LMFDB - Embedded newform 2205.2.d.d.1324.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,2,Mod(1324,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2205.1324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2205.d (of order $2$, degree $1$, 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:	$17.6070136457$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1324.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2205.1324
Dual form			2205.2.d.d.1324.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -3.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} +1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -3.00000i q^{8} +(2.00000 + 1.00000i) q^{10} -2.00000i q^{13} -1.00000 q^{16} -2.00000i q^{17} -6.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} -3.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -2.00000 q^{26} +7.00000 q^{29} +2.00000 q^{31} -5.00000i q^{32} -2.00000 q^{34} -8.00000i q^{37} +6.00000i q^{38} +(6.00000 + 3.00000i) q^{40} -5.00000 q^{41} -7.00000i q^{43} -3.00000 q^{46} +(-4.00000 + 3.00000i) q^{50} -2.00000i q^{52} +6.00000i q^{53} -7.00000i q^{58} +10.0000 q^{59} +7.00000 q^{61} -2.00000i q^{62} -7.00000 q^{64} +(4.00000 + 2.00000i) q^{65} -5.00000i q^{67} -2.00000i q^{68} +2.00000 q^{71} +6.00000i q^{73} -8.00000 q^{74} -6.00000 q^{76} +2.00000 q^{79} +(1.00000 - 2.00000i) q^{80} +5.00000i q^{82} -11.0000i q^{83} +(4.00000 + 2.00000i) q^{85} -7.00000 q^{86} +9.00000 q^{89} -3.00000i q^{92} +(6.00000 - 12.0000i) q^{95} -16.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^4 - 2 * q^5	$ 2 q + 2 q^{4} - 2 q^{5} + 4 q^{10} - 2 q^{16} - 12 q^{19} - 2 q^{20} - 6 q^{25} - 4 q^{26} + 14 q^{29} + 4 q^{31} - 4 q^{34} + 12 q^{40} - 10 q^{41} - 6 q^{46} - 8 q^{50} + 20 q^{59} + 14 q^{61} - 14 q^{64} + 8 q^{65} + 4 q^{71} - 16 q^{74} - 12 q^{76} + 4 q^{79} + 2 q^{80} + 8 q^{85} - 14 q^{86} + 18 q^{89} + 12 q^{95}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^5 + 4 * q^10 - 2 * q^16 - 12 * q^19 - 2 * q^20 - 6 * q^25 - 4 * q^26 + 14 * q^29 + 4 * q^31 - 4 * q^34 + 12 * q^40 - 10 * q^41 - 6 * q^46 - 8 * q^50 + 20 * q^59 + 14 * q^61 - 14 * q^64 + 8 * q^65 + 4 * q^71 - 16 * q^74 - 12 * q^76 + 4 * q^79 + 2 * q^80 + 8 * q^85 - 14 * q^86 + 18 * q^89 + 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times$.

$n$	$442$	$1081$	$1226$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		−	1.00000i		−	0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$		−	1.00000i		−	0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		−	3.00000i		−	1.06066i
$9$		0			0
$10$	2.00000	+	1.00000i	0.632456	+	0.316228i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$		0			0
$15$		0			0
$16$	−1.00000			−0.250000
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$	−1.00000	+	2.00000i	−0.223607	+	0.447214i
$21$		0			0
$22$		0			0
$23$		−	3.00000i		−	0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$		−	3.00000i		−	0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$	−2.00000			−0.392232
$27$		0			0
$28$		0			0
$29$	7.00000			1.29987			0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000			1.29987			0.649934	+	0.759991i	$0.274797\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$		−	5.00000i		−	0.883883i
$33$		0			0
$34$	−2.00000			−0.342997
$35$		0			0
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$			6.00000i			0.973329i
$39$		0			0
$40$	6.00000	+	3.00000i	0.948683	+	0.474342i
$41$	−5.00000			−0.780869			−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000			−0.780869			−0.390434	+	0.920631i	$0.627675\pi$
$42$		0			0
$43$		−	7.00000i		−	1.06749i	−0.845645	−	0.533745i	$-0.820784\pi$
$43$		−	7.00000i		−	1.06749i	0.845645	−	0.533745i	$-0.179216\pi$
$44$		0			0
$45$		0			0
$46$	−3.00000			−0.442326
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$		0			0
$50$	−4.00000	+	3.00000i	−0.565685	+	0.424264i
$51$		0			0
$52$		−	2.00000i		−	0.277350i
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		−	7.00000i		−	0.919145i
$59$	10.0000			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$60$		0			0
$61$	7.00000			0.896258			0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000			0.896258			0.448129	+	0.893969i	$0.352090\pi$
$62$		−	2.00000i		−	0.254000i
$63$		0			0
$64$	−7.00000			−0.875000
$65$	4.00000	+	2.00000i	0.496139	+	0.248069i
$66$		0			0
$67$		−	5.00000i		−	0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$		−	5.00000i		−	0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$		−	2.00000i		−	0.242536i
$69$		0			0
$70$		0			0
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−8.00000			−0.929981
$75$		0			0
$76$	−6.00000			−0.688247
$77$		0			0
$78$		0			0
$79$	2.00000			0.225018			0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000			0.225018			0.112509	+	0.993651i	$0.464111\pi$
$80$	1.00000	−	2.00000i	0.111803	−	0.223607i
$81$		0			0
$82$			5.00000i			0.552158i
$83$		−	11.0000i		−	1.20741i	−0.797209	−	0.603703i	$-0.793691\pi$
$83$		−	11.0000i		−	1.20741i	0.797209	−	0.603703i	$-0.206309\pi$
$84$		0			0
$85$	4.00000	+	2.00000i	0.433861	+	0.216930i
$86$	−7.00000			−0.754829
$87$		0			0
$88$		0			0
$89$	9.00000			0.953998			0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000			0.953998			0.476999	+	0.878904i	$0.341725\pi$
$90$		0			0
$91$		0			0
$92$		−	3.00000i		−	0.312772i
$93$		0			0
$94$		0			0
$95$	6.00000	−	12.0000i	0.615587	−	1.23117i
$96$		0			0
$97$		−	16.0000i		−	1.62455i	−0.583272	−	0.812277i	$-0.698228\pi$
