LMFDB - Embedded newform 6422.2.a.bi.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} - 2x^{8} - 12x^{7} + 14x^{6} + 54x^{5} - 11x^{4} - 84x^{3} - 48x^{2} - 4x + 1 $ x^9 - 2x^8 - 12x^7 + 14x^6 + 54x^5 - 11x^4 - 84x^3 - 48x^2 - 4x + 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			$-1.85435$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -3.07508 q^{3} +1.00000 q^{4} +1.48707 q^{5} +3.07508 q^{6} +5.10133 q^{7} -1.00000 q^{8} +6.45613 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.07508 q^{3} +1.00000 q^{4} +1.48707 q^{5} +3.07508 q^{6} +5.10133 q^{7} -1.00000 q^{8} +6.45613 q^{9} -1.48707 q^{10} +1.12734 q^{11} -3.07508 q^{12} -5.10133 q^{14} -4.57288 q^{15} +1.00000 q^{16} -2.43357 q^{17} -6.45613 q^{18} -1.00000 q^{19} +1.48707 q^{20} -15.6870 q^{21} -1.12734 q^{22} +2.35632 q^{23} +3.07508 q^{24} -2.78861 q^{25} -10.6279 q^{27} +5.10133 q^{28} +2.94671 q^{29} +4.57288 q^{30} -8.81471 q^{31} -1.00000 q^{32} -3.46667 q^{33} +2.43357 q^{34} +7.58606 q^{35} +6.45613 q^{36} +0.877617 q^{37} +1.00000 q^{38} -1.48707 q^{40} -8.75062 q^{41} +15.6870 q^{42} -7.93383 q^{43} +1.12734 q^{44} +9.60074 q^{45} -2.35632 q^{46} +8.12050 q^{47} -3.07508 q^{48} +19.0236 q^{49} +2.78861 q^{50} +7.48341 q^{51} +5.93669 q^{53} +10.6279 q^{54} +1.67644 q^{55} -5.10133 q^{56} +3.07508 q^{57} -2.94671 q^{58} -2.93249 q^{59} -4.57288 q^{60} +14.0758 q^{61} +8.81471 q^{62} +32.9348 q^{63} +1.00000 q^{64} +3.46667 q^{66} +3.76997 q^{67} -2.43357 q^{68} -7.24589 q^{69} -7.58606 q^{70} +14.5371 q^{71} -6.45613 q^{72} -3.16881 q^{73} -0.877617 q^{74} +8.57520 q^{75} -1.00000 q^{76} +5.75094 q^{77} +9.67130 q^{79} +1.48707 q^{80} +13.3132 q^{81} +8.75062 q^{82} -5.47241 q^{83} -15.6870 q^{84} -3.61889 q^{85} +7.93383 q^{86} -9.06138 q^{87} -1.12734 q^{88} -11.6974 q^{89} -9.60074 q^{90} +2.35632 q^{92} +27.1060 q^{93} -8.12050 q^{94} -1.48707 q^{95} +3.07508 q^{96} +6.22762 q^{97} -19.0236 q^{98} +7.27826 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{2} - 5 q^{3} + 9 q^{4} - q^{5} + 5 q^{6} + 13 q^{7} - 9 q^{8} + 10 q^{9}+O(q^{10}) $ 9 * q - 9 * q^2 - 5 * q^3 + 9 * q^4 - q^5 + 5 * q^6 + 13 * q^7 - 9 * q^8 + 10 * q^9	\( 9 q - 9 q^{2} - 5 q^{3} + 9 q^{4} - q^{5} + 5 q^{6} + 13 q^{7} - 9 q^{8} + 10 q^{9} + q^{10} + 13 q^{11} - 5 q^{12} - 13 q^{14} + q^{15} + 9 q^{16} - 12 q^{17} - 10 q^{18} - 9 q^{19} - q^{20} - 18 q^{21} - 13 q^{22} - 22 q^{23} + 5 q^{24} + 4 q^{25} - 26 q^{27} + 13 q^{28} + 12 q^{29} - q^{30} - q^{31} - 9 q^{32} + 28 q^{33} + 12 q^{34} - 18 q^{35} + 10 q^{36} + 25 q^{37} + 9 q^{38} + q^{40} - 11 q^{41} + 18 q^{42} - 10 q^{43} + 13 q^{44} + q^{45} + 22 q^{46} + 12 q^{47} - 5 q^{48} + 2 q^{49} - 4 q^{50} + 35 q^{51} + 9 q^{53} + 26 q^{54} + 18 q^{55} - 13 q^{56} + 5 q^{57} - 12 q^{58} - 10 q^{59} + q^{60} + 32 q^{61} + q^{62} + 63 q^{63} + 9 q^{64} - 28 q^{66} + 73 q^{67} - 12 q^{68} + 2 q^{69} + 18 q^{70} + 51 q^{71} - 10 q^{72} + 14 q^{73} - 25 q^{74} - 49 q^{75} - 9 q^{76} + 18 q^{77} - 28 q^{79} - q^{80} + 29 q^{81} + 11 q^{82} + 22 q^{83} - 18 q^{84} + 51 q^{85} + 10 q^{86} - 20 q^{87} - 13 q^{88} - 3 q^{89} - q^{90} - 22 q^{92} + 59 q^{93} - 12 q^{94} + q^{95} + 5 q^{96} - 2 q^{98} - 25 q^{99}+O(q^{100}) \) 9 * q - 9 * q^2 - 5 * q^3 + 9 * q^4 - q^5 + 5 * q^6 + 13 * q^7 - 9 * q^8 + 10 * q^9 + q^10 + 13 * q^11 - 5 * q^12 - 13 * q^14 + q^15 + 9 * q^16 - 12 * q^17 - 10 * q^18 - 9 * q^19 - q^20 - 18 * q^21 - 13 * q^22 - 22 * q^23 + 5 * q^24 + 4 * q^25 - 26 * q^27 + 13 * q^28 + 12 * q^29 - q^30 - q^31 - 9 * q^32 + 28 * q^33 + 12 * q^34 - 18 * q^35 + 10 * q^36 + 25 * q^37 + 9 * q^38 + q^40 - 11 * q^41 + 18 * q^42 - 10 * q^43 + 13 * q^44 + q^45 + 22 * q^46 + 12 * q^47 - 5 * q^48 + 2 * q^49 - 4 * q^50 + 35 * q^51 + 9 * q^53 + 26 * q^54 + 18 * q^55 - 13 * q^56 + 5 * q^57 - 12 * q^58 - 10 * q^59 + q^60 + 32 * q^61 + q^62 + 63 * q^63 + 9 * q^64 - 28 * q^66 + 73 * q^67 - 12 * q^68 + 2 * q^69 + 18 * q^70 + 51 * q^71 - 10 * q^72 + 14 * q^73 - 25 * q^74 - 49 * q^75 - 9 * q^76 + 18 * q^77 - 28 * q^79 - q^80 + 29 * q^81 + 11 * q^82 + 22 * q^83 - 18 * q^84 + 51 * q^85 + 10 * q^86 - 20 * q^87 - 13 * q^88 - 3 * q^89 - q^90 - 22 * q^92 + 59 * q^93 - 12 * q^94 + q^95 + 5 * q^96 - 2 * 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
$3$	−3.07508	−1.77540	−0.887700	−	0.460423i	$-0.847698\pi$
$3$	−3.07508	−1.77540	−0.887700	+	0.460423i	$0.847698\pi$
$4$	1.00000	0.500000
$5$	1.48707	0.665040	0.332520	−	0.943096i	$-0.392101\pi$
$5$	1.48707	0.665040	0.332520	+	0.943096i	$0.392101\pi$
$6$	3.07508	1.25540
$7$	5.10133	1.92812	0.964061	−	0.265681i	$-0.0855967\pi$
$7$	5.10133	1.92812	0.964061	+	0.265681i	$0.0855967\pi$
$8$	−1.00000	−0.353553
$9$	6.45613	2.15204
$10$	−1.48707	−0.470254
$11$	1.12734	0.339906	0.169953	−	0.985452i	$-0.445638\pi$
$11$	1.12734	0.339906	0.169953	+	0.985452i	$0.445638\pi$
$12$	−3.07508	−0.887700
$13$	0	0
$13$	0	0
$14$	−5.10133	−1.36339
$15$	−4.57288	−1.18071
$16$	1.00000	0.250000
$17$	−2.43357	−0.590226	−0.295113	−	0.955462i	$-0.595357\pi$
$17$	−2.43357	−0.590226	−0.295113	+	0.955462i	$0.595357\pi$
$18$	−6.45613	−1.52172
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.48707	0.332520
$21$	−15.6870	−3.42319
$22$	−1.12734	−0.240350
$23$	2.35632	0.491328	0.245664	−	0.969355i	$-0.420994\pi$
$23$	2.35632	0.491328	0.245664	+	0.969355i	$0.420994\pi$
$24$	3.07508	0.627698
$25$	−2.78861	−0.557722
$26$	0	0
$27$	−10.6279	−2.04533
$28$	5.10133	0.964061
$29$	2.94671	0.547191	0.273595	−	0.961845i	$-0.411787\pi$
$29$	2.94671	0.547191	0.273595	+	0.961845i	$0.411787\pi$
$30$	4.57288	0.834889
$31$	−8.81471	−1.58317	−0.791585	−	0.611060i	$-0.790744\pi$
$31$	−8.81471	−1.58317	−0.791585	+	0.611060i	$0.790744\pi$
$32$	−1.00000	−0.176777
$33$	−3.46667	−0.603469
$34$	2.43357	0.417353
$35$	7.58606	1.28228
$36$	6.45613	1.07602
$37$	0.877617	0.144279	0.0721397	−	0.997395i	$-0.477017\pi$
