LMFDB - Embedded newform 2304.2.k.i.577.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(577,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("2304.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2304.k (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:	$18.3975326257$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			577.4
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2304.577
Dual form			2304.2.k.i.1729.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.44949 - 2.44949i) q^{5} +1.41421i q^{7} +(-3.46410 + 3.46410i) q^{11} +(1.00000 + 1.00000i) q^{13} -4.89898 q^{17} +(4.24264 + 4.24264i) q^{19} +6.92820i q^{23} -7.00000i q^{25} +(2.44949 + 2.44949i) q^{29} +1.41421 q^{31} +(3.46410 + 3.46410i) q^{35} +(-7.00000 + 7.00000i) q^{37} +4.89898i q^{41} +(-4.24264 + 4.24264i) q^{43} -6.92820 q^{47} +5.00000 q^{49} +(-2.44949 + 2.44949i) q^{53} +16.9706i q^{55} +(6.92820 - 6.92820i) q^{59} +(5.00000 + 5.00000i) q^{61} +4.89898 q^{65} +(5.65685 + 5.65685i) q^{67} -13.8564i q^{71} -12.0000i q^{73} +(-4.89898 - 4.89898i) q^{77} +15.5563 q^{79} +(-10.3923 - 10.3923i) q^{83} +(-12.0000 + 12.0000i) q^{85} -9.79796i q^{89} +(-1.41421 + 1.41421i) q^{91} +20.7846 q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{13} - 56 q^{37} + 40 q^{49} + 40 q^{61} - 96 q^{85}+O(q^{100}) $ 8 * q + 8 * q^13 - 56 * q^37 + 40 * q^49 + 40 * q^61 - 96 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.44949	−	2.44949i	1.09545	−	1.09545i	0.100509	−	0.994936i	$-0.467953\pi$
$5$	2.44949	−	2.44949i	1.09545	−	1.09545i	0.994936	−	0.100509i	$-0.0320471\pi$
$6$		0			0
$7$			1.41421i			0.534522i	0.963624	+	0.267261i	$0.0861187\pi$
$7$			1.41421i			0.534522i	−0.963624	+	0.267261i	$0.913881\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.46410	+	3.46410i	−1.04447	+	1.04447i	−0.0455017	+	0.998964i	$0.514489\pi$
$11$	−3.46410	+	3.46410i	−1.04447	+	1.04447i	−0.998964	+	0.0455017i	$0.985511\pi$
$12$		0			0
$13$	1.00000	+	1.00000i	0.277350	+	0.277350i	0.832050	−	0.554700i	$-0.187167\pi$
$13$	1.00000	+	1.00000i	0.277350	+	0.277350i	−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.89898			−1.18818			−0.594089	−	0.804400i	$-0.702487\pi$
$17$	−4.89898			−1.18818			−0.594089	+	0.804400i	$0.702487\pi$
$18$		0			0
$19$	4.24264	+	4.24264i	0.973329	+	0.973329i	0.999653	−	0.0263249i	$-0.00838045\pi$
$19$	4.24264	+	4.24264i	0.973329	+	0.973329i	−0.0263249	+	0.999653i	$0.508380\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.92820i			1.44463i	0.691564	+	0.722315i	$0.256922\pi$
$23$			6.92820i			1.44463i	−0.691564	+	0.722315i	$0.743078\pi$
$24$		0			0
$25$		−	7.00000i		−	1.40000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.44949	+	2.44949i	0.454859	+	0.454859i	0.896963	−	0.442105i	$-0.145768\pi$
$29$	2.44949	+	2.44949i	0.454859	+	0.454859i	−0.442105	+	0.896963i	$0.645768\pi$
$30$		0			0
$31$	1.41421			0.254000			0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421			0.254000			0.127000	+	0.991903i	$0.459465\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.46410	+	3.46410i	0.585540	+	0.585540i
$36$		0			0
$37$	−7.00000	+	7.00000i	−1.15079	+	1.15079i	−0.164399	+	0.986394i	$0.552568\pi$
$37$	−7.00000	+	7.00000i	−1.15079	+	1.15079i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.89898i			0.765092i	0.923936	+	0.382546i	$0.124953\pi$
$41$			4.89898i			0.765092i	−0.923936	+	0.382546i	$0.875047\pi$
$42$		0			0
$43$	−4.24264	+	4.24264i	−0.646997	+	0.646997i	−0.952266	−	0.305269i	$-0.901253\pi$
$43$	−4.24264	+	4.24264i	−0.646997	+	0.646997i	0.305269	+	0.952266i	$0.401253\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.92820			−1.01058			−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820			−1.01058			−0.505291	+	0.862949i	$0.668615\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.44949	+	2.44949i	−0.336463	+	0.336463i	−0.855034	−	0.518571i	$-0.826464\pi$
$53$	−2.44949	+	2.44949i	−0.336463	+	0.336463i	0.518571	+	0.855034i	$0.326464\pi$
$54$		0			0
$55$			16.9706i			2.28831i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.92820	−	6.92820i	0.901975	−	0.901975i	−0.0936317	−	0.995607i	$-0.529848\pi$
$59$	6.92820	−	6.92820i	0.901975	−	0.901975i	0.995607	+	0.0936317i	$0.0298476\pi$
$60$		0			0
$61$	5.00000	+	5.00000i	0.640184	+	0.640184i	0.950601	−	0.310416i	$-0.100468\pi$
$61$	5.00000	+	5.00000i	0.640184	+	0.640184i	−0.310416	+	0.950601i	$0.600468\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.89898			0.607644
$66$		0			0
$67$	5.65685	+	5.65685i	0.691095	+	0.691095i	0.962473	−	0.271378i	$-0.0874794\pi$
$67$	5.65685	+	5.65685i	0.691095	+	0.691095i	−0.271378	+	0.962473i	$0.587479\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	13.8564i		−	1.64445i	−0.569160	−	0.822226i	$-0.692732\pi$
$71$		−	13.8564i		−	1.64445i	0.569160	−	0.822226i	$-0.307268\pi$
$72$		0			0
$73$		−	12.0000i		−	1.40449i	−0.711934	−	0.702247i	$-0.752180\pi$
$73$		−	12.0000i		−	1.40449i	0.711934	−	0.702247i	$-0.247820\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.89898	−	4.89898i	−0.558291	−	0.558291i
$78$		0			0
$79$	15.5563			1.75023			0.875113	−	0.483919i	$-0.160787\pi$
$79$	15.5563			1.75023			0.875113	+	0.483919i	$0.160787\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.3923	−	10.3923i	−1.14070	−	1.14070i	−0.988322	−	0.152382i	$-0.951306\pi$
$83$	−10.3923	−	10.3923i	−1.14070	−	1.14070i	−0.152382	−	0.988322i	$-0.548694\pi$
$84$		0			0
