LMFDB - Embedded newform 3840.2.d.a.2689.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3840,2,Mod(2689,3840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,-2,0,-4,0,0,0,2,0,0,0,-8,0,4,0,0,0,0,0,0,0,0,0,6,0,-2,0, 0,0,-16,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(35)] == traces)

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 480)
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.a.2689.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +(-2.00000 - 1.00000i) q^{5} -4.00000i q^{7} +1.00000 q^{9} -4.00000 q^{13} +(2.00000 + 1.00000i) q^{15} +8.00000i q^{19} +4.00000i q^{21} -4.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000 q^{27} +6.00000i q^{29} -8.00000 q^{31} +(-4.00000 + 8.00000i) q^{35} +4.00000 q^{37} +4.00000 q^{39} -6.00000 q^{41} +4.00000 q^{43} +(-2.00000 - 1.00000i) q^{45} +4.00000i q^{47} -9.00000 q^{49} -12.0000 q^{53} -8.00000i q^{57} -6.00000i q^{61} -4.00000i q^{63} +(8.00000 + 4.00000i) q^{65} +12.0000 q^{67} +4.00000i q^{69} +16.0000 q^{71} +(-3.00000 - 4.00000i) q^{75} +8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -6.00000i q^{87} -10.0000 q^{89} +16.0000i q^{91} +8.00000 q^{93} +(8.00000 - 16.0000i) q^{95} -8.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} - 8 q^{13} + 4 q^{15} + 6 q^{25} - 2 q^{27} - 16 q^{31} - 8 q^{35} + 8 q^{37} + 8 q^{39} - 12 q^{41} + 8 q^{43} - 4 q^{45} - 18 q^{49} - 24 q^{53} + 16 q^{65} + 24 q^{67}+ \cdots + 16 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 - 8 * q^13 + 4 * q^15 + 6 * q^25 - 2 * q^27 - 16 * q^31 - 8 * q^35 + 8 * q^37 + 8 * q^39 - 12 * q^41 + 8 * q^43 - 4 * q^45 - 18 * q^49 - 24 * q^53 + 16 * q^65 + 24 * q^67 + 32 * q^71 - 6 * q^75 + 16 * q^79 + 2 * q^81 + 24 * q^83 - 20 * q^89 + 16 * q^93 + 16 * q^95

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
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$	2.00000	+	1.00000i	0.516398	+	0.258199i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$			8.00000i			1.83533i	0.397360	+	0.917663i	$0.369927\pi$
$19$			8.00000i			1.83533i	−0.397360	+	0.917663i	$0.630073\pi$
$20$		0			0
$21$			4.00000i			0.872872i
$22$		0			0
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$		0			0
$27$	−1.00000			−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
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.00000	+	8.00000i	−0.676123	+	1.35225i
$36$		0			0
$37$	4.00000			0.657596			0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000			0.657596			0.328798	+	0.944400i	$0.393356\pi$
$38$		0			0
$39$	4.00000			0.640513
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$	−2.00000	−	1.00000i	−0.298142	−	0.149071i
$46$		0			0
$47$			4.00000i			0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$			4.00000i			0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$		0			0
$52$		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
$55$		0			0
$56$		0			0
$57$		−	8.00000i		−	1.05963i
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$		−	6.00000i		−	0.768221i	−0.923287	−	0.384111i	$-0.874508\pi$
$61$		−	6.00000i		−	0.768221i	0.923287	−	0.384111i	$-0.125492\pi$
$62$		0			0
$63$		−	4.00000i		−	0.503953i
$64$		0			0
$65$	8.00000	+	4.00000i	0.992278	+	0.496139i
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$		0			0
$69$			4.00000i			0.481543i
$70$		0			0
$71$	16.0000			1.89885			0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000			1.89885			0.949425	+	0.313993i	$0.101667\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−3.00000	−	4.00000i	−0.346410	−	0.461880i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$		0			0
$86$		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
$91$			16.0000i			1.67726i
$92$		0			0
$93$	8.00000			0.829561
$94$		0			0
$95$	8.00000	−	16.0000i	0.820783	−	1.64157i
$96$		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
$99$		0			0
$100$		0			0
$101$		−	14.0000i		−	1.39305i	−0.717532	−	0.696526i	$-0.754728\pi$
$101$		−	14.0000i		−	1.39305i	0.717532	−	0.696526i	$-0.245272\pi$
$102$		0			0
$103$			12.0000i			1.18240i	0.806527	+	0.591198i	$0.201345\pi$
$103$			12.0000i			1.18240i	−0.806527	+	0.591198i	$0.798655\pi$
$104$		0			0
$105$	4.00000	−	8.00000i	0.390360	−	0.780720i
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$			10.0000i			0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$			10.0000i			0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$		0			0
$111$	−4.00000			−0.379663
$112$		0			0
$113$		−	8.00000i		−	0.752577i	−0.926503	−	0.376288i	$-0.877200\pi$
$113$		−	8.00000i		−	0.752577i	0.926503	−	0.376288i	$-0.122800\pi$
$114$		0			0
$115$	−4.00000	+	8.00000i	−0.373002	+	0.746004i
$116$		0			0
$117$	−4.00000			−0.369800