$97$		−	16.0000i		−	1.62455i	0.583272	−	0.812277i	$-0.301772\pi$
$98$		0			0
$99$		0			0
$100$	−3.00000	−	4.00000i	−0.300000	−	0.400000i
$101$	−9.00000			−0.895533			−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000			−0.895533			−0.447767	+	0.894150i	$0.647781\pi$
$102$		0			0
$103$		−	7.00000i		−	0.689730i	−0.938652	−	0.344865i	$-0.887925\pi$
$103$		−	7.00000i		−	0.689730i	0.938652	−	0.344865i	$-0.112075\pi$
$104$	−6.00000			−0.588348
$105$		0			0
$106$	6.00000			0.582772
$107$		−	11.0000i		−	1.06341i	−0.846930	−	0.531705i	$-0.821551\pi$
$107$		−	11.0000i		−	1.06341i	0.846930	−	0.531705i	$-0.178449\pi$
$108$		0			0
$109$	−5.00000			−0.478913			−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000			−0.478913			−0.239457	+	0.970907i	$0.576969\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		0			0
$115$	6.00000	+	3.00000i	0.559503	+	0.279751i
$116$	7.00000			0.649934
$117$		0			0
$118$		−	10.0000i		−	0.920575i
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		−	7.00000i		−	0.633750i
$123$		0			0
$124$	2.00000			0.179605
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		0			0
$127$			16.0000i			1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$			16.0000i			1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$		−	3.00000i		−	0.265165i
$129$		0			0
$130$	2.00000	−	4.00000i	0.175412	−	0.350823i
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$		0			0
$134$	−5.00000			−0.431934
$135$		0			0
$136$	−6.00000			−0.514496
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	−8.00000			−0.678551			−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000			−0.678551			−0.339276	+	0.940687i	$0.610182\pi$
$140$		0			0
$141$		0			0
$142$		−	2.00000i		−	0.167836i
$143$		0			0
$144$		0			0
$145$	−7.00000	+	14.0000i	−0.581318	+	1.16264i
$146$	6.00000			0.496564
$147$		0			0
$148$		−	8.00000i		−	0.657596i
$149$	1.00000			0.0819232			0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000			0.0819232			0.0409616	+	0.999161i	$0.486958\pi$
$150$		0			0
$151$	14.0000			1.13930			0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000			1.13930			0.569652	+	0.821886i	$0.307078\pi$
$152$			18.0000i			1.45999i
$153$		0			0
$154$		0			0
$155$	−2.00000	+	4.00000i	−0.160644	+	0.321288i
$156$		0			0
$157$			12.0000i			0.957704i	0.877896	+	0.478852i	$0.158947\pi$
$157$			12.0000i			0.957704i	−0.877896	+	0.478852i	$0.841053\pi$
$158$		−	2.00000i		−	0.159111i
$159$		0			0
$160$	10.0000	+	5.00000i	0.790569	+	0.395285i
$161$		0			0
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−5.00000			−0.390434
$165$		0			0
$166$	−11.0000			−0.853766
$167$			3.00000i			0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$			3.00000i			0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$	2.00000	−	4.00000i	0.153393	−	0.306786i
$171$		0			0
$172$		−	7.00000i		−	0.533745i
$173$		−	12.0000i		−	0.912343i	−0.889892	−	0.456172i	$-0.849220\pi$
$173$		−	12.0000i		−	0.912343i	0.889892	−	0.456172i	$-0.150780\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		−	9.00000i		−	0.674579i
$179$	−2.00000			−0.149487			−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000			−0.149487			−0.0747435	+	0.997203i	$0.523814\pi$
$180$		0			0
$181$	−3.00000			−0.222988			−0.111494	−	0.993765i	$-0.535564\pi$
$181$	−3.00000			−0.222988			−0.111494	+	0.993765i	$0.535564\pi$
$182$		0			0
$183$		0			0
$184$	−9.00000			−0.663489
$185$	16.0000	+	8.00000i	1.17634	+	0.588172i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−12.0000	−	6.00000i	−0.870572	−	0.435286i
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$			4.00000i			0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$			4.00000i			0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	−16.0000			−1.14873
$195$		0			0
$196$		0			0
$197$			8.00000i			0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$			8.00000i			0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$		0			0
$199$	4.00000			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$200$	−12.0000	+	9.00000i	−0.848528	+	0.636396i
$201$		0			0
$202$			9.00000i			0.633238i
$203$		0			0
$204$		0			0
$205$	5.00000	−	10.0000i	0.349215	−	0.698430i
$206$	−7.00000			−0.487713
$207$		0			0
$208$			2.00000i			0.138675i
$209$		0			0
$210$		0			0
$211$	−10.0000			−0.688428			−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000			−0.688428			−0.344214	+	0.938891i	$0.611855\pi$
$212$			6.00000i			0.412082i
$213$		0			0
$214$	−11.0000			−0.751945
$215$	14.0000	+	7.00000i	0.954792	+	0.477396i
$216$		0			0
$217$		0			0
$218$			5.00000i			0.338643i
$219$		0			0
$220$		0			0
$221$	−4.00000			−0.269069
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$	−6.00000			−0.399114
$227$			20.0000i			1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$			20.0000i			1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$		0			0
$229$	22.0000			1.45380			0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000			1.45380			0.726900	+	0.686743i	$0.240960\pi$
$230$	3.00000	−	6.00000i	0.197814	−	0.395628i
$231$		0			0
$232$		−	21.0000i		−	1.37872i
$233$			14.0000i			0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$			14.0000i			0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$		0			0
$235$		0			0
$236$	10.0000			0.650945
$237$		0			0
$238$		0			0
$239$	−18.0000			−1.16432			−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000			−1.16432			−0.582162	+	0.813073i	$0.697793\pi$
$240$		0			0
