LMFDB - Embedded newform 6422.2.a.m.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x - 1 $ x^3 - x^2 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.37720$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.37720 q^{3} +1.00000 q^{4} +1.65109 q^{5} +2.37720 q^{6} -1.37720 q^{7} -1.00000 q^{8} +2.65109 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.37720 q^{3} +1.00000 q^{4} +1.65109 q^{5} +2.37720 q^{6} -1.37720 q^{7} -1.00000 q^{8} +2.65109 q^{9} -1.65109 q^{10} +0.348907 q^{11} -2.37720 q^{12} +1.37720 q^{14} -3.92498 q^{15} +1.00000 q^{16} +2.75441 q^{17} -2.65109 q^{18} -1.00000 q^{19} +1.65109 q^{20} +3.27389 q^{21} -0.348907 q^{22} +0.170578 q^{23} +2.37720 q^{24} -2.27389 q^{25} +0.829422 q^{27} -1.37720 q^{28} -8.19887 q^{29} +3.92498 q^{30} +10.3305 q^{31} -1.00000 q^{32} -0.829422 q^{33} -2.75441 q^{34} -2.27389 q^{35} +2.65109 q^{36} -5.84997 q^{37} +1.00000 q^{38} -1.65109 q^{40} -2.07502 q^{41} -3.27389 q^{42} +1.75441 q^{43} +0.348907 q^{44} +4.37720 q^{45} -0.170578 q^{46} -0.857718 q^{47} -2.37720 q^{48} -5.10331 q^{49} +2.27389 q^{50} -6.54778 q^{51} +10.3305 q^{53} -0.829422 q^{54} +0.576077 q^{55} +1.37720 q^{56} +2.37720 q^{57} +8.19887 q^{58} +0.00775008 q^{59} -3.92498 q^{60} +3.48052 q^{61} -10.3305 q^{62} -3.65109 q^{63} +1.00000 q^{64} +0.829422 q^{66} +3.84997 q^{67} +2.75441 q^{68} -0.405499 q^{69} +2.27389 q^{70} +3.00000 q^{71} -2.65109 q^{72} -16.6999 q^{73} +5.84997 q^{74} +5.40550 q^{75} -1.00000 q^{76} -0.480515 q^{77} +7.88601 q^{79} +1.65109 q^{80} -9.92498 q^{81} +2.07502 q^{82} -13.9738 q^{83} +3.27389 q^{84} +4.54778 q^{85} -1.75441 q^{86} +19.4904 q^{87} -0.348907 q^{88} +10.8217 q^{89} -4.37720 q^{90} +0.170578 q^{92} -24.5577 q^{93} +0.857718 q^{94} -1.65109 q^{95} +2.37720 q^{96} +3.24559 q^{97} +5.10331 q^{98} +0.924984 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 2 q^{6} + q^{7} - 3 q^{8} + q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 2 * q^6 + q^7 - 3 * q^8 + q^9	$ 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 2 q^{6} + q^{7} - 3 q^{8} + q^{9} + 2 q^{10} + 8 q^{11} - 2 q^{12} - q^{14} - 3 q^{15} + 3 q^{16} - 2 q^{17} - q^{18} - 3 q^{19} - 2 q^{20} + 8 q^{21} - 8 q^{22} + 2 q^{23} + 2 q^{24} - 5 q^{25} + q^{27} + q^{28} - 14 q^{29} + 3 q^{30} + 5 q^{31} - 3 q^{32} - q^{33} + 2 q^{34} - 5 q^{35} + q^{36} + 3 q^{38} + 2 q^{40} - 15 q^{41} - 8 q^{42} - 5 q^{43} + 8 q^{44} + 8 q^{45} - 2 q^{46} + 11 q^{47} - 2 q^{48} - 12 q^{49} + 5 q^{50} - 16 q^{51} + 5 q^{53} - q^{54} - 14 q^{55} - q^{56} + 2 q^{57} + 14 q^{58} + 4 q^{59} - 3 q^{60} + 2 q^{61} - 5 q^{62} - 4 q^{63} + 3 q^{64} + q^{66} - 6 q^{67} - 2 q^{68} + 16 q^{69} + 5 q^{70} + 9 q^{71} - q^{72} - 15 q^{73} - q^{75} - 3 q^{76} + 7 q^{77} - 2 q^{79} - 2 q^{80} - 21 q^{81} + 15 q^{82} - 5 q^{83} + 8 q^{84} + 10 q^{85} + 5 q^{86} + 5 q^{87} - 8 q^{88} + 27 q^{89} - 8 q^{90} + 2 q^{92} - 25 q^{93} - 11 q^{94} + 2 q^{95} + 2 q^{96} + 20 q^{97} + 12 q^{98} - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 2 * q^6 + q^7 - 3 * q^8 + q^9 + 2 * q^10 + 8 * q^11 - 2 * q^12 - q^14 - 3 * q^15 + 3 * q^16 - 2 * q^17 - q^18 - 3 * q^19 - 2 * q^20 + 8 * q^21 - 8 * q^22 + 2 * q^23 + 2 * q^24 - 5 * q^25 + q^27 + q^28 - 14 * q^29 + 3 * q^30 + 5 * q^31 - 3 * q^32 - q^33 + 2 * q^34 - 5 * q^35 + q^36 + 3 * q^38 + 2 * q^40 - 15 * q^41 - 8 * q^42 - 5 * q^43 + 8 * q^44 + 8 * q^45 - 2 * q^46 + 11 * q^47 - 2 * q^48 - 12 * q^49 + 5 * q^50 - 16 * q^51 + 5 * q^53 - q^54 - 14 * q^55 - q^56 + 2 * q^57 + 14 * q^58 + 4 * q^59 - 3 * q^60 + 2 * q^61 - 5 * q^62 - 4 * q^63 + 3 * q^64 + q^66 - 6 * q^67 - 2 * q^68 + 16 * q^69 + 5 * q^70 + 9 * q^71 - q^72 - 15 * q^73 - q^75 - 3 * q^76 + 7 * q^77 - 2 * q^79 - 2 * q^80 - 21 * q^81 + 15 * q^82 - 5 * q^83 + 8 * q^84 + 10 * q^85 + 5 * q^86 + 5 * q^87 - 8 * q^88 + 27 * q^89 - 8 * q^90 + 2 * q^92 - 25 * q^93 - 11 * q^94 + 2 * q^95 + 2 * q^96 + 20 * q^97 + 12 * q^98 - 6 * 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
$3$	−2.37720	−1.37248	−0.686239	−	0.727376i	$-0.740740\pi$
$3$	−2.37720	−1.37248	−0.686239	+	0.727376i	$0.740740\pi$
$4$	1.00000	0.500000
$5$	1.65109	0.738391	0.369196	−	0.929352i	$-0.379633\pi$
$5$	1.65109	0.738391	0.369196	+	0.929352i	$0.379633\pi$
$6$	2.37720	0.970489
$7$	−1.37720	−0.520534	−0.260267	−	0.965537i	$-0.583811\pi$
$7$	−1.37720	−0.520534	−0.260267	+	0.965537i	$0.583811\pi$
$8$	−1.00000	−0.353553
$9$	2.65109	0.883698
$10$	−1.65109	−0.522122
$11$	0.348907	0.105199	0.0525996	−	0.998616i	$-0.483249\pi$
$11$	0.348907	0.105199	0.0525996	+	0.998616i	$0.483249\pi$
$12$	−2.37720	−0.686239
$13$	0	0
$13$	0	0
$14$	1.37720	0.368073
$15$	−3.92498	−1.01343
$16$	1.00000	0.250000
$17$	2.75441	0.668042	0.334021	−	0.942566i	$-0.391594\pi$
$17$	2.75441	0.668042	0.334021	+	0.942566i	$0.391594\pi$
$18$	−2.65109	−0.624869
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.65109	0.369196
$21$	3.27389	0.714421
$22$	−0.348907	−0.0743871
$23$	0.170578	0.0355680	0.0177840	−	0.999842i	$-0.494339\pi$
$23$	0.170578	0.0355680	0.0177840	+	0.999842i	$0.494339\pi$
$24$	2.37720	0.485245
$25$	−2.27389	−0.454778
$26$	0	0
$27$	0.829422	0.159622
$28$	−1.37720	−0.260267
$29$	−8.19887	−1.52249	−0.761246	−	0.648463i	$-0.775412\pi$
$29$	−8.19887	−1.52249	−0.761246	+	0.648463i	$0.775412\pi$
$30$	3.92498	0.716601
$31$	10.3305	1.85541	0.927705	−	0.373315i	$-0.121779\pi$
$31$	10.3305	1.85541	0.927705	+	0.373315i	$0.121779\pi$
$32$	−1.00000	−0.176777
$33$	−0.829422	−0.144384
$34$	−2.75441	−0.472377
$35$	−2.27389	−0.384358
$36$	2.65109	0.441849
$37$	−5.84997	−0.961729	−0.480864	−	0.876795i	$-0.659677\pi$
$37$	−5.84997	−0.961729	−0.480864	+	0.876795i	$0.659677\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−1.65109	−0.261061
$41$	−2.07502	−0.324063	−0.162032	−	0.986786i	$-0.551805\pi$
$41$	−2.07502	−0.324063	−0.162032	+	0.986786i	$0.551805\pi$
$42$	−3.27389	−0.505172
$43$	1.75441	0.267544	0.133772	−	0.991012i	$-0.457291\pi$
