LMFDB - Embedded newform 60.2.d.a.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,2,Mod(49,60)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("60.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	60.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:	$0.479102412128$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	60.49
Dual form			60.2.d.a.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +(-2.00000 - 1.00000i) q^{15} +4.00000i q^{17} +4.00000 q^{21} -4.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +1.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} +4.00000i q^{33} +(8.00000 + 4.00000i) q^{35} -8.00000i q^{37} -10.0000 q^{41} -4.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} -4.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} +12.0000i q^{53} +(-4.00000 + 8.00000i) q^{55} -4.00000 q^{59} +2.00000 q^{61} -4.00000i q^{63} -4.00000i q^{67} -4.00000 q^{69} +8.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} -16.0000i q^{77} +12.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +(8.00000 + 4.00000i) q^{85} -6.00000i q^{87} +10.0000 q^{89} -4.00000i q^{93} +8.00000i q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^9	$ 2 q + 2 q^{5} - 2 q^{9} - 8 q^{11} - 4 q^{15} + 8 q^{21} - 6 q^{25} + 12 q^{29} + 8 q^{31} + 16 q^{35} - 20 q^{41} - 2 q^{45} - 18 q^{49} + 8 q^{51} - 8 q^{55} - 8 q^{59} + 4 q^{61} - 8 q^{69} - 8 q^{75} + 24 q^{79} + 2 q^{81} + 16 q^{85} + 20 q^{89} + 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^9 - 8 * q^11 - 4 * q^15 + 8 * q^21 - 6 * q^25 + 12 * q^29 + 8 * q^31 + 16 * q^35 - 20 * q^41 - 2 * q^45 - 18 * q^49 + 8 * q^51 - 8 * q^55 - 8 * q^59 + 4 * q^61 - 8 * q^69 - 8 * q^75 + 24 * q^79 + 2 * q^81 + 16 * q^85 + 20 * q^89 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$41$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−2.00000	−	1.00000i	−0.516398	−	0.258199i
$16$		0			0
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	4.00000			0.872872
$22$		0			0
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$			4.00000i			0.696311i
$34$		0			0
$35$	8.00000	+	4.00000i	1.35225	+	0.676123i
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$	−1.00000	+	2.00000i	−0.149071	+	0.298142i
$46$		0			0
$47$		−	4.00000i		−	0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$		−	4.00000i		−	0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$	4.00000			0.560112
$52$		0			0
$53$			12.0000i			1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$			12.0000i			1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$		0			0
$55$	−4.00000	+	8.00000i	−0.539360	+	1.07872i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		0			0
$63$		−	4.00000i		−	0.503953i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		0			0
$69$	−4.00000			−0.481543
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$			8.00000i			0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$			8.00000i			0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$		0			0
$75$	−4.00000	+	3.00000i	−0.461880	+	0.346410i
$76$		0			0
$77$		−	16.0000i		−	1.82337i
$78$		0			0
$79$	12.0000			1.35011			0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000			1.35011			0.675053	+	0.737769i	$0.264121\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$		0			0
$85$	8.00000	+	4.00000i	0.867722	+	0.433861i
$86$		0			0
$87$		−	6.00000i		−	0.643268i
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	4.00000i		−	0.414781i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			8.00000i			0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$			8.00000i			0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$		0			0
$99$	4.00000			0.402015
$100$		0			0
$101$	−2.00000			−0.199007			−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000			−0.199007			−0.0995037	+	0.995037i	$0.531726\pi$
$102$		0			0
$103$			4.00000i			0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$			4.00000i			0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$		0			0
$105$	4.00000	−	8.00000i	0.390360	−	0.780720i
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$	−8.00000			−0.759326
$112$		0			0
$113$		−	12.0000i		−	1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$		−	12.0000i		−	1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$		0			0
$115$	−8.00000	−	4.00000i	−0.746004	−	0.373002i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−16.0000			−1.46672
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$			10.0000i			0.901670i
$124$		0			0
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$			4.00000i			0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$			4.00000i			0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$		0			0
$129$	−4.00000			−0.352180
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.00000	+	1.00000i	0.172133	+	0.0860663i
$136$		0			0
