LMFDB - Embedded newform 1170.2.e.d.469.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(469,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, 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("1170.469");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.e (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:	$9.34249703649$
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 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			469.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1170.469
Dual form			1170.2.e.d.469.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} +1.00000i q^{8} +(1.00000 - 2.00000i) q^{10} +6.00000 q^{11} -1.00000i q^{13} +1.00000 q^{16} -6.00000 q^{19} +(-2.00000 - 1.00000i) q^{20} -6.00000i q^{22} +6.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000 q^{26} +2.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +10.0000i q^{37} +6.00000i q^{38} +(-1.00000 + 2.00000i) q^{40} +6.00000 q^{41} -8.00000i q^{43} -6.00000 q^{44} +6.00000 q^{46} -8.00000i q^{47} +7.00000 q^{49} +(4.00000 - 3.00000i) q^{50} +1.00000i q^{52} -6.00000i q^{53} +(12.0000 + 6.00000i) q^{55} -2.00000i q^{58} +10.0000 q^{59} -6.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +(1.00000 - 2.00000i) q^{65} +4.00000i q^{67} +8.00000 q^{71} -6.00000i q^{73} +10.0000 q^{74} +6.00000 q^{76} -16.0000 q^{79} +(2.00000 + 1.00000i) q^{80} -6.00000i q^{82} +4.00000i q^{83} -8.00000 q^{86} +6.00000i q^{88} -10.0000 q^{89} -6.00000i q^{92} -8.00000 q^{94} +(-12.0000 - 6.00000i) q^{95} +2.00000i q^{97} -7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{5}+O(q^{10}) $ 2 * q - 2 * q^4 + 4 * q^5	$ 2 q - 2 q^{4} + 4 q^{5} + 2 q^{10} + 12 q^{11} + 2 q^{16} - 12 q^{19} - 4 q^{20} + 6 q^{25} - 2 q^{26} + 4 q^{29} + 8 q^{31} - 2 q^{40} + 12 q^{41} - 12 q^{44} + 12 q^{46} + 14 q^{49} + 8 q^{50} + 24 q^{55} + 20 q^{59} - 12 q^{61} - 2 q^{64} + 2 q^{65} + 16 q^{71} + 20 q^{74} + 12 q^{76} - 32 q^{79} + 4 q^{80} - 16 q^{86} - 20 q^{89} - 16 q^{94} - 24 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^5 + 2 * q^10 + 12 * q^11 + 2 * q^16 - 12 * q^19 - 4 * q^20 + 6 * q^25 - 2 * q^26 + 4 * q^29 + 8 * q^31 - 2 * q^40 + 12 * q^41 - 12 * q^44 + 12 * q^46 + 14 * q^49 + 8 * q^50 + 24 * q^55 + 20 * q^59 - 12 * q^61 - 2 * q^64 + 2 * q^65 + 16 * q^71 + 20 * q^74 + 12 * q^76 - 32 * q^79 + 4 * q^80 - 16 * q^86 - 20 * q^89 - 16 * q^94 - 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$-1$	$1$

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
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	1.00000	−	2.00000i	0.316228	−	0.632456i
$11$	6.00000			1.80907			0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000			1.80907			0.904534	+	0.426401i	$0.140219\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$	−2.00000	−	1.00000i	−0.447214	−	0.223607i
$21$		0			0
$22$		−	6.00000i		−	1.27920i
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$	−1.00000			−0.196116
$27$		0			0
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			10.0000i			1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$			10.0000i			1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$			6.00000i			0.973329i
$39$		0			0
$40$	−1.00000	+	2.00000i	−0.158114	+	0.316228i
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	−6.00000			−0.904534
$45$		0			0
$46$	6.00000			0.884652
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	4.00000	−	3.00000i	0.565685	−	0.424264i
$51$		0			0
$52$			1.00000i			0.138675i
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$	12.0000	+	6.00000i	1.61808	+	0.809040i
$56$		0			0
$57$		0			0
$58$		−	2.00000i		−	0.262613i
$59$	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
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		−	4.00000i		−	0.508001i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	1.00000	−	2.00000i	0.124035	−	0.248069i
$66$		0			0
$67$			4.00000i			0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$			4.00000i			0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	10.0000			1.16248
$75$		0			0
$76$	6.00000			0.688247
$77$		0			0
$78$		0			0
$79$	−16.0000			−1.80014			−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000			−1.80014			−0.900070	+	0.435745i	$0.856485\pi$
$80$	2.00000	+	1.00000i	0.223607	+	0.111803i
$81$		0			0
$82$		−	6.00000i		−	0.662589i
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$		0			0
$85$		0			0
$86$	−8.00000			−0.862662
$87$		0			0
$88$			6.00000i			0.639602i
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$		0			0
$92$		−	6.00000i		−	0.625543i
$93$		0			0
$94$	−8.00000			−0.825137
$95$	−12.0000	−	6.00000i	−1.23117	−	0.615587i
$96$		0			0
$97$			2.00000i			0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$			2.00000i			0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$		−	7.00000i		−	0.707107i
$99$		0			0
$100$	−3.00000	−	4.00000i	−0.300000	−	0.400000i
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$			10.0000i			0.985329i	0.870219	+	0.492665i	$0.163977\pi$
$103$			10.0000i			0.985329i	−0.870219	+	0.492665i	$0.836023\pi$
