LMFDB - Embedded newform 6422.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.a (trivial)

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:	yes
Analytic conductor:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 12x^{7} + 14x^{6} + 35x^{5} - 35x^{4} - 28x^{3} + 10x^{2} + 4x - 1 $ x^9 - x^8 - 12x^7 + 14x^6 + 35x^5 - 35x^4 - 28x^3 + 10x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.380781$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.66099 q^{3} +1.00000 q^{4} +1.69394 q^{5} +1.66099 q^{6} -0.961642 q^{7} -1.00000 q^{8} -0.241101 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.66099 q^{3} +1.00000 q^{4} +1.69394 q^{5} +1.66099 q^{6} -0.961642 q^{7} -1.00000 q^{8} -0.241101 q^{9} -1.69394 q^{10} +4.42494 q^{11} -1.66099 q^{12} +0.961642 q^{14} -2.81362 q^{15} +1.00000 q^{16} +1.01199 q^{17} +0.241101 q^{18} +1.00000 q^{19} +1.69394 q^{20} +1.59728 q^{21} -4.42494 q^{22} +6.07002 q^{23} +1.66099 q^{24} -2.13058 q^{25} +5.38345 q^{27} -0.961642 q^{28} +6.95493 q^{29} +2.81362 q^{30} -9.73717 q^{31} -1.00000 q^{32} -7.34980 q^{33} -1.01199 q^{34} -1.62896 q^{35} -0.241101 q^{36} +0.678162 q^{37} -1.00000 q^{38} -1.69394 q^{40} +11.2026 q^{41} -1.59728 q^{42} +5.44576 q^{43} +4.42494 q^{44} -0.408410 q^{45} -6.07002 q^{46} -3.91685 q^{47} -1.66099 q^{48} -6.07525 q^{49} +2.13058 q^{50} -1.68091 q^{51} +6.18790 q^{53} -5.38345 q^{54} +7.49557 q^{55} +0.961642 q^{56} -1.66099 q^{57} -6.95493 q^{58} -2.06496 q^{59} -2.81362 q^{60} -10.2252 q^{61} +9.73717 q^{62} +0.231853 q^{63} +1.00000 q^{64} +7.34980 q^{66} +5.84309 q^{67} +1.01199 q^{68} -10.0823 q^{69} +1.62896 q^{70} +14.6895 q^{71} +0.241101 q^{72} -2.12183 q^{73} -0.678162 q^{74} +3.53888 q^{75} +1.00000 q^{76} -4.25521 q^{77} -7.59795 q^{79} +1.69394 q^{80} -8.21857 q^{81} -11.2026 q^{82} -4.74343 q^{83} +1.59728 q^{84} +1.71425 q^{85} -5.44576 q^{86} -11.5521 q^{87} -4.42494 q^{88} +6.32590 q^{89} +0.408410 q^{90} +6.07002 q^{92} +16.1734 q^{93} +3.91685 q^{94} +1.69394 q^{95} +1.66099 q^{96} -4.22409 q^{97} +6.07525 q^{98} -1.06686 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{2} + q^{3} + 9 q^{4} - q^{5} - q^{6} + 13 q^{7} - 9 q^{8} - 2 q^{9}+O(q^{10}) $ 9 * q - 9 * q^2 + q^3 + 9 * q^4 - q^5 - q^6 + 13 * q^7 - 9 * q^8 - 2 * q^9	$ 9 q - 9 q^{2} + q^{3} + 9 q^{4} - q^{5} - q^{6} + 13 q^{7} - 9 q^{8} - 2 q^{9} + q^{10} + 3 q^{11} + q^{12} - 13 q^{14} + 3 q^{15} + 9 q^{16} - 8 q^{17} + 2 q^{18} + 9 q^{19} - q^{20} + 24 q^{21} - 3 q^{22} - 10 q^{23} - q^{24} + 8 q^{25} + 10 q^{27} + 13 q^{28} - 20 q^{29} - 3 q^{30} + q^{31} - 9 q^{32} - 2 q^{33} + 8 q^{34} + 4 q^{35} - 2 q^{36} + 15 q^{37} - 9 q^{38} + q^{40} + 19 q^{41} - 24 q^{42} - 16 q^{43} + 3 q^{44} + 15 q^{45} + 10 q^{46} + 18 q^{47} + q^{48} + 18 q^{49} - 8 q^{50} - 11 q^{51} + 17 q^{53} - 10 q^{54} - 26 q^{55} - 13 q^{56} + q^{57} + 20 q^{58} + 24 q^{59} + 3 q^{60} + 6 q^{61} - q^{62} + q^{63} + 9 q^{64} + 2 q^{66} + 29 q^{67} - 8 q^{68} - 12 q^{69} - 4 q^{70} + 23 q^{71} + 2 q^{72} + 38 q^{73} - 15 q^{74} + 11 q^{75} + 9 q^{76} - 40 q^{77} - 20 q^{79} - q^{80} - 31 q^{81} - 19 q^{82} + 20 q^{83} + 24 q^{84} + 39 q^{85} + 16 q^{86} - 10 q^{87} - 3 q^{88} - 7 q^{89} - 15 q^{90} - 10 q^{92} + 11 q^{93} - 18 q^{94} - q^{95} - q^{96} + 28 q^{97} - 18 q^{98} + 25 q^{99}+O(q^{100}) $ 9 * q - 9 * q^2 + q^3 + 9 * q^4 - q^5 - q^6 + 13 * q^7 - 9 * q^8 - 2 * q^9 + q^10 + 3 * q^11 + q^12 - 13 * q^14 + 3 * q^15 + 9 * q^16 - 8 * q^17 + 2 * q^18 + 9 * q^19 - q^20 + 24 * q^21 - 3 * q^22 - 10 * q^23 - q^24 + 8 * q^25 + 10 * q^27 + 13 * q^28 - 20 * q^29 - 3 * q^30 + q^31 - 9 * q^32 - 2 * q^33 + 8 * q^34 + 4 * q^35 - 2 * q^36 + 15 * q^37 - 9 * q^38 + q^40 + 19 * q^41 - 24 * q^42 - 16 * q^43 + 3 * q^44 + 15 * q^45 + 10 * q^46 + 18 * q^47 + q^48 + 18 * q^49 - 8 * q^50 - 11 * q^51 + 17 * q^53 - 10 * q^54 - 26 * q^55 - 13 * q^56 + q^57 + 20 * q^58 + 24 * q^59 + 3 * q^60 + 6 * q^61 - q^62 + q^63 + 9 * q^64 + 2 * q^66 + 29 * q^67 - 8 * q^68 - 12 * q^69 - 4 * q^70 + 23 * q^71 + 2 * q^72 + 38 * q^73 - 15 * q^74 + 11 * q^75 + 9 * q^76 - 40 * q^77 - 20 * q^79 - q^80 - 31 * q^81 - 19 * q^82 + 20 * q^83 + 24 * q^84 + 39 * q^85 + 16 * q^86 - 10 * q^87 - 3 * q^88 - 7 * q^89 - 15 * q^90 - 10 * q^92 + 11 * q^93 - 18 * q^94 - q^95 - q^96 + 28 * q^97 - 18 * q^98 + 25 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.66099	−0.958975	−0.479487	−	0.877549i	$-0.659177\pi$
$3$	−1.66099	−0.958975	−0.479487	+	0.877549i	$0.659177\pi$
$4$	1.00000	0.500000
$5$	1.69394	0.757551	0.378775	−	0.925489i	$-0.376345\pi$
$5$	1.69394	0.757551	0.378775	+	0.925489i	$0.376345\pi$
$6$	1.66099	0.678098
$7$	−0.961642	−0.363466	−0.181733	−	0.983348i	$-0.558171\pi$
$7$	−0.961642	−0.363466	−0.181733	+	0.983348i	$0.558171\pi$
$8$	−1.00000	−0.353553
$9$	−0.241101	−0.0803671
$10$	−1.69394	−0.535669
$11$	4.42494	1.33417	0.667085	−	0.744982i	$-0.267542\pi$
$11$	4.42494	1.33417	0.667085	+	0.744982i	$0.267542\pi$
$12$	−1.66099	−0.479487
$13$	0	0
$13$	0	0
$14$	0.961642	0.257010
$15$	−2.81362	−0.726472
$16$	1.00000	0.250000
$17$	1.01199	0.245444	0.122722	−	0.992441i	$-0.460838\pi$
$17$	1.01199	0.245444	0.122722	+	0.992441i	$0.460838\pi$
$18$	0.241101	0.0568282
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.69394	0.378775
$21$	1.59728	0.348555
$22$	−4.42494	−0.943401
$23$	6.07002	1.26569	0.632844	−	0.774280i	$-0.281888\pi$
$23$	6.07002	1.26569	0.632844	+	0.774280i	$0.281888\pi$
$24$	1.66099	0.339049
$25$	−2.13058	−0.426117
$26$	0	0
$27$	5.38345	1.03604
$28$	−0.961642	−0.181733
$29$	6.95493	1.29150	0.645749	−	0.763550i	$-0.276545\pi$
$29$	6.95493	1.29150	0.645749	+	0.763550i	$0.276545\pi$
$30$	2.81362	0.513694
$31$	−9.73717	−1.74885	−0.874424	−	0.485163i	$-0.838760\pi$
$31$	−9.73717	−1.74885	−0.874424	+	0.485163i	$0.838760\pi$
$32$	−1.00000	−0.176777
$33$	−7.34980	−1.27944
$34$	−1.01199	−0.173555
$35$	−1.62896	−0.275344
$36$	−0.241101	−0.0401836
$37$	0.678162	0.111489	0.0557446	−	0.998445i	$-0.482247\pi$
