LMFDB - Embedded newform 1152.2.v.d.1009.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(145,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 7, 0]))

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

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

chi := DirichletCharacter("1152.145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.v (of order $8$, degree $4$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$8$ over $\Q(\zeta_{8})$
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			1009.2
Character	$\chi$	$=$	1152.1009
Dual form			1152.2.v.d.145.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.42249 - 1.00343i) q^{5} +(-3.40882 - 3.40882i) q^{7} +O(q^{10})$	$q+(-2.42249 - 1.00343i) q^{5} +(-3.40882 - 3.40882i) q^{7} +(0.847276 - 2.04551i) q^{11} +(-1.82981 + 0.757934i) q^{13} +7.04986i q^{17} +(4.96918 - 2.05830i) q^{19} +(-3.37211 + 3.37211i) q^{23} +(1.32604 + 1.32604i) q^{25} +(2.83715 + 6.84948i) q^{29} -1.94207 q^{31} +(4.83732 + 11.6783i) q^{35} +(4.02656 + 1.66786i) q^{37} +(-0.970078 + 0.970078i) q^{41} +(0.0467259 - 0.112806i) q^{43} +6.49099i q^{47} +16.2401i q^{49} +(0.894399 - 2.15927i) q^{53} +(-4.10503 + 4.10503i) q^{55} +(-7.71288 - 3.19478i) q^{59} +(-4.20083 - 10.1417i) q^{61} +5.19323 q^{65} +(-0.933239 - 2.25304i) q^{67} +(4.81273 + 4.81273i) q^{71} +(-2.31997 + 2.31997i) q^{73} +(-9.86098 + 4.08455i) q^{77} +6.41652i q^{79} +(-6.76145 + 2.80069i) q^{83} +(7.07402 - 17.0782i) q^{85} +(-8.37883 - 8.37883i) q^{89} +(8.82117 + 3.65385i) q^{91} -14.1031 q^{95} -5.80771 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 48 q^{31} - 16 q^{43} - 32 q^{55} - 32 q^{61} - 16 q^{67} + 48 q^{91}+O(q^{100}) $ 32 * q - 48 * q^31 - 16 * q^43 - 32 * q^55 - 32 * q^61 - 16 * q^67 + 48 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{8}\right)$

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
$5$	−2.42249	−	1.00343i	−1.08337	−	0.448746i	−0.231679	−	0.972792i	$-0.574422\pi$
$5$	−2.42249	−	1.00343i	−1.08337	−	0.448746i	−0.851690	+	0.524046i	$0.824422\pi$
$6$		0			0
$7$	−3.40882	−	3.40882i	−1.28841	−	1.28841i	−0.935750	−	0.352664i	$-0.885276\pi$
$7$	−3.40882	−	3.40882i	−1.28841	−	1.28841i	−0.352664	−	0.935750i	$-0.614724\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.847276	−	2.04551i	0.255463	−	0.616743i	−0.743165	−	0.669109i	$-0.766676\pi$
$11$	0.847276	−	2.04551i	0.255463	−	0.616743i	0.998628	+	0.0523656i	$0.0166761\pi$
$12$		0			0
$13$	−1.82981	+	0.757934i	−0.507499	+	0.210213i	−0.621716	−	0.783243i	$-0.713564\pi$
$13$	−1.82981	+	0.757934i	−0.507499	+	0.210213i	0.114217	+	0.993456i	$0.463564\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			7.04986i			1.70984i	0.518758	+	0.854921i	$0.326394\pi$
$17$			7.04986i			1.70984i	−0.518758	+	0.854921i	$0.673606\pi$
$18$		0			0
$19$	4.96918	−	2.05830i	1.14001	−	0.472207i	0.268837	−	0.963186i	$-0.413361\pi$
$19$	4.96918	−	2.05830i	1.14001	−	0.472207i	0.871172	+	0.490979i	$0.163361\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.37211	+	3.37211i	−0.703134	+	0.703134i	−0.965082	−	0.261948i	$-0.915635\pi$
$23$	−3.37211	+	3.37211i	−0.703134	+	0.703134i	0.261948	+	0.965082i	$0.415635\pi$
$24$		0			0
$25$	1.32604	+	1.32604i	0.265208	+	0.265208i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.83715	+	6.84948i	0.526845	+	1.27192i	0.933580	+	0.358370i	$0.116667\pi$
$29$	2.83715	+	6.84948i	0.526845	+	1.27192i	−0.406735	+	0.913546i	$0.633333\pi$
$30$		0			0
$31$	−1.94207			−0.348806			−0.174403	−	0.984674i	$-0.555799\pi$
$31$	−1.94207			−0.348806			−0.174403	+	0.984674i	$0.555799\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.83732	+	11.6783i	0.817657	+	1.97400i
$36$		0			0
$37$	4.02656	+	1.66786i	0.661963	+	0.274194i	0.688264	−	0.725460i	$-0.258373\pi$
$37$	4.02656	+	1.66786i	0.661963	+	0.274194i	−0.0263014	+	0.999654i	$0.508373\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.970078	+	0.970078i	−0.151501	+	0.151501i	−0.778788	−	0.627287i	$-0.784165\pi$
$41$	−0.970078	+	0.970078i	−0.151501	+	0.151501i	0.627287	+	0.778788i	$0.284165\pi$
$42$		0			0
$43$	0.0467259	−	0.112806i	0.00712564	−	0.0172028i	−0.920277	−	0.391268i	$-0.872036\pi$
$43$	0.0467259	−	0.112806i	0.00712564	−	0.0172028i	0.927402	+	0.374065i	$0.122036\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.49099i			0.946808i	0.880845	+	0.473404i	$0.156975\pi$
$47$			6.49099i			0.946808i	−0.880845	+	0.473404i	$0.843025\pi$
$48$		0			0
$49$			16.2401i			2.32002i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.894399	−	2.15927i	0.122855	−	0.296598i	−0.850472	−	0.526020i	$-0.823684\pi$
$53$	0.894399	−	2.15927i	0.122855	−	0.296598i	0.973327	+	0.229422i	$0.0736835\pi$
$54$		0			0
$55$	−4.10503	+	4.10503i	−0.553522	+	0.553522i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.71288	−	3.19478i	−1.00413	−	0.415925i	−0.180821	−	0.983516i	$-0.557875\pi$
$59$	−7.71288	−	3.19478i	−1.00413	−	0.415925i	−0.823310	+	0.567591i	$0.807875\pi$
$60$		0			0
$61$	−4.20083	−	10.1417i	−0.537861	−	1.29851i	−0.926213	−	0.377001i	$-0.876955\pi$
$61$	−4.20083	−	10.1417i	−0.537861	−	1.29851i	0.388352	−	0.921511i	$-0.373045\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.19323			0.644141
$66$		0			0
$67$	−0.933239	−	2.25304i	−0.114013	−	0.275252i	0.856565	−	0.516040i	$-0.172594\pi$
$67$	−0.933239	−	2.25304i	−0.114013	−	0.275252i	−0.970578	+	0.240787i	$0.922594\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.81273	+	4.81273i	0.571166	+	0.571166i	0.932454	−	0.361288i	$-0.117663\pi$
$71$	4.81273	+	4.81273i	0.571166	+	0.571166i	−0.361288	+	0.932454i	$0.617663\pi$
$72$		0			0
$73$	−2.31997	+	2.31997i	−0.271532	+	0.271532i	−0.829717	−	0.558184i	$-0.811498\pi$
$73$	−2.31997	+	2.31997i	−0.271532	+	0.271532i	0.558184	+	0.829717i	$0.311498\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.86098	+	4.08455i	−1.12376	+	0.465478i