$104$	1.00000			0.0980581
$105$		0			0
$106$	−6.00000			−0.582772
$107$		−	16.0000i		−	1.54678i	−0.633932	−	0.773389i	$-0.718560\pi$
$107$		−	16.0000i		−	1.54678i	0.633932	−	0.773389i	$-0.281440\pi$
$108$		0			0
$109$	16.0000			1.53252			0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000			1.53252			0.766261	+	0.642529i	$0.222115\pi$
$110$	6.00000	−	12.0000i	0.572078	−	1.14416i
$111$		0			0
$112$		0			0
$113$			8.00000i			0.752577i	0.926503	+	0.376288i	$0.122800\pi$
$113$			8.00000i			0.752577i	−0.926503	+	0.376288i	$0.877200\pi$
$114$		0			0
$115$	−6.00000	+	12.0000i	−0.559503	+	1.11901i
$116$	−2.00000			−0.185695
$117$		0			0
$118$		−	10.0000i		−	0.920575i
$119$		0			0
$120$		0			0
$121$	25.0000			2.27273
$122$			6.00000i			0.543214i
$123$		0			0
$124$	−4.00000			−0.359211
$125$	2.00000	+	11.0000i	0.178885	+	0.983870i
$126$		0			0
$127$		−	18.0000i		−	1.59724i	−0.601834	−	0.798621i	$-0.705563\pi$
$127$		−	18.0000i		−	1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−2.00000	−	1.00000i	−0.175412	−	0.0877058i
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$		0			0
$134$	4.00000			0.345547
$135$		0			0
$136$		0			0
$137$		−	2.00000i		−	0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$		−	2.00000i		−	0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$		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
$142$		−	8.00000i		−	0.671345i
$143$		−	6.00000i		−	0.501745i
$144$		0			0
$145$	4.00000	+	2.00000i	0.332182	+	0.166091i
$146$	−6.00000			−0.496564
$147$		0			0
$148$		−	10.0000i		−	0.821995i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		−	6.00000i		−	0.486664i
$153$		0			0
$154$		0			0
$155$	8.00000	+	4.00000i	0.642575	+	0.321288i
$156$		0			0
$157$		−	2.00000i		−	0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$		−	2.00000i		−	0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$			16.0000i			1.27289i
$159$		0			0
$160$	1.00000	−	2.00000i	0.0790569	−	0.158114i
$161$		0			0
$162$		0			0
$163$		−	12.0000i		−	0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$		−	12.0000i		−	0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	−6.00000			−0.468521
$165$		0			0
$166$	4.00000			0.310460
$167$		−	8.00000i		−	0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$		−	8.00000i		−	0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$			8.00000i			0.609994i
$173$			18.0000i			1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$			18.0000i			1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$		0			0
$175$		0			0
$176$	6.00000			0.452267
$177$		0			0
$178$			10.0000i			0.749532i
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$		0			0
$184$	−6.00000			−0.442326
$185$	−10.0000	+	20.0000i	−0.735215	+	1.47043i
$186$		0			0
$187$		0			0
$188$			8.00000i			0.583460i
$189$		0			0
$190$	−6.00000	+	12.0000i	−0.435286	+	0.870572i
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$		−	10.0000i		−	0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$		−	10.0000i		−	0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	2.00000			0.143592
$195$		0			0
$196$	−7.00000			−0.500000
$197$			22.0000i			1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$			22.0000i			1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$	−4.00000	+	3.00000i	−0.282843	+	0.212132i
$201$		0			0
$202$		−	2.00000i		−	0.140720i
$203$		0			0
$204$		0			0
$205$	12.0000	+	6.00000i	0.838116	+	0.419058i
$206$	10.0000			0.696733
$207$		0			0
$208$		−	1.00000i		−	0.0693375i
$209$	−36.0000			−2.49017
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$			6.00000i			0.412082i
$213$		0			0
$214$	−16.0000			−1.09374
$215$	8.00000	−	16.0000i	0.545595	−	1.09119i
$216$		0			0
$217$		0			0
$218$		−	16.0000i		−	1.08366i
$219$		0			0
$220$	−12.0000	−	6.00000i	−0.809040	−	0.404520i
$221$		0			0
$222$		0			0
$223$		−	4.00000i		−	0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$		−	4.00000i		−	0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$		0			0
$225$		0			0
$226$	8.00000			0.532152
$227$			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
$229$	−20.0000			−1.32164			−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000			−1.32164			−0.660819	+	0.750546i	$0.729791\pi$
$230$	12.0000	+	6.00000i	0.791257	+	0.395628i
$231$		0			0
$232$			2.00000i			0.131306i
$233$		−	8.00000i		−	0.524097i	−0.965055	−	0.262049i	$-0.915602\pi$
$233$		−	8.00000i		−	0.524097i	0.965055	−	0.262049i	$-0.0843981\pi$
$234$		0			0
$235$	8.00000	−	16.0000i	0.521862	−	1.04372i
$236$	−10.0000			−0.650945
$237$		0			0
$238$		0			0
$239$	−28.0000			−1.81117			−0.905585	−	0.424165i	$-0.860568\pi$
$239$	−28.0000			−1.81117			−0.905585	+	0.424165i	$0.860568\pi$
$240$		0			0
$241$	−22.0000			−1.41714			−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000			−1.41714			−0.708572	+	0.705638i	$0.750660\pi$
$242$		−	25.0000i		−	1.60706i
$243$		0			0
$244$	6.00000			0.384111
$245$	14.0000	+	7.00000i	0.894427	+	0.447214i
$246$		0			0
$247$			6.00000i			0.381771i
$248$			4.00000i			0.254000i
$249$		0			0
$250$	11.0000	−	2.00000i	0.695701	−	0.126491i
