LMFDB - Embedded newform 6422.2.a.bh.1.8

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 4x^{6} + 22x^{5} + 3x^{4} - 28x^{3} + 7x^{2} + 6x - 2 $ x^8 - 4x^7 - 4x^6 + 22x^5 + 3x^4 - 28x^3 + 7x^2 + 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.8
Root			$-1.74849$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.74849 q^{3} +1.00000 q^{4} -2.50715 q^{5} +1.74849 q^{6} +1.18596 q^{7} +1.00000 q^{8} +0.0572044 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.74849 q^{3} +1.00000 q^{4} -2.50715 q^{5} +1.74849 q^{6} +1.18596 q^{7} +1.00000 q^{8} +0.0572044 q^{9} -2.50715 q^{10} +1.65788 q^{11} +1.74849 q^{12} +1.18596 q^{14} -4.38372 q^{15} +1.00000 q^{16} -5.69893 q^{17} +0.0572044 q^{18} -1.00000 q^{19} -2.50715 q^{20} +2.07364 q^{21} +1.65788 q^{22} -1.77865 q^{23} +1.74849 q^{24} +1.28580 q^{25} -5.14544 q^{27} +1.18596 q^{28} -2.45819 q^{29} -4.38372 q^{30} -3.34648 q^{31} +1.00000 q^{32} +2.89878 q^{33} -5.69893 q^{34} -2.97339 q^{35} +0.0572044 q^{36} -4.12283 q^{37} -1.00000 q^{38} -2.50715 q^{40} -4.27802 q^{41} +2.07364 q^{42} -3.06400 q^{43} +1.65788 q^{44} -0.143420 q^{45} -1.77865 q^{46} -2.38078 q^{47} +1.74849 q^{48} -5.59349 q^{49} +1.28580 q^{50} -9.96451 q^{51} +13.1073 q^{53} -5.14544 q^{54} -4.15655 q^{55} +1.18596 q^{56} -1.74849 q^{57} -2.45819 q^{58} +9.86518 q^{59} -4.38372 q^{60} -14.7547 q^{61} -3.34648 q^{62} +0.0678424 q^{63} +1.00000 q^{64} +2.89878 q^{66} +0.571177 q^{67} -5.69893 q^{68} -3.10995 q^{69} -2.97339 q^{70} -11.0399 q^{71} +0.0572044 q^{72} +12.7524 q^{73} -4.12283 q^{74} +2.24821 q^{75} -1.00000 q^{76} +1.96618 q^{77} -1.11970 q^{79} -2.50715 q^{80} -9.16834 q^{81} -4.27802 q^{82} +10.9373 q^{83} +2.07364 q^{84} +14.2881 q^{85} -3.06400 q^{86} -4.29810 q^{87} +1.65788 q^{88} -4.26198 q^{89} -0.143420 q^{90} -1.77865 q^{92} -5.85127 q^{93} -2.38078 q^{94} +2.50715 q^{95} +1.74849 q^{96} -0.161643 q^{97} -5.59349 q^{98} +0.0948380 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} + 8 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^7 + 8 * q^8	$ 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} + 8 q^{8} - 2 q^{10} - 10 q^{11} - 4 q^{12} - 2 q^{14} + 8 q^{16} + 2 q^{17} - 8 q^{19} - 2 q^{20} + 12 q^{21} - 10 q^{22} - 8 q^{23} - 4 q^{24} - 14 q^{25} - 22 q^{27} - 2 q^{28} + 8 q^{29} - 12 q^{31} + 8 q^{32} + 4 q^{33} + 2 q^{34} - 10 q^{35} - 8 q^{38} - 2 q^{40} + 2 q^{41} + 12 q^{42} - 16 q^{43} - 10 q^{44} - 12 q^{45} - 8 q^{46} + 12 q^{47} - 4 q^{48} - 14 q^{49} - 14 q^{50} - 22 q^{51} - 24 q^{53} - 22 q^{54} - 10 q^{55} - 2 q^{56} + 4 q^{57} + 8 q^{58} - 4 q^{59} - 26 q^{61} - 12 q^{62} - 12 q^{63} + 8 q^{64} + 4 q^{66} + 10 q^{67} + 2 q^{68} + 6 q^{69} - 10 q^{70} - 36 q^{71} + 20 q^{73} + 6 q^{75} - 8 q^{76} - 12 q^{77} - 22 q^{79} - 2 q^{80} + 36 q^{81} + 2 q^{82} + 18 q^{83} + 12 q^{84} + 26 q^{85} - 16 q^{86} - 38 q^{87} - 10 q^{88} - 18 q^{89} - 12 q^{90} - 8 q^{92} + 12 q^{93} + 12 q^{94} + 2 q^{95} - 4 q^{96} - 8 q^{97} - 14 q^{98} - 8 q^{99}+O(q^{100}) $ 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^7 + 8 * q^8 - 2 * q^10 - 10 * q^11 - 4 * q^12 - 2 * q^14 + 8 * q^16 + 2 * q^17 - 8 * q^19 - 2 * q^20 + 12 * q^21 - 10 * q^22 - 8 * q^23 - 4 * q^24 - 14 * q^25 - 22 * q^27 - 2 * q^28 + 8 * q^29 - 12 * q^31 + 8 * q^32 + 4 * q^33 + 2 * q^34 - 10 * q^35 - 8 * q^38 - 2 * q^40 + 2 * q^41 + 12 * q^42 - 16 * q^43 - 10 * q^44 - 12 * q^45 - 8 * q^46 + 12 * q^47 - 4 * q^48 - 14 * q^49 - 14 * q^50 - 22 * q^51 - 24 * q^53 - 22 * q^54 - 10 * q^55 - 2 * q^56 + 4 * q^57 + 8 * q^58 - 4 * q^59 - 26 * q^61 - 12 * q^62 - 12 * q^63 + 8 * q^64 + 4 * q^66 + 10 * q^67 + 2 * q^68 + 6 * q^69 - 10 * q^70 - 36 * q^71 + 20 * q^73 + 6 * q^75 - 8 * q^76 - 12 * q^77 - 22 * q^79 - 2 * q^80 + 36 * q^81 + 2 * q^82 + 18 * q^83 + 12 * q^84 + 26 * q^85 - 16 * q^86 - 38 * q^87 - 10 * q^88 - 18 * q^89 - 12 * q^90 - 8 * q^92 + 12 * q^93 + 12 * q^94 + 2 * q^95 - 4 * q^96 - 8 * q^97 - 14 * q^98 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.74849	1.00949	0.504745	−	0.863269i	$-0.331587\pi$
$3$	1.74849	1.00949	0.504745	+	0.863269i	$0.331587\pi$
$4$	1.00000	0.500000
$5$	−2.50715	−1.12123	−0.560616	−	0.828076i	$-0.689436\pi$
$5$	−2.50715	−1.12123	−0.560616	+	0.828076i	$0.689436\pi$
$6$	1.74849	0.713817
$7$	1.18596	0.448252	0.224126	−	0.974560i	$-0.428047\pi$
$7$	1.18596	0.448252	0.224126	+	0.974560i	$0.428047\pi$
$8$	1.00000	0.353553
$9$	0.0572044	0.0190681
$10$	−2.50715	−0.792830
$11$	1.65788	0.499869	0.249934	−	0.968263i	$-0.419591\pi$
$11$	1.65788	0.499869	0.249934	+	0.968263i	$0.419591\pi$
$12$	1.74849	0.504745
$13$	0	0
$13$	0	0
$14$	1.18596	0.316962
$15$	−4.38372	−1.13187
$16$	1.00000	0.250000
$17$	−5.69893	−1.38219	−0.691097	−	0.722762i	$-0.742872\pi$
$17$	−5.69893	−1.38219	−0.691097	+	0.722762i	$0.742872\pi$
$18$	0.0572044	0.0134832
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.50715	−0.560616
$21$	2.07364	0.452505
$22$	1.65788	0.353461
$23$	−1.77865	−0.370874	−0.185437	−	0.982656i	$-0.559370\pi$
$23$	−1.77865	−0.370874	−0.185437	+	0.982656i	$0.559370\pi$
$24$	1.74849	0.356908
$25$	1.28580	0.257160
$26$	0	0
$27$	−5.14544	−0.990240
$28$	1.18596	0.224126
$29$	−2.45819	−0.456474	−0.228237	−	0.973606i	$-0.573296\pi$
$29$	−2.45819	−0.456474	−0.228237	+	0.973606i	$0.573296\pi$
$30$	−4.38372	−0.800354
$31$	−3.34648	−0.601045	−0.300523	−	0.953775i	$-0.597161\pi$
$31$	−3.34648	−0.601045	−0.300523	+	0.953775i	$0.597161\pi$
$32$	1.00000	0.176777
$33$	2.89878	0.504612
$34$	−5.69893	−0.977359
$35$	−2.97339	−0.502594
$36$	0.0572044	0.00953407
$37$	−4.12283	−0.677789	−0.338895	−	0.940824i	$-0.610053\pi$
$37$	−4.12283	−0.677789	−0.338895	+	0.940824i	$0.610053\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−2.50715	−0.396415
$41$	−4.27802	−0.668115	−0.334057	−	0.942553i	$-0.608418\pi$
$41$	−4.27802	−0.668115	−0.334057	+	0.942553i	$0.608418\pi$
$42$	2.07364	0.319970
$43$	−3.06400	−0.467255	−0.233627	−	0.972326i	$-0.575060\pi$