$78$		0			0
$79$			6.41652i			0.721915i	0.932582	+	0.360958i	$0.117550\pi$
$79$			6.41652i			0.721915i	−0.932582	+	0.360958i	$0.882450\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.76145	+	2.80069i	−0.742166	+	0.307415i	−0.721541	−	0.692372i	$-0.756566\pi$
$83$	−6.76145	+	2.80069i	−0.742166	+	0.307415i	−0.0206253	+	0.999787i	$0.506566\pi$
$84$		0			0
$85$	7.07402	−	17.0782i	0.767285	−	1.85239i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.37883	−	8.37883i	−0.888155	−	0.888155i	0.106191	−	0.994346i	$-0.466134\pi$
$89$	−8.37883	−	8.37883i	−0.888155	−	0.888155i	−0.994346	+	0.106191i	$0.966134\pi$
$90$		0			0
$91$	8.82117	+	3.65385i	0.924710	+	0.383028i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−14.1031			−1.44695
$96$		0			0
$97$	−5.80771			−0.589684			−0.294842	−	0.955546i	$-0.595267\pi$
$97$	−5.80771			−0.589684			−0.294842	+	0.955546i	$0.595267\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.6963	+	4.43056i	1.06432	+	0.440857i	0.844984	−	0.534792i	$-0.179610\pi$
$101$	10.6963	+	4.43056i	1.06432	+	0.440857i	0.219339	+	0.975649i	$0.429610\pi$
$102$		0			0
$103$	−1.38962	−	1.38962i	−0.136923	−	0.136923i	0.635323	−	0.772247i	$-0.280867\pi$
$103$	−1.38962	−	1.38962i	−0.136923	−	0.136923i	−0.772247	+	0.635323i	$0.780867\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.62705	+	3.92804i	−0.157293	+	0.379738i	−0.982805	−	0.184646i	$-0.940886\pi$
$107$	−1.62705	+	3.92804i	−0.157293	+	0.379738i	0.825512	+	0.564384i	$0.190886\pi$
$108$		0			0
$109$	6.98533	−	2.89342i	0.669073	−	0.277139i	−0.0221778	−	0.999754i	$-0.507060\pi$
$109$	6.98533	−	2.89342i	0.669073	−	0.277139i	0.691251	+	0.722615i	$0.257060\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	0.767319i		−	0.0721833i	−0.999348	−	0.0360916i	$-0.988509\pi$
$113$		−	0.767319i		−	0.0721833i	0.999348	−	0.0360916i	$-0.0114908\pi$
$114$		0			0
$115$	11.5526	−	4.78523i	1.07728	−	0.446225i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	24.0317	−	24.0317i	2.20298	−	2.20298i
$120$		0			0
$121$	4.31196	+	4.31196i	0.391996	+	0.391996i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.13540	+	7.56953i	0.280439	+	0.677039i
$126$		0			0
$127$	−14.9074			−1.32282			−0.661409	−	0.750026i	$-0.730041\pi$
$127$	−14.9074			−1.32282			−0.661409	+	0.750026i	$0.730041\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.272717	−	0.658398i	−0.0238274	−	0.0575245i	0.911518	−	0.411260i	$-0.134911\pi$
$131$	−0.272717	−	0.658398i	−0.0238274	−	0.0575245i	−0.935345	+	0.353736i	$0.884911\pi$
$132$		0			0
$133$	−23.9555	−	9.92267i	−2.07720	−	0.860405i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.8183	+	12.8183i	−1.09514	+	1.09514i	−0.100173	+	0.994970i	$0.531940\pi$
$137$	−12.8183	+	12.8183i	−1.09514	+	1.09514i	−0.994970	+	0.100173i	$0.968060\pi$
$138$		0			0
$139$	1.25849	−	3.03827i	0.106744	−	0.257703i	−0.861478	−	0.507795i	$-0.830461\pi$
$139$	1.25849	−	3.03827i	0.106744	−	0.257703i	0.968222	+	0.250092i	$0.0804609\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			4.38507i			0.366698i
$144$		0			0
$145$		−	19.4396i		−	1.61437i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.47558	−	13.2192i	0.448577	−	1.08296i	−0.524278	−	0.851547i	$-0.675665\pi$
$149$	5.47558	−	13.2192i	0.448577	−	1.08296i	0.972855	−	0.231414i	$-0.0743351\pi$
$150$		0			0
$151$	−3.66239	+	3.66239i	−0.298041	+	0.298041i	−0.840246	−	0.542205i	$-0.817589\pi$
$151$	−3.66239	+	3.66239i	−0.298041	+	0.298041i	0.542205	+	0.840246i	$0.317589\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.70463	+	1.94872i	0.377885	+	0.156525i
$156$		0			0
$157$	−1.46283	−	3.53159i	−0.116747	−	0.281851i	0.854695	−	0.519131i	$-0.173744\pi$
$157$	−1.46283	−	3.53159i	−0.116747	−	0.281851i	−0.971441	+	0.237280i	$0.923744\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	22.9899			1.81186
$162$		0			0
$163$	8.90310	+	21.4940i	0.697345	+	1.68354i	0.729431	+	0.684054i	$0.239785\pi$
$163$	8.90310	+	21.4940i	0.697345	+	1.68354i	−0.0320863	+	0.999485i	$0.510215\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.56063	+	9.56063i	0.739824	+	0.739824i	0.972544	−	0.232720i	$-0.0747625\pi$
$167$	9.56063	+	9.56063i	0.739824	+	0.739824i	−0.232720	+	0.972544i	$0.574762\pi$
$168$		0			0
$169$	−6.41863	+	6.41863i	−0.493741	+	0.493741i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−16.8480	+	6.97867i	−1.28093	+	0.530578i	−0.916270	−	0.400562i	$-0.868815\pi$
$173$	−16.8480	+	6.97867i	−1.28093	+	0.530578i	−0.364660	+	0.931141i	$0.618815\pi$
$174$		0			0
$175$		−	9.04048i		−	0.683396i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.8067	−	4.47628i	0.807730	−	0.334573i	0.0596821	−	0.998217i	$-0.480991\pi$
$179$	10.8067	−	4.47628i	0.807730	−	0.334573i	0.748048	+	0.663645i	$0.230991\pi$
$180$		0			0
$181$	−7.09844	+	17.1372i	−0.527623	+	1.27379i	0.405453	+	0.914116i	$0.367114\pi$
$181$	−7.09844	+	17.1372i	−0.527623	+	1.27379i	−0.933076	+	0.359679i	$0.882886\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.08072	−	8.08072i	−0.594106	−	0.594106i
$186$		0			0
$187$	14.4205	+	5.97318i	1.05453	+	0.436802i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−25.5817			−1.85102			−0.925512	−	0.378719i	$-0.876365\pi$
$191$	−25.5817			−1.85102			−0.925512	+	0.378719i	$0.876365\pi$
$192$		0			0
$193$	20.4742			1.47377			0.736883	−	0.676021i	$-0.236297\pi$
$193$	20.4742			1.47377			0.736883	+	0.676021i	$0.236297\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.32628	−	0.549364i	−0.0944937	−	0.0391406i	0.334936	−	0.942241i	$-0.391285\pi$
$197$	−1.32628	−	0.549364i	−0.0944937	−	0.0391406i	−0.429430	+	0.903100i	$0.641285\pi$
$198$		0			0
$199$	0.735733	+	0.735733i	0.0521548	+	0.0521548i	0.732703	−	0.680548i	$-0.238258\pi$
$199$	0.735733	+	0.735733i	0.0521548	+	0.0521548i	−0.680548	+	0.732703i	$0.738258\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13.6773	−	33.0200i	0.959960	−	2.31755i
