LMFDB - Embedded newform 185.2.c.a.36.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [185,2,Mod(36,185)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(185, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("185.36");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 185 = 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	185.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.47723243739$
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:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			36.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	185.36
Dual form			185.2.c.a.36.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.00000 q^{4} -1.00000i q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.00000 q^{4} -1.00000i q^{5} +1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} +2.00000 q^{12} -1.00000i q^{15} +4.00000 q^{16} +6.00000i q^{17} -6.00000i q^{19} -2.00000i q^{20} +1.00000 q^{21} +6.00000i q^{23} -1.00000 q^{25} -5.00000 q^{27} +2.00000 q^{28} -6.00000i q^{29} +6.00000i q^{31} -3.00000 q^{33} -1.00000i q^{35} -4.00000 q^{36} +(1.00000 - 6.00000i) q^{37} -3.00000 q^{41} +6.00000i q^{43} -6.00000 q^{44} +2.00000i q^{45} -3.00000 q^{47} +4.00000 q^{48} -6.00000 q^{49} +6.00000i q^{51} -9.00000 q^{53} +3.00000i q^{55} -6.00000i q^{57} -12.0000i q^{59} -2.00000i q^{60} +6.00000i q^{61} -2.00000 q^{63} +8.00000 q^{64} +4.00000 q^{67} +12.0000i q^{68} +6.00000i q^{69} +9.00000 q^{71} +7.00000 q^{73} -1.00000 q^{75} -12.0000i q^{76} -3.00000 q^{77} -12.0000i q^{79} -4.00000i q^{80} +1.00000 q^{81} +15.0000 q^{83} +2.00000 q^{84} +6.00000 q^{85} -6.00000i q^{87} +12.0000i q^{92} +6.00000i q^{93} -6.00000 q^{95} +6.00000i q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{4} + 2 q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 4 * q^4 + 2 * q^7 - 4 * q^9	$ 2 q + 2 q^{3} + 4 q^{4} + 2 q^{7} - 4 q^{9} - 6 q^{11} + 4 q^{12} + 8 q^{16} + 2 q^{21} - 2 q^{25} - 10 q^{27} + 4 q^{28} - 6 q^{33} - 8 q^{36} + 2 q^{37} - 6 q^{41} - 12 q^{44} - 6 q^{47} + 8 q^{48} - 12 q^{49} - 18 q^{53} - 4 q^{63} + 16 q^{64} + 8 q^{67} + 18 q^{71} + 14 q^{73} - 2 q^{75} - 6 q^{77} + 2 q^{81} + 30 q^{83} + 4 q^{84} + 12 q^{85} - 12 q^{95} + 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^4 + 2 * q^7 - 4 * q^9 - 6 * q^11 + 4 * q^12 + 8 * q^16 + 2 * q^21 - 2 * q^25 - 10 * q^27 + 4 * q^28 - 6 * q^33 - 8 * q^36 + 2 * q^37 - 6 * q^41 - 12 * q^44 - 6 * q^47 + 8 * q^48 - 12 * q^49 - 18 * q^53 - 4 * q^63 + 16 * q^64 + 8 * q^67 + 18 * q^71 + 14 * q^73 - 2 * q^75 - 6 * q^77 + 2 * q^81 + 30 * q^83 + 4 * q^84 + 12 * q^85 - 12 * q^95 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/185\mathbb{Z}\right)^\times$.

$n$	$76$	$112$
$\chi(n)$	$-1$	$1$

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		1.00000			$0$
$2$		0			0		−1.00000			$\pi$
$3$	1.00000			0.577350			0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000			0.577350			0.288675	+	0.957427i	$0.406785\pi$
$4$	2.00000			1.00000
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	1.00000			0.377964			0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000			0.377964			0.188982	+	0.981981i	$0.439481\pi$
$8$		0			0
$9$	−2.00000			−0.666667
$10$		0			0
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$	2.00000			0.577350
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$	4.00000			1.00000
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		−	2.00000i		−	0.447214i
$21$	1.00000			0.218218
$22$		0			0
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−5.00000			−0.962250
$28$	2.00000			0.377964
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$			6.00000i			1.07763i	0.842424	+	0.538816i	$0.181128\pi$
$31$			6.00000i			1.07763i	−0.842424	+	0.538816i	$0.818872\pi$
$32$		0			0
$33$	−3.00000			−0.522233
$34$		0			0
$35$		−	1.00000i		−	0.169031i
$36$	−4.00000			−0.666667
$37$	1.00000	−	6.00000i	0.164399	−	0.986394i
$37$	1.00000	−	6.00000i	0.164399	−	0.986394i
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.00000			−0.468521			−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000			−0.468521			−0.234261	+	0.972174i	$0.575267\pi$
$42$		0			0
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	−6.00000			−0.904534
$45$			2.00000i			0.298142i
$46$		0			0
$47$	−3.00000			−0.437595			−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000			−0.437595			−0.218797	+	0.975770i	$0.570213\pi$
$48$	4.00000			0.577350
$49$	−6.00000			−0.857143
$50$		0			0
$51$			6.00000i			0.840168i
$52$		0			0
$53$	−9.00000			−1.23625			−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000			−1.23625			−0.618123	+	0.786082i	$0.712106\pi$
$54$		0			0
$55$			3.00000i			0.404520i
$56$		0			0
$57$		−	6.00000i		−	0.794719i
$58$		0			0
$59$		−	12.0000i		−	1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$		−	12.0000i		−	1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$		−	2.00000i		−	0.258199i
$61$			6.00000i			0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$			6.00000i			0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$		0			0
$63$	−2.00000			−0.251976
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	4.00000			0.488678			0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000			0.488678			0.244339	+	0.969690i	$0.421429\pi$
$68$			12.0000i			1.45521i
$69$			6.00000i			0.722315i
