319i	−0.995565	+	0.995565i	0.00442530	+	0.999990i	$0.498591\pi$
$84$		0			0
$85$	122.327	−	122.327i	1.43914	−	1.43914i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	82.6445			0.928590			0.464295	−	0.885681i	$-0.346308\pi$
$89$	82.6445			0.928590			0.464295	+	0.885681i	$0.346308\pi$
$90$		0			0
$91$	−5.01524	+	5.01524i	−0.0551126	+	0.0551126i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			8.73634i			0.0919614i
$96$		0			0
$97$	48.9478			0.504617			0.252308	−	0.967647i	$-0.418810\pi$
$97$	48.9478			0.504617			0.252308	+	0.967647i	$0.418810\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	41.2778	+	41.2778i	0.408691	+	0.408691i	0.881282	−	0.472591i	$-0.156681\pi$
$101$	41.2778	+	41.2778i	0.408691	+	0.408691i	−0.472591	+	0.881282i	$0.656681\pi$
$102$		0			0
$103$			173.236i			1.68190i	0.541111	+	0.840951i	$0.318004\pi$
$103$			173.236i			1.68190i	−0.541111	+	0.840951i	$0.681996\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−53.2761	−	53.2761i	−0.497907	−	0.497907i	0.412879	−	0.910786i	$-0.364523\pi$
$107$	−53.2761	−	53.2761i	−0.497907	−	0.497907i	−0.910786	+	0.412879i	$0.864523\pi$
$108$		0			0
$109$	93.1537	+	93.1537i	0.854621	+	0.854621i	0.990698	−	0.136077i	$-0.0434494\pi$
$109$	93.1537	+	93.1537i	0.854621	+	0.854621i	−0.136077	+	0.990698i	$0.543449\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			40.8189i			0.361229i	0.983554	+	0.180615i	$0.0578087\pi$
$113$			40.8189i			0.361229i	−0.983554	+	0.180615i	$0.942191\pi$
$114$		0			0
$115$	133.574	+	133.574i	1.16151	+	1.16151i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	24.4049			0.205083
$120$		0			0
$121$		−	84.1749i		−	0.695661i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−76.2204	+	76.2204i	−0.609763	+	0.609763i
$126$		0			0
$127$	88.9136			0.700107			0.350054	−	0.936730i	$-0.386163\pi$
$127$	88.9136			0.700107			0.350054	+	0.936730i	$0.386163\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	22.0108	−	22.0108i	0.168021	−	0.168021i	−0.618088	−	0.786109i	$-0.712092\pi$
$131$	22.0108	−	22.0108i	0.168021	−	0.168021i	0.786109	+	0.618088i	$0.212092\pi$
$132$		0			0
$133$	−0.871475	+	0.871475i	−0.00655245	+	0.00655245i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−110.794			−0.808715			−0.404358	−	0.914601i	$-0.632505\pi$
$137$	−110.794			−0.808715			−0.404358	+	0.914601i	$0.632505\pi$
$138$		0			0
$139$	−30.3862	+	30.3862i	−0.218605	+	0.218605i	−0.807911	−	0.589305i	$-0.799402\pi$
$139$	−30.3862	+	30.3862i	−0.218605	+	0.218605i	0.589305	+	0.807911i	$0.299402\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			55.2684i			0.386493i
$144$		0			0
$145$	68.5029			0.472433
$146$		0			0
$147$		0			0
$148$		0			0
$149$	173.924	+	173.924i	1.16727	+	1.16727i	0.982846	+	0.184428i	$0.0590433\pi$
$149$	173.924	+	173.924i	1.16727	+	1.16727i	0.184428	+	0.982846i	$0.440957\pi$
$150$		0			0
$151$		−	26.2521i		−	0.173855i	−0.996215	−	0.0869274i	$-0.972295\pi$
$151$		−	26.2521i		−	0.173855i	0.996215	−	0.0869274i	$-0.0277048\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−198.492	−	198.492i	−1.28059	−	1.28059i
$156$		0			0
$157$	−75.8493	−	75.8493i	−0.483117	−	0.483117i	0.423009	−	0.906126i	$-0.360974\pi$
$157$	−75.8493	−	75.8493i	−0.483117	−	0.483117i	−0.906126	+	0.423009i	$0.860974\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			26.6488i			0.165520i
$162$		0			0
$163$	−155.285	−	155.285i	−0.952672	−	0.952672i	0.0462578	−	0.998930i	$-0.485270\pi$
$163$	−155.285	−	155.285i	−0.952672	−	0.952672i	−0.998930	+	0.0462578i	$0.985270\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−236.902			−1.41857			−0.709286	−	0.704920i	$-0.750983\pi$
$167$	−236.902			−1.41857			−0.709286	+	0.704920i	$0.750983\pi$
$168$		0			0
$169$		−	86.0511i		−	0.509178i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−68.3708	+	68.3708i	−0.395207	+	0.395207i	−0.876539	−	0.481332i	$-0.840153\pi$
$173$	−68.3708	+	68.3708i	−0.395207	+	0.395207i	0.481332	+	0.876539i	$0.340153\pi$
$174$		0			0
$175$	4.26250			0.0243571
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−129.626	+	129.626i	−0.724170	+	0.724170i	−0.969452	−	0.245282i	$-0.921120\pi$
$179$	−129.626	+	129.626i	−0.724170	+	0.724170i	0.245282	+	0.969452i	$0.421120\pi$
$180$		0			0
$181$	−157.641	+	157.641i	−0.870947	+	0.870947i	−0.992576	−	0.121629i	$-0.961188\pi$
$181$	−157.641	+	157.641i	−0.870947	+	0.870947i	0.121629	+	0.992576i	$0.461188\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−228.791			−1.23671
$186$		0			0
$187$	134.472	−	134.472i	0.719101	−	0.719101i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			58.1217i			0.304302i	0.988357	+	0.152151i	$0.0486200\pi$
$191$			58.1217i			0.304302i	−0.988357	+	0.152151i	$0.951380\pi$
$192$		0			0
$193$	312.491			1.61912			0.809561	−	0.587035i	$-0.199705\pi$