$43$	1.75441	0.267544	0.133772	+	0.991012i	$0.457291\pi$
$44$	0.348907	0.0525996
$45$	4.37720	0.652515
$46$	−0.170578	−0.0251504
$47$	−0.857718	−0.125111	−0.0625555	−	0.998041i	$-0.519925\pi$
$47$	−0.857718	−0.125111	−0.0625555	+	0.998041i	$0.519925\pi$
$48$	−2.37720	−0.343120
$49$	−5.10331	−0.729045
$50$	2.27389	0.321577
$51$	−6.54778	−0.916873
$52$	0	0
$53$	10.3305	1.41900	0.709500	−	0.704705i	$-0.248921\pi$
$53$	10.3305	1.41900	0.709500	+	0.704705i	$0.248921\pi$
$54$	−0.829422	−0.112870
$55$	0.576077	0.0776783
$56$	1.37720	0.184036
$57$	2.37720	0.314868
$58$	8.19887	1.07656
$59$	0.00775008	0.00100897	0.000504487	−	1.00000i	$-0.499839\pi$
$59$	0.00775008	0.00100897	0.000504487		1.00000i	$0.499839\pi$
$60$	−3.92498	−0.506713
$61$	3.48052	0.445634	0.222817	−	0.974860i	$-0.428475\pi$
$61$	3.48052	0.445634	0.222817	+	0.974860i	$0.428475\pi$
$62$	−10.3305	−1.31197
$63$	−3.65109	−0.459995
$64$	1.00000	0.125000
$65$	0	0
$66$	0.829422	0.102095
$67$	3.84997	0.470348	0.235174	−	0.971953i	$-0.424434\pi$
$67$	3.84997	0.470348	0.235174	+	0.971953i	$0.424434\pi$
$68$	2.75441	0.334021
$69$	−0.405499	−0.0488164
$70$	2.27389	0.271782
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	−2.65109	−0.312434
$73$	−16.6999	−1.95458	−0.977290	−	0.211907i	$-0.932033\pi$
$73$	−16.6999	−1.95458	−0.977290	+	0.211907i	$0.932033\pi$
$74$	5.84997	0.680045
$75$	5.40550	0.624173
$76$	−1.00000	−0.114708
$77$	−0.480515	−0.0547598
$78$	0	0
$79$	7.88601	0.887246	0.443623	−	0.896214i	$-0.353693\pi$
$79$	7.88601	0.887246	0.443623	+	0.896214i	$0.353693\pi$
$80$	1.65109	0.184598
$81$	−9.92498	−1.10278
$82$	2.07502	0.229147
$83$	−13.9738	−1.53383	−0.766913	−	0.641751i	$-0.778208\pi$
$83$	−13.9738	−1.53383	−0.766913	+	0.641751i	$0.778208\pi$
$84$	3.27389	0.357211
$85$	4.54778	0.493276
$86$	−1.75441	−0.189182
$87$	19.4904	2.08959
$88$	−0.348907	−0.0371936
$89$	10.8217	1.14709	0.573547	−	0.819172i	$-0.305567\pi$
$89$	10.8217	1.14709	0.573547	+	0.819172i	$0.305567\pi$
$90$	−4.37720	−0.461398
$91$	0	0
$92$	0.170578	0.0177840
$93$	−24.5577	−2.54651
$94$	0.857718	0.0884669
$95$	−1.65109	−0.169399
$96$	2.37720	0.242622
$97$	3.24559	0.329540	0.164770	−	0.986332i	$-0.447312\pi$
$97$	3.24559	0.329540	0.164770	+	0.986332i	$0.447312\pi$
$98$	5.10331	0.515512
$99$	0.924984	0.0929644
$100$	−2.27389	−0.227389
$101$	−1.34116	−0.133450	−0.0667250	−	0.997771i	$-0.521255\pi$
$101$	−1.34116	−0.133450	−0.0667250	+	0.997771i	$0.521255\pi$
$102$	6.54778	0.648327
$103$	−6.13936	−0.604929	−0.302464	−	0.953161i	$-0.597809\pi$
$103$	−6.13936	−0.604929	−0.302464	+	0.953161i	$0.597809\pi$
$104$	0	0
$105$	5.40550	0.527523
$106$	−10.3305	−1.00339
$107$	−0.0750160	−0.00725207	−0.00362604	−	0.999993i	$-0.501154\pi$
$107$	−0.0750160	−0.00725207	−0.00362604	+	0.999993i	$0.501154\pi$
$108$	0.829422	0.0798111
$109$	11.2632	1.07882	0.539410	−	0.842043i	$-0.318647\pi$
$109$	11.2632	1.07882	0.539410	+	0.842043i	$0.318647\pi$
$110$	−0.576077	−0.0549268
$111$	13.9066	1.31995
$112$	−1.37720	−0.130133
$113$	−1.81100	−0.170364	−0.0851822	−	0.996365i	$-0.527147\pi$
$113$	−1.81100	−0.170364	−0.0851822	+	0.996365i	$0.527147\pi$
$114$	−2.37720	−0.222645
$115$	0.281641	0.0262631
$116$	−8.19887	−0.761246
$117$	0	0
$118$	−0.00775008	−0.000713453 0
$119$	−3.79338	−0.347738
$120$	3.92498	0.358300
$121$	−10.8783	−0.988933
$122$	−3.48052	−0.315111
$123$	4.93273	0.444770
$124$	10.3305	0.927705
$125$	−12.0099	−1.07420
$126$	3.65109	0.325265
$127$	2.87826	0.255405	0.127702	−	0.991813i	$-0.459240\pi$
$127$	2.87826	0.255405	0.127702	+	0.991813i	$0.459240\pi$
$128$	−1.00000	−0.0883883
$129$	−4.17058	−0.367199
$130$	0	0
$131$	−14.1033	−1.23221	−0.616106	−	0.787663i	$-0.711291\pi$
$131$	−14.1033	−1.23221	−0.616106	+	0.787663i	$0.711291\pi$
$132$	−0.829422	−0.0721919
$133$	1.37720	0.119419
$134$	−3.84997	−0.332587
$135$	1.36945	0.117864
$136$	−2.75441	−0.236188
$137$	−3.26614	−0.279045	−0.139523	−	0.990219i	$-0.544557\pi$
$137$	−3.26614	−0.279045	−0.139523	+	0.990219i	$0.544557\pi$
$138$	0.405499	0.0345184
$139$	−4.06727	−0.344981	−0.172490	−	0.985011i	$-0.555181\pi$
$139$	−4.06727	−0.344981	−0.172490	+	0.985011i	$0.555181\pi$
$140$	−2.27389	−0.192179
$141$	2.03897	0.171712
$142$	−3.00000	−0.251754
$143$	0	0
$144$	2.65109	0.220924
$145$	−13.5371	−1.12420
$146$	16.6999	1.38210
$147$	12.1316	1.00060
$148$	−5.84997	−0.480864
$149$	14.1706	1.16090	0.580450	−	0.814296i	$-0.302877\pi$
$149$	14.1706	1.16090	0.580450	+	0.814296i	$0.302877\pi$
$150$	−5.40550	−0.441357
$151$	15.1316	1.23139	0.615696	−	0.787983i	$-0.288875\pi$
$151$	15.1316	1.23139	0.615696	+	0.787983i	$0.288875\pi$
$152$	1.00000	0.0811107
$153$	7.30219	0.590347
$154$	0.480515	0.0387210
$155$	17.0566	1.37002
$156$	0	0
$157$	−10.9533	−0.874167	−0.437083	−	0.899421i	$-0.643989\pi$
$157$	−10.9533	−0.874167	−0.437083	+	0.899421i	$0.643989\pi$
$158$	−7.88601	−0.627378
$159$	−24.5577	−1.94755
$160$	−1.65109	−0.130530
$161$	−0.234921	−0.0185144
$162$	9.92498	0.779780
$163$	2.65109	0.207650	0.103825	−	0.994596i	$-0.466892\pi$
$163$	2.65109	0.207650	0.103825	+	0.994596i	$0.466892\pi$
$164$	−2.07502	−0.162032
$165$	−1.36945	−0.106612
$166$	13.9738	1.08458
$167$	14.7077	1.13811	0.569057	−	0.822298i	$-0.307308\pi$
$167$	14.7077	1.13811	0.569057	+	0.822298i	$0.307308\pi$
$168$	−3.27389	−0.252586
$169$	0	0
$170$	−4.54778	−0.348799
$171$	−2.65109	−0.202734
$172$	1.75441	0.133772
$173$	−6.09264	−0.463215	−0.231607	−	0.972809i	$-0.574398\pi$
$173$	−6.09264	−0.463215	−0.231607	+	0.972809i	$0.574398\pi$
$174$	−19.4904	−1.47756
$175$	3.13161	0.236727
$176$	0.348907	0.0262998
$177$	−0.0184235	−0.00138480
$178$	−10.8217	−0.811119
$179$	−6.18045	−0.461949	−0.230974	−	0.972960i	$-0.574191\pi$
$179$	−6.18045	−0.461949	−0.230974	+	0.972960i	$0.574191\pi$
$180$	4.37720	0.326257
$181$	5.81875	0.432504	0.216252	−	0.976338i	$-0.430617\pi$
$181$	5.81875	0.432504	0.216252	+	0.976338i	$0.430617\pi$
$182$	0	0
$183$	−8.27389	−0.611624