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$	−4.00000			−0.336861
$142$		0			0
$143$		0			0
$144$		0			0
$145$	6.00000	−	12.0000i	0.498273	−	0.996546i
$146$		0			0
$147$			9.00000i			0.742307i
$148$		0			0
$149$	2.00000			0.163846			0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000			0.163846			0.0819232	+	0.996639i	$0.473894\pi$
$150$		0			0
$151$	−20.0000			−1.62758			−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000			−1.62758			−0.813788	+	0.581161i	$0.802599\pi$
$152$		0			0
$153$		−	4.00000i		−	0.323381i
$154$		0			0
$155$	4.00000	−	8.00000i	0.321288	−	0.642575i
$156$		0			0
$157$			8.00000i			0.638470i	0.947676	+	0.319235i	$0.103426\pi$
$157$			8.00000i			0.638470i	−0.947676	+	0.319235i	$0.896574\pi$
$158$		0			0
$159$	12.0000			0.951662
$160$		0			0
$161$	16.0000			1.26098
$162$		0			0
$163$		−	20.0000i		−	1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$		−	20.0000i		−	1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$		0			0
$165$	8.00000	+	4.00000i	0.622799	+	0.311400i
$166$		0			0
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$			4.00000i			0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$			4.00000i			0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$		0			0
$175$	16.0000	−	12.0000i	1.20949	−	0.907115i
$176$		0			0
$177$			4.00000i			0.300658i
$178$		0			0
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		−	2.00000i		−	0.147844i
$184$		0			0
$185$	−16.0000	−	8.00000i	−1.17634	−	0.588172i
$186$		0			0
$187$		−	16.0000i		−	1.17004i
$188$		0			0
$189$	−4.00000			−0.290957
$190$		0			0
$191$	24.0000			1.73658			0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000			1.73658			0.868290	+	0.496058i	$0.165220\pi$
$192$		0			0
$193$		−	16.0000i		−	1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$		−	16.0000i		−	1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	4.00000i		−	0.284988i	−0.989796	−	0.142494i	$-0.954488\pi$
$197$		−	4.00000i		−	0.284988i	0.989796	−	0.142494i	$-0.0455122\pi$
$198$		0			0
$199$	12.0000			0.850657			0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000			0.850657			0.425329	+	0.905039i	$0.360158\pi$
$200$		0			0
$201$	−4.00000			−0.282138
$202$		0			0
$203$			24.0000i			1.68447i
$204$		0			0
$205$	−10.0000	+	20.0000i	−0.698430	+	1.39686i
$206$		0			0
$207$			4.00000i			0.278019i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.00000	−	4.00000i	−0.545595	−	0.272798i
$216$		0			0
$217$			16.0000i			1.08615i
$218$		0			0
$219$	8.00000			0.540590
$220$		0			0
$221$		0			0
$222$		0			0
$223$			20.0000i			1.33930i	0.742677	+	0.669650i	$0.233556\pi$
$223$			20.0000i			1.33930i	−0.742677	+	0.669650i	$0.766444\pi$
$224$		0			0
$225$	3.00000	+	4.00000i	0.200000	+	0.266667i
$226$		0			0
$227$		−	20.0000i		−	1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$		−	20.0000i		−	1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$		0			0
$229$	−26.0000			−1.71813			−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000			−1.71813			−0.859064	+	0.511868i	$0.828954\pi$
$230$		0			0
$231$	−16.0000			−1.05272
$232$		0			0
$233$		−	20.0000i		−	1.31024i	−0.755523	−	0.655122i	$-0.772617\pi$
$233$		−	20.0000i		−	1.31024i	0.755523	−	0.655122i	$-0.227383\pi$
$234$		0			0
$235$	−8.00000	−	4.00000i	−0.521862	−	0.260931i
$236$		0			0
$237$		−	12.0000i		−	0.779484i
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−9.00000	+	18.0000i	−0.574989	+	1.14998i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−4.00000			−0.253490
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$			16.0000i			1.00591i
$254$		0			0
$255$	4.00000	−	8.00000i	0.250490	−	0.500979i
$256$		0			0
$257$			12.0000i			0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$			12.0000i			0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$		0			0
$259$	32.0000			1.98838
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$		−	4.00000i		−	0.246651i	−0.992366	−	0.123325i	$-0.960644\pi$
$263$		−	4.00000i		−	0.246651i	0.992366	−	0.123325i	$-0.0393559\pi$
$264$		0			0
$265$	24.0000	+	12.0000i	1.47431	+	0.737154i
$266$		0			0
$267$		−	10.0000i		−	0.611990i
$268$		0			0
$269$	−6.00000			−0.365826			−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000			−0.365826			−0.182913	+	0.983129i	$0.558553\pi$
$270$		0			0
$271$	4.00000			0.242983			0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000			0.242983			0.121491	+	0.992592i	$0.461232\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.0000	+	16.0000i	0.723627	+	0.964836i
$276$		0			0
$277$		−	32.0000i		−	1.92269i	−0.275340	−	0.961347i	$-0.588791\pi$
$277$		−	32.0000i		−	1.92269i	0.275340	−	0.961347i	$-0.411209\pi$
$278$		0			0
$279$	−4.00000			−0.239474
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$			28.0000i			1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$			28.0000i			1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	40.0000i		−	2.36113i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$	8.00000			0.468968