$193$	312.491			1.61912			0.809561	+	0.587035i	$0.199705\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	44.8306	+	44.8306i	0.227567	+	0.227567i	0.811675	−	0.584109i	$-0.198556\pi$
$197$	44.8306	+	44.8306i	0.227567	+	0.227567i	−0.584109	+	0.811675i	$0.698556\pi$
$198$		0			0
$199$			157.131i			0.789601i	0.918767	+	0.394801i	$0.129186\pi$
$199$			157.131i			0.789601i	−0.918767	+	0.394801i	$0.870814\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.83336	+	6.83336i	0.0336619	+	0.0336619i
$204$		0			0
$205$	−122.673	−	122.673i	−0.598406	−	0.598406i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			9.60374i			0.0459509i
$210$		0			0
$211$	68.3433	+	68.3433i	0.323902	+	0.323902i	0.850262	−	0.526360i	$-0.176444\pi$
$211$	68.3433	+	68.3433i	0.323902	+	0.323902i	−0.526360	+	0.850262i	$0.676444\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	436.420			2.02986
$216$		0			0
$217$		−	39.6003i		−	0.182490i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	201.820	−	201.820i	0.913214	−	0.913214i
$222$		0			0
$223$	−3.75135			−0.0168222			−0.00841111	−	0.999965i	$-0.502677\pi$
$223$	−3.75135			−0.0168222			−0.00841111	+	0.999965i	$0.502677\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	224.963	−	224.963i	0.991028	−	0.991028i	−0.00893184	−	0.999960i	$-0.502843\pi$
$227$	224.963	−	224.963i	0.991028	−	0.991028i	0.999960	+	0.00893184i	$0.00284313\pi$
$228$		0			0
$229$	−26.3030	+	26.3030i	−0.114860	+	0.114860i	−0.762201	−	0.647341i	$-0.775881\pi$
$229$	−26.3030	+	26.3030i	−0.114860	+	0.114860i	0.647341	+	0.762201i	$0.275881\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	166.492			0.714557			0.357279	−	0.933998i	$-0.383705\pi$
$233$	166.492			0.714557			0.357279	+	0.933998i	$0.383705\pi$
$234$		0			0
$235$	−103.538	+	103.538i	−0.440587	+	0.440587i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			247.174i			1.03420i	0.855925	+	0.517100i	$0.172989\pi$
$239$			247.174i			1.03420i	−0.855925	+	0.517100i	$0.827011\pi$
$240$		0			0
$241$	14.7552			0.0612247			0.0306124	−	0.999531i	$-0.490254\pi$
$241$	14.7552			0.0612247			0.0306124	+	0.999531i	$0.490254\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−188.901	−	188.901i	−0.771023	−	0.771023i
$246$		0			0
$247$			14.4136i			0.0583548i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−41.5560	−	41.5560i	−0.165562	−	0.165562i	0.619464	−	0.785025i	$-0.287350\pi$
$251$	−41.5560	−	41.5560i	−0.165562	−	0.165562i	−0.785025	+	0.619464i	$0.787350\pi$
$252$		0			0
$253$	146.836	+	146.836i	0.580379	+	0.580379i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	189.265i		−	0.736439i	−0.929739	−	0.368220i	$-0.879967\pi$
$257$		−	189.265i		−	0.736439i	0.929739	−	0.368220i	$-0.120033\pi$
$258$		0			0
$259$	−22.8226	−	22.8226i	−0.0881180	−	0.0881180i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	57.5273			0.218735			0.109368	−	0.994001i	$-0.465117\pi$
$263$	57.5273			0.218735			0.109368	+	0.994001i	$0.465117\pi$
$264$		0			0
$265$			76.2252i			0.287642i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	121.413	−	121.413i	0.451350	−	0.451350i	−0.444452	−	0.895803i	$-0.646602\pi$
$269$	121.413	−	121.413i	0.451350	−	0.451350i	0.895803	+	0.444452i	$0.146602\pi$
$270$		0			0
$271$	−217.759			−0.803539			−0.401769	−	0.915741i	$-0.631605\pi$
$271$	−217.759			−0.803539			−0.401769	+	0.915741i	$0.631605\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	23.4865	−	23.4865i	0.0854056	−	0.0854056i
$276$		0			0
$277$	−56.8037	+	56.8037i	−0.205068	+	0.205068i	−0.802167	−	0.597100i	$-0.796320\pi$
$277$	−56.8037	+	56.8037i	−0.205068	+	0.205068i	0.597100	+	0.802167i	$0.296320\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	58.9828			0.209903			0.104952	−	0.994477i	$-0.466531\pi$
$281$	58.9828			0.209903			0.104952	+	0.994477i	$0.466531\pi$
$282$		0			0
$283$	170.002	−	170.002i	0.600713	−	0.600713i	−0.339789	−	0.940502i	$-0.610356\pi$
$283$	170.002	−	170.002i	0.600713	−	0.600713i	0.940502	+	0.339789i	$0.110356\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	24.4740i		−	0.0852753i
$288$		0			0
$289$	−693.086			−2.39822
$290$		0			0
$291$		0			0
$292$		0			0
$293$	46.1968	+	46.1968i	0.157668	+	0.157668i	0.781533	−	0.623864i	$-0.214438\pi$
$293$	46.1968	+	46.1968i	0.157668	+	0.157668i	−0.623864	+	0.781533i	$0.714438\pi$
$294$		0			0
$295$		−	424.463i		−	1.43886i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	220.377	+	220.377i	0.737046	+	0.737046i
$300$		0			0
$301$	43.5342	+	43.5342i	0.144632	+	0.144632i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	367.784i		−	1.20585i
$306$		0			0
$307$	−361.635	−	361.635i	−1.17796	−	1.17796i	−0.980262	−	0.197701i	$-0.936653\pi$
$307$	−361.635	−	361.635i	−1.17796	−	1.17796i	−0.197701	−	0.980262i	$-0.563347\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	175.216			0.563395			0.281697	−	0.959503i	$-0.409103\pi$
$311$	175.216			0.563395			0.281697	+	0.959503i	$0.409103\pi$