$184$	−0.170578	−0.0125752
$185$	−9.65884	−0.710132
$186$	24.5577	1.80065
$187$	0.961030	0.0702775
$188$	−0.857718	−0.0625555
$189$	−1.14228	−0.0830888
$190$	1.65109	0.119783
$191$	−0.472765	−0.0342081	−0.0171040	−	0.999854i	$-0.505445\pi$
$191$	−0.472765	−0.0342081	−0.0171040	+	0.999854i	$0.505445\pi$
$192$	−2.37720	−0.171560
$193$	8.21942	0.591647	0.295823	−	0.955243i	$-0.404406\pi$
$193$	8.21942	0.591647	0.295823	+	0.955243i	$0.404406\pi$
$194$	−3.24559	−0.233020
$195$	0	0
$196$	−5.10331	−0.364522
$197$	−4.22505	−0.301022	−0.150511	−	0.988608i	$-0.548092\pi$
$197$	−4.22505	−0.301022	−0.150511	+	0.988608i	$0.548092\pi$
$198$	−0.924984	−0.0657357
$199$	24.1444	1.71155	0.855776	−	0.517347i	$-0.173080\pi$
$199$	24.1444	1.71155	0.855776	+	0.517347i	$0.173080\pi$
$200$	2.27389	0.160788
$201$	−9.15215	−0.645543
$202$	1.34116	0.0943634
$203$	11.2915	0.792509
$204$	−6.54778	−0.458436
$205$	−3.42605	−0.239285
$206$	6.13936	0.427749
$207$	0.452219	0.0314314
$208$	0	0
$209$	−0.348907	−0.0241344
$210$	−5.40550	−0.373015
$211$	−26.2165	−1.80482	−0.902409	−	0.430880i	$-0.858203\pi$
$211$	−26.2165	−1.80482	−0.902409	+	0.430880i	$0.858203\pi$
$212$	10.3305	0.709500
$213$	−7.13161	−0.488650
$214$	0.0750160	0.00512799
$215$	2.89669	0.197552
$216$	−0.829422	−0.0564350
$217$	−14.2272	−0.965803
$218$	−11.2632	−0.762841
$219$	39.6991	2.68262
$220$	0.576077	0.0388391
$221$	0	0
$222$	−13.9066	−0.933347
$223$	4.79338	0.320988	0.160494	−	0.987037i	$-0.448691\pi$
$223$	4.79338	0.320988	0.160494	+	0.987037i	$0.448691\pi$
$224$	1.37720	0.0920182
$225$	−6.02830	−0.401886
$226$	1.81100	0.120466
$227$	2.25334	0.149560	0.0747799	−	0.997200i	$-0.476175\pi$
$227$	2.25334	0.149560	0.0747799	+	0.997200i	$0.476175\pi$
$228$	2.37720	0.157434
$229$	16.5860	1.09603	0.548015	−	0.836468i	$-0.315384\pi$
$229$	16.5860	1.09603	0.548015	+	0.836468i	$0.315384\pi$
$230$	−0.281641	−0.0185708
$231$	1.14228	0.0751566
$232$	8.19887	0.538282
$233$	1.23492	0.0809024	0.0404512	−	0.999182i	$-0.487120\pi$
$233$	1.23492	0.0809024	0.0404512	+	0.999182i	$0.487120\pi$
$234$	0	0
$235$	−1.41617	−0.0923809
$236$	0.00775008	0.000504487 0
$237$	−18.7467	−1.21773
$238$	3.79338	0.245888
$239$	28.9837	1.87480	0.937400	−	0.348255i	$-0.113226\pi$
$239$	28.9837	1.87480	0.937400	+	0.348255i	$0.113226\pi$
$240$	−3.92498	−0.253357
$241$	−3.89669	−0.251008	−0.125504	−	0.992093i	$-0.540055\pi$
$241$	−3.89669	−0.251008	−0.125504	+	0.992093i	$0.540055\pi$
$242$	10.8783	0.699281
$243$	21.1054	1.35391
$244$	3.48052	0.222817
$245$	−8.42605	−0.538320
$246$	−4.93273	−0.314500
$247$	0	0
$248$	−10.3305	−0.655986
$249$	33.2186	2.10514
$250$	12.0099	0.759571
$251$	−2.82942	−0.178592	−0.0892958	−	0.996005i	$-0.528462\pi$
$251$	−2.82942	−0.178592	−0.0892958	+	0.996005i	$0.528462\pi$
$252$	−3.65109	−0.229997
$253$	0.0595159	0.00374173
$254$	−2.87826	−0.180598
$255$	−10.8110	−0.677011
$256$	1.00000	0.0625000
$257$	−15.3871	−0.959819	−0.479910	−	0.877318i	$-0.659331\pi$
$257$	−15.3871	−0.959819	−0.479910	+	0.877318i	$0.659331\pi$
$258$	4.17058	0.259649
$259$	8.05659	0.500612
$260$	0	0
$261$	−21.7360	−1.34542
$262$	14.1033	0.871306
$263$	−6.46289	−0.398519	−0.199260	−	0.979947i	$-0.563854\pi$
$263$	−6.46289	−0.398519	−0.199260	+	0.979947i	$0.563854\pi$
$264$	0.829422	0.0510474
$265$	17.0566	1.04778
$266$	−1.37720	−0.0844417
$267$	−25.7253	−1.57436
$268$	3.84997	0.235174
$269$	16.8705	1.02861	0.514307	−	0.857606i	$-0.328049\pi$
$269$	16.8705	1.02861	0.514307	+	0.857606i	$0.328049\pi$
$270$	−1.36945	−0.0833422
$271$	4.21437	0.256005	0.128003	−	0.991774i	$-0.459143\pi$
$271$	4.21437	0.256005	0.128003	+	0.991774i	$0.459143\pi$
$272$	2.75441	0.167010
$273$	0	0
$274$	3.26614	0.197315
$275$	−0.793375	−0.0478423
$276$	−0.405499	−0.0244082
$277$	−23.9816	−1.44091	−0.720457	−	0.693500i	$-0.756068\pi$
$277$	−23.9816	−1.44091	−0.720457	+	0.693500i	$0.756068\pi$
$278$	4.06727	0.243938
$279$	27.3871	1.63962
$280$	2.27389	0.135891
$281$	−22.0643	−1.31625	−0.658124	−	0.752909i	$-0.728650\pi$
$281$	−22.0643	−1.31625	−0.658124	+	0.752909i	$0.728650\pi$
$282$	−2.03897	−0.121419
$283$	13.7926	0.819883	0.409942	−	0.912112i	$-0.365549\pi$
$283$	13.7926	0.819883	0.409942	+	0.912112i	$0.365549\pi$
$284$	3.00000	0.178017
$285$	3.92498	0.232496
$286$	0	0
$287$	2.85772	0.168686
$288$	−2.65109	−0.156217
$289$	−9.41325	−0.553721
$290$	13.5371	0.794926
$291$	−7.71544	−0.452287
$292$	−16.6999	−0.977290
$293$	20.5032	1.19781	0.598904	−	0.800821i	$-0.295603\pi$
$293$	20.5032	1.19781	0.598904	+	0.800821i	$0.295603\pi$
$294$	−12.1316	−0.707530
$295$	0.0127961	0.000745018 0
$296$	5.84997	0.340022
$297$	0.289391	0.0167922
$298$	−14.1706	−0.820880
$299$	0	0
$300$	5.40550	0.312087
$301$	−2.41617	−0.139266
$302$	−15.1316	−0.870726
$303$	3.18820	0.183157
$304$	−1.00000	−0.0573539
$305$	5.74666	0.329053
$306$	−7.30219	−0.417438
$307$	3.81100	0.217505	0.108753	−	0.994069i	$-0.465314\pi$
$307$	3.81100	0.217505	0.108753	+	0.994069i	$0.465314\pi$
$308$	−0.480515	−0.0273799
$309$	14.5945	0.830252
$310$	−17.0566	−0.968749
$311$	−2.43380	−0.138008	−0.0690039	−	0.997616i	$-0.521982\pi$
$311$	−2.43380	−0.138008	−0.0690039	+	0.997616i	$0.521982\pi$
$312$	0	0
$313$	−25.1882	−1.42372	−0.711861	−	0.702321i	$-0.752147\pi$
$313$	−25.1882	−1.42372	−0.711861	+	0.702321i	$0.752147\pi$
$314$	10.9533	0.618129
$315$	−6.02830	−0.339656
$316$	7.88601	0.443623
$317$	−17.5139	−0.983676	−0.491838	−	0.870687i	$-0.663675\pi$
$317$	−17.5139	−0.983676	−0.491838	+	0.870687i	$0.663675\pi$
$318$	24.5577	1.37712
$319$	−2.86064	−0.160165
$320$	1.65109	0.0922989
$321$	0.178328	0.00995332
$322$	0.234921	0.0130916
$323$	−2.75441	−0.153259
$324$	−9.92498	−0.551388
$325$	0	0
$326$	−2.65109	−0.146830
$327$	−26.7750	−1.48066
$328$	2.07502	0.114574
$329$	1.18125	0.0651245
$330$	1.36945	0.0753859
$331$	15.0099	0.825017	0.412509	−	0.910954i	$-0.364653\pi$
$331$	15.0099	0.825017	0.412509	+	0.910954i	$0.364653\pi$
$332$	−13.9738	−0.766913
