LMFDB - Embedded newform 4056.2.c.h.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(337,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, 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("4056.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.c (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:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4056.337
Dual form			4056.2.c.h.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.00000i q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.00000i q^{5} +1.00000 q^{9} +2.00000i q^{15} -2.00000 q^{17} -4.00000i q^{19} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +2.00000i q^{37} +6.00000i q^{41} +12.0000 q^{43} +2.00000i q^{45} +4.00000i q^{47} +7.00000 q^{49} -2.00000 q^{51} +6.00000 q^{53} -4.00000i q^{57} +8.00000i q^{59} -2.00000 q^{61} +4.00000i q^{67} -12.0000i q^{71} +14.0000i q^{73} +1.00000 q^{75} +1.00000 q^{81} +8.00000i q^{83} -4.00000i q^{85} +6.00000 q^{87} +18.0000i q^{89} +8.00000 q^{95} -6.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{9} - 4 q^{17} + 2 q^{25} + 2 q^{27} + 12 q^{29} + 24 q^{43} + 14 q^{49} - 4 q^{51} + 12 q^{53} - 4 q^{61} + 2 q^{75} + 2 q^{81} + 12 q^{87} + 16 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 - 4 * q^17 + 2 * q^25 + 2 * q^27 + 12 * q^29 + 24 * q^43 + 14 * q^49 - 4 * q^51 + 12 * q^53 - 4 * q^61 + 2 * q^75 + 2 * q^81 + 12 * q^87 + 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$.

$n$	$1015$	$2029$	$2705$	$3889$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.00000i	0.516398i
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	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
$21$	0	0
$22$	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
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000i	0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$	6.00000i	0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	0	0
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	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
$49$	7.00000	1.00000
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 4.00000i	− 0.529813i
$58$	0	0
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 12.0000i	− 1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$	− 12.0000i	− 1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	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
$85$	− 4.00000i	− 0.433861i
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	18.0000i	1.90800i	0.299813	+	0.953998i	$0.403076\pi$
$89$	18.0000i	1.90800i	−0.299813	+	0.953998i	$0.596924\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	− 6.00000i	− 0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	0	0
$111$	2.00000i	0.189832i
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	6.00000i	0.541002i
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.00000i	0.172133i
$136$	0	0
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	4.00000i	0.336861i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	12.0000i	0.996546i
$146$	0	0
$147$	7.00000	0.577350
$148$	0	0
$149$	10.0000i	0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$	10.0000i	0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$	0	0
$151$	− 8.00000i	− 0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$	− 8.00000i	− 0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.00000i	0.601317i
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	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
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	4.00000i	0.282138i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	− 12.0000i	− 0.822226i
$214$	0	0
$215$	24.0000i	1.63679i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	14.0000i	0.946032i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	26.0000i	1.71813i	0.511868	+	0.859064i	$0.328954\pi$
$229$	26.0000i	1.71813i	−0.511868	+	0.859064i	$0.671046\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 20.0000i	− 1.29369i	−0.762620	−	0.646846i	$-0.776088\pi$
$239$	− 20.0000i	− 1.29369i	0.762620	−	0.646846i	$-0.223912\pi$
$240$	0	0
$241$	− 26.0000i	− 1.67481i	−0.546585	−	0.837404i	$-0.684072\pi$
$241$	− 26.0000i	− 1.67481i	0.546585	−	0.837404i	$-0.315928\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	14.0000i	0.894427i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	8.00000i	0.506979i
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	− 4.00000i	− 0.250490i
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	12.0000i	0.737154i
$266$	0	0
$267$	18.0000i	1.10158i
$268$	0	0
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	8.00000i	0.485965i	0.970031	+	0.242983i	$0.0781258\pi$
$271$	8.00000i	0.485965i	−0.970031	+	0.242983i	$0.921874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 6.00000i	− 0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$	− 6.00000i	− 0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	− 6.00000i	− 0.351726i
$292$	0	0
$293$	− 10.0000i	− 0.584206i	−0.956387	−	0.292103i	$-0.905645\pi$