$251$	−8.00000			−0.504956			−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000			−0.504956			−0.252478	+	0.967603i	$0.581245\pi$
$252$		0			0
$253$			36.0000i			2.26330i
$254$	−18.0000			−1.12942
$255$		0			0
$256$	1.00000			0.0625000
$257$			24.0000i			1.49708i	0.663090	+	0.748539i	$0.269245\pi$
$257$			24.0000i			1.49708i	−0.663090	+	0.748539i	$0.730755\pi$
$258$		0			0
$259$		0			0
$260$	−1.00000	+	2.00000i	−0.0620174	+	0.124035i
$261$		0			0
$262$			12.0000i			0.741362i
$263$		−	6.00000i		−	0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$		−	6.00000i		−	0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$		0			0
$265$	6.00000	−	12.0000i	0.368577	−	0.737154i
$266$		0			0
$267$		0			0
$268$		−	4.00000i		−	0.244339i
$269$	14.0000			0.853595			0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$		0			0
$274$	−2.00000			−0.120824
$275$	18.0000	+	24.0000i	1.08544	+	1.44725i
$276$		0			0
$277$			26.0000i			1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$			26.0000i			1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$			8.00000i			0.479808i
$279$		0			0
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$			4.00000i			0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$			4.00000i			0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	−8.00000			−0.474713
$285$		0			0
$286$	−6.00000			−0.354787
$287$		0			0
$288$		0			0
$289$	17.0000			1.00000
$290$	2.00000	−	4.00000i	0.117444	−	0.234888i
$291$		0			0
$292$			6.00000i			0.351123i
$293$		−	14.0000i		−	0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$		−	14.0000i		−	0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$		0			0
$295$	20.0000	+	10.0000i	1.16445	+	0.582223i
$296$	−10.0000			−0.581238
$297$		0			0
$298$		0			0
$299$	6.00000			0.346989
$300$		0			0
$301$		0			0
$302$			8.00000i			0.460348i
$303$		0			0
$304$	−6.00000			−0.344124
$305$	−12.0000	−	6.00000i	−0.687118	−	0.343559i
$306$		0			0
$307$		−	28.0000i		−	1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$		−	28.0000i		−	1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$		0			0
$309$		0			0
$310$	4.00000	−	8.00000i	0.227185	−	0.454369i
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$			16.0000i			0.904373i	0.891923	+	0.452187i	$0.149356\pi$
$313$			16.0000i			0.904373i	−0.891923	+	0.452187i	$0.850644\pi$
$314$	−2.00000			−0.112867
$315$		0			0
$316$	16.0000			0.900070
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$	12.0000			0.671871
$320$	−2.00000	−	1.00000i	−0.111803	−	0.0559017i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	4.00000	−	3.00000i	0.221880	−	0.166410i
$326$	−12.0000			−0.664619
$327$		0			0
$328$			6.00000i			0.331295i
$329$		0			0
$330$		0			0
$331$	−26.0000			−1.42909			−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000			−1.42909			−0.714545	+	0.699590i	$0.753366\pi$
$332$		−	4.00000i		−	0.219529i
$333$		0			0
$334$	−8.00000			−0.437741
$335$	−4.00000	+	8.00000i	−0.218543	+	0.437087i
$336$		0			0
$337$			8.00000i			0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$			8.00000i			0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$			1.00000i			0.0543928i
$339$		0			0
$340$		0			0
$341$	24.0000			1.29967
$342$		0			0
$343$		0			0
$344$	8.00000			0.431331
$345$		0			0
$346$	18.0000			0.967686
$347$		−	32.0000i		−	1.71785i	−0.512101	−	0.858925i	$-0.671133\pi$
$347$		−	32.0000i		−	1.71785i	0.512101	−	0.858925i	$-0.328867\pi$
$348$		0			0
$349$	−8.00000			−0.428230			−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000			−0.428230			−0.214115	+	0.976808i	$0.568687\pi$
$350$		0			0
$351$		0			0
$352$		−	6.00000i		−	0.319801i
$353$		−	10.0000i		−	0.532246i	−0.963939	−	0.266123i	$-0.914257\pi$
$353$		−	10.0000i		−	0.532246i	0.963939	−	0.266123i	$-0.0857428\pi$
$354$		0			0
$355$	16.0000	+	8.00000i	0.849192	+	0.424596i
$356$	10.0000			0.529999
$357$		0			0
$358$			12.0000i			0.634220i
$359$	−12.0000			−0.633336			−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000			−0.633336			−0.316668	+	0.948536i	$0.602564\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$			2.00000i			0.105118i
$363$		0			0
$364$		0			0
$365$	6.00000	−	12.0000i	0.314054	−	0.628109i
$366$		0			0
$367$			6.00000i			0.313197i	0.987662	+	0.156599i	$0.0500529\pi$
$367$			6.00000i			0.313197i	−0.987662	+	0.156599i	$0.949947\pi$
$368$			6.00000i			0.312772i
$369$		0			0
$370$	20.0000	+	10.0000i	1.03975	+	0.519875i
$371$		0			0
$372$		0			0
$373$			10.0000i			0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$			10.0000i			0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$		0			0
$375$		0			0
$376$	8.00000			0.412568
$377$		−	2.00000i		−	0.103005i
$378$		0			0
$379$	−22.0000			−1.13006			−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000			−1.13006			−0.565032	+	0.825069i	$0.691136\pi$
$380$	12.0000	+	6.00000i	0.615587	+	0.307794i
$381$		0			0
$382$		−	8.00000i		−	0.409316i
$383$		−	20.0000i		−	1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$		−	20.0000i		−	1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$		0			0
$385$		0			0
