LMFDB - Embedded newform 5700.2.f.l.3649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5700,2,Mod(3649,5700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5700.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5700.f (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:	$45.5147291521$
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 1140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5700.3649
Dual form			5700.2.f.l.3649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} +4.00000 q^{11} -2.00000i q^{17} -1.00000 q^{19} +2.00000 q^{21} +2.00000i q^{23} -1.00000i q^{27} +6.00000 q^{29} +4.00000i q^{33} -8.00000i q^{37} -2.00000 q^{41} +6.00000i q^{43} -2.00000i q^{47} +3.00000 q^{49} +2.00000 q^{51} +4.00000i q^{53} -1.00000i q^{57} -4.00000 q^{59} -10.0000 q^{61} +2.00000i q^{63} -12.0000i q^{67} -2.00000 q^{69} +8.00000 q^{71} +2.00000i q^{73} -8.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} -14.0000i q^{83} +6.00000i q^{87} +2.00000 q^{89} -4.00000i q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 8 q^{11} - 2 q^{19} + 4 q^{21} + 12 q^{29} - 4 q^{41} + 6 q^{49} + 4 q^{51} - 8 q^{59} - 20 q^{61} - 4 q^{69} + 16 q^{71} + 16 q^{79} + 2 q^{81} + 4 q^{89} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 8 * q^11 - 2 * q^19 + 4 * q^21 + 12 * q^29 - 4 * q^41 + 6 * q^49 + 4 * q^51 - 8 * q^59 - 20 * q^61 - 4 * q^69 + 16 * q^71 + 16 * q^79 + 2 * q^81 + 4 * q^89 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5700\mathbb{Z}\right)^\times$.

$n$	$1901$	$2851$	$3877$	$4201$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	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
$33$	4.00000i	0.696311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	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
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	4.00000i	0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$	4.00000i	0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.00000i	− 0.132453i
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 8.00000i	− 0.911685i
$78$	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
$83$	− 14.0000i	− 1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$	− 14.0000i	− 1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000i	0.643268i
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.00000i	− 0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$	− 4.00000i	− 0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	− 4.00000i	− 0.376288i	−0.982141	−	0.188144i	$-0.939753\pi$
$113$	− 4.00000i	− 0.376288i	0.982141	−	0.188144i	$-0.0602472\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	− 2.00000i	− 0.180334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	2.00000i	0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.00000i	0.247436i
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	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
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	− 12.0000i	− 0.912343i	−0.889892	−	0.456172i	$-0.849220\pi$
$173$	− 12.0000i	− 0.912343i	0.889892	−	0.456172i	$-0.150780\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.00000i	− 0.300658i
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	− 10.0000i	− 0.739221i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	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
$201$	12.0000	0.846415
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.00000i	− 0.139010i
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	0	0
$213$	8.00000i	0.548151i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	0	0
$222$	0	0
$223$	24.0000i	1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$	24.0000i	1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.00000i	0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$	8.00000i	0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	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
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	14.0000	0.887214
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	8.00000i	0.502956i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 24.0000i	− 1.49708i	−0.663090	−	0.748539i	$-0.730755\pi$
$257$	− 24.0000i	− 1.49708i	0.663090	−	0.748539i	$-0.269245\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	− 30.0000i	− 1.84988i	−0.380114	−	0.924940i	$-0.624115\pi$
$263$	− 30.0000i	− 1.84988i	0.380114	−	0.924940i	$-0.375885\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.00000i	0.122398i
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	0	0
$274$	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
$279$	0	0
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	2.00000i	0.118888i	0.998232	+	0.0594438i	$0.0189327\pi$
