LMFDB - Embedded newform 1440.2.w.f.1313.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(737,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))

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

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

chi := DirichletCharacter("1440.737");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.w (of order $4$, degree $2$, 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:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 27x^{8} + 107x^{4} + 1 $ x^12 + 27x^8 + 107x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1313.4
Root			$-1.04736 + 1.04736i$ of defining polynomial
Character	$\chi$	$=$	1440.1313
Dual form			1440.2.w.f.737.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.137134 - 2.23186i) q^{5} +(-0.806063 + 0.806063i) q^{7} +O(q^{10})$	$q+(0.137134 - 2.23186i) q^{5} +(-0.806063 + 0.806063i) q^{7} +5.87793i q^{11} +(-0.193937 - 0.193937i) q^{13} +(3.50894 + 3.50894i) q^{17} -2.38787i q^{19} +(-4.18945 + 4.18945i) q^{23} +(-4.96239 - 0.612127i) q^{25} -7.29214 q^{29} +4.31265 q^{31} +(1.68848 + 1.90956i) q^{35} +(1.54420 - 1.54420i) q^{37} +3.32377i q^{41} +(8.31265 + 8.31265i) q^{43} +(6.42647 + 6.42647i) q^{47} +5.70052i q^{49} +(4.64888 - 4.64888i) q^{53} +(13.1187 + 0.806063i) q^{55} -10.4377 q^{59} +7.53690 q^{61} +(-0.459434 + 0.406244i) q^{65} +(4.31265 - 4.31265i) q^{67} +9.47597i q^{71} +(0.0376114 + 0.0376114i) q^{73} +(-4.73799 - 4.73799i) q^{77} -3.68735i q^{79} +(-0.221077 + 0.221077i) q^{83} +(8.31265 - 7.35026i) q^{85} +9.52916 q^{89} +0.312650 q^{91} +(-5.32940 - 0.327458i) q^{95} +(-8.27504 + 8.27504i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{7}+O(q^{10}) $ 12 * q - 8 * q^7	$ 12 q - 8 q^{7} - 4 q^{13} - 16 q^{25} - 32 q^{31} - 20 q^{37} + 16 q^{43} + 72 q^{55} - 32 q^{67} + 44 q^{73} + 16 q^{85} - 80 q^{91} + 28 q^{97}+O(q^{100}) $ 12 * q - 8 * q^7 - 4 * q^13 - 16 * q^25 - 32 * q^31 - 20 * q^37 + 16 * q^43 + 72 * q^55 - 32 * q^67 + 44 * q^73 + 16 * q^85 - 80 * q^91 + 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$1$	$1$

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
$5$	0.137134	−	2.23186i	0.0613281	−	0.998118i
$5$	0.137134	−	2.23186i	0.0613281	−	0.998118i
$6$		0			0
$7$	−0.806063	+	0.806063i	−0.304663	+	0.304663i	−0.842835	−	0.538172i	$-0.819115\pi$
$7$	−0.806063	+	0.806063i	−0.304663	+	0.304663i	0.538172	+	0.842835i	$0.319115\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.87793i			1.77226i	0.463434	+	0.886132i	$0.346617\pi$
$11$			5.87793i			1.77226i	−0.463434	+	0.886132i	$0.653383\pi$
$12$		0			0
$13$	−0.193937	−	0.193937i	−0.0537883	−	0.0537883i	0.679701	−	0.733489i	$-0.262110\pi$
$13$	−0.193937	−	0.193937i	−0.0537883	−	0.0537883i	−0.733489	+	0.679701i	$0.762110\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.50894	+	3.50894i	0.851043	+	0.851043i	0.990262	−	0.139219i	$-0.0444592\pi$
$17$	3.50894	+	3.50894i	0.851043	+	0.851043i	−0.139219	+	0.990262i	$0.544459\pi$
$18$		0			0
$19$		−	2.38787i		−	0.547816i	−0.961756	−	0.273908i	$-0.911684\pi$
$19$		−	2.38787i		−	0.547816i	0.961756	−	0.273908i	$-0.0883163\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.18945	+	4.18945i	−0.873561	+	0.873561i	−0.992859	−	0.119298i	$-0.961936\pi$
$23$	−4.18945	+	4.18945i	−0.873561	+	0.873561i	0.119298	+	0.992859i	$0.461936\pi$
$24$		0			0
$25$	−4.96239	−	0.612127i	−0.992478	−	0.122425i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.29214			−1.35412			−0.677059	−	0.735929i	$-0.736746\pi$
$29$	−7.29214			−1.35412			−0.677059	+	0.735929i	$0.736746\pi$
$30$		0			0
$31$	4.31265			0.774575			0.387287	−	0.921959i	$-0.373412\pi$
$31$	4.31265			0.774575			0.387287	+	0.921959i	$0.373412\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.68848	+	1.90956i	0.285405	+	0.322774i
$36$		0			0
$37$	1.54420	−	1.54420i	0.253865	−	0.253865i	−0.568688	−	0.822553i	$-0.692549\pi$
$37$	1.54420	−	1.54420i	0.253865	−	0.253865i	0.822553	+	0.568688i	$0.192549\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3.32377i			0.519086i	0.965732	+	0.259543i	$0.0835719\pi$
$41$			3.32377i			0.519086i	−0.965732	+	0.259543i	$0.916428\pi$
$42$		0			0
$43$	8.31265	+	8.31265i	1.26767	+	1.26767i	0.947290	+	0.320377i	$0.103810\pi$
$43$	8.31265	+	8.31265i	1.26767	+	1.26767i	0.320377	+	0.947290i	$0.396190\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.42647	+	6.42647i	0.937397	+	0.937397i	0.998153	−	0.0607561i	$-0.0193512\pi$
$47$	6.42647	+	6.42647i	0.937397	+	0.937397i	−0.0607561	+	0.998153i	$0.519351\pi$
$48$		0			0
$49$			5.70052i			0.814360i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.64888	−	4.64888i	0.638573	−	0.638573i	−0.311630	−	0.950203i	$-0.600875\pi$
$53$	4.64888	−	4.64888i	0.638573	−	0.638573i	0.950203	+	0.311630i	$0.100875\pi$
$54$		0			0
$55$	13.1187	+	0.806063i	1.76893	+	0.108690i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.4377			−1.35887			−0.679437	−	0.733734i	$-0.737776\pi$
$59$	−10.4377			−1.35887			−0.679437	+	0.733734i	$0.737776\pi$
$60$		0			0
$61$	7.53690			0.965002			0.482501	−	0.875896i	$-0.339729\pi$
$61$	7.53690			0.965002			0.482501	+	0.875896i	$0.339729\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.459434	+	0.406244i	−0.0569858	+	0.0503883i
$66$		0			0
$67$	4.31265	−	4.31265i	0.526874	−	0.526874i	−0.392765	−	0.919639i	$-0.628481\pi$
$67$	4.31265	−	4.31265i	0.526874	−	0.526874i	0.919639	+	0.392765i	$0.128481\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			9.47597i			1.12459i	0.826936	+	0.562295i	$0.190082\pi$
$71$			9.47597i			1.12459i	−0.826936	+	0.562295i	$0.809918\pi$
$72$		0			0
$73$	0.0376114	+	0.0376114i	0.00440208	+	0.00440208i	0.709304	−	0.704902i	$-0.249009\pi$
