LMFDB - Embedded newform 3840.2.k.y.1921.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(1921,3840)]

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, 0]))

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

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

chi := DirichletCharacter("3840.1921");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.k (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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1921.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3840.1921
Dual form			3840.2.k.y.1921.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} +2.00000i q^{13} -1.00000 q^{15} +6.00000 q^{17} -4.00000i q^{19} +4.00000i q^{21} -1.00000 q^{25} -1.00000i q^{27} -6.00000i q^{29} +8.00000 q^{31} +4.00000i q^{35} -2.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} +4.00000i q^{43} -1.00000i q^{45} +9.00000 q^{49} +6.00000i q^{51} +6.00000i q^{53} +4.00000 q^{57} -10.0000i q^{61} -4.00000 q^{63} -2.00000 q^{65} -4.00000i q^{67} -2.00000 q^{73} -1.00000i q^{75} +8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +6.00000i q^{85} +6.00000 q^{87} -18.0000 q^{89} +8.00000i q^{91} +8.00000i q^{93} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 8 * q^7 - 2 * q^9	$ 2 q + 8 q^{7} - 2 q^{9} - 2 q^{15} + 12 q^{17} - 2 q^{25} + 16 q^{31} - 4 q^{39} + 12 q^{41} + 18 q^{49} + 8 q^{57} - 8 q^{63} - 4 q^{65} - 4 q^{73} + 16 q^{79} + 2 q^{81} + 12 q^{87} - 36 q^{89} + 8 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q + 8 * q^7 - 2 * q^9 - 2 * q^15 + 12 * q^17 - 2 * q^25 + 16 * q^31 - 4 * q^39 + 12 * q^41 + 18 * q^49 + 8 * q^57 - 8 * q^63 - 4 * q^65 - 4 * q^73 + 16 * q^79 + 2 * q^81 + 12 * q^87 - 36 * q^89 + 8 * q^95 + 4 * q^97

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	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
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	4.00000i	0.872872i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 6.00000i	− 1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$	− 6.00000i	− 1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	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
$35$	4.00000i	0.676123i
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	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
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	6.00000i	0.840168i
$52$	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
$57$	4.00000	0.529813
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	− 10.0000i	− 1.28037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	− 10.0000i	− 1.28037i	0.768221	−	0.640184i	$-0.221142\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	0	0
$65$	−2.00000	−0.248069
$66$	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
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	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
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	6.00000i	0.650791i
$86$	0	0
$87$	6.00000	0.643268
$88$	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
$91$	8.00000i	0.838628i
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 18.0000i	− 1.79107i	−0.444994	−	0.895533i	$-0.646794\pi$
$101$	− 18.0000i	− 1.79107i	0.444994	−	0.895533i	$-0.353206\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	− 10.0000i	− 0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$	− 10.0000i	− 0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	24.0000	2.20008
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	6.00000i	0.541002i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	− 16.0000i	− 1.38738i
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	4.00000i	0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$	4.00000i	0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	9.00000i	0.742307i
$148$	0	0
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	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
$153$	−6.00000	−0.485071
$154$	0	0
$155$	8.00000i	0.642575i
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.0000i	1.79384i	0.442189	+	0.896922i	$0.354202\pi$