$312$		0			0
$313$			432.899i			1.38307i	0.722345	+	0.691533i	$0.243064\pi$
$313$			432.899i			1.38307i	−0.722345	+	0.691533i	$0.756936\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−180.916	+	180.916i	−0.570714	+	0.570714i	−0.932328	−	0.361614i	$-0.882226\pi$
$317$	−180.916	+	180.916i	−0.570714	+	0.570714i	0.361614	+	0.932328i	$0.382226\pi$
$318$		0			0
$319$	75.3043			0.236064
$320$		0			0
$321$		0			0
$322$		0			0
$323$	35.0694	−	35.0694i	0.108574	−	0.108574i
$324$		0			0
$325$	35.2495	−	35.2495i	0.108460	−	0.108460i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−20.6564			−0.0627855
$330$		0			0
$331$	−98.3525	+	98.3525i	−0.297137	+	0.297137i	−0.839892	−	0.542754i	$-0.817382\pi$
$331$	−98.3525	+	98.3525i	−0.297137	+	0.297137i	0.542754	+	0.839892i	$0.317382\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	516.297i		−	1.54119i
$336$		0			0
$337$	−384.266			−1.14026			−0.570128	−	0.821556i	$-0.693106\pi$
$337$	−384.266			−1.14026			−0.570128	+	0.821556i	$0.693106\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−218.200	−	218.200i	−0.639881	−	0.639881i
$342$		0			0
$343$		−	75.8459i		−	0.221125i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	380.232	+	380.232i	1.09577	+	1.09577i	0.994900	+	0.100869i	$0.0321623\pi$
$347$	380.232	+	380.232i	1.09577	+	1.09577i	0.100869	+	0.994900i	$0.467838\pi$
$348$		0			0
$349$	−260.199	−	260.199i	−0.745555	−	0.745555i	0.228086	−	0.973641i	$-0.426753\pi$
$349$	−260.199	−	260.199i	−0.745555	−	0.745555i	−0.973641	+	0.228086i	$0.926753\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	97.9162i		−	0.277383i	−0.990336	−	0.138692i	$-0.955710\pi$
$353$		−	97.9162i		−	0.277383i	0.990336	−	0.138692i	$-0.0442897\pi$
$354$		0			0
$355$	296.427	+	296.427i	0.835006	+	0.835006i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−126.839			−0.353311			−0.176656	−	0.984273i	$-0.556528\pi$
$359$	−126.839			−0.353311			−0.176656	+	0.984273i	$0.556528\pi$
$360$		0			0
$361$		−	358.495i		−	0.993062i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	94.4600	−	94.4600i	0.258794	−	0.258794i
$366$		0			0
$367$	−354.084			−0.964807			−0.482404	−	0.875949i	$-0.660236\pi$
$367$	−354.084			−0.964807			−0.482404	+	0.875949i	$0.660236\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.60369	+	7.60369i	−0.0204951	+	0.0204951i
$372$		0			0
$373$	−77.4388	+	77.4388i	−0.207611	+	0.207611i	−0.803251	−	0.595640i	$-0.796898\pi$
$373$	−77.4388	+	77.4388i	−0.207611	+	0.207611i	0.595640	+	0.803251i	$0.296898\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	113.019			0.299786
$378$		0			0
$379$	209.955	−	209.955i	0.553971	−	0.553971i	−0.373614	−	0.927584i	$-0.621881\pi$
$379$	209.955	−	209.955i	0.553971	−	0.553971i	0.927584	+	0.373614i	$0.121881\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	579.620i		−	1.51337i	−0.653781	−	0.756684i	$-0.726818\pi$
$383$		−	579.620i		−	1.51337i	0.653781	−	0.756684i	$-0.273182\pi$
$384$		0			0
$385$	26.0876			0.0677601
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−420.113	−	420.113i	−1.07998	−	1.07998i	−0.996510	−	0.0834708i	$-0.973399\pi$
$389$	−420.113	−	420.113i	−1.07998	−	1.07998i	−0.0834708	−	0.996510i	$-0.526601\pi$
$390$		0			0
$391$		−	1072.38i		−	2.74267i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	314.403	+	314.403i	0.795957	+	0.795957i
$396$		0			0
$397$	−13.0474	−	13.0474i	−0.0328650	−	0.0328650i	0.690483	−	0.723348i	$-0.257398\pi$
$397$	−13.0474	−	13.0474i	−0.0328650	−	0.0328650i	−0.723348	+	0.690483i	$0.757398\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			152.424i			0.380110i	0.981773	+	0.190055i	$0.0608666\pi$
$401$			152.424i			0.380110i	−0.981773	+	0.190055i	$0.939133\pi$
$402$		0			0
$403$	−327.482	−	327.482i	−0.812610	−	0.812610i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−251.507			−0.617952
$408$		0			0
$409$		−	43.2426i		−	0.105728i	−0.998602	−	0.0528639i	$-0.983165\pi$
$409$		−	43.2426i		−	0.105728i	0.998602	−	0.0528639i	$-0.0168349\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	42.3415	−	42.3415i	0.102522	−	0.102522i
$414$		0			0
$415$	645.095			1.55445
$416$		0			0
$417$		0			0
$418$		0			0
$419$	81.3896	−	81.3896i	0.194247	−	0.194247i	−0.603281	−	0.797528i	$-0.706140\pi$
$419$	81.3896	−	81.3896i	0.194247	−	0.194247i	0.797528	+	0.603281i	$0.206140\pi$
$420$		0			0
$421$	−198.570	+	198.570i	−0.471663	+	0.471663i	−0.902452	−	0.430789i	$-0.858235\pi$
$421$	−198.570	+	198.570i	−0.471663	+	0.471663i	0.430789	+	0.902452i	$0.358235\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−171.529			−0.403597
$426$		0			0
$427$	36.6875	−	36.6875i	0.0859193	−	0.0859193i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			395.497i			0.917627i	0.888533	+	0.458813i	$0.151725\pi$
$431$			395.497i			0.917627i	−0.888533	+	0.458813i	$0.848275\pi$
$432$		0			0