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$			11.0000i			0.707107i
$243$		0			0
$244$	7.00000			0.448129
$245$		0			0
$246$		0			0
$247$			12.0000i			0.763542i
$248$		−	6.00000i		−	0.381000i
$249$		0			0
$250$	−2.00000	−	11.0000i	−0.126491	−	0.695701i
$251$	−30.0000			−1.89358			−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000			−1.89358			−0.946792	+	0.321847i	$0.895696\pi$
$252$		0			0
$253$		0			0
$254$	16.0000			1.00393
$255$		0			0
$256$	−17.0000			−1.06250
$257$		−	20.0000i		−	1.24757i	−0.781598	−	0.623783i	$-0.785595\pi$
$257$		−	20.0000i		−	1.24757i	0.781598	−	0.623783i	$-0.214405\pi$
$258$		0			0
$259$		0			0
$260$	4.00000	+	2.00000i	0.248069	+	0.124035i
$261$		0			0
$262$			4.00000i			0.247121i
$263$		−	9.00000i		−	0.554964i	−0.960731	−	0.277482i	$-0.910500\pi$
$263$		−	9.00000i		−	0.554964i	0.960731	−	0.277482i	$-0.0894999\pi$
$264$		0			0
$265$	−12.0000	−	6.00000i	−0.737154	−	0.368577i
$266$		0			0
$267$		0			0
$268$		−	5.00000i		−	0.305424i
$269$	−11.0000			−0.670682			−0.335341	−	0.942097i	$-0.608852\pi$
$269$	−11.0000			−0.670682			−0.335341	+	0.942097i	$0.608852\pi$
$270$		0			0
$271$	−2.00000			−0.121491			−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000			−0.121491			−0.0607457	+	0.998153i	$0.519348\pi$
$272$			2.00000i			0.121268i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$			10.0000i			0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$			10.0000i			0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$			8.00000i			0.479808i
$279$		0			0
$280$		0			0
$281$	14.0000			0.835170			0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000			0.835170			0.417585	+	0.908638i	$0.362877\pi$
$282$		0			0
$283$		−	16.0000i		−	0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$		−	16.0000i		−	0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	2.00000			0.118678
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	13.0000			0.764706
$290$	14.0000	+	7.00000i	0.822108	+	0.411054i
$291$		0			0
$292$			6.00000i			0.351123i
$293$			24.0000i			1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$			24.0000i			1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$		0			0
$295$	−10.0000	+	20.0000i	−0.582223	+	1.16445i
$296$	−24.0000			−1.39497
$297$		0			0
$298$		−	1.00000i		−	0.0579284i
$299$	−6.00000			−0.346989
$300$		0			0
$301$		0			0
$302$		−	14.0000i		−	0.805609i
$303$		0			0
$304$	6.00000			0.344124
$305$	−7.00000	+	14.0000i	−0.400819	+	0.801638i
$306$		0			0
$307$			23.0000i			1.31268i	0.754466	+	0.656340i	$0.227896\pi$
$307$			23.0000i			1.31268i	−0.754466	+	0.656340i	$0.772104\pi$
$308$		0			0
$309$		0			0
$310$	4.00000	+	2.00000i	0.227185	+	0.113592i
$311$	−30.0000			−1.70114			−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000			−1.70114			−0.850572	+	0.525859i	$0.823744\pi$
$312$		0			0
$313$		−	24.0000i		−	1.35656i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$		−	24.0000i		−	1.35656i	0.734803	−	0.678280i	$-0.237274\pi$
$314$	12.0000			0.677199
$315$		0			0
$316$	2.00000			0.112509
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		0			0
$319$		0			0
$320$	7.00000	−	14.0000i	0.391312	−	0.782624i
$321$		0			0
$322$		0			0
$323$			12.0000i			0.667698i
$324$		0			0
$325$	−8.00000	+	6.00000i	−0.443760	+	0.332820i
$326$	4.00000			0.221540
$327$		0			0
$328$			15.0000i			0.828236i
$329$		0			0
$330$		0			0
$331$	6.00000			0.329790			0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000			0.329790			0.164895	+	0.986311i	$0.447272\pi$
$332$		−	11.0000i		−	0.603703i
$333$		0			0
$334$	3.00000			0.164153
$335$	10.0000	+	5.00000i	0.546358	+	0.273179i
$336$		0			0
$337$			18.0000i			0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$			18.0000i			0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$		−	9.00000i		−	0.489535i
$339$		0			0
$340$	4.00000	+	2.00000i	0.216930	+	0.108465i
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−21.0000			−1.13224
$345$		0			0
$346$	−12.0000			−0.645124
$347$		−	15.0000i		−	0.805242i	−0.915367	−	0.402621i	$-0.868099\pi$
$347$		−	15.0000i		−	0.805242i	0.915367	−	0.402621i	$-0.131901\pi$
$348$		0			0
$349$	17.0000			0.909989			0.454995	−	0.890494i	$-0.349641\pi$
$349$	17.0000			0.909989			0.454995	+	0.890494i	$0.349641\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	26.0000i		−	1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$		−	26.0000i		−	1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$		0			0
$355$	−2.00000	+	4.00000i	−0.106149	+	0.212298i
$356$	9.00000			0.476999
$357$		0			0
$358$			2.00000i			0.105703i
$359$	10.0000			0.527780			0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000			0.527780			0.263890	+	0.964553i	$0.414994\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$			3.00000i			0.157676i
$363$		0			0
$364$		0			0
$365$	−12.0000	−	6.00000i	−0.628109	−	0.314054i
$366$		0			0
$367$			27.0000i			1.40939i	0.709511	+	0.704694i	$0.248916\pi$
$367$			27.0000i			1.40939i	−0.709511	+	0.704694i	$0.751084\pi$
$368$			3.00000i			0.156386i
$369$		0			0
$370$	8.00000	−	16.0000i	0.415900	−	0.831800i
$371$		0			0
$372$		0			0
$373$			24.0000i			1.24267i	0.783544	+	0.621336i	$0.213410\pi$
$373$			24.0000i			1.24267i	−0.783544	+	0.621336i	$0.786590\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	14.0000i		−	0.721037i
$378$		0			0
$379$	10.0000			0.513665			0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000			0.513665			0.256833	+	0.966456i	$0.417321\pi$