$73$	0.0376114	+	0.0376114i	0.00440208	+	0.00440208i	−0.704902	+	0.709304i	$0.749009\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.73799	−	4.73799i	−0.539944	−	0.539944i
$78$		0			0
$79$		−	3.68735i		−	0.414859i	−0.978250	−	0.207430i	$-0.933490\pi$
$79$		−	3.68735i		−	0.414859i	0.978250	−	0.207430i	$-0.0665098\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.221077	+	0.221077i	−0.0242664	+	0.0242664i	−0.719136	−	0.694869i	$-0.755462\pi$
$83$	−0.221077	+	0.221077i	−0.0242664	+	0.0242664i	0.694869	+	0.719136i	$0.255462\pi$
$84$		0			0
$85$	8.31265	−	7.35026i	0.901634	−	0.797248i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.52916			1.01009			0.505045	−	0.863093i	$-0.331476\pi$
$89$	9.52916			1.01009			0.505045	+	0.863093i	$0.331476\pi$
$90$		0			0
$91$	0.312650			0.0327747
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.32940	−	0.327458i	−0.546784	−	0.0335965i
$96$		0			0
$97$	−8.27504	+	8.27504i	−0.840203	+	0.840203i	−0.988885	−	0.148682i	$-0.952497\pi$
$97$	−8.27504	+	8.27504i	−0.840203	+	0.840203i	0.148682	+	0.988885i	$0.452497\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	5.82474i		−	0.579583i	−0.957090	−	0.289792i	$-0.906414\pi$
$101$		−	5.82474i		−	0.579583i	0.957090	−	0.289792i	$-0.0935860\pi$
$102$		0			0
$103$	2.41819	+	2.41819i	0.238271	+	0.238271i	0.816134	−	0.577863i	$-0.196113\pi$
$103$	2.41819	+	2.41819i	0.238271	+	0.238271i	−0.577863	+	0.816134i	$0.696113\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.5348	+	11.5348i	1.11511	+	1.11511i	0.992449	+	0.122661i	$0.0391429\pi$
$107$	11.5348	+	11.5348i	1.11511	+	1.11511i	0.122661	+	0.992449i	$0.460857\pi$
$108$		0			0
$109$		−	11.0132i		−	1.05487i	−0.849595	−	0.527435i	$-0.823154\pi$
$109$		−	11.0132i		−	1.05487i	0.849595	−	0.527435i	$-0.176846\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.20304	+	7.20304i	−0.677605	+	0.677605i	−0.959458	−	0.281852i	$-0.909051\pi$
$113$	−7.20304	+	7.20304i	−0.677605	+	0.677605i	0.281852	+	0.959458i	$0.409051\pi$
$114$		0			0
$115$	8.77575	+	9.92478i	0.818343	+	0.925490i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.65685			−0.518563
$120$		0			0
$121$	−23.5501			−2.14092
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.04669	+	10.9914i	−0.183062	+	0.983101i
$126$		0			0
$127$	−8.80606	+	8.80606i	−0.781412	+	0.781412i	−0.980069	−	0.198657i	$-0.936342\pi$
$127$	−8.80606	+	8.80606i	−0.781412	+	0.781412i	0.198657	+	0.980069i	$0.436342\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			13.8147i			1.20699i	0.797365	+	0.603497i	$0.206226\pi$
$131$			13.8147i			1.20699i	−0.797365	+	0.603497i	$0.793774\pi$
$132$		0			0
$133$	1.92478	+	1.92478i	0.166899	+	0.166899i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.64326	+	2.64326i	0.225829	+	0.225829i	0.810948	−	0.585119i	$-0.198952\pi$
$137$	2.64326	+	2.64326i	0.225829	+	0.225829i	−0.585119	+	0.810948i	$0.698952\pi$
$138$		0			0
$139$		−	18.7005i		−	1.58616i	−0.609119	−	0.793079i	$-0.708477\pi$
$139$		−	18.7005i		−	1.58616i	0.609119	−	0.793079i	$-0.291523\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.13995	−	1.13995i	0.0953271	−	0.0953271i
$144$		0			0
$145$	−1.00000	+	16.2750i	−0.0830455	+	1.35157i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.55610			0.619020			0.309510	−	0.950896i	$-0.399835\pi$
$149$	7.55610			0.619020			0.309510	+	0.950896i	$0.399835\pi$
$150$		0			0
$151$	−17.1490			−1.39557			−0.697784	−	0.716308i	$-0.745831\pi$
$151$	−17.1490			−1.39557			−0.697784	+	0.716308i	$0.745831\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.591410	−	9.62523i	0.0475032	−	0.773117i
$156$		0			0
$157$	8.69323	−	8.69323i	0.693795	−	0.693795i	−0.269270	−	0.963065i	$-0.586782\pi$
$157$	8.69323	−	8.69323i	0.693795	−	0.693795i	0.963065	+	0.269270i	$0.0867823\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	6.75393i		−	0.532284i
$162$		0			0
$163$	−4.31265	−	4.31265i	−0.337793	−	0.337793i	0.517743	−	0.855536i	$-0.326772\pi$
$163$	−4.31265	−	4.31265i	−0.337793	−	0.337793i	−0.855536	+	0.517743i	$0.826772\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.90956	−	1.90956i	−0.147766	−	0.147766i	0.629353	−	0.777119i	$-0.283320\pi$
$167$	−1.90956	−	1.90956i	−0.147766	−	0.147766i	−0.777119	+	0.629353i	$0.783320\pi$
$168$		0			0
$169$		−	12.9248i		−	0.994214i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.17586	+	1.17586i	−0.0893987	+	0.0893987i	−0.750392	−	0.660993i	$-0.770135\pi$
$173$	−1.17586	+	1.17586i	−0.0893987	+	0.0893987i	0.660993	+	0.750392i	$0.270135\pi$
$174$		0			0
$175$	4.49341	−	3.50659i	0.339670	−	0.265073i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.81274			−0.658695			−0.329348	−	0.944209i	$-0.606829\pi$
$179$	−8.81274			−0.658695			−0.329348	+	0.944209i	$0.606829\pi$
$180$		0			0
$181$	15.7889			1.17358			0.586791	−	0.809739i	$-0.300391\pi$
$181$	15.7889			1.17358			0.586791	+	0.809739i	$0.300391\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.23467	−	3.65819i	−0.237818	−	0.268956i
$186$		0			0
$187$	−20.6253	+	20.6253i	−1.50827	+	1.50827i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			1.83774i			0.132974i	0.997787	+	0.0664870i	$0.0211791\pi$
$191$			1.83774i			0.132974i	−0.997787	+	0.0664870i	$0.978821\pi$
$192$		0			0
$193$	10.0884	+	10.0884i	0.726179	+	0.726179i	0.969856	−	0.243678i	$-0.0783539\pi$
$193$	10.0884	+	10.0884i	0.726179	+	0.726179i	−0.243678	+	0.969856i	$0.578354\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−17.5979	−	17.5979i	−1.25380	−	1.25380i	−0.954004	−	0.299793i	$-0.903082\pi$
$197$	−17.5979	−	17.5979i	−1.25380	−	1.25380i	−0.299793	−	0.954004i	$-0.596918\pi$
$198$		0			0
$199$		−	21.9248i		−	1.55421i	−0.629373	−	0.777103i	$-0.716688\pi$
$199$		−	21.9248i		−	1.55421i	0.629373	−	0.777103i	$-0.283312\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	5.87793	−	5.87793i	0.412550	−	0.412550i