$433$	−672.683			−1.55354			−0.776770	−	0.629784i	$-0.783143\pi$
$433$	−672.683			−1.55354			−0.776770	+	0.629784i	$0.783143\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	38.2939	+	38.2939i	0.0876290	+	0.0876290i
$438$		0			0
$439$			26.8920i			0.0612574i	0.999531	+	0.0306287i	$0.00975095\pi$
$439$			26.8920i			0.0612574i	−0.999531	+	0.0306287i	$0.990249\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−524.214	−	524.214i	−1.18333	−	1.18333i	−0.978877	−	0.204451i	$-0.934459\pi$
$443$	−524.214	−	524.214i	−1.18333	−	1.18333i	−0.204451	−	0.978877i	$-0.565541\pi$
$444$		0			0
$445$	−322.597	−	322.597i	−0.724937	−	0.724937i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			518.381i			1.15452i	0.816559	+	0.577262i	$0.195879\pi$
$449$			518.381i			1.15452i	−0.816559	+	0.577262i	$0.804121\pi$
$450$		0			0
$451$	−134.853	−	134.853i	−0.299009	−	0.299009i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	39.1533			0.0860511
$456$		0			0
$457$			853.516i			1.86765i	0.357729	+	0.933825i	$0.383551\pi$
$457$			853.516i			1.86765i	−0.357729	+	0.933825i	$0.616449\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−331.810	+	331.810i	−0.719761	+	0.719761i	−0.968556	−	0.248795i	$-0.919966\pi$
$461$	−331.810	+	331.810i	−0.719761	+	0.719761i	0.248795	+	0.968556i	$0.419966\pi$
$462$		0			0
$463$	−164.893			−0.356141			−0.178070	−	0.984018i	$-0.556986\pi$
$463$	−164.893			−0.356141			−0.178070	+	0.984018i	$0.556986\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	105.253	−	105.253i	0.225380	−	0.225380i	−0.585379	−	0.810760i	$-0.699054\pi$
$467$	105.253	−	105.253i	0.225380	−	0.225380i	0.810760	+	0.585379i	$0.199054\pi$
$468$		0			0
$469$	51.5022	−	51.5022i	0.109813	−	0.109813i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	479.751			1.01427
$474$		0			0
$475$	6.12514	−	6.12514i	0.0128950	−	0.0128950i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	678.054i		−	1.41556i	−0.706432	−	0.707781i	$-0.749696\pi$
$479$		−	678.054i		−	1.41556i	0.706432	−	0.707781i	$-0.250304\pi$
$480$		0			0
$481$	−377.470			−0.784762
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−191.064	−	191.064i	−0.393947	−	0.393947i
$486$		0			0
$487$		−	7.72273i		−	0.0158578i	−0.999969	−	0.00792888i	$-0.997476\pi$
$487$		−	7.72273i		−	0.0158578i	0.999969	−	0.00792888i	$-0.00252387\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	296.311	+	296.311i	0.603484	+	0.603484i	0.941235	−	0.337751i	$-0.109666\pi$
$491$	296.311	+	296.311i	0.603484	+	0.603484i	−0.337751	+	0.941235i	$0.609666\pi$
$492$		0			0
$493$	−274.984	−	274.984i	−0.557777	−	0.557777i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			59.1390i			0.118992i
$498$		0			0
$499$	−555.542	−	555.542i	−1.11331	−	1.11331i	−0.992700	−	0.120612i	$-0.961514\pi$
$499$	−555.542	−	555.542i	−1.11331	−	1.11331i	−0.120612	−	0.992700i	$-0.538486\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	322.916			0.641979			0.320990	−	0.947083i	$-0.395985\pi$
$503$	322.916			0.641979			0.320990	+	0.947083i	$0.395985\pi$
$504$		0			0
$505$		−	322.250i		−	0.638118i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	522.344	−	522.344i	1.02622	−	1.02622i	0.0265686	−	0.999647i	$-0.491542\pi$
$509$	522.344	−	522.344i	1.02622	−	1.02622i	0.999647	−	0.0265686i	$-0.00845804\pi$
$510$		0			0
$511$	18.8453			0.0368793
$512$		0			0
$513$		0			0
$514$		0			0
$515$	676.213	−	676.213i	1.31304	−	1.31304i
$516$		0			0
$517$	−113.818	+	113.818i	−0.220150	+	0.220150i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−80.4109			−0.154340			−0.0771698	−	0.997018i	$-0.524588\pi$
$521$	−80.4109			−0.154340			−0.0771698	+	0.997018i	$0.524588\pi$
$522$		0			0
$523$	−133.868	+	133.868i	−0.255962	+	0.255962i	−0.823410	−	0.567447i	$-0.807931\pi$
$523$	−133.868	+	133.868i	−0.255962	+	0.255962i	0.567447	+	0.823410i	$0.307931\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			1593.57i			3.02386i
$528$		0			0
$529$	641.985			1.21358
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−202.392	−	202.392i	−0.379723	−	0.379723i
$534$		0			0
$535$			415.918i			0.777417i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−207.656	−	207.656i	−0.385261	−	0.385261i
$540$		0			0
$541$	643.791	+	643.791i	1.19000	+	1.19000i	0.977066	+	0.212936i	$0.0683025\pi$
$541$	643.791	+	643.791i	1.19000	+	1.19000i	0.212936	+	0.977066i	$0.431698\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	727.237i		−	1.33438i
$546$		0			0
$547$	543.953	+	543.953i	0.994429	+	0.994429i	0.999985	−	0.00555539i	$-0.00176834\pi$
$547$	543.953	+	543.953i	0.994429	+	0.994429i	−0.00555539	+	0.999985i	$0.501768\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	19.6389			0.0356422
$552$		0			0
$553$			62.7252i			0.113427i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	436.430	−	436.430i	0.783537	−	0.783537i	−0.196889	−	0.980426i	$-0.563084\pi$
