LMFDB - Embedded newform 4032.2.v.e.1583.12

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			1583.12
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.e.3599.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.111394 + 0.111394i) q^{5} -1.00000 q^{7} +O(q^{10})$	$q+(0.111394 + 0.111394i) q^{5} -1.00000 q^{7} +(3.61173 - 3.61173i) q^{11} +(-1.94473 - 1.94473i) q^{13} -4.79732i q^{17} +(-3.03275 + 3.03275i) q^{19} +6.58652i q^{23} -4.97518i q^{25} +(-1.53154 + 1.53154i) q^{29} -3.26529i q^{31} +(-0.111394 - 0.111394i) q^{35} +(1.05597 - 1.05597i) q^{37} +1.26613 q^{41} +(-0.484499 - 0.484499i) q^{43} -11.2247 q^{47} +1.00000 q^{49} +(-4.00870 - 4.00870i) q^{53} +0.804648 q^{55} +(-7.61474 + 7.61474i) q^{59} +(5.44215 + 5.44215i) q^{61} -0.433262i q^{65} +(0.897143 - 0.897143i) q^{67} +2.83052i q^{71} -15.7394i q^{73} +(-3.61173 + 3.61173i) q^{77} +15.4151i q^{79} +(-7.57988 - 7.57988i) q^{83} +(0.534392 - 0.534392i) q^{85} -13.1420 q^{89} +(1.94473 + 1.94473i) q^{91} -0.675660 q^{95} -10.4839 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.111394	+	0.111394i	0.0498168	+	0.0498168i	0.731576	−	0.681760i	$-0.238785\pi$
$5$	0.111394	+	0.111394i	0.0498168	+	0.0498168i	−0.681760	+	0.731576i	$0.738785\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.61173	−	3.61173i	1.08898	−	1.08898i	0.0933436	−	0.995634i	$-0.470245\pi$
$11$	3.61173	−	3.61173i	1.08898	−	1.08898i	0.995634	−	0.0933436i	$-0.0297555\pi$
$12$		0			0
$13$	−1.94473	−	1.94473i	−0.539372	−	0.539372i	0.383973	−	0.923344i	$-0.374556\pi$
$13$	−1.94473	−	1.94473i	−0.539372	−	0.539372i	−0.923344	+	0.383973i	$0.874556\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.79732i		−	1.16352i	−0.813360	−	0.581761i	$-0.802364\pi$
$17$		−	4.79732i		−	1.16352i	0.813360	−	0.581761i	$-0.197636\pi$
$18$		0			0
$19$	−3.03275	+	3.03275i	−0.695761	+	0.695761i	−0.963493	−	0.267732i	$-0.913726\pi$
$19$	−3.03275	+	3.03275i	−0.695761	+	0.695761i	0.267732	+	0.963493i	$0.413726\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.58652i			1.37339i	0.726948	+	0.686693i	$0.240938\pi$
$23$			6.58652i			1.37339i	−0.726948	+	0.686693i	$0.759062\pi$
$24$		0			0
$25$		−	4.97518i		−	0.995037i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.53154	+	1.53154i	−0.284399	+	0.284399i	−0.834861	−	0.550462i	$-0.814452\pi$
$29$	−1.53154	+	1.53154i	−0.284399	+	0.284399i	0.550462	+	0.834861i	$0.314452\pi$
$30$		0			0
$31$		−	3.26529i		−	0.586464i	−0.956041	−	0.293232i	$-0.905269\pi$
$31$		−	3.26529i		−	0.586464i	0.956041	−	0.293232i	$-0.0947308\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.111394	−	0.111394i	−0.0188290	−	0.0188290i
$36$		0			0
$37$	1.05597	−	1.05597i	0.173601	−	0.173601i	−0.614959	−	0.788559i	$-0.710827\pi$
$37$	1.05597	−	1.05597i	0.173601	−	0.173601i	0.788559	+	0.614959i	$0.210827\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.26613			0.197737			0.0988684	−	0.995101i	$-0.468478\pi$
$41$	1.26613			0.197737			0.0988684	+	0.995101i	$0.468478\pi$
$42$		0			0
$43$	−0.484499	−	0.484499i	−0.0738855	−	0.0738855i	0.669198	−	0.743084i	$-0.266638\pi$
$43$	−0.484499	−	0.484499i	−0.0738855	−	0.0738855i	−0.743084	+	0.669198i	$0.766638\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−11.2247			−1.63729			−0.818644	−	0.574301i	$-0.805274\pi$
$47$	−11.2247			−1.63729			−0.818644	+	0.574301i	$0.805274\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.00870	−	4.00870i	−0.550637	−	0.550637i	0.375988	−	0.926625i	$-0.377303\pi$
$53$	−4.00870	−	4.00870i	−0.550637	−	0.550637i	−0.926625	+	0.375988i	$0.877303\pi$
$54$		0			0
$55$	0.804648			0.108499
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.61474	+	7.61474i	−0.991354	+	0.991354i	−0.999963	−	0.00860874i	$-0.997260\pi$
$59$	−7.61474	+	7.61474i	−0.991354	+	0.991354i	0.00860874	+	0.999963i	$0.497260\pi$
$60$		0			0
$61$	5.44215	+	5.44215i	0.696796	+	0.696796i	0.963718	−	0.266922i	$-0.0860066\pi$
$61$	5.44215	+	5.44215i	0.696796	+	0.696796i	−0.266922	+	0.963718i	$0.586007\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	0.433262i		−	0.0537395i
$66$		0			0
$67$	0.897143	−	0.897143i	0.109603	−	0.109603i	−0.650178	−	0.759782i	$-0.725306\pi$
$67$	0.897143	−	0.897143i	0.109603	−	0.109603i	0.759782	+	0.650178i	$0.225306\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.83052i			0.335921i	0.985794	+	0.167960i	$0.0537181\pi$
$71$			2.83052i			0.335921i	−0.985794	+	0.167960i	$0.946282\pi$
$72$		0			0
$73$		−	15.7394i		−	1.84216i	−0.389372	−	0.921080i	$-0.627308\pi$
$73$		−	15.7394i		−	1.84216i	0.389372	−	0.921080i	$-0.372692\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.61173	+	3.61173i	−0.411595	+	0.411595i
$78$		0			0
$79$			15.4151i			1.73433i	0.498020	+	0.867165i	$0.334061\pi$
$79$			15.4151i			1.73433i	−0.498020	+	0.867165i	$0.665939\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.57988	−	7.57988i	−0.832000	−	0.832000i	0.155790	−	0.987790i	$-0.450208\pi$
$83$	−7.57988	−	7.57988i	−0.832000	−	0.832000i	−0.987790	+	0.155790i	$0.950208\pi$
$84$		0			0
$85$	0.534392	−	0.534392i	0.0579629	−	0.0579629i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.1420			−1.39305			−0.696525	−	0.717532i	$-0.745272\pi$
$89$	−13.1420			−1.39305			−0.696525	+	0.717532i	$0.745272\pi$
$90$		0			0
$91$	1.94473	+	1.94473i	0.203863	+	0.203863i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.675660			−0.0693212
$96$		0			0
