LMFDB - Embedded newform 2304.2.k.d.1729.1

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:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 768)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1729.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2304.1729
Dual form			2304.2.k.d.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 + 2.00000i) q^{5} -4.24264i q^{7} +O(q^{10})$	$q+(2.00000 + 2.00000i) q^{5} -4.24264i q^{7} +(-2.82843 - 2.82843i) q^{11} +(-3.00000 + 3.00000i) q^{13} +6.00000 q^{17} +(-1.41421 + 1.41421i) q^{19} -2.82843i q^{23} +3.00000i q^{25} +(4.00000 - 4.00000i) q^{29} +4.24264 q^{31} +(8.48528 - 8.48528i) q^{35} +(-3.00000 - 3.00000i) q^{37} -10.0000i q^{41} +(-4.24264 - 4.24264i) q^{43} +2.82843 q^{47} -11.0000 q^{49} +(4.00000 + 4.00000i) q^{53} -11.3137i q^{55} +(-3.00000 + 3.00000i) q^{61} -12.0000 q^{65} +(2.82843 - 2.82843i) q^{67} +2.82843i q^{71} -16.0000i q^{73} +(-12.0000 + 12.0000i) q^{77} -4.24264 q^{79} +(11.3137 - 11.3137i) q^{83} +(12.0000 + 12.0000i) q^{85} -14.0000i q^{89} +(12.7279 + 12.7279i) q^{91} -5.65685 q^{95} -4.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5}+O(q^{10}) $ 4 * q + 8 * q^5	$ 4 q + 8 q^{5} - 12 q^{13} + 24 q^{17} + 16 q^{29} - 12 q^{37} - 44 q^{49} + 16 q^{53} - 12 q^{61} - 48 q^{65} - 48 q^{77} + 48 q^{85} - 16 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 - 12 * q^13 + 24 * q^17 + 16 * q^29 - 12 * q^37 - 44 * q^49 + 16 * q^53 - 12 * q^61 - 48 * q^65 - 48 * q^77 + 48 * q^85 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	2.00000	+	2.00000i	0.894427	+	0.894427i	0.994936	−	0.100509i	$-0.0320471\pi$
$5$	2.00000	+	2.00000i	0.894427	+	0.894427i	−0.100509	+	0.994936i	$0.532047\pi$
$6$		0			0
$7$		−	4.24264i		−	1.60357i	−0.597614	−	0.801784i	$-0.703885\pi$
$7$		−	4.24264i		−	1.60357i	0.597614	−	0.801784i	$-0.296115\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.82843	−	2.82843i	−0.852803	−	0.852803i	0.137675	−	0.990478i	$-0.456037\pi$
$11$	−2.82843	−	2.82843i	−0.852803	−	0.852803i	−0.990478	+	0.137675i	$0.956037\pi$
$12$		0			0
$13$	−3.00000	+	3.00000i	−0.832050	+	0.832050i	−0.987797	−	0.155747i	$-0.950222\pi$
$13$	−3.00000	+	3.00000i	−0.832050	+	0.832050i	0.155747	+	0.987797i	$0.450222\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$	−1.41421	+	1.41421i	−0.324443	+	0.324443i	−0.850469	−	0.526026i	$-0.823682\pi$
$19$	−1.41421	+	1.41421i	−0.324443	+	0.324443i	0.526026	+	0.850469i	$0.323682\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.82843i		−	0.589768i	−0.955533	−	0.294884i	$-0.904719\pi$
$23$		−	2.82843i		−	0.589768i	0.955533	−	0.294884i	$-0.0952810\pi$
$24$		0			0
$25$			3.00000i			0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.00000	−	4.00000i	0.742781	−	0.742781i	−0.230331	−	0.973112i	$-0.573981\pi$
$29$	4.00000	−	4.00000i	0.742781	−	0.742781i	0.973112	+	0.230331i	$0.0739809\pi$
$30$		0			0
$31$	4.24264			0.762001			0.381000	−	0.924575i	$-0.375580\pi$
$31$	4.24264			0.762001			0.381000	+	0.924575i	$0.375580\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.48528	−	8.48528i	1.43427	−	1.43427i
$36$		0			0
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	0.416115	−	0.909312i	$-0.363391\pi$
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	−0.909312	+	0.416115i	$0.863391\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$		0			0
$43$	−4.24264	−	4.24264i	−0.646997	−	0.646997i	0.305269	−	0.952266i	$-0.401253\pi$
$43$	−4.24264	−	4.24264i	−0.646997	−	0.646997i	−0.952266	+	0.305269i	$0.901253\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.82843			0.412568			0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843			0.412568			0.206284	+	0.978492i	$0.433863\pi$
$48$		0			0
$49$	−11.0000			−1.57143
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.00000	+	4.00000i	0.549442	+	0.549442i	0.926279	−	0.376837i	$-0.122988\pi$
$53$	4.00000	+	4.00000i	0.549442	+	0.549442i	−0.376837	+	0.926279i	$0.622988\pi$
$54$		0			0
$55$		−	11.3137i		−	1.52554i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	−3.00000	+	3.00000i	−0.384111	+	0.384111i	−0.872581	−	0.488470i	$-0.837555\pi$
$61$	−3.00000	+	3.00000i	−0.384111	+	0.384111i	0.488470	+	0.872581i	$0.337555\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.0000			−1.48842
$66$		0			0
$67$	2.82843	−	2.82843i	0.345547	−	0.345547i	−0.512901	−	0.858448i	$-0.671429\pi$
$67$	2.82843	−	2.82843i	0.345547	−	0.345547i	0.858448	+	0.512901i	$0.171429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.82843i			0.335673i	0.985815	+	0.167836i	$0.0536780\pi$
$71$			2.82843i			0.335673i	−0.985815	+	0.167836i	$0.946322\pi$
$72$		0			0
$73$		−	16.0000i		−	1.87266i	−0.351123	−	0.936329i	$-0.614200\pi$
$73$		−	16.0000i		−	1.87266i	0.351123	−	0.936329i	$-0.385800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.0000	+	12.0000i	−1.36753	+	1.36753i
$78$		0			0
$79$	−4.24264			−0.477334			−0.238667	−	0.971101i	$-0.576710\pi$
$79$	−4.24264			−0.477334			−0.238667	+	0.971101i	$0.576710\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.3137	−	11.3137i	1.24184	−	1.24184i	0.282604	−	0.959237i	$-0.408802\pi$
$83$	11.3137	−	11.3137i	1.24184	−	1.24184i	0.959237	−	0.282604i	$-0.0911983\pi$
$84$		0			0
$85$	12.0000	+	12.0000i	1.30158	+	1.30158i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	14.0000i		−	1.48400i	−0.670402	−	0.741999i	$-0.733878\pi$
