LMFDB - Embedded newform 4026.2.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4026,2,Mod(1,4026)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4026, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4026.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4026 = 2 \cdot 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4026.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:	$32.1477718538$
Analytic rank:	$0$
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} - 3x^{7} - 22x^{6} + 42x^{5} + 182x^{4} - 111x^{3} - 538x^{2} - 256x - 32 $ x^8 - 3x^7 - 22x^6 + 42x^5 + 182x^4 - 111x^3 - 538x^2 - 256*x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.26126$ of defining polynomial
Character	$\chi$	$=$	4026.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.92862 q^{5} +1.00000 q^{6} -2.26126 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.92862 q^{5} +1.00000 q^{6} -2.26126 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.92862 q^{10} -1.00000 q^{11} +1.00000 q^{12} -0.873201 q^{13} -2.26126 q^{14} -1.92862 q^{15} +1.00000 q^{16} -3.06469 q^{17} +1.00000 q^{18} +7.77911 q^{19} -1.92862 q^{20} -2.26126 q^{21} -1.00000 q^{22} +4.91749 q^{23} +1.00000 q^{24} -1.28044 q^{25} -0.873201 q^{26} +1.00000 q^{27} -2.26126 q^{28} +4.42004 q^{29} -1.92862 q^{30} -0.988877 q^{31} +1.00000 q^{32} -1.00000 q^{33} -3.06469 q^{34} +4.36110 q^{35} +1.00000 q^{36} +7.42838 q^{37} +7.77911 q^{38} -0.873201 q^{39} -1.92862 q^{40} +0.410765 q^{41} -2.26126 q^{42} +4.69011 q^{43} -1.00000 q^{44} -1.92862 q^{45} +4.91749 q^{46} +2.36986 q^{47} +1.00000 q^{48} -1.88673 q^{49} -1.28044 q^{50} -3.06469 q^{51} -0.873201 q^{52} -2.81425 q^{53} +1.00000 q^{54} +1.92862 q^{55} -2.26126 q^{56} +7.77911 q^{57} +4.42004 q^{58} +1.62420 q^{59} -1.92862 q^{60} +1.00000 q^{61} -0.988877 q^{62} -2.26126 q^{63} +1.00000 q^{64} +1.68407 q^{65} -1.00000 q^{66} +10.1876 q^{67} -3.06469 q^{68} +4.91749 q^{69} +4.36110 q^{70} +8.05201 q^{71} +1.00000 q^{72} +1.69404 q^{73} +7.42838 q^{74} -1.28044 q^{75} +7.77911 q^{76} +2.26126 q^{77} -0.873201 q^{78} +5.89360 q^{79} -1.92862 q^{80} +1.00000 q^{81} +0.410765 q^{82} +13.8812 q^{83} -2.26126 q^{84} +5.91062 q^{85} +4.69011 q^{86} +4.42004 q^{87} -1.00000 q^{88} -17.5559 q^{89} -1.92862 q^{90} +1.97453 q^{91} +4.91749 q^{92} -0.988877 q^{93} +2.36986 q^{94} -15.0029 q^{95} +1.00000 q^{96} +16.4295 q^{97} -1.88673 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} + 13 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 5 * q^5 + 8 * q^6 + 13 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} + 13 q^{7} + 8 q^{8} + 8 q^{9} + 5 q^{10} - 8 q^{11} + 8 q^{12} + 10 q^{13} + 13 q^{14} + 5 q^{15} + 8 q^{16} + 4 q^{17} + 8 q^{18} + 11 q^{19} + 5 q^{20} + 13 q^{21} - 8 q^{22} + 2 q^{23} + 8 q^{24} + 23 q^{25} + 10 q^{26} + 8 q^{27} + 13 q^{28} + 10 q^{29} + 5 q^{30} + 9 q^{31} + 8 q^{32} - 8 q^{33} + 4 q^{34} - 3 q^{35} + 8 q^{36} + 9 q^{37} + 11 q^{38} + 10 q^{39} + 5 q^{40} + 3 q^{41} + 13 q^{42} + 16 q^{43} - 8 q^{44} + 5 q^{45} + 2 q^{46} - 16 q^{47} + 8 q^{48} + 17 q^{49} + 23 q^{50} + 4 q^{51} + 10 q^{52} + 7 q^{53} + 8 q^{54} - 5 q^{55} + 13 q^{56} + 11 q^{57} + 10 q^{58} - 14 q^{59} + 5 q^{60} + 8 q^{61} + 9 q^{62} + 13 q^{63} + 8 q^{64} + 22 q^{65} - 8 q^{66} + 8 q^{67} + 4 q^{68} + 2 q^{69} - 3 q^{70} + 11 q^{71} + 8 q^{72} + 14 q^{73} + 9 q^{74} + 23 q^{75} + 11 q^{76} - 13 q^{77} + 10 q^{78} + 22 q^{79} + 5 q^{80} + 8 q^{81} + 3 q^{82} - 16 q^{83} + 13 q^{84} + 3 q^{85} + 16 q^{86} + 10 q^{87} - 8 q^{88} + q^{89} + 5 q^{90} + 15 q^{91} + 2 q^{92} + 9 q^{93} - 16 q^{94} - 