$283$	2.00000i	0.118888i	−0.998232	+	0.0594438i	$0.981067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.00000i	0.236113i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	4.00000	0.234484
$292$	0	0
$293$	4.00000i	0.233682i	0.993151	+	0.116841i	$0.0372769\pi$
$293$	4.00000i	0.233682i	−0.993151	+	0.116841i	$0.962723\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.00000i	− 0.232104i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	2.00000i	0.114897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	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
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	2.00000i	0.111283i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.0000i	0.553001i
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	8.00000i	0.438397i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 28.0000i	− 1.52526i	−0.646837	−	0.762629i	$-0.723908\pi$
$337$	− 28.0000i	− 1.52526i	0.646837	−	0.762629i	$-0.276092\pi$
$338$	0	0
$339$	4.00000	0.217250
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.0000i	0.536828i	0.963304	+	0.268414i	$0.0864995\pi$
$347$	10.0000i	0.536828i	−0.963304	+	0.268414i	$0.913500\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	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
$357$	− 4.00000i	− 0.211702i
$358$	0	0
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.00000i	0.262432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 10.0000i	− 0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$	− 10.0000i	− 0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	8.00000	0.415339
$372$	0	0
$373$	28.0000i	1.44979i	0.688862	+	0.724893i	$0.258111\pi$
$373$	28.0000i	1.44979i	−0.688862	+	0.724893i	$0.741889\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	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
$387$	− 6.00000i	− 0.304997i
$388$	0	0
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	30.0000i	1.50566i	0.658217	+	0.752828i	$0.271311\pi$
$397$	30.0000i	1.50566i	−0.658217	+	0.752828i	$0.728689\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	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
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 32.0000i	− 1.58618i
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	0	0
$413$	8.00000i	0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 8.00000i	− 0.391762i
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	2.00000i	0.0972433i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	20.0000i	0.967868i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	36.0000i	1.73005i	0.501729	+	0.865025i	$0.332697\pi$
$433$	36.0000i	1.73005i	−0.501729	+	0.865025i	$0.667303\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.00000i	− 0.0956730i
$438$	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
$443$	18.0000i	0.855206i	0.903967	+	0.427603i	$0.140642\pi$
$443$	18.0000i	0.855206i	−0.903967	+	0.427603i	$0.859358\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.0000i	0.472984i
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	0	0
$453$	− 8.00000i	− 0.375873i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.0000i	1.21623i	0.793849	+	0.608114i	$0.208074\pi$
$457$	26.0000i	1.21623i	−0.793849	+	0.608114i	$0.791926\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	18.0000i	0.836531i	0.908325	+	0.418265i	$0.137362\pi$
$463$	18.0000i	0.836531i	−0.908325	+	0.418265i	$0.862638\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.00000i	0.0925490i	0.998929	+	0.0462745i	$0.0147349\pi$
$467$	2.00000i	0.0925490i	−0.998929	+	0.0462745i	$0.985265\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	0	0
$471$	−6.00000	−0.276465
$472$	0	0
$473$	24.0000i	1.10352i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4.00000i	− 0.183147i
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	4.00000i	0.182006i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	20.0000i	0.906287i	0.891438	+	0.453143i	$0.149697\pi$
$487$	20.0000i	0.906287i	−0.891438	+	0.453143i	$0.850303\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 16.0000i	− 0.717698i
$498$	0	0
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	− 22.0000i	− 0.980932i	−0.871460	−	0.490466i	$-0.836827\pi$
$503$	− 22.0000i	− 0.980932i	0.871460	−	0.490466i	$-0.163173\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	13.0000i	0.577350i
$508$	0	0
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	1.00000i	0.0441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	19.0000	0.826087
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4.00000i	− 0.172613i
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	6.00000i	0.257485i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 16.0000i	− 0.684111i	−0.939680	−	0.342055i	$-0.888877\pi$
$547$	− 16.0000i	− 0.684111i	0.939680	−	0.342055i	$-0.111123\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.00000i	− 0.0839921i