$292$		0			0
$293$		−	12.0000i		−	0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$		−	12.0000i		−	0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$		0			0
$295$	−4.00000	+	8.00000i	−0.232889	+	0.465778i
$296$		0			0
$297$		−	4.00000i		−	0.232104i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	16.0000			0.922225
$302$		0			0
$303$			2.00000i			0.114897i
$304$		0			0
$305$	2.00000	−	4.00000i	0.114520	−	0.229039i
$306$		0			0
$307$			12.0000i			0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$			12.0000i			0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$		0			0
$309$	4.00000			0.227552
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$		−	16.0000i		−	0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$		−	16.0000i		−	0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$		0			0
$315$	−8.00000	−	4.00000i	−0.450749	−	0.225374i
$316$		0			0
$317$			4.00000i			0.224662i	0.993671	+	0.112331i	$0.0358318\pi$
$317$			4.00000i			0.224662i	−0.993671	+	0.112331i	$0.964168\pi$
$318$		0			0
$319$	−24.0000			−1.34374
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	2.00000i		−	0.110600i
$328$		0			0
$329$	16.0000			0.882109
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		0			0
$333$			8.00000i			0.438397i
$334$		0			0
$335$	−8.00000	−	4.00000i	−0.437087	−	0.218543i
$336$		0			0
$337$		−	8.00000i		−	0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$		−	8.00000i		−	0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$		0			0
$339$	−12.0000			−0.651751
$340$		0			0
$341$	−16.0000			−0.866449
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$	−4.00000	+	8.00000i	−0.215353	+	0.430706i
$346$		0			0
$347$			12.0000i			0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$			12.0000i			0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			28.0000i			1.49029i	0.666903	+	0.745145i	$0.267620\pi$
$353$			28.0000i			1.49029i	−0.666903	+	0.745145i	$0.732380\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			16.0000i			0.846810i
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		−	5.00000i		−	0.262432i
$364$		0			0
$365$	16.0000	+	8.00000i	0.837478	+	0.418739i
$366$		0			0
$367$			4.00000i			0.208798i	0.994535	+	0.104399i	$0.0332919\pi$
$367$			4.00000i			0.208798i	−0.994535	+	0.104399i	$0.966708\pi$
$368$		0			0
$369$	10.0000			0.520579
$370$		0			0
$371$	−48.0000			−2.49204
$372$		0			0
$373$		−	24.0000i		−	1.24267i	−0.783544	−	0.621336i	$-0.786590\pi$
$373$		−	24.0000i		−	1.24267i	0.783544	−	0.621336i	$-0.213410\pi$
$374$		0			0
$375$	2.00000	+	11.0000i	0.103280	+	0.568038i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−8.00000			−0.410932			−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000			−0.410932			−0.205466	+	0.978664i	$0.565871\pi$
$380$		0			0
$381$	4.00000			0.204926
$382$		0			0
$383$			28.0000i			1.43073i	0.698749	+	0.715367i	$0.253740\pi$
$383$			28.0000i			1.43073i	−0.698749	+	0.715367i	$0.746260\pi$
$384$		0			0
$385$	−32.0000	−	16.0000i	−1.63087	−	0.815436i
$386$		0			0
$387$			4.00000i			0.203331i
$388$		0			0
$389$	34.0000			1.72387			0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000			1.72387			0.861934	+	0.507020i	$0.169253\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$		−	12.0000i		−	0.605320i
$394$		0			0
$395$	12.0000	−	24.0000i	0.603786	−	1.20757i
$396$		0			0
$397$		−	8.00000i		−	0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$		−	8.00000i		−	0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−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
$403$		0			0
$404$		0			0
$405$	1.00000	−	2.00000i	0.0496904	−	0.0993808i
$406$		0			0
$407$			32.0000i			1.58618i
$408$		0			0
$409$	−26.0000			−1.28562			−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000			−1.28562			−0.642809	+	0.766027i	$0.722231\pi$
$410$		0			0
$411$	12.0000			0.591916
$412$		0			0
$413$		−	16.0000i		−	0.787309i
$414$		0			0
$415$	−8.00000	−	4.00000i	−0.392705	−	0.196352i
$416$		0			0
$417$			16.0000i			0.783523i
$418$		0			0
$419$	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
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$			4.00000i			0.194487i
$424$		0			0
$425$	16.0000	−	12.0000i	0.776114	−	0.582086i
$426$		0			0
$427$			8.00000i			0.387147i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	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
$433$		−	16.0000i		−	0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$		−	16.0000i		−	0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$		0			0
$435$	−12.0000	−	6.00000i	−0.575356	−	0.287678i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	12.0000			0.572729			0.286364	−	0.958121i	$-0.407553\pi$
$439$	12.0000			0.572729			0.286364	+	0.958121i	$0.407553\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$		0			0