$204$		0			0
$205$	7.41819	+	0.455802i	0.518109	+	0.0318346i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	14.0358			0.970873
$210$		0			0
$211$	−15.4763			−1.06543			−0.532715	−	0.846295i	$-0.678828\pi$
$211$	−15.4763			−1.06543			−0.532715	+	0.846295i	$0.678828\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	19.6926	−	17.4127i	1.34302	−	1.18754i
$216$		0			0
$217$	−3.47627	+	3.47627i	−0.235985	+	0.235985i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	1.36102i		−	0.0915523i
$222$		0			0
$223$	16.2823	+	16.2823i	1.09035	+	1.09035i	0.995491	+	0.0948545i	$0.0302386\pi$
$223$	16.2823	+	16.2823i	1.09035	+	1.09035i	0.0948545	+	0.995491i	$0.469761\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.09901	−	6.09901i	−0.404805	−	0.404805i	0.475117	−	0.879923i	$-0.342406\pi$
$227$	−6.09901	−	6.09901i	−0.404805	−	0.404805i	−0.879923	+	0.475117i	$0.842406\pi$
$228$		0			0
$229$			21.0738i			1.39260i	0.717753	+	0.696298i	$0.245171\pi$
$229$			21.0738i			1.39260i	−0.717753	+	0.696298i	$0.754829\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.5428	+	12.5428i	−0.821703	+	0.821703i	−0.986352	−	0.164649i	$-0.947351\pi$
$233$	−12.5428	+	12.5428i	−0.821703	+	0.821703i	0.164649	+	0.986352i	$0.447351\pi$
$234$		0			0
$235$	15.2243	−	13.4617i	0.993121	−	0.878143i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.4108			−0.802787			−0.401393	−	0.915906i	$-0.631474\pi$
$239$	−12.4108			−0.802787			−0.401393	+	0.915906i	$0.631474\pi$
$240$		0			0
$241$	1.47627			0.0950949			0.0475474	−	0.998869i	$-0.484859\pi$
$241$	1.47627			0.0950949			0.0475474	+	0.998869i	$0.484859\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.7228	+	0.781735i	0.812828	+	0.0499432i
$246$		0			0
$247$	−0.463096	+	0.463096i	−0.0294661	+	0.0294661i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	26.0127i		−	1.64191i	−0.570995	−	0.820953i	$-0.693443\pi$
$251$		−	26.0127i		−	1.64191i	0.570995	−	0.820953i	$-0.306557\pi$
$252$		0			0
$253$	−24.6253	−	24.6253i	−1.54818	−	1.54818i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	18.1464	+	18.1464i	1.13194	+	1.13194i	0.989854	+	0.142089i	$0.0453818\pi$
$257$	18.1464	+	18.1464i	1.13194	+	1.13194i	0.142089	+	0.989854i	$0.454618\pi$
$258$		0			0
$259$			2.48944i			0.154686i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	19.3223	−	19.3223i	1.19146	−	1.19146i	0.214806	−	0.976657i	$-0.431088\pi$
$263$	19.3223	−	19.3223i	1.19146	−	1.19146i	0.976657	−	0.214806i	$-0.0689118\pi$
$264$		0			0
$265$	−9.73813	−	11.0132i	−0.598209	−	0.676534i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−30.6257			−1.86728			−0.933640	−	0.358213i	$-0.883386\pi$
$269$	−30.6257			−1.86728			−0.933640	+	0.358213i	$0.883386\pi$
$270$		0			0
$271$	−12.2520			−0.744257			−0.372128	−	0.928181i	$-0.621372\pi$
$271$	−12.2520			−0.744257			−0.372128	+	0.928181i	$0.621372\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.59804	−	29.1686i	0.216970	−	1.75893i
$276$		0			0
$277$	−1.80606	+	1.80606i	−0.108516	+	0.108516i	−0.759280	−	0.650764i	$-0.774449\pi$
$277$	−1.80606	+	1.80606i	−0.108516	+	0.108516i	0.650764	+	0.759280i	$0.274449\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			16.8110i			1.00286i	0.865198	+	0.501430i	$0.167192\pi$
$281$			16.8110i			1.00286i	−0.865198	+	0.501430i	$0.832808\pi$
$282$		0			0
$283$	−4.62530	−	4.62530i	−0.274946	−	0.274946i	0.556142	−	0.831087i	$-0.312281\pi$
$283$	−4.62530	−	4.62530i	−0.274946	−	0.274946i	−0.831087	+	0.556142i	$0.812281\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.67917	−	2.67917i	−0.158146	−	0.158146i
$288$		0			0
$289$			7.62530i			0.448547i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	22.5141	−	22.5141i	1.31529	−	1.31529i	0.397825	−	0.917461i	$-0.369765\pi$
$293$	22.5141	−	22.5141i	1.31529	−	1.31529i	0.917461	−	0.397825i	$-0.130235\pi$
$294$		0			0
$295$	−1.43136	+	23.2955i	−0.0833372	+	1.35632i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.62498			0.0939747
$300$		0			0
$301$	−13.4010			−0.772424
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.03356	−	16.8213i	0.0591817	−	0.963185i
$306$		0			0
$307$	−10.3879	+	10.3879i	−0.592867	+	0.592867i	−0.938405	−	0.345538i	$-0.887697\pi$
$307$	−10.3879	+	10.3879i	−0.592867	+	0.592867i	0.345538	+	0.938405i	$0.387697\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	8.82105i		−	0.500196i	−0.968221	−	0.250098i	$-0.919537\pi$
$311$		−	8.82105i		−	0.500196i	0.968221	−	0.250098i	$-0.0804629\pi$
$312$		0			0
$313$	−3.23743	−	3.23743i	−0.182990	−	0.182990i	0.609667	−	0.792657i	$-0.291303\pi$
$313$	−3.23743	−	3.23743i	−0.182990	−	0.182990i	−0.792657	+	0.609667i	$0.791303\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.0891011	+	0.0891011i	0.00500442	+	0.00500442i	0.709605	−	0.704600i	$-0.248874\pi$
$317$	0.0891011	+	0.0891011i	0.00500442	+	0.00500442i	−0.704600	+	0.709605i	$0.748874\pi$
$318$		0			0
$319$		−	42.8627i		−	2.39985i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.37890	−	8.37890i	0.466214	−	0.466214i
$324$		0			0
$325$	0.843675	+	1.08110i	0.0467987	+	0.0599688i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−10.3603			−0.571181
$330$		0			0
$331$	−20.6859			−1.13700			−0.568501	−	0.822683i	$-0.692476\pi$
$331$	−20.6859			−1.13700			−0.568501	+	0.822683i	$0.692476\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9.03382	−	10.2166i	−0.493570	−	0.558195i
$336$		0			0
$337$	23.4387	−	23.4387i	1.27679	−	1.27679i	0.334328	−	0.942457i	$-0.391491\pi$
$337$	23.4387	−	23.4387i	1.27679	−	1.27679i	0.942457	−	0.334328i	$-0.108509\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			25.3495i			1.37275i
$342$		0			0
$343$	−10.2374	−	10.2374i	−0.552769	−	0.552769i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.55978	+	4.55978i	0.244782	+	0.244782i	0.818825	−	0.574043i	$-0.194626\pi$