9 q^{95} + 8 q^{96} + 24 q^{97} + 17 q^{98} - 8 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 5 * q^5 + 8 * q^6 + 13 * q^7 + 8 * q^8 + 8 * q^9 + 5 * q^10 - 8 * q^11 + 8 * q^12 + 10 * q^13 + 13 * q^14 + 5 * q^15 + 8 * q^16 + 4 * q^17 + 8 * q^18 + 11 * q^19 + 5 * q^20 + 13 * q^21 - 8 * q^22 + 2 * q^23 + 8 * q^24 + 23 * q^25 + 10 * q^26 + 8 * q^27 + 13 * q^28 + 10 * q^29 + 5 * q^30 + 9 * q^31 + 8 * q^32 - 8 * q^33 + 4 * q^34 - 3 * q^35 + 8 * q^36 + 9 * q^37 + 11 * q^38 + 10 * q^39 + 5 * q^40 + 3 * q^41 + 13 * q^42 + 16 * q^43 - 8 * q^44 + 5 * q^45 + 2 * q^46 - 16 * q^47 + 8 * q^48 + 17 * q^49 + 23 * q^50 + 4 * q^51 + 10 * q^52 + 7 * q^53 + 8 * q^54 - 5 * q^55 + 13 * q^56 + 11 * q^57 + 10 * q^58 - 14 * q^59 + 5 * q^60 + 8 * q^61 + 9 * q^62 + 13 * q^63 + 8 * q^64 + 22 * q^65 - 8 * q^66 + 8 * q^67 + 4 * q^68 + 2 * q^69 - 3 * q^70 + 11 * q^71 + 8 * q^72 + 14 * q^73 + 9 * q^74 + 23 * q^75 + 11 * q^76 - 13 * q^77 + 10 * q^78 + 22 * q^79 + 5 * q^80 + 8 * q^81 + 3 * q^82 - 16 * q^83 + 13 * q^84 + 3 * q^85 + 16 * q^86 + 10 * q^87 - 8 * q^88 + q^89 + 5 * q^90 + 15 * q^91 + 2 * q^92 + 9 * q^93 - 16 * q^94 - 9 * q^95 + 8 * q^96 + 24 * q^97 + 17 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.92862	−0.862504	−0.431252	−	0.902232i	$-0.641928\pi$
$5$	−1.92862	−0.862504	−0.431252	+	0.902232i	$0.641928\pi$
$6$	1.00000	0.408248
$7$	−2.26126	−0.854674	−0.427337	−	0.904092i	$-0.640548\pi$
$7$	−2.26126	−0.854674	−0.427337	+	0.904092i	$0.640548\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.92862	−0.609882
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	−0.873201	−0.242182	−0.121091	−	0.992641i	$-0.538639\pi$
$13$	−0.873201	−0.242182	−0.121091	+	0.992641i	$0.538639\pi$
$14$	−2.26126	−0.604346
$15$	−1.92862	−0.497967
$16$	1.00000	0.250000
$17$	−3.06469	−0.743297	−0.371649	−	0.928373i	$-0.621207\pi$
$17$	−3.06469	−0.743297	−0.371649	+	0.928373i	$0.621207\pi$
$18$	1.00000	0.235702
$19$	7.77911	1.78465	0.892325	−	0.451394i	$-0.149073\pi$
$19$	7.77911	1.78465	0.892325	+	0.451394i	$0.149073\pi$
$20$	−1.92862	−0.431252
$21$	−2.26126	−0.493446
$22$	−1.00000	−0.213201
$23$	4.91749	1.02537	0.512684	−	0.858577i	$-0.328651\pi$
$23$	4.91749	1.02537	0.512684	+	0.858577i	$0.328651\pi$
$24$	1.00000	0.204124
$25$	−1.28044	−0.256087
$26$	−0.873201	−0.171249
$27$	1.00000	0.192450
$28$	−2.26126	−0.427337
$29$	4.42004	0.820781	0.410391	−	0.911910i	$-0.365392\pi$
$29$	4.42004	0.820781	0.410391	+	0.911910i	$0.365392\pi$
$30$	−1.92862	−0.352116
$31$	−0.988877	−0.177608	−0.0888038	−	0.996049i	$-0.528304\pi$
$31$	−0.988877	−0.177608	−0.0888038	+	0.996049i	$0.528304\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−3.06469	−0.525590
$35$	4.36110	0.737160
$36$	1.00000	0.166667
$37$	7.42838	1.22122	0.610609	−	0.791932i	$-0.290925\pi$
$37$	7.42838	1.22122	0.610609	+	0.791932i	$0.290925\pi$
$38$	7.77911	1.26194
$39$	−0.873201	−0.139824
$40$	−1.92862	−0.304941
$41$	0.410765	0.0641507	0.0320753	−	0.999485i	$-0.489788\pi$
$41$	0.410765	0.0641507	0.0320753	+	0.999485i	$0.489788\pi$
$42$	−2.26126	−0.348919
$43$	4.69011	0.715236	0.357618	−	0.933868i	$-0.383589\pi$
$43$	4.69011	0.715236	0.357618	+	0.933868i	$0.383589\pi$
$44$	−1.00000	−0.150756
$45$	−1.92862	−0.287501
$46$	4.91749	0.725045
$47$	2.36986	0.345679	0.172840	−	0.984950i	$-0.444706\pi$
$47$	2.36986	0.345679	0.172840	+	0.984950i	$0.444706\pi$
$48$	1.00000	0.144338
$49$	−1.88673	−0.269532
$50$	−1.28044	−0.181081
$51$	−3.06469	−0.429143
$52$	−0.873201	−0.121091
$53$	−2.81425	−0.386568	−0.193284	−	0.981143i	$-0.561914\pi$
$53$	−2.81425	−0.386568	−0.193284	+	0.981143i	$0.561914\pi$
$54$	1.00000	0.136083