$37$	0.877617	0.144279	0.0721397	+	0.997395i	$0.477017\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−1.48707	−0.235127
$41$	−8.75062	−1.36662	−0.683309	−	0.730129i	$-0.739460\pi$
$41$	−8.75062	−1.36662	−0.683309	+	0.730129i	$0.739460\pi$
$42$	15.6870	2.42056
$43$	−7.93383	−1.20990	−0.604949	−	0.796264i	$-0.706806\pi$
$43$	−7.93383	−1.20990	−0.604949	+	0.796264i	$0.706806\pi$
$44$	1.12734	0.169953
$45$	9.60074	1.43119
$46$	−2.35632	−0.347421
$47$	8.12050	1.18450	0.592248	−	0.805756i	$-0.298241\pi$
$47$	8.12050	1.18450	0.592248	+	0.805756i	$0.298241\pi$
$48$	−3.07508	−0.443850
$49$	19.0236	2.71765
$50$	2.78861	0.394369
$51$	7.48341	1.04789
$52$	0	0
$53$	5.93669	0.815467	0.407733	−	0.913101i	$-0.366319\pi$
$53$	5.93669	0.815467	0.407733	+	0.913101i	$0.366319\pi$
$54$	10.6279	1.44627
$55$	1.67644	0.226051
$56$	−5.10133	−0.681694
$57$	3.07508	0.407304
$58$	−2.94671	−0.386922
$59$	−2.93249	−0.381778	−0.190889	−	0.981612i	$-0.561137\pi$
$59$	−2.93249	−0.381778	−0.190889	+	0.981612i	$0.561137\pi$
$60$	−4.57288	−0.590356
$61$	14.0758	1.80222	0.901108	−	0.433595i	$-0.142755\pi$
$61$	14.0758	1.80222	0.901108	+	0.433595i	$0.142755\pi$
$62$	8.81471	1.11947
$63$	32.9348	4.14940
$64$	1.00000	0.125000
$65$	0	0
$66$	3.46667	0.426717
$67$	3.76997	0.460575	0.230288	−	0.973123i	$-0.426033\pi$
$67$	3.76997	0.460575	0.230288	+	0.973123i	$0.426033\pi$
$68$	−2.43357	−0.295113
$69$	−7.24589	−0.872303
$70$	−7.58606	−0.906708
$71$	14.5371	1.72523	0.862615	−	0.505861i	$-0.168825\pi$
$71$	14.5371	1.72523	0.862615	+	0.505861i	$0.168825\pi$
$72$	−6.45613	−0.760862
$73$	−3.16881	−0.370881	−0.185440	−	0.982656i	$-0.559371\pi$
$73$	−3.16881	−0.370881	−0.185440	+	0.982656i	$0.559371\pi$
$74$	−0.877617	−0.102021
$75$	8.57520	0.990179
$76$	−1.00000	−0.114708
$77$	5.75094	0.655381
$78$	0	0
$79$	9.67130	1.08811	0.544053	−	0.839051i	$-0.316889\pi$
$79$	9.67130	1.08811	0.544053	+	0.839051i	$0.316889\pi$
$80$	1.48707	0.166260
$81$	13.3132	1.47924
$82$	8.75062	0.966345
$83$	−5.47241	−0.600675	−0.300337	−	0.953833i	$-0.597099\pi$
$83$	−5.47241	−0.600675	−0.300337	+	0.953833i	$0.597099\pi$
$84$	−15.6870	−1.71159
$85$	−3.61889	−0.392524
$86$	7.93383	0.855527
$87$	−9.06138	−0.971482
$88$	−1.12734	−0.120175
$89$	−11.6974	−1.23992	−0.619962	−	0.784632i	$-0.712852\pi$
$89$	−11.6974	−1.23992	−0.619962	+	0.784632i	$0.712852\pi$
$90$	−9.60074	−1.01201
$91$	0	0
$92$	2.35632	0.245664
$93$	27.1060	2.81076
$94$	−8.12050	−0.837565
$95$	−1.48707	−0.152571
$96$	3.07508	0.313849
$97$	6.22762	0.632319	0.316159	−	0.948706i	$-0.397607\pi$
$97$	6.22762	0.632319	0.316159	+	0.948706i	$0.397607\pi$
$98$	−19.0236	−1.92167
$99$	7.27826	0.731492
$100$	−2.78861	−0.278861
$101$	−7.82683	−0.778798	−0.389399	−	0.921069i	$-0.627317\pi$
$101$	−7.82683	−0.778798	−0.389399	+	0.921069i	$0.627317\pi$
$102$	−7.48341	−0.740968
$103$	−14.7108	−1.44950	−0.724749	−	0.689013i	$-0.758044\pi$
$103$	−14.7108	−1.44950	−0.724749	+	0.689013i	$0.758044\pi$
$104$	0	0
$105$	−23.3278	−2.27656
$106$	−5.93669	−0.576622
$107$	16.4002	1.58547	0.792735	−	0.609567i	$-0.208657\pi$
$107$	16.4002	1.58547	0.792735	+	0.609567i	$0.208657\pi$
$108$	−10.6279	−1.02267
$109$	8.98211	0.860330	0.430165	−	0.902750i	$-0.358455\pi$
$109$	8.98211	0.860330	0.430165	+	0.902750i	$0.358455\pi$
$110$	−1.67644	−0.159842
$111$	−2.69875	−0.256154
$112$	5.10133	0.482031
$113$	8.74788	0.822931	0.411466	−	0.911425i	$-0.365017\pi$
$113$	8.74788	0.822931	0.411466	+	0.911425i	$0.365017\pi$
$114$	−3.07508	−0.288008
$115$	3.50403	0.326753
$116$	2.94671	0.273595
$117$	0	0
$118$	2.93249	0.269958
$119$	−12.4144	−1.13803
$120$	4.57288	0.417445
$121$	−9.72910	−0.884464
$122$	−14.0758	−1.27436
$123$	26.9089	2.42629
$124$	−8.81471	−0.791585
$125$	−11.5822	−1.03595
$126$	−32.9348	−2.93407
$127$	3.30964	0.293683	0.146841	−	0.989160i	$-0.453089\pi$
$127$	3.30964	0.293683	0.146841	+	0.989160i	$0.453089\pi$
$128$	−1.00000	−0.0883883
$129$	24.3972	2.14805
$130$	0	0
$131$	−12.8639	−1.12393	−0.561963	−	0.827163i	$-0.689954\pi$
$131$	−12.8639	−1.12393	−0.561963	+	0.827163i	$0.689954\pi$
$132$	−3.46667	−0.301735
$133$	−5.10133	−0.442342
$134$	−3.76997	−0.325676
$135$	−15.8044	−1.36023
$136$	2.43357	0.208677
$137$	14.9850	1.28025	0.640126	−	0.768270i	$-0.278882\pi$
$137$	14.9850	1.28025	0.640126	+	0.768270i	$0.278882\pi$
$138$	7.24589	0.616811
$139$	15.9535	1.35316	0.676579	−	0.736370i	$-0.263462\pi$
$139$	15.9535	1.35316	0.676579	+	0.736370i	$0.263462\pi$
$140$	7.58606	0.641139
$141$	−24.9712	−2.10295
$142$	−14.5371	−1.21992
$143$	0	0
$144$	6.45613	0.538011
$145$	4.38198	0.363904
$146$	3.16881	0.262252
$147$	−58.4991	−4.82492
$148$	0.877617	0.0721397
$149$	21.6983	1.77759	0.888797	−	0.458301i	$-0.151542\pi$
$149$	21.6983	1.77759	0.888797	+	0.458301i	$0.151542\pi$
$150$	−8.57520	−0.700162
$151$	1.11577	0.0908005	0.0454002	−	0.998969i	$-0.485544\pi$
$151$	1.11577	0.0908005	0.0454002	+	0.998969i	$0.485544\pi$
$152$	1.00000	0.0811107
$153$	−15.7114	−1.27019
$154$	−5.75094	−0.463424
$155$	−13.1081	−1.05287
$156$	0	0
$157$	21.4663	1.71320	0.856598	−	0.515984i	$-0.172574\pi$
$157$	21.4663	1.71320	0.856598	+	0.515984i	$0.172574\pi$
$158$	−9.67130	−0.769407
$159$	−18.2558	−1.44778
$160$	−1.48707	−0.117564
$161$	12.0204	0.947340
$162$	−13.3132	−1.04598
$163$	18.2074	1.42611	0.713057	−	0.701106i	$-0.247310\pi$
$163$	18.2074	1.42611	0.713057	+	0.701106i	$0.247310\pi$
$164$	−8.75062	−0.683309
$165$	−5.15519	−0.401331
$166$	5.47241	0.424741
$167$	−19.6829	−1.52311	−0.761556	−	0.648099i	$-0.775564\pi$
$167$	−19.6829	−1.52311	−0.761556	+	0.648099i	$0.775564\pi$
$168$	15.6870	1.21028
$169$	0	0
$170$	3.61889	0.277557
$171$	−6.45613	−0.493712
$172$	−7.93383	−0.604949
$173$	−0.489868	−0.0372440	−0.0186220	−	0.999827i	$-0.505928\pi$
$173$	−0.489868	−0.0372440	−0.0186220	+	0.999827i	$0.505928\pi$
$174$	9.06138	0.686941
$175$	−14.2256	−1.07536
$176$	1.12734	0.0849766
$177$	9.01766	0.677809
$178$	11.6974	0.876759