$89$		−	14.0000i		−	1.48400i	0.670402	−	0.741999i	$-0.266122\pi$
$90$		0			0
$91$	12.7279	+	12.7279i	1.33425	+	1.33425i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.65685			−0.580381
$96$		0			0
$97$	−4.00000			−0.406138			−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000			−0.406138			−0.203069	+	0.979164i	$0.565092\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.00000	+	8.00000i	0.796030	+	0.796030i	0.982467	−	0.186437i	$-0.0596941\pi$
$101$	8.00000	+	8.00000i	0.796030	+	0.796030i	−0.186437	+	0.982467i	$0.559694\pi$
$102$		0			0
$103$			18.3848i			1.81151i	0.423806	+	0.905753i	$0.360694\pi$
$103$			18.3848i			1.81151i	−0.423806	+	0.905753i	$0.639306\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.82843	−	2.82843i	−0.273434	−	0.273434i	0.557047	−	0.830481i	$-0.311934\pi$
$107$	−2.82843	−	2.82843i	−0.273434	−	0.273434i	−0.830481	+	0.557047i	$0.811934\pi$
$108$		0			0
$109$	7.00000	−	7.00000i	0.670478	−	0.670478i	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	7.00000	−	7.00000i	0.670478	−	0.670478i	0.957826	+	0.287348i	$0.0927736\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	5.65685	−	5.65685i	0.527504	−	0.527504i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	25.4558i		−	2.33353i
$120$		0			0
$121$			5.00000i			0.454545i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.00000	−	4.00000i	0.357771	−	0.357771i
$126$		0			0
$127$	4.24264			0.376473			0.188237	−	0.982124i	$-0.439723\pi$
$127$	4.24264			0.376473			0.188237	+	0.982124i	$0.439723\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−14.1421	+	14.1421i	−1.23560	+	1.23560i	−0.273824	+	0.961780i	$0.588289\pi$
$131$	−14.1421	+	14.1421i	−1.23560	+	1.23560i	−0.961780	+	0.273824i	$0.911711\pi$
$132$		0			0
$133$	6.00000	+	6.00000i	0.520266	+	0.520266i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			10.0000i			0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$			10.0000i			0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$		0			0
$139$	−8.48528	−	8.48528i	−0.719712	−	0.719712i	0.248834	−	0.968546i	$-0.419953\pi$
$139$	−8.48528	−	8.48528i	−0.719712	−	0.719712i	−0.968546	+	0.248834i	$0.919953\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	16.9706			1.41915
$144$		0			0
$145$	16.0000			1.32873
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.00000	+	6.00000i	0.491539	+	0.491539i	0.908791	−	0.417252i	$-0.137007\pi$
$149$	6.00000	+	6.00000i	0.491539	+	0.491539i	−0.417252	+	0.908791i	$0.637007\pi$
$150$		0			0
$151$		−	1.41421i		−	0.115087i	−0.998343	−	0.0575435i	$-0.981673\pi$
$151$		−	1.41421i		−	0.115087i	0.998343	−	0.0575435i	$-0.0183268\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.48528	+	8.48528i	0.681554	+	0.681554i
$156$		0			0
$157$	−5.00000	+	5.00000i	−0.399043	+	0.399043i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−5.00000	+	5.00000i	−0.399043	+	0.399043i	0.478852	+	0.877896i	$0.341053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$	7.07107	−	7.07107i	0.553849	−	0.553849i	−0.373701	−	0.927549i	$-0.621911\pi$
$163$	7.07107	−	7.07107i	0.553849	−	0.553849i	0.927549	+	0.373701i	$0.121911\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	11.3137i		−	0.875481i	−0.899101	−	0.437741i	$-0.855779\pi$
$167$		−	11.3137i		−	0.875481i	0.899101	−	0.437741i	$-0.144221\pi$
$168$		0			0
$169$		−	5.00000i		−	0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.00000	+	2.00000i	−0.152057	+	0.152057i	−0.779036	−	0.626979i	$-0.784291\pi$
$173$	−2.00000	+	2.00000i	−0.152057	+	0.152057i	0.626979	+	0.779036i	$0.284291\pi$
$174$		0			0
$175$	12.7279			0.962140
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$179$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$180$		0			0
$181$	1.00000	+	1.00000i	0.0743294	+	0.0743294i	0.743294	−	0.668965i	$-0.233262\pi$
$181$	1.00000	+	1.00000i	0.0743294	+	0.0743294i	−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	12.0000i		−	0.882258i
$186$		0			0
$187$	−16.9706	−	16.9706i	−1.24101	−	1.24101i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	22.6274			1.63726			0.818631	−	0.574320i	$-0.194733\pi$
$191$	22.6274			1.63726			0.818631	+	0.574320i	$0.194733\pi$
$192$		0			0
$193$	18.0000			1.29567			0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000			1.29567			0.647834	+	0.761781i	$0.275675\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.00000	+	4.00000i	0.284988	+	0.284988i	0.835095	−	0.550106i	$-0.185413\pi$
$197$	4.00000	+	4.00000i	0.284988	+	0.284988i	−0.550106	+	0.835095i	$0.685413\pi$
$198$		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
$203$	−16.9706	−	16.9706i	−1.19110	−	1.19110i
$204$		0			0
$205$	20.0000	−	20.0000i	1.39686	−	1.39686i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.00000			0.553372
$210$		0			0
$211$	8.48528	−	8.48528i	0.584151	−	0.584151i	−0.351890	−	0.936041i	$-0.614461\pi$
$211$	8.48528	−	8.48528i	0.584151	−	0.584151i	0.936041	+	0.351890i	$0.114461\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	16.9706i		−	1.15738i
$216$		0			0
$217$		−	18.0000i		−	1.22192i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−18.0000	+	18.0000i	−1.21081	+	1.21081i
$222$		0			0
$223$	−21.2132			−1.42054			−0.710271	−	0.703929i	$-0.751427\pi$
$223$	−21.2132			−1.42054			−0.710271	+	0.703929i	$0.751427\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$	−11.0000	−	11.0000i	−0.726900	−	0.726900i	0.243101	−	0.970001i	$-0.421835\pi$
