LMFDB - Embedded newform 6422.2.a.bg.1.5

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			1.5
Root			$0.394661$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -0.394661 q^{3} +1.00000 q^{4} +0.507765 q^{5} +0.394661 q^{6} +1.24955 q^{7} -1.00000 q^{8} -2.84424 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.394661 q^{3} +1.00000 q^{4} +0.507765 q^{5} +0.394661 q^{6} +1.24955 q^{7} -1.00000 q^{8} -2.84424 q^{9} -0.507765 q^{10} -2.69825 q^{11} -0.394661 q^{12} -1.24955 q^{14} -0.200395 q^{15} +1.00000 q^{16} +4.56219 q^{17} +2.84424 q^{18} +1.00000 q^{19} +0.507765 q^{20} -0.493147 q^{21} +2.69825 q^{22} +2.24994 q^{23} +0.394661 q^{24} -4.74218 q^{25} +2.30649 q^{27} +1.24955 q^{28} -9.14046 q^{29} +0.200395 q^{30} -1.74898 q^{31} -1.00000 q^{32} +1.06489 q^{33} -4.56219 q^{34} +0.634476 q^{35} -2.84424 q^{36} +8.71511 q^{37} -1.00000 q^{38} -0.507765 q^{40} +9.55485 q^{41} +0.493147 q^{42} +3.72738 q^{43} -2.69825 q^{44} -1.44421 q^{45} -2.24994 q^{46} -8.96128 q^{47} -0.394661 q^{48} -5.43863 q^{49} +4.74218 q^{50} -1.80052 q^{51} +5.80547 q^{53} -2.30649 q^{54} -1.37007 q^{55} -1.24955 q^{56} -0.394661 q^{57} +9.14046 q^{58} +8.65749 q^{59} -0.200395 q^{60} -0.618453 q^{61} +1.74898 q^{62} -3.55402 q^{63} +1.00000 q^{64} -1.06489 q^{66} -12.2636 q^{67} +4.56219 q^{68} -0.887963 q^{69} -0.634476 q^{70} +0.995275 q^{71} +2.84424 q^{72} -0.150857 q^{73} -8.71511 q^{74} +1.87155 q^{75} +1.00000 q^{76} -3.37159 q^{77} -5.80849 q^{79} +0.507765 q^{80} +7.62245 q^{81} -9.55485 q^{82} +8.15004 q^{83} -0.493147 q^{84} +2.31652 q^{85} -3.72738 q^{86} +3.60738 q^{87} +2.69825 q^{88} -2.44741 q^{89} +1.44421 q^{90} +2.24994 q^{92} +0.690253 q^{93} +8.96128 q^{94} +0.507765 q^{95} +0.394661 q^{96} +5.19401 q^{97} +5.43863 q^{98} +7.67447 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.394661	−0.227857	−0.113929	−	0.993489i	$-0.536344\pi$
$3$	−0.394661	−0.227857	−0.113929	+	0.993489i	$0.536344\pi$
$4$	1.00000	0.500000
$5$	0.507765	0.227079	0.113540	−	0.993533i	$-0.463781\pi$
$5$	0.507765	0.227079	0.113540	+	0.993533i	$0.463781\pi$
$6$	0.394661	0.161120
$7$	1.24955	0.472285	0.236142	−	0.971718i	$-0.424117\pi$
$7$	1.24955	0.472285	0.236142	+	0.971718i	$0.424117\pi$
$8$	−1.00000	−0.353553
$9$	−2.84424	−0.948081
$10$	−0.507765	−0.160569
$11$	−2.69825	−0.813552	−0.406776	−	0.913528i	$-0.633347\pi$
$11$	−2.69825	−0.813552	−0.406776	+	0.913528i	$0.633347\pi$
$12$	−0.394661	−0.113929
$13$	0	0
$13$	0	0
$14$	−1.24955	−0.333956
$15$	−0.200395	−0.0517417
$16$	1.00000	0.250000
$17$	4.56219	1.10649	0.553247	−	0.833017i	$-0.313389\pi$
$17$	4.56219	1.10649	0.553247	+	0.833017i	$0.313389\pi$
$18$	2.84424	0.670395
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.507765	0.113540
$21$	−0.493147	−0.107614
$22$	2.69825	0.575268
$23$	2.24994	0.469145	0.234572	−	0.972099i	$-0.424631\pi$
$23$	2.24994	0.469145	0.234572	+	0.972099i	$0.424631\pi$
$24$	0.394661	0.0805598
$25$	−4.74218	−0.948435
$26$	0	0
$27$	2.30649	0.443885
$28$	1.24955	0.236142
$29$	−9.14046	−1.69734	−0.848671	−	0.528922i	$-0.822597\pi$
$29$	−9.14046	−1.69734	−0.848671	+	0.528922i	$0.822597\pi$
$30$	0.200395	0.0365869
$31$	−1.74898	−0.314126	−0.157063	−	0.987589i	$-0.550203\pi$
$31$	−1.74898	−0.314126	−0.157063	+	0.987589i	$0.550203\pi$
$32$	−1.00000	−0.176777
$33$	1.06489	0.185374
$34$	−4.56219	−0.782409
$35$	0.634476	0.107246
$36$	−2.84424	−0.474041
$37$	8.71511	1.43275	0.716377	−	0.697713i	$-0.245799\pi$
$37$	8.71511	1.43275	0.716377	+	0.697713i	$0.245799\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−0.507765	−0.0802846
$41$	9.55485	1.49222	0.746109	−	0.665824i	$-0.231920\pi$
$41$	9.55485	1.49222	0.746109	+	0.665824i	$0.231920\pi$
$42$	0.493147	0.0760943
$43$	3.72738	0.568421	0.284210	−	0.958762i	$-0.408269\pi$
$43$	3.72738	0.568421	0.284210	+	0.958762i	$0.408269\pi$
$44$	−2.69825	−0.406776
$45$	−1.44421	−0.215289
$46$	−2.24994	−0.331736
$47$	−8.96128	−1.30714	−0.653568	−	0.756867i	$-0.726729\pi$
$47$	−8.96128	−1.30714	−0.653568	+	0.756867i	$0.726729\pi$
$48$	−0.394661	−0.0569644
$49$	−5.43863	−0.776947
$50$	4.74218	0.670645
$51$	−1.80052	−0.252123
$52$	0	0
$53$	5.80547	0.797443	0.398721	−	0.917072i	$-0.369454\pi$
$53$	5.80547	0.797443	0.398721	+	0.917072i	$0.369454\pi$
$54$	−2.30649	−0.313874
$55$	−1.37007	−0.184741
$56$	−1.24955	−0.166978
$57$	−0.394661	−0.0522741
$58$	9.14046	1.20020
$59$	8.65749	1.12711	0.563555	−	0.826079i	$-0.309433\pi$
$59$	8.65749	1.12711	0.563555	+	0.826079i	$0.309433\pi$
$60$	−0.200395	−0.0258708
$61$	−0.618453	−0.0791848	−0.0395924	−	0.999216i	$-0.512606\pi$
$61$	−0.618453	−0.0791848	−0.0395924	+	0.999216i	$0.512606\pi$
$62$	1.74898	0.222120
$63$	−3.55402	−0.447764
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.06489	−0.131079
$67$	−12.2636	−1.49824	−0.749118	−	0.662436i	$-0.769523\pi$
$67$	−12.2636	−1.49824	−0.749118	+	0.662436i	$0.769523\pi$
$68$	4.56219	0.553247
$69$	−0.887963	−0.106898
$70$	−0.634476	−0.0758344
$71$	0.995275	0.118117	0.0590587	−	0.998255i	$-0.481190\pi$
$71$	0.995275	0.118117	0.0590587	+	0.998255i	$0.481190\pi$
$72$	2.84424	0.335197
$73$	−0.150857	−0.0176564	−0.00882821	−	0.999961i	$-0.502810\pi$
$73$	−0.150857	−0.0176564	−0.00882821	+	0.999961i	$0.502810\pi$
$74$	−8.71511	−1.01311
$75$	1.87155	0.216108
$76$	1.00000	0.114708
$77$	−3.37159	−0.384228
$78$	0	0
$79$	−5.80849	−0.653506	−0.326753	−	0.945110i	$-0.605955\pi$
