LMFDB - Embedded newform 4032.2.v.e.3599.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("4032.1583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.v (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3599.14
Character	$\chi$	$=$	4032.3599
Dual form			4032.2.v.e.1583.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.925496 - 0.925496i) q^{5} -1.00000 q^{7} +O(q^{10})$	$q+(0.925496 - 0.925496i) q^{5} -1.00000 q^{7} +(1.72403 + 1.72403i) q^{11} +(0.328273 - 0.328273i) q^{13} -2.34181i q^{17} +(1.77976 + 1.77976i) q^{19} +6.17143i q^{23} +3.28691i q^{25} +(-0.122671 - 0.122671i) q^{29} +1.74700i q^{31} +(-0.925496 + 0.925496i) q^{35} +(-1.68105 - 1.68105i) q^{37} +2.88812 q^{41} +(-2.77330 + 2.77330i) q^{43} -5.92184 q^{47} +1.00000 q^{49} +(-0.973689 + 0.973689i) q^{53} +3.19117 q^{55} +(8.33124 + 8.33124i) q^{59} +(4.28808 - 4.28808i) q^{61} -0.607630i q^{65} +(-1.78259 - 1.78259i) q^{67} +8.57053i q^{71} -6.41750i q^{73} +(-1.72403 - 1.72403i) q^{77} +5.38299i q^{79} +(3.46360 - 3.46360i) q^{83} +(-2.16734 - 2.16734i) q^{85} -1.51391 q^{89} +(-0.328273 + 0.328273i) q^{91} +3.29433 q^{95} +15.7177 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 40 q^{7}+O(q^{10}) $ 40 * q - 40 * q^7	$ 40 q - 40 q^{7} - 24 q^{13} + 32 q^{19} - 8 q^{37} - 32 q^{43} + 40 q^{49} - 48 q^{55} - 24 q^{61} + 64 q^{85} + 24 q^{91} + 64 q^{97}+O(q^{100}) $ 40 * q - 40 * q^7 - 24 * q^13 + 32 * q^19 - 8 * q^37 - 32 * q^43 + 40 * q^49 - 48 * q^55 - 24 * q^61 + 64 * q^85 + 24 * q^91 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.925496	−	0.925496i	0.413894	−	0.413894i	−0.469198	−	0.883093i	$-0.655457\pi$
$5$	0.925496	−	0.925496i	0.413894	−	0.413894i	0.883093	+	0.469198i	$0.155457\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.72403	+	1.72403i	0.519815	+	0.519815i	0.917515	−	0.397701i	$-0.130192\pi$
$11$	1.72403	+	1.72403i	0.519815	+	0.519815i	−0.397701	+	0.917515i	$0.630192\pi$
$12$		0			0
$13$	0.328273	−	0.328273i	0.0910464	−	0.0910464i	−0.660117	−	0.751163i	$-0.729493\pi$
$13$	0.328273	−	0.328273i	0.0910464	−	0.0910464i	0.751163	+	0.660117i	$0.229493\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.34181i		−	0.567973i	−0.958828	−	0.283986i	$-0.908343\pi$
$17$		−	2.34181i		−	0.567973i	0.958828	−	0.283986i	$-0.0916570\pi$
$18$		0			0
$19$	1.77976	+	1.77976i	0.408306	+	0.408306i	0.881147	−	0.472842i	$-0.156772\pi$
$19$	1.77976	+	1.77976i	0.408306	+	0.408306i	−0.472842	+	0.881147i	$0.656772\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.17143i			1.28683i	0.765517	+	0.643416i	$0.222483\pi$
$23$			6.17143i			1.28683i	−0.765517	+	0.643416i	$0.777517\pi$
$24$		0			0
$25$			3.28691i			0.657383i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.122671	−	0.122671i	−0.0227794	−	0.0227794i	0.695625	−	0.718405i	$-0.255127\pi$
$29$	−0.122671	−	0.122671i	−0.0227794	−	0.0227794i	−0.718405	+	0.695625i	$0.755127\pi$
$30$		0			0
$31$			1.74700i			0.313770i	0.987617	+	0.156885i	$0.0501452\pi$
$31$			1.74700i			0.313770i	−0.987617	+	0.156885i	$0.949855\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.925496	+	0.925496i	−0.156437	+	0.156437i
$36$		0			0
$37$	−1.68105	−	1.68105i	−0.276363	−	0.276363i	0.555292	−	0.831655i	$-0.312606\pi$
$37$	−1.68105	−	1.68105i	−0.276363	−	0.276363i	−0.831655	+	0.555292i	$0.812606\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.88812			0.451048			0.225524	−	0.974238i	$-0.427591\pi$
$41$	2.88812			0.451048			0.225524	+	0.974238i	$0.427591\pi$
$42$		0			0
$43$	−2.77330	+	2.77330i	−0.422924	+	0.422924i	−0.886209	−	0.463285i	$-0.846670\pi$
$43$	−2.77330	+	2.77330i	−0.422924	+	0.422924i	0.463285	+	0.886209i	$0.346670\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.92184			−0.863789			−0.431894	−	0.901924i	$-0.642155\pi$
$47$	−5.92184			−0.863789			−0.431894	+	0.901924i	$0.642155\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.973689	+	0.973689i	−0.133746	+	0.133746i	−0.770811	−	0.637064i	$-0.780149\pi$
$53$	−0.973689	+	0.973689i	−0.133746	+	0.133746i	0.637064	+	0.770811i	$0.280149\pi$
$54$		0			0
$55$	3.19117			0.430297
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.33124	+	8.33124i	1.08464	+	1.08464i	0.996070	+	0.0885647i	$0.0282280\pi$
$59$	8.33124	+	8.33124i	1.08464	+	1.08464i	0.0885647	+	0.996070i	$0.471772\pi$
$60$		0			0
$61$	4.28808	−	4.28808i	0.549033	−	0.549033i	−0.377128	−	0.926161i	$-0.623088\pi$
$61$	4.28808	−	4.28808i	0.549033	−	0.549033i	0.926161	+	0.377128i	$0.123088\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	0.607630i		−	0.0753672i
$66$		0			0
$67$	−1.78259	−	1.78259i	−0.217778	−	0.217778i	0.589783	−	0.807561i	$-0.299213\pi$
$67$	−1.78259	−	1.78259i	−0.217778	−	0.217778i	−0.807561	+	0.589783i	$0.799213\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			8.57053i			1.01713i	0.861022	+	0.508567i	$0.169825\pi$
$71$			8.57053i			1.01713i	−0.861022	+	0.508567i	$0.830175\pi$
$72$		0			0
$73$		−	6.41750i		−	0.751112i	−0.926800	−	0.375556i	$-0.877452\pi$
$73$		−	6.41750i		−	0.751112i	0.926800	−	0.375556i	$-0.122548\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.72403	−	1.72403i	−0.196471	−	0.196471i
$78$		0			0
$79$			5.38299i			0.605633i	0.953049	+	0.302817i	$0.0979270\pi$
$79$			5.38299i			0.605633i	−0.953049	+	0.302817i	$0.902073\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.46360	−	3.46360i	0.380179	−	0.380179i	−0.490987	−	0.871167i	$-0.663364\pi$
$83$	3.46360	−	3.46360i	0.380179	−	0.380179i	0.871167	+	0.490987i	$0.163364\pi$
$84$		0			0