$179$	4.81280	0.359726	0.179863	−	0.983692i	$-0.442435\pi$
$179$	4.81280	0.359726	0.179863	+	0.983692i	$0.442435\pi$
$180$	9.60074	0.715597
$181$	13.6875	1.01738	0.508691	−	0.860949i	$-0.330129\pi$
$181$	13.6875	1.01738	0.508691	+	0.860949i	$0.330129\pi$
$182$	0	0
$183$	−43.2841	−3.19965
$184$	−2.35632	−0.173711
$185$	1.30508	0.0959516
$186$	−27.1060	−1.98751
$187$	−2.74346	−0.200622
$188$	8.12050	0.592248
$189$	−54.2163	−3.94365
$190$	1.48707	0.107884
$191$	−24.3725	−1.76353	−0.881765	−	0.471689i	$-0.843644\pi$
$191$	−24.3725	−1.76353	−0.881765	+	0.471689i	$0.843644\pi$
$192$	−3.07508	−0.221925
$193$	11.5703	0.832848	0.416424	−	0.909170i	$-0.363283\pi$
$193$	11.5703	0.832848	0.416424	+	0.909170i	$0.363283\pi$
$194$	−6.22762	−0.447117
$195$	0	0
$196$	19.0236	1.35883
$197$	8.55288	0.609368	0.304684	−	0.952454i	$-0.401449\pi$
$197$	8.55288	0.609368	0.304684	+	0.952454i	$0.401449\pi$
$198$	−7.27826	−0.517243
$199$	3.24298	0.229888	0.114944	−	0.993372i	$-0.463331\pi$
$199$	3.24298	0.229888	0.114944	+	0.993372i	$0.463331\pi$
$200$	2.78861	0.197184
$201$	−11.5930	−0.817705
$202$	7.82683	0.550693
$203$	15.0322	1.05505
$204$	7.48341	0.523944
$205$	−13.0128	−0.908856
$206$	14.7108	1.02495
$207$	15.2127	1.05736
$208$	0	0
$209$	−1.12734	−0.0779798
$210$	23.3278	1.60977
$211$	−3.26431	−0.224724	−0.112362	−	0.993667i	$-0.535842\pi$
$211$	−3.26431	−0.224724	−0.112362	+	0.993667i	$0.535842\pi$
$212$	5.93669	0.407733
$213$	−44.7026	−3.06297
$214$	−16.4002	−1.12110
$215$	−11.7982	−0.804630
$216$	10.6279	0.723135
$217$	−44.9668	−3.05254
$218$	−8.98211	−0.608345
$219$	9.74434	0.658461
$220$	1.67644	0.113026
$221$	0	0
$222$	2.69875	0.181128
$223$	−10.0657	−0.674047	−0.337023	−	0.941496i	$-0.609420\pi$
$223$	−10.0657	−0.674047	−0.337023	+	0.941496i	$0.609420\pi$
$224$	−5.10133	−0.340847
$225$	−18.0036	−1.20024
$226$	−8.74788	−0.581900
$227$	−3.02146	−0.200541	−0.100271	−	0.994960i	$-0.531971\pi$
$227$	−3.02146	−0.200541	−0.100271	+	0.994960i	$0.531971\pi$
$228$	3.07508	0.203652
$229$	16.7115	1.10433	0.552164	−	0.833736i	$-0.313802\pi$
$229$	16.7115	1.10433	0.552164	+	0.833736i	$0.313802\pi$
$230$	−3.50403	−0.231049
$231$	−17.6846	−1.16356
$232$	−2.94671	−0.193461
$233$	−25.7957	−1.68993	−0.844966	−	0.534820i	$-0.820379\pi$
$233$	−25.7957	−1.68993	−0.844966	+	0.534820i	$0.820379\pi$
$234$	0	0
$235$	12.0758	0.787737
$236$	−2.93249	−0.190889
$237$	−29.7400	−1.93182
$238$	12.4144	0.804708
$239$	−6.89332	−0.445892	−0.222946	−	0.974831i	$-0.571567\pi$
$239$	−6.89332	−0.445892	−0.222946	+	0.974831i	$0.571567\pi$
$240$	−4.57288	−0.295178
$241$	18.0294	1.16137	0.580687	−	0.814127i	$-0.302784\pi$
$241$	18.0294	1.16137	0.580687	+	0.814127i	$0.302784\pi$
$242$	9.72910	0.625410
$243$	−9.05552	−0.580912
$244$	14.0758	0.901108
$245$	28.2895	1.80735
$246$	−26.9089	−1.71565
$247$	0	0
$248$	8.81471	0.559735
$249$	16.8281	1.06644
$250$	11.5822	0.732525
$251$	−26.7560	−1.68882	−0.844411	−	0.535696i	$-0.820049\pi$
$251$	−26.7560	−1.68882	−0.844411	+	0.535696i	$0.820049\pi$
$252$	32.9348	2.07470
$253$	2.65638	0.167005
$254$	−3.30964	−0.207665
$255$	11.1284	0.696887
$256$	1.00000	0.0625000
$257$	1.50803	0.0940685	0.0470342	−	0.998893i	$-0.485023\pi$
$257$	1.50803	0.0940685	0.0470342	+	0.998893i	$0.485023\pi$
$258$	−24.3972	−1.51890
$259$	4.47702	0.278188
$260$	0	0
$261$	19.0243	1.17758
$262$	12.8639	0.794735
$263$	−9.51919	−0.586979	−0.293489	−	0.955962i	$-0.594817\pi$
$263$	−9.51919	−0.586979	−0.293489	+	0.955962i	$0.594817\pi$
$264$	3.46667	0.213359
$265$	8.82830	0.542318
$266$	5.10133	0.312783
$267$	35.9705	2.20136
$268$	3.76997	0.230288
$269$	15.4627	0.942775	0.471387	−	0.881926i	$-0.343753\pi$
$269$	15.4627	0.942775	0.471387	+	0.881926i	$0.343753\pi$
$270$	15.8044	0.961827
$271$	30.2959	1.84034	0.920172	−	0.391514i	$-0.128048\pi$
$271$	30.2959	1.84034	0.920172	+	0.391514i	$0.128048\pi$
$272$	−2.43357	−0.147557
$273$	0	0
$274$	−14.9850	−0.905275
$275$	−3.14371	−0.189573
$276$	−7.24589	−0.436151
$277$	10.2202	0.614075	0.307038	−	0.951697i	$-0.400662\pi$
$277$	10.2202	0.614075	0.307038	+	0.951697i	$0.400662\pi$
$278$	−15.9535	−0.956827
$279$	−56.9089	−3.40705
$280$	−7.58606	−0.453354
$281$	−19.5591	−1.16680	−0.583399	−	0.812186i	$-0.698278\pi$
$281$	−19.5591	−1.16680	−0.583399	+	0.812186i	$0.698278\pi$
$282$	24.9712	1.48701
$283$	4.81914	0.286468	0.143234	−	0.989689i	$-0.454250\pi$
$283$	4.81914	0.286468	0.143234	+	0.989689i	$0.454250\pi$
$284$	14.5371	0.862615
$285$	4.57288	0.270874
$286$	0	0
$287$	−44.6398	−2.63501
$288$	−6.45613	−0.380431
$289$	−11.0778	−0.651633
$290$	−4.38198	−0.257319
$291$	−19.1504	−1.12262
$292$	−3.16881	−0.185440
$293$	1.28168	0.0748766	0.0374383	−	0.999299i	$-0.488080\pi$
$293$	1.28168	0.0748766	0.0374383	+	0.999299i	$0.488080\pi$
$294$	58.4991	3.41174
$295$	−4.36084	−0.253898
$296$	−0.877617	−0.0510105
$297$	−11.9812	−0.695222
$298$	−21.6983	−1.25695
$299$	0	0
$300$	8.57520	0.495089
$301$	−40.4731	−2.33283
$302$	−1.11577	−0.0642056
$303$	24.0681	1.38268
$304$	−1.00000	−0.0573539
$305$	20.9317	1.19855
$306$	15.7114	0.898161
$307$	14.7032	0.839157	0.419579	−	0.907719i	$-0.362178\pi$
$307$	14.7032	0.839157	0.419579	+	0.907719i	$0.362178\pi$
$308$	5.75094	0.327690
$309$	45.2369	2.57344
$310$	13.1081	0.744492
$311$	−14.9108	−0.845515	−0.422757	−	0.906243i	$-0.638938\pi$
$311$	−14.9108	−0.845515	−0.422757	+	0.906243i	$0.638938\pi$
$312$	0	0
$313$	23.6699	1.33790	0.668952	−	0.743306i	$-0.266743\pi$
$313$	23.6699	1.33790	0.668952	+	0.743306i	$0.266743\pi$
$314$	−21.4663	−1.21141
$315$	48.9766	2.75952
$316$	9.67130	0.544053
$317$	17.4182	0.978301	0.489151	−	0.872199i	$-0.337307\pi$
$317$	17.4182	0.978301	0.489151	+	0.872199i	$0.337307\pi$
$318$	18.2558	1.02373
$319$	3.32195	0.185993
$320$	1.48707	0.0831300
$321$	−50.4320	−2.81484
$322$	−12.0204	−0.669870
$323$	2.43357	0.135407
$324$	13.3132	0.739621
$325$	0	0
$326$	−18.2074	−1.00841
$327$	−27.6207	−1.52743
