LMFDB - Embedded newform 1920.2.f.c.769.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(769,1920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1920.769");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$15.3312771881$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1920.769
Dual form			1920.2.f.c.769.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} -6.00000i q^{13} +(1.00000 - 2.00000i) q^{15} +2.00000i q^{17} -6.00000 q^{19} +4.00000 q^{21} +6.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000i q^{27} -8.00000 q^{29} +8.00000 q^{31} +(-4.00000 + 8.00000i) q^{35} +10.0000i q^{37} +6.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +(2.00000 + 1.00000i) q^{45} -2.00000i q^{47} -9.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -6.00000i q^{57} +12.0000 q^{59} -14.0000 q^{61} +4.00000i q^{63} +(-6.00000 + 12.0000i) q^{65} -4.00000i q^{67} -6.00000 q^{69} +8.00000 q^{71} -4.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} -12.0000 q^{79} +1.00000 q^{81} +8.00000i q^{83} +(2.00000 - 4.00000i) q^{85} -8.00000i q^{87} -6.00000 q^{89} -24.0000 q^{91} +8.00000i q^{93} +(12.0000 + 6.00000i) q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^5 - 2 * q^9	$ 2 q - 4 q^{5} - 2 q^{9} + 2 q^{15} - 12 q^{19} + 8 q^{21} + 6 q^{25} - 16 q^{29} + 16 q^{31} - 8 q^{35} + 12 q^{39} - 12 q^{41} + 4 q^{45} - 18 q^{49} - 4 q^{51} + 24 q^{59} - 28 q^{61} - 12 q^{65} - 12 q^{69} + 16 q^{71} - 8 q^{75} - 24 q^{79} + 2 q^{81} + 4 q^{85} - 12 q^{89} - 48 q^{91} + 24 q^{95}+O(q^{100}) $ 2 * q - 4 * q^5 - 2 * q^9 + 2 * q^15 - 12 * q^19 + 8 * q^21 + 6 * q^25 - 16 * q^29 + 16 * q^31 - 8 * q^35 + 12 * q^39 - 12 * q^41 + 4 * q^45 - 18 * q^49 - 4 * q^51 + 24 * q^59 - 28 * q^61 - 12 * q^65 - 12 * q^69 + 16 * q^71 - 8 * q^75 - 24 * q^79 + 2 * q^81 + 4 * q^85 - 12 * q^89 - 48 * q^91 + 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		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			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$		−	6.00000i		−	1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$		−	6.00000i		−	1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$		0			0
$15$	1.00000	−	2.00000i	0.258199	−	0.516398i
$16$		0			0
$17$			2.00000i			0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$			2.00000i			0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		0			0
$21$	4.00000			0.872872
$22$		0			0
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−8.00000			−1.48556			−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000			−1.48556			−0.742781	+	0.669534i	$0.766494\pi$
$30$		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
$33$		0			0
$34$		0			0
$35$	−4.00000	+	8.00000i	−0.676123	+	1.35225i
$36$		0			0
$37$			10.0000i			1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$			10.0000i			1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$		0			0
$39$	6.00000			0.960769
$40$		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
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		0			0
$45$	2.00000	+	1.00000i	0.298142	+	0.149071i
$46$		0			0
$47$		−	2.00000i		−	0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$		−	2.00000i		−	0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	6.00000i		−	0.794719i
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$	−14.0000			−1.79252			−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000			−1.79252			−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$			4.00000i			0.503953i
$64$		0			0
$65$	−6.00000	+	12.0000i	−0.744208	+	1.48842i
$66$		0			0
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$		−	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
$75$	−4.00000	+	3.00000i	−0.461880	+	0.346410i
$76$		0			0
$77$		0			0
$78$		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
$83$			8.00000i			0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$			8.00000i			0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$		0			0
$85$	2.00000	−	4.00000i	0.216930	−	0.433861i
$86$		0			0
$87$		−	8.00000i		−	0.857690i
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	−24.0000			−2.51588
$92$		0			0
$93$			8.00000i			0.829561i
$94$		0			0
$95$	12.0000	+	6.00000i	1.23117	+	0.615587i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$		0			0
$105$	−8.00000	−	4.00000i	−0.780720	−	0.390360i
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$	−10.0000			−0.949158
$112$		0			0
$113$			10.0000i			0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$			10.0000i			0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$		0			0
$115$	6.00000	−	12.0000i	0.559503	−	1.11901i
$116$		0			0
$117$			6.00000i			0.554700i
$118$		0			0
$119$	8.00000			0.733359