$37$	0.678162	0.111489	0.0557446	+	0.998445i	$0.482247\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−1.69394	−0.267835
$41$	11.2026	1.74955	0.874777	−	0.484526i	$-0.161008\pi$
$41$	11.2026	1.74955	0.874777	+	0.484526i	$0.161008\pi$
$42$	−1.59728	−0.246466
$43$	5.44576	0.830470	0.415235	−	0.909714i	$-0.363699\pi$
$43$	5.44576	0.830470	0.415235	+	0.909714i	$0.363699\pi$
$44$	4.42494	0.667085
$45$	−0.408410	−0.0608822
$46$	−6.07002	−0.894976
$47$	−3.91685	−0.571331	−0.285666	−	0.958329i	$-0.592215\pi$
$47$	−3.91685	−0.571331	−0.285666	+	0.958329i	$0.592215\pi$
$48$	−1.66099	−0.239744
$49$	−6.07525	−0.867892
$50$	2.13058	0.301310
$51$	−1.68091	−0.235374
$52$	0	0
$53$	6.18790	0.849973	0.424986	−	0.905200i	$-0.360279\pi$
$53$	6.18790	0.849973	0.424986	+	0.905200i	$0.360279\pi$
$54$	−5.38345	−0.732594
$55$	7.49557	1.01070
$56$	0.961642	0.128505
$57$	−1.66099	−0.220004
$58$	−6.95493	−0.913227
$59$	−2.06496	−0.268835	−0.134418	−	0.990925i	$-0.542916\pi$
$59$	−2.06496	−0.268835	−0.134418	+	0.990925i	$0.542916\pi$
$60$	−2.81362	−0.363236
$61$	−10.2252	−1.30920	−0.654602	−	0.755974i	$-0.727164\pi$
$61$	−10.2252	−1.30920	−0.654602	+	0.755974i	$0.727164\pi$
$62$	9.73717	1.23662
$63$	0.231853	0.0292108
$64$	1.00000	0.125000
$65$	0	0
$66$	7.34980	0.904698
$67$	5.84309	0.713847	0.356924	−	0.934134i	$-0.383826\pi$
$67$	5.84309	0.713847	0.356924	+	0.934134i	$0.383826\pi$
$68$	1.01199	0.122722
$69$	−10.0823	−1.21376
$70$	1.62896	0.194698
$71$	14.6895	1.74332	0.871662	−	0.490108i	$-0.163043\pi$
$71$	14.6895	1.74332	0.871662	+	0.490108i	$0.163043\pi$
$72$	0.241101	0.0284141
$73$	−2.12183	−0.248341	−0.124171	−	0.992261i	$-0.539627\pi$
$73$	−2.12183	−0.248341	−0.124171	+	0.992261i	$0.539627\pi$
$74$	−0.678162	−0.0788347
$75$	3.53888	0.408635
$76$	1.00000	0.114708
$77$	−4.25521	−0.484926
$78$	0	0
$79$	−7.59795	−0.854837	−0.427418	−	0.904054i	$-0.640577\pi$
$79$	−7.59795	−0.854837	−0.427418	+	0.904054i	$0.640577\pi$
$80$	1.69394	0.189388
$81$	−8.21857	−0.913174
$82$	−11.2026	−1.23712
$83$	−4.74343	−0.520659	−0.260330	−	0.965520i	$-0.583831\pi$
$83$	−4.74343	−0.520659	−0.260330	+	0.965520i	$0.583831\pi$
$84$	1.59728	0.174278
$85$	1.71425	0.185936
$86$	−5.44576	−0.587231
$87$	−11.5521	−1.23851
$88$	−4.42494	−0.471700
$89$	6.32590	0.670544	0.335272	−	0.942121i	$-0.391172\pi$
$89$	6.32590	0.670544	0.335272	+	0.942121i	$0.391172\pi$
$90$	0.408410	0.0430502
$91$	0	0
$92$	6.07002	0.632844
$93$	16.1734	1.67710
$94$	3.91685	0.403992
$95$	1.69394	0.173794
$96$	1.66099	0.169524
$97$	−4.22409	−0.428892	−0.214446	−	0.976736i	$-0.568795\pi$
$97$	−4.22409	−0.428892	−0.214446	+	0.976736i	$0.568795\pi$
$98$	6.07525	0.613692
$99$	−1.06686	−0.107223
$100$	−2.13058	−0.213058
$101$	−5.33989	−0.531339	−0.265669	−	0.964064i	$-0.585593\pi$
$101$	−5.33989	−0.531339	−0.265669	+	0.964064i	$0.585593\pi$
$102$	1.68091	0.166435
$103$	−5.98387	−0.589608	−0.294804	−	0.955558i	$-0.595254\pi$
$103$	−5.98387	−0.589608	−0.294804	+	0.955558i	$0.595254\pi$
$104$	0	0
$105$	2.70569	0.264048
$106$	−6.18790	−0.601022
$107$	0.456797	0.0441603	0.0220801	−	0.999756i	$-0.492971\pi$
$107$	0.456797	0.0441603	0.0220801	+	0.999756i	$0.492971\pi$
$108$	5.38345	0.518022
$109$	13.2279	1.26700	0.633500	−	0.773743i	$-0.281618\pi$
$109$	13.2279	1.26700	0.633500	+	0.773743i	$0.281618\pi$
$110$	−7.49557	−0.714674
$111$	−1.12642	−0.106915
$112$	−0.961642	−0.0908666
$113$	11.5596	1.08743	0.543716	−	0.839269i	$-0.317017\pi$
$113$	11.5596	1.08743	0.543716	+	0.839269i	$0.317017\pi$
$114$	1.66099	0.155566
$115$	10.2822	0.958822
$116$	6.95493	0.645749
$117$	0	0
$118$	2.06496	0.190095
$119$	−0.973172	−0.0892105
$120$	2.81362	0.256847
$121$	8.58011	0.780010
$122$	10.2252	0.925747
$123$	−18.6075	−1.67778
$124$	−9.73717	−0.874424
$125$	−12.0787	−1.08036
$126$	−0.231853	−0.0206551
$127$	−2.75202	−0.244202	−0.122101	−	0.992518i	$-0.538963\pi$
$127$	−2.75202	−0.244202	−0.122101	+	0.992518i	$0.538963\pi$
$128$	−1.00000	−0.0883883
$129$	−9.04537	−0.796400
$130$	0	0
$131$	−11.9707	−1.04589	−0.522943	−	0.852368i	$-0.675166\pi$
$131$	−11.9707	−1.04589	−0.522943	+	0.852368i	$0.675166\pi$
$132$	−7.34980	−0.639718
$133$	−0.961642	−0.0833849
$134$	−5.84309	−0.504766
$135$	9.11921	0.784857
$136$	−1.01199	−0.0867774
$137$	21.3846	1.82701	0.913505	−	0.406827i	$-0.133365\pi$
$137$	21.3846	1.82701	0.913505	+	0.406827i	$0.133365\pi$
$138$	10.0823	0.858259
$139$	−12.8882	−1.09316	−0.546580	−	0.837407i	$-0.684070\pi$
$139$	−12.8882	−1.09316	−0.546580	+	0.837407i	$0.684070\pi$
$140$	−1.62896	−0.137672
$141$	6.50586	0.547892
$142$	−14.6895	−1.23272
$143$	0	0
$144$	−0.241101	−0.0200918
$145$	11.7812	0.978375
$146$	2.12183	0.175604
$147$	10.0909	0.832287
$148$	0.678162	0.0557446
$149$	−23.2522	−1.90489	−0.952445	−	0.304709i	$-0.901441\pi$
$149$	−23.2522	−1.90489	−0.952445	+	0.304709i	$0.901441\pi$
$150$	−3.53888	−0.288949
$151$	24.5628	1.99889	0.999446	−	0.0332861i	$-0.0105972\pi$
$151$	24.5628	1.99889	0.999446	+	0.0332861i	$0.0105972\pi$
$152$	−1.00000	−0.0811107
$153$	−0.243992	−0.0197256
$154$	4.25521	0.342894
$155$	−16.4941	−1.32484
$156$	0	0
$157$	9.81548	0.783360	0.391680	−	0.920101i	$-0.371894\pi$
$157$	9.81548	0.783360	0.391680	+	0.920101i	$0.371894\pi$
$158$	7.59795	0.604461
$159$	−10.2781	−0.815103
$160$	−1.69394	−0.133917
$161$	−5.83719	−0.460035
$162$	8.21857	0.645712
$163$	−9.15174	−0.716820	−0.358410	−	0.933564i	$-0.616681\pi$
$163$	−9.15174	−0.716820	−0.358410	+	0.933564i	$0.616681\pi$
$164$	11.2026	0.874777
$165$	−12.4501	−0.969238
$166$	4.74343	0.368162
$167$	16.3959	1.26875	0.634376	−	0.773024i	$-0.281257\pi$
$167$	16.3959	1.26875	0.634376	+	0.773024i	$0.281257\pi$
$168$	−1.59728	−0.123233
$169$	0	0
$170$	−1.71425	−0.131477
$171$	−0.241101	−0.0184375
$172$	5.44576	0.415235
$173$	7.00746	0.532768	0.266384	−	0.963867i	$-0.414171\pi$
$173$	7.00746	0.532768	0.266384	+	0.963867i	$0.414171\pi$
$174$	11.5521	0.875761
$175$	2.04886	0.154879
$176$	4.42494	0.333543
$177$	3.42989	0.257806
