LMFDB - Embedded newform 1152.3.j.b.161.10

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			161.10
Character	$\chi$	$=$	1152.161
Dual form			1152.3.j.b.737.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.66372 + 1.66372i) q^{5} +13.3224i q^{7} +O(q^{10})$	$q+(1.66372 + 1.66372i) q^{5} +13.3224i q^{7} +(7.81788 + 7.81788i) q^{11} +(10.4955 + 10.4955i) q^{13} -10.0304i q^{17} +(5.07048 + 5.07048i) q^{19} +29.7810 q^{23} -19.4640i q^{25} +(-18.6855 + 18.6855i) q^{29} -27.1537 q^{31} +(-22.1648 + 22.1648i) q^{35} +(-13.3368 + 13.3368i) q^{37} -34.9561 q^{41} +(7.29114 - 7.29114i) q^{43} -51.5850i q^{47} -128.487 q^{49} +(68.4354 + 68.4354i) q^{53} +26.0136i q^{55} +(31.5812 + 31.5812i) q^{59} +(-72.6595 - 72.6595i) q^{61} +34.9233i q^{65} +(60.3054 + 60.3054i) q^{67} +3.09226 q^{71} -2.05367i q^{73} +(-104.153 + 104.153i) q^{77} -53.3986 q^{79} +(21.7960 - 21.7960i) q^{83} +(16.6877 - 16.6877i) q^{85} +137.585 q^{89} +(-139.826 + 139.826i) q^{91} +16.8718i q^{95} -17.1890 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.66372	+	1.66372i	0.332745	+	0.332745i	0.853628	−	0.520883i	$-0.174397\pi$
$5$	1.66372	+	1.66372i	0.332745	+	0.332745i	−0.520883	+	0.853628i	$0.674397\pi$
$6$		0			0
$7$			13.3224i			1.90320i	0.307337	+	0.951601i	$0.400562\pi$
$7$			13.3224i			1.90320i	−0.307337	+	0.951601i	$0.599438\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	7.81788	+	7.81788i	0.710716	+	0.710716i	0.966685	−	0.255969i	$-0.0823944\pi$
$11$	7.81788	+	7.81788i	0.710716	+	0.710716i	−0.255969	+	0.966685i	$0.582394\pi$
$12$		0			0
$13$	10.4955	+	10.4955i	0.807347	+	0.807347i	0.984232	−	0.176884i	$-0.0566019\pi$
$13$	10.4955	+	10.4955i	0.807347	+	0.807347i	−0.176884	+	0.984232i	$0.556602\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	10.0304i		−	0.590021i	−0.955494	−	0.295011i	$-0.904677\pi$
$17$		−	10.0304i		−	0.590021i	0.955494	−	0.295011i	$-0.0953232\pi$
$18$		0			0
$19$	5.07048	+	5.07048i	0.266867	+	0.266867i	0.827837	−	0.560969i	$-0.189571\pi$
$19$	5.07048	+	5.07048i	0.266867	+	0.266867i	−0.560969	+	0.827837i	$0.689571\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	29.7810			1.29483			0.647413	−	0.762140i	$-0.275851\pi$
$23$	29.7810			1.29483			0.647413	+	0.762140i	$0.275851\pi$
$24$		0			0
$25$		−	19.4640i		−	0.778562i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−18.6855	+	18.6855i	−0.644329	+	0.644329i	−0.951617	−	0.307288i	$-0.900579\pi$
$29$	−18.6855	+	18.6855i	−0.644329	+	0.644329i	0.307288	+	0.951617i	$0.400579\pi$
$30$		0			0
$31$	−27.1537			−0.875926			−0.437963	−	0.898993i	$-0.644300\pi$
$31$	−27.1537			−0.875926			−0.437963	+	0.898993i	$0.644300\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−22.1648	+	22.1648i	−0.633280	+	0.633280i
$36$		0			0
$37$	−13.3368	+	13.3368i	−0.360453	+	0.360453i	−0.863980	−	0.503526i	$-0.832036\pi$
$37$	−13.3368	+	13.3368i	−0.360453	+	0.360453i	0.503526	+	0.863980i	$0.332036\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−34.9561			−0.852587			−0.426293	−	0.904585i	$-0.640181\pi$
$41$	−34.9561			−0.852587			−0.426293	+	0.904585i	$0.640181\pi$
$42$		0			0
$43$	7.29114	−	7.29114i	0.169561	−	0.169561i	−0.617225	−	0.786787i	$-0.711743\pi$
$43$	7.29114	−	7.29114i	0.169561	−	0.169561i	0.786787	+	0.617225i	$0.211743\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	51.5850i		−	1.09755i	−0.835969	−	0.548777i	$-0.815094\pi$
$47$		−	51.5850i		−	1.09755i	0.835969	−	0.548777i	$-0.184906\pi$
$48$		0			0
$49$	−128.487			−2.62217
$50$		0			0
$51$		0			0
$52$		0			0
$53$	68.4354	+	68.4354i	1.29123	+	1.29123i	0.934024	+	0.357209i	$0.116272\pi$
$53$	68.4354	+	68.4354i	1.29123	+	1.29123i	0.357209	+	0.934024i	$0.383728\pi$
$54$		0			0
$55$			26.0136i			0.472974i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	31.5812	+	31.5812i	0.535275	+	0.535275i	0.922138	−	0.386862i	$-0.126441\pi$
$59$	31.5812	+	31.5812i	0.535275	+	0.535275i	−0.386862	+	0.922138i	$0.626441\pi$
$60$		0			0
$61$	−72.6595	−	72.6595i	−1.19114	−	1.19114i	−0.976748	−	0.214392i	$-0.931223\pi$
$61$	−72.6595	−	72.6595i	−1.19114	−	1.19114i	−0.214392	−	0.976748i	$-0.568777\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			34.9233i			0.537281i
$66$		0			0
$67$	60.3054	+	60.3054i	0.900080	+	0.900080i	0.995443	−	0.0953625i	$-0.0304010\pi$
$67$	60.3054	+	60.3054i	0.900080	+	0.900080i	−0.0953625	+	0.995443i	$0.530401\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.09226			0.0435530			0.0217765	−	0.999763i	$-0.493068\pi$
$71$	3.09226			0.0435530			0.0217765	+	0.999763i	$0.493068\pi$
$72$		0			0
$73$		−	2.05367i		−	0.0281324i	−0.999901	−	0.0140662i	$-0.995522\pi$
$73$		−	2.05367i		−	0.0281324i	0.999901	−	0.0140662i	$-0.00447757\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−104.153	+	104.153i	−1.35264	+	1.35264i
$78$		0			0
$79$	−53.3986			−0.675932			−0.337966	−	0.941158i	$-0.609739\pi$
$79$	−53.3986			−0.675932			−0.337966	+	0.941158i	$0.609739\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	21.7960	−	21.7960i	0.262602	−	0.262602i	−0.563508	−	0.826110i	$-0.690549\pi$
$83$	21.7960	−	21.7960i	0.262602	−	0.262602i	0.826110	+	0.563508i	$0.190549\pi$
$84$		0			0
$85$	16.6877	−	16.6877i	0.196326	−	0.196326i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	137.585			1.54590			0.772948	−	0.634470i	$-0.218782\pi$
$89$	137.585			1.54590			0.772948	+	0.634470i	$0.218782\pi$
$90$		0			0
$91$	−139.826	+	139.826i	−1.53654	+	1.53654i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			16.8718i			0.177597i