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		−	6.00000i		−	0.541002i
$124$		0			0
$125$	−2.00000	−	11.0000i	−0.178885	−	0.983870i
$126$		0			0
$127$		−	8.00000i		−	0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$		−	8.00000i		−	0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$		0			0
$129$	−4.00000			−0.352180
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$			24.0000i			2.08106i
$134$		0			0
$135$	−1.00000	+	2.00000i	−0.0860663	+	0.172133i
$136$		0			0
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	2.00000			0.169638			0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000			0.169638			0.0848189	+	0.996396i	$0.472969\pi$
$140$		0			0
$141$	2.00000			0.168430
$142$		0			0
$143$		0			0
$144$		0			0
$145$	16.0000	+	8.00000i	1.32873	+	0.664364i
$146$		0			0
$147$		−	9.00000i		−	0.742307i
$148$		0			0
$149$	−12.0000			−0.983078			−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000			−0.983078			−0.491539	+	0.870855i	$0.663566\pi$
$150$		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
$153$		−	2.00000i		−	0.161690i
$154$		0			0
$155$	−16.0000	−	8.00000i	−1.28515	−	0.642575i
$156$		0			0
$157$		−	2.00000i		−	0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$		−	2.00000i		−	0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$	24.0000			1.89146
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	2.00000i		−	0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$167$		−	2.00000i		−	0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$		0			0
$171$	6.00000			0.458831
$172$		0			0
$173$		−	18.0000i		−	1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$		−	18.0000i		−	1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$		0			0
$175$	16.0000	−	12.0000i	1.20949	−	0.907115i
$176$		0			0
$177$			12.0000i			0.901975i
$178$		0			0
$179$	−16.0000			−1.19590			−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000			−1.19590			−0.597948	+	0.801535i	$0.704017\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		−	14.0000i		−	1.03491i
$184$		0			0
$185$	10.0000	−	20.0000i	0.735215	−	1.47043i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−4.00000			−0.290957
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		−	4.00000i		−	0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$		−	4.00000i		−	0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$		0			0
$195$	−12.0000	−	6.00000i	−0.859338	−	0.429669i
$196$		0			0
$197$		−	22.0000i		−	1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$		−	22.0000i		−	1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$	4.00000			0.282138
$202$		0			0
$203$			32.0000i			2.24596i
$204$		0			0
$205$	12.0000	+	6.00000i	0.838116	+	0.419058i
$206$		0			0
$207$		−	6.00000i		−	0.417029i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	6.00000			0.413057			0.206529	−	0.978441i	$-0.433783\pi$
$211$	6.00000			0.413057			0.206529	+	0.978441i	$0.433783\pi$
$212$		0			0
$213$			8.00000i			0.548151i
$214$		0			0
$215$	4.00000	−	8.00000i	0.272798	−	0.545595i
$216$		0			0
$217$		−	32.0000i		−	2.17230i
$218$		0			0
$219$	4.00000			0.270295
$220$		0			0
$221$	12.0000			0.807207
$222$		0			0
$223$		−	4.00000i		−	0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$		−	4.00000i		−	0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$		0			0
$225$	−3.00000	−	4.00000i	−0.200000	−	0.266667i
$226$		0			0
$227$			24.0000i			1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$			24.0000i			1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$		0			0
$229$	2.00000			0.132164			0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000			0.132164			0.0660819	+	0.997814i	$0.478950\pi$
$230$		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
$235$	−2.00000	+	4.00000i	−0.130466	+	0.260931i
$236$		0			0
$237$		−	12.0000i		−	0.779484i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		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
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	18.0000	+	9.00000i	1.14998	+	0.574989i
$246$		0			0
$247$			36.0000i			2.29063i
$248$		0			0
$249$	−8.00000			−0.506979
$250$		0			0
$251$	4.00000			0.252478			0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000			0.252478			0.126239	+	0.992000i	$0.459709\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	4.00000	+	2.00000i	0.250490	+	0.125245i
$256$		0			0
$257$			22.0000i			1.37232i	0.727450	+	0.686161i	$0.240706\pi$
$257$			22.0000i			1.37232i	−0.727450	+	0.686161i	$0.759294\pi$
$258$		0			0
$259$	40.0000			2.48548
$260$		0			0
$261$	8.00000			0.495188
$262$		0			0
$263$			30.0000i			1.84988i	0.380114	+	0.924940i	$0.375885\pi$
$263$			30.0000i			1.84988i	−0.380114	+	0.924940i	$0.624115\pi$
