LMFDB - Embedded newform 3840.2.d.be.2689.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(2689,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("3840.2689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.d (of order $2$, degree $1$, not 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:	$30.6625543762$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2689.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3840.2689
Dual form			3840.2.d.be.2689.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +(2.00000 - 1.00000i) q^{5} -4.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +(2.00000 - 1.00000i) q^{5} -4.00000i q^{7} +1.00000 q^{9} -4.00000i q^{11} +(2.00000 - 1.00000i) q^{15} -4.00000i q^{17} -4.00000i q^{21} +4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +1.00000 q^{27} +6.00000i q^{29} -4.00000 q^{31} -4.00000i q^{33} +(-4.00000 - 8.00000i) q^{35} +8.00000 q^{37} +10.0000 q^{41} -4.00000 q^{43} +(2.00000 - 1.00000i) q^{45} -4.00000i q^{47} -9.00000 q^{49} -4.00000i q^{51} -12.0000 q^{53} +(-4.00000 - 8.00000i) q^{55} -4.00000i q^{59} +2.00000i q^{61} -4.00000i q^{63} +4.00000 q^{67} +4.00000i q^{69} +8.00000i q^{73} +(3.00000 - 4.00000i) q^{75} -16.0000 q^{77} -12.0000 q^{79} +1.00000 q^{81} +4.00000 q^{83} +(-4.00000 - 8.00000i) q^{85} +6.00000i q^{87} -10.0000 q^{89} -4.00000 q^{93} -8.00000i q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 4 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9} + 4 q^{15} + 6 q^{25} + 2 q^{27} - 8 q^{31} - 8 q^{35} + 16 q^{37} + 20 q^{41} - 8 q^{43} + 4 q^{45} - 18 q^{49} - 24 q^{53} - 8 q^{55} + 8 q^{67} + 6 q^{75} - 32 q^{77} - 24 q^{79} + 2 q^{81} + 8 q^{83} - 8 q^{85} - 20 q^{89} - 8 q^{93}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^5 + 2 * q^9 + 4 * q^15 + 6 * q^25 + 2 * q^27 - 8 * q^31 - 8 * q^35 + 16 * q^37 + 20 * q^41 - 8 * q^43 + 4 * q^45 - 18 * q^49 - 24 * q^53 - 8 * q^55 + 8 * q^67 + 6 * q^75 - 32 * q^77 - 24 * q^79 + 2 * q^81 + 8 * q^83 - 8 * q^85 - 20 * q^89 - 8 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$.

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	4.00000i		−	1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$		−	4.00000i		−	1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	2.00000	−	1.00000i	0.516398	−	0.258199i
$16$		0			0
$17$		−	4.00000i		−	0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$		−	4.00000i		−	0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		−	4.00000i		−	0.872872i
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		−	4.00000i		−	0.696311i
$34$		0			0
$35$	−4.00000	−	8.00000i	−0.676123	−	1.35225i
$36$		0			0
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.0000			1.56174			0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000			1.56174			0.780869	+	0.624695i	$0.214777\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$	2.00000	−	1.00000i	0.298142	−	0.149071i
$46$		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
$51$		−	4.00000i		−	0.560112i
$52$		0			0
$53$	−12.0000			−1.64833			−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000			−1.64833			−0.824163	+	0.566352i	$0.808354\pi$
$54$		0			0
$55$	−4.00000	−	8.00000i	−0.539360	−	1.07872i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		0			0
$61$			2.00000i			0.256074i	0.991769	+	0.128037i	$0.0408676\pi$
$61$			2.00000i			0.256074i	−0.991769	+	0.128037i	$0.959132\pi$
$62$		0			0
$63$		−	4.00000i		−	0.503953i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.00000			0.488678			0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000			0.488678			0.244339	+	0.969690i	$0.421429\pi$
$68$		0			0
$69$			4.00000i			0.481543i
$70$		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
$75$	3.00000	−	4.00000i	0.346410	−	0.461880i
$76$		0			0
$77$	−16.0000			−1.82337
$78$		0			0
$79$	−12.0000			−1.35011			−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000			−1.35011			−0.675053	+	0.737769i	$0.735879\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$		0			0
$85$	−4.00000	−	8.00000i	−0.433861	−	0.867722i
$86$		0			0
$87$			6.00000i			0.643268i
$88$		0			0
$89$	−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
$92$		0			0
$93$	−4.00000			−0.414781
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	8.00000i		−	0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$		−	8.00000i		−	0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$		0			0
$99$		−	4.00000i		−	0.402015i
$100$		0			0
$101$			2.00000i			0.199007i	0.995037	+	0.0995037i	$0.0317255\pi$
$101$			2.00000i			0.199007i	−0.995037	+	0.0995037i	$0.968274\pi$
$102$		0			0
