LMFDB - Embedded newform 3840.2.d.bb.2689.2

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:	$ 2 $
Twist minimal:	no (minimal twist has level 1920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +(1.00000 + 2.00000i) q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +(1.00000 + 2.00000i) q^{5} +1.00000 q^{9} +4.00000i q^{11} +2.00000 q^{13} +(1.00000 + 2.00000i) q^{15} -2.00000i q^{17} -2.00000i q^{19} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +1.00000 q^{27} +8.00000i q^{29} +8.00000 q^{31} +4.00000i q^{33} +6.00000 q^{37} +2.00000 q^{39} -2.00000 q^{41} -4.00000 q^{43} +(1.00000 + 2.00000i) q^{45} +6.00000i q^{47} +7.00000 q^{49} -2.00000i q^{51} +10.0000 q^{53} +(-8.00000 + 4.00000i) q^{55} -2.00000i q^{57} +8.00000i q^{59} +2.00000i q^{61} +(2.00000 + 4.00000i) q^{65} +4.00000 q^{67} -6.00000i q^{69} -4.00000i q^{73} +(-3.00000 + 4.00000i) q^{75} -4.00000 q^{79} +1.00000 q^{81} -16.0000 q^{83} +(4.00000 - 2.00000i) q^{85} +8.00000i q^{87} -18.0000 q^{89} +8.00000 q^{93} +(4.00000 - 2.00000i) q^{95} +16.0000i q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 6 q^{25} + 2 q^{27} + 16 q^{31} + 12 q^{37} + 4 q^{39} - 4 q^{41} - 8 q^{43} + 2 q^{45} + 14 q^{49} + 20 q^{53} - 16 q^{55} + 4 q^{65} + 8 q^{67} - 6 q^{75} - 8 q^{79} + 2 q^{81} - 32 q^{83} + 8 q^{85} - 36 q^{89} + 16 q^{93} + 8 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 6 * q^25 + 2 * q^27 + 16 * q^31 + 12 * q^37 + 4 * q^39 - 4 * q^41 - 8 * q^43 + 2 * q^45 + 14 * q^49 + 20 * q^53 - 16 * q^55 + 4 * q^65 + 8 * q^67 - 6 * q^75 - 8 * q^79 + 2 * q^81 - 32 * q^83 + 8 * q^85 - 36 * q^89 + 16 * q^93 + 8 * q^95

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
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			4.00000i			1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$			4.00000i			1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$	1.00000	+	2.00000i	0.258199	+	0.516398i
$16$		0			0
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$			8.00000i			1.48556i	0.669534	+	0.742781i	$0.266494\pi$
$29$			8.00000i			1.48556i	−0.669534	+	0.742781i	$0.733506\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$			4.00000i			0.696311i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$		0			0
$39$	2.00000			0.320256
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		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
$45$	1.00000	+	2.00000i	0.149071	+	0.298142i
$46$		0			0
$47$			6.00000i			0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$			6.00000i			0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		−	2.00000i		−	0.280056i
$52$		0			0
$53$	10.0000			1.37361			0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000			1.37361			0.686803	+	0.726844i	$0.259014\pi$
$54$		0			0
$55$	−8.00000	+	4.00000i	−1.07872	+	0.539360i
$56$		0			0
$57$		−	2.00000i		−	0.264906i
$58$		0			0
$59$			8.00000i			1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$			8.00000i			1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$		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
$63$		0			0
$64$		0			0
$65$	2.00000	+	4.00000i	0.248069	+	0.496139i
$66$		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
$69$		−	6.00000i		−	0.722315i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$		0			0
$75$	−3.00000	+	4.00000i	−0.346410	+	0.461880i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$	4.00000	−	2.00000i	0.433861	−	0.216930i
$86$		0			0
$87$			8.00000i			0.857690i
$88$		0			0
$89$	−18.0000			−1.90800			−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000			−1.90800			−0.953998	+	0.299813i	$0.903076\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	8.00000			0.829561
$94$		0			0
$95$	4.00000	−	2.00000i	0.410391	−	0.205196i
$96$		0			0
$97$			16.0000i			1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$			16.0000i			1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$		0			0
$99$			4.00000i			0.402015i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		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
$105$		0			0
$106$		0			0
$107$	−8.00000			−0.773389			−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000			−0.773389			−0.386695	+	0.922208i	$0.626383\pi$
$108$		0			0
$109$			14.0000i			1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$			14.0000i			1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$		0			0
$111$	6.00000			0.569495
$112$		0			0
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$	12.0000	−	6.00000i	1.11901	−	0.559503i
$116$		0			0
$117$	2.00000			0.184900
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$	−2.00000			−0.180334
$124$		0			0
$125$	−11.0000	−	2.00000i	−0.983870	−	0.178885i
$126$		0			0