$347$	4.55978	+	4.55978i	0.244782	+	0.244782i	−0.574043	+	0.818825i	$0.694626\pi$
$348$		0			0
$349$			8.46310i			0.453019i	0.974009	+	0.226510i	$0.0727315\pi$
$349$			8.46310i			0.453019i	−0.974009	+	0.226510i	$0.927269\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.9690	−	10.9690i	0.583819	−	0.583819i	−0.352131	−	0.935951i	$-0.614543\pi$
$353$	10.9690	−	10.9690i	0.583819	−	0.583819i	0.935951	+	0.352131i	$0.114543\pi$
$354$		0			0
$355$	21.1490	+	1.29948i	1.12247	+	0.0689691i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−24.9073			−1.31456			−0.657279	−	0.753648i	$-0.728293\pi$
$359$	−24.9073			−1.31456			−0.657279	+	0.753648i	$0.728293\pi$
$360$		0			0
$361$	13.2981			0.699898
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.0891011	−	0.0787855i	0.00466377	−	0.00412382i
$366$		0			0
$367$	−21.7440	+	21.7440i	−1.13503	+	1.13503i	−0.145699	+	0.989329i	$0.546543\pi$
$367$	−21.7440	+	21.7440i	−1.13503	+	1.13503i	−0.989329	+	0.145699i	$0.953457\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			7.49459i			0.389100i
$372$		0			0
$373$	16.1695	+	16.1695i	0.837225	+	0.837225i	0.988493	−	0.151268i	$-0.0483356\pi$
$373$	16.1695	+	16.1695i	0.837225	+	0.837225i	−0.151268	+	0.988493i	$0.548336\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.41421	+	1.41421i	0.0728357	+	0.0728357i
$378$		0			0
$379$		−	18.7005i		−	0.960581i	−0.877109	−	0.480291i	$-0.840531\pi$
$379$		−	18.7005i		−	0.960581i	0.877109	−	0.480291i	$-0.159469\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	25.0220	−	25.0220i	1.27856	−	1.27856i	0.337094	−	0.941471i	$-0.390556\pi$
$383$	25.0220	−	25.0220i	1.27856	−	1.27856i	0.941471	−	0.337094i	$-0.109444\pi$
$384$		0			0
$385$	−11.2243	+	9.92478i	−0.572041	+	0.505813i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−16.1478			−0.818724			−0.409362	−	0.912372i	$-0.634249\pi$
$389$	−16.1478			−0.818724			−0.409362	+	0.912372i	$0.634249\pi$
$390$		0			0
$391$	−29.4010			−1.48687
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.22964	−	0.505660i	−0.414078	−	0.0254425i
$396$		0			0
$397$	−15.4690	+	15.4690i	−0.776366	+	0.776366i	−0.979211	−	0.202845i	$-0.934981\pi$
$397$	−15.4690	+	15.4690i	−0.776366	+	0.776366i	0.202845	+	0.979211i	$0.434981\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	25.1181i		−	1.25434i	−0.778884	−	0.627168i	$-0.784214\pi$
$401$		−	25.1181i		−	1.25434i	0.778884	−	0.627168i	$-0.215786\pi$
$402$		0			0
$403$	−0.836381	−	0.836381i	−0.0416631	−	0.0416631i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.07669	+	9.07669i	0.449915	+	0.449915i
$408$		0			0
$409$		−	9.92478i		−	0.490749i	−0.969428	−	0.245374i	$-0.921089\pi$
$409$		−	9.92478i		−	0.490749i	0.969428	−	0.245374i	$-0.0789109\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.41346	−	8.41346i	0.413999	−	0.413999i
$414$		0			0
$415$	0.463096	+	0.523730i	0.0227325	+	0.0257089i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.06075			0.344940			0.172470	−	0.985015i	$-0.444825\pi$
$419$	7.06075			0.344940			0.172470	+	0.985015i	$0.444825\pi$
$420$		0			0
$421$	10.7757			0.525178			0.262589	−	0.964908i	$-0.415424\pi$
$421$	10.7757			0.525178			0.262589	+	0.964908i	$0.415424\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−15.2648	−	19.5606i	−0.740452	−	0.948830i
$426$		0			0
$427$	−6.07522	+	6.07522i	−0.294001	+	0.294001i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	2.19414i		−	0.105688i	−0.998603	−	0.0528440i	$-0.983171\pi$
$431$		−	2.19414i		−	0.105688i	0.998603	−	0.0528440i	$-0.0168286\pi$
$432$		0			0
$433$	16.4617	+	16.4617i	0.791098	+	0.791098i	0.981673	−	0.190575i	$-0.0610352\pi$
$433$	16.4617	+	16.4617i	0.791098	+	0.791098i	−0.190575	+	0.981673i	$0.561035\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	10.0039	+	10.0039i	0.478550	+	0.478550i
$438$		0			0
$439$			26.3996i			1.25999i	0.776601	+	0.629993i	$0.216942\pi$
$439$			26.3996i			1.25999i	−0.776601	+	0.629993i	$0.783058\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.37696	−	3.37696i	0.160444	−	0.160444i	−0.622319	−	0.782764i	$-0.713809\pi$
$443$	3.37696	−	3.37696i	0.160444	−	0.160444i	0.782764	+	0.622319i	$0.213809\pi$
$444$		0			0
$445$	1.30677	−	21.2677i	0.0619469	−	1.00819i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	21.5490			1.01696			0.508480	−	0.861074i	$-0.330208\pi$
$449$	21.5490			1.01696			0.508480	+	0.861074i	$0.330208\pi$
$450$		0			0
$451$	−19.5369			−0.919957
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.0428749	−	0.697791i	0.00201001	−	0.0327130i
$456$		0			0
$457$	14.9003	−	14.9003i	0.697008	−	0.697008i	−0.266756	−	0.963764i	$-0.585952\pi$
$457$	14.9003	−	14.9003i	0.697008	−	0.697008i	0.963764	+	0.266756i	$0.0859517\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			13.0208i			0.606440i	0.952921	+	0.303220i	$0.0980617\pi$
$461$			13.0208i			0.606440i	−0.952921	+	0.303220i	$0.901938\pi$
$462$		0			0
$463$	−8.28233	−	8.28233i	−0.384913	−	0.384913i	0.487956	−	0.872868i	$-0.337743\pi$
$463$	−8.28233	−	8.28233i	−0.384913	−	0.384913i	−0.872868	+	0.487956i	$0.837743\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9.25489	+	9.25489i	0.428265	+	0.428265i	0.888037	−	0.459772i	$-0.152069\pi$
$467$	9.25489	+	9.25489i	0.428265	+	0.428265i	−0.459772	+	0.888037i	$0.652069\pi$
$468$		0			0
$469$			6.95254i			0.321038i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−48.8612	+	48.8612i	−2.24664	+	2.24664i
$474$		0			0
$475$	−1.46168	+	11.8496i	−0.0670665	+	0.543695i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−14.0358			−0.641310			−0.320655	−	0.947196i	$-0.603903\pi$
$479$	−14.0358			−0.641310			−0.320655	+	0.947196i	$0.603903\pi$
$480$		0			0
$481$	−0.598953			−0.0273099
$482$		0			0