$96$		0			0
$97$	−17.1890			−0.177206			−0.0886030	−	0.996067i	$-0.528240\pi$
$97$	−17.1890			−0.177206			−0.0886030	+	0.996067i	$0.528240\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−89.4355	−	89.4355i	−0.885500	−	0.885500i	0.108587	−	0.994087i	$-0.465368\pi$
$101$	−89.4355	−	89.4355i	−0.885500	−	0.885500i	−0.994087	+	0.108587i	$0.965368\pi$
$102$		0			0
$103$			124.313i			1.20693i	0.797391	+	0.603463i	$0.206213\pi$
$103$			124.313i			1.20693i	−0.797391	+	0.603463i	$0.793787\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	117.662	+	117.662i	1.09964	+	1.09964i	0.994452	+	0.105191i	$0.0335454\pi$
$107$	117.662	+	117.662i	1.09964	+	1.09964i	0.105191	+	0.994452i	$0.466455\pi$
$108$		0			0
$109$	−1.84962	−	1.84962i	−0.0169690	−	0.0169690i	0.698571	−	0.715540i	$-0.253819\pi$
$109$	−1.84962	−	1.84962i	−0.0169690	−	0.0169690i	−0.715540	+	0.698571i	$0.753819\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	28.3900i		−	0.251239i	−0.992079	−	0.125619i	$-0.959908\pi$
$113$		−	28.3900i		−	0.251239i	0.992079	−	0.125619i	$-0.0400918\pi$
$114$		0			0
$115$	49.5473	+	49.5473i	0.430846	+	0.430846i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	133.629			1.12293
$120$		0			0
$121$			1.23847i			0.0102353i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	73.9759	−	73.9759i	0.591807	−	0.591807i
$126$		0			0
$127$	−31.7899			−0.250314			−0.125157	−	0.992137i	$-0.539944\pi$
$127$	−31.7899			−0.250314			−0.125157	+	0.992137i	$0.539944\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	160.832	−	160.832i	1.22772	−	1.22772i	0.262902	−	0.964823i	$-0.415320\pi$
$131$	160.832	−	160.832i	1.22772	−	1.22772i	0.964823	−	0.262902i	$-0.0846796\pi$
$132$		0			0
$133$	−67.5510	+	67.5510i	−0.507902	+	0.507902i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−224.414			−1.63805			−0.819027	−	0.573754i	$-0.805486\pi$
$137$	−224.414			−1.63805			−0.819027	+	0.573754i	$0.805486\pi$
$138$		0			0
$139$	−55.3627	+	55.3627i	−0.398292	+	0.398292i	−0.877630	−	0.479338i	$-0.840877\pi$
$139$	−55.3627	+	55.3627i	−0.398292	+	0.398292i	0.479338	+	0.877630i	$0.340877\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			164.105i			1.14759i
$144$		0			0
$145$	−62.1752			−0.428794
$146$		0			0
$147$		0			0
$148$		0			0
$149$	99.5497	+	99.5497i	0.668119	+	0.668119i	0.957280	−	0.289162i	$-0.0933765\pi$
$149$	99.5497	+	99.5497i	0.668119	+	0.668119i	−0.289162	+	0.957280i	$0.593377\pi$
$150$		0			0
$151$			130.380i			0.863442i	0.902007	+	0.431721i	$0.142094\pi$
$151$			130.380i			0.863442i	−0.902007	+	0.431721i	$0.857906\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−45.1763	−	45.1763i	−0.291460	−	0.291460i
$156$		0			0
$157$	−145.961	−	145.961i	−0.929686	−	0.929686i	0.0679992	−	0.997685i	$-0.478338\pi$
$157$	−145.961	−	145.961i	−0.929686	−	0.929686i	−0.997685	+	0.0679992i	$0.978338\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			396.755i			2.46431i
$162$		0			0
$163$	−41.4323	−	41.4323i	−0.254186	−	0.254186i	0.568498	−	0.822684i	$-0.307525\pi$
$163$	−41.4323	−	41.4323i	−0.254186	−	0.254186i	−0.822684	+	0.568498i	$0.807525\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	13.8976			0.0832192			0.0416096	−	0.999134i	$-0.486751\pi$
$167$	13.8976			0.0832192			0.0416096	+	0.999134i	$0.486751\pi$
$168$		0			0
$169$			51.3116i			0.303619i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−43.8507	+	43.8507i	−0.253473	+	0.253473i	−0.822393	−	0.568920i	$-0.807361\pi$
$173$	−43.8507	+	43.8507i	−0.253473	+	0.253473i	0.568920	+	0.822393i	$0.307361\pi$
$174$		0			0
$175$	259.308			1.48176
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−154.797	+	154.797i	−0.864788	+	0.864788i	−0.991890	−	0.127102i	$-0.959432\pi$
$179$	−154.797	+	154.797i	−0.864788	+	0.864788i	0.127102	+	0.991890i	$0.459432\pi$
$180$		0			0
$181$	−105.914	+	105.914i	−0.585160	+	0.585160i	−0.936317	−	0.351157i	$-0.885788\pi$
$181$	−105.914	+	105.914i	−0.585160	+	0.585160i	0.351157	+	0.936317i	$0.385788\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−44.3774			−0.239878
$186$		0			0
$187$	78.4162	−	78.4162i	0.419338	−	0.419338i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			69.4546i			0.363637i	0.983332	+	0.181818i	$0.0581983\pi$
$191$			69.4546i			0.363637i	−0.983332	+	0.181818i	$0.941802\pi$
$192$		0			0
$193$	35.5238			0.184061			0.0920306	−	0.995756i	$-0.470664\pi$
$193$	35.5238			0.184061			0.0920306	+	0.995756i	$0.470664\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−7.74617	−	7.74617i	−0.0393206	−	0.0393206i	0.687173	−	0.726494i	$-0.258851\pi$
$197$	−7.74617	−	7.74617i	−0.0393206	−	0.0393206i	−0.726494	+	0.687173i	$0.758851\pi$
$198$		0			0
$199$		−	158.474i		−	0.796352i	−0.917309	−	0.398176i	$-0.869643\pi$
$199$		−	158.474i		−	0.796352i	0.917309	−	0.398176i	$-0.130357\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−248.936	−	248.936i	−1.22629	−	1.22629i
$204$		0			0
$205$	−58.1572	−	58.1572i	−0.283694	−	0.283694i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			79.2808i			0.379334i
$210$		0			0
$211$	−168.889	−	168.889i	−0.800422	−	0.800422i	0.182739	−	0.983161i	$-0.441504\pi$
$211$	−168.889	−	168.889i	−0.800422	−	0.800422i	−0.983161	+	0.182739i	$0.941504\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	24.2609			0.112841
$216$		0			0
$217$		−	361.753i		−	1.66706i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	105.274	−	105.274i	0.476352	−	0.476352i
$222$		0			0