$55$	1.92862	0.260055
$56$	−2.26126	−0.302173
$57$	7.77911	1.03037
$58$	4.42004	0.580380
$59$	1.62420	0.211452	0.105726	−	0.994395i	$-0.466283\pi$
$59$	1.62420	0.211452	0.105726	+	0.994395i	$0.466283\pi$
$60$	−1.92862	−0.248983
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−0.988877	−0.125587
$63$	−2.26126	−0.284891
$64$	1.00000	0.125000
$65$	1.68407	0.208883
$66$	−1.00000	−0.123091
$67$	10.1876	1.24461	0.622304	−	0.782775i	$-0.286197\pi$
$67$	10.1876	1.24461	0.622304	+	0.782775i	$0.286197\pi$
$68$	−3.06469	−0.371649
$69$	4.91749	0.591997
$70$	4.36110	0.521251
$71$	8.05201	0.955597	0.477799	−	0.878469i	$-0.341435\pi$
$71$	8.05201	0.955597	0.477799	+	0.878469i	$0.341435\pi$
$72$	1.00000	0.117851
$73$	1.69404	0.198272	0.0991361	−	0.995074i	$-0.468392\pi$
$73$	1.69404	0.198272	0.0991361	+	0.995074i	$0.468392\pi$
$74$	7.42838	0.863531
$75$	−1.28044	−0.147852
$76$	7.77911	0.892325
$77$	2.26126	0.257694
$78$	−0.873201	−0.0988705
$79$	5.89360	0.663082	0.331541	−	0.943441i	$-0.392431\pi$
$79$	5.89360	0.663082	0.331541	+	0.943441i	$0.392431\pi$
$80$	−1.92862	−0.215626
$81$	1.00000	0.111111
$82$	0.410765	0.0453614
$83$	13.8812	1.52366	0.761832	−	0.647775i	$-0.224300\pi$
$83$	13.8812	1.52366	0.761832	+	0.647775i	$0.224300\pi$
$84$	−2.26126	−0.246723
$85$	5.91062	0.641097
$86$	4.69011	0.505748
$87$	4.42004	0.473878
$88$	−1.00000	−0.106600
$89$	−17.5559	−1.86092	−0.930459	−	0.366396i	$-0.880592\pi$
$89$	−17.5559	−1.86092	−0.930459	+	0.366396i	$0.880592\pi$
$90$	−1.92862	−0.203294
$91$	1.97453	0.206987
$92$	4.91749	0.512684
$93$	−0.988877	−0.102542
$94$	2.36986	0.244432
$95$	−15.0029	−1.53927
$96$	1.00000	0.102062
$97$	16.4295	1.66816	0.834080	−	0.551643i	$-0.185999\pi$
$97$	16.4295	1.66816	0.834080	+	0.551643i	$0.185999\pi$
$98$	−1.88673	−0.190588
$99$	−1.00000	−0.100504
$100$	−1.28044	−0.128044
$101$	8.58883	0.854621	0.427310	−	0.904105i	$-0.359461\pi$
$101$	8.58883	0.854621	0.427310	+	0.904105i	$0.359461\pi$
$102$	−3.06469	−0.303450
$103$	8.13013	0.801086	0.400543	−	0.916278i	$-0.368822\pi$
$103$	8.13013	0.801086	0.400543	+	0.916278i	$0.368822\pi$
$104$	−0.873201	−0.0856244
$105$	4.36110	0.425599
$106$	−2.81425	−0.273345
$107$	−14.9345	−1.44377	−0.721885	−	0.692013i	$-0.756724\pi$
$107$	−14.9345	−1.44377	−0.721885	+	0.692013i	$0.756724\pi$
$108$	1.00000	0.0962250
$109$	−6.73374	−0.644975	−0.322488	−	0.946574i	$-0.604519\pi$
$109$	−6.73374	−0.644975	−0.322488	+	0.946574i	$0.604519\pi$
$110$	1.92862	0.183886
$111$	7.42838	0.705070
$112$	−2.26126	−0.213669
$113$	4.31004	0.405454	0.202727	−	0.979235i	$-0.435020\pi$
$113$	4.31004	0.405454	0.202727	+	0.979235i	$0.435020\pi$
$114$	7.77911	0.728580
$115$	−9.48396	−0.884384
$116$	4.42004	0.410391
$117$	−0.873201	−0.0807275
$118$	1.62420	0.149519
$119$	6.93005	0.635277
$120$	−1.92862	−0.176058
$121$	1.00000	0.0909091
$122$	1.00000	0.0905357
$123$	0.410765	0.0370374
$124$	−0.988877	−0.0888038
$125$	12.1126	1.08338
$126$	−2.26126	−0.201449
$127$	−1.91106	−0.169579	−0.0847894	−	0.996399i	$-0.527022\pi$
$127$	−1.91106	−0.169579	−0.0847894	+	0.996399i	$0.527022\pi$
$128$	1.00000	0.0883883
$129$	4.69011	0.412941
$130$	1.68407	0.147703
$131$	7.76942	0.678817	0.339409	−	0.940639i	$-0.389773\pi$
$131$	7.76942	0.678817	0.339409	+	0.940639i	$0.389773\pi$
$132$	−1.00000	−0.0870388
$133$	−17.5905	−1.52529
$134$	10.1876	0.880071
$135$	−1.92862	−0.165989
$136$	−3.06469	−0.262795
$137$	−13.9439	−1.19130	−0.595652	−	0.803242i	$-0.703106\pi$
$137$	−13.9439	−1.19130	−0.595652	+	0.803242i	$0.703106\pi$
$138$	4.91749	0.418605
$139$	−6.28321	−0.532935	−0.266467	−	0.963844i	$-0.585856\pi$