$229$	−11.0000	−	11.0000i	−0.726900	−	0.726900i	−0.970001	+	0.243101i	$0.921835\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			18.0000i			1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$			18.0000i			1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$		0			0
$235$	5.65685	+	5.65685i	0.369012	+	0.369012i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.3137			−0.731823			−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137			−0.731823			−0.365911	+	0.930650i	$0.619243\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−22.0000	−	22.0000i	−1.40553	−	1.40553i
$246$		0			0
$247$		−	8.48528i		−	0.539906i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.82843	+	2.82843i	0.178529	+	0.178529i	0.790714	−	0.612185i	$-0.209709\pi$
$251$	2.82843	+	2.82843i	0.178529	+	0.178529i	−0.612185	+	0.790714i	$0.709709\pi$
$252$		0			0
$253$	−8.00000	+	8.00000i	−0.502956	+	0.502956i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	14.0000			0.873296			0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000			0.873296			0.436648	+	0.899632i	$0.356166\pi$
$258$		0			0
$259$	−12.7279	+	12.7279i	−0.790875	+	0.790875i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			22.6274i			1.39527i	0.716455	+	0.697633i	$0.245763\pi$
$263$			22.6274i			1.39527i	−0.716455	+	0.697633i	$0.754237\pi$
$264$		0			0
$265$			16.0000i			0.982872i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$269$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$270$		0			0
$271$	−1.41421			−0.0859074			−0.0429537	−	0.999077i	$-0.513677\pi$
$271$	−1.41421			−0.0859074			−0.0429537	+	0.999077i	$0.513677\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.48528	−	8.48528i	0.511682	−	0.511682i
$276$		0			0
$277$	−1.00000	−	1.00000i	−0.0600842	−	0.0600842i	0.676426	−	0.736510i	$-0.263528\pi$
$277$	−1.00000	−	1.00000i	−0.0600842	−	0.0600842i	−0.736510	+	0.676426i	$0.763528\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	6.00000i		−	0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$		−	6.00000i		−	0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$		0			0
$283$	−2.82843	−	2.82843i	−0.168133	−	0.168133i	0.618026	−	0.786158i	$-0.287933\pi$
$283$	−2.82843	−	2.82843i	−0.168133	−	0.168133i	−0.786158	+	0.618026i	$0.787933\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−42.4264			−2.50435
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.00000	+	4.00000i	0.233682	+	0.233682i	0.814228	−	0.580545i	$-0.197161\pi$
$293$	4.00000	+	4.00000i	0.233682	+	0.233682i	−0.580545	+	0.814228i	$0.697161\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.48528	+	8.48528i	0.490716	+	0.490716i
$300$		0			0
$301$	−18.0000	+	18.0000i	−1.03750	+	1.03750i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.0000			−0.687118
$306$		0			0
$307$	2.82843	−	2.82843i	0.161427	−	0.161427i	−0.621772	−	0.783199i	$-0.713587\pi$
$307$	2.82843	−	2.82843i	0.161427	−	0.161427i	0.783199	+	0.621772i	$0.213587\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			28.2843i			1.60385i	0.597422	+	0.801927i	$0.296192\pi$
$311$			28.2843i			1.60385i	−0.597422	+	0.801927i	$0.703808\pi$
$312$		0			0
$313$			26.0000i			1.46961i	0.678280	+	0.734803i	$0.262726\pi$
$313$			26.0000i			1.46961i	−0.678280	+	0.734803i	$0.737274\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.0000	−	12.0000i	0.673987	−	0.673987i	−0.284646	−	0.958633i	$-0.591876\pi$
$317$	12.0000	−	12.0000i	0.673987	−	0.673987i	0.958633	+	0.284646i	$0.0918759\pi$
$318$		0			0
$319$	−22.6274			−1.26689
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.48528	+	8.48528i	−0.472134	+	0.472134i
$324$		0			0
$325$	−9.00000	−	9.00000i	−0.499230	−	0.499230i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	12.0000i		−	0.661581i
$330$		0			0
$331$	2.82843	+	2.82843i	0.155464	+	0.155464i	0.780553	−	0.625089i	$-0.214937\pi$
$331$	2.82843	+	2.82843i	0.155464	+	0.155464i	−0.625089	+	0.780553i	$0.714937\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	11.3137			0.618134
$336$		0			0
$337$	−8.00000			−0.435788			−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000			−0.435788			−0.217894	+	0.975972i	$0.569919\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−12.0000	−	12.0000i	−0.649836	−	0.649836i
$342$		0			0
$343$			16.9706i			0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	16.9706	+	16.9706i	0.911028	+	0.911028i	0.996353	−	0.0853256i	$-0.0271930\pi$
$347$	16.9706	+	16.9706i	0.911028	+	0.911028i	−0.0853256	+	0.996353i	$0.527193\pi$
$348$		0			0
$349$	3.00000	−	3.00000i	0.160586	−	0.160586i	−0.622240	−	0.782826i	$-0.713777\pi$
$349$	3.00000	−	3.00000i	0.160586	−	0.160586i	0.782826	+	0.622240i	$0.213777\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$	−5.65685	+	5.65685i	−0.300235	+	0.300235i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			2.82843i			0.149279i	0.997211	+	0.0746393i	$0.0237806\pi$
$359$			2.82843i			0.149279i	−0.997211	+	0.0746393i	$0.976219\pi$
$360$		0			0
$361$			15.0000i			0.789474i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	32.0000	−	32.0000i	1.67496	−	1.67496i
$366$		0			0
$367$	18.3848			0.959678			0.479839	−	0.877357i	$-0.340695\pi$
$367$	18.3848			0.959678			0.479839	+	0.877357i	$0.340695\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	16.9706	−	16.9706i	0.881068	−	0.881068i
$372$		0			0