$127$		−	4.00000i		−	0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$		−	4.00000i		−	0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$		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
$135$	1.00000	+	2.00000i	0.0860663	+	0.172133i
$136$		0			0
$137$			22.0000i			1.87959i	0.341743	+	0.939793i	$0.388983\pi$
$137$			22.0000i			1.87959i	−0.341743	+	0.939793i	$0.611017\pi$
$138$		0			0
$139$		−	14.0000i		−	1.18746i	−0.804663	−	0.593732i	$-0.797654\pi$
$139$		−	14.0000i		−	1.18746i	0.804663	−	0.593732i	$-0.202346\pi$
$140$		0			0
$141$			6.00000i			0.505291i
$142$		0			0
$143$			8.00000i			0.668994i
$144$		0			0
$145$	−16.0000	+	8.00000i	−1.32873	+	0.664364i
$146$		0			0
$147$	7.00000			0.577350
$148$		0			0
$149$		−	4.00000i		−	0.327693i	−0.986486	−	0.163846i	$-0.947610\pi$
$149$		−	4.00000i		−	0.327693i	0.986486	−	0.163846i	$-0.0523901\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		−	2.00000i		−	0.161690i
$154$		0			0
$155$	8.00000	+	16.0000i	0.642575	+	1.28515i
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$		0			0
$159$	10.0000			0.793052
$160$		0			0
$161$		0			0
$162$		0			0
$163$	12.0000			0.939913			0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000			0.939913			0.469956	+	0.882690i	$0.344270\pi$
$164$		0			0
$165$	−8.00000	+	4.00000i	−0.622799	+	0.311400i
$166$		0			0
$167$		−	14.0000i		−	1.08335i	−0.840587	−	0.541676i	$-0.817790\pi$
$167$		−	14.0000i		−	1.08335i	0.840587	−	0.541676i	$-0.182210\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$		−	2.00000i		−	0.152944i
$172$		0			0
$173$	14.0000			1.06440			0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000			1.06440			0.532200	+	0.846619i	$0.321365\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			8.00000i			0.601317i
$178$		0			0
$179$			20.0000i			1.49487i	0.664335	+	0.747435i	$0.268715\pi$
$179$			20.0000i			1.49487i	−0.664335	+	0.747435i	$0.731285\pi$
$180$		0			0
$181$		−	2.00000i		−	0.148659i	−0.997234	−	0.0743294i	$-0.976318\pi$
$181$		−	2.00000i		−	0.148659i	0.997234	−	0.0743294i	$-0.0236816\pi$
$182$		0			0
$183$			2.00000i			0.147844i
$184$		0			0
$185$	6.00000	+	12.0000i	0.441129	+	0.882258i
$186$		0			0
$187$	8.00000			0.585018
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$			4.00000i			0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$			4.00000i			0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$		0			0
$195$	2.00000	+	4.00000i	0.143223	+	0.286446i
$196$		0			0
$197$	22.0000			1.56744			0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000			1.56744			0.783718	+	0.621117i	$0.213321\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$	4.00000			0.282138
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−2.00000	−	4.00000i	−0.139686	−	0.279372i
$206$		0			0
$207$		−	6.00000i		−	0.417029i
$208$		0			0
$209$	8.00000			0.553372
$210$		0			0
$211$		−	22.0000i		−	1.51454i	−0.653101	−	0.757271i	$-0.726532\pi$
$211$		−	22.0000i		−	1.51454i	0.653101	−	0.757271i	$-0.273468\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.00000	−	8.00000i	−0.272798	−	0.545595i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	4.00000i		−	0.270295i
$220$		0			0
$221$		−	4.00000i		−	0.269069i
$222$		0			0
$223$		−	8.00000i		−	0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$		−	8.00000i		−	0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$		0			0
$225$	−3.00000	+	4.00000i	−0.200000	+	0.266667i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		−	2.00000i		−	0.132164i	−0.997814	−	0.0660819i	$-0.978950\pi$
$229$		−	2.00000i		−	0.132164i	0.997814	−	0.0660819i	$-0.0210498\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			6.00000i			0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$			6.00000i			0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$		0			0
$235$	−12.0000	+	6.00000i	−0.782794	+	0.391397i
$236$		0			0
$237$	−4.00000			−0.259828
$238$		0			0
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$		0			0
$241$	−26.0000			−1.67481			−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000			−1.67481			−0.837404	+	0.546585i	$0.815928\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	7.00000	+	14.0000i	0.447214	+	0.894427i
$246$		0			0
$247$		−	4.00000i		−	0.254514i
$248$		0			0
$249$	−16.0000			−1.01396
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	24.0000			1.50887
$254$		0			0
$255$	4.00000	−	2.00000i	0.250490	−	0.125245i
$256$		0			0
$257$		−	6.00000i		−	0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$		−	6.00000i		−	0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$			8.00000i			0.495188i
$262$		0			0
$263$			2.00000i			0.123325i	0.998097	+	0.0616626i	$0.0196403\pi$