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	6.00000			0.541002
$124$		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
$131$		−	16.0000i		−	1.39793i	−0.715158	−	0.698963i	$-0.753645\pi$
$131$		−	16.0000i		−	1.39793i	0.715158	−	0.698963i	$-0.246355\pi$
$132$		0			0
$133$	32.0000			2.77475
$134$		0			0
$135$	2.00000	+	1.00000i	0.172133	+	0.0860663i
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$			8.00000i			0.678551i	0.940687	+	0.339276i	$0.110182\pi$
$139$			8.00000i			0.678551i	−0.940687	+	0.339276i	$0.889818\pi$
$140$		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
$149$			14.0000i			1.14692i	0.819232	+	0.573462i	$0.194400\pi$
$149$			14.0000i			1.14692i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.0000	+	8.00000i	1.28515	+	0.642575i
$156$		0			0
$157$	4.00000			0.319235			0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000			0.319235			0.159617	+	0.987179i	$0.448974\pi$
$158$		0			0
$159$	12.0000			0.951662
$160$		0			0
$161$	−16.0000			−1.26098
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.00000i		−	0.309529i	−0.987951	−	0.154765i	$-0.950538\pi$
$167$		−	4.00000i		−	0.309529i	0.987951	−	0.154765i	$-0.0494619\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$			8.00000i			0.611775i
$172$		0			0
$173$	−4.00000			−0.304114			−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000			−0.304114			−0.152057	+	0.988372i	$0.548590\pi$
$174$		0			0
$175$	16.0000	−	12.0000i	1.20949	−	0.907115i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			16.0000i			1.19590i	0.801535	+	0.597948i	$0.204017\pi$
$179$			16.0000i			1.19590i	−0.801535	+	0.597948i	$0.795983\pi$
$180$		0			0
$181$		−	14.0000i		−	1.04061i	−0.853980	−	0.520306i	$-0.825818\pi$
$181$		−	14.0000i		−	1.04061i	0.853980	−	0.520306i	$-0.174182\pi$
$182$		0			0
$183$			6.00000i			0.443533i
$184$		0			0
$185$	−8.00000	−	4.00000i	−0.588172	−	0.294086i
$186$		0			0
$187$		0			0
$188$		0			0
$189$			4.00000i			0.290957i
$190$		0			0
$191$	16.0000			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$192$		0			0
$193$			24.0000i			1.72756i	0.503871	+	0.863779i	$0.331909\pi$
$193$			24.0000i			1.72756i	−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$	−8.00000	−	4.00000i	−0.572892	−	0.286446i
$196$		0			0
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$		0			0
$201$	−12.0000			−0.846415
$202$		0			0
$203$	24.0000			1.68447
$204$		0			0
$205$	12.0000	+	6.00000i	0.838116	+	0.419058i
$206$		0			0
$207$		−	4.00000i		−	0.278019i
$208$		0			0
$209$		0			0
$210$		0			0
$211$			8.00000i			0.550743i	0.961338	+	0.275371i	$0.0888008\pi$
$211$			8.00000i			0.550743i	−0.961338	+	0.275371i	$0.911199\pi$
$212$		0			0
$213$	−16.0000			−1.09630
$214$		0			0
$215$	−8.00000	−	4.00000i	−0.545595	−	0.272798i
$216$		0			0
$217$			32.0000i			2.17230i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			4.00000i			0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$			4.00000i			0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$		0			0
$225$	3.00000	+	4.00000i	0.200000	+	0.266667i
$226$		0			0
$227$	−4.00000			−0.265489			−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000			−0.265489			−0.132745	+	0.991150i	$0.542379\pi$
$228$		0			0
$229$			2.00000i			0.132164i	0.997814	+	0.0660819i	$0.0210498\pi$
$229$			2.00000i			0.132164i	−0.997814	+	0.0660819i	$0.978950\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	8.00000i		−	0.524097i	−0.965055	−	0.262049i	$-0.915602\pi$
$233$		−	8.00000i		−	0.524097i	0.965055	−	0.262049i	$-0.0843981\pi$
$234$		0			0
$235$	4.00000	−	8.00000i	0.260931	−	0.521862i
$236$		0			0
$237$	−8.00000			−0.519656
$238$		0			0
$239$	16.0000			1.03495			0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$		0			0
$241$	14.0000			0.901819			0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000			0.901819			0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	18.0000	+	9.00000i	1.14998	+	0.574989i
$246$		0			0
$247$		−	32.0000i		−	2.03611i
$248$		0			0
$249$	−12.0000			−0.760469
$250$		0			0
$251$			16.0000i			1.00991i	0.863145	+	0.504956i	$0.168491\pi$
$251$			16.0000i			1.00991i	−0.863145	+	0.504956i	$0.831509\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$			24.0000i			1.49708i	0.663090	+	0.748539i	$0.269245\pi$
$257$			24.0000i			1.49708i	−0.663090	+	0.748539i	$0.730755\pi$
$258$		0			0
$259$		−	16.0000i		−	0.994192i
$260$		0			0
$261$			6.00000i			0.371391i
$262$		0			0
$263$			12.0000i			0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$			12.0000i			0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$		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
$271$	24.0000			1.45790			0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000			1.45790			0.728948	+	0.684569i	$0.240010\pi$
