LMFDB - Embedded newform 837.2.c.b.836.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [837,2,Mod(836,837)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(837, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("837.836");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 837 = 3^{3} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	837.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:	$6.68347864918$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			836.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	837.836
Dual form			837.2.c.b.836.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{2} -1.00000 q^{4} -3.46410i q^{5} +2.00000 q^{7} -1.73205i q^{8} +O(q^{10})$	$q-1.73205i q^{2} -1.00000 q^{4} -3.46410i q^{5} +2.00000 q^{7} -1.73205i q^{8} -6.00000 q^{10} +6.00000 q^{11} -3.46410i q^{14} -5.00000 q^{16} -3.00000 q^{17} +4.00000 q^{19} +3.46410i q^{20} -10.3923i q^{22} +6.00000 q^{23} -7.00000 q^{25} -2.00000 q^{28} -6.00000 q^{29} +(2.00000 + 5.19615i) q^{31} +5.19615i q^{32} +5.19615i q^{34} -6.92820i q^{35} -6.92820i q^{38} -6.00000 q^{40} +3.46410i q^{41} +5.19615i q^{43} -6.00000 q^{44} -10.3923i q^{46} +5.19615i q^{47} -3.00000 q^{49} +12.1244i q^{50} -9.00000 q^{53} -20.7846i q^{55} -3.46410i q^{56} +10.3923i q^{58} +1.73205i q^{59} +(9.00000 - 3.46410i) q^{62} -1.00000 q^{64} -14.0000 q^{67} +3.00000 q^{68} -12.0000 q^{70} -1.73205i q^{71} -10.3923i q^{73} -4.00000 q^{76} +12.0000 q^{77} +15.5885i q^{79} +17.3205i q^{80} +6.00000 q^{82} +6.00000 q^{83} +10.3923i q^{85} +9.00000 q^{86} -10.3923i q^{88} -15.0000 q^{89} -6.00000 q^{92} +9.00000 q^{94} -13.8564i q^{95} +11.0000 q^{97} +5.19615i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^4 + 4 * q^7	$ 2 q - 2 q^{4} + 4 q^{7} - 12 q^{10} + 12 q^{11} - 10 q^{16} - 6 q^{17} + 8 q^{19} + 12 q^{23} - 14 q^{25} - 4 q^{28} - 12 q^{29} + 4 q^{31} - 12 q^{40} - 12 q^{44} - 6 q^{49} - 18 q^{53} + 18 q^{62} - 2 q^{64} - 28 q^{67} + 6 q^{68} - 24 q^{70} - 8 q^{76} + 24 q^{77} + 12 q^{82} + 12 q^{83} + 18 q^{86} - 30 q^{89} - 12 q^{92} + 18 q^{94} + 22 q^{97}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^7 - 12 * q^10 + 12 * q^11 - 10 * q^16 - 6 * q^17 + 8 * q^19 + 12 * q^23 - 14 * q^25 - 4 * q^28 - 12 * q^29 + 4 * q^31 - 12 * q^40 - 12 * q^44 - 6 * q^49 - 18 * q^53 + 18 * q^62 - 2 * q^64 - 28 * q^67 + 6 * q^68 - 24 * q^70 - 8 * q^76 + 24 * q^77 + 12 * q^82 + 12 * q^83 + 18 * q^86 - 30 * q^89 - 12 * q^92 + 18 * q^94 + 22 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/837\mathbb{Z}\right)^\times$.

$n$	$218$	$406$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	1.73205i		−	1.22474i	−0.790569	−	0.612372i	$-0.790215\pi$
$2$		−	1.73205i		−	1.22474i	0.790569	−	0.612372i	$-0.209785\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		−	3.46410i		−	1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$		−	3.46410i		−	1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$		0			0
$7$	2.00000			0.755929			0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000			0.755929			0.377964	+	0.925820i	$0.376624\pi$
$8$		−	1.73205i		−	0.612372i
$9$		0			0
$10$	−6.00000			−1.89737
$11$	6.00000			1.80907			0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000			1.80907			0.904534	+	0.426401i	$0.140219\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		−	3.46410i		−	0.925820i
$15$		0			0
$16$	−5.00000			−1.25000
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$			3.46410i			0.774597i
$21$		0			0
$22$		−	10.3923i		−	2.21565i
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$		0			0
$25$	−7.00000			−1.40000
$26$		0			0
$27$		0			0
$28$	−2.00000			−0.377964
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	2.00000	+	5.19615i	0.359211	+	0.933257i
$31$	2.00000	+	5.19615i	0.359211	+	0.933257i
$32$			5.19615i			0.918559i
$33$		0			0
$34$			5.19615i			0.891133i
$35$		−	6.92820i		−	1.17108i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		−	6.92820i		−	1.12390i
$39$		0			0
$40$	−6.00000			−0.948683
$41$			3.46410i			0.541002i	0.962720	+	0.270501i	$0.0871893\pi$
$41$			3.46410i			0.541002i	−0.962720	+	0.270501i	$0.912811\pi$
$42$		0			0
$43$			5.19615i			0.792406i	0.918163	+	0.396203i	$0.129672\pi$
$43$			5.19615i			0.792406i	−0.918163	+	0.396203i	$0.870328\pi$
$44$	−6.00000			−0.904534
$45$		0			0
$46$		−	10.3923i		−	1.53226i
$47$			5.19615i			0.757937i	0.925410	+	0.378968i	$0.123721\pi$
$47$			5.19615i			0.757937i	−0.925410	+	0.378968i	$0.876279\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$			12.1244i			1.71464i
$51$		0			0
$52$		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
$55$		−	20.7846i		−	2.80260i
$56$		−	3.46410i		−	0.462910i
$57$		0			0
$58$			10.3923i			1.36458i
$59$			1.73205i			0.225494i	0.993624	+	0.112747i	$0.0359649\pi$
$59$			1.73205i			0.225494i	−0.993624	+	0.112747i	$0.964035\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	9.00000	−	3.46410i	1.14300	−	0.439941i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	−14.0000			−1.71037			−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000			−1.71037			−0.855186	+	0.518321i	$0.826557\pi$
$68$	3.00000			0.363803
$69$		0			0
$70$	−12.0000			−1.43427
$71$		−	1.73205i		−	0.205557i	−0.994704	−	0.102778i	$-0.967227\pi$