$43$	−3.06400	−0.467255	−0.233627	+	0.972326i	$0.575060\pi$
$44$	1.65788	0.249934
$45$	−0.143420	−0.0213798
$46$	−1.77865	−0.262248
$47$	−2.38078	−0.347272	−0.173636	−	0.984810i	$-0.555552\pi$
$47$	−2.38078	−0.347272	−0.173636	+	0.984810i	$0.555552\pi$
$48$	1.74849	0.252372
$49$	−5.59349	−0.799070
$50$	1.28580	0.181840
$51$	−9.96451	−1.39531
$52$	0	0
$53$	13.1073	1.80042	0.900212	−	0.435451i	$-0.143411\pi$
$53$	13.1073	1.80042	0.900212	+	0.435451i	$0.143411\pi$
$54$	−5.14544	−0.700205
$55$	−4.15655	−0.560469
$56$	1.18596	0.158481
$57$	−1.74849	−0.231593
$58$	−2.45819	−0.322776
$59$	9.86518	1.28434	0.642168	−	0.766564i	$-0.278035\pi$
$59$	9.86518	1.28434	0.642168	+	0.766564i	$0.278035\pi$
$60$	−4.38372	−0.565936
$61$	−14.7547	−1.88915	−0.944574	−	0.328298i	$-0.893525\pi$
$61$	−14.7547	−1.88915	−0.944574	+	0.328298i	$0.893525\pi$
$62$	−3.34648	−0.425003
$63$	0.0678424	0.00854733
$64$	1.00000	0.125000
$65$	0	0
$66$	2.89878	0.356815
$67$	0.571177	0.0697804	0.0348902	−	0.999391i	$-0.488892\pi$
$67$	0.571177	0.0697804	0.0348902	+	0.999391i	$0.488892\pi$
$68$	−5.69893	−0.691097
$69$	−3.10995	−0.374394
$70$	−2.97339	−0.355388
$71$	−11.0399	−1.31019	−0.655096	−	0.755546i	$-0.727372\pi$
$71$	−11.0399	−1.31019	−0.655096	+	0.755546i	$0.727372\pi$
$72$	0.0572044	0.00674161
$73$	12.7524	1.49255	0.746277	−	0.665635i	$-0.231839\pi$
$73$	12.7524	1.49255	0.746277	+	0.665635i	$0.231839\pi$
$74$	−4.12283	−0.479269
$75$	2.24821	0.259601
$76$	−1.00000	−0.114708
$77$	1.96618	0.224067
$78$	0	0
$79$	−1.11970	−0.125976	−0.0629880	−	0.998014i	$-0.520063\pi$
$79$	−1.11970	−0.125976	−0.0629880	+	0.998014i	$0.520063\pi$
$80$	−2.50715	−0.280308
$81$	−9.16834	−1.01870
$82$	−4.27802	−0.472428
$83$	10.9373	1.20053	0.600263	−	0.799802i	$-0.295062\pi$
$83$	10.9373	1.20053	0.600263	+	0.799802i	$0.295062\pi$
$84$	2.07364	0.226253
$85$	14.2881	1.54976
$86$	−3.06400	−0.330399
$87$	−4.29810	−0.460805
$88$	1.65788	0.176730
$89$	−4.26198	−0.451769	−0.225885	−	0.974154i	$-0.572527\pi$
$89$	−4.26198	−0.451769	−0.225885	+	0.974154i	$0.572527\pi$
$90$	−0.143420	−0.0151178
$91$	0	0
$92$	−1.77865	−0.185437
$93$	−5.85127	−0.606749
$94$	−2.38078	−0.245559
$95$	2.50715	0.257228
$96$	1.74849	0.178454
$97$	−0.161643	−0.0164123	−0.00820617	−	0.999966i	$-0.502612\pi$
$97$	−0.161643	−0.0164123	−0.00820617	+	0.999966i	$0.502612\pi$
$98$	−5.59349	−0.565028
$99$	0.0948380	0.00953158
$100$	1.28580	0.128580
$101$	2.97676	0.296199	0.148100	−	0.988972i	$-0.452684\pi$
$101$	2.97676	0.296199	0.148100	+	0.988972i	$0.452684\pi$
$102$	−9.96451	−0.986633
$103$	9.09960	0.896611	0.448305	−	0.893881i	$-0.352028\pi$
$103$	9.09960	0.896611	0.448305	+	0.893881i	$0.352028\pi$
$104$	0	0
$105$	−5.19893	−0.507363
$106$	13.1073	1.27309
$107$	−13.1687	−1.27307	−0.636534	−	0.771249i	$-0.719633\pi$
$107$	−13.1687	−1.27307	−0.636534	+	0.771249i	$0.719633\pi$
$108$	−5.14544	−0.495120
$109$	−4.15792	−0.398257	−0.199128	−	0.979973i	$-0.563811\pi$
$109$	−4.15792	−0.398257	−0.199128	+	0.979973i	$0.563811\pi$
$110$	−4.15655	−0.396311
$111$	−7.20871	−0.684221
$112$	1.18596	0.112063
$113$	−6.61266	−0.622067	−0.311033	−	0.950399i	$-0.600675\pi$
$113$	−6.61266	−0.622067	−0.311033	+	0.950399i	$0.600675\pi$
$114$	−1.74849	−0.163761
$115$	4.45935	0.415836
$116$	−2.45819	−0.228237
$117$	0	0
$118$	9.86518	0.908163
$119$	−6.75872	−0.619571
$120$	−4.38372	−0.400177
$121$	−8.25144	−0.750131
$122$	−14.7547	−1.33583
$123$	−7.48006	−0.674454
$124$	−3.34648	−0.300523
$125$	9.31205	0.832895
$126$	0.0678424	0.00604388
$127$	11.1835	0.992376	0.496188	−	0.868215i	$-0.334733\pi$
$127$	11.1835	0.992376	0.496188	+	0.868215i	$0.334733\pi$
$128$	1.00000	0.0883883
$129$	−5.35735	−0.471689
$130$	0	0
$131$	−3.62320	−0.316560	−0.158280	−	0.987394i	$-0.550595\pi$
$131$	−3.62320	−0.316560	−0.158280	+	0.987394i	$0.550595\pi$
$132$	2.89878	0.252306
$133$	−1.18596	−0.102836
$134$	0.571177	0.0493422
$135$	12.9004	1.11029
$136$	−5.69893	−0.488680
$137$	0.286896	0.0245112	0.0122556	−	0.999925i	$-0.496099\pi$
$137$	0.286896	0.0245112	0.0122556	+	0.999925i	$0.496099\pi$
$138$	−3.10995	−0.264736
$139$	−11.9839	−1.01646	−0.508232	−	0.861220i	$-0.669701\pi$
$139$	−11.9839	−1.01646	−0.508232	+	0.861220i	$0.669701\pi$
$140$	−2.97339	−0.251297
$141$	−4.16276	−0.350568
$142$	−11.0399	−0.926445
$143$	0	0
$144$	0.0572044	0.00476704
$145$	6.16304	0.511813
$146$	12.7524	1.05540
$147$	−9.78014	−0.806653
$148$	−4.12283	−0.338895
$149$	0.660806	0.0541353	0.0270677	−	0.999634i	$-0.491383\pi$
$149$	0.660806	0.0541353	0.0270677	+	0.999634i	$0.491383\pi$
$150$	2.24821	0.183565
$151$	−5.62805	−0.458004	−0.229002	−	0.973426i	$-0.573546\pi$
$151$	−5.62805	−0.458004	−0.229002	+	0.973426i	$0.573546\pi$
$152$	−1.00000	−0.0811107
$153$	−0.326004	−0.0263559
$154$	1.96618	0.158439
$155$	8.39012	0.673911
$156$	0	0
$157$	−14.8599	−1.18595	−0.592973	−	0.805223i	$-0.702046\pi$
$157$	−14.8599	−1.18595	−0.592973	+	0.805223i	$0.702046\pi$
$158$	−1.11970	−0.0890785
$159$	22.9179	1.81751
$160$	−2.50715	−0.198208
$161$	−2.10941	−0.166245
$162$	−9.16834	−0.720333
$163$	15.5601	1.21876	0.609381	−	0.792878i	$-0.291418\pi$
$163$	15.5601	1.21876	0.609381	+	0.792878i	$0.291418\pi$
$164$	−4.27802	−0.334057
$165$	−7.26767	−0.565787
$166$	10.9373	0.848901
$167$	20.7297	1.60412	0.802058	−	0.597247i	$-0.203739\pi$
$167$	20.7297	1.60412	0.802058	+	0.597247i	$0.203739\pi$
$168$	2.07364	0.159985
$169$	0	0
$170$	14.2881	1.09585
$171$	−0.0572044	−0.00437453
$172$	−3.06400	−0.233627
$173$	1.44324	0.109728	0.0548638	−	0.998494i	$-0.482528\pi$
$173$	1.44324	0.109728	0.0548638	+	0.998494i	$0.482528\pi$
$174$	−4.29810	−0.325838
$175$	1.52491	0.115273
$176$	1.65788	0.124967
$177$	17.2491	1.29652
$178$	−4.26198	−0.319449
$179$	−23.4293	−1.75119	−0.875594	−	0.483048i	$-0.839530\pi$
$179$	−23.4293	−1.75119	−0.875594	+	0.483048i	$0.839530\pi$
$180$	−0.143420	−0.0106899
$181$	−18.9196	−1.40628	−0.703140	−	0.711051i	$-0.748220\pi$