$443$		−	36.0000i		−	1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$		−	36.0000i		−	1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$		0			0
$445$	10.0000	−	20.0000i	0.474045	−	0.948091i
$446$		0			0
$447$		−	2.00000i		−	0.0945968i
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	40.0000			1.88353
$452$		0			0
$453$			20.0000i			0.939682i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			40.0000i			1.87112i	0.353166	+	0.935561i	$0.385105\pi$
$457$			40.0000i			1.87112i	−0.353166	+	0.935561i	$0.614895\pi$
$458$		0			0
$459$	−4.00000			−0.186704
$460$		0			0
$461$	6.00000			0.279448			0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000			0.279448			0.139724	+	0.990190i	$0.455378\pi$
$462$		0			0
$463$		−	12.0000i		−	0.557687i	−0.960337	−	0.278844i	$-0.910049\pi$
$463$		−	12.0000i		−	0.557687i	0.960337	−	0.278844i	$-0.0899511\pi$
$464$		0			0
$465$	−8.00000	−	4.00000i	−0.370991	−	0.185496i
$466$		0			0
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$		0			0
$471$	8.00000			0.368621
$472$		0			0
$473$			16.0000i			0.735681i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	12.0000i		−	0.549442i
$478$		0			0
$479$	16.0000			0.731059			0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000			0.731059			0.365529	+	0.930800i	$0.380888\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		−	16.0000i		−	0.728025i
$484$		0			0
$485$	16.0000	+	8.00000i	0.726523	+	0.363261i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		0			0
$489$	−20.0000			−0.904431
$490$		0			0
$491$	−36.0000			−1.62466			−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000			−1.62466			−0.812329	+	0.583200i	$0.801800\pi$
$492$		0			0
$493$			24.0000i			1.08091i
$494$		0			0
$495$	4.00000	−	8.00000i	0.179787	−	0.359573i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−24.0000			−1.07439			−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000			−1.07439			−0.537194	+	0.843459i	$0.680516\pi$
$500$		0			0
$501$	12.0000			0.536120
$502$		0			0
$503$		−	36.0000i		−	1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$		−	36.0000i		−	1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$		0			0
$505$	−2.00000	+	4.00000i	−0.0889988	+	0.177998i
$506$		0			0
$507$		−	13.0000i		−	0.577350i
$508$		0			0
$509$	−42.0000			−1.86162			−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000			−1.86162			−0.930809	+	0.365507i	$0.880896\pi$
$510$		0			0
$511$	−32.0000			−1.41560
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.00000	+	4.00000i	0.352522	+	0.176261i
$516$		0			0
$517$			16.0000i			0.703679i
$518$		0			0
$519$	4.00000			0.175581
$520$		0			0
$521$	−38.0000			−1.66481			−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000			−1.66481			−0.832405	+	0.554168i	$0.813037\pi$
$522$		0			0
$523$			28.0000i			1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$			28.0000i			1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$		0			0
$525$	−12.0000	−	16.0000i	−0.523723	−	0.698297i
$526$		0			0
$527$			16.0000i			0.696971i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$		0			0
$534$		0			0
$535$	24.0000	+	12.0000i	1.03761	+	0.518805i
$536$		0			0
$537$			4.00000i			0.172613i
$538$		0			0
$539$	36.0000			1.55063
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$			10.0000i			0.429141i
$544$		0			0
$545$	2.00000	−	4.00000i	0.0856706	−	0.171341i
$546$		0			0
$547$			28.0000i			1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$			28.0000i			1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$		0			0
$549$	−2.00000			−0.0853579
$550$		0			0
$551$		0			0
$552$		0			0
$553$			48.0000i			2.04117i
$554$		0			0
$555$	−8.00000	+	16.0000i	−0.339581	+	0.679162i
$556$		0			0
$557$		−	12.0000i		−	0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$		−	12.0000i		−	0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−16.0000			−0.675521
$562$		0			0
$563$		−	20.0000i		−	0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$		−	20.0000i		−	0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$		0			0
$565$	−24.0000	−	12.0000i	−1.00969	−	0.504844i
$566$		0			0
$567$			4.00000i			0.167984i
$568$		0			0
$569$	−26.0000			−1.08998			−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000			−1.08998			−0.544988	+	0.838444i	$0.683466\pi$
$570$		0			0
$571$	40.0000			1.67395			0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000			1.67395			0.836974	+	0.547243i	$0.184323\pi$
$572$		0			0
$573$		−	24.0000i		−	1.00261i
$574$		0			0
$575$	−16.0000	+	12.0000i	−0.667246	+	0.500435i
$576$		0			0
$577$		−	32.0000i		−	1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$		−	32.0000i		−	1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$		0			0
$579$	−16.0000			−0.664937
$580$		0			0
$581$	16.0000			0.663792
$582$		0			0