$386$	−10.0000			−0.508987
$387$		0			0
$388$		−	2.00000i		−	0.101535i
$389$	−14.0000			−0.709828			−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000			−0.709828			−0.354914	+	0.934899i	$0.615490\pi$
$390$		0			0
$391$		0			0
$392$			7.00000i			0.353553i
$393$		0			0
$394$	22.0000			1.10834
$395$	−32.0000	−	16.0000i	−1.61009	−	0.805047i
$396$		0			0
$397$		−	18.0000i		−	0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$		−	18.0000i		−	0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$			16.0000i			0.802008i
$399$		0			0
$400$	3.00000	+	4.00000i	0.150000	+	0.200000i
$401$	−14.0000			−0.699127			−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000			−0.699127			−0.349563	+	0.936913i	$0.613670\pi$
$402$		0			0
$403$		−	4.00000i		−	0.199254i
$404$	−2.00000			−0.0995037
$405$		0			0
$406$		0			0
$407$			60.0000i			2.97409i
$408$		0			0
$409$	−30.0000			−1.48340			−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000			−1.48340			−0.741702	+	0.670729i	$0.765981\pi$
$410$	6.00000	−	12.0000i	0.296319	−	0.592638i
$411$		0			0
$412$		−	10.0000i		−	0.492665i
$413$		0			0
$414$		0			0
$415$	−4.00000	+	8.00000i	−0.196352	+	0.392705i
$416$	−1.00000			−0.0490290
$417$		0			0
$418$			36.0000i			1.76082i
$419$	28.0000			1.36789			0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000			1.36789			0.683945	+	0.729534i	$0.260263\pi$
$420$		0			0
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		−	4.00000i		−	0.194717i
$423$		0			0
$424$	6.00000			0.291386
$425$		0			0
$426$		0			0
$427$		0			0
$428$			16.0000i			0.773389i
$429$		0			0
$430$	−16.0000	−	8.00000i	−0.771589	−	0.385794i
$431$	8.00000			0.385346			0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000			0.385346			0.192673	+	0.981263i	$0.438284\pi$
$432$		0			0
$433$		−	24.0000i		−	1.15337i	−0.816968	−	0.576683i	$-0.804347\pi$
$433$		−	24.0000i		−	1.15337i	0.816968	−	0.576683i	$-0.195653\pi$
$434$		0			0
$435$		0			0
$436$	−16.0000			−0.766261
$437$		−	36.0000i		−	1.72211i
$438$		0			0
$439$	−16.0000			−0.763638			−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000			−0.763638			−0.381819	+	0.924237i	$0.624702\pi$
$440$	−6.00000	+	12.0000i	−0.286039	+	0.572078i
$441$		0			0
$442$		0			0
$443$		−	32.0000i		−	1.52037i	−0.649709	−	0.760183i	$-0.725109\pi$
$443$		−	32.0000i		−	1.52037i	0.649709	−	0.760183i	$-0.274891\pi$
$444$		0			0
$445$	−20.0000	−	10.0000i	−0.948091	−	0.474045i
$446$	−4.00000			−0.189405
$447$		0			0
$448$		0			0
$449$	−34.0000			−1.60456			−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000			−1.60456			−0.802280	+	0.596948i	$0.796380\pi$
$450$		0			0
$451$	36.0000			1.69517
$452$		−	8.00000i		−	0.376288i
$453$		0			0
$454$	20.0000			0.938647
$455$		0			0
$456$		0			0
$457$			18.0000i			0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$			18.0000i			0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$			20.0000i			0.934539i
$459$		0			0
$460$	6.00000	−	12.0000i	0.279751	−	0.559503i
$461$	−36.0000			−1.67669			−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000			−1.67669			−0.838344	+	0.545142i	$0.816476\pi$
$462$		0			0
$463$			16.0000i			0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$			16.0000i			0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	2.00000			0.0928477
$465$		0			0
$466$	−8.00000			−0.370593
$467$			40.0000i			1.85098i	0.378773	+	0.925490i	$0.376346\pi$
$467$			40.0000i			1.85098i	−0.378773	+	0.925490i	$0.623654\pi$
$468$		0			0
$469$		0			0
$470$	−16.0000	−	8.00000i	−0.738025	−	0.369012i
$471$		0			0
$472$			10.0000i			0.460287i
$473$		−	48.0000i		−	2.20704i
$474$		0			0
$475$	−18.0000	−	24.0000i	−0.825897	−	1.10120i
$476$		0			0
$477$		0			0
$478$			28.0000i			1.28069i
$479$	20.0000			0.913823			0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000			0.913823			0.456912	+	0.889512i	$0.348956\pi$
$480$		0			0
$481$	10.0000			0.455961
$482$			22.0000i			1.00207i
$483$		0			0
$484$	−25.0000			−1.13636
$485$	−2.00000	+	4.00000i	−0.0908153	+	0.181631i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		−	6.00000i		−	0.271607i
$489$		0			0
$490$	7.00000	−	14.0000i	0.316228	−	0.632456i
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$		0			0
$494$	6.00000			0.269953
$495$		0			0
$496$	4.00000			0.179605
$497$		0			0
$498$		0			0
$499$	6.00000			0.268597			0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000			0.268597			0.134298	+	0.990941i	$0.457122\pi$
$500$	−2.00000	−	11.0000i	−0.0894427	−	0.491935i
$501$		0			0
$502$			8.00000i			0.357057i
$503$		−	26.0000i		−	1.15928i	−0.814872	−	0.579641i	$-0.803193\pi$
$503$		−	26.0000i		−	1.15928i	0.814872	−	0.579641i	$-0.196807\pi$
$504$		0			0
$505$	4.00000	+	2.00000i	0.177998	+	0.0889988i
$506$	36.0000			1.60040
$507$		0			0
$508$			18.0000i			0.798621i
$509$	24.0000			1.06378			0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000			1.06378			0.531891	+	0.846813i	$0.321482\pi$
$510$		0			0
$511$		0			0
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	24.0000			1.05859
$515$	−10.0000	+	20.0000i	−0.440653	+	0.881305i
$516$		0			0