$70$		0			0
$71$	9.00000			1.06810			0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000			1.06810			0.534052	+	0.845452i	$0.320669\pi$
$72$		0			0
$73$	7.00000			0.819288			0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000			0.819288			0.409644	+	0.912245i	$0.365653\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		−	12.0000i		−	1.37649i
$77$	−3.00000			−0.341882
$78$		0			0
$79$		−	12.0000i		−	1.35011i	−0.737769	−	0.675053i	$-0.764121\pi$
$79$		−	12.0000i		−	1.35011i	0.737769	−	0.675053i	$-0.235879\pi$
$80$		−	4.00000i		−	0.447214i
$81$	1.00000			0.111111
$82$		0			0
$83$	15.0000			1.64646			0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000			1.64646			0.823232	+	0.567705i	$0.192169\pi$
$84$	2.00000			0.218218
$85$	6.00000			0.650791
$86$		0			0
$87$		−	6.00000i		−	0.643268i
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$			12.0000i			1.25109i
$93$			6.00000i			0.622171i
$94$		0			0
$95$	−6.00000			−0.615587
$96$		0			0
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$	−2.00000			−0.200000
$101$	−3.00000			−0.298511			−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000			−0.298511			−0.149256	+	0.988799i	$0.547688\pi$
$102$		0			0
$103$		−	6.00000i		−	0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$		−	6.00000i		−	0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$		0			0
$105$		−	1.00000i		−	0.0975900i
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$	−10.0000			−0.962250
$109$			12.0000i			1.14939i	0.818367	+	0.574696i	$0.194880\pi$
$109$			12.0000i			1.14939i	−0.818367	+	0.574696i	$0.805120\pi$
$110$		0			0
$111$	1.00000	−	6.00000i	0.0949158	−	0.569495i
$112$	4.00000			0.377964
$113$		−	12.0000i		−	1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$		−	12.0000i		−	1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$		0			0
$115$	6.00000			0.559503
$116$		−	12.0000i		−	1.11417i
$117$		0			0
$118$		0			0
$119$			6.00000i			0.550019i
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$	−3.00000			−0.270501
$124$			12.0000i			1.07763i
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	11.0000			0.976092			0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000			0.976092			0.488046	+	0.872818i	$0.337710\pi$
$128$		0			0
$129$			6.00000i			0.528271i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	−6.00000			−0.522233
$133$		−	6.00000i		−	0.520266i
$134$		0			0
$135$			5.00000i			0.430331i
$136$		0			0
$137$	−6.00000			−0.512615			−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000			−0.512615			−0.256307	+	0.966595i	$0.582506\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		−	2.00000i		−	0.169031i
$141$	−3.00000			−0.252646
$142$		0			0
$143$		0			0
$144$	−8.00000			−0.666667
$145$	−6.00000			−0.498273
$146$		0			0
$147$	−6.00000			−0.494872
$148$	2.00000	−	12.0000i	0.164399	−	0.986394i
$149$	3.00000			0.245770			0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000			0.245770			0.122885	+	0.992421i	$0.460785\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$		−	12.0000i		−	0.970143i
$154$		0			0
$155$	6.00000			0.481932
$156$		0			0
$157$	−23.0000			−1.83560			−0.917800	−	0.397043i	$-0.870036\pi$
$157$	−23.0000			−1.83560			−0.917800	+	0.397043i	$0.870036\pi$
$158$		0			0
$159$	−9.00000			−0.713746
$160$		0			0
$161$			6.00000i			0.472866i
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	−6.00000			−0.468521
$165$			3.00000i			0.233550i
$166$		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
$169$	13.0000			1.00000
$170$		0			0
$171$			12.0000i			0.917663i
$172$			12.0000i			0.914991i
$173$	−15.0000			−1.14043			−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000			−1.14043			−0.570214	+	0.821496i	$0.693140\pi$
$174$		0			0
$175$	−1.00000			−0.0755929
$176$	−12.0000			−0.904534
$177$		−	12.0000i		−	0.901975i
$178$		0			0
$179$			6.00000i			0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$			6.00000i			0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$			4.00000i			0.298142i
$181$	−11.0000			−0.817624			−0.408812	−	0.912619i	$-0.634057\pi$
$181$	−11.0000			−0.817624			−0.408812	+	0.912619i	$0.634057\pi$
$182$		0			0
$183$			6.00000i			0.443533i
$184$		0			0
$185$	−6.00000	−	1.00000i	−0.441129	−	0.0735215i
$186$		0			0
$187$		−	18.0000i		−	1.31629i
$188$	−6.00000			−0.437595
$189$	−5.00000			−0.363696
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	8.00000			0.577350
$193$		−	12.0000i		−	0.863779i	−0.901927	−	0.431889i	$-0.857847\pi$
$193$		−	12.0000i		−	0.863779i	0.901927	−	0.431889i	$-0.142153\pi$
$194$		0			0
$195$		0			0
$196$	−12.0000			−0.857143
$197$	21.0000			1.49619			0.748094	−	0.663593i	$-0.230969\pi$
$197$	21.0000			1.49619			0.748094	+	0.663593i	$0.230969\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	4.00000			0.282138
$202$		0			0
$203$		−	6.00000i		−	0.421117i
$204$			12.0000i			0.840168i
$205$			3.00000i			0.209529i
$206$		0			0
$207$		−	12.0000i		−	0.834058i
$208$		0			0
$209$			18.0000i			1.24509i
$210$		0			0
$211$	23.0000			1.58339			0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000			1.58339			0.791693	+	0.610920i	$0.209200\pi$
$212$	−18.0000			−1.23625
$213$	9.00000			0.616670
$214$		0			0
$215$	6.00000			0.409197
$216$		0			0
$217$			6.00000i			0.407307i
$218$		0			0
$219$	7.00000			0.473016
$220$			6.00000i			0.404520i