$79$	−5.80849	−0.653506	−0.326753	+	0.945110i	$0.605955\pi$
$80$	0.507765	0.0567698
$81$	7.62245	0.846939
$82$	−9.55485	−1.05516
$83$	8.15004	0.894583	0.447291	−	0.894388i	$-0.352389\pi$
$83$	8.15004	0.894583	0.447291	+	0.894388i	$0.352389\pi$
$84$	−0.493147	−0.0538068
$85$	2.31652	0.251262
$86$	−3.72738	−0.401934
$87$	3.60738	0.386752
$88$	2.69825	0.287634
$89$	−2.44741	−0.259424	−0.129712	−	0.991552i	$-0.541405\pi$
$89$	−2.44741	−0.259424	−0.129712	+	0.991552i	$0.541405\pi$
$90$	1.44421	0.152233
$91$	0	0
$92$	2.24994	0.234572
$93$	0.690253	0.0715759
$94$	8.96128	0.924285
$95$	0.507765	0.0520955
$96$	0.394661	0.0402799
$97$	5.19401	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$97$	5.19401	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$98$	5.43863	0.549385
$99$	7.67447	0.771313
$100$	−4.74218	−0.474218
$101$	−16.9227	−1.68387	−0.841936	−	0.539577i	$-0.818584\pi$
$101$	−16.9227	−1.68387	−0.841936	+	0.539577i	$0.818584\pi$
$102$	1.80052	0.178278
$103$	4.09263	0.403259	0.201629	−	0.979462i	$-0.435376\pi$
$103$	4.09263	0.403259	0.201629	+	0.979462i	$0.435376\pi$
$104$	0	0
$105$	−0.250403	−0.0244368
$106$	−5.80547	−0.563877
$107$	17.5881	1.70031	0.850154	−	0.526534i	$-0.176509\pi$
$107$	17.5881	1.70031	0.850154	+	0.526534i	$0.176509\pi$
$108$	2.30649	0.221942
$109$	−19.6811	−1.88510	−0.942552	−	0.334059i	$-0.891581\pi$
$109$	−19.6811	−1.88510	−0.942552	+	0.334059i	$0.891581\pi$
$110$	1.37007	0.130631
$111$	−3.43951	−0.326464
$112$	1.24955	0.118071
$113$	−10.0150	−0.942128	−0.471064	−	0.882099i	$-0.656130\pi$
$113$	−10.0150	−0.942128	−0.471064	+	0.882099i	$0.656130\pi$
$114$	0.394661	0.0369634
$115$	1.14244	0.106533
$116$	−9.14046	−0.848671
$117$	0	0
$118$	−8.65749	−0.796987
$119$	5.70067	0.522580
$120$	0.200395	0.0182934
$121$	−3.71947	−0.338133
$122$	0.618453	0.0559921
$123$	−3.77092	−0.340013
$124$	−1.74898	−0.157063
$125$	−4.94673	−0.442449
$126$	3.55402	0.316617
$127$	−22.1440	−1.96497	−0.982483	−	0.186352i	$-0.940333\pi$
$127$	−22.1440	−1.96497	−0.982483	+	0.186352i	$0.940333\pi$
$128$	−1.00000	−0.0883883
$129$	−1.47105	−0.129519
$130$	0	0
$131$	−19.4396	−1.69845	−0.849223	−	0.528034i	$-0.822929\pi$
$131$	−19.4396	−1.69845	−0.849223	+	0.528034i	$0.822929\pi$
$132$	1.06489	0.0926869
$133$	1.24955	0.108350
$134$	12.2636	1.05941
$135$	1.17116	0.100797
$136$	−4.56219	−0.391205
$137$	5.54764	0.473966	0.236983	−	0.971514i	$-0.423841\pi$
$137$	5.54764	0.473966	0.236983	+	0.971514i	$0.423841\pi$
$138$	0.887963	0.0755884
$139$	11.9267	1.01161	0.505806	−	0.862647i	$-0.331195\pi$
$139$	11.9267	1.01161	0.505806	+	0.862647i	$0.331195\pi$
$140$	0.634476	0.0536230
$141$	3.53666	0.297841
$142$	−0.995275	−0.0835216
$143$	0	0
$144$	−2.84424	−0.237020
$145$	−4.64120	−0.385431
$146$	0.150857	0.0124850
$147$	2.14641	0.177033
$148$	8.71511	0.716377
$149$	15.2350	1.24810	0.624052	−	0.781383i	$-0.285485\pi$
$149$	15.2350	1.24810	0.624052	+	0.781383i	$0.285485\pi$
$150$	−1.87155	−0.152811
$151$	−16.5771	−1.34902	−0.674512	−	0.738264i	$-0.735646\pi$
$151$	−16.5771	−1.34902	−0.674512	+	0.738264i	$0.735646\pi$
$152$	−1.00000	−0.0811107
$153$	−12.9760	−1.04905
$154$	3.37159	0.271690
$155$	−0.888069	−0.0713314
$156$	0	0
$157$	0.699811	0.0558510	0.0279255	−	0.999610i	$-0.491110\pi$
$157$	0.699811	0.0558510	0.0279255	+	0.999610i	$0.491110\pi$
$158$	5.80849	0.462099
$159$	−2.29119	−0.181703
$160$	−0.507765	−0.0401423
$161$	2.81141	0.221570
$162$	−7.62245	−0.598876
$163$	23.5897	1.84769	0.923844	−	0.382770i	$-0.125030\pi$
$163$	23.5897	1.84769	0.923844	+	0.382770i	$0.125030\pi$
$164$	9.55485	0.746109
$165$	0.540714	0.0420945
$166$	−8.15004	−0.632565
$167$	9.79491	0.757953	0.378976	−	0.925406i	$-0.376276\pi$
$167$	9.79491	0.757953	0.378976	+	0.925406i	$0.376276\pi$
$168$	0.493147	0.0380471
$169$	0	0
$170$	−2.31652	−0.177669
$171$	−2.84424	−0.217505
$172$	3.72738	0.284210
$173$	−7.53389	−0.572791	−0.286396	−	0.958111i	$-0.592457\pi$
$173$	−7.53389	−0.572791	−0.286396	+	0.958111i	$0.592457\pi$
$174$	−3.60738	−0.273475
$175$	−5.92557	−0.447931
$176$	−2.69825	−0.203388
$177$	−3.41677	−0.256820
$178$	2.44741	0.183441
$179$	−5.33482	−0.398743	−0.199372	−	0.979924i	$-0.563890\pi$
$179$	−5.33482	−0.398743	−0.199372	+	0.979924i	$0.563890\pi$
$180$	−1.44421	−0.107645
$181$	−24.1292	−1.79351	−0.896755	−	0.442527i	$-0.854082\pi$
$181$	−24.1292	−1.79351	−0.896755	+	0.442527i	$0.854082\pi$
$182$	0	0
$183$	0.244079	0.0180428
$184$	−2.24994	−0.165868
$185$	4.42522	0.325349
$186$	−0.690253	−0.0506118
$187$	−12.3099	−0.900190
$188$	−8.96128	−0.653568
$189$	2.88207	0.209640
$190$	−0.507765	−0.0368371
$191$	−20.7322	−1.50013	−0.750064	−	0.661365i	$-0.769978\pi$
$191$	−20.7322	−1.50013	−0.750064	+	0.661365i	$0.769978\pi$
$192$	−0.394661	−0.0284822
$193$	9.89665	0.712376	0.356188	−	0.934414i	$-0.384076\pi$
$193$	9.89665	0.712376	0.356188	+	0.934414i	$0.384076\pi$
$194$	−5.19401	−0.372908
$195$	0	0
$196$	−5.43863	−0.388474
$197$	−5.46053	−0.389046	−0.194523	−	0.980898i	$-0.562316\pi$
$197$	−5.46053	−0.389046	−0.194523	+	0.980898i	$0.562316\pi$
$198$	−7.67447	−0.545401
$199$	−15.8738	−1.12526	−0.562632	−	0.826708i	$-0.690211\pi$
$199$	−15.8738	−1.12526	−0.562632	+	0.826708i	$0.690211\pi$
$200$	4.74218	0.335322
$201$	4.83996	0.341384
$202$	16.9227	1.19068
$203$	−11.4214	−0.801628
$204$	−1.80052	−0.126061
$205$	4.85162	0.338852
$206$	−4.09263	−0.285147
$207$	−6.39938	−0.444787
$208$	0	0
$209$	−2.69825	−0.186642
$210$	0.250403	0.0172794