$264$		0			0
$265$	6.00000	−	12.0000i	0.368577	−	0.737154i
$266$		0			0
$267$		−	6.00000i		−	0.367194i
$268$		0			0
$269$	−32.0000			−1.95107			−0.975537	−	0.219834i	$-0.929448\pi$
$269$	−32.0000			−1.95107			−0.975537	+	0.219834i	$0.929448\pi$
$270$		0			0
$271$	20.0000			1.21491			0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000			1.21491			0.607457	+	0.794353i	$0.292190\pi$
$272$		0			0
$273$		−	24.0000i		−	1.45255i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			10.0000i			0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$			10.0000i			0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$		0			0
$279$	−8.00000			−0.478947
$280$		0			0
$281$	22.0000			1.31241			0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000			1.31241			0.656205	+	0.754583i	$0.272161\pi$
$282$		0			0
$283$		−	20.0000i		−	1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$		−	20.0000i		−	1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$		0			0
$285$	−6.00000	+	12.0000i	−0.355409	+	0.710819i
$286$		0			0
$287$			24.0000i			1.41668i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$			10.0000i			0.584206i	0.956387	+	0.292103i	$0.0943550\pi$
$293$			10.0000i			0.584206i	−0.956387	+	0.292103i	$0.905645\pi$
$294$		0			0
$295$	−24.0000	−	12.0000i	−1.39733	−	0.698667i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	36.0000			2.08193
$300$		0			0
$301$	16.0000			0.922225
$302$		0			0
$303$		0			0
$304$		0			0
$305$	28.0000	+	14.0000i	1.60328	+	0.801638i
$306$		0			0
$307$			12.0000i			0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$			12.0000i			0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$			4.00000i			0.226093i	0.993590	+	0.113047i	$0.0360610\pi$
$313$			4.00000i			0.226093i	−0.993590	+	0.113047i	$0.963939\pi$
$314$		0			0
$315$	4.00000	−	8.00000i	0.225374	−	0.450749i
$316$		0			0
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$	24.0000	−	18.0000i	1.33128	−	0.998460i
$326$		0			0
$327$		−	2.00000i		−	0.110600i
$328$		0			0
$329$	−8.00000			−0.441054
$330$		0			0
$331$	−10.0000			−0.549650			−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000			−0.549650			−0.274825	+	0.961494i	$0.588620\pi$
$332$		0			0
$333$		−	10.0000i		−	0.547997i
$334$		0			0
$335$	−4.00000	+	8.00000i	−0.218543	+	0.437087i
$336$		0			0
$337$			20.0000i			1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$			20.0000i			1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$		0			0
$339$	−10.0000			−0.543125
$340$		0			0
$341$		0			0
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$	12.0000	+	6.00000i	0.646058	+	0.323029i
$346$		0			0
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$	−6.00000			−0.320256
$352$		0			0
$353$		−	18.0000i		−	0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$		−	18.0000i		−	0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$		0			0
$355$	−16.0000	−	8.00000i	−0.849192	−	0.424596i
$356$		0			0
$357$			8.00000i			0.423405i
$358$		0			0
$359$	−32.0000			−1.68890			−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000			−1.68890			−0.844448	+	0.535638i	$0.820071\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		−	11.0000i		−	0.577350i
$364$		0			0
$365$	−4.00000	+	8.00000i	−0.209370	+	0.418739i
$366$		0			0
$367$			8.00000i			0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$			8.00000i			0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$	24.0000			1.24602
$372$		0			0
$373$		−	6.00000i		−	0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$		−	6.00000i		−	0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$		0			0
$375$	11.0000	−	2.00000i	0.568038	−	0.103280i
$376$		0			0
$377$			48.0000i			2.47213i
$378$		0			0
$379$	2.00000			0.102733			0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000			0.102733			0.0513665	+	0.998680i	$0.483642\pi$
$380$		0			0
$381$	8.00000			0.409852
$382$		0			0
$383$			6.00000i			0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$			6.00000i			0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	4.00000i		−	0.203331i
$388$		0			0
$389$	−4.00000			−0.202808			−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000			−0.202808			−0.101404	+	0.994845i	$0.532333\pi$
$390$		0			0
$391$	−12.0000			−0.606866
$392$		0			0
$393$		0			0
$394$		0			0
$395$	24.0000	+	12.0000i	1.20757	+	0.603786i
$396$		0			0
$397$			10.0000i			0.501886i	0.968002	+	0.250943i	$0.0807406\pi$
$397$			10.0000i			0.501886i	−0.968002	+	0.250943i	$0.919259\pi$
$398$		0			0
$399$	−24.0000			−1.20150
$400$		0			0
$401$	−10.0000			−0.499376			−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000			−0.499376			−0.249688	+	0.968326i	$0.580328\pi$
$402$		0			0