$483$		0			0
$484$		0			0
$485$	17.3339	+	19.6035i	0.787093	+	0.890149i
$486$		0			0
$487$	11.0435	−	11.0435i	0.500428	−	0.500428i	−0.411143	−	0.911571i	$-0.634870\pi$
$487$	11.0435	−	11.0435i	0.500428	−	0.500428i	0.911571	+	0.411143i	$0.134870\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			8.07207i			0.364288i	0.983272	+	0.182144i	$0.0583036\pi$
$491$			8.07207i			0.364288i	−0.983272	+	0.182144i	$0.941696\pi$
$492$		0			0
$493$	−25.5877	−	25.5877i	−1.15241	−	1.15241i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.63823	−	7.63823i	−0.342622	−	0.342622i
$498$		0			0
$499$		−	42.0870i		−	1.88407i	−0.335512	−	0.942036i	$-0.608909\pi$
$499$		−	42.0870i		−	1.88407i	0.335512	−	0.942036i	$-0.391091\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−26.1191	+	26.1191i	−1.16459	+	1.16459i	−0.181134	+	0.983459i	$0.557977\pi$
$503$	−26.1191	+	26.1191i	−1.16459	+	1.16459i	−0.983459	+	0.181134i	$0.942023\pi$
$504$		0			0
$505$	−13.0000	−	0.798769i	−0.578492	−	0.0355448i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.7362			0.564524			0.282262	−	0.959337i	$-0.408915\pi$
$509$	12.7362			0.564524			0.282262	+	0.959337i	$0.408915\pi$
$510$		0			0
$511$	−0.0606343			−0.00268231
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.72868	−	5.06544i	0.252436	−	0.223210i
$516$		0			0
$517$	−37.7743	+	37.7743i	−1.66131	+	1.66131i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	29.9279i		−	1.31116i	−0.755124	−	0.655582i	$-0.772423\pi$
$521$		−	29.9279i		−	1.31116i	0.755124	−	0.655582i	$-0.227577\pi$
$522$		0			0
$523$	16.3733	+	16.3733i	0.715954	+	0.715954i	0.967774	−	0.251820i	$-0.0810292\pi$
$523$	16.3733	+	16.3733i	0.715954	+	0.715954i	−0.251820	+	0.967774i	$0.581029\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	15.1328	+	15.1328i	0.659196	+	0.659196i
$528$		0			0
$529$		−	12.1030i		−	0.526217i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.644601	−	0.644601i	0.0279208	−	0.0279208i
$534$		0			0
$535$	27.3258	−	24.1622i	1.18140	−	1.04462i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−33.5073			−1.44326
$540$		0			0
$541$	−19.6385			−0.844324			−0.422162	−	0.906520i	$-0.638729\pi$
$541$	−19.6385			−0.844324			−0.422162	+	0.906520i	$0.638729\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−24.5799	−	1.51028i	−1.05289	−	0.0646932i
$546$		0			0
$547$	−15.0738	+	15.0738i	−0.644509	+	0.644509i	−0.951661	−	0.307151i	$-0.900624\pi$
$547$	−15.0738	+	15.0738i	−0.644509	+	0.644509i	0.307151	+	0.951661i	$0.400624\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			17.4127i			0.741807i
$552$		0			0
$553$	2.97224	+	2.97224i	0.126392	+	0.126392i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	13.8609	+	13.8609i	0.587305	+	0.587305i	0.936901	−	0.349595i	$-0.113681\pi$
$557$	13.8609	+	13.8609i	0.587305	+	0.587305i	−0.349595	+	0.936901i	$0.613681\pi$
$558$		0			0
$559$		−	3.22425i		−	0.136371i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.81912	+	3.81912i	−0.160957	+	0.160957i	−0.782990	−	0.622034i	$-0.786307\pi$
$563$	−3.81912	+	3.81912i	−0.160957	+	0.160957i	0.622034	+	0.782990i	$0.286307\pi$
$564$		0			0
$565$	15.0884	+	17.0640i	0.634774	+	0.717886i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	14.3735			0.602570			0.301285	−	0.953534i	$-0.402585\pi$
$569$	14.3735			0.602570			0.301285	+	0.953534i	$0.402585\pi$
$570$		0			0
$571$	21.3112			0.891847			0.445924	−	0.895071i	$-0.352875\pi$
$571$	21.3112			0.891847			0.445924	+	0.895071i	$0.352875\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	23.3542	−	18.2252i	0.973936	−	0.760044i
$576$		0			0
$577$	8.55149	−	8.55149i	0.356003	−	0.356003i	−0.506334	−	0.862337i	$-0.669000\pi$
$577$	8.55149	−	8.55149i	0.356003	−	0.356003i	0.862337	+	0.506334i	$0.169000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	0.356404i		−	0.0147861i
$582$		0			0
$583$	27.3258	+	27.3258i	1.13172	+	1.13172i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.7176	−	12.7176i	−0.524912	−	0.524912i	0.394139	−	0.919051i	$-0.371043\pi$
$587$	−12.7176	−	12.7176i	−0.524912	−	0.524912i	−0.919051	+	0.394139i	$0.871043\pi$
$588$		0			0
$589$		−	10.2981i		−	0.424324i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	23.5905	−	23.5905i	0.968746	−	0.968746i	−0.0307800	−	0.999526i	$-0.509799\pi$
$593$	23.5905	−	23.5905i	0.968746	−	0.968746i	0.999526	+	0.0307800i	$0.00979912\pi$
$594$		0			0
$595$	−0.775746	+	12.6253i	−0.0318025	+	0.517587i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	25.7059			1.05031			0.525157	−	0.851006i	$-0.324007\pi$
$599$	25.7059			1.05031			0.525157	+	0.851006i	$0.324007\pi$
$600$		0			0
$601$	7.74798			0.316047			0.158023	−	0.987435i	$-0.449488\pi$
$601$	7.74798			0.316047			0.158023	+	0.987435i	$0.449488\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.22951	+	52.5605i	−0.131298	+	2.13689i
$606$		0			0
$607$	3.04349	−	3.04349i	0.123531	−	0.123531i	−0.642638	−	0.766170i	$-0.722160\pi$
$607$	3.04349	−	3.04349i	0.123531	−	0.123531i	0.766170	+	0.642638i	$0.222160\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	2.49265i		−	0.100842i
$612$		0			0
$613$	8.24472	+	8.24472i	0.333001	+	0.333001i	0.853725	−	0.520724i	$-0.174338\pi$
$613$	8.24472	+	8.24472i	0.333001	+	0.333001i	−0.520724	+	0.853725i	$0.674338\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−27.4628	−	27.4628i	−1.10561	−	1.10561i	−0.993720	−	0.111891i	$-0.964309\pi$
$617$	−27.4628	−	27.4628i	−1.10561	−	1.10561i	−0.111891	−	0.993720i	$-0.535691\pi$
$618$		0			0
$619$			24.7269i			0.993857i	0.867792	+	0.496928i	$0.165539\pi$
$619$			24.7269i			0.993857i	−0.867792	+	0.496928i	$0.834461\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.68111	+	7.68111i	−0.307737	+	0.307737i
$624$		0			0