$373$	−13.0000	−	13.0000i	−0.673114	−	0.673114i	0.285318	−	0.958433i	$-0.407901\pi$
$373$	−13.0000	−	13.0000i	−0.673114	−	0.673114i	−0.958433	+	0.285318i	$0.907901\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			24.0000i			1.23606i
$378$		0			0
$379$	26.8701	+	26.8701i	1.38022	+	1.38022i	0.844211	+	0.536011i	$0.180070\pi$
$379$	26.8701	+	26.8701i	1.38022	+	1.38022i	0.536011	+	0.844211i	$0.319930\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−28.2843			−1.44526			−0.722629	−	0.691236i	$-0.757067\pi$
$383$	−28.2843			−1.44526			−0.722629	+	0.691236i	$0.757067\pi$
$384$		0			0
$385$	−48.0000			−2.44631
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.00000	−	2.00000i	−0.101404	−	0.101404i	0.654585	−	0.755989i	$-0.272844\pi$
$389$	−2.00000	−	2.00000i	−0.101404	−	0.101404i	−0.755989	+	0.654585i	$0.772844\pi$
$390$		0			0
$391$		−	16.9706i		−	0.858238i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.48528	−	8.48528i	−0.426941	−	0.426941i
$396$		0			0
$397$	−23.0000	+	23.0000i	−1.15434	+	1.15434i	−0.168663	+	0.985674i	$0.553945\pi$
$397$	−23.0000	+	23.0000i	−1.15434	+	1.15434i	−0.985674	+	0.168663i	$0.946055\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10.0000			0.499376			0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000			0.499376			0.249688	+	0.968326i	$0.419672\pi$
$402$		0			0
$403$	−12.7279	+	12.7279i	−0.634023	+	0.634023i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			16.9706i			0.841200i
$408$		0			0
$409$		−	28.0000i		−	1.38451i	−0.721653	−	0.692255i	$-0.756617\pi$
$409$		−	28.0000i		−	1.38451i	0.721653	−	0.692255i	$-0.243383\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	45.2548			2.22147
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11.3137	+	11.3137i	−0.552711	+	0.552711i	−0.927222	−	0.374511i	$-0.877810\pi$
$419$	−11.3137	+	11.3137i	−0.552711	+	0.552711i	0.374511	+	0.927222i	$0.377810\pi$
$420$		0			0
$421$	25.0000	+	25.0000i	1.21843	+	1.21843i	0.968183	+	0.250242i	$0.0805102\pi$
$421$	25.0000	+	25.0000i	1.21843	+	1.21843i	0.250242	+	0.968183i	$0.419490\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			18.0000i			0.873128i
$426$		0			0
$427$	12.7279	+	12.7279i	0.615947	+	0.615947i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	36.7696			1.77113			0.885564	−	0.464518i	$-0.153773\pi$
$431$	36.7696			1.77113			0.885564	+	0.464518i	$0.153773\pi$
$432$		0			0
$433$	−8.00000			−0.384455			−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000			−0.384455			−0.192228	+	0.981350i	$0.561571\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.00000	+	4.00000i	0.191346	+	0.191346i
$438$		0			0
$439$			1.41421i			0.0674967i	0.999430	+	0.0337484i	$0.0107445\pi$
$439$			1.41421i			0.0674967i	−0.999430	+	0.0337484i	$0.989256\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−16.9706	−	16.9706i	−0.806296	−	0.806296i	0.177775	−	0.984071i	$-0.443110\pi$
$443$	−16.9706	−	16.9706i	−0.806296	−	0.806296i	−0.984071	+	0.177775i	$0.943110\pi$
$444$		0			0
$445$	28.0000	−	28.0000i	1.32733	−	1.32733i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	26.0000			1.22702			0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000			1.22702			0.613508	+	0.789689i	$0.289758\pi$
$450$		0			0
$451$	−28.2843	+	28.2843i	−1.33185	+	1.33185i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			50.9117i			2.38678i
$456$		0			0
$457$		−	12.0000i		−	0.561336i	−0.959805	−	0.280668i	$-0.909444\pi$
$457$		−	12.0000i		−	0.561336i	0.959805	−	0.280668i	$-0.0905560\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.00000	−	6.00000i	0.279448	−	0.279448i	−0.553441	−	0.832889i	$-0.686685\pi$
$461$	6.00000	−	6.00000i	0.279448	−	0.279448i	0.832889	+	0.553441i	$0.186685\pi$
$462$		0			0
$463$	12.7279			0.591517			0.295758	−	0.955263i	$-0.404428\pi$
$463$	12.7279			0.591517			0.295758	+	0.955263i	$0.404428\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.9706	+	16.9706i	−0.785304	+	0.785304i	−0.980720	−	0.195416i	$-0.937394\pi$
$467$	−16.9706	+	16.9706i	−0.785304	+	0.785304i	0.195416	+	0.980720i	$0.437394\pi$
$468$		0			0
$469$	−12.0000	−	12.0000i	−0.554109	−	0.554109i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			24.0000i			1.10352i
$474$		0			0
$475$	−4.24264	−	4.24264i	−0.194666	−	0.194666i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−25.4558			−1.16311			−0.581554	−	0.813508i	$-0.697555\pi$
$479$	−25.4558			−1.16311			−0.581554	+	0.813508i	$0.697555\pi$
$480$		0			0
$481$	18.0000			0.820729
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.00000	−	8.00000i	−0.363261	−	0.363261i
$486$		0			0
$487$			43.8406i			1.98661i	0.115529	+	0.993304i	$0.463144\pi$
$487$			43.8406i			1.98661i	−0.115529	+	0.993304i	$0.536856\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.65685	−	5.65685i	−0.255290	−	0.255290i	0.567845	−	0.823135i	$-0.307777\pi$
$491$	−5.65685	−	5.65685i	−0.255290	−	0.255290i	−0.823135	+	0.567845i	$0.807777\pi$
$492$		0			0
$493$	24.0000	−	24.0000i	1.08091	−	1.08091i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.0000			0.538274
$498$		0			0
$499$	−14.1421	+	14.1421i	−0.633089	+	0.633089i	−0.948842	−	0.315753i	$-0.897743\pi$
$499$	−14.1421	+	14.1421i	−0.633089	+	0.633089i	0.315753	+	0.948842i	$0.397743\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			14.1421i			0.630567i	0.948998	+	0.315283i	$0.102100\pi$
$503$			14.1421i			0.630567i	−0.948998	+	0.315283i	$0.897900\pi$
$504$		0			0