$403$		−	48.0000i		−	2.39105i
$404$		0			0
$405$	−2.00000	−	1.00000i	−0.0993808	−	0.0496904i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	14.0000			0.692255			0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000			0.692255			0.346128	+	0.938187i	$0.387496\pi$
$410$		0			0
$411$	−6.00000			−0.295958
$412$		0			0
$413$		−	48.0000i		−	2.36193i
$414$		0			0
$415$	8.00000	−	16.0000i	0.392705	−	0.785409i
$416$		0			0
$417$			2.00000i			0.0979404i
$418$		0			0
$419$	36.0000			1.75872			0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000			1.75872			0.879358	+	0.476162i	$0.157972\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$			2.00000i			0.0972433i
$424$		0			0
$425$	−8.00000	+	6.00000i	−0.388057	+	0.291043i
$426$		0			0
$427$			56.0000i			2.71003i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−32.0000			−1.54139			−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000			−1.54139			−0.770693	+	0.637207i	$0.780090\pi$
$432$		0			0
$433$		−	20.0000i		−	0.961139i	−0.876957	−	0.480569i	$-0.840430\pi$
$433$		−	20.0000i		−	0.961139i	0.876957	−	0.480569i	$-0.159570\pi$
$434$		0			0
$435$	−8.00000	+	16.0000i	−0.383571	+	0.767141i
$436$		0			0
$437$		−	36.0000i		−	1.72211i
$438$		0			0
$439$	−36.0000			−1.71819			−0.859093	−	0.511819i	$-0.828972\pi$
$439$	−36.0000			−1.71819			−0.859093	+	0.511819i	$0.828972\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$		0			0
$443$		−	24.0000i		−	1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$		−	24.0000i		−	1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$		0			0
$445$	12.0000	+	6.00000i	0.568855	+	0.284427i
$446$		0			0
$447$		−	12.0000i		−	0.567581i
$448$		0			0
$449$	−38.0000			−1.79333			−0.896665	−	0.442709i	$-0.854018\pi$
$449$	−38.0000			−1.79333			−0.896665	+	0.442709i	$0.854018\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			8.00000i			0.375873i
$454$		0			0
$455$	48.0000	+	24.0000i	2.25027	+	1.12514i
$456$		0			0
$457$		−	32.0000i		−	1.49690i	−0.663193	−	0.748448i	$-0.730799\pi$
$457$		−	32.0000i		−	1.49690i	0.663193	−	0.748448i	$-0.269201\pi$
$458$		0			0
$459$	2.00000			0.0933520
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$	8.00000	−	16.0000i	0.370991	−	0.741982i
$466$		0			0
$467$		−	28.0000i		−	1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$		−	28.0000i		−	1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$	2.00000			0.0921551
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−18.0000	−	24.0000i	−0.825897	−	1.10120i
$476$		0			0
$477$		−	6.00000i		−	0.274721i
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$	60.0000			2.73576
$482$		0			0
$483$			24.0000i			1.09204i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	16.0000i		−	0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$		−	16.0000i		−	0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$		−	16.0000i		−	0.720604i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	32.0000i		−	1.43540i
$498$		0			0
$499$	42.0000			1.88018			0.940089	−	0.340929i	$-0.110742\pi$
$499$	42.0000			1.88018			0.940089	+	0.340929i	$0.110742\pi$
$500$		0			0
$501$	2.00000			0.0893534
$502$		0			0
$503$		−	30.0000i		−	1.33763i	−0.743427	−	0.668817i	$-0.766801\pi$
$503$		−	30.0000i		−	1.33763i	0.743427	−	0.668817i	$-0.233199\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	23.0000i		−	1.02147i
$508$		0			0
$509$	40.0000			1.77297			0.886484	−	0.462758i	$-0.153140\pi$
$509$	40.0000			1.77297			0.886484	+	0.462758i	$0.153140\pi$
$510$		0			0
$511$	−16.0000			−0.707798
$512$		0			0
$513$			6.00000i			0.264906i
$514$		0			0
$515$	−8.00000	+	16.0000i	−0.352522	+	0.705044i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	18.0000			0.790112
$520$		0			0
$521$	30.0000			1.31432			0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000			1.31432			0.657162	+	0.753749i	$0.271757\pi$
$522$		0			0
$523$			4.00000i			0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$			4.00000i			0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$		0			0
$525$	12.0000	+	16.0000i	0.523723	+	0.698297i
$526$		0			0
$527$			16.0000i			0.696971i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$			36.0000i			1.55933i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	16.0000i		−	0.690451i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	34.0000			1.46177			0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000			1.46177			0.730887	+	0.682498i	$0.239107\pi$
$542$		0			0
$543$			2.00000i			0.0858282i
$544$		0			0
$545$	4.00000	+	2.00000i	0.171341	+	0.0856706i
$546$		0			0
$547$		−	20.0000i		−	0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$		−	20.0000i		−	0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$		0			0