$583$		−	48.0000i		−	1.98796i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	36.0000i		−	1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$		−	36.0000i		−	1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−4.00000			−0.164538
$592$		0			0
$593$			36.0000i			1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$			36.0000i			1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$		0			0
$595$	−16.0000	+	32.0000i	−0.655936	+	1.31187i
$596$		0			0
$597$		−	12.0000i		−	0.491127i
$598$		0			0
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$		0			0
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$		0			0
$603$			4.00000i			0.162893i
$604$		0			0
$605$	5.00000	−	10.0000i	0.203279	−	0.406558i
$606$		0			0
$607$		−	28.0000i		−	1.13648i	−0.822861	−	0.568242i	$-0.807624\pi$
$607$		−	28.0000i		−	1.13648i	0.822861	−	0.568242i	$-0.192376\pi$
$608$		0			0
$609$	24.0000			0.972529
$610$		0			0
$611$		0			0
$612$		0			0
$613$			8.00000i			0.323117i	0.986863	+	0.161558i	$0.0516520\pi$
$613$			8.00000i			0.323117i	−0.986863	+	0.161558i	$0.948348\pi$
$614$		0			0
$615$	20.0000	+	10.0000i	0.806478	+	0.403239i
$616$		0			0
$617$		−	44.0000i		−	1.77137i	−0.464283	−	0.885687i	$-0.653688\pi$
$617$		−	44.0000i		−	1.77137i	0.464283	−	0.885687i	$-0.346312\pi$
$618$		0			0
$619$	16.0000			0.643094			0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000			0.643094			0.321547	+	0.946894i	$0.395797\pi$
$620$		0			0
$621$	4.00000			0.160514
$622$		0			0
$623$			40.0000i			1.60257i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	32.0000			1.27592
$630$		0			0
$631$	−28.0000			−1.11466			−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000			−1.11466			−0.557331	+	0.830290i	$0.688175\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.00000	+	4.00000i	0.317470	+	0.158735i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	14.0000			0.552967			0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000			0.552967			0.276483	+	0.961019i	$0.410831\pi$
$642$		0			0
$643$		−	36.0000i		−	1.41970i	−0.704352	−	0.709851i	$-0.748762\pi$
$643$		−	36.0000i		−	1.41970i	0.704352	−	0.709851i	$-0.251238\pi$
$644$		0			0
$645$	−4.00000	+	8.00000i	−0.157500	+	0.315000i
$646$		0			0
$647$			12.0000i			0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$			12.0000i			0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$		0			0
$649$	16.0000			0.628055
$650$		0			0
$651$	16.0000			0.627089
$652$		0			0
$653$		−	4.00000i		−	0.156532i	−0.996933	−	0.0782660i	$-0.975062\pi$
$653$		−	4.00000i		−	0.156532i	0.996933	−	0.0782660i	$-0.0249384\pi$
$654$		0			0
$655$	12.0000	−	24.0000i	0.468879	−	0.937758i
$656$		0			0
$657$		−	8.00000i		−	0.312110i
$658$		0			0
$659$	−28.0000			−1.09073			−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000			−1.09073			−0.545363	+	0.838200i	$0.683608\pi$
$660$		0			0
$661$	−22.0000			−0.855701			−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000			−0.855701			−0.427850	+	0.903850i	$0.640729\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	24.0000i		−	0.929284i
$668$		0			0
$669$	20.0000			0.773245
$670$		0			0
$671$	−8.00000			−0.308837
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	4.00000	−	3.00000i	0.153960	−	0.115470i
$676$		0			0
$677$			4.00000i			0.153732i	0.997041	+	0.0768662i	$0.0244914\pi$
$677$			4.00000i			0.153732i	−0.997041	+	0.0768662i	$0.975509\pi$
$678$		0			0
$679$	−32.0000			−1.22805
$680$		0			0
$681$	−20.0000			−0.766402
$682$		0			0
$683$			44.0000i			1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$			44.0000i			1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$		0			0
$685$	24.0000	+	12.0000i	0.916993	+	0.458496i
$686$		0			0
$687$			26.0000i			0.991962i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−32.0000			−1.21734			−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000			−1.21734			−0.608669	+	0.793424i	$0.708296\pi$
$692$		0			0
$693$			16.0000i			0.607790i
$694$		0			0
$695$	−16.0000	+	32.0000i	−0.606915	+	1.21383i
$696$		0			0
$697$		−	40.0000i		−	1.51511i
$698$		0			0
$699$	−20.0000			−0.756469
$700$		0			0
$701$	26.0000			0.982006			0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000			0.982006			0.491003	+	0.871158i	$0.336630\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−4.00000	+	8.00000i	−0.150649	+	0.301297i
$706$		0			0
$707$		−	8.00000i		−	0.300871i
$708$		0			0
$709$	6.00000			0.225335			0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000			0.225335			0.112667	+	0.993633i	$0.464061\pi$
$710$		0			0
$711$	−12.0000			−0.450035
$712$		0			0
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			8.00000i			0.298765i
$718$		0			0
$719$	−8.00000			−0.298350			−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000			−0.298350			−0.149175	+	0.988811i	$0.547662\pi$
$720$		0			0
$721$	−16.0000			−0.595871
$722$		0			0
$723$			2.00000i			0.0743808i
$724$		0			0
$725$	−18.0000	−	24.0000i	−0.668503	−	0.891338i
$726$		0			0