$263$			2.00000i			0.123325i	−0.998097	+	0.0616626i	$0.980360\pi$
$264$		0			0
$265$	10.0000	+	20.0000i	0.614295	+	1.22859i
$266$		0			0
$267$	−18.0000			−1.10158
$268$		0			0
$269$		−	16.0000i		−	0.975537i	−0.872973	−	0.487769i	$-0.837811\pi$
$269$		−	16.0000i		−	0.975537i	0.872973	−	0.487769i	$-0.162189\pi$
$270$		0			0
$271$	−20.0000			−1.21491			−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000			−1.21491			−0.607457	+	0.794353i	$0.707810\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−16.0000	−	12.0000i	−0.964836	−	0.723627i
$276$		0			0
$277$	−26.0000			−1.56219			−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000			−1.56219			−0.781094	+	0.624413i	$0.785338\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		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
$285$	4.00000	−	2.00000i	0.236940	−	0.118470i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$			16.0000i			0.937937i
$292$		0			0
$293$	22.0000			1.28525			0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000			1.28525			0.642627	+	0.766179i	$0.277845\pi$
$294$		0			0
$295$	−16.0000	+	8.00000i	−0.931556	+	0.465778i
$296$		0			0
$297$			4.00000i			0.232104i
$298$		0			0
$299$		−	12.0000i		−	0.693978i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.00000	+	2.00000i	−0.229039	+	0.114520i
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		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
$313$		−	12.0000i		−	0.678280i	−0.940736	−	0.339140i	$-0.889864\pi$
$313$		−	12.0000i		−	0.678280i	0.940736	−	0.339140i	$-0.110136\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−18.0000			−1.01098			−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000			−1.01098			−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$	−32.0000			−1.79166
$320$		0			0
$321$	−8.00000			−0.446516
$322$		0			0
$323$	−4.00000			−0.222566
$324$		0			0
$325$	−6.00000	+	8.00000i	−0.332820	+	0.443760i
$326$		0			0
$327$			14.0000i			0.774202i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	34.0000i		−	1.86881i	−0.356214	−	0.934405i	$-0.615932\pi$
$331$		−	34.0000i		−	1.86881i	0.356214	−	0.934405i	$-0.384068\pi$
$332$		0			0
$333$	6.00000			0.328798
$334$		0			0
$335$	4.00000	+	8.00000i	0.218543	+	0.437087i
$336$		0			0
$337$			12.0000i			0.653682i	0.945079	+	0.326841i	$0.105984\pi$
$337$			12.0000i			0.653682i	−0.945079	+	0.326841i	$0.894016\pi$
$338$		0			0
$339$			6.00000i			0.325875i
$340$		0			0
$341$			32.0000i			1.73290i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	12.0000	−	6.00000i	0.646058	−	0.323029i
$346$		0			0
$347$	28.0000			1.50312			0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000			1.50312			0.751559	+	0.659665i	$0.229302\pi$
$348$		0			0
$349$			30.0000i			1.60586i	0.596071	+	0.802932i	$0.296728\pi$
$349$			30.0000i			1.60586i	−0.596071	+	0.802932i	$0.703272\pi$
$350$		0			0
$351$	2.00000			0.106752
$352$		0			0
$353$		−	30.0000i		−	1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$		−	30.0000i		−	1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	32.0000			1.68890			0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000			1.68890			0.844448	+	0.535638i	$0.179929\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$	−5.00000			−0.262432
$364$		0			0
$365$	8.00000	−	4.00000i	0.418739	−	0.209370i
$366$		0			0
$367$		−	4.00000i		−	0.208798i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$		−	4.00000i		−	0.208798i	0.994535	−	0.104399i	$-0.0332919\pi$
$368$		0			0
$369$	−2.00000			−0.104116
$370$		0			0
$371$		0			0
$372$		0			0
$373$	38.0000			1.96757			0.983783	−	0.179364i	$-0.0574041\pi$
$373$	38.0000			1.96757			0.983783	+	0.179364i	$0.0574041\pi$
$374$		0			0
$375$	−11.0000	−	2.00000i	−0.568038	−	0.103280i
$376$		0			0
$377$			16.0000i			0.824042i
$378$		0			0
$379$		−	30.0000i		−	1.54100i	−0.637442	−	0.770498i	$-0.720007\pi$
$379$		−	30.0000i		−	1.54100i	0.637442	−	0.770498i	$-0.279993\pi$
$380$		0			0
$381$		−	4.00000i		−	0.204926i
$382$		0			0
$383$			30.0000i			1.53293i	0.642287	+	0.766464i	$0.277986\pi$
$383$			30.0000i			1.53293i	−0.642287	+	0.766464i	$0.722014\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.00000			−0.203331
$388$		0			0
$389$		−	12.0000i		−	0.608424i	−0.952604	−	0.304212i	$-0.901607\pi$
$389$		−	12.0000i		−	0.608424i	0.952604	−	0.304212i	$-0.0983931\pi$
$390$		0			0
$391$	−12.0000			−0.606866
$392$		0			0
$393$		−	12.0000i		−	0.605320i
$394$		0			0
$395$	−4.00000	−	8.00000i	−0.201262	−	0.402524i
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.00000			−0.0998752			−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000			−0.0998752			−0.0499376	+	0.998752i	$0.515902\pi$
$402$		0			0
$403$	16.0000			0.797017
$404$		0			0