$333$	−15.5088	−0.849878
$334$	−14.7077	−0.804769
$335$	6.35666	0.347301
$336$	3.27389	0.178605
$337$	−5.67939	−0.309376	−0.154688	−	0.987963i	$-0.549437\pi$
$337$	−5.67939	−0.309376	−0.154688	+	0.987963i	$0.549437\pi$
$338$	0	0
$339$	4.30511	0.233821
$340$	4.54778	0.246638
$341$	3.60437	0.195188
$342$	2.65109	0.143355
$343$	16.6687	0.900026
$344$	−1.75441	−0.0945912
$345$	−0.669517	−0.0360456
$346$	6.09264	0.327542
$347$	−28.0304	−1.50475	−0.752376	−	0.658734i	$-0.771092\pi$
$347$	−28.0304	−1.50475	−0.752376	+	0.658734i	$0.771092\pi$
$348$	19.4904	1.04479
$349$	−34.4252	−1.84274	−0.921371	−	0.388685i	$-0.872929\pi$
$349$	−34.4252	−1.84274	−0.921371	+	0.388685i	$0.872929\pi$
$350$	−3.13161	−0.167392
$351$	0	0
$352$	−0.348907	−0.0185968
$353$	4.03817	0.214930	0.107465	−	0.994209i	$-0.465727\pi$
$353$	4.03817	0.214930	0.107465	+	0.994209i	$0.465727\pi$
$354$	0.0184235	0.000979199 0
$355$	4.95328	0.262893
$356$	10.8217	0.573547
$357$	9.01762	0.477263
$358$	6.18045	0.326647
$359$	8.80113	0.464506	0.232253	−	0.972655i	$-0.425390\pi$
$359$	8.80113	0.464506	0.232253	+	0.972655i	$0.425390\pi$
$360$	−4.37720	−0.230699
$361$	1.00000	0.0526316
$362$	−5.81875	−0.305827
$363$	25.8598	1.35729
$364$	0	0
$365$	−27.5732	−1.44324
$366$	8.27389	0.432483
$367$	−20.9610	−1.09416	−0.547078	−	0.837081i	$-0.684260\pi$
$367$	−20.9610	−1.09416	−0.547078	+	0.837081i	$0.684260\pi$
$368$	0.170578	0.00889201
$369$	−5.50106	−0.286374
$370$	9.65884	0.502139
$371$	−14.2272	−0.738638
$372$	−24.5577	−1.27326
$373$	−21.2186	−1.09866	−0.549329	−	0.835606i	$-0.685117\pi$
$373$	−21.2186	−1.09866	−0.549329	+	0.835606i	$0.685117\pi$
$374$	−0.961030	−0.0496937
$375$	28.5499	1.47431
$376$	0.857718	0.0442334
$377$	0	0
$378$	1.14228	0.0587526
$379$	2.32544	0.119450	0.0597248	−	0.998215i	$-0.480978\pi$
$379$	2.32544	0.119450	0.0597248	+	0.998215i	$0.480978\pi$
$380$	−1.65109	−0.0846993
$381$	−6.84222	−0.350537
$382$	0.472765	0.0241888
$383$	−36.5547	−1.86786	−0.933930	−	0.357457i	$-0.883644\pi$
$383$	−36.5547	−1.86786	−0.933930	+	0.357457i	$0.883644\pi$
$384$	2.37720	0.121311
$385$	−0.793375	−0.0404342
$386$	−8.21942	−0.418357
$387$	4.65109	0.236428
$388$	3.24559	0.164770
$389$	19.1628	0.971594	0.485797	−	0.874072i	$-0.338529\pi$
$389$	19.1628	0.971594	0.485797	+	0.874072i	$0.338529\pi$
$390$	0	0
$391$	0.469842	0.0237609
$392$	5.10331	0.257756
$393$	33.5264	1.69118
$394$	4.22505	0.212855
$395$	13.0205	0.655135
$396$	0.924984	0.0464822
$397$	−20.9298	−1.05044	−0.525219	−	0.850967i	$-0.676016\pi$
$397$	−20.9298	−1.05044	−0.525219	+	0.850967i	$0.676016\pi$
$398$	−24.1444	−1.21025
$399$	−3.27389	−0.163900
$400$	−2.27389	−0.113695
$401$	−11.2066	−0.559632	−0.279816	−	0.960054i	$-0.590273\pi$
$401$	−11.2066	−0.559632	−0.279816	+	0.960054i	$0.590273\pi$
$402$	9.15215	0.456468
$403$	0	0
$404$	−1.34116	−0.0667250
$405$	−16.3871	−0.814280
$406$	−11.2915	−0.560388
$407$	−2.04109	−0.101173
$408$	6.54778	0.324163
$409$	−29.5315	−1.46024	−0.730119	−	0.683320i	$-0.760535\pi$
$409$	−29.5315	−1.46024	−0.730119	+	0.683320i	$0.760535\pi$
$410$	3.42605	0.169200
$411$	7.76428	0.382984
$412$	−6.13936	−0.302464
$413$	−0.0106734	−0.000525205 0
$414$	−0.452219	−0.0222253
$415$	−23.0721	−1.13256
$416$	0	0
$417$	9.66872	0.473479
$418$	0.348907	0.0170656
$419$	11.7176	0.572440	0.286220	−	0.958164i	$-0.407601\pi$
$419$	11.7176	0.572440	0.286220	+	0.958164i	$0.407601\pi$
$420$	5.40550	0.263761
$421$	−15.7184	−0.766066	−0.383033	−	0.923735i	$-0.625120\pi$
$421$	−15.7184	−0.766066	−0.383033	+	0.923735i	$0.625120\pi$
$422$	26.2165	1.27620
$423$	−2.27389	−0.110560
$424$	−10.3305	−0.501693
$425$	−6.26322	−0.303811
$426$	7.13161	0.345528
$427$	−4.79338	−0.231968
$428$	−0.0750160	−0.00362604
$429$	0	0
$430$	−2.89669	−0.139691
$431$	8.21942	0.395916	0.197958	−	0.980211i	$-0.436569\pi$
$431$	8.21942	0.395916	0.197958	+	0.980211i	$0.436569\pi$
$432$	0.829422	0.0399056
$433$	−26.8187	−1.28883	−0.644413	−	0.764677i	$-0.722898\pi$
$433$	−26.8187	−1.28883	−0.644413	+	0.764677i	$0.722898\pi$
$434$	14.2272	0.682926
$435$	32.1805	1.54293
$436$	11.2632	0.539410
$437$	−0.170578	−0.00815986
$438$	−39.6991	−1.89690
$439$	1.12094	0.0534993	0.0267497	−	0.999642i	$-0.491484\pi$
$439$	1.12094	0.0534993	0.0267497	+	0.999642i	$0.491484\pi$
$440$	−0.576077	−0.0274634
$441$	−13.5294	−0.644255
$442$	0	0
$443$	−38.0459	−1.80762	−0.903808	−	0.427938i	$-0.859240\pi$
$443$	−38.0459	−1.80762	−0.903808	+	0.427938i	$0.859240\pi$
$444$	13.9066	0.659976
$445$	17.8676	0.847005
$446$	−4.79338	−0.226973
$447$	−33.6863	−1.59331
$448$	−1.37720	−0.0650667
$449$	−27.3969	−1.29294	−0.646471	−	0.762939i	$-0.723756\pi$
$449$	−27.3969	−1.29294	−0.646471	+	0.762939i	$0.723756\pi$
$450$	6.02830	0.284177
$451$	−0.723987	−0.0340912
$452$	−1.81100	−0.0851822
$453$	−35.9709	−1.69006
$454$	−2.25334	−0.105755
$455$	0	0
$456$	−2.37720	−0.111323
$457$	39.9525	1.86890	0.934449	−	0.356097i	$-0.115893\pi$
$457$	39.9525	1.86890	0.934449	+	0.356097i	$0.115893\pi$
$458$	−16.5860	−0.775011
$459$	2.28456	0.106634
$460$	0.281641	0.0131316
$461$	−32.1650	−1.49807	−0.749036	−	0.662530i	$-0.769483\pi$
$461$	−32.1650	−1.49807	−0.749036	+	0.662530i	$0.769483\pi$
$462$	−1.14228	−0.0531438
$463$	26.5860	1.23555	0.617777	−	0.786353i	$-0.288033\pi$
$463$	26.5860	1.23555	0.617777	+	0.786353i	$0.288033\pi$
$464$	−8.19887	−0.380623
$465$	−40.5470	−1.88032
$466$	−1.23492	−0.0572066
$467$	13.4776	0.623669	0.311834	−	0.950137i	$-0.399057\pi$
$467$	13.4776	0.623669	0.311834	+	0.950137i	$0.399057\pi$
$468$	0	0
$469$	−5.30219	−0.244832
$470$	1.41617	0.0653232
$471$	26.0382	1.19978
$472$	−0.00775008	−0.000356726 0
$473$	0.612124	0.0281455
$474$	18.7467	0.861062
$475$	2.27389	0.104333
$476$	−3.79338	−0.173869
$477$	27.3871	1.25397
$478$	−28.9837	−1.32568
$479$	3.94341	0.180179	0.0900894	−	0.995934i	$-0.471285\pi$
$479$	3.94341	0.180179	0.0900894	+	0.995934i	$0.471285\pi$
$480$	3.92498	0.179150
$481$	0	0
$482$	3.89669	0.177489