$625$	24.2506	+	6.07522i	0.970024	+	0.243009i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	10.8370			0.432099
$630$		0			0
$631$	40.7875			1.62372			0.811862	−	0.583849i	$-0.198454\pi$
$631$	40.7875			1.62372			0.811862	+	0.583849i	$0.198454\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.4463	+	20.8615i	0.732018	+	0.827863i
$636$		0			0
$637$	1.10554	−	1.10554i	0.0438031	−	0.0438031i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	42.0747i		−	1.66185i	−0.556384	−	0.830925i	$-0.687812\pi$
$641$		−	42.0747i		−	1.66185i	0.556384	−	0.830925i	$-0.312188\pi$
$642$		0			0
$643$	15.8496	+	15.8496i	0.625045	+	0.625045i	0.946817	−	0.321772i	$-0.104278\pi$
$643$	15.8496	+	15.8496i	0.625045	+	0.625045i	−0.321772	+	0.946817i	$0.604278\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.95243	−	1.95243i	−0.0767581	−	0.0767581i	0.667685	−	0.744444i	$-0.267285\pi$
$647$	−1.95243	−	1.95243i	−0.0767581	−	0.0767581i	−0.744444	+	0.667685i	$0.767285\pi$
$648$		0			0
$649$		−	61.3522i		−	2.40828i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	23.1587	−	23.1587i	0.906269	−	0.906269i	−0.0896999	−	0.995969i	$-0.528591\pi$
$653$	23.1587	−	23.1587i	0.906269	−	0.906269i	0.995969	+	0.0896999i	$0.0285908\pi$
$654$		0			0
$655$	30.8324	+	1.89446i	1.20472	+	0.0740227i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4.12594			−0.160724			−0.0803620	−	0.996766i	$-0.525608\pi$
$659$	−4.12594			−0.160724			−0.0803620	+	0.996766i	$0.525608\pi$
$660$		0			0
$661$	14.9116			0.579994			0.289997	−	0.957028i	$-0.406346\pi$
$661$	14.9116			0.579994			0.289997	+	0.957028i	$0.406346\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.55978	−	4.03188i	0.176821	−	0.156350i
$666$		0			0
$667$	30.5501	−	30.5501i	1.18290	−	1.18290i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			44.3014i			1.71024i
$672$		0			0
$673$	29.9380	+	29.9380i	1.15402	+	1.15402i	0.985738	+	0.168285i	$0.0538228\pi$
$673$	29.9380	+	29.9380i	1.15402	+	1.15402i	0.168285	+	0.985738i	$0.446177\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−24.9964	−	24.9964i	−0.960690	−	0.960690i	0.0385660	−	0.999256i	$-0.487721\pi$
$677$	−24.9964	−	24.9964i	−0.960690	−	0.960690i	−0.999256	+	0.0385660i	$0.987721\pi$
$678$		0			0
$679$		−	13.3404i		−	0.511958i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9.34064	+	9.34064i	−0.357410	+	0.357410i	−0.862857	−	0.505448i	$-0.831327\pi$
$683$	−9.34064	+	9.34064i	−0.357410	+	0.357410i	0.505448	+	0.862857i	$0.331327\pi$
$684$		0			0
$685$	6.26187	−	5.53690i	0.239254	−	0.211554i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.80318			−0.0686956
$690$		0			0
$691$	22.2374			0.845952			0.422976	−	0.906141i	$-0.360986\pi$
$691$	22.2374			0.845952			0.422976	+	0.906141i	$0.360986\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−41.7369	−	2.56447i	−1.58317	−	0.0972761i
$696$		0			0
$697$	−11.6629	+	11.6629i	−0.441764	+	0.441764i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			21.1358i			0.798287i	0.916888	+	0.399143i	$0.130692\pi$
$701$			21.1358i			0.798287i	−0.916888	+	0.399143i	$0.869308\pi$
$702$		0			0
$703$	−3.68735	−	3.68735i	−0.139071	−	0.139071i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.69511	+	4.69511i	0.176578	+	0.176578i
$708$		0			0
$709$			38.4749i			1.44495i	0.691395	+	0.722477i	$0.256996\pi$
$709$			38.4749i			1.44495i	−0.691395	+	0.722477i	$0.743004\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−18.0676	+	18.0676i	−0.676638	+	0.676638i
$714$		0			0
$715$	−2.38787	−	2.70052i	−0.0893014	−	0.100994i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.81912			0.142429			0.0712145	−	0.997461i	$-0.477313\pi$
$719$	3.81912			0.142429			0.0712145	+	0.997461i	$0.477313\pi$
$720$		0			0
$721$	−3.89843			−0.145185
$722$		0			0
$723$		0			0
$724$		0			0
$725$	36.1865	+	4.46372i	1.34393	+	0.165778i
$726$		0			0
$727$	−30.8930	+	30.8930i	−1.14576	+	1.14576i	−0.158382	+	0.987378i	$0.550628\pi$
$727$	−30.8930	+	30.8930i	−1.14576	+	1.14576i	−0.987378	+	0.158382i	$0.949372\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			58.3372i			2.15768i
$732$		0			0
$733$	−25.6556	−	25.6556i	−0.947612	−	0.947612i	0.0510826	−	0.998694i	$-0.483733\pi$
$733$	−25.6556	−	25.6556i	−0.947612	−	0.947612i	−0.998694	+	0.0510826i	$0.983733\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	25.3495	+	25.3495i	0.933759	+	0.933759i
$738$		0			0
$739$		−	25.9657i		−	0.955164i	−0.878587	−	0.477582i	$-0.841513\pi$
$739$		−	25.9657i		−	0.955164i	0.878587	−	0.477582i	$-0.158487\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−11.4284	+	11.4284i	−0.419268	+	0.419268i	−0.884951	−	0.465684i	$-0.845808\pi$
$743$	−11.4284	+	11.4284i	−0.419268	+	0.419268i	0.465684	+	0.884951i	$0.345808\pi$
$744$		0			0
$745$	1.03620	−	16.8641i	0.0379633	−	0.617854i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−18.5955			−0.679466
$750$		0			0
$751$	20.9380			0.764037			0.382018	−	0.924155i	$-0.375229\pi$
$751$	20.9380			0.764037			0.382018	+	0.924155i	$0.375229\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.35171	+	38.2742i	−0.0855876	+	1.39294i
$756$		0			0
$757$	−16.1187	+	16.1187i	−0.585845	+	0.585845i	−0.936503	−	0.350659i	$-0.885958\pi$
$757$	−16.1187	+	16.1187i	−0.585845	+	0.585845i	0.350659	+	0.936503i	$0.385958\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			31.4877i			1.14143i	0.821148	+	0.570715i	$0.193334\pi$
$761$			31.4877i			1.14143i	−0.821148	+	0.570715i	$0.806666\pi$
$762$		0			0
$763$	8.87732	+	8.87732i	0.321380	+	0.321380i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.02425	+	2.02425i	0.0730916	+	0.0730916i
$768$		0			0
$769$			3.47627i			0.125357i	0.998034	+	0.0626787i	$0.0199644\pi$
$769$			3.47627i			0.125357i	−0.998034	+	0.0626787i	$0.980036\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.1089	+	12.1089i	−0.435527	+	0.435527i	−0.890504	−	0.454976i	$-0.849648\pi$