$179$	24.0000i	1.79384i	−0.442189	+	0.896922i	$0.645798\pi$
$180$	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
$183$	10.0000	0.739221
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 4.00000i	− 0.290957i
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	− 2.00000i	− 0.143223i
$196$	0	0
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	− 24.0000i	− 1.68447i
$204$	0	0
$205$	6.00000i	0.419058i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.0000i	1.37686i	0.725304	+	0.688428i	$0.241699\pi$
$211$	20.0000i	1.37686i	−0.725304	+	0.688428i	$0.758301\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	32.0000	2.17230
$218$	0	0
$219$	− 2.00000i	− 0.135147i
$220$	0	0
$221$	12.0000i	0.807207i
$222$	0	0
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	10.0000i	0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$	10.0000i	0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000i	0.519656i
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	9.00000i	0.574989i
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	24.0000i	1.51487i	0.652913	+	0.757433i	$0.273547\pi$
$251$	24.0000i	1.51487i	−0.652913	+	0.757433i	$0.726453\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−6.00000	−0.375735
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	− 8.00000i	− 0.497096i
$260$	0	0
$261$	6.00000i	0.371391i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	− 18.0000i	− 1.10158i
$268$	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
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	28.0000i	1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$	28.0000i	1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$	0	0
$285$	4.00000i	0.236940i
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	2.00000i	0.117242i
$292$	0	0
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	16.0000i	0.922225i
$302$	0	0
$303$	18.0000	1.03407
$304$	0	0
$305$	10.0000	0.572598
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	4.00000i	0.227552i
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	0	0
$315$	− 4.00000i	− 0.225374i
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	− 2.00000i	− 0.110940i
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	0	0
$330$	0	0
$331$	28.0000i	1.53902i	0.638635	+	0.769510i	$0.279499\pi$
$331$	28.0000i	1.53902i	−0.638635	+	0.769510i	$0.720501\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	− 18.0000i	− 0.977626i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	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
$349$	− 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
$351$	2.00000	0.106752
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	24.0000i	1.27021i
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	11.0000i	0.577350i
$364$	0	0
$365$	− 2.00000i	− 0.104685i
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	24.0000i	1.24602i
$372$	0	0
$373$	− 26.0000i	− 1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$	− 26.0000i	− 1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	4.00000i	0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$	4.00000i	0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	0	0
$381$	20.0000i	1.02463i
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.00000i	− 0.203331i
$388$	0	0
$389$	6.00000i	0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$	6.00000i	0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.00000i	0.402524i
$396$	0	0
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	0	0
$399$	16.0000	0.801002
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	16.0000i	0.797017i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	− 6.00000i	− 0.295958i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	− 40.0000i	− 1.93574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	6.00000i	0.287678i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	− 18.0000i	− 0.853282i
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 8.00000i	− 0.375873i
$454$	0	0
$455$	−8.00000	−0.375046
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	0	0
$459$	− 6.00000i	− 0.280056i
