LMFDB - Embedded newform 390.2.b.d.181.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(181,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("390.181");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	390.b (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:	$3.11416567883$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.3
Root			$-1.56155i$ of defining polynomial
Character	$\chi$	$=$	390.181
Dual form			390.2.b.d.181.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.12311i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.12311i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.12311i q^{11} -1.00000 q^{12} +(3.56155 + 0.561553i) q^{13} +3.12311 q^{14} -1.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000i q^{18} +6.00000i q^{19} +1.00000i q^{20} -3.12311i q^{21} +5.12311 q^{22} -5.12311 q^{23} -1.00000i q^{24} -1.00000 q^{25} +(-0.561553 + 3.56155i) q^{26} +1.00000 q^{27} +3.12311i q^{28} +2.00000 q^{29} +1.00000 q^{30} -3.12311i q^{31} +1.00000i q^{32} -5.12311i q^{33} +2.00000i q^{34} -3.12311 q^{35} -1.00000 q^{36} +5.12311i q^{37} -6.00000 q^{38} +(3.56155 + 0.561553i) q^{39} -1.00000 q^{40} -0.876894i q^{41} +3.12311 q^{42} +6.24621 q^{43} +5.12311i q^{44} -1.00000i q^{45} -5.12311i q^{46} +6.24621i q^{47} +1.00000 q^{48} -2.75379 q^{49} -1.00000i q^{50} +2.00000 q^{51} +(-3.56155 - 0.561553i) q^{52} -13.3693 q^{53} +1.00000i q^{54} -5.12311 q^{55} -3.12311 q^{56} +6.00000i q^{57} +2.00000i q^{58} +1.12311i q^{59} +1.00000i q^{60} +10.0000 q^{61} +3.12311 q^{62} -3.12311i q^{63} -1.00000 q^{64} +(0.561553 - 3.56155i) q^{65} +5.12311 q^{66} -4.87689i q^{67} -2.00000 q^{68} -5.12311 q^{69} -3.12311i q^{70} -10.2462i q^{71} -1.00000i q^{72} +13.1231i q^{73} -5.12311 q^{74} -1.00000 q^{75} -6.00000i q^{76} -16.0000 q^{77} +(-0.561553 + 3.56155i) q^{78} +8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +0.876894 q^{82} +6.24621i q^{83} +3.12311i q^{84} -2.00000i q^{85} +6.24621i q^{86} +2.00000 q^{87} -5.12311 q^{88} -3.12311i q^{89} +1.00000 q^{90} +(1.75379 - 11.1231i) q^{91} +5.12311 q^{92} -3.12311i q^{93} -6.24621 q^{94} +6.00000 q^{95} +1.00000i q^{96} +13.1231i q^{97} -2.75379i q^{98} -5.12311i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} + 4 q^{10} - 4 q^{12} + 6 q^{13} - 4 q^{14} + 4 q^{16} + 8 q^{17} + 4 q^{22} - 4 q^{23} - 4 q^{25} + 6 q^{26} + 4 q^{27} + 8 q^{29} + 4 q^{30} + 4 q^{35} - 4 q^{36} - 24 q^{38} + 6 q^{39} - 4 q^{40} - 4 q^{42} - 8 q^{43} + 4 q^{48} - 44 q^{49} + 8 q^{51} - 6 q^{52} - 4 q^{53} - 4 q^{55} + 4 q^{56} + 40 q^{61} - 4 q^{62} - 4 q^{64} - 6 q^{65} + 4 q^{66} - 8 q^{68} - 4 q^{69} - 4 q^{74} - 4 q^{75} - 64 q^{77} + 6 q^{78} + 32 q^{79} + 4 q^{81} + 20 q^{82} + 8 q^{87} - 4 q^{88} + 4 q^{90} + 40 q^{91} + 4 q^{92} + 8 q^{94} + 24 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 + 4 * q^10 - 4 * q^12 + 6 * q^13 - 4 * q^14 + 4 * q^16 + 8 * q^17 + 4 * q^22 - 4 * q^23 - 4 * q^25 + 6 * q^26 + 4 * q^27 + 8 * q^29 + 4 * q^30 + 4 * q^35 - 4 * q^36 - 24 * q^38 + 6 * q^39 - 4 * q^40 - 4 * q^42 - 8 * q^43 + 4 * q^48 - 44 * q^49 + 8 * q^51 - 6 * q^52 - 4 * q^53 - 4 * q^55 + 4 * q^56 + 40 * q^61 - 4 * q^62 - 4 * q^64 - 6 * q^65 + 4 * q^66 - 8 * q^68 - 4 * q^69 - 4 * q^74 - 4 * q^75 - 64 * q^77 + 6 * q^78 + 32 * q^79 + 4 * q^81 + 20 * q^82 + 8 * q^87 - 4 * q^88 + 4 * q^90 + 40 * q^91 + 4 * q^92 + 8 * q^94 + 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$.

$n$	$131$	$157$	$301$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	−1.00000			−0.500000
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$			1.00000i			0.408248i
$7$		−	3.12311i		−	1.18042i	−0.807249	−	0.590211i	$-0.799044\pi$
$7$		−	3.12311i		−	1.18042i	0.807249	−	0.590211i	$-0.200956\pi$
$8$		−	1.00000i		−	0.353553i
$9$	1.00000			0.333333
$10$	1.00000			0.316228
$11$		−	5.12311i		−	1.54467i	−0.635213	−	0.772337i	$-0.719088\pi$
$11$		−	5.12311i		−	1.54467i	0.635213	−	0.772337i	$-0.280912\pi$
$12$	−1.00000			−0.288675
$13$	3.56155	+	0.561553i	0.987797	+	0.155747i
$13$	3.56155	+	0.561553i	0.987797	+	0.155747i
$14$	3.12311			0.834685
$15$		−	1.00000i		−	0.258199i
$16$	1.00000			0.250000
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$			1.00000i			0.235702i
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$			1.00000i			0.223607i
$21$		−	3.12311i		−	0.681518i
$22$	5.12311			1.09225
$23$	−5.12311			−1.06824			−0.534121	−	0.845408i	$-0.679357\pi$
$23$	−5.12311			−1.06824			−0.534121	+	0.845408i	$0.679357\pi$
$24$		−	1.00000i		−	0.204124i
$25$	−1.00000			−0.200000
$26$	−0.561553	+	3.56155i	−0.110130	+	0.698478i
$27$	1.00000			0.192450
$28$			3.12311i			0.590211i
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$	1.00000			0.182574
$31$		−	3.12311i		−	0.560926i	−0.959865	−	0.280463i	$-0.909512\pi$
$31$		−	3.12311i		−	0.560926i	0.959865	−	0.280463i	$-0.0904881\pi$
$32$			1.00000i			0.176777i
$33$		−	5.12311i		−	0.891818i
$34$			2.00000i			0.342997i
$35$	−3.12311			−0.527901
$36$	−1.00000			−0.166667
$37$			5.12311i			0.842233i	0.907006	+	0.421117i	$0.138362\pi$
$37$			5.12311i			0.842233i	−0.907006	+	0.421117i	$0.861638\pi$
$38$	−6.00000			−0.973329
$39$	3.56155	+	0.561553i	0.570305	+	0.0899204i
$40$	−1.00000			−0.158114
$41$		−	0.876894i		−	0.136948i	−0.997653	−	0.0684739i	$-0.978187\pi$
$41$		−	0.876894i		−	0.136948i	0.997653	−	0.0684739i	$-0.0218130\pi$