$223$	−83.6618			−0.375165			−0.187582	−	0.982249i	$-0.560065\pi$
$223$	−83.6618			−0.375165			−0.187582	+	0.982249i	$0.560065\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	72.4600	−	72.4600i	0.319207	−	0.319207i	−0.529255	−	0.848463i	$-0.677529\pi$
$227$	72.4600	−	72.4600i	0.319207	−	0.319207i	0.848463	+	0.529255i	$0.177529\pi$
$228$		0			0
$229$	−119.027	+	119.027i	−0.519767	+	0.519767i	−0.917501	−	0.397734i	$-0.869797\pi$
$229$	−119.027	+	119.027i	−0.519767	+	0.519767i	0.397734	+	0.917501i	$0.369797\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	105.840			0.454248			0.227124	−	0.973866i	$-0.427068\pi$
$233$	105.840			0.454248			0.227124	+	0.973866i	$0.427068\pi$
$234$		0			0
$235$	85.8232	−	85.8232i	0.365205	−	0.365205i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			149.734i			0.626502i	0.949670	+	0.313251i	$0.101418\pi$
$239$			149.734i			0.626502i	−0.949670	+	0.313251i	$0.898582\pi$
$240$		0			0
$241$	158.414			0.657321			0.328661	−	0.944448i	$-0.393403\pi$
$241$	158.414			0.657321			0.328661	+	0.944448i	$0.393403\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−213.766	−	213.766i	−0.872515	−	0.872515i
$246$		0			0
$247$			106.435i			0.430909i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.32722	+	8.32722i	0.0331762	+	0.0331762i	0.723500	−	0.690324i	$-0.242532\pi$
$251$	8.32722	+	8.32722i	0.0331762	+	0.0331762i	−0.690324	+	0.723500i	$0.742532\pi$
$252$		0			0
$253$	232.824	+	232.824i	0.920254	+	0.920254i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	352.499i		−	1.37159i	−0.727794	−	0.685796i	$-0.759454\pi$
$257$		−	352.499i		−	1.37159i	0.727794	−	0.685796i	$-0.240546\pi$
$258$		0			0
$259$	−177.678	−	177.678i	−0.686015	−	0.686015i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−301.404			−1.14602			−0.573011	−	0.819548i	$-0.694225\pi$
$263$	−301.404			−1.14602			−0.573011	+	0.819548i	$0.694225\pi$
$264$		0			0
$265$			227.715i			0.859302i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	216.782	−	216.782i	0.805879	−	0.805879i	−0.178128	−	0.984007i	$-0.557004\pi$
$269$	216.782	−	216.782i	0.805879	−	0.805879i	0.984007	+	0.178128i	$0.0570041\pi$
$270$		0			0
$271$	92.3711			0.340853			0.170426	−	0.985370i	$-0.445486\pi$
$271$	92.3711			0.340853			0.170426	+	0.985370i	$0.445486\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	152.168	−	152.168i	0.553337	−	0.553337i
$276$		0			0
$277$	300.909	−	300.909i	1.08631	−	1.08631i	0.0904098	−	0.995905i	$-0.471182\pi$
$277$	300.909	−	300.909i	1.08631	−	1.08631i	0.995905	−	0.0904098i	$-0.0288177\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−85.6163			−0.304684			−0.152342	−	0.988328i	$-0.548682\pi$
$281$	−85.6163			−0.304684			−0.152342	+	0.988328i	$0.548682\pi$
$282$		0			0
$283$	−99.4025	+	99.4025i	−0.351246	+	0.351246i	−0.860573	−	0.509327i	$-0.829894\pi$
$283$	−99.4025	+	99.4025i	−0.351246	+	0.351246i	0.509327	+	0.860573i	$0.329894\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	465.699i		−	1.62264i
$288$		0			0
$289$	188.392			0.651875
$290$		0			0
$291$		0			0
$292$		0			0
$293$	130.268	+	130.268i	0.444602	+	0.444602i	0.893555	−	0.448953i	$-0.148203\pi$
$293$	130.268	+	130.268i	0.444602	+	0.444602i	−0.448953	+	0.893555i	$0.648203\pi$
$294$		0			0
$295$			105.085i			0.356220i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	312.567	+	312.567i	1.04537	+	1.04537i
$300$		0			0
$301$	97.1356	+	97.1356i	0.322709	+	0.322709i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	241.771i		−	0.792691i
$306$		0			0
$307$	425.171	+	425.171i	1.38492	+	1.38492i	0.835635	+	0.549286i	$0.185100\pi$
$307$	425.171	+	425.171i	1.38492	+	1.38492i	0.549286	+	0.835635i	$0.314900\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	28.3005			0.0909985			0.0454993	−	0.998964i	$-0.485512\pi$
$311$	28.3005			0.0909985			0.0454993	+	0.998964i	$0.485512\pi$
$312$		0			0
$313$			501.098i			1.60095i	0.599366	+	0.800475i	$0.295420\pi$
$313$			501.098i			1.60095i	−0.599366	+	0.800475i	$0.704580\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−72.5630	+	72.5630i	−0.228905	+	0.228905i	−0.812235	−	0.583330i	$-0.801749\pi$
$317$	−72.5630	+	72.5630i	−0.228905	+	0.228905i	0.583330	+	0.812235i	$0.301749\pi$
$318$		0			0
$319$	−292.163			−0.915871
$320$		0			0
$321$		0			0
$322$		0			0
$323$	50.8588	−	50.8588i	0.157457	−	0.157457i
$324$		0			0
$325$	204.285	−	204.285i	0.628570	−	0.628570i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	687.237			2.08887
$330$		0			0
$331$	128.092	−	128.092i	0.386985	−	0.386985i	−0.486626	−	0.873610i	$-0.661773\pi$
$331$	128.092	−	128.092i	0.386985	−	0.386985i	0.873610	+	0.486626i	$0.161773\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			200.663i			0.598994i
$336$		0			0
$337$	429.921			1.27573			0.637865	−	0.770148i	$-0.279818\pi$
$337$	429.921			1.27573			0.637865	+	0.770148i	$0.279818\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−212.284	−	212.284i	−0.622535	−	0.622535i
$342$		0			0
$343$		−	1058.95i		−	3.08733i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−360.070	−	360.070i	−1.03766	−	1.03766i	−0.999262	−	0.0384019i	$-0.987773\pi$
$347$	−360.070	−	360.070i	−1.03766	−	1.03766i	−0.0384019	−	0.999262i	$-0.512227\pi$
$348$		0			0
$349$	276.674	+	276.674i	0.792763	+	0.792763i	0.981942	−	0.189180i	$-0.0605829\pi$
$349$	276.674	+	276.674i	0.792763	+	0.792763i	−0.189180	+	0.981942i	$0.560583\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	453.982i		−	1.28607i	−0.765837	−	0.643035i	$-0.777675\pi$