$97$	−10.4839			−1.06447			−0.532237	−	0.846595i	$-0.678648\pi$
$97$	−10.4839			−1.06447			−0.532237	+	0.846595i	$0.678648\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.472060	−	0.472060i	−0.0469717	−	0.0469717i	0.683231	−	0.730202i	$-0.260574\pi$
$101$	−0.472060	−	0.472060i	−0.0469717	−	0.0469717i	−0.730202	+	0.683231i	$0.760574\pi$
$102$		0			0
$103$	16.5554			1.63125			0.815624	−	0.578583i	$-0.196394\pi$
$103$	16.5554			1.63125			0.815624	+	0.578583i	$0.196394\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.98789	+	7.98789i	−0.772218	+	0.772218i	−0.978494	−	0.206276i	$-0.933866\pi$
$107$	−7.98789	+	7.98789i	−0.772218	+	0.772218i	0.206276	+	0.978494i	$0.433866\pi$
$108$		0			0
$109$	−6.01886	−	6.01886i	−0.576502	−	0.576502i	0.357436	−	0.933938i	$-0.383651\pi$
$109$	−6.01886	−	6.01886i	−0.576502	−	0.576502i	−0.933938	+	0.357436i	$0.883651\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			7.81071i			0.734770i	0.930069	+	0.367385i	$0.119747\pi$
$113$			7.81071i			0.734770i	−0.930069	+	0.367385i	$0.880253\pi$
$114$		0			0
$115$	−0.733698	+	0.733698i	−0.0684177	+	0.0684177i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			4.79732i			0.439770i
$120$		0			0
$121$		−	15.0892i		−	1.37174i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.11117	−	1.11117i	0.0993863	−	0.0993863i
$126$		0			0
$127$		−	6.59439i		−	0.585157i	−0.956241	−	0.292579i	$-0.905487\pi$
$127$		−	6.59439i		−	0.585157i	0.956241	−	0.292579i	$-0.0945133\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.27211	+	6.27211i	0.547997	+	0.547997i	0.925861	−	0.377864i	$-0.123341\pi$
$131$	6.27211	+	6.27211i	0.547997	+	0.547997i	−0.377864	+	0.925861i	$0.623341\pi$
$132$		0			0
$133$	3.03275	−	3.03275i	0.262973	−	0.262973i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	22.4152			1.91506			0.957530	−	0.288333i	$-0.0931010\pi$
$137$	22.4152			1.91506			0.957530	+	0.288333i	$0.0931010\pi$
$138$		0			0
$139$	−9.44120	−	9.44120i	−0.800792	−	0.800792i	0.182427	−	0.983219i	$-0.441605\pi$
$139$	−9.44120	−	9.44120i	−0.800792	−	0.800792i	−0.983219	+	0.182427i	$0.941605\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−14.0477			−1.17473
$144$		0			0
$145$	−0.341207			−0.0283357
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.6024	−	10.6024i	−0.868581	−	0.868581i	0.123735	−	0.992315i	$-0.460513\pi$
$149$	−10.6024	−	10.6024i	−0.868581	−	0.868581i	−0.992315	+	0.123735i	$0.960513\pi$
$150$		0			0
$151$	0.651929			0.0530532			0.0265266	−	0.999648i	$-0.491555\pi$
$151$	0.651929			0.0530532			0.0265266	+	0.999648i	$0.491555\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.363733	−	0.363733i	0.0292157	−	0.0292157i
$156$		0			0
$157$	−15.6768	−	15.6768i	−1.25114	−	1.25114i	−0.955209	−	0.295934i	$-0.904369\pi$
$157$	−15.6768	−	15.6768i	−1.25114	−	1.25114i	−0.295934	−	0.955209i	$-0.595631\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	6.58652i		−	0.519091i
$162$		0			0
$163$	4.07211	−	4.07211i	0.318952	−	0.318952i	−0.529412	−	0.848365i	$-0.677588\pi$
$163$	4.07211	−	4.07211i	0.318952	−	0.318952i	0.848365	+	0.529412i	$0.177588\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	8.04509i		−	0.622548i	−0.950320	−	0.311274i	$-0.899244\pi$
$167$		−	8.04509i		−	0.622548i	0.950320	−	0.311274i	$-0.100756\pi$
$168$		0			0
$169$		−	5.43603i		−	0.418156i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.546907	−	0.546907i	0.0415806	−	0.0415806i	−0.686011	−	0.727591i	$-0.740640\pi$
$173$	0.546907	−	0.546907i	0.0415806	−	0.0415806i	0.727591	+	0.686011i	$0.240640\pi$
$174$		0			0
$175$			4.97518i			0.376088i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.4071	−	15.4071i	−1.15158	−	1.15158i	−0.986236	−	0.165345i	$-0.947126\pi$
$179$	−15.4071	−	15.4071i	−1.15158	−	1.15158i	−0.165345	−	0.986236i	$-0.552874\pi$
$180$		0			0
$181$	−4.10925	+	4.10925i	−0.305438	+	0.305438i	−0.843137	−	0.537699i	$-0.819294\pi$
$181$	−4.10925	+	4.10925i	−0.305438	+	0.305438i	0.537699	+	0.843137i	$0.319294\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.235257			0.0172965
$186$		0			0
$187$	−17.3266	−	17.3266i	−1.26705	−	1.26705i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.56461			−0.547356			−0.273678	−	0.961821i	$-0.588240\pi$
$191$	−7.56461			−0.547356			−0.273678	+	0.961821i	$0.588240\pi$
$192$		0			0
$193$	14.0355			1.01030			0.505149	−	0.863032i	$-0.331438\pi$
$193$	14.0355			1.01030			0.505149	+	0.863032i	$0.331438\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.59894	−	9.59894i	−0.683896	−	0.683896i	0.276980	−	0.960876i	$-0.410666\pi$
$197$	−9.59894	−	9.59894i	−0.683896	−	0.683896i	−0.960876	+	0.276980i	$0.910666\pi$
$198$		0			0
$199$	12.5176			0.887349			0.443675	−	0.896188i	$-0.353675\pi$
$199$	12.5176			0.887349			0.443675	+	0.896188i	$0.353675\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.53154	−	1.53154i	0.107493	−	0.107493i
$204$		0			0
$205$	0.141039	+	0.141039i	0.00985062	+	0.00985062i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			21.9070i			1.51534i
$210$		0			0
$211$	−6.19384	+	6.19384i	−0.426401	+	0.426401i	−0.887401	−	0.460999i	$-0.847491\pi$
$211$	−6.19384	+	6.19384i	−0.426401	+	0.426401i	0.460999	+	0.887401i	$0.347491\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	0.107940i		−	0.00736148i
$216$		0			0
$217$			3.26529i			0.221662i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−9.32951	+	9.32951i	−0.627571	+	0.627571i
$222$		0			0