$181$	−18.9196	−1.40628	−0.703140	+	0.711051i	$0.748220\pi$
$182$	0	0
$183$	−25.7984	−1.90707
$184$	−1.77865	−0.131124
$185$	10.3366	0.759959
$186$	−5.85127	−0.429036
$187$	−9.44814	−0.690916
$188$	−2.38078	−0.173636
$189$	−6.10230	−0.443877
$190$	2.50715	0.181888
$191$	−5.67838	−0.410873	−0.205436	−	0.978670i	$-0.565861\pi$
$191$	−5.67838	−0.410873	−0.205436	+	0.978670i	$0.565861\pi$
$192$	1.74849	0.126186
$193$	17.5233	1.26135	0.630677	−	0.776046i	$-0.282777\pi$
$193$	17.5233	1.26135	0.630677	+	0.776046i	$0.282777\pi$
$194$	−0.161643	−0.0116053
$195$	0	0
$196$	−5.59349	−0.399535
$197$	−17.0594	−1.21543	−0.607717	−	0.794153i	$-0.707915\pi$
$197$	−17.0594	−1.21543	−0.607717	+	0.794153i	$0.707915\pi$
$198$	0.0948380	0.00673984
$199$	−1.98089	−0.140421	−0.0702107	−	0.997532i	$-0.522367\pi$
$199$	−1.98089	−0.140421	−0.0702107	+	0.997532i	$0.522367\pi$
$200$	1.28580	0.0909199
$201$	0.998696	0.0704426
$202$	2.97676	0.209444
$203$	−2.91532	−0.204615
$204$	−9.96451	−0.697655
$205$	10.7256	0.749111
$206$	9.09960	0.633999
$207$	−0.101747	−0.00707189
$208$	0	0
$209$	−1.65788	−0.114678
$210$	−5.19893	−0.358760
$211$	−13.7028	−0.943342	−0.471671	−	0.881775i	$-0.656349\pi$
$211$	−13.7028	−0.943342	−0.471671	+	0.881775i	$0.656349\pi$
$212$	13.1073	0.900212
$213$	−19.3031	−1.32262
$214$	−13.1687	−0.900195
$215$	7.68190	0.523901
$216$	−5.14544	−0.350103
$217$	−3.96880	−0.269420
$218$	−4.15792	−0.281610
$219$	22.2974	1.50672
$220$	−4.15655	−0.280234
$221$	0	0
$222$	−7.20871	−0.483817
$223$	−0.222390	−0.0148923	−0.00744617	−	0.999972i	$-0.502370\pi$
$223$	−0.222390	−0.0148923	−0.00744617	+	0.999972i	$0.502370\pi$
$224$	1.18596	0.0792405
$225$	0.0735536	0.00490357
$226$	−6.61266	−0.439867
$227$	7.01122	0.465351	0.232675	−	0.972554i	$-0.425252\pi$
$227$	7.01122	0.465351	0.232675	+	0.972554i	$0.425252\pi$
$228$	−1.74849	−0.115796
$229$	−12.7985	−0.845749	−0.422874	−	0.906188i	$-0.638979\pi$
$229$	−12.7985	−0.845749	−0.422874	+	0.906188i	$0.638979\pi$
$230$	4.45935	0.294041
$231$	3.43784	0.226193
$232$	−2.45819	−0.161388
$233$	−4.04194	−0.264797	−0.132398	−	0.991197i	$-0.542268\pi$
$233$	−4.04194	−0.264797	−0.132398	+	0.991197i	$0.542268\pi$
$234$	0	0
$235$	5.96897	0.389373
$236$	9.86518	0.642168
$237$	−1.95778	−0.127171
$238$	−6.75872	−0.438103
$239$	19.3219	1.24983	0.624915	−	0.780693i	$-0.285134\pi$
$239$	19.3219	1.24983	0.624915	+	0.780693i	$0.285134\pi$
$240$	−4.38372	−0.282968
$241$	−2.54260	−0.163783	−0.0818915	−	0.996641i	$-0.526096\pi$
$241$	−2.54260	−0.163783	−0.0818915	+	0.996641i	$0.526096\pi$
$242$	−8.25144	−0.530423
$243$	−0.594406	−0.0381311
$244$	−14.7547	−0.944574
$245$	14.0237	0.895943
$246$	−7.48006	−0.476911
$247$	0	0
$248$	−3.34648	−0.212502
$249$	19.1238	1.21192
$250$	9.31205	0.588946
$251$	13.0556	0.824065	0.412032	−	0.911169i	$-0.364819\pi$
$251$	13.0556	0.824065	0.412032	+	0.911169i	$0.364819\pi$
$252$	0.0678424	0.00427367
$253$	−2.94879	−0.185389
$254$	11.1835	0.701716
$255$	24.9825	1.56447
$256$	1.00000	0.0625000
$257$	12.3545	0.770650	0.385325	−	0.922781i	$-0.374089\pi$
$257$	12.3545	0.770650	0.385325	+	0.922781i	$0.374089\pi$
$258$	−5.35735	−0.333534
$259$	−4.88952	−0.303820
$260$	0	0
$261$	−0.140619	−0.00870411
$262$	−3.62320	−0.223842
$263$	28.8232	1.77731	0.888656	−	0.458574i	$-0.151640\pi$
$263$	28.8232	1.77731	0.888656	+	0.458574i	$0.151640\pi$
$264$	2.89878	0.178407
$265$	−32.8619	−2.01869
$266$	−1.18596	−0.0727161
$267$	−7.45202	−0.456056
$268$	0.571177	0.0348902
$269$	−5.50200	−0.335463	−0.167731	−	0.985833i	$-0.553644\pi$
$269$	−5.50200	−0.335463	−0.167731	+	0.985833i	$0.553644\pi$
$270$	12.9004	0.785092
$271$	−4.10727	−0.249499	−0.124750	−	0.992188i	$-0.539813\pi$
$271$	−4.10727	−0.249499	−0.124750	+	0.992188i	$0.539813\pi$
$272$	−5.69893	−0.345549
$273$	0	0
$274$	0.286896	0.0173320
$275$	2.13170	0.128546
$276$	−3.10995	−0.187197
$277$	−25.9330	−1.55816	−0.779082	−	0.626922i	$-0.784315\pi$
$277$	−25.9330	−1.55816	−0.779082	+	0.626922i	$0.784315\pi$
$278$	−11.9839	−0.718749
$279$	−0.191433	−0.0114608
$280$	−2.97339	−0.177694
$281$	−25.1058	−1.49769	−0.748844	−	0.662746i	$-0.769391\pi$
$281$	−25.1058	−1.49769	−0.748844	+	0.662746i	$0.769391\pi$
$282$	−4.16276	−0.247889
$283$	−2.64839	−0.157431	−0.0787153	−	0.996897i	$-0.525082\pi$
$283$	−2.64839	−0.157431	−0.0787153	+	0.996897i	$0.525082\pi$
$284$	−11.0399	−0.655096
$285$	4.38372	0.259669
$286$	0	0
$287$	−5.07357	−0.299484
$288$	0.0572044	0.00337080
$289$	15.4778	0.910461
$290$	6.16304	0.361906
$291$	−0.282630	−0.0165681
$292$	12.7524	0.746277
$293$	6.67140	0.389748	0.194874	−	0.980828i	$-0.437570\pi$
$293$	6.67140	0.389748	0.194874	+	0.980828i	$0.437570\pi$
$294$	−9.78014	−0.570390
$295$	−24.7335	−1.44004
$296$	−4.12283	−0.239635
$297$	−8.53051	−0.494990
$298$	0.660806	0.0382794
$299$	0	0
$300$	2.24821	0.129800
$301$	−3.63379	−0.209448
$302$	−5.62805	−0.323858
$303$	5.20483	0.299010
$304$	−1.00000	−0.0573539
$305$	36.9923	2.11817
$306$	−0.326004	−0.0186364
$307$	4.48857	0.256176	0.128088	−	0.991763i	$-0.459116\pi$
$307$	4.48857	0.256176	0.128088	+	0.991763i	$0.459116\pi$
$308$	1.96618	0.112034
$309$	15.9105	0.905119
$310$	8.39012	0.476527
$311$	23.4166	1.32783	0.663916	−	0.747808i	$-0.268893\pi$
$311$	23.4166	1.32783	0.663916	+	0.747808i	$0.268893\pi$
$312$	0	0
$313$	−1.84917	−0.104521	−0.0522606	−	0.998633i	$-0.516643\pi$
$313$	−1.84917	−0.104521	−0.0522606	+	0.998633i	$0.516643\pi$
$314$	−14.8599	−0.838590
$315$	−0.170091	−0.00958354
$316$	−1.11970	−0.0629880
$317$	9.28882	0.521712	0.260856	−	0.965378i	$-0.415995\pi$
$317$	9.28882	0.521712	0.260856	+	0.965378i	$0.415995\pi$
$318$	22.9179	1.28517
$319$	−4.07537	−0.228177
$320$	−2.50715	−0.140154
$321$	−23.0253	−1.28515
$322$	−2.10941	−0.117553
$323$	5.69893	0.317097
$324$	−9.16834	−0.509352
$325$	0	0
$326$	15.5601	0.861794
$327$	−7.27007	−0.402036
$328$	−4.27802	−0.236214
$329$	−2.82352	−0.155665
$330$	−7.26767	−0.400072