$211$	−19.8984	−1.36986	−0.684931	−	0.728608i	$-0.740167\pi$
$211$	−19.8984	−1.36986	−0.684931	+	0.728608i	$0.740167\pi$
$212$	5.80547	0.398721
$213$	−0.392796	−0.0269139
$214$	−17.5881	−1.20230
$215$	1.89263	0.129076
$216$	−2.30649	−0.156937
$217$	−2.18543	−0.148357
$218$	19.6811	1.33297
$219$	0.0595371	0.00402315
$220$	−1.37007	−0.0923703
$221$	0	0
$222$	3.43951	0.230845
$223$	−10.3506	−0.693126	−0.346563	−	0.938027i	$-0.612651\pi$
$223$	−10.3506	−0.693126	−0.346563	+	0.938027i	$0.612651\pi$
$224$	−1.24955	−0.0834889
$225$	13.4879	0.899193
$226$	10.0150	0.666185
$227$	6.04541	0.401248	0.200624	−	0.979668i	$-0.435703\pi$
$227$	6.04541	0.401248	0.200624	+	0.979668i	$0.435703\pi$
$228$	−0.394661	−0.0261370
$229$	11.3666	0.751125	0.375563	−	0.926797i	$-0.377449\pi$
$229$	11.3666	0.751125	0.375563	+	0.926797i	$0.377449\pi$
$230$	−1.14244	−0.0753302
$231$	1.33063	0.0875492
$232$	9.14046	0.600101
$233$	−19.6434	−1.28688	−0.643441	−	0.765496i	$-0.722494\pi$
$233$	−19.6434	−1.28688	−0.643441	+	0.765496i	$0.722494\pi$
$234$	0	0
$235$	−4.55022	−0.296824
$236$	8.65749	0.563555
$237$	2.29238	0.148906
$238$	−5.70067	−0.369520
$239$	8.02294	0.518961	0.259481	−	0.965748i	$-0.416449\pi$
$239$	8.02294	0.518961	0.259481	+	0.965748i	$0.416449\pi$
$240$	−0.200395	−0.0129354
$241$	−13.5781	−0.874640	−0.437320	−	0.899306i	$-0.644072\pi$
$241$	−13.5781	−0.874640	−0.437320	+	0.899306i	$0.644072\pi$
$242$	3.71947	0.239096
$243$	−9.92776	−0.636866
$244$	−0.618453	−0.0395924
$245$	−2.76154	−0.176429
$246$	3.77092	0.240425
$247$	0	0
$248$	1.74898	0.111060
$249$	−3.21650	−0.203837
$250$	4.94673	0.312859
$251$	−21.7978	−1.37587	−0.687933	−	0.725774i	$-0.741482\pi$
$251$	−21.7978	−1.37587	−0.687933	+	0.725774i	$0.741482\pi$
$252$	−3.55402	−0.223882
$253$	−6.07089	−0.381674
$254$	22.1440	1.38944
$255$	−0.914239	−0.0572518
$256$	1.00000	0.0625000
$257$	21.9326	1.36812	0.684059	−	0.729427i	$-0.260213\pi$
$257$	21.9326	1.36812	0.684059	+	0.729427i	$0.260213\pi$
$258$	1.47105	0.0915836
$259$	10.8899	0.676668
$260$	0	0
$261$	25.9977	1.60922
$262$	19.4396	1.20098
$263$	8.68002	0.535233	0.267616	−	0.963526i	$-0.413764\pi$
$263$	8.68002	0.535233	0.267616	+	0.963526i	$0.413764\pi$
$264$	−1.06489	−0.0655395
$265$	2.94781	0.181083
$266$	−1.24955	−0.0766147
$267$	0.965894	0.0591118
$268$	−12.2636	−0.749118
$269$	10.2644	0.625832	0.312916	−	0.949781i	$-0.398694\pi$
$269$	10.2644	0.625832	0.312916	+	0.949781i	$0.398694\pi$
$270$	−1.17116	−0.0712742
$271$	19.1893	1.16566	0.582832	−	0.812593i	$-0.301945\pi$
$271$	19.1893	1.16566	0.582832	+	0.812593i	$0.301945\pi$
$272$	4.56219	0.276623
$273$	0	0
$274$	−5.54764	−0.335145
$275$	12.7956	0.771601
$276$	−0.887963	−0.0534491
$277$	3.03422	0.182309	0.0911543	−	0.995837i	$-0.470944\pi$
$277$	3.03422	0.182309	0.0911543	+	0.995837i	$0.470944\pi$
$278$	−11.9267	−0.715318
$279$	4.97452	0.297817
$280$	−0.634476	−0.0379172
$281$	6.21368	0.370677	0.185339	−	0.982675i	$-0.440662\pi$
$281$	6.21368	0.370677	0.185339	+	0.982675i	$0.440662\pi$
$282$	−3.53666	−0.210605
$283$	−12.5355	−0.745157	−0.372578	−	0.928001i	$-0.621526\pi$
$283$	−12.5355	−0.745157	−0.372578	+	0.928001i	$0.621526\pi$
$284$	0.995275	0.0590587
$285$	−0.200395	−0.0118704
$286$	0	0
$287$	11.9392	0.704751
$288$	2.84424	0.167599
$289$	3.81359	0.224329
$290$	4.64120	0.272541
$291$	−2.04987	−0.120166
$292$	−0.150857	−0.00882821
$293$	14.9894	0.875689	0.437845	−	0.899051i	$-0.355742\pi$
$293$	14.9894	0.875689	0.437845	+	0.899051i	$0.355742\pi$
$294$	−2.14641	−0.125181
$295$	4.39597	0.255943
$296$	−8.71511	−0.506555
$297$	−6.22348	−0.361123
$298$	−15.2350	−0.882542
$299$	0	0
$300$	1.87155	0.108054
$301$	4.65754	0.268456
$302$	16.5771	0.953904
$303$	6.67873	0.383683
$304$	1.00000	0.0573539
$305$	−0.314029	−0.0179812
$306$	12.9760	0.741787
$307$	−29.9417	−1.70886	−0.854431	−	0.519564i	$-0.826094\pi$
$307$	−29.9417	−1.70886	−0.854431	+	0.519564i	$0.826094\pi$
$308$	−3.37159	−0.192114
$309$	−1.61520	−0.0918855
$310$	0.888069	0.0504389
$311$	−4.40422	−0.249740	−0.124870	−	0.992173i	$-0.539851\pi$
$311$	−4.40422	−0.249740	−0.124870	+	0.992173i	$0.539851\pi$
$312$	0	0
$313$	10.8316	0.612238	0.306119	−	0.951993i	$-0.400969\pi$
$313$	10.8316	0.612238	0.306119	+	0.951993i	$0.400969\pi$
$314$	−0.699811	−0.0394926
$315$	−1.80460	−0.101678
$316$	−5.80849	−0.326753
$317$	−27.7571	−1.55900	−0.779498	−	0.626405i	$-0.784526\pi$
$317$	−27.7571	−1.55900	−0.779498	+	0.626405i	$0.784526\pi$
$318$	2.29119	0.128484
$319$	24.6632	1.38088
$320$	0.507765	0.0283849
$321$	−6.94134	−0.387428
$322$	−2.81141	−0.156674
$323$	4.56219	0.253847
$324$	7.62245	0.423469
$325$	0	0
$326$	−23.5897	−1.30651
$327$	7.76734	0.429535
$328$	−9.55485	−0.527579
$329$	−11.1975	−0.617341
$330$	−0.540714	−0.0297653
$331$	15.6936	0.862600	0.431300	−	0.902209i	$-0.358055\pi$
$331$	15.6936	0.862600	0.431300	+	0.902209i	$0.358055\pi$
$332$	8.15004	0.447291
$333$	−24.7879	−1.35837
$334$	−9.79491	−0.535953
$335$	−6.22702	−0.340218
$336$	−0.493147	−0.0269034
$337$	20.3658	1.10940	0.554698	−	0.832052i	$-0.312834\pi$
$337$	20.3658	1.10940	0.554698	+	0.832052i	$0.312834\pi$
$338$	0	0
$339$	3.95251	0.214671
$340$	2.31652	0.125631
$341$	4.71917	0.255558
$342$	2.84424	0.153799
$343$	−15.5427	−0.839225
$344$	−3.72738	−0.200967
$345$	−0.450876	−0.0242743
$346$	7.53389	0.405025
$347$	20.8429	1.11890	0.559451	−	0.828863i	$-0.311012\pi$
$347$	20.8429	1.11890	0.559451	+	0.828863i	$0.311012\pi$