$517$		−	48.0000i		−	2.11104i
$518$		0			0
$519$		0			0
$520$	2.00000	+	1.00000i	0.0877058	+	0.0438529i
$521$	10.0000			0.438108			0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000			0.438108			0.219054	+	0.975713i	$0.429703\pi$
$522$		0			0
$523$		−	40.0000i		−	1.74908i	−0.484955	−	0.874539i	$-0.661164\pi$
$523$		−	40.0000i		−	1.74908i	0.484955	−	0.874539i	$-0.338836\pi$
$524$	12.0000			0.524222
$525$		0			0
$526$	−6.00000			−0.261612
$527$		0			0
$528$		0			0
$529$	−13.0000			−0.565217
$530$	−12.0000	−	6.00000i	−0.521247	−	0.260623i
$531$		0			0
$532$		0			0
$533$		−	6.00000i		−	0.259889i
$534$		0			0
$535$	16.0000	−	32.0000i	0.691740	−	1.38348i
$536$	−4.00000			−0.172774
$537$		0			0
$538$		−	14.0000i		−	0.603583i
$539$	42.0000			1.80907
$540$		0			0
$541$	4.00000			0.171973			0.0859867	−	0.996296i	$-0.472596\pi$
$541$	4.00000			0.171973			0.0859867	+	0.996296i	$0.472596\pi$
$542$			16.0000i			0.687259i
$543$		0			0
$544$		0			0
$545$	32.0000	+	16.0000i	1.37073	+	0.685365i
$546$		0			0
$547$			12.0000i			0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$			12.0000i			0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$			2.00000i			0.0854358i
$549$		0			0
$550$	24.0000	−	18.0000i	1.02336	−	0.767523i
$551$	−12.0000			−0.511217
$552$		0			0
$553$		0			0
$554$	26.0000			1.10463
$555$		0			0
$556$	8.00000			0.339276
$557$			2.00000i			0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$			2.00000i			0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		−	10.0000i		−	0.421825i
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$		0			0
$565$	−8.00000	+	16.0000i	−0.336563	+	0.673125i
$566$	4.00000			0.168133
$567$		0			0
$568$			8.00000i			0.335673i
$569$	22.0000			0.922288			0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000			0.922288			0.461144	+	0.887325i	$0.347439\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$			6.00000i			0.250873i
$573$		0			0
$574$		0			0
$575$	−24.0000	+	18.0000i	−1.00087	+	0.750652i
$576$		0			0
$577$		−	2.00000i		−	0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$		−	2.00000i		−	0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$		−	17.0000i		−	0.707107i
$579$		0			0
$580$	−4.00000	−	2.00000i	−0.166091	−	0.0830455i
$581$		0			0
$582$		0			0
$583$		−	36.0000i		−	1.49097i
$584$	6.00000			0.248282
$585$		0			0
$586$	−14.0000			−0.578335
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$		0			0
$589$	−24.0000			−0.988903
$590$	10.0000	−	20.0000i	0.411693	−	0.823387i
$591$		0			0
$592$			10.0000i			0.410997i
$593$			6.00000i			0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$			6.00000i			0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		−	6.00000i		−	0.245358i
$599$	16.0000			0.653742			0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000			0.653742			0.326871	+	0.945069i	$0.394006\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$		0			0
$604$	8.00000			0.325515
$605$	50.0000	+	25.0000i	2.03279	+	1.01639i
$606$		0			0
$607$			2.00000i			0.0811775i	0.999176	+	0.0405887i	$0.0129233\pi$
$607$			2.00000i			0.0811775i	−0.999176	+	0.0405887i	$0.987077\pi$
$608$			6.00000i			0.243332i
$609$		0			0
$610$	−6.00000	+	12.0000i	−0.242933	+	0.485866i
$611$	−8.00000			−0.323645
$612$		0			0
$613$			38.0000i			1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$			38.0000i			1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$	−28.0000			−1.12999
$615$		0			0
$616$		0			0
$617$			18.0000i			0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$			18.0000i			0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$		0			0
$619$	−38.0000			−1.52735			−0.763674	−	0.645601i	$-0.776607\pi$
$619$	−38.0000			−1.52735			−0.763674	+	0.645601i	$0.776607\pi$
$620$	−8.00000	−	4.00000i	−0.321288	−	0.160644i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	16.0000			0.639489
$627$		0			0
$628$			2.00000i			0.0798087i
$629$		0			0
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$		−	16.0000i		−	0.636446i
$633$		0			0
$634$	−18.0000			−0.714871
$635$	18.0000	−	36.0000i	0.714308	−	1.42862i
$636$		0			0
$637$		−	7.00000i		−	0.277350i
$638$		−	12.0000i		−	0.475085i
$639$		0			0
$640$	−1.00000	+	2.00000i	−0.0395285	+	0.0790569i
$641$	42.0000			1.65890			0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000			1.65890			0.829450	+	0.558581i	$0.188654\pi$
$642$		0			0
$643$		−	44.0000i		−	1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$		−	44.0000i		−	1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	14.0000i		−	0.550397i	−0.961387	−	0.275198i	$-0.911256\pi$
$647$		−	14.0000i		−	0.550397i	0.961387	−	0.275198i	$-0.0887435\pi$
$648$		0			0
$649$	60.0000			2.35521
$650$	−3.00000	−	4.00000i	−0.117670	−	0.156893i
$651$		0			0
$652$			12.0000i			0.469956i
$653$			6.00000i			0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$			6.00000i			0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$		0			0
$655$	−24.0000	−	12.0000i	−0.937758	−	0.468879i
$656$	6.00000			0.234261
$657$		0			0
$658$		0			0
