LMFDB - Embedded newform 1152.3.m.e.991.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(415,1152)]

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, 0]))

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

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

chi := DirichletCharacter("1152.415");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.m (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:	$16$
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} - 6x^{14} + 10x^{12} + 88x^{10} - 752x^{8} + 1408x^{6} + 2560x^{4} - 24576x^{2} + 65536 $ x^16 - 6x^14 + 10x^12 + 88x^10 - 752x^8 + 1408x^6 + 2560x^4 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			991.3
Root			$1.64663 + 1.13516i$ of defining polynomial
Character	$\chi$	$=$	1152.991
Dual form			1152.3.m.e.415.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.41234 + 2.41234i) q^{5} -11.8718 q^{7} +O(q^{10})$	$q+(-2.41234 + 2.41234i) q^{5} -11.8718 q^{7} +(11.9421 + 11.9421i) q^{11} +(-2.08177 - 2.08177i) q^{13} -23.1512 q^{17} +(6.77297 - 6.77297i) q^{19} +3.92781 q^{23} +13.3613i q^{25} +(-0.782548 - 0.782548i) q^{29} -2.65295i q^{31} +(28.6389 - 28.6389i) q^{35} +(-37.2078 + 37.2078i) q^{37} -69.1259i q^{41} +(29.2978 + 29.2978i) q^{43} -68.0631i q^{47} +91.9405 q^{49} +(40.3674 - 40.3674i) q^{53} -57.6167 q^{55} +(-23.5544 - 23.5544i) q^{59} +(-65.9304 - 65.9304i) q^{61} +10.0438 q^{65} +(40.9626 - 40.9626i) q^{67} +98.1885 q^{71} -74.2864i q^{73} +(-141.774 - 141.774i) q^{77} -0.779953i q^{79} +(91.7887 - 91.7887i) q^{83} +(55.8486 - 55.8486i) q^{85} -20.1568i q^{89} +(24.7144 + 24.7144i) q^{91} +32.6774i q^{95} -86.6476 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 32 q^{19} - 96 q^{37} + 32 q^{43} + 112 q^{49} - 256 q^{55} + 32 q^{61} + 256 q^{67} - 160 q^{85} - 288 q^{91}+O(q^{100}) $ 16 * q + 32 * q^19 - 96 * q^37 + 32 * q^43 + 112 * q^49 - 256 * q^55 + 32 * q^61 + 256 * q^67 - 160 * q^85 - 288 * 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
$5$	−2.41234	+	2.41234i	−0.482467	+	0.482467i	−0.905919	−	0.423451i	$-0.860818\pi$
$5$	−2.41234	+	2.41234i	−0.482467	+	0.482467i	0.423451	+	0.905919i	$0.360818\pi$
$6$		0			0
$7$	−11.8718			−1.69598			−0.847988	−	0.530015i	$-0.822186\pi$
$7$	−11.8718			−1.69598			−0.847988	+	0.530015i	$0.822186\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	11.9421	+	11.9421i	1.08564	+	1.08564i	0.995971	+	0.0896729i	$0.0285822\pi$
$11$	11.9421	+	11.9421i	1.08564	+	1.08564i	0.0896729	+	0.995971i	$0.471418\pi$
$12$		0			0
$13$	−2.08177	−	2.08177i	−0.160136	−	0.160136i	0.622491	−	0.782627i	$-0.286121\pi$
$13$	−2.08177	−	2.08177i	−0.160136	−	0.160136i	−0.782627	+	0.622491i	$0.786121\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−23.1512			−1.36184			−0.680919	−	0.732359i	$-0.738419\pi$
$17$	−23.1512			−1.36184			−0.680919	+	0.732359i	$0.738419\pi$
$18$		0			0
$19$	6.77297	−	6.77297i	0.356472	−	0.356472i	−0.506039	−	0.862511i	$-0.668891\pi$
$19$	6.77297	−	6.77297i	0.356472	−	0.356472i	0.862511	+	0.506039i	$0.168891\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.92781			0.170774			0.0853872	−	0.996348i	$-0.472787\pi$
$23$	3.92781			0.170774			0.0853872	+	0.996348i	$0.472787\pi$
$24$		0			0
$25$			13.3613i			0.534450i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.782548	−	0.782548i	−0.0269844	−	0.0269844i	0.693486	−	0.720470i	$-0.256074\pi$
$29$	−0.782548	−	0.782548i	−0.0269844	−	0.0269844i	−0.720470	+	0.693486i	$0.756074\pi$
$30$		0			0
$31$		−	2.65295i		−	0.0855792i	−0.999084	−	0.0427896i	$-0.986375\pi$
$31$		−	2.65295i		−	0.0855792i	0.999084	−	0.0427896i	$-0.0136245\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	28.6389	−	28.6389i	0.818253	−	0.818253i
$36$		0			0
$37$	−37.2078	+	37.2078i	−1.00562	+	1.00562i	−0.00563338	+	0.999984i	$0.501793\pi$
$37$	−37.2078	+	37.2078i	−1.00562	+	1.00562i	−0.999984	+	0.00563338i	$0.998207\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	69.1259i		−	1.68600i	−0.537916	−	0.842998i	$-0.680788\pi$
$41$		−	69.1259i		−	1.68600i	0.537916	−	0.842998i	$-0.319212\pi$
$42$		0			0
$43$	29.2978	+	29.2978i	0.681343	+	0.681343i	0.960303	−	0.278960i	$-0.0899895\pi$
$43$	29.2978	+	29.2978i	0.681343	+	0.681343i	−0.278960	+	0.960303i	$0.589990\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	68.0631i		−	1.44815i	−0.689720	−	0.724076i	$-0.742266\pi$
$47$		−	68.0631i		−	1.44815i	0.689720	−	0.724076i	$-0.257734\pi$
$48$		0			0
$49$	91.9405			1.87634
$50$		0			0
$51$		0			0
$52$		0			0
$53$	40.3674	−	40.3674i	0.761649	−	0.761649i	−0.214971	−	0.976620i	$-0.568966\pi$
$53$	40.3674	−	40.3674i	0.761649	−	0.761649i	0.976620	+	0.214971i	$0.0689658\pi$
$54$		0			0
$55$	−57.6167			−1.04758
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−23.5544	−	23.5544i	−0.399227	−	0.399227i	0.478734	−	0.877960i	$-0.341096\pi$
$59$	−23.5544	−	23.5544i	−0.399227	−	0.399227i	−0.877960	+	0.478734i	$0.841096\pi$
$60$		0			0
$61$	−65.9304	−	65.9304i	−1.08083	−	1.08083i	−0.996433	−	0.0843932i	$-0.973105\pi$
$61$	−65.9304	−	65.9304i	−1.08083	−	1.08083i	−0.0843932	−	0.996433i	$-0.526895\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.0438			0.154521
$66$		0			0
$67$	40.9626	−	40.9626i	0.611381	−	0.611381i	−0.331925	−	0.943306i	$-0.607698\pi$
$67$	40.9626	−	40.9626i	0.611381	−	0.611381i	0.943306	+	0.331925i	$0.107698\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	98.1885			1.38294			0.691468	−	0.722407i	$-0.256964\pi$
$71$	98.1885			1.38294			0.691468	+	0.722407i	$0.256964\pi$
$72$		0			0
$73$		−	74.2864i		−	1.01762i	−0.860878	−	0.508811i	$-0.830085\pi$