$178$	−6.32590	−0.474146
$179$	16.0201	1.19740	0.598698	−	0.800975i	$-0.295685\pi$
$179$	16.0201	1.19740	0.598698	+	0.800975i	$0.295685\pi$
$180$	−0.408410	−0.0304411
$181$	2.81392	0.209157	0.104578	−	0.994517i	$-0.466651\pi$
$181$	2.81392	0.209157	0.104578	+	0.994517i	$0.466651\pi$
$182$	0	0
$183$	16.9840	1.25549
$184$	−6.07002	−0.447488
$185$	1.14876	0.0844587
$186$	−16.1734	−1.18589
$187$	4.47800	0.327464
$188$	−3.91685	−0.285666
$189$	−5.17695	−0.376568
$190$	−1.69394	−0.122891
$191$	−7.85491	−0.568361	−0.284181	−	0.958771i	$-0.591721\pi$
$191$	−7.85491	−0.568361	−0.284181	+	0.958771i	$0.591721\pi$
$192$	−1.66099	−0.119872
$193$	8.38465	0.603540	0.301770	−	0.953381i	$-0.402422\pi$
$193$	8.38465	0.603540	0.301770	+	0.953381i	$0.402422\pi$
$194$	4.22409	0.303272
$195$	0	0
$196$	−6.07525	−0.433946
$197$	10.1478	0.723001	0.361501	−	0.932372i	$-0.382265\pi$
$197$	10.1478	0.723001	0.361501	+	0.932372i	$0.382265\pi$
$198$	1.06686	0.0758184
$199$	1.80755	0.128134	0.0640670	−	0.997946i	$-0.479593\pi$
$199$	1.80755	0.128134	0.0640670	+	0.997946i	$0.479593\pi$
$200$	2.13058	0.150655
$201$	−9.70533	−0.684561
$202$	5.33989	0.375713
$203$	−6.68815	−0.469416
$204$	−1.68091	−0.117687
$205$	18.9765	1.32538
$206$	5.98387	0.416916
$207$	−1.46349	−0.101720
$208$	0	0
$209$	4.42494	0.306080
$210$	−2.70569	−0.186710
$211$	−27.7630	−1.91128	−0.955642	−	0.294532i	$-0.904836\pi$
$211$	−27.7630	−1.91128	−0.955642	+	0.294532i	$0.904836\pi$
$212$	6.18790	0.424986
$213$	−24.3992	−1.67180
$214$	−0.456797	−0.0312260
$215$	9.22476	0.629123
$216$	−5.38345	−0.366297
$217$	9.36367	0.635647
$218$	−13.2279	−0.895904
$219$	3.52435	0.238153
$220$	7.49557	0.505351
$221$	0	0
$222$	1.12642	0.0756005
$223$	−7.00880	−0.469344	−0.234672	−	0.972075i	$-0.575402\pi$
$223$	−7.00880	−0.469344	−0.234672	+	0.972075i	$0.575402\pi$
$224$	0.961642	0.0642524
$225$	0.513687	0.0342458
$226$	−11.5596	−0.768930
$227$	−20.8321	−1.38267	−0.691337	−	0.722532i	$-0.742978\pi$
$227$	−20.8321	−1.38267	−0.691337	+	0.722532i	$0.742978\pi$
$228$	−1.66099	−0.110002
$229$	−6.04193	−0.399262	−0.199631	−	0.979871i	$-0.563974\pi$
$229$	−6.04193	−0.399262	−0.199631	+	0.979871i	$0.563974\pi$
$230$	−10.2822	−0.677990
$231$	7.06787	0.465032
$232$	−6.95493	−0.456613
$233$	−3.21677	−0.210737	−0.105369	−	0.994433i	$-0.533602\pi$
$233$	−3.21677	−0.210737	−0.105369	+	0.994433i	$0.533602\pi$
$234$	0	0
$235$	−6.63489	−0.432813
$236$	−2.06496	−0.134418
$237$	12.6201	0.819767
$238$	0.973172	0.0630814
$239$	−7.65203	−0.494969	−0.247484	−	0.968892i	$-0.579604\pi$
$239$	−7.65203	−0.494969	−0.247484	+	0.968892i	$0.579604\pi$
$240$	−2.81362	−0.181618
$241$	−9.27475	−0.597439	−0.298720	−	0.954341i	$-0.596560\pi$
$241$	−9.27475	−0.597439	−0.298720	+	0.954341i	$0.596560\pi$
$242$	−8.58011	−0.551550
$243$	−2.49936	−0.160334
$244$	−10.2252	−0.654602
$245$	−10.2911	−0.657473
$246$	18.6075	1.18637
$247$	0	0
$248$	9.73717	0.618311
$249$	7.87881	0.499299
$250$	12.0787	0.763927
$251$	−1.09472	−0.0690984	−0.0345492	−	0.999403i	$-0.511000\pi$
$251$	−1.09472	−0.0690984	−0.0345492	+	0.999403i	$0.511000\pi$
$252$	0.231853	0.0146054
$253$	26.8595	1.68864
$254$	2.75202	0.172677
$255$	−2.84735	−0.178308
$256$	1.00000	0.0625000
$257$	1.33399	0.0832118	0.0416059	−	0.999134i	$-0.486753\pi$
$257$	1.33399	0.0832118	0.0416059	+	0.999134i	$0.486753\pi$
$258$	9.04537	0.563140
$259$	−0.652149	−0.0405226
$260$	0	0
$261$	−1.67684	−0.103794
$262$	11.9707	0.739553
$263$	−20.5528	−1.26734	−0.633669	−	0.773604i	$-0.718452\pi$
$263$	−20.5528	−1.26734	−0.633669	+	0.773604i	$0.718452\pi$
$264$	7.34980	0.452349
$265$	10.4819	0.643898
$266$	0.961642	0.0589620
$267$	−10.5073	−0.643035
$268$	5.84309	0.356924
$269$	−20.8405	−1.27067	−0.635333	−	0.772238i	$-0.719137\pi$
$269$	−20.8405	−1.27067	−0.635333	+	0.772238i	$0.719137\pi$
$270$	−9.11921	−0.554978
$271$	5.43640	0.330238	0.165119	−	0.986274i	$-0.447199\pi$
$271$	5.43640	0.330238	0.165119	+	0.986274i	$0.447199\pi$
$272$	1.01199	0.0613609
$273$	0	0
$274$	−21.3846	−1.29189
$275$	−9.42770	−0.568512
$276$	−10.0823	−0.606881
$277$	24.6539	1.48131	0.740653	−	0.671888i	$-0.234516\pi$
$277$	24.6539	1.48131	0.740653	+	0.671888i	$0.234516\pi$
$278$	12.8882	0.772980
$279$	2.34765	0.140550
$280$	1.62896	0.0973489
$281$	25.5274	1.52284	0.761419	−	0.648260i	$-0.224503\pi$
$281$	25.5274	1.52284	0.761419	+	0.648260i	$0.224503\pi$
$282$	−6.50586	−0.387418
$283$	−16.6966	−0.992510	−0.496255	−	0.868177i	$-0.665292\pi$
$283$	−16.6966	−0.992510	−0.496255	+	0.868177i	$0.665292\pi$
$284$	14.6895	0.871662
$285$	−2.81362	−0.166664
$286$	0	0
$287$	−10.7729	−0.635904
$288$	0.241101	0.0142070
$289$	−15.9759	−0.939757
$290$	−11.7812	−0.691816
$291$	7.01619	0.411297
$292$	−2.12183	−0.124171
$293$	21.2827	1.24335	0.621676	−	0.783275i	$-0.286452\pi$
$293$	21.2827	1.24335	0.621676	+	0.783275i	$0.286452\pi$
$294$	−10.0909	−0.588516
$295$	−3.49791	−0.203656
$296$	−0.678162	−0.0394174
$297$	23.8214	1.38226
$298$	23.2522	1.34696
$299$	0	0
$300$	3.53888	0.204318
$301$	−5.23687	−0.301848
$302$	−24.5628	−1.41343
$303$	8.86952	0.509541
$304$	1.00000	0.0573539
$305$	−17.3208	−0.991789
$306$	0.243992	0.0139481
$307$	4.70636	0.268606	0.134303	−	0.990940i	$-0.457120\pi$
$307$	4.70636	0.268606	0.134303	+	0.990940i	$0.457120\pi$
$308$	−4.25521	−0.242463
$309$	9.93917	0.565420
$310$	16.4941	0.936804
$311$	4.94400	0.280348	0.140174	−	0.990127i	$-0.455234\pi$
$311$	4.94400	0.280348	0.140174	+	0.990127i	$0.455234\pi$
$312$	0	0
$313$	12.7603	0.721253	0.360627	−	0.932710i	$-0.382563\pi$
$313$	12.7603	0.721253	0.360627	+	0.932710i	$0.382563\pi$
$314$	−9.81548	−0.553919
$315$	0.392744	0.0221286
$316$	−7.59795	−0.427418
$317$	−13.6955	−0.769218	−0.384609	−	0.923080i	$-0.625664\pi$
$317$	−13.6955	−0.769218	−0.384609	+	0.923080i	$0.625664\pi$
$318$	10.2781	0.576365
$319$	30.7751	1.72308
$320$	1.69394	0.0946939
$321$	−0.758738	−0.0423486
$322$	5.83719	0.325294
$323$	1.01199	0.0563086
$324$	−8.21857	−0.456587