$557$	436.430	−	436.430i	0.783537	−	0.783537i	0.980426	+	0.196889i	$0.0630838\pi$
$558$		0			0
$559$	720.027			1.28806
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−525.339	+	525.339i	−0.933107	+	0.933107i	−0.997899	−	0.0647918i	$-0.979362\pi$
$563$	−525.339	+	525.339i	−0.933107	+	0.933107i	0.0647918	+	0.997899i	$0.479362\pi$
$564$		0			0
$565$	159.334	−	159.334i	0.282006	−	0.282006i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−848.985			−1.49207			−0.746033	−	0.665909i	$-0.768044\pi$
$569$	−848.985			−1.49207			−0.746033	+	0.665909i	$0.768044\pi$
$570$		0			0
$571$	−503.454	+	503.454i	−0.881706	+	0.881706i	−0.993708	−	0.112002i	$-0.964274\pi$
$571$	−503.454	+	503.454i	−0.881706	+	0.881706i	0.112002	+	0.993708i	$0.464274\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	187.300i		−	0.325739i
$576$		0			0
$577$	891.580			1.54520			0.772600	−	0.634894i	$-0.218956\pi$
$577$	891.580			1.54520			0.772600	+	0.634894i	$0.218956\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	64.3501	+	64.3501i	0.110758	+	0.110758i
$582$		0			0
$583$			83.7934i			0.143728i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	121.737	+	121.737i	0.207388	+	0.207388i	0.803156	−	0.595768i	$-0.203152\pi$
$587$	121.737	+	121.737i	0.207388	+	0.207388i	−0.595768	+	0.803156i	$0.703152\pi$
$588$		0			0
$589$	−56.9050	−	56.9050i	−0.0966129	−	0.0966129i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	207.925i		−	0.350632i	−0.984512	−	0.175316i	$-0.943905\pi$
$593$		−	207.925i		−	0.350632i	0.984512	−	0.175316i	$-0.0560947\pi$
$594$		0			0
$595$	−95.2626	−	95.2626i	−0.160105	−	0.160105i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	689.528			1.15113			0.575566	−	0.817755i	$-0.304782\pi$
$599$	689.528			1.15113			0.575566	+	0.817755i	$0.304782\pi$
$600$		0			0
$601$		−	542.421i		−	0.902531i	−0.892390	−	0.451266i	$-0.850973\pi$
$601$		−	542.421i		−	0.902531i	0.892390	−	0.451266i	$-0.149027\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−328.571	+	328.571i	−0.543092	+	0.543092i
$606$		0			0
$607$	−1018.93			−1.67864			−0.839318	−	0.543641i	$-0.817045\pi$
$607$	−1018.93			−1.67864			−0.839318	+	0.543641i	$0.817045\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−170.822	+	170.822i	−0.279578	+	0.279578i
$612$		0			0
$613$	−169.481	+	169.481i	−0.276478	+	0.276478i	−0.831701	−	0.555223i	$-0.812633\pi$
$613$	−169.481	+	169.481i	−0.276478	+	0.276478i	0.555223	+	0.831701i	$0.312633\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	834.597			1.35267			0.676335	−	0.736594i	$-0.263567\pi$
$617$	834.597			1.35267			0.676335	+	0.736594i	$0.263567\pi$
$618$		0			0
$619$	322.772	−	322.772i	0.521441	−	0.521441i	−0.396565	−	0.918007i	$-0.629798\pi$
$619$	322.772	−	322.772i	0.521441	−	0.521441i	0.918007	+	0.396565i	$0.129798\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	64.3600i		−	0.103307i
$624$		0			0
$625$	731.878			1.17100
$626$		0			0
$627$		0			0
$628$		0			0
$629$	918.411	+	918.411i	1.46011	+	1.46011i
$630$		0			0
$631$			1006.06i			1.59439i	0.603723	+	0.797194i	$0.293683\pi$
$631$			1006.06i			1.59439i	−0.603723	+	0.797194i	$0.706317\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−347.068	−	347.068i	−0.546563	−	0.546563i
$636$		0			0
$637$	−311.658	−	311.658i	−0.489258	−	0.489258i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			257.206i			0.401257i	0.979667	+	0.200628i	$0.0642984\pi$
$641$			257.206i			0.401257i	−0.979667	+	0.200628i	$0.935702\pi$
$642$		0			0
$643$	311.891	+	311.891i	0.485056	+	0.485056i	0.906742	−	0.421686i	$-0.138562\pi$
$643$	311.891	+	311.891i	0.485056	+	0.485056i	−0.421686	+	0.906742i	$0.638562\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−855.255			−1.32188			−0.660939	−	0.750440i	$-0.729842\pi$
$647$	−855.255			−1.32188			−0.660939	+	0.750440i	$0.729842\pi$
$648$		0			0
$649$		−	466.607i		−	0.718962i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−171.197	+	171.197i	−0.262170	+	0.262170i	−0.825935	−	0.563765i	$-0.809352\pi$
$653$	−171.197	+	171.197i	−0.262170	+	0.262170i	0.563765	+	0.825935i	$0.309352\pi$
$654$		0			0
$655$	−171.835			−0.262343
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−510.283	+	510.283i	−0.774330	+	0.774330i	−0.978860	−	0.204530i	$-0.934433\pi$
$659$	−510.283	+	510.283i	−0.774330	+	0.774330i	0.204530	+	0.978860i	$0.434433\pi$
$660$		0			0
$661$	−491.063	+	491.063i	−0.742910	+	0.742910i	−0.973137	−	0.230227i	$-0.926053\pi$
$661$	−491.063	+	491.063i	−0.742910	+	0.742910i	0.230227	+	0.973137i	$0.426053\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.80348			0.0102308
$666$		0			0
$667$	300.268	−	300.268i	0.450176	−	0.450176i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	404.300i		−	0.602534i
$672$		0			0
$673$	−200.282			−0.297596			−0.148798	−	0.988868i	$-0.547540\pi$