$103$		−	4.00000i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$		−	4.00000i		−	0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$		0			0
$105$	−4.00000	−	8.00000i	−0.390360	−	0.780720i
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$			2.00000i			0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$			2.00000i			0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$		0			0
$111$	8.00000			0.759326
$112$		0			0
$113$			12.0000i			1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$			12.0000i			1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$		0			0
$115$	4.00000	+	8.00000i	0.373002	+	0.746004i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−16.0000			−1.46672
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$	10.0000			0.901670
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		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
$131$		−	12.0000i		−	1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$		−	12.0000i		−	1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$		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
$139$		−	16.0000i		−	1.35710i	−0.734553	−	0.678551i	$-0.762608\pi$
$139$		−	16.0000i		−	1.35710i	0.734553	−	0.678551i	$-0.237392\pi$
$140$		0			0
$141$		−	4.00000i		−	0.336861i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	6.00000	+	12.0000i	0.498273	+	0.996546i
$146$		0			0
$147$	−9.00000			−0.742307
$148$		0			0
$149$		−	2.00000i		−	0.163846i	−0.996639	−	0.0819232i	$-0.973894\pi$
$149$		−	2.00000i		−	0.163846i	0.996639	−	0.0819232i	$-0.0261062\pi$
$150$		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
$155$	−8.00000	+	4.00000i	−0.642575	+	0.321288i
$156$		0			0
$157$	8.00000			0.638470			0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000			0.638470			0.319235	+	0.947676i	$0.396574\pi$
$158$		0			0
$159$	−12.0000			−0.951662
$160$		0			0
$161$	16.0000			1.26098
$162$		0			0
$163$	20.0000			1.56652			0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000			1.56652			0.783260	+	0.621694i	$0.213555\pi$
$164$		0			0
$165$	−4.00000	−	8.00000i	−0.311400	−	0.622799i
$166$		0			0
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.00000			0.304114			0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000			0.304114			0.152057	+	0.988372i	$0.451410\pi$
$174$		0			0
$175$	−16.0000	−	12.0000i	−1.20949	−	0.907115i
$176$		0			0
$177$		−	4.00000i		−	0.300658i
$178$		0			0
$179$			4.00000i			0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$			4.00000i			0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$		0			0
$181$			10.0000i			0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$			10.0000i			0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$		0			0
$183$			2.00000i			0.147844i
$184$		0			0
$185$	16.0000	−	8.00000i	1.17634	−	0.588172i
$186$		0			0
$187$	−16.0000			−1.17004
$188$		0			0
$189$		−	4.00000i		−	0.290957i
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.00000			0.284988			0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000			0.284988			0.142494	+	0.989796i	$0.454488\pi$
$198$		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
$203$	24.0000			1.68447
$204$		0			0
$205$	20.0000	−	10.0000i	1.39686	−	0.698430i
$206$		0			0
$207$			4.00000i			0.278019i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		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
$219$			8.00000i			0.540590i
$220$		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
$227$	20.0000			1.32745			0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000			1.32745			0.663723	+	0.747978i	$0.268975\pi$
$228$		0			0
$229$			26.0000i			1.71813i	0.511868	+	0.859064i	$0.328954\pi$
$229$			26.0000i			1.71813i	−0.511868	+	0.859064i	$0.671046\pi$
$230$		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
$235$	−4.00000	−	8.00000i	−0.260931	−	0.521862i
$236$		0			0
$237$	−12.0000			−0.779484
$238$		0			0
$239$	8.00000			0.517477			0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000			0.517477			0.258738	+	0.965947i	$0.416693\pi$
$240$		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
$243$	1.00000			0.0641500
$244$		0			0
$245$	−18.0000	+	9.00000i	−1.14998	+	0.574989i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	4.00000			0.253490
$250$		0			0
$251$		−	12.0000i		−	0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$		−	12.0000i		−	0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$		0			0
$253$	16.0000			1.00591
$254$		0			0
$255$	−4.00000	−	8.00000i	−0.250490	−	0.500979i
$256$		0			0
$257$		−	12.0000i		−	0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$		−	12.0000i		−	0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$		0			0