$380$	6.00000	−	12.0000i	0.307794	−	0.615587i
$381$		0			0
$382$			12.0000i			0.613973i
$383$			5.00000i			0.255488i	0.991807	+	0.127744i	$0.0407736\pi$
$383$			5.00000i			0.255488i	−0.991807	+	0.127744i	$0.959226\pi$
$384$		0			0
$385$		0			0
$386$	4.00000			0.203595
$387$		0			0
$388$		−	16.0000i		−	0.812277i
$389$	14.0000			0.709828			0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000			0.709828			0.354914	+	0.934899i	$0.384510\pi$
$390$		0			0
$391$	−6.00000			−0.303433
$392$		0			0
$393$		0			0
$394$	8.00000			0.403034
$395$	−2.00000	+	4.00000i	−0.100631	+	0.201262i
$396$		0			0
$397$		−	16.0000i		−	0.803017i	−0.915855	−	0.401508i	$-0.868486\pi$
$397$		−	16.0000i		−	0.803017i	0.915855	−	0.401508i	$-0.131514\pi$
$398$		−	4.00000i		−	0.200502i
$399$		0			0
$400$	3.00000	+	4.00000i	0.150000	+	0.200000i
$401$	−3.00000			−0.149813			−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000			−0.149813			−0.0749064	+	0.997191i	$0.523866\pi$
$402$		0			0
$403$		−	4.00000i		−	0.199254i
$404$	−9.00000			−0.447767
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	25.0000			1.23617			0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000			1.23617			0.618085	+	0.786111i	$0.287909\pi$
$410$	−10.0000	−	5.00000i	−0.493865	−	0.246932i
$411$		0			0
$412$		−	7.00000i		−	0.344865i
$413$		0			0
$414$		0			0
$415$	22.0000	+	11.0000i	1.07994	+	0.539969i
$416$	−10.0000			−0.490290
$417$		0			0
$418$		0			0
$419$	24.0000			1.17248			0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000			1.17248			0.586238	+	0.810139i	$0.300608\pi$
$420$		0			0
$421$	15.0000			0.731055			0.365528	−	0.930800i	$-0.380889\pi$
$421$	15.0000			0.731055			0.365528	+	0.930800i	$0.380889\pi$
$422$			10.0000i			0.486792i
$423$		0			0
$424$	18.0000			0.874157
$425$	−8.00000	+	6.00000i	−0.388057	+	0.291043i
$426$		0			0
$427$		0			0
$428$		−	11.0000i		−	0.531705i
$429$		0			0
$430$	7.00000	−	14.0000i	0.337570	−	0.675140i
$431$	32.0000			1.54139			0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000			1.54139			0.770693	+	0.637207i	$0.219910\pi$
$432$		0			0
$433$			2.00000i			0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$			2.00000i			0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$		0			0
$435$		0			0
$436$	−5.00000			−0.239457
$437$			18.0000i			0.861057i
$438$		0			0
$439$	−24.0000			−1.14546			−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000			−1.14546			−0.572729	+	0.819745i	$0.694115\pi$
$440$		0			0
$441$		0			0
$442$			4.00000i			0.190261i
$443$			31.0000i			1.47285i	0.676517	+	0.736427i	$0.263489\pi$
$443$			31.0000i			1.47285i	−0.676517	+	0.736427i	$0.736511\pi$
$444$		0			0
$445$	−9.00000	+	18.0000i	−0.426641	+	0.853282i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	31.0000			1.46298			0.731490	−	0.681852i	$-0.238825\pi$
$449$	31.0000			1.46298			0.731490	+	0.681852i	$0.238825\pi$
$450$		0			0
$451$		0			0
$452$		−	6.00000i		−	0.282216i
$453$		0			0
$454$	20.0000			0.938647
$455$		0			0
$456$		0			0
$457$			32.0000i			1.49690i	0.663193	+	0.748448i	$0.269201\pi$
$457$			32.0000i			1.49690i	−0.663193	+	0.748448i	$0.730799\pi$
$458$		−	22.0000i		−	1.02799i
$459$		0			0
$460$	6.00000	+	3.00000i	0.279751	+	0.139876i
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$		−	3.00000i		−	0.139422i	−0.997567	−	0.0697109i	$-0.977792\pi$
$463$		−	3.00000i		−	0.139422i	0.997567	−	0.0697109i	$-0.0222077\pi$
$464$	−7.00000			−0.324967
$465$		0			0
$466$	14.0000			0.648537
$467$		−	3.00000i		−	0.138823i	−0.997588	−	0.0694117i	$-0.977888\pi$
$467$		−	3.00000i		−	0.138823i	0.997588	−	0.0694117i	$-0.0221122\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		−	30.0000i		−	1.38086i
$473$		0			0
$474$		0			0
$475$	18.0000	+	24.0000i	0.825897	+	1.10120i
$476$		0			0
$477$		0			0
$478$			18.0000i			0.823301i
$479$	42.0000			1.91903			0.959514	−	0.281659i	$-0.0908848\pi$
$479$	42.0000			1.91903			0.959514	+	0.281659i	$0.0908848\pi$
$480$		0			0
$481$	−16.0000			−0.729537
$482$			14.0000i			0.637683i
$483$		0			0
$484$	−11.0000			−0.500000
$485$	32.0000	+	16.0000i	1.45305	+	0.726523i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		−	21.0000i		−	0.950625i
$489$		0			0
$490$		0			0
$491$	26.0000			1.17336			0.586682	−	0.809818i	$-0.300434\pi$
$491$	26.0000			1.17336			0.586682	+	0.809818i	$0.300434\pi$
$492$		0			0
$493$		−	14.0000i		−	0.630528i
$494$	12.0000			0.539906
$495$		0			0
$496$	−2.00000			−0.0898027
$497$		0			0
$498$		0			0
$499$	−16.0000			−0.716258			−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000			−0.716258			−0.358129	+	0.933672i	$0.616585\pi$
$500$	11.0000	−	2.00000i	0.491935	−	0.0894427i
$501$		0			0
$502$			30.0000i			1.33897i
$503$		−	15.0000i		−	0.668817i	−0.942428	−	0.334408i	$-0.891463\pi$
$503$		−	15.0000i		−	0.668817i	0.942428	−	0.334408i	$-0.108537\pi$
$504$		0			0
$505$	9.00000	−	18.0000i	0.400495	−	0.800989i
$506$		0			0
$507$		0			0
$508$			16.0000i			0.709885i
$509$	27.0000			1.19675			0.598377	−	0.801215i	$-0.295813\pi$
$509$	27.0000			1.19675			0.598377	+	0.801215i	$0.295813\pi$
$510$		0			0
$511$		0			0
$512$			11.0000i			0.486136i
$513$		0			0
$514$	−20.0000			−0.882162
$515$	14.0000	+	7.00000i	0.616914	+	0.308457i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	6.00000	−	12.0000i	0.263117	−	0.526235i
$521$	−14.0000			−0.613351			−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000			−0.613351			−0.306676	+	0.951814i	$0.599217\pi$
$522$		0			0