$139$	−6.28321	−0.532935	−0.266467	+	0.963844i	$0.585856\pi$
$140$	4.36110	0.368580
$141$	2.36986	0.199578
$142$	8.05201	0.675709
$143$	0.873201	0.0730207
$144$	1.00000	0.0833333
$145$	−8.52457	−0.707927
$146$	1.69404	0.140200
$147$	−1.88673	−0.155615
$148$	7.42838	0.610609
$149$	−9.08077	−0.743926	−0.371963	−	0.928248i	$-0.621315\pi$
$149$	−9.08077	−0.743926	−0.371963	+	0.928248i	$0.621315\pi$
$150$	−1.28044	−0.104547
$151$	12.9000	1.04979	0.524895	−	0.851167i	$-0.324105\pi$
$151$	12.9000	1.04979	0.524895	+	0.851167i	$0.324105\pi$
$152$	7.77911	0.630969
$153$	−3.06469	−0.247766
$154$	2.26126	0.182217
$155$	1.90717	0.153187
$156$	−0.873201	−0.0699120
$157$	9.80139	0.782236	0.391118	−	0.920341i	$-0.372088\pi$
$157$	9.80139	0.782236	0.391118	+	0.920341i	$0.372088\pi$
$158$	5.89360	0.468870
$159$	−2.81425	−0.223185
$160$	−1.92862	−0.152471
$161$	−11.1197	−0.876356
$162$	1.00000	0.0785674
$163$	−0.451975	−0.0354014	−0.0177007	−	0.999843i	$-0.505635\pi$
$163$	−0.451975	−0.0354014	−0.0177007	+	0.999843i	$0.505635\pi$
$164$	0.410765	0.0320753
$165$	1.92862	0.150143
$166$	13.8812	1.07739
$167$	15.2873	1.18296	0.591482	−	0.806318i	$-0.298543\pi$
$167$	15.2873	1.18296	0.591482	+	0.806318i	$0.298543\pi$
$168$	−2.26126	−0.174460
$169$	−12.2375	−0.941348
$170$	5.91062	0.453324
$171$	7.77911	0.594883
$172$	4.69011	0.357618
$173$	18.0507	1.37237	0.686186	−	0.727426i	$-0.259284\pi$
$173$	18.0507	1.37237	0.686186	+	0.727426i	$0.259284\pi$
$174$	4.42004	0.335082
$175$	2.89539	0.218871
$176$	−1.00000	−0.0753778
$177$	1.62420	0.122082
$178$	−17.5559	−1.31587
$179$	−1.82203	−0.136185	−0.0680926	−	0.997679i	$-0.521691\pi$
$179$	−1.82203	−0.136185	−0.0680926	+	0.997679i	$0.521691\pi$
$180$	−1.92862	−0.143751
$181$	17.3994	1.29329	0.646643	−	0.762793i	$-0.276172\pi$
$181$	17.3994	1.29329	0.646643	+	0.762793i	$0.276172\pi$
$182$	1.97453	0.146362
$183$	1.00000	0.0739221
$184$	4.91749	0.362522
$185$	−14.3265	−1.05330
$186$	−0.988877	−0.0725080
$187$	3.06469	0.224113
$188$	2.36986	0.172840
$189$	−2.26126	−0.164482
$190$	−15.0029	−1.08843
$191$	−23.8388	−1.72491	−0.862456	−	0.506131i	$-0.831075\pi$
$191$	−23.8388	−1.72491	−0.862456	+	0.506131i	$0.831075\pi$
$192$	1.00000	0.0721688
$193$	−6.05697	−0.435990	−0.217995	−	0.975950i	$-0.569952\pi$
$193$	−6.05697	−0.435990	−0.217995	+	0.975950i	$0.569952\pi$
$194$	16.4295	1.17957
$195$	1.68407	0.120599
$196$	−1.88673	−0.134766
$197$	3.37431	0.240410	0.120205	−	0.992749i	$-0.461645\pi$
$197$	3.37431	0.240410	0.120205	+	0.992749i	$0.461645\pi$
$198$	−1.00000	−0.0710669
$199$	−17.9859	−1.27499	−0.637493	−	0.770456i	$-0.720029\pi$
$199$	−17.9859	−1.27499	−0.637493	+	0.770456i	$0.720029\pi$
$200$	−1.28044	−0.0905405
$201$	10.1876	0.718575
$202$	8.58883	0.604308
$203$	−9.99484	−0.701500
$204$	−3.06469	−0.214571
$205$	−0.792208	−0.0553302
$206$	8.13013	0.566453
$207$	4.91749	0.341789
$208$	−0.873201	−0.0605456
$209$	−7.77911	−0.538092
$210$	4.36110	0.300944
$211$	−18.3538	−1.26353	−0.631764	−	0.775161i	$-0.717669\pi$
$211$	−18.3538	−1.26353	−0.631764	+	0.775161i	$0.717669\pi$
$212$	−2.81425	−0.193284
$213$	8.05201	0.551714
$214$	−14.9345	−1.02090
$215$	−9.04543	−0.616893
$216$	1.00000	0.0680414
$217$	2.23610	0.151797
$218$	−6.73374	−0.456066
$219$	1.69404	0.114473
$220$	1.92862	0.130027
$221$	2.67609	0.180013
$222$	7.42838	0.498560
$223$	−19.7968	−1.32569	−0.662845	−	0.748757i	$-0.730651\pi$
$223$	−19.7968	−1.32569	−0.662845	+	0.748757i	$0.730651\pi$
$224$	−2.26126	−0.151086
$225$	−1.28044	−0.0853624
$226$	4.31004	0.286699
$227$	−25.4316	−1.68796	−0.843978	−	0.536378i	$-0.819792\pi$
$227$	−25.4316	−1.68796	−0.843978	+	0.536378i	$0.819792\pi$