$325$	0	0
$326$	9.15174	0.506868
$327$	−21.9714	−1.21502
$328$	−11.2026	−0.618561
$329$	3.76661	0.207660
$330$	12.4501	0.685355
$331$	−6.78053	−0.372692	−0.186346	−	0.982484i	$-0.559664\pi$
$331$	−6.78053	−0.372692	−0.186346	+	0.982484i	$0.559664\pi$
$332$	−4.74343	−0.260330
$333$	−0.163506	−0.00896006
$334$	−16.3959	−0.897143
$335$	9.89782	0.540775
$336$	1.59728	0.0871388
$337$	18.0123	0.981195	0.490597	−	0.871386i	$-0.336779\pi$
$337$	18.0123	0.981195	0.490597	+	0.871386i	$0.336779\pi$
$338$	0	0
$339$	−19.2003	−1.04282
$340$	1.71425	0.0929680
$341$	−43.0864	−2.33326
$342$	0.241101	0.0130373
$343$	12.5737	0.678916
$344$	−5.44576	−0.293616
$345$	−17.0787	−0.919487
$346$	−7.00746	−0.376724
$347$	24.5390	1.31732	0.658661	−	0.752440i	$-0.271123\pi$
$347$	24.5390	1.31732	0.658661	+	0.752440i	$0.271123\pi$
$348$	−11.5521	−0.619257
$349$	4.97210	0.266151	0.133075	−	0.991106i	$-0.457515\pi$
$349$	4.97210	0.266151	0.133075	+	0.991106i	$0.457515\pi$
$350$	−2.04886	−0.109516
$351$	0	0
$352$	−4.42494	−0.235850
$353$	13.5038	0.718734	0.359367	−	0.933196i	$-0.382993\pi$
$353$	13.5038	0.718734	0.359367	+	0.933196i	$0.382993\pi$
$354$	−3.42989	−0.182296
$355$	24.8831	1.32066
$356$	6.32590	0.335272
$357$	1.61643	0.0855507
$358$	−16.0201	−0.846687
$359$	35.4154	1.86915	0.934576	−	0.355764i	$-0.115779\pi$
$359$	35.4154	1.86915	0.934576	+	0.355764i	$0.115779\pi$
$360$	0.408410	0.0215251
$361$	1.00000	0.0526316
$362$	−2.81392	−0.147896
$363$	−14.2515	−0.748010
$364$	0	0
$365$	−3.59424	−0.188131
$366$	−16.9840	−0.887768
$367$	−35.1014	−1.83228	−0.916140	−	0.400858i	$-0.868712\pi$
$367$	−35.1014	−1.83228	−0.916140	+	0.400858i	$0.868712\pi$
$368$	6.07002	0.316422
$369$	−2.70096	−0.140607
$370$	−1.14876	−0.0597213
$371$	−5.95054	−0.308937
$372$	16.1734	0.838551
$373$	−5.02537	−0.260204	−0.130102	−	0.991501i	$-0.541530\pi$
$373$	−5.02537	−0.260204	−0.130102	+	0.991501i	$0.541530\pi$
$374$	−4.47800	−0.231552
$375$	20.0627	1.03603
$376$	3.91685	0.201996
$377$	0	0
$378$	5.17695	0.266273
$379$	19.8743	1.02088	0.510438	−	0.859915i	$-0.329483\pi$
$379$	19.8743	1.02088	0.510438	+	0.859915i	$0.329483\pi$
$380$	1.69394	0.0868971
$381$	4.57108	0.234184
$382$	7.85491	0.401892
$383$	13.1631	0.672602	0.336301	−	0.941754i	$-0.390824\pi$
$383$	13.1631	0.672602	0.336301	+	0.941754i	$0.390824\pi$
$384$	1.66099	0.0847622
$385$	−7.20805	−0.367356
$386$	−8.38465	−0.426767
$387$	−1.31298	−0.0667425
$388$	−4.22409	−0.214446
$389$	17.9823	0.911739	0.455869	−	0.890047i	$-0.349328\pi$
$389$	17.9823	0.911739	0.455869	+	0.890047i	$0.349328\pi$
$390$	0	0
$391$	6.14280	0.310655
$392$	6.07525	0.306846
$393$	19.8833	1.00298
$394$	−10.1478	−0.511239
$395$	−12.8704	−0.647582
$396$	−1.06686	−0.0536117
$397$	10.7231	0.538176	0.269088	−	0.963116i	$-0.413278\pi$
$397$	10.7231	0.538176	0.269088	+	0.963116i	$0.413278\pi$
$398$	−1.80755	−0.0906044
$399$	1.59728	0.0799640
$400$	−2.13058	−0.106529
$401$	−23.5965	−1.17835	−0.589176	−	0.808004i	$-0.700548\pi$
$401$	−23.5965	−1.17835	−0.589176	+	0.808004i	$0.700548\pi$
$402$	9.70533	0.484058
$403$	0	0
$404$	−5.33989	−0.265669
$405$	−13.9217	−0.691776
$406$	6.68815	0.331927
$407$	3.00083	0.148745
$408$	1.68091	0.0832174
$409$	2.70488	0.133748	0.0668739	−	0.997761i	$-0.478697\pi$
$409$	2.70488	0.133748	0.0668739	+	0.997761i	$0.478697\pi$
$410$	−18.9765	−0.937182
$411$	−35.5197	−1.75206
$412$	−5.98387	−0.294804
$413$	1.98575	0.0977125
$414$	1.46349	0.0719267
$415$	−8.03507	−0.394426
$416$	0	0
$417$	21.4071	1.04831
$418$	−4.42494	−0.216431
$419$	16.1618	0.789554	0.394777	−	0.918777i	$-0.370822\pi$
$419$	16.1618	0.789554	0.394777	+	0.918777i	$0.370822\pi$
$420$	2.70569	0.132024
$421$	16.8012	0.818839	0.409419	−	0.912346i	$-0.365731\pi$
$421$	16.8012	0.818839	0.409419	+	0.912346i	$0.365731\pi$
$422$	27.7630	1.35148
$423$	0.944358	0.0459163
$424$	−6.18790	−0.300511
$425$	−2.15613	−0.104588
$426$	24.3992	1.18214
$427$	9.83299	0.475852
$428$	0.456797	0.0220801
$429$	0	0
$430$	−9.22476	−0.444857
$431$	33.9067	1.63323	0.816615	−	0.577183i	$-0.195848\pi$
$431$	33.9067	1.63323	0.816615	+	0.577183i	$0.195848\pi$
$432$	5.38345	0.259011
$433$	20.3139	0.976223	0.488111	−	0.872781i	$-0.337686\pi$
$433$	20.3139	0.976223	0.488111	+	0.872781i	$0.337686\pi$
$434$	−9.36367	−0.449471
$435$	−19.5685	−0.938237
$436$	13.2279	0.633500
$437$	6.07002	0.290369
$438$	−3.52435	−0.168400
$439$	−23.5505	−1.12400	−0.562001	−	0.827136i	$-0.689968\pi$
$439$	−23.5505	−1.12400	−0.562001	+	0.827136i	$0.689968\pi$
$440$	−7.49557	−0.357337
$441$	1.46475	0.0697500
$442$	0	0
$443$	26.0674	1.23850	0.619250	−	0.785194i	$-0.287437\pi$
$443$	26.0674	1.23850	0.619250	+	0.785194i	$0.287437\pi$
$444$	−1.12642	−0.0534576
$445$	10.7157	0.507971
$446$	7.00880	0.331876
$447$	38.6217	1.82674
$448$	−0.961642	−0.0454333
$449$	−6.84222	−0.322904	−0.161452	−	0.986881i	$-0.551618\pi$
$449$	−6.84222	−0.322904	−0.161452	+	0.986881i	$0.551618\pi$
$450$	−0.513687	−0.0242154
$451$	49.5709	2.33420
$452$	11.5596	0.543716
$453$	−40.7986	−1.91689
$454$	20.8321	0.977698
$455$	0	0
$456$	1.66099	0.0777831
$457$	24.4501	1.14373	0.571865	−	0.820348i	$-0.306220\pi$
$457$	24.4501	1.14373	0.571865	+	0.820348i	$0.306220\pi$
$458$	6.04193	0.282321
$459$	5.44800	0.254291
$460$	10.2822	0.479411
$461$	−1.80454	−0.0840456	−0.0420228	−	0.999117i	$-0.513380\pi$
$461$	−1.80454	−0.0840456	−0.0420228	+	0.999117i	$0.513380\pi$
$462$	−7.06787	−0.328827
$463$	−4.62667	−0.215020	−0.107510	−	0.994204i	$-0.534288\pi$
$463$	−4.62667	−0.215020	−0.107510	+	0.994204i	$0.534288\pi$
$464$	6.95493	0.322874
$465$	27.3967	1.27049
$466$	3.21677	0.149014
$467$	10.7837	0.499009	0.249504	−	0.968374i	$-0.419732\pi$
$467$	10.7837	0.499009	0.249504	+	0.968374i	$0.419732\pi$
$468$	0	0
$469$	−5.61896	−0.259459
$470$	6.63489	0.306045
$471$	−16.3034	−0.751223
$472$	2.06496	0.0950475
$473$	24.0972	1.10799
$474$	−12.6201	−0.579663
$475$	−2.13058	−0.0977578
$476$	−0.973172	−0.0446053
$477$	−1.49191	−0.0683099