$259$		−	32.0000i		−	1.98838i
$260$		0			0
$261$			6.00000i			0.371391i
$262$		0			0
$263$			4.00000i			0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$			4.00000i			0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$		0			0
$265$	−24.0000	+	12.0000i	−1.47431	+	0.737154i
$266$		0			0
$267$	−10.0000			−0.611990
$268$		0			0
$269$		−	6.00000i		−	0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$269$		−	6.00000i		−	0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$270$		0			0
$271$	−4.00000			−0.242983			−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000			−0.242983			−0.121491	+	0.992592i	$0.538768\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−16.0000	−	12.0000i	−0.964836	−	0.723627i
$276$		0			0
$277$	32.0000			1.92269			0.961347	−	0.275340i	$-0.0887905\pi$
$277$	32.0000			1.92269			0.961347	+	0.275340i	$0.0887905\pi$
$278$		0			0
$279$	−4.00000			−0.239474
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	28.0000			1.66443			0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000			1.66443			0.832214	+	0.554455i	$0.187073\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	40.0000i		−	2.36113i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$		−	8.00000i		−	0.468968i
$292$		0			0
$293$	12.0000			0.701047			0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000			0.701047			0.350524	+	0.936554i	$0.386004\pi$
$294$		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
$301$			16.0000i			0.922225i
$302$		0			0
$303$			2.00000i			0.114897i
$304$		0			0
$305$	2.00000	+	4.00000i	0.114520	+	0.229039i
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$		0			0
$309$		−	4.00000i		−	0.227552i
$310$		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
$315$	−4.00000	−	8.00000i	−0.225374	−	0.450749i
$316$		0			0
$317$	4.00000			0.224662			0.112331	−	0.993671i	$-0.464168\pi$
$317$	4.00000			0.224662			0.112331	+	0.993671i	$0.464168\pi$
$318$		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
$331$		−	8.00000i		−	0.439720i	−0.975531	−	0.219860i	$-0.929440\pi$
$331$		−	8.00000i		−	0.439720i	0.975531	−	0.219860i	$-0.0705600\pi$
$332$		0			0
$333$	8.00000			0.438397
$334$		0			0
$335$	8.00000	−	4.00000i	0.437087	−	0.218543i
$336$		0			0
$337$			8.00000i			0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$			8.00000i			0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$		0			0
$339$			12.0000i			0.651751i
$340$		0			0
$341$			16.0000i			0.866449i
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$	4.00000	+	8.00000i	0.215353	+	0.430706i
$346$		0			0
$347$	12.0000			0.644194			0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000			0.644194			0.322097	+	0.946707i	$0.395612\pi$
$348$		0			0
$349$			14.0000i			0.749403i	0.927146	+	0.374701i	$0.122255\pi$
$349$			14.0000i			0.749403i	−0.927146	+	0.374701i	$0.877745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	28.0000i		−	1.49029i	−0.666903	−	0.745145i	$-0.732380\pi$
$353$		−	28.0000i		−	1.49029i	0.666903	−	0.745145i	$-0.267620\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−16.0000			−0.846810
$358$		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
$363$	−5.00000			−0.262432
$364$		0			0
$365$	8.00000	+	16.0000i	0.418739	+	0.837478i
$366$		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
$371$			48.0000i			2.49204i
$372$		0			0
$373$	24.0000			1.24267			0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000			1.24267			0.621336	+	0.783544i	$0.286590\pi$
$374$		0			0
$375$	2.00000	−	11.0000i	0.103280	−	0.568038i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	8.00000i		−	0.410932i	−0.978664	−	0.205466i	$-0.934129\pi$
$379$		−	8.00000i		−	0.410932i	0.978664	−	0.205466i	$-0.0658711\pi$
$380$		0			0
$381$			4.00000i			0.204926i
$382$		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
$387$	−4.00000			−0.203331
$388$		0			0
$389$		−	34.0000i		−	1.72387i	−0.507020	−	0.861934i	$-0.669253\pi$
$389$		−	34.0000i		−	1.72387i	0.507020	−	0.861934i	$-0.330747\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$		−	12.0000i		−	0.605320i
$394$		0			0
$395$	−24.0000	+	12.0000i	−1.20757	+	0.603786i
$396$		0			0
$397$	−8.00000			−0.401508			−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000			−0.401508			−0.200754	+	0.979642i	$0.564339\pi$
$398$		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
$405$	2.00000	−	1.00000i	0.0993808	−	0.0496904i
$406$		0			0
$407$		−	32.0000i		−	1.58618i
$408$		0			0
$409$	26.0000			1.28562			0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000			1.28562			0.642809	+	0.766027i	$0.277769\pi$
$410$		0			0
$411$			12.0000i			0.591916i
$412$		0			0
$413$	−16.0000			−0.787309