$331$	−23.4772	−1.29042	−0.645212	−	0.764003i	$-0.723231\pi$
$331$	−23.4772	−1.29042	−0.645212	+	0.764003i	$0.723231\pi$
$332$	10.9373	0.600263
$333$	−0.235844	−0.0129242
$334$	20.7297	1.13428
$335$	−1.43203	−0.0782400
$336$	2.07364	0.113126
$337$	−0.828390	−0.0451253	−0.0225626	−	0.999745i	$-0.507183\pi$
$337$	−0.828390	−0.0451253	−0.0225626	+	0.999745i	$0.507183\pi$
$338$	0	0
$339$	−11.5621	−0.627969
$340$	14.2881	0.774880
$341$	−5.54805	−0.300444
$342$	−0.0572044	−0.00309326
$343$	−14.9354	−0.806437
$344$	−3.06400	−0.165200
$345$	7.79711	0.419782
$346$	1.44324	0.0775892
$347$	9.08526	0.487722	0.243861	−	0.969810i	$-0.421586\pi$
$347$	9.08526	0.487722	0.243861	+	0.969810i	$0.421586\pi$
$348$	−4.29810	−0.230403
$349$	−17.3650	−0.929529	−0.464765	−	0.885434i	$-0.653861\pi$
$349$	−17.3650	−0.929529	−0.464765	+	0.885434i	$0.653861\pi$
$350$	1.52491	0.0815100
$351$	0	0
$352$	1.65788	0.0883652
$353$	−28.9603	−1.54140	−0.770700	−	0.637198i	$-0.780093\pi$
$353$	−28.9603	−1.54140	−0.770700	+	0.637198i	$0.780093\pi$
$354$	17.2491	0.916781
$355$	27.6786	1.46903
$356$	−4.26198	−0.225885
$357$	−11.8175	−0.625450
$358$	−23.4293	−1.23828
$359$	−22.8698	−1.20702	−0.603511	−	0.797355i	$-0.706232\pi$
$359$	−22.8698	−1.20702	−0.603511	+	0.797355i	$0.706232\pi$
$360$	−0.143420	−0.00755890
$361$	1.00000	0.0526316
$362$	−18.9196	−0.994391
$363$	−14.4275	−0.757249
$364$	0	0
$365$	−31.9722	−1.67350
$366$	−25.7984	−1.34851
$367$	34.8185	1.81751	0.908755	−	0.417330i	$-0.137034\pi$
$367$	34.8185	1.81751	0.908755	+	0.417330i	$0.137034\pi$
$368$	−1.77865	−0.0927186
$369$	−0.244722	−0.0127397
$370$	10.3366	0.537372
$371$	15.5448	0.807044
$372$	−5.85127	−0.303374
$373$	−9.37051	−0.485187	−0.242593	−	0.970128i	$-0.577998\pi$
$373$	−9.37051	−0.485187	−0.242593	+	0.970128i	$0.577998\pi$
$374$	−9.44814	−0.488551
$375$	16.2820	0.840799
$376$	−2.38078	−0.122779
$377$	0	0
$378$	−6.10230	−0.313868
$379$	−17.9547	−0.922271	−0.461136	−	0.887330i	$-0.652558\pi$
$379$	−17.9547	−0.922271	−0.461136	+	0.887330i	$0.652558\pi$
$380$	2.50715	0.128614
$381$	19.5542	1.00179
$382$	−5.67838	−0.290531
$383$	1.57714	0.0805882	0.0402941	−	0.999188i	$-0.487171\pi$
$383$	1.57714	0.0805882	0.0402941	+	0.999188i	$0.487171\pi$
$384$	1.74849	0.0892271
$385$	−4.92951	−0.251231
$386$	17.5233	0.891912
$387$	−0.175274	−0.00890969
$388$	−0.161643	−0.00820617
$389$	10.0604	0.510084	0.255042	−	0.966930i	$-0.417911\pi$
$389$	10.0604	0.510084	0.255042	+	0.966930i	$0.417911\pi$
$390$	0	0
$391$	10.1364	0.512621
$392$	−5.59349	−0.282514
$393$	−6.33511	−0.319564
$394$	−17.0594	−0.859442
$395$	2.80725	0.141248
$396$	0.0948380	0.00476579
$397$	19.9122	0.999365	0.499683	−	0.866209i	$-0.333450\pi$
$397$	19.9122	0.999365	0.499683	+	0.866209i	$0.333450\pi$
$398$	−1.98089	−0.0992929
$399$	−2.07364	−0.103812
$400$	1.28580	0.0642901
$401$	−21.6420	−1.08075	−0.540375	−	0.841424i	$-0.681718\pi$
$401$	−21.6420	−1.08075	−0.540375	+	0.841424i	$0.681718\pi$
$402$	0.998696	0.0498104
$403$	0	0
$404$	2.97676	0.148100
$405$	22.9864	1.14220
$406$	−2.91532	−0.144685
$407$	−6.83515	−0.338806
$408$	−9.96451	−0.493317
$409$	−29.2032	−1.44400	−0.722002	−	0.691891i	$-0.756778\pi$
$409$	−29.2032	−1.44400	−0.722002	+	0.691891i	$0.756778\pi$
$410$	10.7256	0.529702
$411$	0.501634	0.0247438
$412$	9.09960	0.448305
$413$	11.6997	0.575706
$414$	−0.101747	−0.00500058
$415$	−27.4215	−1.34607
$416$	0	0
$417$	−20.9538	−1.02611
$418$	−1.65788	−0.0810895
$419$	25.6492	1.25304	0.626522	−	0.779404i	$-0.284478\pi$
$419$	25.6492	1.25304	0.626522	+	0.779404i	$0.284478\pi$
$420$	−5.19893	−0.253682
$421$	18.9201	0.922108	0.461054	−	0.887372i	$-0.347471\pi$
$421$	18.9201	0.922108	0.461054	+	0.887372i	$0.347471\pi$
$422$	−13.7028	−0.667043
$423$	−0.136191	−0.00662184
$424$	13.1073	0.636546
$425$	−7.32770	−0.355446
$426$	−19.3031	−0.935236
$427$	−17.4985	−0.846814
$428$	−13.1687	−0.636534
$429$	0	0
$430$	7.68190	0.370454
$431$	28.9147	1.39277	0.696386	−	0.717667i	$-0.254790\pi$
$431$	28.9147	1.39277	0.696386	+	0.717667i	$0.254790\pi$
$432$	−5.14544	−0.247560
$433$	−25.5746	−1.22904	−0.614518	−	0.788903i	$-0.710649\pi$
$433$	−25.5746	−1.22904	−0.614518	+	0.788903i	$0.710649\pi$
$434$	−3.96880	−0.190508
$435$	10.7760	0.516669
$436$	−4.15792	−0.199128
$437$	1.77865	0.0850844
$438$	22.2974	1.06541
$439$	38.2331	1.82477	0.912384	−	0.409335i	$-0.134239\pi$
$439$	38.2331	1.82477	0.912384	+	0.409335i	$0.134239\pi$
$440$	−4.15655	−0.198156
$441$	−0.319973	−0.0152368
$442$	0	0
$443$	−8.95028	−0.425241	−0.212620	−	0.977135i	$-0.568200\pi$
$443$	−8.95028	−0.425241	−0.212620	+	0.977135i	$0.568200\pi$
$444$	−7.20871	−0.342110
$445$	10.6854	0.506538
$446$	−0.222390	−0.0105305
$447$	1.15541	0.0546490
$448$	1.18596	0.0560315
$449$	25.1153	1.18526	0.592632	−	0.805474i	$-0.298089\pi$
$449$	25.1153	1.18526	0.592632	+	0.805474i	$0.298089\pi$
$450$	0.0735536	0.00346735
$451$	−7.09244	−0.333970
$452$	−6.61266	−0.311033
$453$	−9.84056	−0.462350
$454$	7.01122	0.329053
$455$	0	0
$456$	−1.74849	−0.0818804
$457$	19.5261	0.913392	0.456696	−	0.889623i	$-0.349033\pi$
$457$	19.5261	0.913392	0.456696	+	0.889623i	$0.349033\pi$
$458$	−12.7985	−0.598035
$459$	29.3235	1.36870
$460$	4.45935	0.207918
$461$	−30.5217	−1.42154	−0.710768	−	0.703426i	$-0.751653\pi$
$461$	−30.5217	−1.42154	−0.710768	+	0.703426i	$0.751653\pi$
$462$	3.43784	0.159943
$463$	−38.0611	−1.76885	−0.884425	−	0.466682i	$-0.845449\pi$
$463$	−38.0611	−1.76885	−0.884425	+	0.466682i	$0.845449\pi$
$464$	−2.45819	−0.114118
$465$	14.6700	0.680306
$466$	−4.04194	−0.187239
$467$	1.62087	0.0750050	0.0375025	−	0.999297i	$-0.488060\pi$
$467$	1.62087	0.0750050	0.0375025	+	0.999297i	$0.488060\pi$
$468$	0	0
$469$	0.677395	0.0312792
$470$	5.96897	0.275328
$471$	−25.9822	−1.19720
$472$	9.86518	0.454082
$473$	−5.07973	−0.233566
$474$	−1.95778	−0.0899237
$475$	−1.28580	−0.0589966
$476$	−6.75872	−0.309786
$477$	0.749795	0.0343308
$478$	19.3219	0.883764
$479$	36.2061	1.65430	0.827150	−	0.561981i	$-0.189961\pi$