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$	−28.0000			−1.08907			−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000			−1.08907			−0.544537	+	0.838737i	$0.683295\pi$
$662$			26.0000i			1.01052i
$663$		0			0
$664$	−4.00000			−0.155230
$665$		0			0
$666$		0			0
$667$			12.0000i			0.464642i
$668$			8.00000i			0.309529i
$669$		0			0
$670$	8.00000	+	4.00000i	0.309067	+	0.154533i
$671$	−36.0000			−1.38976
$672$		0			0
$673$		−	44.0000i		−	1.69608i	−0.529936	−	0.848038i	$-0.677784\pi$
$673$		−	44.0000i		−	1.69608i	0.529936	−	0.848038i	$-0.322216\pi$
$674$	8.00000			0.308148
$675$		0			0
$676$	1.00000			0.0384615
$677$		−	38.0000i		−	1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$		−	38.0000i		−	1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		−	24.0000i		−	0.919007i
$683$		−	12.0000i		−	0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$		−	12.0000i		−	0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$		0			0
$685$	2.00000	−	4.00000i	0.0764161	−	0.152832i
$686$		0			0
$687$		0			0
$688$		−	8.00000i		−	0.304997i
$689$	−6.00000			−0.228582
$690$		0			0
$691$	22.0000			0.836919			0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000			0.836919			0.418460	+	0.908235i	$0.362570\pi$
$692$		−	18.0000i		−	0.684257i
$693$		0			0
$694$	−32.0000			−1.21470
$695$	−16.0000	−	8.00000i	−0.606915	−	0.303457i
$696$		0			0
$697$		0			0
$698$			8.00000i			0.302804i
$699$		0			0
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$		−	60.0000i		−	2.26294i
$704$	−6.00000			−0.226134
$705$		0			0
$706$	−10.0000			−0.376355
$707$		0			0
$708$		0			0
$709$	44.0000			1.65245			0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000			1.65245			0.826227	+	0.563337i	$0.190483\pi$
$710$	8.00000	−	16.0000i	0.300235	−	0.600469i
$711$		0			0
$712$		−	10.0000i		−	0.374766i
$713$			24.0000i			0.898807i
$714$		0			0
$715$	6.00000	−	12.0000i	0.224387	−	0.448775i
$716$	12.0000			0.448461
$717$		0			0
$718$			12.0000i			0.447836i
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$		0			0
$722$		−	17.0000i		−	0.632674i
$723$		0			0
$724$	2.00000			0.0743294
$725$	6.00000	+	8.00000i	0.222834	+	0.297113i
$726$		0			0
$727$			38.0000i			1.40934i	0.709534	+	0.704671i	$0.248905\pi$
$727$			38.0000i			1.40934i	−0.709534	+	0.704671i	$0.751095\pi$
$728$		0			0
$729$		0			0
$730$	−12.0000	−	6.00000i	−0.444140	−	0.222070i
$731$		0			0
$732$		0			0
$733$			26.0000i			0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$			26.0000i			0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	6.00000			0.221464
$735$		0			0
$736$	6.00000			0.221163
$737$			24.0000i			0.884051i
$738$		0			0
$739$	22.0000			0.809283			0.404642	−	0.914475i	$-0.367396\pi$
$739$	22.0000			0.809283			0.404642	+	0.914475i	$0.367396\pi$
$740$	10.0000	−	20.0000i	0.367607	−	0.735215i
$741$		0			0
$742$		0			0
$743$			52.0000i			1.90769i	0.300291	+	0.953847i	$0.402916\pi$
$743$			52.0000i			1.90769i	−0.300291	+	0.953847i	$0.597084\pi$
$744$		0			0
$745$		0			0
$746$	10.0000			0.366126
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$		−	8.00000i		−	0.291730i
$753$		0			0
$754$	−2.00000			−0.0728357
$755$	−16.0000	−	8.00000i	−0.582300	−	0.291150i
$756$		0			0
$757$			10.0000i			0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$			10.0000i			0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$			22.0000i			0.799076i
$759$		0			0
$760$	6.00000	−	12.0000i	0.217643	−	0.435286i
$761$	−30.0000			−1.08750			−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000			−1.08750			−0.543750	+	0.839248i	$0.682996\pi$
$762$		0			0
$763$		0			0
$764$	−8.00000			−0.289430
$765$		0			0
$766$	−20.0000			−0.722629
$767$		−	10.0000i		−	0.361079i
$768$		0			0
$769$	54.0000			1.94729			0.973645	−	0.228069i	$-0.0732413\pi$
$769$	54.0000			1.94729			0.973645	+	0.228069i	$0.0732413\pi$
$770$		0			0
$771$		0			0
$772$			10.0000i			0.359908i
$773$		−	18.0000i		−	0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$		−	18.0000i		−	0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$		0			0
$775$	12.0000	+	16.0000i	0.431053	+	0.574737i
$776$	−2.00000			−0.0717958
$777$		0			0
$778$			14.0000i			0.501924i
$779$	−36.0000			−1.28983
$780$		0			0
$781$	48.0000			1.71758
$782$		0			0
$783$		0			0
$784$	7.00000			0.250000
$785$	2.00000	−	4.00000i	0.0713831	−	0.142766i
$786$		0			0
$787$		−	12.0000i		−	0.427754i	−0.976861	−	0.213877i	$-0.931391\pi$
$787$		−	12.0000i		−	0.427754i	0.976861	−	0.213877i	$-0.0686091\pi$
$788$		−	22.0000i		−	0.783718i
$789$		0			0
$790$	−16.0000	+	32.0000i	−0.569254	+	1.13851i
$791$		0			0
$792$		0			0
$793$			6.00000i			0.213066i
$794$	−18.0000			−0.638796
$795$		0			0
$796$	16.0000			0.567105
$797$		−	42.0000i		−	1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$		−	42.0000i		−	1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$		0			0
$799$		0			0
$800$	4.00000	−	3.00000i	0.141421	−	0.106066i
$801$		0			0
$802$			14.0000i			0.494357i
$803$		−	36.0000i		−	1.27041i
$804$		0			0
$805$		0			0
$806$	−4.00000			−0.140894
$807$		0			0
$808$			2.00000i			0.0703598i