$42$	3.12311			0.481906
$43$	6.24621			0.952538			0.476269	−	0.879300i	$-0.341989\pi$
$43$	6.24621			0.952538			0.476269	+	0.879300i	$0.341989\pi$
$44$			5.12311i			0.772337i
$45$		−	1.00000i		−	0.149071i
$46$		−	5.12311i		−	0.755361i
$47$			6.24621i			0.911104i	0.890209	+	0.455552i	$0.150558\pi$
$47$			6.24621i			0.911104i	−0.890209	+	0.455552i	$0.849442\pi$
$48$	1.00000			0.144338
$49$	−2.75379			−0.393398
$50$		−	1.00000i		−	0.141421i
$51$	2.00000			0.280056
$52$	−3.56155	−	0.561553i	−0.493899	−	0.0778734i
$53$	−13.3693			−1.83642			−0.918208	−	0.396098i	$-0.870364\pi$
$53$	−13.3693			−1.83642			−0.918208	+	0.396098i	$0.870364\pi$
$54$			1.00000i			0.136083i
$55$	−5.12311			−0.690799
$56$	−3.12311			−0.417343
$57$			6.00000i			0.794719i
$58$			2.00000i			0.262613i
$59$			1.12311i			0.146216i	0.997324	+	0.0731079i	$0.0232918\pi$
$59$			1.12311i			0.146216i	−0.997324	+	0.0731079i	$0.976708\pi$
$60$			1.00000i			0.129099i
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$	3.12311			0.396635
$63$		−	3.12311i		−	0.393474i
$64$	−1.00000			−0.125000
$65$	0.561553	−	3.56155i	0.0696521	−	0.441756i
$66$	5.12311			0.630611
$67$		−	4.87689i		−	0.595807i	−0.954596	−	0.297904i	$-0.903713\pi$
$67$		−	4.87689i		−	0.595807i	0.954596	−	0.297904i	$-0.0962874\pi$
$68$	−2.00000			−0.242536
$69$	−5.12311			−0.616749
$70$		−	3.12311i		−	0.373283i
$71$		−	10.2462i		−	1.21600i	−0.793936	−	0.608001i	$-0.791972\pi$
$71$		−	10.2462i		−	1.21600i	0.793936	−	0.608001i	$-0.208028\pi$
$72$		−	1.00000i		−	0.117851i
$73$			13.1231i			1.53594i	0.640484	+	0.767972i	$0.278734\pi$
$73$			13.1231i			1.53594i	−0.640484	+	0.767972i	$0.721266\pi$
$74$	−5.12311			−0.595549
$75$	−1.00000			−0.115470
$76$		−	6.00000i		−	0.688247i
$77$	−16.0000			−1.82337
$78$	−0.561553	+	3.56155i	−0.0635833	+	0.403266i
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		−	1.00000i		−	0.111803i
$81$	1.00000			0.111111
$82$	0.876894			0.0968368
$83$			6.24621i			0.685611i	0.939406	+	0.342805i	$0.111377\pi$
$83$			6.24621i			0.685611i	−0.939406	+	0.342805i	$0.888623\pi$
$84$			3.12311i			0.340759i
$85$		−	2.00000i		−	0.216930i
$86$			6.24621i			0.673546i
$87$	2.00000			0.214423
$88$	−5.12311			−0.546125
$89$		−	3.12311i		−	0.331049i	−0.986206	−	0.165524i	$-0.947068\pi$
$89$		−	3.12311i		−	0.331049i	0.986206	−	0.165524i	$-0.0529316\pi$
$90$	1.00000			0.105409
$91$	1.75379	−	11.1231i	0.183847	−	1.16602i
$92$	5.12311			0.534121
$93$		−	3.12311i		−	0.323851i
$94$	−6.24621			−0.644247
$95$	6.00000			0.615587
$96$			1.00000i			0.102062i
$97$			13.1231i			1.33245i	0.745751	+	0.666225i	$0.232091\pi$
$97$			13.1231i			1.33245i	−0.745751	+	0.666225i	$0.767909\pi$
$98$		−	2.75379i		−	0.278175i
$99$		−	5.12311i		−	0.514891i
$100$	1.00000			0.100000
$101$	−12.2462			−1.21854			−0.609272	−	0.792961i	$-0.708538\pi$
$101$	−12.2462			−1.21854			−0.609272	+	0.792961i	$0.708538\pi$
$102$			2.00000i			0.198030i
$103$	−13.1231			−1.29306			−0.646529	−	0.762889i	$-0.723780\pi$
$103$	−13.1231			−1.29306			−0.646529	+	0.762889i	$0.723780\pi$
$104$	0.561553	−	3.56155i	0.0550648	−	0.349239i
$105$	−3.12311			−0.304784
$106$		−	13.3693i		−	1.29854i
$107$	−8.00000			−0.773389			−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000			−0.773389			−0.386695	+	0.922208i	$0.626383\pi$
$108$	−1.00000			−0.0962250
$109$		−	2.87689i		−	0.275557i	−0.990463	−	0.137778i	$-0.956004\pi$
$109$		−	2.87689i		−	0.275557i	0.990463	−	0.137778i	$-0.0439961\pi$
$110$		−	5.12311i		−	0.488469i
$111$			5.12311i			0.486264i
$112$		−	3.12311i		−	0.295106i
$113$	12.2462			1.15203			0.576013	−	0.817440i	$-0.304608\pi$
$113$	12.2462			1.15203			0.576013	+	0.817440i	$0.304608\pi$
$114$	−6.00000			−0.561951
$115$			5.12311i			0.477732i
$116$	−2.00000			−0.185695
$117$	3.56155	+	0.561553i	0.329266	+	0.0519156i
$118$	−1.12311			−0.103390
$119$		−	6.24621i		−	0.572589i
$120$	−1.00000			−0.0912871
$121$	−15.2462			−1.38602
$122$			10.0000i			0.905357i
$123$		−	0.876894i		−	0.0790669i
$124$			3.12311i			0.280463i
$125$			1.00000i			0.0894427i
$126$	3.12311			0.278228
$127$	−13.1231			−1.16449			−0.582244	−	0.813014i	$-0.697825\pi$
$127$	−13.1231			−1.16449			−0.582244	+	0.813014i	$0.697825\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	6.24621			0.549948
$130$	3.56155	+	0.561553i	0.312369	+	0.0492514i
$131$	4.00000			0.349482			0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000			0.349482			0.174741	+	0.984614i	$0.444091\pi$
$132$			5.12311i			0.445909i
$133$	18.7386			1.62485
$134$	4.87689			0.421300
$135$		−	1.00000i		−	0.0860663i
$136$		−	2.00000i		−	0.171499i
$137$			10.4924i			0.896428i	0.893926	+	0.448214i	$0.147940\pi$
$137$			10.4924i			0.896428i	−0.893926	+	0.448214i	$0.852060\pi$
$138$		−	5.12311i		−	0.436108i
$139$	−16.4924			−1.39887			−0.699435	−	0.714697i	$-0.746565\pi$
$139$	−16.4924			−1.39887			−0.699435	+	0.714697i	$0.746565\pi$
$140$	3.12311			0.263951
$141$			6.24621i			0.526026i
$142$	10.2462			0.859843
$143$	2.87689	−	18.2462i	0.240578	−	1.52582i
$144$	1.00000			0.0833333
$145$		−	2.00000i		−	0.166091i
$146$	−13.1231			−1.08608
$147$	−2.75379			−0.227129
$148$		−	5.12311i		−	0.421117i
$149$			14.0000i			1.14692i	0.819232	+	0.573462i	$0.194400\pi$
$149$			14.0000i			1.14692i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		−	1.00000i		−	0.0816497i