$479$	36.2061	1.65430	0.827150	+	0.561981i	$0.189961\pi$
$480$	−4.38372	−0.200088
$481$	0	0
$482$	−2.54260	−0.115812
$483$	−3.68828	−0.167823
$484$	−8.25144	−0.375066
$485$	0.405263	0.0184020
$486$	−0.594406	−0.0269628
$487$	−4.63099	−0.209850	−0.104925	−	0.994480i	$-0.533460\pi$
$487$	−4.63099	−0.209850	−0.104925	+	0.994480i	$0.533460\pi$
$488$	−14.7547	−0.667915
$489$	27.2066	1.23033
$490$	14.0237	0.633527
$491$	−20.2251	−0.912748	−0.456374	−	0.889788i	$-0.650852\pi$
$491$	−20.2251	−0.912748	−0.456374	+	0.889788i	$0.650852\pi$
$492$	−7.48006	−0.337227
$493$	14.0090	0.630935
$494$	0	0
$495$	−0.237773	−0.0106871
$496$	−3.34648	−0.150261
$497$	−13.0929	−0.587296
$498$	19.1238	0.856956
$499$	−0.330827	−0.0148098	−0.00740492	−	0.999973i	$-0.502357\pi$
$499$	−0.330827	−0.0148098	−0.00740492	+	0.999973i	$0.502357\pi$
$500$	9.31205	0.416448
$501$	36.2457	1.61934
$502$	13.0556	0.582702
$503$	4.35784	0.194307	0.0971533	−	0.995269i	$-0.469026\pi$
$503$	4.35784	0.194307	0.0971533	+	0.995269i	$0.469026\pi$
$504$	0.0678424	0.00302194
$505$	−7.46319	−0.332108
$506$	−2.94879	−0.131090
$507$	0	0
$508$	11.1835	0.496188
$509$	31.9081	1.41430	0.707150	−	0.707063i	$-0.249980\pi$
$509$	31.9081	1.41430	0.707150	+	0.707063i	$0.249980\pi$
$510$	24.9825	1.10624
$511$	15.1239	0.669040
$512$	1.00000	0.0441942
$513$	5.14544	0.227177
$514$	12.3545	0.544932
$515$	−22.8141	−1.00531
$516$	−5.35735	−0.235844
$517$	−3.94704	−0.173591
$518$	−4.88952	−0.214833
$519$	2.52349	0.110769
$520$	0	0
$521$	33.7395	1.47815	0.739076	−	0.673622i	$-0.235262\pi$
$521$	33.7395	1.47815	0.739076	+	0.673622i	$0.235262\pi$
$522$	−0.140619	−0.00615473
$523$	7.62982	0.333629	0.166814	−	0.985988i	$-0.446652\pi$
$523$	7.62982	0.333629	0.166814	+	0.985988i	$0.446652\pi$
$524$	−3.62320	−0.158280
$525$	2.66629	0.116366
$526$	28.8232	1.25675
$527$	19.0714	0.830761
$528$	2.89878	0.126153
$529$	−19.8364	−0.862452
$530$	−32.8619	−1.42743
$531$	0.564332	0.0244899
$532$	−1.18596	−0.0514180
$533$	0	0
$534$	−7.45202	−0.322480
$535$	33.0160	1.42740
$536$	0.571177	0.0246711
$537$	−40.9658	−1.76780
$538$	−5.50200	−0.237208
$539$	−9.27333	−0.399430
$540$	12.9004	0.555144
$541$	−41.5590	−1.78676	−0.893381	−	0.449300i	$-0.851673\pi$
$541$	−41.5590	−1.78676	−0.893381	+	0.449300i	$0.851673\pi$
$542$	−4.10727	−0.176422
$543$	−33.0806	−1.41963
$544$	−5.69893	−0.244340
$545$	10.4245	0.446538
$546$	0	0
$547$	41.2753	1.76480	0.882402	−	0.470496i	$-0.155925\pi$
$547$	41.2753	1.76480	0.882402	+	0.470496i	$0.155925\pi$
$548$	0.286896	0.0122556
$549$	−0.844035	−0.0360226
$550$	2.13170	0.0908961
$551$	2.45819	0.104722
$552$	−3.10995	−0.132368
$553$	−1.32792	−0.0564690
$554$	−25.9330	−1.10179
$555$	18.0733	0.767170
$556$	−11.9839	−0.508232
$557$	19.7360	0.836240	0.418120	−	0.908392i	$-0.362689\pi$
$557$	19.7360	0.836240	0.418120	+	0.908392i	$0.362689\pi$
$558$	−0.191433	−0.00810402
$559$	0	0
$560$	−2.97339	−0.125649
$561$	−16.5199	−0.697472
$562$	−25.1058	−1.05903
$563$	5.78881	0.243969	0.121985	−	0.992532i	$-0.461074\pi$
$563$	5.78881	0.243969	0.121985	+	0.992532i	$0.461074\pi$
$564$	−4.16276	−0.175284
$565$	16.5789	0.697481
$566$	−2.64839	−0.111320
$567$	−10.8733	−0.456636
$568$	−11.0399	−0.463223
$569$	−32.7552	−1.37317	−0.686585	−	0.727050i	$-0.740891\pi$
$569$	−32.7552	−1.37317	−0.686585	+	0.727050i	$0.740891\pi$
$570$	4.38372	0.183614
$571$	16.3057	0.682371	0.341186	−	0.939996i	$-0.389172\pi$
$571$	16.3057	0.682371	0.341186	+	0.939996i	$0.389172\pi$
$572$	0	0
$573$	−9.92856	−0.414772
$574$	−5.07357	−0.211767
$575$	−2.28699	−0.0953742
$576$	0.0572044	0.00238352
$577$	29.4023	1.22403	0.612017	−	0.790845i	$-0.290359\pi$
$577$	29.4023	1.22403	0.612017	+	0.790845i	$0.290359\pi$
$578$	15.4778	0.643794
$579$	30.6392	1.27332
$580$	6.16304	0.255906
$581$	12.9713	0.538138
$582$	−0.282630	−0.0117154
$583$	21.7303	0.899976
$584$	12.7524	0.527698
$585$	0	0
$586$	6.67140	0.275593
$587$	11.2376	0.463825	0.231912	−	0.972737i	$-0.425502\pi$
$587$	11.2376	0.463825	0.231912	+	0.972737i	$0.425502\pi$
$588$	−9.78014	−0.403326
$589$	3.34648	0.137889
$590$	−24.7335	−1.01826
$591$	−29.8282	−1.22697
$592$	−4.12283	−0.169447
$593$	−20.0471	−0.823238	−0.411619	−	0.911356i	$-0.635036\pi$
$593$	−20.0471	−0.823238	−0.411619	+	0.911356i	$0.635036\pi$
$594$	−8.53051	−0.350011
$595$	16.9451	0.694683
$596$	0.660806	0.0270677
$597$	−3.46355	−0.141754
$598$	0	0
$599$	13.2845	0.542788	0.271394	−	0.962468i	$-0.412515\pi$
$599$	13.2845	0.542788	0.271394	+	0.962468i	$0.412515\pi$
$600$	2.24821	0.0917826
$601$	32.6265	1.33086	0.665431	−	0.746459i	$-0.268248\pi$
$601$	32.6265	1.33086	0.665431	+	0.746459i	$0.268248\pi$
$602$	−3.63379	−0.148102
$603$	0.0326739	0.00133058
$604$	−5.62805	−0.229002
$605$	20.6876	0.841071
$606$	5.20483	0.211432
$607$	37.7162	1.53085	0.765427	−	0.643523i	$-0.222528\pi$
$607$	37.7162	1.53085	0.765427	+	0.643523i	$0.222528\pi$
$608$	−1.00000	−0.0405554
$609$	−5.09739	−0.206557
$610$	36.9923	1.49777
$611$	0	0
$612$	−0.326004	−0.0131779
$613$	−9.17890	−0.370732	−0.185366	−	0.982670i	$-0.559347\pi$
$613$	−9.17890	−0.370732	−0.185366	+	0.982670i	$0.559347\pi$
$614$	4.48857	0.181144
$615$	18.7536	0.756220
$616$	1.96618	0.0792197
$617$	32.9967	1.32840	0.664198	−	0.747557i	$-0.268773\pi$
$617$	32.9967	1.32840	0.664198	+	0.747557i	$0.268773\pi$
$618$	15.9105	0.640015
$619$	41.2830	1.65930	0.829651	−	0.558282i	$-0.188539\pi$
$619$	41.2830	1.65930	0.829651	+	0.558282i	$0.188539\pi$
$620$	8.39012	0.336955
$621$	9.15194	0.367255
$622$	23.4166	0.938918
$623$	−5.05455	−0.202506
$624$	0	0
$625$	−29.7757	−1.19103
$626$	−1.84917	−0.0739077
$627$	−2.89878	−0.115766
$628$	−14.8599	−0.592973
$629$	23.4957	0.936837
$630$	−0.170091	−0.00677659
$631$	10.7778	0.429056	0.214528	−	0.976718i	$-0.431179\pi$
$631$	10.7778	0.429056	0.214528	+	0.976718i	$0.431179\pi$
$632$	−1.11970	−0.0445392
$633$	−23.9592	−0.952293
$634$	9.28882	0.368906
$635$	−28.0387	−1.11268