$71$		−	1.73205i		−	0.205557i	0.994704	−	0.102778i	$-0.0327732\pi$
$72$		0			0
$73$		−	10.3923i		−	1.21633i	−0.793812	−	0.608164i	$-0.791906\pi$
$73$		−	10.3923i		−	1.21633i	0.793812	−	0.608164i	$-0.208094\pi$
$74$		0			0
$75$		0			0
$76$	−4.00000			−0.458831
$77$	12.0000			1.36753
$78$		0			0
$79$			15.5885i			1.75384i	0.480638	+	0.876919i	$0.340405\pi$
$79$			15.5885i			1.75384i	−0.480638	+	0.876919i	$0.659595\pi$
$80$			17.3205i			1.93649i
$81$		0			0
$82$	6.00000			0.662589
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$			10.3923i			1.12720i
$86$	9.00000			0.970495
$87$		0			0
$88$		−	10.3923i		−	1.10782i
$89$	−15.0000			−1.59000			−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000			−1.59000			−0.794998	+	0.606612i	$0.792528\pi$
$90$		0			0
$91$		0			0
$92$	−6.00000			−0.625543
$93$		0			0
$94$	9.00000			0.928279
$95$		−	13.8564i		−	1.42164i
$96$		0			0
$97$	11.0000			1.11688			0.558440	−	0.829545i	$-0.311400\pi$
$97$	11.0000			1.11688			0.558440	+	0.829545i	$0.311400\pi$
$98$			5.19615i			0.524891i
$99$		0			0
$100$	7.00000			0.700000
$101$			3.46410i			0.344691i	0.985037	+	0.172345i	$0.0551346\pi$
$101$			3.46410i			0.344691i	−0.985037	+	0.172345i	$0.944865\pi$
$102$		0			0
$103$	10.0000			0.985329			0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000			0.985329			0.492665	+	0.870219i	$0.336023\pi$
$104$		0			0
$105$		0			0
$106$			15.5885i			1.51408i
$107$		−	19.0526i		−	1.84188i	−0.389704	−	0.920940i	$-0.627423\pi$
$107$		−	19.0526i		−	1.84188i	0.389704	−	0.920940i	$-0.372577\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$	−36.0000			−3.43247
$111$		0			0
$112$	−10.0000			−0.944911
$113$			13.8564i			1.30350i	0.758433	+	0.651751i	$0.225965\pi$
$113$			13.8564i			1.30350i	−0.758433	+	0.651751i	$0.774035\pi$
$114$		0			0
$115$		−	20.7846i		−	1.93817i
$116$	6.00000			0.557086
$117$		0			0
$118$	3.00000			0.276172
$119$	−6.00000			−0.550019
$120$		0			0
$121$	25.0000			2.27273
$122$		0			0
$123$		0			0
$124$	−2.00000	−	5.19615i	−0.179605	−	0.466628i
$125$			6.92820i			0.619677i
$126$		0			0
$127$			15.5885i			1.38325i	0.722256	+	0.691626i	$0.243105\pi$
$127$			15.5885i			1.38325i	−0.722256	+	0.691626i	$0.756895\pi$
$128$			12.1244i			1.07165i
$129$		0			0
$130$		0			0
$131$		−	5.19615i		−	0.453990i	−0.973896	−	0.226995i	$-0.927110\pi$
$131$		−	5.19615i		−	0.453990i	0.973896	−	0.226995i	$-0.0728901\pi$
$132$		0			0
$133$	8.00000			0.693688
$134$			24.2487i			2.09477i
$135$		0			0
$136$			5.19615i			0.445566i
$137$	18.0000			1.53784			0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000			1.53784			0.768922	+	0.639343i	$0.220793\pi$
$138$		0			0
$139$			10.3923i			0.881464i	0.897639	+	0.440732i	$0.145281\pi$
$139$			10.3923i			0.881464i	−0.897639	+	0.440732i	$0.854719\pi$
$140$			6.92820i			0.585540i
$141$		0			0
$142$	−3.00000			−0.251754
$143$		0			0
$144$		0			0
$145$			20.7846i			1.72607i
$146$	−18.0000			−1.48969
$147$		0			0
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$			10.3923i			0.845714i	0.906196	+	0.422857i	$0.138973\pi$
$151$			10.3923i			0.845714i	−0.906196	+	0.422857i	$0.861027\pi$
$152$		−	6.92820i		−	0.561951i
$153$		0			0
$154$		−	20.7846i		−	1.67487i
$155$	18.0000	−	6.92820i	1.44579	−	0.556487i
$156$		0			0
$157$	1.00000			0.0798087			0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000			0.0798087			0.0399043	+	0.999204i	$0.487295\pi$
$158$	27.0000			2.14800
$159$		0			0
$160$	18.0000			1.42302
$161$	12.0000			0.945732
$162$		0			0
$163$	14.0000			1.09656			0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000			1.09656			0.548282	+	0.836293i	$0.315282\pi$
$164$		−	3.46410i		−	0.270501i
$165$		0			0
$166$		−	10.3923i		−	0.806599i
$167$	−18.0000			−1.39288			−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000			−1.39288			−0.696441	+	0.717614i	$0.745234\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$	18.0000			1.38054
$171$		0			0
$172$		−	5.19615i		−	0.396203i
$173$		−	20.7846i		−	1.58022i	−0.612962	−	0.790112i	$-0.710022\pi$
$173$		−	20.7846i		−	1.58022i	0.612962	−	0.790112i	$-0.289978\pi$
$174$		0			0
$175$	−14.0000			−1.05830
$176$	−30.0000			−2.26134
$177$		0			0
$178$			25.9808i			1.94734i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		−	10.3923i		−	0.772454i	−0.922404	−	0.386227i	$-0.873778\pi$
$181$		−	10.3923i		−	0.772454i	0.922404	−	0.386227i	$-0.126222\pi$
$182$		0			0
$183$		0			0
$184$		−	10.3923i		−	0.766131i
$185$		0			0
$186$		0			0
$187$	−18.0000			−1.31629
$188$		−	5.19615i		−	0.378968i
$189$		0			0
$190$	−24.0000			−1.74114
$191$			12.1244i			0.877288i	0.898661	+	0.438644i	$0.144541\pi$
$191$			12.1244i			0.877288i	−0.898661	+	0.438644i	$0.855459\pi$
$192$		0			0
$193$	−17.0000			−1.22369			−0.611843	−	0.790979i	$-0.709572\pi$
$193$	−17.0000			−1.22369			−0.611843	+	0.790979i	$0.709572\pi$
$194$		−	19.0526i		−	1.36789i
$195$		0			0
$196$	3.00000			0.214286
$197$	−9.00000			−0.641223			−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000			−0.641223			−0.320612	+	0.947211i	$0.603888\pi$
$198$		0			0
$199$		−	25.9808i		−	1.84173i	−0.389885	−	0.920864i	$-0.627485\pi$
$199$		−	25.9808i		−	1.84173i	0.389885	−	0.920864i	$-0.372515\pi$
$200$			12.1244i			0.857321i
$201$		0			0
$202$	6.00000			0.422159
$203$	−12.0000			−0.842235