$348$	3.60738	0.193376
$349$	−1.36583	−0.0731113	−0.0365556	−	0.999332i	$-0.511639\pi$
$349$	−1.36583	−0.0731113	−0.0365556	+	0.999332i	$0.511639\pi$
$350$	5.92557	0.316735
$351$	0	0
$352$	2.69825	0.143817
$353$	1.42712	0.0759581	0.0379791	−	0.999279i	$-0.487908\pi$
$353$	1.42712	0.0759581	0.0379791	+	0.999279i	$0.487908\pi$
$354$	3.41677	0.181599
$355$	0.505365	0.0268220
$356$	−2.44741	−0.129712
$357$	−2.24983	−0.119074
$358$	5.33482	0.281954
$359$	27.3665	1.44435	0.722173	−	0.691712i	$-0.243143\pi$
$359$	27.3665	1.44435	0.722173	+	0.691712i	$0.243143\pi$
$360$	1.44421	0.0761163
$361$	1.00000	0.0526316
$362$	24.1292	1.26820
$363$	1.46793	0.0770462
$364$	0	0
$365$	−0.0765996	−0.00400941
$366$	−0.244079	−0.0127582
$367$	24.9590	1.30285	0.651424	−	0.758714i	$-0.274172\pi$
$367$	24.9590	1.30285	0.651424	+	0.758714i	$0.274172\pi$
$368$	2.24994	0.117286
$369$	−27.1763	−1.41474
$370$	−4.42522	−0.230056
$371$	7.25421	0.376620
$372$	0.690253	0.0357879
$373$	−17.5861	−0.910573	−0.455286	−	0.890345i	$-0.650463\pi$
$373$	−17.5861	−0.910573	−0.455286	+	0.890345i	$0.650463\pi$
$374$	12.3099	0.636531
$375$	1.95228	0.100815
$376$	8.96128	0.462143
$377$	0	0
$378$	−2.88207	−0.148238
$379$	−14.2698	−0.732992	−0.366496	−	0.930420i	$-0.619443\pi$
$379$	−14.2698	−0.732992	−0.366496	+	0.930420i	$0.619443\pi$
$380$	0.507765	0.0260478
$381$	8.73938	0.447732
$382$	20.7322	1.06075
$383$	−12.8820	−0.658241	−0.329121	−	0.944288i	$-0.606752\pi$
$383$	−12.8820	−0.658241	−0.329121	+	0.944288i	$0.606752\pi$
$384$	0.394661	0.0201399
$385$	−1.71197	−0.0872502
$386$	−9.89665	−0.503726
$387$	−10.6016	−0.538909
$388$	5.19401	0.263686
$389$	4.94578	0.250761	0.125380	−	0.992109i	$-0.459985\pi$
$389$	4.94578	0.250761	0.125380	+	0.992109i	$0.459985\pi$
$390$	0	0
$391$	10.2647	0.519106
$392$	5.43863	0.274692
$393$	7.67205	0.387003
$394$	5.46053	0.275097
$395$	−2.94934	−0.148398
$396$	7.67447	0.385657
$397$	−22.0346	−1.10589	−0.552943	−	0.833219i	$-0.686495\pi$
$397$	−22.0346	−1.10589	−0.552943	+	0.833219i	$0.686495\pi$
$398$	15.8738	0.795681
$399$	−0.493147	−0.0246882
$400$	−4.74218	−0.237109
$401$	2.94800	0.147216	0.0736082	−	0.997287i	$-0.476549\pi$
$401$	2.94800	0.147216	0.0736082	+	0.997287i	$0.476549\pi$
$402$	−4.83996	−0.241395
$403$	0	0
$404$	−16.9227	−0.841936
$405$	3.87041	0.192322
$406$	11.4214	0.566837
$407$	−23.5155	−1.16562
$408$	1.80052	0.0891389
$409$	−18.3674	−0.908211	−0.454105	−	0.890948i	$-0.650041\pi$
$409$	−18.3674	−0.908211	−0.454105	+	0.890948i	$0.650041\pi$
$410$	−4.85162	−0.239604
$411$	−2.18943	−0.107997
$412$	4.09263	0.201629
$413$	10.8179	0.532316
$414$	6.39938	0.314512
$415$	4.13830	0.203141
$416$	0	0
$417$	−4.70701	−0.230503
$418$	2.69825	0.131976
$419$	−7.18949	−0.351229	−0.175615	−	0.984459i	$-0.556191\pi$
$419$	−7.18949	−0.351229	−0.175615	+	0.984459i	$0.556191\pi$
$420$	−0.250403	−0.0122184
$421$	−13.4228	−0.654189	−0.327095	−	0.944992i	$-0.606070\pi$
$421$	−13.4228	−0.654189	−0.327095	+	0.944992i	$0.606070\pi$
$422$	19.8984	0.968638
$423$	25.4881	1.23927
$424$	−5.80547	−0.281939
$425$	−21.6347	−1.04944
$426$	0.392796	0.0190310
$427$	−0.772787	−0.0373978
$428$	17.5881	0.850154
$429$	0	0
$430$	−1.89263	−0.0912709
$431$	−19.5840	−0.943328	−0.471664	−	0.881778i	$-0.656347\pi$
$431$	−19.5840	−0.943328	−0.471664	+	0.881778i	$0.656347\pi$
$432$	2.30649	0.110971
$433$	−33.5004	−1.60993	−0.804964	−	0.593324i	$-0.797815\pi$
$433$	−33.5004	−1.60993	−0.804964	+	0.593324i	$0.797815\pi$
$434$	2.18543	0.104904
$435$	1.83170	0.0878233
$436$	−19.6811	−0.942552
$437$	2.24994	0.107629
$438$	−0.0595371	−0.00284479
$439$	−4.74365	−0.226402	−0.113201	−	0.993572i	$-0.536110\pi$
$439$	−4.74365	−0.226402	−0.113201	+	0.993572i	$0.536110\pi$
$440$	1.37007	0.0653157
$441$	15.4688	0.736609
$442$	0	0
$443$	31.0710	1.47623	0.738114	−	0.674676i	$-0.235717\pi$
$443$	31.0710	1.47623	0.738114	+	0.674676i	$0.235717\pi$
$444$	−3.43951	−0.163232
$445$	−1.24271	−0.0589099
$446$	10.3506	0.490114
$447$	−6.01267	−0.284390
$448$	1.24955	0.0590356
$449$	−15.7887	−0.745117	−0.372558	−	0.928009i	$-0.621519\pi$
$449$	−15.7887	−0.745117	−0.372558	+	0.928009i	$0.621519\pi$
$450$	−13.4879	−0.635826
$451$	−25.7813	−1.21400
$452$	−10.0150	−0.471064
$453$	6.54232	0.307385
$454$	−6.04541	−0.283725
$455$	0	0
$456$	0.394661	0.0184817
$457$	−1.77056	−0.0828232	−0.0414116	−	0.999142i	$-0.513185\pi$
$457$	−1.77056	−0.0828232	−0.0414116	+	0.999142i	$0.513185\pi$
$458$	−11.3666	−0.531126
$459$	10.5227	0.491156
$460$	1.14244	0.0532665
$461$	−11.8636	−0.552545	−0.276272	−	0.961079i	$-0.589099\pi$
$461$	−11.8636	−0.552545	−0.276272	+	0.961079i	$0.589099\pi$
$462$	−1.33063	−0.0619066
$463$	4.13269	0.192063	0.0960313	−	0.995378i	$-0.469385\pi$
$463$	4.13269	0.192063	0.0960313	+	0.995378i	$0.469385\pi$
$464$	−9.14046	−0.424335
$465$	0.350486	0.0162534
$466$	19.6434	0.909962
$467$	−18.5126	−0.856662	−0.428331	−	0.903622i	$-0.640898\pi$
$467$	−18.5126	−0.856662	−0.428331	+	0.903622i	$0.640898\pi$
$468$	0	0
$469$	−15.3239	−0.707594
$470$	4.55022	0.209886
$471$	−0.276188	−0.0127261
$472$	−8.65749	−0.398493
$473$	−10.0574	−0.462440
$474$	−2.29238	−0.105293
$475$	−4.74218	−0.217586
$476$	5.70067	0.261290
$477$	−16.5122	−0.756040
$478$	−8.02294	−0.366961
$479$	−18.4074	−0.841055	−0.420527	−	0.907280i	$-0.638155\pi$
$479$	−18.4074	−0.841055	−0.420527	+	0.907280i	$0.638155\pi$
$480$	0.200395	0.00914672
$481$	0	0
$482$	13.5781	0.618464