$73$		−	74.2864i		−	1.01762i	0.860878	−	0.508811i	$-0.169915\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−141.774	−	141.774i	−1.84123	−	1.84123i
$78$		0			0
$79$		−	0.779953i		−	0.00987283i	−0.999988	−	0.00493641i	$-0.998429\pi$
$79$		−	0.779953i		−	0.00987283i	0.999988	−	0.00493641i	$-0.00157132\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	91.7887	−	91.7887i	1.10589	−	1.10589i	0.112202	−	0.993685i	$-0.464210\pi$
$83$	91.7887	−	91.7887i	1.10589	−	1.10589i	0.993685	−	0.112202i	$-0.0357904\pi$
$84$		0			0
$85$	55.8486	−	55.8486i	0.657042	−	0.657042i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	20.1568i		−	0.226481i	−0.993568	−	0.113240i	$-0.963877\pi$
$89$		−	20.1568i		−	0.226481i	0.993568	−	0.113240i	$-0.0361230\pi$
$90$		0			0
$91$	24.7144	+	24.7144i	0.271587	+	0.271587i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			32.6774i			0.343972i
$96$		0			0
$97$	−86.6476			−0.893275			−0.446637	−	0.894715i	$-0.647379\pi$
$97$	−86.6476			−0.893275			−0.446637	+	0.894715i	$0.647379\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−58.6920	+	58.6920i	−0.581109	+	0.581109i	−0.935208	−	0.354099i	$-0.884788\pi$
$101$	−58.6920	+	58.6920i	−0.581109	+	0.581109i	0.354099	+	0.935208i	$0.384788\pi$
$102$		0			0
$103$	34.6530			0.336436			0.168218	−	0.985750i	$-0.446199\pi$
$103$	34.6530			0.336436			0.168218	+	0.985750i	$0.446199\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	46.7828	+	46.7828i	0.437223	+	0.437223i	0.891076	−	0.453854i	$-0.149951\pi$
$107$	46.7828	+	46.7828i	0.437223	+	0.437223i	−0.453854	+	0.891076i	$0.649951\pi$
$108$		0			0
$109$	93.0565	+	93.0565i	0.853729	+	0.853729i	0.990590	−	0.136861i	$-0.0437015\pi$
$109$	93.0565	+	93.0565i	0.853729	+	0.853729i	−0.136861	+	0.990590i	$0.543701\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−143.982			−1.27418			−0.637088	−	0.770791i	$-0.719861\pi$
$113$	−143.982			−1.27418			−0.637088	+	0.770791i	$0.719861\pi$
$114$		0			0
$115$	−9.47521	+	9.47521i	−0.0823931	+	0.0823931i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	274.848			2.30965
$120$		0			0
$121$			164.227i			1.35725i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−92.5403	−	92.5403i	−0.740322	−	0.740322i
$126$		0			0
$127$			47.3047i			0.372478i	0.982504	+	0.186239i	$0.0596299\pi$
$127$			47.3047i			0.372478i	−0.982504	+	0.186239i	$0.940370\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	26.5360	−	26.5360i	0.202565	−	0.202565i	−0.598533	−	0.801098i	$-0.704250\pi$
$131$	26.5360	−	26.5360i	0.202565	−	0.202565i	0.801098	+	0.598533i	$0.204250\pi$
$132$		0			0
$133$	−80.4076	+	80.4076i	−0.604568	+	0.604568i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	122.806i		−	0.896391i	−0.893936	−	0.448196i	$-0.852067\pi$
$137$		−	122.806i		−	0.896391i	0.893936	−	0.448196i	$-0.147933\pi$
$138$		0			0
$139$	−44.5369	−	44.5369i	−0.320410	−	0.320410i	0.528515	−	0.848924i	$-0.322749\pi$
$139$	−44.5369	−	44.5369i	−0.320410	−	0.320410i	−0.848924	+	0.528515i	$0.822749\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	49.7212i		−	0.347701i
$144$		0			0
$145$	3.77554			0.0260382
$146$		0			0
$147$		0			0
$148$		0			0
$149$	27.5232	−	27.5232i	0.184719	−	0.184719i	−0.608689	−	0.793409i	$-0.708304\pi$
$149$	27.5232	−	27.5232i	0.184719	−	0.184719i	0.793409	+	0.608689i	$0.208304\pi$
$150$		0			0
$151$	59.6723			0.395181			0.197590	−	0.980285i	$-0.436688\pi$
$151$	59.6723			0.395181			0.197590	+	0.980285i	$0.436688\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.39982	+	6.39982i	0.0412892	+	0.0412892i
$156$		0			0
$157$	−23.4479	−	23.4479i	−0.149350	−	0.149350i	0.628478	−	0.777827i	$-0.283678\pi$
$157$	−23.4479	−	23.4479i	−0.149350	−	0.149350i	−0.777827	+	0.628478i	$0.783678\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−46.6304			−0.289630
$162$		0			0
$163$	−119.673	+	119.673i	−0.734189	+	0.734189i	−0.971447	−	0.237258i	$-0.923751\pi$
$163$	−119.673	+	119.673i	−0.734189	+	0.734189i	0.237258	+	0.971447i	$0.423751\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−225.083			−1.34780			−0.673902	−	0.738820i	$-0.735383\pi$
$167$	−225.083			−1.34780			−0.673902	+	0.738820i	$0.735383\pi$
$168$		0			0
$169$		−	160.333i		−	0.948713i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	85.4327	+	85.4327i	0.493830	+	0.493830i	0.909511	−	0.415680i	$-0.136456\pi$
$173$	85.4327	+	85.4327i	0.493830	+	0.493830i	−0.415680	+	0.909511i	$0.636456\pi$
$174$		0			0
$175$		−	158.623i		−	0.906415i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	159.342	−	159.342i	0.890180	−	0.890180i	−0.104360	−	0.994540i	$-0.533279\pi$
$179$	159.342	−	159.342i	0.890180	−	0.890180i	0.994540	+	0.104360i	$0.0332793\pi$
$180$		0			0
$181$	127.296	−	127.296i	0.703295	−	0.703295i	−0.261821	−	0.965116i	$-0.584323\pi$
$181$	127.296	−	127.296i	0.703295	−	0.703295i	0.965116	+	0.261821i	$0.0843230\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	179.516i		−	0.970355i
$186$		0			0
$187$	−276.474	−	276.474i	−1.47847	−	1.47847i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	269.623i		−	1.41164i	−0.708392	−	0.705819i	$-0.750579\pi$
$191$		−	269.623i		−	1.41164i	0.708392	−	0.705819i	$-0.249421\pi$
$192$		0			0
$193$	−245.619			−1.27264			−0.636318	−	0.771427i	$-0.719544\pi$
$193$	−245.619			−1.27264			−0.636318	+	0.771427i	$0.719544\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	220.753	−	220.753i	1.12057	−	1.12057i	0.128917	−	0.991655i	$-0.458850\pi$
$197$	220.753	−	220.753i	1.12057	−	1.12057i	0.991655	−	0.128917i	$-0.0411499\pi$
$198$		0			0
$199$	231.414			1.16288			0.581441	−	0.813588i	$-0.302489\pi$
$199$	231.414			1.16288			0.581441	+	0.813588i	$0.302489\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.29029	+	9.29029i	0.0457650	+	0.0457650i