$353$		−	453.982i		−	1.28607i	0.765837	−	0.643035i	$-0.222325\pi$
$354$		0			0
$355$	5.14467	+	5.14467i	0.0144920	+	0.0144920i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−247.893			−0.690510			−0.345255	−	0.938509i	$-0.612208\pi$
$359$	−247.893			−0.690510			−0.345255	+	0.938509i	$0.612208\pi$
$360$		0			0
$361$		−	309.580i		−	0.857564i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.41674	−	3.41674i	0.00936092	−	0.00936092i
$366$		0			0
$367$	195.121			0.531665			0.265832	−	0.964019i	$-0.414353\pi$
$367$	195.121			0.531665			0.265832	+	0.964019i	$0.414353\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−911.724	+	911.724i	−2.45748	+	2.45748i
$372$		0			0
$373$	146.937	−	146.937i	0.393932	−	0.393932i	−0.482154	−	0.876086i	$-0.660145\pi$
$373$	146.937	−	146.937i	0.393932	−	0.393932i	0.876086	+	0.482154i	$0.160145\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−392.229			−1.04039
$378$		0			0
$379$	80.0235	−	80.0235i	0.211144	−	0.211144i	−0.593609	−	0.804753i	$-0.702298\pi$
$379$	80.0235	−	80.0235i	0.211144	−	0.211144i	0.804753	+	0.593609i	$0.202298\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	412.277i		−	1.07644i	−0.842804	−	0.538221i	$-0.819097\pi$
$383$		−	412.277i		−	1.07644i	0.842804	−	0.538221i	$-0.180903\pi$
$384$		0			0
$385$	−346.563			−0.900165
$386$		0			0
$387$		0			0
$388$		0			0
$389$	177.471	+	177.471i	0.456223	+	0.456223i	0.897414	−	0.441190i	$-0.145444\pi$
$389$	177.471	+	177.471i	0.456223	+	0.456223i	−0.441190	+	0.897414i	$0.645444\pi$
$390$		0			0
$391$		−	298.714i		−	0.763975i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−88.8405	−	88.8405i	−0.224913	−	0.224913i
$396$		0			0
$397$	−51.6079	−	51.6079i	−0.129995	−	0.129995i	0.639116	−	0.769110i	$-0.279300\pi$
$397$	−51.6079	−	51.6079i	−0.129995	−	0.129995i	−0.769110	+	0.639116i	$0.779300\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			87.1607i			0.217358i	0.994077	+	0.108679i	$0.0346621\pi$
$401$			87.1607i			0.217358i	−0.994077	+	0.108679i	$0.965338\pi$
$402$		0			0
$403$	−284.992	−	284.992i	−0.707177	−	0.707177i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−208.531			−0.512360
$408$		0			0
$409$			367.848i			0.899383i	0.893184	+	0.449691i	$0.148466\pi$
$409$			367.848i			0.899383i	−0.893184	+	0.449691i	$0.851534\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−420.738	+	420.738i	−1.01874	+	1.01874i
$414$		0			0
$415$	72.5249			0.174759
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−20.1974	+	20.1974i	−0.0482037	+	0.0482037i	−0.730798	−	0.682594i	$-0.760852\pi$
$419$	−20.1974	+	20.1974i	−0.0482037	+	0.0482037i	0.682594	+	0.730798i	$0.260852\pi$
$420$		0			0
$421$	353.549	−	353.549i	0.839784	−	0.839784i	−0.149046	−	0.988830i	$-0.547620\pi$
$421$	353.549	−	353.549i	0.839784	−	0.839784i	0.988830	+	0.149046i	$0.0476204\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−195.231			−0.459368
$426$		0			0
$427$	968.000	−	968.000i	2.26698	−	2.26698i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	534.507i		−	1.24015i	−0.784541	−	0.620077i	$-0.787101\pi$
$431$		−	534.507i		−	1.24015i	0.784541	−	0.620077i	$-0.212899\pi$
$432$		0			0
$433$	102.953			0.237768			0.118884	−	0.992908i	$-0.462068\pi$
$433$	102.953			0.237768			0.118884	+	0.992908i	$0.462068\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	151.004	+	151.004i	0.345547	+	0.345547i
$438$		0			0
$439$		−	323.949i		−	0.737925i	−0.929444	−	0.368963i	$-0.879713\pi$
$439$		−	323.949i		−	0.737925i	0.929444	−	0.368963i	$-0.120287\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	36.0517	+	36.0517i	0.0813809	+	0.0813809i	0.746625	−	0.665245i	$-0.231673\pi$
$443$	36.0517	+	36.0517i	0.0813809	+	0.0813809i	−0.665245	+	0.746625i	$0.731673\pi$
$444$		0			0
$445$	228.903	+	228.903i	0.514388	+	0.514388i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	556.262i		−	1.23889i	−0.785040	−	0.619445i	$-0.787358\pi$
$449$		−	556.262i		−	1.23889i	0.785040	−	0.619445i	$-0.212642\pi$
$450$		0			0
$451$	−273.282	−	273.282i	−0.605947	−	0.605947i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−465.262			−1.02255
$456$		0			0
$457$		−	118.453i		−	0.259196i	−0.991567	−	0.129598i	$-0.958631\pi$
$457$		−	118.453i		−	0.259196i	0.991567	−	0.129598i	$-0.0413687\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	231.411	−	231.411i	0.501977	−	0.501977i	−0.410075	−	0.912052i	$-0.634497\pi$
$461$	231.411	−	231.411i	0.501977	−	0.501977i	0.912052	+	0.410075i	$0.134497\pi$
$462$		0			0
$463$	−256.298			−0.553559			−0.276780	−	0.960933i	$-0.589267\pi$
$463$	−256.298			−0.553559			−0.276780	+	0.960933i	$0.589267\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	529.492	−	529.492i	1.13382	−	1.13382i	0.144278	−	0.989537i	$-0.453914\pi$
$467$	529.492	−	529.492i	1.13382	−	1.13382i	0.989537	−	0.144278i	$-0.0460859\pi$
$468$		0			0
$469$	−803.413	+	803.413i	−1.71303	+	1.71303i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	114.003			0.241020
$474$		0			0
$475$	98.6921	−	98.6921i	0.207773	−	0.207773i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			315.124i			0.657879i	0.944351	+	0.328939i	$0.106691\pi$
$479$			315.124i			0.657879i	−0.944351	+	0.328939i	$0.893309\pi$
$480$		0			0
$481$	−279.953			−0.582022
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−28.5977	−	28.5977i	−0.0589644	−	0.0589644i
$486$		0			0
$487$			490.083i			1.00633i	0.864190	+	0.503165i	$0.167831\pi$