$85$	−2.16734	−	2.16734i	−0.235081	−	0.235081i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.51391			−0.160474			−0.0802368	−	0.996776i	$-0.525568\pi$
$89$	−1.51391			−0.160474			−0.0802368	+	0.996776i	$0.525568\pi$
$90$		0			0
$91$	−0.328273	+	0.328273i	−0.0344123	+	0.0344123i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.29433			0.337991
$96$		0			0
$97$	15.7177			1.59589			0.797943	−	0.602732i	$-0.205921\pi$
$97$	15.7177			1.59589			0.797943	+	0.602732i	$0.205921\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−13.6438	+	13.6438i	−1.35760	+	1.35760i	−0.480744	+	0.876861i	$0.659633\pi$
$101$	−13.6438	+	13.6438i	−1.35760	+	1.35760i	−0.876861	+	0.480744i	$0.840367\pi$
$102$		0			0
$103$	11.9238			1.17489			0.587445	−	0.809264i	$-0.300134\pi$
$103$	11.9238			1.17489			0.587445	+	0.809264i	$0.300134\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.6108	+	13.6108i	1.31580	+	1.31580i	0.917064	+	0.398740i	$0.130553\pi$
$107$	13.6108	+	13.6108i	1.31580	+	1.31580i	0.398740	+	0.917064i	$0.369447\pi$
$108$		0			0
$109$	4.97865	−	4.97865i	0.476868	−	0.476868i	−0.427260	−	0.904129i	$-0.640521\pi$
$109$	4.97865	−	4.97865i	0.476868	−	0.476868i	0.904129	+	0.427260i	$0.140521\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			0.0768901i			0.00723322i	0.999993	+	0.00361661i	$0.00115120\pi$
$113$			0.0768901i			0.00723322i	−0.999993	+	0.00361661i	$0.998849\pi$
$114$		0			0
$115$	5.71163	+	5.71163i	0.532612	+	0.532612i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2.34181i			0.214673i
$120$		0			0
$121$		−	5.05544i		−	0.459585i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.66951	+	7.66951i	0.685981	+	0.685981i
$126$		0			0
$127$		−	15.0611i		−	1.33646i	−0.743956	−	0.668229i	$-0.767053\pi$
$127$		−	15.0611i		−	1.33646i	0.743956	−	0.668229i	$-0.232947\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.87027	+	3.87027i	−0.338147	+	0.338147i	−0.855670	−	0.517522i	$-0.826854\pi$
$131$	−3.87027	+	3.87027i	−0.338147	+	0.338147i	0.517522	+	0.855670i	$0.326854\pi$
$132$		0			0
$133$	−1.77976	−	1.77976i	−0.154325	−	0.154325i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.47491			−0.467753			−0.233877	−	0.972266i	$-0.575141\pi$
$137$	−5.47491			−0.467753			−0.233877	+	0.972266i	$0.575141\pi$
$138$		0			0
$139$	−11.1117	+	11.1117i	−0.942478	+	0.942478i	−0.998433	−	0.0559554i	$-0.982180\pi$
$139$	−11.1117	+	11.1117i	−0.942478	+	0.942478i	0.0559554	+	0.998433i	$0.482180\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.13190			0.0946545
$144$		0			0
$145$	−0.227063			−0.0188565
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.2234	+	10.2234i	−0.837535	+	0.837535i	−0.988534	−	0.150999i	$-0.951751\pi$
$149$	−10.2234	+	10.2234i	−0.837535	+	0.837535i	0.150999	+	0.988534i	$0.451751\pi$
$150$		0			0
$151$	−5.98993			−0.487454			−0.243727	−	0.969844i	$-0.578370\pi$
$151$	−5.98993			−0.487454			−0.243727	+	0.969844i	$0.578370\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.61684	+	1.61684i	0.129868	+	0.129868i
$156$		0			0
$157$	9.57922	−	9.57922i	0.764505	−	0.764505i	−0.212628	−	0.977133i	$-0.568202\pi$
$157$	9.57922	−	9.57922i	0.764505	−	0.764505i	0.977133	+	0.212628i	$0.0682023\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	6.17143i		−	0.486377i
$162$		0			0
$163$	7.81085	+	7.81085i	0.611793	+	0.611793i	0.943413	−	0.331620i	$-0.107595\pi$
$163$	7.81085	+	7.81085i	0.611793	+	0.611793i	−0.331620	+	0.943413i	$0.607595\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	5.28108i		−	0.408662i	−0.978902	−	0.204331i	$-0.934498\pi$
$167$		−	5.28108i		−	0.408662i	0.978902	−	0.204331i	$-0.0655019\pi$
$168$		0			0
$169$			12.7845i			0.983421i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.8893	+	11.8893i	0.903929	+	0.903929i	0.995773	−	0.0918444i	$-0.0292762\pi$
$173$	11.8893	+	11.8893i	0.903929	+	0.903929i	−0.0918444	+	0.995773i	$0.529276\pi$
$174$		0			0
$175$		−	3.28691i		−	0.248467i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.38837	+	6.38837i	−0.477489	+	0.477489i	−0.904328	−	0.426839i	$-0.859627\pi$
$179$	−6.38837	+	6.38837i	−0.477489	+	0.477489i	0.426839	+	0.904328i	$0.359627\pi$
$180$		0			0
$181$	5.56367	+	5.56367i	0.413545	+	0.413545i	0.882971	−	0.469427i	$-0.155539\pi$
$181$	5.56367	+	5.56367i	0.413545	+	0.413545i	−0.469427	+	0.882971i	$0.655539\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.11161			−0.228770
$186$		0			0
$187$	4.03735	−	4.03735i	0.295240	−	0.295240i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.43241			−0.176003			−0.0880014	−	0.996120i	$-0.528048\pi$
$191$	−2.43241			−0.176003			−0.0880014	+	0.996120i	$0.528048\pi$
$192$		0			0
$193$	24.8507			1.78879			0.894397	−	0.447273i	$-0.147605\pi$
$193$	24.8507			1.78879			0.894397	+	0.447273i	$0.147605\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.77151	−	1.77151i	0.126215	−	0.126215i	−0.641178	−	0.767393i	$-0.721554\pi$
$197$	1.77151	−	1.77151i	0.126215	−	0.126215i	0.767393	+	0.641178i	$0.221554\pi$
$198$		0			0
$199$	13.8970			0.985130			0.492565	−	0.870276i	$-0.336059\pi$
$199$	13.8970			0.985130			0.492565	+	0.870276i	$0.336059\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.122671	+	0.122671i	0.00860980	+	0.00860980i
$204$		0			0
$205$	2.67294	−	2.67294i	0.186686	−	0.186686i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			6.13673i			0.424487i
$210$		0			0
$211$	0.243974	+	0.243974i	0.0167959	+	0.0167959i	0.715455	−	0.698659i	$-0.246220\pi$
$211$	0.243974	+	0.243974i	0.0167959	+	0.0167959i	−0.698659	+	0.715455i	$0.746220\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			5.13335i			0.350091i