$204$		0			0
$205$	166.755	+	166.755i	0.813438	+	0.813438i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	161.767			0.774004
$210$		0			0
$211$	−226.721	+	226.721i	−1.07451	+	1.07451i	−0.0775144	+	0.996991i	$0.524698\pi$
$211$	−226.721	+	226.721i	−1.07451	+	1.07451i	−0.996991	+	0.0775144i	$0.975302\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−141.352			−0.657452
$216$		0			0
$217$			31.4954i			0.145140i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	48.1955	+	48.1955i	0.218079	+	0.218079i
$222$		0			0
$223$			205.578i			0.921874i	0.887433	+	0.460937i	$0.152487\pi$
$223$			205.578i			0.921874i	−0.887433	+	0.460937i	$0.847513\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−170.299	+	170.299i	−0.750215	+	0.750215i	−0.974519	−	0.224304i	$-0.927989\pi$
$227$	−170.299	+	170.299i	−0.750215	+	0.750215i	0.224304	+	0.974519i	$0.427989\pi$
$228$		0			0
$229$	140.673	−	140.673i	0.614293	−	0.614293i	−0.329769	−	0.944062i	$-0.606971\pi$
$229$	140.673	−	140.673i	0.614293	−	0.614293i	0.944062	+	0.329769i	$0.106971\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			140.555i			0.603238i	0.953428	+	0.301619i	$0.0975271\pi$
$233$			140.555i			0.603238i	−0.953428	+	0.301619i	$0.902473\pi$
$234$		0			0
$235$	164.191	+	164.191i	0.698686	+	0.698686i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	48.3886i		−	0.202463i	−0.994863	−	0.101231i	$-0.967722\pi$
$239$		−	48.3886i		−	0.202463i	0.994863	−	0.101231i	$-0.0322783\pi$
$240$		0			0
$241$	−129.557			−0.537582			−0.268791	−	0.963198i	$-0.586624\pi$
$241$	−129.557			−0.537582			−0.268791	+	0.963198i	$0.586624\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−221.791	+	221.791i	−0.905271	+	0.905271i
$246$		0			0
$247$	−28.1995			−0.114168
$248$		0			0
$249$		0			0
$250$		0			0
$251$	61.7026	+	61.7026i	0.245827	+	0.245827i	0.819256	−	0.573428i	$-0.194387\pi$
$251$	61.7026	+	61.7026i	0.245827	+	0.245827i	−0.573428	+	0.819256i	$0.694387\pi$
$252$		0			0
$253$	46.9063	+	46.9063i	0.185400	+	0.185400i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−242.911			−0.945181			−0.472590	−	0.881282i	$-0.656681\pi$
$257$	−242.911			−0.945181			−0.472590	+	0.881282i	$0.656681\pi$
$258$		0			0
$259$	441.725	−	441.725i	1.70550	−	1.70550i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.18268			0.0197060			0.00985301	−	0.999951i	$-0.496864\pi$
$263$	5.18268			0.0197060			0.00985301	+	0.999951i	$0.496864\pi$
$264$		0			0
$265$			194.760i			0.734942i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	202.350	+	202.350i	0.752230	+	0.752230i	0.974895	−	0.222665i	$-0.0714755\pi$
$269$	202.350	+	202.350i	0.752230	+	0.752230i	−0.222665	+	0.974895i	$0.571476\pi$
$270$		0			0
$271$		−	42.0616i		−	0.155209i	−0.996984	−	0.0776044i	$-0.975273\pi$
$271$		−	42.0616i		−	0.155209i	0.996984	−	0.0776044i	$-0.0247271\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−159.561	+	159.561i	−0.580223	+	0.580223i
$276$		0			0
$277$	−117.767	+	117.767i	−0.425151	+	0.425151i	−0.886973	−	0.461822i	$-0.847196\pi$
$277$	−117.767	+	117.767i	−0.425151	+	0.425151i	0.461822	+	0.886973i	$0.347196\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			186.670i			0.664307i	0.943225	+	0.332153i	$0.107775\pi$
$281$			186.670i			0.664307i	−0.943225	+	0.332153i	$0.892225\pi$
$282$		0			0
$283$	233.633	+	233.633i	0.825557	+	0.825557i	0.986899	−	0.161342i	$-0.0515821\pi$
$283$	233.633	+	233.633i	0.825557	+	0.825557i	−0.161342	+	0.986899i	$0.551582\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			820.651i			2.85941i
$288$		0			0
$289$	246.980			0.854603
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−189.454	+	189.454i	−0.646602	+	0.646602i	−0.952170	−	0.305568i	$-0.901154\pi$
$293$	−189.454	+	189.454i	−0.646602	+	0.646602i	0.305568	+	0.952170i	$0.401154\pi$
$294$		0			0
$295$	113.642			0.385228
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.17678	−	8.17678i	−0.0273471	−	0.0273471i
$300$		0			0
$301$	−347.818	−	347.818i	−1.15554	−	1.15554i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	318.093			1.04293
$306$		0			0
$307$	−37.1123	+	37.1123i	−0.120887	+	0.120887i	−0.764962	−	0.644075i	$-0.777242\pi$
$307$	−37.1123	+	37.1123i	−0.120887	+	0.120887i	0.644075	+	0.764962i	$0.277242\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	18.2210			0.0585884			0.0292942	−	0.999571i	$-0.490674\pi$
$311$	18.2210			0.0585884			0.0292942	+	0.999571i	$0.490674\pi$
$312$		0			0
$313$		−	481.905i		−	1.53963i	−0.638265	−	0.769816i	$-0.720348\pi$
$313$		−	481.905i		−	1.53963i	0.638265	−	0.769816i	$-0.279652\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−63.1197	−	63.1197i	−0.199116	−	0.199116i	0.600505	−	0.799621i	$-0.294966\pi$
$317$	−63.1197	−	63.1197i	−0.199116	−	0.199116i	−0.799621	+	0.600505i	$0.794966\pi$
$318$		0			0
$319$		−	18.6905i		−	0.0585910i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−156.803	+	156.803i	−0.485457	+	0.485457i
$324$		0			0
$325$	27.8150	−	27.8150i	0.0855846	−	0.0855846i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			808.034i			2.45603i
$330$		0			0
$331$	−21.7753	−	21.7753i	−0.0657865	−	0.0657865i	0.673448	−	0.739235i	$-0.264813\pi$
$331$	−21.7753	−	21.7753i	−0.0657865	−	0.0657865i	−0.739235	+	0.673448i	$0.764813\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			197.631i			0.589943i
$336$		0			0
$337$	91.1781			0.270558			0.135279	−	0.990808i	$-0.456807\pi$
$337$	91.1781			0.270558			0.135279	+	0.990808i	$0.456807\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	31.6818	−	31.6818i	0.0929085	−	0.0929085i
$342$		0			0
$343$	−509.782			−1.48625