$85$	−12.0000	+	12.0000i	−1.30158	+	1.30158i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	9.79796i		−	1.03858i	−0.854598	−	0.519291i	$-0.826196\pi$
$89$		−	9.79796i		−	1.03858i	0.854598	−	0.519291i	$-0.173804\pi$
$90$		0			0
$91$	−1.41421	+	1.41421i	−0.148250	+	0.148250i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	20.7846			2.13246
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.34847	+	7.34847i	−0.731200	+	0.731200i	−0.970858	−	0.239657i	$-0.922965\pi$
$101$	−7.34847	+	7.34847i	−0.731200	+	0.731200i	0.239657	+	0.970858i	$0.422965\pi$
$102$		0			0
$103$			7.07107i			0.696733i	0.937358	+	0.348367i	$0.113264\pi$
$103$			7.07107i			0.696733i	−0.937358	+	0.348367i	$0.886736\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$	−1.00000	−	1.00000i	−0.0957826	−	0.0957826i	0.657592	−	0.753374i	$-0.271575\pi$
$109$	−1.00000	−	1.00000i	−0.0957826	−	0.0957826i	−0.753374	+	0.657592i	$0.771575\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	19.5959			1.84343			0.921714	−	0.387869i	$-0.126789\pi$
$113$	19.5959			1.84343			0.921714	+	0.387869i	$0.126789\pi$
$114$		0			0
$115$	16.9706	+	16.9706i	1.58251	+	1.58251i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	6.92820i		−	0.635107i
$120$		0			0
$121$		−	13.0000i		−	1.18182i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.89898	−	4.89898i	−0.438178	−	0.438178i
$126$		0			0
$127$	−9.89949			−0.878438			−0.439219	−	0.898380i	$-0.644745\pi$
$127$	−9.89949			−0.878438			−0.439219	+	0.898380i	$0.644745\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.92820	+	6.92820i	0.605320	+	0.605320i	0.941719	−	0.336399i	$-0.109209\pi$
$131$	6.92820	+	6.92820i	0.605320	+	0.605320i	−0.336399	+	0.941719i	$0.609209\pi$
$132$		0			0
$133$	−6.00000	+	6.00000i	−0.520266	+	0.520266i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	14.6969i		−	1.25564i	−0.778357	−	0.627822i	$-0.783947\pi$
$137$		−	14.6969i		−	1.25564i	0.778357	−	0.627822i	$-0.216053\pi$
$138$		0			0
$139$	−11.3137	+	11.3137i	−0.959616	+	0.959616i	−0.999216	−	0.0395994i	$-0.987392\pi$
$139$	−11.3137	+	11.3137i	−0.959616	+	0.959616i	0.0395994	+	0.999216i	$0.487392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.92820			−0.579365
$144$		0			0
$145$	12.0000			0.996546
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.44949	−	2.44949i	0.200670	−	0.200670i	−0.599617	−	0.800287i	$-0.704680\pi$
$149$	2.44949	−	2.44949i	0.200670	−	0.200670i	0.800287	+	0.599617i	$0.204680\pi$
$150$		0			0
$151$			15.5563i			1.26596i	0.774169	+	0.632979i	$0.218168\pi$
$151$			15.5563i			1.26596i	−0.774169	+	0.632979i	$0.781832\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.46410	−	3.46410i	0.278243	−	0.278243i
$156$		0			0
$157$	7.00000	+	7.00000i	0.558661	+	0.558661i	0.928926	−	0.370265i	$-0.120733\pi$
$157$	7.00000	+	7.00000i	0.558661	+	0.558661i	−0.370265	+	0.928926i	$0.620733\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.79796			−0.772187
$162$		0			0
$163$	−4.24264	−	4.24264i	−0.332309	−	0.332309i	0.521154	−	0.853463i	$-0.325502\pi$
$163$	−4.24264	−	4.24264i	−0.332309	−	0.332309i	−0.853463	+	0.521154i	$0.825502\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			6.92820i			0.536120i	0.963402	+	0.268060i	$0.0863826\pi$
$167$			6.92820i			0.536120i	−0.963402	+	0.268060i	$0.913617\pi$
$168$		0			0
$169$		−	11.0000i		−	0.846154i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12.2474	+	12.2474i	0.931156	+	0.931156i	0.997778	−	0.0666220i	$-0.0212222\pi$
$173$	12.2474	+	12.2474i	0.931156	+	0.931156i	−0.0666220	+	0.997778i	$0.521222\pi$
$174$		0			0
$175$	9.89949			0.748331
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$179$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$180$		0			0
$181$	1.00000	−	1.00000i	0.0743294	−	0.0743294i	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	1.00000	−	1.00000i	0.0743294	−	0.0743294i	0.743294	+	0.668965i	$0.233262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			34.2929i			2.52126i
$186$		0			0
$187$	16.9706	−	16.9706i	1.24101	−	1.24101i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−20.7846			−1.50392			−0.751961	−	0.659208i	$-0.770892\pi$
$191$	−20.7846			−1.50392			−0.751961	+	0.659208i	$0.770892\pi$
$192$		0			0
$193$	10.0000			0.719816			0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000			0.719816			0.359908	+	0.932988i	$0.382808\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.44949	−	2.44949i	0.174519	−	0.174519i	−0.614443	−	0.788962i	$-0.710619\pi$
$197$	2.44949	−	2.44949i	0.174519	−	0.174519i	0.788962	+	0.614443i	$0.210619\pi$
$198$		0			0
$199$		−	7.07107i		−	0.501255i	−0.968084	−	0.250627i	$-0.919363\pi$
$199$		−	7.07107i		−	0.501255i	0.968084	−	0.250627i	$-0.0806369\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.46410	+	3.46410i	−0.243132	+	0.243132i
$204$		0			0
$205$	12.0000	+	12.0000i	0.838116	+	0.838116i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−29.3939			−2.03322
$210$		0			0
$211$	11.3137	+	11.3137i	0.778868	+	0.778868i	0.979638	−	0.200770i	$-0.0643445\pi$
$211$	11.3137	+	11.3137i	0.778868	+	0.778868i	−0.200770	+	0.979638i	$0.564345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			20.7846i			1.41750i
$216$		0			0
$217$			2.00000i			0.135769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.89898	−	4.89898i	−0.329541	−	0.329541i
$222$		0			0
$223$	−7.07107			−0.473514			−0.236757	−	0.971569i	$-0.576084\pi$