$405$	1.00000	+	2.00000i	0.0496904	+	0.0993808i
$406$		0			0
$407$			24.0000i			1.18964i
$408$		0			0
$409$	18.0000			0.890043			0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000			0.890043			0.445021	+	0.895520i	$0.353196\pi$
$410$		0			0
$411$			22.0000i			1.08518i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−16.0000	−	32.0000i	−0.785409	−	1.57082i
$416$		0			0
$417$		−	14.0000i		−	0.685583i
$418$		0			0
$419$			24.0000i			1.17248i	0.810139	+	0.586238i	$0.199392\pi$
$419$			24.0000i			1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$		0			0
$421$			18.0000i			0.877266i	0.898666	+	0.438633i	$0.144537\pi$
$421$			18.0000i			0.877266i	−0.898666	+	0.438633i	$0.855463\pi$
$422$		0			0
$423$			6.00000i			0.291730i
$424$		0			0
$425$	8.00000	+	6.00000i	0.388057	+	0.291043i
$426$		0			0
$427$		0			0
$428$		0			0
$429$			8.00000i			0.386244i
$430$		0			0
$431$	32.0000			1.54139			0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000			1.54139			0.770693	+	0.637207i	$0.219910\pi$
$432$		0			0
$433$		−	28.0000i		−	1.34559i	−0.739827	−	0.672797i	$-0.765093\pi$
$433$		−	28.0000i		−	1.34559i	0.739827	−	0.672797i	$-0.234907\pi$
$434$		0			0
$435$	−16.0000	+	8.00000i	−0.767141	+	0.383571i
$436$		0			0
$437$	−12.0000			−0.574038
$438$		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
$441$	7.00000			0.333333
$442$		0			0
$443$	32.0000			1.52037			0.760183	−	0.649709i	$-0.225109\pi$
$443$	32.0000			1.52037			0.760183	+	0.649709i	$0.225109\pi$
$444$		0			0
$445$	−18.0000	−	36.0000i	−0.853282	−	1.70656i
$446$		0			0
$447$		−	4.00000i		−	0.189194i
$448$		0			0
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$		0			0
$451$		−	8.00000i		−	0.376705i
$452$		0			0
$453$	−8.00000			−0.375873
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	16.0000i		−	0.748448i	−0.927338	−	0.374224i	$-0.877909\pi$
$457$		−	16.0000i		−	0.748448i	0.927338	−	0.374224i	$-0.122091\pi$
$458$		0			0
$459$		−	2.00000i		−	0.0933520i
$460$		0			0
$461$		−	12.0000i		−	0.558896i	−0.960161	−	0.279448i	$-0.909849\pi$
$461$		−	12.0000i		−	0.558896i	0.960161	−	0.279448i	$-0.0901514\pi$
$462$		0			0
$463$		−	28.0000i		−	1.30127i	−0.759390	−	0.650635i	$-0.774503\pi$
$463$		−	28.0000i		−	1.30127i	0.759390	−	0.650635i	$-0.225497\pi$
$464$		0			0
$465$	8.00000	+	16.0000i	0.370991	+	0.741982i
$466$		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
$469$		0			0
$470$		0			0
$471$	14.0000			0.645086
$472$		0			0
$473$		−	16.0000i		−	0.735681i
$474$		0			0
$475$	8.00000	+	6.00000i	0.367065	+	0.275299i
$476$		0			0
$477$	10.0000			0.457869
$478$		0			0
$479$	−32.0000			−1.46212			−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000			−1.46212			−0.731059	+	0.682315i	$0.760973\pi$
$480$		0			0
$481$	12.0000			0.547153
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−32.0000	+	16.0000i	−1.45305	+	0.726523i
$486$		0			0
$487$		−	44.0000i		−	1.99383i	−0.0784867	−	0.996915i	$-0.525009\pi$
$487$		−	44.0000i		−	1.99383i	0.0784867	−	0.996915i	$-0.474991\pi$
$488$		0			0
$489$	12.0000			0.542659
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	16.0000			0.720604
$494$		0			0
$495$	−8.00000	+	4.00000i	−0.359573	+	0.179787i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			14.0000i			0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$			14.0000i			0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$		0			0
$501$		−	14.0000i		−	0.625474i
$502$		0			0
$503$			6.00000i			0.267527i	0.991013	+	0.133763i	$0.0427062\pi$
$503$			6.00000i			0.267527i	−0.991013	+	0.133763i	$0.957294\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−9.00000			−0.399704
$508$		0			0
$509$			24.0000i			1.06378i	0.846813	+	0.531891i	$0.178518\pi$
$509$			24.0000i			1.06378i	−0.846813	+	0.531891i	$0.821482\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	2.00000i		−	0.0883022i
$514$		0			0
$515$	8.00000	−	4.00000i	0.352522	−	0.176261i
$516$		0			0
$517$	−24.0000			−1.05552
$518$		0			0
$519$	14.0000			0.614532
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$	−44.0000			−1.92399			−0.961993	−	0.273075i	$-0.911959\pi$
$523$	−44.0000			−1.92399			−0.961993	+	0.273075i	$0.911959\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	16.0000i		−	0.696971i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$			8.00000i			0.347170i
$532$		0			0
$533$	−4.00000			−0.173259
$534$		0			0
$535$	−8.00000	−	16.0000i	−0.345870	−	0.691740i
$536$		0			0
$537$			20.0000i			0.863064i
$538$		0			0
$539$			28.0000i			1.20605i
$540$		0			0
$541$		−	30.0000i		−	1.28980i	−0.764267	−	0.644900i	$-0.776899\pi$
$541$		−	30.0000i		−	1.28980i	0.764267	−	0.644900i	$-0.223101\pi$
$542$		0			0
$543$		−	2.00000i		−	0.0858282i