$523$		−	4.00000i		−	0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$		−	4.00000i		−	0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	−4.00000			−0.174741
$525$		0			0
$526$	−9.00000			−0.392419
$527$		−	4.00000i		−	0.174243i
$528$		0			0
$529$	14.0000			0.608696
$530$	−6.00000	+	12.0000i	−0.260623	+	0.521247i
$531$		0			0
$532$		0			0
$533$			10.0000i			0.433148i
$534$		0			0
$535$	22.0000	+	11.0000i	0.951143	+	0.475571i
$536$	−15.0000			−0.647901
$537$		0			0
$538$			11.0000i			0.474244i
$539$		0			0
$540$		0			0
$541$	−3.00000			−0.128980			−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000			−0.128980			−0.0644900	+	0.997918i	$0.520542\pi$
$542$			2.00000i			0.0859074i
$543$		0			0
$544$	−10.0000			−0.428746
$545$	5.00000	−	10.0000i	0.214176	−	0.428353i
$546$		0			0
$547$		−	17.0000i		−	0.726868i	−0.931620	−	0.363434i	$-0.881604\pi$
$547$		−	17.0000i		−	0.726868i	0.931620	−	0.363434i	$-0.118396\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−42.0000			−1.78926
$552$		0			0
$553$		0			0
$554$	10.0000			0.424859
$555$		0			0
$556$	−8.00000			−0.339276
$557$			44.0000i			1.86434i	0.362021	+	0.932170i	$0.382087\pi$
$557$			44.0000i			1.86434i	−0.362021	+	0.932170i	$0.617913\pi$
$558$		0			0
$559$	−14.0000			−0.592137
$560$		0			0
$561$		0			0
$562$		−	14.0000i		−	0.590554i
$563$			33.0000i			1.39078i	0.718631	+	0.695392i	$0.244769\pi$
$563$			33.0000i			1.39078i	−0.718631	+	0.695392i	$0.755231\pi$
$564$		0			0
$565$	12.0000	+	6.00000i	0.504844	+	0.252422i
$566$	−16.0000			−0.672530
$567$		0			0
$568$		−	6.00000i		−	0.251754i
$569$	−18.0000			−0.754599			−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000			−0.754599			−0.377300	+	0.926091i	$0.623147\pi$
$570$		0			0
$571$	−18.0000			−0.753277			−0.376638	−	0.926360i	$-0.622920\pi$
$571$	−18.0000			−0.753277			−0.376638	+	0.926360i	$0.622920\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12.0000	+	9.00000i	−0.500435	+	0.375326i
$576$		0			0
$577$		−	32.0000i		−	1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$		−	32.0000i		−	1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$		−	13.0000i		−	0.540729i
$579$		0			0
$580$	−7.00000	+	14.0000i	−0.290659	+	0.581318i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	18.0000			0.744845
$585$		0			0
$586$	24.0000			0.991431
$587$			36.0000i			1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$			36.0000i			1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$	20.0000	+	10.0000i	0.823387	+	0.411693i
$591$		0			0
$592$			8.00000i			0.328798i
$593$			22.0000i			0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$			22.0000i			0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$		0			0
$595$		0			0
$596$	1.00000			0.0409616
$597$		0			0
$598$			6.00000i			0.245358i
$599$	−44.0000			−1.79779			−0.898896	−	0.438163i	$-0.855629\pi$
$599$	−44.0000			−1.79779			−0.898896	+	0.438163i	$0.855629\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$		0			0
$604$	14.0000			0.569652
$605$	11.0000	−	22.0000i	0.447214	−	0.894427i
$606$		0			0
$607$		−	33.0000i		−	1.33943i	−0.742619	−	0.669714i	$-0.766417\pi$
$607$		−	33.0000i		−	1.33943i	0.742619	−	0.669714i	$-0.233583\pi$
$608$			30.0000i			1.21666i
$609$		0			0
$610$	14.0000	+	7.00000i	0.566843	+	0.283422i
$611$		0			0
$612$		0			0
$613$			34.0000i			1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$			34.0000i			1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	23.0000			0.928204
$615$		0			0
$616$		0			0
$617$			4.00000i			0.161034i	0.996753	+	0.0805170i	$0.0256571\pi$
$617$			4.00000i			0.161034i	−0.996753	+	0.0805170i	$0.974343\pi$
$618$		0			0
$619$	−28.0000			−1.12542			−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000			−1.12542			−0.562708	+	0.826656i	$0.690240\pi$
$620$	−2.00000	+	4.00000i	−0.0803219	+	0.160644i
$621$		0			0
$622$			30.0000i			1.20289i
$623$		0			0
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−24.0000			−0.959233
$627$		0			0
$628$			12.0000i			0.478852i
$629$	−16.0000			−0.637962
$630$		0			0
$631$	40.0000			1.59237			0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000			1.59237			0.796187	+	0.605050i	$0.206847\pi$
$632$		−	6.00000i		−	0.238667i
$633$		0			0
$634$	6.00000			0.238290
$635$	−32.0000	−	16.0000i	−1.26988	−	0.634941i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	6.00000	+	3.00000i	0.237171	+	0.118585i
$641$	11.0000			0.434474			0.217237	−	0.976119i	$-0.430296\pi$
$641$	11.0000			0.434474			0.217237	+	0.976119i	$0.430296\pi$
$642$		0			0
$643$			4.00000i			0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$			4.00000i			0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$		0			0
$645$		0			0
$646$	12.0000			0.472134
$647$		−	47.0000i		−	1.84776i	−0.382682	−	0.923880i	$-0.624999\pi$
$647$		−	47.0000i		−	1.84776i	0.382682	−	0.923880i	$-0.375001\pi$
$648$		0			0
$649$		0			0
$650$	6.00000	+	8.00000i	0.235339	+	0.313786i
$651$		0			0
$652$			4.00000i			0.156652i
$653$		−	26.0000i		−	1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$		−	26.0000i		−	1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$		0			0
$655$	4.00000	−	8.00000i	0.156293	−	0.312586i
$656$	5.00000			0.195217
$657$		0			0
$658$		0			0
$659$	6.00000			0.233727			0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000			0.233727			0.116863	+	0.993148i	$0.462716\pi$
$660$		0			0
$661$	−9.00000			−0.350059			−0.175030	−	0.984563i	$-0.556002\pi$