$460$	0	0
$461$	− 30.0000i	− 1.39724i	−0.715493	−	0.698620i	$-0.753798\pi$
$461$	− 30.0000i	− 1.39724i	0.715493	−	0.698620i	$-0.246202\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	0	0
$469$	− 16.0000i	− 0.738811i
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.00000i	0.183533i
$476$	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
$481$	4.00000	0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.00000i	0.0908153i
$486$	0	0
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	− 24.0000i	− 1.08310i	−0.840667	−	0.541552i	$-0.817837\pi$
$491$	− 24.0000i	− 1.08310i	0.840667	−	0.541552i	$-0.182163\pi$
$492$	0	0
$493$	− 36.0000i	− 1.62136i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 4.00000i	− 0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$	− 4.00000i	− 0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	0	0
$507$	9.00000i	0.399704i
$508$	0	0
$509$	− 6.00000i	− 0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$	− 6.00000i	− 0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	4.00000i	0.176261i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	0	0
$525$	− 4.00000i	− 0.174574i
$526$	0	0
$527$	48.0000	2.09091
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	−24.0000	−1.03568
$538$	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
$543$	14.0000	0.600798
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	0	0
$549$	10.0000i	0.426790i
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	32.0000	1.36078
$554$	0	0
$555$	2.00000i	0.0848953i
$556$	0	0
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	− 18.0000i	− 0.757266i
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	− 20.0000i	− 0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$	− 20.0000i	− 0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$	0	0
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	− 22.0000i	− 0.914289i
$580$	0	0
$581$	48.0000i	1.99138i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	− 32.0000i	− 1.31854i
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	24.0000i	0.983904i
$596$	0	0
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	11.0000i	0.447214i
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	24.0000	0.972529
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	− 44.0000i	− 1.76851i	−0.467005	−	0.884255i	$-0.654667\pi$
$619$	− 44.0000i	− 1.76851i	0.467005	−	0.884255i	$-0.345333\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−72.0000	−2.88462
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 12.0000i	− 0.478471i
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	20.0000i	0.793676i
$636$	0	0
$637$	18.0000i	0.713186i
$638$	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
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	− 4.00000i	− 0.157500i
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	32.0000i	1.25418i
$652$	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
$657$	2.00000	0.0780274
$658$	0	0
$659$	48.0000i	1.86981i	0.354892	+	0.934907i	$0.384518\pi$
$659$	48.0000i	1.86981i	−0.354892	+	0.934907i	$0.615482\pi$
$660$	0	0
$661$	− 14.0000i	− 0.544537i	−0.962221	−	0.272268i	$-0.912226\pi$
$661$	− 14.0000i	− 0.544537i	0.962221	−	0.272268i	$-0.0877739\pi$
$662$	0	0
$663$	−12.0000	−0.466041
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	0	0
$668$	0	0
$669$	20.0000i	0.773245i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	− 6.00000i	− 0.229248i
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	44.0000i	1.67384i	0.547326	+	0.836919i	$0.315646\pi$
$691$	44.0000i	1.67384i	−0.547326	+	0.836919i	$0.684354\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	36.0000	1.36360
$698$	0	0
$699$	18.0000i	0.680823i
$700$	0	0
$701$	− 6.00000i	− 0.226617i	−0.993560	−	0.113308i	$-0.963855\pi$
$701$	− 6.00000i	− 0.226617i	0.993560	−	0.113308i	$-0.0361448\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 72.0000i	− 2.70784i
$708$	0	0
$709$	− 38.0000i	− 1.42712i	−0.700594	−	0.713560i	$-0.747082\pi$
$709$	− 38.0000i	− 1.42712i	0.700594	−	0.713560i	$-0.252918\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.0000i	0.896296i