$216$		0			0
$217$		−	1.74700i		−	0.118594i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.768752	−	0.768752i	−0.0517119	−	0.0517119i
$222$		0			0
$223$		−	7.07187i		−	0.473568i	−0.971562	−	0.236784i	$-0.923907\pi$
$223$		−	7.07187i		−	0.473568i	0.971562	−	0.236784i	$-0.0760933\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.15348	−	2.15348i	0.142932	−	0.142932i	−0.632020	−	0.774952i	$-0.717774\pi$
$227$	2.15348	−	2.15348i	0.142932	−	0.142932i	0.774952	+	0.632020i	$0.217774\pi$
$228$		0			0
$229$	−14.4509	−	14.4509i	−0.954943	−	0.954943i	0.0440845	−	0.999028i	$-0.485963\pi$
$229$	−14.4509	−	14.4509i	−0.954943	−	0.954943i	−0.999028	+	0.0440845i	$0.985963\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	17.7969			1.16591			0.582955	−	0.812504i	$-0.301896\pi$
$233$	17.7969			1.16591			0.582955	+	0.812504i	$0.301896\pi$
$234$		0			0
$235$	−5.48063	+	5.48063i	−0.357517	+	0.357517i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.62147			−0.622361			−0.311180	−	0.950351i	$-0.600724\pi$
$239$	−9.62147			−0.622361			−0.311180	+	0.950351i	$0.600724\pi$
$240$		0			0
$241$	27.3786			1.76361			0.881805	−	0.471614i	$-0.156329\pi$
$241$	27.3786			1.76361			0.881805	+	0.471614i	$0.156329\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.925496	−	0.925496i	0.0591278	−	0.0591278i
$246$		0			0
$247$	1.16849			0.0743495
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.56293	−	6.56293i	−0.414249	−	0.414249i	0.468967	−	0.883216i	$-0.344626\pi$
$251$	−6.56293	−	6.56293i	−0.414249	−	0.414249i	−0.883216	+	0.468967i	$0.844626\pi$
$252$		0			0
$253$	−10.6397	+	10.6397i	−0.668914	+	0.668914i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			13.6808i			0.853383i	0.904397	+	0.426691i	$0.140321\pi$
$257$			13.6808i			0.853383i	−0.904397	+	0.426691i	$0.859679\pi$
$258$		0			0
$259$	1.68105	+	1.68105i	0.104455	+	0.104455i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	26.5064i		−	1.63445i	−0.576317	−	0.817226i	$-0.695511\pi$
$263$		−	26.5064i		−	1.63445i	0.576317	−	0.817226i	$-0.304489\pi$
$264$		0			0
$265$			1.80229i			0.110714i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	11.7053	+	11.7053i	0.713684	+	0.713684i	0.967304	−	0.253620i	$-0.0816212\pi$
$269$	11.7053	+	11.7053i	0.713684	+	0.713684i	−0.253620	+	0.967304i	$0.581621\pi$
$270$		0			0
$271$		−	26.7904i		−	1.62740i	−0.581286	−	0.813700i	$-0.697450\pi$
$271$		−	26.7904i		−	1.62740i	0.581286	−	0.813700i	$-0.302550\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.66674	+	5.66674i	−0.341717	+	0.341717i
$276$		0			0
$277$	−0.376534	−	0.376534i	−0.0226237	−	0.0226237i	0.695704	−	0.718328i	$-0.255092\pi$
$277$	−0.376534	−	0.376534i	−0.0226237	−	0.0226237i	−0.718328	+	0.695704i	$0.755092\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.8174			0.645314			0.322657	−	0.946516i	$-0.395424\pi$
$281$	10.8174			0.645314			0.322657	+	0.946516i	$0.395424\pi$
$282$		0			0
$283$	−0.954650	+	0.954650i	−0.0567481	+	0.0567481i	−0.734911	−	0.678163i	$-0.762776\pi$
$283$	−0.954650	+	0.954650i	−0.0567481	+	0.0567481i	0.678163	+	0.734911i	$0.262776\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.88812			−0.170480
$288$		0			0
$289$	11.5159			0.677407
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−14.4106	+	14.4106i	−0.841877	+	0.841877i	−0.989103	−	0.147226i	$-0.952965\pi$
$293$	−14.4106	+	14.4106i	−0.841877	+	0.841877i	0.147226	+	0.989103i	$0.452965\pi$
$294$		0			0
$295$	15.4211			0.897849
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.02591	+	2.02591i	0.117161	+	0.117161i
$300$		0			0
$301$	2.77330	−	2.77330i	0.159850	−	0.159850i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	7.93721i		−	0.454483i
$306$		0			0
$307$	−15.9801	−	15.9801i	−0.912034	−	0.912034i	0.0843977	−	0.996432i	$-0.473103\pi$
$307$	−15.9801	−	15.9801i	−0.912034	−	0.912034i	−0.996432	+	0.0843977i	$0.973103\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			7.61653i			0.431894i	0.976405	+	0.215947i	$0.0692838\pi$
$311$			7.61653i			0.431894i	−0.976405	+	0.215947i	$0.930716\pi$
$312$		0			0
$313$			30.7549i			1.73837i	0.494490	+	0.869184i	$0.335355\pi$
$313$			30.7549i			1.73837i	−0.494490	+	0.869184i	$0.664645\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.52751	−	4.52751i	−0.254291	−	0.254291i	0.568437	−	0.822727i	$-0.307548\pi$
$317$	−4.52751	−	4.52751i	−0.254291	−	0.254291i	−0.822727	+	0.568437i	$0.807548\pi$
$318$		0			0
$319$		−	0.422976i		−	0.0236821i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.16787	−	4.16787i	0.231906	−	0.231906i
$324$		0			0
$325$	1.07900	+	1.07900i	0.0598524	+	0.0598524i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5.92184			0.326481
$330$		0			0
$331$	−11.4312	+	11.4312i	−0.628318	+	0.628318i	−0.947645	−	0.319326i	$-0.896543\pi$
$331$	−11.4312	+	11.4312i	−0.628318	+	0.628318i	0.319326	+	0.947645i	$0.396543\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.29956			−0.180274
$336$		0			0
$337$	−10.7569			−0.585964			−0.292982	−	0.956118i	$-0.594648\pi$
$337$	−10.7569			−0.585964			−0.292982	+	0.956118i	$0.594648\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.01187	+	3.01187i	−0.163102	+	0.163102i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	16.3589	+	16.3589i	0.878193	+	0.878193i	0.993348	−	0.115154i	$-0.0367362\pi$
$347$	16.3589	+	16.3589i	0.878193	+	0.878193i	−0.115154	+	0.993348i	$0.536736\pi$
$348$		0			0
$349$	−25.8389	+	25.8389i	−1.38312	+	1.38312i	−0.544112	+	0.839013i	$0.683133\pi$