$661$	−9.00000			−0.350059			−0.175030	+	0.984563i	$0.556002\pi$
$662$		−	6.00000i		−	0.233197i
$663$		0			0
$664$	−33.0000			−1.28065
$665$		0			0
$666$		0			0
$667$		−	21.0000i		−	0.813123i
$668$			3.00000i			0.116073i
$669$		0			0
$670$	5.00000	−	10.0000i	0.193167	−	0.386334i
$671$		0			0
$672$		0			0
$673$			36.0000i			1.38770i	0.720121	+	0.693849i	$0.244086\pi$
$673$			36.0000i			1.38770i	−0.720121	+	0.693849i	$0.755914\pi$
$674$	18.0000			0.693334
$675$		0			0
$676$	9.00000			0.346154
$677$			14.0000i			0.538064i	0.963131	+	0.269032i	$0.0867037\pi$
$677$			14.0000i			0.538064i	−0.963131	+	0.269032i	$0.913296\pi$
$678$		0			0
$679$		0			0
$680$	6.00000	−	12.0000i	0.230089	−	0.460179i
$681$		0			0
$682$		0			0
$683$			35.0000i			1.33924i	0.742705	+	0.669619i	$0.233543\pi$
$683$			35.0000i			1.33924i	−0.742705	+	0.669619i	$0.766457\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$			7.00000i			0.266872i
$689$	12.0000			0.457164
$690$		0			0
$691$	−20.0000			−0.760836			−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000			−0.760836			−0.380418	+	0.924815i	$0.624220\pi$
$692$		−	12.0000i		−	0.456172i
$693$		0			0
$694$	−15.0000			−0.569392
$695$	8.00000	−	16.0000i	0.303457	−	0.606915i
$696$		0			0
$697$			10.0000i			0.378777i
$698$		−	17.0000i		−	0.643459i
$699$		0			0
$700$		0			0
$701$	1.00000			0.0377695			0.0188847	−	0.999822i	$-0.493988\pi$
$701$	1.00000			0.0377695			0.0188847	+	0.999822i	$0.493988\pi$
$702$		0			0
$703$			48.0000i			1.81035i
$704$		0			0
$705$		0			0
$706$	−26.0000			−0.978523
$707$		0			0
$708$		0			0
$709$	49.0000			1.84023			0.920117	−	0.391644i	$-0.128094\pi$
$709$	49.0000			1.84023			0.920117	+	0.391644i	$0.128094\pi$
$710$	4.00000	+	2.00000i	0.150117	+	0.0750587i
$711$		0			0
$712$		−	27.0000i		−	1.01187i
$713$		−	6.00000i		−	0.224702i
$714$		0			0
$715$		0			0
$716$	−2.00000			−0.0747435
$717$		0			0
$718$		−	10.0000i		−	0.373197i
$719$	36.0000			1.34257			0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000			1.34257			0.671287	+	0.741198i	$0.265742\pi$
$720$		0			0
$721$		0			0
$722$		−	17.0000i		−	0.632674i
$723$		0			0
$724$	−3.00000			−0.111494
$725$	−21.0000	−	28.0000i	−0.779920	−	1.03989i
$726$		0			0
$727$		−	47.0000i		−	1.74313i	−0.490277	−	0.871567i	$-0.663104\pi$
$727$		−	47.0000i		−	1.74313i	0.490277	−	0.871567i	$-0.336896\pi$
$728$		0			0
$729$		0			0
$730$	−6.00000	+	12.0000i	−0.222070	+	0.444140i
$731$	−14.0000			−0.517809
$732$		0			0
$733$			16.0000i			0.590973i	0.955347	+	0.295487i	$0.0954818\pi$
$733$			16.0000i			0.590973i	−0.955347	+	0.295487i	$0.904518\pi$
$734$	27.0000			0.996588
$735$		0			0
$736$	−15.0000			−0.552907
$737$		0			0
$738$		0			0
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$	16.0000	+	8.00000i	0.588172	+	0.294086i
$741$		0			0
$742$		0			0
$743$		−	49.0000i		−	1.79764i	−0.438322	−	0.898818i	$-0.644427\pi$
$743$		−	49.0000i		−	1.79764i	0.438322	−	0.898818i	$-0.355573\pi$
$744$		0			0
$745$	−1.00000	+	2.00000i	−0.0366372	+	0.0732743i
$746$	24.0000			0.878702
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−12.0000			−0.437886			−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000			−0.437886			−0.218943	+	0.975738i	$0.570261\pi$
$752$		0			0
$753$		0			0
$754$	−14.0000			−0.509850
$755$	−14.0000	+	28.0000i	−0.509512	+	1.01902i
$756$		0			0
$757$			22.0000i			0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$			22.0000i			0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$		−	10.0000i		−	0.363216i
$759$		0			0
$760$	−36.0000	−	18.0000i	−1.30586	−	0.652929i
$761$	−14.0000			−0.507500			−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000			−0.507500			−0.253750	+	0.967270i	$0.581664\pi$
$762$		0			0
$763$		0			0
$764$	−12.0000			−0.434145
$765$		0			0
$766$	5.00000			0.180657
$767$		−	20.0000i		−	0.722158i
$768$		0			0
$769$	30.0000			1.08183			0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000			1.08183			0.540914	+	0.841078i	$0.318079\pi$
$770$		0			0
$771$		0			0
$772$			4.00000i			0.143963i
$773$			18.0000i			0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$			18.0000i			0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$		0			0
$775$	−6.00000	−	8.00000i	−0.215526	−	0.287368i
$776$	−48.0000			−1.72310
$777$		0			0
$778$		−	14.0000i		−	0.501924i
$779$	30.0000			1.07486
$780$		0			0
$781$		0			0
$782$			6.00000i			0.214560i
$783$		0			0
$784$		0			0
$785$	−24.0000	−	12.0000i	−0.856597	−	0.428298i
$786$		0			0
$787$		−	17.0000i		−	0.605985i	−0.952993	−	0.302992i	$-0.902014\pi$
$787$		−	17.0000i		−	0.605985i	0.952993	−	0.302992i	$-0.0979856\pi$
$788$			8.00000i			0.284988i
$789$		0			0
$790$	4.00000	+	2.00000i	0.142314	+	0.0711568i
$791$		0			0
$792$		0			0
$793$		−	14.0000i		−	0.497155i
$794$	−16.0000			−0.567819
$795$		0			0
$796$	4.00000			0.141776
$797$		−	40.0000i		−	1.41687i	−0.705775	−	0.708436i	$-0.749401\pi$
$797$		−	40.0000i		−	1.41687i	0.705775	−	0.708436i	$-0.250599\pi$
$798$		0			0
$799$		0			0
$800$	−20.0000	+	15.0000i	−0.707107	+	0.530330i
$801$		0			0
$802$			3.00000i			0.105934i
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−4.00000			−0.140894
$807$		0			0
$808$			27.0000i			0.949857i
$809$	9.00000			0.316423			0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000			0.316423			0.158212	+	0.987405i	$0.449427\pi$
$810$		0			0
$811$	−2.00000			−0.0702295			−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000			−0.0702295			−0.0351147	+	0.999383i	$0.511180\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8.00000	−	4.00000i	−0.280228	−	0.140114i