$505$			32.0000i			1.42398i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	16.0000	−	16.0000i	0.709188	−	0.709188i	−0.257177	−	0.966364i	$-0.582792\pi$
$509$	16.0000	−	16.0000i	0.709188	−	0.709188i	0.966364	+	0.257177i	$0.0827923\pi$
$510$		0			0
$511$	−67.8823			−3.00293
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−36.7696	+	36.7696i	−1.62026	+	1.62026i
$516$		0			0
$517$	−8.00000	−	8.00000i	−0.351840	−	0.351840i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	22.0000i		−	0.963837i	−0.876216	−	0.481919i	$-0.839940\pi$
$521$		−	22.0000i		−	0.963837i	0.876216	−	0.481919i	$-0.160060\pi$
$522$		0			0
$523$	−15.5563	−	15.5563i	−0.680232	−	0.680232i	0.279821	−	0.960052i	$-0.409725\pi$
$523$	−15.5563	−	15.5563i	−0.680232	−	0.680232i	−0.960052	+	0.279821i	$0.909725\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	25.4558			1.10887
$528$		0			0
$529$	15.0000			0.652174
$530$		0			0
$531$		0			0
$532$		0			0
$533$	30.0000	+	30.0000i	1.29944	+	1.29944i
$534$		0			0
$535$		−	11.3137i		−	0.489134i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	31.1127	+	31.1127i	1.34012	+	1.34012i
$540$		0			0
$541$	−29.0000	+	29.0000i	−1.24681	+	1.24681i	−0.289685	+	0.957122i	$0.593551\pi$
$541$	−29.0000	+	29.0000i	−1.24681	+	1.24681i	−0.957122	+	0.289685i	$0.906449\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	28.0000			1.19939
$546$		0			0
$547$	−24.0416	+	24.0416i	−1.02795	+	1.02795i	−0.0283478	+	0.999598i	$0.509025\pi$
$547$	−24.0416	+	24.0416i	−1.02795	+	1.02795i	−0.999598	+	0.0283478i	$0.990975\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			11.3137i			0.481980i
$552$		0			0
$553$			18.0000i			0.765438i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14.0000	+	14.0000i	−0.593199	+	0.593199i	−0.938494	−	0.345295i	$-0.887779\pi$
$557$	−14.0000	+	14.0000i	−0.593199	+	0.593199i	0.345295	+	0.938494i	$0.387779\pi$
$558$		0			0
$559$	25.4558			1.07667
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.82843	+	2.82843i	−0.119204	+	0.119204i	−0.764192	−	0.644988i	$-0.776862\pi$
$563$	−2.82843	+	2.82843i	−0.119204	+	0.119204i	0.644988	+	0.764192i	$0.276862\pi$
$564$		0			0
$565$	−12.0000	−	12.0000i	−0.504844	−	0.504844i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			18.0000i			0.754599i	0.926091	+	0.377300i	$0.123147\pi$
$569$			18.0000i			0.754599i	−0.926091	+	0.377300i	$0.876853\pi$
$570$		0			0
$571$	14.1421	+	14.1421i	0.591830	+	0.591830i	0.938125	−	0.346296i	$-0.112561\pi$
$571$	14.1421	+	14.1421i	0.591830	+	0.591830i	−0.346296	+	0.938125i	$0.612561\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.48528			0.353861
$576$		0			0
$577$	4.00000			0.166522			0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000			0.166522			0.0832611	+	0.996528i	$0.473466\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−48.0000	−	48.0000i	−1.99138	−	1.99138i
$582$		0			0
$583$		−	22.6274i		−	0.937132i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.9706	+	16.9706i	0.700450	+	0.700450i	0.964507	−	0.264057i	$-0.0850607\pi$
$587$	16.9706	+	16.9706i	0.700450	+	0.700450i	−0.264057	+	0.964507i	$0.585061\pi$
$588$		0			0
$589$	−6.00000	+	6.00000i	−0.247226	+	0.247226i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−14.0000			−0.574911			−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000			−0.574911			−0.287456	+	0.957794i	$0.592809\pi$
$594$		0			0
$595$	50.9117	−	50.9117i	2.08718	−	2.08718i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	48.0833i		−	1.96463i	−0.187239	−	0.982314i	$-0.559954\pi$
$599$		−	48.0833i		−	1.96463i	0.187239	−	0.982314i	$-0.440046\pi$
$600$		0			0
$601$			28.0000i			1.14214i	0.820900	+	0.571072i	$0.193472\pi$
$601$			28.0000i			1.14214i	−0.820900	+	0.571072i	$0.806528\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.0000	+	10.0000i	−0.406558	+	0.406558i
$606$		0			0
$607$	4.24264			0.172203			0.0861017	−	0.996286i	$-0.472559\pi$
$607$	4.24264			0.172203			0.0861017	+	0.996286i	$0.472559\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.48528	+	8.48528i	−0.343278	+	0.343278i
$612$		0			0
$613$	13.0000	+	13.0000i	0.525065	+	0.525065i	0.919097	−	0.394032i	$-0.128920\pi$
$613$	13.0000	+	13.0000i	0.525065	+	0.525065i	−0.394032	+	0.919097i	$0.628920\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			22.0000i			0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$			22.0000i			0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$		0			0
$619$	−25.4558	−	25.4558i	−1.02316	−	1.02316i	−0.999725	−	0.0234313i	$-0.992541\pi$
$619$	−25.4558	−	25.4558i	−1.02316	−	1.02316i	−0.0234313	−	0.999725i	$-0.507459\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−59.3970			−2.37969
$624$		0			0
$625$	31.0000			1.24000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−18.0000	−	18.0000i	−0.717707	−	0.717707i
$630$		0			0
$631$			35.3553i			1.40747i	0.710461	+	0.703737i	$0.248487\pi$
$631$			35.3553i			1.40747i	−0.710461	+	0.703737i	$0.751513\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.48528	+	8.48528i	0.336728	+	0.336728i
$636$		0			0
$637$	33.0000	−	33.0000i	1.30751	−	1.30751i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$	9.89949	−	9.89949i	0.390398	−	0.390398i	−0.484431	−	0.874829i	$-0.660973\pi$
$643$	9.89949	−	9.89949i	0.390398	−	0.390398i	0.874829	+	0.484431i	$0.160973\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			14.1421i			0.555985i	0.960583	+	0.277992i	$0.0896690\pi$