$272$		0			0
$273$		−	16.0000i		−	0.968364i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	4.00000			0.240337			0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000			0.240337			0.120168	+	0.992754i	$0.461657\pi$
$278$		0			0
$279$	−8.00000			−0.478947
$280$		0			0
$281$	26.0000			1.55103			0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000			1.55103			0.775515	+	0.631329i	$0.217490\pi$
$282$		0			0
$283$	20.0000			1.18888			0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000			1.18888			0.594438	+	0.804141i	$0.297374\pi$
$284$		0			0
$285$	−8.00000	+	16.0000i	−0.473879	+	0.947758i
$286$		0			0
$287$			24.0000i			1.41668i
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$			8.00000i			0.468968i
$292$		0			0
$293$	−20.0000			−1.16841			−0.584206	−	0.811605i	$-0.698594\pi$
$293$	−20.0000			−1.16841			−0.584206	+	0.811605i	$0.698594\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$			16.0000i			0.925304i
$300$		0			0
$301$		−	16.0000i		−	0.922225i
$302$		0			0
$303$			14.0000i			0.804279i
$304$		0			0
$305$	−6.00000	+	12.0000i	−0.343559	+	0.687118i
$306$		0			0
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$		−	12.0000i		−	0.682656i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$			8.00000i			0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$			8.00000i			0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$		0			0
$315$	−4.00000	+	8.00000i	−0.225374	+	0.450749i
$316$		0			0
$317$	−12.0000			−0.673987			−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000			−0.673987			−0.336994	+	0.941507i	$0.609410\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−12.0000	−	16.0000i	−0.665640	−	0.887520i
$326$		0			0
$327$		−	10.0000i		−	0.553001i
$328$		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
$333$	4.00000			0.219199
$334$		0			0
$335$	−24.0000	−	12.0000i	−1.31126	−	0.655630i
$336$		0			0
$337$			16.0000i			0.871576i	0.900049	+	0.435788i	$0.143530\pi$
$337$			16.0000i			0.871576i	−0.900049	+	0.435788i	$0.856470\pi$
$338$		0			0
$339$			8.00000i			0.434500i
$340$		0			0
$341$		0			0
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$	4.00000	−	8.00000i	0.215353	−	0.430706i
$346$		0			0
$347$	−28.0000			−1.50312			−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000			−1.50312			−0.751559	+	0.659665i	$0.770698\pi$
$348$		0			0
$349$		−	10.0000i		−	0.535288i	−0.963518	−	0.267644i	$-0.913755\pi$
$349$		−	10.0000i		−	0.535288i	0.963518	−	0.267644i	$-0.0862451\pi$
$350$		0			0
$351$	4.00000			0.213504
$352$		0			0
$353$		−	24.0000i		−	1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$		−	24.0000i		−	1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$		0			0
$355$	−32.0000	−	16.0000i	−1.69838	−	0.849192i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−45.0000			−2.36842
$362$		0			0
$363$	−11.0000			−0.577350
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	28.0000i		−	1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$		−	28.0000i		−	1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$		0			0
$369$	−6.00000			−0.312348
$370$		0			0
$371$			48.0000i			2.49204i
$372$		0			0
$373$	20.0000			1.03556			0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000			1.03556			0.517780	+	0.855514i	$0.326758\pi$
$374$		0			0
$375$	2.00000	+	11.0000i	0.103280	+	0.568038i
$376$		0			0
$377$		−	24.0000i		−	1.23606i
$378$		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
$383$		−	12.0000i		−	0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$		−	12.0000i		−	0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.00000			0.203331
$388$		0			0
$389$			30.0000i			1.52106i	0.649303	+	0.760530i	$0.275061\pi$
$389$			30.0000i			1.52106i	−0.649303	+	0.760530i	$0.724939\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			16.0000i			0.807093i
$394$		0			0
$395$	−16.0000	−	8.00000i	−0.805047	−	0.402524i
$396$		0			0
$397$	28.0000			1.40528			0.702640	−	0.711546i	$-0.252005\pi$
$397$	28.0000			1.40528			0.702640	+	0.711546i	$0.252005\pi$
$398$		0			0
$399$	−32.0000			−1.60200
$400$		0			0
$401$	−30.0000			−1.49813			−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000			−1.49813			−0.749064	+	0.662497i	$0.769497\pi$
$402$		0			0
$403$	32.0000			1.59403
$404$		0			0
$405$	−2.00000	−	1.00000i	−0.0993808	−	0.0496904i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−24.0000	−	12.0000i	−1.17811	−	0.589057i
$416$		0			0
$417$		−	8.00000i		−	0.391762i
$418$		0			0
$419$		−	16.0000i		−	0.781651i	−0.920465	−	0.390826i	$-0.872190\pi$
$419$		−	16.0000i		−	0.781651i	0.920465	−	0.390826i	$-0.127810\pi$
$420$		0			0
$421$			14.0000i			0.682318i	0.940006	+	0.341159i	$0.110819\pi$
$421$			14.0000i			0.682318i	−0.940006	+	0.341159i	$0.889181\pi$