$568$	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
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	4.00000i	0.167102i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	34.0000i	1.41544i	0.706494	+	0.707719i	$0.250276\pi$
$577$	34.0000i	1.41544i	−0.706494	+	0.707719i	$0.749724\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	−28.0000	−1.16164
$582$	0	0
$583$	16.0000i	0.662652i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 38.0000i	− 1.56843i	−0.620491	−	0.784214i	$-0.713066\pi$
$587$	− 38.0000i	− 1.56843i	0.620491	−	0.784214i	$-0.286934\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	− 10.0000i	− 0.410651i	−0.978694	−	0.205325i	$-0.934175\pi$
$593$	− 10.0000i	− 0.410651i	0.978694	−	0.205325i	$-0.0658253\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 28.0000i	− 1.13648i	−0.822861	−	0.568242i	$-0.807624\pi$
$607$	− 28.0000i	− 1.13648i	0.822861	−	0.568242i	$-0.192376\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	− 4.00000i	− 0.160257i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4.00000i	− 0.159745i
$628$	0	0
$629$	−16.0000	−0.637962
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	28.0000i	1.11290i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	− 22.0000i	− 0.867595i	−0.901010	−	0.433798i	$-0.857173\pi$
$643$	− 22.0000i	− 0.867595i	0.901010	−	0.433798i	$-0.142827\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.0000i	1.65119i	0.564263	+	0.825595i	$0.309160\pi$
$647$	42.0000i	1.65119i	−0.564263	+	0.825595i	$0.690840\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.0000i	0.464642i
$668$	0	0
$669$	−24.0000	−0.927894
$670$	0	0
$671$	−40.0000	−1.54418
$672$	0	0
$673$	− 32.0000i	− 1.23351i	−0.787155	−	0.616755i	$-0.788447\pi$
$673$	− 32.0000i	− 1.23351i	0.787155	−	0.616755i	$-0.211553\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 12.0000i	− 0.461197i	−0.973049	−	0.230599i	$-0.925932\pi$
$677$	− 12.0000i	− 0.461197i	0.973049	−	0.230599i	$-0.0740685\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	−8.00000	−0.306561
$682$	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
$687$	− 10.0000i	− 0.381524i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	0	0
$693$	8.00000i	0.303895i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.00000i	0.151511i
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	8.00000i	0.301726i
$704$	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	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	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
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	22.0000i	0.818189i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	14.0000i	0.519231i	0.965712	+	0.259616i	$0.0835959\pi$
$727$	14.0000i	0.519231i	−0.965712	+	0.259616i	$0.916404\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	− 10.0000i	− 0.369358i	−0.982799	−	0.184679i	$-0.940875\pi$
$733$	− 10.0000i	− 0.369358i	0.982799	−	0.184679i	$-0.0591246\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 48.0000i	− 1.76810i
$738$	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
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	14.0000i	0.512233i
$748$	0	0
$749$	16.0000	0.584627
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	24.0000i	0.874609i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 22.0000i	− 0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$	− 22.0000i	− 0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	0	0
$763$	− 20.0000i	− 0.724049i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 16.0000i	− 0.573997i
$778$	0	0
$779$	2.00000	0.0716574
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	− 6.00000i	− 0.214423i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 8.00000i	− 0.285169i	−0.989783	−	0.142585i	$-0.954459\pi$
$787$	− 8.00000i	− 0.285169i	0.989783	−	0.142585i	$-0.0455413\pi$
$788$	0	0
$789$	30.0000	1.06803
$790$	0	0
$791$	−8.00000	−0.284447
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 28.0000i	− 0.991811i	−0.868377	−	0.495905i	$-0.834836\pi$
$797$	− 28.0000i	− 0.991811i	0.868377	−	0.495905i	$-0.165164\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	8.00000i	0.282314i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 14.0000i	− 0.492823i
$808$	0	0
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	0	0
$813$	8.00000i	0.280572i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 6.00000i	− 0.209913i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	14.0000i	0.488009i	0.969774	+	0.244005i	$0.0784612\pi$
$823$	14.0000i	0.488009i	−0.969774	+	0.244005i	$0.921539\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	− 26.0000i	− 0.895488i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	0	0
$849$	−2.00000	−0.0686398
$850$	0	0
$851$	16.0000	0.548473
$852$	0	0