$544$		0			0
$545$	−28.0000	+	14.0000i	−1.19939	+	0.599694i
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$		0			0
$549$			2.00000i			0.0853579i
$550$		0			0
$551$	16.0000			0.681623
$552$		0			0
$553$		0			0
$554$		0			0
$555$	6.00000	+	12.0000i	0.254686	+	0.509372i
$556$		0			0
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$	8.00000			0.337760
$562$		0			0
$563$	4.00000			0.168580			0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000			0.168580			0.0842900	+	0.996441i	$0.473138\pi$
$564$		0			0
$565$	−12.0000	+	6.00000i	−0.504844	+	0.252422i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10.0000			−0.419222			−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000			−0.419222			−0.209611	+	0.977785i	$0.567220\pi$
$570$		0			0
$571$			6.00000i			0.251092i	0.992088	+	0.125546i	$0.0400683\pi$
$571$			6.00000i			0.251092i	−0.992088	+	0.125546i	$0.959932\pi$
$572$		0			0
$573$	8.00000			0.334205
$574$		0			0
$575$	24.0000	+	18.0000i	1.00087	+	0.750652i
$576$		0			0
$577$			16.0000i			0.666089i	0.942911	+	0.333044i	$0.108076\pi$
$577$			16.0000i			0.666089i	−0.942911	+	0.333044i	$0.891924\pi$
$578$		0			0
$579$			4.00000i			0.166234i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			40.0000i			1.65663i
$584$		0			0
$585$	2.00000	+	4.00000i	0.0826898	+	0.165380i
$586$		0			0
$587$	−24.0000			−0.990586			−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000			−0.990586			−0.495293	+	0.868726i	$0.664939\pi$
$588$		0			0
$589$		−	16.0000i		−	0.659269i
$590$		0			0
$591$	22.0000			0.904959
$592$		0			0
$593$		−	42.0000i		−	1.72473i	−0.506284	−	0.862367i	$-0.668981\pi$
$593$		−	42.0000i		−	1.72473i	0.506284	−	0.862367i	$-0.331019\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−4.00000			−0.163709
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	14.0000			0.571072			0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000			0.571072			0.285536	+	0.958368i	$0.407828\pi$
$602$		0			0
$603$	4.00000			0.162893
$604$		0			0
$605$	−5.00000	−	10.0000i	−0.203279	−	0.406558i
$606$		0			0
$607$			32.0000i			1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$			32.0000i			1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			12.0000i			0.485468i
$612$		0			0
$613$	2.00000			0.0807792			0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000			0.0807792			0.0403896	+	0.999184i	$0.487140\pi$
$614$		0			0
$615$	−2.00000	−	4.00000i	−0.0806478	−	0.161296i
$616$		0			0
$617$		−	6.00000i		−	0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$		−	6.00000i		−	0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$		0			0
$619$		−	46.0000i		−	1.84890i	−0.381308	−	0.924448i	$-0.624526\pi$
$619$		−	46.0000i		−	1.84890i	0.381308	−	0.924448i	$-0.375474\pi$
$620$		0			0
$621$		−	6.00000i		−	0.240772i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$	8.00000			0.319489
$628$		0			0
$629$		−	12.0000i		−	0.478471i
$630$		0			0
$631$	−36.0000			−1.43314			−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000			−1.43314			−0.716569	+	0.697517i	$0.754288\pi$
$632$		0			0
$633$		−	22.0000i		−	0.874421i
$634$		0			0
$635$	8.00000	−	4.00000i	0.317470	−	0.158735i
$636$		0			0
$637$	14.0000			0.554700
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−30.0000			−1.18493			−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000			−1.18493			−0.592464	+	0.805597i	$0.701845\pi$
$642$		0			0
$643$	−4.00000			−0.157745			−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000			−0.157745			−0.0788723	+	0.996885i	$0.525132\pi$
$644$		0			0
$645$	−4.00000	−	8.00000i	−0.157500	−	0.315000i
$646$		0			0
$647$			22.0000i			0.864909i	0.901656	+	0.432455i	$0.142352\pi$
$647$			22.0000i			0.864909i	−0.901656	+	0.432455i	$0.857648\pi$
$648$		0			0
$649$	−32.0000			−1.25611
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$	24.0000	−	12.0000i	0.937758	−	0.468879i
$656$		0			0
$657$		−	4.00000i		−	0.156055i
$658$		0			0
$659$			8.00000i			0.311636i	0.987786	+	0.155818i	$0.0498013\pi$
$659$			8.00000i			0.311636i	−0.987786	+	0.155818i	$0.950199\pi$
$660$		0			0
$661$		−	10.0000i		−	0.388955i	−0.980907	−	0.194477i	$-0.937699\pi$
$661$		−	10.0000i		−	0.388955i	0.980907	−	0.194477i	$-0.0623011\pi$
$662$		0			0
$663$		−	4.00000i		−	0.155347i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	48.0000			1.85857
$668$		0			0
$669$		−	8.00000i		−	0.309298i
$670$		0			0
$671$	−8.00000			−0.308837
$672$		0			0
$673$		−	20.0000i		−	0.770943i	−0.922720	−	0.385472i	$-0.874039\pi$
$673$		−	20.0000i		−	0.770943i	0.922720	−	0.385472i	$-0.125961\pi$
$674$		0			0
$675$	−3.00000	+	4.00000i	−0.115470	+	0.153960i
$676$		0			0