$414$		0			0
$415$	8.00000	−	4.00000i	0.392705	−	0.196352i
$416$		0			0
$417$		−	16.0000i		−	0.783523i
$418$		0			0
$419$		−	28.0000i		−	1.36789i	−0.729534	−	0.683945i	$-0.760263\pi$
$419$		−	28.0000i		−	1.36789i	0.729534	−	0.683945i	$-0.239737\pi$
$420$		0			0
$421$		−	10.0000i		−	0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$		−	10.0000i		−	0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$		0			0
$423$		−	4.00000i		−	0.194487i
$424$		0			0
$425$	−16.0000	−	12.0000i	−0.776114	−	0.582086i
$426$		0			0
$427$	8.00000			0.387147
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.00000			−0.385346			−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000			−0.385346			−0.192673	+	0.981263i	$0.561716\pi$
$432$		0			0
$433$			16.0000i			0.768911i	0.923144	+	0.384455i	$0.125611\pi$
$433$			16.0000i			0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$		0			0
$435$	6.00000	+	12.0000i	0.287678	+	0.575356i
$436$		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
$443$	−36.0000			−1.71041			−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000			−1.71041			−0.855206	+	0.518289i	$0.826569\pi$
$444$		0			0
$445$	−20.0000	+	10.0000i	−0.948091	+	0.474045i
$446$		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
$451$		−	40.0000i		−	1.88353i
$452$		0			0
$453$	−20.0000			−0.939682
$454$		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
$459$		−	4.00000i		−	0.186704i
$460$		0			0
$461$			6.00000i			0.279448i	0.990190	+	0.139724i	$0.0446215\pi$
$461$			6.00000i			0.279448i	−0.990190	+	0.139724i	$0.955378\pi$
$462$		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
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$		−	16.0000i		−	0.738811i
$470$		0			0
$471$	8.00000			0.368621
$472$		0			0
$473$			16.0000i			0.735681i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−12.0000			−0.549442
$478$		0			0
$479$	−16.0000			−0.731059			−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000			−0.731059			−0.365529	+	0.930800i	$0.619112\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	16.0000			0.728025
$484$		0			0
$485$	−8.00000	−	16.0000i	−0.363261	−	0.726523i
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		0			0
$489$	20.0000			0.904431
$490$		0			0
$491$		−	36.0000i		−	1.62466i	−0.583200	−	0.812329i	$-0.698200\pi$
$491$		−	36.0000i		−	1.62466i	0.583200	−	0.812329i	$-0.301800\pi$
$492$		0			0
$493$	24.0000			1.08091
$494$		0			0
$495$	−4.00000	−	8.00000i	−0.179787	−	0.359573i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			24.0000i			1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$			24.0000i			1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$		0			0
$501$		−	12.0000i		−	0.536120i
$502$		0			0
$503$			36.0000i			1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$			36.0000i			1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$		0			0
$505$	2.00000	+	4.00000i	0.0889988	+	0.177998i
$506$		0			0
$507$	−13.0000			−0.577350
$508$		0			0
$509$		−	42.0000i		−	1.86162i	−0.365507	−	0.930809i	$-0.619104\pi$
$509$		−	42.0000i		−	1.86162i	0.365507	−	0.930809i	$-0.380896\pi$
$510$		0			0
$511$	32.0000			1.41560
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.00000	−	8.00000i	−0.176261	−	0.352522i
$516$		0			0
$517$	−16.0000			−0.703679
$518$		0			0
$519$	4.00000			0.175581
$520$		0			0
$521$	38.0000			1.66481			0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000			1.66481			0.832405	+	0.554168i	$0.186963\pi$
$522$		0			0
$523$	28.0000			1.22435			0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000			1.22435			0.612177	+	0.790721i	$0.290294\pi$
$524$		0			0
$525$	−16.0000	−	12.0000i	−0.698297	−	0.523723i
$526$		0			0
$527$			16.0000i			0.696971i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$		−	4.00000i		−	0.173585i
$532$		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
$539$			36.0000i			1.55063i
$540$		0			0
$541$		−	2.00000i		−	0.0859867i	−0.999075	−	0.0429934i	$-0.986311\pi$
$541$		−	2.00000i		−	0.0859867i	0.999075	−	0.0429934i	$-0.0136894\pi$
$542$		0			0
$543$			10.0000i			0.429141i
$544$		0			0
$545$	2.00000	+	4.00000i	0.0856706	+	0.171341i
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$		0			0
$549$			2.00000i			0.0853579i
$550$		0			0
$551$		0			0
$552$		0			0
$553$			48.0000i			2.04117i
$554$		0			0
$555$	16.0000	−	8.00000i	0.679162	−	0.339581i
$556$		0			0
$557$	−12.0000			−0.508456			−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000			−0.508456			−0.254228	+	0.967144i	$0.581821\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−16.0000			−0.675521
$562$		0			0