$647$			14.1421i			0.555985i	−0.960583	+	0.277992i	$0.910331\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.00000	+	6.00000i	−0.234798	+	0.234798i	−0.814692	−	0.579894i	$-0.803094\pi$
$653$	−6.00000	+	6.00000i	−0.234798	+	0.234798i	0.579894	+	0.814692i	$0.303094\pi$
$654$		0			0
$655$	−56.5685			−2.21032
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$659$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$660$		0			0
$661$	9.00000	+	9.00000i	0.350059	+	0.350059i	0.860132	−	0.510072i	$-0.170381\pi$
$661$	9.00000	+	9.00000i	0.350059	+	0.350059i	−0.510072	+	0.860132i	$0.670381\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			24.0000i			0.930680i
$666$		0			0
$667$	−11.3137	−	11.3137i	−0.438069	−	0.438069i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	16.9706			0.655141
$672$		0			0
$673$	−40.0000			−1.54189			−0.770943	−	0.636904i	$-0.780215\pi$
$673$	−40.0000			−1.54189			−0.770943	+	0.636904i	$0.780215\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−26.0000	−	26.0000i	−0.999261	−	0.999261i	0.000738553	−	1.00000i	$-0.499765\pi$
$677$	−26.0000	−	26.0000i	−0.999261	−	0.999261i	−1.00000		0.000738553i	$0.999765\pi$
$678$		0			0
$679$			16.9706i			0.651270i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−16.9706	−	16.9706i	−0.649361	−	0.649361i	0.303478	−	0.952838i	$-0.401852\pi$
$683$	−16.9706	−	16.9706i	−0.649361	−	0.649361i	−0.952838	+	0.303478i	$0.901852\pi$
$684$		0			0
$685$	−20.0000	+	20.0000i	−0.764161	+	0.764161i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−24.0000			−0.914327
$690$		0			0
$691$	15.5563	−	15.5563i	0.591791	−	0.591791i	−0.346324	−	0.938115i	$-0.612570\pi$
$691$	15.5563	−	15.5563i	0.591791	−	0.591791i	0.938115	+	0.346324i	$0.112570\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	33.9411i		−	1.28746i
$696$		0			0
$697$		−	60.0000i		−	2.27266i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−4.00000	+	4.00000i	−0.151078	+	0.151078i	−0.778599	−	0.627521i	$-0.784069\pi$
$701$	−4.00000	+	4.00000i	−0.151078	+	0.151078i	0.627521	+	0.778599i	$0.284069\pi$
$702$		0			0
$703$	8.48528			0.320028
$704$		0			0
$705$		0			0
$706$		0			0
$707$	33.9411	−	33.9411i	1.27649	−	1.27649i
$708$		0			0
$709$	−13.0000	−	13.0000i	−0.488225	−	0.488225i	0.419521	−	0.907746i	$-0.362198\pi$
$709$	−13.0000	−	13.0000i	−0.488225	−	0.488225i	−0.907746	+	0.419521i	$0.862198\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	12.0000i		−	0.449404i
$714$		0			0
$715$	33.9411	+	33.9411i	1.26933	+	1.26933i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.48528			−0.316448			−0.158224	−	0.987403i	$-0.550577\pi$
$719$	−8.48528			−0.316448			−0.158224	+	0.987403i	$0.550577\pi$
$720$		0			0
$721$	78.0000			2.90487
$722$		0			0
$723$		0			0
$724$		0			0
$725$	12.0000	+	12.0000i	0.445669	+	0.445669i
$726$		0			0
$727$		−	26.8701i		−	0.996555i	−0.867018	−	0.498278i	$-0.833966\pi$
$727$		−	26.8701i		−	0.996555i	0.867018	−	0.498278i	$-0.166034\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−25.4558	−	25.4558i	−0.941518	−	0.941518i
$732$		0			0
$733$	25.0000	−	25.0000i	0.923396	−	0.923396i	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	25.0000	−	25.0000i	0.923396	−	0.923396i	0.997268	+	0.0738717i	$0.0235355\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−16.0000			−0.589368
$738$		0			0
$739$	25.4558	−	25.4558i	0.936408	−	0.936408i	−0.0616872	−	0.998096i	$-0.519648\pi$
$739$	25.4558	−	25.4558i	0.936408	−	0.936408i	0.998096	+	0.0616872i	$0.0196481\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	22.6274i		−	0.830119i	−0.909794	−	0.415060i	$-0.863761\pi$
$743$		−	22.6274i		−	0.830119i	0.909794	−	0.415060i	$-0.136239\pi$
$744$		0			0
$745$			24.0000i			0.879292i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12.0000	+	12.0000i	−0.438470	+	0.438470i
$750$		0			0
$751$	−12.7279			−0.464448			−0.232224	−	0.972662i	$-0.574600\pi$
$751$	−12.7279			−0.464448			−0.232224	+	0.972662i	$0.574600\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.82843	−	2.82843i	0.102937	−	0.102937i
$756$		0			0
$757$	−27.0000	−	27.0000i	−0.981332	−	0.981332i	0.0184972	−	0.999829i	$-0.494112\pi$
$757$	−27.0000	−	27.0000i	−0.981332	−	0.981332i	−0.999829	+	0.0184972i	$0.994112\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			34.0000i			1.23250i	0.787551	+	0.616250i	$0.211349\pi$
$761$			34.0000i			1.23250i	−0.787551	+	0.616250i	$0.788651\pi$
$762$		0			0
$763$	−29.6985	−	29.6985i	−1.07516	−	1.07516i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.00000	−	4.00000i	−0.143870	−	0.143870i	0.631503	−	0.775373i	$-0.282438\pi$
$773$	−4.00000	−	4.00000i	−0.143870	−	0.143870i	−0.775373	+	0.631503i	$0.782438\pi$
$774$		0			0
$775$			12.7279i			0.457200i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14.1421	+	14.1421i	0.506695	+	0.506695i
$780$		0			0
$781$	8.00000	−	8.00000i	0.286263	−	0.286263i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−20.0000			−0.713831
$786$		0			0
$787$	−26.8701	+	26.8701i	−0.957814	+	0.957814i	−0.999145	−	0.0413314i	$-0.986840\pi$
$787$	−26.8701	+	26.8701i	−0.957814	+	0.957814i	0.0413314	+	0.999145i	$0.486840\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			25.4558i			0.905106i
$792$		0			0
$793$		−	18.0000i		−	0.639199i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−22.0000	+	22.0000i	−0.779280	+	0.779280i	−0.979708	−	0.200428i	$-0.935767\pi$