$228$	7.77911	0.515184
$229$	24.6913	1.63165	0.815824	−	0.578300i	$-0.196284\pi$
$229$	24.6913	1.63165	0.815824	+	0.578300i	$0.196284\pi$
$230$	−9.48396	−0.625354
$231$	2.26126	0.148780
$232$	4.42004	0.290190
$233$	−15.7394	−1.03112	−0.515560	−	0.856853i	$-0.672416\pi$
$233$	−15.7394	−1.03112	−0.515560	+	0.856853i	$0.672416\pi$
$234$	−0.873201	−0.0570829
$235$	−4.57055	−0.298150
$236$	1.62420	0.105726
$237$	5.89360	0.382831
$238$	6.93005	0.449209
$239$	10.0399	0.649427	0.324714	−	0.945812i	$-0.394732\pi$
$239$	10.0399	0.649427	0.324714	+	0.945812i	$0.394732\pi$
$240$	−1.92862	−0.124492
$241$	2.32941	0.150051	0.0750253	−	0.997182i	$-0.476096\pi$
$241$	2.32941	0.150051	0.0750253	+	0.997182i	$0.476096\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	1.00000	0.0640184
$245$	3.63877	0.232473
$246$	0.410765	0.0261894
$247$	−6.79272	−0.432211
$248$	−0.988877	−0.0627937
$249$	13.8812	0.879688
$250$	12.1126	0.766065
$251$	14.0398	0.886181	0.443091	−	0.896477i	$-0.353882\pi$
$251$	14.0398	0.886181	0.443091	+	0.896477i	$0.353882\pi$
$252$	−2.26126	−0.142446
$253$	−4.91749	−0.309160
$254$	−1.91106	−0.119910
$255$	5.91062	0.370137
$256$	1.00000	0.0625000
$257$	6.37262	0.397513	0.198757	−	0.980049i	$-0.436310\pi$
$257$	6.37262	0.397513	0.198757	+	0.980049i	$0.436310\pi$
$258$	4.69011	0.291994
$259$	−16.7975	−1.04374
$260$	1.68407	0.104442
$261$	4.42004	0.273594
$262$	7.76942	0.479996
$263$	12.5964	0.776728	0.388364	−	0.921506i	$-0.373040\pi$
$263$	12.5964	0.776728	0.388364	+	0.921506i	$0.373040\pi$
$264$	−1.00000	−0.0615457
$265$	5.42762	0.333416
$266$	−17.5905	−1.07855
$267$	−17.5559	−1.07440
$268$	10.1876	0.622304
$269$	−6.49463	−0.395984	−0.197992	−	0.980204i	$-0.563442\pi$
$269$	−6.49463	−0.395984	−0.197992	+	0.980204i	$0.563442\pi$
$270$	−1.92862	−0.117372
$271$	−0.898941	−0.0546068	−0.0273034	−	0.999627i	$-0.508692\pi$
$271$	−0.898941	−0.0546068	−0.0273034	+	0.999627i	$0.508692\pi$
$272$	−3.06469	−0.185824
$273$	1.97453	0.119504
$274$	−13.9439	−0.842380
$275$	1.28044	0.0772132
$276$	4.91749	0.295998
$277$	1.14137	0.0685784	0.0342892	−	0.999412i	$-0.489083\pi$
$277$	1.14137	0.0685784	0.0342892	+	0.999412i	$0.489083\pi$
$278$	−6.28321	−0.376842
$279$	−0.988877	−0.0592025
$280$	4.36110	0.260625
$281$	3.14846	0.187821	0.0939106	−	0.995581i	$-0.470063\pi$
$281$	3.14846	0.187821	0.0939106	+	0.995581i	$0.470063\pi$
$282$	2.36986	0.141123
$283$	31.5485	1.87536	0.937681	−	0.347499i	$-0.112969\pi$
$283$	31.5485	1.87536	0.937681	+	0.347499i	$0.112969\pi$
$284$	8.05201	0.477799
$285$	−15.0029	−0.888696
$286$	0.873201	0.0516335
$287$	−0.928844	−0.0548279
$288$	1.00000	0.0589256
$289$	−7.60766	−0.447509
$290$	−8.52457	−0.500580
$291$	16.4295	0.963113
$292$	1.69404	0.0991361
$293$	−5.45862	−0.318896	−0.159448	−	0.987206i	$-0.550971\pi$
$293$	−5.45862	−0.318896	−0.159448	+	0.987206i	$0.550971\pi$
$294$	−1.88673	−0.110036
$295$	−3.13245	−0.182378
$296$	7.42838	0.431766
$297$	−1.00000	−0.0580259
$298$	−9.08077	−0.526035
$299$	−4.29396	−0.248326
$300$	−1.28044	−0.0739260
$301$	−10.6055	−0.611293
$302$	12.9000	0.742314
$303$	8.58883	0.493416
$304$	7.77911	0.446162
$305$	−1.92862	−0.110432
$306$	−3.06469	−0.175197
$307$	18.4307	1.05189	0.525947	−	0.850518i	$-0.323711\pi$
$307$	18.4307	1.05189	0.525947	+	0.850518i	$0.323711\pi$
$308$	2.26126	0.128847
$309$	8.13013	0.462507
$310$	1.90717	0.108320
$311$	21.0613	1.19427	0.597137	−	0.802139i	$-0.296305\pi$
$311$	21.0613	1.19427	0.597137	+	0.802139i	$0.296305\pi$
$312$	−0.873201	−0.0494353
$313$	−15.9547	−0.901812	−0.450906	−	0.892571i	$-0.648899\pi$
$313$	−15.9547	−0.901812	−0.450906	+	0.892571i	$0.648899\pi$