$422$		0			0
$423$			4.00000i			0.194487i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−24.0000			−1.16144
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.0000			−0.770693			−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000			−0.770693			−0.385346	+	0.922772i	$0.625918\pi$
$432$		0			0
$433$			40.0000i			1.92228i	0.276066	+	0.961139i	$0.410969\pi$
$433$			40.0000i			1.92228i	−0.276066	+	0.961139i	$0.589031\pi$
$434$		0			0
$435$	−6.00000	+	12.0000i	−0.287678	+	0.575356i
$436$		0			0
$437$	32.0000			1.53077
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$		0			0
$441$	−9.00000			−0.428571
$442$		0			0
$443$	20.0000			0.950229			0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000			0.950229			0.475114	+	0.879924i	$0.342407\pi$
$444$		0			0
$445$	20.0000	+	10.0000i	0.948091	+	0.474045i
$446$		0			0
$447$		−	14.0000i		−	0.662177i
$448$		0			0
$449$	30.0000			1.41579			0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000			1.41579			0.707894	+	0.706319i	$0.249646\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−8.00000			−0.375873
$454$		0			0
$455$	16.0000	−	32.0000i	0.750092	−	1.50018i
$456$		0			0
$457$		−	8.00000i		−	0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$		−	8.00000i		−	0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			22.0000i			1.02464i	0.858794	+	0.512321i	$0.171214\pi$
$461$			22.0000i			1.02464i	−0.858794	+	0.512321i	$0.828786\pi$
$462$		0			0
$463$			20.0000i			0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$			20.0000i			0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$		0			0
$465$	−16.0000	−	8.00000i	−0.741982	−	0.370991i
$466$		0			0
$467$	−4.00000			−0.185098			−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000			−0.185098			−0.0925490	+	0.995708i	$0.529501\pi$
$468$		0			0
$469$		−	48.0000i		−	2.21643i
$470$		0			0
$471$	−4.00000			−0.184310
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−32.0000	+	24.0000i	−1.46826	+	1.10120i
$476$		0			0
$477$	−12.0000			−0.549442
$478$		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
$481$	−16.0000			−0.729537
$482$		0			0
$483$	16.0000			0.728025
$484$		0			0
$485$	−8.00000	+	16.0000i	−0.363261	+	0.726523i
$486$		0			0
$487$		−	4.00000i		−	0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$		−	4.00000i		−	0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$		−	32.0000i		−	1.44414i	−0.691820	−	0.722070i	$-0.743191\pi$
$491$		−	32.0000i		−	1.44414i	0.691820	−	0.722070i	$-0.256809\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	64.0000i		−	2.87079i
$498$		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
$501$			4.00000i			0.178707i
$502$		0			0
$503$			28.0000i			1.24846i	0.781241	+	0.624229i	$0.214587\pi$
$503$			28.0000i			1.24846i	−0.781241	+	0.624229i	$0.785413\pi$
$504$		0			0
$505$	−14.0000	+	28.0000i	−0.622992	+	1.24598i
$506$		0			0
$507$	−3.00000			−0.133235
$508$		0			0
$509$			6.00000i			0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$			6.00000i			0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	8.00000i		−	0.353209i
$514$		0			0
$515$	12.0000	−	24.0000i	0.528783	−	1.05757i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	4.00000			0.175581
$520$		0			0
$521$	−10.0000			−0.438108			−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000			−0.438108			−0.219054	+	0.975713i	$0.570297\pi$
$522$		0			0
$523$	−12.0000			−0.524723			−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000			−0.524723			−0.262362	+	0.964970i	$0.584501\pi$
$524$		0			0
$525$	−16.0000	+	12.0000i	−0.698297	+	0.523723i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$	24.0000			1.03956
$534$		0			0
$535$	24.0000	+	12.0000i	1.03761	+	0.518805i
$536$		0			0
$537$		−	16.0000i		−	0.690451i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	10.0000i		−	0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$		−	10.0000i		−	0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$		0			0
$543$			14.0000i			0.600798i
$544$		0			0
$545$	10.0000	−	20.0000i	0.428353	−	0.856706i
$546$		0			0
$547$	−36.0000			−1.53925			−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000			−1.53925			−0.769624	+	0.638497i	$0.779557\pi$
$548$		0			0
$549$		−	6.00000i		−	0.256074i
$550$		0			0
$551$	−48.0000			−2.04487
$552$		0			0
$553$		−	32.0000i		−	1.36078i
$554$		0			0
$555$	8.00000	+	4.00000i	0.339581	+	0.169791i
$556$		0			0
$557$	−4.00000			−0.169485			−0.0847427	−	0.996403i	$-0.527007\pi$
$557$	−4.00000			−0.169485			−0.0847427	+	0.996403i	$0.527007\pi$
$558$		0			0
$559$	−16.0000			−0.676728
$560$		0			0
$561$		0			0
$562$		0			0
$563$	28.0000			1.18006			0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000			1.18006			0.590030	+	0.807382i	$0.299116\pi$