$151$		−	13.3693i		−	1.08798i	−0.839092	−	0.543990i	$-0.816913\pi$
$151$		−	13.3693i		−	1.08798i	0.839092	−	0.543990i	$-0.183087\pi$
$152$	6.00000			0.486664
$153$	2.00000			0.161690
$154$		−	16.0000i		−	1.28932i
$155$	−3.12311			−0.250854
$156$	−3.56155	−	0.561553i	−0.285152	−	0.0449602i
$157$	21.3693			1.70546			0.852729	−	0.522354i	$-0.174946\pi$
$157$	21.3693			1.70546			0.852729	+	0.522354i	$0.174946\pi$
$158$			8.00000i			0.636446i
$159$	−13.3693			−1.06026
$160$	1.00000			0.0790569
$161$			16.0000i			1.26098i
$162$			1.00000i			0.0785674i
$163$			7.12311i			0.557925i	0.960302	+	0.278962i	$0.0899905\pi$
$163$			7.12311i			0.557925i	−0.960302	+	0.278962i	$0.910010\pi$
$164$			0.876894i			0.0684739i
$165$	−5.12311			−0.398833
$166$	−6.24621			−0.484800
$167$			22.2462i			1.72146i	0.509059	+	0.860732i	$0.329994\pi$
$167$			22.2462i			1.72146i	−0.509059	+	0.860732i	$0.670006\pi$
$168$	−3.12311			−0.240953
$169$	12.3693	+	4.00000i	0.951486	+	0.307692i
$170$	2.00000			0.153393
$171$			6.00000i			0.458831i
$172$	−6.24621			−0.476269
$173$	23.1231			1.75802			0.879009	−	0.476806i	$-0.158206\pi$
$173$	23.1231			1.75802			0.879009	+	0.476806i	$0.158206\pi$
$174$			2.00000i			0.151620i
$175$			3.12311i			0.236085i
$176$		−	5.12311i		−	0.386169i
$177$			1.12311i			0.0844178i
$178$	3.12311			0.234087
$179$	16.4924			1.23270			0.616351	−	0.787472i	$-0.288610\pi$
$179$	16.4924			1.23270			0.616351	+	0.787472i	$0.288610\pi$
$180$			1.00000i			0.0745356i
$181$	−20.2462			−1.50489			−0.752445	−	0.658656i	$-0.771125\pi$
$181$	−20.2462			−1.50489			−0.752445	+	0.658656i	$0.771125\pi$
$182$	11.1231	+	1.75379i	0.824499	+	0.129999i
$183$	10.0000			0.739221
$184$			5.12311i			0.377680i
$185$	5.12311			0.376658
$186$	3.12311			0.228997
$187$		−	10.2462i		−	0.749277i
$188$		−	6.24621i		−	0.455552i
$189$		−	3.12311i		−	0.227173i
$190$			6.00000i			0.435286i
$191$	16.4924			1.19335			0.596675	−	0.802483i	$-0.296488\pi$
$191$	16.4924			1.19335			0.596675	+	0.802483i	$0.296488\pi$
$192$	−1.00000			−0.0721688
$193$			25.1231i			1.80840i	0.427109	+	0.904200i	$0.359532\pi$
$193$			25.1231i			1.80840i	−0.427109	+	0.904200i	$0.640468\pi$
$194$	−13.1231			−0.942184
$195$	0.561553	−	3.56155i	0.0402136	−	0.255048i
$196$	2.75379			0.196699
$197$		−	16.2462i		−	1.15749i	−0.815507	−	0.578747i	$-0.803542\pi$
$197$		−	16.2462i		−	1.15749i	0.815507	−	0.578747i	$-0.196458\pi$
$198$	5.12311			0.364083
$199$	18.2462			1.29344			0.646720	−	0.762728i	$-0.276140\pi$
$199$	18.2462			1.29344			0.646720	+	0.762728i	$0.276140\pi$
$200$			1.00000i			0.0707107i
$201$		−	4.87689i		−	0.343990i
$202$		−	12.2462i		−	0.861640i
$203$		−	6.24621i		−	0.438398i
$204$	−2.00000			−0.140028
$205$	−0.876894			−0.0612450
$206$		−	13.1231i		−	0.914330i
$207$	−5.12311			−0.356080
$208$	3.56155	+	0.561553i	0.246949	+	0.0389367i
$209$	30.7386			2.12624
$210$		−	3.12311i		−	0.215515i
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$	13.3693			0.918208
$213$		−	10.2462i		−	0.702059i
$214$		−	8.00000i		−	0.546869i
$215$		−	6.24621i		−	0.425988i
$216$		−	1.00000i		−	0.0680414i
$217$	−9.75379			−0.662130
$218$	2.87689			0.194848
$219$			13.1231i			0.886777i
$220$	5.12311			0.345400
$221$	7.12311	+	1.12311i	0.479152	+	0.0755483i
$222$	−5.12311			−0.343840
$223$		−	9.36932i		−	0.627416i	−0.949520	−	0.313708i	$-0.898429\pi$
$223$		−	9.36932i		−	0.627416i	0.949520	−	0.313708i	$-0.101571\pi$
$224$	3.12311			0.208671
$225$	−1.00000			−0.0666667
$226$			12.2462i			0.814606i
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		−	6.00000i		−	0.397360i
$229$			5.12311i			0.338544i	0.985569	+	0.169272i	$0.0541417\pi$
$229$			5.12311i			0.338544i	−0.985569	+	0.169272i	$0.945858\pi$
$230$	−5.12311			−0.337808
$231$	−16.0000			−1.05272
$232$		−	2.00000i		−	0.131306i
$233$	−7.75379			−0.507968			−0.253984	−	0.967208i	$-0.581741\pi$
$233$	−7.75379			−0.507968			−0.253984	+	0.967208i	$0.581741\pi$
$234$	−0.561553	+	3.56155i	−0.0367099	+	0.232826i
$235$	6.24621			0.407458
$236$		−	1.12311i		−	0.0731079i
$237$	8.00000			0.519656
$238$	6.24621			0.404882
$239$		−	4.49242i		−	0.290591i	−0.989388	−	0.145295i	$-0.953587\pi$
$239$		−	4.49242i		−	0.290591i	0.989388	−	0.145295i	$-0.0464132\pi$
$240$		−	1.00000i		−	0.0645497i
$241$			14.2462i			0.917679i	0.888519	+	0.458840i	$0.151735\pi$
$241$			14.2462i			0.917679i	−0.888519	+	0.458840i	$0.848265\pi$
$242$		−	15.2462i		−	0.980064i
$243$	1.00000			0.0641500
$244$	−10.0000			−0.640184
$245$			2.75379i			0.175933i
$246$	0.876894			0.0559087
$247$	−3.36932	+	21.3693i	−0.214384	+	1.35970i
$248$	−3.12311			−0.198317
$249$			6.24621i			0.395838i
$250$	−1.00000			−0.0632456
$251$	26.2462			1.65665			0.828323	−	0.560251i	$-0.189295\pi$
$251$	26.2462			1.65665			0.828323	+	0.560251i	$0.189295\pi$
$252$			3.12311i			0.196737i
$253$			26.2462i			1.65009i
$254$		−	13.1231i		−	0.823417i
$255$		−	2.00000i		−	0.125245i
$256$	1.00000			0.0625000
$257$	−12.2462			−0.763898			−0.381949	−	0.924183i	$-0.624747\pi$
$257$	−12.2462			−0.763898			−0.381949	+	0.924183i	$0.624747\pi$
$258$			6.24621i			0.388872i
$259$	16.0000			0.994192
$260$	−0.561553	+	3.56155i	−0.0348260	+	0.220878i
$261$	2.00000			0.123797
$262$			4.00000i			0.247121i
$263$	−27.3693			−1.68766			−0.843832	−	0.536607i	$-0.819706\pi$
$263$	−27.3693			−1.68766			−0.843832	+	0.536607i	$0.819706\pi$