$727$		−	12.0000i		−	0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$727$		−	12.0000i		−	0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	16.0000			0.591781
$732$		0			0
$733$			16.0000i			0.590973i	0.955347	+	0.295487i	$0.0954818\pi$
$733$			16.0000i			0.590973i	−0.955347	+	0.295487i	$0.904518\pi$
$734$		0			0
$735$	18.0000	+	9.00000i	0.663940	+	0.331970i
$736$		0			0
$737$			16.0000i			0.589368i
$738$		0			0
$739$	−40.0000			−1.47142			−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000			−1.47142			−0.735712	+	0.677295i	$0.763152\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	36.0000i		−	1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$		−	36.0000i		−	1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$		0			0
$745$	2.00000	−	4.00000i	0.0732743	−	0.146549i
$746$		0			0
$747$			4.00000i			0.146352i
$748$		0			0
$749$	−48.0000			−1.75388
$750$		0			0
$751$	−28.0000			−1.02173			−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000			−1.02173			−0.510867	+	0.859660i	$0.670676\pi$
$752$		0			0
$753$			12.0000i			0.437304i
$754$		0			0
$755$	−20.0000	+	40.0000i	−0.727875	+	1.45575i
$756$		0			0
$757$			16.0000i			0.581530i	0.956795	+	0.290765i	$0.0939098\pi$
$757$			16.0000i			0.581530i	−0.956795	+	0.290765i	$0.906090\pi$
$758$		0			0
$759$	16.0000			0.580763
$760$		0			0
$761$	42.0000			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$762$		0			0
$763$			8.00000i			0.289619i
$764$		0			0
$765$	−8.00000	−	4.00000i	−0.289241	−	0.144620i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	18.0000			0.649097			0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000			0.649097			0.324548	+	0.945869i	$0.394788\pi$
$770$		0			0
$771$	12.0000			0.432169
$772$		0			0
$773$			12.0000i			0.431610i	0.976436	+	0.215805i	$0.0692376\pi$
$773$			12.0000i			0.431610i	−0.976436	+	0.215805i	$0.930762\pi$
$774$		0			0
$775$	−12.0000	−	16.0000i	−0.431053	−	0.574737i
$776$		0			0
$777$		−	32.0000i		−	1.14799i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$			6.00000i			0.214423i
$784$		0			0
$785$	16.0000	+	8.00000i	0.571064	+	0.285532i
$786$		0			0
$787$			28.0000i			0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$			28.0000i			0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$		0			0
$789$	−4.00000			−0.142404
$790$		0			0
$791$	48.0000			1.70668
$792$		0			0
$793$		0			0
$794$		0			0
$795$	12.0000	−	24.0000i	0.425596	−	0.851192i
$796$		0			0
$797$			28.0000i			0.991811i	0.868377	+	0.495905i	$0.165164\pi$
$797$			28.0000i			0.991811i	−0.868377	+	0.495905i	$0.834836\pi$
$798$		0			0
$799$	16.0000			0.566039
$800$		0			0
$801$	−10.0000			−0.353333
$802$		0			0
$803$		−	32.0000i		−	1.12926i
$804$		0			0
$805$	16.0000	−	32.0000i	0.563926	−	1.12785i
$806$		0			0
$807$			6.00000i			0.211210i
$808$		0			0
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	40.0000			1.40459			0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000			1.40459			0.702295	+	0.711886i	$0.252159\pi$
$812$		0			0
$813$		−	4.00000i		−	0.140286i
$814$		0			0
$815$	−40.0000	−	20.0000i	−1.40114	−	0.700569i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−14.0000			−0.488603			−0.244302	−	0.969699i	$-0.578559\pi$
$821$	−14.0000			−0.488603			−0.244302	+	0.969699i	$0.578559\pi$
$822$		0			0
$823$		−	12.0000i		−	0.418294i	−0.977884	−	0.209147i	$-0.932931\pi$
$823$		−	12.0000i		−	0.418294i	0.977884	−	0.209147i	$-0.0670687\pi$
$824$		0			0
$825$	16.0000	−	12.0000i	0.557048	−	0.417786i
$826$		0			0
$827$			28.0000i			0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$			28.0000i			0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$		0			0
$829$	18.0000			0.625166			0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000			0.625166			0.312583	+	0.949890i	$0.398806\pi$
$830$		0			0
$831$	−32.0000			−1.11007
$832$		0			0
$833$		−	36.0000i		−	1.24733i
$834$		0			0
$835$	24.0000	+	12.0000i	0.830554	+	0.415277i
$836$		0			0
$837$			4.00000i			0.138260i
$838$		0			0
$839$	56.0000			1.93333			0.966667	−	0.256036i	$-0.0824164\pi$
$839$	56.0000			1.93333			0.966667	+	0.256036i	$0.0824164\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		−	6.00000i		−	0.206651i
$844$		0			0
$845$	13.0000	−	26.0000i	0.447214	−	0.894427i
$846$		0			0
$847$			20.0000i			0.687208i
$848$		0			0
$849$	28.0000			0.960958
$850$		0			0
$851$	−32.0000			−1.09695
$852$		0			0
$853$		−	8.00000i		−	0.273915i	−0.990577	−	0.136957i	$-0.956268\pi$
$853$		−	8.00000i		−	0.273915i	0.990577	−	0.136957i	$-0.0437323\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	12.0000i		−	0.409912i	−0.978771	−	0.204956i	$-0.934295\pi$
$857$		−	12.0000i		−	0.409912i	0.978771	−	0.204956i	$-0.0657052\pi$
$858$		0			0
$859$	40.0000			1.36478			0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000			1.36478			0.682391	+	0.730987i	$0.260940\pi$
$860$		0			0
$861$	−40.0000			−1.36320