$564$		0			0
$565$	−8.00000	+	16.0000i	−0.336563	+	0.673125i
$566$		0			0
$567$		−	4.00000i		−	0.167984i
$568$		0			0
$569$	10.0000			0.419222			0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000			0.419222			0.209611	+	0.977785i	$0.432780\pi$
$570$		0			0
$571$		−	40.0000i		−	1.67395i	−0.547243	−	0.836974i	$-0.684323\pi$
$571$		−	40.0000i		−	1.67395i	0.547243	−	0.836974i	$-0.315677\pi$
$572$		0			0
$573$	−16.0000			−0.668410
$574$		0			0
$575$	16.0000	−	12.0000i	0.667246	−	0.500435i
$576$		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
$579$		−	24.0000i		−	0.997406i
$580$		0			0
$581$		−	48.0000i		−	1.99138i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	8.00000	+	4.00000i	0.330759	+	0.165380i
$586$		0			0
$587$	−28.0000			−1.15568			−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000			−1.15568			−0.577842	+	0.816149i	$0.696105\pi$
$588$		0			0
$589$		−	64.0000i		−	2.63707i
$590$		0			0
$591$	12.0000			0.493614
$592$		0			0
$593$		−	8.00000i		−	0.328521i	−0.986417	−	0.164260i	$-0.947476\pi$
$593$		−	8.00000i		−	0.328521i	0.986417	−	0.164260i	$-0.0525237\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−8.00000			−0.327418
$598$		0			0
$599$	32.0000			1.30748			0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000			1.30748			0.653742	+	0.756717i	$0.273198\pi$
$600$		0			0
$601$	26.0000			1.06056			0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000			1.06056			0.530281	+	0.847822i	$0.322086\pi$
$602$		0			0
$603$	12.0000			0.488678
$604$		0			0
$605$	−22.0000	−	11.0000i	−0.894427	−	0.447214i
$606$		0			0
$607$			20.0000i			0.811775i	0.913923	+	0.405887i	$0.133038\pi$
$607$			20.0000i			0.811775i	−0.913923	+	0.405887i	$0.866962\pi$
$608$		0			0
$609$	−24.0000			−0.972529
$610$		0			0
$611$		−	16.0000i		−	0.647291i
$612$		0			0
$613$	−20.0000			−0.807792			−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000			−0.807792			−0.403896	+	0.914805i	$0.632344\pi$
$614$		0			0
$615$	−12.0000	−	6.00000i	−0.483887	−	0.241943i
$616$		0			0
$617$		−	8.00000i		−	0.322068i	−0.986949	−	0.161034i	$-0.948517\pi$
$617$		−	8.00000i		−	0.322068i	0.986949	−	0.161034i	$-0.0514829\pi$
$618$		0			0
$619$		−	24.0000i		−	0.964641i	−0.875995	−	0.482321i	$-0.839794\pi$
$619$		−	24.0000i		−	0.964641i	0.875995	−	0.482321i	$-0.160206\pi$
$620$		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
$629$		0			0
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		0			0
$633$		−	8.00000i		−	0.317971i
$634$		0			0
$635$	4.00000	−	8.00000i	0.158735	−	0.317470i
$636$		0			0
$637$	36.0000			1.42637
$638$		0			0
$639$	16.0000			0.632950
$640$		0			0
$641$	−34.0000			−1.34292			−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000			−1.34292			−0.671460	+	0.741041i	$0.734332\pi$
$642$		0			0
$643$	12.0000			0.473234			0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000			0.473234			0.236617	+	0.971603i	$0.423961\pi$
$644$		0			0
$645$	8.00000	+	4.00000i	0.315000	+	0.157500i
$646$		0			0
$647$			12.0000i			0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$			12.0000i			0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		−	32.0000i		−	1.25418i
$652$		0			0
$653$	20.0000			0.782660			0.391330	−	0.920250i	$-0.372015\pi$
$653$	20.0000			0.782660			0.391330	+	0.920250i	$0.372015\pi$
$654$		0			0
$655$	−16.0000	+	32.0000i	−0.625172	+	1.25034i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			30.0000i			1.16686i	0.812162	+	0.583432i	$0.198291\pi$
$661$			30.0000i			1.16686i	−0.812162	+	0.583432i	$0.801709\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−64.0000	−	32.0000i	−2.48181	−	1.24091i
$666$		0			0
$667$	24.0000			0.929284
$668$		0			0
$669$		−	4.00000i		−	0.154649i
$670$		0			0
$671$		0			0
$672$		0			0
$673$			8.00000i			0.308377i	0.988041	+	0.154189i	$0.0492764\pi$
$673$			8.00000i			0.308377i	−0.988041	+	0.154189i	$0.950724\pi$
$674$		0			0
$675$	−3.00000	−	4.00000i	−0.115470	−	0.153960i
$676$		0			0
$677$	−12.0000			−0.461197			−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000			−0.461197			−0.230599	+	0.973049i	$0.574068\pi$
$678$		0			0
$679$	−32.0000			−1.22805
$680$		0			0
$681$	4.00000			0.153280
$682$		0			0
$683$	36.0000			1.37750			0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000			1.37750			0.688751	+	0.724998i	$0.258159\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	2.00000i		−	0.0763048i
$688$		0			0
$689$	48.0000			1.82865
$690$		0			0
$691$			8.00000i			0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$			8.00000i			0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.00000	−	16.0000i	0.303457	−	0.606915i
$696$		0			0
$697$		0			0
$698$		0			0
$699$			8.00000i			0.302588i
$700$		0			0