$221$		0			0
$222$		0			0
$223$	−19.0000			−1.27233			−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000			−1.27233			−0.636167	+	0.771551i	$0.719481\pi$
$224$		0			0
$225$	2.00000			0.133333
$226$		0			0
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		−	12.0000i		−	0.794719i
$229$	−13.0000			−0.859064			−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000			−0.859064			−0.429532	+	0.903052i	$0.641321\pi$
$230$		0			0
$231$	−3.00000			−0.197386
$232$		0			0
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$			3.00000i			0.195698i
$236$		−	24.0000i		−	1.56227i
$237$		−	12.0000i		−	0.779484i
$238$		0			0
$239$			12.0000i			0.776215i	0.921614	+	0.388108i	$0.126871\pi$
$239$			12.0000i			0.776215i	−0.921614	+	0.388108i	$0.873129\pi$
$240$		−	4.00000i		−	0.258199i
$241$			18.0000i			1.15948i	0.814801	+	0.579741i	$0.196846\pi$
$241$			18.0000i			1.15948i	−0.814801	+	0.579741i	$0.803154\pi$
$242$		0			0
$243$	16.0000			1.02640
$244$			12.0000i			0.768221i
$245$			6.00000i			0.383326i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	15.0000			0.950586
$250$		0			0
$251$		−	24.0000i		−	1.51487i	−0.652913	−	0.757433i	$-0.726453\pi$
$251$		−	24.0000i		−	1.51487i	0.652913	−	0.757433i	$-0.273547\pi$
$252$	−4.00000			−0.251976
$253$		−	18.0000i		−	1.13165i
$254$		0			0
$255$	6.00000			0.375735
$256$	16.0000			1.00000
$257$			18.0000i			1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$			18.0000i			1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$		0			0
$259$	1.00000	−	6.00000i	0.0621370	−	0.372822i
$260$		0			0
$261$			12.0000i			0.742781i
$262$		0			0
$263$	3.00000			0.184988			0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000			0.184988			0.0924940	+	0.995713i	$0.470516\pi$
$264$		0			0
$265$			9.00000i			0.552866i
$266$		0			0
$267$		0			0
$268$	8.00000			0.488678
$269$	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
$271$	−11.0000			−0.668202			−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000			−0.668202			−0.334101	+	0.942537i	$0.608433\pi$
$272$			24.0000i			1.45521i
$273$		0			0
$274$		0			0
$275$	3.00000			0.180907
$276$			12.0000i			0.722315i
$277$		−	30.0000i		−	1.80253i	−0.433273	−	0.901263i	$-0.642641\pi$
$277$		−	30.0000i		−	1.80253i	0.433273	−	0.901263i	$-0.357359\pi$
$278$		0			0
$279$		−	12.0000i		−	0.718421i
$280$		0			0
$281$		−	18.0000i		−	1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$		−	18.0000i		−	1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$		0			0
$283$			24.0000i			1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$			24.0000i			1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$	18.0000			1.06810
$285$	−6.00000			−0.355409
$286$		0			0
$287$	−3.00000			−0.177084
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$			6.00000i			0.351726i
$292$	14.0000			0.819288
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		0			0
$295$	−12.0000			−0.698667
$296$		0			0
$297$	15.0000			0.870388
$298$		0			0
$299$		0			0
$300$	−2.00000			−0.115470
$301$			6.00000i			0.345834i
$302$		0			0
$303$	−3.00000			−0.172345
$304$		−	24.0000i		−	1.37649i
$305$	6.00000			0.343559
$306$		0			0
$307$	11.0000			0.627803			0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000			0.627803			0.313902	+	0.949456i	$0.398364\pi$
$308$	−6.00000			−0.341882
$309$		−	6.00000i		−	0.341328i
$310$		0			0
$311$		−	18.0000i		−	1.02069i	−0.859971	−	0.510343i	$-0.829518\pi$
$311$		−	18.0000i		−	1.02069i	0.859971	−	0.510343i	$-0.170482\pi$
$312$		0			0
$313$			18.0000i			1.01742i	0.860938	+	0.508710i	$0.169877\pi$
$313$			18.0000i			1.01742i	−0.860938	+	0.508710i	$0.830123\pi$
$314$		0			0
$315$			2.00000i			0.112687i
$316$		−	24.0000i		−	1.35011i
$317$	−18.0000			−1.01098			−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000			−1.01098			−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$			18.0000i			1.00781i
$320$		−	8.00000i		−	0.447214i
$321$	12.0000			0.669775
$322$		0			0
$323$	36.0000			2.00309
$324$	2.00000			0.111111
$325$		0			0
$326$		0			0
$327$			12.0000i			0.663602i
$328$		0			0
$329$	−3.00000			−0.165395
$330$		0			0
$331$			30.0000i			1.64895i	0.565899	+	0.824475i	$0.308529\pi$
$331$			30.0000i			1.64895i	−0.565899	+	0.824475i	$0.691471\pi$
$332$	30.0000			1.64646
$333$	−2.00000	+	12.0000i	−0.109599	+	0.657596i
$334$		0			0
$335$		−	4.00000i		−	0.218543i
$336$	4.00000			0.218218
$337$	−13.0000			−0.708155			−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000			−0.708155			−0.354078	+	0.935216i	$0.615205\pi$
$338$		0			0
$339$		−	12.0000i		−	0.651751i
$340$	12.0000			0.650791
$341$		−	18.0000i		−	0.974755i
$342$		0			0
$343$	−13.0000			−0.701934
$344$		0			0
$345$	6.00000			0.323029
$346$		0			0
$347$			6.00000i			0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$			6.00000i			0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$		−	12.0000i		−	0.643268i
$349$	−26.0000			−1.39175			−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000			−1.39175			−0.695874	+	0.718164i	$0.744983\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	24.0000i		−	1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$		−	24.0000i		−	1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$		0			0
$355$		−	9.00000i		−	0.477670i
$356$		0			0
$357$			6.00000i			0.317554i
$358$		0			0