$204$		0			0
$205$	3.32340	−	1.37660i	0.232117	−	0.0961458i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	11.9084i		−	0.823724i
$210$		0			0
$211$	−16.2602	+	6.73519i	−1.11940	+	0.463670i	−0.864164	−	0.503210i	$-0.832152\pi$
$211$	−16.2602	+	6.73519i	−1.11940	+	0.463670i	−0.255233	+	0.966879i	$0.582152\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.226386	+	0.226386i	−0.0154394	+	0.0154394i
$216$		0			0
$217$	6.62016	+	6.62016i	0.449406	+	0.449406i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.34333	−	12.8999i	−0.359431	−	0.867743i
$222$		0			0
$223$	−4.15983			−0.278563			−0.139282	−	0.990253i	$-0.544479\pi$
$223$	−4.15983			−0.278563			−0.139282	+	0.990253i	$0.544479\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.08559	+	5.03507i	0.138426	+	0.334189i	0.977856	−	0.209278i	$-0.0671113\pi$
$227$	2.08559	+	5.03507i	0.138426	+	0.334189i	−0.839431	+	0.543467i	$0.817111\pi$
$228$		0			0
$229$	18.0768	+	7.48766i	1.19455	+	0.494798i	0.889234	−	0.457453i	$-0.151238\pi$
$229$	18.0768	+	7.48766i	1.19455	+	0.494798i	0.305315	+	0.952251i	$0.401238\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.22175	−	6.22175i	0.407601	−	0.407601i	−0.473300	−	0.880901i	$-0.656937\pi$
$233$	6.22175	−	6.22175i	0.407601	−	0.407601i	0.880901	+	0.473300i	$0.156937\pi$
$234$		0			0
$235$	6.51323	−	15.7243i	0.424876	−	1.02574i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	11.4462i		−	0.740393i	−0.928954	−	0.370196i	$-0.879290\pi$
$239$		−	11.4462i		−	0.740393i	0.928954	−	0.370196i	$-0.120710\pi$
$240$		0			0
$241$			16.1867i			1.04268i	0.853350	+	0.521338i	$0.174567\pi$
$241$			16.1867i			1.04268i	−0.853350	+	0.521338i	$0.825433\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	16.2958	−	39.3415i	1.04110	−	2.51344i
$246$		0			0
$247$	−7.53262	+	7.53262i	−0.479289	+	0.479289i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−26.3506	−	10.9148i	−1.66324	−	0.688935i	−0.664920	−	0.746915i	$-0.731534\pi$
$251$	−26.3506	−	10.9148i	−1.66324	−	0.688935i	−0.998318	+	0.0579794i	$0.981534\pi$
$252$		0			0
$253$	4.04056	+	9.75478i	0.254028	+	0.613278i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.3518			0.708104			0.354052	−	0.935226i	$-0.384804\pi$
$257$	11.3518			0.708104			0.354052	+	0.935226i	$0.384804\pi$
$258$		0			0
$259$	−8.04041	−	19.4113i	−0.499607	−	1.20616i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−17.7982	−	17.7982i	−1.09748	−	1.09748i	−0.994704	−	0.102779i	$-0.967227\pi$
$263$	−17.7982	−	17.7982i	−1.09748	−	1.09748i	−0.102779	−	0.994704i	$-0.532773\pi$
$264$		0			0
$265$	−4.33334	+	4.33334i	−0.266195	+	0.266195i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	11.0623	−	4.58215i	0.674480	−	0.279379i	−0.0190371	−	0.999819i	$-0.506060\pi$
$269$	11.0623	−	4.58215i	0.674480	−	0.279379i	0.693517	+	0.720440i	$0.256060\pi$
$270$		0			0
$271$		−	13.6375i		−	0.828420i	−0.910181	−	0.414210i	$-0.864058\pi$
$271$		−	13.6375i		−	0.828420i	0.910181	−	0.414210i	$-0.135942\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.83595	−	1.58890i	0.231316	−	0.0958144i
$276$		0			0
$277$	4.97491	−	12.0105i	0.298914	−	0.721641i	−0.701050	−	0.713112i	$-0.747285\pi$
$277$	4.97491	−	12.0105i	0.298914	−	0.721641i	0.999964	−	0.00852911i	$-0.00271493\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.2443	−	12.2443i	−0.730434	−	0.730434i	0.240272	−	0.970706i	$-0.422763\pi$
$281$	−12.2443	−	12.2443i	−0.730434	−	0.730434i	−0.970706	+	0.240272i	$0.922763\pi$
$282$		0			0
$283$	−17.1942	−	7.12206i	−1.02209	−	0.423363i	−0.192237	−	0.981349i	$-0.561574\pi$
$283$	−17.1942	−	7.12206i	−1.02209	−	0.423363i	−0.829850	+	0.557986i	$0.811574\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.61365			0.390391
$288$		0			0
$289$	−32.7005			−1.92356
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−18.9164	−	7.83544i	−1.10511	−	0.457751i	−0.245858	−	0.969306i	$-0.579070\pi$
$293$	−18.9164	−	7.83544i	−1.10511	−	0.457751i	−0.859251	+	0.511555i	$0.829070\pi$
$294$		0			0
$295$	15.4786	+	15.4786i	0.901200	+	0.901200i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.61450	−	8.72618i	0.209032	−	0.504648i
$300$		0			0
$301$	−0.543817	+	0.225257i	−0.0313451	+	0.0129836i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			28.7834i			1.64813i
$306$		0			0
$307$	−8.91242	+	3.69164i	−0.508658	+	0.210693i	−0.622227	−	0.782837i	$-0.713772\pi$
$307$	−8.91242	+	3.69164i	−0.508658	+	0.210693i	0.113568	+	0.993530i	$0.463772\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.45365	−	4.45365i	0.252543	−	0.252543i	−0.569469	−	0.822013i	$-0.692851\pi$
$311$	4.45365	−	4.45365i	0.252543	−	0.252543i	0.822013	+	0.569469i	$0.192851\pi$
$312$		0			0
$313$	0.518243	+	0.518243i	0.0292929	+	0.0292929i	0.721602	−	0.692309i	$-0.243406\pi$
$313$	0.518243	+	0.518243i	0.0292929	+	0.0292929i	−0.692309	+	0.721602i	$0.743406\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.56624	+	13.4381i	0.312631	+	0.754758i	0.999606	+	0.0280769i	$0.00893833\pi$
$317$	5.56624	+	13.4381i	0.312631	+	0.754758i	−0.686975	+	0.726681i	$0.741062\pi$
$318$		0			0
$319$	16.4145			0.919035
$320$		0			0
$321$		0			0
$322$		0			0
$323$	14.5107	+	35.0320i	0.807399	+	1.94923i
$324$		0			0
$325$	−3.43146	−	1.42136i	−0.190343	−	0.0788428i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	22.1266	−	22.1266i	1.21988	−	1.21988i
$330$		0			0
$331$	1.14627	−	2.76734i	0.0630046	−	0.152107i	−0.889242	−	0.457438i	$-0.848767\pi$
$331$	1.14627	−	2.76734i	0.0630046	−	0.152107i	0.952246	+	0.305331i	$0.0987672\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			6.39439i			0.349363i
$336$		0			0
$337$		−	9.71758i		−	0.529350i	−0.964338	−	0.264675i	$-0.914735\pi$
$337$		−	9.71758i		−	0.529350i	0.964338	−	0.264675i	$-0.0852647\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.64547	+	3.97251i	−0.0891070	+	0.215123i
$342$		0			0