$701$			42.0000i			1.58632i	0.609015	+	0.793159i	$0.291565\pi$
$701$			42.0000i			1.58632i	−0.609015	+	0.793159i	$0.708435\pi$
$702$		0			0
$703$			32.0000i			1.20690i
$704$		0			0
$705$	−4.00000	+	8.00000i	−0.150649	+	0.301297i
$706$		0			0
$707$	−56.0000			−2.10610
$708$		0			0
$709$			2.00000i			0.0751116i	0.999295	+	0.0375558i	$0.0119572\pi$
$709$			2.00000i			0.0751116i	−0.999295	+	0.0375558i	$0.988043\pi$
$710$		0			0
$711$	8.00000			0.300023
$712$		0			0
$713$			32.0000i			1.19841i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−16.0000			−0.597531
$718$		0			0
$719$	16.0000			0.596699			0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000			0.596699			0.298350	+	0.954457i	$0.403564\pi$
$720$		0			0
$721$	48.0000			1.78761
$722$		0			0
$723$	−14.0000			−0.520666
$724$		0			0
$725$	−24.0000	+	18.0000i	−0.891338	+	0.668503i
$726$		0			0
$727$			28.0000i			1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$			28.0000i			1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		0			0
$732$		0			0
$733$	4.00000			0.147743			0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000			0.147743			0.0738717	+	0.997268i	$0.476464\pi$
$734$		0			0
$735$	−18.0000	−	9.00000i	−0.663940	−	0.331970i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	40.0000i		−	1.47142i	−0.677295	−	0.735712i	$-0.736848\pi$
$739$		−	40.0000i		−	1.47142i	0.677295	−	0.735712i	$-0.263152\pi$
$740$		0			0
$741$			32.0000i			1.17555i
$742$		0			0
$743$			12.0000i			0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$			12.0000i			0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$		0			0
$745$	14.0000	−	28.0000i	0.512920	−	1.02584i
$746$		0			0
$747$	12.0000			0.439057
$748$		0			0
$749$			48.0000i			1.75388i
$750$		0			0
$751$	−8.00000			−0.291924			−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000			−0.291924			−0.145962	+	0.989290i	$0.546628\pi$
$752$		0			0
$753$		−	16.0000i		−	0.583072i
$754$		0			0
$755$	−16.0000	−	8.00000i	−0.582300	−	0.291150i
$756$		0			0
$757$	−12.0000			−0.436147			−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000			−0.436147			−0.218074	+	0.975932i	$0.569977\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$	40.0000			1.44810
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$		−	24.0000i		−	0.864339i
$772$		0			0
$773$	−28.0000			−1.00709			−0.503545	−	0.863969i	$-0.667971\pi$
$773$	−28.0000			−1.00709			−0.503545	+	0.863969i	$0.667971\pi$
$774$		0			0
$775$	−24.0000	−	32.0000i	−0.862105	−	1.14947i
$776$		0			0
$777$			16.0000i			0.573997i
$778$		0			0
$779$		−	48.0000i		−	1.71978i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	6.00000i		−	0.214423i
$784$		0			0
$785$	−8.00000	−	4.00000i	−0.285532	−	0.142766i
$786$		0			0
$787$	12.0000			0.427754			0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000			0.427754			0.213877	+	0.976861i	$0.431391\pi$
$788$		0			0
$789$		−	12.0000i		−	0.427211i
$790$		0			0
$791$	−32.0000			−1.13779
$792$		0			0
$793$			24.0000i			0.852265i
$794$		0			0
$795$	−24.0000	−	12.0000i	−0.851192	−	0.425596i
$796$		0			0
$797$	−12.0000			−0.425062			−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000			−0.425062			−0.212531	+	0.977154i	$0.568171\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−10.0000			−0.353333
$802$		0			0
$803$		0			0
$804$		0			0
$805$	32.0000	+	16.0000i	1.12785	+	0.563926i
$806$		0			0
$807$			6.00000i			0.211210i
$808$		0			0
$809$	22.0000			0.773479			0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000			0.773479			0.386739	+	0.922189i	$0.373601\pi$
$810$		0			0
$811$		−	8.00000i		−	0.280918i	−0.990086	−	0.140459i	$-0.955142\pi$
$811$		−	8.00000i		−	0.280918i	0.990086	−	0.140459i	$-0.0448578\pi$
$812$		0			0
$813$	−24.0000			−0.841717
$814$		0			0
$815$	8.00000	+	4.00000i	0.280228	+	0.140114i
$816$		0			0
$817$			32.0000i			1.11954i
$818$		0			0
$819$			16.0000i			0.559085i
$820$		0			0
$821$		−	18.0000i		−	0.628204i	−0.949389	−	0.314102i	$-0.898297\pi$
$821$		−	18.0000i		−	0.628204i	0.949389	−	0.314102i	$-0.101703\pi$
$822$		0			0
$823$			44.0000i			1.53374i	0.641800	+	0.766872i	$0.278188\pi$
$823$			44.0000i			1.53374i	−0.641800	+	0.766872i	$0.721812\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	36.0000			1.25184			0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000			1.25184			0.625921	+	0.779886i	$0.284723\pi$
$828$		0			0
$829$			26.0000i			0.903017i	0.892267	+	0.451509i	$0.149114\pi$
$829$			26.0000i			0.903017i	−0.892267	+	0.451509i	$0.850886\pi$
$830$		0			0
$831$	−4.00000			−0.138758
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−4.00000	+	8.00000i	−0.138426	+	0.276851i
$836$		0			0
$837$	8.00000			0.276520
$838$		0			0