$204$		0			0
$205$	12.0000			0.838116
$206$		−	17.3205i		−	1.20678i
$207$		0			0
$208$		0			0
$209$	24.0000			1.66011
$210$		0			0
$211$	26.0000			1.78991			0.894957	−	0.446153i	$-0.147206\pi$
$211$	26.0000			1.78991			0.894957	+	0.446153i	$0.147206\pi$
$212$	9.00000			0.618123
$213$		0			0
$214$	−33.0000			−2.25583
$215$	18.0000			1.22759
$216$		0			0
$217$	4.00000	+	10.3923i	0.271538	+	0.705476i
$218$		−	8.66025i		−	0.586546i
$219$		0			0
$220$			20.7846i			1.40130i
$221$		0			0
$222$		0			0
$223$		−	5.19615i		−	0.347960i	−0.984749	−	0.173980i	$-0.944337\pi$
$223$		−	5.19615i		−	0.347960i	0.984749	−	0.173980i	$-0.0556628\pi$
$224$			10.3923i			0.694365i
$225$		0			0
$226$	24.0000			1.59646
$227$		−	25.9808i		−	1.72440i	−0.506565	−	0.862202i	$-0.669085\pi$
$227$		−	25.9808i		−	1.72440i	0.506565	−	0.862202i	$-0.330915\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$	−36.0000			−2.37377
$231$		0			0
$232$			10.3923i			0.682288i
$233$			6.92820i			0.453882i	0.973909	+	0.226941i	$0.0728724\pi$
$233$			6.92820i			0.453882i	−0.973909	+	0.226941i	$0.927128\pi$
$234$		0			0
$235$	18.0000			1.17419
$236$		−	1.73205i		−	0.112747i
$237$		0			0
$238$			10.3923i			0.673633i
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$			10.3923i			0.669427i	0.942320	+	0.334714i	$0.108640\pi$
$241$			10.3923i			0.669427i	−0.942320	+	0.334714i	$0.891360\pi$
$242$		−	43.3013i		−	2.78351i
$243$		0			0
$244$		0			0
$245$			10.3923i			0.663940i
$246$		0			0
$247$		0			0
$248$	9.00000	−	3.46410i	0.571501	−	0.219971i
$249$		0			0
$250$	12.0000			0.758947
$251$	6.00000			0.378717			0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000			0.378717			0.189358	+	0.981908i	$0.439359\pi$
$252$		0			0
$253$	36.0000			2.26330
$254$	27.0000			1.69413
$255$		0			0
$256$	19.0000			1.18750
$257$		−	24.2487i		−	1.51259i	−0.654229	−	0.756297i	$-0.727007\pi$
$257$		−	24.2487i		−	1.51259i	0.654229	−	0.756297i	$-0.272993\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−9.00000			−0.556022
$263$	12.0000			0.739952			0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000			0.739952			0.369976	+	0.929041i	$0.379366\pi$
$264$		0			0
$265$			31.1769i			1.91518i
$266$		−	13.8564i		−	0.849591i
$267$		0			0
$268$	14.0000			0.855186
$269$	−21.0000			−1.28039			−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000			−1.28039			−0.640196	+	0.768211i	$0.721147\pi$
$270$		0			0
$271$		−	5.19615i		−	0.315644i	−0.987468	−	0.157822i	$-0.949553\pi$
$271$		−	5.19615i		−	0.315644i	0.987468	−	0.157822i	$-0.0504472\pi$
$272$	15.0000			0.909509
$273$		0			0
$274$		−	31.1769i		−	1.88347i
$275$	−42.0000			−2.53270
$276$		0			0
$277$		−	10.3923i		−	0.624413i	−0.950014	−	0.312207i	$-0.898932\pi$
$277$		−	10.3923i		−	0.624413i	0.950014	−	0.312207i	$-0.101068\pi$
$278$	18.0000			1.07957
$279$		0			0
$280$	−12.0000			−0.717137
$281$			17.3205i			1.03325i	0.856210	+	0.516627i	$0.172813\pi$
$281$			17.3205i			1.03325i	−0.856210	+	0.516627i	$0.827187\pi$
$282$		0			0
$283$	14.0000			0.832214			0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000			0.832214			0.416107	+	0.909316i	$0.363394\pi$
$284$			1.73205i			0.102778i
$285$		0			0
$286$		0			0
$287$			6.92820i			0.408959i
$288$		0			0
$289$	−8.00000			−0.470588
$290$	36.0000			2.11399
$291$		0			0
$292$			10.3923i			0.608164i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$	6.00000			0.349334
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$			10.3923i			0.599002i
$302$	18.0000			1.03578
$303$		0			0
$304$	−20.0000			−1.14708
$305$		0			0
$306$		0			0
$307$	−32.0000			−1.82634			−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000			−1.82634			−0.913168	+	0.407583i	$0.866372\pi$
$308$	−12.0000			−0.683763
$309$		0			0
$310$	−12.0000	−	31.1769i	−0.681554	−	1.77073i
$311$		−	1.73205i		−	0.0982156i	−0.998793	−	0.0491078i	$-0.984362\pi$
$311$		−	1.73205i		−	0.0982156i	0.998793	−	0.0491078i	$-0.0156378\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		−	1.73205i		−	0.0977453i
$315$		0			0
$316$		−	15.5885i		−	0.876919i
$317$			13.8564i			0.778253i	0.921184	+	0.389127i	$0.127223\pi$
$317$			13.8564i			0.778253i	−0.921184	+	0.389127i	$0.872777\pi$
$318$		0			0
$319$	−36.0000			−2.01561
$320$			3.46410i			0.193649i
$321$		0			0
$322$		−	20.7846i		−	1.15828i
$323$	−12.0000			−0.667698
$324$		0			0
$325$		0			0
$326$		−	24.2487i		−	1.34301i
$327$		0			0
$328$	6.00000			0.331295
$329$			10.3923i			0.572946i
$330$		0			0
$331$			25.9808i			1.42803i	0.700129	+	0.714016i	$0.253126\pi$
$331$			25.9808i			1.42803i	−0.700129	+	0.714016i	$0.746874\pi$
$332$	−6.00000			−0.329293
$333$		0			0
$334$			31.1769i			1.70592i
$335$			48.4974i			2.64970i
$336$		0			0
$337$		−	31.1769i		−	1.69831i	−0.528140	−	0.849157i	$-0.677110\pi$
$337$		−	31.1769i		−	1.69831i	0.528140	−	0.849157i	$-0.322890\pi$
$338$		−	22.5167i		−	1.22474i
$339$		0			0
$340$		−	10.3923i		−	0.563602i
$341$	12.0000	+	31.1769i	0.649836	+	1.68832i
$342$		0			0
$343$	−20.0000			−1.07990
$344$	9.00000			0.485247
$345$		0			0
$346$	−36.0000			−1.93537
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		0			0
$349$	1.00000			0.0535288			0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000			0.0535288			0.0267644	+	0.999642i	$0.491480\pi$