$673$	−200.282			−0.297596			−0.148798	+	0.988868i	$0.547540\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	818.346	+	818.346i	1.20878	+	1.20878i	0.971422	+	0.237361i	$0.0762823\pi$
$677$	818.346	+	818.346i	1.20878	+	1.20878i	0.237361	+	0.971422i	$0.423718\pi$
$678$		0			0
$679$		−	38.1184i		−	0.0561391i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	559.041	+	559.041i	0.818508	+	0.818508i	0.985892	−	0.167383i	$-0.0535318\pi$
$683$	559.041	+	559.041i	0.818508	+	0.818508i	−0.167383	+	0.985892i	$0.553532\pi$
$684$		0			0
$685$	432.476	+	432.476i	0.631352	+	0.631352i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			125.760i			0.182526i
$690$		0			0
$691$	558.744	+	558.744i	0.808602	+	0.808602i	0.984422	−	0.175820i	$-0.0562577\pi$
$691$	558.744	+	558.744i	0.808602	+	0.808602i	−0.175820	+	0.984422i	$0.556258\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	237.220			0.341324
$696$		0			0
$697$			984.869i			1.41301i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−452.428	+	452.428i	−0.645404	+	0.645404i	−0.951879	−	0.306475i	$-0.900851\pi$
$701$	−452.428	+	452.428i	−0.645404	+	0.645404i	0.306475	+	0.951879i	$0.400851\pi$
$702$		0			0
$703$	−65.5913			−0.0933019
$704$		0			0
$705$		0			0
$706$		0			0
$707$	32.1454	−	32.1454i	0.0454673	−	0.0454673i
$708$		0			0
$709$	693.360	−	693.360i	0.977941	−	0.977941i	−0.0218205	−	0.999762i	$-0.506946\pi$
$709$	693.360	−	693.360i	0.977941	−	0.977941i	0.999762	+	0.0218205i	$0.00694622\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1740.09			−2.44052
$714$		0			0
$715$	215.736	−	215.736i	0.301729	−	0.301729i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			641.528i			0.892251i	0.894970	+	0.446125i	$0.147196\pi$
$719$			641.528i			0.892251i	−0.894970	+	0.446125i	$0.852804\pi$
$720$		0			0
$721$	134.909			0.187113
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−48.0281	−	48.0281i	−0.0662456	−	0.0662456i
$726$		0			0
$727$			644.665i			0.886747i	0.896337	+	0.443373i	$0.146218\pi$
$727$			644.665i			0.886747i	−0.896337	+	0.443373i	$0.853782\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1751.88	−	1751.88i	−2.39655	−	2.39655i
$732$		0			0
$733$	748.983	+	748.983i	1.02180	+	1.02180i	0.999757	+	0.0220473i	$0.00701846\pi$
$733$	748.983	+	748.983i	1.02180	+	1.02180i	0.0220473	+	0.999757i	$0.492982\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	567.559i		−	0.770093i
$738$		0			0
$739$	387.245	+	387.245i	0.524013	+	0.524013i	0.918781	−	0.394768i	$-0.129175\pi$
$739$	387.245	+	387.245i	0.524013	+	0.524013i	−0.394768	+	0.918781i	$0.629175\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−245.165			−0.329966			−0.164983	−	0.986296i	$-0.552757\pi$
$743$	−245.165			−0.329966			−0.164983	+	0.986296i	$0.552757\pi$
$744$		0			0
$745$		−	1357.80i		−	1.82255i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−41.4891	+	41.4891i	−0.0553926	+	0.0553926i
$750$		0			0
$751$	−547.753			−0.729365			−0.364683	−	0.931132i	$-0.618822\pi$
$751$	−547.753			−0.729365			−0.364683	+	0.931132i	$0.618822\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−102.473	+	102.473i	−0.135726	+	0.135726i
$756$		0			0
$757$	876.755	−	876.755i	1.15820	−	1.15820i	0.173334	−	0.984863i	$-0.444546\pi$
$757$	876.755	−	876.755i	1.15820	−	1.15820i	0.984863	−	0.173334i	$-0.0554540\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	876.682			1.15201			0.576007	−	0.817445i	$-0.304610\pi$
$761$	876.682			1.15201			0.576007	+	0.817445i	$0.304610\pi$
$762$		0			0
$763$	72.5441	−	72.5441i	0.0950774	−	0.0950774i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	700.300i		−	0.913038i
$768$		0			0
$769$	−155.364			−0.202034			−0.101017	−	0.994885i	$-0.532210\pi$
$769$	−155.364			−0.202034			−0.101017	+	0.994885i	$0.532210\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	336.932	+	336.932i	0.435876	+	0.435876i	0.890621	−	0.454746i	$-0.150270\pi$
$773$	336.932	+	336.932i	0.435876	+	0.435876i	−0.454746	+	0.890621i	$0.650270\pi$
$774$		0			0
$775$			278.330i			0.359135i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−35.1688	−	35.1688i	−0.0451460	−	0.0451460i
$780$		0			0
$781$	325.858	+	325.858i	0.417232	+	0.417232i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			592.144i			0.754324i
$786$		0			0
$787$	−201.318	−	201.318i	−0.255804	−	0.255804i	0.567541	−	0.823345i	$-0.307895\pi$
$787$	−201.318	−	201.318i	−0.255804	−	0.255804i	−0.823345	+	0.567541i	$0.807895\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	31.7880			0.0401871
$792$		0			0
$793$		−	606.788i		−	0.765181i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−571.213	+	571.213i	−0.716704	+	0.716704i	−0.967929	−	0.251225i	$-0.919167\pi$
$797$	−571.213	+	571.213i	−0.716704	+	0.716704i	0.251225	+	0.967929i	$0.419167\pi$
$798$		0			0
$799$	831.243			1.04035
$800$		0			0
$801$		0			0
$802$		0			0
$803$	103.839	−	103.839i	0.129313	−	0.129313i
$804$		0			0