$223$	−7.07107			−0.473514			−0.236757	+	0.971569i	$0.576084\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.46410	−	3.46410i	−0.229920	−	0.229920i	0.582739	−	0.812659i	$-0.301981\pi$
$227$	−3.46410	−	3.46410i	−0.229920	−	0.229920i	−0.812659	+	0.582739i	$0.801981\pi$
$228$		0			0
$229$	1.00000	−	1.00000i	0.0660819	−	0.0660819i	−0.673293	−	0.739375i	$-0.735121\pi$
$229$	1.00000	−	1.00000i	0.0660819	−	0.0660819i	0.739375	+	0.673293i	$0.235121\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			9.79796i			0.641886i	0.947099	+	0.320943i	$0.104000\pi$
$233$			9.79796i			0.641886i	−0.947099	+	0.320943i	$0.896000\pi$
$234$		0			0
$235$	−16.9706	+	16.9706i	−1.10704	+	1.10704i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.92820			0.448148			0.224074	−	0.974572i	$-0.428064\pi$
$239$	6.92820			0.448148			0.224074	+	0.974572i	$0.428064\pi$
$240$		0			0
$241$	−22.0000			−1.41714			−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000			−1.41714			−0.708572	+	0.705638i	$0.750660\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.2474	−	12.2474i	0.782461	−	0.782461i
$246$		0			0
$247$			8.48528i			0.539906i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−17.3205	+	17.3205i	−1.09326	+	1.09326i	−0.0980825	+	0.995178i	$0.531271\pi$
$251$	−17.3205	+	17.3205i	−1.09326	+	1.09326i	−0.995178	+	0.0980825i	$0.968729\pi$
$252$		0			0
$253$	−24.0000	−	24.0000i	−1.50887	−	1.50887i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.79796			−0.611180			−0.305590	−	0.952163i	$-0.598854\pi$
$257$	−9.79796			−0.611180			−0.305590	+	0.952163i	$0.598854\pi$
$258$		0			0
$259$	−9.89949	−	9.89949i	−0.615125	−	0.615125i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			13.8564i			0.854423i	0.904152	+	0.427211i	$0.140504\pi$
$263$			13.8564i			0.854423i	−0.904152	+	0.427211i	$0.859496\pi$
$264$		0			0
$265$			12.0000i			0.737154i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.34847	+	7.34847i	0.448044	+	0.448044i	0.894704	−	0.446660i	$-0.147387\pi$
$269$	7.34847	+	7.34847i	0.448044	+	0.448044i	−0.446660	+	0.894704i	$0.647387\pi$
$270$		0			0
$271$	7.07107			0.429537			0.214768	−	0.976665i	$-0.431100\pi$
$271$	7.07107			0.429537			0.214768	+	0.976665i	$0.431100\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	24.2487	+	24.2487i	1.46225	+	1.46225i
$276$		0			0
$277$	−1.00000	+	1.00000i	−0.0600842	+	0.0600842i	−0.736510	−	0.676426i	$-0.763528\pi$
$277$	−1.00000	+	1.00000i	−0.0600842	+	0.0600842i	0.676426	+	0.736510i	$0.263528\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	29.3939i		−	1.75349i	−0.480954	−	0.876746i	$-0.659710\pi$
$281$		−	29.3939i		−	1.75349i	0.480954	−	0.876746i	$-0.340290\pi$
$282$		0			0
$283$	5.65685	−	5.65685i	0.336265	−	0.336265i	−0.518695	−	0.854960i	$-0.673582\pi$
$283$	5.65685	−	5.65685i	0.336265	−	0.336265i	0.854960	+	0.518695i	$0.173582\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.92820			−0.408959
$288$		0			0
$289$	7.00000			0.411765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.2474	−	12.2474i	0.715504	−	0.715504i	−0.252177	−	0.967681i	$-0.581147\pi$
$293$	12.2474	−	12.2474i	0.715504	−	0.715504i	0.967681	+	0.252177i	$0.0811467\pi$
$294$		0			0
$295$		−	33.9411i		−	1.97613i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.92820	+	6.92820i	−0.400668	+	0.400668i
$300$		0			0
$301$	−6.00000	−	6.00000i	−0.345834	−	0.345834i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	24.4949			1.40257
$306$		0			0
$307$	−5.65685	−	5.65685i	−0.322854	−	0.322854i	0.527007	−	0.849861i	$-0.323314\pi$
$307$	−5.65685	−	5.65685i	−0.322854	−	0.322854i	−0.849861	+	0.527007i	$0.823314\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	13.8564i		−	0.785725i	−0.919597	−	0.392862i	$-0.871485\pi$
$311$		−	13.8564i		−	0.785725i	0.919597	−	0.392862i	$-0.128515\pi$
$312$		0			0
$313$			14.0000i			0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$			14.0000i			0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.34847	+	7.34847i	0.412731	+	0.412731i	0.882689	−	0.469958i	$-0.155731\pi$
$317$	7.34847	+	7.34847i	0.412731	+	0.412731i	−0.469958	+	0.882689i	$0.655731\pi$
$318$		0			0
$319$	−16.9706			−0.950169
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−20.7846	−	20.7846i	−1.15649	−	1.15649i
$324$		0			0
$325$	7.00000	−	7.00000i	0.388290	−	0.388290i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	9.79796i		−	0.540179i
$330$		0			0
$331$	11.3137	−	11.3137i	0.621858	−	0.621858i	−0.324149	−	0.946006i	$-0.605078\pi$
$331$	11.3137	−	11.3137i	0.621858	−	0.621858i	0.946006	+	0.324149i	$0.105078\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	27.7128			1.51411
$336$		0			0
$337$	−12.0000			−0.653682			−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000			−0.653682			−0.326841	+	0.945079i	$0.605984\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.89898	+	4.89898i	−0.265295	+	0.265295i
$342$		0			0
$343$			16.9706i			0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.46410	+	3.46410i	−0.185963	+	0.185963i	−0.793948	−	0.607985i	$-0.791978\pi$
$347$	−3.46410	+	3.46410i	−0.185963	+	0.185963i	0.607985	+	0.793948i	$0.291978\pi$
$348$		0			0
$349$	19.0000	+	19.0000i	1.01705	+	1.01705i	0.999852	+	0.0171945i	$0.00547346\pi$
$349$	19.0000	+	19.0000i	1.01705	+	1.01705i	0.0171945	+	0.999852i	$0.494527\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.79796			0.521493			0.260746	−	0.965407i	$-0.416031\pi$
$353$	9.79796			0.521493			0.260746	+	0.965407i	$0.416031\pi$
$354$		0			0