$677$	−6.00000			−0.230599			−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000			−0.230599			−0.115299	+	0.993331i	$0.536783\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−36.0000			−1.37750			−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000			−1.37750			−0.688751	+	0.724998i	$0.741841\pi$
$684$		0			0
$685$	−44.0000	+	22.0000i	−1.68115	+	0.840577i
$686$		0			0
$687$		−	2.00000i		−	0.0763048i
$688$		0			0
$689$	20.0000			0.761939
$690$		0			0
$691$		−	10.0000i		−	0.380418i	−0.981744	−	0.190209i	$-0.939083\pi$
$691$		−	10.0000i		−	0.380418i	0.981744	−	0.190209i	$-0.0609166\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	28.0000	−	14.0000i	1.06210	−	0.531050i
$696$		0			0
$697$			4.00000i			0.151511i
$698$		0			0
$699$			6.00000i			0.226941i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		−	12.0000i		−	0.452589i
$704$		0			0
$705$	−12.0000	+	6.00000i	−0.451946	+	0.225973i
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	34.0000i		−	1.27690i	−0.769665	−	0.638448i	$-0.779577\pi$
$709$		−	34.0000i		−	1.27690i	0.769665	−	0.638448i	$-0.220423\pi$
$710$		0			0
$711$	−4.00000			−0.150012
$712$		0			0
$713$		−	48.0000i		−	1.79761i
$714$		0			0
$715$	−16.0000	+	8.00000i	−0.598366	+	0.299183i
$716$		0			0
$717$	−24.0000			−0.896296
$718$		0			0
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−26.0000			−0.966950
$724$		0			0
$725$	−32.0000	−	24.0000i	−1.18845	−	0.891338i
$726$		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
$731$			8.00000i			0.295891i
$732$		0			0
$733$	−34.0000			−1.25582			−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000			−1.25582			−0.627909	+	0.778287i	$0.716089\pi$
$734$		0			0
$735$	7.00000	+	14.0000i	0.258199	+	0.516398i
$736$		0			0
$737$			16.0000i			0.589368i
$738$		0			0
$739$		−	2.00000i		−	0.0735712i	−0.999323	−	0.0367856i	$-0.988288\pi$
$739$		−	2.00000i		−	0.0735712i	0.999323	−	0.0367856i	$-0.0117119\pi$
$740$		0			0
$741$		−	4.00000i		−	0.146944i
$742$		0			0
$743$			6.00000i			0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$			6.00000i			0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$		0			0
$745$	8.00000	−	4.00000i	0.293097	−	0.146549i
$746$		0			0
$747$	−16.0000			−0.585409
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−24.0000			−0.875772			−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000			−0.875772			−0.437886	+	0.899030i	$0.644273\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.00000	−	16.0000i	−0.291150	−	0.582300i
$756$		0			0
$757$	−34.0000			−1.23575			−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000			−1.23575			−0.617876	+	0.786276i	$0.712006\pi$
$758$		0			0
$759$	24.0000			0.871145
$760$		0			0
$761$	18.0000			0.652499			0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000			0.652499			0.326250	+	0.945284i	$0.394215\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	4.00000	−	2.00000i	0.144620	−	0.0723102i
$766$		0			0
$767$			16.0000i			0.577727i
$768$		0			0
$769$	50.0000			1.80305			0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000			1.80305			0.901523	+	0.432731i	$0.142450\pi$
$770$		0			0
$771$		−	6.00000i		−	0.216085i
$772$		0			0
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$		0			0
$775$	−24.0000	+	32.0000i	−0.862105	+	1.14947i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			4.00000i			0.143315i
$780$		0			0
$781$		0			0
$782$		0			0
$783$			8.00000i			0.285897i
$784$		0			0
$785$	14.0000	+	28.0000i	0.499681	+	0.999363i
$786$		0			0
$787$	−44.0000			−1.56843			−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000			−1.56843			−0.784215	+	0.620489i	$0.786934\pi$
$788$		0			0
$789$			2.00000i			0.0712019i
$790$		0			0
$791$		0			0
$792$		0			0
$793$			4.00000i			0.142044i
$794$		0			0
$795$	10.0000	+	20.0000i	0.354663	+	0.709327i
$796$		0			0
$797$	−42.0000			−1.48772			−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000			−1.48772			−0.743858	+	0.668338i	$0.767006\pi$
$798$		0			0
$799$	12.0000			0.424529
$800$		0			0
$801$	−18.0000			−0.635999
$802$		0			0
$803$	16.0000			0.564628
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	16.0000i		−	0.563227i
$808$		0			0
$809$	−14.0000			−0.492214			−0.246107	−	0.969243i	$-0.579151\pi$
$809$	−14.0000			−0.492214			−0.246107	+	0.969243i	$0.579151\pi$
$810$		0			0
$811$			10.0000i			0.351147i	0.984466	+	0.175574i	$0.0561780\pi$
$811$			10.0000i			0.351147i	−0.984466	+	0.175574i	$0.943822\pi$
$812$		0			0
$813$	−20.0000			−0.701431
$814$		0			0
$815$	12.0000	+	24.0000i	0.420342	+	0.840683i
$816$		0			0
$817$			8.00000i			0.279885i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	40.0000i		−	1.39601i	−0.716093	−	0.698005i	$-0.754071\pi$
$821$		−	40.0000i		−	1.39601i	0.716093	−	0.698005i	$-0.245929\pi$