$223$		−	23.4809i		−	1.57240i	−0.617972	−	0.786200i	$-0.712045\pi$
$223$		−	23.4809i		−	1.57240i	0.617972	−	0.786200i	$-0.287955\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.8869	−	13.8869i	−0.921707	−	0.921707i	0.0754430	−	0.997150i	$-0.475963\pi$
$227$	−13.8869	−	13.8869i	−0.921707	−	0.921707i	−0.997150	+	0.0754430i	$0.975963\pi$
$228$		0			0
$229$	−2.96200	+	2.96200i	−0.195735	+	0.195735i	−0.798169	−	0.602434i	$-0.794198\pi$
$229$	−2.96200	+	2.96200i	−0.195735	+	0.195735i	0.602434	+	0.798169i	$0.294198\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.94090			−0.127153			−0.0635763	−	0.997977i	$-0.520251\pi$
$233$	−1.94090			−0.127153			−0.0635763	+	0.997977i	$0.520251\pi$
$234$		0			0
$235$	−1.25036	−	1.25036i	−0.0815645	−	0.0815645i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.73260			0.241442			0.120721	−	0.992686i	$-0.461479\pi$
$239$	3.73260			0.241442			0.120721	+	0.992686i	$0.461479\pi$
$240$		0			0
$241$	−6.72519			−0.433208			−0.216604	−	0.976260i	$-0.569498\pi$
$241$	−6.72519			−0.433208			−0.216604	+	0.976260i	$0.569498\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.111394	+	0.111394i	0.00711669	+	0.00711669i
$246$		0			0
$247$	11.7958			0.750548
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.29389	−	7.29389i	0.460387	−	0.460387i	−0.438396	−	0.898782i	$-0.644453\pi$
$251$	7.29389	−	7.29389i	0.460387	−	0.460387i	0.898782	+	0.438396i	$0.144453\pi$
$252$		0			0
$253$	23.7887	+	23.7887i	1.49559	+	1.49559i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			3.33778i			0.208205i	0.994567	+	0.104103i	$0.0331970\pi$
$257$			3.33778i			0.208205i	−0.994567	+	0.104103i	$0.966803\pi$
$258$		0			0
$259$	−1.05597	+	1.05597i	−0.0656149	+	0.0656149i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			16.2784i			1.00377i	0.864934	+	0.501886i	$0.167360\pi$
$263$			16.2784i			1.00377i	−0.864934	+	0.501886i	$0.832640\pi$
$264$		0			0
$265$		−	0.893087i		−	0.0548619i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−16.6895	+	16.6895i	−1.01758	+	1.01758i	−0.0177322	+	0.999843i	$0.505645\pi$
$269$	−16.6895	+	16.6895i	−1.01758	+	1.01758i	−0.999843	+	0.0177322i	$0.994355\pi$
$270$		0			0
$271$		−	19.4291i		−	1.18023i	−0.807318	−	0.590116i	$-0.799082\pi$
$271$		−	19.4291i		−	1.18023i	0.807318	−	0.590116i	$-0.200918\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−17.9690	−	17.9690i	−1.08357	−	1.08357i
$276$		0			0
$277$	−20.2943	+	20.2943i	−1.21937	+	1.21937i	−0.251511	+	0.967854i	$0.580927\pi$
$277$	−20.2943	+	20.2943i	−1.21937	+	1.21937i	−0.967854	+	0.251511i	$0.919073\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.67930			−0.517763			−0.258882	−	0.965909i	$-0.583354\pi$
$281$	−8.67930			−0.517763			−0.258882	+	0.965909i	$0.583354\pi$
$282$		0			0
$283$	5.11240	+	5.11240i	0.303901	+	0.303901i	0.842538	−	0.538637i	$-0.181061\pi$
$283$	5.11240	+	5.11240i	0.303901	+	0.303901i	−0.538637	+	0.842538i	$0.681061\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.26613			−0.0747375
$288$		0			0
$289$	−6.01430			−0.353782
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.21225	+	4.21225i	0.246082	+	0.246082i	0.819361	−	0.573279i	$-0.194329\pi$
$293$	4.21225	+	4.21225i	0.246082	+	0.246082i	−0.573279	+	0.819361i	$0.694329\pi$
$294$		0			0
$295$	−1.69647			−0.0987722
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.8090	−	12.8090i	0.740765	−	0.740765i
$300$		0			0
$301$	0.484499	+	0.484499i	0.0279261	+	0.0279261i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			1.21244i			0.0694243i
$306$		0			0
$307$	−5.47769	+	5.47769i	−0.312628	+	0.312628i	−0.845927	−	0.533299i	$-0.820952\pi$
$307$	−5.47769	+	5.47769i	−0.312628	+	0.312628i	0.533299	+	0.845927i	$0.320952\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			10.8058i			0.612740i	0.951913	+	0.306370i	$0.0991144\pi$
$311$			10.8058i			0.612740i	−0.951913	+	0.306370i	$0.900886\pi$
$312$		0			0
$313$		−	26.1747i		−	1.47948i	−0.672893	−	0.739740i	$-0.734949\pi$
$313$		−	26.1747i		−	1.47948i	0.672893	−	0.739740i	$-0.265051\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.7908	+	14.7908i	−0.830735	+	0.830735i	−0.987617	−	0.156883i	$-0.949856\pi$
$317$	−14.7908	+	14.7908i	−0.830735	+	0.830735i	0.156883	+	0.987617i	$0.449856\pi$
$318$		0			0
$319$			11.0630i			0.619408i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	14.5491	+	14.5491i	0.809533	+	0.809533i
$324$		0			0
$325$	−9.67540	+	9.67540i	−0.536695	+	0.536695i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	11.2247			0.618837
$330$		0			0
$331$	−12.8239	−	12.8239i	−0.704868	−	0.704868i	0.260583	−	0.965451i	$-0.416085\pi$
$331$	−12.8239	−	12.8239i	−0.704868	−	0.704868i	−0.965451	+	0.260583i	$0.916085\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.199872			0.0109202
$336$		0			0
$337$	11.5576			0.629582			0.314791	−	0.949161i	$-0.398066\pi$
$337$	11.5576			0.629582			0.314791	+	0.949161i	$0.398066\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−11.7934	−	11.7934i	−0.638646	−	0.638646i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.89704	+	2.89704i	−0.155521	+	0.155521i	−0.780579	−	0.625058i	$-0.785076\pi$
$347$	−2.89704	+	2.89704i	−0.155521	+	0.155521i	0.625058	+	0.780579i	$0.285076\pi$
$348$		0			0
$349$	8.35296	+	8.35296i	0.447124	+	0.447124i	0.894397	−	0.447273i	$-0.147605\pi$
$349$	8.35296	+	8.35296i	0.447124	+	0.447124i	−0.447273	+	0.894397i	$0.647605\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			11.6486i			0.619993i	0.950738	+	0.309996i	$0.100328\pi$