$483$	−1.10955	−0.0504863
$484$	−3.71947	−0.169067
$485$	2.63733	0.119755
$486$	9.92776	0.450332
$487$	−1.99282	−0.0903034	−0.0451517	−	0.998980i	$-0.514377\pi$
$487$	−1.99282	−0.0903034	−0.0451517	+	0.998980i	$0.514377\pi$
$488$	0.618453	0.0279961
$489$	−9.30992	−0.421009
$490$	2.76154	0.124754
$491$	−34.5469	−1.55908	−0.779540	−	0.626353i	$-0.784547\pi$
$491$	−34.5469	−1.55908	−0.779540	+	0.626353i	$0.784547\pi$
$492$	−3.77092	−0.170006
$493$	−41.7005	−1.87810
$494$	0	0
$495$	3.89682	0.175149
$496$	−1.74898	−0.0785314
$497$	1.24364	0.0557850
$498$	3.21650	0.144135
$499$	−24.5129	−1.09735	−0.548675	−	0.836036i	$-0.684867\pi$
$499$	−24.5129	−1.09735	−0.548675	+	0.836036i	$0.684867\pi$
$500$	−4.94673	−0.221225
$501$	−3.86566	−0.172705
$502$	21.7978	0.972884
$503$	31.8996	1.42233	0.711165	−	0.703025i	$-0.248168\pi$
$503$	31.8996	1.42233	0.711165	+	0.703025i	$0.248168\pi$
$504$	3.55402	0.158308
$505$	−8.59275	−0.382372
$506$	6.07089	0.269884
$507$	0	0
$508$	−22.1440	−0.982483
$509$	−40.5559	−1.79761	−0.898804	−	0.438351i	$-0.855563\pi$
$509$	−40.5559	−1.79761	−0.898804	+	0.438351i	$0.855563\pi$
$510$	0.914239	0.0404832
$511$	−0.188502	−0.00833885
$512$	−1.00000	−0.0441942
$513$	2.30649	0.101834
$514$	−21.9326	−0.967405
$515$	2.07809	0.0915717
$516$	−1.47105	−0.0647594
$517$	24.1797	1.06342
$518$	−10.8899	−0.478477
$519$	2.97333	0.130515
$520$	0	0
$521$	−33.5725	−1.47084	−0.735419	−	0.677613i	$-0.763015\pi$
$521$	−33.5725	−1.47084	−0.735419	+	0.677613i	$0.763015\pi$
$522$	−25.9977	−1.13789
$523$	−37.0407	−1.61968	−0.809838	−	0.586653i	$-0.800445\pi$
$523$	−37.0407	−1.61968	−0.809838	+	0.586653i	$0.800445\pi$
$524$	−19.4396	−0.849223
$525$	2.33859	0.102064
$526$	−8.68002	−0.378467
$527$	−7.97917	−0.347578
$528$	1.06489	0.0463435
$529$	−17.9378	−0.779903
$530$	−2.94781	−0.128045
$531$	−24.6240	−1.06859
$532$	1.24955	0.0541748
$533$	0	0
$534$	−0.965894	−0.0417983
$535$	8.93063	0.386105
$536$	12.2636	0.529707
$537$	2.10544	0.0908566
$538$	−10.2644	−0.442530
$539$	14.6748	0.632087
$540$	1.17116	0.0503985
$541$	−38.3168	−1.64737	−0.823684	−	0.567050i	$-0.808085\pi$
$541$	−38.3168	−1.64737	−0.823684	+	0.567050i	$0.808085\pi$
$542$	−19.1893	−0.824249
$543$	9.52285	0.408665
$544$	−4.56219	−0.195602
$545$	−9.99335	−0.428068
$546$	0	0
$547$	32.6170	1.39460	0.697302	−	0.716778i	$-0.254384\pi$
$547$	32.6170	1.39460	0.697302	+	0.716778i	$0.254384\pi$
$548$	5.54764	0.236983
$549$	1.75903	0.0750736
$550$	−12.7956	−0.545604
$551$	−9.14046	−0.389397
$552$	0.887963	0.0377942
$553$	−7.25798	−0.308641
$554$	−3.03422	−0.128912
$555$	−1.74646	−0.0741331
$556$	11.9267	0.505806
$557$	14.1172	0.598164	0.299082	−	0.954227i	$-0.403320\pi$
$557$	14.1172	0.598164	0.299082	+	0.954227i	$0.403320\pi$
$558$	−4.97452	−0.210588
$559$	0	0
$560$	0.634476	0.0268115
$561$	4.85824	0.205115
$562$	−6.21368	−0.262108
$563$	7.41776	0.312621	0.156311	−	0.987708i	$-0.450040\pi$
$563$	7.41776	0.312621	0.156311	+	0.987708i	$0.450040\pi$
$564$	3.53666	0.148920
$565$	−5.08524	−0.213938
$566$	12.5355	0.526905
$567$	9.52461	0.399996
$568$	−0.995275	−0.0417608
$569$	−8.69486	−0.364508	−0.182254	−	0.983252i	$-0.558339\pi$
$569$	−8.69486	−0.364508	−0.182254	+	0.983252i	$0.558339\pi$
$570$	0.200395	0.00839361
$571$	24.1793	1.01187	0.505936	−	0.862571i	$-0.331147\pi$
$571$	24.1793	1.01187	0.505936	+	0.862571i	$0.331147\pi$
$572$	0	0
$573$	8.18218	0.341815
$574$	−11.9392	−0.498334
$575$	−10.6696	−0.444953
$576$	−2.84424	−0.118510
$577$	−18.3652	−0.764554	−0.382277	−	0.924048i	$-0.624860\pi$
$577$	−18.3652	−0.764554	−0.382277	+	0.924048i	$0.624860\pi$
$578$	−3.81359	−0.158624
$579$	−3.90582	−0.162320
$580$	−4.64120	−0.192715
$581$	10.1839	0.422498
$582$	2.04987	0.0849699
$583$	−15.6646	−0.648761
$584$	0.150857	0.00624249
$585$	0	0
$586$	−14.9894	−0.619206
$587$	−19.5310	−0.806130	−0.403065	−	0.915171i	$-0.632055\pi$
$587$	−19.5310	−0.806130	−0.403065	+	0.915171i	$0.632055\pi$
$588$	2.14641	0.0885166
$589$	−1.74898	−0.0720654
$590$	−4.39597	−0.180979
$591$	2.15505	0.0886471
$592$	8.71511	0.358189
$593$	−13.0686	−0.536663	−0.268331	−	0.963327i	$-0.586472\pi$
$593$	−13.0686	−0.536663	−0.268331	+	0.963327i	$0.586472\pi$
$594$	6.22348	0.255353
$595$	2.89460	0.118667
$596$	15.2350	0.624052
$597$	6.26476	0.256400
$598$	0	0
$599$	1.29989	0.0531120	0.0265560	−	0.999647i	$-0.491546\pi$
$599$	1.29989	0.0531120	0.0265560	+	0.999647i	$0.491546\pi$
$600$	−1.87155	−0.0764057
$601$	−6.84573	−0.279243	−0.139622	−	0.990205i	$-0.544589\pi$
$601$	−6.84573	−0.279243	−0.139622	+	0.990205i	$0.544589\pi$
$602$	−4.65754	−0.189827
$603$	34.8806	1.42045
$604$	−16.5771	−0.674512
$605$	−1.88861	−0.0767831
$606$	−6.67873	−0.271305
$607$	−18.2932	−0.742499	−0.371249	−	0.928533i	$-0.621070\pi$
$607$	−18.2932	−0.742499	−0.371249	+	0.928533i	$0.621070\pi$
$608$	−1.00000	−0.0405554
$609$	4.50759	0.182657
$610$	0.314029	0.0127146
$611$	0	0
$612$	−12.9760	−0.524523
$613$	−1.71604	−0.0693101	−0.0346550	−	0.999399i	$-0.511033\pi$
$613$	−1.71604	−0.0693101	−0.0346550	+	0.999399i	$0.511033\pi$
$614$	29.9417	1.20835
$615$	−1.91474	−0.0772098
$616$	3.37159	0.135845
$617$	−22.6741	−0.912823	−0.456412	−	0.889769i	$-0.650866\pi$
$617$	−22.6741	−0.912823	−0.456412	+	0.889769i	$0.650866\pi$
$618$	1.61520	0.0649729
$619$	2.65268	0.106620	0.0533101	−	0.998578i	$-0.483023\pi$
$619$	2.65268	0.106620	0.0533101	+	0.998578i	$0.483023\pi$
$620$	−0.888069	−0.0356657
$621$	5.18947	0.208246