$549$	14.0000			0.597505
$550$		0			0
$551$	48.0000			2.04487
$552$		0			0
$553$			48.0000i			2.04117i
$554$		0			0
$555$	20.0000	+	10.0000i	0.848953	+	0.424476i
$556$		0			0
$557$			46.0000i			1.94908i	0.224208	+	0.974541i	$0.428020\pi$
$557$			46.0000i			1.94908i	−0.224208	+	0.974541i	$0.571980\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$			20.0000i			0.842900i	0.906852	+	0.421450i	$0.138479\pi$
$563$			20.0000i			0.842900i	−0.906852	+	0.421450i	$0.861521\pi$
$564$		0			0
$565$	10.0000	−	20.0000i	0.420703	−	0.841406i
$566$		0			0
$567$		−	4.00000i		−	0.167984i
$568$		0			0
$569$	2.00000			0.0838444			0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000			0.0838444			0.0419222	+	0.999121i	$0.486652\pi$
$570$		0			0
$571$	−2.00000			−0.0836974			−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000			−0.0836974			−0.0418487	+	0.999124i	$0.513325\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.0000	+	18.0000i	−1.00087	+	0.750652i
$576$		0			0
$577$		−	32.0000i		−	1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$		−	32.0000i		−	1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$		0			0
$579$	4.00000			0.166234
$580$		0			0
$581$	32.0000			1.32758
$582$		0			0
$583$		0			0
$584$		0			0
$585$	6.00000	−	12.0000i	0.248069	−	0.496139i
$586$		0			0
$587$			16.0000i			0.660391i	0.943913	+	0.330195i	$0.107115\pi$
$587$			16.0000i			0.660391i	−0.943913	+	0.330195i	$0.892885\pi$
$588$		0			0
$589$	−48.0000			−1.97781
$590$		0			0
$591$	22.0000			0.904959
$592$		0			0
$593$		−	6.00000i		−	0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$		−	6.00000i		−	0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$		0			0
$595$	−16.0000	−	8.00000i	−0.655936	−	0.327968i
$596$		0			0
$597$		−	20.0000i		−	0.818546i
$598$		0			0
$599$	−32.0000			−1.30748			−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000			−1.30748			−0.653742	+	0.756717i	$0.726802\pi$
$600$		0			0
$601$	−14.0000			−0.571072			−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000			−0.571072			−0.285536	+	0.958368i	$0.592172\pi$
$602$		0			0
$603$			4.00000i			0.162893i
$604$		0			0
$605$	22.0000	+	11.0000i	0.894427	+	0.447214i
$606$		0			0
$607$			4.00000i			0.162355i	0.996700	+	0.0811775i	$0.0258681\pi$
$607$			4.00000i			0.162355i	−0.996700	+	0.0811775i	$0.974132\pi$
$608$		0			0
$609$	−32.0000			−1.29671
$610$		0			0
$611$	−12.0000			−0.485468
$612$		0			0
$613$		−	26.0000i		−	1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$		−	26.0000i		−	1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$		0			0
$615$	−6.00000	+	12.0000i	−0.241943	+	0.483887i
$616$		0			0
$617$		−	38.0000i		−	1.52982i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$		−	38.0000i		−	1.52982i	0.644136	−	0.764911i	$-0.277217\pi$
$618$		0			0
$619$	−14.0000			−0.562708			−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000			−0.562708			−0.281354	+	0.959604i	$0.590783\pi$
$620$		0			0
$621$	6.00000			0.240772
$622$		0			0
$623$			24.0000i			0.961540i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−20.0000			−0.797452
$630$		0			0
$631$	−20.0000			−0.796187			−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000			−0.796187			−0.398094	+	0.917345i	$0.630328\pi$
$632$		0			0
$633$			6.00000i			0.238479i
$634$		0			0
$635$	−8.00000	+	16.0000i	−0.317470	+	0.634941i
$636$		0			0
$637$			54.0000i			2.13956i
$638$		0			0
$639$	−8.00000			−0.316475
$640$		0			0
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$			36.0000i			1.41970i	0.704352	+	0.709851i	$0.251238\pi$
$643$			36.0000i			1.41970i	−0.704352	+	0.709851i	$0.748762\pi$
$644$		0			0
$645$	8.00000	+	4.00000i	0.315000	+	0.157500i
$646$		0			0
$647$		−	30.0000i		−	1.17942i	−0.807614	−	0.589711i	$-0.799242\pi$
$647$		−	30.0000i		−	1.17942i	0.807614	−	0.589711i	$-0.200758\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	32.0000			1.25418
$652$		0			0
$653$			18.0000i			0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$			18.0000i			0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			4.00000i			0.156055i
$658$		0			0
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$	26.0000			1.01128			0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000			1.01128			0.505641	+	0.862744i	$0.331256\pi$
$662$		0			0
$663$			12.0000i			0.466041i
$664$		0			0
$665$	24.0000	−	48.0000i	0.930680	−	1.86136i
$666$		0			0
$667$		−	48.0000i		−	1.85857i
$668$		0			0
$669$	4.00000			0.154649
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	28.0000i		−	1.07932i	−0.841883	−	0.539660i	$-0.818553\pi$
$673$		−	28.0000i		−	1.07932i	0.841883	−	0.539660i	$-0.181447\pi$
$674$		0			0