$773$	−12.1089	+	12.1089i	−0.435527	+	0.435527i	0.454976	+	0.890504i	$0.349648\pi$
$774$		0			0
$775$	−21.4010	−	2.63989i	−0.768748	−	0.0948276i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7.93675			0.284363
$780$		0			0
$781$	−55.6991			−1.99307
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−18.2099	−	20.5942i	−0.649940	−	0.735038i
$786$		0			0
$787$	19.7889	−	19.7889i	0.705399	−	0.705399i	−0.260165	−	0.965564i	$-0.583777\pi$
$787$	19.7889	−	19.7889i	0.705399	−	0.705399i	0.965564	+	0.260165i	$0.0837771\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	11.6122i		−	0.412883i
$792$		0			0
$793$	−1.46168	−	1.46168i	−0.0519058	−	0.0519058i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12.4896	−	12.4896i	−0.442403	−	0.442403i	0.450416	−	0.892819i	$-0.351276\pi$
$797$	−12.4896	−	12.4896i	−0.442403	−	0.442403i	−0.892819	+	0.450416i	$0.851276\pi$
$798$		0			0
$799$			45.1002i			1.59553i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.221077	+	0.221077i	−0.00780164	+	0.00780164i
$804$		0			0
$805$	−15.0738	−	0.926192i	−0.531282	−	0.0326440i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.50860			−0.228830			−0.114415	−	0.993433i	$-0.536499\pi$
$809$	−6.50860			−0.228830			−0.114415	+	0.993433i	$0.536499\pi$
$810$		0			0
$811$	−20.3733			−0.715403			−0.357701	−	0.933836i	$-0.616439\pi$
$811$	−20.3733			−0.715403			−0.357701	+	0.933836i	$0.616439\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10.2166	+	9.03382i	−0.357873	+	0.316441i
$816$		0			0
$817$	19.8496	−	19.8496i	0.694448	−	0.694448i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			14.7867i			0.516061i	0.966137	+	0.258030i	$0.0830735\pi$
$821$			14.7867i			0.516061i	−0.966137	+	0.258030i	$0.916927\pi$
$822$		0			0
$823$	−18.5198	−	18.5198i	−0.645558	−	0.645558i	0.306358	−	0.951916i	$-0.400890\pi$
$823$	−18.5198	−	18.5198i	−0.645558	−	0.645558i	−0.951916	+	0.306358i	$0.900890\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	27.6294	+	27.6294i	0.960767	+	0.960767i	0.999259	−	0.0384921i	$-0.0122554\pi$
$827$	27.6294	+	27.6294i	0.960767	+	0.960767i	−0.0384921	+	0.999259i	$0.512255\pi$
$828$		0			0
$829$			49.4617i			1.71788i	0.512080	+	0.858938i	$0.328875\pi$
$829$			49.4617i			1.71788i	−0.512080	+	0.858938i	$0.671125\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−20.0028	+	20.0028i	−0.693055	+	0.693055i
$834$		0			0
$835$	−4.52373	+	4.00000i	−0.156550	+	0.138426i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.39558			0.0481809			0.0240904	−	0.999710i	$-0.492331\pi$
$839$	1.39558			0.0481809			0.0240904	+	0.999710i	$0.492331\pi$
$840$		0			0
$841$	24.1754			0.833634
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−28.8463	−	1.77242i	−0.992342	−	0.0609733i
$846$		0			0
$847$	18.9829	−	18.9829i	0.652259	−	0.652259i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			12.9387i			0.443532i
$852$		0			0
$853$	−15.9927	−	15.9927i	−0.547580	−	0.547580i	0.378160	−	0.925740i	$-0.376557\pi$
$853$	−15.9927	−	15.9927i	−0.547580	−	0.547580i	−0.925740	+	0.378160i	$0.876557\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.1829	+	25.1829i	0.860232	+	0.860232i	0.991365	−	0.131132i	$-0.0418613\pi$
$857$	25.1829	+	25.1829i	0.860232	+	0.860232i	−0.131132	+	0.991365i	$0.541861\pi$
$858$		0			0
$859$			14.9722i			0.510846i	0.966829	+	0.255423i	$0.0822148\pi$
$859$			14.9722i			0.510846i	−0.966829	+	0.255423i	$0.917785\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−37.4757	+	37.4757i	−1.27569	+	1.27569i	−0.332628	+	0.943058i	$0.607935\pi$
$863$	−37.4757	+	37.4757i	−1.27569	+	1.27569i	−0.943058	+	0.332628i	$0.892065\pi$
$864$		0			0
$865$	2.46310	+	2.78560i	0.0837478	+	0.0947131i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	21.6740			0.735240
$870$		0			0
$871$	−1.67276			−0.0566794
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−7.21001	−	10.5095i	−0.243743	−	0.355287i
$876$		0			0
$877$	9.60483	−	9.60483i	0.324332	−	0.324332i	−0.526094	−	0.850426i	$-0.676344\pi$
$877$	9.60483	−	9.60483i	0.324332	−	0.324332i	0.850426	+	0.526094i	$0.176344\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	3.23802i		−	0.109092i	−0.998511	−	0.0545459i	$-0.982629\pi$
$881$		−	3.23802i		−	0.109092i	0.998511	−	0.0545459i	$-0.0173711\pi$
$882$		0			0
$883$	15.5369	+	15.5369i	0.522858	+	0.522858i	0.918434	−	0.395575i	$-0.129455\pi$
$883$	15.5369	+	15.5369i	0.522858	+	0.522858i	−0.395575	+	0.918434i	$0.629455\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	19.7644	+	19.7644i	0.663625	+	0.663625i	0.956233	−	0.292608i	$-0.0945231\pi$
$887$	19.7644	+	19.7644i	0.663625	+	0.663625i	−0.292608	+	0.956233i	$0.594523\pi$
$888$		0			0
$889$		−	14.1965i		−	0.476135i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	15.3456	−	15.3456i	0.513521	−	0.513521i
$894$		0			0
$895$	−1.20853	+	19.6688i	−0.0403965	+	0.657455i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−31.4485			−1.04887
$900$		0			0
$901$	32.6253			1.08691
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.16520	−	35.2386i	0.0719735	−	1.17137i
$906$		0			0
$907$	−0.836381	+	0.836381i	−0.0277716	+	0.0277716i	−0.720856	−	0.693085i	$-0.756251\pi$
$907$	−0.836381	+	0.836381i	−0.0277716	+	0.0277716i	0.693085	+	0.720856i	$0.256251\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	3.90487i		−	0.129374i	−0.997906	−	0.0646870i	$-0.979395\pi$
$911$		−	3.90487i		−	0.129374i	0.997906	−	0.0646870i	$-0.0206049\pi$
$912$		0			0
$913$	−1.29948	−	1.29948i	−0.0430064	−	0.0430064i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−11.1355	−	11.1355i	−0.367727	−	0.367727i
$918$		0			0
$919$		−	19.3865i		−	0.639500i	−0.947502	−	0.319750i	$-0.896401\pi$