$862$		0			0
$863$			12.0000i			0.408485i	0.978920	+	0.204242i	$0.0654731\pi$
$863$			12.0000i			0.408485i	−0.978920	+	0.204242i	$0.934527\pi$
$864$		0			0
$865$	8.00000	+	4.00000i	0.272008	+	0.136004i
$866$		0			0
$867$		−	1.00000i		−	0.0339618i
$868$		0			0
$869$	−48.0000			−1.62829
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	8.00000i		−	0.270759i
$874$		0			0
$875$	−8.00000	−	44.0000i	−0.270449	−	1.48747i
$876$		0			0
$877$			56.0000i			1.89099i	0.325643	+	0.945493i	$0.394419\pi$
$877$			56.0000i			1.89099i	−0.325643	+	0.945493i	$0.605581\pi$
$878$		0			0
$879$	−12.0000			−0.404750
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$			12.0000i			0.403832i	0.979403	+	0.201916i	$0.0647168\pi$
$883$			12.0000i			0.403832i	−0.979403	+	0.201916i	$0.935283\pi$
$884$		0			0
$885$	8.00000	+	4.00000i	0.268917	+	0.134459i
$886$		0			0
$887$		−	36.0000i		−	1.20876i	−0.796696	−	0.604381i	$-0.793421\pi$
$887$		−	36.0000i		−	1.20876i	0.796696	−	0.604381i	$-0.206579\pi$
$888$		0			0
$889$	−16.0000			−0.536623
$890$		0			0
$891$	−4.00000			−0.134005
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−4.00000	+	8.00000i	−0.133705	+	0.267411i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	24.0000			0.800445
$900$		0			0
$901$	−48.0000			−1.59911
$902$		0			0
$903$		−	16.0000i		−	0.532447i
$904$		0			0
$905$	−10.0000	+	20.0000i	−0.332411	+	0.664822i
$906$		0			0
$907$			12.0000i			0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$			12.0000i			0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$		0			0
$909$	2.00000			0.0663358
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$			16.0000i			0.529523i
$914$		0			0
$915$	−4.00000	−	2.00000i	−0.132236	−	0.0661180i
$916$		0			0
$917$			48.0000i			1.58510i
$918$		0			0
$919$	12.0000			0.395843			0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000			0.395843			0.197922	+	0.980218i	$0.436581\pi$
$920$		0			0
$921$	12.0000			0.395413
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−32.0000	+	24.0000i	−1.05215	+	0.789115i
$926$		0			0
$927$		−	4.00000i		−	0.131377i
$928$		0			0
$929$	−18.0000			−0.590561			−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000			−0.590561			−0.295280	+	0.955411i	$0.595413\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		−	24.0000i		−	0.785725i
$934$		0			0
$935$	−32.0000	−	16.0000i	−1.04651	−	0.523256i
$936$		0			0
$937$			32.0000i			1.04539i	0.852518	+	0.522697i	$0.175074\pi$
$937$			32.0000i			1.04539i	−0.852518	+	0.522697i	$0.824926\pi$
$938$		0			0
$939$	−16.0000			−0.522140
$940$		0			0
$941$	−10.0000			−0.325991			−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000			−0.325991			−0.162995	+	0.986627i	$0.552116\pi$
$942$		0			0
$943$			40.0000i			1.30258i
$944$		0			0
$945$	−4.00000	+	8.00000i	−0.130120	+	0.260240i
$946$		0			0
$947$			12.0000i			0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$			12.0000i			0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	4.00000			0.129709
$952$		0			0
$953$			4.00000i			0.129573i	0.997899	+	0.0647864i	$0.0206366\pi$
$953$			4.00000i			0.129573i	−0.997899	+	0.0647864i	$0.979363\pi$
$954$		0			0
$955$	24.0000	−	48.0000i	0.776622	−	1.55324i
$956$		0			0
$957$			24.0000i			0.775810i
$958$		0			0
$959$	−48.0000			−1.55000
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		−	12.0000i		−	0.386695i
$964$		0			0
$965$	−32.0000	−	16.0000i	−1.03012	−	0.515058i
$966$		0			0
$967$			20.0000i			0.643157i	0.946883	+	0.321578i	$0.104213\pi$
$967$			20.0000i			0.643157i	−0.946883	+	0.321578i	$0.895787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$		−	64.0000i		−	2.05175i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	12.0000i		−	0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$		−	12.0000i		−	0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$		0			0
$979$	−40.0000			−1.27841
$980$		0			0
$981$	−2.00000			−0.0638551
$982$		0			0
$983$			28.0000i			0.893061i	0.894768	+	0.446531i	$0.147341\pi$
$983$			28.0000i			0.893061i	−0.894768	+	0.446531i	$0.852659\pi$
$984$		0			0
$985$	−8.00000	−	4.00000i	−0.254901	−	0.127451i
$986$		0			0
$987$		−	16.0000i		−	0.509286i
$988$		0			0
$989$	−16.0000			−0.508770
$990$		0			0
$991$	28.0000			0.889449			0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000			0.889449			0.444725	+	0.895667i	$0.353302\pi$
$992$		0			0
$993$			8.00000i			0.253872i
$994$		0			0
$995$	12.0000	−	24.0000i	0.380426	−	0.760851i
$996$		0			0
$997$		−	24.0000i		−	0.760088i	−0.924968	−	0.380044i	$-0.875909\pi$
$997$		−	24.0000i		−	0.760088i	0.924968	−	0.380044i	$-0.124091\pi$
$998$		0			0
$999$	8.00000			0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.2.d.a.49.1	✓	2
3.2	odd	2		180.2.d.a.109.2		2
4.3	odd	2		240.2.f.b.49.2		2
5.2	odd	4		300.2.a.a.1.1		1
5.3	odd	4		300.2.a.d.1.1		1
5.4	even	2	inner	60.2.d.a.49.2	yes	2
7.2	even	3		2940.2.bb.d.949.2		4