$350$			24.2487i			1.29615i
$351$		0			0
$352$			31.1769i			1.66174i
$353$	−18.0000			−0.958043			−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000			−0.958043			−0.479022	+	0.877803i	$0.659008\pi$
$354$		0			0
$355$	−6.00000			−0.318447
$356$	15.0000			0.794998
$357$		0			0
$358$		0			0
$359$			19.0526i			1.00556i	0.864416	+	0.502778i	$0.167689\pi$
$359$			19.0526i			1.00556i	−0.864416	+	0.502778i	$0.832311\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$	−18.0000			−0.946059
$363$		0			0
$364$		0			0
$365$	−36.0000			−1.88433
$366$		0			0
$367$		−	10.3923i		−	0.542474i	−0.962513	−	0.271237i	$-0.912567\pi$
$367$		−	10.3923i		−	0.542474i	0.962513	−	0.271237i	$-0.0874327\pi$
$368$	−30.0000			−1.56386
$369$		0			0
$370$		0			0
$371$	−18.0000			−0.934513
$372$		0			0
$373$	17.0000			0.880227			0.440113	−	0.897942i	$-0.354938\pi$
$373$	17.0000			0.880227			0.440113	+	0.897942i	$0.354938\pi$
$374$			31.1769i			1.61212i
$375$		0			0
$376$	9.00000			0.464140
$377$		0			0
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$			13.8564i			0.710819i
$381$		0			0
$382$	21.0000			1.07445
$383$	−12.0000			−0.613171			−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000			−0.613171			−0.306586	+	0.951843i	$0.599187\pi$
$384$		0			0
$385$		−	41.5692i		−	2.11856i
$386$			29.4449i			1.49870i
$387$		0			0
$388$	−11.0000			−0.558440
$389$	−21.0000			−1.06474			−0.532371	−	0.846511i	$-0.678699\pi$
$389$	−21.0000			−1.06474			−0.532371	+	0.846511i	$0.678699\pi$
$390$		0			0
$391$	−18.0000			−0.910299
$392$			5.19615i			0.262445i
$393$		0			0
$394$			15.5885i			0.785335i
$395$	54.0000			2.71703
$396$		0			0
$397$	5.00000			0.250943			0.125471	−	0.992097i	$-0.459956\pi$
$397$	5.00000			0.250943			0.125471	+	0.992097i	$0.459956\pi$
$398$	−45.0000			−2.25565
$399$		0			0
$400$	35.0000			1.75000
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$		0			0
$403$		0			0
$404$		−	3.46410i		−	0.172345i
$405$		0			0
$406$			20.7846i			1.03152i
$407$		0			0
$408$		0			0
$409$		−	10.3923i		−	0.513866i	−0.966429	−	0.256933i	$-0.917288\pi$
$409$		−	10.3923i		−	0.513866i	0.966429	−	0.256933i	$-0.0827120\pi$
$410$		−	20.7846i		−	1.02648i
$411$		0			0
$412$	−10.0000			−0.492665
$413$			3.46410i			0.170457i
$414$		0			0
$415$		−	20.7846i		−	1.02028i
$416$		0			0
$417$		0			0
$418$		−	41.5692i		−	2.03322i
$419$			24.2487i			1.18463i	0.805708	+	0.592314i	$0.201785\pi$
$419$			24.2487i			1.18463i	−0.805708	+	0.592314i	$0.798215\pi$
$420$		0			0
$421$	−11.0000			−0.536107			−0.268054	−	0.963404i	$-0.586380\pi$
$421$	−11.0000			−0.536107			−0.268054	+	0.963404i	$0.586380\pi$
$422$		−	45.0333i		−	2.19219i
$423$		0			0
$424$			15.5885i			0.757042i
$425$	21.0000			1.01865
$426$		0			0
$427$		0			0
$428$			19.0526i			0.920940i
$429$		0			0
$430$		−	31.1769i		−	1.50348i
$431$			31.1769i			1.50174i	0.660451	+	0.750870i	$0.270365\pi$
$431$			31.1769i			1.50174i	−0.660451	+	0.750870i	$0.729635\pi$
$432$		0			0
$433$			10.3923i			0.499422i	0.968320	+	0.249711i	$0.0803357\pi$
$433$			10.3923i			0.499422i	−0.968320	+	0.249711i	$0.919664\pi$
$434$	18.0000	−	6.92820i	0.864028	−	0.332564i
$435$		0			0
$436$	−5.00000			−0.239457
$437$	24.0000			1.14808
$438$		0			0
$439$	−20.0000			−0.954548			−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000			−0.954548			−0.477274	+	0.878755i	$0.658375\pi$
$440$	−36.0000			−1.71623
$441$		0			0
$442$		0			0
$443$			17.3205i			0.822922i	0.911427	+	0.411461i	$0.134981\pi$
$443$			17.3205i			0.822922i	−0.911427	+	0.411461i	$0.865019\pi$
$444$		0			0
$445$			51.9615i			2.46321i
$446$	−9.00000			−0.426162
$447$		0			0
$448$	−2.00000			−0.0944911
$449$	21.0000			0.991051			0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000			0.991051			0.495526	+	0.868593i	$0.334975\pi$
$450$		0			0
$451$			20.7846i			0.978709i
$452$		−	13.8564i		−	0.651751i
$453$		0			0
$454$	−45.0000			−2.11195
$455$		0			0
$456$		0			0
$457$		−	20.7846i		−	0.972263i	−0.873886	−	0.486132i	$-0.838408\pi$
$457$		−	20.7846i		−	0.972263i	0.873886	−	0.486132i	$-0.161592\pi$
$458$		0			0
$459$		0			0
$460$			20.7846i			0.969087i
$461$	−33.0000			−1.53696			−0.768482	−	0.639872i	$-0.778987\pi$
$461$	−33.0000			−1.53696			−0.768482	+	0.639872i	$0.778987\pi$
$462$		0			0
$463$			10.3923i			0.482971i	0.970404	+	0.241486i	$0.0776347\pi$
$463$			10.3923i			0.482971i	−0.970404	+	0.241486i	$0.922365\pi$
$464$	30.0000			1.39272
$465$		0			0
$466$	12.0000			0.555889
$467$		−	12.1244i		−	0.561048i	−0.959847	−	0.280524i	$-0.909492\pi$
$467$		−	12.1244i		−	0.561048i	0.959847	−	0.280524i	$-0.0905083\pi$
$468$		0			0
$469$	−28.0000			−1.29292
$470$		−	31.1769i		−	1.43808i
$471$		0			0
$472$	3.00000			0.138086
$473$			31.1769i			1.43352i
$474$		0			0
$475$	−28.0000			−1.28473
$476$	6.00000			0.275010
$477$		0			0
$478$			10.3923i			0.475333i
$479$			17.3205i			0.791394i	0.918381	+	0.395697i	$0.129497\pi$
$479$			17.3205i			0.791394i	−0.918381	+	0.395697i	$0.870503\pi$
$480$		0			0
$481$		0			0
$482$	18.0000			0.819878
$483$		0			0
$484$	−25.0000			−1.13636
$485$		−	38.1051i		−	1.73026i
$486$		0			0
$487$		−	25.9808i		−	1.17730i	−0.808388	−	0.588650i	$-0.799659\pi$
$487$		−	25.9808i		−	1.17730i	0.808388	−	0.588650i	$-0.200341\pi$
$488$		0			0
$489$		0			0