$349$	−25.8389	+	25.8389i	−1.38312	+	1.38312i	−0.839013	+	0.544112i	$0.816867\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			24.5770i			1.30810i	0.756451	+	0.654051i	$0.226932\pi$
$353$			24.5770i			1.30810i	−0.756451	+	0.654051i	$0.773068\pi$
$354$		0			0
$355$	7.93199	+	7.93199i	0.420986	+	0.420986i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			17.5485i			0.926176i	0.886312	+	0.463088i	$0.153259\pi$
$359$			17.5485i			0.926176i	−0.886312	+	0.463088i	$0.846741\pi$
$360$		0			0
$361$		−	12.6649i		−	0.666573i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.93937	−	5.93937i	−0.310881	−	0.310881i
$366$		0			0
$367$		−	15.7849i		−	0.823964i	−0.911192	−	0.411982i	$-0.864837\pi$
$367$		−	15.7849i		−	0.823964i	0.911192	−	0.411982i	$-0.135163\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.973689	−	0.973689i	0.0505514	−	0.0505514i
$372$		0			0
$373$	−11.5837	−	11.5837i	−0.599781	−	0.599781i	0.340473	−	0.940254i	$-0.389413\pi$
$373$	−11.5837	−	11.5837i	−0.599781	−	0.599781i	−0.940254	+	0.340473i	$0.889413\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.0805389			−0.00414796
$378$		0			0
$379$	1.84371	−	1.84371i	0.0947049	−	0.0947049i	−0.658167	−	0.752872i	$-0.728668\pi$
$379$	1.84371	−	1.84371i	0.0947049	−	0.0947049i	0.752872	+	0.658167i	$0.228668\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.04073			−0.206472			−0.103236	−	0.994657i	$-0.532920\pi$
$383$	−4.04073			−0.206472			−0.103236	+	0.994657i	$0.532920\pi$
$384$		0			0
$385$	−3.19117			−0.162637
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.92444	−	2.92444i	0.148275	−	0.148275i	−0.629072	−	0.777347i	$-0.716565\pi$
$389$	2.92444	−	2.92444i	0.148275	−	0.148275i	0.777347	+	0.629072i	$0.216565\pi$
$390$		0			0
$391$	14.4523			0.730885
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.98193	+	4.98193i	0.250668	+	0.250668i
$396$		0			0
$397$	−21.9555	+	21.9555i	−1.10191	+	1.10191i	−0.107735	+	0.994180i	$0.534360\pi$
$397$	−21.9555	+	21.9555i	−1.10191	+	1.10191i	−0.994180	+	0.107735i	$0.965640\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			3.18398i			0.159000i	0.996835	+	0.0795001i	$0.0253324\pi$
$401$			3.18398i			0.159000i	−0.996835	+	0.0795001i	$0.974668\pi$
$402$		0			0
$403$	0.573491	+	0.573491i	0.0285676	+	0.0285676i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	5.79636i		−	0.287315i
$408$		0			0
$409$		−	15.1001i		−	0.746650i	−0.927701	−	0.373325i	$-0.878218\pi$
$409$		−	15.1001i		−	0.746650i	0.927701	−	0.373325i	$-0.121782\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8.33124	−	8.33124i	−0.409954	−	0.409954i
$414$		0			0
$415$		−	6.41109i		−	0.314708i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.19389	−	2.19389i	0.107178	−	0.107178i	−0.651484	−	0.758662i	$-0.725853\pi$
$419$	2.19389	−	2.19389i	0.107178	−	0.107178i	0.758662	+	0.651484i	$0.225853\pi$
$420$		0			0
$421$	−16.8539	−	16.8539i	−0.821410	−	0.821410i	0.164900	−	0.986310i	$-0.447270\pi$
$421$	−16.8539	−	16.8539i	−0.821410	−	0.821410i	−0.986310	+	0.164900i	$0.947270\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.69733			0.373376
$426$		0			0
$427$	−4.28808	+	4.28808i	−0.207515	+	0.207515i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−19.9661			−0.961735			−0.480868	−	0.876793i	$-0.659678\pi$
$431$	−19.9661			−0.961735			−0.480868	+	0.876793i	$0.659678\pi$
$432$		0			0
$433$	−3.83212			−0.184160			−0.0920800	−	0.995752i	$-0.529352\pi$
$433$	−3.83212			−0.184160			−0.0920800	+	0.995752i	$0.529352\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.9837	+	10.9837i	−0.525421	+	0.525421i
$438$		0			0
$439$	23.0120			1.09830			0.549151	−	0.835723i	$-0.314951\pi$
$439$	23.0120			1.09830			0.549151	+	0.835723i	$0.314951\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.6982	+	13.6982i	0.650821	+	0.650821i	0.953191	−	0.302370i	$-0.0977778\pi$
$443$	13.6982	+	13.6982i	0.650821	+	0.650821i	−0.302370	+	0.953191i	$0.597778\pi$
$444$		0			0
$445$	−1.40111	+	1.40111i	−0.0664191	+	0.0664191i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	27.3194i		−	1.28928i	−0.764486	−	0.644640i	$-0.777007\pi$
$449$		−	27.3194i		−	1.28928i	0.764486	−	0.644640i	$-0.222993\pi$
$450$		0			0
$451$	4.97920	+	4.97920i	0.234462	+	0.234462i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			0.607630i			0.0284861i
$456$		0			0
$457$			30.4516i			1.42447i	0.701942	+	0.712234i	$0.252316\pi$
$457$			30.4516i			1.42447i	−0.701942	+	0.712234i	$0.747684\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.8016	+	17.8016i	0.829101	+	0.829101i	0.987392	−	0.158291i	$-0.0505984\pi$
$461$	17.8016	+	17.8016i	0.829101	+	0.829101i	−0.158291	+	0.987392i	$0.550598\pi$
$462$		0			0
$463$		−	8.43314i		−	0.391921i	−0.980612	−	0.195961i	$-0.937218\pi$
$463$		−	8.43314i		−	0.391921i	0.980612	−	0.195961i	$-0.0627825\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.17711	−	3.17711i	0.147019	−	0.147019i	−0.629766	−	0.776785i	$-0.716849\pi$
$467$	3.17711	−	3.17711i	0.147019	−	0.147019i	0.776785	+	0.629766i	$0.216849\pi$
$468$		0			0
$469$	1.78259	+	1.78259i	0.0823124	+	0.0823124i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−9.56249			−0.439684
$474$		0			0
$475$	−5.84993	+	5.84993i	−0.268413	+	0.268413i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	0.550076			0.0251336			0.0125668	−	0.999921i	$-0.496000\pi$
$479$	0.550076			0.0251336			0.0125668	+	0.999921i	$0.496000\pi$
$480$		0			0
$481$	−1.10368			−0.0503237
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.5466	−	14.5466i	0.660528	−	0.660528i