$853$	30.0000i	1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$	30.0000i	1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.0000i	− 0.819824i	−0.912125	−	0.409912i	$-0.865559\pi$
$857$	− 24.0000i	− 0.819824i	0.912125	−	0.409912i	$-0.134441\pi$
$858$	0	0
$859$	−52.0000	−1.77422	−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000	−1.77422	−0.887109	+	0.461561i	$0.847290\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	0	0
$863$	− 20.0000i	− 0.680808i	−0.940279	−	0.340404i	$-0.889436\pi$
$863$	− 20.0000i	− 0.680808i	0.940279	−	0.340404i	$-0.110564\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	0	0
$872$	0	0
$873$	4.00000i	0.135379i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.0000i	0.675352i	0.941262	+	0.337676i	$0.109641\pi$
$877$	20.0000i	0.675352i	−0.941262	+	0.337676i	$0.890359\pi$
$878$	0	0
$879$	−4.00000	−0.134917
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	− 42.0000i	− 1.41341i	−0.707507	−	0.706706i	$-0.750180\pi$
$883$	− 42.0000i	− 1.41341i	0.707507	−	0.706706i	$-0.249820\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	2.00000i	0.0669274i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	8.00000	0.266519
$902$	0	0
$903$	12.0000i	0.399335i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 8.00000i	− 0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$	− 8.00000i	− 0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$	0	0
$909$	−2.00000	−0.0663358
$910$	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
$913$	− 56.0000i	− 1.85333i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	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
$931$	−3.00000	−0.0983210
$932$	0	0
$933$	16.0000i	0.523816i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.0000i	0.457360i	0.973502	+	0.228680i	$0.0734410\pi$
$937$	14.0000i	0.457360i	−0.973502	+	0.228680i	$0.926559\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	−26.0000	−0.847576	−0.423788	−	0.905761i	$-0.639300\pi$
$941$	−26.0000	−0.847576	−0.423788	+	0.905761i	$0.639300\pi$
$942$	0	0
$943$	− 4.00000i	− 0.130258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.0000i	1.10485i	0.833562	+	0.552426i	$0.186298\pi$
$947$	34.0000i	1.10485i	−0.833562	+	0.552426i	$0.813702\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 24.0000i	− 0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$	− 24.0000i	− 0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	24.0000i	0.775810i
$958$	0	0
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 8.00000i	− 0.257796i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	42.0000i	1.35063i	0.737530	+	0.675314i	$0.235992\pi$
$967$	42.0000i	1.35063i	−0.737530	+	0.675314i	$0.764008\pi$
$968$	0	0
$969$	−2.00000	−0.0642493
$970$	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
$973$	16.0000i	0.512936i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	− 12.0000i	− 0.382741i	−0.981518	−	0.191370i	$-0.938707\pi$
$983$	− 12.0000i	− 0.382741i	0.981518	−	0.191370i	$-0.0612931\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4.00000i	− 0.127321i
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	12.0000i	0.380808i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.0000i	0.443384i	0.975117	+	0.221692i	$0.0711580\pi$
$997$	14.0000i	0.443384i	−0.975117	+	0.221692i	$0.928842\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5700.2.f.l.3649.2		2
5.2	odd	4		5700.2.a.r.1.1		1
5.3	odd	4		1140.2.a.a.1.1	✓	1
5.4	even	2	inner	5700.2.f.l.3649.1		2
15.8	even	4		3420.2.a.b.1.1		1
20.3	even	4		4560.2.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.a.1.1	✓	1	5.3	odd	4
3420.2.a.b.1.1		1	15.8	even	4
4560.2.a.u.1.1		1	20.3	even	4
5700.2.a.r.1.1		1	5.2	odd	4
5700.2.f.l.3649.1		2	5.4	even	2	inner
5700.2.f.l.3649.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5700.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} +4.00000 q^{11} -2.00000i q^{17} -1.00000 q^{19} +2.00000 q^{21} +2.00000i q^{23} -1.00000i q^{27} +6.00000 q^{29} +4.00000i q^{33} -8.00000i q^{37} -2.00000 q^{41} +6.00000i q^{43} -2.00000i q^{47} +3.00000 q^{49} +2.00000 q^{51} +4.00000i q^{53} -1.00000i q^{57} -4.00000 q^{59} -10.0000 q^{61} +2.00000i q^{63} -12.0000i q^{67} -2.00000 q^{69} +8.00000 q^{71} +2.00000i q^{73} -8.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} -14.0000i q^{83} +6.00000i q^{87} +2.00000 q^{89} -4.00000i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 8 q^{11} - 2 q^{19} + 4 q^{21} + 12 q^{29} - 4 q^{41} + 6 q^{49} + 4 q^{51} - 8 q^{59} - 20 q^{61} - 4 q^{69} + 16 q^{71} + 16 q^{79} + 2 q^{81} + 4 q^{89} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 8 * q^11 - 2 * q^19 + 4 * q^21 + 12 * q^29 - 4 * q^41 + 6 * q^49 + 4 * q^51 - 8 * q^59 - 20 * q^61 - 4 * q^69 + 16 * q^71 + 16 * q^79 + 2 * q^81 + 4 * q^89 - 8 * q^99

Introduction

L-functions

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