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−375.508	−	375.508i	−1.08216	−	1.08216i	−0.996308	−	0.0858482i	$-0.972640\pi$
$347$	−375.508	−	375.508i	−1.08216	−	1.08216i	−0.0858482	−	0.996308i	$-0.527360\pi$
$348$		0			0
$349$	175.613	+	175.613i	0.503189	+	0.503189i	0.912428	−	0.409238i	$-0.134206\pi$
$349$	175.613	+	175.613i	0.503189	+	0.503189i	−0.409238	+	0.912428i	$0.634206\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	322.470			0.913514			0.456757	−	0.889592i	$-0.349011\pi$
$353$	322.470			0.913514			0.456757	+	0.889592i	$0.349011\pi$
$354$		0			0
$355$	−236.864	+	236.864i	−0.667222	+	0.667222i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−100.748			−0.280636			−0.140318	−	0.990107i	$-0.544812\pi$
$359$	−100.748			−0.280636			−0.140318	+	0.990107i	$0.544812\pi$
$360$		0			0
$361$			269.254i			0.745855i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	179.204	+	179.204i	0.490969	+	0.490969i
$366$		0			0
$367$		−	220.387i		−	0.600509i	−0.953859	−	0.300255i	$-0.902928\pi$
$367$		−	220.387i		−	0.600509i	0.953859	−	0.300255i	$-0.0970717\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−479.235	+	479.235i	−1.29174	+	1.29174i
$372$		0			0
$373$	−257.247	+	257.247i	−0.689671	+	0.689671i	−0.962159	−	0.272488i	$-0.912154\pi$
$373$	−257.247	+	257.247i	−0.689671	+	0.689671i	0.272488	+	0.962159i	$0.412154\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3.25816i			0.00864234i
$378$		0			0
$379$	−466.048	−	466.048i	−1.22968	−	1.22968i	−0.964086	−	0.265592i	$-0.914433\pi$
$379$	−466.048	−	466.048i	−1.22968	−	1.22968i	−0.265592	−	0.964086i	$-0.585567\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	416.164i		−	1.08659i	−0.839541	−	0.543296i	$-0.817176\pi$
$383$		−	416.164i		−	1.08659i	0.839541	−	0.543296i	$-0.182824\pi$
$384$		0			0
$385$	684.016			1.77666
$386$		0			0
$387$		0			0
$388$		0			0
$389$	80.1982	−	80.1982i	0.206165	−	0.206165i	−0.596470	−	0.802635i	$-0.703431\pi$
$389$	80.1982	−	80.1982i	0.206165	−	0.206165i	0.802635	+	0.596470i	$0.203431\pi$
$390$		0			0
$391$	−90.9338			−0.232567
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.88151	+	1.88151i	0.00476332	+	0.00476332i
$396$		0			0
$397$	−347.220	−	347.220i	−0.874610	−	0.874610i	0.118360	−	0.992971i	$-0.462236\pi$
$397$	−347.220	−	347.220i	−0.874610	−	0.874610i	−0.992971	+	0.118360i	$0.962236\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	464.444			1.15821			0.579107	−	0.815251i	$-0.303401\pi$
$401$	464.444			1.15821			0.579107	+	0.815251i	$0.303401\pi$
$402$		0			0
$403$	−5.52283	+	5.52283i	−0.0137043	+	0.0137043i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−888.679			−2.18349
$408$		0			0
$409$			93.6191i			0.228898i	0.993429	+	0.114449i	$0.0365102\pi$
$409$			93.6191i			0.228898i	−0.993429	+	0.114449i	$0.963490\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	279.634	+	279.634i	0.677079	+	0.677079i
$414$		0			0
$415$			442.850i			1.06711i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	166.970	−	166.970i	0.398497	−	0.398497i	−0.479206	−	0.877703i	$-0.659075\pi$
$419$	166.970	−	166.970i	0.398497	−	0.398497i	0.877703	+	0.479206i	$0.159075\pi$
$420$		0			0
$421$	−201.491	+	201.491i	−0.478600	+	0.478600i	−0.904684	−	0.426083i	$-0.859893\pi$
$421$	−201.491	+	201.491i	−0.478600	+	0.478600i	0.426083	+	0.904684i	$0.359893\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	309.330i		−	0.727835i
$426$		0			0
$427$	782.715	+	782.715i	1.83306	+	1.83306i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			158.438i			0.367606i	0.982963	+	0.183803i	$0.0588409\pi$
$431$			158.438i			0.367606i	−0.982963	+	0.183803i	$0.941159\pi$
$432$		0			0
$433$	−178.566			−0.412393			−0.206197	−	0.978511i	$-0.566109\pi$
$433$	−178.566			−0.412393			−0.206197	+	0.978511i	$0.566109\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	26.6030	−	26.6030i	0.0608763	−	0.0608763i
$438$		0			0
$439$	−702.502			−1.60023			−0.800116	−	0.599845i	$-0.795229\pi$
$439$	−702.502			−1.60023			−0.800116	+	0.599845i	$0.795229\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−118.984	−	118.984i	−0.268587	−	0.268587i	0.559943	−	0.828531i	$-0.310823\pi$
$443$	−118.984	−	118.984i	−0.268587	−	0.268587i	−0.828531	+	0.559943i	$0.810823\pi$
$444$		0			0
$445$	48.6249	+	48.6249i	0.109270	+	0.109270i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	504.107			1.12273			0.561366	−	0.827567i	$-0.310276\pi$
$449$	504.107			1.12273			0.561366	+	0.827567i	$0.310276\pi$
$450$		0			0
$451$	825.507	−	825.507i	1.83039	−	1.83039i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−119.239			−0.262063
$456$		0			0
$457$			392.141i			0.858077i	0.903286	+	0.429039i	$0.141148\pi$
$457$			392.141i			0.858077i	−0.903286	+	0.429039i	$0.858852\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	174.727	+	174.727i	0.379017	+	0.379017i	0.870747	−	0.491731i	$-0.163635\pi$
$461$	174.727	+	174.727i	0.379017	+	0.379017i	−0.491731	+	0.870747i	$0.663635\pi$
$462$		0			0
$463$		−	769.387i		−	1.66174i	−0.556464	−	0.830872i	$-0.687842\pi$
$463$		−	769.387i		−	1.66174i	0.556464	−	0.830872i	$-0.312158\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−238.009	+	238.009i	−0.509656	+	0.509656i	−0.914421	−	0.404765i	$-0.867353\pi$
$467$	−238.009	+	238.009i	−0.509656	+	0.509656i	0.404765	+	0.914421i	$0.367353\pi$
$468$		0			0
$469$	−486.301	+	486.301i	−1.03689	+	1.03689i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			699.753i			1.47939i
$474$		0			0
$475$	90.4954	+	90.4954i	0.190517	+	0.190517i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			98.2671i			0.205151i	0.994725	+	0.102575i	$0.0327083\pi$
$479$			98.2671i			0.205151i	−0.994725	+	0.102575i	$0.967292\pi$