$490$	18.0000			0.813157
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$	18.0000			0.810679
$494$		0			0
$495$		0			0
$496$	−10.0000	−	25.9808i	−0.449013	−	1.16657i
$497$		−	3.46410i		−	0.155386i
$498$		0			0
$499$		−	36.3731i		−	1.62828i	−0.580667	−	0.814141i	$-0.697208\pi$
$499$		−	36.3731i		−	1.62828i	0.580667	−	0.814141i	$-0.302792\pi$
$500$		−	6.92820i		−	0.309839i
$501$		0			0
$502$		−	10.3923i		−	0.463831i
$503$			17.3205i			0.772283i	0.922440	+	0.386142i	$0.126192\pi$
$503$			17.3205i			0.772283i	−0.922440	+	0.386142i	$0.873808\pi$
$504$		0			0
$505$	12.0000			0.533993
$506$		−	62.3538i		−	2.77197i
$507$		0			0
$508$		−	15.5885i		−	0.691626i
$509$	30.0000			1.32973			0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000			1.32973			0.664863	+	0.746965i	$0.268490\pi$
$510$		0			0
$511$		−	20.7846i		−	0.919457i
$512$		−	8.66025i		−	0.382733i
$513$		0			0
$514$	−42.0000			−1.85254
$515$		−	34.6410i		−	1.52647i
$516$		0			0
$517$			31.1769i			1.37116i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	20.7846i		−	0.910590i	−0.890341	−	0.455295i	$-0.849534\pi$
$521$		−	20.7846i		−	0.910590i	0.890341	−	0.455295i	$-0.150466\pi$
$522$		0			0
$523$		−	31.1769i		−	1.36327i	−0.731692	−	0.681636i	$-0.761269\pi$
$523$		−	31.1769i		−	1.36327i	0.731692	−	0.681636i	$-0.238731\pi$
$524$			5.19615i			0.226995i
$525$		0			0
$526$		−	20.7846i		−	0.906252i
$527$	−6.00000	−	15.5885i	−0.261364	−	0.679044i
$528$		0			0
$529$	13.0000			0.565217
$530$	54.0000			2.34561
$531$		0			0
$532$	−8.00000			−0.346844
$533$		0			0
$534$		0			0
$535$	−66.0000			−2.85343
$536$			24.2487i			1.04738i
$537$		0			0
$538$			36.3731i			1.56815i
$539$	−18.0000			−0.775315
$540$		0			0
$541$	13.0000			0.558914			0.279457	−	0.960158i	$-0.409846\pi$
$541$	13.0000			0.558914			0.279457	+	0.960158i	$0.409846\pi$
$542$	−9.00000			−0.386583
$543$		0			0
$544$		−	15.5885i		−	0.668350i
$545$		−	17.3205i		−	0.741929i
$546$		0			0
$547$	34.0000			1.45374			0.726868	−	0.686778i	$-0.240975\pi$
$547$	34.0000			1.45374			0.726868	+	0.686778i	$0.240975\pi$
$548$	−18.0000			−0.768922
$549$		0			0
$550$			72.7461i			3.10191i
$551$	−24.0000			−1.02243
$552$		0			0
$553$			31.1769i			1.32578i
$554$	−18.0000			−0.764747
$555$		0			0
$556$		−	10.3923i		−	0.440732i
$557$	3.00000			0.127114			0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000			0.127114			0.0635570	+	0.997978i	$0.479756\pi$
$558$		0			0
$559$		0			0
$560$			34.6410i			1.46385i
$561$		0			0
$562$	30.0000			1.26547
$563$			3.46410i			0.145994i	0.997332	+	0.0729972i	$0.0232564\pi$
$563$			3.46410i			0.145994i	−0.997332	+	0.0729972i	$0.976744\pi$
$564$		0			0
$565$	48.0000			2.01938
$566$		−	24.2487i		−	1.01925i
$567$		0			0
$568$	−3.00000			−0.125877
$569$	−15.0000			−0.628833			−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000			−0.628833			−0.314416	+	0.949285i	$0.601809\pi$
$570$		0			0
$571$		−	36.3731i		−	1.52217i	−0.648655	−	0.761083i	$-0.724668\pi$
$571$		−	36.3731i		−	1.52217i	0.648655	−	0.761083i	$-0.275332\pi$
$572$		0			0
$573$		0			0
$574$	12.0000			0.500870
$575$	−42.0000			−1.75152
$576$		0			0
$577$	7.00000			0.291414			0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000			0.291414			0.145707	+	0.989328i	$0.453454\pi$
$578$			13.8564i			0.576351i
$579$		0			0
$580$		−	20.7846i		−	0.863034i
$581$	12.0000			0.497844
$582$		0			0
$583$	−54.0000			−2.23645
$584$	−18.0000			−0.744845
$585$		0			0
$586$		0			0
$587$	42.0000			1.73353			0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000			1.73353			0.866763	+	0.498721i	$0.166197\pi$
$588$		0			0
$589$	8.00000	+	20.7846i	0.329634	+	0.856415i
$590$		−	10.3923i		−	0.427844i
$591$		0			0
$592$		0			0
$593$		−	34.6410i		−	1.42254i	−0.702921	−	0.711268i	$-0.748121\pi$
$593$		−	34.6410i		−	1.42254i	0.702921	−	0.711268i	$-0.251879\pi$
$594$		0			0
$595$			20.7846i			0.852086i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	3.46410i		−	0.141539i	−0.997493	−	0.0707697i	$-0.977454\pi$
$599$		−	3.46410i		−	0.141539i	0.997493	−	0.0707697i	$-0.0225455\pi$
$600$		0			0
$601$		−	20.7846i		−	0.847822i	−0.905704	−	0.423911i	$-0.860657\pi$
$601$		−	20.7846i		−	0.847822i	0.905704	−	0.423911i	$-0.139343\pi$
$602$	18.0000			0.733625
$603$		0			0
$604$		−	10.3923i		−	0.422857i
$605$		−	86.6025i		−	3.52089i
$606$		0			0
$607$	22.0000			0.892952			0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000			0.892952			0.446476	+	0.894795i	$0.352679\pi$
$608$			20.7846i			0.842927i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			31.1769i			1.25922i	0.776910	+	0.629612i	$0.216786\pi$
$613$			31.1769i			1.25922i	−0.776910	+	0.629612i	$0.783214\pi$
$614$			55.4256i			2.23680i
$615$		0			0
$616$		−	20.7846i		−	0.837436i
$617$		−	10.3923i		−	0.418378i	−0.977875	−	0.209189i	$-0.932918\pi$
$617$		−	10.3923i		−	0.418378i	0.977875	−	0.209189i	$-0.0670825\pi$
$618$		0			0
$619$			5.19615i			0.208851i	0.994533	+	0.104425i	$0.0333004\pi$
$619$			5.19615i			0.208851i	−0.994533	+	0.104425i	$0.966700\pi$
$620$	−18.0000	+	6.92820i	−0.722897	+	0.278243i
$621$		0			0
$622$	−3.00000			−0.120289
$623$	−30.0000			−1.20192
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$		0			0
$628$	−1.00000			−0.0399043
$629$		0			0
$630$		0			0