$328$	8.75062	0.483172
$329$	41.4253	2.28385
$330$	5.15519	0.283784
$331$	−21.1765	−1.16396	−0.581982	−	0.813202i	$-0.697723\pi$
$331$	−21.1765	−1.16396	−0.581982	+	0.813202i	$0.697723\pi$
$332$	−5.47241	−0.300337
$333$	5.66601	0.310495
$334$	19.6829	1.07700
$335$	5.60623	0.306301
$336$	−15.6870	−0.855797
$337$	17.8159	0.970494	0.485247	−	0.874377i	$-0.338730\pi$
$337$	17.8159	0.970494	0.485247	+	0.874377i	$0.338730\pi$
$338$	0	0
$339$	−26.9004	−1.46103
$340$	−3.61889	−0.196262
$341$	−9.93719	−0.538129
$342$	6.45613	0.349107
$343$	61.3363	3.31185
$344$	7.93383	0.427763
$345$	−10.7752	−0.580116
$346$	0.489868	0.0263355
$347$	12.4370	0.667655	0.333827	−	0.942634i	$-0.391660\pi$
$347$	12.4370	0.667655	0.333827	+	0.942634i	$0.391660\pi$
$348$	−9.06138	−0.485741
$349$	0.667333	0.0357215	0.0178608	−	0.999840i	$-0.494314\pi$
$349$	0.667333	0.0357215	0.0178608	+	0.999840i	$0.494314\pi$
$350$	14.2256	0.760391
$351$	0	0
$352$	−1.12734	−0.0600875
$353$	−6.28783	−0.334667	−0.167334	−	0.985900i	$-0.553516\pi$
$353$	−6.28783	−0.334667	−0.167334	+	0.985900i	$0.553516\pi$
$354$	−9.01766	−0.479283
$355$	21.6177	1.14735
$356$	−11.6974	−0.619962
$357$	38.1754	2.02046
$358$	−4.81280	−0.254364
$359$	−8.65052	−0.456557	−0.228278	−	0.973596i	$-0.573310\pi$
$359$	−8.65052	−0.456557	−0.228278	+	0.973596i	$0.573310\pi$
$360$	−9.60074	−0.506004
$361$	1.00000	0.0526316
$362$	−13.6875	−0.719398
$363$	29.9178	1.57028
$364$	0	0
$365$	−4.71225	−0.246651
$366$	43.2841	2.26250
$367$	18.5295	0.967231	0.483615	−	0.875281i	$-0.339323\pi$
$367$	18.5295	0.967231	0.483615	+	0.875281i	$0.339323\pi$
$368$	2.35632	0.122832
$369$	−56.4951	−2.94102
$370$	−1.30508	−0.0678480
$371$	30.2850	1.57232
$372$	27.1060	1.40538
$373$	−17.4523	−0.903646	−0.451823	−	0.892108i	$-0.649226\pi$
$373$	−17.4523	−0.903646	−0.451823	+	0.892108i	$0.649226\pi$
$374$	2.74346	0.141861
$375$	35.6163	1.83922
$376$	−8.12050	−0.418783
$377$	0	0
$378$	54.2163	2.78858
$379$	31.9627	1.64181	0.820906	−	0.571064i	$-0.193469\pi$
$379$	31.9627	1.64181	0.820906	+	0.571064i	$0.193469\pi$
$380$	−1.48707	−0.0762853
$381$	−10.1774	−0.521404
$382$	24.3725	1.24700
$383$	10.8220	0.552980	0.276490	−	0.961017i	$-0.410829\pi$
$383$	10.8220	0.552980	0.276490	+	0.961017i	$0.410829\pi$
$384$	3.07508	0.156925
$385$	8.55208	0.435854
$386$	−11.5703	−0.588913
$387$	−51.2218	−2.60375
$388$	6.22762	0.316159
$389$	30.4623	1.54450	0.772249	−	0.635320i	$-0.219132\pi$
$389$	30.4623	1.54450	0.772249	+	0.635320i	$0.219132\pi$
$390$	0	0
$391$	−5.73427	−0.289995
$392$	−19.0236	−0.960836
$393$	39.5576	1.99542
$394$	−8.55288	−0.430888
$395$	14.3819	0.723634
$396$	7.27826	0.365746
$397$	22.4399	1.12623	0.563114	−	0.826379i	$-0.309603\pi$
$397$	22.4399	1.12623	0.563114	+	0.826379i	$0.309603\pi$
$398$	−3.24298	−0.162556
$399$	15.6870	0.785333
$400$	−2.78861	−0.139430
$401$	−8.62169	−0.430547	−0.215273	−	0.976554i	$-0.569064\pi$
$401$	−8.62169	−0.430547	−0.215273	+	0.976554i	$0.569064\pi$
$402$	11.5930	0.578205
$403$	0	0
$404$	−7.82683	−0.389399
$405$	19.7977	0.983756
$406$	−15.0322	−0.746033
$407$	0.989374	0.0490415
$408$	−7.48341	−0.370484
$409$	14.0179	0.693138	0.346569	−	0.938024i	$-0.387347\pi$
$409$	14.0179	0.693138	0.346569	+	0.938024i	$0.387347\pi$
$410$	13.0128	0.642658
$411$	−46.0800	−2.27296
$412$	−14.7108	−0.724749
$413$	−14.9596	−0.736115
$414$	−15.2127	−0.747665
$415$	−8.13788	−0.399473
$416$	0	0
$417$	−49.0583	−2.40239
$418$	1.12734	0.0551401
$419$	−27.4326	−1.34017	−0.670085	−	0.742285i	$-0.733742\pi$
$419$	−27.4326	−1.34017	−0.670085	+	0.742285i	$0.733742\pi$
$420$	−23.3278	−1.13828
$421$	−32.8186	−1.59948	−0.799741	−	0.600345i	$-0.795030\pi$
$421$	−32.8186	−1.59948	−0.799741	+	0.600345i	$0.795030\pi$
$422$	3.26431	0.158904
$423$	52.4270	2.54909
$424$	−5.93669	−0.288311
$425$	6.78626	0.329182
$426$	44.7026	2.16585
$427$	71.8051	3.47489
$428$	16.4002	0.792735
$429$	0	0
$430$	11.7982	0.568960
$431$	21.1256	1.01758	0.508792	−	0.860890i	$-0.330092\pi$
$431$	21.1256	1.01758	0.508792	+	0.860890i	$0.330092\pi$
$432$	−10.6279	−0.511334
$433$	22.8202	1.09667	0.548334	−	0.836259i	$-0.315262\pi$
$433$	22.8202	1.09667	0.548334	+	0.836259i	$0.315262\pi$
$434$	44.9668	2.15847
$435$	−13.4749	−0.646074
$436$	8.98211	0.430165
$437$	−2.35632	−0.112718
$438$	−9.74434	−0.465602
$439$	−7.76870	−0.370780	−0.185390	−	0.982665i	$-0.559355\pi$
$439$	−7.76870	−0.370780	−0.185390	+	0.982665i	$0.559355\pi$
$440$	−1.67644	−0.0799212
$441$	122.819	5.84851
$442$	0	0
$443$	12.6726	0.602092	0.301046	−	0.953610i	$-0.402664\pi$
$443$	12.6726	0.602092	0.301046	+	0.953610i	$0.402664\pi$
$444$	−2.69875	−0.128077
$445$	−17.3949	−0.824600
$446$	10.0657	0.476623
$447$	−66.7241	−3.15594
$448$	5.10133	0.241015
$449$	11.0057	0.519389	0.259695	−	0.965691i	$-0.416378\pi$
$449$	11.0057	0.519389	0.259695	+	0.965691i	$0.416378\pi$
$450$	18.0036	0.848698
$451$	−9.86494	−0.464522
$452$	8.74788	0.411466
$453$	−3.43110	−0.161207
$454$	3.02146	0.141804
$455$	0	0
$456$	−3.07508	−0.144004
$457$	12.4216	0.581056	0.290528	−	0.956866i	$-0.406169\pi$
$457$	12.4216	0.581056	0.290528	+	0.956866i	$0.406169\pi$
$458$	−16.7115	−0.780878
$459$	25.8636	1.20721
$460$	3.50403	0.163376
$461$	−27.7593	−1.29288	−0.646439	−	0.762966i	$-0.723742\pi$
$461$	−27.7593	−1.29288	−0.646439	+	0.762966i	$0.723742\pi$
$462$	17.6846	0.822763
$463$	11.2619	0.523384	0.261692	−	0.965151i	$-0.415720\pi$
$463$	11.2619	0.523384	0.261692	+	0.965151i	$0.415720\pi$
$464$	2.94671	0.136798
$465$	40.3086	1.86927
$466$	25.7957	1.19496
$467$	−11.7718	−0.544736	−0.272368	−	0.962193i	$-0.587807\pi$
$467$	−11.7718	−0.544736	−0.272368	+	0.962193i	$0.587807\pi$
$468$	0	0
$469$	19.2319	0.888045
$470$	−12.0758	−0.557014
$471$	−66.0106	−3.04161
$472$	2.93249	0.134979
$473$	−8.94413	−0.411252
$474$	29.7400	1.36601
$475$	2.78861	0.127950
$476$	−12.4144	−0.569014
$477$	38.3280	1.75492
$478$	6.89332	0.315293