$822$		0			0
$823$		−	40.0000i		−	1.39431i	−0.716919	−	0.697156i	$-0.754448\pi$
$823$		−	40.0000i		−	1.39431i	0.716919	−	0.697156i	$-0.245552\pi$
$824$		0			0
$825$	−16.0000	−	12.0000i	−0.557048	−	0.417786i
$826$		0			0
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$		−	10.0000i		−	0.347314i	−0.984806	−	0.173657i	$-0.944442\pi$
$829$		−	10.0000i		−	0.347314i	0.984806	−	0.173657i	$-0.0555585\pi$
$830$		0			0
$831$	−26.0000			−0.901930
$832$		0			0
$833$		−	14.0000i		−	0.485071i
$834$		0			0
$835$	28.0000	−	14.0000i	0.968980	−	0.484490i
$836$		0			0
$837$	8.00000			0.276520
$838$		0			0
$839$	−40.0000			−1.38095			−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000			−1.38095			−0.690477	+	0.723355i	$0.742599\pi$
$840$		0			0
$841$	−35.0000			−1.20690
$842$		0			0
$843$	−6.00000			−0.206651
$844$		0			0
$845$	−9.00000	−	18.0000i	−0.309609	−	0.619219i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	28.0000			0.960958
$850$		0			0
$851$		−	36.0000i		−	1.23406i
$852$		0			0
$853$	26.0000			0.890223			0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000			0.890223			0.445112	+	0.895475i	$0.353164\pi$
$854$		0			0
$855$	4.00000	−	2.00000i	0.136797	−	0.0683986i
$856$		0			0
$857$		−	30.0000i		−	1.02478i	−0.858753	−	0.512390i	$-0.828760\pi$
$857$		−	30.0000i		−	1.02478i	0.858753	−	0.512390i	$-0.171240\pi$
$858$		0			0
$859$			18.0000i			0.614152i	0.951685	+	0.307076i	$0.0993506\pi$
$859$			18.0000i			0.614152i	−0.951685	+	0.307076i	$0.900649\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	46.0000i		−	1.56586i	−0.622111	−	0.782929i	$-0.713725\pi$
$863$		−	46.0000i		−	1.56586i	0.622111	−	0.782929i	$-0.286275\pi$
$864$		0			0
$865$	14.0000	+	28.0000i	0.476014	+	0.952029i
$866$		0			0
$867$	13.0000			0.441503
$868$		0			0
$869$		−	16.0000i		−	0.542763i
$870$		0			0
$871$	8.00000			0.271070
$872$		0			0
$873$			16.0000i			0.541518i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−18.0000			−0.607817			−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000			−0.607817			−0.303908	+	0.952701i	$0.598292\pi$
$878$		0			0
$879$	22.0000			0.742042
$880$		0			0
$881$	54.0000			1.81931			0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000			1.81931			0.909653	+	0.415369i	$0.136347\pi$
$882$		0			0
$883$	4.00000			0.134611			0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000			0.134611			0.0673054	+	0.997732i	$0.478560\pi$
$884$		0			0
$885$	−16.0000	+	8.00000i	−0.537834	+	0.268917i
$886$		0			0
$887$			26.0000i			0.872995i	0.899706	+	0.436497i	$0.143781\pi$
$887$			26.0000i			0.872995i	−0.899706	+	0.436497i	$0.856219\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$			4.00000i			0.134005i
$892$		0			0
$893$	12.0000			0.401565
$894$		0			0
$895$	−40.0000	+	20.0000i	−1.33705	+	0.668526i
$896$		0			0
$897$		−	12.0000i		−	0.400668i
$898$		0			0
$899$			64.0000i			2.13452i
$900$		0			0
$901$		−	20.0000i		−	0.666297i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.00000	−	2.00000i	0.132964	−	0.0664822i
$906$		0			0
$907$	20.0000			0.664089			0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000			0.664089			0.332045	+	0.943264i	$0.392262\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	24.0000			0.795155			0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000			0.795155			0.397578	+	0.917568i	$0.369851\pi$
$912$		0			0
$913$		−	64.0000i		−	2.11809i
$914$		0			0
$915$	−4.00000	+	2.00000i	−0.132236	+	0.0661180i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−20.0000			−0.659022
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−18.0000	+	24.0000i	−0.591836	+	0.789115i
$926$		0			0
$927$		−	4.00000i		−	0.131377i
$928$		0			0
$929$	2.00000			0.0656179			0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000			0.0656179			0.0328089	+	0.999462i	$0.489555\pi$
$930$		0			0
$931$		−	14.0000i		−	0.458831i
$932$		0			0
$933$	24.0000			0.785725
$934$		0			0
$935$	8.00000	+	16.0000i	0.261628	+	0.523256i
$936$		0			0
$937$		−	48.0000i		−	1.56809i	−0.620703	−	0.784046i	$-0.713153\pi$
$937$		−	48.0000i		−	1.56809i	0.620703	−	0.784046i	$-0.286847\pi$
$938$		0			0
$939$		−	12.0000i		−	0.391605i
$940$		0			0
$941$			12.0000i			0.391189i	0.980685	+	0.195594i	$0.0626636\pi$
$941$			12.0000i			0.391189i	−0.980685	+	0.195594i	$0.937336\pi$
$942$		0			0
$943$			12.0000i			0.390774i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24.0000			0.779895			0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000			0.779895			0.389948	+	0.920837i	$0.372493\pi$
$948$		0			0
$949$		−	8.00000i		−	0.259691i
$950$		0			0
$951$	−18.0000			−0.583690
$952$		0			0
$953$		−	30.0000i		−	0.971795i	−0.874016	−	0.485898i	$-0.838493\pi$
