LMFDB - Embedded newform 6600.2.d.y.1849.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6600,2,Mod(1849,6600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6600.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6600.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$52.7012653340$
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 1320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	6600.1849
Dual form			6600.2.d.y.1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} +8.00000i q^{17} +8.00000 q^{19} -2.00000 q^{21} +4.00000i q^{23} -1.00000i q^{27} +6.00000 q^{29} +1.00000i q^{33} -6.00000i q^{37} -2.00000 q^{41} +2.00000i q^{43} +4.00000i q^{47} +3.00000 q^{49} -8.00000 q^{51} -2.00000i q^{53} +8.00000i q^{57} +12.0000 q^{59} -6.00000 q^{61} -2.00000i q^{63} -8.00000i q^{67} -4.00000 q^{69} -8.00000i q^{73} +2.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +6.00000i q^{87} +10.0000 q^{89} +10.0000i q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 2 q^{11} + 16 q^{19} - 4 q^{21} + 12 q^{29} - 4 q^{41} + 6 q^{49} - 16 q^{51} + 24 q^{59} - 12 q^{61} - 8 q^{69} - 8 q^{79} + 2 q^{81} + 20 q^{89} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 2 * q^11 + 16 * q^19 - 4 * q^21 + 12 * q^29 - 4 * q^41 + 6 * q^49 - 16 * q^51 + 24 * q^59 - 12 * q^61 - 8 * q^69 - 8 * q^79 + 2 * q^81 + 20 * q^89 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6600\mathbb{Z}\right)^\times$.

$n$	$1201$	$2201$	$2377$	$3301$	$4951$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	8.00000i	1.94029i	0.242536	+	0.970143i	$0.422021\pi$
$17$	8.00000i	1.94029i	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	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		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	1.00000i	0.174078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	0	0
$46$	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
$49$	3.00000	0.428571
$50$	0	0
$51$	−8.00000	−1.12022
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.00000i	1.05963i
$58$	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
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	− 2.00000i	− 0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 8.00000i	− 0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$	− 8.00000i	− 0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000i	0.643268i
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 10.0000i	− 0.966736i	−0.875417	−	0.483368i	$-0.839413\pi$
$107$	− 10.0000i	− 0.966736i	0.875417	−	0.483368i	$-0.160587\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−16.0000	−1.46672
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 2.00000i	− 0.180334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.00000i	− 0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$	− 2.00000i	− 0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	16.0000i	1.38738i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.00000i	0.247436i
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	− 8.00000i	− 0.646762i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	−8.00000	−0.630488
$162$	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
$167$	− 10.0000i	− 0.773823i	−0.922117	−	0.386912i	$-0.873542\pi$
$167$	− 10.0000i	− 0.773823i	0.922117	−	0.386912i	$-0.126458\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	−8.00000	−0.611775
$172$	0	0
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0000i	0.901975i
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	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
$183$	− 6.00000i	− 0.443533i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	24.0000i	1.72756i	0.503871	+	0.863779i	$0.331909\pi$
$193$	24.0000i	1.72756i	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.0000i	1.42494i	0.701702	+	0.712470i	$0.252424\pi$
$197$	20.0000i	1.42494i	−0.701702	+	0.712470i	$0.747576\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	12.0000i	0.842235i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 4.00000i	− 0.278019i
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	8.00000	0.540590
$220$	0	0
$221$	0	0
$222$	0	0
$223$	12.0000i	0.803579i	0.915732	+	0.401790i	$0.131612\pi$
$223$	12.0000i	0.803579i	−0.915732	+	0.401790i	$0.868388\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 30.0000i	− 1.99117i	−0.0938647	−	0.995585i	$-0.529922\pi$
$227$	− 30.0000i	− 1.99117i	0.0938647	−	0.995585i	$-0.470078\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	20.0000i	1.31024i	0.755523	+	0.655122i	$0.227383\pi$
$233$	20.0000i	1.31024i	−0.755523	+	0.655122i	$0.772617\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4.00000i	− 0.259828i
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	4.00000i	0.251478i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	6.00000i	0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$	6.00000i	0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10.0000i	0.611990i
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.0000	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$282$	0	0