$343$	31.4980	−	31.4980i	1.70073	−	1.70073i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.669714	−	0.277404i	−0.0359521	−	0.0148918i	0.364635	−	0.931151i	$-0.381194\pi$
$347$	−0.669714	−	0.277404i	−0.0359521	−	0.0148918i	−0.400587	+	0.916259i	$0.631194\pi$
$348$		0			0
$349$	2.03674	+	4.91712i	0.109024	+	0.263207i	0.968970	−	0.247177i	$-0.0795030\pi$
$349$	2.03674	+	4.91712i	0.109024	+	0.263207i	−0.859946	+	0.510385i	$0.829503\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.88404			−0.526074			−0.263037	−	0.964786i	$-0.584724\pi$
$353$	−9.88404			−0.526074			−0.263037	+	0.964786i	$0.584724\pi$
$354$		0			0
$355$	−6.82955	−	16.4880i	−0.362475	−	0.875092i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	17.4997	+	17.4997i	0.923599	+	0.923599i	0.997282	−	0.0736824i	$-0.0234751\pi$
$359$	17.4997	+	17.4997i	0.923599	+	0.923599i	−0.0736824	+	0.997282i	$0.523475\pi$
$360$		0			0
$361$	7.02114	−	7.02114i	0.369534	−	0.369534i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.94803	−	3.29218i	0.416019	−	0.172321i
$366$		0			0
$367$			16.9225i			0.883345i	0.897176	+	0.441672i	$0.145615\pi$
$367$			16.9225i			0.883345i	−0.897176	+	0.441672i	$0.854385\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−10.4094	+	4.31172i	−0.540430	+	0.223853i
$372$		0			0
$373$	−7.97974	+	19.2648i	−0.413175	+	0.997494i	0.571104	+	0.820878i	$0.306515\pi$
$373$	−7.97974	+	19.2648i	−0.413175	+	0.997494i	−0.984280	+	0.176616i	$0.943485\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−10.3829	−	10.3829i	−0.534747	−	0.534747i
$378$		0			0
$379$	−31.0714	−	12.8702i	−1.59603	−	0.661097i	−0.605183	−	0.796087i	$-0.706900\pi$
$379$	−31.0714	−	12.8702i	−1.59603	−	0.661097i	−0.990847	+	0.134990i	$0.956900\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	15.6267			0.798487			0.399243	−	0.916845i	$-0.369273\pi$
$383$	15.6267			0.798487			0.399243	+	0.916845i	$0.369273\pi$
$384$		0			0
$385$	27.9866			1.42633
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.14019	−	1.30071i	−0.159214	−	0.0659485i	0.301653	−	0.953418i	$-0.402462\pi$
$389$	−3.14019	−	1.30071i	−0.159214	−	0.0659485i	−0.460867	+	0.887469i	$0.652462\pi$
$390$		0			0
$391$	−23.7729	−	23.7729i	−1.20225	−	1.20225i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.43851	−	15.5439i	0.323957	−	0.782101i
$396$		0			0
$397$	4.00985	−	1.66094i	0.201249	−	0.0833600i	−0.279782	−	0.960063i	$-0.590262\pi$
$397$	4.00985	−	1.66094i	0.201249	−	0.0833600i	0.481031	+	0.876703i	$0.340262\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			10.1004i			0.504391i	0.967676	+	0.252196i	$0.0811527\pi$
$401$			10.1004i			0.504391i	−0.967676	+	0.252196i	$0.918847\pi$
$402$		0			0
$403$	3.55362	−	1.47196i	0.177019	−	0.0733235i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.82322	−	6.82322i	0.338214	−	0.338214i
$408$		0			0
$409$	−20.7146	−	20.7146i	−1.02427	−	1.02427i	−0.999698	−	0.0245731i	$-0.992177\pi$
$409$	−20.7146	−	20.7146i	−1.02427	−	1.02427i	−0.0245731	−	0.999698i	$-0.507823\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	15.4014	+	37.1823i	0.757853	+	1.82962i
$414$		0			0
$415$	19.1898			0.941991
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.64991	−	13.6401i	−0.276016	−	0.666363i	0.723702	−	0.690113i	$-0.242439\pi$
$419$	−5.64991	−	13.6401i	−0.276016	−	0.666363i	−0.999718	+	0.0237506i	$0.992439\pi$
$420$		0			0
$421$	12.2423	+	5.07093i	0.596654	+	0.247142i	0.660510	−	0.750817i	$-0.270340\pi$
$421$	12.2423	+	5.07093i	0.596654	+	0.247142i	−0.0638563	+	0.997959i	$0.520340\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−9.34841	+	9.34841i	−0.453464	+	0.453464i
$426$		0			0
$427$	−20.2514	+	48.8911i	−0.980033	+	2.36601i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	2.26090i		−	0.108904i	−0.998516	−	0.0544519i	$-0.982659\pi$
$431$		−	2.26090i		−	0.108904i	0.998516	−	0.0544519i	$-0.0173412\pi$
$432$		0			0
$433$		−	14.3037i		−	0.687391i	−0.939081	−	0.343695i	$-0.888321\pi$
$433$		−	14.3037i		−	0.687391i	0.939081	−	0.343695i	$-0.111679\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−9.81582	+	23.6975i	−0.469554	+	1.13360i
$438$		0			0
$439$	−11.8230	+	11.8230i	−0.564282	+	0.564282i	−0.930521	−	0.366239i	$-0.880645\pi$
$439$	−11.8230	+	11.8230i	−0.564282	+	0.564282i	0.366239	+	0.930521i	$0.380645\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.6974	−	8.15892i	−0.935851	−	0.387642i	−0.137956	−	0.990438i	$-0.544053\pi$
$443$	−19.6974	−	8.15892i	−0.935851	−	0.387642i	−0.797895	+	0.602796i	$0.794053\pi$
$444$		0			0
$445$	11.8901	+	28.7052i	0.563643	+	1.36076i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.93202			0.0911774			0.0455887	−	0.998960i	$-0.485484\pi$
$449$	1.93202			0.0911774			0.0455887	+	0.998960i	$0.485484\pi$
$450$		0			0
$451$	1.16238	+	2.80622i	0.0547341	+	0.132140i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−17.7028	−	17.7028i	−0.829920	−	0.829920i
$456$		0			0
$457$	25.1357	−	25.1357i	1.17580	−	1.17580i	0.194997	−	0.980804i	$-0.437530\pi$
$457$	25.1357	−	25.1357i	1.17580	−	1.17580i	0.980804	−	0.194997i	$-0.0624696\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−21.7563	+	9.01174i	−1.01329	+	0.419719i	−0.826654	−	0.562711i	$-0.809759\pi$
$461$	−21.7563	+	9.01174i	−1.01329	+	0.419719i	−0.186636	+	0.982429i	$0.559759\pi$
$462$		0			0
$463$			10.6911i			0.496857i	0.968650	+	0.248428i	$0.0799141\pi$
$463$			10.6911i			0.496857i	−0.968650	+	0.248428i	$0.920086\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.75029	−	3.21027i	0.358641	−	0.148554i	−0.196086	−	0.980587i	$-0.562823\pi$
$467$	7.75029	−	3.21027i	0.358641	−	0.148554i	0.554726	+	0.832033i	$0.312823\pi$
$468$		0			0
$469$	−4.49896	+	10.8615i	−0.207743	+	0.501535i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.191156	−	0.191156i	−0.00878938	−	0.00878938i
$474$		0			0