$953$		−	30.0000i		−	0.971795i	0.874016	−	0.485898i	$-0.161507\pi$
$954$		0			0
$955$	8.00000	+	16.0000i	0.258874	+	0.517748i
$956$		0			0
$957$	−32.0000			−1.03441
$958$		0			0
$959$		0			0
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	−8.00000			−0.257796
$964$		0			0
$965$	−8.00000	+	4.00000i	−0.257529	+	0.128765i
$966$		0			0
$967$		−	56.0000i		−	1.80084i	−0.435023	−	0.900419i	$-0.643260\pi$
$967$		−	56.0000i		−	1.80084i	0.435023	−	0.900419i	$-0.356740\pi$
$968$		0			0
$969$	−4.00000			−0.128499
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−6.00000	+	8.00000i	−0.192154	+	0.256205i
$976$		0			0
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$		0			0
$979$		−	72.0000i		−	2.30113i
$980$		0			0
$981$			14.0000i			0.446986i
$982$		0			0
$983$		−	34.0000i		−	1.08443i	−0.840239	−	0.542216i	$-0.817586\pi$
$983$		−	34.0000i		−	1.08443i	0.840239	−	0.542216i	$-0.182414\pi$
$984$		0			0
$985$	22.0000	+	44.0000i	0.700978	+	1.40196i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			24.0000i			0.763156i
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$		0			0
$993$		−	34.0000i		−	1.07896i
$994$		0			0
$995$	−4.00000	−	8.00000i	−0.126809	−	0.253617i
$996$		0			0
$997$	−46.0000			−1.45683			−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000			−1.45683			−0.728417	+	0.685134i	$0.759744\pi$
$998$		0			0
$999$	6.00000			0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.d.bb.2689.2		2
4.3	odd	2		3840.2.d.l.2689.2		2
5.4	even	2		3840.2.d.f.2689.2		2
8.3	odd	2		3840.2.d.t.2689.1		2
8.5	even	2		3840.2.d.f.2689.1		2
16.3	odd	4		1920.2.f.i.769.1	yes	2
16.5	even	4		1920.2.f.a.769.1	✓	2
16.11	odd	4		1920.2.f.d.769.2	yes	2
16.13	even	4		1920.2.f.l.769.2	yes	2
20.19	odd	2		3840.2.d.t.2689.2		2
40.19	odd	2		3840.2.d.l.2689.1		2
40.29	even	2	inner	3840.2.d.bb.2689.1		2
80.3	even	4		9600.2.a.bl.1.1		1
80.13	odd	4		9600.2.a.r.1.1		1
80.19	odd	4		1920.2.f.i.769.2	yes	2
80.27	even	4		9600.2.a.bt.1.1		1
80.29	even	4		1920.2.f.l.769.1	yes	2
80.37	odd	4		9600.2.a.j.1.1		1
80.43	even	4		9600.2.a.s.1.1		1
80.53	odd	4		9600.2.a.bm.1.1		1
80.59	odd	4		1920.2.f.d.769.1	yes	2
80.67	even	4		9600.2.a.k.1.1		1
80.69	even	4		1920.2.f.a.769.2	yes	2
80.77	odd	4		9600.2.a.bu.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.f.a.769.1	✓	2	16.5	even	4
1920.2.f.a.769.2	yes	2	80.69	even	4
1920.2.f.d.769.1	yes	2	80.59	odd	4
1920.2.f.d.769.2	yes	2	16.11	odd	4
1920.2.f.i.769.1	yes	2	16.3	odd	4
1920.2.f.i.769.2	yes	2	80.19	odd	4
1920.2.f.l.769.1	yes	2	80.29	even	4
1920.2.f.l.769.2	yes	2	16.13	even	4
3840.2.d.f.2689.1		2	8.5	even	2
3840.2.d.f.2689.2		2	5.4	even	2
3840.2.d.l.2689.1		2	40.19	odd	2
3840.2.d.l.2689.2		2	4.3	odd	2
3840.2.d.t.2689.1		2	8.3	odd	2
3840.2.d.t.2689.2		2	20.19	odd	2
3840.2.d.bb.2689.1		2	40.29	even	2	inner
3840.2.d.bb.2689.2		2	1.1	even	1	trivial
9600.2.a.j.1.1		1	80.37	odd	4
9600.2.a.k.1.1		1	80.67	even	4
9600.2.a.r.1.1		1	80.13	odd	4
9600.2.a.s.1.1		1	80.43	even	4
9600.2.a.bl.1.1		1	80.3	even	4
9600.2.a.bm.1.1		1	80.53	odd	4
9600.2.a.bt.1.1		1	80.27	even	4
9600.2.a.bu.1.1		1	80.77	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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +(1.00000 + 2.00000i) q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +(1.00000 + 2.00000i) q^{5} +1.00000 q^{9} +4.00000i q^{11} +2.00000 q^{13} +(1.00000 + 2.00000i) q^{15} -2.00000i q^{17} -2.00000i q^{19} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +1.00000 q^{27} +8.00000i q^{29} +8.00000 q^{31} +4.00000i q^{33} +6.00000 q^{37} +2.00000 q^{39} -2.00000 q^{41} -4.00000 q^{43} +(1.00000 + 2.00000i) q^{45} +6.00000i q^{47} +7.00000 q^{49} -2.00000i q^{51} +10.0000 q^{53} +(-8.00000 + 4.00000i) q^{55} -2.00000i q^{57} +8.00000i q^{59} +2.00000i q^{61} +(2.00000 + 4.00000i) q^{65} +4.00000 q^{67} -6.00000i q^{69} -4.00000i q^{73} +(-3.00000 + 4.00000i) q^{75} -4.00000 q^{79} +1.00000 q^{81} -16.0000 q^{83} +(4.00000 - 2.00000i) q^{85} +8.00000i q^{87} -18.0000 q^{89} +8.00000 q^{93} +(4.00000 - 2.00000i) q^{95} +16.0000i q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 6 q^{25} + 2 q^{27} + 16 q^{31} + 12 q^{37} + 4 q^{39} - 4 q^{41} - 8 q^{43} + 2 q^{45} + 14 q^{49} + 20 q^{53} - 16 q^{55} + 4 q^{65} + 8 q^{67} - 6 q^{75} - 8 q^{79} + 2 q^{81} - 32 q^{83} + 8 q^{85} - 36 q^{89} + 16 q^{93} + 8 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 6 * q^25 + 2 * q^27 + 16 * q^31 + 12 * q^37 + 4 * q^39 - 4 * q^41 - 8 * q^43 + 2 * q^45 + 14 * q^49 + 20 * q^53 - 16 * q^55 + 4 * q^65 + 8 * q^67 - 6 * q^75 - 8 * q^79 + 2 * q^81 - 32 * q^83 + 8 * q^85 - 36 * q^89 + 16 * q^93 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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