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	2.00000i	0.0743808i
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.0000i	0.887672i
$732$	0	0
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	0	0
$735$	−9.00000	−0.331970
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 52.0000i	− 1.91285i	−0.291977	−	0.956425i	$-0.594313\pi$
$739$	− 52.0000i	− 1.91285i	0.291977	−	0.956425i	$-0.405687\pi$
$740$	0	0
$741$	8.00000i	0.293887i
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	48.0000i	1.75388i
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	−24.0000	−0.874609
$754$	0	0
$755$	− 8.00000i	− 0.291150i
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	− 40.0000i	− 1.44810i
$764$	0	0
$765$	− 6.00000i	− 0.216930i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	− 18.0000i	− 0.648254i
$772$	0	0
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−72.0000	−2.56003
$792$	0	0
$793$	20.0000	0.710221
$794$	0	0
$795$	− 6.00000i	− 0.212798i
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	0	0
$811$	4.00000i	0.140459i	0.997531	+	0.0702295i	$0.0223732\pi$
$811$	4.00000i	0.140459i	−0.997531	+	0.0702295i	$0.977627\pi$
$812$	0	0
$813$	− 16.0000i	− 0.561144i
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	− 8.00000i	− 0.279543i
$820$	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
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	38.0000i	1.31979i	0.751356	+	0.659897i	$0.229400\pi$
$829$	38.0000i	1.31979i	−0.751356	+	0.659897i	$0.770600\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	54.0000	1.87099
$834$	0	0
$835$	0	0
$836$	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
$841$	−7.00000	−0.241379
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	9.00000i	0.309609i
$846$	0	0
$847$	44.0000	1.51186
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	0	0
$852$	0	0
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	4.00000i	0.136478i	0.997669	+	0.0682391i	$0.0217381\pi$
$859$	4.00000i	0.136478i	−0.997669	+	0.0682391i	$0.978262\pi$
$860$	0	0
$861$	24.0000i	0.817918i
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	19.0000i	0.645274i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	− 4.00000i	− 0.135225i
$876$	0	0
$877$	2.00000i	0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$	2.00000i	0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	0	0
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	80.0000	2.68311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 48.0000i	− 1.60089i
$900$	0	0
$901$	36.0000i	1.19933i
$902$	0	0
$903$	−16.0000	−0.532447
$904$	0	0
$905$	14.0000	0.465376
$906$	0	0
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	0	0
$909$	18.0000i	0.597022i
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	10.0000i	0.330590i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	0	0
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	− 36.0000i	− 1.17985i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	− 2.00000i	− 0.0652675i
$940$	0	0
$941$	18.0000i	0.586783i	0.955992	+	0.293392i	$0.0947840\pi$
$941$	18.0000i	0.586783i	−0.955992	+	0.293392i	$0.905216\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	0	0
$949$	− 4.00000i	− 0.129845i
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	− 24.0000i	− 0.776622i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−24.0000	−0.775000
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	0	0
$965$	− 22.0000i	− 0.708205i
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	− 24.0000i	− 0.770197i	−0.922876	−	0.385098i	$-0.874168\pi$
$971$	− 24.0000i	− 0.770197i	0.922876	−	0.385098i	$-0.125832\pi$
$972$	0	0
$973$	16.0000i	0.512936i
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	10.0000i	0.319275i
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	0	0
$995$	− 8.00000i	− 0.253617i
$996$	0	0
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.k.y.1921.2		2
4.3	odd	2		3840.2.k.f.1921.1		2
8.3	odd	2		3840.2.k.f.1921.2		2
8.5	even	2	inner	3840.2.k.y.1921.1		2
16.3	odd	4		960.2.a.p.1.1		1
16.5	even	4		30.2.a.a.1.1	✓	1