$675$	4.00000	−	3.00000i	0.153960	−	0.115470i
$676$		0			0
$677$		−	26.0000i		−	0.999261i	−0.866239	−	0.499631i	$-0.833469\pi$
$677$		−	26.0000i		−	0.999261i	0.866239	−	0.499631i	$-0.166531\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−24.0000			−0.919682
$682$		0			0
$683$			4.00000i			0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$			4.00000i			0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$		0			0
$685$	6.00000	−	12.0000i	0.229248	−	0.458496i
$686$		0			0
$687$			2.00000i			0.0763048i
$688$		0			0
$689$	36.0000			1.37149
$690$		0			0
$691$	−14.0000			−0.532585			−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000			−0.532585			−0.266293	+	0.963892i	$0.585799\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.00000	−	2.00000i	−0.151729	−	0.0758643i
$696$		0			0
$697$		−	12.0000i		−	0.454532i
$698$		0			0
$699$	−6.00000			−0.226941
$700$		0			0
$701$	−16.0000			−0.604312			−0.302156	−	0.953259i	$-0.597706\pi$
$701$	−16.0000			−0.604312			−0.302156	+	0.953259i	$0.597706\pi$
$702$		0			0
$703$		−	60.0000i		−	2.26294i
$704$		0			0
$705$	−4.00000	−	2.00000i	−0.150649	−	0.0753244i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−30.0000			−1.12667			−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000			−1.12667			−0.563337	+	0.826227i	$0.690483\pi$
$710$		0			0
$711$	12.0000			0.450035
$712$		0			0
$713$			48.0000i			1.79761i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		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
$721$	−32.0000			−1.19174
$722$		0			0
$723$		−	26.0000i		−	0.966950i
$724$		0			0
$725$	−24.0000	−	32.0000i	−0.891338	−	1.18845i
$726$		0			0
$727$			24.0000i			0.890111i	0.895503	+	0.445055i	$0.146816\pi$
$727$			24.0000i			0.890111i	−0.895503	+	0.445055i	$0.853184\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−8.00000			−0.295891
$732$		0			0
$733$		−	34.0000i		−	1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$		−	34.0000i		−	1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$		0			0
$735$	−9.00000	+	18.0000i	−0.331970	+	0.663940i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	10.0000			0.367856			0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000			0.367856			0.183928	+	0.982940i	$0.441119\pi$
$740$		0			0
$741$	−36.0000			−1.32249
$742$		0			0
$743$		−	30.0000i		−	1.10059i	−0.834969	−	0.550297i	$-0.814515\pi$
$743$		−	30.0000i		−	1.10059i	0.834969	−	0.550297i	$-0.185485\pi$
$744$		0			0
$745$	24.0000	+	12.0000i	0.879292	+	0.439646i
$746$		0			0
$747$		−	8.00000i		−	0.292705i
$748$		0			0
$749$		0			0
$750$		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
$753$			4.00000i			0.145768i
$754$		0			0
$755$	−16.0000	−	8.00000i	−0.582300	−	0.291150i
$756$		0			0
$757$			34.0000i			1.23575i	0.786276	+	0.617876i	$0.212006\pi$
$757$			34.0000i			1.23575i	−0.786276	+	0.617876i	$0.787994\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−50.0000			−1.81250			−0.906249	−	0.422744i	$-0.861067\pi$
$761$	−50.0000			−1.81250			−0.906249	+	0.422744i	$0.861067\pi$
$762$		0			0
$763$			8.00000i			0.289619i
$764$		0			0
$765$	−2.00000	+	4.00000i	−0.0723102	+	0.144620i
$766$		0			0
$767$		−	72.0000i		−	2.59977i
$768$		0			0
$769$	−30.0000			−1.08183			−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000			−1.08183			−0.540914	+	0.841078i	$0.681921\pi$
$770$		0			0
$771$	−22.0000			−0.792311
$772$		0			0
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$		0			0
$775$	24.0000	+	32.0000i	0.862105	+	1.14947i
$776$		0			0
$777$			40.0000i			1.43499i
$778$		0			0
$779$	36.0000			1.28983
$780$		0			0
$781$		0			0
$782$		0			0
$783$			8.00000i			0.285897i
$784$		0			0
$785$	−2.00000	+	4.00000i	−0.0713831	+	0.142766i
$786$		0			0
$787$			28.0000i			0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$			28.0000i			0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$		0			0
$789$	−30.0000			−1.06803
$790$		0			0
$791$	40.0000			1.42224
$792$		0			0
$793$			84.0000i			2.98293i
$794$		0			0
$795$	12.0000	+	6.00000i	0.425596	+	0.212798i
$796$		0			0
$797$			38.0000i			1.34603i	0.739629	+	0.673015i	$0.235001\pi$
$797$			38.0000i			1.34603i	−0.739629	+	0.673015i	$0.764999\pi$
$798$		0			0
$799$	4.00000			0.141510
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−48.0000	−	24.0000i	−1.69178	−	0.845889i
$806$		0			0
$807$		−	32.0000i		−	1.12645i
$808$		0			0
$809$	−18.0000			−0.632846			−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000			−0.632846			−0.316423	+	0.948618i	$0.602482\pi$
$810$		0			0
$811$	26.0000			0.912983			0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000			0.912983			0.456492	+	0.889728i	$0.349106\pi$
$812$		0			0
$813$			20.0000i			0.701431i
$814$		0			0