$314$	9.80139	0.553125
$315$	4.36110	0.245720
$316$	5.89360	0.331541
$317$	−27.5581	−1.54782	−0.773910	−	0.633296i	$-0.781702\pi$
$317$	−27.5581	−1.54782	−0.773910	+	0.633296i	$0.781702\pi$
$318$	−2.81425	−0.157816
$319$	−4.42004	−0.247475
$320$	−1.92862	−0.107813
$321$	−14.9345	−0.833562
$322$	−11.1197	−0.619677
$323$	−23.8406	−1.32653
$324$	1.00000	0.0555556
$325$	1.11808	0.0620198
$326$	−0.451975	−0.0250326
$327$	−6.73374	−0.372377
$328$	0.410765	0.0226807
$329$	−5.35885	−0.295443
$330$	1.92862	0.106167
$331$	8.33039	0.457880	0.228940	−	0.973441i	$-0.426474\pi$
$331$	8.33039	0.457880	0.228940	+	0.973441i	$0.426474\pi$
$332$	13.8812	0.761832
$333$	7.42838	0.407073
$334$	15.2873	0.836482
$335$	−19.6479	−1.07348
$336$	−2.26126	−0.123362
$337$	−5.68929	−0.309915	−0.154958	−	0.987921i	$-0.549524\pi$
$337$	−5.68929	−0.309915	−0.154958	+	0.987921i	$0.549524\pi$
$338$	−12.2375	−0.665633
$339$	4.31004	0.234089
$340$	5.91062	0.320548
$341$	0.988877	0.0535507
$342$	7.77911	0.420646
$343$	20.0952	1.08504
$344$	4.69011	0.252874
$345$	−9.48396	−0.510599
$346$	18.0507	0.970413
$347$	−25.6170	−1.37519	−0.687597	−	0.726092i	$-0.741335\pi$
$347$	−25.6170	−1.37519	−0.687597	+	0.726092i	$0.741335\pi$
$348$	4.42004	0.236939
$349$	6.52564	0.349309	0.174655	−	0.984630i	$-0.444119\pi$
$349$	6.52564	0.349309	0.174655	+	0.984630i	$0.444119\pi$
$350$	2.89539	0.154765
$351$	−0.873201	−0.0466080
$352$	−1.00000	−0.0533002
$353$	−28.3757	−1.51029	−0.755143	−	0.655560i	$-0.772433\pi$
$353$	−28.3757	−1.51029	−0.755143	+	0.655560i	$0.772433\pi$
$354$	1.62420	0.0863251
$355$	−15.5292	−0.824206
$356$	−17.5559	−0.930459
$357$	6.93005	0.366777
$358$	−1.82203	−0.0962974
$359$	−33.2373	−1.75420	−0.877098	−	0.480312i	$-0.840523\pi$
$359$	−33.2373	−1.75420	−0.877098	+	0.480312i	$0.840523\pi$
$360$	−1.92862	−0.101647
$361$	41.5145	2.18497
$362$	17.3994	0.914492
$363$	1.00000	0.0524864
$364$	1.97453	0.103493
$365$	−3.26715	−0.171011
$366$	1.00000	0.0522708
$367$	0.752222	0.0392657	0.0196328	−	0.999807i	$-0.493750\pi$
$367$	0.752222	0.0392657	0.0196328	+	0.999807i	$0.493750\pi$
$368$	4.91749	0.256342
$369$	0.410765	0.0213836
$370$	−14.3265	−0.744799
$371$	6.36375	0.330389
$372$	−0.988877	−0.0512709
$373$	14.4260	0.746951	0.373475	−	0.927640i	$-0.378166\pi$
$373$	14.4260	0.746951	0.373475	+	0.927640i	$0.378166\pi$
$374$	3.06469	0.158471
$375$	12.1126	0.625490
$376$	2.36986	0.122216
$377$	−3.85958	−0.198779
$378$	−2.26126	−0.116306
$379$	4.02822	0.206916	0.103458	−	0.994634i	$-0.467009\pi$
$379$	4.02822	0.206916	0.103458	+	0.994634i	$0.467009\pi$
$380$	−15.0029	−0.769634
$381$	−1.91106	−0.0979064
$382$	−23.8388	−1.21970
$383$	−36.8520	−1.88305	−0.941525	−	0.336943i	$-0.890607\pi$
$383$	−36.8520	−1.88305	−0.941525	+	0.336943i	$0.890607\pi$
$384$	1.00000	0.0510310
$385$	−4.36110	−0.222262
$386$	−6.05697	−0.308292
$387$	4.69011	0.238412
$388$	16.4295	0.834080
$389$	−24.1552	−1.22472	−0.612359	−	0.790579i	$-0.709779\pi$
$389$	−24.1552	−1.22472	−0.612359	+	0.790579i	$0.709779\pi$
$390$	1.68407	0.0852762
$391$	−15.0706	−0.762153
$392$	−1.88673	−0.0952940
$393$	7.76942	0.391915
$394$	3.37431	0.169996
$395$	−11.3665	−0.571911
$396$	−1.00000	−0.0502519
$397$	−5.63136	−0.282630	−0.141315	−	0.989965i	$-0.545133\pi$
$397$	−5.63136	−0.282630	−0.141315	+	0.989965i	$0.545133\pi$
$398$	−17.9859	−0.901552
$399$	−17.5905	−0.880629
$400$	−1.28044	−0.0640218
$401$	20.6288	1.03016	0.515078	−	0.857144i	$-0.327763\pi$
$401$	20.6288	1.03016	0.515078	+	0.857144i	$0.327763\pi$
$402$	10.1876	0.508109
$403$	0.863488	0.0430134
$404$	8.58883	0.427310
$405$	−1.92862	−0.0958338