$353$			11.6486i			0.619993i	−0.950738	+	0.309996i	$0.899672\pi$
$354$		0			0
$355$	−0.315302	+	0.315302i	−0.0167345	+	0.0167345i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			24.0137i			1.26739i	0.773582	+	0.633696i	$0.218463\pi$
$359$			24.0137i			1.26739i	−0.773582	+	0.633696i	$0.781537\pi$
$360$		0			0
$361$			0.604810i			0.0318321i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.75327	−	1.75327i	0.0917706	−	0.0917706i
$366$		0			0
$367$		−	12.1938i		−	0.636510i	−0.948005	−	0.318255i	$-0.896903\pi$
$367$		−	12.1938i		−	0.636510i	0.948005	−	0.318255i	$-0.103097\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.00870	+	4.00870i	0.208121	+	0.208121i
$372$		0			0
$373$	17.3599	−	17.3599i	0.898859	−	0.898859i	−0.0964762	−	0.995335i	$-0.530757\pi$
$373$	17.3599	−	17.3599i	0.898859	−	0.898859i	0.995335	+	0.0964762i	$0.0307572\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.95685			0.306794
$378$		0			0
$379$	−16.2388	−	16.2388i	−0.834130	−	0.834130i	0.153949	−	0.988079i	$-0.450801\pi$
$379$	−16.2388	−	16.2388i	−0.834130	−	0.834130i	−0.988079	+	0.153949i	$0.950801\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.0190			0.614141			0.307070	−	0.951687i	$-0.400651\pi$
$383$	12.0190			0.614141			0.307070	+	0.951687i	$0.400651\pi$
$384$		0			0
$385$	−0.804648			−0.0410087
$386$		0			0
$387$		0			0
$388$		0			0
$389$	13.2292	+	13.2292i	0.670749	+	0.670749i	0.957889	−	0.287140i	$-0.0927044\pi$
$389$	13.2292	+	13.2292i	0.670749	+	0.670749i	−0.287140	+	0.957889i	$0.592704\pi$
$390$		0			0
$391$	31.5977			1.59796
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.71714	+	1.71714i	−0.0863988	+	0.0863988i
$396$		0			0
$397$	21.8846	+	21.8846i	1.09836	+	1.09836i	0.994603	+	0.103756i	$0.0330860\pi$
$397$	21.8846	+	21.8846i	1.09836	+	1.09836i	0.103756	+	0.994603i	$0.466914\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	27.8646i		−	1.39149i	−0.718288	−	0.695746i	$-0.755074\pi$
$401$		−	27.8646i		−	1.39149i	0.718288	−	0.695746i	$-0.244926\pi$
$402$		0			0
$403$	−6.35012	+	6.35012i	−0.316322	+	0.316322i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	7.62778i		−	0.378095i
$408$		0			0
$409$		−	9.57933i		−	0.473667i	−0.971550	−	0.236834i	$-0.923890\pi$
$409$		−	9.57933i		−	0.473667i	0.971550	−	0.236834i	$-0.0761097\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.61474	−	7.61474i	0.374697	−	0.374697i
$414$		0			0
$415$		−	1.68870i		−	0.0828952i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.53866	−	6.53866i	−0.319435	−	0.319435i	0.529115	−	0.848550i	$-0.322524\pi$
$419$	−6.53866	−	6.53866i	−0.319435	−	0.319435i	−0.848550	+	0.529115i	$0.822524\pi$
$420$		0			0
$421$	−17.8801	+	17.8801i	−0.871424	+	0.871424i	−0.992628	−	0.121204i	$-0.961325\pi$
$421$	−17.8801	+	17.8801i	−0.871424	+	0.871424i	0.121204	+	0.992628i	$0.461325\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−23.8676			−1.15775
$426$		0			0
$427$	−5.44215	−	5.44215i	−0.263364	−	0.263364i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3.02619			0.145766			0.0728832	−	0.997340i	$-0.476780\pi$
$431$	3.02619			0.145766			0.0728832	+	0.997340i	$0.476780\pi$
$432$		0			0
$433$	33.3339			1.60192			0.800962	−	0.598715i	$-0.204322\pi$
$433$	33.3339			1.60192			0.800962	+	0.598715i	$0.204322\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−19.9753	−	19.9753i	−0.955549	−	0.955549i
$438$		0			0
$439$	−25.0388			−1.19504			−0.597519	−	0.801855i	$-0.703847\pi$
$439$	−25.0388			−1.19504			−0.597519	+	0.801855i	$0.703847\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.5230	−	11.5230i	0.547473	−	0.547473i	−0.378236	−	0.925709i	$-0.623469\pi$
$443$	11.5230	−	11.5230i	0.547473	−	0.547473i	0.925709	+	0.378236i	$0.123469\pi$
$444$		0			0
$445$	−1.46394	−	1.46394i	−0.0693973	−	0.0693973i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	24.9666i		−	1.17825i	−0.808042	−	0.589124i	$-0.799473\pi$
$449$		−	24.9666i		−	1.17825i	0.808042	−	0.589124i	$-0.200527\pi$
$450$		0			0
$451$	4.57293	−	4.57293i	0.215331	−	0.215331i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			0.433262i			0.0203116i
$456$		0			0
$457$		−	15.0783i		−	0.705333i	−0.935749	−	0.352666i	$-0.885275\pi$
$457$		−	15.0783i		−	0.705333i	0.935749	−	0.352666i	$-0.114725\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.98842	−	1.98842i	0.0926101	−	0.0926101i	−0.659284	−	0.751894i	$-0.729140\pi$
$461$	1.98842	−	1.98842i	0.0926101	−	0.0926101i	0.751894	+	0.659284i	$0.229140\pi$
$462$		0			0
$463$			13.8131i			0.641949i	0.947088	+	0.320975i	$0.104010\pi$
$463$			13.8131i			0.641949i	−0.947088	+	0.320975i	$0.895990\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	27.8035	+	27.8035i	1.28659	+	1.28659i	0.936842	+	0.349753i	$0.113734\pi$
$467$	27.8035	+	27.8035i	1.28659	+	1.28659i	0.349753	+	0.936842i	$0.386266\pi$
$468$		0			0
$469$	−0.897143	+	0.897143i	−0.0414262	+	0.0414262i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.49976			−0.160919
$474$		0			0
$475$	15.0885	+	15.0885i	0.692308	+	0.692308i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−21.4026			−0.977909			−0.488954	−	0.872309i	$-0.662622\pi$
$479$	−21.4026			−0.977909			−0.488954	+	0.872309i	$0.662622\pi$
$480$		0			0
$481$	−4.10717			−0.187271
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.16784	−	1.16784i	−0.0530287	−	0.0530287i
$486$		0			0