$487$			490.083i			1.00633i	−0.864190	+	0.503165i	$0.832169\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	583.373	+	583.373i	1.18813	+	1.18813i	0.977583	+	0.210550i	$0.0675254\pi$
$491$	583.373	+	583.373i	1.18813	+	1.18813i	0.210550	+	0.977583i	$0.432475\pi$
$492$		0			0
$493$	187.423	+	187.423i	0.380168	+	0.380168i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			41.1964i			0.0828901i
$498$		0			0
$499$	310.729	+	310.729i	0.622703	+	0.622703i	0.946222	−	0.323519i	$-0.104866\pi$
$499$	310.729	+	310.729i	0.622703	+	0.622703i	−0.323519	+	0.946222i	$0.604866\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	722.400			1.43618			0.718091	−	0.695949i	$-0.245016\pi$
$503$	722.400			1.43618			0.718091	+	0.695949i	$0.245016\pi$
$504$		0			0
$505$		−	297.592i		−	0.589291i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	162.106	−	162.106i	0.318479	−	0.318479i	−0.529704	−	0.848183i	$-0.677697\pi$
$509$	162.106	−	162.106i	0.318479	−	0.318479i	0.848183	+	0.529704i	$0.177697\pi$
$510$		0			0
$511$	27.3598			0.0535417
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−206.823	+	206.823i	−0.401598	+	0.401598i
$516$		0			0
$517$	403.285	−	403.285i	0.780049	−	0.780049i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	184.583			0.354285			0.177143	−	0.984185i	$-0.443315\pi$
$521$	184.583			0.354285			0.177143	+	0.984185i	$0.443315\pi$
$522$		0			0
$523$	−593.324	+	593.324i	−1.13446	+	1.13446i	−0.145035	+	0.989426i	$0.546330\pi$
$523$	−593.324	+	593.324i	−1.13446	+	1.13446i	−0.989426	+	0.145035i	$0.953670\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			272.362i			0.516815i
$528$		0			0
$529$	357.907			0.676573
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−366.882	−	366.882i	−0.688334	−	0.688334i
$534$		0			0
$535$			391.513i			0.731801i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1004.49	−	1004.49i	−1.86362	−	1.86362i
$540$		0			0
$541$	550.411	+	550.411i	1.01739	+	1.01739i	0.999846	+	0.0175489i	$0.00558627\pi$
$541$	550.411	+	550.411i	1.01739	+	1.01739i	0.0175489	+	0.999846i	$0.494414\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	6.15452i		−	0.0112927i
$546$		0			0
$547$	−177.395	−	177.395i	−0.324305	−	0.324305i	0.526111	−	0.850416i	$-0.323650\pi$
$547$	−177.395	−	177.395i	−0.324305	−	0.324305i	−0.850416	+	0.526111i	$0.823650\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−189.489			−0.343901
$552$		0			0
$553$		−	711.398i		−	1.28643i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	374.619	−	374.619i	0.672566	−	0.672566i	−0.285741	−	0.958307i	$-0.592240\pi$
$557$	374.619	−	374.619i	0.672566	−	0.672566i	0.958307	+	0.285741i	$0.0922397\pi$
$558$		0			0
$559$	153.049			0.273790
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−370.252	+	370.252i	−0.657640	+	0.657640i	−0.954821	−	0.297181i	$-0.903954\pi$
$563$	−370.252	+	370.252i	−0.657640	+	0.657640i	0.297181	+	0.954821i	$0.403954\pi$
$564$		0			0
$565$	47.2331	−	47.2331i	0.0835984	−	0.0835984i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	30.3419			0.0533249			0.0266625	−	0.999644i	$-0.491512\pi$
$569$	30.3419			0.0533249			0.0266625	+	0.999644i	$0.491512\pi$
$570$		0			0
$571$	−336.964	+	336.964i	−0.590130	+	0.590130i	−0.937666	−	0.347537i	$-0.887018\pi$
$571$	−336.964	+	336.964i	−0.590130	+	0.590130i	0.347537	+	0.937666i	$0.387018\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	579.659i		−	1.00810i
$576$		0			0
$577$	959.056			1.66214			0.831071	−	0.556166i	$-0.187728\pi$
$577$	959.056			1.66214			0.831071	+	0.556166i	$0.187728\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	290.375	+	290.375i	0.499784	+	0.499784i
$582$		0			0
$583$			1070.04i			1.83540i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	242.869	+	242.869i	0.413747	+	0.413747i	0.883042	−	0.469295i	$-0.155492\pi$
$587$	242.869	+	242.869i	0.413747	+	0.413747i	−0.469295	+	0.883042i	$0.655492\pi$
$588$		0			0
$589$	−137.682	−	137.682i	−0.233756	−	0.233756i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			280.646i			0.473265i	0.971599	+	0.236633i	$0.0760438\pi$
$593$			280.646i			0.473265i	−0.971599	+	0.236633i	$0.923956\pi$
$594$		0			0
$595$	222.321	+	222.321i	0.373649	+	0.373649i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	945.632			1.57868			0.789342	−	0.613953i	$-0.210422\pi$
$599$	945.632			1.57868			0.789342	+	0.613953i	$0.210422\pi$
$600$		0			0
$601$			414.239i			0.689250i	0.938740	+	0.344625i	$0.111994\pi$
$601$			414.239i			0.689250i	−0.938740	+	0.344625i	$0.888006\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.06047	+	2.06047i	−0.00340573	+	0.00340573i
$606$		0			0
$607$	−214.881			−0.354005			−0.177003	−	0.984210i	$-0.556640\pi$
$607$	−214.881			−0.354005			−0.177003	+	0.984210i	$0.556640\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	541.411	−	541.411i	0.886107	−	0.886107i
$612$		0			0
$613$	−238.136	+	238.136i	−0.388476	+	0.388476i	−0.874143	−	0.485668i	$-0.838576\pi$
$613$	−238.136	+	238.136i	−0.388476	+	0.388476i	0.485668	+	0.874143i	$0.338576\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−223.561			−0.362335			−0.181167	−	0.983452i	$-0.557988\pi$
$617$	−223.561			−0.362335			−0.181167	+	0.983452i	$0.557988\pi$
$618$		0			0
$619$	35.0122	−	35.0122i	0.0565626	−	0.0565626i	−0.678260	−	0.734822i	$-0.737266\pi$
$619$	35.0122	−	35.0122i	0.0565626	−	0.0565626i	0.734822	+	0.678260i	$0.237266\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1832.96i			2.94215i
$624$		0			0
$625$	−240.450			−0.384721
$626$		0			0
$627$		0			0
$628$		0			0
$629$	133.773	+	133.773i	0.212675	+	0.212675i