$563$	20.0000			0.842900			0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000			0.842900			0.421450	+	0.906852i	$0.361521\pi$
$564$		0			0
$565$	12.0000	+	24.0000i	0.504844	+	1.00969i
$566$		0			0
$567$		−	4.00000i		−	0.167984i
$568$		0			0
$569$	26.0000			1.08998			0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000			1.08998			0.544988	+	0.838444i	$0.316534\pi$
$570$		0			0
$571$			40.0000i			1.67395i	0.547243	+	0.836974i	$0.315677\pi$
$571$			40.0000i			1.67395i	−0.547243	+	0.836974i	$0.684323\pi$
$572$		0			0
$573$	−24.0000			−1.00261
$574$		0			0
$575$	16.0000	+	12.0000i	0.667246	+	0.500435i
$576$		0			0
$577$			32.0000i			1.33218i	0.745873	+	0.666089i	$0.232033\pi$
$577$			32.0000i			1.33218i	−0.745873	+	0.666089i	$0.767967\pi$
$578$		0			0
$579$			16.0000i			0.664937i
$580$		0			0
$581$		−	16.0000i		−	0.663792i
$582$		0			0
$583$			48.0000i			1.98796i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−36.0000			−1.48588			−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000			−1.48588			−0.742940	+	0.669359i	$0.766569\pi$
$588$		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	$-0.735217\pi$
$593$		−	36.0000i		−	1.47834i	0.673517	−	0.739171i	$-0.264783\pi$
$594$		0			0
$595$	−32.0000	+	16.0000i	−1.31187	+	0.655936i
$596$		0			0
$597$	12.0000			0.491127
$598$		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	$-0.782263\pi$
$601$	−38.0000			−1.55005			−0.775026	+	0.631929i	$0.782263\pi$
$602$		0			0
$603$	4.00000			0.162893
$604$		0			0
$605$	−10.0000	+	5.00000i	−0.406558	+	0.203279i
$606$		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
$613$	−8.00000			−0.323117			−0.161558	−	0.986863i	$-0.551652\pi$
$613$	−8.00000			−0.323117			−0.161558	+	0.986863i	$0.551652\pi$
$614$		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
$619$			16.0000i			0.643094i	0.946894	+	0.321547i	$0.104203\pi$
$619$			16.0000i			0.643094i	−0.946894	+	0.321547i	$0.895797\pi$
$620$		0			0
$621$			4.00000i			0.160514i
$622$		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
$629$		−	32.0000i		−	1.27592i
$630$		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
$635$	4.00000	+	8.00000i	0.158735	+	0.317470i
$636$		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
$643$	36.0000			1.41970			0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000			1.41970			0.709851	+	0.704352i	$0.248762\pi$
$644$		0			0
$645$	−8.00000	+	4.00000i	−0.315000	+	0.157500i
$646$		0			0
$647$		−	12.0000i		−	0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$		−	12.0000i		−	0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$		0			0
$649$	−16.0000			−0.628055
$650$		0			0
$651$			16.0000i			0.627089i
$652$		0			0
$653$	−4.00000			−0.156532			−0.0782660	−	0.996933i	$-0.524938\pi$
$653$	−4.00000			−0.156532			−0.0782660	+	0.996933i	$0.524938\pi$
$654$		0			0
$655$	−12.0000	−	24.0000i	−0.468879	−	0.937758i
$656$		0			0
$657$			8.00000i			0.312110i
$658$		0			0
$659$			28.0000i			1.09073i	0.838200	+	0.545363i	$0.183608\pi$
$659$			28.0000i			1.09073i	−0.838200	+	0.545363i	$0.816392\pi$
$660$		0			0
$661$			22.0000i			0.855701i	0.903850	+	0.427850i	$0.140729\pi$
$661$			22.0000i			0.855701i	−0.903850	+	0.427850i	$0.859271\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−24.0000			−0.929284
$668$		0			0
$669$			20.0000i			0.773245i
$670$		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
$675$	3.00000	−	4.00000i	0.115470	−	0.153960i
$676$		0			0
$677$	−4.00000			−0.153732			−0.0768662	−	0.997041i	$-0.524491\pi$
$677$	−4.00000			−0.153732			−0.0768662	+	0.997041i	$0.524491\pi$
$678$		0			0
$679$	−32.0000			−1.22805
$680$		0			0
$681$	20.0000			0.766402
$682$		0			0
$683$	44.0000			1.68361			0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000			1.68361			0.841807	+	0.539779i	$0.181492\pi$
$684$		0			0
$685$	12.0000	+	24.0000i	0.458496	+	0.916993i
$686$		0			0
$687$			26.0000i			0.991962i
$688$		0			0
$689$		0			0
$690$		0			0
$691$			32.0000i			1.21734i	0.793424	+	0.608669i	$0.208296\pi$
$691$			32.0000i			1.21734i	−0.793424	+	0.608669i	$0.791704\pi$
$692$		0			0
$693$	−16.0000			−0.607790
$694$		0			0
$695$	−16.0000	−	32.0000i	−0.606915	−	1.21383i
$696$		0			0
$697$		−	40.0000i		−	1.51511i
$698$		0			0
$699$		−	20.0000i		−	0.756469i
$700$		0			0
$701$			26.0000i			0.982006i	0.871158	+	0.491003i	$0.163370\pi$
$701$			26.0000i			0.982006i	−0.871158	+	0.491003i	$0.836630\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−4.00000	−	8.00000i	−0.150649	−	0.301297i
$706$		0			0
$707$	8.00000			0.300871
$708$		0			0
$709$		−	6.00000i		−	0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$		0			0
$711$	−12.0000			−0.450035
$712$		0			0