$487$	27.9883			1.26827			0.634136	−	0.773222i	$-0.281356\pi$
$487$	27.9883			1.26827			0.634136	+	0.773222i	$0.281356\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	22.0491	−	22.0491i	0.995063	−	0.995063i	−0.00492464	−	0.999988i	$-0.501568\pi$
$491$	22.0491	−	22.0491i	0.995063	−	0.995063i	0.999988	+	0.00492464i	$0.00156757\pi$
$492$		0			0
$493$	7.34727	+	7.34727i	0.330904	+	0.330904i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	2.83052i		−	0.126966i
$498$		0			0
$499$	−5.60480	+	5.60480i	−0.250905	+	0.250905i	−0.821342	−	0.570436i	$-0.806774\pi$
$499$	−5.60480	+	5.60480i	−0.250905	+	0.250905i	0.570436	+	0.821342i	$0.306774\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			5.02371i			0.223996i	0.993708	+	0.111998i	$0.0357250\pi$
$503$			5.02371i			0.223996i	−0.993708	+	0.111998i	$0.964275\pi$
$504$		0			0
$505$		−	0.105169i		−	0.00467996i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−26.3372	+	26.3372i	−1.16738	+	1.16738i	−0.184555	+	0.982822i	$0.559085\pi$
$509$	−26.3372	+	26.3372i	−1.16738	+	1.16738i	−0.982822	+	0.184555i	$0.940915\pi$
$510$		0			0
$511$			15.7394i			0.696271i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.84416	+	1.84416i	0.0812635	+	0.0812635i
$516$		0			0
$517$	−40.5405	+	40.5405i	−1.78297	+	1.78297i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.453686			−0.0198763			−0.00993817	−	0.999951i	$-0.503163\pi$
$521$	−0.453686			−0.0198763			−0.00993817	+	0.999951i	$0.503163\pi$
$522$		0			0
$523$	−26.5974	−	26.5974i	−1.16302	−	1.16302i	−0.983810	−	0.179214i	$-0.942645\pi$
$523$	−26.5974	−	26.5974i	−1.16302	−	1.16302i	−0.179214	−	0.983810i	$-0.557355\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−15.6647			−0.682363
$528$		0			0
$529$	−20.3823			−0.886187
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.46229	−	2.46229i	−0.106654	−	0.106654i
$534$		0			0
$535$	−1.77960			−0.0769389
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.61173	−	3.61173i	0.155568	−	0.155568i
$540$		0			0
$541$	−17.5559	−	17.5559i	−0.754785	−	0.754785i	0.220583	−	0.975368i	$-0.429204\pi$
$541$	−17.5559	−	17.5559i	−0.754785	−	0.754785i	−0.975368	+	0.220583i	$0.929204\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1.34093i		−	0.0574390i
$546$		0			0
$547$	−6.76713	+	6.76713i	−0.289342	+	0.289342i	−0.836820	−	0.547478i	$-0.815588\pi$
$547$	−6.76713	+	6.76713i	−0.289342	+	0.289342i	0.547478	+	0.836820i	$0.315588\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	9.28954i		−	0.395748i
$552$		0			0
$553$		−	15.4151i		−	0.655515i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	29.6889	−	29.6889i	1.25796	−	1.25796i	0.305893	−	0.952066i	$-0.401045\pi$
$557$	29.6889	−	29.6889i	1.25796	−	1.25796i	0.952066	−	0.305893i	$-0.0989548\pi$
$558$		0			0
$559$			1.88444i			0.0797035i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−25.1503	−	25.1503i	−1.05996	−	1.05996i	−0.998084	−	0.0618753i	$-0.980292\pi$
$563$	−25.1503	−	25.1503i	−1.05996	−	1.05996i	−0.0618753	−	0.998084i	$-0.519708\pi$
$564$		0			0
$565$	−0.870065	+	0.870065i	−0.0366039	+	0.0366039i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.4528			0.731660			0.365830	−	0.930682i	$-0.380785\pi$
$569$	17.4528			0.731660			0.365830	+	0.930682i	$0.380785\pi$
$570$		0			0
$571$	22.4569	+	22.4569i	0.939793	+	0.939793i	0.998288	−	0.0584947i	$-0.0186301\pi$
$571$	22.4569	+	22.4569i	0.939793	+	0.939793i	−0.0584947	+	0.998288i	$0.518630\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	32.7692			1.36657
$576$		0			0
$577$	40.6265			1.69130			0.845652	−	0.533734i	$-0.179212\pi$
$577$	40.6265			1.69130			0.845652	+	0.533734i	$0.179212\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.57988	+	7.57988i	0.314466	+	0.314466i
$582$		0			0
$583$	−28.9567			−1.19926
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.79577	+	1.79577i	−0.0741194	+	0.0741194i	−0.743195	−	0.669075i	$-0.766690\pi$
$587$	−1.79577	+	1.79577i	−0.0741194	+	0.0741194i	0.669075	+	0.743195i	$0.266690\pi$
$588$		0			0
$589$	9.90283	+	9.90283i	0.408039	+	0.408039i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			11.4046i			0.468331i	0.972197	+	0.234165i	$0.0752357\pi$
$593$			11.4046i			0.468331i	−0.972197	+	0.234165i	$0.924764\pi$
$594$		0			0
$595$	−0.534392	+	0.534392i	−0.0219079	+	0.0219079i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	7.81649i		−	0.319373i	−0.987168	−	0.159687i	$-0.948952\pi$
$599$		−	7.81649i		−	0.319373i	0.987168	−	0.159687i	$-0.0510484\pi$
$600$		0			0
$601$			22.2752i			0.908625i	0.890842	+	0.454313i	$0.150115\pi$
$601$			22.2752i			0.908625i	−0.890842	+	0.454313i	$0.849885\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.68084	−	1.68084i	0.0683359	−	0.0683359i
$606$		0			0
$607$		−	11.5129i		−	0.467293i	−0.972322	−	0.233646i	$-0.924934\pi$
$607$		−	11.5129i		−	0.467293i	0.972322	−	0.233646i	$-0.0750658\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	21.8290	+	21.8290i	0.883107	+	0.883107i
$612$		0			0
$613$	24.6256	−	24.6256i	0.994618	−	0.994618i	−0.00536742	−	0.999986i	$-0.501709\pi$
$613$	24.6256	−	24.6256i	0.994618	−	0.994618i	0.999986	+	0.00536742i	$0.00170851\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	29.3630			1.18211			0.591054	−	0.806632i	$-0.298712\pi$
$617$	29.3630			1.18211			0.591054	+	0.806632i	$0.298712\pi$
$618$		0			0
$619$	−32.5316	−	32.5316i	−1.30756	−	1.30756i	−0.923173	−	0.384384i	$-0.874414\pi$