$630$		0			0
$631$		−	613.896i		−	0.972894i	−0.873710	−	0.486447i	$-0.838293\pi$
$631$		−	613.896i		−	0.972894i	0.873710	−	0.486447i	$-0.161707\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−52.8896	−	52.8896i	−0.0832908	−	0.0832908i
$636$		0			0
$637$	−1348.53	−	1348.53i	−2.11701	−	2.11701i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			340.013i			0.530442i	0.964188	+	0.265221i	$0.0854449\pi$
$641$			340.013i			0.530442i	−0.964188	+	0.265221i	$0.914555\pi$
$642$		0			0
$643$	−516.730	−	516.730i	−0.803624	−	0.803624i	0.180036	−	0.983660i	$-0.442379\pi$
$643$	−516.730	−	516.730i	−0.803624	−	0.803624i	−0.983660	+	0.180036i	$0.942379\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−541.491			−0.836926			−0.418463	−	0.908234i	$-0.637431\pi$
$647$	−541.491			−0.836926			−0.418463	+	0.908234i	$0.637431\pi$
$648$		0			0
$649$			493.797i			0.760858i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	329.446	−	329.446i	0.504512	−	0.504512i	−0.408325	−	0.912837i	$-0.633887\pi$
$653$	329.446	−	329.446i	0.504512	−	0.504512i	0.912837	+	0.408325i	$0.133887\pi$
$654$		0			0
$655$	535.160			0.817038
$656$		0			0
$657$		0			0
$658$		0			0
$659$	70.8742	−	70.8742i	0.107548	−	0.107548i	−0.651285	−	0.758833i	$-0.725770\pi$
$659$	70.8742	−	70.8742i	0.107548	−	0.107548i	0.758833	+	0.651285i	$0.225770\pi$
$660$		0			0
$661$	192.446	−	192.446i	0.291143	−	0.291143i	−0.546388	−	0.837532i	$-0.683998\pi$
$661$	192.446	−	192.446i	0.291143	−	0.291143i	0.837532	+	0.546388i	$0.183998\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−224.772			−0.338004
$666$		0			0
$667$	−556.474	+	556.474i	−0.834294	+	0.834294i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1136.09i		−	1.69313i
$672$		0			0
$673$	−672.958			−0.999937			−0.499968	−	0.866044i	$-0.666655\pi$
$673$	−672.958			−0.999937			−0.499968	+	0.866044i	$0.666655\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−569.590	−	569.590i	−0.841344	−	0.841344i	0.147690	−	0.989034i	$-0.452816\pi$
$677$	−569.590	−	569.590i	−0.841344	−	0.841344i	−0.989034	+	0.147690i	$0.952816\pi$
$678$		0			0
$679$		−	228.999i		−	0.337259i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	474.354	+	474.354i	0.694515	+	0.694515i	0.963222	−	0.268707i	$-0.0865963\pi$
$683$	474.354	+	474.354i	0.694515	+	0.694515i	−0.268707	+	0.963222i	$0.586596\pi$
$684$		0			0
$685$	−373.362	−	373.362i	−0.545054	−	0.545054i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1436.53i			2.08495i
$690$		0			0
$691$	399.972	+	399.972i	0.578831	+	0.578831i	0.934581	−	0.355750i	$-0.115775\pi$
$691$	399.972	+	399.972i	0.578831	+	0.578831i	−0.355750	+	0.934581i	$0.615775\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−184.216			−0.265059
$696$		0			0
$697$			350.622i			0.503045i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−218.780	+	218.780i	−0.312098	+	0.312098i	−0.845722	−	0.533624i	$-0.820830\pi$
$701$	−218.780	+	218.780i	−0.312098	+	0.312098i	0.533624	+	0.845722i	$0.320830\pi$
$702$		0			0
$703$	−135.248			−0.192386
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1191.50	−	1191.50i	1.68529	−	1.68529i
$708$		0			0
$709$	−733.592	+	733.592i	−1.03469	+	1.03469i	−0.0353089	+	0.999376i	$0.511241\pi$
$709$	−733.592	+	733.592i	−1.03469	+	1.03469i	−0.999376	+	0.0353089i	$0.988759\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−808.664			−1.13417
$714$		0			0
$715$	−273.026	+	273.026i	−0.381854	+	0.381854i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			773.403i			1.07567i	0.843052	+	0.537833i	$0.180757\pi$
$719$			773.403i			1.07567i	−0.843052	+	0.537833i	$0.819243\pi$
$720$		0			0
$721$	−1656.15			−2.29702
$722$		0			0
$723$		0			0
$724$		0			0
$725$	363.696	+	363.696i	0.501650	+	0.501650i
$726$		0			0
$727$		−	158.236i		−	0.217656i	−0.994061	−	0.108828i	$-0.965290\pi$
$727$		−	158.236i		−	0.217656i	0.994061	−	0.108828i	$-0.0347098\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−73.1328	−	73.1328i	−0.100045	−	0.100045i
$732$		0			0
$733$	−925.166	−	925.166i	−1.26216	−	1.26216i	−0.950042	−	0.312122i	$-0.898960\pi$
$733$	−925.166	−	925.166i	−1.26216	−	1.26216i	−0.312122	−	0.950042i	$-0.601040\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			942.920i			1.27940i
$738$		0			0
$739$	313.893	+	313.893i	0.424754	+	0.424754i	0.886837	−	0.462083i	$-0.152898\pi$
$739$	313.893	+	313.893i	0.424754	+	0.424754i	−0.462083	+	0.886837i	$0.652898\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−116.397			−0.156658			−0.0783291	−	0.996928i	$-0.524958\pi$
$743$	−116.397			−0.156658			−0.0783291	+	0.996928i	$0.524958\pi$
$744$		0			0
$745$			331.246i			0.444626i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1567.54	+	1567.54i	−2.09284	+	2.09284i
$750$		0			0
$751$	700.738			0.933074			0.466537	−	0.884502i	$-0.345501\pi$
$751$	700.738			0.933074			0.466537	+	0.884502i	$0.345501\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−216.916	+	216.916i	−0.287306	+	0.287306i
$756$		0			0
$757$	−248.839	+	248.839i	−0.328717	+	0.328717i	−0.852099	−	0.523381i	$-0.824670\pi$
$757$	−248.839	+	248.839i	−0.328717	+	0.328717i	0.523381	+	0.852099i	$0.324670\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1269.42			1.66809			0.834046	−	0.551695i	$-0.186019\pi$
$761$	1269.42			1.66809			0.834046	+	0.551695i	$0.186019\pi$
$762$		0			0
$763$	24.6414	−	24.6414i	0.0322955	−	0.0322955i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			662.923i			0.864306i
$768$		0			0