$636$	22.9179	0.908754
$637$	0	0
$638$	−4.07537	−0.161345
$639$	−0.631530	−0.0249829
$640$	−2.50715	−0.0991038
$641$	−18.2000	−0.718857	−0.359429	−	0.933173i	$-0.617028\pi$
$641$	−18.2000	−0.718857	−0.359429	+	0.933173i	$0.617028\pi$
$642$	−23.0253	−0.908737
$643$	8.03533	0.316882	0.158441	−	0.987368i	$-0.449353\pi$
$643$	8.03533	0.316882	0.158441	+	0.987368i	$0.449353\pi$
$644$	−2.10941	−0.0831226
$645$	13.4317	0.528872
$646$	5.69893	0.224222
$647$	−29.1964	−1.14783	−0.573915	−	0.818915i	$-0.694576\pi$
$647$	−29.1964	−1.14783	−0.573915	+	0.818915i	$0.694576\pi$
$648$	−9.16834	−0.360166
$649$	16.3553	0.642000
$650$	0	0
$651$	−6.93939	−0.271976
$652$	15.5601	0.609381
$653$	42.6126	1.66756	0.833779	−	0.552098i	$-0.186172\pi$
$653$	42.6126	1.66756	0.833779	+	0.552098i	$0.186172\pi$
$654$	−7.27007	−0.284282
$655$	9.08390	0.354937
$656$	−4.27802	−0.167029
$657$	0.729493	0.0284602
$658$	−2.82352	−0.110072
$659$	25.7580	1.00339	0.501695	−	0.865045i	$-0.332710\pi$
$659$	25.7580	1.00339	0.501695	+	0.865045i	$0.332710\pi$
$660$	−7.26767	−0.282894
$661$	15.3786	0.598158	0.299079	−	0.954228i	$-0.403321\pi$
$661$	15.3786	0.598158	0.299079	+	0.954228i	$0.403321\pi$
$662$	−23.4772	−0.912468
$663$	0	0
$664$	10.9373	0.424450
$665$	2.97339	0.115303
$666$	−0.235844	−0.00913878
$667$	4.37225	0.169294
$668$	20.7297	0.802058
$669$	−0.388846	−0.0150337
$670$	−1.43203	−0.0553240
$671$	−24.4615	−0.944326
$672$	2.07364	0.0799924
$673$	−49.4252	−1.90520	−0.952600	−	0.304226i	$-0.901602\pi$
$673$	−49.4252	−1.90520	−0.952600	+	0.304226i	$0.901602\pi$
$674$	−0.828390	−0.0319084
$675$	−6.61601	−0.254650
$676$	0	0
$677$	−46.3503	−1.78139	−0.890693	−	0.454606i	$-0.849780\pi$
$677$	−46.3503	−1.78139	−0.890693	+	0.454606i	$0.849780\pi$
$678$	−11.5621	−0.444041
$679$	−0.191702	−0.00735686
$680$	14.2881	0.547923
$681$	12.2590	0.469766
$682$	−5.54805	−0.212446
$683$	−32.8922	−1.25859	−0.629293	−	0.777168i	$-0.716655\pi$
$683$	−32.8922	−1.25859	−0.629293	+	0.777168i	$0.716655\pi$
$684$	−0.0572044	−0.00218727
$685$	−0.719292	−0.0274827
$686$	−14.9354	−0.570237
$687$	−22.3780	−0.853774
$688$	−3.06400	−0.116814
$689$	0	0
$690$	7.79711	0.296831
$691$	25.4600	0.968544	0.484272	−	0.874917i	$-0.339084\pi$
$691$	25.4600	0.968544	0.484272	+	0.874917i	$0.339084\pi$
$692$	1.44324	0.0548638
$693$	0.112474	0.00427255
$694$	9.08526	0.344872
$695$	30.0455	1.13969
$696$	−4.29810	−0.162919
$697$	24.3802	0.923464
$698$	−17.3650	−0.657277
$699$	−7.06728	−0.267309
$700$	1.52491	0.0576363
$701$	6.09722	0.230289	0.115144	−	0.993349i	$-0.463267\pi$
$701$	6.09722	0.230289	0.115144	+	0.993349i	$0.463267\pi$
$702$	0	0
$703$	4.12283	0.155496
$704$	1.65788	0.0624836
$705$	10.4367	0.393067
$706$	−28.9603	−1.08993
$707$	3.53033	0.132772
$708$	17.2491	0.648262
$709$	−39.2960	−1.47579	−0.737896	−	0.674914i	$-0.764181\pi$
$709$	−39.2960	−1.47579	−0.737896	+	0.674914i	$0.764181\pi$
$710$	27.6786	1.03876
$711$	−0.0640518	−0.00240213
$712$	−4.26198	−0.159725
$713$	5.95222	0.222912
$714$	−11.8175	−0.442260
$715$	0	0
$716$	−23.4293	−0.875594
$717$	33.7841	1.26169
$718$	−22.8698	−0.853493
$719$	38.2175	1.42527	0.712636	−	0.701534i	$-0.247501\pi$
$719$	38.2175	1.42527	0.712636	+	0.701534i	$0.247501\pi$
$720$	−0.143420	−0.00534495
$721$	10.7918	0.401907
$722$	1.00000	0.0372161
$723$	−4.44569	−0.165337
$724$	−18.9196	−0.703140
$725$	−3.16074	−0.117387
$726$	−14.4275	−0.535456
$727$	−15.3174	−0.568091	−0.284045	−	0.958811i	$-0.591677\pi$
$727$	−15.3174	−0.568091	−0.284045	+	0.958811i	$0.591677\pi$
$728$	0	0
$729$	26.4657	0.980212
$730$	−31.9722	−1.18334
$731$	17.4615	0.645837
$732$	−25.7984	−0.953537
$733$	7.42216	0.274144	0.137072	−	0.990561i	$-0.456231\pi$
$733$	7.42216	0.274144	0.137072	+	0.990561i	$0.456231\pi$
$734$	34.8185	1.28517
$735$	24.5203	0.904444
$736$	−1.77865	−0.0655620
$737$	0.946942	0.0348811
$738$	−0.244722	−0.00900834
$739$	42.3225	1.55686	0.778429	−	0.627733i	$-0.216017\pi$
$739$	42.3225	1.55686	0.778429	+	0.627733i	$0.216017\pi$
$740$	10.3366	0.379979
$741$	0	0
$742$	15.5448	0.570666
$743$	−24.7907	−0.909484	−0.454742	−	0.890623i	$-0.650268\pi$
$743$	−24.7907	−0.909484	−0.454742	+	0.890623i	$0.650268\pi$
$744$	−5.85127	−0.214518
$745$	−1.65674	−0.0606982
$746$	−9.37051	−0.343079
$747$	0.625663	0.0228918
$748$	−9.44814	−0.345458
$749$	−15.6176	−0.570655
$750$	16.2820	0.594534
$751$	−2.67235	−0.0975155	−0.0487577	−	0.998811i	$-0.515526\pi$
$751$	−2.67235	−0.0975155	−0.0487577	+	0.998811i	$0.515526\pi$
$752$	−2.38078	−0.0868181
$753$	22.8276	0.831885
$754$	0	0
$755$	14.1104	0.513528
$756$	−6.10230	−0.221938
$757$	−28.8089	−1.04708	−0.523539	−	0.852001i	$-0.675389\pi$
$757$	−28.8089	−1.04708	−0.523539	+	0.852001i	$0.675389\pi$
$758$	−17.9547	−0.652144
$759$	−5.15591	−0.187148
$760$	2.50715	0.0909439
$761$	−31.9370	−1.15771	−0.578857	−	0.815429i	$-0.696501\pi$
$761$	−31.9370	−1.15771	−0.578857	+	0.815429i	$0.696501\pi$
$762$	19.5542	0.708374
$763$	−4.93114	−0.178519
$764$	−5.67838	−0.205436
$765$	0.817342	0.0295511
$766$	1.57714	0.0569844
$767$	0	0
$768$	1.74849	0.0630931
$769$	15.4789	0.558184	0.279092	−	0.960264i	$-0.409967\pi$
$769$	15.4789	0.558184	0.279092	+	0.960264i	$0.409967\pi$
$770$	−4.92951	−0.177647
$771$	21.6016	0.777962
$772$	17.5233	0.630677
$773$	24.4576	0.879679	0.439839	−	0.898076i	$-0.355035\pi$
$773$	24.4576	0.879679	0.439839	+	0.898076i	$0.355035\pi$
$774$	−0.175274	−0.00630010
$775$	−4.30291	−0.154565
$776$	−0.161643	−0.00580264
$777$	−8.54927	−0.306703
$778$	10.0604	0.360684
$779$	4.27802	0.153276
$780$	0	0
$781$	−18.3028	−0.654924
$782$	10.1364	0.362478
$783$	12.6484	0.452018
$784$	−5.59349	−0.199768
$785$	37.2559	1.32972
$786$	−6.33511	−0.225966
$787$	−8.44850	−0.301156	−0.150578	−	0.988598i	$-0.548114\pi$
$787$	−8.44850	−0.301156	−0.150578	+	0.988598i	$0.548114\pi$
$788$	−17.0594	−0.607717
$789$	50.3969	1.79418
$790$	2.80725	0.0998776
$791$	−7.84237	−0.278842
$792$	0.0948380	0.00336992