$359$	−3.00000			−0.158334			−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000			−0.158334			−0.0791670	+	0.996861i	$0.525226\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$	−2.00000			−0.104973
$364$		0			0
$365$		−	7.00000i		−	0.366397i
$366$		0			0
$367$	−28.0000			−1.46159			−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000			−1.46159			−0.730794	+	0.682598i	$0.760850\pi$
$368$			24.0000i			1.25109i
$369$	6.00000			0.312348
$370$		0			0
$371$	−9.00000			−0.467257
$372$			12.0000i			0.622171i
$373$	−23.0000			−1.19089			−0.595447	−	0.803394i	$-0.703025\pi$
$373$	−23.0000			−1.19089			−0.595447	+	0.803394i	$0.703025\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	7.00000			0.359566			0.179783	−	0.983706i	$-0.442460\pi$
$379$	7.00000			0.359566			0.179783	+	0.983706i	$0.442460\pi$
$380$	−12.0000			−0.615587
$381$	11.0000			0.563547
$382$		0			0
$383$			30.0000i			1.53293i	0.642287	+	0.766464i	$0.277986\pi$
$383$			30.0000i			1.53293i	−0.642287	+	0.766464i	$0.722014\pi$
$384$		0			0
$385$			3.00000i			0.152894i
$386$		0			0
$387$		−	12.0000i		−	0.609994i
$388$			12.0000i			0.609208i
$389$			36.0000i			1.82527i	0.408773	+	0.912636i	$0.365957\pi$
$389$			36.0000i			1.82527i	−0.408773	+	0.912636i	$0.634043\pi$
$390$		0			0
$391$	−36.0000			−1.82060
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−12.0000			−0.603786
$396$	12.0000			0.603023
$397$	−29.0000			−1.45547			−0.727734	−	0.685859i	$-0.759427\pi$
$397$	−29.0000			−1.45547			−0.727734	+	0.685859i	$0.759427\pi$
$398$		0			0
$399$		−	6.00000i		−	0.300376i
$400$	−4.00000			−0.200000
$401$		−	18.0000i		−	0.898877i	−0.893311	−	0.449439i	$-0.851624\pi$
$401$		−	18.0000i		−	0.898877i	0.893311	−	0.449439i	$-0.148376\pi$
$402$		0			0
$403$		0			0
$404$	−6.00000			−0.298511
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$	−3.00000	+	18.0000i	−0.148704	+	0.892227i
$408$		0			0
$409$		−	36.0000i		−	1.78009i	−0.455877	−	0.890043i	$-0.650674\pi$
$409$		−	36.0000i		−	1.78009i	0.455877	−	0.890043i	$-0.349326\pi$
$410$		0			0
$411$	−6.00000			−0.295958
$412$		−	12.0000i		−	0.591198i
$413$		−	12.0000i		−	0.590481i
$414$		0			0
$415$		−	15.0000i		−	0.736321i
$416$		0			0
$417$	4.00000			0.195881
$418$		0			0
$419$	−21.0000			−1.02592			−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000			−1.02592			−0.512959	+	0.858413i	$0.671451\pi$
$420$		−	2.00000i		−	0.0975900i
$421$			30.0000i			1.46211i	0.682318	+	0.731055i	$0.260972\pi$
$421$			30.0000i			1.46211i	−0.682318	+	0.731055i	$0.739028\pi$
$422$		0			0
$423$	6.00000			0.291730
$424$		0			0
$425$		−	6.00000i		−	0.291043i
$426$		0			0
$427$			6.00000i			0.290360i
$428$	24.0000			1.16008
$429$		0			0
$430$		0			0
$431$			12.0000i			0.578020i	0.957326	+	0.289010i	$0.0933260\pi$
$431$			12.0000i			0.578020i	−0.957326	+	0.289010i	$0.906674\pi$
$432$	−20.0000			−0.962250
$433$	−29.0000			−1.39365			−0.696826	−	0.717241i	$-0.745405\pi$
$433$	−29.0000			−1.39365			−0.696826	+	0.717241i	$0.745405\pi$
$434$		0			0
$435$	−6.00000			−0.287678
$436$			24.0000i			1.14939i
$437$	36.0000			1.72211
$438$		0			0
$439$		−	12.0000i		−	0.572729i	−0.958121	−	0.286364i	$-0.907553\pi$
$439$		−	12.0000i		−	0.572729i	0.958121	−	0.286364i	$-0.0924468\pi$
$440$		0			0
$441$	12.0000			0.571429
$442$		0			0
$443$	−27.0000			−1.28281			−0.641404	−	0.767203i	$-0.721648\pi$
$443$	−27.0000			−1.28281			−0.641404	+	0.767203i	$0.721648\pi$
$444$	2.00000	−	12.0000i	0.0949158	−	0.569495i
$445$		0			0
$446$		0			0
$447$	3.00000			0.141895
$448$	8.00000			0.377964
$449$			12.0000i			0.566315i	0.959073	+	0.283158i	$0.0913819\pi$
$449$			12.0000i			0.566315i	−0.959073	+	0.283158i	$0.908618\pi$
$450$		0			0
$451$	9.00000			0.423793
$452$		−	24.0000i		−	1.12887i
$453$	8.00000			0.375873
$454$		0			0
$455$		0			0
$456$		0			0
$457$			18.0000i			0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$			18.0000i			0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$		0			0
$459$		−	30.0000i		−	1.40028i
$460$	12.0000			0.559503
$461$			6.00000i			0.279448i	0.990190	+	0.139724i	$0.0446215\pi$
$461$			6.00000i			0.279448i	−0.990190	+	0.139724i	$0.955378\pi$
$462$		0			0
$463$			24.0000i			1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$			24.0000i			1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$		−	24.0000i		−	1.11417i
$465$	6.00000			0.278243
$466$		0			0
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$		0			0
$471$	−23.0000			−1.05978
$472$		0			0
$473$		−	18.0000i		−	0.827641i
$474$		0			0
$475$			6.00000i			0.275299i
$476$			12.0000i			0.550019i
$477$	18.0000			0.824163
$478$		0			0
$479$		−	6.00000i		−	0.274147i	−0.990561	−	0.137073i	$-0.956230\pi$
$479$		−	6.00000i		−	0.274147i	0.990561	−	0.137073i	$-0.0437697\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$			6.00000i			0.273009i
$484$	−4.00000			−0.181818
$485$	6.00000			0.272446
$486$		0			0
$487$		−	18.0000i		−	0.815658i	−0.913058	−	0.407829i	$-0.866286\pi$
$487$		−	18.0000i		−	0.815658i	0.913058	−	0.407829i	$-0.133714\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	36.0000			1.62466			0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000			1.62466			0.812329	+	0.583200i	$0.198200\pi$
$492$	−6.00000			−0.270501
$493$	36.0000			1.62136
$494$		0			0
$495$		−	6.00000i		−	0.269680i
$496$			24.0000i			1.07763i
$497$	9.00000			0.403705
$498$		0			0