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.00000i	− 0.236113i
$288$	0	0
$289$	−47.0000	−2.76471
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	− 32.0000i	− 1.86946i	−0.355359	−	0.934730i	$-0.615641\pi$
$293$	− 32.0000i	− 1.86946i	0.355359	−	0.934730i	$-0.384359\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.00000i	− 0.0580259i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	− 2.00000i	− 0.114897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 10.0000i	− 0.570730i	−0.958419	−	0.285365i	$-0.907885\pi$
$307$	− 10.0000i	− 0.570730i	0.958419	−	0.285365i	$-0.0921148\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	0	0
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 26.0000i	− 1.46031i	−0.683284	−	0.730153i	$-0.739449\pi$
$317$	− 26.0000i	− 1.46031i	0.683284	−	0.730153i	$-0.260551\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	10.0000	0.558146
$322$	0	0
$323$	64.0000i	3.56106i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 14.0000i	− 0.774202i
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	6.00000i	0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	32.0000i	1.74315i	0.490261	+	0.871576i	$0.336901\pi$
$337$	32.0000i	1.74315i	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.0000i	1.61048i	0.592946	+	0.805242i	$0.297965\pi$
$347$	30.0000i	1.61048i	−0.592946	+	0.805242i	$0.702035\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 16.0000i	− 0.846810i
$358$	0	0
$359$	28.0000	1.47778	0.738892	−	0.673824i	$-0.235349\pi$
$359$	28.0000	1.47778	0.738892	+	0.673824i	$0.235349\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	0	0
$363$	1.00000i	0.0524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 28.0000i	− 1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$	− 28.0000i	− 1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	0	0
$383$	8.00000i	0.408781i	0.978889	+	0.204390i	$0.0655212\pi$
$383$	8.00000i	0.408781i	−0.978889	+	0.204390i	$0.934479\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.00000i	− 0.101666i
$388$	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
$391$	−32.0000	−1.61831
$392$	0	0
$393$	− 8.00000i	− 0.403547i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	−16.0000	−0.801002
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.00000i	− 0.297409i
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	0	0
$413$	24.0000i	1.18096i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 12.0000i	− 0.587643i
$418$	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
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	− 4.00000i	− 0.194487i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 12.0000i	− 0.580721i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	32.0000i	1.53077i
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	18.0000i	0.851371i
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	12.0000i	0.563809i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 28.0000i	− 1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$	− 28.0000i	− 1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	38.0000	1.76984	0.884918	−	0.465746i	$-0.154214\pi$
$461$	38.0000	1.76984	0.884918	+	0.465746i	$0.154214\pi$
$462$	0	0
$463$	− 20.0000i	− 0.929479i	−0.885448	−	0.464739i	$-0.846148\pi$
$463$	− 20.0000i	− 0.929479i	0.885448	−	0.464739i	$-0.153852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.00000i	0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$	8.00000i	0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	2.00000i	0.0919601i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.00000i	0.0915737i
$478$	0	0
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 8.00000i	− 0.364013i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 12.0000i	− 0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$	− 12.0000i	− 0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	48.0000i	2.16181i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	10.0000	0.446767
$502$	0	0
$503$	18.0000i	0.802580i	0.915951	+	0.401290i	$0.131438\pi$
$503$	18.0000i	0.802580i	−0.915951	+	0.401290i	$0.868562\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	13.0000i	0.577350i
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	0	0
$513$	− 8.00000i	− 0.353209i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.00000i	0.175920i
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	0	0
$523$	− 18.0000i	− 0.787085i	−0.919306	−	0.393543i	$-0.871249\pi$
$523$	− 18.0000i	− 0.787085i	0.919306	−	0.393543i	$-0.128751\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.00000i	0.172613i
$538$	0	0
$539$	3.00000	0.129219
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	2.00000i	0.0858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	26.0000i	1.11168i	0.831289	+	0.555840i	$0.187603\pi$
$547$	26.0000i	1.11168i	−0.831289	+	0.555840i	$0.812397\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	48.0000	2.04487
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.0000i	0.847427i	0.905796	+	0.423714i	$0.139274\pi$
$557$	20.0000i	0.847427i	−0.905796	+	0.423714i	$0.860726\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	30.0000i	1.26435i	0.774826	+	0.632175i	$0.217837\pi$