$293$	− 10.0000i	− 0.584206i	0.956387	−	0.292103i	$-0.0943550\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−14.0000	−0.804279
$304$	0	0
$305$	− 4.00000i	− 0.229039i
$306$	0	0
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.00000i	− 0.110600i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 36.0000i	− 1.97874i	−0.145424	−	0.989369i	$-0.546455\pi$
$331$	− 36.0000i	− 1.97874i	0.145424	−	0.989369i	$-0.453545\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	0	0
$349$	− 22.0000i	− 1.17763i	−0.808267	−	0.588817i	$-0.799594\pi$
$349$	− 22.0000i	− 1.17763i	0.808267	−	0.588817i	$-0.200406\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	0	0
$355$	24.0000	1.27379
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.0000i	0.633336i	0.948536	+	0.316668i	$0.102564\pi$
$359$	12.0000i	0.633336i	−0.948536	+	0.316668i	$0.897436\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	−28.0000	−1.46559
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	6.00000i	0.312348i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	12.0000i	0.619677i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 28.0000i	− 1.43826i	−0.694874	−	0.719132i	$-0.744540\pi$
$379$	− 28.0000i	− 1.43826i	0.694874	−	0.719132i	$-0.255460\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	4.00000i	0.204390i	0.994764	+	0.102195i	$0.0325866\pi$
$383$	4.00000i	0.204390i	−0.994764	+	0.102195i	$0.967413\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.0000	0.609994
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	0	0
$396$	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
$399$	0	0
$400$	0	0
$401$	− 6.00000i	− 0.299626i	−0.988714	−	0.149813i	$-0.952133\pi$
$401$	− 6.00000i	− 0.299626i	0.988714	−	0.149813i	$-0.0478671\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.00000i	0.0993808i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 30.0000i	− 1.48340i	−0.670729	−	0.741702i	$-0.734019\pi$
$409$	− 30.0000i	− 1.48340i	0.670729	−	0.741702i	$-0.265981\pi$
$410$	0	0
$411$	− 6.00000i	− 0.295958i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−16.0000	−0.785409
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	6.00000i	0.292422i	0.989253	+	0.146211i	$0.0467079\pi$
$421$	6.00000i	0.292422i	−0.989253	+	0.146211i	$0.953292\pi$
$422$	0	0
$423$	4.00000i	0.194487i
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.00000i	0.192673i	0.995349	+	0.0963366i	$0.0307125\pi$
$431$	4.00000i	0.192673i	−0.995349	+	0.0963366i	$0.969287\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	12.0000i	0.575356i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	7.00000	0.333333
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	−36.0000	−1.70656
$446$	0	0
$447$	10.0000i	0.472984i
$448$	0	0
$449$	− 6.00000i	− 0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$	− 6.00000i	− 0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 8.00000i	− 0.375873i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	34.0000i	1.59045i	0.606313	+	0.795226i	$0.292648\pi$
$457$	34.0000i	1.59045i	−0.606313	+	0.795226i	$0.707352\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	42.0000i	1.95614i	0.208288	+	0.978068i	$0.433211\pi$
$461$	42.0000i	1.95614i	−0.208288	+	0.978068i	$0.566789\pi$
$462$	0	0
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	− 36.0000i	− 1.64488i	−0.568850	−	0.822441i	$-0.692612\pi$
$479$	− 36.0000i	− 1.64488i	0.568850	−	0.822441i	$-0.307388\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.0000	0.544892
$486$	0	0
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	0	0
$489$	− 12.0000i	− 0.542659i
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	0	0
$501$	12.0000i	0.536120i
$502$	0	0
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	0	0
$505$	− 28.0000i	− 1.24598i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 22.0000i	− 0.975133i	−0.873086	−	0.487566i	$-0.837885\pi$
$509$	− 22.0000i	− 0.975133i	0.873086	−	0.487566i	$-0.162115\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 4.00000i	− 0.176604i
$514$	0	0
$515$	− 32.0000i	− 1.41009i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	8.00000i	0.347170i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	8.00000i	0.345870i
$536$	0	0
$537$	−20.0000	−0.863064
$538$	0	0
$539$	0	0
$540$	0	0
$541$	34.0000i	1.46177i	0.682498	+	0.730887i	$0.260893\pi$
$541$	34.0000i	1.46177i	−0.682498	+	0.730887i	$0.739107\pi$
$542$	0	0
$543$	−22.0000	−0.944110
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	− 24.0000i	− 1.02243i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	− 12.0000i	− 0.504844i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	16.0000	0.668410
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.0000i	0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$	18.0000i	0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$	0	0
$579$	− 2.00000i	− 0.0831172i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 16.0000i	− 0.660391i	−0.943913	−	0.330195i	$-0.892885\pi$