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	8.00000			0.298765
$718$		0			0
$719$	8.00000			0.298350			0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		0			0
$721$	−16.0000			−0.595871
$722$		0			0
$723$	−2.00000			−0.0743808
$724$		0			0
$725$	24.0000	+	18.0000i	0.891338	+	0.668503i
$726$		0			0
$727$			12.0000i			0.445055i	0.974926	+	0.222528i	$0.0714308\pi$
$727$			12.0000i			0.445055i	−0.974926	+	0.222528i	$0.928569\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			16.0000i			0.591781i
$732$		0			0
$733$	16.0000			0.590973			0.295487	−	0.955347i	$-0.404518\pi$
$733$	16.0000			0.590973			0.295487	+	0.955347i	$0.404518\pi$
$734$		0			0
$735$	−18.0000	+	9.00000i	−0.663940	+	0.331970i
$736$		0			0
$737$		−	16.0000i		−	0.589368i
$738$		0			0
$739$			40.0000i			1.47142i	0.677295	+	0.735712i	$0.263152\pi$
$739$			40.0000i			1.47142i	−0.677295	+	0.735712i	$0.736848\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			36.0000i			1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$			36.0000i			1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$		0			0
$745$	−2.00000	−	4.00000i	−0.0732743	−	0.146549i
$746$		0			0
$747$	4.00000			0.146352
$748$		0			0
$749$		−	48.0000i		−	1.75388i
$750$		0			0
$751$	28.0000			1.02173			0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000			1.02173			0.510867	+	0.859660i	$0.329324\pi$
$752$		0			0
$753$		−	12.0000i		−	0.437304i
$754$		0			0
$755$	−40.0000	+	20.0000i	−1.45575	+	0.727875i
$756$		0			0
$757$	−16.0000			−0.581530			−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000			−0.581530			−0.290765	+	0.956795i	$0.593910\pi$
$758$		0			0
$759$	16.0000			0.580763
$760$		0			0
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$	8.00000			0.289619
$764$		0			0
$765$	−4.00000	−	8.00000i	−0.144620	−	0.289241i
$766$		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
$771$		−	12.0000i		−	0.432169i
$772$		0			0
$773$	−12.0000			−0.431610			−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000			−0.431610			−0.215805	+	0.976436i	$0.569238\pi$
$774$		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
$787$	−28.0000			−0.998092			−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000			−0.998092			−0.499046	+	0.866575i	$0.666316\pi$
$788$		0			0
$789$			4.00000i			0.142404i
$790$		0			0
$791$	48.0000			1.70668
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−24.0000	+	12.0000i	−0.851192	+	0.425596i
$796$		0			0
$797$	28.0000			0.991811			0.495905	−	0.868377i	$-0.334836\pi$
$797$	28.0000			0.991811			0.495905	+	0.868377i	$0.334836\pi$
$798$		0			0
$799$	−16.0000			−0.566039
$800$		0			0
$801$	−10.0000			−0.353333
$802$		0			0
$803$	32.0000			1.12926
$804$		0			0
$805$	32.0000	−	16.0000i	1.12785	−	0.563926i
$806$		0			0
$807$		−	6.00000i		−	0.211210i
$808$		0			0
$809$	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
$811$			40.0000i			1.40459i	0.711886	+	0.702295i	$0.247841\pi$
$811$			40.0000i			1.40459i	−0.711886	+	0.702295i	$0.752159\pi$
$812$		0			0
$813$	−4.00000			−0.140286
$814$		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
$821$			14.0000i			0.488603i	0.969699	+	0.244302i	$0.0785587\pi$
$821$			14.0000i			0.488603i	−0.969699	+	0.244302i	$0.921441\pi$
$822$		0			0
$823$			12.0000i			0.418294i	0.977884	+	0.209147i	$0.0670687\pi$
$823$			12.0000i			0.418294i	−0.977884	+	0.209147i	$0.932931\pi$
$824$		0			0
$825$	−16.0000	−	12.0000i	−0.557048	−	0.417786i
$826$		0			0
$827$	28.0000			0.973655			0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000			0.973655			0.486828	+	0.873498i	$0.338154\pi$
$828$		0			0
$829$			18.0000i			0.625166i	0.949890	+	0.312583i	$0.101194\pi$
$829$			18.0000i			0.625166i	−0.949890	+	0.312583i	$0.898806\pi$
$830$		0			0
$831$	32.0000			1.11007
$832$		0			0
$833$			36.0000i			1.24733i
$834$		0			0
$835$	−12.0000	−	24.0000i	−0.415277	−	0.830554i
$836$		0			0
$837$	−4.00000			−0.138260
$838$		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
$843$	−6.00000			−0.206651
$844$		0			0
$845$	−26.0000	+	13.0000i	−0.894427	+	0.447214i
$846$		0			0
$847$			20.0000i			0.687208i
$848$		0			0
$849$	28.0000			0.960958
$850$		0			0
$851$			32.0000i			1.09695i
$852$		0			0
$853$	8.00000			0.273915			0.136957	−	0.990577i	$-0.456268\pi$
$853$	8.00000			0.273915			0.136957	+	0.990577i	$0.456268\pi$
$854$		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
$859$			40.0000i			1.36478i	0.730987	+	0.682391i	$0.239060\pi$
$859$			40.0000i			1.36478i	−0.730987	+	0.682391i	$0.760940\pi$
$860$		0			0
$861$		−	40.0000i		−	1.36320i
$862$		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
$867$	1.00000			0.0339618