$479$	−23.3639	−1.06752	−0.533762	−	0.845634i	$-0.679222\pi$
$479$	−23.3639	−1.06752	−0.533762	+	0.845634i	$0.679222\pi$
$480$	4.57288	0.208722
$481$	0	0
$482$	−18.0294	−0.821216
$483$	−36.9637	−1.68191
$484$	−9.72910	−0.442232
$485$	9.26093	0.420517
$486$	9.05552	0.410767
$487$	22.1847	1.00529	0.502643	−	0.864494i	$-0.332361\pi$
$487$	22.1847	1.00529	0.502643	+	0.864494i	$0.332361\pi$
$488$	−14.0758	−0.637180
$489$	−55.9893	−2.53192
$490$	−28.2895	−1.27799
$491$	−6.24016	−0.281614	−0.140807	−	0.990037i	$-0.544970\pi$
$491$	−6.24016	−0.281614	−0.140807	+	0.990037i	$0.544970\pi$
$492$	26.9089	1.21315
$493$	−7.17102	−0.322966
$494$	0	0
$495$	10.8233	0.486472
$496$	−8.81471	−0.395792
$497$	74.1583	3.32646
$498$	−16.8281	−0.754085
$499$	−6.07304	−0.271866	−0.135933	−	0.990718i	$-0.543403\pi$
$499$	−6.07304	−0.271866	−0.135933	+	0.990718i	$0.543403\pi$
$500$	−11.5822	−0.517974
$501$	60.5267	2.70413
$502$	26.7560	1.19418
$503$	−8.66112	−0.386180	−0.193090	−	0.981181i	$-0.561851\pi$
$503$	−8.66112	−0.386180	−0.193090	+	0.981181i	$0.561851\pi$
$504$	−32.9348	−1.46703
$505$	−11.6391	−0.517932
$506$	−2.65638	−0.118091
$507$	0	0
$508$	3.30964	0.146841
$509$	23.5060	1.04189	0.520943	−	0.853592i	$-0.325580\pi$
$509$	23.5060	1.04189	0.520943	+	0.853592i	$0.325580\pi$
$510$	−11.1284	−0.492774
$511$	−16.1651	−0.715103
$512$	−1.00000	−0.0441942
$513$	10.6279	0.469232
$514$	−1.50803	−0.0665165
$515$	−21.8760	−0.963974
$516$	24.3972	1.07403
$517$	9.15457	0.402618
$518$	−4.47702	−0.196709
$519$	1.50638	0.0661229
$520$	0	0
$521$	1.21784	0.0533544	0.0266772	−	0.999644i	$-0.491507\pi$
$521$	1.21784	0.0533544	0.0266772	+	0.999644i	$0.491507\pi$
$522$	−19.0243	−0.832673
$523$	−0.500722	−0.0218951	−0.0109475	−	0.999940i	$-0.503485\pi$
$523$	−0.500722	−0.0218951	−0.0109475	+	0.999940i	$0.503485\pi$
$524$	−12.8639	−0.561963
$525$	43.7449	1.90919
$526$	9.51919	0.415057
$527$	21.4512	0.934428
$528$	−3.46667	−0.150867
$529$	−17.4477	−0.758597
$530$	−8.82830	−0.383477
$531$	−18.9326	−0.821603
$532$	−5.10133	−0.221171
$533$	0	0
$534$	−35.9705	−1.55660
$535$	24.3884	1.05440
$536$	−3.76997	−0.162838
$537$	−14.7998	−0.638657
$538$	−15.4627	−0.666642
$539$	21.4461	0.923748
$540$	−15.8044	−0.680115
$541$	40.4861	1.74063	0.870317	−	0.492493i	$-0.163914\pi$
$541$	40.4861	1.74063	0.870317	+	0.492493i	$0.163914\pi$
$542$	−30.2959	−1.30132
$543$	−42.0901	−1.80626
$544$	2.43357	0.104338
$545$	13.3571	0.572154
$546$	0	0
$547$	−5.15647	−0.220475	−0.110237	−	0.993905i	$-0.535161\pi$
$547$	−5.15647	−0.220475	−0.110237	+	0.993905i	$0.535161\pi$
$548$	14.9850	0.640126
$549$	90.8749	3.87844
$550$	3.14371	0.134048
$551$	−2.94671	−0.125534
$552$	7.24589	0.308406
$553$	49.3365	2.09800
$554$	−10.2202	−0.434217
$555$	−4.01324	−0.170352
$556$	15.9535	0.676579
$557$	39.5827	1.67717	0.838586	−	0.544769i	$-0.183383\pi$
$557$	39.5827	1.67717	0.838586	+	0.544769i	$0.183383\pi$
$558$	56.9089	2.40915
$559$	0	0
$560$	7.58606	0.320570
$561$	8.43636	0.356183
$562$	19.5591	0.825051
$563$	−4.06419	−0.171285	−0.0856427	−	0.996326i	$-0.527294\pi$
$563$	−4.06419	−0.171285	−0.0856427	+	0.996326i	$0.527294\pi$
$564$	−24.9712	−1.05148
$565$	13.0087	0.547282
$566$	−4.81914	−0.202563
$567$	67.9150	2.85216
$568$	−14.5371	−0.609961
$569$	−2.41470	−0.101229	−0.0506147	−	0.998718i	$-0.516118\pi$
$569$	−2.41470	−0.101229	−0.0506147	+	0.998718i	$0.516118\pi$
$570$	−4.57288	−0.191537
$571$	4.10083	0.171614	0.0858071	−	0.996312i	$-0.472653\pi$
$571$	4.10083	0.171614	0.0858071	+	0.996312i	$0.472653\pi$
$572$	0	0
$573$	74.9473	3.13097
$574$	44.6398	1.86323
$575$	−6.57087	−0.274024
$576$	6.45613	0.269005
$577$	−22.4028	−0.932640	−0.466320	−	0.884616i	$-0.654421\pi$
$577$	−22.4028	−0.932640	−0.466320	+	0.884616i	$0.654421\pi$
$578$	11.0778	0.460774
$579$	−35.5796	−1.47864
$580$	4.38198	0.181952
$581$	−27.9166	−1.15817
$582$	19.1504	0.793811
$583$	6.69267	0.277182
$584$	3.16881	0.131126
$585$	0	0
$586$	−1.28168	−0.0529457
$587$	−14.2212	−0.586972	−0.293486	−	0.955963i	$-0.594815\pi$
$587$	−14.2212	−0.586972	−0.293486	+	0.955963i	$0.594815\pi$
$588$	−58.4991	−2.41246
$589$	8.81471	0.363204
$590$	4.36084	0.179533
$591$	−26.3008	−1.08187
$592$	0.877617	0.0360699
$593$	12.8302	0.526872	0.263436	−	0.964677i	$-0.415144\pi$
$593$	12.8302	0.526872	0.263436	+	0.964677i	$0.415144\pi$
$594$	11.9812	0.491596
$595$	−18.4612	−0.756835
$596$	21.6983	0.888797
$597$	−9.97241	−0.408144
$598$	0	0
$599$	−2.88187	−0.117750	−0.0588750	−	0.998265i	$-0.518751\pi$
$599$	−2.88187	−0.117750	−0.0588750	+	0.998265i	$0.518751\pi$
$600$	−8.57520	−0.350081
$601$	−4.06380	−0.165766	−0.0828829	−	0.996559i	$-0.526413\pi$
$601$	−4.06380	−0.165766	−0.0828829	+	0.996559i	$0.526413\pi$
$602$	40.4731	1.64956
$603$	24.3394	0.991177
$604$	1.11577	0.0454002
$605$	−14.4679	−0.588204
$606$	−24.0681	−0.977701
$607$	25.4761	1.03404	0.517021	−	0.855973i	$-0.327041\pi$
$607$	25.4761	1.03404	0.517021	+	0.855973i	$0.327041\pi$
$608$	1.00000	0.0405554
$609$	−46.2251	−1.87314
$610$	−20.9317	−0.847500
$611$	0	0
$612$	−15.7114	−0.635096
$613$	−12.0150	−0.485282	−0.242641	−	0.970116i	$-0.578014\pi$
$613$	−12.0150	−0.485282	−0.242641	+	0.970116i	$0.578014\pi$
$614$	−14.7032	−0.593374
$615$	40.0155	1.61358
$616$	−5.75094	−0.231712
$617$	−27.4244	−1.10406	−0.552032	−	0.833823i	$-0.686148\pi$
$617$	−27.4244	−1.10406	−0.552032	+	0.833823i	$0.686148\pi$
$618$	−45.2369	−1.81969
$619$	−40.9094	−1.64429	−0.822144	−	0.569279i	$-0.807222\pi$
$619$	−40.9094	−1.64429	−0.822144	+	0.569279i	$0.807222\pi$
$620$	−13.1081	−0.526435
$621$	−25.0427	−1.00493
$622$	14.9108	0.597869
$623$	−59.6725	−2.39073
$624$	0	0
$625$	−3.28062	−0.131225
$626$	−23.6699	−0.946041
$627$	3.46667	0.138445
$628$	21.4663	0.856598
$629$	−2.13574	−0.0851575
$630$	−48.9766	−1.95127
$631$	−20.3378	−0.809634	−0.404817	−	0.914398i	$-0.632665\pi$
$631$	−20.3378	−0.809634	−0.404817	+	0.914398i	$0.632665\pi$
$632$	−9.67130	−0.384704