$480$		0			0
$481$	154.916			0.322071
$482$		0			0
$483$		0			0
$484$		0			0
$485$	209.023	−	209.023i	0.430976	−	0.430976i
$486$		0			0
$487$	385.471			0.791522			0.395761	−	0.918353i	$-0.370481\pi$
$487$	385.471			0.791522			0.395761	+	0.918353i	$0.370481\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−118.148	−	118.148i	−0.240627	−	0.240627i	0.576483	−	0.817109i	$-0.304425\pi$
$491$	−118.148	−	118.148i	−0.240627	−	0.240627i	−0.817109	+	0.576483i	$0.804425\pi$
$492$		0			0
$493$	18.1170	+	18.1170i	0.0367484	+	0.0367484i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1165.68			−2.34543
$498$		0			0
$499$	−215.550	+	215.550i	−0.431964	+	0.431964i	−0.889296	−	0.457332i	$-0.848805\pi$
$499$	−215.550	+	215.550i	−0.431964	+	0.431964i	0.457332	+	0.889296i	$0.348805\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	58.9526			0.117202			0.0586010	−	0.998281i	$-0.481336\pi$
$503$	58.9526			0.117202			0.0586010	+	0.998281i	$0.481336\pi$
$504$		0			0
$505$		−	283.170i		−	0.560733i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	612.268	+	612.268i	1.20288	+	1.20288i	0.973285	+	0.229600i	$0.0737417\pi$
$509$	612.268	+	612.268i	1.20288	+	1.20288i	0.229600	+	0.973285i	$0.426258\pi$
$510$		0			0
$511$			881.916i			1.72586i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−83.5946	+	83.5946i	−0.162320	+	0.162320i
$516$		0			0
$517$	812.816	−	812.816i	1.57218	−	1.57218i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	562.971i		−	1.08056i	−0.841486	−	0.540279i	$-0.818319\pi$
$521$		−	562.971i		−	1.08056i	0.841486	−	0.540279i	$-0.181681\pi$
$522$		0			0
$523$	−104.076	−	104.076i	−0.198998	−	0.198998i	0.600572	−	0.799571i	$-0.294940\pi$
$523$	−104.076	−	104.076i	−0.198998	−	0.198998i	−0.799571	+	0.600572i	$0.794940\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			61.4192i			0.116545i
$528$		0			0
$529$	−513.572			−0.970836
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−143.904	+	143.904i	−0.269988	+	0.269988i
$534$		0			0
$535$	−225.712			−0.421891
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1097.96	+	1097.96i	2.03703	+	2.03703i
$540$		0			0
$541$	209.794	+	209.794i	0.387788	+	0.387788i	0.873898	−	0.486109i	$-0.161584\pi$
$541$	209.794	+	209.794i	0.387788	+	0.387788i	−0.486109	+	0.873898i	$0.661584\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−448.967			−0.823793
$546$		0			0
$547$	−283.251	+	283.251i	−0.517827	+	0.517827i	−0.916913	−	0.399086i	$-0.869327\pi$
$547$	−283.251	+	283.251i	−0.517827	+	0.517827i	0.399086	+	0.916913i	$0.369327\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.6004			−0.0192384
$552$		0			0
$553$			9.25948i			0.0167441i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−316.798	−	316.798i	−0.568758	−	0.568758i	0.363022	−	0.931781i	$-0.381745\pi$
$557$	−316.798	−	316.798i	−0.568758	−	0.568758i	−0.931781	+	0.363022i	$0.881745\pi$
$558$		0			0
$559$		−	121.982i		−	0.218215i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	298.317	−	298.317i	0.529871	−	0.529871i	−0.390663	−	0.920534i	$-0.627754\pi$
$563$	298.317	−	298.317i	0.529871	−	0.529871i	0.920534	+	0.390663i	$0.127754\pi$
$564$		0			0
$565$	347.333	−	347.333i	0.614748	−	0.614748i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			24.9852i			0.0439107i	0.999759	+	0.0219553i	$0.00698916\pi$
$569$			24.9852i			0.0439107i	−0.999759	+	0.0219553i	$0.993011\pi$
$570$		0			0
$571$	−739.901	−	739.901i	−1.29580	−	1.29580i	−0.931142	−	0.364657i	$-0.881186\pi$
$571$	−739.901	−	739.901i	−1.29580	−	1.29580i	−0.364657	−	0.931142i	$-0.618814\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			52.4805i			0.0912705i
$576$		0			0
$577$	−120.741			−0.209256			−0.104628	−	0.994511i	$-0.533365\pi$
$577$	−120.741			−0.209256			−0.104628	+	0.994511i	$0.533365\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1089.70	+	1089.70i	−1.87556	+	1.87556i
$582$		0			0
$583$	964.142			1.65376
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−83.1502	−	83.1502i	−0.141653	−	0.141653i	0.632724	−	0.774377i	$-0.281937\pi$
$587$	−83.1502	−	83.1502i	−0.141653	−	0.141653i	−0.774377	+	0.632724i	$0.781937\pi$
$588$		0			0
$589$	−17.9684	−	17.9684i	−0.0305066	−	0.0305066i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	135.037			0.227719			0.113859	−	0.993497i	$-0.463679\pi$
$593$	135.037			0.227719			0.113859	+	0.993497i	$0.463679\pi$
$594$		0			0
$595$	−663.025	+	663.025i	−1.11433	+	1.11433i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	244.844			0.408755			0.204377	−	0.978892i	$-0.434483\pi$
$599$	244.844			0.408755			0.204377	+	0.978892i	$0.434483\pi$
$600$		0			0
$601$		−	765.274i		−	1.27333i	−0.771139	−	0.636667i	$-0.780313\pi$
$601$		−	765.274i		−	1.27333i	0.771139	−	0.636667i	$-0.219687\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−396.171	−	396.171i	−0.654827	−	0.654827i
$606$		0			0
$607$		−	546.253i		−	0.899923i	−0.893048	−	0.449961i	$-0.851438\pi$
$607$		−	546.253i		−	0.899923i	0.893048	−	0.449961i	$-0.148562\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−141.691	+	141.691i	−0.231901	+	0.231901i
$612$		0			0
$613$	−11.8818	+	11.8818i	−0.0193830	+	0.0193830i	−0.716732	−	0.697349i	$-0.754363\pi$
$613$	−11.8818	+	11.8818i	−0.0193830	+	0.0193830i	0.697349	+	0.716732i	$0.254363\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	72.2000i		−	0.117018i	−0.998287	−	0.0585089i	$-0.981365\pi$
$617$		−	72.2000i		−	0.117018i	0.998287	−	0.0585089i	$-0.0186346\pi$
$618$		0			0
$619$	265.473	+	265.473i	0.428874	+	0.428874i	0.888245	−	0.459371i	$-0.151925\pi$
$619$	265.473	+	265.473i	0.428874	+	0.428874i	−0.459371	+	0.888245i	$0.651925\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			239.298i			0.384106i