$406$	−9.99484	−0.496036
$407$	−7.42838	−0.368211
$408$	−3.06469	−0.151725
$409$	15.5884	0.770797	0.385398	−	0.922750i	$-0.374064\pi$
$409$	15.5884	0.770797	0.385398	+	0.922750i	$0.374064\pi$
$410$	−0.792208	−0.0391244
$411$	−13.9439	−0.687800
$412$	8.13013	0.400543
$413$	−3.67272	−0.180723
$414$	4.91749	0.241682
$415$	−26.7716	−1.31417
$416$	−0.873201	−0.0428122
$417$	−6.28321	−0.307690
$418$	−7.77911	−0.380489
$419$	−9.23481	−0.451150	−0.225575	−	0.974226i	$-0.572426\pi$
$419$	−9.23481	−0.451150	−0.225575	+	0.974226i	$0.572426\pi$
$420$	4.36110	0.212800
$421$	9.90944	0.482956	0.241478	−	0.970406i	$-0.422368\pi$
$421$	9.90944	0.482956	0.241478	+	0.970406i	$0.422368\pi$
$422$	−18.3538	−0.893449
$423$	2.36986	0.115226
$424$	−2.81425	−0.136672
$425$	3.92414	0.190349
$426$	8.05201	0.390121
$427$	−2.26126	−0.109430
$428$	−14.9345	−0.721885
$429$	0.873201	0.0421585
$430$	−9.04543	−0.436210
$431$	−17.7159	−0.853343	−0.426671	−	0.904407i	$-0.640314\pi$
$431$	−17.7159	−0.853343	−0.426671	+	0.904407i	$0.640314\pi$
$432$	1.00000	0.0481125
$433$	20.8748	1.00318	0.501591	−	0.865105i	$-0.332748\pi$
$433$	20.8748	1.00318	0.501591	+	0.865105i	$0.332748\pi$
$434$	2.23610	0.107336
$435$	−8.52457	−0.408722
$436$	−6.73374	−0.322488
$437$	38.2537	1.82992
$438$	1.69404	0.0809443
$439$	−5.42028	−0.258696	−0.129348	−	0.991599i	$-0.541288\pi$
$439$	−5.42028	−0.258696	−0.129348	+	0.991599i	$0.541288\pi$
$440$	1.92862	0.0919432
$441$	−1.88673	−0.0898441
$442$	2.67609	0.127289
$443$	9.40677	0.446929	0.223464	−	0.974712i	$-0.428263\pi$
$443$	9.40677	0.446929	0.223464	+	0.974712i	$0.428263\pi$
$444$	7.42838	0.352535
$445$	33.8585	1.60505
$446$	−19.7968	−0.937404
$447$	−9.08077	−0.429506
$448$	−2.26126	−0.106834
$449$	−8.39029	−0.395962	−0.197981	−	0.980206i	$-0.563438\pi$
$449$	−8.39029	−0.395962	−0.197981	+	0.980206i	$0.563438\pi$
$450$	−1.28044	−0.0603603
$451$	−0.410765	−0.0193422
$452$	4.31004	0.202727
$453$	12.9000	0.606097
$454$	−25.4316	−1.19357
$455$	−3.80811	−0.178527
$456$	7.77911	0.364290
$457$	−40.6310	−1.90064	−0.950319	−	0.311279i	$-0.899243\pi$
$457$	−40.6310	−1.90064	−0.950319	+	0.311279i	$0.899243\pi$
$458$	24.6913	1.15375
$459$	−3.06469	−0.143048
$460$	−9.48396	−0.442192
$461$	19.9901	0.931033	0.465517	−	0.885039i	$-0.345868\pi$
$461$	19.9901	0.931033	0.465517	+	0.885039i	$0.345868\pi$
$462$	2.26126	0.105203
$463$	1.22887	0.0571106	0.0285553	−	0.999592i	$-0.490909\pi$
$463$	1.22887	0.0571106	0.0285553	+	0.999592i	$0.490909\pi$
$464$	4.42004	0.205195
$465$	1.90717	0.0884427
$466$	−15.7394	−0.729112
$467$	−37.0365	−1.71385	−0.856924	−	0.515444i	$-0.827627\pi$
$467$	−37.0365	−1.71385	−0.856924	+	0.515444i	$0.827627\pi$
$468$	−0.873201	−0.0403637
$469$	−23.0367	−1.06373
$470$	−4.57055	−0.210824
$471$	9.80139	0.451624
$472$	1.62420	0.0747597
$473$	−4.69011	−0.215652
$474$	5.89360	0.270702
$475$	−9.96065	−0.457026
$476$	6.93005	0.317638
$477$	−2.81425	−0.128856
$478$	10.0399	0.459215
$479$	21.5676	0.985448	0.492724	−	0.870186i	$-0.336001\pi$
$479$	21.5676	0.985448	0.492724	+	0.870186i	$0.336001\pi$
$480$	−1.92862	−0.0880289
$481$	−6.48647	−0.295757
$482$	2.32941	0.106102
$483$	−11.1197	−0.505964
$484$	1.00000	0.0454545
$485$	−31.6862	−1.43880
$486$	1.00000	0.0453609
$487$	−32.0318	−1.45150	−0.725751	−	0.687958i	$-0.758507\pi$
$487$	−32.0318	−1.45150	−0.725751	+	0.687958i	$0.758507\pi$
$488$	1.00000	0.0452679
$489$	−0.451975	−0.0204390
$490$	3.63877	0.164383
$491$	31.8127	1.43569	0.717844	−	0.696204i	$-0.245129\pi$
$491$	31.8127	1.43569	0.717844	+	0.696204i	$0.245129\pi$
$492$	0.410765	0.0185187
$493$	−13.5461	−0.610084
$494$	−6.79272	−0.305619