$563$	30.0000i	1.26435i	−0.774826	+	0.632175i	$0.782163\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.00000i	0.0839921i
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	0	0
$573$	8.00000i	0.334205i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	0	0
$579$	−24.0000	−0.997406
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	− 2.00000i	− 0.0828315i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−20.0000	−0.822690
$592$	0	0
$593$	16.0000i	0.657041i	0.944497	+	0.328521i	$0.106550\pi$
$593$	16.0000i	0.657041i	−0.944497	+	0.328521i	$0.893450\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 24.0000i	− 0.982255i
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	8.00000i	0.325785i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 18.0000i	− 0.730597i	−0.930890	−	0.365299i	$-0.880967\pi$
$607$	− 18.0000i	− 0.730597i	0.930890	−	0.365299i	$-0.119033\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 4.00000i	− 0.161558i	−0.996732	−	0.0807792i	$-0.974259\pi$
$613$	− 4.00000i	− 0.161558i	0.996732	−	0.0807792i	$-0.0257409\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.00000i	0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$	2.00000i	0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	20.0000i	0.801283i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.00000i	0.319489i
$628$	0	0
$629$	48.0000	1.91389
$630$	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
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	− 36.0000i	− 1.41970i	−0.704352	−	0.709851i	$-0.748762\pi$
$643$	− 36.0000i	− 1.41970i	0.704352	−	0.709851i	$-0.251238\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.00000i	0.312110i
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	24.0000i	0.929284i
$668$	0	0
$669$	−12.0000	−0.463947
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	8.00000i	0.308377i	0.988041	+	0.154189i	$0.0492764\pi$
$673$	8.00000i	0.308377i	−0.988041	+	0.154189i	$0.950724\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	48.0000i	1.84479i	0.386248	+	0.922395i	$0.373771\pi$
$677$	48.0000i	1.84479i	−0.386248	+	0.922395i	$0.626229\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	30.0000	1.14960
$682$	0	0
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 6.00000i	− 0.228914i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	0	0
$693$	− 2.00000i	− 0.0759737i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 16.0000i	− 0.606043i
$698$	0	0
$699$	−20.0000	−0.756469
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	− 48.0000i	− 1.81035i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.00000i	− 0.150435i
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 12.0000i	− 0.448148i
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	0	0
$723$	− 10.0000i	− 0.371904i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	20.0000i	0.738717i	0.929287	+	0.369358i	$0.120423\pi$
$733$	20.0000i	0.738717i	−0.929287	+	0.369358i	$0.879577\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.00000i	− 0.294684i
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.00000i	0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$	6.00000i	0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.00000i	0.219529i
$748$	0	0
$749$	20.0000	0.730784
$750$	0	0
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	0	0
$753$	− 20.0000i	− 0.728841i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 18.0000i	− 0.654221i	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	− 18.0000i	− 0.654221i	0.944986	−	0.327111i	$-0.106075\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	0	0
$763$	− 28.0000i	− 1.01367i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	42.0000	1.51456	0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000	1.51456	0.757279	+	0.653091i	$0.226528\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	22.0000i	0.791285i	0.918405	+	0.395643i	$0.129478\pi$
$773$	22.0000i	0.791285i	−0.918405	+	0.395643i	$0.870522\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	12.0000i	0.430498i
$778$	0	0
$779$	−16.0000	−0.573259
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 6.00000i	− 0.214423i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 18.0000i	− 0.641631i	−0.947142	−	0.320815i	$-0.896043\pi$
$787$	− 18.0000i	− 0.641631i	0.947142	−	0.320815i	$-0.103957\pi$
$788$	0	0
$789$	−6.00000	−0.213606
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	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
$799$	−32.0000	−1.13208
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	− 8.00000i	− 0.282314i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18.0000i	0.633630i
$808$	0	0
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	− 28.0000i	− 0.982003i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	0	0
$823$	− 40.0000i	− 1.39431i	−0.716919	−	0.697156i	$-0.754448\pi$
$823$	− 40.0000i	− 1.39431i	0.716919	−	0.697156i	$-0.245552\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.0000i	0.486828i	0.969923	+	0.243414i	$0.0782673\pi$