$631$			36.3731i			1.44799i	0.689806	+	0.723994i	$0.257696\pi$
$631$			36.3731i			1.44799i	−0.689806	+	0.723994i	$0.742304\pi$
$632$	27.0000			1.07400
$633$		0			0
$634$	24.0000			0.953162
$635$	54.0000			2.14292
$636$		0			0
$637$		0			0
$638$			62.3538i			2.46861i
$639$		0			0
$640$	42.0000			1.66020
$641$	9.00000			0.355479			0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000			0.355479			0.177739	+	0.984078i	$0.443122\pi$
$642$		0			0
$643$			25.9808i			1.02458i	0.858812	+	0.512291i	$0.171203\pi$
$643$			25.9808i			1.02458i	−0.858812	+	0.512291i	$0.828797\pi$
$644$	−12.0000			−0.472866
$645$		0			0
$646$			20.7846i			0.817760i
$647$	−6.00000			−0.235884			−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000			−0.235884			−0.117942	+	0.993020i	$0.537630\pi$
$648$		0			0
$649$			10.3923i			0.407934i
$650$		0			0
$651$		0			0
$652$	−14.0000			−0.548282
$653$		−	31.1769i		−	1.22005i	−0.792383	−	0.610023i	$-0.791160\pi$
$653$		−	31.1769i		−	1.22005i	0.792383	−	0.610023i	$-0.208840\pi$
$654$		0			0
$655$	−18.0000			−0.703318
$656$		−	17.3205i		−	0.676252i
$657$		0			0
$658$	18.0000			0.701713
$659$			22.5167i			0.877125i	0.898701	+	0.438562i	$0.144512\pi$
$659$			22.5167i			0.877125i	−0.898701	+	0.438562i	$0.855488\pi$
$660$		0			0
$661$	−19.0000			−0.739014			−0.369507	−	0.929228i	$-0.620473\pi$
$661$	−19.0000			−0.739014			−0.369507	+	0.929228i	$0.620473\pi$
$662$	45.0000			1.74897
$663$		0			0
$664$		−	10.3923i		−	0.403300i
$665$		−	27.7128i		−	1.07466i
$666$		0			0
$667$	−36.0000			−1.39393
$668$	18.0000			0.696441
$669$		0			0
$670$	84.0000			3.24520
$671$		0			0
$672$		0			0
$673$		−	10.3923i		−	0.400594i	−0.979735	−	0.200297i	$-0.935809\pi$
$673$		−	10.3923i		−	0.400594i	0.979735	−	0.200297i	$-0.0641907\pi$
$674$	−54.0000			−2.08000
$675$		0			0
$676$	−13.0000			−0.500000
$677$	−30.0000			−1.15299			−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000			−1.15299			−0.576497	+	0.817099i	$0.695581\pi$
$678$		0			0
$679$	22.0000			0.844283
$680$	18.0000			0.690268
$681$		0			0
$682$	54.0000	−	20.7846i	2.06777	−	0.795884i
$683$		−	24.2487i		−	0.927851i	−0.885874	−	0.463926i	$-0.846441\pi$
$683$		−	24.2487i		−	0.927851i	0.885874	−	0.463926i	$-0.153559\pi$
$684$		0			0
$685$		−	62.3538i		−	2.38242i
$686$			34.6410i			1.32260i
$687$		0			0
$688$		−	25.9808i		−	0.990507i
$689$		0			0
$690$		0			0
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$			20.7846i			0.790112i
$693$		0			0
$694$			20.7846i			0.788973i
$695$	36.0000			1.36556
$696$		0			0
$697$		−	10.3923i		−	0.393637i
$698$		−	1.73205i		−	0.0655591i
$699$		0			0
$700$	14.0000			0.529150
$701$			24.2487i			0.915861i	0.888988	+	0.457931i	$0.151409\pi$
$701$			24.2487i			0.915861i	−0.888988	+	0.457931i	$0.848591\pi$
$702$		0			0
$703$		0			0
$704$	−6.00000			−0.226134
$705$		0			0
$706$			31.1769i			1.17336i
$707$			6.92820i			0.260562i
$708$		0			0
$709$		−	41.5692i		−	1.56116i	−0.625053	−	0.780582i	$-0.714923\pi$
$709$		−	41.5692i		−	1.56116i	0.625053	−	0.780582i	$-0.285077\pi$
$710$			10.3923i			0.390016i
$711$		0			0
$712$			25.9808i			0.973670i
$713$	12.0000	+	31.1769i	0.449404	+	1.16758i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	33.0000			1.23155
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$	20.0000			0.744839
$722$			5.19615i			0.193381i
$723$		0			0
$724$			10.3923i			0.386227i
$725$	42.0000			1.55984
$726$		0			0
$727$	−52.0000			−1.92857			−0.964287	−	0.264861i	$-0.914674\pi$
$727$	−52.0000			−1.92857			−0.964287	+	0.264861i	$0.914674\pi$
$728$		0			0
$729$		0			0
$730$			62.3538i			2.30782i
$731$		−	15.5885i		−	0.576560i
$732$		0			0
$733$	1.00000			0.0369358			0.0184679	−	0.999829i	$-0.494121\pi$
$733$	1.00000			0.0369358			0.0184679	+	0.999829i	$0.494121\pi$
$734$	−18.0000			−0.664392
$735$		0			0
$736$			31.1769i			1.14920i
$737$	−84.0000			−3.09418
$738$		0			0
$739$		−	31.1769i		−	1.14686i	−0.819254	−	0.573431i	$-0.805612\pi$
$739$		−	31.1769i		−	1.14686i	0.819254	−	0.573431i	$-0.194388\pi$
$740$		0			0
$741$		0			0
$742$			31.1769i			1.14454i
$743$	12.0000			0.440237			0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000			0.440237			0.220119	+	0.975473i	$0.429356\pi$
$744$		0			0
$745$		0			0
$746$		−	29.4449i		−	1.07805i
$747$		0			0
$748$	18.0000			0.658145
$749$		−	38.1051i		−	1.39233i
$750$		0			0
$751$	−8.00000			−0.291924			−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000			−0.291924			−0.145962	+	0.989290i	$0.546628\pi$
$752$		−	25.9808i		−	0.947421i
$753$		0			0
$754$		0			0
$755$	36.0000			1.31017
$756$		0			0
$757$			51.9615i			1.88857i	0.329124	+	0.944287i	$0.393247\pi$
$757$			51.9615i			1.88857i	−0.329124	+	0.944287i	$0.606753\pi$
$758$			27.7128i			1.00657i
$759$		0			0
$760$	−24.0000			−0.870572
$761$	−27.0000			−0.978749			−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000			−0.978749			−0.489375	+	0.872074i	$0.662775\pi$
$762$		0			0
$763$	10.0000			0.362024
$764$		−	12.1244i		−	0.438644i
$765$		0			0
$766$			20.7846i			0.750978i
$767$		0			0
$768$		0			0
$769$	−17.0000			−0.613036			−0.306518	−	0.951865i	$-0.599164\pi$
$769$	−17.0000			−0.613036			−0.306518	+	0.951865i	$0.599164\pi$
$770$	−72.0000			−2.59470
$771$		0			0
$772$	17.0000			0.611843