$919$		−	19.3865i		−	0.639500i	0.947502	−	0.319750i	$-0.103599\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.83774	−	1.83774i	0.0604899	−	0.0604899i
$924$		0			0
$925$	−8.60816	+	6.71767i	−0.283034	+	0.220876i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	16.6328			0.545704			0.272852	−	0.962056i	$-0.412033\pi$
$929$	16.6328			0.545704			0.272852	+	0.962056i	$0.412033\pi$
$930$		0			0
$931$	13.6121			0.446119
$932$		0			0
$933$		0			0
$934$		0			0
$935$	43.2043	+	48.8612i	1.41293	+	1.59793i
$936$		0			0
$937$	40.1754	−	40.1754i	1.31247	−	1.31247i	0.392885	−	0.919588i	$-0.371477\pi$
$937$	40.1754	−	40.1754i	1.31247	−	1.31247i	0.919588	−	0.392885i	$-0.128523\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	10.2988i		−	0.335730i	−0.985810	−	0.167865i	$-0.946313\pi$
$941$		−	10.2988i		−	0.335730i	0.985810	−	0.167865i	$-0.0536873\pi$
$942$		0			0
$943$	−13.9248	−	13.9248i	−0.453453	−	0.453453i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	3.90487	+	3.90487i	0.126891	+	0.126891i	0.767700	−	0.640809i	$-0.221401\pi$
$947$	3.90487	+	3.90487i	0.126891	+	0.126891i	−0.640809	+	0.767700i	$0.721401\pi$
$948$		0			0
$949$		−	0.0145884i		−	0.000473561i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	10.2815	−	10.2815i	0.333050	−	0.333050i	−0.520693	−	0.853744i	$-0.674327\pi$
$953$	10.2815	−	10.2815i	0.333050	−	0.333050i	0.853744	+	0.520693i	$0.174327\pi$
$954$		0			0
$955$	4.10157	+	0.252016i	0.132724	+	0.00815505i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.26127			−0.137604
$960$		0			0
$961$	−12.4010			−0.400034
$962$		0			0
$963$		0			0
$964$		0			0
$965$	23.8993	−	21.1324i	0.769347	−	0.680277i
$966$		0			0
$967$	25.8945	−	25.8945i	0.832710	−	0.832710i	−0.155177	−	0.987887i	$-0.549595\pi$
$967$	25.8945	−	25.8945i	0.832710	−	0.832710i	0.987887	+	0.155177i	$0.0495948\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	43.1960i		−	1.38623i	−0.720829	−	0.693113i	$-0.756239\pi$
$971$		−	43.1960i		−	1.38623i	0.720829	−	0.693113i	$-0.243761\pi$
$972$		0			0
$973$	15.0738	+	15.0738i	0.483244	+	0.483244i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	21.5766	+	21.5766i	0.690296	+	0.690296i	0.962297	−	0.272001i	$-0.0876855\pi$
$977$	21.5766	+	21.5766i	0.690296	+	0.690296i	−0.272001	+	0.962297i	$0.587685\pi$
$978$		0			0
$979$			56.0118i			1.79014i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.9230	−	18.9230i	0.603550	−	0.603550i	−0.337703	−	0.941253i	$-0.609650\pi$
$983$	18.9230	−	18.9230i	0.603550	−	0.603550i	0.941253	+	0.337703i	$0.109650\pi$
$984$		0			0
$985$	−41.6893	+	36.8627i	−1.32833	+	1.17454i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−69.6509			−2.21477
$990$		0			0
$991$	−17.6531			−0.560768			−0.280384	−	0.959888i	$-0.590462\pi$
$991$	−17.6531			−0.560768			−0.280384	+	0.959888i	$0.590462\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−48.9330	−	3.00663i	−1.55128	−	0.0953166i
$996$		0			0
$997$	2.38058	−	2.38058i	0.0753937	−	0.0753937i	−0.668404	−	0.743798i	$-0.733022\pi$
$997$	2.38058	−	2.38058i	0.0753937	−	0.0753937i	0.743798	+	0.668404i	$0.233022\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.w.f.1313.4	yes	12
3.2	odd	2	inner	1440.2.w.f.1313.3	yes	12
4.3	odd	2		1440.2.w.g.1313.4	yes	12
5.2	odd	4	inner	1440.2.w.f.737.3	✓	12
8.3	odd	2		2880.2.w.q.2753.3		12
8.5	even	2		2880.2.w.p.2753.3		12
12.11	even	2		1440.2.w.g.1313.3	yes	12
15.2	even	4	inner	1440.2.w.f.737.4	yes	12
20.7	even	4		1440.2.w.g.737.3	yes	12
24.5	odd	2		2880.2.w.p.2753.4		12
24.11	even	2		2880.2.w.q.2753.4		12
40.27	even	4		2880.2.w.q.2177.4		12
40.37	odd	4		2880.2.w.p.2177.4		12
60.47	odd	4		1440.2.w.g.737.4	yes	12
120.77	even	4		2880.2.w.p.2177.3		12
120.107	odd	4		2880.2.w.q.2177.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.w.f.737.3	✓	12	5.2	odd	4	inner
1440.2.w.f.737.4	yes	12	15.2	even	4	inner
1440.2.w.f.1313.3	yes	12	3.2	odd	2	inner
1440.2.w.f.1313.4	yes	12	1.1	even	1	trivial
1440.2.w.g.737.3	yes	12	20.7	even	4
1440.2.w.g.737.4	yes	12	60.47	odd	4
1440.2.w.g.1313.3	yes	12	12.11	even	2
1440.2.w.g.1313.4	yes	12	4.3	odd	2
2880.2.w.p.2177.3		12	120.77	even	4
2880.2.w.p.2177.4		12	40.37	odd	4
2880.2.w.p.2753.3		12	8.5	even	2
2880.2.w.p.2753.4		12	24.5	odd	2
2880.2.w.q.2177.3		12	120.107	odd	4
2880.2.w.q.2177.4		12	40.27	even	4
2880.2.w.q.2753.3		12	8.3	odd	2
2880.2.w.q.2753.4		12	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.w (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.137134 - 2.23186i) q^{5} +(-0.806063 + 0.806063i) q^{7} +O(q^{10})\)	\(q+(0.137134 - 2.23186i) q^{5} +(-0.806063 + 0.806063i) q^{7} +5.87793i q^{11} +(-0.193937 - 0.193937i) q^{13} +(3.50894 + 3.50894i) q^{17} -2.38787i q^{19} +(-4.18945 + 4.18945i) q^{23} +(-4.96239 - 0.612127i) q^{25} -7.29214 q^{29} +4.31265 q^{31} +(1.68848 + 1.90956i) q^{35} +(1.54420 - 1.54420i) q^{37} +3.32377i q^{41} +(8.31265 + 8.31265i) q^{43} +(6.42647 + 6.42647i) q^{47} +5.70052i q^{49} +(4.64888 - 4.64888i) q^{53} +(13.1187 + 0.806063i) q^{55} -10.4377 q^{59} +7.53690 q^{61} +(-0.459434 + 0.406244i) q^{65} +(4.31265 - 4.31265i) q^{67} +9.47597i q^{71} +(0.0376114 + 0.0376114i) q^{73} +(-4.73799 - 4.73799i) q^{77} -3.68735i q^{79} +(-0.221077 + 0.221077i) q^{83} +(8.31265 - 7.35026i) q^{85} +9.52916 q^{89} +0.312650 q^{91} +(-5.32940 - 0.327458i) q^{95} +(-8.27504 + 8.27504i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{7}+O(q^{10}) \) 12 * q - 8 * q^7	\( 12 q - 8 q^{7} - 4 q^{13} - 16 q^{25} - 32 q^{31} - 20 q^{37} + 16 q^{43} + 72 q^{55} - 32 q^{67} + 44 q^{73} + 16 q^{85} - 80 q^{91} + 28 q^{97}+O(q^{100}) \) 12 * q - 8 * q^7 - 4 * q^13 - 16 * q^25 - 32 * q^31 - 20 * q^37 + 16 * q^43 + 72 * q^55 - 32 * q^67 + 44 * q^73 + 16 * q^85 - 80 * q^91 + 28 * q^97

Introduction

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