$264$	−5.12311			−0.315305
$265$			13.3693i			0.821271i
$266$			18.7386i			1.14894i
$267$		−	3.12311i		−	0.191131i
$268$			4.87689i			0.297904i
$269$	−16.2462			−0.990549			−0.495274	−	0.868737i	$-0.664933\pi$
$269$	−16.2462			−0.990549			−0.495274	+	0.868737i	$0.664933\pi$
$270$	1.00000			0.0608581
$271$		−	23.1231i		−	1.40463i	−0.711867	−	0.702314i	$-0.752150\pi$
$271$		−	23.1231i		−	1.40463i	0.711867	−	0.702314i	$-0.247850\pi$
$272$	2.00000			0.121268
$273$	1.75379	−	11.1231i	0.106144	−	0.673201i
$274$	−10.4924			−0.633870
$275$			5.12311i			0.308935i
$276$	5.12311			0.308375
$277$	−29.8617			−1.79422			−0.897109	−	0.441809i	$-0.854337\pi$
$277$	−29.8617			−1.79422			−0.897109	+	0.441809i	$0.854337\pi$
$278$		−	16.4924i		−	0.989150i
$279$		−	3.12311i		−	0.186975i
$280$			3.12311i			0.186641i
$281$			3.12311i			0.186309i	0.995652	+	0.0931544i	$0.0296950\pi$
$281$			3.12311i			0.186309i	−0.995652	+	0.0931544i	$0.970305\pi$
$282$	−6.24621			−0.371956
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$			10.2462i			0.608001i
$285$	6.00000			0.355409
$286$	18.2462	+	2.87689i	1.07892	+	0.170114i
$287$	−2.73863			−0.161656
$288$			1.00000i			0.0589256i
$289$	−13.0000			−0.764706
$290$	2.00000			0.117444
$291$			13.1231i			0.769290i
$292$		−	13.1231i		−	0.767972i
$293$		−	28.7386i		−	1.67893i	−0.543415	−	0.839464i	$-0.682869\pi$
$293$		−	28.7386i		−	1.67893i	0.543415	−	0.839464i	$-0.317131\pi$
$294$		−	2.75379i		−	0.160604i
$295$	1.12311			0.0653897
$296$	5.12311			0.297774
$297$		−	5.12311i		−	0.297273i
$298$	−14.0000			−0.810998
$299$	−18.2462	−	2.87689i	−1.05521	−	0.166375i
$300$	1.00000			0.0577350
$301$		−	19.5076i		−	1.12440i
$302$	13.3693			0.769318
$303$	−12.2462			−0.703526
$304$			6.00000i			0.344124i
$305$		−	10.0000i		−	0.572598i
$306$			2.00000i			0.114332i
$307$		−	31.1231i		−	1.77629i	−0.459564	−	0.888145i	$-0.651994\pi$
$307$		−	31.1231i		−	1.77629i	0.459564	−	0.888145i	$-0.348006\pi$
$308$	16.0000			0.911685
$309$	−13.1231			−0.746547
$310$		−	3.12311i		−	0.177380i
$311$	8.49242			0.481561			0.240781	−	0.970580i	$-0.422597\pi$
$311$	8.49242			0.481561			0.240781	+	0.970580i	$0.422597\pi$
$312$	0.561553	−	3.56155i	0.0317917	−	0.201633i
$313$	16.2462			0.918290			0.459145	−	0.888361i	$-0.348156\pi$
$313$	16.2462			0.918290			0.459145	+	0.888361i	$0.348156\pi$
$314$			21.3693i			1.20594i
$315$	−3.12311			−0.175967
$316$	−8.00000			−0.450035
$317$		−	6.00000i		−	0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$		−	6.00000i		−	0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$		−	13.3693i		−	0.749714i
$319$		−	10.2462i		−	0.573678i
$320$			1.00000i			0.0559017i
$321$	−8.00000			−0.446516
$322$	−16.0000			−0.891645
$323$			12.0000i			0.667698i
$324$	−1.00000			−0.0555556
$325$	−3.56155	−	0.561553i	−0.197559	−	0.0311493i
$326$	−7.12311			−0.394512
$327$		−	2.87689i		−	0.159093i
$328$	−0.876894			−0.0484184
$329$	19.5076			1.07549
$330$		−	5.12311i		−	0.282018i
$331$			7.75379i			0.426187i	0.977032	+	0.213093i	$0.0683539\pi$
$331$			7.75379i			0.426187i	−0.977032	+	0.213093i	$0.931646\pi$
$332$		−	6.24621i		−	0.342805i
$333$			5.12311i			0.280744i
$334$	−22.2462			−1.21726
$335$	−4.87689			−0.266453
$336$		−	3.12311i		−	0.170379i
$337$	6.00000			0.326841			0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000			0.326841			0.163420	+	0.986557i	$0.447747\pi$
$338$	−4.00000	+	12.3693i	−0.217571	+	0.672802i
$339$	12.2462			0.665123
$340$			2.00000i			0.108465i
$341$	−16.0000			−0.866449
$342$	−6.00000			−0.324443
$343$		−	13.2614i		−	0.716046i
$344$		−	6.24621i		−	0.336773i
$345$			5.12311i			0.275819i
$346$			23.1231i			1.24311i
$347$	−14.2462			−0.764777			−0.382388	−	0.924002i	$-0.624898\pi$
$347$	−14.2462			−0.764777			−0.382388	+	0.924002i	$0.624898\pi$
$348$	−2.00000			−0.107211
$349$		−	19.3693i		−	1.03682i	−0.855133	−	0.518408i	$-0.826525\pi$
$349$		−	19.3693i		−	1.03682i	0.855133	−	0.518408i	$-0.173475\pi$
$350$	−3.12311			−0.166937
$351$	3.56155	+	0.561553i	0.190102	+	0.0299735i
$352$	5.12311			0.273062
$353$			12.2462i			0.651800i	0.945404	+	0.325900i	$0.105667\pi$
$353$			12.2462i			0.651800i	−0.945404	+	0.325900i	$0.894333\pi$
$354$	−1.12311			−0.0596924
$355$	−10.2462			−0.543812
$356$			3.12311i			0.165524i
$357$		−	6.24621i		−	0.330585i
$358$			16.4924i			0.871652i
$359$		−	17.7538i		−	0.937009i	−0.883461	−	0.468505i	$-0.844793\pi$
$359$		−	17.7538i		−	0.937009i	0.883461	−	0.468505i	$-0.155207\pi$
$360$	−1.00000			−0.0527046
$361$	−17.0000			−0.894737
$362$		−	20.2462i		−	1.06412i
$363$	−15.2462			−0.800219
$364$	−1.75379	+	11.1231i	−0.0919235	+	0.583009i
$365$	13.1231			0.686895
$366$			10.0000i			0.522708i
$367$	8.63068			0.450518			0.225259	−	0.974299i	$-0.427677\pi$
$367$	8.63068			0.450518			0.225259	+	0.974299i	$0.427677\pi$
$368$	−5.12311			−0.267060
$369$		−	0.876894i		−	0.0456493i
$370$			5.12311i			0.266338i
$371$			41.7538i			2.16775i
$372$			3.12311i			0.161925i
$373$	7.12311			0.368820			0.184410	−	0.982849i	$-0.440963\pi$
$373$	7.12311			0.368820			0.184410	+	0.982849i	$0.440963\pi$
$374$	10.2462			0.529819
$375$			1.00000i			0.0516398i
$376$	6.24621			0.322124
$377$	7.12311	+	1.12311i	0.366859	+	0.0578429i
$378$	3.12311			0.160635
$379$			6.00000i			0.308199i	0.988055	+	0.154100i	$0.0492477\pi$