$793$	0	0
$794$	19.9122	0.706658
$795$	−57.4587	−2.03785
$796$	−1.98089	−0.0702107
$797$	35.8144	1.26861	0.634306	−	0.773082i	$-0.281286\pi$
$797$	35.8144	1.26861	0.634306	+	0.773082i	$0.281286\pi$
$798$	−2.07364	−0.0734061
$799$	13.5679	0.479998
$800$	1.28580	0.0454599
$801$	−0.243804	−0.00861440
$802$	−21.6420	−0.764206
$803$	21.1419	0.746082
$804$	0.998696	0.0352213
$805$	5.28862	0.186399
$806$	0	0
$807$	−9.62017	−0.338646
$808$	2.97676	0.104722
$809$	8.81295	0.309847	0.154923	−	0.987926i	$-0.450487\pi$
$809$	8.81295	0.309847	0.154923	+	0.987926i	$0.450487\pi$
$810$	22.9864	0.807660
$811$	−13.7793	−0.483858	−0.241929	−	0.970294i	$-0.577780\pi$
$811$	−13.7793	−0.483858	−0.241929	+	0.970294i	$0.577780\pi$
$812$	−2.91532	−0.102308
$813$	−7.18151	−0.251867
$814$	−6.83515	−0.239572
$815$	−39.0115	−1.36651
$816$	−9.96451	−0.348828
$817$	3.06400	0.107196
$818$	−29.2032	−1.02106
$819$	0	0
$820$	10.7256	0.374556
$821$	42.2212	1.47353	0.736764	−	0.676150i	$-0.236353\pi$
$821$	42.2212	1.47353	0.736764	+	0.676150i	$0.236353\pi$
$822$	0.501634	0.0174965
$823$	−4.77272	−0.166367	−0.0831833	−	0.996534i	$-0.526509\pi$
$823$	−4.77272	−0.166367	−0.0831833	+	0.996534i	$0.526509\pi$
$824$	9.09960	0.317000
$825$	3.72725	0.129766
$826$	11.6997	0.407086
$827$	19.1656	0.666454	0.333227	−	0.942847i	$-0.391862\pi$
$827$	19.1656	0.666454	0.333227	+	0.942847i	$0.391862\pi$
$828$	−0.101747	−0.00353594
$829$	−3.19847	−0.111087	−0.0555437	−	0.998456i	$-0.517689\pi$
$829$	−3.19847	−0.111087	−0.0555437	+	0.998456i	$0.517689\pi$
$830$	−27.4215	−0.951814
$831$	−45.3435	−1.57295
$832$	0	0
$833$	31.8769	1.10447
$834$	−20.9538	−0.725569
$835$	−51.9726	−1.79858
$836$	−1.65788	−0.0573389
$837$	17.2191	0.595179
$838$	25.6492	0.886036
$839$	−27.5869	−0.952406	−0.476203	−	0.879335i	$-0.657987\pi$
$839$	−27.5869	−0.952406	−0.476203	+	0.879335i	$0.657987\pi$
$840$	−5.19893	−0.179380
$841$	−22.9573	−0.791632
$842$	18.9201	0.652029
$843$	−43.8972	−1.51190
$844$	−13.7028	−0.471671
$845$	0	0
$846$	−0.136191	−0.00468235
$847$	−9.78590	−0.336248
$848$	13.1073	0.450106
$849$	−4.63068	−0.158924
$850$	−7.32770	−0.251338
$851$	7.33308	0.251375
$852$	−19.3031	−0.661312
$853$	−1.82744	−0.0625705	−0.0312853	−	0.999510i	$-0.509960\pi$
$853$	−1.82744	−0.0625705	−0.0312853	+	0.999510i	$0.509960\pi$
$854$	−17.4985	−0.598788
$855$	0.143420	0.00490487
$856$	−13.1687	−0.450098
$857$	−19.3861	−0.662216	−0.331108	−	0.943593i	$-0.607422\pi$
$857$	−19.3861	−0.662216	−0.331108	+	0.943593i	$0.607422\pi$
$858$	0	0
$859$	−48.5959	−1.65807	−0.829035	−	0.559196i	$-0.811110\pi$
$859$	−48.5959	−1.65807	−0.829035	+	0.559196i	$0.811110\pi$
$860$	7.68190	0.261951
$861$	−8.87108	−0.302325
$862$	28.9147	0.984839
$863$	−19.4830	−0.663209	−0.331605	−	0.943418i	$-0.607590\pi$
$863$	−19.4830	−0.663209	−0.331605	+	0.943418i	$0.607590\pi$
$864$	−5.14544	−0.175051
$865$	−3.61842	−0.123030
$866$	−25.5746	−0.869059
$867$	27.0628	0.919101
$868$	−3.96880	−0.134710
$869$	−1.85632	−0.0629715
$870$	10.7760	0.365340
$871$	0	0
$872$	−4.15792	−0.140805
$873$	−0.00924668	−0.000312953 0
$874$	1.77865	0.0601638
$875$	11.0437	0.373347
$876$	22.2974	0.753359
$877$	−33.5836	−1.13404	−0.567018	−	0.823705i	$-0.691903\pi$
$877$	−33.5836	−1.13404	−0.567018	+	0.823705i	$0.691903\pi$
$878$	38.2331	1.29031
$879$	11.6649	0.393446
$880$	−4.15655	−0.140117
$881$	41.9449	1.41316	0.706580	−	0.707633i	$-0.250237\pi$
$881$	41.9449	1.41316	0.706580	+	0.707633i	$0.250237\pi$
$882$	−0.319973	−0.0107740
$883$	−31.3360	−1.05454	−0.527270	−	0.849698i	$-0.676784\pi$
$883$	−31.3360	−1.05454	−0.527270	+	0.849698i	$0.676784\pi$
$884$	0	0
$885$	−43.2461	−1.45370
$886$	−8.95028	−0.300691
$887$	12.2914	0.412706	0.206353	−	0.978478i	$-0.433841\pi$
$887$	12.2914	0.412706	0.206353	+	0.978478i	$0.433841\pi$
$888$	−7.20871	−0.241909
$889$	13.2632	0.444834
$890$	10.6854	0.358176
$891$	−15.2000	−0.509219
$892$	−0.222390	−0.00744617
$893$	2.38078	0.0796697
$894$	1.15541	0.0386427
$895$	58.7408	1.96349
$896$	1.18596	0.0396202
$897$	0	0
$898$	25.1153	0.838108
$899$	8.22626	0.274361
$900$	0.0735536	0.00245179
$901$	−74.6976	−2.48854
$902$	−7.09244	−0.236152
$903$	−6.35362	−0.211435
$904$	−6.61266	−0.219934
$905$	47.4342	1.57677
$906$	−9.84056	−0.326931
$907$	−55.8500	−1.85447	−0.927234	−	0.374481i	$-0.877821\pi$
$907$	−55.8500	−1.85447	−0.927234	+	0.374481i	$0.877821\pi$
$908$	7.01122	0.232675
$909$	0.170284	0.00564797
$910$	0	0
$911$	23.0564	0.763893	0.381947	−	0.924184i	$-0.375254\pi$
$911$	23.0564	0.763893	0.381947	+	0.924184i	$0.375254\pi$
$912$	−1.74849	−0.0578982
$913$	18.1327	0.600106
$914$	19.5261	0.645865
$915$	64.6805	2.13827
$916$	−12.7985	−0.422874
$917$	−4.29698	−0.141899
$918$	29.3235	0.967820
$919$	7.52970	0.248382	0.124191	−	0.992258i	$-0.460366\pi$
$919$	7.52970	0.248382	0.124191	+	0.992258i	$0.460366\pi$
$920$	4.45935	0.147020
$921$	7.84821	0.258607
$922$	−30.5217	−1.00518
$923$	0	0
$924$	3.43784	0.113097
$925$	−5.30114	−0.174300
$926$	−38.0611	−1.25077
$927$	0.520538	0.0170967
$928$	−2.45819	−0.0806939
$929$	−25.6365	−0.841105	−0.420552	−	0.907268i	$-0.638164\pi$
$929$	−25.6365	−0.841105	−0.420552	+	0.907268i	$0.638164\pi$
$930$	14.6700	0.481049
$931$	5.59349	0.183319
$932$	−4.04194	−0.132398
$933$	40.9435	1.34043
$934$	1.62087	0.0530366
$935$	23.6879	0.774677
$936$	0	0
$937$	−8.05192	−0.263045	−0.131522	−	0.991313i	$-0.541986\pi$
$937$	−8.05192	−0.263045	−0.131522	+	0.991313i	$0.541986\pi$
$938$	0.677395	0.0221177
$939$	−3.23325	−0.105513
$940$	5.96897	0.194686
$941$	30.5487	0.995860	0.497930	−	0.867217i	$-0.334094\pi$
$941$	30.5487	0.995860	0.497930	+	0.867217i	$0.334094\pi$
$942$	−25.9822	−0.846547
$943$	7.60911	0.247787
$944$	9.86518	0.321084
$945$	15.2994	0.497689
$946$	−5.07973	−0.165156
$947$	16.2870	0.529255	0.264627	−	0.964351i	$-0.414751\pi$
$947$	16.2870	0.529255	0.264627	+	0.964351i	$0.414751\pi$
$948$	−1.95778	−0.0635857
$949$	0	0
$950$	−1.28580	−0.0417169