$499$		−	42.0000i		−	1.88018i	−0.340929	−	0.940089i	$-0.610742\pi$
$499$		−	42.0000i		−	1.88018i	0.340929	−	0.940089i	$-0.389258\pi$
$500$			2.00000i			0.0894427i
$501$			12.0000i			0.536120i
$502$		0			0
$503$		−	18.0000i		−	0.802580i	−0.915951	−	0.401290i	$-0.868562\pi$
$503$		−	18.0000i		−	0.802580i	0.915951	−	0.401290i	$-0.131438\pi$
$504$		0			0
$505$			3.00000i			0.133498i
$506$		0			0
$507$	13.0000			0.577350
$508$	22.0000			0.976092
$509$	15.0000			0.664863			0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000			0.664863			0.332432	+	0.943127i	$0.392131\pi$
$510$		0			0
$511$	7.00000			0.309662
$512$		0			0
$513$			30.0000i			1.32453i
$514$		0			0
$515$	−6.00000			−0.264392
$516$			12.0000i			0.528271i
$517$	9.00000			0.395820
$518$		0			0
$519$	−15.0000			−0.658427
$520$		0			0
$521$	33.0000			1.44576			0.722878	−	0.690976i	$-0.242819\pi$
$521$	33.0000			1.44576			0.722878	+	0.690976i	$0.242819\pi$
$522$		0			0
$523$			6.00000i			0.262362i	0.991358	+	0.131181i	$0.0418769\pi$
$523$			6.00000i			0.262362i	−0.991358	+	0.131181i	$0.958123\pi$
$524$		0			0
$525$	−1.00000			−0.0436436
$526$		0			0
$527$	−36.0000			−1.56818
$528$	−12.0000			−0.522233
$529$	−13.0000			−0.565217
$530$		0			0
$531$			24.0000i			1.04151i
$532$		−	12.0000i		−	0.520266i
$533$		0			0
$534$		0			0
$535$		−	12.0000i		−	0.518805i
$536$		0			0
$537$			6.00000i			0.258919i
$538$		0			0
$539$	18.0000			0.775315
$540$			10.0000i			0.430331i
$541$		−	6.00000i		−	0.257960i	−0.991647	−	0.128980i	$-0.958830\pi$
$541$		−	6.00000i		−	0.257960i	0.991647	−	0.128980i	$-0.0411703\pi$
$542$		0			0
$543$	−11.0000			−0.472055
$544$		0			0
$545$	12.0000			0.514024
$546$		0			0
$547$		−	12.0000i		−	0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$		−	12.0000i		−	0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$	−12.0000			−0.512615
$549$		−	12.0000i		−	0.512148i
$550$		0			0
$551$	−36.0000			−1.53365
$552$		0			0
$553$		−	12.0000i		−	0.510292i
$554$		0			0
$555$	−6.00000	−	1.00000i	−0.254686	−	0.0424476i
$556$	8.00000			0.339276
$557$		−	6.00000i		−	0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$		−	6.00000i		−	0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$		0			0
$559$		0			0
$560$		−	4.00000i		−	0.169031i
$561$		−	18.0000i		−	0.759961i
$562$		0			0
$563$			36.0000i			1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$			36.0000i			1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	−6.00000			−0.252646
$565$	−12.0000			−0.504844
$566$		0			0
$567$	1.00000			0.0419961
$568$		0			0
$569$			6.00000i			0.251533i	0.992060	+	0.125767i	$0.0401390\pi$
$569$			6.00000i			0.251533i	−0.992060	+	0.125767i	$0.959861\pi$
$570$		0			0
$571$	−5.00000			−0.209243			−0.104622	−	0.994512i	$-0.533363\pi$
$571$	−5.00000			−0.209243			−0.104622	+	0.994512i	$0.533363\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	6.00000i		−	0.250217i
$576$	−16.0000			−0.666667
$577$			6.00000i			0.249783i	0.992170	+	0.124892i	$0.0398583\pi$
$577$			6.00000i			0.249783i	−0.992170	+	0.124892i	$0.960142\pi$
$578$		0			0
$579$		−	12.0000i		−	0.498703i
$580$	−12.0000			−0.498273
$581$	15.0000			0.622305
$582$		0			0
$583$	27.0000			1.11823
$584$		0			0
$585$		0			0
$586$		0			0
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	−12.0000			−0.494872
$589$	36.0000			1.48335
$590$		0			0
$591$	21.0000			0.863825
$592$	4.00000	−	24.0000i	0.164399	−	0.986394i
$593$	9.00000			0.369586			0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000			0.369586			0.184793	+	0.982777i	$0.440839\pi$
$594$		0			0
$595$	6.00000			0.245976
$596$	6.00000			0.245770
$597$		0			0
$598$		0			0
$599$	−3.00000			−0.122577			−0.0612883	−	0.998120i	$-0.519521\pi$
$599$	−3.00000			−0.122577			−0.0612883	+	0.998120i	$0.519521\pi$
$600$		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
$603$	−8.00000			−0.325785
$604$	16.0000			0.651031
$605$			2.00000i			0.0813116i
$606$		0			0
$607$			30.0000i			1.21766i	0.793300	+	0.608831i	$0.208361\pi$
$607$			30.0000i			1.21766i	−0.793300	+	0.608831i	$0.791639\pi$
$608$		0			0
$609$		−	6.00000i		−	0.243132i
$610$		0			0
$611$		0			0
$612$		−	24.0000i		−	0.970143i
$613$	47.0000			1.89831			0.949156	−	0.314806i	$-0.101939\pi$
$613$	47.0000			1.89831			0.949156	+	0.314806i	$0.101939\pi$
$614$		0			0
$615$			3.00000i			0.120972i
$616$		0			0
$617$	33.0000			1.32853			0.664265	−	0.747497i	$-0.268745\pi$
$617$	33.0000			1.32853			0.664265	+	0.747497i	$0.268745\pi$
$618$		0			0
$619$	−19.0000			−0.763674			−0.381837	−	0.924230i	$-0.624709\pi$
$619$	−19.0000			−0.763674			−0.381837	+	0.924230i	$0.624709\pi$
$620$	12.0000			0.481932
$621$		−	30.0000i		−	1.20386i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			18.0000i			0.718851i
$628$	−46.0000			−1.83560
$629$	36.0000	+	6.00000i	1.43541	+	0.239236i
$630$		0			0
$631$		−	36.0000i		−	1.43314i	−0.697517	−	0.716569i	$-0.745712\pi$
$631$		−	36.0000i		−	1.43314i	0.697517	−	0.716569i	$-0.254288\pi$
$632$		0			0
$633$	23.0000			0.914168
$634$		0			0
$635$		−	11.0000i		−	0.436522i
$636$	−18.0000			−0.713746
$637$		0			0
$638$		0			0
$639$	−18.0000			−0.712069
$640$		0			0
$641$	−45.0000			−1.77739			−0.888697	−	0.458496i	$-0.848388\pi$
$641$	−45.0000			−1.77739			−0.888697	+	0.458496i	$0.848388\pi$
$642$		0			0
$643$			18.0000i			0.709851i	0.934895	+	0.354925i	$0.115494\pi$