$379$			6.00000i			0.308199i	−0.988055	+	0.154100i	$0.950752\pi$
$380$	−6.00000			−0.307794
$381$	−13.1231			−0.672317
$382$			16.4924i			0.843826i
$383$		−	1.75379i		−	0.0896144i	−0.998996	−	0.0448072i	$-0.985733\pi$
$383$		−	1.75379i		−	0.0896144i	0.998996	−	0.0448072i	$-0.0142674\pi$
$384$		−	1.00000i		−	0.0510310i
$385$			16.0000i			0.815436i
$386$	−25.1231			−1.27873
$387$	6.24621			0.317513
$388$		−	13.1231i		−	0.666225i
$389$	−0.246211			−0.0124834			−0.00624170	−	0.999981i	$-0.501987\pi$
$389$	−0.246211			−0.0124834			−0.00624170	+	0.999981i	$0.501987\pi$
$390$	3.56155	+	0.561553i	0.180346	+	0.0284353i
$391$	−10.2462			−0.518173
$392$			2.75379i			0.139087i
$393$	4.00000			0.201773
$394$	16.2462			0.818472
$395$		−	8.00000i		−	0.402524i
$396$			5.12311i			0.257446i
$397$			15.3693i			0.771364i	0.922632	+	0.385682i	$0.126034\pi$
$397$			15.3693i			0.771364i	−0.922632	+	0.385682i	$0.873966\pi$
$398$			18.2462i			0.914600i
$399$	18.7386			0.938105
$400$	−1.00000			−0.0500000
$401$			1.36932i			0.0683804i	0.999415	+	0.0341902i	$0.0108852\pi$
$401$			1.36932i			0.0683804i	−0.999415	+	0.0341902i	$0.989115\pi$
$402$	4.87689			0.243237
$403$	1.75379	−	11.1231i	0.0873624	−	0.554081i
$404$	12.2462			0.609272
$405$		−	1.00000i		−	0.0496904i
$406$	6.24621			0.309994
$407$	26.2462			1.30098
$408$		−	2.00000i		−	0.0990148i
$409$		−	8.49242i		−	0.419923i	−0.977710	−	0.209962i	$-0.932666\pi$
$409$		−	8.49242i		−	0.419923i	0.977710	−	0.209962i	$-0.0673339\pi$
$410$		−	0.876894i		−	0.0433067i
$411$			10.4924i			0.517553i
$412$	13.1231			0.646529
$413$	3.50758			0.172597
$414$		−	5.12311i		−	0.251787i
$415$	6.24621			0.306614
$416$	−0.561553	+	3.56155i	−0.0275324	+	0.174619i
$417$	−16.4924			−0.807637
$418$			30.7386i			1.50348i
$419$	−28.0000			−1.36789			−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000			−1.36789			−0.683945	+	0.729534i	$0.739737\pi$
$420$	3.12311			0.152392
$421$			0.630683i			0.0307376i	0.999882	+	0.0153688i	$0.00489224\pi$
$421$			0.630683i			0.0307376i	−0.999882	+	0.0153688i	$0.995108\pi$
$422$			4.00000i			0.194717i
$423$			6.24621i			0.303701i
$424$			13.3693i			0.649271i
$425$	−2.00000			−0.0970143
$426$	10.2462			0.496431
$427$		−	31.2311i		−	1.51138i
$428$	8.00000			0.386695
$429$	2.87689	−	18.2462i	0.138898	−	0.880935i
$430$	6.24621			0.301219
$431$			32.4924i			1.56510i	0.622585	+	0.782552i	$0.286083\pi$
$431$			32.4924i			1.56510i	−0.622585	+	0.782552i	$0.713917\pi$
$432$	1.00000			0.0481125
$433$	−18.0000			−0.865025			−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000			−0.865025			−0.432512	+	0.901628i	$0.642373\pi$
$434$		−	9.75379i		−	0.468197i
$435$		−	2.00000i		−	0.0958927i
$436$			2.87689i			0.137778i
$437$		−	30.7386i		−	1.47043i
$438$	−13.1231			−0.627046
$439$	−36.4924			−1.74169			−0.870844	−	0.491559i	$-0.836427\pi$
$439$	−36.4924			−1.74169			−0.870844	+	0.491559i	$0.836427\pi$
$440$			5.12311i			0.244234i
$441$	−2.75379			−0.131133
$442$	−1.12311	+	7.12311i	−0.0534207	+	0.338812i
$443$	−3.50758			−0.166650			−0.0833250	−	0.996522i	$-0.526554\pi$
$443$	−3.50758			−0.166650			−0.0833250	+	0.996522i	$0.526554\pi$
$444$		−	5.12311i		−	0.243132i
$445$	−3.12311			−0.148049
$446$	9.36932			0.443650
$447$			14.0000i			0.662177i
$448$			3.12311i			0.147553i
$449$			28.8769i			1.36278i	0.731918	+	0.681392i	$0.238625\pi$
$449$			28.8769i			1.36278i	−0.731918	+	0.681392i	$0.761375\pi$
$450$		−	1.00000i		−	0.0471405i
$451$	−4.49242			−0.211540
$452$	−12.2462			−0.576013
$453$		−	13.3693i		−	0.628145i
$454$	8.00000			0.375459
$455$	−11.1231	−	1.75379i	−0.521459	−	0.0822189i
$456$	6.00000			0.280976
$457$		−	31.3693i		−	1.46739i	−0.679476	−	0.733697i	$-0.737793\pi$
$457$		−	31.3693i		−	1.46739i	0.679476	−	0.733697i	$-0.262207\pi$
$458$	−5.12311			−0.239387
$459$	2.00000			0.0933520
$460$		−	5.12311i		−	0.238866i
$461$		−	18.4924i		−	0.861278i	−0.902524	−	0.430639i	$-0.858288\pi$
$461$		−	18.4924i		−	0.861278i	0.902524	−	0.430639i	$-0.141712\pi$
$462$		−	16.0000i		−	0.744387i
$463$			21.8617i			1.01600i	0.861357	+	0.508001i	$0.169615\pi$
$463$			21.8617i			1.01600i	−0.861357	+	0.508001i	$0.830385\pi$
$464$	2.00000			0.0928477
$465$	−3.12311			−0.144831
$466$		−	7.75379i		−	0.359187i
$467$	−22.2462			−1.02943			−0.514716	−	0.857361i	$-0.672103\pi$
$467$	−22.2462			−1.02943			−0.514716	+	0.857361i	$0.672103\pi$
$468$	−3.56155	−	0.561553i	−0.164633	−	0.0259578i
$469$	−15.2311			−0.703305
$470$			6.24621i			0.288116i
$471$	21.3693			0.984646
$472$	1.12311			0.0516951
$473$		−	32.0000i		−	1.47136i
$474$			8.00000i			0.367452i
$475$		−	6.00000i		−	0.275299i
$476$			6.24621i			0.286295i
$477$	−13.3693			−0.612139
$478$	4.49242			0.205479
$479$			12.4924i			0.570793i	0.958409	+	0.285397i	$0.0921253\pi$
$479$			12.4924i			0.570793i	−0.958409	+	0.285397i	$0.907875\pi$
$480$	1.00000			0.0456435
$481$	−2.87689	+	18.2462i	−0.131175	+	0.831956i
$482$	−14.2462			−0.648897
$483$			16.0000i			0.728025i
$484$	15.2462			0.693010
$485$	13.1231			0.595890
$486$			1.00000i			0.0453609i
$487$		−	17.3693i		−	0.787079i	−0.919308	−	0.393539i	$-0.871250\pi$
$487$		−	17.3693i		−	0.787079i	0.919308	−	0.393539i	$-0.128750\pi$
$488$		−	10.0000i		−	0.452679i
$489$			7.12311i			0.322118i
$490$	−2.75379			−0.124403
$491$	38.7386			1.74825			0.874125	−	0.485701i	$-0.161436\pi$