$624$		0			0
$625$	112.445			0.179912
$626$		0			0
$627$		0			0
$628$		0			0
$629$	861.408	−	861.408i	1.36949	−	1.36949i
$630$		0			0
$631$	343.644			0.544602			0.272301	−	0.962212i	$-0.412215\pi$
$631$	343.644			0.544602			0.272301	+	0.962212i	$0.412215\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−114.115	−	114.115i	−0.179709	−	0.179709i
$636$		0			0
$637$	−191.398	−	191.398i	−0.300469	−	0.300469i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	158.821			0.247771			0.123885	−	0.992297i	$-0.460465\pi$
$641$	158.821			0.247771			0.123885	+	0.992297i	$0.460465\pi$
$642$		0			0
$643$	576.120	−	576.120i	0.895988	−	0.895988i	−0.0990902	−	0.995078i	$-0.531593\pi$
$643$	576.120	−	576.120i	0.895988	−	0.895988i	0.995078	+	0.0990902i	$0.0315932\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	750.114			1.15937			0.579686	−	0.814840i	$-0.303175\pi$
$647$	750.114			1.15937			0.579686	+	0.814840i	$0.303175\pi$
$648$		0			0
$649$		−	562.577i		−	0.866836i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−631.097	−	631.097i	−0.966458	−	0.966458i	0.0329977	−	0.999455i	$-0.489495\pi$
$653$	−631.097	−	631.097i	−0.966458	−	0.966458i	−0.999455	+	0.0329977i	$0.989495\pi$
$654$		0			0
$655$			128.027i			0.195462i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	555.313	−	555.313i	0.842659	−	0.842659i	−0.146545	−	0.989204i	$-0.546815\pi$
$659$	555.313	−	555.313i	0.842659	−	0.842659i	0.989204	+	0.146545i	$0.0468152\pi$
$660$		0			0
$661$	−75.6151	+	75.6151i	−0.114395	+	0.114395i	−0.761987	−	0.647592i	$-0.775776\pi$
$661$	−75.6151	+	75.6151i	−0.114395	+	0.114395i	0.647592	+	0.761987i	$0.275776\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	387.940i		−	0.583369i
$666$		0			0
$667$	−3.07370	−	3.07370i	−0.00460825	−	0.00460825i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1574.69i		−	2.34678i
$672$		0			0
$673$	−317.869			−0.472317			−0.236158	−	0.971715i	$-0.575888\pi$
$673$	−317.869			−0.472317			−0.236158	+	0.971715i	$0.575888\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	621.736	−	621.736i	0.918370	−	0.918370i	−0.0785409	−	0.996911i	$-0.525026\pi$
$677$	621.736	−	621.736i	0.918370	−	0.918370i	0.996911	+	0.0785409i	$0.0250261\pi$
$678$		0			0
$679$	1028.67			1.51497
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−438.781	−	438.781i	−0.642431	−	0.642431i	0.308721	−	0.951153i	$-0.400099\pi$
$683$	−438.781	−	438.781i	−0.642431	−	0.642431i	−0.951153	+	0.308721i	$0.900099\pi$
$684$		0			0
$685$	296.248	+	296.248i	0.432480	+	0.432480i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−168.071			−0.243935
$690$		0			0
$691$	−418.978	+	418.978i	−0.606336	+	0.606336i	−0.941987	−	0.335651i	$-0.891044\pi$
$691$	−418.978	+	418.978i	−0.606336	+	0.606336i	0.335651	+	0.941987i	$0.391044\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	214.876			0.309174
$696$		0			0
$697$			1600.35i			2.29605i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−244.043	−	244.043i	−0.348136	−	0.348136i	0.511279	−	0.859415i	$-0.329172\pi$
$701$	−244.043	−	244.043i	−0.348136	−	0.348136i	−0.859415	+	0.511279i	$0.829172\pi$
$702$		0			0
$703$			504.015i			0.716949i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	696.782	−	696.782i	0.985548	−	0.985548i
$708$		0			0
$709$	−580.549	+	580.549i	−0.818828	+	0.818828i	−0.985938	−	0.167111i	$-0.946556\pi$
$709$	−580.549	+	580.549i	−0.818828	+	0.818828i	0.167111	+	0.985938i	$0.446556\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	10.4203i		−	0.0146147i
$714$		0			0
$715$	119.944	+	119.944i	0.167754	+	0.167754i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			404.510i			0.562600i	0.959620	+	0.281300i	$0.0907657\pi$
$719$			404.510i			0.562600i	−0.959620	+	0.281300i	$0.909234\pi$
$720$		0			0
$721$	−411.394			−0.570588
$722$		0			0
$723$		0			0
$724$		0			0
$725$	10.4558	−	10.4558i	0.0144218	−	0.0144218i
$726$		0			0
$727$	−449.531			−0.618337			−0.309169	−	0.951007i	$-0.600051\pi$
$727$	−449.531			−0.618337			−0.309169	+	0.951007i	$0.600051\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−678.280	−	678.280i	−0.927879	−	0.927879i
$732$		0			0
$733$	418.626	+	418.626i	0.571113	+	0.571113i	0.932440	−	0.361326i	$-0.117676\pi$
$733$	418.626	+	418.626i	0.571113	+	0.571113i	−0.361326	+	0.932440i	$0.617676\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	978.357			1.32749
$738$		0			0
$739$	54.2122	−	54.2122i	0.0733588	−	0.0733588i	−0.669475	−	0.742834i	$-0.733481\pi$
$739$	54.2122	−	54.2122i	0.0733588	−	0.0733588i	0.742834	+	0.669475i	$0.233481\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	119.012			0.160178			0.0800891	−	0.996788i	$-0.474480\pi$
$743$	119.012			0.160178			0.0800891	+	0.996788i	$0.474480\pi$
$744$		0			0
$745$			132.790i			0.178242i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−555.398	−	555.398i	−0.741519	−	0.741519i
$750$		0			0
$751$		−	758.619i		−	1.01015i	−0.863077	−	0.505073i	$-0.831466\pi$
$751$		−	758.619i		−	1.01015i	0.863077	−	0.505073i	$-0.168534\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−143.950	+	143.950i	−0.190662	+	0.190662i
$756$		0			0
$757$	980.581	−	980.581i	1.29535	−	1.29535i	0.363922	−	0.931429i	$-0.381437\pi$
$757$	980.581	−	980.581i	1.29535	−	1.29535i	0.931429	−	0.363922i	$-0.118563\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1119.28i		−	1.47080i	−0.677634	−	0.735399i	$-0.736995\pi$
$761$		−	1119.28i		−	1.47080i	0.677634	−	0.735399i	$-0.263005\pi$
$762$		0			0
$763$	−1104.75	−	1104.75i	−1.44790	−	1.44790i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			98.0693i			0.127861i
$768$		0			0