$769$	261.949			0.340636			0.170318	−	0.985389i	$-0.445520\pi$
$769$	261.949			0.340636			0.170318	+	0.985389i	$0.445520\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.513628	+	0.513628i	0.000664461	+	0.000664461i	0.707439	−	0.706774i	$-0.249850\pi$
$773$	0.513628	+	0.513628i	0.000664461	+	0.000664461i	−0.706774	+	0.707439i	$0.749850\pi$
$774$		0			0
$775$			528.521i			0.681963i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−177.244	−	177.244i	−0.227528	−	0.227528i
$780$		0			0
$781$	24.1749	+	24.1749i	0.0309538	+	0.0309538i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	485.677i		−	0.618696i
$786$		0			0
$787$	−40.5482	−	40.5482i	−0.0515226	−	0.0515226i	0.680876	−	0.732399i	$-0.261599\pi$
$787$	−40.5482	−	40.5482i	−0.0515226	−	0.0515226i	−0.732399	+	0.680876i	$0.761599\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	378.223			0.478158
$792$		0			0
$793$		−	1525.20i		−	1.92333i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	316.113	−	316.113i	0.396629	−	0.396629i	−0.480413	−	0.877042i	$-0.659513\pi$
$797$	316.113	−	316.113i	0.396629	−	0.396629i	0.877042	+	0.480413i	$0.159513\pi$
$798$		0			0
$799$	−517.416			−0.647580
$800$		0			0
$801$		0			0
$802$		0			0
$803$	16.0553	−	16.0553i	0.0199942	−	0.0199942i
$804$		0			0
$805$	−660.090	+	660.090i	−0.819987	+	0.819987i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	701.360			0.866947			0.433473	−	0.901166i	$-0.357288\pi$
$809$	701.360			0.866947			0.433473	+	0.901166i	$0.357288\pi$
$810$		0			0
$811$	−266.653	+	266.653i	−0.328795	+	0.328795i	−0.852128	−	0.523333i	$-0.824688\pi$
$811$	−266.653	+	266.653i	−0.328795	+	0.328795i	0.523333	+	0.852128i	$0.324688\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	137.864i		−	0.169158i
$816$		0			0
$817$	73.9392			0.0905008
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−468.141	−	468.141i	−0.570209	−	0.570209i	0.361978	−	0.932187i	$-0.382102\pi$
$821$	−468.141	−	468.141i	−0.570209	−	0.570209i	−0.932187	+	0.361978i	$0.882102\pi$
$822$		0			0
$823$			285.382i			0.346758i	0.984855	+	0.173379i	$0.0554685\pi$
$823$			285.382i			0.346758i	−0.984855	+	0.173379i	$0.944531\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	560.223	+	560.223i	0.677416	+	0.677416i	0.959415	−	0.281999i	$-0.0909974\pi$
$827$	560.223	+	560.223i	0.677416	+	0.677416i	−0.281999	+	0.959415i	$0.590997\pi$
$828$		0			0
$829$	350.009	+	350.009i	0.422206	+	0.422206i	0.885963	−	0.463757i	$-0.153499\pi$
$829$	350.009	+	350.009i	0.422206	+	0.422206i	−0.463757	+	0.885963i	$0.653499\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1288.77i			1.54714i
$834$		0			0
$835$	23.1218	+	23.1218i	0.0276908	+	0.0276908i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	635.294			0.757204			0.378602	−	0.925560i	$-0.376405\pi$
$839$	635.294			0.757204			0.378602	+	0.925560i	$0.376405\pi$
$840$		0			0
$841$			142.701i			0.169680i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−85.3683	+	85.3683i	−0.101028	+	0.101028i
$846$		0			0
$847$	−16.4994			−0.0194798
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−397.182	+	397.182i	−0.466724	+	0.466724i
$852$		0			0
$853$	729.331	−	729.331i	0.855018	−	0.855018i	−0.135728	−	0.990746i	$-0.543337\pi$
$853$	729.331	−	729.331i	0.855018	−	0.855018i	0.990746	+	0.135728i	$0.0433373\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1560.63			−1.82104			−0.910522	−	0.413461i	$-0.864320\pi$
$857$	−1560.63			−1.82104			−0.910522	+	0.413461i	$0.864320\pi$
$858$		0			0
$859$	−880.955	+	880.955i	−1.02556	+	1.02556i	−0.0258940	+	0.999665i	$0.508243\pi$
$859$	−880.955	+	880.955i	−1.02556	+	1.02556i	−0.999665	+	0.0258940i	$0.991757\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			314.655i			0.364606i	0.983242	+	0.182303i	$0.0583551\pi$
$863$			314.655i			0.364606i	−0.983242	+	0.182303i	$0.941645\pi$
$864$		0			0
$865$	−145.911			−0.168683
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−417.464	−	417.464i	−0.480396	−	0.480396i
$870$		0			0
$871$			1265.87i			1.45335i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	985.537	+	985.537i	1.12633	+	1.12633i
$876$		0			0
$877$	377.957	+	377.957i	0.430966	+	0.430966i	0.888957	−	0.457991i	$-0.151431\pi$
$877$	377.957	+	377.957i	0.430966	+	0.430966i	−0.457991	+	0.888957i	$0.651431\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1208.58i		−	1.37183i	−0.727681	−	0.685916i	$-0.759402\pi$
$881$		−	1208.58i		−	1.37183i	0.727681	−	0.685916i	$-0.240598\pi$
$882$		0			0
$883$	840.437	+	840.437i	0.951797	+	0.951797i	0.998890	−	0.0470934i	$-0.0149958\pi$
$883$	840.437	+	840.437i	0.951797	+	0.951797i	−0.0470934	+	0.998890i	$0.514996\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	343.674			0.387456			0.193728	−	0.981055i	$-0.437942\pi$
$887$	343.674			0.387456			0.193728	+	0.981055i	$0.437942\pi$
$888$		0			0
$889$		−	423.518i		−	0.476399i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	261.561	−	261.561i	0.292901	−	0.292901i
$894$		0			0
$895$	−515.079			−0.575507
$896$		0			0
$897$		0			0
$898$		0			0
$899$	507.382	−	507.382i	0.564385	−	0.564385i
$900$		0			0
$901$	686.432	−	686.432i	0.761855	−	0.761855i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−352.423			−0.389417
$906$		0			0
$907$	319.782	−	319.782i	0.352571	−	0.352571i	−0.508494	−	0.861065i	$-0.669798\pi$
$907$	319.782	−	319.782i	0.352571	−	0.352571i	0.861065	+	0.508494i	$0.169798\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			444.391i			0.487806i	0.969800	+	0.243903i	$0.0784279\pi$