$816$		0			0
$817$			42.0000i			1.46939i
$818$		−	25.0000i		−	0.874105i
$819$		0			0
$820$	5.00000	−	10.0000i	0.174608	−	0.349215i
$821$	−22.0000			−0.767805			−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000			−0.767805			−0.383903	+	0.923374i	$0.625420\pi$
$822$		0			0
$823$		−	25.0000i		−	0.871445i	−0.900081	−	0.435723i	$-0.856493\pi$
$823$		−	25.0000i		−	0.871445i	0.900081	−	0.435723i	$-0.143507\pi$
$824$	−21.0000			−0.731570
$825$		0			0
$826$		0			0
$827$			39.0000i			1.35616i	0.734987	+	0.678081i	$0.237188\pi$
$827$			39.0000i			1.35616i	−0.734987	+	0.678081i	$0.762812\pi$
$828$		0			0
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$	11.0000	−	22.0000i	0.381816	−	0.763631i
$831$		0			0
$832$			14.0000i			0.485363i
$833$		0			0
$834$		0			0
$835$	−6.00000	−	3.00000i	−0.207639	−	0.103819i
$836$		0			0
$837$		0			0
$838$		−	24.0000i		−	0.829066i
$839$	48.0000			1.65714			0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000			1.65714			0.828572	+	0.559883i	$0.189154\pi$
$840$		0			0
$841$	20.0000			0.689655
$842$		−	15.0000i		−	0.516934i
$843$		0			0
$844$	−10.0000			−0.344214
$845$	−9.00000	+	18.0000i	−0.309609	+	0.619219i
$846$		0			0
$847$		0			0
$848$		−	6.00000i		−	0.206041i
$849$		0			0
$850$	6.00000	+	8.00000i	0.205798	+	0.274398i
$851$	−24.0000			−0.822709
$852$		0			0
$853$			6.00000i			0.205436i	0.994711	+	0.102718i	$0.0327539\pi$
$853$			6.00000i			0.205436i	−0.994711	+	0.102718i	$0.967246\pi$
$854$		0			0
$855$		0			0
$856$	−33.0000			−1.12792
$857$			12.0000i			0.409912i	0.978771	+	0.204956i	$0.0657052\pi$
$857$			12.0000i			0.409912i	−0.978771	+	0.204956i	$0.934295\pi$
$858$		0			0
$859$	32.0000			1.09183			0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000			1.09183			0.545913	+	0.837842i	$0.316183\pi$
$860$	14.0000	+	7.00000i	0.477396	+	0.238698i
$861$		0			0
$862$		−	32.0000i		−	1.08992i
$863$		−	9.00000i		−	0.306364i	−0.988198	−	0.153182i	$-0.951048\pi$
$863$		−	9.00000i		−	0.306364i	0.988198	−	0.153182i	$-0.0489520\pi$
$864$		0			0
$865$	24.0000	+	12.0000i	0.816024	+	0.408012i
$866$	2.00000			0.0679628
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−10.0000			−0.338837
$872$			15.0000i			0.507964i
$873$		0			0
$874$	18.0000			0.608859
$875$		0			0
$876$		0			0
$877$		−	2.00000i		−	0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$		−	2.00000i		−	0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$			24.0000i			0.809961i
$879$		0			0
$880$		0			0
$881$	43.0000			1.44871			0.724353	−	0.689429i	$-0.242138\pi$
$881$	43.0000			1.44871			0.724353	+	0.689429i	$0.242138\pi$
$882$		0			0
$883$		−	36.0000i		−	1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$		−	36.0000i		−	1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$	−4.00000			−0.134535
$885$		0			0
$886$	31.0000			1.04147
$887$		−	43.0000i		−	1.44380i	−0.691998	−	0.721899i	$-0.743269\pi$
$887$		−	43.0000i		−	1.44380i	0.691998	−	0.721899i	$-0.256731\pi$
$888$		0			0
$889$		0			0
$890$	18.0000	+	9.00000i	0.603361	+	0.301681i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	2.00000	−	4.00000i	0.0668526	−	0.133705i
$896$		0			0
$897$		0			0
$898$		−	31.0000i		−	1.03448i
$899$	14.0000			0.466926
$900$		0			0
$901$	12.0000			0.399778
$902$		0			0
$903$		0			0
$904$	−18.0000			−0.598671
$905$	3.00000	−	6.00000i	0.0997234	−	0.199447i
$906$		0			0
$907$		−	19.0000i		−	0.630885i	−0.948945	−	0.315442i	$-0.897847\pi$
$907$		−	19.0000i		−	0.630885i	0.948945	−	0.315442i	$-0.102153\pi$
$908$			20.0000i			0.663723i
$909$		0			0
$910$		0			0
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$		0			0
$913$		0			0
$914$	32.0000			1.05847
$915$		0			0
$916$	22.0000			0.726900
$917$		0			0
$918$		0			0
$919$	−34.0000			−1.12156			−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000			−1.12156			−0.560778	+	0.827966i	$0.689498\pi$
$920$	9.00000	−	18.0000i	0.296721	−	0.593442i
$921$		0			0
$922$			18.0000i			0.592798i
$923$		−	4.00000i		−	0.131662i
$924$		0			0
$925$	−32.0000	+	24.0000i	−1.05215	+	0.789115i
$926$	−3.00000			−0.0985861
$927$		0			0
$928$		−	35.0000i		−	1.14893i
$929$	−29.0000			−0.951459			−0.475730	−	0.879592i	$-0.657816\pi$
$929$	−29.0000			−0.951459			−0.475730	+	0.879592i	$0.657816\pi$
$930$		0			0
$931$		0			0
$932$			14.0000i			0.458585i
$933$		0			0
$934$	−3.00000			−0.0981630
$935$		0			0
$936$		0			0
$937$		−	8.00000i		−	0.261349i	−0.991425	−	0.130674i	$-0.958286\pi$
$937$		−	8.00000i		−	0.261349i	0.991425	−	0.130674i	$-0.0417142\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	18.0000			0.586783			0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000			0.586783			0.293392	+	0.955992i	$0.405216\pi$
$942$		0			0
$943$			15.0000i			0.488467i
$944$	−10.0000			−0.325472
$945$		0			0
$946$		0			0
$947$			57.0000i			1.85225i	0.377215	+	0.926126i	$0.376882\pi$
$947$			57.0000i			1.85225i	−0.377215	+	0.926126i	$0.623118\pi$
$948$		0			0
$949$	12.0000			0.389536
$950$	24.0000	−	18.0000i	0.778663	−	0.583997i
$951$		0			0
$952$		0			0
$953$			60.0000i			1.94359i	0.235826	+	0.971795i	$0.424220\pi$
$953$			60.0000i			1.94359i	−0.235826	+	0.971795i	$0.575780\pi$
$954$		0			0
$955$	12.0000	−	24.0000i	0.388311	−	0.776622i
$956$	−18.0000			−0.582162
$957$		0			0
$958$		−	42.0000i		−	1.35696i
$959$		0			0
$960$		0			0
$961$	−27.0000			−0.870968
$962$			16.0000i			0.515861i
$963$		0			0