$951$	16.2414	0.526662
$952$	−6.75872	−0.219052
$953$	−35.4767	−1.14920	−0.574601	−	0.818433i	$-0.694843\pi$
$953$	−35.4767	−1.14920	−0.574601	+	0.818433i	$0.694843\pi$
$954$	0.749795	0.0242755
$955$	14.2365	0.460684
$956$	19.3219	0.624915
$957$	−7.12573	−0.230342
$958$	36.2061	1.16977
$959$	0.340248	0.0109872
$960$	−4.38372	−0.141484
$961$	−19.8011	−0.638745
$962$	0	0
$963$	−0.753309	−0.0242751
$964$	−2.54260	−0.0818915
$965$	−43.9335	−1.41427
$966$	−3.68828	−0.118669
$967$	4.20251	0.135144	0.0675719	−	0.997714i	$-0.478475\pi$
$967$	4.20251	0.135144	0.0675719	+	0.997714i	$0.478475\pi$
$968$	−8.25144	−0.265211
$969$	9.96451	0.320106
$970$	0.405263	0.0130122
$971$	42.9254	1.37754	0.688771	−	0.724979i	$-0.258150\pi$
$971$	42.9254	1.37754	0.688771	+	0.724979i	$0.258150\pi$
$972$	−0.594406	−0.0190656
$973$	−14.2125	−0.455632
$974$	−4.63099	−0.148387
$975$	0	0
$976$	−14.7547	−0.472287
$977$	−48.0339	−1.53674	−0.768370	−	0.640005i	$-0.778932\pi$
$977$	−48.0339	−1.53674	−0.768370	+	0.640005i	$0.778932\pi$
$978$	27.2066	0.869972
$979$	−7.06585	−0.225825
$980$	14.0237	0.447971
$981$	−0.237852	−0.00759402
$982$	−20.2251	−0.645410
$983$	−3.13291	−0.0999244	−0.0499622	−	0.998751i	$-0.515910\pi$
$983$	−3.13291	−0.0999244	−0.0499622	+	0.998751i	$0.515910\pi$
$984$	−7.48006	−0.238456
$985$	42.7706	1.36278
$986$	14.0090	0.446139
$987$	−4.93688	−0.157143
$988$	0	0
$989$	5.44978	0.173293
$990$	−0.237773	−0.00755692
$991$	52.5296	1.66866	0.834329	−	0.551267i	$-0.185856\pi$
$991$	52.5296	1.66866	0.834329	+	0.551267i	$0.185856\pi$
$992$	−3.34648	−0.106251
$993$	−41.0496	−1.30267
$994$	−13.0929	−0.415281
$995$	4.96638	0.157445
$996$	19.1238	0.605959
$997$	−0.884782	−0.0280213	−0.0140107	−	0.999902i	$-0.504460\pi$
$997$	−0.884782	−0.0280213	−0.0140107	+	0.999902i	$0.504460\pi$
$998$	−0.330827	−0.0104721
$999$	21.2138	0.671174

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bh.1.8		8
13.2	odd	12		494.2.m.a.381.5	yes	16
13.7	odd	12		494.2.m.a.153.5	✓	16
13.12	even	2		6422.2.a.bg.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.m.a.153.5	✓	16	13.7	odd	12
494.2.m.a.381.5	yes	16	13.2	odd	12
6422.2.a.bg.1.8		8	13.12	even	2
6422.2.a.bh.1.8		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.74849 q^{3} +1.00000 q^{4} -2.50715 q^{5} +1.74849 q^{6} +1.18596 q^{7} +1.00000 q^{8} +0.0572044 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.74849 q^{3} +1.00000 q^{4} -2.50715 q^{5} +1.74849 q^{6} +1.18596 q^{7} +1.00000 q^{8} +0.0572044 q^{9} -2.50715 q^{10} +1.65788 q^{11} +1.74849 q^{12} +1.18596 q^{14} -4.38372 q^{15} +1.00000 q^{16} -5.69893 q^{17} +0.0572044 q^{18} -1.00000 q^{19} -2.50715 q^{20} +2.07364 q^{21} +1.65788 q^{22} -1.77865 q^{23} +1.74849 q^{24} +1.28580 q^{25} -5.14544 q^{27} +1.18596 q^{28} -2.45819 q^{29} -4.38372 q^{30} -3.34648 q^{31} +1.00000 q^{32} +2.89878 q^{33} -5.69893 q^{34} -2.97339 q^{35} +0.0572044 q^{36} -4.12283 q^{37} -1.00000 q^{38} -2.50715 q^{40} -4.27802 q^{41} +2.07364 q^{42} -3.06400 q^{43} +1.65788 q^{44} -0.143420 q^{45} -1.77865 q^{46} -2.38078 q^{47} +1.74849 q^{48} -5.59349 q^{49} +1.28580 q^{50} -9.96451 q^{51} +13.1073 q^{53} -5.14544 q^{54} -4.15655 q^{55} +1.18596 q^{56} -1.74849 q^{57} -2.45819 q^{58} +9.86518 q^{59} -4.38372 q^{60} -14.7547 q^{61} -3.34648 q^{62} +0.0678424 q^{63} +1.00000 q^{64} +2.89878 q^{66} +0.571177 q^{67} -5.69893 q^{68} -3.10995 q^{69} -2.97339 q^{70} -11.0399 q^{71} +0.0572044 q^{72} +12.7524 q^{73} -4.12283 q^{74} +2.24821 q^{75} -1.00000 q^{76} +1.96618 q^{77} -1.11970 q^{79} -2.50715 q^{80} -9.16834 q^{81} -4.27802 q^{82} +10.9373 q^{83} +2.07364 q^{84} +14.2881 q^{85} -3.06400 q^{86} -4.29810 q^{87} +1.65788 q^{88} -4.26198 q^{89} -0.143420 q^{90} -1.77865 q^{92} -5.85127 q^{93} -2.38078 q^{94} +2.50715 q^{95} +1.74849 q^{96} -0.161643 q^{97} -5.59349 q^{98} +0.0948380 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} + 8 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^7 + 8 * q^8	\( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} + 8 q^{8} - 2 q^{10} - 10 q^{11} - 4 q^{12} - 2 q^{14} + 8 q^{16} + 2 q^{17} - 8 q^{19} - 2 q^{20} + 12 q^{21} - 10 q^{22} - 8 q^{23} - 4 q^{24} - 14 q^{25} - 22 q^{27} - 2 q^{28} + 8 q^{29} - 12 q^{31} + 8 q^{32} + 4 q^{33} + 2 q^{34} - 10 q^{35} - 8 q^{38} - 2 q^{40} + 2 q^{41} + 12 q^{42} - 16 q^{43} - 10 q^{44} - 12 q^{45} - 8 q^{46} + 12 q^{47} - 4 q^{48} - 14 q^{49} - 14 q^{50} - 22 q^{51} - 24 q^{53} - 22 q^{54} - 10 q^{55} - 2 q^{56} + 4 q^{57} + 8 q^{58} - 4 q^{59} - 26 q^{61} - 12 q^{62} - 12 q^{63} + 8 q^{64} + 4 q^{66} + 10 q^{67} + 2 q^{68} + 6 q^{69} - 10 q^{70} - 36 q^{71} + 20 q^{73} + 6 q^{75} - 8 q^{76} - 12 q^{77} - 22 q^{79} - 2 q^{80} + 36 q^{81} + 2 q^{82} + 18 q^{83} + 12 q^{84} + 26 q^{85} - 16 q^{86} - 38 q^{87} - 10 q^{88} - 18 q^{89} - 12 q^{90} - 8 q^{92} + 12 q^{93} + 12 q^{94} + 2 q^{95} - 4 q^{96} - 8 q^{97} - 14 q^{98} - 8 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^7 + 8 * q^8 - 2 * q^10 - 10 * q^11 - 4 * q^12 - 2 * q^14 + 8 * q^16 + 2 * q^17 - 8 * q^19 - 2 * q^20 + 12 * q^21 - 10 * q^22 - 8 * q^23 - 4 * q^24 - 14 * q^25 - 22 * q^27 - 2 * q^28 + 8 * q^29 - 12 * q^31 + 8 * q^32 + 4 * q^33 + 2 * q^34 - 10 * q^35 - 8 * q^38 - 2 * q^40 + 2 * q^41 + 12 * q^42 - 16 * q^43 - 10 * q^44 - 12 * q^45 - 8 * q^46 + 12 * q^47 - 4 * q^48 - 14 * q^49 - 14 * q^50 - 22 * q^51 - 24 * q^53 - 22 * q^54 - 10 * q^55 - 2 * q^56 + 4 * q^57 + 8 * q^58 - 4 * q^59 - 26 * q^61 - 12 * q^62 - 12 * q^63 + 8 * q^64 + 4 * q^66 + 10 * q^67 + 2 * q^68 + 6 * q^69 - 10 * q^70 - 36 * q^71 + 20 * q^73 + 6 * q^75 - 8 * q^76 - 12 * q^77 - 22 * q^79 - 2 * q^80 + 36 * q^81 + 2 * q^82 + 18 * q^83 + 12 * q^84 + 26 * q^85 - 16 * q^86 - 38 * q^87 - 10 * q^88 - 18 * q^89 - 12 * q^90 - 8 * q^92 + 12 * q^93 + 12 * q^94 + 2 * q^95 - 4 * q^96 - 8 * q^97 - 14 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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