$486$		0			0
$487$	−18.1678			−0.823262			−0.411631	−	0.911351i	$-0.635041\pi$
$487$	−18.1678			−0.823262			−0.411631	+	0.911351i	$0.635041\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.4565	−	12.4565i	−0.562156	−	0.562156i	0.367764	−	0.929919i	$-0.380124\pi$
$491$	−12.4565	−	12.4565i	−0.562156	−	0.562156i	−0.929919	+	0.367764i	$0.880124\pi$
$492$		0			0
$493$	−0.287272	+	0.287272i	−0.0129381	+	0.0129381i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	8.57053i		−	0.384441i
$498$		0			0
$499$	23.1835	+	23.1835i	1.03784	+	1.03784i	0.999256	+	0.0385798i	$0.0122834\pi$
$499$	23.1835	+	23.1835i	1.03784	+	1.03784i	0.0385798	+	0.999256i	$0.487717\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	39.4375i		−	1.75843i	−0.476426	−	0.879215i	$-0.658068\pi$
$503$		−	39.4375i		−	1.75843i	0.476426	−	0.879215i	$-0.341932\pi$
$504$		0			0
$505$			25.2545i			1.12381i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.26193	−	1.26193i	−0.0559340	−	0.0559340i	0.678587	−	0.734520i	$-0.262593\pi$
$509$	−1.26193	−	1.26193i	−0.0559340	−	0.0559340i	−0.734520	+	0.678587i	$0.762593\pi$
$510$		0			0
$511$			6.41750i			0.283894i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	11.0355	−	11.0355i	0.486280	−	0.486280i
$516$		0			0
$517$	−10.2094	−	10.2094i	−0.449010	−	0.449010i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	27.2672			1.19460			0.597300	−	0.802018i	$-0.296240\pi$
$521$	27.2672			1.19460			0.597300	+	0.802018i	$0.296240\pi$
$522$		0			0
$523$	31.1608	−	31.1608i	1.36257	−	1.36257i	0.491933	−	0.870633i	$-0.336291\pi$
$523$	31.1608	−	31.1608i	1.36257	−	1.36257i	0.870633	−	0.491933i	$-0.163709\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.09113			0.178213
$528$		0			0
$529$	−15.0865			−0.655936
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.948090	−	0.948090i	0.0410663	−	0.0410663i
$534$		0			0
$535$	25.1935			1.08921
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.72403	+	1.72403i	0.0742592	+	0.0742592i
$540$		0			0
$541$	12.9117	−	12.9117i	0.555119	−	0.555119i	−0.372795	−	0.927914i	$-0.621600\pi$
$541$	12.9117	−	12.9117i	0.555119	−	0.555119i	0.927914	+	0.372795i	$0.121600\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	9.21544i		−	0.394746i
$546$		0			0
$547$	17.6998	+	17.6998i	0.756790	+	0.756790i	0.975737	−	0.218947i	$-0.0702622\pi$
$547$	17.6998	+	17.6998i	0.756790	+	0.756790i	−0.218947	+	0.975737i	$0.570262\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	0.436650i		−	0.0186019i
$552$		0			0
$553$		−	5.38299i		−	0.228908i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.90087	+	3.90087i	0.165285	+	0.165285i	0.784903	−	0.619618i	$-0.212713\pi$
$557$	3.90087	+	3.90087i	0.165285	+	0.165285i	−0.619618	+	0.784903i	$0.712713\pi$
$558$		0			0
$559$			1.82079i			0.0770114i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−27.4905	+	27.4905i	−1.15859	+	1.15859i	−0.173807	+	0.984780i	$0.555607\pi$
$563$	−27.4905	+	27.4905i	−1.15859	+	1.15859i	−0.984780	+	0.173807i	$0.944393\pi$
$564$		0			0
$565$	0.0711615	+	0.0711615i	0.00299379	+	0.00299379i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19.2030			−0.805031			−0.402516	−	0.915413i	$-0.631864\pi$
$569$	−19.2030			−0.805031			−0.402516	+	0.915413i	$0.631864\pi$
$570$		0			0
$571$	−23.2363	+	23.2363i	−0.972408	+	0.972408i	−0.999629	−	0.0272217i	$-0.991334\pi$
$571$	−23.2363	+	23.2363i	−0.972408	+	0.972408i	0.0272217	+	0.999629i	$0.491334\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−20.2850			−0.845941
$576$		0			0
$577$	25.0676			1.04358			0.521789	−	0.853074i	$-0.325265\pi$
$577$	25.0676			1.04358			0.521789	+	0.853074i	$0.325265\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.46360	+	3.46360i	−0.143694	+	0.143694i
$582$		0			0
$583$	−3.35734			−0.139047
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.53626	−	6.53626i	−0.269780	−	0.269780i	0.559231	−	0.829012i	$-0.311096\pi$
$587$	−6.53626	−	6.53626i	−0.269780	−	0.269780i	−0.829012	+	0.559231i	$0.811096\pi$
$588$		0			0
$589$	−3.10924	+	3.10924i	−0.128114	+	0.128114i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	15.1162i		−	0.620749i	−0.950614	−	0.310375i	$-0.899545\pi$
$593$		−	15.1162i		−	0.620749i	0.950614	−	0.310375i	$-0.100455\pi$
$594$		0			0
$595$	2.16734	+	2.16734i	0.0888521	+	0.0888521i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	29.1264i		−	1.19007i	−0.803700	−	0.595035i	$-0.797138\pi$
$599$		−	29.1264i		−	1.19007i	0.803700	−	0.595035i	$-0.202862\pi$
$600$		0			0
$601$		−	14.2202i		−	0.580053i	−0.957019	−	0.290026i	$-0.906336\pi$
$601$		−	14.2202i		−	0.580053i	0.957019	−	0.290026i	$-0.0936641\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.67879	−	4.67879i	−0.190220	−	0.190220i
$606$		0			0
$607$		−	26.8584i		−	1.09015i	−0.838387	−	0.545075i	$-0.816501\pi$
$607$		−	26.8584i		−	1.09015i	0.838387	−	0.545075i	$-0.183499\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.94398	+	1.94398i	−0.0786448	+	0.0786448i
$612$		0			0
$613$	−28.2483	−	28.2483i	−1.14094	−	1.14094i	−0.988279	−	0.152659i	$-0.951216\pi$
$613$	−28.2483	−	28.2483i	−1.14094	−	1.14094i	−0.152659	−	0.988279i	$-0.548784\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	31.4617			1.26660			0.633300	−	0.773906i	$-0.281700\pi$
$617$	31.4617			1.26660			0.633300	+	0.773906i	$0.281700\pi$
$618$		0			0
$619$	−13.3846	+	13.3846i	−0.537973	+	0.537973i	−0.922933	−	0.384961i	$-0.874215\pi$
$619$	−13.3846	+	13.3846i	−0.537973	+	0.537973i	0.384961	+	0.922933i	$0.374215\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.51391			0.0606533