$868$		0			0
$869$			48.0000i			1.62829i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	8.00000i		−	0.270759i
$874$		0			0
$875$	−44.0000	−	8.00000i	−1.48747	−	0.270449i
$876$		0			0
$877$	56.0000			1.89099			0.945493	−	0.325643i	$-0.105581\pi$
$877$	56.0000			1.89099			0.945493	+	0.325643i	$0.105581\pi$
$878$		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
$883$	−12.0000			−0.403832			−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000			−0.403832			−0.201916	+	0.979403i	$0.564717\pi$
$884$		0			0
$885$	−4.00000	−	8.00000i	−0.134459	−	0.268917i
$886$		0			0
$887$			36.0000i			1.20876i	0.796696	+	0.604381i	$0.206579\pi$
$887$			36.0000i			1.20876i	−0.796696	+	0.604381i	$0.793421\pi$
$888$		0			0
$889$	16.0000			0.536623
$890$		0			0
$891$		−	4.00000i		−	0.134005i
$892$		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
$899$		−	24.0000i		−	0.800445i
$900$		0			0
$901$			48.0000i			1.59911i
$902$		0			0
$903$			16.0000i			0.532447i
$904$		0			0
$905$	10.0000	+	20.0000i	0.332411	+	0.664822i
$906$		0			0
$907$	12.0000			0.398453			0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000			0.398453			0.199227	+	0.979953i	$0.436157\pi$
$908$		0			0
$909$			2.00000i			0.0663358i
$910$		0			0
$911$	−48.0000			−1.59031			−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000			−1.59031			−0.795155	+	0.606406i	$0.792611\pi$
$912$		0			0
$913$		−	16.0000i		−	0.529523i
$914$		0			0
$915$	2.00000	+	4.00000i	0.0661180	+	0.132236i
$916$		0			0
$917$	−48.0000			−1.58510
$918$		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
$925$	24.0000	−	32.0000i	0.789115	−	1.05215i
$926$		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
$933$	24.0000			0.785725
$934$		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
$939$		−	16.0000i		−	0.522140i
$940$		0			0
$941$		−	10.0000i		−	0.325991i	−0.986627	−	0.162995i	$-0.947884\pi$
$941$		−	10.0000i		−	0.325991i	0.986627	−	0.162995i	$-0.0521156\pi$
$942$		0			0
$943$			40.0000i			1.30258i
$944$		0			0
$945$	−4.00000	−	8.00000i	−0.130120	−	0.260240i
$946$		0			0
$947$	−12.0000			−0.389948			−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000			−0.389948			−0.194974	+	0.980808i	$0.562462\pi$
$948$		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
$955$	−48.0000	+	24.0000i	−1.55324	+	0.776622i
$956$		0			0
$957$	24.0000			0.775810
$958$		0			0
$959$	48.0000			1.55000
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$	12.0000			0.386695
$964$		0			0
$965$	16.0000	+	32.0000i	0.515058	+	1.03012i
$966$		0			0
$967$		−	20.0000i		−	0.643157i	−0.946883	−	0.321578i	$-0.895787\pi$
$967$		−	20.0000i		−	0.643157i	0.946883	−	0.321578i	$-0.104213\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	12.0000i		−	0.385098i	−0.981287	−	0.192549i	$-0.938325\pi$
$971$		−	12.0000i		−	0.385098i	0.981287	−	0.192549i	$-0.0616755\pi$
$972$		0			0
$973$	−64.0000			−2.05175
$974$		0			0
$975$		0			0
$976$		0			0
$977$			12.0000i			0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$			12.0000i			0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$		0			0
$979$			40.0000i			1.27841i
$980$		0			0
$981$			2.00000i			0.0638551i
$982$		0			0
$983$		−	28.0000i		−	0.893061i	−0.894768	−	0.446531i	$-0.852659\pi$
$983$		−	28.0000i		−	0.893061i	0.894768	−	0.446531i	$-0.147341\pi$
$984$		0			0
$985$	8.00000	−	4.00000i	0.254901	−	0.127451i
$986$		0			0
$987$	−16.0000			−0.509286
$988$		0			0
$989$		−	16.0000i		−	0.508770i
$990$		0			0
$991$	−28.0000			−0.889449			−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000			−0.889449			−0.444725	+	0.895667i	$0.646698\pi$
$992$		0			0
$993$		−	8.00000i		−	0.253872i
$994$		0			0
$995$	24.0000	−	12.0000i	0.760851	−	0.380426i
$996$		0			0
$997$	24.0000			0.760088			0.380044	−	0.924968i	$-0.375909\pi$
$997$	24.0000			0.760088			0.380044	+	0.924968i	$0.375909\pi$
$998$		0			0
$999$	8.00000			0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.d.be.2689.1		2
4.3	odd	2		3840.2.d.o.2689.1		2
5.4	even	2		3840.2.d.b.2689.1		2
8.3	odd	2		3840.2.d.r.2689.2		2
8.5	even	2		3840.2.d.b.2689.2		2
16.3	odd	4		960.2.f.f.769.1		2
16.5	even	4		240.2.f.b.49.1		2
16.11	odd	4		60.2.d.a.49.2	yes	2
16.13	even	4		960.2.f.c.769.2		2
20.19	odd	2		3840.2.d.r.2689.1		2
40.19	odd	2		3840.2.d.o.2689.2		2
40.29	even	2	inner	3840.2.d.be.2689.2		2
48.5	odd	4		720.2.f.c.289.1		2
48.11	even	4		180.2.d.a.109.1		2
48.29	odd	4		2880.2.f.p.1729.2		2
48.35	even	4		2880.2.f.l.1729.2		2
80.3	even	4		4800.2.a.bn.1.1		1
80.13	odd	4		4800.2.a.bf.1.1		1
80.19	odd	4		960.2.f.f.769.2		2
80.27	even	4		300.2.a.d.1.1		1
80.29	even	4		960.2.f.c.769.1		2
80.37	odd	4		1200.2.a.a.1.1		1