$839$	−32.0000			−1.10476			−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000			−1.10476			−0.552381	+	0.833592i	$0.686281\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$	−26.0000			−0.895488
$844$		0			0
$845$	−6.00000	−	3.00000i	−0.206406	−	0.103203i
$846$		0			0
$847$		−	44.0000i		−	1.51186i
$848$		0			0
$849$	−20.0000			−0.686398
$850$		0			0
$851$		−	16.0000i		−	0.548473i
$852$		0			0
$853$	12.0000			0.410872			0.205436	−	0.978671i	$-0.434139\pi$
$853$	12.0000			0.410872			0.205436	+	0.978671i	$0.434139\pi$
$854$		0			0
$855$	8.00000	−	16.0000i	0.273594	−	0.547188i
$856$		0			0
$857$			48.0000i			1.63965i	0.572615	+	0.819824i	$0.305929\pi$
$857$			48.0000i			1.63965i	−0.572615	+	0.819824i	$0.694071\pi$
$858$		0			0
$859$			56.0000i			1.91070i	0.295484	+	0.955348i	$0.404519\pi$
$859$			56.0000i			1.91070i	−0.295484	+	0.955348i	$0.595481\pi$
$860$		0			0
$861$		−	24.0000i		−	0.817918i
$862$		0			0
$863$			36.0000i			1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$			36.0000i			1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$		0			0
$865$	8.00000	+	4.00000i	0.272008	+	0.136004i
$866$		0			0
$867$	−17.0000			−0.577350
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−48.0000			−1.62642
$872$		0			0
$873$		−	8.00000i		−	0.270759i
$874$		0			0
$875$	−44.0000	+	8.00000i	−1.48747	+	0.270449i
$876$		0			0
$877$	−28.0000			−0.945493			−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000			−0.945493			−0.472746	+	0.881199i	$0.656737\pi$
$878$		0			0
$879$	20.0000			0.674583
$880$		0			0
$881$	−2.00000			−0.0673817			−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000			−0.0673817			−0.0336909	+	0.999432i	$0.510726\pi$
$882$		0			0
$883$	−4.00000			−0.134611			−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000			−0.134611			−0.0673054	+	0.997732i	$0.521440\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			12.0000i			0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$			12.0000i			0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$		0			0
$889$	16.0000			0.536623
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−32.0000			−1.07084
$894$		0			0
$895$	16.0000	−	32.0000i	0.534821	−	1.06964i
$896$		0			0
$897$		−	16.0000i		−	0.534224i
$898$		0			0
$899$		−	48.0000i		−	1.60089i
$900$		0			0
$901$		0			0
$902$		0			0
$903$			16.0000i			0.532447i
$904$		0			0
$905$	−14.0000	+	28.0000i	−0.465376	+	0.930751i
$906$		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
$909$		−	14.0000i		−	0.464351i
$910$		0			0
$911$	−32.0000			−1.06021			−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000			−1.06021			−0.530104	+	0.847933i	$0.677847\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	6.00000	−	12.0000i	0.198354	−	0.396708i
$916$		0			0
$917$	−64.0000			−2.11347
$918$		0			0
$919$	−24.0000			−0.791687			−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000			−0.791687			−0.395843	+	0.918318i	$0.629548\pi$
$920$		0			0
$921$	−12.0000			−0.395413
$922$		0			0
$923$	−64.0000			−2.10659
$924$		0			0
$925$	12.0000	+	16.0000i	0.394558	+	0.526077i
$926$		0			0
$927$			12.0000i			0.394132i
$928$		0			0
$929$	−2.00000			−0.0656179			−0.0328089	−	0.999462i	$-0.510445\pi$
$929$	−2.00000			−0.0656179			−0.0328089	+	0.999462i	$0.510445\pi$
$930$		0			0
$931$		−	72.0000i		−	2.35970i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			16.0000i			0.522697i	0.965244	+	0.261349i	$0.0841672\pi$
$937$			16.0000i			0.522697i	−0.965244	+	0.261349i	$0.915833\pi$
$938$		0			0
$939$		−	8.00000i		−	0.261070i
$940$		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
$943$			24.0000i			0.781548i
$944$		0			0
$945$	4.00000	−	8.00000i	0.130120	−	0.260240i
$946$		0			0
$947$	−20.0000			−0.649913			−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000			−0.649913			−0.324956	+	0.945729i	$0.605350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	12.0000			0.389127
$952$		0			0
$953$		−	48.0000i		−	1.55487i	−0.628962	−	0.777436i	$-0.716520\pi$
$953$		−	48.0000i		−	1.55487i	0.628962	−	0.777436i	$-0.283480\pi$
$954$		0			0
$955$	−32.0000	−	16.0000i	−1.03550	−	0.517748i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	−12.0000			−0.386695
$964$		0			0
$965$	24.0000	−	48.0000i	0.772587	−	1.54517i
$966$		0			0
$967$			12.0000i			0.385894i	0.981209	+	0.192947i	$0.0618045\pi$
$967$			12.0000i			0.385894i	−0.981209	+	0.192947i	$0.938195\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			16.0000i			0.513464i	0.966483	+	0.256732i	$0.0826458\pi$
$971$			16.0000i			0.513464i	−0.966483	+	0.256732i	$0.917354\pi$
$972$		0			0
$973$	32.0000			1.02587
$974$		0			0
$975$	12.0000	+	16.0000i	0.384308	+	0.512410i
$976$		0			0
$977$			48.0000i			1.53566i	0.640656	+	0.767828i	$0.278662\pi$