$827$	14.0000i	0.486828i	−0.969923	+	0.243414i	$0.921733\pi$
$828$	0	0
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	0	0
$833$	24.0000i	0.831551i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	14.0000i	0.482186i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.00000i	0.0687208i
$848$	0	0
$849$	14.0000	0.480479
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	− 44.0000i	− 1.50653i	−0.657716	−	0.753266i	$-0.728477\pi$
$853$	− 44.0000i	− 1.50653i	0.657716	−	0.753266i	$-0.271523\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 56.0000i	− 1.91292i	−0.291858	−	0.956462i	$-0.594273\pi$
$857$	− 56.0000i	− 1.91292i	0.291858	−	0.956462i	$-0.405727\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	0	0
$863$	− 32.0000i	− 1.08929i	−0.838666	−	0.544646i	$-0.816664\pi$
$863$	− 32.0000i	− 1.08929i	0.838666	−	0.544646i	$-0.183336\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 47.0000i	− 1.59620i
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44.0000i	1.48577i	0.669417	+	0.742887i	$0.266544\pi$
$877$	44.0000i	1.48577i	−0.669417	+	0.742887i	$0.733456\pi$
$878$	0	0
$879$	32.0000	1.07933
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 42.0000i	− 1.41022i	−0.709097	−	0.705111i	$-0.750897\pi$
$887$	− 42.0000i	− 1.41022i	0.709097	−	0.705111i	$-0.249103\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	16.0000	0.533037
$902$	0	0
$903$	− 4.00000i	− 0.133112i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.0000i	1.06254i	0.847202	+	0.531271i	$0.178286\pi$
$907$	32.0000i	1.06254i	−0.847202	+	0.531271i	$0.821714\pi$
$908$	0	0
$909$	2.00000	0.0663358
$910$	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
$913$	− 6.00000i	− 0.198571i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 16.0000i	− 0.528367i
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	10.0000	0.329511
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 16.0000i	− 0.525509i
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	0	0
$933$	− 32.0000i	− 1.04763i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	52.0000i	1.69877i	0.527777	+	0.849383i	$0.323026\pi$
$937$	52.0000i	1.69877i	−0.527777	+	0.849383i	$0.676974\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	0	0
$943$	− 8.00000i	− 0.260516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 16.0000i	− 0.519930i	−0.965618	−	0.259965i	$-0.916289\pi$
$947$	− 16.0000i	− 0.519930i	0.965618	−	0.259965i	$-0.0837111\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	26.0000	0.843108
$952$	0	0
$953$	− 12.0000i	− 0.388718i	−0.980930	−	0.194359i	$-0.937737\pi$
$953$	− 12.0000i	− 0.388718i	0.980930	−	0.194359i	$-0.0622627\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.00000i	0.193952i
$958$	0	0
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	10.0000i	0.322245i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	22.0000i	0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$967$	22.0000i	0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$968$	0	0
$969$	−64.0000	−2.05598
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	− 24.0000i	− 0.769405i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 46.0000i	− 1.47167i	−0.677161	−	0.735835i	$-0.736790\pi$
$977$	− 46.0000i	− 1.47167i	0.677161	−	0.735835i	$-0.263210\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	0	0
$993$	4.00000i	0.126936i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	36.0000i	1.14013i	0.821599	+	0.570066i	$0.193082\pi$
$997$	36.0000i	1.14013i	−0.821599	+	0.570066i	$0.806918\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6600.2.d.y.1849.2		2
5.2	odd	4		1320.2.a.k.1.1	✓	1
5.3	odd	4		6600.2.a.n.1.1		1
5.4	even	2	inner	6600.2.d.y.1849.1		2
15.2	even	4		3960.2.a.k.1.1		1
20.7	even	4		2640.2.a.c.1.1		1
60.47	odd	4		7920.2.a.bj.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.a.k.1.1	✓	1	5.2	odd	4
2640.2.a.c.1.1		1	20.7	even	4
3960.2.a.k.1.1		1	15.2	even	4
6600.2.a.n.1.1		1	5.3	odd	4
6600.2.d.y.1849.1		2	5.4	even	2	inner
6600.2.d.y.1849.2		2	1.1	even	1	trivial
7920.2.a.bj.1.1		1	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6600.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} +8.00000i q^{17} +8.00000 q^{19} -2.00000 q^{21} +4.00000i q^{23} -1.00000i q^{27} +6.00000 q^{29} +1.00000i q^{33} -6.00000i q^{37} -2.00000 q^{41} +2.00000i q^{43} +4.00000i q^{47} +3.00000 q^{49} -8.00000 q^{51} -2.00000i q^{53} +8.00000i q^{57} +12.0000 q^{59} -6.00000 q^{61} -2.00000i q^{63} -8.00000i q^{67} -4.00000 q^{69} -8.00000i q^{73} +2.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +6.00000i q^{87} +10.0000 q^{89} +10.0000i q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 2 q^{11} + 16 q^{19} - 4 q^{21} + 12 q^{29} - 4 q^{41} + 6 q^{49} - 16 q^{51} + 24 q^{59} - 12 q^{61} - 8 q^{69} - 8 q^{79} + 2 q^{81} + 20 q^{89} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 2 * q^11 + 16 * q^19 - 4 * q^21 + 12 * q^29 - 4 * q^41 + 6 * q^49 - 16 * q^51 + 24 * q^59 - 12 * q^61 - 8 * q^69 - 8 * q^79 + 2 * q^81 + 20 * q^89 - 2 * q^99

Introduction

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