$624$		0			0
$625$	−2.23838			−0.0895353
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.93670	+	3.93670i	−0.156966	+	0.156966i
$630$		0			0
$631$	−15.5279			−0.618154			−0.309077	−	0.951037i	$-0.600020\pi$
$631$	−15.5279			−0.618154			−0.309077	+	0.951037i	$0.600020\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13.9390	−	13.9390i	−0.553152	−	0.553152i
$636$		0			0
$637$	0.328273	−	0.328273i	0.0130066	−	0.0130066i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	33.1261i		−	1.30840i	−0.756320	−	0.654201i	$-0.773005\pi$
$641$		−	33.1261i		−	1.30840i	0.756320	−	0.654201i	$-0.226995\pi$
$642$		0			0
$643$	−28.5683	−	28.5683i	−1.12662	−	1.12662i	−0.990722	−	0.135902i	$-0.956607\pi$
$643$	−28.5683	−	28.5683i	−1.12662	−	1.12662i	−0.135902	−	0.990722i	$-0.543393\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			48.7052i			1.91480i	0.288765	+	0.957400i	$0.406755\pi$
$647$			48.7052i			1.91480i	−0.288765	+	0.957400i	$0.593245\pi$
$648$		0			0
$649$			28.7266i			1.12762i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.67453	+	2.67453i	0.104662	+	0.104662i	0.757499	−	0.652836i	$-0.226421\pi$
$653$	2.67453	+	2.67453i	0.104662	+	0.104662i	−0.652836	+	0.757499i	$0.726421\pi$
$654$		0			0
$655$			7.16384i			0.279915i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	20.0589	−	20.0589i	0.781384	−	0.781384i	−0.198680	−	0.980064i	$-0.563666\pi$
$659$	20.0589	−	20.0589i	0.781384	−	0.781384i	0.980064	+	0.198680i	$0.0636655\pi$
$660$		0			0
$661$	3.44403	+	3.44403i	0.133957	+	0.133957i	0.770906	−	0.636949i	$-0.219804\pi$
$661$	3.44403	+	3.44403i	0.133957	+	0.133957i	−0.636949	+	0.770906i	$0.719804\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.29433			−0.127749
$666$		0			0
$667$	0.757054	−	0.757054i	0.0293132	−	0.0293132i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.7856			0.570791
$672$		0			0
$673$	−32.1701			−1.24007			−0.620033	−	0.784576i	$-0.712881\pi$
$673$	−32.1701			−1.24007			−0.620033	+	0.784576i	$0.712881\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.59090	−	3.59090i	0.138009	−	0.138009i	−0.634727	−	0.772736i	$-0.718887\pi$
$677$	3.59090	−	3.59090i	0.138009	−	0.138009i	0.772736	+	0.634727i	$0.218887\pi$
$678$		0			0
$679$	−15.7177			−0.603188
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4.35732	−	4.35732i	−0.166728	−	0.166728i	0.618811	−	0.785540i	$-0.287615\pi$
$683$	−4.35732	−	4.35732i	−0.166728	−	0.166728i	−0.785540	+	0.618811i	$0.787615\pi$
$684$		0			0
$685$	−5.06701	+	5.06701i	−0.193600	+	0.193600i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			0.639271i			0.0243543i
$690$		0			0
$691$	9.12084	+	9.12084i	0.346973	+	0.346973i	0.858981	−	0.512008i	$-0.171098\pi$
$691$	9.12084	+	9.12084i	0.346973	+	0.346973i	−0.512008	+	0.858981i	$0.671098\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			20.5676i			0.780173i
$696$		0			0
$697$		−	6.76343i		−	0.256183i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−13.1832	−	13.1832i	−0.497923	−	0.497923i	0.412868	−	0.910791i	$-0.364527\pi$
$701$	−13.1832	−	13.1832i	−0.497923	−	0.497923i	−0.910791	+	0.412868i	$0.864527\pi$
$702$		0			0
$703$		−	5.98374i		−	0.225681i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	13.6438	−	13.6438i	0.513126	−	0.513126i
$708$		0			0
$709$	5.95604	+	5.95604i	0.223684	+	0.223684i	0.810048	−	0.586364i	$-0.199441\pi$
$709$	5.95604	+	5.95604i	0.223684	+	0.223684i	−0.586364	+	0.810048i	$0.699441\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−10.7815			−0.403769
$714$		0			0
$715$	1.04757	−	1.04757i	0.0391770	−	0.0391770i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−38.3917			−1.43177			−0.715884	−	0.698220i	$-0.753976\pi$
$719$	−38.3917			−1.43177			−0.715884	+	0.698220i	$0.753976\pi$
$720$		0			0
$721$	−11.9238			−0.444067
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.403208	−	0.403208i	0.0149748	−	0.0149748i
$726$		0			0
$727$	5.05200			0.187368			0.0936842	−	0.995602i	$-0.470136\pi$
$727$	5.05200			0.187368			0.0936842	+	0.995602i	$0.470136\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.49454	+	6.49454i	0.240209	+	0.240209i
$732$		0			0
$733$	2.73691	−	2.73691i	0.101090	−	0.101090i	−0.654753	−	0.755843i	$-0.727227\pi$
$733$	2.73691	−	2.73691i	0.101090	−	0.101090i	0.755843	+	0.654753i	$0.227227\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	6.14648i		−	0.226408i
$738$		0			0
$739$	−18.5031	−	18.5031i	−0.680648	−	0.680648i	0.279498	−	0.960146i	$-0.409832\pi$
$739$	−18.5031	−	18.5031i	−0.680648	−	0.680648i	−0.960146	+	0.279498i	$0.909832\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	41.1110i		−	1.50822i	−0.656751	−	0.754108i	$-0.728070\pi$
$743$		−	41.1110i		−	1.50822i	0.656751	−	0.754108i	$-0.271930\pi$
$744$		0			0
$745$			18.9235i			0.693302i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−13.6108	−	13.6108i	−0.497327	−	0.497327i
$750$		0			0
$751$		−	27.5241i		−	1.00437i	−0.864760	−	0.502185i	$-0.832530\pi$
$751$		−	27.5241i		−	1.00437i	0.864760	−	0.502185i	$-0.167470\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−5.54366	+	5.54366i	−0.201754	+	0.201754i
$756$		0			0
$757$	6.27349	+	6.27349i	0.228014	+	0.228014i	0.811863	−	0.583849i	$-0.198454\pi$
$757$	6.27349	+	6.27349i	0.228014	+	0.228014i	−0.583849	+	0.811863i	$0.698454\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	23.6884			0.858704			0.429352	−	0.903137i	$-0.358742\pi$
$761$	23.6884			0.858704			0.429352	+	0.903137i	$0.358742\pi$