$633$	10.0380	0.398976
$634$	−17.4182	−0.691763
$635$	4.92168	0.195311
$636$	−18.2558	−0.723889
$637$	0	0
$638$	−3.32195	−0.131517
$639$	93.8530	3.71277
$640$	−1.48707	−0.0587818
$641$	−14.1911	−0.560515	−0.280258	−	0.959925i	$-0.590420\pi$
$641$	−14.1911	−0.560515	−0.280258	+	0.959925i	$0.590420\pi$
$642$	50.4320	1.99039
$643$	8.49426	0.334981	0.167490	−	0.985874i	$-0.446434\pi$
$643$	8.49426	0.334981	0.167490	+	0.985874i	$0.446434\pi$
$644$	12.0204	0.473670
$645$	36.2804	1.42854
$646$	−2.43357	−0.0957474
$647$	−13.4982	−0.530670	−0.265335	−	0.964156i	$-0.585483\pi$
$647$	−13.4982	−0.530670	−0.265335	+	0.964156i	$0.585483\pi$
$648$	−13.3132	−0.522991
$649$	−3.30592	−0.129769
$650$	0	0
$651$	138.276	5.41948
$652$	18.2074	0.713057
$653$	34.3917	1.34585	0.672925	−	0.739711i	$-0.265038\pi$
$653$	34.3917	1.34585	0.672925	+	0.739711i	$0.265038\pi$
$654$	27.6207	1.08006
$655$	−19.1296	−0.747455
$656$	−8.75062	−0.341654
$657$	−20.4582	−0.798151
$658$	−41.4253	−1.61493
$659$	−24.8668	−0.968675	−0.484337	−	0.874881i	$-0.660939\pi$
$659$	−24.8668	−0.968675	−0.484337	+	0.874881i	$0.660939\pi$
$660$	−5.15519	−0.200666
$661$	−24.5126	−0.953429	−0.476715	−	0.879058i	$-0.658172\pi$
$661$	−24.5126	−0.953429	−0.476715	+	0.879058i	$0.658172\pi$
$662$	21.1765	0.823047
$663$	0	0
$664$	5.47241	0.212371
$665$	−7.58606	−0.294175
$666$	−5.66601	−0.219553
$667$	6.94341	0.268850
$668$	−19.6829	−0.761556
$669$	30.9527	1.19670
$670$	−5.60623	−0.216588
$671$	15.8682	0.612584
$672$	15.6870	0.605140
$673$	2.73224	0.105320	0.0526601	−	0.998612i	$-0.483230\pi$
$673$	2.73224	0.105320	0.0526601	+	0.998612i	$0.483230\pi$
$674$	−17.8159	−0.686243
$675$	29.6370	1.14073
$676$	0	0
$677$	21.8985	0.841629	0.420815	−	0.907147i	$-0.361744\pi$
$677$	21.8985	0.841629	0.420815	+	0.907147i	$0.361744\pi$
$678$	26.9004	1.03310
$679$	31.7691	1.21919
$680$	3.61889	0.138778
$681$	9.29124	0.356041
$682$	9.93719	0.380515
$683$	−26.8837	−1.02868	−0.514338	−	0.857587i	$-0.671962\pi$
$683$	−26.8837	−1.02868	−0.514338	+	0.857587i	$0.671962\pi$
$684$	−6.45613	−0.246856
$685$	22.2838	0.851419
$686$	−61.3363	−2.34183
$687$	−51.3893	−1.96062
$688$	−7.93383	−0.302474
$689$	0	0
$690$	10.7752	0.410204
$691$	17.4805	0.664988	0.332494	−	0.943105i	$-0.392110\pi$
$691$	17.4805	0.664988	0.332494	+	0.943105i	$0.392110\pi$
$692$	−0.489868	−0.0186220
$693$	37.1288	1.41041
$694$	−12.4370	−0.472103
$695$	23.7240	0.899904
$696$	9.06138	0.343471
$697$	21.2952	0.806614
$698$	−0.667333	−0.0252589
$699$	79.3239	3.00030
$700$	−14.2256	−0.537678
$701$	16.0447	0.605999	0.302999	−	0.952991i	$-0.402012\pi$
$701$	16.0447	0.605999	0.302999	+	0.952991i	$0.402012\pi$
$702$	0	0
$703$	−0.877617	−0.0331000
$704$	1.12734	0.0424883
$705$	−37.1340	−1.39855
$706$	6.28783	0.236645
$707$	−39.9272	−1.50162
$708$	9.01766	0.338904
$709$	−47.2048	−1.77281	−0.886406	−	0.462908i	$-0.846806\pi$
$709$	−47.2048	−1.77281	−0.886406	+	0.462908i	$0.846806\pi$
$710$	−21.6177	−0.811297
$711$	62.4391	2.34165
$712$	11.6974	0.438380
$713$	−20.7703	−0.777855
$714$	−38.1754	−1.42868
$715$	0	0
$716$	4.81280	0.179863
$717$	21.1975	0.791636
$718$	8.65052	0.322834
$719$	2.49883	0.0931906	0.0465953	−	0.998914i	$-0.485163\pi$
$719$	2.49883	0.0931906	0.0465953	+	0.998914i	$0.485163\pi$
$720$	9.60074	0.357799
$721$	−75.0446	−2.79481
$722$	−1.00000	−0.0372161
$723$	−55.4418	−2.06190
$724$	13.6875	0.508691
$725$	−8.21722	−0.305180
$726$	−29.9178	−1.11035
$727$	−21.3036	−0.790107	−0.395054	−	0.918658i	$-0.629274\pi$
$727$	−21.3036	−0.790107	−0.395054	+	0.918658i	$0.629274\pi$
$728$	0	0
$729$	−12.0931	−0.447892
$730$	4.71225	0.174408
$731$	19.3075	0.714114
$732$	−43.2841	−1.59983
$733$	18.7589	0.692877	0.346438	−	0.938073i	$-0.387391\pi$
$733$	18.7589	0.692877	0.346438	+	0.938073i	$0.387391\pi$
$734$	−18.5295	−0.683935
$735$	−86.9925	−3.20877
$736$	−2.35632	−0.0868553
$737$	4.25004	0.156552
$738$	56.4951	2.07961
$739$	−47.1743	−1.73534	−0.867668	−	0.497145i	$-0.834382\pi$
$739$	−47.1743	−1.73534	−0.867668	+	0.497145i	$0.834382\pi$
$740$	1.30508	0.0479758
$741$	0	0
$742$	−30.2850	−1.11180
$743$	−30.3546	−1.11360	−0.556801	−	0.830646i	$-0.687972\pi$
$743$	−30.3546	−1.11360	−0.556801	+	0.830646i	$0.687972\pi$
$744$	−27.1060	−0.993753
$745$	32.2670	1.18217
$746$	17.4523	0.638974
$747$	−35.3306	−1.29268
$748$	−2.74346	−0.100311
$749$	83.6630	3.05698
$750$	−35.6163	−1.30052
$751$	28.3623	1.03496	0.517478	−	0.855697i	$-0.326871\pi$
$751$	28.3623	1.03496	0.517478	+	0.855697i	$0.326871\pi$
$752$	8.12050	0.296124
$753$	82.2768	2.99833
$754$	0	0
$755$	1.65924	0.0603859
$756$	−54.2163	−1.97183
$757$	−9.78062	−0.355483	−0.177741	−	0.984077i	$-0.556879\pi$
$757$	−9.78062	−0.355483	−0.177741	+	0.984077i	$0.556879\pi$
$758$	−31.9627	−1.16094
$759$	−8.16859	−0.296501
$760$	1.48707	0.0539419
$761$	48.8114	1.76941	0.884705	−	0.466151i	$-0.154360\pi$
$761$	48.8114	1.76941	0.884705	+	0.466151i	$0.154360\pi$
$762$	10.1774	0.368689
$763$	45.8207	1.65882
$764$	−24.3725	−0.881765
$765$	−23.3640	−0.844729
$766$	−10.8220	−0.391016
$767$	0	0
$768$	−3.07508	−0.110962
$769$	4.79554	0.172931	0.0864657	−	0.996255i	$-0.472443\pi$
$769$	4.79554	0.172931	0.0864657	+	0.996255i	$0.472443\pi$
$770$	−8.55208	−0.308196
$771$	−4.63732	−0.167009
$772$	11.5703	0.416424
$773$	−29.0168	−1.04366	−0.521831	−	0.853049i	$-0.674751\pi$
$773$	−29.0168	−1.04366	−0.521831	+	0.853049i	$0.674751\pi$
$774$	51.2218	1.84113
$775$	24.5808	0.882968
$776$	−6.22762	−0.223558
$777$	−13.7672	−0.493895
$778$	−30.4623	−1.09212
$779$	8.75062	0.313524
$780$	0	0
$781$	16.3882	0.586417
$782$	5.73427	0.205057
$783$	−31.3173	−1.11919
$784$	19.0236	0.679414
$785$	31.9220	1.13934
$786$	−39.5576	−1.41097
$787$	11.3153	0.403347	0.201673	−	0.979453i	$-0.435362\pi$
$787$	11.3153	0.403347	0.201673	+	0.979453i	$0.435362\pi$
$788$	8.55288	0.304684
$789$	29.2723	1.04212
$790$	−14.3819	−0.511687
$791$	44.6258	1.58671
$792$	−7.27826	−0.258622