$773$	18.0000			0.647415			0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000			0.647415			0.323708	+	0.946157i	$0.395071\pi$
$774$		0			0
$775$	−14.0000	−	36.3731i	−0.502895	−	1.30656i
$776$		−	19.0526i		−	0.683947i
$777$		0			0
$778$			36.3731i			1.30404i
$779$			13.8564i			0.496457i
$780$		0			0
$781$		−	10.3923i		−	0.371866i
$782$			31.1769i			1.11488i
$783$		0			0
$784$	15.0000			0.535714
$785$		−	3.46410i		−	0.123639i
$786$		0			0
$787$			25.9808i			0.926114i	0.886328	+	0.463057i	$0.153248\pi$
$787$			25.9808i			0.926114i	−0.886328	+	0.463057i	$0.846752\pi$
$788$	9.00000			0.320612
$789$		0			0
$790$		−	93.5307i		−	3.32767i
$791$			27.7128i			0.985354i
$792$		0			0
$793$		0			0
$794$		−	8.66025i		−	0.307341i
$795$		0			0
$796$			25.9808i			0.920864i
$797$	15.0000			0.531327			0.265664	−	0.964066i	$-0.414409\pi$
$797$	15.0000			0.531327			0.265664	+	0.964066i	$0.414409\pi$
$798$		0			0
$799$		−	15.5885i		−	0.551480i
$800$		−	36.3731i		−	1.28598i
$801$		0			0
$802$		−	31.1769i		−	1.10090i
$803$		−	62.3538i		−	2.20042i
$804$		0			0
$805$		−	41.5692i		−	1.46512i
$806$		0			0
$807$		0			0
$808$	6.00000			0.211079
$809$	−27.0000			−0.949269			−0.474635	−	0.880183i	$-0.657420\pi$
$809$	−27.0000			−0.949269			−0.474635	+	0.880183i	$0.657420\pi$
$810$		0			0
$811$	16.0000			0.561836			0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000			0.561836			0.280918	+	0.959732i	$0.409361\pi$
$812$	12.0000			0.421117
$813$		0			0
$814$		0			0
$815$		−	48.4974i		−	1.69879i
$816$		0			0
$817$			20.7846i			0.727161i
$818$	−18.0000			−0.629355
$819$		0			0
$820$	−12.0000			−0.419058
$821$	27.0000			0.942306			0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000			0.942306			0.471153	+	0.882051i	$0.343838\pi$
$822$		0			0
$823$		−	5.19615i		−	0.181126i	−0.995891	−	0.0905632i	$-0.971133\pi$
$823$		−	5.19615i		−	0.181126i	0.995891	−	0.0905632i	$-0.0288667\pi$
$824$		−	17.3205i		−	0.603388i
$825$		0			0
$826$	6.00000			0.208767
$827$	−18.0000			−0.625921			−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000			−0.625921			−0.312961	+	0.949766i	$0.601321\pi$
$828$		0			0
$829$			10.3923i			0.360940i	0.983581	+	0.180470i	$0.0577618\pi$
$829$			10.3923i			0.360940i	−0.983581	+	0.180470i	$0.942238\pi$
$830$	−36.0000			−1.24958
$831$		0			0
$832$		0			0
$833$	9.00000			0.311832
$834$		0			0
$835$			62.3538i			2.15784i
$836$	−24.0000			−0.830057
$837$		0			0
$838$	42.0000			1.45087
$839$			31.1769i			1.07635i	0.842834	+	0.538173i	$0.180885\pi$
$839$			31.1769i			1.07635i	−0.842834	+	0.538173i	$0.819115\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$			19.0526i			0.656595i
$843$		0			0
$844$	−26.0000			−0.894957
$845$		−	45.0333i		−	1.54919i
$846$		0			0
$847$	50.0000			1.71802
$848$	45.0000			1.54531
$849$		0			0
$850$		−	36.3731i		−	1.24759i
$851$		0			0
$852$		0			0
$853$	10.0000			0.342393			0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000			0.342393			0.171197	+	0.985237i	$0.445237\pi$
$854$		0			0
$855$		0			0
$856$	−33.0000			−1.12792
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		−	36.3731i		−	1.24103i	−0.784193	−	0.620517i	$-0.786923\pi$
$859$		−	36.3731i		−	1.24103i	0.784193	−	0.620517i	$-0.213077\pi$
$860$	−18.0000			−0.613795
$861$		0			0
$862$	54.0000			1.83925
$863$	−6.00000			−0.204242			−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000			−0.204242			−0.102121	+	0.994772i	$0.532563\pi$
$864$		0			0
$865$	−72.0000			−2.44807
$866$	18.0000			0.611665
$867$		0			0
$868$	−4.00000	−	10.3923i	−0.135769	−	0.352738i
$869$			93.5307i			3.17281i
$870$		0			0
$871$		0			0
$872$		−	8.66025i		−	0.293273i
$873$		0			0
$874$		−	41.5692i		−	1.40610i
$875$			13.8564i			0.468432i
$876$		0			0
$877$	1.00000			0.0337676			0.0168838	−	0.999857i	$-0.494625\pi$
$877$	1.00000			0.0337676			0.0168838	+	0.999857i	$0.494625\pi$
$878$			34.6410i			1.16908i
$879$		0			0
$880$			103.923i			3.50325i
$881$	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
$883$			10.3923i			0.349729i	0.984593	+	0.174864i	$0.0559487\pi$
$883$			10.3923i			0.349729i	−0.984593	+	0.174864i	$0.944051\pi$
$884$		0			0
$885$		0			0
$886$	30.0000			1.00787
$887$			19.0526i			0.639722i	0.947464	+	0.319861i	$0.103636\pi$
$887$			19.0526i			0.639722i	−0.947464	+	0.319861i	$0.896364\pi$
$888$		0			0
$889$			31.1769i			1.04564i
$890$	90.0000			3.01681
$891$		0			0
$892$			5.19615i			0.173980i
$893$			20.7846i			0.695530i
$894$		0			0
$895$		0			0
$896$			24.2487i			0.810093i
$897$		0			0
$898$		−	36.3731i		−	1.21378i
$899$	−12.0000	−	31.1769i	−0.400222	−	1.03981i
$900$		0			0
$901$	27.0000			0.899500
$902$	36.0000			1.19867
$903$		0			0
$904$	24.0000			0.798228
$905$	−36.0000			−1.19668
$906$		0			0
$907$	52.0000			1.72663			0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000			1.72663			0.863316	+	0.504664i	$0.168384\pi$
$908$			25.9808i			0.862202i
$909$		0			0
$910$		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
$913$	36.0000			1.19143
$914$	−36.0000			−1.19077
$915$		0			0
$916$		0			0
$917$		−	10.3923i		−	0.343184i
$918$		0			0
$919$	−38.0000			−1.25350			−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000			−1.25350			−0.626752	+	0.779219i	$0.715616\pi$
$920$	−36.0000			−1.18688
$921$		0			0
$922$			57.1577i			1.88239i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	18.0000			0.591517