$475$	9.31874	+	3.85995i	0.427573	+	0.177107i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.07960			−0.186402			−0.0932008	−	0.995647i	$-0.529710\pi$
$479$	−4.07960			−0.186402			−0.0932008	+	0.995647i	$0.529710\pi$
$480$		0			0
$481$	−8.63199			−0.393585
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.0691	+	5.82761i	0.638845	+	0.264618i
$486$		0			0
$487$	18.4969	+	18.4969i	0.838174	+	0.838174i	0.988619	−	0.150444i	$-0.0480704\pi$
$487$	18.4969	+	18.4969i	0.838174	+	0.838174i	−0.150444	+	0.988619i	$0.548070\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−13.5020	+	32.5966i	−0.609335	+	1.47106i	0.254390	+	0.967102i	$0.418125\pi$
$491$	−13.5020	+	32.5966i	−0.609335	+	1.47106i	−0.863725	+	0.503963i	$0.831875\pi$
$492$		0			0
$493$	−48.2879	+	20.0015i	−2.17478	+	0.900822i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	32.8115i		−	1.47180i
$498$		0			0
$499$	32.2644	−	13.3643i	1.44435	−	0.598270i	0.483503	−	0.875343i	$-0.339364\pi$
$499$	32.2644	−	13.3643i	1.44435	−	0.598270i	0.960849	+	0.277072i	$0.0893641\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−7.89059	+	7.89059i	−0.351824	+	0.351824i	−0.860788	−	0.508964i	$-0.830029\pi$
$503$	−7.89059	+	7.89059i	−0.351824	+	0.351824i	0.508964	+	0.860788i	$0.330029\pi$
$504$		0			0
$505$	−21.4659	−	21.4659i	−0.955221	−	0.955221i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.75033	−	23.5394i	−0.432176	−	1.04336i	−0.978584	−	0.205846i	$-0.934005\pi$
$509$	−9.75033	−	23.5394i	−0.432176	−	1.04336i	0.546409	−	0.837519i	$-0.315995\pi$
$510$		0			0
$511$	15.8168			0.699692
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.97196	+	4.76072i	0.0868948	+	0.209783i
$516$		0			0
$517$	13.2773	+	5.49966i	0.583937	+	0.241875i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−24.9521	+	24.9521i	−1.09317	+	1.09317i	−0.0979840	+	0.995188i	$0.531239\pi$
$521$	−24.9521	+	24.9521i	−1.09317	+	1.09317i	−0.995188	+	0.0979840i	$0.968761\pi$
$522$		0			0
$523$	−10.1770	+	24.5696i	−0.445011	+	1.07435i	0.529156	+	0.848525i	$0.322509\pi$
$523$	−10.1770	+	24.5696i	−0.445011	+	1.07435i	−0.974167	+	0.225828i	$0.927491\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	13.6913i		−	0.596402i
$528$		0			0
$529$			0.257710i			0.0112048i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.03981	−	2.51032i	0.0450391	−	0.108734i
$534$		0			0
$535$	7.88301	−	7.88301i	0.340812	−	0.340812i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	33.2193	+	13.7599i	1.43086	+	0.592680i
$540$		0			0
$541$	−4.54509	−	10.9728i	−0.195409	−	0.471758i	0.795556	−	0.605880i	$-0.207179\pi$
$541$	−4.54509	−	10.9728i	−0.195409	−	0.471758i	−0.990965	+	0.134122i	$0.957179\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−19.8252			−0.849218
$546$		0			0
$547$	15.0871	+	36.4236i	0.645079	+	1.55736i	0.819743	+	0.572731i	$0.194116\pi$
$547$	15.0871	+	36.4236i	0.645079	+	1.55736i	−0.174664	+	0.984628i	$0.555884\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	28.1966	+	28.1966i	1.20122	+	1.20122i
$552$		0			0
$553$	21.8728	−	21.8728i	0.930126	−	0.930126i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.22423	+	0.507092i	−0.0518722	+	0.0214862i	−0.408469	−	0.912772i	$-0.633937\pi$
$557$	−1.22423	+	0.507092i	−0.0518722	+	0.0214862i	0.356597	+	0.934258i	$0.383937\pi$
$558$		0			0
$559$			0.241830i			0.0102283i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−38.1362	+	15.7965i	−1.60725	+	0.665744i	−0.992417	−	0.122915i	$-0.960776\pi$
$563$	−38.1362	+	15.7965i	−1.60725	+	0.665744i	−0.614831	+	0.788659i	$0.710776\pi$
$564$		0			0
$565$	−0.769948	+	1.85882i	−0.0323920	+	0.0782011i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23.2115	+	23.2115i	0.973076	+	0.973076i	0.999647	−	0.0265706i	$-0.00845867\pi$
$569$	23.2115	+	23.2115i	0.973076	+	0.973076i	−0.0265706	+	0.999647i	$0.508459\pi$
$570$		0			0
$571$	13.1318	+	5.43938i	0.549549	+	0.227631i	0.640141	−	0.768257i	$-0.278876\pi$
$571$	13.1318	+	5.43938i	0.549549	+	0.227631i	−0.0905917	+	0.995888i	$0.528876\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.94313			−0.372954
$576$		0			0
$577$	−20.3054			−0.845326			−0.422663	−	0.906287i	$-0.638905\pi$
$577$	−20.3054			−0.845326			−0.422663	+	0.906287i	$0.638905\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	32.5956	+	13.5016i	1.35229	+	0.560139i
$582$		0			0
$583$	−3.65899	−	3.65899i	−0.151540	−	0.151540i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.5239	+	30.2353i	−0.516916	+	1.24794i	0.422873	+	0.906189i	$0.361022\pi$
$587$	−12.5239	+	30.2353i	−0.516916	+	1.24794i	−0.939789	+	0.341756i	$0.888978\pi$
$588$		0			0
$589$	−9.65049	+	3.99736i	−0.397641	+	0.164708i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			8.64651i			0.355070i	0.984115	+	0.177535i	$0.0568122\pi$
$593$			8.64651i			0.355070i	−0.984115	+	0.177535i	$0.943188\pi$
$594$		0			0
$595$	−82.3306	+	34.1024i	−3.37522	+	1.39806i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−18.3874	+	18.3874i	−0.751290	+	0.751290i	−0.974720	−	0.223430i	$-0.928275\pi$
$599$	−18.3874	+	18.3874i	−0.751290	+	0.751290i	0.223430	+	0.974720i	$0.428275\pi$
$600$		0			0
$601$	0.462374	+	0.462374i	0.0188606	+	0.0188606i	0.716474	−	0.697614i	$-0.245755\pi$
$601$	0.462374	+	0.462374i	0.0188606	+	0.0188606i	−0.697614	+	0.716474i	$0.745755\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.11893	−	14.7724i	−0.248770	−	0.600584i
$606$		0			0
$607$	17.2953			0.701996			0.350998	−	0.936376i	$-0.385842\pi$
$607$	17.2953			0.701996			0.350998	+	0.936376i	$0.385842\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.91974	−	11.8773i	−0.199031	−	0.480504i
$612$		0			0
$613$	12.4951	+	5.17565i	0.504674	+	0.209043i	0.620470	−	0.784230i	$-0.286942\pi$
$613$	12.4951	+	5.17565i	0.504674	+	0.209043i	−0.115796	+	0.993273i	$0.536942\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−11.5500	+	11.5500i	−0.464986	+	0.464986i	−0.900286	−	0.435300i	$-0.856642\pi$