$483$	0.558455	0.0254106
$484$	−10.8783	−0.494467
$485$	5.35878	0.243330
$486$	−21.1054	−0.957362
$487$	1.13453	0.0514105	0.0257053	−	0.999670i	$-0.491817\pi$
$487$	1.13453	0.0514105	0.0257053	+	0.999670i	$0.491817\pi$
$488$	−3.48052	−0.157556
$489$	−6.30219	−0.284995
$490$	8.42605	0.380650
$491$	30.5315	1.37787	0.688933	−	0.724825i	$-0.258079\pi$
$491$	30.5315	1.37787	0.688933	+	0.724825i	$0.258079\pi$
$492$	4.93273	0.222385
$493$	−22.5830	−1.01709
$494$	0	0
$495$	1.52723	0.0686441
$496$	10.3305	0.463852
$497$	−4.13161	−0.185328
$498$	−33.2186	−1.48856
$499$	1.75441	0.0785380	0.0392690	−	0.999229i	$-0.487497\pi$
$499$	1.75441	0.0785380	0.0392690	+	0.999229i	$0.487497\pi$
$500$	−12.0099	−0.537098
$501$	−34.9632	−1.56204
$502$	2.82942	0.126283
$503$	−41.1874	−1.83646	−0.918228	−	0.396053i	$-0.870380\pi$
$503$	−41.1874	−1.83646	−0.918228	+	0.396053i	$0.870380\pi$
$504$	3.65109	0.162633
$505$	−2.21437	−0.0985384
$506$	−0.0595159	−0.00264580
$507$	0	0
$508$	2.87826	0.127702
$509$	29.1570	1.29236	0.646180	−	0.763185i	$-0.276365\pi$
$509$	29.1570	1.29236	0.646180	+	0.763185i	$0.276365\pi$
$510$	10.8110	0.478719
$511$	22.9992	1.01742
$512$	−1.00000	−0.0441942
$513$	−0.829422	−0.0366199
$514$	15.3871	0.678695
$515$	−10.1367	−0.446674
$516$	−4.17058	−0.183599
$517$	−0.299263	−0.0131616
$518$	−8.05659	−0.353986
$519$	14.4834	0.635752
$520$	0	0
$521$	−10.5761	−0.463346	−0.231673	−	0.972794i	$-0.574420\pi$
$521$	−10.5761	−0.463346	−0.231673	+	0.972794i	$0.574420\pi$
$522$	21.7360	0.951358
$523$	−39.0558	−1.70779	−0.853895	−	0.520445i	$-0.825766\pi$
$523$	−39.0558	−1.70779	−0.853895	+	0.520445i	$0.825766\pi$
$524$	−14.1033	−0.616106
$525$	−7.44447	−0.324903
$526$	6.46289	0.281795
$527$	28.4543	1.23949
$528$	−0.829422	−0.0360959
$529$	−22.9709	−0.998735
$530$	−17.0566	−0.740891
$531$	0.0205462	0.000891628 0
$532$	1.37720	0.0597093
$533$	0	0
$534$	25.7253	1.11324
$535$	−0.123858	−0.00535487
$536$	−3.84997	−0.166293
$537$	14.6922	0.634015
$538$	−16.8705	−0.727340
$539$	−1.78058	−0.0766950
$540$	1.36945	0.0589319
$541$	−36.1492	−1.55418	−0.777088	−	0.629391i	$-0.783304\pi$
$541$	−36.1492	−1.55418	−0.777088	+	0.629391i	$0.783304\pi$
$542$	−4.21437	−0.181023
$543$	−13.8323	−0.593603
$544$	−2.75441	−0.118094
$545$	18.5966	0.796592
$546$	0	0
$547$	6.19032	0.264679	0.132340	−	0.991204i	$-0.457751\pi$
$547$	6.19032	0.264679	0.132340	+	0.991204i	$0.457751\pi$
$548$	−3.26614	−0.139523
$549$	9.22717	0.393806
$550$	0.793375	0.0338296
$551$	8.19887	0.349284
$552$	0.405499	0.0172592
$553$	−10.8606	−0.461841
$554$	23.9816	1.01888
$555$	22.9610	0.974641
$556$	−4.06727	−0.172490
$557$	−7.22505	−0.306135	−0.153068	−	0.988216i	$-0.548915\pi$
$557$	−7.22505	−0.306135	−0.153068	+	0.988216i	$0.548915\pi$
$558$	−27.3871	−1.15939
$559$	0	0
$560$	−2.27389	−0.0960894
$561$	−2.28456	−0.0964544
$562$	22.0643	0.930728
$563$	−41.3708	−1.74357	−0.871785	−	0.489888i	$-0.837038\pi$
$563$	−41.3708	−1.74357	−0.871785	+	0.489888i	$0.837038\pi$
$564$	2.03897	0.0858561
$565$	−2.99013	−0.125796
$566$	−13.7926	−0.579745
$567$	13.6687	0.574032
$568$	−3.00000	−0.125877
$569$	−28.9504	−1.21366	−0.606831	−	0.794831i	$-0.707560\pi$
$569$	−28.9504	−1.21366	−0.606831	+	0.794831i	$0.707560\pi$
$570$	−3.92498	−0.164399
$571$	−7.80325	−0.326556	−0.163278	−	0.986580i	$-0.552207\pi$
$571$	−7.80325	−0.326556	−0.163278	+	0.986580i	$0.552207\pi$
$572$	0	0
$573$	1.12386	0.0469499
$574$	−2.85772	−0.119279
$575$	−0.387876	−0.0161756
$576$	2.65109	0.110462
$577$	4.29736	0.178901	0.0894507	−	0.995991i	$-0.471489\pi$
$577$	4.29736	0.178901	0.0894507	+	0.995991i	$0.471489\pi$
$578$	9.41325	0.391540
$579$	−19.5392	−0.812023
$580$	−13.5371	−0.562098
$581$	19.2448	0.798409
$582$	7.71544	0.319815
$583$	3.60437	0.149278
$584$	16.6999	0.691048
$585$	0	0
$586$	−20.5032	−0.846979
$587$	−40.9602	−1.69061	−0.845305	−	0.534284i	$-0.820581\pi$
$587$	−40.9602	−1.69061	−0.845305	+	0.534284i	$0.820581\pi$
$588$	12.1316	0.500299
$589$	−10.3305	−0.425660
$590$	−0.0127961	−0.000526807 0
$591$	10.0438	0.413147
$592$	−5.84997	−0.240432
$593$	−23.8655	−0.980037	−0.490019	−	0.871712i	$-0.663010\pi$
$593$	−23.8655	−0.980037	−0.490019	+	0.871712i	$0.663010\pi$
$594$	−0.289391	−0.0118738
$595$	−6.26322	−0.256767
$596$	14.1706	0.580450
$597$	−57.3961	−2.34907
$598$	0	0
$599$	−44.6631	−1.82488	−0.912442	−	0.409206i	$-0.865806\pi$
$599$	−44.6631	−1.82488	−0.912442	+	0.409206i	$0.865806\pi$
$600$	−5.40550	−0.220679
$601$	−40.7480	−1.66214	−0.831072	−	0.556164i	$-0.812273\pi$
$601$	−40.7480	−1.66214	−0.831072	+	0.556164i	$0.812273\pi$
$602$	2.41617	0.0984758
$603$	10.2066	0.415646
$604$	15.1316	0.615696
$605$	−17.9610	−0.730220
$606$	−3.18820	−0.129512
$607$	−28.0283	−1.13763	−0.568817	−	0.822464i	$-0.692599\pi$
$607$	−28.0283	−1.13763	−0.568817	+	0.822464i	$0.692599\pi$
$608$	1.00000	0.0405554
$609$	−26.8422	−1.08770
$610$	−5.74666	−0.232675
$611$	0	0
$612$	7.30219	0.295173
$613$	−37.0013	−1.49447	−0.747235	−	0.664560i	$-0.768619\pi$
$613$	−37.0013	−1.49447	−0.747235	+	0.664560i	$0.768619\pi$
$614$	−3.81100	−0.153799
$615$	8.14440	0.328414
$616$	0.480515	0.0193605
$617$	39.5470	1.59210	0.796051	−	0.605230i	$-0.206919\pi$
$617$	39.5470	1.59210	0.796051	+	0.605230i	$0.206919\pi$
$618$	−14.5945	−0.587077
$619$	37.8422	1.52101	0.760504	−	0.649334i	$-0.224952\pi$
$619$	37.8422	1.52101	0.760504	+	0.649334i	$0.224952\pi$
$620$	17.0566	0.685009
$621$	0.141481	0.00567745
$622$	2.43380	0.0975863
$623$	−14.9036	−0.597102
$624$	0	0
$625$	−8.45997	−0.338399
$626$	25.1882	1.00672
$627$	0.829422	0.0331239
$628$	−10.9533	−0.437083
$629$	−16.1132	−0.642475
$630$	6.02830	0.240173
$631$	−25.2066	−1.00346	−0.501730	−	0.865024i	$-0.667303\pi$
$631$	−25.2066	−1.00346	−0.501730	+	0.865024i	$0.667303\pi$
$632$	−7.88601	−0.313689
$633$	62.3219	2.47707
$634$	17.5139	0.695564
$635$	4.75228	0.188589
$636$	−24.5577	−0.973774
$637$	0	0
$638$	2.86064	0.113254
$639$	7.95328	0.314627