7.3	odd	6		2940.2.bb.e.1549.2		4
7.4	even	3		2940.2.bb.d.1549.1		4
7.5	odd	6		2940.2.bb.e.949.1		4
7.6	odd	2		2940.2.k.c.589.2		2
8.3	odd	2		960.2.f.c.769.1		2
8.5	even	2		960.2.f.f.769.2		2
9.2	odd	6		1620.2.r.d.109.1		4
9.4	even	3		1620.2.r.c.1189.1		4
9.5	odd	6		1620.2.r.d.1189.2		4
9.7	even	3		1620.2.r.c.109.2		4
12.11	even	2		720.2.f.c.289.2		2
15.2	even	4		900.2.a.a.1.1		1
15.8	even	4		900.2.a.h.1.1		1
15.14	odd	2		180.2.d.a.109.1		2
16.3	odd	4		3840.2.d.b.2689.1		2
16.5	even	4		3840.2.d.o.2689.2		2
16.11	odd	4		3840.2.d.be.2689.2		2
16.13	even	4		3840.2.d.r.2689.1		2
20.3	even	4		1200.2.a.a.1.1		1
20.7	even	4		1200.2.a.s.1.1		1
20.19	odd	2		240.2.f.b.49.1		2
24.5	odd	2		2880.2.f.l.1729.1		2
24.11	even	2		2880.2.f.p.1729.1		2
35.4	even	6		2940.2.bb.d.1549.2		4
35.9	even	6		2940.2.bb.d.949.1		4
35.19	odd	6		2940.2.bb.e.949.2		4
35.24	odd	6		2940.2.bb.e.1549.1		4
35.34	odd	2		2940.2.k.c.589.1		2
40.3	even	4		4800.2.a.bk.1.1		1
40.13	odd	4		4800.2.a.bj.1.1		1
40.19	odd	2		960.2.f.c.769.2		2
40.27	even	4		4800.2.a.bf.1.1		1
40.29	even	2		960.2.f.f.769.1		2
40.37	odd	4		4800.2.a.bn.1.1		1
45.4	even	6		1620.2.r.c.1189.2		4
45.14	odd	6		1620.2.r.d.1189.1		4
45.29	odd	6		1620.2.r.d.109.2		4
45.34	even	6		1620.2.r.c.109.1		4
60.23	odd	4		3600.2.a.d.1.1		1
60.47	odd	4		3600.2.a.bm.1.1		1
60.59	even	2		720.2.f.c.289.1		2
80.19	odd	4		3840.2.d.be.2689.1		2
80.29	even	4		3840.2.d.o.2689.1		2
80.59	odd	4		3840.2.d.b.2689.2		2
80.69	even	4		3840.2.d.r.2689.2		2
120.29	odd	2		2880.2.f.l.1729.2		2
120.59	even	2		2880.2.f.p.1729.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.2.d.a.49.1	✓	2	1.1	even	1	trivial
60.2.d.a.49.2	yes	2	5.4	even	2	inner
180.2.d.a.109.1		2	15.14	odd	2
180.2.d.a.109.2		2	3.2	odd	2
240.2.f.b.49.1		2	20.19	odd	2
240.2.f.b.49.2		2	4.3	odd	2
300.2.a.a.1.1		1	5.2	odd	4
300.2.a.d.1.1		1	5.3	odd	4
720.2.f.c.289.1		2	60.59	even	2
720.2.f.c.289.2		2	12.11	even	2
900.2.a.a.1.1		1	15.2	even	4
900.2.a.h.1.1		1	15.8	even	4
960.2.f.c.769.1		2	8.3	odd	2
960.2.f.c.769.2		2	40.19	odd	2
960.2.f.f.769.1		2	40.29	even	2
960.2.f.f.769.2		2	8.5	even	2
1200.2.a.a.1.1		1	20.3	even	4
1200.2.a.s.1.1		1	20.7	even	4
1620.2.r.c.109.1		4	45.34	even	6
1620.2.r.c.109.2		4	9.7	even	3
1620.2.r.c.1189.1		4	9.4	even	3
1620.2.r.c.1189.2		4	45.4	even	6
1620.2.r.d.109.1		4	9.2	odd	6
1620.2.r.d.109.2		4	45.29	odd	6
1620.2.r.d.1189.1		4	45.14	odd	6
1620.2.r.d.1189.2		4	9.5	odd	6
2880.2.f.l.1729.1		2	24.5	odd	2
2880.2.f.l.1729.2		2	120.29	odd	2
2880.2.f.p.1729.1		2	24.11	even	2
2880.2.f.p.1729.2		2	120.59	even	2
2940.2.k.c.589.1		2	35.34	odd	2
2940.2.k.c.589.2		2	7.6	odd	2
2940.2.bb.d.949.1		4	35.9	even	6
2940.2.bb.d.949.2		4	7.2	even	3
2940.2.bb.d.1549.1		4	7.4	even	3
2940.2.bb.d.1549.2		4	35.4	even	6
2940.2.bb.e.949.1		4	7.5	odd	6
2940.2.bb.e.949.2		4	35.19	odd	6
2940.2.bb.e.1549.1		4	35.24	odd	6
2940.2.bb.e.1549.2		4	7.3	odd	6
3600.2.a.d.1.1		1	60.23	odd	4
3600.2.a.bm.1.1		1	60.47	odd	4
3840.2.d.b.2689.1		2	16.3	odd	4
3840.2.d.b.2689.2		2	80.59	odd	4
3840.2.d.o.2689.1		2	80.29	even	4
3840.2.d.o.2689.2		2	16.5	even	4
3840.2.d.r.2689.1		2	16.13	even	4
3840.2.d.r.2689.2		2	80.69	even	4
3840.2.d.be.2689.1		2	80.19	odd	4
3840.2.d.be.2689.2		2	16.11	odd	4
4800.2.a.bf.1.1		1	40.27	even	4
4800.2.a.bj.1.1		1	40.13	odd	4
4800.2.a.bk.1.1		1	40.3	even	4
4800.2.a.bn.1.1		1	40.37	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:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	60.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +(-2.00000 - 1.00000i) q^{15} +4.00000i q^{17} +4.00000 q^{21} -4.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +1.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} +4.00000i q^{33} +(8.00000 + 4.00000i) q^{35} -8.00000i q^{37} -10.0000 q^{41} -4.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} -4.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} +12.0000i q^{53} +(-4.00000 + 8.00000i) q^{55} -4.00000 q^{59} +2.00000 q^{61} -4.00000i q^{63} -4.00000i q^{67} -4.00000 q^{69} +8.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} -16.0000i q^{77} +12.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +(8.00000 + 4.00000i) q^{85} -6.00000i q^{87} +10.0000 q^{89} -4.00000i q^{93} +8.00000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^9	\( 2 q + 2 q^{5} - 2 q^{9} - 8 q^{11} - 4 q^{15} + 8 q^{21} - 6 q^{25} + 12 q^{29} + 8 q^{31} + 16 q^{35} - 20 q^{41} - 2 q^{45} - 18 q^{49} + 8 q^{51} - 8 q^{55} - 8 q^{59} + 4 q^{61} - 8 q^{69} - 8 q^{75} + 24 q^{79} + 2 q^{81} + 16 q^{85} + 20 q^{89} + 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^9 - 8 * q^11 - 4 * q^15 + 8 * q^21 - 6 * q^25 + 12 * q^29 + 8 * q^31 + 16 * q^35 - 20 * q^41 - 2 * q^45 - 18 * q^49 + 8 * q^51 - 8 * q^55 - 8 * q^59 + 4 * q^61 - 8 * q^69 - 8 * q^75 + 24 * q^79 + 2 * q^81 + 16 * q^85 + 20 * q^89 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more