$619$	−32.5316	−	32.5316i	−1.30756	−	1.30756i	−0.384384	−	0.923173i	$-0.625586\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	13.1420			0.526524
$624$		0			0
$625$	−24.6284			−0.985134
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.06584	−	5.06584i	−0.201988	−	0.201988i
$630$		0			0
$631$	10.4307			0.415240			0.207620	−	0.978210i	$-0.433428\pi$
$631$	10.4307			0.415240			0.207620	+	0.978210i	$0.433428\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.734574	−	0.734574i	0.0291507	−	0.0291507i
$636$		0			0
$637$	−1.94473	−	1.94473i	−0.0770531	−	0.0770531i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			26.3287i			1.03992i	0.854191	+	0.519960i	$0.174053\pi$
$641$			26.3287i			1.03992i	−0.854191	+	0.519960i	$0.825947\pi$
$642$		0			0
$643$	−10.1662	+	10.1662i	−0.400917	+	0.400917i	−0.878556	−	0.477639i	$-0.841493\pi$
$643$	−10.1662	+	10.1662i	−0.400917	+	0.400917i	0.477639	+	0.878556i	$0.341493\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			30.4560i			1.19735i	0.800992	+	0.598675i	$0.204306\pi$
$647$			30.4560i			1.19735i	−0.800992	+	0.598675i	$0.795694\pi$
$648$		0			0
$649$			55.0047i			2.15912i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.424352	+	0.424352i	−0.0166062	+	0.0166062i	−0.715361	−	0.698755i	$-0.753738\pi$
$653$	−0.424352	+	0.424352i	−0.0166062	+	0.0166062i	0.698755	+	0.715361i	$0.253738\pi$
$654$		0			0
$655$			1.39735i			0.0545989i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	21.4421	+	21.4421i	0.835268	+	0.835268i	0.988232	−	0.152964i	$-0.0488819\pi$
$659$	21.4421	+	21.4421i	0.835268	+	0.835268i	−0.152964	+	0.988232i	$0.548882\pi$
$660$		0			0
$661$	29.6529	−	29.6529i	1.15337	−	1.15337i	0.167492	−	0.985873i	$-0.446433\pi$
$661$	29.6529	−	29.6529i	1.15337	−	1.15337i	0.985873	−	0.167492i	$-0.0535668\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.675660			0.0262010
$666$		0			0
$667$	−10.0875	−	10.0875i	−0.390589	−	0.390589i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	39.3112			1.51759
$672$		0			0
$673$	−33.5720			−1.29410			−0.647052	−	0.762446i	$-0.723998\pi$
$673$	−33.5720			−1.29410			−0.647052	+	0.762446i	$0.723998\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−23.4845	−	23.4845i	−0.902584	−	0.902584i	0.0930751	−	0.995659i	$-0.470330\pi$
$677$	−23.4845	−	23.4845i	−0.902584	−	0.902584i	−0.995659	+	0.0930751i	$0.970330\pi$
$678$		0			0
$679$	10.4839			0.402333
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4.90491	+	4.90491i	−0.187681	+	0.187681i	−0.794693	−	0.607012i	$-0.792368\pi$
$683$	−4.90491	+	4.90491i	−0.187681	+	0.187681i	0.607012	+	0.794693i	$0.292368\pi$
$684$		0			0
$685$	2.49691	+	2.49691i	0.0954022	+	0.0954022i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			15.5917i			0.593996i
$690$		0			0
$691$	19.1177	−	19.1177i	0.727272	−	0.727272i	−0.242804	−	0.970075i	$-0.578067\pi$
$691$	19.1177	−	19.1177i	0.727272	−	0.727272i	0.970075	+	0.242804i	$0.0780671\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	2.10338i		−	0.0797858i
$696$		0			0
$697$		−	6.07405i		−	0.230071i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.01355	+	6.01355i	−0.227129	+	0.227129i	−0.811492	−	0.584363i	$-0.801344\pi$
$701$	−6.01355	+	6.01355i	−0.227129	+	0.227129i	0.584363	+	0.811492i	$0.301344\pi$
$702$		0			0
$703$			6.40501i			0.241570i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.472060	+	0.472060i	0.0177536	+	0.0177536i
$708$		0			0
$709$	−6.33448	+	6.33448i	−0.237896	+	0.237896i	−0.815979	−	0.578082i	$-0.803801\pi$
$709$	−6.33448	+	6.33448i	−0.237896	+	0.237896i	0.578082	+	0.815979i	$0.303801\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	21.5069			0.805441
$714$		0			0
$715$	−1.56483	−	1.56483i	−0.0585212	−	0.0585212i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	19.8643			0.740814			0.370407	−	0.928869i	$-0.379218\pi$
$719$	19.8643			0.740814			0.370407	+	0.928869i	$0.379218\pi$
$720$		0			0
$721$	−16.5554			−0.616554
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.61967	+	7.61967i	0.282987	+	0.282987i
$726$		0			0
$727$	17.0880			0.633759			0.316879	−	0.948466i	$-0.397365\pi$
$727$	17.0880			0.633759			0.316879	+	0.948466i	$0.397365\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.32430	+	2.32430i	−0.0859673	+	0.0859673i
$732$		0			0
$733$	27.4896	+	27.4896i	1.01535	+	1.01535i	0.999880	+	0.0154724i	$0.00492520\pi$
$733$	27.4896	+	27.4896i	1.01535	+	1.01535i	0.0154724	+	0.999880i	$0.495075\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	6.48048i		−	0.238711i
$738$		0			0
$739$	−33.4666	+	33.4666i	−1.23109	+	1.23109i	−0.267542	+	0.963546i	$0.586212\pi$
$739$	−33.4666	+	33.4666i	−1.23109	+	1.23109i	−0.963546	+	0.267542i	$0.913788\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	0.720828i		−	0.0264446i	−0.999913	−	0.0132223i	$-0.995791\pi$
$743$		−	0.720828i		−	0.0264446i	0.999913	−	0.0132223i	$-0.00420892\pi$
$744$		0			0
$745$		−	2.36208i		−	0.0865398i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7.98789	−	7.98789i	0.291871	−	0.291871i
$750$		0			0
$751$			8.54158i			0.311687i	0.987782	+	0.155843i	$0.0498095\pi$
$751$			8.54158i			0.311687i	−0.987782	+	0.155843i	$0.950190\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.0726208	+	0.0726208i	0.00264294	+	0.00264294i
$756$		0			0
$757$	5.80744	−	5.80744i	0.211075	−	0.211075i	−0.593649	−	0.804724i	$-0.702313\pi$