$478$	7.65203	0.349996
$479$	−0.0903531	−0.00412834	−0.00206417	−	0.999998i	$-0.500657\pi$
$479$	−0.0903531	−0.00412834	−0.00206417	+	0.999998i	$0.500657\pi$
$480$	2.81362	0.128423
$481$	0	0
$482$	9.27475	0.422453
$483$	9.69553	0.441162
$484$	8.58011	0.390005
$485$	−7.15534	−0.324907
$486$	2.49936	0.113373
$487$	16.1552	0.732064	0.366032	−	0.930602i	$-0.380716\pi$
$487$	16.1552	0.732064	0.366032	+	0.930602i	$0.380716\pi$
$488$	10.2252	0.462873
$489$	15.2010	0.687412
$490$	10.2911	0.464903
$491$	4.49239	0.202739	0.101369	−	0.994849i	$-0.467678\pi$
$491$	4.49239	0.202739	0.101369	+	0.994849i	$0.467678\pi$
$492$	−18.6075	−0.838889
$493$	7.03832	0.316990
$494$	0	0
$495$	−1.80719	−0.0812272
$496$	−9.73717	−0.437212
$497$	−14.1260	−0.633640
$498$	−7.87881	−0.353058
$499$	10.3831	0.464810	0.232405	−	0.972619i	$-0.425341\pi$
$499$	10.3831	0.464810	0.232405	+	0.972619i	$0.425341\pi$
$500$	−12.0787	−0.540178
$501$	−27.2335	−1.21670
$502$	1.09472	0.0488599
$503$	32.2709	1.43889	0.719444	−	0.694551i	$-0.244397\pi$
$503$	32.2709	1.43889	0.719444	+	0.694551i	$0.244397\pi$
$504$	−0.231853	−0.0103276
$505$	−9.04543	−0.402516
$506$	−26.8595	−1.19405
$507$	0	0
$508$	−2.75202	−0.122101
$509$	2.97775	0.131986	0.0659931	−	0.997820i	$-0.478978\pi$
$509$	2.97775	0.131986	0.0659931	+	0.997820i	$0.478978\pi$
$510$	2.84735	0.126083
$511$	2.04044	0.0902638
$512$	−1.00000	−0.0441942
$513$	5.38345	0.237685
$514$	−1.33399	−0.0588397
$515$	−10.1363	−0.446658
$516$	−9.04537	−0.398200
$517$	−17.3318	−0.762253
$518$	0.652149	0.0286538
$519$	−11.6394	−0.510911
$520$	0	0
$521$	3.76919	0.165131	0.0825657	−	0.996586i	$-0.473689\pi$
$521$	3.76919	0.165131	0.0825657	+	0.996586i	$0.473689\pi$
$522$	1.67684	0.0733934
$523$	4.65115	0.203381	0.101690	−	0.994816i	$-0.467575\pi$
$523$	4.65115	0.203381	0.101690	+	0.994816i	$0.467575\pi$
$524$	−11.9707	−0.522943
$525$	−3.40314	−0.148525
$526$	20.5528	0.896143
$527$	−9.85392	−0.429244
$528$	−7.34980	−0.319859
$529$	13.8452	0.601964
$530$	−10.4819	−0.455305
$531$	0.497865	0.0216055
$532$	−0.961642	−0.0416925
$533$	0	0
$534$	10.5073	0.454694
$535$	0.773785	0.0334537
$536$	−5.84309	−0.252383
$537$	−26.6092	−1.14827
$538$	20.8405	0.898497
$539$	−26.8826	−1.15792
$540$	9.11921	0.392428
$541$	26.4333	1.13646	0.568229	−	0.822871i	$-0.307629\pi$
$541$	26.4333	1.13646	0.568229	+	0.822871i	$0.307629\pi$
$542$	−5.43640	−0.233513
$543$	−4.67390	−0.200576
$544$	−1.01199	−0.0433887
$545$	22.4071	0.959817
$546$	0	0
$547$	−38.2590	−1.63584	−0.817919	−	0.575333i	$-0.804872\pi$
$547$	−38.2590	−1.63584	−0.817919	+	0.575333i	$0.804872\pi$
$548$	21.3846	0.913505
$549$	2.46531	0.105217
$550$	9.42770	0.401999
$551$	6.95493	0.296290
$552$	10.0823	0.429130
$553$	7.30651	0.310704
$554$	−24.6539	−1.04744
$555$	−1.90809	−0.0809938
$556$	−12.8882	−0.546580
$557$	23.5589	0.998221	0.499111	−	0.866538i	$-0.333660\pi$
$557$	23.5589	0.998221	0.499111	+	0.866538i	$0.333660\pi$
$558$	−2.34765	−0.0993838
$559$	0	0
$560$	−1.62896	−0.0688361
$561$	−7.43792	−0.314029
$562$	−25.5274	−1.07681
$563$	−5.38431	−0.226922	−0.113461	−	0.993542i	$-0.536194\pi$
$563$	−5.38431	−0.226922	−0.113461	+	0.993542i	$0.536194\pi$
$564$	6.50586	0.273946
$565$	19.5811	0.823785
$566$	16.6966	0.701811
$567$	7.90332	0.331908
$568$	−14.6895	−0.616358
$569$	15.1819	0.636457	0.318229	−	0.948014i	$-0.396912\pi$
$569$	15.1819	0.636457	0.318229	+	0.948014i	$0.396912\pi$
$570$	2.81362	0.117849
$571$	1.16095	0.0485842	0.0242921	−	0.999705i	$-0.492267\pi$
$571$	1.16095	0.0485842	0.0242921	+	0.999705i	$0.492267\pi$
$572$	0	0
$573$	13.0469	0.545044
$574$	10.7729	0.449652
$575$	−12.9327	−0.539330
$576$	−0.241101	−0.0100459
$577$	−5.68941	−0.236853	−0.118427	−	0.992963i	$-0.537785\pi$
$577$	−5.68941	−0.236853	−0.118427	+	0.992963i	$0.537785\pi$
$578$	15.9759	0.664509
$579$	−13.9268	−0.578780
$580$	11.7812	0.489188
$581$	4.56148	0.189242
$582$	−7.01619	−0.290831
$583$	27.3811	1.13401
$584$	2.12183	0.0878020
$585$	0	0
$586$	−21.2827	−0.879182
$587$	−41.9965	−1.73338	−0.866691	−	0.498845i	$-0.833758\pi$
$587$	−41.9965	−1.73338	−0.866691	+	0.498845i	$0.833758\pi$
$588$	10.0909	0.416143
$589$	−9.73717	−0.401213
$590$	3.49791	0.144007
$591$	−16.8554	−0.693340
$592$	0.678162	0.0278723
$593$	−9.17552	−0.376793	−0.188397	−	0.982093i	$-0.560329\pi$
$593$	−9.17552	−0.376793	−0.188397	+	0.982093i	$0.560329\pi$
$594$	−23.8214	−0.977406
$595$	−1.64849	−0.0675815
$596$	−23.2522	−0.952445
$597$	−3.00233	−0.122877
$598$	0	0
$599$	−38.4504	−1.57104	−0.785520	−	0.618837i	$-0.787604\pi$
$599$	−38.4504	−1.57104	−0.785520	+	0.618837i	$0.787604\pi$
$600$	−3.53888	−0.144474
$601$	17.4977	0.713748	0.356874	−	0.934153i	$-0.383843\pi$
$601$	17.4977	0.713748	0.356874	+	0.934153i	$0.383843\pi$
$602$	5.23687	0.213439
$603$	−1.40878	−0.0573698
$604$	24.5628	0.999446
$605$	14.5341	0.590897
$606$	−8.86952	−0.360300
$607$	−22.8352	−0.926851	−0.463426	−	0.886136i	$-0.653380\pi$
$607$	−22.8352	−0.926851	−0.463426	+	0.886136i	$0.653380\pi$
$608$	−1.00000	−0.0405554
$609$	11.1090	0.450158
$610$	17.3208	0.701300
$611$	0	0
$612$	−0.243992	−0.00986280
$613$	44.8074	1.80975	0.904877	−	0.425674i	$-0.139963\pi$
$613$	44.8074	1.80975	0.904877	+	0.425674i	$0.139963\pi$
$614$	−4.70636	−0.189933
$615$	−31.5198	−1.27100
$616$	4.25521	0.171447
$617$	12.3912	0.498850	0.249425	−	0.968394i	$-0.419758\pi$
$617$	12.3912	0.498850	0.249425	+	0.968394i	$0.419758\pi$
$618$	−9.93917	−0.399812
$619$	44.6247	1.79362	0.896809	−	0.442418i	$-0.145879\pi$
$619$	44.6247	1.79362	0.896809	+	0.442418i	$0.145879\pi$
$620$	−16.4941	−0.662421
$621$	32.6776	1.31131
$622$	−4.94400	−0.198236
$623$	−6.08325	−0.243720
$624$	0	0
$625$	−9.80770	−0.392308
$626$	−12.7603	−0.510003
$627$	−7.34980	−0.293523
$628$	9.81548	0.391680
$629$	0.686293	0.0273643
$630$	−0.392744	−0.0156473
$631$	−21.6787	−0.863014	−0.431507	−	0.902110i	$-0.642018\pi$
$631$	−21.6787	−0.863014	−0.431507	+	0.902110i	$0.642018\pi$
$632$	7.59795	0.302230
$633$	46.1142	1.83287