16.11	odd	4		240.2.a.b.1.1		1
16.13	even	4		960.2.a.e.1.1		1
48.5	odd	4		90.2.a.c.1.1		1
48.11	even	4		720.2.a.j.1.1		1
48.29	odd	4		2880.2.a.a.1.1		1
48.35	even	4		2880.2.a.q.1.1		1
80.3	even	4		4800.2.f.w.3649.2		2
80.13	odd	4		4800.2.f.p.3649.1		2
80.19	odd	4		4800.2.a.d.1.1		1
80.27	even	4		1200.2.f.e.49.2		2
80.29	even	4		4800.2.a.cq.1.1		1
80.37	odd	4		150.2.c.a.49.1		2
80.43	even	4		1200.2.f.e.49.1		2
80.53	odd	4		150.2.c.a.49.2		2
80.59	odd	4		1200.2.a.k.1.1		1
80.67	even	4		4800.2.f.w.3649.1		2
80.69	even	4		150.2.a.b.1.1		1
80.77	odd	4		4800.2.f.p.3649.2		2
112.5	odd	12		1470.2.i.q.361.1		2
112.37	even	12		1470.2.i.o.361.1		2
112.53	even	12		1470.2.i.o.961.1		2
112.69	odd	4		1470.2.a.d.1.1		1
112.101	odd	12		1470.2.i.q.961.1		2
144.5	odd	12		810.2.e.b.541.1		2
144.85	even	12		810.2.e.l.541.1		2
144.101	odd	12		810.2.e.b.271.1		2
144.133	even	12		810.2.e.l.271.1		2
176.21	odd	4		3630.2.a.w.1.1		1
208.5	odd	4		5070.2.b.k.1351.2		2
208.21	odd	4		5070.2.b.k.1351.1		2
208.181	even	4		5070.2.a.w.1.1		1
240.53	even	4		450.2.c.b.199.1		2
240.59	even	4		3600.2.a.f.1.1		1
240.107	odd	4		3600.2.f.i.2449.2		2
240.149	odd	4		450.2.a.d.1.1		1
240.197	even	4		450.2.c.b.199.2		2
240.203	odd	4		3600.2.f.i.2449.1		2
272.101	even	4		8670.2.a.g.1.1		1
336.293	even	4		4410.2.a.z.1.1		1
560.69	odd	4		7350.2.a.ct.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.2.a.a.1.1	✓	1	16.5	even	4
90.2.a.c.1.1		1	48.5	odd	4
150.2.a.b.1.1		1	80.69	even	4
150.2.c.a.49.1		2	80.37	odd	4
150.2.c.a.49.2		2	80.53	odd	4
240.2.a.b.1.1		1	16.11	odd	4
450.2.a.d.1.1		1	240.149	odd	4
450.2.c.b.199.1		2	240.53	even	4
450.2.c.b.199.2		2	240.197	even	4
720.2.a.j.1.1		1	48.11	even	4
810.2.e.b.271.1		2	144.101	odd	12
810.2.e.b.541.1		2	144.5	odd	12
810.2.e.l.271.1		2	144.133	even	12
810.2.e.l.541.1		2	144.85	even	12
960.2.a.e.1.1		1	16.13	even	4
960.2.a.p.1.1		1	16.3	odd	4
1200.2.a.k.1.1		1	80.59	odd	4
1200.2.f.e.49.1		2	80.43	even	4
1200.2.f.e.49.2		2	80.27	even	4
1470.2.a.d.1.1		1	112.69	odd	4
1470.2.i.o.361.1		2	112.37	even	12
1470.2.i.o.961.1		2	112.53	even	12
1470.2.i.q.361.1		2	112.5	odd	12
1470.2.i.q.961.1		2	112.101	odd	12
2880.2.a.a.1.1		1	48.29	odd	4
2880.2.a.q.1.1		1	48.35	even	4
3600.2.a.f.1.1		1	240.59	even	4
3600.2.f.i.2449.1		2	240.203	odd	4
3600.2.f.i.2449.2		2	240.107	odd	4
3630.2.a.w.1.1		1	176.21	odd	4
3840.2.k.f.1921.1		2	4.3	odd	2
3840.2.k.f.1921.2		2	8.3	odd	2
3840.2.k.y.1921.1		2	8.5	even	2	inner
3840.2.k.y.1921.2		2	1.1	even	1	trivial
4410.2.a.z.1.1		1	336.293	even	4
4800.2.a.d.1.1		1	80.19	odd	4
4800.2.a.cq.1.1		1	80.29	even	4
4800.2.f.p.3649.1		2	80.13	odd	4
4800.2.f.p.3649.2		2	80.77	odd	4
4800.2.f.w.3649.1		2	80.67	even	4
4800.2.f.w.3649.2		2	80.3	even	4
5070.2.a.w.1.1		1	208.181	even	4
5070.2.b.k.1351.1		2	208.21	odd	4
5070.2.b.k.1351.2		2	208.5	odd	4
7350.2.a.ct.1.1		1	560.69	odd	4
8670.2.a.g.1.1		1	272.101	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} +2.00000i q^{13} -1.00000 q^{15} +6.00000 q^{17} -4.00000i q^{19} +4.00000i q^{21} -1.00000 q^{25} -1.00000i q^{27} -6.00000i q^{29} +8.00000 q^{31} +4.00000i q^{35} -2.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} +4.00000i q^{43} -1.00000i q^{45} +9.00000 q^{49} +6.00000i q^{51} +6.00000i q^{53} +4.00000 q^{57} -10.0000i q^{61} -4.00000 q^{63} -2.00000 q^{65} -4.00000i q^{67} -2.00000 q^{73} -1.00000i q^{75} +8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +6.00000i q^{85} +6.00000 q^{87} -18.0000 q^{89} +8.00000i q^{91} +8.00000i q^{93} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 8 * q^7 - 2 * q^9	\( 2 q + 8 q^{7} - 2 q^{9} - 2 q^{15} + 12 q^{17} - 2 q^{25} + 16 q^{31} - 4 q^{39} + 12 q^{41} + 18 q^{49} + 8 q^{57} - 8 q^{63} - 4 q^{65} - 4 q^{73} + 16 q^{79} + 2 q^{81} + 12 q^{87} - 36 q^{89} + 8 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q + 8 * q^7 - 2 * q^9 - 2 * q^15 + 12 * q^17 - 2 * q^25 + 16 * q^31 - 4 * q^39 + 12 * q^41 + 18 * q^49 + 8 * q^57 - 8 * q^63 - 4 * q^65 - 4 * q^73 + 16 * q^79 + 2 * q^81 + 12 * q^87 - 36 * q^89 + 8 * q^95 + 4 * q^97

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