$757$	5.80744	−	5.80744i	0.211075	−	0.211075i	0.804724	+	0.593649i	$0.202313\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	19.8670			0.720179			0.360090	−	0.932918i	$-0.382746\pi$
$761$	19.8670			0.720179			0.360090	+	0.932918i	$0.382746\pi$
$762$		0			0
$763$	6.01886	+	6.01886i	0.217897	+	0.217897i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	29.6172			1.06942
$768$		0			0
$769$	51.4781			1.85635			0.928173	−	0.372148i	$-0.121379\pi$
$769$	51.4781			1.85635			0.928173	+	0.372148i	$0.121379\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−17.8748	−	17.8748i	−0.642913	−	0.642913i	0.308358	−	0.951271i	$-0.400221\pi$
$773$	−17.8748	−	17.8748i	−0.642913	−	0.642913i	−0.951271	+	0.308358i	$0.900221\pi$
$774$		0			0
$775$	−16.2454			−0.583553
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.83987	+	3.83987i	−0.137578	+	0.137578i
$780$		0			0
$781$	10.2231	+	10.2231i	0.365810	+	0.365810i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	3.49259i		−	0.124656i
$786$		0			0
$787$	9.93955	−	9.93955i	0.354307	−	0.354307i	−0.507403	−	0.861709i	$-0.669394\pi$
$787$	9.93955	−	9.93955i	0.354307	−	0.354307i	0.861709	+	0.507403i	$0.169394\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	7.81071i		−	0.277717i
$792$		0			0
$793$		−	21.1671i		−	0.751664i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.30890	−	1.30890i	0.0463636	−	0.0463636i	−0.683545	−	0.729908i	$-0.739563\pi$
$797$	1.30890	−	1.30890i	0.0463636	−	0.0463636i	0.729908	+	0.683545i	$0.239563\pi$
$798$		0			0
$799$			53.8484i			1.90502i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−56.8466	−	56.8466i	−2.00607	−	2.00607i
$804$		0			0
$805$	0.733698	−	0.733698i	0.0258594	−	0.0258594i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	38.4014			1.35012			0.675060	−	0.737763i	$-0.264118\pi$
$809$	38.4014			1.35012			0.675060	+	0.737763i	$0.264118\pi$
$810$		0			0
$811$	32.4504	+	32.4504i	1.13949	+	1.13949i	0.988542	+	0.150946i	$0.0482319\pi$
$811$	32.4504	+	32.4504i	1.13949	+	1.13949i	0.150946	+	0.988542i	$0.451768\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.907215			0.0317784
$816$		0			0
$817$	2.93873			0.102813
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13.9023	+	13.9023i	0.485194	+	0.485194i	0.906786	−	0.421592i	$-0.138528\pi$
$821$	13.9023	+	13.9023i	0.485194	+	0.485194i	−0.421592	+	0.906786i	$0.638528\pi$
$822$		0			0
$823$	15.7710			0.549744			0.274872	−	0.961481i	$-0.411365\pi$
$823$	15.7710			0.549744			0.274872	+	0.961481i	$0.411365\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.4573	−	25.4573i	0.885237	−	0.885237i	−0.108824	−	0.994061i	$-0.534709\pi$
$827$	25.4573	−	25.4573i	0.885237	−	0.885237i	0.994061	+	0.108824i	$0.0347085\pi$
$828$		0			0
$829$	−18.0084	−	18.0084i	−0.625459	−	0.625459i	0.321463	−	0.946922i	$-0.395825\pi$
$829$	−18.0084	−	18.0084i	−0.625459	−	0.625459i	−0.946922	+	0.321463i	$0.895825\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	4.79732i		−	0.166217i
$834$		0			0
$835$	0.896173	−	0.896173i	0.0310133	−	0.0310133i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	6.91080i		−	0.238587i	−0.992859	−	0.119294i	$-0.961937\pi$
$839$		−	6.91080i		−	0.238587i	0.992859	−	0.119294i	$-0.0380630\pi$
$840$		0			0
$841$			24.3088i			0.838234i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.605540	−	0.605540i	0.0208312	−	0.0208312i
$846$		0			0
$847$			15.0892i			0.518471i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.95519	+	6.95519i	0.238421	+	0.238421i
$852$		0			0
$853$	−8.91240	+	8.91240i	−0.305155	+	0.305155i	−0.843027	−	0.537872i	$-0.819228\pi$
$853$	−8.91240	+	8.91240i	−0.305155	+	0.305155i	0.537872	+	0.843027i	$0.319228\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−29.4916			−1.00741			−0.503706	−	0.863875i	$-0.668031\pi$
$857$	−29.4916			−1.00741			−0.503706	+	0.863875i	$0.668031\pi$
$858$		0			0
$859$	11.9280	+	11.9280i	0.406977	+	0.406977i	0.880683	−	0.473706i	$-0.157084\pi$
$859$	11.9280	+	11.9280i	0.406977	+	0.406977i	−0.473706	+	0.880683i	$0.657084\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−45.3343			−1.54320			−0.771598	−	0.636110i	$-0.780542\pi$
$863$	−45.3343			−1.54320			−0.771598	+	0.636110i	$0.780542\pi$
$864$		0			0
$865$	0.121844			0.00414282
$866$		0			0
$867$		0			0
$868$		0			0
$869$	55.6751	+	55.6751i	1.88865	+	1.88865i
$870$		0			0
$871$	−3.48941			−0.118234
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.11117	+	1.11117i	−0.0375645	+	0.0375645i
$876$		0			0
$877$	31.3609	+	31.3609i	1.05898	+	1.05898i	0.998148	+	0.0608358i	$0.0193766\pi$
$877$	31.3609	+	31.3609i	1.05898	+	1.05898i	0.0608358	+	0.998148i	$0.480623\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			51.6409i			1.73983i	0.493204	+	0.869913i	$0.335825\pi$
$881$			51.6409i			1.73983i	−0.493204	+	0.869913i	$0.664175\pi$
$882$		0			0
$883$	21.8785	−	21.8785i	0.736269	−	0.736269i	−0.235585	−	0.971854i	$-0.575701\pi$
$883$	21.8785	−	21.8785i	0.736269	−	0.736269i	0.971854	+	0.235585i	$0.0757006\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			19.7592i			0.663449i	0.943376	+	0.331725i	$0.107630\pi$
$887$			19.7592i			0.663449i	−0.943376	+	0.331725i	$0.892370\pi$
$888$		0			0
$889$			6.59439i			0.221169i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	34.0417	−	34.0417i	1.13916	−	1.13916i
$894$		0			0
$895$		−	3.43251i		−	0.114736i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.00091	+	5.00091i	0.166790	+	0.166790i
$900$		0			0