$491$	38.7386			1.74825			0.874125	+	0.485701i	$0.161436\pi$
$492$			0.876894i			0.0395335i
$493$	4.00000			0.180151
$494$	−21.3693	−	3.36932i	−0.961451	−	0.151593i
$495$	−5.12311			−0.230266
$496$		−	3.12311i		−	0.140232i
$497$	−32.0000			−1.43540
$498$	−6.24621			−0.279899
$499$		−	34.4924i		−	1.54409i	−0.635566	−	0.772046i	$-0.719233\pi$
$499$		−	34.4924i		−	1.54409i	0.635566	−	0.772046i	$-0.280767\pi$
$500$		−	1.00000i		−	0.0447214i
$501$			22.2462i			0.993887i
$502$			26.2462i			1.17143i
$503$	35.3693			1.57704			0.788520	−	0.615009i	$-0.210848\pi$
$503$	35.3693			1.57704			0.788520	+	0.615009i	$0.210848\pi$
$504$	−3.12311			−0.139114
$505$			12.2462i			0.544949i
$506$	−26.2462			−1.16679
$507$	12.3693	+	4.00000i	0.549341	+	0.177646i
$508$	13.1231			0.582244
$509$		−	32.2462i		−	1.42929i	−0.699488	−	0.714644i	$-0.746589\pi$
$509$		−	32.2462i		−	1.42929i	0.699488	−	0.714644i	$-0.253411\pi$
$510$	2.00000			0.0885615
$511$	40.9848			1.81306
$512$			1.00000i			0.0441942i
$513$			6.00000i			0.264906i
$514$		−	12.2462i		−	0.540157i
$515$			13.1231i			0.578273i
$516$	−6.24621			−0.274974
$517$	32.0000			1.40736
$518$			16.0000i			0.703000i
$519$	23.1231			1.01499
$520$	−3.56155	−	0.561553i	−0.156184	−	0.0246257i
$521$	0.246211			0.0107867			0.00539336	−	0.999985i	$-0.498283\pi$
$521$	0.246211			0.0107867			0.00539336	+	0.999985i	$0.498283\pi$
$522$			2.00000i			0.0875376i
$523$	−26.7386			−1.16920			−0.584599	−	0.811322i	$-0.698748\pi$
$523$	−26.7386			−1.16920			−0.584599	+	0.811322i	$0.698748\pi$
$524$	−4.00000			−0.174741
$525$			3.12311i			0.136304i
$526$		−	27.3693i		−	1.19336i
$527$		−	6.24621i		−	0.272089i
$528$		−	5.12311i		−	0.222955i
$529$	3.24621			0.141140
$530$	−13.3693			−0.580726
$531$			1.12311i			0.0487386i
$532$	−18.7386			−0.812423
$533$	0.492423	−	3.12311i	0.0213292	−	0.135277i
$534$	3.12311			0.135150
$535$			8.00000i			0.345870i
$536$	−4.87689			−0.210650
$537$	16.4924			0.711701
$538$		−	16.2462i		−	0.700424i
$539$			14.1080i			0.607672i
$540$			1.00000i			0.0430331i
$541$		−	10.8769i		−	0.467634i	−0.972281	−	0.233817i	$-0.924878\pi$
$541$		−	10.8769i		−	0.467634i	0.972281	−	0.233817i	$-0.0751217\pi$
$542$	23.1231			0.993222
$543$	−20.2462			−0.868848
$544$			2.00000i			0.0857493i
$545$	−2.87689			−0.123233
$546$	11.1231	+	1.75379i	0.476025	+	0.0750552i
$547$	20.9848			0.897247			0.448624	−	0.893721i	$-0.351914\pi$
$547$	20.9848			0.897247			0.448624	+	0.893721i	$0.351914\pi$
$548$		−	10.4924i		−	0.448214i
$549$	10.0000			0.426790
$550$	−5.12311			−0.218450
$551$			12.0000i			0.511217i
$552$			5.12311i			0.218054i
$553$		−	24.9848i		−	1.06246i
$554$		−	29.8617i		−	1.26870i
$555$	5.12311			0.217464
$556$	16.4924			0.699435
$557$		−	11.7538i		−	0.498024i	−0.968500	−	0.249012i	$-0.919894\pi$
$557$		−	11.7538i		−	0.498024i	0.968500	−	0.249012i	$-0.0801059\pi$
$558$	3.12311			0.132212
$559$	22.2462	+	3.50758i	0.940914	+	0.148355i
$560$	−3.12311			−0.131975
$561$		−	10.2462i		−	0.432595i
$562$	−3.12311			−0.131740
$563$	−32.9848			−1.39015			−0.695073	−	0.718939i	$-0.744628\pi$
$563$	−32.9848			−1.39015			−0.695073	+	0.718939i	$0.744628\pi$
$564$		−	6.24621i		−	0.263013i
$565$		−	12.2462i		−	0.515202i
$566$		−	4.00000i		−	0.168133i
$567$		−	3.12311i		−	0.131158i
$568$	−10.2462			−0.429921
$569$	−12.7386			−0.534031			−0.267016	−	0.963692i	$-0.586038\pi$
$569$	−12.7386			−0.534031			−0.267016	+	0.963692i	$0.586038\pi$
$570$			6.00000i			0.251312i
$571$	−8.49242			−0.355397			−0.177698	−	0.984085i	$-0.556865\pi$
$571$	−8.49242			−0.355397			−0.177698	+	0.984085i	$0.556865\pi$
$572$	−2.87689	+	18.2462i	−0.120289	+	0.762912i
$573$	16.4924			0.688981
$574$		−	2.73863i		−	0.114308i
$575$	5.12311			0.213648
$576$	−1.00000			−0.0416667
$577$		−	27.3693i		−	1.13940i	−0.821853	−	0.569700i	$-0.807059\pi$
$577$		−	27.3693i		−	1.13940i	0.821853	−	0.569700i	$-0.192941\pi$
$578$		−	13.0000i		−	0.540729i
$579$			25.1231i			1.04408i
$580$			2.00000i			0.0830455i
$581$	19.5076			0.809311
$582$	−13.1231			−0.543970
$583$			68.4924i			2.83667i
$584$	13.1231			0.543038
$585$	0.561553	−	3.56155i	0.0232174	−	0.147252i
$586$	28.7386			1.18718
$587$			16.4924i			0.680715i	0.940296	+	0.340358i	$0.110548\pi$
$587$			16.4924i			0.680715i	−0.940296	+	0.340358i	$0.889452\pi$
$588$	2.75379			0.113564
$589$	18.7386			0.772112
$590$			1.12311i			0.0462375i
$591$		−	16.2462i		−	0.668280i
$592$			5.12311i			0.210558i
$593$			5.50758i			0.226169i	0.993585	+	0.113085i	$0.0360731\pi$
$593$			5.50758i			0.226169i	−0.993585	+	0.113085i	$0.963927\pi$
$594$	5.12311			0.210204
$595$	−6.24621			−0.256070
$596$		−	14.0000i		−	0.573462i
$597$	18.2462			0.746768
$598$	2.87689	−	18.2462i	0.117645	−	0.746143i
$599$	−36.4924			−1.49104			−0.745520	−	0.666483i	$-0.767799\pi$
$599$	−36.4924			−1.49104			−0.745520	+	0.666483i	$0.767799\pi$
$600$			1.00000i			0.0408248i
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$	19.5076			0.795070
$603$		−	4.87689i		−	0.198602i
$604$			13.3693i			0.543990i
$605$			15.2462i			0.619847i
$606$		−	12.2462i		−	0.497468i
$607$	15.3693			0.623821			0.311911	−	0.950111i	$-0.399031\pi$
$607$	15.3693			0.623821			0.311911	+	0.950111i	$0.399031\pi$
$608$	−6.00000			−0.243332
$609$		−	6.24621i		−	0.253109i
$610$	10.0000			0.404888
$611$	−3.50758	+	22.2462i	−0.141901	+	0.899985i
$612$	−2.00000			−0.0808452