$640$	−1.65109	−0.0652652
$641$	−16.7184	−0.660335	−0.330168	−	0.943922i	$-0.607105\pi$
$641$	−16.7184	−0.660335	−0.330168	+	0.943922i	$0.607105\pi$
$642$	−0.178328	−0.00703806
$643$	−21.4629	−0.846414	−0.423207	−	0.906033i	$-0.639096\pi$
$643$	−21.4629	−0.846414	−0.423207	+	0.906033i	$0.639096\pi$
$644$	−0.234921	−0.00925718
$645$	−6.88601	−0.271137
$646$	2.75441	0.108371
$647$	31.9370	1.25557	0.627786	−	0.778386i	$-0.283961\pi$
$647$	31.9370	1.25557	0.627786	+	0.778386i	$0.283961\pi$
$648$	9.92498	0.389890
$649$	0.00270405	0.000106143 0
$650$	0	0
$651$	33.8209	1.32554
$652$	2.65109	0.103825
$653$	35.5315	1.39045	0.695227	−	0.718790i	$-0.255304\pi$
$653$	35.5315	1.39045	0.695227	+	0.718790i	$0.255304\pi$
$654$	26.7750	1.04698
$655$	−23.2859	−0.909855
$656$	−2.07502	−0.0810158
$657$	−44.2731	−1.72726
$658$	−1.18125	−0.0460500
$659$	−23.1981	−0.903669	−0.451834	−	0.892102i	$-0.649230\pi$
$659$	−23.1981	−0.903669	−0.451834	+	0.892102i	$0.649230\pi$
$660$	−1.36945	−0.0533059
$661$	21.1436	0.822391	0.411195	−	0.911547i	$-0.365111\pi$
$661$	21.1436	0.822391	0.411195	+	0.911547i	$0.365111\pi$
$662$	−15.0099	−0.583375
$663$	0	0
$664$	13.9738	0.542290
$665$	2.27389	0.0881777
$666$	15.5088	0.600954
$667$	−1.39855	−0.0541521
$668$	14.7077	0.569057
$669$	−11.3948	−0.440549
$670$	−6.35666	−0.245579
$671$	1.21437	0.0468804
$672$	−3.27389	−0.126293
$673$	34.5499	1.33180	0.665900	−	0.746041i	$-0.268048\pi$
$673$	34.5499	1.33180	0.665900	+	0.746041i	$0.268048\pi$
$674$	5.67939	0.218762
$675$	−1.88601	−0.0725927
$676$	0	0
$677$	19.3043	0.741925	0.370962	−	0.928648i	$-0.379028\pi$
$677$	19.3043	0.741925	0.370962	+	0.928648i	$0.379028\pi$
$678$	−4.30511	−0.165337
$679$	−4.46984	−0.171537
$680$	−4.54778	−0.174399
$681$	−5.35666	−0.205268
$682$	−3.60437	−0.138019
$683$	37.4338	1.43236	0.716182	−	0.697913i	$-0.245888\pi$
$683$	37.4338	1.43236	0.716182	+	0.697913i	$0.245888\pi$
$684$	−2.65109	−0.101367
$685$	−5.39270	−0.206045
$686$	−16.6687	−0.636415
$687$	−39.4282	−1.50428
$688$	1.75441	0.0668861
$689$	0	0
$690$	0.669517	0.0254881
$691$	1.77495	0.0675224	0.0337612	−	0.999430i	$-0.489251\pi$
$691$	1.77495	0.0675224	0.0337612	+	0.999430i	$0.489251\pi$
$692$	−6.09264	−0.231607
$693$	−1.27389	−0.0483911
$694$	28.0304	1.06402
$695$	−6.71544	−0.254731
$696$	−19.4904	−0.738781
$697$	−5.71544	−0.216488
$698$	34.4252	1.30301
$699$	−2.93566	−0.111037
$700$	3.13161	0.118364
$701$	41.5547	1.56950	0.784750	−	0.619812i	$-0.212791\pi$
$701$	41.5547	1.56950	0.784750	+	0.619812i	$0.212791\pi$
$702$	0	0
$703$	5.84997	0.220636
$704$	0.348907	0.0131499
$705$	3.36653	0.126791
$706$	−4.03817	−0.151978
$707$	1.84704	0.0694653
$708$	−0.0184235	−0.000692398 0
$709$	16.1444	0.606316	0.303158	−	0.952940i	$-0.401959\pi$
$709$	16.1444	0.606316	0.303158	+	0.952940i	$0.401959\pi$
$710$	−4.95328	−0.185893
$711$	20.9066	0.784057
$712$	−10.8217	−0.405559
$713$	1.76216	0.0659933
$714$	−9.01762	−0.337476
$715$	0	0
$716$	−6.18045	−0.230974
$717$	−68.9001	−2.57312
$718$	−8.80113	−0.328455
$719$	−19.1962	−0.715896	−0.357948	−	0.933741i	$-0.616524\pi$
$719$	−19.1962	−0.715896	−0.357948	+	0.933741i	$0.616524\pi$
$720$	4.37720	0.163129
$721$	8.45514	0.314886
$722$	−1.00000	−0.0372161
$723$	9.26322	0.344503
$724$	5.81875	0.216252
$725$	18.6433	0.692396
$726$	−25.8598	−0.959749
$727$	−10.6540	−0.395136	−0.197568	−	0.980289i	$-0.563304\pi$
$727$	−10.6540	−0.395136	−0.197568	+	0.980289i	$0.563304\pi$
$728$	0	0
$729$	−20.3969	−0.755443
$730$	27.5732	1.02053
$731$	4.83235	0.178731
$732$	−8.27389	−0.305812
$733$	−50.6220	−1.86977	−0.934883	−	0.354956i	$-0.884496\pi$
$733$	−50.6220	−1.86977	−0.934883	+	0.354956i	$0.884496\pi$
$734$	20.9610	0.773686
$735$	20.0304	0.738833
$736$	−0.170578	−0.00628760
$737$	1.34328	0.0494803
$738$	5.50106	0.202497
$739$	−2.94823	−0.108453	−0.0542263	−	0.998529i	$-0.517269\pi$
$739$	−2.94823	−0.108453	−0.0542263	+	0.998529i	$0.517269\pi$
$740$	−9.65884	−0.355066
$741$	0	0
$742$	14.2272	0.522296
$743$	−18.4132	−0.675517	−0.337758	−	0.941233i	$-0.609669\pi$
$743$	−18.4132	−0.675517	−0.337758	+	0.941233i	$0.609669\pi$
$744$	24.5577	0.900327
$745$	23.3969	0.857198
$746$	21.2186	0.776869
$747$	−37.0459	−1.35544
$748$	0.961030	0.0351387
$749$	0.103312	0.00377495
$750$	−28.5499	−1.04250
$751$	−20.3927	−0.744140	−0.372070	−	0.928205i	$-0.621352\pi$
$751$	−20.3927	−0.744140	−0.372070	+	0.928205i	$0.621352\pi$
$752$	−0.857718	−0.0312778
$753$	6.72611	0.245113
$754$	0	0
$755$	24.9837	0.909250
$756$	−1.14228	−0.0415444
$757$	−25.7261	−0.935031	−0.467516	−	0.883985i	$-0.654851\pi$
$757$	−25.7261	−0.935031	−0.467516	+	0.883985i	$0.654851\pi$
$758$	−2.32544	−0.0844637
$759$	−0.141481	−0.00513545
$760$	1.65109	0.0598915
$761$	−8.84917	−0.320782	−0.160391	−	0.987054i	$-0.551276\pi$
$761$	−8.84917	−0.320782	−0.160391	+	0.987054i	$0.551276\pi$
$762$	6.84222	0.247867
$763$	−15.5117	−0.561563
$764$	−0.472765	−0.0171040
$765$	12.0566	0.435907
$766$	36.5547	1.32078
$767$	0	0
$768$	−2.37720	−0.0857799
$769$	51.2773	1.84911	0.924554	−	0.381051i	$-0.124438\pi$
$769$	51.2773	1.84911	0.924554	+	0.381051i	$0.124438\pi$
$770$	0.793375	0.0285913
$771$	36.5782	1.31733
$772$	8.21942	0.295823
$773$	−28.0742	−1.00976	−0.504880	−	0.863190i	$-0.668463\pi$
$773$	−28.0742	−1.00976	−0.504880	+	0.863190i	$0.668463\pi$
$774$	−4.65109	−0.167180
$775$	−23.4904	−0.843800
$776$	−3.24559	−0.116510
$777$	−19.1522	−0.687080
$778$	−19.1628	−0.687021
$779$	2.07502	0.0743452
$780$	0	0
$781$	1.04672	0.0374546
$782$	−0.469842	−0.0168015
$783$	−6.80032	−0.243024
$784$	−5.10331	−0.182261
$785$	−18.0849	−0.645477
$786$	−33.5264	−1.19585
$787$	−17.7312	−0.632047	−0.316024	−	0.948751i	$-0.602348\pi$
$787$	−17.7312	−0.632047	−0.316024	+	0.948751i	$0.602348\pi$
$788$	−4.22505	−0.150511
$789$	15.3636	0.546959
$790$	−13.0205	−0.463250
$791$	2.49411	0.0886804
$792$	−0.924984	−0.0328679
$793$	0	0
$794$	20.9298	0.742771
$795$	−40.5470	−1.43805
$796$	24.1444	0.855776