$643$			18.0000i			0.709851i	−0.934895	+	0.354925i	$0.884506\pi$
$644$			12.0000i			0.472866i
$645$	6.00000			0.236250
$646$		0			0
$647$			24.0000i			0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$			24.0000i			0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$		0			0
$649$			36.0000i			1.41312i
$650$		0			0
$651$			6.00000i			0.235159i
$652$		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
$656$	−12.0000			−0.468521
$657$	−14.0000			−0.546192
$658$		0			0
$659$	−33.0000			−1.28550			−0.642749	−	0.766077i	$-0.722206\pi$
$659$	−33.0000			−1.28550			−0.642749	+	0.766077i	$0.722206\pi$
$660$			6.00000i			0.233550i
$661$		−	12.0000i		−	0.466746i	−0.972387	−	0.233373i	$-0.925024\pi$
$661$		−	12.0000i		−	0.466746i	0.972387	−	0.233373i	$-0.0749763\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.00000			−0.232670
$666$		0			0
$667$	36.0000			1.39393
$668$			24.0000i			0.928588i
$669$	−19.0000			−0.734582
$670$		0			0
$671$		−	18.0000i		−	0.694882i
$672$		0			0
$673$	1.00000			0.0385472			0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000			0.0385472			0.0192736	+	0.999814i	$0.493865\pi$
$674$		0			0
$675$	5.00000			0.192450
$676$	26.0000			1.00000
$677$	−15.0000			−0.576497			−0.288248	−	0.957556i	$-0.593073\pi$
$677$	−15.0000			−0.576497			−0.288248	+	0.957556i	$0.593073\pi$
$678$		0			0
$679$			6.00000i			0.230259i
$680$		0			0
$681$		−	12.0000i		−	0.459841i
$682$		0			0
$683$		−	24.0000i		−	0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$		−	24.0000i		−	0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$			24.0000i			0.917663i
$685$			6.00000i			0.229248i
$686$		0			0
$687$	−13.0000			−0.495981
$688$			24.0000i			0.914991i
$689$		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	$0.321221\pi$
$692$	−30.0000			−1.14043
$693$	6.00000			0.227921
$694$		0			0
$695$		−	4.00000i		−	0.151729i
$696$		0			0
$697$		−	18.0000i		−	0.681799i
$698$		0			0
$699$	6.00000			0.226941
$700$	−2.00000			−0.0755929
$701$		−	48.0000i		−	1.81293i	−0.422276	−	0.906467i	$-0.638769\pi$
$701$		−	48.0000i		−	1.81293i	0.422276	−	0.906467i	$-0.361231\pi$
$702$		0			0
$703$	−36.0000	−	6.00000i	−1.35777	−	0.226294i
$704$	−24.0000			−0.904534
$705$			3.00000i			0.112987i
$706$		0			0
$707$	−3.00000			−0.112827
$708$		−	24.0000i		−	0.901975i
$709$			30.0000i			1.12667i	0.826227	+	0.563337i	$0.190483\pi$
$709$			30.0000i			1.12667i	−0.826227	+	0.563337i	$0.809517\pi$
$710$		0			0
$711$			24.0000i			0.900070i
$712$		0			0
$713$	−36.0000			−1.34821
$714$		0			0
$715$		0			0
$716$			12.0000i			0.448461i
$717$			12.0000i			0.448148i
$718$		0			0
$719$	3.00000			0.111881			0.0559406	−	0.998434i	$-0.482184\pi$
$719$	3.00000			0.111881			0.0559406	+	0.998434i	$0.482184\pi$
$720$			8.00000i			0.298142i
$721$		−	6.00000i		−	0.223452i
$722$		0			0
$723$			18.0000i			0.669427i
$724$	−22.0000			−0.817624
$725$			6.00000i			0.222834i
$726$		0			0
$727$		−	12.0000i		−	0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$727$		−	12.0000i		−	0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−36.0000			−1.33151
$732$			12.0000i			0.443533i
$733$	−13.0000			−0.480166			−0.240083	−	0.970752i	$-0.577175\pi$
$733$	−13.0000			−0.480166			−0.240083	+	0.970752i	$0.577175\pi$
$734$		0			0
$735$			6.00000i			0.221313i
$736$		0			0
$737$	−12.0000			−0.442026
$738$		0			0
$739$	29.0000			1.06678			0.533391	−	0.845869i	$-0.320917\pi$
$739$	29.0000			1.06678			0.533391	+	0.845869i	$0.320917\pi$
$740$	−12.0000	−	2.00000i	−0.441129	−	0.0735215i
$741$		0			0
$742$		0			0
$743$	−27.0000			−0.990534			−0.495267	−	0.868741i	$-0.664930\pi$
$743$	−27.0000			−0.990534			−0.495267	+	0.868741i	$0.664930\pi$
$744$		0			0
$745$		−	3.00000i		−	0.109911i
$746$		0			0
$747$	−30.0000			−1.09764
$748$		−	36.0000i		−	1.31629i
$749$	12.0000			0.438470
$750$		0			0
$751$	−5.00000			−0.182453			−0.0912263	−	0.995830i	$-0.529079\pi$
$751$	−5.00000			−0.182453			−0.0912263	+	0.995830i	$0.529079\pi$
$752$	−12.0000			−0.437595
$753$		−	24.0000i		−	0.874609i
$754$		0			0
$755$		−	8.00000i		−	0.291150i
$756$	−10.0000			−0.363696
$757$			42.0000i			1.52652i	0.646094	+	0.763258i	$0.276401\pi$
$757$			42.0000i			1.52652i	−0.646094	+	0.763258i	$0.723599\pi$
$758$		0			0
$759$		−	18.0000i		−	0.653359i
$760$		0			0
$761$	−45.0000			−1.63125			−0.815624	−	0.578582i	$-0.803606\pi$
$761$	−45.0000			−1.63125			−0.815624	+	0.578582i	$0.803606\pi$
$762$		0			0
$763$			12.0000i			0.434429i
$764$		0			0
$765$	−12.0000			−0.433861
$766$		0			0
$767$		0			0
$768$	16.0000			0.577350
$769$		−	18.0000i		−	0.649097i	−0.945869	−	0.324548i	$-0.894788\pi$
$769$		−	18.0000i		−	0.649097i	0.945869	−	0.324548i	$-0.105212\pi$
$770$		0			0
$771$			18.0000i			0.648254i
$772$		−	24.0000i		−	0.863779i
$773$	21.0000			0.755318			0.377659	−	0.925945i	$-0.376729\pi$
$773$	21.0000			0.755318			0.377659	+	0.925945i	$0.376729\pi$
$774$		0			0
$775$		−	6.00000i		−	0.215526i
$776$		0			0
$777$	1.00000	−	6.00000i	0.0358748	−	0.215249i
$778$		0			0
$779$			18.0000i			0.644917i
$780$		0			0
$781$	−27.0000			−0.966136
$782$		0			0
$783$			30.0000i			1.07211i
$784$	−24.0000			−0.857143
$785$			23.0000i			0.820905i
$786$		0			0
$787$	5.00000			0.178231			0.0891154	−	0.996021i	$-0.471596\pi$
$787$	5.00000			0.178231			0.0891154	+	0.996021i	$0.471596\pi$
$788$	42.0000			1.49619