$622$	4.40422	0.176593
$623$	−3.05815	−0.122522
$624$	0	0
$625$	21.1991	0.847964
$626$	−10.8316	−0.432917
$627$	1.06489	0.0425277
$628$	0.699811	0.0279255
$629$	39.7600	1.58533
$630$	1.80460	0.0718971
$631$	12.5657	0.500234	0.250117	−	0.968216i	$-0.419531\pi$
$631$	12.5657	0.500234	0.250117	+	0.968216i	$0.419531\pi$
$632$	5.80849	0.231049
$633$	7.85311	0.312133
$634$	27.7571	1.10238
$635$	−11.2440	−0.446203
$636$	−2.29119	−0.0908516
$637$	0	0
$638$	−24.6632	−0.976426
$639$	−2.83080	−0.111985
$640$	−0.507765	−0.0200712
$641$	28.6389	1.13117	0.565583	−	0.824691i	$-0.308651\pi$
$641$	28.6389	1.13117	0.565583	+	0.824691i	$0.308651\pi$
$642$	6.94134	0.273953
$643$	30.2543	1.19311	0.596556	−	0.802571i	$-0.296535\pi$
$643$	30.2543	1.19311	0.596556	+	0.802571i	$0.296535\pi$
$644$	2.81141	0.110785
$645$	−0.746948	−0.0294110
$646$	−4.56219	−0.179497
$647$	−2.12073	−0.0833745	−0.0416873	−	0.999131i	$-0.513273\pi$
$647$	−2.12073	−0.0833745	−0.0416873	+	0.999131i	$0.513273\pi$
$648$	−7.62245	−0.299438
$649$	−23.3600	−0.916962
$650$	0	0
$651$	0.862504	0.0338042
$652$	23.5897	0.923844
$653$	−30.6229	−1.19837	−0.599183	−	0.800612i	$-0.704508\pi$
$653$	−30.6229	−1.19837	−0.599183	+	0.800612i	$0.704508\pi$
$654$	−7.76734	−0.303727
$655$	−9.87074	−0.385682
$656$	9.55485	0.373054
$657$	0.429073	0.0167397
$658$	11.1975	0.436526
$659$	10.1419	0.395074	0.197537	−	0.980295i	$-0.436706\pi$
$659$	10.1419	0.395074	0.197537	+	0.980295i	$0.436706\pi$
$660$	0.540714	0.0210473
$661$	−15.0540	−0.585535	−0.292767	−	0.956184i	$-0.594576\pi$
$661$	−15.0540	−0.585535	−0.292767	+	0.956184i	$0.594576\pi$
$662$	−15.6936	−0.609950
$663$	0	0
$664$	−8.15004	−0.316283
$665$	0.634476	0.0246039
$666$	24.7879	0.960511
$667$	−20.5655	−0.796299
$668$	9.79491	0.378976
$669$	4.08497	0.157934
$670$	6.22702	0.240571
$671$	1.66874	0.0644210
$672$	0.493147	0.0190236
$673$	−30.7053	−1.18360	−0.591800	−	0.806085i	$-0.701583\pi$
$673$	−30.7053	−1.18360	−0.591800	+	0.806085i	$0.701583\pi$
$674$	−20.3658	−0.784462
$675$	−10.9378	−0.420996
$676$	0	0
$677$	1.75471	0.0674389	0.0337194	−	0.999431i	$-0.489265\pi$
$677$	1.75471	0.0674389	0.0337194	+	0.999431i	$0.489265\pi$
$678$	−3.95251	−0.151795
$679$	6.49016	0.249070
$680$	−2.31652	−0.0888344
$681$	−2.38588	−0.0914273
$682$	−4.71917	−0.180707
$683$	−1.88736	−0.0722178	−0.0361089	−	0.999348i	$-0.511496\pi$
$683$	−1.88736	−0.0722178	−0.0361089	+	0.999348i	$0.511496\pi$
$684$	−2.84424	−0.108752
$685$	2.81689	0.107628
$686$	15.5427	0.593422
$687$	−4.48595	−0.171149
$688$	3.72738	0.142105
$689$	0	0
$690$	0.450876	0.0171646
$691$	16.5400	0.629211	0.314605	−	0.949223i	$-0.398128\pi$
$691$	16.5400	0.629211	0.314605	+	0.949223i	$0.398128\pi$
$692$	−7.53389	−0.286396
$693$	9.58961	0.364279
$694$	−20.8429	−0.791184
$695$	6.05597	0.229716
$696$	−3.60738	−0.136737
$697$	43.5911	1.65113
$698$	1.36583	0.0516975
$699$	7.75247	0.293225
$700$	−5.92557	−0.223966
$701$	−5.08097	−0.191906	−0.0959529	−	0.995386i	$-0.530590\pi$
$701$	−5.08097	−0.191906	−0.0959529	+	0.995386i	$0.530590\pi$
$702$	0	0
$703$	8.71511	0.328697
$704$	−2.69825	−0.101694
$705$	1.79579	0.0676335
$706$	−1.42712	−0.0537105
$707$	−21.1457	−0.795267
$708$	−3.41677	−0.128410
$709$	−4.19444	−0.157525	−0.0787627	−	0.996893i	$-0.525097\pi$
$709$	−4.19444	−0.157525	−0.0787627	+	0.996893i	$0.525097\pi$
$710$	−0.505365	−0.0189660
$711$	16.5208	0.619577
$712$	2.44741	0.0917204
$713$	−3.93510	−0.147370
$714$	2.24983	0.0841978
$715$	0	0
$716$	−5.33482	−0.199372
$717$	−3.16634	−0.118249
$718$	−27.3665	−1.02131
$719$	25.3410	0.945059	0.472529	−	0.881315i	$-0.343341\pi$
$719$	25.3410	0.945059	0.472529	+	0.881315i	$0.343341\pi$
$720$	−1.44421	−0.0538224
$721$	5.11394	0.190453
$722$	−1.00000	−0.0372161
$723$	5.35873	0.199293
$724$	−24.1292	−0.896755
$725$	43.3457	1.60982
$726$	−1.46793	−0.0544799
$727$	15.9473	0.591454	0.295727	−	0.955272i	$-0.404438\pi$
$727$	15.9473	0.591454	0.295727	+	0.955272i	$0.404438\pi$
$728$	0	0
$729$	−18.9492	−0.701824
$730$	0.0765996	0.00283508
$731$	17.0050	0.628954
$732$	0.244079	0.00902142
$733$	13.2154	0.488123	0.244062	−	0.969760i	$-0.421520\pi$
$733$	13.2154	0.488123	0.244062	+	0.969760i	$0.421520\pi$
$734$	−24.9590	−0.921253
$735$	1.08987	0.0402006
$736$	−2.24994	−0.0829339
$737$	33.0902	1.21889
$738$	27.1763	1.00037
$739$	−18.2589	−0.671665	−0.335832	−	0.941922i	$-0.609018\pi$
$739$	−18.2589	−0.671665	−0.335832	+	0.941922i	$0.609018\pi$
$740$	4.42522	0.162674
$741$	0	0
$742$	−7.25421	−0.266311
$743$	17.4411	0.639852	0.319926	−	0.947443i	$-0.396342\pi$
$743$	17.4411	0.639852	0.319926	+	0.947443i	$0.396342\pi$
$744$	−0.690253	−0.0253059
$745$	7.73581	0.283418
$746$	17.5861	0.643872
$747$	−23.1807	−0.848137
$748$	−12.3099	−0.450095
$749$	21.9772	0.803029
$750$	−1.95228	−0.0712872
$751$	−10.8938	−0.397519	−0.198759	−	0.980048i	$-0.563691\pi$
$751$	−10.8938	−0.397519	−0.198759	+	0.980048i	$0.563691\pi$
$752$	−8.96128	−0.326784
$753$	8.60274	0.313501
$754$	0	0
$755$	−8.41726	−0.306335
$756$	2.88207	0.104820
$757$	−18.7457	−0.681325	−0.340662	−	0.940186i	$-0.610651\pi$
$757$	−18.7457	−0.681325	−0.340662	+	0.940186i	$0.610651\pi$
$758$	14.2698	0.518303
$759$	2.39594	0.0869672
$760$	−0.507765	−0.0184186
$761$	32.0451	1.16164	0.580818	−	0.814034i	$-0.302733\pi$
$761$	32.0451	1.16164	0.580818	+	0.814034i	$0.302733\pi$
$762$	−8.73938	−0.316594
$763$	−24.5924	−0.890306
$764$	−20.7322	−0.750064
$765$	−6.58874	−0.238216