$762$		0			0
$763$	−4.97865	+	4.97865i	−0.180239	+	0.180239i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.46983			0.197504
$768$		0			0
$769$	−4.74756			−0.171201			−0.0856006	−	0.996330i	$-0.527281\pi$
$769$	−4.74756			−0.171201			−0.0856006	+	0.996330i	$0.527281\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18.2666	−	18.2666i	0.657005	−	0.657005i	−0.297666	−	0.954670i	$-0.596208\pi$
$773$	18.2666	−	18.2666i	0.657005	−	0.657005i	0.954670	+	0.297666i	$0.0962080\pi$
$774$		0			0
$775$	−5.74223			−0.206267
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.14017	+	5.14017i	0.184166	+	0.184166i
$780$		0			0
$781$	−14.7759	+	14.7759i	−0.528722	+	0.528722i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	17.7311i		−	0.632849i
$786$		0			0
$787$	−28.1972	−	28.1972i	−1.00512	−	1.00512i	−0.999987	−	0.00513510i	$-0.998365\pi$
$787$	−28.1972	−	28.1972i	−1.00512	−	1.00512i	−0.00513510	−	0.999987i	$-0.501635\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	0.0768901i		−	0.00273390i
$792$		0			0
$793$		−	2.81532i		−	0.0999749i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−24.8697	−	24.8697i	−0.880930	−	0.880930i	0.112699	−	0.993629i	$-0.464050\pi$
$797$	−24.8697	−	24.8697i	−0.880930	−	0.880930i	−0.993629	+	0.112699i	$0.964050\pi$
$798$		0			0
$799$			13.8678i			0.490608i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	11.0640	−	11.0640i	0.390439	−	0.390439i
$804$		0			0
$805$	−5.71163	−	5.71163i	−0.201309	−	0.201309i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−44.9141			−1.57910			−0.789548	−	0.613688i	$-0.789685\pi$
$809$	−44.9141			−1.57910			−0.789548	+	0.613688i	$0.789685\pi$
$810$		0			0
$811$	−13.1446	+	13.1446i	−0.461570	+	0.461570i	−0.899170	−	0.437600i	$-0.855829\pi$
$811$	−13.1446	+	13.1446i	−0.461570	+	0.461570i	0.437600	+	0.899170i	$0.355829\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.4578			0.506435
$816$		0			0
$817$	−9.87162			−0.345364
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.0363	+	15.0363i	−0.524772	+	0.524772i	−0.919009	−	0.394237i	$-0.871009\pi$
$821$	−15.0363	+	15.0363i	−0.524772	+	0.524772i	0.394237	+	0.919009i	$0.371009\pi$
$822$		0			0
$823$	−12.3186			−0.429399			−0.214699	−	0.976680i	$-0.568877\pi$
$823$	−12.3186			−0.429399			−0.214699	+	0.976680i	$0.568877\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−29.2509	−	29.2509i	−1.01715	−	1.01715i	−0.999850	−	0.0173016i	$-0.994492\pi$
$827$	−29.2509	−	29.2509i	−1.01715	−	1.01715i	−0.0173016	−	0.999850i	$-0.505508\pi$
$828$		0			0
$829$	3.68627	−	3.68627i	0.128029	−	0.128029i	−0.640189	−	0.768218i	$-0.721144\pi$
$829$	3.68627	−	3.68627i	0.128029	−	0.128029i	0.768218	+	0.640189i	$0.221144\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	2.34181i		−	0.0811389i
$834$		0			0
$835$	−4.88761	−	4.88761i	−0.169143	−	0.169143i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	26.5841i		−	0.917786i	−0.888492	−	0.458893i	$-0.848246\pi$
$839$		−	26.5841i		−	0.917786i	0.888492	−	0.458893i	$-0.151754\pi$
$840$		0			0
$841$		−	28.9699i		−	0.998962i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11.8320	+	11.8320i	0.407032	+	0.407032i
$846$		0			0
$847$			5.05544i			0.173707i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10.3745	−	10.3745i	0.355632	−	0.355632i
$852$		0			0
$853$	8.95460	+	8.95460i	0.306600	+	0.306600i	0.843589	−	0.536989i	$-0.180438\pi$
$853$	8.95460	+	8.95460i	0.306600	+	0.306600i	−0.536989	+	0.843589i	$0.680438\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−36.9763			−1.26309			−0.631543	−	0.775341i	$-0.717578\pi$
$857$	−36.9763			−1.26309			−0.631543	+	0.775341i	$0.717578\pi$
$858$		0			0
$859$	−1.95338	+	1.95338i	−0.0666483	+	0.0666483i	−0.739645	−	0.672997i	$-0.765007\pi$
$859$	−1.95338	+	1.95338i	−0.0666483	+	0.0666483i	0.672997	+	0.739645i	$0.265007\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−11.1583			−0.379835			−0.189917	−	0.981800i	$-0.560822\pi$
$863$	−11.1583			−0.379835			−0.189917	+	0.981800i	$0.560822\pi$
$864$		0			0
$865$	22.0071			0.748262
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.28043	+	9.28043i	−0.314817	+	0.314817i
$870$		0			0
$871$	−1.17035			−0.0396558
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−7.66951	−	7.66951i	−0.259277	−	0.259277i
$876$		0			0
$877$	34.5959	−	34.5959i	1.16822	−	1.16822i	0.185595	−	0.982626i	$-0.440579\pi$
$877$	34.5959	−	34.5959i	1.16822	−	1.16822i	0.982626	−	0.185595i	$-0.0594213\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	51.6167i		−	1.73901i	−0.493924	−	0.869505i	$-0.664438\pi$
$881$		−	51.6167i		−	1.73901i	0.493924	−	0.869505i	$-0.335562\pi$
$882$		0			0
$883$	−6.62392	−	6.62392i	−0.222913	−	0.222913i	0.586811	−	0.809724i	$-0.300383\pi$
$883$	−6.62392	−	6.62392i	−0.222913	−	0.222913i	−0.809724	+	0.586811i	$0.800383\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	41.8546i		−	1.40534i	−0.711515	−	0.702671i	$-0.751991\pi$
$887$		−	41.8546i		−	1.40534i	0.711515	−	0.702671i	$-0.248009\pi$
$888$		0			0
$889$			15.0611i			0.505134i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−10.5395	−	10.5395i	−0.352690	−	0.352690i
$894$		0			0
$895$			11.8248i			0.395260i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	0.214305	−	0.214305i	0.00714748	−	0.00714748i
$900$		0			0
$901$	2.28020	+	2.28020i	0.0759643	+	0.0759643i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	10.2983			0.342328
$906$		0			0
$907$	−20.3030	+	20.3030i	−0.674150	+	0.674150i	−0.958670	−	0.284520i	$-0.908166\pi$