$911$			444.391i			0.487806i	−0.969800	+	0.243903i	$0.921572\pi$
$912$		0			0
$913$	340.796			0.373271
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2142.67	+	2142.67i	2.33661	+	2.33661i
$918$		0			0
$919$			1096.81i			1.19348i	0.802435	+	0.596739i	$0.203537\pi$
$919$			1096.81i			1.19348i	−0.802435	+	0.596739i	$0.796463\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	32.4549	+	32.4549i	0.0351624	+	0.0351624i
$924$		0			0
$925$	259.588	+	259.588i	0.280635	+	0.280635i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	362.291i		−	0.389979i	−0.980805	−	0.194990i	$-0.937533\pi$
$929$		−	362.291i		−	0.389979i	0.980805	−	0.194990i	$-0.0624673\pi$
$930$		0			0
$931$	−651.489	−	651.489i	−0.699773	−	0.699773i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	260.926			0.279065
$936$		0			0
$937$			1339.48i			1.42954i	0.699362	+	0.714768i	$0.253468\pi$
$937$			1339.48i			1.42954i	−0.699362	+	0.714768i	$0.746532\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−28.2194	+	28.2194i	−0.0299887	+	0.0299887i	−0.721942	−	0.691953i	$-0.756750\pi$
$941$	−28.2194	+	28.2194i	−0.0299887	+	0.0299887i	0.691953	+	0.721942i	$0.256750\pi$
$942$		0			0
$943$	−1041.03			−1.10395
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−632.413	+	632.413i	−0.667807	+	0.667807i	−0.957208	−	0.289401i	$-0.906544\pi$
$947$	−632.413	+	632.413i	−0.667807	+	0.667807i	0.289401	+	0.957208i	$0.406544\pi$
$948$		0			0
$949$	21.5543	−	21.5543i	0.0227126	−	0.0227126i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1203.56			1.26291			0.631456	−	0.775411i	$-0.282457\pi$
$953$	1203.56			1.26291			0.631456	+	0.775411i	$0.282457\pi$
$954$		0			0
$955$	−115.553	+	115.553i	−0.120998	+	0.120998i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	2989.73i		−	3.11755i
$960$		0			0
$961$	−223.676			−0.232753
$962$		0			0
$963$		0			0
$964$		0			0
$965$	59.1018	+	59.1018i	0.0612454	+	0.0612454i
$966$		0			0
$967$			1155.76i			1.19521i	0.801792	+	0.597603i	$0.203880\pi$
$967$			1155.76i			1.19521i	−0.801792	+	0.597603i	$0.796120\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	177.985	+	177.985i	0.183301	+	0.183301i	0.792793	−	0.609492i	$-0.208626\pi$
$971$	177.985	+	177.985i	0.183301	+	0.183301i	−0.609492	+	0.792793i	$0.708626\pi$
$972$		0			0
$973$	−737.564	−	737.564i	−0.758031	−	0.758031i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1662.86i		−	1.70200i	−0.525163	−	0.851002i	$-0.675996\pi$
$977$		−	1662.86i		−	1.70200i	0.525163	−	0.851002i	$-0.324004\pi$
$978$		0			0
$979$	1075.62	+	1075.62i	1.09869	+	1.09869i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1467.82			1.49320			0.746600	−	0.665273i	$-0.231685\pi$
$983$	1467.82			1.49320			0.746600	+	0.665273i	$0.231685\pi$
$984$		0			0
$985$		−	25.7750i		−	0.0261675i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	217.137	−	217.137i	0.219552	−	0.219552i
$990$		0			0
$991$	−685.863			−0.692091			−0.346046	−	0.938218i	$-0.612476\pi$
$991$	−685.863			−0.692091			−0.346046	+	0.938218i	$0.612476\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	263.657	−	263.657i	0.264982	−	0.264982i
$996$		0			0
$997$	−1209.62	+	1209.62i	−1.21326	+	1.21326i	−0.243317	+	0.969947i	$0.578235\pi$
$997$	−1209.62	+	1209.62i	−1.21326	+	1.21326i	−0.969947	+	0.243317i	$0.921765\pi$
$998$		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.161.10		32
3.2	odd	2	inner	1152.3.j.b.161.7		32
4.3	odd	2		1152.3.j.a.161.10		32
8.3	odd	2		144.3.j.a.125.7	yes	32
8.5	even	2		576.3.j.a.17.7		32
12.11	even	2		1152.3.j.a.161.7		32
16.3	odd	4		144.3.j.a.53.10	yes	32
16.5	even	4	inner	1152.3.j.b.737.7		32
16.11	odd	4		1152.3.j.a.737.7		32
16.13	even	4		576.3.j.a.305.10		32
24.5	odd	2		576.3.j.a.17.10		32
24.11	even	2		144.3.j.a.125.10	yes	32
48.5	odd	4	inner	1152.3.j.b.737.10		32
48.11	even	4		1152.3.j.a.737.10		32
48.29	odd	4		576.3.j.a.305.7		32
48.35	even	4		144.3.j.a.53.7	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.j.a.53.7	✓	32	48.35	even	4
144.3.j.a.53.10	yes	32	16.3	odd	4
144.3.j.a.125.7	yes	32	8.3	odd	2
144.3.j.a.125.10	yes	32	24.11	even	2
576.3.j.a.17.7		32	8.5	even	2
576.3.j.a.17.10		32	24.5	odd	2
576.3.j.a.305.7		32	48.29	odd	4
576.3.j.a.305.10		32	16.13	even	4
1152.3.j.a.161.7		32	12.11	even	2
1152.3.j.a.161.10		32	4.3	odd	2
1152.3.j.a.737.7		32	16.11	odd	4
1152.3.j.a.737.10		32	48.11	even	4
1152.3.j.b.161.7		32	3.2	odd	2	inner
1152.3.j.b.161.10		32	1.1	even	1	trivial
1152.3.j.b.737.7		32	16.5	even	4	inner
1152.3.j.b.737.10		32	48.5	odd	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)

\(f(q)\)	\(=\)	\(q+(1.66372 + 1.66372i) q^{5} +13.3224i q^{7} +O(q^{10})\)	\(q+(1.66372 + 1.66372i) q^{5} +13.3224i q^{7} +(7.81788 + 7.81788i) q^{11} +(10.4955 + 10.4955i) q^{13} -10.0304i q^{17} +(5.07048 + 5.07048i) q^{19} +29.7810 q^{23} -19.4640i q^{25} +(-18.6855 + 18.6855i) q^{29} -27.1537 q^{31} +(-22.1648 + 22.1648i) q^{35} +(-13.3368 + 13.3368i) q^{37} -34.9561 q^{41} +(7.29114 - 7.29114i) q^{43} -51.5850i q^{47} -128.487 q^{49} +(68.4354 + 68.4354i) q^{53} +26.0136i q^{55} +(31.5812 + 31.5812i) q^{59} +(-72.6595 - 72.6595i) q^{61} +34.9233i q^{65} +(60.3054 + 60.3054i) q^{67} +3.09226 q^{71} -2.05367i q^{73} +(-104.153 + 104.153i) q^{77} -53.3986 q^{79} +(21.7960 - 21.7960i) q^{83} +(16.6877 - 16.6877i) q^{85} +137.585 q^{89} +(-139.826 + 139.826i) q^{91} +16.8718i q^{95} -17.1890 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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