$815$	4.00000	−	8.00000i	0.140114	−	0.280228i
$816$		0			0
$817$		−	24.0000i		−	0.839654i
$818$		0			0
$819$	24.0000			0.838628
$820$		0			0
$821$	−24.0000			−0.837606			−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000			−0.837606			−0.418803	+	0.908077i	$0.637550\pi$
$822$		0			0
$823$		−	12.0000i		−	0.418294i	−0.977884	−	0.209147i	$-0.932931\pi$
$823$		−	12.0000i		−	0.418294i	0.977884	−	0.209147i	$-0.0670687\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	36.0000i		−	1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$		−	36.0000i		−	1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$		0			0
$829$	6.00000			0.208389			0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000			0.208389			0.104194	+	0.994557i	$0.466774\pi$
$830$		0			0
$831$	−10.0000			−0.346896
$832$		0			0
$833$		−	18.0000i		−	0.623663i
$834$		0			0
$835$	−2.00000	+	4.00000i	−0.0692129	+	0.138426i
$836$		0			0
$837$		−	8.00000i		−	0.276520i
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		0			0
$843$			22.0000i			0.757720i
$844$		0			0
$845$	46.0000	+	23.0000i	1.58245	+	0.791224i
$846$		0			0
$847$			44.0000i			1.51186i
$848$		0			0
$849$	20.0000			0.686398
$850$		0			0
$851$	−60.0000			−2.05677
$852$		0			0
$853$			14.0000i			0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$			14.0000i			0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$		0			0
$855$	−12.0000	−	6.00000i	−0.410391	−	0.205196i
$856$		0			0
$857$			18.0000i			0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$			18.0000i			0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$		0			0
$859$	−46.0000			−1.56950			−0.784750	−	0.619813i	$-0.787209\pi$
$859$	−46.0000			−1.56950			−0.784750	+	0.619813i	$0.787209\pi$
$860$		0			0
$861$	−24.0000			−0.817918
$862$		0			0
$863$			18.0000i			0.612727i	0.951915	+	0.306364i	$0.0991123\pi$
$863$			18.0000i			0.612727i	−0.951915	+	0.306364i	$0.900888\pi$
$864$		0			0
$865$	−18.0000	+	36.0000i	−0.612018	+	1.22404i
$866$		0			0
$867$			13.0000i			0.441503i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−24.0000			−0.813209
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−44.0000	+	8.00000i	−1.48747	+	0.270449i
$876$		0			0
$877$		−	2.00000i		−	0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$		−	2.00000i		−	0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$		0			0
$879$	−10.0000			−0.337292
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$		−	36.0000i		−	1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$		−	36.0000i		−	1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$		0			0
$885$	12.0000	−	24.0000i	0.403376	−	0.806751i
$886$		0			0
$887$			22.0000i			0.738688i	0.929293	+	0.369344i	$0.120418\pi$
$887$			22.0000i			0.738688i	−0.929293	+	0.369344i	$0.879582\pi$
$888$		0			0
$889$	−32.0000			−1.07325
$890$		0			0
$891$		0			0
$892$		0			0
$893$			12.0000i			0.401565i
$894$		0			0
$895$	32.0000	+	16.0000i	1.06964	+	0.534821i
$896$		0			0
$897$			36.0000i			1.20201i
$898$		0			0
$899$	−64.0000			−2.13452
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$			16.0000i			0.532447i
$904$		0			0
$905$	−4.00000	−	2.00000i	−0.132964	−	0.0664822i
$906$		0			0
$907$			44.0000i			1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$			44.0000i			1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	40.0000			1.32526			0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000			1.32526			0.662630	+	0.748947i	$0.269440\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−14.0000	+	28.0000i	−0.462826	+	0.925651i
$916$		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
$921$	−12.0000			−0.395413
$922$		0			0
$923$		−	48.0000i		−	1.57994i
$924$		0			0
$925$	−40.0000	+	30.0000i	−1.31519	+	0.986394i
$926$		0			0
$927$			8.00000i			0.262754i
$928$		0			0
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$	54.0000			1.76978
$932$		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
$939$	−4.00000			−0.130535
$940$		0			0
$941$	28.0000			0.912774			0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000			0.912774			0.456387	+	0.889781i	$0.349143\pi$
$942$		0			0
$943$		−	36.0000i		−	1.17232i
$944$		0			0
$945$	8.00000	+	4.00000i	0.260240	+	0.130120i
$946$		0			0
$947$		−	48.0000i		−	1.55979i	−0.625910	−	0.779895i	$-0.715272\pi$
$947$		−	48.0000i		−	1.55979i	0.625910	−	0.779895i	$-0.284728\pi$
$948$		0			0
$949$	−24.0000			−0.779073
$950$		0			0
$951$	18.0000			0.583690
$952$		0			0
$953$			18.0000i			0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$			18.0000i			0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	24.0000			0.775000
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.00000	+	8.00000i	−0.128765	+	0.257529i