$617$	−11.5500	+	11.5500i	−0.464986	+	0.464986i	0.435300	+	0.900286i	$0.356642\pi$
$618$		0			0
$619$	16.9010	−	40.8026i	0.679309	−	1.64000i	−0.0859701	−	0.996298i	$-0.527399\pi$
$619$	16.9010	−	40.8026i	0.679309	−	1.64000i	0.765279	−	0.643699i	$-0.222601\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			57.1239i			2.28862i
$624$		0			0
$625$		−	30.8598i		−	1.23439i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−11.7582	+	28.3867i	−0.468828	+	1.13185i
$630$		0			0
$631$	5.44778	−	5.44778i	0.216873	−	0.216873i	−0.590306	−	0.807179i	$-0.700993\pi$
$631$	5.44778	−	5.44778i	0.216873	−	0.216873i	0.807179	+	0.590306i	$0.200993\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	36.1130	+	14.9585i	1.43310	+	0.593609i
$636$		0			0
$637$	−12.3090	−	29.7164i	−0.487699	−	1.17741i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	31.4403			1.24182			0.620909	−	0.783883i	$-0.286764\pi$
$641$	31.4403			1.24182			0.620909	+	0.783883i	$0.286764\pi$
$642$		0			0
$643$	−14.2661	−	34.4415i	−0.562601	−	1.35824i	−0.907679	−	0.419665i	$-0.862148\pi$
$643$	−14.2661	−	34.4415i	−0.562601	−	1.35824i	0.345078	−	0.938574i	$-0.387852\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.49866	+	4.49866i	0.176861	+	0.176861i	0.789986	−	0.613125i	$-0.210088\pi$
$647$	4.49866	+	4.49866i	0.176861	+	0.176861i	−0.613125	+	0.789986i	$0.710088\pi$
$648$		0			0
$649$	−13.0699	+	13.0699i	−0.513037	+	0.513037i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.51087	+	1.45425i	−0.137391	+	0.0569092i	−0.450320	−	0.892867i	$-0.648690\pi$
$653$	−3.51087	+	1.45425i	−0.137391	+	0.0569092i	0.312929	+	0.949777i	$0.398690\pi$
$654$		0			0
$655$			1.86861i			0.0730127i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	31.6320	−	13.1024i	1.23221	−	0.510398i	0.330938	−	0.943653i	$-0.392635\pi$
$659$	31.6320	−	13.1024i	1.23221	−	0.510398i	0.901272	+	0.433255i	$0.142635\pi$
$660$		0			0
$661$	−3.01417	+	7.27686i	−0.117238	+	0.283037i	−0.971595	−	0.236649i	$-0.923951\pi$
$661$	−3.01417	+	7.27686i	−0.117238	+	0.283037i	0.854358	+	0.519686i	$0.173951\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	48.0751	+	48.0751i	1.86427	+	1.86427i
$666$		0			0
$667$	−32.6644	−	13.5300i	−1.26477	−	0.523885i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−24.3042			−0.938252
$672$		0			0
$673$	24.0467			0.926931			0.463465	−	0.886115i	$-0.346606\pi$
$673$	24.0467			0.926931			0.463465	+	0.886115i	$0.346606\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−17.7623	−	7.35739i	−0.682661	−	0.282767i	0.0142777	−	0.999898i	$-0.495455\pi$
$677$	−17.7623	−	7.35739i	−0.682661	−	0.282767i	−0.696939	+	0.717131i	$0.745455\pi$
$678$		0			0
$679$	19.7975	+	19.7975i	0.759757	+	0.759757i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.1328	+	29.2913i	−0.464250	+	1.12080i	0.502386	+	0.864644i	$0.332456\pi$
$683$	−12.1328	+	29.2913i	−0.464250	+	1.12080i	−0.966636	+	0.256155i	$0.917544\pi$
$684$		0			0
$685$	43.9145	−	18.1900i	1.67789	−	0.695003i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			4.62896i			0.176349i
$690$		0			0
$691$	25.9512	−	10.7493i	0.987231	−	0.408925i	0.170132	−	0.985421i	$-0.445581\pi$
$691$	25.9512	−	10.7493i	0.987231	−	0.408925i	0.817099	+	0.576497i	$0.195581\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.09736	+	6.09736i	−0.231286	+	0.231286i
$696$		0			0
$697$	−6.83891	−	6.83891i	−0.259042	−	0.259042i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−5.57892	−	13.4687i	−0.210713	−	0.508706i	0.782820	−	0.622248i	$-0.213780\pi$
$701$	−5.57892	−	13.4687i	−0.210713	−	0.508706i	−0.993533	+	0.113542i	$0.963780\pi$
$702$		0			0
$703$	23.4417			0.884120
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−21.3588	−	51.5648i	−0.803282	−	1.93929i
$708$		0			0
$709$	6.08885	+	2.52208i	0.228672	+	0.0947189i	0.494077	−	0.869418i	$-0.335506\pi$
$709$	6.08885	+	2.52208i	0.228672	+	0.0947189i	−0.265406	+	0.964137i	$0.585506\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.54887	−	6.54887i	0.245257	−	0.245257i
$714$		0			0
$715$	4.40010	−	10.6228i	0.164554	−	0.397269i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			18.5446i			0.691595i	0.938309	+	0.345798i	$0.112392\pi$
$719$			18.5446i			0.691595i	−0.938309	+	0.345798i	$0.887608\pi$
$720$		0			0
$721$			9.47395i			0.352828i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.32052	+	12.8449i	−0.197599	+	0.477047i
$726$		0			0
$727$	2.54824	−	2.54824i	0.0945090	−	0.0945090i	−0.658272	−	0.752781i	$-0.728712\pi$
$727$	2.54824	−	2.54824i	0.0945090	−	0.0945090i	0.752781	+	0.658272i	$0.228712\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.795269	+	0.329411i	0.0294141	+	0.0121837i
$732$		0			0
$733$	11.8033	+	28.4957i	0.435965	+	1.05251i	0.977330	+	0.211724i	$0.0679077\pi$
$733$	11.8033	+	28.4957i	0.435965	+	1.05251i	−0.541365	+	0.840788i	$0.682092\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.39931			−0.198886
$738$		0			0
$739$	−1.88672	−	4.55495i	−0.0694042	−	0.167557i	0.885371	−	0.464885i	$-0.153904\pi$
$739$	−1.88672	−	4.55495i	−0.0694042	−	0.167557i	−0.954775	+	0.297329i	$0.903904\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−22.8209	−	22.8209i	−0.837218	−	0.837218i	0.151274	−	0.988492i	$-0.451663\pi$
$743$	−22.8209	−	22.8209i	−0.837218	−	0.837218i	−0.988492	+	0.151274i	$0.951663\pi$
$744$		0			0
$745$	−26.5290	+	26.5290i	−0.971949	+	0.971949i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.9363	−	7.84368i	0.691918	−	0.286602i
$750$		0			0
$751$		−	5.12686i		−	0.187082i	−0.995615	−	0.0935409i	$-0.970181\pi$
$751$		−	5.12686i		−	0.187082i	0.995615	−	0.0935409i	$-0.0298186\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.5470	−	5.19714i	0.456633	−	0.189143i
$756$		0			0
$757$	11.1063	−	26.8129i	0.403664	−	0.974531i	−0.583105	−	0.812397i	$-0.698162\pi$