$634$	13.6955	0.543919
$635$	−4.66174	−0.184996
$636$	−10.2781	−0.407551
$637$	0	0
$638$	−30.7751	−1.21840
$639$	−3.54166	−0.140106
$640$	−1.69394	−0.0669587
$641$	1.22156	0.0482485	0.0241243	−	0.999709i	$-0.492320\pi$
$641$	1.22156	0.0482485	0.0241243	+	0.999709i	$0.492320\pi$
$642$	0.758738	0.0299450
$643$	−20.1091	−0.793024	−0.396512	−	0.918030i	$-0.629779\pi$
$643$	−20.1091	−0.793024	−0.396512	+	0.918030i	$0.629779\pi$
$644$	−5.83719	−0.230017
$645$	−15.3223	−0.603314
$646$	−1.01199	−0.0398162
$647$	33.4789	1.31619	0.658095	−	0.752935i	$-0.271362\pi$
$647$	33.4789	1.31619	0.658095	+	0.752935i	$0.271362\pi$
$648$	8.21857	0.322856
$649$	−9.13733	−0.358672
$650$	0	0
$651$	−15.5530	−0.609570
$652$	−9.15174	−0.358410
$653$	17.1761	0.672153	0.336077	−	0.941835i	$-0.390900\pi$
$653$	17.1761	0.672153	0.336077	+	0.941835i	$0.390900\pi$
$654$	21.9714	0.859149
$655$	−20.2776	−0.792312
$656$	11.2026	0.437388
$657$	0.511576	0.0199585
$658$	−3.76661	−0.146838
$659$	−27.2804	−1.06269	−0.531347	−	0.847155i	$-0.678314\pi$
$659$	−27.2804	−1.06269	−0.531347	+	0.847155i	$0.678314\pi$
$660$	−12.4501	−0.484619
$661$	45.5635	1.77221	0.886107	−	0.463481i	$-0.153400\pi$
$661$	45.5635	1.77221	0.886107	+	0.463481i	$0.153400\pi$
$662$	6.78053	0.263533
$663$	0	0
$664$	4.74343	0.184081
$665$	−1.62896	−0.0631683
$666$	0.163506	0.00633572
$667$	42.2166	1.63463
$668$	16.3959	0.634376
$669$	11.6416	0.450089
$670$	−9.89782	−0.382386
$671$	−45.2459	−1.74670
$672$	−1.59728	−0.0616164
$673$	28.5965	1.10232	0.551158	−	0.834401i	$-0.314186\pi$
$673$	28.5965	1.10232	0.551158	+	0.834401i	$0.314186\pi$
$674$	−18.0123	−0.693809
$675$	−11.4699	−0.441476
$676$	0	0
$677$	29.9014	1.14921	0.574603	−	0.818433i	$-0.305157\pi$
$677$	29.9014	1.14921	0.574603	+	0.818433i	$0.305157\pi$
$678$	19.2003	0.737385
$679$	4.06207	0.155888
$680$	−1.71425	−0.0657383
$681$	34.6020	1.32595
$682$	43.0864	1.64986
$683$	32.4539	1.24181	0.620907	−	0.783884i	$-0.286764\pi$
$683$	32.4539	1.24181	0.620907	+	0.783884i	$0.286764\pi$
$684$	−0.241101	−0.00921874
$685$	36.2242	1.38405
$686$	−12.5737	−0.480066
$687$	10.0356	0.382883
$688$	5.44576	0.207618
$689$	0	0
$690$	17.0787	0.650175
$691$	15.0559	0.572752	0.286376	−	0.958117i	$-0.407549\pi$
$691$	15.0559	0.572752	0.286376	+	0.958117i	$0.407549\pi$
$692$	7.00746	0.266384
$693$	1.02594	0.0389721
$694$	−24.5390	−0.931487
$695$	−21.8317	−0.828124
$696$	11.5521	0.437881
$697$	11.3369	0.429417
$698$	−4.97210	−0.188197
$699$	5.34303	0.202092
$700$	2.04886	0.0774395
$701$	32.2158	1.21677	0.608387	−	0.793641i	$-0.291817\pi$
$701$	32.2158	1.21677	0.608387	+	0.793641i	$0.291817\pi$
$702$	0	0
$703$	0.678162	0.0255774
$704$	4.42494	0.166771
$705$	11.0205	0.415056
$706$	−13.5038	−0.508222
$707$	5.13506	0.193124
$708$	3.42989	0.128903
$709$	−27.6450	−1.03823	−0.519116	−	0.854704i	$-0.673739\pi$
$709$	−27.6450	−1.03823	−0.519116	+	0.854704i	$0.673739\pi$
$710$	−24.8831	−0.933845
$711$	1.83188	0.0687008
$712$	−6.32590	−0.237073
$713$	−59.1048	−2.21349
$714$	−1.61643	−0.0604934
$715$	0	0
$716$	16.0201	0.598698
$717$	12.7100	0.474663
$718$	−35.4154	−1.32169
$719$	8.31650	0.310153	0.155077	−	0.987902i	$-0.450438\pi$
$719$	8.31650	0.310153	0.155077	+	0.987902i	$0.450438\pi$
$720$	−0.408410	−0.0152206
$721$	5.75434	0.214303
$722$	−1.00000	−0.0372161
$723$	15.4053	0.572929
$724$	2.81392	0.104578
$725$	−14.8180	−0.550328
$726$	14.2515	0.528923
$727$	20.4351	0.757897	0.378949	−	0.925418i	$-0.376286\pi$
$727$	20.4351	0.757897	0.378949	+	0.925418i	$0.376286\pi$
$728$	0	0
$729$	28.8071	1.06693
$730$	3.59424	0.133029
$731$	5.51105	0.203834
$732$	16.9840	0.627747
$733$	42.1710	1.55762	0.778810	−	0.627260i	$-0.215824\pi$
$733$	42.1710	1.55762	0.778810	+	0.627260i	$0.215824\pi$
$734$	35.1014	1.29562
$735$	17.0934	0.630500
$736$	−6.07002	−0.223744
$737$	25.8553	0.952393
$738$	2.70096	0.0994239
$739$	−27.6948	−1.01877	−0.509384	−	0.860539i	$-0.670127\pi$
$739$	−27.6948	−1.01877	−0.509384	+	0.860539i	$0.670127\pi$
$740$	1.14876	0.0422293
$741$	0	0
$742$	5.95054	0.218451
$743$	7.94485	0.291468	0.145734	−	0.989324i	$-0.453446\pi$
$743$	7.94485	0.291468	0.145734	+	0.989324i	$0.453446\pi$
$744$	−16.1734	−0.592945
$745$	−39.3876	−1.44305
$746$	5.02537	0.183992
$747$	1.14365	0.0418439
$748$	4.47800	0.163732
$749$	−0.439276	−0.0160508
$750$	−20.0627	−0.732587
$751$	32.1477	1.17309	0.586543	−	0.809918i	$-0.300489\pi$
$751$	32.1477	1.17309	0.586543	+	0.809918i	$0.300489\pi$
$752$	−3.91685	−0.142833
$753$	1.81833	0.0662636
$754$	0	0
$755$	41.6078	1.51426
$756$	−5.17695	−0.188284
$757$	−3.80527	−0.138305	−0.0691525	−	0.997606i	$-0.522030\pi$
$757$	−3.80527	−0.138305	−0.0691525	+	0.997606i	$0.522030\pi$
$758$	−19.8743	−0.721868
$759$	−44.6134	−1.61937
$760$	−1.69394	−0.0614455
$761$	27.4468	0.994944	0.497472	−	0.867480i	$-0.334262\pi$
$761$	27.4468	0.994944	0.497472	+	0.867480i	$0.334262\pi$
$762$	−4.57108	−0.165593
$763$	−12.7205	−0.460512
$764$	−7.85491	−0.284181
$765$	−0.413307	−0.0149431
$766$	−13.1631	−0.475602
$767$	0	0
$768$	−1.66099	−0.0599359
$769$	−28.9834	−1.04517	−0.522584	−	0.852588i	$-0.675032\pi$
$769$	−28.9834	−1.04517	−0.522584	+	0.852588i	$0.675032\pi$
$770$	7.20805	0.259760
$771$	−2.21574	−0.0797981
$772$	8.38465	0.301770
$773$	−38.8125	−1.39599	−0.697995	−	0.716103i	$-0.745924\pi$
$773$	−38.8125	−1.39599	−0.697995	+	0.716103i	$0.745924\pi$
$774$	1.31298	0.0471941
$775$	20.7459	0.745213
$776$	4.22409	0.151636
$777$	1.08321	0.0388601
$778$	−17.9823	−0.644697
$779$	11.2026	0.401375
$780$	0	0
$781$	65.0002	2.32589
$782$	−6.14280	−0.219666
$783$	37.4415	1.33805
$784$	−6.07525	−0.216973
$785$	16.6268	0.593435
$786$	−19.8833	−0.709213
$787$	41.3374	1.47352	0.736760	−	0.676154i	$-0.236355\pi$
$787$	41.3374	1.47352	0.736760	+	0.676154i	$0.236355\pi$
$788$	10.1478	0.361501
$789$	34.1380	1.21534
$790$	12.8704	0.457910
$791$	−11.1162	−0.395245
$792$	1.06686	0.0379092
$793$	0	0
$794$	−10.7231	−0.380548
$795$	−17.4104	−0.617482
$796$	1.80755	0.0640670