$793$	0	0
$794$	−22.4399	−0.796364
$795$	−27.1477	−0.962831
$796$	3.24298	0.114944
$797$	27.1634	0.962179	0.481089	−	0.876672i	$-0.340241\pi$
$797$	27.1634	0.962179	0.481089	+	0.876672i	$0.340241\pi$
$798$	−15.6870	−0.555314
$799$	−19.7618	−0.699121
$800$	2.78861	0.0985922
$801$	−75.5201	−2.66837
$802$	8.62169	0.304443
$803$	−3.57233	−0.126065
$804$	−11.5930	−0.408852
$805$	17.8752	0.630019
$806$	0	0
$807$	−47.5489	−1.67380
$808$	7.82683	0.275347
$809$	−31.6866	−1.11404	−0.557020	−	0.830499i	$-0.688055\pi$
$809$	−31.6866	−1.11404	−0.557020	+	0.830499i	$0.688055\pi$
$810$	−19.7977	−0.695620
$811$	12.9214	0.453732	0.226866	−	0.973926i	$-0.427152\pi$
$811$	12.9214	0.453732	0.226866	+	0.973926i	$0.427152\pi$
$812$	15.0322	0.527525
$813$	−93.1624	−3.26735
$814$	−0.989374	−0.0346776
$815$	27.0758	0.948423
$816$	7.48341	0.261972
$817$	7.93383	0.277570
$818$	−14.0179	−0.490123
$819$	0	0
$820$	−13.0128	−0.454428
$821$	−29.1735	−1.01816	−0.509080	−	0.860719i	$-0.670014\pi$
$821$	−29.1735	−1.01816	−0.509080	+	0.860719i	$0.670014\pi$
$822$	46.0800	1.60723
$823$	21.7761	0.759066	0.379533	−	0.925178i	$-0.376085\pi$
$823$	21.7761	0.759066	0.379533	+	0.925178i	$0.376085\pi$
$824$	14.7108	0.512475
$825$	9.66718	0.336568
$826$	14.9596	0.520512
$827$	20.1121	0.699366	0.349683	−	0.936868i	$-0.386289\pi$
$827$	20.1121	0.699366	0.349683	+	0.936868i	$0.386289\pi$
$828$	15.2127	0.528679
$829$	−32.0440	−1.11293	−0.556467	−	0.830870i	$-0.687843\pi$
$829$	−32.0440	−1.11293	−0.556467	+	0.830870i	$0.687843\pi$
$830$	8.13788	0.282470
$831$	−31.4281	−1.09023
$832$	0	0
$833$	−46.2951	−1.60403
$834$	49.0583	1.69875
$835$	−29.2700	−1.01293
$836$	−1.12734	−0.0389899
$837$	93.6816	3.23811
$838$	27.4326	0.947643
$839$	34.3402	1.18556	0.592778	−	0.805366i	$-0.298031\pi$
$839$	34.3402	1.18556	0.592778	+	0.805366i	$0.298031\pi$
$840$	23.3278	0.804884
$841$	−20.3169	−0.700582
$842$	32.8186	1.13100
$843$	60.1458	2.07153
$844$	−3.26431	−0.112362
$845$	0	0
$846$	−52.4270	−1.80248
$847$	−49.6314	−1.70535
$848$	5.93669	0.203867
$849$	−14.8192	−0.508595
$850$	−6.78626	−0.232767
$851$	2.06795	0.0708885
$852$	−44.7026	−1.53149
$853$	22.0045	0.753421	0.376710	−	0.926331i	$-0.377055\pi$
$853$	22.0045	0.753421	0.376710	+	0.926331i	$0.377055\pi$
$854$	−71.8051	−2.45712
$855$	−9.60074	−0.328338
$856$	−16.4002	−0.560548
$857$	3.25576	0.111215	0.0556074	−	0.998453i	$-0.482290\pi$
$857$	3.25576	0.111215	0.0556074	+	0.998453i	$0.482290\pi$
$858$	0	0
$859$	−38.9601	−1.32930	−0.664651	−	0.747154i	$-0.731420\pi$
$859$	−38.9601	−1.32930	−0.664651	+	0.747154i	$0.731420\pi$
$860$	−11.7982	−0.402315
$861$	137.271	4.67819
$862$	−21.1256	−0.719540
$863$	22.0396	0.750236	0.375118	−	0.926977i	$-0.377602\pi$
$863$	22.0396	0.750236	0.375118	+	0.926977i	$0.377602\pi$
$864$	10.6279	0.361567
$865$	−0.728470	−0.0247687
$866$	−22.8202	−0.775462
$867$	34.0650	1.15691
$868$	−44.9668	−1.52627
$869$	10.9029	0.369854
$870$	13.4749	0.456844
$871$	0	0
$872$	−8.98211	−0.304173
$873$	40.2063	1.36078
$874$	2.35632	0.0797039
$875$	−59.0849	−1.99743
$876$	9.74434	0.329231
$877$	−16.7065	−0.564138	−0.282069	−	0.959394i	$-0.591021\pi$
$877$	−16.7065	−0.564138	−0.282069	+	0.959394i	$0.591021\pi$
$878$	7.76870	0.262181
$879$	−3.94127	−0.132936
$880$	1.67644	0.0565128
$881$	−1.97865	−0.0666625	−0.0333313	−	0.999444i	$-0.510612\pi$
$881$	−1.97865	−0.0666625	−0.0333313	+	0.999444i	$0.510612\pi$
$882$	−122.819	−4.13552
$883$	−9.60749	−0.323318	−0.161659	−	0.986847i	$-0.551684\pi$
$883$	−9.60749	−0.323318	−0.161659	+	0.986847i	$0.551684\pi$
$884$	0	0
$885$	13.4099	0.450770
$886$	−12.6726	−0.425743
$887$	−24.9360	−0.837270	−0.418635	−	0.908155i	$-0.637491\pi$
$887$	−24.9360	−0.837270	−0.418635	+	0.908155i	$0.637491\pi$
$888$	2.69875	0.0905640
$889$	16.8836	0.566257
$890$	17.3949	0.583080
$891$	15.0085	0.502804
$892$	−10.0657	−0.337023
$893$	−8.12050	−0.271742
$894$	66.7241	2.23159
$895$	7.15700	0.239232
$896$	−5.10133	−0.170424
$897$	0	0
$898$	−11.0057	−0.367264
$899$	−25.9744	−0.866295
$900$	−18.0036	−0.600120
$901$	−14.4473	−0.481310
$902$	9.86494	0.328467
$903$	124.458	4.14171
$904$	−8.74788	−0.290950
$905$	20.3543	0.676600
$906$	3.43110	0.113991
$907$	4.58782	0.152336	0.0761680	−	0.997095i	$-0.475731\pi$
$907$	4.58782	0.152336	0.0761680	+	0.997095i	$0.475731\pi$
$908$	−3.02146	−0.100271
$909$	−50.5310	−1.67601
$910$	0	0
$911$	−27.1065	−0.898078	−0.449039	−	0.893512i	$-0.648233\pi$
$911$	−27.1065	−0.898078	−0.449039	+	0.893512i	$0.648233\pi$
$912$	3.07508	0.101826
$913$	−6.16927	−0.204173
$914$	−12.4216	−0.410869
$915$	−64.3667	−2.12790
$916$	16.7115	0.552164
$917$	−65.6231	−2.16707
$918$	−25.8636	−0.853627
$919$	−33.2901	−1.09814	−0.549070	−	0.835777i	$-0.685018\pi$
$919$	−33.2901	−1.09814	−0.549070	+	0.835777i	$0.685018\pi$
$920$	−3.50403	−0.115524
$921$	−45.2136	−1.48984
$922$	27.7593	0.914203
$923$	0	0
$924$	−17.6846	−0.581781
$925$	−2.44733	−0.0804678
$926$	−11.2619	−0.370088
$927$	−94.9747	−3.11938
$928$	−2.94671	−0.0967305
$929$	60.1149	1.97231	0.986153	−	0.165836i	$-0.0530321\pi$
$929$	60.1149	1.97231	0.986153	+	0.165836i	$0.0530321\pi$
$930$	−40.3086	−1.32177
$931$	−19.0236	−0.623473
$932$	−25.7957	−0.844966
$933$	45.8520	1.50113
$934$	11.7718	0.385186
$935$	−4.07973	−0.133421
$936$	0	0
$937$	2.99498	0.0978418	0.0489209	−	0.998803i	$-0.484422\pi$
$937$	2.99498	0.0978418	0.0489209	+	0.998803i	$0.484422\pi$
$938$	−19.2319	−0.627943
$939$	−72.7870	−2.37531
$940$	12.0758	0.393869
$941$	24.3114	0.792527	0.396264	−	0.918137i	$-0.370307\pi$
$941$	24.3114	0.792527	0.396264	+	0.918137i	$0.370307\pi$
$942$	66.0106	2.15074
$943$	−20.6193	−0.671457
$944$	−2.93249	−0.0954446
$945$	−80.6237	−2.62269
$946$	8.94413	0.290799
$947$	−43.3641	−1.40914	−0.704571	−	0.709633i	$-0.748861\pi$
$947$	−43.3641	−1.40914	−0.704571	+	0.709633i	$0.748861\pi$
$948$	−29.7400	−0.965912
$949$	0	0
$950$	−2.78861	−0.0904744
$951$	−53.5623	−1.73688