$797$	44.2079	1.56593	0.782963	−	0.622068i	$-0.213707\pi$
$797$	44.2079	1.56593	0.782963	+	0.622068i	$0.213707\pi$
$798$	3.27389	0.115894
$799$	−2.36250	−0.0835794
$800$	2.27389	0.0803942
$801$	28.6893	1.01369
$802$	11.2066	0.395720
$803$	−5.82672	−0.205620
$804$	−9.15215	−0.322772
$805$	−0.387876	−0.0136708
$806$	0	0
$807$	−40.1046	−1.41175
$808$	1.34116	0.0471817
$809$	15.9533	0.560887	0.280444	−	0.959870i	$-0.409518\pi$
$809$	15.9533	0.560887	0.280444	+	0.959870i	$0.409518\pi$
$810$	16.3871	0.575783
$811$	−23.2349	−0.815888	−0.407944	−	0.913007i	$-0.633754\pi$
$811$	−23.2349	−0.815888	−0.407944	+	0.913007i	$0.633754\pi$
$812$	11.2915	0.396254
$813$	−10.0184	−0.351361
$814$	2.04109	0.0715403
$815$	4.37720	0.153327
$816$	−6.54778	−0.229218
$817$	−1.75441	−0.0613789
$818$	29.5315	1.03254
$819$	0	0
$820$	−3.42605	−0.119643
$821$	8.48556	0.296148	0.148074	−	0.988976i	$-0.452693\pi$
$821$	8.48556	0.296148	0.148074	+	0.988976i	$0.452693\pi$
$822$	−7.76428	−0.270810
$823$	−40.3588	−1.40682	−0.703409	−	0.710785i	$-0.748340\pi$
$823$	−40.3588	−1.40682	−0.703409	+	0.710785i	$0.748340\pi$
$824$	6.13936	0.213875
$825$	1.88601	0.0656626
$826$	0.0106734	0.000371376 0
$827$	30.3227	1.05442	0.527212	−	0.849734i	$-0.323237\pi$
$827$	30.3227	1.05442	0.527212	+	0.849734i	$0.323237\pi$
$828$	0.452219	0.0157157
$829$	−12.9143	−0.448533	−0.224266	−	0.974528i	$-0.571999\pi$
$829$	−12.9143	−0.448533	−0.224266	+	0.974528i	$0.571999\pi$
$830$	23.0721	0.800844
$831$	57.0091	1.97762
$832$	0	0
$833$	−14.0566	−0.487032
$834$	−9.66872	−0.334800
$835$	24.2838	0.840374
$836$	−0.348907	−0.0120672
$837$	8.56833	0.296165
$838$	−11.7176	−0.404776
$839$	−13.7029	−0.473075	−0.236538	−	0.971622i	$-0.576013\pi$
$839$	−13.7029	−0.473075	−0.236538	+	0.971622i	$0.576013\pi$
$840$	−5.40550	−0.186507
$841$	38.2215	1.31798
$842$	15.7184	0.541690
$843$	52.4514	1.80652
$844$	−26.2165	−0.902409
$845$	0	0
$846$	2.27389	0.0781780
$847$	14.9816	0.514773
$848$	10.3305	0.354750
$849$	−32.7877	−1.12527
$850$	6.26322	0.214827
$851$	−0.997877	−0.0342068
$852$	−7.13161	−0.244325
$853$	−48.1174	−1.64751	−0.823755	−	0.566946i	$-0.808125\pi$
$853$	−48.1174	−1.64751	−0.823755	+	0.566946i	$0.808125\pi$
$854$	4.79338	0.164026
$855$	−4.37720	−0.149697
$856$	0.0750160	0.00256400
$857$	37.6941	1.28761	0.643803	−	0.765191i	$-0.277356\pi$
$857$	37.6941	1.28761	0.643803	+	0.765191i	$0.277356\pi$
$858$	0	0
$859$	−34.1153	−1.16400	−0.582000	−	0.813189i	$-0.697729\pi$
$859$	−34.1153	−1.16400	−0.582000	+	0.813189i	$0.697729\pi$
$860$	2.89669	0.0987762
$861$	−6.79338	−0.231518
$862$	−8.21942	−0.279955
$863$	0.518684	0.0176562	0.00882811	−	0.999961i	$-0.497190\pi$
$863$	0.518684	0.0176562	0.00882811	+	0.999961i	$0.497190\pi$
$864$	−0.829422	−0.0282175
$865$	−10.0595	−0.342034
$866$	26.8187	0.911338
$867$	22.3772	0.759970
$868$	−14.2272	−0.482902
$869$	2.75148	0.0933376
$870$	−32.1805	−1.09102
$871$	0	0
$872$	−11.2632	−0.381421
$873$	8.60437	0.291214
$874$	0.170578	0.00576990
$875$	16.5400	0.559155
$876$	39.6991	1.34131
$877$	−39.7798	−1.34327	−0.671634	−	0.740883i	$-0.734407\pi$
$877$	−39.7798	−1.34327	−0.671634	+	0.740883i	$0.734407\pi$
$878$	−1.12094	−0.0378297
$879$	−48.7402	−1.64397
$880$	0.576077	0.0194196
$881$	17.6842	0.595796	0.297898	−	0.954598i	$-0.403714\pi$
$881$	17.6842	0.595796	0.297898	+	0.954598i	$0.403714\pi$
$882$	13.5294	0.455557
$883$	−8.21942	−0.276606	−0.138303	−	0.990390i	$-0.544165\pi$
$883$	−8.21942	−0.276606	−0.138303	+	0.990390i	$0.544165\pi$
$884$	0	0
$885$	−0.0304189	−0.00102252
$886$	38.0459	1.27818
$887$	22.0828	0.741467	0.370733	−	0.928739i	$-0.379106\pi$
$887$	22.0828	0.741467	0.370733	+	0.928739i	$0.379106\pi$
$888$	−13.9066	−0.466674
$889$	−3.96395	−0.132947
$890$	−17.8676	−0.598923
$891$	−3.46289	−0.116011
$892$	4.79338	0.160494
$893$	0.857718	0.0287024
$894$	33.6863	1.12664
$895$	−10.2045	−0.341099
$896$	1.37720	0.0460091
$897$	0	0
$898$	27.3969	0.914248
$899$	−84.6983	−2.82485
$900$	−6.02830	−0.200943
$901$	28.4543	0.947952
$902$	0.723987	0.0241061
$903$	5.74373	0.191139
$904$	1.81100	0.0602329
$905$	9.60730	0.319357
$906$	35.9709	1.19505
$907$	−35.5109	−1.17912	−0.589561	−	0.807724i	$-0.700699\pi$
$907$	−35.5109	−1.17912	−0.589561	+	0.807724i	$0.700699\pi$
$908$	2.25334	0.0747799
$909$	−3.55553	−0.117930
$910$	0	0
$911$	55.1690	1.82783	0.913915	−	0.405906i	$-0.133044\pi$
$911$	55.1690	1.82783	0.913915	+	0.405906i	$0.133044\pi$
$912$	2.37720	0.0787171
$913$	−4.87556	−0.161357
$914$	−39.9525	−1.32151
$915$	−13.6610	−0.451618
$916$	16.5860	0.548015
$917$	19.4231	0.641408
$918$	−2.28456	−0.0754018
$919$	−30.6278	−1.01032	−0.505160	−	0.863026i	$-0.668566\pi$
$919$	−30.6278	−1.01032	−0.505160	+	0.863026i	$0.668566\pi$
$920$	−0.281641	−0.00928542
$921$	−9.05952	−0.298521
$922$	32.1650	1.05930
$923$	0	0
$924$	1.14228	0.0375783
$925$	13.3022	0.437373
$926$	−26.5860	−0.873669
$927$	−16.2760	−0.534574
$928$	8.19887	0.269141
$929$	6.64605	0.218050	0.109025	−	0.994039i	$-0.465227\pi$
$929$	6.64605	0.218050	0.109025	+	0.994039i	$0.465227\pi$
$930$	40.5470	1.32959
$931$	5.10331	0.167254
$932$	1.23492	0.0404512
$933$	5.78563	0.189413
$934$	−13.4776	−0.441000
$935$	1.58675	0.0518923
$936$	0	0
$937$	−23.3638	−0.763263	−0.381631	−	0.924315i	$-0.624638\pi$
$937$	−23.3638	−0.763263	−0.381631	+	0.924315i	$0.624638\pi$
$938$	5.30219	0.173123
$939$	59.8775	1.95403
$940$	−1.41617	−0.0461905
$941$	−17.0262	−0.555037	−0.277519	−	0.960720i	$-0.589512\pi$
$941$	−17.0262	−0.555037	−0.277519	+	0.960720i	$0.589512\pi$
$942$	−26.0382	−0.848369
$943$	−0.353953	−0.0115263
$944$	0.00775008	0.000252244 0
$945$	−1.88601	−0.0613520
$946$	−0.612124	−0.0199019
$947$	−23.0918	−0.750384	−0.375192	−	0.926947i	$-0.622423\pi$
$947$	−23.0918	−0.750384	−0.375192	+	0.926947i	$0.622423\pi$
$948$	−18.7467	−0.608863
$949$	0	0
$950$	−2.27389	−0.0737748
$951$	41.6340	1.35007
$952$	3.79338	0.122944
$953$	−32.6949	−1.05909	−0.529546	−	0.848281i	$-0.677638\pi$