$355$	−33.9411	−	33.9411i	−1.80141	−	1.80141i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$			17.0000i			0.894737i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−29.3939	−	29.3939i	−1.53855	−	1.53855i
$366$		0			0
$367$	32.5269			1.69789			0.848945	−	0.528480i	$-0.177238\pi$
$367$	32.5269			1.69789			0.848945	+	0.528480i	$0.177238\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.46410	−	3.46410i	−0.179847	−	0.179847i
$372$		0			0
$373$	19.0000	−	19.0000i	0.983783	−	0.983783i	−0.0160879	−	0.999871i	$-0.505121\pi$
$373$	19.0000	−	19.0000i	0.983783	−	0.983783i	0.999871	+	0.0160879i	$0.00512115\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			4.89898i			0.252310i
$378$		0			0
$379$	4.24264	−	4.24264i	0.217930	−	0.217930i	−0.589696	−	0.807626i	$-0.700752\pi$
$379$	4.24264	−	4.24264i	0.217930	−	0.217930i	0.807626	+	0.589696i	$0.200752\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.92820			−0.354015			−0.177007	−	0.984210i	$-0.556642\pi$
$383$	−6.92820			−0.354015			−0.177007	+	0.984210i	$0.556642\pi$
$384$		0			0
$385$	−24.0000			−1.22315
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.1464	+	17.1464i	−0.869358	+	0.869358i	−0.992401	−	0.123043i	$-0.960735\pi$
$389$	−17.1464	+	17.1464i	−0.869358	+	0.869358i	0.123043	+	0.992401i	$0.460735\pi$
$390$		0			0
$391$		−	33.9411i		−	1.71648i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	38.1051	−	38.1051i	1.91728	−	1.91728i
$396$		0			0
$397$	−7.00000	−	7.00000i	−0.351320	−	0.351320i	0.509281	−	0.860601i	$-0.329912\pi$
$397$	−7.00000	−	7.00000i	−0.351320	−	0.351320i	−0.860601	+	0.509281i	$0.829912\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−14.6969			−0.733930			−0.366965	−	0.930235i	$-0.619603\pi$
$401$	−14.6969			−0.733930			−0.366965	+	0.930235i	$0.619603\pi$
$402$		0			0
$403$	1.41421	+	1.41421i	0.0704470	+	0.0704470i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	48.4974i		−	2.40393i
$408$		0			0
$409$			24.0000i			1.18672i	0.804936	+	0.593362i	$0.202200\pi$
$409$			24.0000i			1.18672i	−0.804936	+	0.593362i	$0.797800\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.79796	+	9.79796i	0.482126	+	0.482126i
$414$		0			0
$415$	−50.9117			−2.49916
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10.3923	−	10.3923i	−0.507697	−	0.507697i	0.406122	−	0.913819i	$-0.366881\pi$
$419$	−10.3923	−	10.3923i	−0.507697	−	0.507697i	−0.913819	+	0.406122i	$0.866881\pi$
$420$		0			0
$421$	25.0000	−	25.0000i	1.21843	−	1.21843i	0.250242	−	0.968183i	$-0.419490\pi$
$421$	25.0000	−	25.0000i	1.21843	−	1.21843i	0.968183	−	0.250242i	$-0.0805102\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			34.2929i			1.66345i
$426$		0			0
$427$	−7.07107	+	7.07107i	−0.342193	+	0.342193i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.92820			0.333720			0.166860	−	0.985981i	$-0.446637\pi$
$431$	6.92820			0.333720			0.166860	+	0.985981i	$0.446637\pi$
$432$		0			0
$433$	24.0000			1.15337			0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000			1.15337			0.576683	+	0.816968i	$0.304347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−29.3939	+	29.3939i	−1.40610	+	1.40610i
$438$		0			0
$439$		−	26.8701i		−	1.28244i	−0.767358	−	0.641219i	$-0.778429\pi$
$439$		−	26.8701i		−	1.28244i	0.767358	−	0.641219i	$-0.221571\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.3923	−	10.3923i	0.493753	−	0.493753i	−0.415733	−	0.909487i	$-0.636475\pi$
$443$	10.3923	−	10.3923i	0.493753	−	0.493753i	0.909487	+	0.415733i	$0.136475\pi$
$444$		0			0
$445$	−24.0000	−	24.0000i	−1.13771	−	1.13771i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	34.2929			1.61838			0.809190	−	0.587547i	$-0.199906\pi$
$449$	34.2929			1.61838			0.809190	+	0.587547i	$0.199906\pi$
$450$		0			0
$451$	−16.9706	−	16.9706i	−0.799113	−	0.799113i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			6.92820i			0.324799i
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.44949	−	2.44949i	−0.114084	−	0.114084i	0.647760	−	0.761844i	$-0.275706\pi$
$461$	−2.44949	−	2.44949i	−0.114084	−	0.114084i	−0.761844	+	0.647760i	$0.775706\pi$
$462$		0			0
$463$	−24.0416			−1.11731			−0.558655	−	0.829400i	$-0.688682\pi$
$463$	−24.0416			−1.11731			−0.558655	+	0.829400i	$0.688682\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.46410	+	3.46410i	0.160300	+	0.160300i	0.782699	−	0.622400i	$-0.213842\pi$
$467$	3.46410	+	3.46410i	0.160300	+	0.160300i	−0.622400	+	0.782699i	$0.713842\pi$
$468$		0			0
$469$	−8.00000	+	8.00000i	−0.369406	+	0.369406i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	29.3939i		−	1.35153i
$474$		0			0
$475$	29.6985	−	29.6985i	1.36266	−	1.36266i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−27.7128			−1.26623			−0.633115	−	0.774057i	$-0.718224\pi$
$479$	−27.7128			−1.26623			−0.633115	+	0.774057i	$0.718224\pi$
$480$		0			0
$481$	−14.0000			−0.638345
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	35.3553i		−	1.60210i	−0.598595	−	0.801052i	$-0.704274\pi$
$487$		−	35.3553i		−	1.60210i	0.598595	−	0.801052i	$-0.295726\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	20.7846	−	20.7846i	0.937996	−	0.937996i	−0.0601906	−	0.998187i	$-0.519171\pi$
$491$	20.7846	−	20.7846i	0.937996	−	0.937996i	0.998187	+	0.0601906i	$0.0191708\pi$
$492$		0			0
$493$	−12.0000	−	12.0000i	−0.540453	−	0.540453i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	19.5959			0.878997
$498$		0			0
$499$	5.65685	+	5.65685i	0.253236	+	0.253236i	0.822296	−	0.569060i	$-0.192693\pi$