$769$	−1268.90			−1.65006			−0.825032	−	0.565087i	$-0.808843\pi$
$769$	−1268.90			−1.65006			−0.825032	+	0.565087i	$0.808843\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	213.869	−	213.869i	0.276674	−	0.276674i	−0.555106	−	0.831780i	$-0.687322\pi$
$773$	213.869	−	213.869i	0.276674	−	0.276674i	0.831780	+	0.555106i	$0.187322\pi$
$774$		0			0
$775$	35.4468			0.0457378
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−468.187	−	468.187i	−0.601011	−	0.601011i
$780$		0			0
$781$	1172.58	+	1172.58i	1.50138	+	1.50138i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	113.128			0.144113
$786$		0			0
$787$	−760.645	+	760.645i	−0.966512	+	0.966512i	−0.999457	−	0.0329453i	$-0.989511\pi$
$787$	−760.645	+	760.645i	−0.966512	+	0.966512i	0.0329453	+	0.999457i	$0.489511\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1709.33			2.16097
$792$		0			0
$793$			274.503i			0.346158i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−801.178	−	801.178i	−1.00524	−	1.00524i	−0.999986	−	0.00525555i	$-0.998327\pi$
$797$	−801.178	−	801.178i	−1.00524	−	1.00524i	−0.00525555	−	0.999986i	$-0.501673\pi$
$798$		0			0
$799$			1575.75i			1.97215i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	887.134	−	887.134i	1.10477	−	1.10477i
$804$		0			0
$805$	112.488	−	112.488i	0.139737	−	0.139737i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1311.28i		−	1.62087i	−0.585831	−	0.810433i	$-0.699232\pi$
$809$		−	1311.28i		−	1.62087i	0.585831	−	0.810433i	$-0.300768\pi$
$810$		0			0
$811$	358.552	+	358.552i	0.442112	+	0.442112i	0.892721	−	0.450610i	$-0.148793\pi$
$811$	358.552	+	358.552i	0.442112	+	0.442112i	−0.450610	+	0.892721i	$0.648793\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	577.382i		−	0.708444i
$816$		0			0
$817$	396.866			0.485760
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−526.464	+	526.464i	−0.641247	+	0.641247i	−0.950862	−	0.309615i	$-0.899800\pi$
$821$	−526.464	+	526.464i	−0.641247	+	0.641247i	0.309615	+	0.950862i	$0.399800\pi$
$822$		0			0
$823$	−947.410			−1.15117			−0.575583	−	0.817743i	$-0.695225\pi$
$823$	−947.410			−1.15117			−0.575583	+	0.817743i	$0.695225\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	856.014	+	856.014i	1.03508	+	1.03508i	0.999362	+	0.0357214i	$0.0113729\pi$
$827$	856.014	+	856.014i	1.03508	+	1.03508i	0.0357214	+	0.999362i	$0.488627\pi$
$828$		0			0
$829$	−741.590	−	741.590i	−0.894560	−	0.894560i	0.100389	−	0.994948i	$-0.467991\pi$
$829$	−741.590	−	741.590i	−0.894560	−	0.894560i	−0.994948	+	0.100389i	$0.967991\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2128.54			−2.55527
$834$		0			0
$835$	542.977	−	542.977i	0.650272	−	0.650272i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	607.471			0.724042			0.362021	−	0.932170i	$-0.382087\pi$
$839$	607.471			0.724042			0.362021	+	0.932170i	$0.382087\pi$
$840$		0			0
$841$		−	839.775i		−	0.998544i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	386.776	+	386.776i	0.457723	+	0.457723i
$846$		0			0
$847$		−	1949.67i		−	2.30186i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−146.145	+	146.145i	−0.171734	+	0.171734i
$852$		0			0
$853$	156.942	−	156.942i	0.183989	−	0.183989i	−0.609103	−	0.793091i	$-0.708470\pi$
$853$	156.942	−	156.942i	0.183989	−	0.183989i	0.793091	+	0.609103i	$0.208470\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			915.604i			1.06838i	0.845364	+	0.534191i	$0.179384\pi$
$857$			915.604i			1.06838i	−0.845364	+	0.534191i	$0.820616\pi$
$858$		0			0
$859$	−379.829	−	379.829i	−0.442176	−	0.442176i	0.450567	−	0.892743i	$-0.351222\pi$
$859$	−379.829	−	379.829i	−0.442176	−	0.442176i	−0.892743	+	0.450567i	$0.851222\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	49.7576i		−	0.0576565i	−0.999584	−	0.0288283i	$-0.990822\pi$
$863$		−	49.7576i		−	0.0576565i	0.999584	−	0.0288283i	$-0.00917759\pi$
$864$		0			0
$865$	−412.185			−0.476514
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9.31427	−	9.31427i	0.0107184	−	0.0107184i
$870$		0			0
$871$	−170.549			−0.195808
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1098.62	+	1098.62i	1.25557	+	1.25557i
$876$		0			0
$877$	−963.055	−	963.055i	−1.09812	−	1.09812i	−0.994630	−	0.103494i	$-0.966998\pi$
$877$	−963.055	−	963.055i	−1.09812	−	1.09812i	−0.103494	−	0.994630i	$-0.533002\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−159.518			−0.181064			−0.0905322	−	0.995894i	$-0.528857\pi$
$881$	−159.518			−0.181064			−0.0905322	+	0.995894i	$0.528857\pi$
$882$		0			0
$883$	53.9632	−	53.9632i	0.0611134	−	0.0611134i	−0.675890	−	0.737003i	$-0.736240\pi$
$883$	53.9632	−	53.9632i	0.0611134	−	0.0611134i	0.737003	+	0.675890i	$0.236240\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1215.94			−1.37085			−0.685424	−	0.728144i	$-0.740383\pi$
$887$	−1215.94			−1.37085			−0.685424	+	0.728144i	$0.740383\pi$
$888$		0			0
$889$		−	561.594i		−	0.631714i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−460.990	−	460.990i	−0.516226	−	0.516226i
$894$		0			0
$895$			768.774i			0.858966i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.07607	+	2.07607i	−0.00230931	+	0.00230931i
$900$		0			0
$901$	−934.556	+	934.556i	−1.03724	+	1.03724i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			614.164i			0.678634i
$906$		0			0
$907$	780.678	+	780.678i	0.860726	+	0.860726i	0.991422	−	0.130697i	$-0.0417214\pi$
$907$	780.678	+	780.678i	0.860726	+	0.860726i	−0.130697	+	0.991422i	$0.541721\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	760.487i		−	0.834782i	−0.908727	−	0.417391i	$-0.862945\pi$
$911$		−	760.487i		−	0.834782i	0.908727	−	0.417391i	$-0.137055\pi$
$912$		0			0
$913$	2192.30			2.40120