$966$		0			0
$967$			52.0000i			1.67221i	0.548572	+	0.836104i	$0.315172\pi$
$967$			52.0000i			1.67221i	−0.548572	+	0.836104i	$0.684828\pi$
$968$		0			0
$969$	12.0000			0.385496
$970$		0			0
$971$	20.0000			0.641831			0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000			0.641831			0.320915	+	0.947108i	$0.396010\pi$
$972$		0			0
$973$		−	8.00000i		−	0.256468i
$974$		0			0
$975$	18.0000	+	24.0000i	0.576461	+	0.768615i
$976$		0			0
$977$			18.0000i			0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$			18.0000i			0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	2.00000			0.0638551
$982$		0			0
$983$		−	6.00000i		−	0.191370i	−0.995412	−	0.0956851i	$-0.969496\pi$
$983$		−	6.00000i		−	0.191370i	0.995412	−	0.0956851i	$-0.0305042\pi$
$984$		0			0
$985$	−22.0000	+	44.0000i	−0.700978	+	1.40196i
$986$		0			0
$987$		−	8.00000i		−	0.254643i
$988$		0			0
$989$	−24.0000			−0.763156
$990$		0			0
$991$	44.0000			1.39771			0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000			1.39771			0.698853	+	0.715265i	$0.253694\pi$
$992$		0			0
$993$		−	10.0000i		−	0.317340i
$994$		0			0
$995$	40.0000	+	20.0000i	1.26809	+	0.634043i
$996$		0			0
$997$			6.00000i			0.190022i	0.995476	+	0.0950110i	$0.0302886\pi$
$997$			6.00000i			0.190022i	−0.995476	+	0.0950110i	$0.969711\pi$
$998$		0			0
$999$	10.0000			0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.f.c.769.2	yes	2
4.3	odd	2		1920.2.f.b.769.1	✓	2
5.2	odd	4		9600.2.a.ca.1.1		1
5.3	odd	4		9600.2.a.d.1.1		1
5.4	even	2	inner	1920.2.f.c.769.1	yes	2
8.3	odd	2		1920.2.f.j.769.2	yes	2
8.5	even	2		1920.2.f.k.769.1	yes	2
16.3	odd	4		3840.2.d.w.2689.2		2
16.5	even	4		3840.2.d.z.2689.1		2
16.11	odd	4		3840.2.d.i.2689.1		2
16.13	even	4		3840.2.d.h.2689.2		2
20.3	even	4		9600.2.a.cb.1.1		1
20.7	even	4		9600.2.a.c.1.1		1
20.19	odd	2		1920.2.f.b.769.2	yes	2
40.3	even	4		9600.2.a.ba.1.1		1
40.13	odd	4		9600.2.a.bc.1.1		1
40.19	odd	2		1920.2.f.j.769.1	yes	2
40.27	even	4		9600.2.a.bd.1.1		1
40.29	even	2		1920.2.f.k.769.2	yes	2
40.37	odd	4		9600.2.a.bb.1.1		1
80.19	odd	4		3840.2.d.i.2689.2		2
80.29	even	4		3840.2.d.z.2689.2		2
80.59	odd	4		3840.2.d.w.2689.1		2
80.69	even	4		3840.2.d.h.2689.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.f.b.769.1	✓	2	4.3	odd	2
1920.2.f.b.769.2	yes	2	20.19	odd	2
1920.2.f.c.769.1	yes	2	5.4	even	2	inner
1920.2.f.c.769.2	yes	2	1.1	even	1	trivial
1920.2.f.j.769.1	yes	2	40.19	odd	2
1920.2.f.j.769.2	yes	2	8.3	odd	2
1920.2.f.k.769.1	yes	2	8.5	even	2
1920.2.f.k.769.2	yes	2	40.29	even	2
3840.2.d.h.2689.1		2	80.69	even	4
3840.2.d.h.2689.2		2	16.13	even	4
3840.2.d.i.2689.1		2	16.11	odd	4
3840.2.d.i.2689.2		2	80.19	odd	4
3840.2.d.w.2689.1		2	80.59	odd	4
3840.2.d.w.2689.2		2	16.3	odd	4
3840.2.d.z.2689.1		2	16.5	even	4
3840.2.d.z.2689.2		2	80.29	even	4
9600.2.a.c.1.1		1	20.7	even	4
9600.2.a.d.1.1		1	5.3	odd	4
9600.2.a.ba.1.1		1	40.3	even	4
9600.2.a.bb.1.1		1	40.37	odd	4
9600.2.a.bc.1.1		1	40.13	odd	4
9600.2.a.bd.1.1		1	40.27	even	4
9600.2.a.ca.1.1		1	5.2	odd	4
9600.2.a.cb.1.1		1	20.3	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:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} -6.00000i q^{13} +(1.00000 - 2.00000i) q^{15} +2.00000i q^{17} -6.00000 q^{19} +4.00000 q^{21} +6.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000i q^{27} -8.00000 q^{29} +8.00000 q^{31} +(-4.00000 + 8.00000i) q^{35} +10.0000i q^{37} +6.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +(2.00000 + 1.00000i) q^{45} -2.00000i q^{47} -9.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -6.00000i q^{57} +12.0000 q^{59} -14.0000 q^{61} +4.00000i q^{63} +(-6.00000 + 12.0000i) q^{65} -4.00000i q^{67} -6.00000 q^{69} +8.00000 q^{71} -4.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} -12.0000 q^{79} +1.00000 q^{81} +8.00000i q^{83} +(2.00000 - 4.00000i) q^{85} -8.00000i q^{87} -6.00000 q^{89} -24.0000 q^{91} +8.00000i q^{93} +(12.0000 + 6.00000i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^5 - 2 * q^9	\( 2 q - 4 q^{5} - 2 q^{9} + 2 q^{15} - 12 q^{19} + 8 q^{21} + 6 q^{25} - 16 q^{29} + 16 q^{31} - 8 q^{35} + 12 q^{39} - 12 q^{41} + 4 q^{45} - 18 q^{49} - 4 q^{51} + 24 q^{59} - 28 q^{61} - 12 q^{65} - 12 q^{69} + 16 q^{71} - 8 q^{75} - 24 q^{79} + 2 q^{81} + 4 q^{85} - 12 q^{89} - 48 q^{91} + 24 q^{95}+O(q^{100}) \) 2 * q - 4 * q^5 - 2 * q^9 + 2 * q^15 - 12 * q^19 + 8 * q^21 + 6 * q^25 - 16 * q^29 + 16 * q^31 - 8 * q^35 + 12 * q^39 - 12 * q^41 + 4 * q^45 - 18 * q^49 - 4 * q^51 + 24 * q^59 - 28 * q^61 - 12 * q^65 - 12 * q^69 + 16 * q^71 - 8 * q^75 - 24 * q^79 + 2 * q^81 + 4 * q^85 - 12 * q^89 - 48 * q^91 + 24 * q^95

Introduction

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