$499$	5.65685	+	5.65685i	0.253236	+	0.253236i	−0.569060	+	0.822296i	$0.692693\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$			36.0000i			1.60198i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.34847	+	7.34847i	0.325715	+	0.325715i	0.850955	−	0.525239i	$-0.176024\pi$
$509$	7.34847	+	7.34847i	0.325715	+	0.325715i	−0.525239	+	0.850955i	$0.676024\pi$
$510$		0			0
$511$	16.9706			0.750733
$512$		0			0
$513$		0			0
$514$		0			0
$515$	17.3205	+	17.3205i	0.763233	+	0.763233i
$516$		0			0
$517$	24.0000	−	24.0000i	1.05552	−	1.05552i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	14.6969i		−	0.643885i	−0.946759	−	0.321942i	$-0.895664\pi$
$521$		−	14.6969i		−	0.643885i	0.946759	−	0.321942i	$-0.104336\pi$
$522$		0			0
$523$	−21.2132	+	21.2132i	−0.927589	+	0.927589i	−0.997550	−	0.0699611i	$-0.977712\pi$
$523$	−21.2132	+	21.2132i	−0.927589	+	0.927589i	0.0699611	+	0.997550i	$0.477712\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.92820			−0.301797
$528$		0			0
$529$	−25.0000			−1.08696
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.89898	+	4.89898i	−0.212198	+	0.212198i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−17.3205	+	17.3205i	−0.746047	+	0.746047i
$540$		0			0
$541$	11.0000	+	11.0000i	0.472927	+	0.472927i	0.902861	−	0.429934i	$-0.141463\pi$
$541$	11.0000	+	11.0000i	0.472927	+	0.472927i	−0.429934	+	0.902861i	$0.641463\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.89898			−0.209849
$546$		0			0
$547$	−12.7279	−	12.7279i	−0.544207	−	0.544207i	0.380553	−	0.924759i	$-0.375734\pi$
$547$	−12.7279	−	12.7279i	−0.544207	−	0.544207i	−0.924759	+	0.380553i	$0.875734\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			20.7846i			0.885454i
$552$		0			0
$553$			22.0000i			0.935535i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.34847	+	7.34847i	0.311365	+	0.311365i	0.845438	−	0.534073i	$-0.179339\pi$
$557$	7.34847	+	7.34847i	0.311365	+	0.311365i	−0.534073	+	0.845438i	$0.679339\pi$
$558$		0			0
$559$	−8.48528			−0.358889
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.46410	−	3.46410i	−0.145994	−	0.145994i	0.630332	−	0.776326i	$-0.282919\pi$
$563$	−3.46410	−	3.46410i	−0.145994	−	0.145994i	−0.776326	+	0.630332i	$0.782919\pi$
$564$		0			0
$565$	48.0000	−	48.0000i	2.01938	−	2.01938i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			14.6969i			0.616128i	0.951366	+	0.308064i	$0.0996810\pi$
$569$			14.6969i			0.616128i	−0.951366	+	0.308064i	$0.900319\pi$
$570$		0			0
$571$	28.2843	−	28.2843i	1.18366	−	1.18366i	0.204871	−	0.978789i	$-0.434323\pi$
$571$	28.2843	−	28.2843i	1.18366	−	1.18366i	0.978789	−	0.204871i	$-0.0656775\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	48.4974			2.02248
$576$		0			0
$577$	12.0000			0.499567			0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000			0.499567			0.249783	+	0.968302i	$0.419641\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	14.6969	−	14.6969i	0.609732	−	0.609732i
$582$		0			0
$583$		−	16.9706i		−	0.702849i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−27.7128	+	27.7128i	−1.14383	+	1.14383i	−0.156087	+	0.987743i	$0.549888\pi$
$587$	−27.7128	+	27.7128i	−1.14383	+	1.14383i	−0.987743	+	0.156087i	$0.950112\pi$
$588$		0			0
$589$	6.00000	+	6.00000i	0.247226	+	0.247226i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−39.1918			−1.60942			−0.804708	−	0.593671i	$-0.797678\pi$
$593$	−39.1918			−1.60942			−0.804708	+	0.593671i	$0.797678\pi$
$594$		0			0
$595$	−16.9706	−	16.9706i	−0.695725	−	0.695725i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	6.92820i		−	0.283079i	−0.989933	−	0.141539i	$-0.954795\pi$
$599$		−	6.92820i		−	0.283079i	0.989933	−	0.141539i	$-0.0452052\pi$
$600$		0			0
$601$		−	12.0000i		−	0.489490i	−0.969587	−	0.244745i	$-0.921296\pi$
$601$		−	12.0000i		−	0.489490i	0.969587	−	0.244745i	$-0.0787043\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−31.8434	−	31.8434i	−1.29462	−	1.29462i
$606$		0			0
$607$	−9.89949			−0.401808			−0.200904	−	0.979611i	$-0.564388\pi$
$607$	−9.89949			−0.401808			−0.200904	+	0.979611i	$0.564388\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6.92820	−	6.92820i	−0.280285	−	0.280285i
$612$		0			0
$613$	−19.0000	+	19.0000i	−0.767403	+	0.767403i	−0.977649	−	0.210246i	$-0.932574\pi$
$613$	−19.0000	+	19.0000i	−0.767403	+	0.767403i	0.210246	+	0.977649i	$0.432574\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	19.5959i		−	0.788902i	−0.918917	−	0.394451i	$-0.870935\pi$
$617$		−	19.5959i		−	0.788902i	0.918917	−	0.394451i	$-0.129065\pi$
$618$		0			0
$619$	11.3137	−	11.3137i	0.454736	−	0.454736i	−0.442187	−	0.896923i	$-0.645797\pi$
$619$	11.3137	−	11.3137i	0.454736	−	0.454736i	0.896923	+	0.442187i	$0.145797\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	13.8564			0.555145
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	34.2929	−	34.2929i	1.36735	−	1.36735i
$630$		0			0
$631$			1.41421i			0.0562990i	0.999604	+	0.0281495i	$0.00896144\pi$
$631$			1.41421i			0.0562990i	−0.999604	+	0.0281495i	$0.991039\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.2487	+	24.2487i	−0.962281	+	0.962281i
$636$		0			0
$637$	5.00000	+	5.00000i	0.198107	+	0.198107i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14.6969			−0.580494			−0.290247	−	0.956952i	$-0.593737\pi$
$641$	−14.6969			−0.580494			−0.290247	+	0.956952i	$0.593737\pi$
$642$		0			0
$643$	−29.6985	−	29.6985i	−1.17119	−	1.17119i	−0.981925	−	0.189269i	$-0.939388\pi$