$953$	−32.6949	−1.05909	−0.529546	+	0.848281i	$0.677638\pi$
$954$	−27.3871	−0.886689
$955$	−0.780579	−0.0252590
$956$	28.9837	0.937400
$957$	6.80032	0.219823
$958$	−3.94341	−0.127406
$959$	4.49814	0.145252
$960$	−3.92498	−0.126678
$961$	75.7189	2.44254
$962$	0	0
$963$	−0.198875	−0.00640864
$964$	−3.89669	−0.125504
$965$	13.5710	0.436867
$966$	−0.558455	−0.0179680
$967$	−13.6305	−0.438329	−0.219164	−	0.975688i	$-0.570333\pi$
$967$	−13.6305	−0.438329	−0.219164	+	0.975688i	$0.570333\pi$
$968$	10.8783	0.349641
$969$	6.54778	0.210345
$970$	−5.35878	−0.172060
$971$	26.8860	0.862813	0.431407	−	0.902158i	$-0.358017\pi$
$971$	26.8860	0.862813	0.431407	+	0.902158i	$0.358017\pi$
$972$	21.1054	0.676957
$973$	5.60145	0.179574
$974$	−1.13453	−0.0363527
$975$	0	0
$976$	3.48052	0.111409
$977$	−9.89398	−0.316537	−0.158268	−	0.987396i	$-0.550591\pi$
$977$	−9.89398	−0.316537	−0.158268	+	0.987396i	$0.550591\pi$
$978$	6.30219	0.201522
$979$	3.77575	0.120674
$980$	−8.42605	−0.269160
$981$	29.8598	0.953351
$982$	−30.5315	−0.974299
$983$	−24.8492	−0.792565	−0.396283	−	0.918129i	$-0.629700\pi$
$983$	−24.8492	−0.792565	−0.396283	+	0.918129i	$0.629700\pi$
$984$	−4.93273	−0.157250
$985$	−6.97595	−0.222272
$986$	22.5830	0.719190
$987$	−2.80807	−0.0893820
$988$	0	0
$989$	0.299263	0.00951602
$990$	−1.52723	−0.0485387
$991$	−29.7488	−0.945001	−0.472501	−	0.881330i	$-0.656649\pi$
$991$	−29.7488	−0.945001	−0.472501	+	0.881330i	$0.656649\pi$
$992$	−10.3305	−0.327993
$993$	−35.6815	−1.13232
$994$	4.13161	0.131047
$995$	39.8647	1.26379
$996$	33.2186	1.05257
$997$	−32.3588	−1.02481	−0.512406	−	0.858743i	$-0.671246\pi$
$997$	−32.3588	−1.02481	−0.512406	+	0.858743i	$0.671246\pi$
$998$	−1.75441	−0.0555347
$999$	−4.85209	−0.153513

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.m.1.1		3
13.4	even	6		494.2.g.b.419.3	yes	6
13.10	even	6		494.2.g.b.191.3	✓	6
13.12	even	2		6422.2.a.u.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.b.191.3	✓	6	13.10	even	6
494.2.g.b.419.3	yes	6	13.4	even	6
6422.2.a.m.1.1		3	1.1	even	1	trivial
6422.2.a.u.1.1		3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.37720 q^{3} +1.00000 q^{4} +1.65109 q^{5} +2.37720 q^{6} -1.37720 q^{7} -1.00000 q^{8} +2.65109 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.37720 q^{3} +1.00000 q^{4} +1.65109 q^{5} +2.37720 q^{6} -1.37720 q^{7} -1.00000 q^{8} +2.65109 q^{9} -1.65109 q^{10} +0.348907 q^{11} -2.37720 q^{12} +1.37720 q^{14} -3.92498 q^{15} +1.00000 q^{16} +2.75441 q^{17} -2.65109 q^{18} -1.00000 q^{19} +1.65109 q^{20} +3.27389 q^{21} -0.348907 q^{22} +0.170578 q^{23} +2.37720 q^{24} -2.27389 q^{25} +0.829422 q^{27} -1.37720 q^{28} -8.19887 q^{29} +3.92498 q^{30} +10.3305 q^{31} -1.00000 q^{32} -0.829422 q^{33} -2.75441 q^{34} -2.27389 q^{35} +2.65109 q^{36} -5.84997 q^{37} +1.00000 q^{38} -1.65109 q^{40} -2.07502 q^{41} -3.27389 q^{42} +1.75441 q^{43} +0.348907 q^{44} +4.37720 q^{45} -0.170578 q^{46} -0.857718 q^{47} -2.37720 q^{48} -5.10331 q^{49} +2.27389 q^{50} -6.54778 q^{51} +10.3305 q^{53} -0.829422 q^{54} +0.576077 q^{55} +1.37720 q^{56} +2.37720 q^{57} +8.19887 q^{58} +0.00775008 q^{59} -3.92498 q^{60} +3.48052 q^{61} -10.3305 q^{62} -3.65109 q^{63} +1.00000 q^{64} +0.829422 q^{66} +3.84997 q^{67} +2.75441 q^{68} -0.405499 q^{69} +2.27389 q^{70} +3.00000 q^{71} -2.65109 q^{72} -16.6999 q^{73} +5.84997 q^{74} +5.40550 q^{75} -1.00000 q^{76} -0.480515 q^{77} +7.88601 q^{79} +1.65109 q^{80} -9.92498 q^{81} +2.07502 q^{82} -13.9738 q^{83} +3.27389 q^{84} +4.54778 q^{85} -1.75441 q^{86} +19.4904 q^{87} -0.348907 q^{88} +10.8217 q^{89} -4.37720 q^{90} +0.170578 q^{92} -24.5577 q^{93} +0.857718 q^{94} -1.65109 q^{95} +2.37720 q^{96} +3.24559 q^{97} +5.10331 q^{98} +0.924984 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 2 q^{6} + q^{7} - 3 q^{8} + q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 2 * q^6 + q^7 - 3 * q^8 + q^9	\( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 2 q^{6} + q^{7} - 3 q^{8} + q^{9} + 2 q^{10} + 8 q^{11} - 2 q^{12} - q^{14} - 3 q^{15} + 3 q^{16} - 2 q^{17} - q^{18} - 3 q^{19} - 2 q^{20} + 8 q^{21} - 8 q^{22} + 2 q^{23} + 2 q^{24} - 5 q^{25} + q^{27} + q^{28} - 14 q^{29} + 3 q^{30} + 5 q^{31} - 3 q^{32} - q^{33} + 2 q^{34} - 5 q^{35} + q^{36} + 3 q^{38} + 2 q^{40} - 15 q^{41} - 8 q^{42} - 5 q^{43} + 8 q^{44} + 8 q^{45} - 2 q^{46} + 11 q^{47} - 2 q^{48} - 12 q^{49} + 5 q^{50} - 16 q^{51} + 5 q^{53} - q^{54} - 14 q^{55} - q^{56} + 2 q^{57} + 14 q^{58} + 4 q^{59} - 3 q^{60} + 2 q^{61} - 5 q^{62} - 4 q^{63} + 3 q^{64} + q^{66} - 6 q^{67} - 2 q^{68} + 16 q^{69} + 5 q^{70} + 9 q^{71} - q^{72} - 15 q^{73} - q^{75} - 3 q^{76} + 7 q^{77} - 2 q^{79} - 2 q^{80} - 21 q^{81} + 15 q^{82} - 5 q^{83} + 8 q^{84} + 10 q^{85} + 5 q^{86} + 5 q^{87} - 8 q^{88} + 27 q^{89} - 8 q^{90} + 2 q^{92} - 25 q^{93} - 11 q^{94} + 2 q^{95} + 2 q^{96} + 20 q^{97} + 12 q^{98} - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 2 * q^6 + q^7 - 3 * q^8 + q^9 + 2 * q^10 + 8 * q^11 - 2 * q^12 - q^14 - 3 * q^15 + 3 * q^16 - 2 * q^17 - q^18 - 3 * q^19 - 2 * q^20 + 8 * q^21 - 8 * q^22 + 2 * q^23 + 2 * q^24 - 5 * q^25 + q^27 + q^28 - 14 * q^29 + 3 * q^30 + 5 * q^31 - 3 * q^32 - q^33 + 2 * q^34 - 5 * q^35 + q^36 + 3 * q^38 + 2 * q^40 - 15 * q^41 - 8 * q^42 - 5 * q^43 + 8 * q^44 + 8 * q^45 - 2 * q^46 + 11 * q^47 - 2 * q^48 - 12 * q^49 + 5 * q^50 - 16 * q^51 + 5 * q^53 - q^54 - 14 * q^55 - q^56 + 2 * q^57 + 14 * q^58 + 4 * q^59 - 3 * q^60 + 2 * q^61 - 5 * q^62 - 4 * q^63 + 3 * q^64 + q^66 - 6 * q^67 - 2 * q^68 + 16 * q^69 + 5 * q^70 + 9 * q^71 - q^72 - 15 * q^73 - q^75 - 3 * q^76 + 7 * q^77 - 2 * q^79 - 2 * q^80 - 21 * q^81 + 15 * q^82 - 5 * q^83 + 8 * q^84 + 10 * q^85 + 5 * q^86 + 5 * q^87 - 8 * q^88 + 27 * q^89 - 8 * q^90 + 2 * q^92 - 25 * q^93 - 11 * q^94 + 2 * q^95 + 2 * q^96 + 20 * q^97 + 12 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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