$766$	12.8820	0.465447
$767$	0	0
$768$	−0.394661	−0.0142411
$769$	16.7370	0.603551	0.301776	−	0.953379i	$-0.402421\pi$
$769$	16.7370	0.603551	0.301776	+	0.953379i	$0.402421\pi$
$770$	1.71197	0.0616952
$771$	−8.65593	−0.311736
$772$	9.89665	0.356188
$773$	−24.8714	−0.894562	−0.447281	−	0.894393i	$-0.647608\pi$
$773$	−24.8714	−0.894562	−0.447281	+	0.894393i	$0.647608\pi$
$774$	10.6016	0.381066
$775$	8.29396	0.297928
$776$	−5.19401	−0.186454
$777$	−4.29783	−0.154184
$778$	−4.94578	−0.177315
$779$	9.55485	0.342338
$780$	0	0
$781$	−2.68550	−0.0960946
$782$	−10.2647	−0.367063
$783$	−21.0824	−0.753424
$784$	−5.43863	−0.194237
$785$	0.355339	0.0126826
$786$	−7.67205	−0.273653
$787$	−27.9235	−0.995367	−0.497683	−	0.867359i	$-0.665816\pi$
$787$	−27.9235	−0.995367	−0.497683	+	0.867359i	$0.665816\pi$
$788$	−5.46053	−0.194523
$789$	−3.42566	−0.121957
$790$	2.94934	0.104933
$791$	−12.5142	−0.444952
$792$	−7.67447	−0.272700
$793$	0	0
$794$	22.0346	0.781980
$795$	−1.16339	−0.0412610
$796$	−15.8738	−0.562632
$797$	−33.6046	−1.19034	−0.595168	−	0.803602i	$-0.702914\pi$
$797$	−33.6046	−1.19034	−0.595168	+	0.803602i	$0.702914\pi$
$798$	0.493147	0.0174572
$799$	−40.8831	−1.44634
$800$	4.74218	0.167661
$801$	6.96101	0.245955
$802$	−2.94800	−0.104098
$803$	0.407048	0.0143644
$804$	4.83996	0.170692
$805$	1.42753	0.0503139
$806$	0	0
$807$	−4.05096	−0.142601
$808$	16.9227	0.595339
$809$	−16.0538	−0.564420	−0.282210	−	0.959353i	$-0.591068\pi$
$809$	−16.0538	−0.564420	−0.282210	+	0.959353i	$0.591068\pi$
$810$	−3.87041	−0.135992
$811$	4.73984	0.166438	0.0832191	−	0.996531i	$-0.473480\pi$
$811$	4.73984	0.166438	0.0832191	+	0.996531i	$0.473480\pi$
$812$	−11.4214	−0.400814
$813$	−7.57324	−0.265605
$814$	23.5155	0.824218
$815$	11.9780	0.419571
$816$	−1.80052	−0.0630307
$817$	3.72738	0.130405
$818$	18.3674	0.642202
$819$	0	0
$820$	4.85162	0.169426
$821$	1.82332	0.0636344	0.0318172	−	0.999494i	$-0.489871\pi$
$821$	1.82332	0.0636344	0.0318172	+	0.999494i	$0.489871\pi$
$822$	2.18943	0.0763652
$823$	−12.2522	−0.427086	−0.213543	−	0.976934i	$-0.568500\pi$
$823$	−12.2522	−0.427086	−0.213543	+	0.976934i	$0.568500\pi$
$824$	−4.09263	−0.142574
$825$	−5.04990	−0.175815
$826$	−10.8179	−0.376404
$827$	32.2781	1.12242	0.561209	−	0.827674i	$-0.310336\pi$
$827$	32.2781	1.12242	0.561209	+	0.827674i	$0.310336\pi$
$828$	−6.39938	−0.222394
$829$	−21.8654	−0.759415	−0.379708	−	0.925107i	$-0.623975\pi$
$829$	−21.8654	−0.759415	−0.379708	+	0.925107i	$0.623975\pi$
$830$	−4.13830	−0.143642
$831$	−1.19749	−0.0415404
$832$	0	0
$833$	−24.8121	−0.859687
$834$	4.70701	0.162990
$835$	4.97351	0.172115
$836$	−2.69825	−0.0933208
$837$	−4.03401	−0.139436
$838$	7.18949	0.248357
$839$	−24.0252	−0.829442	−0.414721	−	0.909949i	$-0.636121\pi$
$839$	−24.0252	−0.829442	−0.414721	+	0.909949i	$0.636121\pi$
$840$	0.250403	0.00863971
$841$	54.5481	1.88097
$842$	13.4228	0.462582
$843$	−2.45230	−0.0844616
$844$	−19.8984	−0.684931
$845$	0	0
$846$	−25.4881	−0.876297
$847$	−4.64765	−0.159695
$848$	5.80547	0.199361
$849$	4.94726	0.169790
$850$	21.6347	0.742064
$851$	19.6085	0.672170
$852$	−0.392796	−0.0134570
$853$	−46.1729	−1.58093	−0.790465	−	0.612508i	$-0.790161\pi$
$853$	−46.1729	−1.58093	−0.790465	+	0.612508i	$0.790161\pi$
$854$	0.772787	0.0264442
$855$	−1.44421	−0.0493908
$856$	−17.5881	−0.601150
$857$	28.7012	0.980413	0.490207	−	0.871606i	$-0.336921\pi$
$857$	28.7012	0.980413	0.490207	+	0.871606i	$0.336921\pi$
$858$	0	0
$859$	29.4439	1.00461	0.502307	−	0.864689i	$-0.332485\pi$
$859$	29.4439	1.00461	0.502307	+	0.864689i	$0.332485\pi$
$860$	1.89263	0.0645382
$861$	−4.71195	−0.160583
$862$	19.5840	0.667034
$863$	−55.2739	−1.88155	−0.940773	−	0.339038i	$-0.889898\pi$
$863$	−55.2739	−1.88155	−0.940773	+	0.339038i	$0.889898\pi$
$864$	−2.30649	−0.0784685
$865$	−3.82544	−0.130069
$866$	33.5004	1.13839
$867$	−1.50507	−0.0511149
$868$	−2.18543	−0.0741784
$869$	15.6727	0.531661
$870$	−1.83170	−0.0621004
$871$	0	0
$872$	19.6811	0.666485
$873$	−14.7730	−0.499991
$874$	−2.24994	−0.0761053
$875$	−6.18118	−0.208962
$876$	0.0595371	0.00201157
$877$	7.70433	0.260157	0.130078	−	0.991504i	$-0.458477\pi$
$877$	7.70433	0.260157	0.130078	+	0.991504i	$0.458477\pi$
$878$	4.74365	0.160090
$879$	−5.91572	−0.199532
$880$	−1.37007	−0.0461852
$881$	40.9945	1.38114	0.690569	−	0.723266i	$-0.257360\pi$
$881$	40.9945	1.38114	0.690569	+	0.723266i	$0.257360\pi$
$882$	−15.4688	−0.520861
$883$	−50.0831	−1.68543	−0.842715	−	0.538360i	$-0.819044\pi$
$883$	−50.0831	−1.68543	−0.842715	+	0.538360i	$0.819044\pi$
$884$	0	0
$885$	−1.73491	−0.0583185
$886$	−31.0710	−1.04385
$887$	36.7002	1.23227	0.616136	−	0.787639i	$-0.288697\pi$
$887$	36.7002	1.23227	0.616136	+	0.787639i	$0.288697\pi$
$888$	3.43951	0.115422
$889$	−27.6700	−0.928023
$890$	1.24271	0.0416556
$891$	−20.5672	−0.689028
$892$	−10.3506	−0.346563
$893$	−8.96128	−0.299878
$894$	6.01267	0.201094
$895$	−2.70883	−0.0905463
$896$	−1.24955	−0.0417445
$897$	0	0
$898$	15.7887	0.526877
$899$	15.9865	0.533179
$900$	13.4879	0.449597
$901$	26.4857	0.882366
$902$	25.7813	0.858425
$903$	−1.83815	−0.0611697
$904$	10.0150	0.333093
$905$	−12.2520	−0.407269
$906$	−6.54232	−0.217354
$907$	23.3382	0.774931	0.387466	−	0.921884i	$-0.373351\pi$
$907$	23.3382	0.774931	0.387466	+	0.921884i	$0.373351\pi$
$908$	6.04541	0.200624
$909$	48.1323	1.59645
$910$	0	0
$911$	54.7035	1.81241	0.906204	−	0.422841i	$-0.138967\pi$
$911$	54.7035	1.81241	0.906204	+	0.422841i	$0.138967\pi$