$797$	−22.0000	+	22.0000i	−0.779280	+	0.779280i	0.200428	+	0.979708i	$0.435767\pi$
$798$		0			0
$799$	16.9706			0.600375
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−45.2548	+	45.2548i	−1.59701	+	1.59701i
$804$		0			0
$805$	−24.0000	−	24.0000i	−0.845889	−	0.845889i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	30.0000i		−	1.05474i	−0.849635	−	0.527372i	$-0.823177\pi$
$809$		−	30.0000i		−	1.05474i	0.849635	−	0.527372i	$-0.176823\pi$
$810$		0			0
$811$	−1.41421	−	1.41421i	−0.0496598	−	0.0496598i	0.681841	−	0.731501i	$-0.261180\pi$
$811$	−1.41421	−	1.41421i	−0.0496598	−	0.0496598i	−0.731501	+	0.681841i	$0.761180\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	28.2843			0.990755
$816$		0			0
$817$	12.0000			0.419827
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−20.0000	−	20.0000i	−0.698005	−	0.698005i	0.265975	−	0.963980i	$-0.414306\pi$
$821$	−20.0000	−	20.0000i	−0.698005	−	0.698005i	−0.963980	+	0.265975i	$0.914306\pi$
$822$		0			0
$823$		−	4.24264i		−	0.147889i	−0.997262	−	0.0739446i	$-0.976441\pi$
$823$		−	4.24264i		−	0.147889i	0.997262	−	0.0739446i	$-0.0235588\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.4558	+	25.4558i	0.885186	+	0.885186i	0.994056	−	0.108870i	$-0.0347231\pi$
$827$	25.4558	+	25.4558i	0.885186	+	0.885186i	−0.108870	+	0.994056i	$0.534723\pi$
$828$		0			0
$829$	−15.0000	+	15.0000i	−0.520972	+	0.520972i	−0.917865	−	0.396893i	$-0.870088\pi$
$829$	−15.0000	+	15.0000i	−0.520972	+	0.520972i	0.396893	+	0.917865i	$0.370088\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−66.0000			−2.28676
$834$		0			0
$835$	22.6274	−	22.6274i	0.783054	−	0.783054i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			31.1127i			1.07413i	0.843541	+	0.537065i	$0.180467\pi$
$839$			31.1127i			1.07413i	−0.843541	+	0.537065i	$0.819533\pi$
$840$		0			0
$841$		−	3.00000i		−	0.103448i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.0000	−	10.0000i	0.344010	−	0.344010i
$846$		0			0
$847$	21.2132			0.728894
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.48528	+	8.48528i	−0.290872	+	0.290872i
$852$		0			0
$853$	15.0000	+	15.0000i	0.513590	+	0.513590i	0.915625	−	0.402034i	$-0.131697\pi$
$853$	15.0000	+	15.0000i	0.513590	+	0.513590i	−0.402034	+	0.915625i	$0.631697\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	30.0000i		−	1.02478i	−0.858753	−	0.512390i	$-0.828760\pi$
$857$		−	30.0000i		−	1.02478i	0.858753	−	0.512390i	$-0.171240\pi$
$858$		0			0
$859$	7.07107	+	7.07107i	0.241262	+	0.241262i	0.817372	−	0.576110i	$-0.195430\pi$
$859$	7.07107	+	7.07107i	0.241262	+	0.241262i	−0.576110	+	0.817372i	$0.695430\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−28.2843			−0.962808			−0.481404	−	0.876499i	$-0.659873\pi$
$863$	−28.2843			−0.962808			−0.481404	+	0.876499i	$0.659873\pi$
$864$		0			0
$865$	−8.00000			−0.272008
$866$		0			0
$867$		0			0
$868$		0			0
$869$	12.0000	+	12.0000i	0.407072	+	0.407072i
$870$		0			0
$871$			16.9706i			0.575026i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−16.9706	−	16.9706i	−0.573710	−	0.573710i
$876$		0			0
$877$	11.0000	−	11.0000i	0.371444	−	0.371444i	−0.496559	−	0.868003i	$-0.665403\pi$
$877$	11.0000	−	11.0000i	0.371444	−	0.371444i	0.868003	+	0.496559i	$0.165403\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	58.0000			1.95407			0.977035	−	0.213080i	$-0.0683494\pi$
$881$	58.0000			1.95407			0.977035	+	0.213080i	$0.0683494\pi$
$882$		0			0
$883$	−24.0416	+	24.0416i	−0.809065	+	0.809065i	−0.984492	−	0.175427i	$-0.943869\pi$
$883$	−24.0416	+	24.0416i	−0.809065	+	0.809065i	0.175427	+	0.984492i	$0.443869\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	16.9706i		−	0.569816i	−0.958555	−	0.284908i	$-0.908037\pi$
$887$		−	16.9706i		−	0.569816i	0.958555	−	0.284908i	$-0.0919630\pi$
$888$		0			0
$889$		−	18.0000i		−	0.603701i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.00000	+	4.00000i	−0.133855	+	0.133855i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	16.9706	−	16.9706i	0.566000	−	0.566000i
$900$		0			0
$901$	24.0000	+	24.0000i	0.799556	+	0.799556i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			4.00000i			0.132964i
$906$		0			0
$907$	−41.0122	−	41.0122i	−1.36179	−	1.36179i	−0.871639	−	0.490149i	$-0.836942\pi$
$907$	−41.0122	−	41.0122i	−1.36179	−	1.36179i	−0.490149	−	0.871639i	$-0.663058\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	22.6274			0.749680			0.374840	−	0.927090i	$-0.377698\pi$
$911$	22.6274			0.749680			0.374840	+	0.927090i	$0.377698\pi$
$912$		0			0
$913$	−64.0000			−2.11809
$914$		0			0
$915$		0			0
$916$		0			0
$917$	60.0000	+	60.0000i	1.98137	+	1.98137i
$918$		0			0
$919$			18.3848i			0.606458i	0.952918	+	0.303229i	$0.0980647\pi$
$919$			18.3848i			0.606458i	−0.952918	+	0.303229i	$0.901935\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8.48528	−	8.48528i	−0.279296	−	0.279296i
$924$		0			0
$925$	9.00000	−	9.00000i	0.295918	−	0.295918i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	46.0000			1.50921			0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000			1.50921			0.754606	+	0.656179i	$0.227828\pi$
$930$		0			0
$931$	15.5563	−	15.5563i	0.509839	−	0.509839i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	67.8823i		−	2.21999i
$936$		0			0
$937$		−	2.00000i		−	0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$		−	2.00000i		−	0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	26.0000	−	26.0000i	0.847576	−	0.847576i	−0.142254	−	0.989830i	$-0.545435\pi$