$901$	−19.2310	+	19.2310i	−0.640678	+	0.640678i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.915490			−0.0304319
$906$		0			0
$907$	31.5036	+	31.5036i	1.04606	+	1.04606i	0.998887	+	0.0471723i	$0.0150210\pi$
$907$	31.5036	+	31.5036i	1.04606	+	1.04606i	0.0471723	+	0.998887i	$0.484979\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−15.7479			−0.521752			−0.260876	−	0.965372i	$-0.584011\pi$
$911$	−15.7479			−0.521752			−0.260876	+	0.965372i	$0.584011\pi$
$912$		0			0
$913$	−54.7530			−1.81206
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.27211	−	6.27211i	−0.207123	−	0.207123i
$918$		0			0
$919$	31.5809			1.04176			0.520880	−	0.853630i	$-0.325604\pi$
$919$	31.5809			1.04176			0.520880	+	0.853630i	$0.325604\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.50460	−	5.50460i	0.181186	−	0.181186i
$924$		0			0
$925$	−5.25366	−	5.25366i	−0.172739	−	0.172739i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	31.4061i		−	1.03040i	−0.857070	−	0.515200i	$-0.827718\pi$
$929$		−	31.4061i		−	1.03040i	0.857070	−	0.515200i	$-0.172282\pi$
$930$		0			0
$931$	−3.03275	+	3.03275i	−0.0993945	+	0.0993945i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	3.86016i		−	0.126241i
$936$		0			0
$937$			14.9239i			0.487544i	0.969833	+	0.243772i	$0.0783848\pi$
$937$			14.9239i			0.487544i	−0.969833	+	0.243772i	$0.921615\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.4870	+	13.4870i	−0.439664	+	0.439664i	−0.891899	−	0.452235i	$-0.850627\pi$
$941$	−13.4870	+	13.4870i	−0.439664	+	0.439664i	0.452235	+	0.891899i	$0.350627\pi$
$942$		0			0
$943$			8.33942i			0.271569i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	28.6554	+	28.6554i	0.931174	+	0.931174i	0.997779	−	0.0666054i	$-0.0212169\pi$
$947$	28.6554	+	28.6554i	0.931174	+	0.931174i	−0.0666054	+	0.997779i	$0.521217\pi$
$948$		0			0
$949$	−30.6090	+	30.6090i	−0.993609	+	0.993609i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	48.9378			1.58525			0.792626	−	0.609709i	$-0.208714\pi$
$953$	48.9378			1.58525			0.792626	+	0.609709i	$0.208714\pi$
$954$		0			0
$955$	−0.842651	−	0.842651i	−0.0272675	−	0.0272675i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−22.4152			−0.723825
$960$		0			0
$961$	20.3379			0.656060
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.56347	+	1.56347i	0.0503298	+	0.0503298i
$966$		0			0
$967$	8.50595			0.273533			0.136767	−	0.990603i	$-0.456329\pi$
$967$	8.50595			0.273533			0.136767	+	0.990603i	$0.456329\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	8.52407	−	8.52407i	0.273550	−	0.273550i	−0.556977	−	0.830528i	$-0.688039\pi$
$971$	8.52407	−	8.52407i	0.273550	−	0.273550i	0.830528	+	0.556977i	$0.188039\pi$
$972$		0			0
$973$	9.44120	+	9.44120i	0.302671	+	0.302671i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	30.2469i		−	0.967685i	−0.875155	−	0.483842i	$-0.839241\pi$
$977$		−	30.2469i		−	0.967685i	0.875155	−	0.483842i	$-0.160759\pi$
$978$		0			0
$979$	−47.4654	+	47.4654i	−1.51700	+	1.51700i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	49.8151i		−	1.58886i	−0.607358	−	0.794428i	$-0.707771\pi$
$983$		−	49.8151i		−	1.58886i	0.607358	−	0.794428i	$-0.292229\pi$
$984$		0			0
$985$		−	2.13852i		−	0.0681390i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.19117	−	3.19117i	0.101473	−	0.101473i
$990$		0			0
$991$			26.1968i			0.832167i	0.909326	+	0.416084i	$0.136598\pi$
$991$			26.1968i			0.832167i	−0.909326	+	0.416084i	$0.863402\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.39438	+	1.39438i	0.0442049	+	0.0442049i
$996$		0			0
$997$	−5.13429	+	5.13429i	−0.162604	+	0.162604i	−0.783719	−	0.621115i	$-0.786680\pi$
$997$	−5.13429	+	5.13429i	−0.162604	+	0.162604i	0.621115	+	0.783719i	$0.286680\pi$
$998$		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.1583.12		40
3.2	odd	2	inner	4032.2.v.e.1583.9		40
4.3	odd	2		1008.2.v.e.323.3	✓	40
12.11	even	2		1008.2.v.e.323.18	yes	40
16.5	even	4		1008.2.v.e.827.18	yes	40
16.11	odd	4	inner	4032.2.v.e.3599.9		40
48.5	odd	4		1008.2.v.e.827.3	yes	40
48.11	even	4	inner	4032.2.v.e.3599.12		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.e.323.3	✓	40	4.3	odd	2
1008.2.v.e.323.18	yes	40	12.11	even	2
1008.2.v.e.827.3	yes	40	48.5	odd	4
1008.2.v.e.827.18	yes	40	16.5	even	4
4032.2.v.e.1583.9		40	3.2	odd	2	inner
4032.2.v.e.1583.12		40	1.1	even	1	trivial
4032.2.v.e.3599.9		40	16.11	odd	4	inner
4032.2.v.e.3599.12		40	48.11	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.111394 + 0.111394i) q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+(0.111394 + 0.111394i) q^{5} -1.00000 q^{7} +(3.61173 - 3.61173i) q^{11} +(-1.94473 - 1.94473i) q^{13} -4.79732i q^{17} +(-3.03275 + 3.03275i) q^{19} +6.58652i q^{23} -4.97518i q^{25} +(-1.53154 + 1.53154i) q^{29} -3.26529i q^{31} +(-0.111394 - 0.111394i) q^{35} +(1.05597 - 1.05597i) q^{37} +1.26613 q^{41} +(-0.484499 - 0.484499i) q^{43} -11.2247 q^{47} +1.00000 q^{49} +(-4.00870 - 4.00870i) q^{53} +0.804648 q^{55} +(-7.61474 + 7.61474i) q^{59} +(5.44215 + 5.44215i) q^{61} -0.433262i q^{65} +(0.897143 - 0.897143i) q^{67} +2.83052i q^{71} -15.7394i q^{73} +(-3.61173 + 3.61173i) q^{77} +15.4151i q^{79} +(-7.57988 - 7.57988i) q^{83} +(0.534392 - 0.534392i) q^{85} -13.1420 q^{89} +(1.94473 + 1.94473i) q^{91} -0.675660 q^{95} -10.4839 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{7}+O(q^{10}) \) 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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