$587$	− 16.0000i	− 0.660391i	0.943913	−	0.330195i	$-0.107115\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	2.00000i	0.0822690i
$592$	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
$595$	0	0
$596$	0	0
$597$	24.0000	0.982255
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	22.0000i	0.894427i
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 34.0000i	− 1.37325i	−0.727013	−	0.686624i	$-0.759092\pi$
$613$	− 34.0000i	− 1.37325i	0.727013	−	0.686624i	$-0.240908\pi$
$614$	0	0
$615$	−12.0000	−0.483887
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	0	0
$619$	20.0000i	0.803868i	0.915669	+	0.401934i	$0.131662\pi$
$619$	20.0000i	0.803868i	−0.915669	+	0.401934i	$0.868338\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 4.00000i	− 0.159490i
$630$	0	0
$631$	8.00000i	0.318475i	0.987240	+	0.159237i	$0.0509036\pi$
$631$	8.00000i	0.318475i	−0.987240	+	0.159237i	$0.949096\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	− 32.0000i	− 1.26988i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 12.0000i	− 0.474713i
$640$	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
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	24.0000i	0.944999i
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	24.0000i	0.937758i
$656$	0	0
$657$	14.0000i	0.546192i
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	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
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	16.0000i	0.618596i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	30.0000	1.15642	0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000	1.15642	0.578208	+	0.815890i	$0.303752\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−10.0000	−0.384331	−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000	−0.384331	−0.192166	+	0.981363i	$0.561551\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16.0000i	0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$	16.0000i	0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	26.0000i	0.991962i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 28.0000i	− 1.06517i	−0.846376	−	0.532585i	$-0.821221\pi$
$691$	− 28.0000i	− 1.06517i	0.846376	−	0.532585i	$-0.178779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.00000i	0.303457i
$696$	0	0
$697$	− 12.0000i	− 0.454532i
$698$	0	0
$699$	14.0000	0.529529
$700$	0	0
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	0	0
$708$	0	0
$709$	10.0000i	0.375558i	0.982211	+	0.187779i	$0.0601289\pi$
$709$	10.0000i	0.375558i	−0.982211	+	0.187779i	$0.939871\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 20.0000i	− 0.746914i
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 26.0000i	− 0.966950i
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	− 18.0000i	− 0.664845i	−0.943131	−	0.332423i	$-0.892134\pi$
$733$	− 18.0000i	− 0.664845i	0.943131	−	0.332423i	$-0.107866\pi$
$734$	0	0
$735$	14.0000i	0.516398i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 4.00000i	− 0.147142i	−0.997290	−	0.0735712i	$-0.976560\pi$
$739$	− 4.00000i	− 0.147142i	0.997290	−	0.0735712i	$-0.0234396\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 44.0000i	− 1.61420i	−0.590412	−	0.807102i	$-0.701035\pi$
$743$	− 44.0000i	− 1.61420i	0.590412	−	0.807102i	$-0.298965\pi$
$744$	0	0
$745$	−20.0000	−0.732743
$746$	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	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	16.0000	0.582300
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.0000i	0.362500i	0.983437	+	0.181250i	$0.0580143\pi$
$761$	10.0000i	0.362500i	−0.983437	+	0.181250i	$0.941986\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 4.00000i	− 0.144620i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 14.0000i	− 0.504853i	−0.967616	−	0.252426i	$-0.918771\pi$
$769$	− 14.0000i	− 0.504853i	0.967616	−	0.252426i	$-0.0812286\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	28.0000i	0.999363i
$786$	0	0
$787$	− 52.0000i	− 1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$	− 52.0000i	− 1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	12.0000i	0.425596i
$796$	0	0
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	− 8.00000i	− 0.283020i
$800$	0	0
$801$	18.0000i	0.635999i
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−26.0000	−0.915243
$808$	0	0
$809$	34.0000	1.19538	0.597688	−	0.801729i	$-0.296086\pi$
$809$	34.0000	1.19538	0.597688	+	0.801729i	$0.296086\pi$
$810$	0	0
$811$	20.0000i	0.702295i	0.936320	+	0.351147i	$0.114208\pi$
$811$	20.0000i	0.702295i	−0.936320	+	0.351147i	$0.885792\pi$
$812$	0	0
$813$	8.00000i	0.280572i
$814$	0	0
$815$	24.0000	0.840683
$816$	0	0
$817$	− 48.0000i	− 1.67931i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 22.0000i	− 0.767805i	−0.923374	−	0.383903i	$-0.874580\pi$
$821$	− 22.0000i	− 0.767805i	0.923374	−	0.383903i	$-0.125420\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.0000i	1.11275i	0.830932	+	0.556375i	$0.187808\pi$
$827$	32.0000i	1.11275i	−0.830932	+	0.556375i	$0.812192\pi$
$828$	0	0
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	−14.0000	−0.485071
$834$	0	0
$835$	−24.0000	−0.830554