$613$		−	39.3693i		−	1.59011i	−0.606536	−	0.795056i	$-0.707441\pi$
$613$		−	39.3693i		−	1.59011i	0.606536	−	0.795056i	$-0.292559\pi$
$614$	31.1231			1.25603
$615$	−0.876894			−0.0353598
$616$			16.0000i			0.644658i
$617$			40.7386i			1.64008i	0.572309	+	0.820038i	$0.306048\pi$
$617$			40.7386i			1.64008i	−0.572309	+	0.820038i	$0.693952\pi$
$618$		−	13.1231i		−	0.527889i
$619$		−	38.9848i		−	1.56693i	−0.621434	−	0.783467i	$-0.713450\pi$
$619$		−	38.9848i		−	1.56693i	0.621434	−	0.783467i	$-0.286550\pi$
$620$	3.12311			0.125427
$621$	−5.12311			−0.205583
$622$			8.49242i			0.340515i
$623$	−9.75379			−0.390777
$624$	3.56155	+	0.561553i	0.142576	+	0.0224801i
$625$	1.00000			0.0400000
$626$			16.2462i			0.649329i
$627$	30.7386			1.22758
$628$	−21.3693			−0.852729
$629$			10.2462i			0.408543i
$630$		−	3.12311i		−	0.124428i
$631$			35.6155i			1.41783i	0.705293	+	0.708916i	$0.250815\pi$
$631$			35.6155i			1.41783i	−0.705293	+	0.708916i	$0.749185\pi$
$632$		−	8.00000i		−	0.318223i
$633$	4.00000			0.158986
$634$	6.00000			0.238290
$635$			13.1231i			0.520775i
$636$	13.3693			0.530128
$637$	−9.80776	−	1.54640i	−0.388598	−	0.0612705i
$638$	10.2462			0.405651
$639$		−	10.2462i		−	0.405334i
$640$	−1.00000			−0.0395285
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$		−	8.00000i		−	0.315735i
$643$		−	17.3693i		−	0.684979i	−0.939522	−	0.342489i	$-0.888730\pi$
$643$		−	17.3693i		−	0.684979i	0.939522	−	0.342489i	$-0.111270\pi$
$644$		−	16.0000i		−	0.630488i
$645$		−	6.24621i		−	0.245944i
$646$	−12.0000			−0.472134
$647$	29.6155			1.16431			0.582153	−	0.813079i	$-0.302210\pi$
$647$	29.6155			1.16431			0.582153	+	0.813079i	$0.302210\pi$
$648$		−	1.00000i		−	0.0392837i
$649$	5.75379			0.225856
$650$	0.561553	−	3.56155i	0.0220259	−	0.139696i
$651$	−9.75379			−0.382281
$652$		−	7.12311i		−	0.278962i
$653$	−13.8617			−0.542452			−0.271226	−	0.962516i	$-0.587429\pi$
$653$	−13.8617			−0.542452			−0.271226	+	0.962516i	$0.587429\pi$
$654$	2.87689			0.112495
$655$		−	4.00000i		−	0.156293i
$656$		−	0.876894i		−	0.0342370i
$657$			13.1231i			0.511981i
$658$			19.5076i			0.760485i
$659$	−21.7538			−0.847407			−0.423704	−	0.905801i	$-0.639270\pi$
$659$	−21.7538			−0.847407			−0.423704	+	0.905801i	$0.639270\pi$
$660$	5.12311			0.199417
$661$		−	9.12311i		−	0.354848i	−0.984135	−	0.177424i	$-0.943224\pi$
$661$		−	9.12311i		−	0.354848i	0.984135	−	0.177424i	$-0.0567763\pi$
$662$	−7.75379			−0.301360
$663$	7.12311	+	1.12311i	0.276638	+	0.0436178i
$664$	6.24621			0.242400
$665$		−	18.7386i		−	0.726653i
$666$	−5.12311			−0.198516
$667$	−10.2462			−0.396735
$668$		−	22.2462i		−	0.860732i
$669$		−	9.36932i		−	0.362239i
$670$		−	4.87689i		−	0.188411i
$671$		−	51.2311i		−	1.97775i
$672$	3.12311			0.120476
$673$	26.9848			1.04019			0.520095	−	0.854109i	$-0.325897\pi$
$673$	26.9848			1.04019			0.520095	+	0.854109i	$0.325897\pi$
$674$			6.00000i			0.231111i
$675$	−1.00000			−0.0384900
$676$	−12.3693	−	4.00000i	−0.475743	−	0.153846i
$677$	12.8769			0.494899			0.247450	−	0.968901i	$-0.420408\pi$
$677$	12.8769			0.494899			0.247450	+	0.968901i	$0.420408\pi$
$678$			12.2462i			0.470313i
$679$	40.9848			1.57285
$680$	−2.00000			−0.0766965
$681$		−	8.00000i		−	0.306561i
$682$		−	16.0000i		−	0.612672i
$683$			16.9848i			0.649907i	0.945730	+	0.324954i	$0.105349\pi$
$683$			16.9848i			0.649907i	−0.945730	+	0.324954i	$0.894651\pi$
$684$		−	6.00000i		−	0.229416i
$685$	10.4924			0.400895
$686$	13.2614			0.506321
$687$			5.12311i			0.195459i
$688$	6.24621			0.238135
$689$	−47.6155	−	7.50758i	−1.81401	−	0.286016i
$690$	−5.12311			−0.195033
$691$		−	28.7386i		−	1.09327i	−0.837371	−	0.546635i	$-0.815909\pi$
$691$		−	28.7386i		−	1.09327i	0.837371	−	0.546635i	$-0.184091\pi$
$692$	−23.1231			−0.879009
$693$	−16.0000			−0.607790
$694$		−	14.2462i		−	0.540779i
$695$			16.4924i			0.625593i
$696$		−	2.00000i		−	0.0758098i
$697$		−	1.75379i		−	0.0664295i
$698$	19.3693			0.733139
$699$	−7.75379			−0.293275
$700$		−	3.12311i		−	0.118042i
$701$	−14.0000			−0.528773			−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000			−0.528773			−0.264386	+	0.964417i	$0.585169\pi$
$702$	−0.561553	+	3.56155i	−0.0211944	+	0.134422i
$703$	−30.7386			−1.15933
$704$			5.12311i			0.193084i
$705$	6.24621			0.235246
$706$	−12.2462			−0.460892
$707$			38.2462i			1.43840i
$708$		−	1.12311i		−	0.0422089i
$709$			1.61553i			0.0606724i	0.999540	+	0.0303362i	$0.00965780\pi$
$709$			1.61553i			0.0606724i	−0.999540	+	0.0303362i	$0.990342\pi$
$710$		−	10.2462i		−	0.384533i
$711$	8.00000			0.300023
$712$	−3.12311			−0.117043
$713$			16.0000i			0.599205i
$714$	6.24621			0.233759
$715$	−18.2462	−	2.87689i	−0.682370	−	0.107590i
$716$	−16.4924			−0.616351
$717$		−	4.49242i		−	0.167773i
$718$	17.7538			0.662566
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		−	1.00000i		−	0.0372678i
$721$			40.9848i			1.52636i
$722$		−	17.0000i		−	0.632674i
$723$			14.2462i			0.529822i
$724$	20.2462			0.752445
$725$	−2.00000			−0.0742781
$726$		−	15.2462i		−	0.565840i
$727$	19.8617			0.736631			0.368316	−	0.929701i	$-0.379935\pi$
$727$	19.8617			0.736631			0.368316	+	0.929701i	$0.379935\pi$
$728$	−11.1231	−	1.75379i	−0.412250	−	0.0649997i
$729$	1.00000			0.0370370
$730$			13.1231i			0.485708i
$731$	12.4924			0.462049
$732$	−10.0000			−0.369611
$733$		−	35.3693i		−	1.30640i	−0.757187	−	0.653198i	$-0.773427\pi$