$952$	12.4144	0.402354
$953$	42.6875	1.38278	0.691392	−	0.722480i	$-0.256998\pi$
$953$	42.6875	1.38278	0.691392	+	0.722480i	$0.256998\pi$
$954$	−38.3280	−1.24091
$955$	−36.2437	−1.17282
$956$	−6.89332	−0.222946
$957$	−10.2153	−0.330213
$958$	23.3639	0.754854
$959$	76.4433	2.46848
$960$	−4.57288	−0.147589
$961$	46.6992	1.50642
$962$	0	0
$963$	105.882	3.41200
$964$	18.0294	0.580687
$965$	17.2059	0.553878
$966$	36.9637	1.18929
$967$	−15.9059	−0.511500	−0.255750	−	0.966743i	$-0.582322\pi$
$967$	−15.9059	−0.511500	−0.255750	+	0.966743i	$0.582322\pi$
$968$	9.72910	0.312705
$969$	−7.48341	−0.240402
$970$	−9.26093	−0.297351
$971$	−20.1704	−0.647300	−0.323650	−	0.946177i	$-0.604910\pi$
$971$	−20.1704	−0.647300	−0.323650	+	0.946177i	$0.604910\pi$
$972$	−9.05552	−0.290456
$973$	81.3840	2.60905
$974$	−22.1847	−0.710845
$975$	0	0
$976$	14.0758	0.450554
$977$	−39.0516	−1.24937	−0.624686	−	0.780876i	$-0.714773\pi$
$977$	−39.0516	−1.24937	−0.624686	+	0.780876i	$0.714773\pi$
$978$	55.9893	1.79034
$979$	−13.1870	−0.421458
$980$	28.2895	0.903675
$981$	57.9896	1.85147
$982$	6.24016	0.199131
$983$	44.0123	1.40377	0.701887	−	0.712289i	$-0.252341\pi$
$983$	44.0123	1.40377	0.701887	+	0.712289i	$0.252341\pi$
$984$	−26.9089	−0.857824
$985$	12.7188	0.405254
$986$	7.17102	0.228372
$987$	−127.386	−4.05475
$988$	0	0
$989$	−18.6947	−0.594456
$990$	−10.8233	−0.343988
$991$	−32.0291	−1.01744	−0.508719	−	0.860933i	$-0.669881\pi$
$991$	−32.0291	−1.01744	−0.508719	+	0.860933i	$0.669881\pi$
$992$	8.81471	0.279867
$993$	65.1194	2.06650
$994$	−74.1583	−2.35216
$995$	4.82255	0.152885
$996$	16.8281	0.533219
$997$	−8.49717	−0.269108	−0.134554	−	0.990906i	$-0.542960\pi$
$997$	−8.49717	−0.269108	−0.134554	+	0.990906i	$0.542960\pi$
$998$	6.07304	0.192239
$999$	−9.32720	−0.295100

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bi.1.2	✓	9	1.1	even	1	trivial
6422.2.a.bk.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.07508 q^{3} +1.00000 q^{4} +1.48707 q^{5} +3.07508 q^{6} +5.10133 q^{7} -1.00000 q^{8} +6.45613 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.07508 q^{3} +1.00000 q^{4} +1.48707 q^{5} +3.07508 q^{6} +5.10133 q^{7} -1.00000 q^{8} +6.45613 q^{9} -1.48707 q^{10} +1.12734 q^{11} -3.07508 q^{12} -5.10133 q^{14} -4.57288 q^{15} +1.00000 q^{16} -2.43357 q^{17} -6.45613 q^{18} -1.00000 q^{19} +1.48707 q^{20} -15.6870 q^{21} -1.12734 q^{22} +2.35632 q^{23} +3.07508 q^{24} -2.78861 q^{25} -10.6279 q^{27} +5.10133 q^{28} +2.94671 q^{29} +4.57288 q^{30} -8.81471 q^{31} -1.00000 q^{32} -3.46667 q^{33} +2.43357 q^{34} +7.58606 q^{35} +6.45613 q^{36} +0.877617 q^{37} +1.00000 q^{38} -1.48707 q^{40} -8.75062 q^{41} +15.6870 q^{42} -7.93383 q^{43} +1.12734 q^{44} +9.60074 q^{45} -2.35632 q^{46} +8.12050 q^{47} -3.07508 q^{48} +19.0236 q^{49} +2.78861 q^{50} +7.48341 q^{51} +5.93669 q^{53} +10.6279 q^{54} +1.67644 q^{55} -5.10133 q^{56} +3.07508 q^{57} -2.94671 q^{58} -2.93249 q^{59} -4.57288 q^{60} +14.0758 q^{61} +8.81471 q^{62} +32.9348 q^{63} +1.00000 q^{64} +3.46667 q^{66} +3.76997 q^{67} -2.43357 q^{68} -7.24589 q^{69} -7.58606 q^{70} +14.5371 q^{71} -6.45613 q^{72} -3.16881 q^{73} -0.877617 q^{74} +8.57520 q^{75} -1.00000 q^{76} +5.75094 q^{77} +9.67130 q^{79} +1.48707 q^{80} +13.3132 q^{81} +8.75062 q^{82} -5.47241 q^{83} -15.6870 q^{84} -3.61889 q^{85} +7.93383 q^{86} -9.06138 q^{87} -1.12734 q^{88} -11.6974 q^{89} -9.60074 q^{90} +2.35632 q^{92} +27.1060 q^{93} -8.12050 q^{94} -1.48707 q^{95} +3.07508 q^{96} +6.22762 q^{97} -19.0236 q^{98} +7.27826 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{2} - 5 q^{3} + 9 q^{4} - q^{5} + 5 q^{6} + 13 q^{7} - 9 q^{8} + 10 q^{9}+O(q^{10}) \) 9 * q - 9 * q^2 - 5 * q^3 + 9 * q^4 - q^5 + 5 * q^6 + 13 * q^7 - 9 * q^8 + 10 * q^9	\( 9 q - 9 q^{2} - 5 q^{3} + 9 q^{4} - q^{5} + 5 q^{6} + 13 q^{7} - 9 q^{8} + 10 q^{9} + q^{10} + 13 q^{11} - 5 q^{12} - 13 q^{14} + q^{15} + 9 q^{16} - 12 q^{17} - 10 q^{18} - 9 q^{19} - q^{20} - 18 q^{21} - 13 q^{22} - 22 q^{23} + 5 q^{24} + 4 q^{25} - 26 q^{27} + 13 q^{28} + 12 q^{29} - q^{30} - q^{31} - 9 q^{32} + 28 q^{33} + 12 q^{34} - 18 q^{35} + 10 q^{36} + 25 q^{37} + 9 q^{38} + q^{40} - 11 q^{41} + 18 q^{42} - 10 q^{43} + 13 q^{44} + q^{45} + 22 q^{46} + 12 q^{47} - 5 q^{48} + 2 q^{49} - 4 q^{50} + 35 q^{51} + 9 q^{53} + 26 q^{54} + 18 q^{55} - 13 q^{56} + 5 q^{57} - 12 q^{58} - 10 q^{59} + q^{60} + 32 q^{61} + q^{62} + 63 q^{63} + 9 q^{64} - 28 q^{66} + 73 q^{67} - 12 q^{68} + 2 q^{69} + 18 q^{70} + 51 q^{71} - 10 q^{72} + 14 q^{73} - 25 q^{74} - 49 q^{75} - 9 q^{76} + 18 q^{77} - 28 q^{79} - q^{80} + 29 q^{81} + 11 q^{82} + 22 q^{83} - 18 q^{84} + 51 q^{85} + 10 q^{86} - 20 q^{87} - 13 q^{88} - 3 q^{89} - q^{90} - 22 q^{92} + 59 q^{93} - 12 q^{94} + q^{95} + 5 q^{96} - 2 q^{98} - 25 q^{99}+O(q^{100}) \) 9 * q - 9 * q^2 - 5 * q^3 + 9 * q^4 - q^5 + 5 * q^6 + 13 * q^7 - 9 * q^8 + 10 * q^9 + q^10 + 13 * q^11 - 5 * q^12 - 13 * q^14 + q^15 + 9 * q^16 - 12 * q^17 - 10 * q^18 - 9 * q^19 - q^20 - 18 * q^21 - 13 * q^22 - 22 * q^23 + 5 * q^24 + 4 * q^25 - 26 * q^27 + 13 * q^28 + 12 * q^29 - q^30 - q^31 - 9 * q^32 + 28 * q^33 + 12 * q^34 - 18 * q^35 + 10 * q^36 + 25 * q^37 + 9 * q^38 + q^40 - 11 * q^41 + 18 * q^42 - 10 * q^43 + 13 * q^44 + q^45 + 22 * q^46 + 12 * q^47 - 5 * q^48 + 2 * q^49 - 4 * q^50 + 35 * q^51 + 9 * q^53 + 26 * q^54 + 18 * q^55 - 13 * q^56 + 5 * q^57 - 12 * q^58 - 10 * q^59 + q^60 + 32 * q^61 + q^62 + 63 * q^63 + 9 * q^64 - 28 * q^66 + 73 * q^67 - 12 * q^68 + 2 * q^69 + 18 * q^70 + 51 * q^71 - 10 * q^72 + 14 * q^73 - 25 * q^74 - 49 * q^75 - 9 * q^76 + 18 * q^77 - 28 * q^79 - q^80 + 29 * q^81 + 11 * q^82 + 22 * q^83 - 18 * q^84 + 51 * q^85 + 10 * q^86 - 20 * q^87 - 13 * q^88 - 3 * q^89 - q^90 - 22 * q^92 + 59 * q^93 - 12 * q^94 + q^95 + 5 * q^96 - 2 * q^98 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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