$907$	−20.3030	+	20.3030i	−0.674150	+	0.674150i	0.284520	+	0.958670i	$0.408166\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.0607824			−0.00201381			−0.00100691	−	0.999999i	$-0.500321\pi$
$911$	−0.0607824			−0.00201381			−0.00100691	+	0.999999i	$0.500321\pi$
$912$		0			0
$913$	11.9427			0.395246
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.87027	−	3.87027i	0.127808	−	0.127808i
$918$		0			0
$919$	−41.3187			−1.36298			−0.681489	−	0.731828i	$-0.738667\pi$
$919$	−41.3187			−1.36298			−0.681489	+	0.731828i	$0.738667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.81347	+	2.81347i	0.0926065	+	0.0926065i
$924$		0			0
$925$	5.52546	−	5.52546i	0.181676	−	0.181676i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			41.2586i			1.35365i	0.736143	+	0.676826i	$0.236645\pi$
$929$			41.2586i			1.35365i	−0.736143	+	0.676826i	$0.763355\pi$
$930$		0			0
$931$	1.77976	+	1.77976i	0.0583294	+	0.0583294i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	7.47311i		−	0.244397i
$936$		0			0
$937$		−	53.0095i		−	1.73175i	−0.500264	−	0.865873i	$-0.666764\pi$
$937$		−	53.0095i		−	1.73175i	0.500264	−	0.865873i	$-0.333236\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	24.3743	+	24.3743i	0.794580	+	0.794580i	0.982235	−	0.187655i	$-0.0600887\pi$
$941$	24.3743	+	24.3743i	0.794580	+	0.794580i	−0.187655	+	0.982235i	$0.560089\pi$
$942$		0			0
$943$			17.8238i			0.580423i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	22.4434	−	22.4434i	0.729314	−	0.729314i	−0.241169	−	0.970483i	$-0.577531\pi$
$947$	22.4434	−	22.4434i	0.729314	−	0.729314i	0.970483	+	0.241169i	$0.0775309\pi$
$948$		0			0
$949$	−2.10669	−	2.10669i	−0.0683861	−	0.0683861i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−11.7067			−0.379218			−0.189609	−	0.981860i	$-0.560722\pi$
$953$	−11.7067			−0.379218			−0.189609	+	0.981860i	$0.560722\pi$
$954$		0			0
$955$	−2.25118	+	2.25118i	−0.0728466	+	0.0728466i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.47491			0.176794
$960$		0			0
$961$	27.9480			0.901549
$962$		0			0
$963$		0			0
$964$		0			0
$965$	22.9992	−	22.9992i	0.740372	−	0.740372i
$966$		0			0
$967$	−53.7319			−1.72790			−0.863951	−	0.503576i	$-0.832017\pi$
$967$	−53.7319			−1.72790			−0.863951	+	0.503576i	$0.832017\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.49392	+	1.49392i	0.0479423	+	0.0479423i	0.730671	−	0.682729i	$-0.239207\pi$
$971$	1.49392	+	1.49392i	0.0479423	+	0.0479423i	−0.682729	+	0.730671i	$0.739207\pi$
$972$		0			0
$973$	11.1117	−	11.1117i	0.356223	−	0.356223i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	35.4073i		−	1.13278i	−0.824138	−	0.566390i	$-0.808340\pi$
$977$		−	35.4073i		−	1.13278i	0.824138	−	0.566390i	$-0.191660\pi$
$978$		0			0
$979$	−2.61002	−	2.61002i	−0.0834166	−	0.0834166i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			17.8771i			0.570191i	0.958499	+	0.285096i	$0.0920254\pi$
$983$			17.8771i			0.570191i	−0.958499	+	0.285096i	$0.907975\pi$
$984$		0			0
$985$		−	3.27905i		−	0.104479i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−17.1152	−	17.1152i	−0.544232	−	0.544232i
$990$		0			0
$991$			54.7193i			1.73822i	0.494623	+	0.869108i	$0.335306\pi$
$991$			54.7193i			1.73822i	−0.494623	+	0.869108i	$0.664694\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	12.8616	−	12.8616i	0.407740	−	0.407740i
$996$		0			0
$997$	38.1277	+	38.1277i	1.20751	+	1.20751i	0.971829	+	0.235686i	$0.0757336\pi$
$997$	38.1277	+	38.1277i	1.20751	+	1.20751i	0.235686	+	0.971829i	$0.424266\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.v.e.3599.14		40
3.2	odd	2	inner	4032.2.v.e.3599.7		40
4.3	odd	2		1008.2.v.e.827.5	yes	40
12.11	even	2		1008.2.v.e.827.16	yes	40
16.3	odd	4	inner	4032.2.v.e.1583.7		40
16.13	even	4		1008.2.v.e.323.16	yes	40
48.29	odd	4		1008.2.v.e.323.5	✓	40
48.35	even	4	inner	4032.2.v.e.1583.14		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.e.323.5	✓	40	48.29	odd	4
1008.2.v.e.323.16	yes	40	16.13	even	4
1008.2.v.e.827.5	yes	40	4.3	odd	2
1008.2.v.e.827.16	yes	40	12.11	even	2
4032.2.v.e.1583.7		40	16.3	odd	4	inner
4032.2.v.e.1583.14		40	48.35	even	4	inner
4032.2.v.e.3599.7		40	3.2	odd	2	inner
4032.2.v.e.3599.14		40	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.925496 - 0.925496i) q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+(0.925496 - 0.925496i) q^{5} -1.00000 q^{7} +(1.72403 + 1.72403i) q^{11} +(0.328273 - 0.328273i) q^{13} -2.34181i q^{17} +(1.77976 + 1.77976i) q^{19} +6.17143i q^{23} +3.28691i q^{25} +(-0.122671 - 0.122671i) q^{29} +1.74700i q^{31} +(-0.925496 + 0.925496i) q^{35} +(-1.68105 - 1.68105i) q^{37} +2.88812 q^{41} +(-2.77330 + 2.77330i) q^{43} -5.92184 q^{47} +1.00000 q^{49} +(-0.973689 + 0.973689i) q^{53} +3.19117 q^{55} +(8.33124 + 8.33124i) q^{59} +(4.28808 - 4.28808i) q^{61} -0.607630i q^{65} +(-1.78259 - 1.78259i) q^{67} +8.57053i q^{71} -6.41750i q^{73} +(-1.72403 - 1.72403i) q^{77} +5.38299i q^{79} +(3.46360 - 3.46360i) q^{83} +(-2.16734 - 2.16734i) q^{85} -1.51391 q^{89} +(-0.328273 + 0.328273i) q^{91} +3.29433 q^{95} +15.7177 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{7}+O(q^{10}) \) 40 * q - 40 * q^7	\( 40 q - 40 q^{7} - 24 q^{13} + 32 q^{19} - 8 q^{37} - 32 q^{43} + 40 q^{49} - 48 q^{55} - 24 q^{61} + 64 q^{85} + 24 q^{91} + 64 q^{97}+O(q^{100}) \) 40 * q - 40 * q^7 - 24 * q^13 + 32 * q^19 - 8 * q^37 - 32 * q^43 + 40 * q^49 - 48 * q^55 - 24 * q^61 + 64 * q^85 + 24 * q^91 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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