$789$	3.00000			0.106803
$790$		0			0
$791$		−	12.0000i		−	0.426671i
$792$		0			0
$793$		0			0
$794$		0			0
$795$			9.00000i			0.319197i
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		−	18.0000i		−	0.636794i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−21.0000			−0.741074
$804$	8.00000			0.282138
$805$	6.00000			0.211472
$806$		0			0
$807$	18.0000			0.633630
$808$		0			0
$809$			30.0000i			1.05474i	0.849635	+	0.527372i	$0.176823\pi$
$809$			30.0000i			1.05474i	−0.849635	+	0.527372i	$0.823177\pi$
$810$		0			0
$811$	43.0000			1.50993			0.754967	−	0.655763i	$-0.227653\pi$
$811$	43.0000			1.50993			0.754967	+	0.655763i	$0.227653\pi$
$812$		−	12.0000i		−	0.421117i
$813$	−11.0000			−0.385787
$814$		0			0
$815$		0			0
$816$			24.0000i			0.840168i
$817$	36.0000			1.25948
$818$		0			0
$819$		0			0
$820$			6.00000i			0.209529i
$821$	33.0000			1.15171			0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000			1.15171			0.575854	+	0.817553i	$0.304670\pi$
$822$		0			0
$823$	32.0000			1.11545			0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000			1.11545			0.557725	+	0.830026i	$0.311674\pi$
$824$		0			0
$825$	3.00000			0.104447
$826$		0			0
$827$		−	42.0000i		−	1.46048i	−0.683189	−	0.730242i	$-0.739408\pi$
$827$		−	42.0000i		−	1.46048i	0.683189	−	0.730242i	$-0.260592\pi$
$828$		−	24.0000i		−	0.834058i
$829$			48.0000i			1.66711i	0.552437	+	0.833554i	$0.313698\pi$
$829$			48.0000i			1.66711i	−0.552437	+	0.833554i	$0.686302\pi$
$830$		0			0
$831$		−	30.0000i		−	1.04069i
$832$		0			0
$833$		−	36.0000i		−	1.24733i
$834$		0			0
$835$	12.0000			0.415277
$836$			36.0000i			1.24509i
$837$		−	30.0000i		−	1.03695i
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$		−	18.0000i		−	0.619953i
$844$	46.0000			1.58339
$845$		−	13.0000i		−	0.447214i
$846$		0			0
$847$	−2.00000			−0.0687208
$848$	−36.0000			−1.23625
$849$			24.0000i			0.823678i
$850$		0			0
$851$	36.0000	+	6.00000i	1.23406	+	0.205677i
$852$	18.0000			0.616670
$853$		−	54.0000i		−	1.84892i	−0.381273	−	0.924462i	$-0.624514\pi$
$853$		−	54.0000i		−	1.84892i	0.381273	−	0.924462i	$-0.375486\pi$
$854$		0			0
$855$	12.0000			0.410391
$856$		0			0
$857$			48.0000i			1.63965i	0.572615	+	0.819824i	$0.305929\pi$
$857$			48.0000i			1.63965i	−0.572615	+	0.819824i	$0.694071\pi$
$858$		0			0
$859$			6.00000i			0.204717i	0.994748	+	0.102359i	$0.0326389\pi$
$859$			6.00000i			0.204717i	−0.994748	+	0.102359i	$0.967361\pi$
$860$	12.0000			0.409197
$861$	−3.00000			−0.102240
$862$		0			0
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$		0			0
$865$			15.0000i			0.510015i
$866$		0			0
$867$	−19.0000			−0.645274
$868$			12.0000i			0.407307i
$869$			36.0000i			1.22122i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	12.0000i		−	0.406138i
$874$		0			0
$875$			1.00000i			0.0338062i
$876$	14.0000			0.473016
$877$	22.0000			0.742887			0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000			0.742887			0.371444	+	0.928456i	$0.378863\pi$
$878$		0			0
$879$	−6.00000			−0.202375
$880$			12.0000i			0.404520i
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$		−	48.0000i		−	1.61533i	−0.589643	−	0.807664i	$-0.700731\pi$
$883$		−	48.0000i		−	1.61533i	0.589643	−	0.807664i	$-0.299269\pi$
$884$		0			0
$885$	−12.0000			−0.403376
$886$		0			0
$887$	33.0000			1.10803			0.554016	−	0.832506i	$-0.313095\pi$
$887$	33.0000			1.10803			0.554016	+	0.832506i	$0.313095\pi$
$888$		0			0
$889$	11.0000			0.368928
$890$		0			0
$891$	−3.00000			−0.100504
$892$	−38.0000			−1.27233
$893$			18.0000i			0.602347i
$894$		0			0
$895$	6.00000			0.200558
$896$		0			0
$897$		0			0
$898$		0			0
$899$	36.0000			1.20067
$900$	4.00000			0.133333
$901$		−	54.0000i		−	1.79900i
$902$		0			0
$903$			6.00000i			0.199667i
$904$		0			0
$905$			11.0000i			0.365652i
$906$		0			0
$907$		−	6.00000i		−	0.199227i	−0.995026	−	0.0996134i	$-0.968239\pi$
$907$		−	6.00000i		−	0.199227i	0.995026	−	0.0996134i	$-0.0317606\pi$
$908$		−	24.0000i		−	0.796468i
$909$	6.00000			0.199007
$910$		0			0
$911$		−	6.00000i		−	0.198789i	−0.995048	−	0.0993944i	$-0.968309\pi$
$911$		−	6.00000i		−	0.198789i	0.995048	−	0.0993944i	$-0.0316906\pi$
$912$		−	24.0000i		−	0.794719i
$913$	−45.0000			−1.48928
$914$		0			0
$915$	6.00000			0.198354
$916$	−26.0000			−0.859064
$917$		0			0
$918$		0			0
$919$			24.0000i			0.791687i	0.918318	+	0.395843i	$0.129548\pi$
$919$			24.0000i			0.791687i	−0.918318	+	0.395843i	$0.870452\pi$
$920$		0			0
$921$	11.0000			0.362462
$922$		0			0
$923$		0			0
$924$	−6.00000			−0.197386
$925$	−1.00000	+	6.00000i	−0.0328798	+	0.197279i
$926$		0			0
$927$			12.0000i			0.394132i
$928$		0			0
$929$	−6.00000			−0.196854			−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000			−0.196854			−0.0984268	+	0.995144i	$0.531381\pi$
$930$		0			0
$931$			36.0000i			1.17985i
$932$	12.0000			0.393073
$933$		−	18.0000i		−	0.589294i
$934$		0			0
$935$	−18.0000			−0.588663
$936$		0			0
$937$	−25.0000			−0.816714			−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000			−0.816714			−0.408357	+	0.912822i	$0.633898\pi$
$938$		0			0
$939$			18.0000i			0.587408i
$940$			6.00000i			0.195698i
$941$	18.0000			0.586783			0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000			0.586783			0.293392	+	0.955992i	$0.405216\pi$
$942$		0			0
$943$		−	18.0000i		−	0.586161i
$944$		−	48.0000i		−	1.56227i
$945$			5.00000i			0.162650i