$805$	104.022	−	104.022i	0.129219	−	0.129219i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	554.891			0.685898			0.342949	−	0.939354i	$-0.388574\pi$
$809$	554.891			0.685898			0.342949	+	0.939354i	$0.388574\pi$
$810$		0			0
$811$	−570.087	+	570.087i	−0.702944	+	0.702944i	−0.965041	−	0.262098i	$-0.915586\pi$
$811$	−570.087	+	570.087i	−0.702944	+	0.702944i	0.262098	+	0.965041i	$0.415586\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			1212.29i			1.48747i
$816$		0			0
$817$	125.116			0.153141
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−150.698	−	150.698i	−0.183554	−	0.183554i	0.609348	−	0.792903i	$-0.291431\pi$
$821$	−150.698	−	150.698i	−0.183554	−	0.183554i	−0.792903	+	0.609348i	$0.791431\pi$
$822$		0			0
$823$			742.019i			0.901603i	0.892624	+	0.450801i	$0.148862\pi$
$823$			742.019i			0.901603i	−0.892624	+	0.450801i	$0.851138\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−248.150	−	248.150i	−0.300061	−	0.300061i	0.540977	−	0.841037i	$-0.318055\pi$
$827$	−248.150	−	248.150i	−0.300061	−	0.300061i	−0.841037	+	0.540977i	$0.818055\pi$
$828$		0			0
$829$	−373.174	−	373.174i	−0.450150	−	0.450150i	0.445255	−	0.895404i	$-0.353113\pi$
$829$	−373.174	−	373.174i	−0.450150	−	0.450150i	−0.895404	+	0.445255i	$0.853113\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1516.57i			1.82061i
$834$		0			0
$835$	924.728	+	924.728i	1.10746	+	1.10746i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1227.49			−1.46304			−0.731520	−	0.681820i	$-0.761189\pi$
$839$	−1227.49			−1.46304			−0.731520	+	0.681820i	$0.761189\pi$
$840$		0			0
$841$			687.009i			0.816895i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−335.894	+	335.894i	−0.397508	+	0.397508i
$846$		0			0
$847$	−65.5518			−0.0773929
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1002.86	+	1002.86i	−1.17844	+	1.17844i
$852$		0			0
$853$	−24.4613	+	24.4613i	−0.0286768	+	0.0286768i	−0.721300	−	0.692623i	$-0.756455\pi$
$853$	−24.4613	+	24.4613i	−0.0286768	+	0.0286768i	0.692623	+	0.721300i	$0.256455\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−839.732			−0.979850			−0.489925	−	0.871764i	$-0.662976\pi$
$857$	−839.732			−0.979850			−0.489925	+	0.871764i	$0.662976\pi$
$858$		0			0
$859$	483.229	−	483.229i	0.562548	−	0.562548i	−0.367482	−	0.930031i	$-0.619780\pi$
$859$	483.229	−	483.229i	0.562548	−	0.562548i	0.930031	+	0.367482i	$0.119780\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			21.1881i			0.0245517i	0.999925	+	0.0122758i	$0.00390761\pi$
$863$			21.1881i			0.0245517i	−0.999925	+	0.0122758i	$0.996092\pi$
$864$		0			0
$865$	533.761			0.617064
$866$		0			0
$867$		0			0
$868$		0			0
$869$	345.619	+	345.619i	0.397720	+	0.397720i
$870$		0			0
$871$		−	851.813i		−	0.977971i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	59.3571	+	59.3571i	0.0678367	+	0.0678367i
$876$		0			0
$877$	−350.528	−	350.528i	−0.399690	−	0.399690i	0.478434	−	0.878124i	$-0.341205\pi$
$877$	−350.528	−	350.528i	−0.399690	−	0.399690i	−0.878124	+	0.478434i	$0.841205\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			472.491i			0.536312i	0.963375	+	0.268156i	$0.0864143\pi$
$881$			472.491i			0.536312i	−0.963375	+	0.268156i	$0.913586\pi$
$882$		0			0
$883$	−294.850	−	294.850i	−0.333919	−	0.333919i	0.520154	−	0.854073i	$-0.325875\pi$
$883$	−294.850	−	294.850i	−0.333919	−	0.333919i	−0.854073	+	0.520154i	$0.825875\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−189.405			−0.213535			−0.106767	−	0.994284i	$-0.534050\pi$
$887$	−189.405			−0.213535			−0.106767	+	0.994284i	$0.534050\pi$
$888$		0			0
$889$		−	69.2421i		−	0.0778876i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−29.6829	+	29.6829i	−0.0332396	+	0.0332396i
$894$		0			0
$895$	1011.97			1.13070
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−446.200	+	446.200i	−0.496330	+	0.496330i
$900$		0			0
$901$	305.983	−	305.983i	0.339604	−	0.339604i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1230.68			1.35987
$906$		0			0
$907$	−1025.20	+	1025.20i	−1.13032	+	1.13032i	−0.140200	+	0.990123i	$0.544774\pi$
$907$	−1025.20	+	1025.20i	−1.13032	+	1.13032i	−0.990123	+	0.140200i	$0.955226\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1115.84i		−	1.22485i	−0.790528	−	0.612426i	$-0.790194\pi$
$911$		−	1115.84i		−	1.22485i	0.790528	−	0.612426i	$-0.209806\pi$
$912$		0			0
$913$	709.144			0.776719
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−17.1410	−	17.1410i	−0.0186925	−	0.0186925i
$918$		0			0
$919$			1169.50i			1.27258i	0.771450	+	0.636290i	$0.219532\pi$
$919$			1169.50i			1.27258i	−0.771450	+	0.636290i	$0.780468\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	489.060	+	489.060i	0.529859	+	0.529859i
$924$		0			0
$925$	160.408	+	160.408i	0.173414	+	0.173414i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			115.506i			0.124334i	0.998066	+	0.0621671i	$0.0198012\pi$