$809$	6.00000			0.210949			0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000			0.210949			0.105474	+	0.994422i	$0.466364\pi$
$810$		0			0
$811$	2.00000			0.0702295			0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000			0.0702295			0.0351147	+	0.999383i	$0.488820\pi$
$812$		0			0
$813$		0			0
$814$	60.0000			2.10300
$815$	12.0000	−	24.0000i	0.420342	−	0.840683i
$816$		0			0
$817$			48.0000i			1.67931i
$818$			30.0000i			1.04893i
$819$		0			0
$820$	−12.0000	−	6.00000i	−0.419058	−	0.209529i
$821$	−4.00000			−0.139601			−0.0698005	−	0.997561i	$-0.522236\pi$
$821$	−4.00000			−0.139601			−0.0698005	+	0.997561i	$0.522236\pi$
$822$		0			0
$823$			54.0000i			1.88232i	0.337959	+	0.941161i	$0.390263\pi$
$823$			54.0000i			1.88232i	−0.337959	+	0.941161i	$0.609737\pi$
$824$	−10.0000			−0.348367
$825$		0			0
$826$		0			0
$827$			20.0000i			0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$			20.0000i			0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$		0			0
$829$	−26.0000			−0.903017			−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000			−0.903017			−0.451509	+	0.892267i	$0.649114\pi$
$830$	8.00000	+	4.00000i	0.277684	+	0.138842i
$831$		0			0
$832$			1.00000i			0.0346688i
$833$		0			0
$834$		0			0
$835$	8.00000	−	16.0000i	0.276851	−	0.553703i
$836$	36.0000			1.24509
$837$		0			0
$838$		−	28.0000i		−	0.967244i
$839$	−16.0000			−0.552381			−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000			−0.552381			−0.276191	+	0.961103i	$0.589072\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$			8.00000i			0.275698i
$843$		0			0
$844$	−4.00000			−0.137686
$845$	−2.00000	−	1.00000i	−0.0688021	−	0.0344010i
$846$		0			0
$847$		0			0
$848$		−	6.00000i		−	0.206041i
$849$		0			0
$850$		0			0
$851$	−60.0000			−2.05677
$852$		0			0
$853$		−	14.0000i		−	0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$		−	14.0000i		−	0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$		0			0
$855$		0			0
$856$	16.0000			0.546869
$857$			52.0000i			1.77629i	0.459567	+	0.888143i	$0.348005\pi$
$857$			52.0000i			1.77629i	−0.459567	+	0.888143i	$0.651995\pi$
$858$		0			0
$859$	−12.0000			−0.409435			−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000			−0.409435			−0.204717	+	0.978821i	$0.565628\pi$
$860$	−8.00000	+	16.0000i	−0.272798	+	0.545595i
$861$		0			0
$862$		−	8.00000i		−	0.272481i
$863$		−	32.0000i		−	1.08929i	−0.838666	−	0.544646i	$-0.816664\pi$
$863$		−	32.0000i		−	1.08929i	0.838666	−	0.544646i	$-0.183336\pi$
$864$		0			0
$865$	−18.0000	+	36.0000i	−0.612018	+	1.22404i
$866$	−24.0000			−0.815553
$867$		0			0
$868$		0			0
$869$	−96.0000			−3.25658
$870$		0			0
$871$	4.00000			0.135535
$872$			16.0000i			0.541828i
$873$		0			0
$874$	−36.0000			−1.21772
$875$		0			0
$876$		0			0
$877$		−	22.0000i		−	0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$		−	22.0000i		−	0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$			16.0000i			0.539974i
$879$		0			0
$880$	12.0000	+	6.00000i	0.404520	+	0.202260i
$881$	54.0000			1.81931			0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000			1.81931			0.909653	+	0.415369i	$0.136347\pi$
$882$		0			0
$883$			16.0000i			0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$			16.0000i			0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$		0			0
$885$		0			0
$886$	−32.0000			−1.07506
$887$			26.0000i			0.872995i	0.899706	+	0.436497i	$0.143781\pi$
$887$			26.0000i			0.872995i	−0.899706	+	0.436497i	$0.856219\pi$
$888$		0			0
$889$		0			0
$890$	−10.0000	+	20.0000i	−0.335201	+	0.670402i
$891$		0			0
$892$			4.00000i			0.133930i
$893$			48.0000i			1.60626i
$894$		0			0
$895$	−24.0000	−	12.0000i	−0.802232	−	0.401116i
$896$		0			0
$897$		0			0
$898$			34.0000i			1.13459i
$899$	8.00000			0.266815
$900$		0			0
$901$		0			0
$902$		−	36.0000i		−	1.19867i
$903$		0			0
$904$	−8.00000			−0.266076
$905$	−4.00000	−	2.00000i	−0.132964	−	0.0664822i
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		−	20.0000i		−	0.663723i
$909$		0			0
$910$		0			0
$911$	16.0000			0.530104			0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000			0.530104			0.265052	+	0.964234i	$0.414611\pi$
$912$		0			0
$913$			24.0000i			0.794284i
$914$	18.0000			0.595387
$915$		0			0
$916$	20.0000			0.660819
$917$		0			0
$918$		0			0
$919$	24.0000			0.791687			0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000			0.791687			0.395843	+	0.918318i	$0.370452\pi$
$920$	−12.0000	−	6.00000i	−0.395628	−	0.197814i
$921$		0			0
$922$			36.0000i			1.18560i
$923$		−	8.00000i		−	0.263323i
$924$		0			0
$925$	−40.0000	+	30.0000i	−1.31519	+	0.986394i
$926$	16.0000			0.525793
$927$		0			0
$928$		−	2.00000i		−	0.0656532i
$929$	6.00000			0.196854			0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000			0.196854			0.0984268	+	0.995144i	$0.468619\pi$
$930$		0			0
$931$	−42.0000			−1.37649
$932$			8.00000i			0.262049i
$933$		0			0
$934$	40.0000			1.30884
$935$		0			0
$936$		0			0
$937$			16.0000i			0.522697i	0.965244	+	0.261349i	$0.0841672\pi$
$937$			16.0000i			0.522697i	−0.965244	+	0.261349i	$0.915833\pi$
$938$		0			0
$939$		0			0
$940$	−8.00000	+	16.0000i	−0.260931	+	0.521862i