$946$		0			0
$947$		−	18.0000i		−	0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$		−	18.0000i		−	0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$		−	24.0000i		−	0.779484i
$949$		0			0
$950$		0			0
$951$	−18.0000			−0.583690
$952$		0			0
$953$	57.0000			1.84641			0.923206	−	0.384307i	$-0.125559\pi$
$953$	57.0000			1.84641			0.923206	+	0.384307i	$0.125559\pi$
$954$		0			0
$955$		0			0
$956$			24.0000i			0.776215i
$957$			18.0000i			0.581857i
$958$		0			0
$959$	−6.00000			−0.193750
$960$		−	8.00000i		−	0.258199i
$961$	−5.00000			−0.161290
$962$		0			0
$963$	−24.0000			−0.773389
$964$			36.0000i			1.15948i
$965$	−12.0000			−0.386294
$966$		0			0
$967$			48.0000i			1.54358i	0.635880	+	0.771788i	$0.280637\pi$
$967$			48.0000i			1.54358i	−0.635880	+	0.771788i	$0.719363\pi$
$968$		0			0
$969$	36.0000			1.15649
$970$		0			0
$971$	−48.0000			−1.54039			−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000			−1.54039			−0.770197	+	0.637806i	$0.779842\pi$
$972$	32.0000			1.02640
$973$	4.00000			0.128234
$974$		0			0
$975$		0			0
$976$			24.0000i			0.768221i
$977$		−	48.0000i		−	1.53566i	−0.640656	−	0.767828i	$-0.721338\pi$
$977$		−	48.0000i		−	1.53566i	0.640656	−	0.767828i	$-0.278662\pi$
$978$		0			0
$979$		0			0
$980$			12.0000i			0.383326i
$981$		−	24.0000i		−	0.766261i
$982$		0			0
$983$	−3.00000			−0.0956851			−0.0478426	−	0.998855i	$-0.515235\pi$
$983$	−3.00000			−0.0956851			−0.0478426	+	0.998855i	$0.515235\pi$
$984$		0			0
$985$		−	21.0000i		−	0.669116i
$986$		0			0
$987$	−3.00000			−0.0954911
$988$		0			0
$989$	−36.0000			−1.14473
$990$		0			0
$991$			30.0000i			0.952981i	0.879180	+	0.476491i	$0.158091\pi$
$991$			30.0000i			0.952981i	−0.879180	+	0.476491i	$0.841909\pi$
$992$		0			0
$993$			30.0000i			0.952021i
$994$		0			0
$995$		0			0
$996$	30.0000			0.950586
$997$		−	54.0000i		−	1.71020i	−0.518465	−	0.855099i	$-0.673497\pi$
$997$		−	54.0000i		−	1.71020i	0.518465	−	0.855099i	$-0.326503\pi$
$998$		0			0
$999$	−5.00000	+	30.0000i	−0.158193	+	0.949158i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	185.2.c.a.36.1	✓	2
3.2	odd	2		1665.2.e.b.406.2		2
4.3	odd	2		2960.2.p.c.961.1		2
5.2	odd	4		925.2.d.b.924.1		2
5.3	odd	4		925.2.d.c.924.2		2
5.4	even	2		925.2.c.a.776.1		2
37.6	odd	4		6845.2.a.d.1.1		1
37.31	odd	4		6845.2.a.c.1.1		1
37.36	even	2	inner	185.2.c.a.36.2	yes	2
111.110	odd	2		1665.2.e.b.406.1		2
148.147	odd	2		2960.2.p.c.961.2		2
185.73	odd	4		925.2.d.b.924.2		2
185.147	odd	4		925.2.d.c.924.1		2
185.184	even	2		925.2.c.a.776.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.c.a.36.1	✓	2	1.1	even	1	trivial
185.2.c.a.36.2	yes	2	37.36	even	2	inner
925.2.c.a.776.1		2	5.4	even	2
925.2.c.a.776.2		2	185.184	even	2
925.2.d.b.924.1		2	5.2	odd	4
925.2.d.b.924.2		2	185.73	odd	4
925.2.d.c.924.1		2	185.147	odd	4
925.2.d.c.924.2		2	5.3	odd	4
1665.2.e.b.406.1		2	111.110	odd	2
1665.2.e.b.406.2		2	3.2	odd	2
2960.2.p.c.961.1		2	4.3	odd	2
2960.2.p.c.961.2		2	148.147	odd	2
6845.2.a.c.1.1		1	37.31	odd	4
6845.2.a.d.1.1		1	37.6	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:	\( N \)	\(=\)	\( 185 = 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	185.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.00000 q^{4} -1.00000i q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.00000 q^{4} -1.00000i q^{5} +1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} +2.00000 q^{12} -1.00000i q^{15} +4.00000 q^{16} +6.00000i q^{17} -6.00000i q^{19} -2.00000i q^{20} +1.00000 q^{21} +6.00000i q^{23} -1.00000 q^{25} -5.00000 q^{27} +2.00000 q^{28} -6.00000i q^{29} +6.00000i q^{31} -3.00000 q^{33} -1.00000i q^{35} -4.00000 q^{36} +(1.00000 - 6.00000i) q^{37} -3.00000 q^{41} +6.00000i q^{43} -6.00000 q^{44} +2.00000i q^{45} -3.00000 q^{47} +4.00000 q^{48} -6.00000 q^{49} +6.00000i q^{51} -9.00000 q^{53} +3.00000i q^{55} -6.00000i q^{57} -12.0000i q^{59} -2.00000i q^{60} +6.00000i q^{61} -2.00000 q^{63} +8.00000 q^{64} +4.00000 q^{67} +12.0000i q^{68} +6.00000i q^{69} +9.00000 q^{71} +7.00000 q^{73} -1.00000 q^{75} -12.0000i q^{76} -3.00000 q^{77} -12.0000i q^{79} -4.00000i q^{80} +1.00000 q^{81} +15.0000 q^{83} +2.00000 q^{84} +6.00000 q^{85} -6.00000i q^{87} +12.0000i q^{92} +6.00000i q^{93} -6.00000 q^{95} +6.00000i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{4} + 2 q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 4 * q^4 + 2 * q^7 - 4 * q^9	\( 2 q + 2 q^{3} + 4 q^{4} + 2 q^{7} - 4 q^{9} - 6 q^{11} + 4 q^{12} + 8 q^{16} + 2 q^{21} - 2 q^{25} - 10 q^{27} + 4 q^{28} - 6 q^{33} - 8 q^{36} + 2 q^{37} - 6 q^{41} - 12 q^{44} - 6 q^{47} + 8 q^{48} - 12 q^{49} - 18 q^{53} - 4 q^{63} + 16 q^{64} + 8 q^{67} + 18 q^{71} + 14 q^{73} - 2 q^{75} - 6 q^{77} + 2 q^{81} + 30 q^{83} + 4 q^{84} + 12 q^{85} - 12 q^{95} + 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^4 + 2 * q^7 - 4 * q^9 - 6 * q^11 + 4 * q^12 + 8 * q^16 + 2 * q^21 - 2 * q^25 - 10 * q^27 + 4 * q^28 - 6 * q^33 - 8 * q^36 + 2 * q^37 - 6 * q^41 - 12 * q^44 - 6 * q^47 + 8 * q^48 - 12 * q^49 - 18 * q^53 - 4 * q^63 + 16 * q^64 + 8 * q^67 + 18 * q^71 + 14 * q^73 - 2 * q^75 - 6 * q^77 + 2 * q^81 + 30 * q^83 + 4 * q^84 + 12 * q^85 - 12 * q^95 + 12 * q^99

Introduction

L-functions

Modular forms

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Database

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