$733$		−	35.3693i		−	1.30640i	0.757187	−	0.653198i	$-0.226573\pi$
$734$			8.63068i			0.318564i
$735$			2.75379i			0.101575i
$736$		−	5.12311i		−	0.188840i
$737$	−24.9848			−0.920329
$738$	0.876894			0.0322789
$739$			37.2311i			1.36957i	0.728747	+	0.684783i	$0.240103\pi$
$739$			37.2311i			1.36957i	−0.728747	+	0.684783i	$0.759897\pi$
$740$	−5.12311			−0.188329
$741$	−3.36932	+	21.3693i	−0.123775	+	0.785021i
$742$	−41.7538			−1.53283
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	−3.12311			−0.114499
$745$	14.0000			0.512920
$746$			7.12311i			0.260795i
$747$			6.24621i			0.228537i
$748$			10.2462i			0.374639i
$749$			24.9848i			0.912926i
$750$	−1.00000			−0.0365148
$751$	−26.2462			−0.957738			−0.478869	−	0.877886i	$-0.658953\pi$
$751$	−26.2462			−0.957738			−0.478869	+	0.877886i	$0.658953\pi$
$752$			6.24621i			0.227776i
$753$	26.2462			0.956465
$754$	−1.12311	+	7.12311i	−0.0409011	+	0.259408i
$755$	−13.3693			−0.486559
$756$			3.12311i			0.113586i
$757$	−3.12311			−0.113511			−0.0567556	−	0.998388i	$-0.518076\pi$
$757$	−3.12311			−0.113511			−0.0567556	+	0.998388i	$0.518076\pi$
$758$	−6.00000			−0.217930
$759$			26.2462i			0.952677i
$760$		−	6.00000i		−	0.217643i
$761$			3.12311i			0.113212i	0.998397	+	0.0566062i	$0.0180280\pi$
$761$			3.12311i			0.113212i	−0.998397	+	0.0566062i	$0.981972\pi$
$762$		−	13.1231i		−	0.475400i
$763$	−8.98485			−0.325273
$764$	−16.4924			−0.596675
$765$		−	2.00000i		−	0.0723102i
$766$	1.75379			0.0633670
$767$	−0.630683	+	4.00000i	−0.0227726	+	0.144432i
$768$	1.00000			0.0360844
$769$			32.9848i			1.18946i	0.803924	+	0.594732i	$0.202742\pi$
$769$			32.9848i			1.18946i	−0.803924	+	0.594732i	$0.797258\pi$
$770$	−16.0000			−0.576600
$771$	−12.2462			−0.441037
$772$		−	25.1231i		−	0.904200i
$773$			16.2462i			0.584336i	0.956367	+	0.292168i	$0.0943766\pi$
$773$			16.2462i			0.584336i	−0.956367	+	0.292168i	$0.905623\pi$
$774$			6.24621i			0.224515i
$775$			3.12311i			0.112185i
$776$	13.1231			0.471092
$777$	16.0000			0.573997
$778$		−	0.246211i		−	0.00882710i
$779$	5.26137			0.188508
$780$	−0.561553	+	3.56155i	−0.0201068	+	0.127524i
$781$	−52.4924			−1.87833
$782$		−	10.2462i		−	0.366404i
$783$	2.00000			0.0714742
$784$	−2.75379			−0.0983496
$785$		−	21.3693i

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:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.12311i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.12311i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.12311i q^{11} -1.00000 q^{12} +(3.56155 + 0.561553i) q^{13} +3.12311 q^{14} -1.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000i q^{18} +6.00000i q^{19} +1.00000i q^{20} -3.12311i q^{21} +5.12311 q^{22} -5.12311 q^{23} -1.00000i q^{24} -1.00000 q^{25} +(-0.561553 + 3.56155i) q^{26} +1.00000 q^{27} +3.12311i q^{28} +2.00000 q^{29} +1.00000 q^{30} -3.12311i q^{31} +1.00000i q^{32} -5.12311i q^{33} +2.00000i q^{34} -3.12311 q^{35} -1.00000 q^{36} +5.12311i q^{37} -6.00000 q^{38} +(3.56155 + 0.561553i) q^{39} -1.00000 q^{40} -0.876894i q^{41} +3.12311 q^{42} +6.24621 q^{43} +5.12311i q^{44} -1.00000i q^{45} -5.12311i q^{46} +6.24621i q^{47} +1.00000 q^{48} -2.75379 q^{49} -1.00000i q^{50} +2.00000 q^{51} +(-3.56155 - 0.561553i) q^{52} -13.3693 q^{53} +1.00000i q^{54} -5.12311 q^{55} -3.12311 q^{56} +6.00000i q^{57} +2.00000i q^{58} +1.12311i q^{59} +1.00000i q^{60} +10.0000 q^{61} +3.12311 q^{62} -3.12311i q^{63} -1.00000 q^{64} +(0.561553 - 3.56155i) q^{65} +5.12311 q^{66} -4.87689i q^{67} -2.00000 q^{68} -5.12311 q^{69} -3.12311i q^{70} -10.2462i q^{71} -1.00000i q^{72} +13.1231i q^{73} -5.12311 q^{74} -1.00000 q^{75} -6.00000i q^{76} -16.0000 q^{77} +(-0.561553 + 3.56155i) q^{78} +8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +0.876894 q^{82} +6.24621i q^{83} +3.12311i q^{84} -2.00000i q^{85} +6.24621i q^{86} +2.00000 q^{87} -5.12311 q^{88} -3.12311i q^{89} +1.00000 q^{90} +(1.75379 - 11.1231i) q^{91} +5.12311 q^{92} -3.12311i q^{93} -6.24621 q^{94} +6.00000 q^{95} +1.00000i q^{96} +13.1231i q^{97} -2.75379i q^{98} -5.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} + 4 q^{10} - 4 q^{12} + 6 q^{13} - 4 q^{14} + 4 q^{16} + 8 q^{17} + 4 q^{22} - 4 q^{23} - 4 q^{25} + 6 q^{26} + 4 q^{27} + 8 q^{29} + 4 q^{30} + 4 q^{35} - 4 q^{36} - 24 q^{38} + 6 q^{39} - 4 q^{40} - 4 q^{42} - 8 q^{43} + 4 q^{48} - 44 q^{49} + 8 q^{51} - 6 q^{52} - 4 q^{53} - 4 q^{55} + 4 q^{56} + 40 q^{61} - 4 q^{62} - 4 q^{64} - 6 q^{65} + 4 q^{66} - 8 q^{68} - 4 q^{69} - 4 q^{74} - 4 q^{75} - 64 q^{77} + 6 q^{78} + 32 q^{79} + 4 q^{81} + 20 q^{82} + 8 q^{87} - 4 q^{88} + 4 q^{90} + 40 q^{91} + 4 q^{92} + 8 q^{94} + 24 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 + 4 * q^10 - 4 * q^12 + 6 * q^13 - 4 * q^14 + 4 * q^16 + 8 * q^17 + 4 * q^22 - 4 * q^23 - 4 * q^25 + 6 * q^26 + 4 * q^27 + 8 * q^29 + 4 * q^30 + 4 * q^35 - 4 * q^36 - 24 * q^38 + 6 * q^39 - 4 * q^40 - 4 * q^42 - 8 * q^43 + 4 * q^48 - 44 * q^49 + 8 * q^51 - 6 * q^52 - 4 * q^53 - 4 * q^55 + 4 * q^56 + 40 * q^61 - 4 * q^62 - 4 * q^64 - 6 * q^65 + 4 * q^66 - 8 * q^68 - 4 * q^69 - 4 * q^74 - 4 * q^75 - 64 * q^77 + 6 * q^78 + 32 * q^79 + 4 * q^81 + 20 * q^82 + 8 * q^87 - 4 * q^88 + 4 * q^90 + 40 * q^91 + 4 * q^92 + 8 * q^94 + 24 * q^95

Introduction

L-functions

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