$757$	11.1063	−	26.8129i	0.403664	−	0.974531i	0.986769	−	0.162134i	$-0.0518377\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12.7301	−	12.7301i	−0.461467	−	0.461467i	0.437669	−	0.899136i	$-0.355804\pi$
$761$	−12.7301	−	12.7301i	−0.461467	−	0.461467i	−0.899136	+	0.437669i	$0.855804\pi$
$762$		0			0
$763$	−33.6749	−	13.9486i	−1.21911	−	0.504973i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	16.5346			0.597028
$768$		0			0
$769$	−30.0514			−1.08368			−0.541841	−	0.840481i	$-0.682273\pi$
$769$	−30.0514			−1.08368			−0.541841	+	0.840481i	$0.682273\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8.38463	−	3.47303i	−0.301574	−	0.124916i	0.226765	−	0.973949i	$-0.427185\pi$
$773$	−8.38463	−	3.47303i	−0.301574	−	0.124916i	−0.528339	+	0.849033i	$0.677185\pi$
$774$		0			0
$775$	−2.57526	−	2.57526i	−0.0925062	−	0.0925062i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.82378	+	6.81721i	−0.101172	+	0.244252i
$780$		0			0
$781$	13.9222	−	5.76675i	0.498175	−	0.206351i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			10.0231i			0.357739i
$786$		0			0
$787$	−1.44711	+	0.599414i	−0.0515840	+	0.0213668i	−0.408326	−	0.912836i	$-0.633887\pi$
$787$	−1.44711	+	0.599414i	−0.0515840	+	0.0213668i	0.356742	+	0.934203i	$0.383887\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.61565	+	2.61565i	−0.0930019	+	0.0930019i
$792$		0			0
$793$	15.3735	+	15.3735i	0.545928	+	0.545928i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15.8381	−	38.2365i	−0.561013	−	1.35441i	−0.908957	−	0.416890i	$-0.863120\pi$
$797$	−15.8381	−	38.2365i	−0.561013	−	1.35441i	0.347944	−	0.937515i	$-0.386880\pi$
$798$		0			0
$799$	−45.7605			−1.61889
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.77986	+	6.71118i	0.0980992	+	0.236832i
$804$		0			0
$805$	−55.6926	−	23.0687i	−1.96291	−	0.813063i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.53975	−	5.53975i	0.194767	−	0.194767i	−0.602985	−	0.797752i	$-0.706022\pi$
$809$	5.53975	−	5.53975i	0.194767	−	0.194767i	0.797752	+	0.602985i	$0.206022\pi$
$810$		0			0
$811$	−9.48319	+	22.8945i	−0.333000	+	0.803933i	0.665351	+	0.746530i	$0.268282\pi$
$811$	−9.48319	+	22.8945i	−0.333000	+	0.803933i	−0.998351	+	0.0574024i	$0.981718\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	61.0025i		−	2.13683i
$816$		0			0
$817$		−	0.656732i		−	0.0229761i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12.2661	+	29.6131i	−0.428091	+	1.03350i	0.551801	+	0.833976i	$0.313941\pi$
$821$	−12.2661	+	29.6131i	−0.428091	+	1.03350i	−0.979892	+	0.199527i	$0.936059\pi$
$822$		0			0
$823$	−17.0586	+	17.0586i	−0.594625	+	0.594625i	−0.938877	−	0.344252i	$-0.888133\pi$
$823$	−17.0586	+	17.0586i	−0.594625	+	0.594625i	0.344252	+	0.938877i	$0.388133\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14.5571	+	6.02974i	0.506199	+	0.209674i	0.621143	−	0.783698i	$-0.286669\pi$
$827$	14.5571	+	6.02974i	0.506199	+	0.209674i	−0.114944	+	0.993372i	$0.536669\pi$
$828$		0			0
$829$	14.6824	+	35.4465i	0.509942	+	1.23111i	0.943917	+	0.330184i	$0.107111\pi$
$829$	14.6824	+	35.4465i	0.509942	+	1.23111i	−0.433975	+	0.900925i	$0.642889\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−114.491			−3.96687
$834$		0			0
$835$	−13.5671	−	32.7539i	−0.469509	−	1.13350i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.5711	−	12.5711i	−0.434004	−	0.434004i	0.455984	−	0.889988i	$-0.349287\pi$
$839$	−12.5711	−	12.5711i	−0.434004	−	0.434004i	−0.889988	+	0.455984i	$0.849287\pi$
$840$		0			0
$841$	−18.3599	+	18.3599i	−0.633099	+	0.633099i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	21.9897	−	9.10842i	0.756468	−	0.313339i
$846$		0			0
$847$		−	29.3974i		−	1.01011i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−19.2022	+	7.95382i	−0.658244	+	0.272654i
$852$		0			0
$853$	−14.1621	+	34.1902i	−0.484900	+	1.17065i	0.472356	+	0.881408i	$0.343404\pi$
$853$	−14.1621	+	34.1902i	−0.484900	+	1.17065i	−0.957255	+	0.289244i	$0.906596\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.37799	−	8.37799i	−0.286187	−	0.286187i	0.549384	−	0.835570i	$-0.314863\pi$
$857$	−8.37799	−	8.37799i	−0.286187	−	0.286187i	−0.835570	+	0.549384i	$0.814863\pi$
$858$		0			0
$859$	41.0639	+	17.0092i	1.40108	+	0.580347i	0.950032	−	0.312152i	$-0.101050\pi$
$859$	41.0639	+	17.0092i	1.40108	+	0.580347i	0.451049	+	0.892499i	$0.351050\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	54.9639			1.87099			0.935497	−	0.353335i	$-0.114952\pi$
$863$	54.9639			1.87099			0.935497	+	0.353335i	$0.114952\pi$
$864$		0			0
$865$	47.8166			1.62581
$866$		0			0
$867$		0			0
$868$		0			0
$869$	13.1250	+	5.43657i	0.445236	+	0.184423i
$870$		0			0
$871$	3.41531	+	3.41531i	0.115723	+	0.115723i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	15.1152	−	36.4912i	0.510985	−	1.23363i
$876$		0			0
$877$	9.45863	−	3.91789i	0.319395	−	0.132298i	−0.217225	−	0.976121i	$-0.569701\pi$
$877$	9.45863	−	3.91789i	0.319395	−	0.132298i	0.536621	+	0.843824i	$0.319701\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	27.2316i		−	0.917456i	−0.888577	−	0.458728i	$-0.848305\pi$
$881$		−	27.2316i		−	0.917456i	0.888577	−	0.458728i	$-0.151695\pi$
$882$		0			0
$883$	23.2873	−	9.64591i	0.783680	−	0.324611i	0.0452804	−	0.998974i	$-0.485582\pi$
$883$	23.2873	−	9.64591i	0.783680	−	0.324611i	0.738400	+	0.674363i	$0.235582\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−26.9018	+	26.9018i	−0.903273	+	0.903273i	−0.995718	−	0.0924444i	$-0.970532\pi$
$887$	−26.9018	+	26.9018i	−0.903273	+	0.903273i	0.0924444	+	0.995718i	$0.470532\pi$
$888$		0			0
$889$	50.8167	+	50.8167i	1.70434	+	1.70434i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	13.3604	+	32.2549i	0.447089	+	1.07937i
$894$		0			0
$895$	−30.6707			−1.02521
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5.50993	−	13.3022i	−0.183767	−	0.443652i
$900$		0			0
$901$	15.2225	+	6.30538i	0.507137	+	0.210063i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	34.3918	−	34.3918i	1.14322	−	1.14322i