$941$	26.0000	−	26.0000i	0.847576	−	0.847576i	0.989830	+	0.142254i	$0.0454351\pi$
$942$		0			0
$943$	−28.2843			−0.921063
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$	48.0000	+	48.0000i	1.55815	+	1.55815i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	18.0000i		−	0.583077i	−0.956559	−	0.291539i	$-0.905833\pi$
$953$		−	18.0000i		−	0.583077i	0.956559	−	0.291539i	$-0.0941672\pi$
$954$		0			0
$955$	45.2548	+	45.2548i	1.46441	+	1.46441i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	42.4264			1.37002
$960$		0			0
$961$	−13.0000			−0.419355
$962$		0			0
$963$		0			0
$964$		0			0
$965$	36.0000	+	36.0000i	1.15888	+	1.15888i
$966$		0			0
$967$			35.3553i			1.13695i	0.822700	+	0.568476i	$0.192467\pi$
$967$			35.3553i			1.13695i	−0.822700	+	0.568476i	$0.807533\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−22.6274	−	22.6274i	−0.726148	−	0.726148i	0.243702	−	0.969850i	$-0.421638\pi$
$971$	−22.6274	−	22.6274i	−0.726148	−	0.726148i	−0.969850	+	0.243702i	$0.921638\pi$
$972$		0			0
$973$	−36.0000	+	36.0000i	−1.15411	+	1.15411i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.00000			0.0639857			0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000			0.0639857			0.0319928	+	0.999488i	$0.489815\pi$
$978$		0			0
$979$	−39.5980	+	39.5980i	−1.26556	+	1.26556i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	22.6274i		−	0.721703i	−0.932623	−	0.360851i	$-0.882486\pi$
$983$		−	22.6274i		−	0.721703i	0.932623	−	0.360851i	$-0.117514\pi$
$984$		0			0
$985$			16.0000i			0.509802i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.0000	+	12.0000i	−0.381578	+	0.381578i
$990$		0			0
$991$	46.6690			1.48249			0.741246	−	0.671234i	$-0.234235\pi$
$991$	46.6690			1.48249			0.741246	+	0.671234i	$0.234235\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	14.1421	−	14.1421i	0.448336	−	0.448336i
$996$		0			0
$997$	33.0000	+	33.0000i	1.04512	+	1.04512i	0.998933	+	0.0461877i	$0.0147072\pi$
$997$	33.0000	+	33.0000i	1.04512	+	1.04512i	0.0461877	+	0.998933i	$0.485293\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.k.d.1729.1		4
3.2	odd	2		768.2.j.a.193.1	✓	4
4.3	odd	2	inner	2304.2.k.d.1729.2		4
8.3	odd	2		2304.2.k.a.1729.2		4
8.5	even	2		2304.2.k.a.1729.1		4
12.11	even	2		768.2.j.a.193.2	yes	4
16.3	odd	4	inner	2304.2.k.d.577.1		4
16.5	even	4		2304.2.k.a.577.2		4
16.11	odd	4		2304.2.k.a.577.1		4
16.13	even	4	inner	2304.2.k.d.577.2		4
24.5	odd	2		768.2.j.d.193.2	yes	4
24.11	even	2		768.2.j.d.193.1	yes	4
32.3	odd	8		9216.2.a.q.1.1		2
32.13	even	8		9216.2.a.q.1.2		2
32.19	odd	8		9216.2.a.e.1.2		2
32.29	even	8		9216.2.a.e.1.1		2
48.5	odd	4		768.2.j.d.577.2	yes	4
48.11	even	4		768.2.j.d.577.1	yes	4
48.29	odd	4		768.2.j.a.577.1	yes	4
48.35	even	4		768.2.j.a.577.2	yes	4
96.5	odd	8		3072.2.d.d.1537.1		4
96.11	even	8		3072.2.d.d.1537.2		4
96.29	odd	8		3072.2.a.d.1.2		2
96.35	even	8		3072.2.a.f.1.2		2
96.53	odd	8		3072.2.d.d.1537.4		4
96.59	even	8		3072.2.d.d.1537.3		4
96.77	odd	8		3072.2.a.f.1.1		2
96.83	even	8		3072.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.2.j.a.193.1	✓	4	3.2	odd	2
768.2.j.a.193.2	yes	4	12.11	even	2
768.2.j.a.577.1	yes	4	48.29	odd	4
768.2.j.a.577.2	yes	4	48.35	even	4
768.2.j.d.193.1	yes	4	24.11	even	2
768.2.j.d.193.2	yes	4	24.5	odd	2
768.2.j.d.577.1	yes	4	48.11	even	4
768.2.j.d.577.2	yes	4	48.5	odd	4
2304.2.k.a.577.1		4	16.11	odd	4
2304.2.k.a.577.2		4	16.5	even	4
2304.2.k.a.1729.1		4	8.5	even	2
2304.2.k.a.1729.2		4	8.3	odd	2
2304.2.k.d.577.1		4	16.3	odd	4	inner
2304.2.k.d.577.2		4	16.13	even	4	inner
2304.2.k.d.1729.1		4	1.1	even	1	trivial
2304.2.k.d.1729.2		4	4.3	odd	2	inner
3072.2.a.d.1.1		2	96.83	even	8
3072.2.a.d.1.2		2	96.29	odd	8
3072.2.a.f.1.1		2	96.77	odd	8
3072.2.a.f.1.2		2	96.35	even	8
3072.2.d.d.1537.1		4	96.5	odd	8
3072.2.d.d.1537.2		4	96.11	even	8
3072.2.d.d.1537.3		4	96.59	even	8
3072.2.d.d.1537.4		4	96.53	odd	8
9216.2.a.e.1.1		2	32.29	even	8
9216.2.a.e.1.2		2	32.19	odd	8
9216.2.a.q.1.1		2	32.3	odd	8
9216.2.a.q.1.2		2	32.13	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)

\(f(q)\)	\(=\)	\(q+(2.00000 + 2.00000i) q^{5} -4.24264i q^{7} +O(q^{10})\)	\(q+(2.00000 + 2.00000i) q^{5} -4.24264i q^{7} +(-2.82843 - 2.82843i) q^{11} +(-3.00000 + 3.00000i) q^{13} +6.00000 q^{17} +(-1.41421 + 1.41421i) q^{19} -2.82843i q^{23} +3.00000i q^{25} +(4.00000 - 4.00000i) q^{29} +4.24264 q^{31} +(8.48528 - 8.48528i) q^{35} +(-3.00000 - 3.00000i) q^{37} -10.0000i q^{41} +(-4.24264 - 4.24264i) q^{43} +2.82843 q^{47} -11.0000 q^{49} +(4.00000 + 4.00000i) q^{53} -11.3137i q^{55} +(-3.00000 + 3.00000i) q^{61} -12.0000 q^{65} +(2.82843 - 2.82843i) q^{67} +2.82843i q^{71} -16.0000i q^{73} +(-12.0000 + 12.0000i) q^{77} -4.24264 q^{79} +(11.3137 - 11.3137i) q^{83} +(12.0000 + 12.0000i) q^{85} -14.0000i q^{89} +(12.7279 + 12.7279i) q^{91} -5.65685 q^{95} -4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5}+O(q^{10}) \) 4 * q + 8 * q^5	\( 4 q + 8 q^{5} - 12 q^{13} + 24 q^{17} + 16 q^{29} - 12 q^{37} - 44 q^{49} + 16 q^{53} - 12 q^{61} - 48 q^{65} - 48 q^{77} + 48 q^{85} - 16 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 - 12 * q^13 + 24 * q^17 + 16 * q^29 - 12 * q^37 - 44 * q^49 + 16 * q^53 - 12 * q^61 - 48 * q^65 - 48 * q^77 + 48 * q^85 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more