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 12.0000i	− 0.414286i	−0.978311	−	0.207143i	$-0.933583\pi$
$839$	− 12.0000i	− 0.414286i	0.978311	−	0.207143i	$-0.0664165\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	− 6.00000i	− 0.206651i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 46.0000i	− 1.57501i	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	− 46.0000i	− 1.57501i	0.616308	−	0.787505i	$-0.288628\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	0	0
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 44.0000i	− 1.49778i	−0.662696	−	0.748889i	$-0.730588\pi$
$863$	− 44.0000i	− 1.49778i	0.662696	−	0.748889i	$-0.269412\pi$
$864$	0	0
$865$	− 12.0000i	− 0.408012i
$866$	0	0
$867$	−13.0000	−0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 6.00000i	− 0.203069i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38.0000i	1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$	38.0000i	1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$	0	0
$879$	− 10.0000i	− 0.337292i
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	−16.0000	−0.537834
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.0000	0.535420
$894$	0	0
$895$	− 40.0000i	− 1.33705i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 44.0000i	− 1.46261i
$906$	0	0
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	0	0
$909$	−14.0000	−0.464351
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 4.00000i	− 0.132236i
$916$	0	0
$917$	0	0
$918$	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
$921$	− 20.0000i	− 0.659022i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	14.0000i	0.459325i	0.973270	+	0.229663i	$0.0737623\pi$
$929$	14.0000i	0.459325i	−0.973270	+	0.229663i	$0.926238\pi$
$930$	0	0
$931$	− 28.0000i	− 0.917663i
$932$	0	0
$933$	16.0000	0.523816
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	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
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 6.00000i	− 0.194563i
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	0	0
$955$	32.0000i	1.03550i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	4.00000	0.128765
$966$	0	0
$967$	40.0000i	1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$	40.0000i	1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$	0	0
$969$	8.00000i	0.256997i
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.00000i	0.191957i	0.995383	+	0.0959785i	$0.0305980\pi$
$977$	6.00000i	0.191957i	−0.995383	+	0.0959785i	$0.969402\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 2.00000i	− 0.0638551i
$982$	0	0
$983$	20.0000i	0.637901i	0.947771	+	0.318950i	$0.103330\pi$
$983$	20.0000i	0.637901i	−0.947771	+	0.318950i	$0.896670\pi$
$984$	0	0
$985$	−4.00000	−0.127451
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	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
$993$	− 36.0000i	− 1.14243i
$994$	0	0
$995$	48.0000i	1.52170i
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	0	0
$999$	2.00000i	0.0632772i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.h.337.2		2
13.5	odd	4		312.2.a.f.1.1	✓	1
13.8	odd	4		4056.2.a.m.1.1		1
13.12	even	2	inner	4056.2.c.h.337.1		2
39.5	even	4		936.2.a.b.1.1		1
52.31	even	4		624.2.a.d.1.1		1
52.47	even	4		8112.2.a.f.1.1		1
65.44	odd	4		7800.2.a.d.1.1		1
104.5	odd	4		2496.2.a.c.1.1		1
104.83	even	4		2496.2.a.s.1.1		1
156.83	odd	4		1872.2.a.e.1.1		1
312.5	even	4		7488.2.a.bs.1.1		1
312.83	odd	4		7488.2.a.br.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.a.f.1.1	✓	1	13.5	odd	4
624.2.a.d.1.1		1	52.31	even	4
936.2.a.b.1.1		1	39.5	even	4
1872.2.a.e.1.1		1	156.83	odd	4
2496.2.a.c.1.1		1	104.5	odd	4
2496.2.a.s.1.1		1	104.83	even	4
4056.2.a.m.1.1		1	13.8	odd	4
4056.2.c.h.337.1		2	13.12	even	2	inner
4056.2.c.h.337.2		2	1.1	even	1	trivial
7488.2.a.br.1.1		1	312.83	odd	4
7488.2.a.bs.1.1		1	312.5	even	4
7800.2.a.d.1.1		1	65.44	odd	4
8112.2.a.f.1.1		1	52.47	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:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.00000i q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.00000i q^{5} +1.00000 q^{9} +2.00000i q^{15} -2.00000 q^{17} -4.00000i q^{19} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +2.00000i q^{37} +6.00000i q^{41} +12.0000 q^{43} +2.00000i q^{45} +4.00000i q^{47} +7.00000 q^{49} -2.00000 q^{51} +6.00000 q^{53} -4.00000i q^{57} +8.00000i q^{59} -2.00000 q^{61} +4.00000i q^{67} -12.0000i q^{71} +14.0000i q^{73} +1.00000 q^{75} +1.00000 q^{81} +8.00000i q^{83} -4.00000i q^{85} +6.00000 q^{87} +18.0000i q^{89} +8.00000 q^{95} -6.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} - 4 q^{17} + 2 q^{25} + 2 q^{27} + 12 q^{29} + 24 q^{43} + 14 q^{49} - 4 q^{51} + 12 q^{53} - 4 q^{61} + 2 q^{75} + 2 q^{81} + 12 q^{87} + 16 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 - 4 * q^17 + 2 * q^25 + 2 * q^27 + 12 * q^29 + 24 * q^43 + 14 * q^49 - 4 * q^51 + 12 * q^53 - 4 * q^61 + 2 * q^75 + 2 * q^81 + 12 * q^87 + 16 * q^95

Introduction

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