$941$	24.0000			0.782378			0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000			0.782378			0.391189	+	0.920310i	$0.372064\pi$
$942$		0			0
$943$			36.0000i			1.17232i
$944$	10.0000			0.325472
$945$		0			0
$946$	−48.0000			−1.56061
$947$		−	4.00000i		−	0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$		−	4.00000i		−	0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$		0			0
$949$	−6.00000			−0.194768
$950$	−24.0000	+	18.0000i	−0.778663	+	0.583997i
$951$		0			0
$952$		0			0
$953$			44.0000i			1.42530i	0.701520	+	0.712650i	$0.252505\pi$
$953$			44.0000i			1.42530i	−0.701520	+	0.712650i	$0.747495\pi$
$954$		0			0
$955$	16.0000	+	8.00000i	0.517748	+	0.258874i
$956$	28.0000			0.905585
$957$		0			0
$958$		−	20.0000i		−	0.646171i
$959$		0			0
$960$		0			0
$961$	−15.0000			−0.483871
$962$		−	10.0000i		−	0.322413i
$963$		0			0
$964$	22.0000			0.708572
$965$	10.0000	−	20.0000i	0.321911	−	0.643823i
$966$		0			0
$967$			32.0000i			1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$			32.0000i			1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$			25.0000i			0.803530i
$969$		0			0
$970$	4.00000	+	2.00000i	0.128432	+	0.0642161i
$971$	−40.0000			−1.28366			−0.641831	−	0.766846i	$-0.721825\pi$
$971$	−40.0000			−1.28366			−0.641831	+	0.766846i	$0.721825\pi$
$972$		0			0
$973$		0			0
$974$	−12.0000			−0.384505
$975$		0			0
$976$	−6.00000			−0.192055
$977$		−	50.0000i		−	1.59964i	−0.600239	−	0.799821i	$-0.704928\pi$
$977$		−	50.0000i		−	1.59964i	0.600239	−	0.799821i	$-0.295072\pi$
$978$		0			0
$979$	−60.0000			−1.91761
$980$	−14.0000	−	7.00000i	−0.447214	−	0.223607i
$981$		0			0
$982$		−	12.0000i		−	0.382935i
$983$			12.0000i			0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$			12.0000i			0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$		0			0
$985$	−22.0000	+	44.0000i	−0.700978	+	1.40196i
$986$		0			0
$987$		0			0
$988$		−	6.00000i		−	0.190885i
$989$	48.0000			1.52631
$990$		0			0
$991$	8.00000			0.254128			0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000			0.254128			0.127064	+	0.991894i	$0.459445\pi$
$992$		−	4.00000i		−	0.127000i
$993$		0			0
$994$		0			0
$995$	−32.0000	−	16.0000i	−1.01447	−	0.507234i
$996$		0			0
$997$			14.0000i			0.443384i	0.975117	+	0.221692i	$0.0711580\pi$
$997$			14.0000i			0.443384i	−0.975117	+	0.221692i	$0.928842\pi$
$998$		−	6.00000i		−	0.189927i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.e.d.469.1		2
3.2	odd	2		390.2.e.a.79.2	yes	2
5.2	odd	4		5850.2.a.bs.1.1		1
5.3	odd	4		5850.2.a.o.1.1		1
5.4	even	2	inner	1170.2.e.d.469.2		2
12.11	even	2		3120.2.l.a.1249.1		2
15.2	even	4		1950.2.a.i.1.1		1
15.8	even	4		1950.2.a.r.1.1		1
15.14	odd	2		390.2.e.a.79.1	✓	2
60.59	even	2		3120.2.l.a.1249.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.a.79.1	✓	2	15.14	odd	2
390.2.e.a.79.2	yes	2	3.2	odd	2
1170.2.e.d.469.1		2	1.1	even	1	trivial
1170.2.e.d.469.2		2	5.4	even	2	inner
1950.2.a.i.1.1		1	15.2	even	4
1950.2.a.r.1.1		1	15.8	even	4
3120.2.l.a.1249.1		2	12.11	even	2
3120.2.l.a.1249.2		2	60.59	even	2
5850.2.a.o.1.1		1	5.3	odd	4
5850.2.a.bs.1.1		1	5.2	odd	4

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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} +1.00000i q^{8} +(1.00000 - 2.00000i) q^{10} +6.00000 q^{11} -1.00000i q^{13} +1.00000 q^{16} -6.00000 q^{19} +(-2.00000 - 1.00000i) q^{20} -6.00000i q^{22} +6.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000 q^{26} +2.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +10.0000i q^{37} +6.00000i q^{38} +(-1.00000 + 2.00000i) q^{40} +6.00000 q^{41} -8.00000i q^{43} -6.00000 q^{44} +6.00000 q^{46} -8.00000i q^{47} +7.00000 q^{49} +(4.00000 - 3.00000i) q^{50} +1.00000i q^{52} -6.00000i q^{53} +(12.0000 + 6.00000i) q^{55} -2.00000i q^{58} +10.0000 q^{59} -6.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +(1.00000 - 2.00000i) q^{65} +4.00000i q^{67} +8.00000 q^{71} -6.00000i q^{73} +10.0000 q^{74} +6.00000 q^{76} -16.0000 q^{79} +(2.00000 + 1.00000i) q^{80} -6.00000i q^{82} +4.00000i q^{83} -8.00000 q^{86} +6.00000i q^{88} -10.0000 q^{89} -6.00000i q^{92} -8.00000 q^{94} +(-12.0000 - 6.00000i) q^{95} +2.00000i q^{97} -7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{5}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^5	\( 2 q - 2 q^{4} + 4 q^{5} + 2 q^{10} + 12 q^{11} + 2 q^{16} - 12 q^{19} - 4 q^{20} + 6 q^{25} - 2 q^{26} + 4 q^{29} + 8 q^{31} - 2 q^{40} + 12 q^{41} - 12 q^{44} + 12 q^{46} + 14 q^{49} + 8 q^{50} + 24 q^{55} + 20 q^{59} - 12 q^{61} - 2 q^{64} + 2 q^{65} + 16 q^{71} + 20 q^{74} + 12 q^{76} - 32 q^{79} + 4 q^{80} - 16 q^{86} - 20 q^{89} - 16 q^{94} - 24 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^5 + 2 * q^10 + 12 * q^11 + 2 * q^16 - 12 * q^19 - 4 * q^20 + 6 * q^25 - 2 * q^26 + 4 * q^29 + 8 * q^31 - 2 * q^40 + 12 * q^41 - 12 * q^44 + 12 * q^46 + 14 * q^49 + 8 * q^50 + 24 * q^55 + 20 * q^59 - 12 * q^61 - 2 * q^64 + 2 * q^65 + 16 * q^71 + 20 * q^74 + 12 * q^76 - 32 * q^79 + 4 * q^80 - 16 * q^86 - 20 * q^89 - 16 * q^94 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more