$929$			115.506i			0.124334i	−0.998066	+	0.0621671i	$0.980199\pi$
$930$		0			0
$931$	−54.1553	−	54.1553i	−0.0581689	−	0.0581689i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1049.80			−1.12278
$936$		0			0
$937$			666.120i			0.710907i	0.934694	+	0.355454i	$0.115674\pi$
$937$			666.120i			0.710907i	−0.934694	+	0.355454i	$0.884326\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	559.735	−	559.735i	0.594830	−	0.594830i	−0.344102	−	0.938932i	$-0.611817\pi$
$941$	559.735	−	559.735i	0.594830	−	0.594830i	0.938932	+	0.344102i	$0.111817\pi$
$942$		0			0
$943$	−1075.42			−1.14043
$944$		0			0
$945$		0			0
$946$		0			0
$947$	871.120	−	871.120i	0.919873	−	0.919873i	−0.0771466	−	0.997020i	$-0.524581\pi$
$947$	871.120	−	871.120i	0.919873	−	0.919873i	0.997020	+	0.0771466i	$0.0245809\pi$
$948$		0			0
$949$	155.845	−	155.845i	0.164220	−	0.164220i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−264.384			−0.277423			−0.138711	−	0.990333i	$-0.544296\pi$
$953$	−264.384			−0.277423			−0.138711	+	0.990333i	$0.544296\pi$
$954$		0			0
$955$	226.874	−	226.874i	0.237564	−	0.237564i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			86.2816i			0.0899703i
$960$		0			0
$961$	1624.80			1.69073
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1219.78	−	1219.78i	−1.26402	−	1.26402i
$966$		0			0
$967$		−	92.8445i		−	0.0960129i	−0.998847	−	0.0480065i	$-0.984713\pi$
$967$		−	92.8445i		−	0.0960129i	0.998847	−	0.0480065i	$-0.0152868\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1086.25	+	1086.25i	1.11870	+	1.11870i	0.991933	+	0.126764i	$0.0404592\pi$
$971$	1086.25	+	1086.25i	1.11870	+	1.11870i	0.126764	+	0.991933i	$0.459541\pi$
$972$		0			0
$973$	23.6634	+	23.6634i	0.0243201	+	0.0243201i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	626.473i		−	0.641222i	−0.947211	−	0.320611i	$-0.896112\pi$
$977$		−	626.473i		−	0.641222i	0.947211	−	0.320611i	$-0.103888\pi$
$978$		0			0
$979$	−354.626	−	354.626i	−0.362233	−	0.362233i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	204.267			0.207800			0.103900	−	0.994588i	$-0.466868\pi$
$983$	204.267			0.207800			0.103900	+	0.994588i	$0.466868\pi$
$984$		0			0
$985$		−	349.986i		−	0.355316i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1912.95	−	1912.95i	1.93423	−	1.93423i
$990$		0			0
$991$	1147.30			1.15772			0.578859	−	0.815428i	$-0.303498\pi$
$991$	1147.30			1.15772			0.578859	+	0.815428i	$0.303498\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	613.348	−	613.348i	0.616430	−	0.616430i
$996$		0			0
$997$	−514.690	+	514.690i	−0.516239	+	0.516239i	−0.916431	−	0.400192i	$-0.868943\pi$
$997$	−514.690	+	514.690i	−0.516239	+	0.516239i	0.400192	+	0.916431i	$0.368943\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.j.a.161.4		32
3.2	odd	2	inner	1152.3.j.a.161.13		32
4.3	odd	2		1152.3.j.b.161.4		32
8.3	odd	2		576.3.j.a.17.13		32
8.5	even	2		144.3.j.a.125.5	yes	32
12.11	even	2		1152.3.j.b.161.13		32
16.3	odd	4		576.3.j.a.305.4		32
16.5	even	4	inner	1152.3.j.a.737.13		32
16.11	odd	4		1152.3.j.b.737.13		32
16.13	even	4		144.3.j.a.53.12	yes	32
24.5	odd	2		144.3.j.a.125.12	yes	32
24.11	even	2		576.3.j.a.17.4		32
48.5	odd	4	inner	1152.3.j.a.737.4		32
48.11	even	4		1152.3.j.b.737.4		32
48.29	odd	4		144.3.j.a.53.5	✓	32
48.35	even	4		576.3.j.a.305.13		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.j.a.53.5	✓	32	48.29	odd	4
144.3.j.a.53.12	yes	32	16.13	even	4
144.3.j.a.125.5	yes	32	8.5	even	2
144.3.j.a.125.12	yes	32	24.5	odd	2
576.3.j.a.17.4		32	24.11	even	2
576.3.j.a.17.13		32	8.3	odd	2
576.3.j.a.305.4		32	16.3	odd	4
576.3.j.a.305.13		32	48.35	even	4
1152.3.j.a.161.4		32	1.1	even	1	trivial
1152.3.j.a.161.13		32	3.2	odd	2	inner
1152.3.j.a.737.4		32	48.5	odd	4	inner
1152.3.j.a.737.13		32	16.5	even	4	inner
1152.3.j.b.161.4		32	4.3	odd	2
1152.3.j.b.161.13		32	12.11	even	2
1152.3.j.b.737.4		32	48.11	even	4
1152.3.j.b.737.13		32	16.11	odd	4

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)

\(f(q)\)	\(=\)	\(q+(-3.90343 - 3.90343i) q^{5} -0.778757i q^{7} +O(q^{10})\)	\(q+(-3.90343 - 3.90343i) q^{5} -0.778757i q^{7} +(-4.29098 - 4.29098i) q^{11} +(-6.44007 - 6.44007i) q^{13} +31.3383i q^{17} +(-1.11906 - 1.11906i) q^{19} -34.2197 q^{23} +5.47346i q^{25} +(-8.77471 + 8.77471i) q^{29} +50.8507 q^{31} +(-3.03982 + 3.03982i) q^{35} +(29.3064 - 29.3064i) q^{37} +31.4271 q^{41} +(-55.9022 + 55.9022i) q^{43} -26.5249i q^{47} +48.3935 q^{49} +(-9.76389 - 9.76389i) q^{53} +33.4991i q^{55} +(54.3706 + 54.3706i) q^{59} +(47.1104 + 47.1104i) q^{61} +50.2766i q^{65} +(66.1339 + 66.1339i) q^{67} -75.9403 q^{71} +24.1992i q^{73} +(-3.34163 + 3.34163i) q^{77} -80.5454 q^{79} +(-82.6319 + 82.6319i) q^{83} +(122.327 - 122.327i) q^{85} +82.6445 q^{89} +(-5.01524 + 5.01524i) q^{91} +8.73634i q^{95} +48.9478 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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