$906$		0			0
$907$	10.0120	−	24.1711i	0.332443	−	0.802588i	−0.665954	−	0.745993i	$-0.731975\pi$
$907$	10.0120	−	24.1711i	0.332443	−	0.802588i	0.998397	−	0.0565957i	$-0.0180246\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	33.5125i		−	1.11032i	−0.831744	−	0.555159i	$-0.812657\pi$
$911$		−	33.5125i		−	1.11032i	0.831744	−	0.555159i	$-0.187343\pi$
$912$		0			0
$913$			16.2035i			0.536259i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.31472	+	3.17401i	−0.0434158	+	0.104815i
$918$		0			0
$919$	40.3944	−	40.3944i	1.33249	−	1.33249i	0.429349	−	0.903139i	$-0.358743\pi$
$919$	40.3944	−	40.3944i	1.33249	−	1.33249i	0.903139	−	0.429349i	$-0.141257\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−12.4541	−	5.15867i	−0.409933	−	0.169800i
$924$		0			0
$925$	3.12774	+	7.55104i	0.102840	+	0.248277i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	28.9134			0.948618			0.474309	−	0.880358i	$-0.342698\pi$
$929$	28.9134			0.948618			0.474309	+	0.880358i	$0.342698\pi$
$930$		0			0
$931$	33.4271	+	80.7003i	1.09553	+	2.64484i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−28.9399	−	28.9399i	−0.946435	−	0.946435i
$936$		0			0
$937$	3.10146	−	3.10146i	0.101320	−	0.101320i	−0.654629	−	0.755950i	$-0.727175\pi$
$937$	3.10146	−	3.10146i	0.101320	−	0.101320i	0.755950	+	0.654629i	$0.227175\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	15.8775	−	6.57666i	0.517590	−	0.214393i	−0.108568	−	0.994089i	$-0.534626\pi$
$941$	15.8775	−	6.57666i	0.517590	−	0.214393i	0.626158	+	0.779696i	$0.284626\pi$
$942$		0			0
$943$		−	6.54243i		−	0.213051i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	40.8972	−	16.9402i	1.32898	−	0.550481i	0.398615	−	0.917118i	$-0.369491\pi$
$947$	40.8972	−	16.9402i	1.32898	−	0.550481i	0.930364	+	0.366637i	$0.119491\pi$
$948$		0			0
$949$	2.48673	−	6.00351i	0.0807228	−	0.194882i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	24.4591	+	24.4591i	0.792308	+	0.792308i	0.981869	−	0.189561i	$-0.0607065\pi$
$953$	24.4591	+	24.4591i	0.792308	+	0.792308i	−0.189561	+	0.981869i	$0.560706\pi$
$954$		0			0
$955$	61.9712	+	25.6693i	2.00534	+	0.830640i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	87.3908			2.82200
$960$		0			0
$961$	−27.2284			−0.878335
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−49.5985	−	20.5444i	−1.59663	−	0.661347i
$966$		0			0
$967$	−26.1329	−	26.1329i	−0.840378	−	0.840378i	0.148530	−	0.988908i	$-0.452546\pi$
$967$	−26.1329	−	26.1329i	−0.840378	−	0.840378i	−0.988908	+	0.148530i	$0.952546\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.53031	+	8.52292i	−0.113293	+	0.273514i	−0.970348	−	0.241711i	$-0.922291\pi$
$971$	−3.53031	+	8.52292i	−0.113293	+	0.273514i	0.857055	+	0.515225i	$0.172291\pi$
$972$		0			0
$973$	−14.6469	+	6.06694i	−0.469558	+	0.194497i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	35.9963i		−	1.15162i	−0.817582	−	0.575812i	$-0.804686\pi$
$977$		−	35.9963i		−	1.15162i	0.817582	−	0.575812i	$-0.195314\pi$
$978$		0			0
$979$	−24.2381	+	10.0398i	−0.774654	+	0.320872i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−27.6960	+	27.6960i	−0.883366	+	0.883366i	−0.993875	−	0.110509i	$-0.964752\pi$
$983$	−27.6960	+	27.6960i	−0.883366	+	0.883366i	0.110509	+	0.993875i	$0.464752\pi$
$984$		0			0
$985$	2.66165	+	2.66165i	0.0848073	+	0.0848073i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.222831	+	0.537961i	0.00708561	+	0.0171062i
$990$		0			0
$991$	−33.2769			−1.05708			−0.528538	−	0.848910i	$-0.677259\pi$
$991$	−33.2769			−1.05708			−0.528538	+	0.848910i	$0.677259\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.04405	−	2.52056i	−0.0330986	−	0.0799071i
$996$		0			0
$997$	−45.1656	−	18.7082i	−1.43041	−	0.592494i	−0.472955	−	0.881086i	$-0.656813\pi$
$997$	−45.1656	−	18.7082i	−1.43041	−	0.592494i	−0.957452	+	0.288592i	$0.906813\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.2.v.d.1009.2		32
3.2	odd	2	inner	1152.2.v.d.1009.7		32
4.3	odd	2		288.2.v.c.37.4	✓	32
12.11	even	2		288.2.v.c.37.5	yes	32
32.13	even	8	inner	1152.2.v.d.145.2		32
32.19	odd	8		288.2.v.c.109.4	yes	32
96.77	odd	8	inner	1152.2.v.d.145.7		32
96.83	even	8		288.2.v.c.109.5	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.v.c.37.4	✓	32	4.3	odd	2
288.2.v.c.37.5	yes	32	12.11	even	2
288.2.v.c.109.4	yes	32	32.19	odd	8
288.2.v.c.109.5	yes	32	96.83	even	8
1152.2.v.d.145.2		32	32.13	even	8	inner
1152.2.v.d.145.7		32	96.77	odd	8	inner
1152.2.v.d.1009.2		32	1.1	even	1	trivial
1152.2.v.d.1009.7		32	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.v (of order \(8\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.42249 - 1.00343i) q^{5} +(-3.40882 - 3.40882i) q^{7} +O(q^{10})\)	\(q+(-2.42249 - 1.00343i) q^{5} +(-3.40882 - 3.40882i) q^{7} +(0.847276 - 2.04551i) q^{11} +(-1.82981 + 0.757934i) q^{13} +7.04986i q^{17} +(4.96918 - 2.05830i) q^{19} +(-3.37211 + 3.37211i) q^{23} +(1.32604 + 1.32604i) q^{25} +(2.83715 + 6.84948i) q^{29} -1.94207 q^{31} +(4.83732 + 11.6783i) q^{35} +(4.02656 + 1.66786i) q^{37} +(-0.970078 + 0.970078i) q^{41} +(0.0467259 - 0.112806i) q^{43} +6.49099i q^{47} +16.2401i q^{49} +(0.894399 - 2.15927i) q^{53} +(-4.10503 + 4.10503i) q^{55} +(-7.71288 - 3.19478i) q^{59} +(-4.20083 - 10.1417i) q^{61} +5.19323 q^{65} +(-0.933239 - 2.25304i) q^{67} +(4.81273 + 4.81273i) q^{71} +(-2.31997 + 2.31997i) q^{73} +(-9.86098 + 4.08455i) q^{77} +6.41652i q^{79} +(-6.76145 + 2.80069i) q^{83} +(7.07402 - 17.0782i) q^{85} +(-8.37883 - 8.37883i) q^{89} +(8.82117 + 3.65385i) q^{91} -14.1031 q^{95} -5.80771 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 48 q^{31} - 16 q^{43} - 32 q^{55} - 32 q^{61} - 16 q^{67} + 48 q^{91}+O(q^{100}) \) 32 * q - 48 * q^31 - 16 * q^43 - 32 * q^55 - 32 * q^61 - 16 * q^67 + 48 * q^91

Introduction

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