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−315.031	+	315.031i	−0.343545	+	0.343545i
$918$		0			0
$919$	−905.928			−0.985776			−0.492888	−	0.870093i	$-0.664059\pi$
$919$	−905.928			−0.985776			−0.492888	+	0.870093i	$0.664059\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−204.405	−	204.405i	−0.221458	−	0.221458i
$924$		0			0
$925$	−497.144	−	497.144i	−0.537453	−	0.537453i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−932.848			−1.00414			−0.502071	−	0.864826i	$-0.667428\pi$
$929$	−932.848			−1.00414			−0.502071	+	0.864826i	$0.667428\pi$
$930$		0			0
$931$	622.710	−	622.710i	0.668862	−	0.668862i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1333.90			1.42663
$936$		0			0
$937$		−	26.1861i		−	0.0279468i	−0.999902	−	0.0139734i	$-0.995552\pi$
$937$		−	26.1861i		−	0.0279468i	0.999902	−	0.0139734i	$-0.00444801\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−313.838	−	313.838i	−0.333516	−	0.333516i	0.520404	−	0.853920i	$-0.325781\pi$
$941$	−313.838	−	313.838i	−0.333516	−	0.333516i	−0.853920	+	0.520404i	$0.825781\pi$
$942$		0			0
$943$		−	271.514i		−	0.287925i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1091.06	−	1091.06i	1.15213	−	1.15213i	0.166001	−	0.986126i	$-0.446915\pi$
$947$	1091.06	−	1091.06i	1.15213	−	1.15213i	0.986126	−	0.166001i	$-0.0530854\pi$
$948$		0			0
$949$	−154.647	+	154.647i	−0.162958	+	0.162958i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1305.60i		−	1.36999i	−0.728546	−	0.684997i	$-0.759803\pi$
$953$		−	1305.60i		−	1.36999i	0.728546	−	0.684997i	$-0.240197\pi$
$954$		0			0
$955$	650.421	+	650.421i	0.681069	+	0.681069i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1457.93i			1.52026i
$960$		0			0
$961$	953.962			0.992676
$962$		0			0
$963$		0			0
$964$		0			0
$965$	592.515	−	592.515i	0.614006	−	0.614006i
$966$		0			0
$967$	1005.88			1.04021			0.520103	−	0.854104i	$-0.325894\pi$
$967$	1005.88			1.04021			0.520103	+	0.854104i	$0.325894\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1310.71	+	1310.71i	1.34985	+	1.34985i	0.885815	+	0.464039i	$0.153600\pi$
$971$	1310.71	+	1310.71i	1.34985	+	1.34985i	0.464039	+	0.885815i	$0.346400\pi$
$972$		0			0
$973$	528.735	+	528.735i	0.543407	+	0.543407i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−257.041			−0.263092			−0.131546	−	0.991310i	$-0.541994\pi$
$977$	−257.041			−0.263092			−0.131546	+	0.991310i	$0.541994\pi$
$978$		0			0
$979$	240.714	−	240.714i	0.245877	−	0.245877i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−405.819			−0.412837			−0.206419	−	0.978464i	$-0.566181\pi$
$983$	−405.819			−0.412837			−0.206419	+	0.978464i	$0.566181\pi$
$984$		0			0
$985$			1065.06i			1.08128i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	115.076	+	115.076i	0.116356	+	0.116356i
$990$		0			0
$991$			143.499i			0.144802i	0.997376	+	0.0724010i	$0.0230661\pi$
$991$			143.499i			0.144802i	−0.997376	+	0.0724010i	$0.976934\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−558.248	+	558.248i	−0.561053	+	0.561053i
$996$		0			0
$997$	−595.333	+	595.333i	−0.597124	+	0.597124i	−0.939546	−	0.342422i	$-0.888753\pi$
$997$	−595.333	+	595.333i	−0.597124	+	0.597124i	0.342422	+	0.939546i	$0.388753\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.m.e.991.3		16
3.2	odd	2	inner	1152.3.m.e.991.6		16
4.3	odd	2		1152.3.m.d.991.3		16
8.3	odd	2		576.3.m.b.559.6		16
8.5	even	2		144.3.m.b.19.2	✓	16
12.11	even	2		1152.3.m.d.991.6		16
16.3	odd	4		144.3.m.b.91.2	yes	16
16.5	even	4		1152.3.m.d.415.3		16
16.11	odd	4	inner	1152.3.m.e.415.3		16
16.13	even	4		576.3.m.b.271.6		16
24.5	odd	2		144.3.m.b.19.7	yes	16
24.11	even	2		576.3.m.b.559.3		16
48.5	odd	4		1152.3.m.d.415.6		16
48.11	even	4	inner	1152.3.m.e.415.6		16
48.29	odd	4		576.3.m.b.271.3		16
48.35	even	4		144.3.m.b.91.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.m.b.19.2	✓	16	8.5	even	2
144.3.m.b.19.7	yes	16	24.5	odd	2
144.3.m.b.91.2	yes	16	16.3	odd	4
144.3.m.b.91.7	yes	16	48.35	even	4
576.3.m.b.271.3		16	48.29	odd	4
576.3.m.b.271.6		16	16.13	even	4
576.3.m.b.559.3		16	24.11	even	2
576.3.m.b.559.6		16	8.3	odd	2
1152.3.m.d.415.3		16	16.5	even	4
1152.3.m.d.415.6		16	48.5	odd	4
1152.3.m.d.991.3		16	4.3	odd	2
1152.3.m.d.991.6		16	12.11	even	2
1152.3.m.e.415.3		16	16.11	odd	4	inner
1152.3.m.e.415.6		16	48.11	even	4	inner
1152.3.m.e.991.3		16	1.1	even	1	trivial
1152.3.m.e.991.6		16	3.2	odd	2	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.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.41234 + 2.41234i) q^{5} -11.8718 q^{7} +O(q^{10})\)	\(q+(-2.41234 + 2.41234i) q^{5} -11.8718 q^{7} +(11.9421 + 11.9421i) q^{11} +(-2.08177 - 2.08177i) q^{13} -23.1512 q^{17} +(6.77297 - 6.77297i) q^{19} +3.92781 q^{23} +13.3613i q^{25} +(-0.782548 - 0.782548i) q^{29} -2.65295i q^{31} +(28.6389 - 28.6389i) q^{35} +(-37.2078 + 37.2078i) q^{37} -69.1259i q^{41} +(29.2978 + 29.2978i) q^{43} -68.0631i q^{47} +91.9405 q^{49} +(40.3674 - 40.3674i) q^{53} -57.6167 q^{55} +(-23.5544 - 23.5544i) q^{59} +(-65.9304 - 65.9304i) q^{61} +10.0438 q^{65} +(40.9626 - 40.9626i) q^{67} +98.1885 q^{71} -74.2864i q^{73} +(-141.774 - 141.774i) q^{77} -0.779953i q^{79} +(91.7887 - 91.7887i) q^{83} +(55.8486 - 55.8486i) q^{85} -20.1568i q^{89} +(24.7144 + 24.7144i) q^{91} +32.6774i q^{95} -86.6476 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 32 q^{19} - 96 q^{37} + 32 q^{43} + 112 q^{49} - 256 q^{55} + 32 q^{61} + 256 q^{67} - 160 q^{85} - 288 q^{91}+O(q^{100}) \) 16 * q + 32 * q^19 - 96 * q^37 + 32 * q^43 + 112 * q^49 - 256 * q^55 + 32 * q^61 + 256 * q^67 - 160 * q^85 - 288 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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