$797$	41.7974	1.48054	0.740270	−	0.672309i	$-0.234698\pi$
$797$	41.7974	1.48054	0.740270	+	0.672309i	$0.234698\pi$
$798$	−1.59728	−0.0565431
$799$	−3.96381	−0.140230
$800$	2.13058	0.0753275
$801$	−1.52518	−0.0538897
$802$	23.5965	0.833221
$803$	−9.38898	−0.331330
$804$	−9.70533	−0.342281
$805$	−9.88782	−0.348500
$806$	0	0
$807$	34.6159	1.21854
$808$	5.33989	0.187857
$809$	−15.4613	−0.543592	−0.271796	−	0.962355i	$-0.587618\pi$
$809$	−15.4613	−0.543592	−0.271796	+	0.962355i	$0.587618\pi$
$810$	13.9217	0.489159
$811$	−40.5158	−1.42270	−0.711351	−	0.702837i	$-0.751916\pi$
$811$	−40.5158	−1.42270	−0.711351	+	0.702837i	$0.751916\pi$
$812$	−6.68815	−0.234708
$813$	−9.02982	−0.316690
$814$	−3.00083	−0.105179
$815$	−15.5025	−0.543027
$816$	−1.68091	−0.0588436
$817$	5.44576	0.190523
$818$	−2.70488	−0.0945739
$819$	0	0
$820$	18.9765	0.662688
$821$	16.2439	0.566915	0.283457	−	0.958985i	$-0.408519\pi$
$821$	16.2439	0.566915	0.283457	+	0.958985i	$0.408519\pi$
$822$	35.5197	1.23889
$823$	−22.0517	−0.768674	−0.384337	−	0.923193i	$-0.625570\pi$
$823$	−22.0517	−0.768674	−0.384337	+	0.923193i	$0.625570\pi$
$824$	5.98387	0.208458
$825$	15.6594	0.545189
$826$	−1.98575	−0.0690932
$827$	−20.1988	−0.702380	−0.351190	−	0.936304i	$-0.614223\pi$
$827$	−20.1988	−0.702380	−0.351190	+	0.936304i	$0.614223\pi$
$828$	−1.46349	−0.0508598
$829$	−4.60677	−0.160000	−0.0799998	−	0.996795i	$-0.525492\pi$
$829$	−4.60677	−0.160000	−0.0799998	+	0.996795i	$0.525492\pi$
$830$	8.03507	0.278901
$831$	−40.9499	−1.42054
$832$	0	0
$833$	−6.14809	−0.213019
$834$	−21.4071	−0.741269
$835$	27.7736	0.961145
$836$	4.42494	0.153040
$837$	−52.4196	−1.81188
$838$	−16.1618	−0.558299
$839$	17.0875	0.589925	0.294962	−	0.955509i	$-0.404693\pi$
$839$	17.0875	0.589925	0.294962	+	0.955509i	$0.404693\pi$
$840$	−2.70569	−0.0933552
$841$	19.3710	0.667965
$842$	−16.8012	−0.579006
$843$	−42.4009	−1.46036
$844$	−27.7630	−0.955642
$845$	0	0
$846$	−0.944358	−0.0324677
$847$	−8.25099	−0.283507
$848$	6.18790	0.212493
$849$	27.7329	0.951792
$850$	2.15613	0.0739546
$851$	4.11646	0.141110
$852$	−24.3992	−0.835902
$853$	42.6641	1.46079	0.730396	−	0.683024i	$-0.239335\pi$
$853$	42.6641	1.46079	0.730396	+	0.683024i	$0.239335\pi$
$854$	−9.83299	−0.336478
$855$	−0.408410	−0.0139673
$856$	−0.456797	−0.0156130
$857$	38.0767	1.30068	0.650338	−	0.759645i	$-0.274627\pi$
$857$	38.0767	1.30068	0.650338	+	0.759645i	$0.274627\pi$
$858$	0	0
$859$	−12.9531	−0.441955	−0.220978	−	0.975279i	$-0.570925\pi$
$859$	−12.9531	−0.441955	−0.220978	+	0.975279i	$0.570925\pi$
$860$	9.22476	0.314562
$861$	17.8937	0.609816
$862$	−33.9067	−1.15487
$863$	−35.8320	−1.21974	−0.609868	−	0.792503i	$-0.708778\pi$
$863$	−35.8320	−1.21974	−0.609868	+	0.792503i	$0.708778\pi$
$864$	−5.38345	−0.183149
$865$	11.8702	0.403599
$866$	−20.3139	−0.690294
$867$	26.5358	0.901204
$868$	9.36367	0.317824
$869$	−33.6205	−1.14050
$870$	19.5685	0.663434
$871$	0	0
$872$	−13.2279	−0.447952
$873$	1.01844	0.0344688
$874$	−6.07002	−0.205322
$875$	11.6154	0.392673
$876$	3.52435	0.119077
$877$	−20.1701	−0.681095	−0.340548	−	0.940227i	$-0.610612\pi$
$877$	−20.1701	−0.681095	−0.340548	+	0.940227i	$0.610612\pi$
$878$	23.5505	0.794790
$879$	−35.3505	−1.19234
$880$	7.49557	0.252675
$881$	13.5626	0.456934	0.228467	−	0.973552i	$-0.426629\pi$
$881$	13.5626	0.456934	0.228467	+	0.973552i	$0.426629\pi$
$882$	−1.46475	−0.0493207
$883$	−24.7191	−0.831863	−0.415932	−	0.909396i	$-0.636544\pi$
$883$	−24.7191	−0.831863	−0.415932	+	0.909396i	$0.636544\pi$
$884$	0	0
$885$	5.81001	0.195301
$886$	−26.0674	−0.875752
$887$	19.4900	0.654409	0.327204	−	0.944954i	$-0.393893\pi$
$887$	19.4900	0.654409	0.327204	+	0.944954i	$0.393893\pi$
$888$	1.12642	0.0378003
$889$	2.64646	0.0887593
$890$	−10.7157	−0.359190
$891$	−36.3667	−1.21833
$892$	−7.00880	−0.234672
$893$	−3.91685	−0.131072
$894$	−38.6217	−1.29170
$895$	27.1370	0.907089
$896$	0.961642	0.0321262
$897$	0	0
$898$	6.84222	0.228328
$899$	−67.7213	−2.25863
$900$	0.513687	0.0171229
$901$	6.26209	0.208620
$902$	−49.5709	−1.65053
$903$	8.69840	0.289465
$904$	−11.5596	−0.384465
$905$	4.76660	0.158447
$906$	40.7986	1.35544
$907$	31.2071	1.03621	0.518107	−	0.855316i	$-0.326637\pi$
$907$	31.2071	1.03621	0.518107	+	0.855316i	$0.326637\pi$
$908$	−20.8321	−0.691337
$909$	1.28746	0.0427022
$910$	0	0
$911$	13.0732	0.433135	0.216567	−	0.976268i	$-0.430514\pi$
$911$	13.0732	0.433135	0.216567	+	0.976268i	$0.430514\pi$
$912$	−1.66099	−0.0550010
$913$	−20.9894	−0.694648
$914$	−24.4501	−0.808739
$915$	28.7698	0.951100
$916$	−6.04193	−0.199631
$917$	11.5115	0.380144
$918$	−5.44800	−0.179811
$919$	−56.2420	−1.85525	−0.927627	−	0.373508i	$-0.878155\pi$
$919$	−56.2420	−1.85525	−0.927627	+	0.373508i	$0.878155\pi$
$920$	−10.2822	−0.338995
$921$	−7.81722	−0.257586
$922$	1.80454	0.0594292
$923$	0	0
$924$	7.06787	0.232516
$925$	−1.44488	−0.0475074
$926$	4.62667	0.152042
$927$	1.44272	0.0473851
$928$	−6.95493	−0.228307
$929$	29.1740	0.957169	0.478585	−	0.878041i	$-0.341150\pi$
$929$	29.1740	0.957169	0.478585	+	0.878041i	$0.341150\pi$
$930$	−27.3967	−0.898372
$931$	−6.07525	−0.199108
$932$	−3.21677	−0.105369
$933$	−8.21195	−0.268847
$934$	−10.7837	−0.352853
$935$	7.58544	0.248070
$936$	0	0
$937$	−41.5894	−1.35867	−0.679333	−	0.733830i	$-0.737731\pi$
$937$	−41.5894	−1.35867	−0.679333	+	0.733830i	$0.737731\pi$
$938$	5.61896	0.183466
$939$	−21.1947	−0.691664
$940$	−6.63489	−0.216406
$941$	−28.2701	−0.921578	−0.460789	−	0.887510i	$-0.652433\pi$
$941$	−28.2701	−0.921578	−0.460789	+	0.887510i	$0.652433\pi$
$942$	16.3034	0.531195
$943$	68.0001	2.21439
$944$	−2.06496	−0.0672088
$945$	−8.76942	−0.285269
$946$	−24.0972	−0.783466
$947$	44.0304	1.43079	0.715397	−	0.698718i	$-0.246246\pi$
$947$	44.0304	1.43079	0.715397	+	0.698718i	$0.246246\pi$
$948$	12.6201	0.409883
$949$	0	0
$950$	2.13058	0.0691252
$951$	22.7482	0.737660
$952$	0.973172	0.0315407
$953$	−52.4548	−1.69918	−0.849589	−	0.527445i	$-0.823150\pi$