$927$		0			0
$928$		−	31.1769i		−	1.02343i
$929$	−3.00000			−0.0984268			−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000			−0.0984268			−0.0492134	+	0.998788i	$0.515671\pi$
$930$		0			0
$931$	−12.0000			−0.393284
$932$		−	6.92820i		−	0.226941i
$933$		0			0
$934$	−21.0000			−0.687141
$935$			62.3538i			2.03919i
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$			48.4974i			1.58350i
$939$		0			0
$940$	−18.0000			−0.587095
$941$	−9.00000			−0.293392			−0.146696	−	0.989182i	$-0.546864\pi$
$941$	−9.00000			−0.293392			−0.146696	+	0.989182i	$0.546864\pi$
$942$		0			0
$943$			20.7846i			0.676840i
$944$		−	8.66025i		−	0.281867i
$945$		0			0
$946$	54.0000			1.75569
$947$	18.0000			0.584921			0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000			0.584921			0.292461	+	0.956278i	$0.405526\pi$
$948$		0			0
$949$		0			0
$950$			48.4974i			1.57346i
$951$		0			0
$952$			10.3923i			0.336817i
$953$	45.0000			1.45769			0.728846	−	0.684677i	$-0.240057\pi$
$953$	45.0000			1.45769			0.728846	+	0.684677i	$0.240057\pi$
$954$		0			0
$955$	42.0000			1.35909
$956$	6.00000			0.194054
$957$		0			0
$958$	30.0000			0.969256
$959$	36.0000			1.16250
$960$		0			0
$961$	−23.0000	+	20.7846i	−0.741935	+	0.670471i
$962$		0			0
$963$		0			0
$964$		−	10.3923i		−	0.334714i
$965$			58.8897i			1.89573i
$966$		0			0
$967$		−	25.9808i		−	0.835485i	−0.908565	−	0.417742i	$-0.862821\pi$
$967$		−	25.9808i		−	0.835485i	0.908565	−	0.417742i	$-0.137179\pi$
$968$		−	43.3013i		−	1.39176i
$969$		0			0
$970$	−66.0000			−2.11913
$971$			57.1577i			1.83428i	0.398568	+	0.917139i	$0.369507\pi$
$971$			57.1577i			1.83428i	−0.398568	+	0.917139i	$0.630493\pi$
$972$		0			0
$973$			20.7846i			0.666324i
$974$	−45.0000			−1.44189
$975$		0			0
$976$		0			0
$977$			13.8564i			0.443306i	0.975126	+	0.221653i	$0.0711452\pi$
$977$			13.8564i			0.443306i	−0.975126	+	0.221653i	$0.928855\pi$
$978$		0			0
$979$	−90.0000			−2.87641
$980$		−	10.3923i		−	0.331970i
$981$		0			0
$982$		−	20.7846i		−	0.663264i
$983$	24.0000			0.765481			0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000			0.765481			0.382741	+	0.923856i	$0.374980\pi$
$984$		0			0
$985$			31.1769i			0.993379i
$986$		−	31.1769i		−	0.992875i
$987$		0			0
$988$		0			0
$989$			31.1769i			0.991368i
$990$		0			0
$991$		−	46.7654i		−	1.48555i	−0.669541	−	0.742775i	$-0.733509\pi$
$991$		−	46.7654i		−	1.48555i	0.669541	−	0.742775i	$-0.266491\pi$
$992$	−27.0000	+	10.3923i	−0.857251	+	0.329956i
$993$		0			0
$994$	−6.00000			−0.190308
$995$	−90.0000			−2.85319
$996$		0			0
$997$	−43.0000			−1.36182			−0.680912	−	0.732365i	$-0.738416\pi$
$997$	−43.0000			−1.36182			−0.680912	+	0.732365i	$0.738416\pi$
$998$	−63.0000			−1.99423
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	837.2.c.b.836.1	yes	2
3.2	odd	2		837.2.c.a.836.2	yes	2
31.30	odd	2		837.2.c.a.836.1	✓	2
93.92	even	2	inner	837.2.c.b.836.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
837.2.c.a.836.1	✓	2	31.30	odd	2
837.2.c.a.836.2	yes	2	3.2	odd	2
837.2.c.b.836.1	yes	2	1.1	even	1	trivial
837.2.c.b.836.2	yes	2	93.92	even	2	inner

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:	\( N \)	\(=\)	\( 837 = 3^{3} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	837.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{2} -1.00000 q^{4} -3.46410i q^{5} +2.00000 q^{7} -1.73205i q^{8} +O(q^{10})\)	\(q-1.73205i q^{2} -1.00000 q^{4} -3.46410i q^{5} +2.00000 q^{7} -1.73205i q^{8} -6.00000 q^{10} +6.00000 q^{11} -3.46410i q^{14} -5.00000 q^{16} -3.00000 q^{17} +4.00000 q^{19} +3.46410i q^{20} -10.3923i q^{22} +6.00000 q^{23} -7.00000 q^{25} -2.00000 q^{28} -6.00000 q^{29} +(2.00000 + 5.19615i) q^{31} +5.19615i q^{32} +5.19615i q^{34} -6.92820i q^{35} -6.92820i q^{38} -6.00000 q^{40} +3.46410i q^{41} +5.19615i q^{43} -6.00000 q^{44} -10.3923i q^{46} +5.19615i q^{47} -3.00000 q^{49} +12.1244i q^{50} -9.00000 q^{53} -20.7846i q^{55} -3.46410i q^{56} +10.3923i q^{58} +1.73205i q^{59} +(9.00000 - 3.46410i) q^{62} -1.00000 q^{64} -14.0000 q^{67} +3.00000 q^{68} -12.0000 q^{70} -1.73205i q^{71} -10.3923i q^{73} -4.00000 q^{76} +12.0000 q^{77} +15.5885i q^{79} +17.3205i q^{80} +6.00000 q^{82} +6.00000 q^{83} +10.3923i q^{85} +9.00000 q^{86} -10.3923i q^{88} -15.0000 q^{89} -6.00000 q^{92} +9.00000 q^{94} -13.8564i q^{95} +11.0000 q^{97} +5.19615i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^7	\( 2 q - 2 q^{4} + 4 q^{7} - 12 q^{10} + 12 q^{11} - 10 q^{16} - 6 q^{17} + 8 q^{19} + 12 q^{23} - 14 q^{25} - 4 q^{28} - 12 q^{29} + 4 q^{31} - 12 q^{40} - 12 q^{44} - 6 q^{49} - 18 q^{53} + 18 q^{62} - 2 q^{64} - 28 q^{67} + 6 q^{68} - 24 q^{70} - 8 q^{76} + 24 q^{77} + 12 q^{82} + 12 q^{83} + 18 q^{86} - 30 q^{89} - 12 q^{92} + 18 q^{94} + 22 q^{97}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^7 - 12 * q^10 + 12 * q^11 - 10 * q^16 - 6 * q^17 + 8 * q^19 + 12 * q^23 - 14 * q^25 - 4 * q^28 - 12 * q^29 + 4 * q^31 - 12 * q^40 - 12 * q^44 - 6 * q^49 - 18 * q^53 + 18 * q^62 - 2 * q^64 - 28 * q^67 + 6 * q^68 - 24 * q^70 - 8 * q^76 + 24 * q^77 + 12 * q^82 + 12 * q^83 + 18 * q^86 - 30 * q^89 - 12 * q^92 + 18 * q^94 + 22 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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