80.43	even	4		300.2.a.a.1.1		1
80.53	odd	4		1200.2.a.s.1.1		1
80.59	odd	4		60.2.d.a.49.1	✓	2
80.67	even	4		4800.2.a.bj.1.1		1
80.69	even	4		240.2.f.b.49.2		2
80.77	odd	4		4800.2.a.bk.1.1		1
112.11	odd	12		2940.2.bb.d.1549.2		4
112.27	even	4		2940.2.k.c.589.1		2
112.59	even	12		2940.2.bb.e.1549.1		4
112.75	even	12		2940.2.bb.e.949.2		4
112.107	odd	12		2940.2.bb.d.949.1		4
144.11	even	12		1620.2.r.d.109.2		4
144.43	odd	12		1620.2.r.c.109.1		4
144.59	even	12		1620.2.r.d.1189.1		4
144.139	odd	12		1620.2.r.c.1189.2		4
240.29	odd	4		2880.2.f.p.1729.1		2
240.53	even	4		3600.2.a.bm.1.1		1
240.59	even	4		180.2.d.a.109.2		2
240.107	odd	4		900.2.a.h.1.1		1
240.149	odd	4		720.2.f.c.289.2		2
240.179	even	4		2880.2.f.l.1729.1		2
240.197	even	4		3600.2.a.d.1.1		1
240.203	odd	4		900.2.a.a.1.1		1
560.59	even	12		2940.2.bb.e.1549.2		4
560.139	even	4		2940.2.k.c.589.2		2
560.219	odd	12		2940.2.bb.d.949.2		4
560.299	even	12		2940.2.bb.e.949.1		4
560.459	odd	12		2940.2.bb.d.1549.1		4
720.59	even	12		1620.2.r.d.1189.2		4
720.139	odd	12		1620.2.r.c.1189.1		4
720.299	even	12		1620.2.r.d.109.1		4
720.619	odd	12		1620.2.r.c.109.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.2.d.a.49.1	✓	2	80.59	odd	4
60.2.d.a.49.2	yes	2	16.11	odd	4
180.2.d.a.109.1		2	48.11	even	4
180.2.d.a.109.2		2	240.59	even	4
240.2.f.b.49.1		2	16.5	even	4
240.2.f.b.49.2		2	80.69	even	4
300.2.a.a.1.1		1	80.43	even	4
300.2.a.d.1.1		1	80.27	even	4
720.2.f.c.289.1		2	48.5	odd	4
720.2.f.c.289.2		2	240.149	odd	4
900.2.a.a.1.1		1	240.203	odd	4
900.2.a.h.1.1		1	240.107	odd	4
960.2.f.c.769.1		2	80.29	even	4
960.2.f.c.769.2		2	16.13	even	4
960.2.f.f.769.1		2	16.3	odd	4
960.2.f.f.769.2		2	80.19	odd	4
1200.2.a.a.1.1		1	80.37	odd	4
1200.2.a.s.1.1		1	80.53	odd	4
1620.2.r.c.109.1		4	144.43	odd	12
1620.2.r.c.109.2		4	720.619	odd	12
1620.2.r.c.1189.1		4	720.139	odd	12
1620.2.r.c.1189.2		4	144.139	odd	12
1620.2.r.d.109.1		4	720.299	even	12
1620.2.r.d.109.2		4	144.11	even	12
1620.2.r.d.1189.1		4	144.59	even	12
1620.2.r.d.1189.2		4	720.59	even	12
2880.2.f.l.1729.1		2	240.179	even	4
2880.2.f.l.1729.2		2	48.35	even	4
2880.2.f.p.1729.1		2	240.29	odd	4
2880.2.f.p.1729.2		2	48.29	odd	4
2940.2.k.c.589.1		2	112.27	even	4
2940.2.k.c.589.2		2	560.139	even	4
2940.2.bb.d.949.1		4	112.107	odd	12
2940.2.bb.d.949.2		4	560.219	odd	12
2940.2.bb.d.1549.1		4	560.459	odd	12
2940.2.bb.d.1549.2		4	112.11	odd	12
2940.2.bb.e.949.1		4	560.299	even	12
2940.2.bb.e.949.2		4	112.75	even	12
2940.2.bb.e.1549.1		4	112.59	even	12
2940.2.bb.e.1549.2		4	560.59	even	12
3600.2.a.d.1.1		1	240.197	even	4
3600.2.a.bm.1.1		1	240.53	even	4
3840.2.d.b.2689.1		2	5.4	even	2
3840.2.d.b.2689.2		2	8.5	even	2
3840.2.d.o.2689.1		2	4.3	odd	2
3840.2.d.o.2689.2		2	40.19	odd	2
3840.2.d.r.2689.1		2	20.19	odd	2
3840.2.d.r.2689.2		2	8.3	odd	2
3840.2.d.be.2689.1		2	1.1	even	1	trivial
3840.2.d.be.2689.2		2	40.29	even	2	inner
4800.2.a.bf.1.1		1	80.13	odd	4
4800.2.a.bj.1.1		1	80.67	even	4
4800.2.a.bk.1.1		1	80.77	odd	4
4800.2.a.bn.1.1		1	80.3	even	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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +(2.00000 - 1.00000i) q^{5} -4.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +(2.00000 - 1.00000i) q^{5} -4.00000i q^{7} +1.00000 q^{9} -4.00000i q^{11} +(2.00000 - 1.00000i) q^{15} -4.00000i q^{17} -4.00000i q^{21} +4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +1.00000 q^{27} +6.00000i q^{29} -4.00000 q^{31} -4.00000i q^{33} +(-4.00000 - 8.00000i) q^{35} +8.00000 q^{37} +10.0000 q^{41} -4.00000 q^{43} +(2.00000 - 1.00000i) q^{45} -4.00000i q^{47} -9.00000 q^{49} -4.00000i q^{51} -12.0000 q^{53} +(-4.00000 - 8.00000i) q^{55} -4.00000i q^{59} +2.00000i q^{61} -4.00000i q^{63} +4.00000 q^{67} +4.00000i q^{69} +8.00000i q^{73} +(3.00000 - 4.00000i) q^{75} -16.0000 q^{77} -12.0000 q^{79} +1.00000 q^{81} +4.00000 q^{83} +(-4.00000 - 8.00000i) q^{85} +6.00000i q^{87} -10.0000 q^{89} -4.00000 q^{93} -8.00000i q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 4 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9} + 4 q^{15} + 6 q^{25} + 2 q^{27} - 8 q^{31} - 8 q^{35} + 16 q^{37} + 20 q^{41} - 8 q^{43} + 4 q^{45} - 18 q^{49} - 24 q^{53} - 8 q^{55} + 8 q^{67} + 6 q^{75} - 32 q^{77} - 24 q^{79} + 2 q^{81} + 8 q^{83} - 8 q^{85} - 20 q^{89} - 8 q^{93}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^5 + 2 * q^9 + 4 * q^15 + 6 * q^25 + 2 * q^27 - 8 * q^31 - 8 * q^35 + 16 * q^37 + 20 * q^41 - 8 * q^43 + 4 * q^45 - 18 * q^49 - 24 * q^53 - 8 * q^55 + 8 * q^67 + 6 * q^75 - 32 * q^77 - 24 * q^79 + 2 * q^81 + 8 * q^83 - 8 * q^85 - 20 * q^89 - 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more