$912$	−0.394661	−0.0130685
$913$	−21.9908	−0.727789
$914$	1.77056	0.0585648
$915$	0.123935	0.00409716
$916$	11.3666	0.375563
$917$	−24.2907	−0.802150
$918$	−10.5227	−0.347300
$919$	38.4249	1.26752	0.633760	−	0.773530i	$-0.281511\pi$
$919$	38.4249	1.26752	0.633760	+	0.773530i	$0.281511\pi$
$920$	−1.14244	−0.0376651
$921$	11.8168	0.389377
$922$	11.8636	0.390708
$923$	0	0
$924$	1.33063	0.0437746
$925$	−41.3286	−1.35887
$926$	−4.13269	−0.135809
$927$	−11.6404	−0.382322
$928$	9.14046	0.300050
$929$	13.2117	0.433462	0.216731	−	0.976231i	$-0.430461\pi$
$929$	13.2117	0.433462	0.216731	+	0.976231i	$0.430461\pi$
$930$	−0.350486	−0.0114929
$931$	−5.43863	−0.178244
$932$	−19.6434	−0.643441
$933$	1.73817	0.0569052
$934$	18.5126	0.605751
$935$	−6.25054	−0.204414
$936$	0	0
$937$	−42.6152	−1.39218	−0.696089	−	0.717955i	$-0.745078\pi$
$937$	−42.6152	−1.39218	−0.696089	+	0.717955i	$0.745078\pi$
$938$	15.3239	0.500344
$939$	−4.27480	−0.139503
$940$	−4.55022	−0.148412
$941$	53.3461	1.73903	0.869517	−	0.493902i	$-0.164430\pi$
$941$	53.3461	1.73903	0.869517	+	0.493902i	$0.164430\pi$
$942$	0.276188	0.00899869
$943$	21.4978	0.700066
$944$	8.65749	0.281777
$945$	1.46341	0.0476049
$946$	10.0574	0.326994
$947$	−1.29330	−0.0420265	−0.0210133	−	0.999779i	$-0.506689\pi$
$947$	−1.29330	−0.0420265	−0.0210133	+	0.999779i	$0.506689\pi$
$948$	2.29238	0.0744531
$949$	0	0
$950$	4.74218	0.153856
$951$	10.9546	0.355229
$952$	−5.70067	−0.184760
$953$	6.22671	0.201703	0.100851	−	0.994901i	$-0.467843\pi$
$953$	6.22671	0.201703	0.100851	+	0.994901i	$0.467843\pi$
$954$	16.5122	0.534601
$955$	−10.5271	−0.340648
$956$	8.02294	0.259481
$957$	−9.73360	−0.314643
$958$	18.4074	0.594715
$959$	6.93203	0.223847
$960$	−0.200395	−0.00646771
$961$	−27.9411	−0.901325
$962$	0	0
$963$	−50.0249	−1.61203
$964$	−13.5781	−0.437320
$965$	5.02517	0.161766
$966$	1.10955	0.0356992
$967$	34.4051	1.10639	0.553197	−	0.833050i	$-0.313408\pi$
$967$	34.4051	1.10639	0.553197	+	0.833050i	$0.313408\pi$
$968$	3.71947	0.119548
$969$	−1.80052	−0.0578409
$970$	−2.63733	−0.0846797
$971$	−23.2656	−0.746629	−0.373315	−	0.927705i	$-0.621779\pi$
$971$	−23.2656	−0.746629	−0.373315	+	0.927705i	$0.621779\pi$
$972$	−9.92776	−0.318433
$973$	14.9030	0.477769
$974$	1.99282	0.0638542
$975$	0	0
$976$	−0.618453	−0.0197962
$977$	9.97729	0.319202	0.159601	−	0.987182i	$-0.448979\pi$
$977$	9.97729	0.319202	0.159601	+	0.987182i	$0.448979\pi$
$978$	9.30992	0.297699
$979$	6.60370	0.211055
$980$	−2.76154	−0.0882143
$981$	55.9777	1.78723
$982$	34.5469	1.10244
$983$	7.96318	0.253986	0.126993	−	0.991904i	$-0.459467\pi$
$983$	7.96318	0.253986	0.126993	+	0.991904i	$0.459467\pi$
$984$	3.77092	0.120213
$985$	−2.77266	−0.0883443
$986$	41.7005	1.32802
$987$	4.41923	0.140666
$988$	0	0
$989$	8.38639	0.266672
$990$	−3.89682	−0.123849
$991$	−44.6566	−1.41856	−0.709281	−	0.704926i	$-0.750980\pi$
$991$	−44.6566	−1.41856	−0.709281	+	0.704926i	$0.750980\pi$
$992$	1.74898	0.0555301
$993$	−6.19365	−0.196550
$994$	−1.24364	−0.0394460
$995$	−8.06015	−0.255524
$996$	−3.21650	−0.101919
$997$	−26.0993	−0.826572	−0.413286	−	0.910601i	$-0.635619\pi$
$997$	−26.0993	−0.826572	−0.413286	+	0.910601i	$0.635619\pi$
$998$	24.5129	0.775943
$999$	20.1013	0.635978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bg.1.5		8
13.2	odd	12		494.2.m.a.381.2	yes	16
13.7	odd	12		494.2.m.a.153.2	✓	16
13.12	even	2		6422.2.a.bh.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.m.a.153.2	✓	16	13.7	odd	12
494.2.m.a.381.2	yes	16	13.2	odd	12
6422.2.a.bg.1.5		8	1.1	even	1	trivial
6422.2.a.bh.1.5		8	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.394661 q^{3} +1.00000 q^{4} +0.507765 q^{5} +0.394661 q^{6} +1.24955 q^{7} -1.00000 q^{8} -2.84424 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.394661 q^{3} +1.00000 q^{4} +0.507765 q^{5} +0.394661 q^{6} +1.24955 q^{7} -1.00000 q^{8} -2.84424 q^{9} -0.507765 q^{10} -2.69825 q^{11} -0.394661 q^{12} -1.24955 q^{14} -0.200395 q^{15} +1.00000 q^{16} +4.56219 q^{17} +2.84424 q^{18} +1.00000 q^{19} +0.507765 q^{20} -0.493147 q^{21} +2.69825 q^{22} +2.24994 q^{23} +0.394661 q^{24} -4.74218 q^{25} +2.30649 q^{27} +1.24955 q^{28} -9.14046 q^{29} +0.200395 q^{30} -1.74898 q^{31} -1.00000 q^{32} +1.06489 q^{33} -4.56219 q^{34} +0.634476 q^{35} -2.84424 q^{36} +8.71511 q^{37} -1.00000 q^{38} -0.507765 q^{40} +9.55485 q^{41} +0.493147 q^{42} +3.72738 q^{43} -2.69825 q^{44} -1.44421 q^{45} -2.24994 q^{46} -8.96128 q^{47} -0.394661 q^{48} -5.43863 q^{49} +4.74218 q^{50} -1.80052 q^{51} +5.80547 q^{53} -2.30649 q^{54} -1.37007 q^{55} -1.24955 q^{56} -0.394661 q^{57} +9.14046 q^{58} +8.65749 q^{59} -0.200395 q^{60} -0.618453 q^{61} +1.74898 q^{62} -3.55402 q^{63} +1.00000 q^{64} -1.06489 q^{66} -12.2636 q^{67} +4.56219 q^{68} -0.887963 q^{69} -0.634476 q^{70} +0.995275 q^{71} +2.84424 q^{72} -0.150857 q^{73} -8.71511 q^{74} +1.87155 q^{75} +1.00000 q^{76} -3.37159 q^{77} -5.80849 q^{79} +0.507765 q^{80} +7.62245 q^{81} -9.55485 q^{82} +8.15004 q^{83} -0.493147 q^{84} +2.31652 q^{85} -3.72738 q^{86} +3.60738 q^{87} +2.69825 q^{88} -2.44741 q^{89} +1.44421 q^{90} +2.24994 q^{92} +0.690253 q^{93} +8.96128 q^{94} +0.507765 q^{95} +0.394661 q^{96} +5.19401 q^{97} +5.43863 q^{98} +7.67447 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} - 8 q^{8}+O(q^{10}) \) 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

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