$964$	−14.0000			−0.450910
$965$	−8.00000	−	4.00000i	−0.257529	−	0.128765i
$966$		0			0
$967$		−	13.0000i		−	0.418052i	−0.977910	−	0.209026i	$-0.932971\pi$
$967$		−	13.0000i		−	0.418052i	0.977910	−	0.209026i	$-0.0670293\pi$
$968$			33.0000i			1.06066i
$969$		0			0
$970$	16.0000	−	32.0000i	0.513729	−	1.02746i
$971$	32.0000			1.02693			0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000			1.02693			0.513464	+	0.858111i	$0.328362\pi$
$972$		0			0
$973$		0			0
$974$	−12.0000			−0.384505
$975$		0			0
$976$	−7.00000			−0.224065
$977$		−	60.0000i		−	1.91957i	−0.280736	−	0.959785i	$-0.590579\pi$
$977$		−	60.0000i		−	1.91957i	0.280736	−	0.959785i	$-0.409421\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		−	26.0000i		−	0.829693i
$983$			3.00000i			0.0956851i	0.998855	+	0.0478426i	$0.0152346\pi$
$983$			3.00000i			0.0956851i	−0.998855	+	0.0478426i	$0.984765\pi$
$984$		0			0
$985$	−16.0000	−	8.00000i	−0.509802	−	0.254901i
$986$	−14.0000			−0.445851
$987$		0			0
$988$			12.0000i			0.381771i
$989$	−21.0000			−0.667761
$990$		0			0
$991$	−50.0000			−1.58830			−0.794151	−	0.607720i	$-0.792084\pi$
$991$	−50.0000			−1.58830			−0.794151	+	0.607720i	$0.792084\pi$
$992$		−	10.0000i		−	0.317500i
$993$		0			0
$994$		0			0
$995$	−4.00000	+	8.00000i	−0.126809	+	0.253617i
$996$		0			0
$997$		−	2.00000i		−	0.0633406i	−0.999498	−	0.0316703i	$-0.989917\pi$
$997$		−	2.00000i		−	0.0633406i	0.999498	−	0.0316703i	$-0.0100827\pi$
$998$			16.0000i			0.506471i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.d.d.1324.1		2
3.2	odd	2		245.2.b.c.99.2		2
5.4	even	2	inner	2205.2.d.d.1324.2		2
7.2	even	3		315.2.bf.a.109.1		4
7.4	even	3		315.2.bf.a.289.2		4
7.6	odd	2		2205.2.d.e.1324.1		2
15.2	even	4		1225.2.a.d.1.1		1
15.8	even	4		1225.2.a.f.1.1		1
15.14	odd	2		245.2.b.c.99.1		2
21.2	odd	6		35.2.j.a.4.2	yes	4
21.5	even	6		245.2.j.c.214.2		4
21.11	odd	6		35.2.j.a.9.1	yes	4
21.17	even	6		245.2.j.c.79.1		4
21.20	even	2		245.2.b.b.99.2		2
35.4	even	6		315.2.bf.a.289.1		4
35.9	even	6		315.2.bf.a.109.2		4
35.34	odd	2		2205.2.d.e.1324.2		2
84.11	even	6		560.2.bw.b.289.1		4
84.23	even	6		560.2.bw.b.529.2		4
105.2	even	12		175.2.e.b.151.1		2
105.23	even	12		175.2.e.a.151.1		2
105.32	even	12		175.2.e.b.51.1		2
105.44	odd	6		35.2.j.a.4.1	✓	4
105.53	even	12		175.2.e.a.51.1		2
105.59	even	6		245.2.j.c.79.2		4
105.62	odd	4		1225.2.a.b.1.1		1
105.74	odd	6		35.2.j.a.9.2	yes	4
105.83	odd	4		1225.2.a.g.1.1		1
105.89	even	6		245.2.j.c.214.1		4
105.104	even	2		245.2.b.b.99.1		2
420.179	even	6		560.2.bw.b.289.2		4
420.359	even	6		560.2.bw.b.529.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.j.a.4.1	✓	4	105.44	odd	6
35.2.j.a.4.2	yes	4	21.2	odd	6
35.2.j.a.9.1	yes	4	21.11	odd	6
35.2.j.a.9.2	yes	4	105.74	odd	6
175.2.e.a.51.1		2	105.53	even	12
175.2.e.a.151.1		2	105.23	even	12
175.2.e.b.51.1		2	105.32	even	12
175.2.e.b.151.1		2	105.2	even	12
245.2.b.b.99.1		2	105.104	even	2
245.2.b.b.99.2		2	21.20	even	2
245.2.b.c.99.1		2	15.14	odd	2
245.2.b.c.99.2		2	3.2	odd	2
245.2.j.c.79.1		4	21.17	even	6
245.2.j.c.79.2		4	105.59	even	6
245.2.j.c.214.1		4	105.89	even	6
245.2.j.c.214.2		4	21.5	even	6
315.2.bf.a.109.1		4	7.2	even	3
315.2.bf.a.109.2		4	35.9	even	6
315.2.bf.a.289.1		4	35.4	even	6
315.2.bf.a.289.2		4	7.4	even	3
560.2.bw.b.289.1		4	84.11	even	6
560.2.bw.b.289.2		4	420.179	even	6
560.2.bw.b.529.1		4	420.359	even	6
560.2.bw.b.529.2		4	84.23	even	6
1225.2.a.b.1.1		1	105.62	odd	4
1225.2.a.d.1.1		1	15.2	even	4
1225.2.a.f.1.1		1	15.8	even	4
1225.2.a.g.1.1		1	105.83	odd	4
2205.2.d.d.1324.1		2	1.1	even	1	trivial
2205.2.d.d.1324.2		2	5.4	even	2	inner
2205.2.d.e.1324.1		2	7.6	odd	2
2205.2.d.e.1324.2		2	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -3.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -3.00000i q^{8} +(2.00000 + 1.00000i) q^{10} -2.00000i q^{13} -1.00000 q^{16} -2.00000i q^{17} -6.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} -3.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -2.00000 q^{26} +7.00000 q^{29} +2.00000 q^{31} -5.00000i q^{32} -2.00000 q^{34} -8.00000i q^{37} +6.00000i q^{38} +(6.00000 + 3.00000i) q^{40} -5.00000 q^{41} -7.00000i q^{43} -3.00000 q^{46} +(-4.00000 + 3.00000i) q^{50} -2.00000i q^{52} +6.00000i q^{53} -7.00000i q^{58} +10.0000 q^{59} +7.00000 q^{61} -2.00000i q^{62} -7.00000 q^{64} +(4.00000 + 2.00000i) q^{65} -5.00000i q^{67} -2.00000i q^{68} +2.00000 q^{71} +6.00000i q^{73} -8.00000 q^{74} -6.00000 q^{76} +2.00000 q^{79} +(1.00000 - 2.00000i) q^{80} +5.00000i q^{82} -11.0000i q^{83} +(4.00000 + 2.00000i) q^{85} -7.00000 q^{86} +9.00000 q^{89} -3.00000i q^{92} +(6.00000 - 12.0000i) q^{95} -16.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^5	\( 2 q + 2 q^{4} - 2 q^{5} + 4 q^{10} - 2 q^{16} - 12 q^{19} - 2 q^{20} - 6 q^{25} - 4 q^{26} + 14 q^{29} + 4 q^{31} - 4 q^{34} + 12 q^{40} - 10 q^{41} - 6 q^{46} - 8 q^{50} + 20 q^{59} + 14 q^{61} - 14 q^{64} + 8 q^{65} + 4 q^{71} - 16 q^{74} - 12 q^{76} + 4 q^{79} + 2 q^{80} + 8 q^{85} - 14 q^{86} + 18 q^{89} + 12 q^{95}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^5 + 4 * q^10 - 2 * q^16 - 12 * q^19 - 2 * q^20 - 6 * q^25 - 4 * q^26 + 14 * q^29 + 4 * q^31 - 4 * q^34 + 12 * q^40 - 10 * q^41 - 6 * q^46 - 8 * q^50 + 20 * q^59 + 14 * q^61 - 14 * q^64 + 8 * q^65 + 4 * q^71 - 16 * q^74 - 12 * q^76 + 4 * q^79 + 2 * q^80 + 8 * q^85 - 14 * q^86 + 18 * q^89 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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