$953$	−52.4548	−1.69918	−0.849589	+	0.527445i	$0.823150\pi$
$954$	1.49191	0.0483024
$955$	−13.3057	−0.430562
$956$	−7.65203	−0.247484
$957$	−51.1173	−1.65239
$958$	0.0903531	0.00291918
$959$	−20.5643	−0.664057
$960$	−2.81362	−0.0908090
$961$	63.8125	2.05847
$962$	0	0
$963$	−0.110135	−0.00354904
$964$	−9.27475	−0.298720
$965$	14.2031	0.457213
$966$	−9.69553	−0.311948
$967$	−8.38307	−0.269581	−0.134791	−	0.990874i	$-0.543036\pi$
$967$	−8.38307	−0.269581	−0.134791	+	0.990874i	$0.543036\pi$
$968$	−8.58011	−0.275775
$969$	−1.68091	−0.0539986
$970$	7.15534	0.229744
$971$	−20.7629	−0.666312	−0.333156	−	0.942872i	$-0.608114\pi$
$971$	−20.7629	−0.666312	−0.333156	+	0.942872i	$0.608114\pi$
$972$	−2.49936	−0.0801670
$973$	12.3938	0.397327
$974$	−16.1552	−0.517647
$975$	0	0
$976$	−10.2252	−0.327301
$977$	58.2524	1.86366	0.931830	−	0.362895i	$-0.118212\pi$
$977$	58.2524	1.86366	0.931830	+	0.362895i	$0.118212\pi$
$978$	−15.2010	−0.486074
$979$	27.9917	0.894620
$980$	−10.2911	−0.328736
$981$	−3.18926	−0.101825
$982$	−4.49239	−0.143358
$983$	2.89705	0.0924015	0.0462007	−	0.998932i	$-0.485289\pi$
$983$	2.89705	0.0924015	0.0462007	+	0.998932i	$0.485289\pi$
$984$	18.6075	0.593184
$985$	17.1897	0.547710
$986$	−7.03832	−0.224146
$987$	−6.25631	−0.199141
$988$	0	0
$989$	33.0559	1.05112
$990$	1.80719	0.0574363
$991$	8.67737	0.275646	0.137823	−	0.990457i	$-0.455990\pi$
$991$	8.67737	0.275646	0.137823	+	0.990457i	$0.455990\pi$
$992$	9.73717	0.309156
$993$	11.2624	0.357402
$994$	14.1260	0.448051
$995$	3.06188	0.0970680
$996$	7.87881	0.249650
$997$	37.0456	1.17324	0.586622	−	0.809861i	$-0.300457\pi$
$997$	37.0456	1.17324	0.586622	+	0.809861i	$0.300457\pi$
$998$	−10.3831	−0.328670
$999$	3.65085	0.115508

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bj.1.2	✓	9
13.12	even	2		6422.2.a.bl.1.2	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bj.1.2	✓	9	1.1	even	1	trivial
6422.2.a.bl.1.2	yes	9	13.12	even	2

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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.66099 q^{3} +1.00000 q^{4} +1.69394 q^{5} +1.66099 q^{6} -0.961642 q^{7} -1.00000 q^{8} -0.241101 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.66099 q^{3} +1.00000 q^{4} +1.69394 q^{5} +1.66099 q^{6} -0.961642 q^{7} -1.00000 q^{8} -0.241101 q^{9} -1.69394 q^{10} +4.42494 q^{11} -1.66099 q^{12} +0.961642 q^{14} -2.81362 q^{15} +1.00000 q^{16} +1.01199 q^{17} +0.241101 q^{18} +1.00000 q^{19} +1.69394 q^{20} +1.59728 q^{21} -4.42494 q^{22} +6.07002 q^{23} +1.66099 q^{24} -2.13058 q^{25} +5.38345 q^{27} -0.961642 q^{28} +6.95493 q^{29} +2.81362 q^{30} -9.73717 q^{31} -1.00000 q^{32} -7.34980 q^{33} -1.01199 q^{34} -1.62896 q^{35} -0.241101 q^{36} +0.678162 q^{37} -1.00000 q^{38} -1.69394 q^{40} +11.2026 q^{41} -1.59728 q^{42} +5.44576 q^{43} +4.42494 q^{44} -0.408410 q^{45} -6.07002 q^{46} -3.91685 q^{47} -1.66099 q^{48} -6.07525 q^{49} +2.13058 q^{50} -1.68091 q^{51} +6.18790 q^{53} -5.38345 q^{54} +7.49557 q^{55} +0.961642 q^{56} -1.66099 q^{57} -6.95493 q^{58} -2.06496 q^{59} -2.81362 q^{60} -10.2252 q^{61} +9.73717 q^{62} +0.231853 q^{63} +1.00000 q^{64} +7.34980 q^{66} +5.84309 q^{67} +1.01199 q^{68} -10.0823 q^{69} +1.62896 q^{70} +14.6895 q^{71} +0.241101 q^{72} -2.12183 q^{73} -0.678162 q^{74} +3.53888 q^{75} +1.00000 q^{76} -4.25521 q^{77} -7.59795 q^{79} +1.69394 q^{80} -8.21857 q^{81} -11.2026 q^{82} -4.74343 q^{83} +1.59728 q^{84} +1.71425 q^{85} -5.44576 q^{86} -11.5521 q^{87} -4.42494 q^{88} +6.32590 q^{89} +0.408410 q^{90} +6.07002 q^{92} +16.1734 q^{93} +3.91685 q^{94} +1.69394 q^{95} +1.66099 q^{96} -4.22409 q^{97} +6.07525 q^{98} -1.06686 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{2} + q^{3} + 9 q^{4} - q^{5} - q^{6} + 13 q^{7} - 9 q^{8} - 2 q^{9}+O(q^{10}) \) 9 * q - 9 * q^2 + q^3 + 9 * q^4 - q^5 - q^6 + 13 * q^7 - 9 * q^8 - 2 * q^9	\( 9 q - 9 q^{2} + q^{3} + 9 q^{4} - q^{5} - q^{6} + 13 q^{7} - 9 q^{8} - 2 q^{9} + q^{10} + 3 q^{11} + q^{12} - 13 q^{14} + 3 q^{15} + 9 q^{16} - 8 q^{17} + 2 q^{18} + 9 q^{19} - q^{20} + 24 q^{21} - 3 q^{22} - 10 q^{23} - q^{24} + 8 q^{25} + 10 q^{27} + 13 q^{28} - 20 q^{29} - 3 q^{30} + q^{31} - 9 q^{32} - 2 q^{33} + 8 q^{34} + 4 q^{35} - 2 q^{36} + 15 q^{37} - 9 q^{38} + q^{40} + 19 q^{41} - 24 q^{42} - 16 q^{43} + 3 q^{44} + 15 q^{45} + 10 q^{46} + 18 q^{47} + q^{48} + 18 q^{49} - 8 q^{50} - 11 q^{51} + 17 q^{53} - 10 q^{54} - 26 q^{55} - 13 q^{56} + q^{57} + 20 q^{58} + 24 q^{59} + 3 q^{60} + 6 q^{61} - q^{62} + q^{63} + 9 q^{64} + 2 q^{66} + 29 q^{67} - 8 q^{68} - 12 q^{69} - 4 q^{70} + 23 q^{71} + 2 q^{72} + 38 q^{73} - 15 q^{74} + 11 q^{75} + 9 q^{76} - 40 q^{77} - 20 q^{79} - q^{80} - 31 q^{81} - 19 q^{82} + 20 q^{83} + 24 q^{84} + 39 q^{85} + 16 q^{86} - 10 q^{87} - 3 q^{88} - 7 q^{89} - 15 q^{90} - 10 q^{92} + 11 q^{93} - 18 q^{94} - q^{95} - q^{96} + 28 q^{97} - 18 q^{98} + 25 q^{99}+O(q^{100}) \) 9 * q - 9 * q^2 + q^3 + 9 * q^4 - q^5 - q^6 + 13 * q^7 - 9 * q^8 - 2 * q^9 + q^10 + 3 * q^11 + q^12 - 13 * q^14 + 3 * q^15 + 9 * q^16 - 8 * q^17 + 2 * q^18 + 9 * q^19 - q^20 + 24 * q^21 - 3 * q^22 - 10 * q^23 - q^24 + 8 * q^25 + 10 * q^27 + 13 * q^28 - 20 * q^29 - 3 * q^30 + q^31 - 9 * q^32 - 2 * q^33 + 8 * q^34 + 4 * q^35 - 2 * q^36 + 15 * q^37 - 9 * q^38 + q^40 + 19 * q^41 - 24 * q^42 - 16 * q^43 + 3 * q^44 + 15 * q^45 + 10 * q^46 + 18 * q^47 + q^48 + 18 * q^49 - 8 * q^50 - 11 * q^51 + 17 * q^53 - 10 * q^54 - 26 * q^55 - 13 * q^56 + q^57 + 20 * q^58 + 24 * q^59 + 3 * q^60 + 6 * q^61 - q^62 + q^63 + 9 * q^64 + 2 * q^66 + 29 * q^67 - 8 * q^68 - 12 * q^69 - 4 * q^70 + 23 * q^71 + 2 * q^72 + 38 * q^73 - 15 * q^74 + 11 * q^75 + 9 * q^76 - 40 * q^77 - 20 * q^79 - q^80 - 31 * q^81 - 19 * q^82 + 20 * q^83 + 24 * q^84 + 39 * q^85 + 16 * q^86 - 10 * q^87 - 3 * q^88 - 7 * q^89 - 15 * q^90 - 10 * q^92 + 11 * q^93 - 18 * q^94 - q^95 - q^96 + 28 * q^97 - 18 * q^98 + 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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