$643$	−29.6985	−	29.6985i	−1.17119	−	1.17119i	−0.189269	−	0.981925i	$-0.560612\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			6.92820i			0.272376i	0.990683	+	0.136188i	$0.0434851\pi$
$647$			6.92820i			0.272376i	−0.990683	+	0.136188i	$0.956515\pi$
$648$		0			0
$649$			48.0000i			1.88416i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	26.9444	+	26.9444i	1.05442	+	1.05442i	0.998432	+	0.0559837i	$0.0178295\pi$
$653$	26.9444	+	26.9444i	1.05442	+	1.05442i	0.0559837	+	0.998432i	$0.482171\pi$
$654$		0			0
$655$	33.9411			1.32619
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.92820	−	6.92820i	−0.269884	−	0.269884i	0.559169	−	0.829054i	$-0.311120\pi$
$659$	−6.92820	−	6.92820i	−0.269884	−	0.269884i	−0.829054	+	0.559169i	$0.811120\pi$
$660$		0			0
$661$	−19.0000	+	19.0000i	−0.739014	+	0.739014i	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−19.0000	+	19.0000i	−0.739014	+	0.739014i	0.233373	+	0.972387i	$0.425024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			29.3939i			1.13985i
$666$		0			0
$667$	−16.9706	+	16.9706i	−0.657103	+	0.657103i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−34.6410			−1.33730
$672$		0			0
$673$	−24.0000			−0.925132			−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000			−0.925132			−0.462566	+	0.886585i	$0.653071\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	26.9444	−	26.9444i	1.03556	−	1.03556i	0.0362128	−	0.999344i	$-0.488471\pi$
$677$	26.9444	−	26.9444i	1.03556	−	1.03556i	0.999344	−	0.0362128i	$-0.0115294\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−17.3205	+	17.3205i	−0.662751	+	0.662751i	−0.956028	−	0.293277i	$-0.905254\pi$
$683$	−17.3205	+	17.3205i	−0.662751	+	0.662751i	0.293277	+	0.956028i	$0.405254\pi$
$684$		0			0
$685$	−36.0000	−	36.0000i	−1.37549	−	1.37549i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.89898			−0.186636
$690$		0			0
$691$	4.24264	+	4.24264i	0.161398	+	0.161398i	0.783186	−	0.621788i	$-0.213593\pi$
$691$	4.24264	+	4.24264i	0.161398	+	0.161398i	−0.621788	+	0.783186i	$0.713593\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			55.4256i			2.10241i
$696$		0			0
$697$		−	24.0000i		−	0.909065i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	2.44949	+	2.44949i	0.0925160	+	0.0925160i	0.751850	−	0.659334i	$-0.229162\pi$
$701$	2.44949	+	2.44949i	0.0925160	+	0.0925160i	−0.659334	+	0.751850i	$0.729162\pi$
$702$		0			0
$703$	−59.3970			−2.24020
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.3923	−	10.3923i	−0.390843	−	0.390843i
$708$		0			0
$709$	−1.00000	+	1.00000i	−0.0375558	+	0.0375558i	−0.725635	−	0.688080i	$-0.758454\pi$
$709$	−1.00000	+	1.00000i	−0.0375558	+	0.0375558i	0.688080	+	0.725635i	$0.258454\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			9.79796i			0.366936i
$714$		0			0
$715$	−16.9706	+	16.9706i	−0.634663	+	0.634663i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−41.5692			−1.55027			−0.775135	−	0.631795i	$-0.782318\pi$
$719$	−41.5692			−1.55027			−0.775135	+	0.631795i	$0.782318\pi$
$720$		0			0
$721$	−10.0000			−0.372419
$722$		0			0
$723$		0			0
$724$		0			0
$725$	17.1464	−	17.1464i	0.636802	−	0.636802i
$726$		0			0
$727$		−	15.5563i		−	0.576953i	−0.957487	−	0.288477i	$-0.906851\pi$
$727$		−	15.5563i		−	0.576953i	0.957487	−	0.288477i	$-0.0931487\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	20.7846	−	20.7846i	0.768747	−	0.768747i
$732$		0			0
$733$	−11.0000	−	11.0000i	−0.406294	−	0.406294i	0.474150	−	0.880444i	$-0.342755\pi$
$733$	−11.0000	−	11.0000i	−0.406294	−	0.406294i	−0.880444	+	0.474150i	$0.842755\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−39.1918			−1.44365
$738$		0			0
$739$	5.65685	+	5.65685i	0.208091	+	0.208091i	0.803456	−	0.595365i	$-0.202992\pi$
$739$	5.65685	+	5.65685i	0.208091	+	0.208091i	−0.595365	+	0.803456i	$0.702992\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	34.6410i		−	1.27086i	−0.772160	−	0.635428i	$-0.780824\pi$
$743$		−	34.6410i		−	1.27086i	0.772160	−	0.635428i	$-0.219176\pi$
$744$		0			0
$745$		−	12.0000i		−	0.439646i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	18.3848			0.670870			0.335435	−	0.942063i	$-0.391117\pi$
$751$	18.3848			0.670870			0.335435	+	0.942063i	$0.391117\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	38.1051	+	38.1051i	1.38679	+	1.38679i
$756$		0			0
$757$	25.0000	−	25.0000i	0.908640	−	0.908640i	−0.0875221	−	0.996163i	$-0.527895\pi$
$757$	25.0000	−	25.0000i	0.908640	−	0.908640i	0.996163	+	0.0875221i	$0.0278948\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			4.89898i			0.177588i	0.996050	+	0.0887939i	$0.0283013\pi$
$761$			4.89898i			0.177588i	−0.996050	+	0.0887939i	$0.971699\pi$
$762$		0			0
$763$	1.41421	−	1.41421i	0.0511980	−	0.0511980i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	13.8564			0.500326
$768$		0			0
$769$	−10.0000			−0.360609			−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000			−0.360609			−0.180305	+	0.983611i	$0.557708\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−26.9444	+	26.9444i	−0.969122	+	0.969122i	−0.999537	−	0.0304151i	$-0.990317\pi$
$773$	−26.9444	+	26.9444i	−0.969122	+	0.969122i	0.0304151	+	0.999537i	$0.490317\pi$
$774$		0			0
$775$		−	9.89949i		−	0.355600i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−20.7846	+	20.7846i	−0.744686	+	0.744686i
$780$		0			0
$781$	48.0000	+	48.0000i	1.71758	+	1.71758i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	34.2929			1.22396
$786$		0			0