$977$			48.0000i			1.53566i	−0.640656	+	0.767828i	$0.721338\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			10.0000i			0.319275i
$982$		0			0
$983$			12.0000i			0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$			12.0000i			0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$		0			0
$985$	24.0000	+	12.0000i	0.764704	+	0.382352i
$986$		0			0
$987$	−16.0000			−0.509286
$988$		0			0
$989$		−	16.0000i		−	0.508770i
$990$		0			0
$991$	−24.0000			−0.762385			−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000			−0.762385			−0.381193	+	0.924496i	$0.624487\pi$
$992$		0			0
$993$			8.00000i			0.253872i
$994$		0			0
$995$	−16.0000	−	8.00000i	−0.507234	−	0.253617i
$996$		0			0
$997$	28.0000			0.886769			0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000			0.886769			0.443384	+	0.896332i	$0.353778\pi$
$998$		0			0
$999$	−4.00000			−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.d.a.2689.1		2
4.3	odd	2		3840.2.d.q.2689.1		2
5.4	even	2		3840.2.d.bf.2689.1		2
8.3	odd	2		3840.2.d.p.2689.2		2
8.5	even	2		3840.2.d.bf.2689.2		2
16.3	odd	4		960.2.f.d.769.2		2
16.5	even	4		480.2.f.d.289.2	yes	2
16.11	odd	4		480.2.f.c.289.1	✓	2
16.13	even	4		960.2.f.e.769.1		2
20.19	odd	2		3840.2.d.p.2689.1		2
40.19	odd	2		3840.2.d.q.2689.2		2
40.29	even	2	inner	3840.2.d.a.2689.2		2
48.5	odd	4		1440.2.f.d.289.2		2
48.11	even	4		1440.2.f.b.289.2		2
48.29	odd	4		2880.2.f.m.1729.1		2
48.35	even	4		2880.2.f.o.1729.1		2
80.3	even	4		4800.2.a.e.1.1		1
80.13	odd	4		4800.2.a.cr.1.1		1
80.19	odd	4		960.2.f.d.769.1		2
80.27	even	4		2400.2.a.p.1.1		1
80.29	even	4		960.2.f.e.769.2		2
80.37	odd	4		2400.2.a.t.1.1		1
80.43	even	4		2400.2.a.s.1.1		1
80.53	odd	4		2400.2.a.o.1.1		1
80.59	odd	4		480.2.f.c.289.2	yes	2
80.67	even	4		4800.2.a.cp.1.1		1
80.69	even	4		480.2.f.d.289.1	yes	2
80.77	odd	4		4800.2.a.c.1.1		1
240.29	odd	4		2880.2.f.m.1729.2		2
240.53	even	4		7200.2.a.by.1.1		1
240.59	even	4		1440.2.f.b.289.1		2
240.107	odd	4		7200.2.a.ca.1.1		1
240.149	odd	4		1440.2.f.d.289.1		2
240.179	even	4		2880.2.f.o.1729.2		2
240.197	even	4		7200.2.a.c.1.1		1
240.203	odd	4		7200.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.f.c.289.1	✓	2	16.11	odd	4
480.2.f.c.289.2	yes	2	80.59	odd	4
480.2.f.d.289.1	yes	2	80.69	even	4
480.2.f.d.289.2	yes	2	16.5	even	4
960.2.f.d.769.1		2	80.19	odd	4
960.2.f.d.769.2		2	16.3	odd	4
960.2.f.e.769.1		2	16.13	even	4
960.2.f.e.769.2		2	80.29	even	4
1440.2.f.b.289.1		2	240.59	even	4
1440.2.f.b.289.2		2	48.11	even	4
1440.2.f.d.289.1		2	240.149	odd	4
1440.2.f.d.289.2		2	48.5	odd	4
2400.2.a.o.1.1		1	80.53	odd	4
2400.2.a.p.1.1		1	80.27	even	4
2400.2.a.s.1.1		1	80.43	even	4
2400.2.a.t.1.1		1	80.37	odd	4
2880.2.f.m.1729.1		2	48.29	odd	4
2880.2.f.m.1729.2		2	240.29	odd	4
2880.2.f.o.1729.1		2	48.35	even	4
2880.2.f.o.1729.2		2	240.179	even	4
3840.2.d.a.2689.1		2	1.1	even	1	trivial
3840.2.d.a.2689.2		2	40.29	even	2	inner
3840.2.d.p.2689.1		2	20.19	odd	2
3840.2.d.p.2689.2		2	8.3	odd	2
3840.2.d.q.2689.1		2	4.3	odd	2
3840.2.d.q.2689.2		2	40.19	odd	2
3840.2.d.bf.2689.1		2	5.4	even	2
3840.2.d.bf.2689.2		2	8.5	even	2
4800.2.a.c.1.1		1	80.77	odd	4
4800.2.a.e.1.1		1	80.3	even	4
4800.2.a.cp.1.1		1	80.67	even	4
4800.2.a.cr.1.1		1	80.13	odd	4
7200.2.a.a.1.1		1	240.203	odd	4
7200.2.a.c.1.1		1	240.197	even	4
7200.2.a.by.1.1		1	240.53	even	4
7200.2.a.ca.1.1		1	240.107	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +(-2.00000 - 1.00000i) q^{5} -4.00000i q^{7} +1.00000 q^{9} -4.00000 q^{13} +(2.00000 + 1.00000i) q^{15} +8.00000i q^{19} +4.00000i q^{21} -4.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000 q^{27} +6.00000i q^{29} -8.00000 q^{31} +(-4.00000 + 8.00000i) q^{35} +4.00000 q^{37} +4.00000 q^{39} -6.00000 q^{41} +4.00000 q^{43} +(-2.00000 - 1.00000i) q^{45} +4.00000i q^{47} -9.00000 q^{49} -12.0000 q^{53} -8.00000i q^{57} -6.00000i q^{61} -4.00000i q^{63} +(8.00000 + 4.00000i) q^{65} +12.0000 q^{67} +4.00000i q^{69} +16.0000 q^{71} +(-3.00000 - 4.00000i) q^{75} +8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -6.00000i q^{87} -10.0000 q^{89} +16.0000i q^{91} +8.00000 q^{93} +(8.00000 - 16.0000i) q^{95} -8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} - 8 q^{13} + 4 q^{15} + 6 q^{25} - 2 q^{27} - 16 q^{31} - 8 q^{35} + 8 q^{37} + 8 q^{39} - 12 q^{41} + 8 q^{43} - 4 q^{45} - 18 q^{49} - 24 q^{53} + 16 q^{65} + 24 q^{67}+ \cdots + 16 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 - 8 * q^13 + 4 * q^15 + 6 * q^25 - 2 * q^27 - 16 * q^31 - 8 * q^35 + 8 * q^37 + 8 * q^39 - 12 * q^41 + 8 * q^43 - 4 * q^45 - 18 * q^49 - 24 * q^53 + 16 * q^65 + 24 * q^67 + 32 * q^71 - 6 * q^75 + 16 * q^79 + 2 * q^81 + 24 * q^83 - 20 * q^89 + 16 * q^93 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more