$787$	12.7279	+	12.7279i	0.453701	+	0.453701i	0.896581	−	0.442880i	$-0.146043\pi$
$787$	12.7279	+	12.7279i	0.453701	+	0.453701i	−0.442880	+	0.896581i	$0.646043\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			27.7128i			0.985354i
$792$		0			0
$793$			10.0000i			0.355110i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	36.7423	+	36.7423i	1.30148	+	1.30148i	0.927394	+	0.374087i	$0.122044\pi$
$797$	36.7423	+	36.7423i	1.30148	+	1.30148i	0.374087	+	0.927394i	$0.377956\pi$
$798$		0			0
$799$	33.9411			1.20075
$800$		0			0
$801$		0			0
$802$		0			0
$803$	41.5692	+	41.5692i	1.46695	+	1.46695i
$804$		0			0
$805$	−24.0000	+	24.0000i	−0.845889	+	0.845889i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	4.89898i		−	0.172239i	−0.996285	−	0.0861195i	$-0.972553\pi$
$809$		−	4.89898i		−	0.172239i	0.996285	−	0.0861195i	$-0.0274467\pi$
$810$		0			0
$811$	4.24264	−	4.24264i	0.148979	−	0.148979i	−0.628683	−	0.777662i	$-0.716406\pi$
$811$	4.24264	−	4.24264i	0.148979	−	0.148979i	0.777662	+	0.628683i	$0.216406\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−20.7846			−0.728053
$816$		0			0
$817$	−36.0000			−1.25948
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.34847	+	7.34847i	−0.256463	+	0.256463i	−0.823614	−	0.567151i	$-0.808046\pi$
$821$	−7.34847	+	7.34847i	−0.256463	+	0.256463i	0.567151	+	0.823614i	$0.308046\pi$
$822$		0			0
$823$			35.3553i			1.23241i	0.787586	+	0.616205i	$0.211331\pi$
$823$			35.3553i			1.23241i	−0.787586	+	0.616205i	$0.788669\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	34.6410	−	34.6410i	1.20459	−	1.20459i	0.231830	−	0.972756i	$-0.425529\pi$
$827$	34.6410	−	34.6410i	1.20459	−	1.20459i	0.972756	−	0.231830i	$-0.0744712\pi$
$828$		0			0
$829$	−23.0000	−	23.0000i	−0.798823	−	0.798823i	0.184087	−	0.982910i	$-0.441067\pi$
$829$	−23.0000	−	23.0000i	−0.798823	−	0.798823i	−0.982910	+	0.184087i	$0.941067\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−24.4949			−0.848698
$834$		0			0
$835$	16.9706	+	16.9706i	0.587291	+	0.587291i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			13.8564i			0.478376i	0.970973	+	0.239188i	$0.0768813\pi$
$839$			13.8564i			0.478376i	−0.970973	+	0.239188i	$0.923119\pi$
$840$		0			0
$841$		−	17.0000i		−	0.586207i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−26.9444	−	26.9444i	−0.926915	−	0.926915i
$846$		0			0
$847$	18.3848			0.631708
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−48.4974	−	48.4974i	−1.66247	−	1.66247i
$852$		0			0
$853$	−29.0000	+	29.0000i	−0.992941	+	0.992941i	−0.999975	−	0.00703417i	$-0.997761\pi$
$853$	−29.0000	+	29.0000i	−0.992941	+	0.992941i	0.00703417	+	0.999975i	$0.497761\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			24.4949i			0.836730i	0.908279	+	0.418365i	$0.137397\pi$
$857$			24.4949i			0.836730i	−0.908279	+	0.418365i	$0.862603\pi$
$858$		0			0
$859$	12.7279	−	12.7279i	0.434271	−	0.434271i	−0.455807	−	0.890078i	$-0.650649\pi$
$859$	12.7279	−	12.7279i	0.434271	−	0.434271i	0.890078	+	0.455807i	$0.150649\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6.92820			0.235839			0.117919	−	0.993023i	$-0.462378\pi$
$863$	6.92820			0.235839			0.117919	+	0.993023i	$0.462378\pi$
$864$		0			0
$865$	60.0000			2.04006
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−53.8888	+	53.8888i	−1.82805	+	1.82805i
$870$		0			0
$871$			11.3137i			0.383350i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6.92820	−	6.92820i	0.234216	−	0.234216i
$876$		0			0
$877$	−5.00000	−	5.00000i	−0.168838	−	0.168838i	0.617630	−	0.786468i	$-0.288093\pi$
$877$	−5.00000	−	5.00000i	−0.168838	−	0.168838i	−0.786468	+	0.617630i	$0.788093\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9.79796			0.330102			0.165051	−	0.986285i	$-0.447221\pi$
$881$	9.79796			0.330102			0.165051	+	0.986285i	$0.447221\pi$
$882$		0			0
$883$	−12.7279	−	12.7279i	−0.428329	−	0.428329i	0.459730	−	0.888059i	$-0.347946\pi$
$883$	−12.7279	−	12.7279i	−0.428329	−	0.428329i	−0.888059	+	0.459730i	$0.847946\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			27.7128i			0.930505i	0.885178	+	0.465253i	$0.154037\pi$
$887$			27.7128i			0.930505i	−0.885178	+	0.465253i	$0.845963\pi$
$888$		0			0
$889$		−	14.0000i		−	0.469545i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−29.3939	−	29.3939i	−0.983629	−	0.983629i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.46410	+	3.46410i	0.115534	+	0.115534i
$900$		0			0
$901$	12.0000	−	12.0000i	0.399778	−	0.399778i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	4.89898i		−	0.162848i
$906$		0			0
$907$	21.2132	−	21.2132i	0.704373	−	0.704373i	−0.260973	−	0.965346i	$-0.584043\pi$
$907$	21.2132	−	21.2132i	0.704373	−	0.704373i	0.965346	+	0.260973i	$0.0840434\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	34.6410			1.14771			0.573854	−	0.818958i	$-0.305448\pi$
$911$	34.6410			1.14771			0.573854	+	0.818958i	$0.305448\pi$
$912$		0			0
$913$	72.0000			2.38285
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.79796	+	9.79796i	−0.323557	+	0.323557i
$918$		0			0
$919$			35.3553i			1.16627i	0.812377	+	0.583133i	$0.198173\pi$
$919$			35.3553i			1.16627i	−0.812377	+	0.583133i	$0.801827\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	13.8564	−	13.8564i	0.456089	−	0.456089i
$924$		0			0
$925$	49.0000	+	49.0000i	1.61111	+	1.61111i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	24.4949			0.803652			0.401826	−	0.915716i	$-0.368376\pi$
$929$	24.4949			0.803652			0.401826	+	0.915716i	$0.368376\pi$
$930$		0			0
$931$	21.2132	+	21.2132i	0.695235	+	0.695235i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	83.1384i		−	2.71892i
$936$		0			0
$937$		−	22.0000i		−	0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$		−	22.0000i		−	0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	36.7423	+	36.7423i	1.19777	+	1.19777i	0.974834	+	0.222932i	$0.0715629\pi$
$941$	36.7423	+	36.7423i	1.19777	+	1.19777i	0.222932	+	0.974834i	$0.428437\pi$
$942$		0			0
$943$	−33.9411			−1.10528
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27.7128	−	27.7128i	−0.900545	−	0.900545i	0.0949378	−	0.995483i	$-0.469735\pi$
$947$	−27.7128	−	27.7128i	−0.900545	−	0.900545i	−0.995483	+	0.0949378i	$0.969735\pi$
$948$		0			0
$949$	12.0000	−	12.0000i	0.389536	−	0.389536i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	44.0908i		−	1.42824i	−0.700022	−	0.714121i	$-0.746827\pi$
$953$		−	44.0908i		−	1.42824i	0.700022	−	0.714121i	$-0.253173\pi$
$954$		0			0
$955$	−50.9117	+	50.9117i	−1.64746	+	1.64746i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	20.7846			0.671170
$960$		0			0
$961$	−29.0000			−0.935484
$962$		0			0
$963$		0			0
$964$		0			0
$965$	24.4949	−	24.4949i	0.788519	−	0.788519i
$966$		0			0
$967$			24.0416i			0.773127i	0.922263	+	0.386563i	$0.126338\pi$
$967$			24.0416i			0.773127i	−0.922263	+	0.386563i	$0.873662\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−10.3923	+	10.3923i	−0.333505	+	0.333505i	−0.853916	−	0.520411i	$-0.825779\pi$
$971$	−10.3923	+	10.3923i	−0.333505	+	0.333505i	0.520411	+	0.853916i	$0.325779\pi$
$972$		0			0
$973$	−16.0000	−	16.0000i	−0.512936	−	0.512936i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	53.8888			1.72405			0.862027	−	0.506862i	$-0.169195\pi$
$977$	53.8888			1.72405			0.862027	+	0.506862i	$0.169195\pi$
$978$		0			0
$979$	33.9411	+	33.9411i	1.08476	+	1.08476i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	55.4256i		−	1.76780i	−0.467673	−	0.883901i	$-0.654908\pi$
$983$		−	55.4256i		−	1.76780i	0.467673	−	0.883901i	$-0.345092\pi$
$984$		0			0
$985$		−	12.0000i		−	0.382352i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−29.3939	−	29.3939i	−0.934671	−	0.934671i
$990$		0			0
$991$	15.5563			0.494164			0.247082	−	0.968995i	$-0.420528\pi$
$991$	15.5563			0.494164			0.247082	+	0.968995i	$0.420528\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−17.3205	−	17.3205i	−0.549097	−	0.549097i
$996$		0			0
$997$	−7.00000	+	7.00000i	−0.221692	+	0.221692i	−0.809211	−	0.587519i	$-0.800105\pi$
$997$	−7.00000	+	7.00000i	−0.221692	+	0.221692i	0.587519	+	0.809211i	$0.300105\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.k.i.577.4	yes	8
3.2	odd	2	inner	2304.2.k.i.577.2	yes	8
4.3	odd	2	inner	2304.2.k.i.577.3	yes	8
8.3	odd	2		2304.2.k.h.577.1	✓	8
8.5	even	2		2304.2.k.h.577.2	yes	8
12.11	even	2	inner	2304.2.k.i.577.1	yes	8
16.3	odd	4		2304.2.k.h.1729.2	yes	8
16.5	even	4	inner	2304.2.k.i.1729.3	yes	8
16.11	odd	4	inner	2304.2.k.i.1729.4	yes	8
16.13	even	4		2304.2.k.h.1729.1	yes	8
24.5	odd	2		2304.2.k.h.577.4	yes	8
24.11	even	2		2304.2.k.h.577.3	yes	8
32.5	even	8		9216.2.a.bh.1.3		4
32.11	odd	8		9216.2.a.bh.1.2		4
32.21	even	8		9216.2.a.be.1.1		4
32.27	odd	8		9216.2.a.be.1.4		4
48.5	odd	4	inner	2304.2.k.i.1729.1	yes	8
48.11	even	4	inner	2304.2.k.i.1729.2	yes	8
48.29	odd	4		2304.2.k.h.1729.3	yes	8
48.35	even	4		2304.2.k.h.1729.4	yes	8
96.5	odd	8		9216.2.a.bh.1.1		4
96.11	even	8		9216.2.a.bh.1.4		4
96.53	odd	8		9216.2.a.be.1.3		4
96.59	even	8		9216.2.a.be.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2304.2.k.h.577.1	✓	8	8.3	odd	2
2304.2.k.h.577.2	yes	8	8.5	even	2
2304.2.k.h.577.3	yes	8	24.11	even	2
2304.2.k.h.577.4	yes	8	24.5	odd	2
2304.2.k.h.1729.1	yes	8	16.13	even	4
2304.2.k.h.1729.2	yes	8	16.3	odd	4
2304.2.k.h.1729.3	yes	8	48.29	odd	4
2304.2.k.h.1729.4	yes	8	48.35	even	4
2304.2.k.i.577.1	yes	8	12.11	even	2	inner
2304.2.k.i.577.2	yes	8	3.2	odd	2	inner
2304.2.k.i.577.3	yes	8	4.3	odd	2	inner
2304.2.k.i.577.4	yes	8	1.1	even	1	trivial
2304.2.k.i.1729.1	yes	8	48.5	odd	4	inner
2304.2.k.i.1729.2	yes	8	48.11	even	4	inner
2304.2.k.i.1729.3	yes	8	16.5	even	4	inner
2304.2.k.i.1729.4	yes	8	16.11	odd	4	inner
9216.2.a.be.1.1		4	32.21	even	8
9216.2.a.be.1.2		4	96.59	even	8
9216.2.a.be.1.3		4	96.53	odd	8
9216.2.a.be.1.4		4	32.27	odd	8
9216.2.a.bh.1.1		4	96.5	odd	8
9216.2.a.bh.1.2		4	32.11	odd	8
9216.2.a.bh.1.3		4	32.5	even	8
9216.2.a.bh.1.4		4	96.11	even	8

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.44949 - 2.44949i) q^{5} +1.41421i q^{7} +(-3.46410 + 3.46410i) q^{11} +(1.00000 + 1.00000i) q^{13} -4.89898 q^{17} +(4.24264 + 4.24264i) q^{19} +6.92820i q^{23} -7.00000i q^{25} +(2.44949 + 2.44949i) q^{29} +1.41421 q^{31} +(3.46410 + 3.46410i) q^{35} +(-7.00000 + 7.00000i) q^{37} +4.89898i q^{41} +(-4.24264 + 4.24264i) q^{43} -6.92820 q^{47} +5.00000 q^{49} +(-2.44949 + 2.44949i) q^{53} +16.9706i q^{55} +(6.92820 - 6.92820i) q^{59} +(5.00000 + 5.00000i) q^{61} +4.89898 q^{65} +(5.65685 + 5.65685i) q^{67} -13.8564i q^{71} -12.0000i q^{73} +(-4.89898 - 4.89898i) q^{77} +15.5563 q^{79} +(-10.3923 - 10.3923i) q^{83} +(-12.0000 + 12.0000i) q^{85} -9.79796i q^{89} +(-1.41421 + 1.41421i) q^{91} +20.7846 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{13} - 56 q^{37} + 40 q^{49} + 40 q^{61} - 96 q^{85}+O(q^{100}) \) 8 * q + 8 * q^13 - 56 * q^37 + 40 * q^49 + 40 * q^61 - 96 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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