$495$	1.92862	0.0866849
$496$	−0.988877	−0.0444019
$497$	−18.2076	−0.816724
$498$	13.8812	0.622033
$499$	10.1537	0.454542	0.227271	−	0.973832i	$-0.427020\pi$
$499$	10.1537	0.454542	0.227271	+	0.973832i	$0.427020\pi$
$500$	12.1126	0.541690
$501$	15.2873	0.682985
$502$	14.0398	0.626625
$503$	29.2908	1.30601	0.653006	−	0.757352i	$-0.273507\pi$
$503$	29.2908	1.30601	0.653006	+	0.757352i	$0.273507\pi$
$504$	−2.26126	−0.100724
$505$	−16.5646	−0.737114
$506$	−4.91749	−0.218609
$507$	−12.2375	−0.543487
$508$	−1.91106	−0.0847894
$509$	10.1333	0.449151	0.224575	−	0.974457i	$-0.427901\pi$
$509$	10.1333	0.449151	0.224575	+	0.974457i	$0.427901\pi$
$510$	5.91062	0.261727
$511$	−3.83065	−0.169458
$512$	1.00000	0.0441942
$513$	7.77911	0.343456
$514$	6.37262	0.281084
$515$	−15.6799	−0.690939
$516$	4.69011	0.206471
$517$	−2.36986	−0.104226
$518$	−16.7975	−0.738038
$519$	18.0507	0.792339
$520$	1.68407	0.0738514
$521$	−8.82351	−0.386565	−0.193282	−	0.981143i	$-0.561913\pi$
$521$	−8.82351	−0.386565	−0.193282	+	0.981143i	$0.561913\pi$
$522$	4.42004	0.193460
$523$	−29.8481	−1.30517	−0.652584	−	0.757716i	$-0.726315\pi$
$523$	−29.8481	−1.30517	−0.652584	+	0.757716i	$0.726315\pi$
$524$	7.76942	0.339409
$525$	2.89539	0.126365
$526$	12.5964	0.549229
$527$	3.03060	0.132015
$528$	−1.00000	−0.0435194
$529$	1.18175	0.0513804
$530$	5.42762	0.235761
$531$	1.62420	0.0704841
$532$	−17.5905	−0.762647
$533$	−0.358680	−0.0155362
$534$	−17.5559	−0.759717
$535$	28.8029	1.24526
$536$	10.1876	0.440036
$537$	−1.82203	−0.0786265
$538$	−6.49463	−0.280003
$539$	1.88673	0.0812670
$540$	−1.92862	−0.0829945
$541$	16.1126	0.692734	0.346367	−	0.938099i	$-0.387415\pi$
$541$	16.1126	0.692734	0.346367	+	0.938099i	$0.387415\pi$
$542$	−0.898941	−0.0386128
$543$	17.3994	0.746679
$544$	−3.06469	−0.131398
$545$	12.9868	0.556294
$546$	1.97453	0.0845021
$547$	33.7619	1.44356	0.721778	−	0.692124i	$-0.243325\pi$
$547$	33.7619	1.44356	0.721778	+	0.692124i	$0.243325\pi$
$548$	−13.9439	−0.595652
$549$	1.00000	0.0426790
$550$	1.28044	0.0545980
$551$	34.3840	1.46481
$552$	4.91749	0.209302
$553$	−13.3269	−0.566719
$554$	1.14137	0.0484922
$555$	−14.3265	−0.608126
$556$	−6.28321	−0.266467
$557$	1.91098	0.0809707	0.0404853	−	0.999180i	$-0.487110\pi$
$557$	1.91098	0.0809707	0.0404853	+	0.999180i	$0.487110\pi$
$558$	−0.988877	−0.0418625
$559$	−4.09541	−0.173217
$560$	4.36110	0.184290
$561$	3.06469	0.129391
$562$	3.14846	0.132810
$563$	−1.53187	−0.0645606	−0.0322803	−	0.999479i	$-0.510277\pi$
$563$	−1.53187	−0.0645606	−0.0322803	+	0.999479i	$0.510277\pi$
$564$	2.36986	0.0997890
$565$	−8.31241	−0.349706
$566$	31.5485	1.32608
$567$	−2.26126	−0.0949638
$568$	8.05201	0.337855
$569$	33.5997	1.40857	0.704285	−	0.709917i	$-0.251268\pi$
$569$	33.5997	1.40857	0.704285	+	0.709917i	$0.251268\pi$
$570$	−15.0029	−0.628403
$571$	−8.50600	−0.355965	−0.177982	−	0.984034i	$-0.556957\pi$
$571$	−8.50600	−0.355965	−0.177982	+	0.984034i	$0.556957\pi$
$572$	0.873201	0.0365104
$573$	−23.8388	−0.995879
$574$	−0.928844	−0.0387692
$575$	−6.29654	−0.262584
$576$	1.00000	0.0416667
$577$	11.0525	0.460122	0.230061	−	0.973176i	$-0.426107\pi$
$577$	11.0525	0.460122	0.230061	+	0.973176i	$0.426107\pi$
$578$	−7.60766	−0.316437
$579$	−6.05697	−0.251719
$580$	−8.52457	−0.353963
$581$	−31.3890	−1.30224
$582$	16.4295	0.681024
$583$	2.81425	0.116555
$584$	1.69404	0.0700998
$585$	1.68407	0.0696277
$586$	−5.45862	−0.225494
$587$	−23.4251	−0.966857	−0.483429	−	0.875384i	$-0.660609\pi$
$587$	−23.4251	−0.966857	−0.483429	+	0.875384i	$0.660609\pi$
$588$	−1.88673	−0.0778073
$589$	−7.69258	−0.316967
$590$	−3.13245	−0.128961
$591$	3.37431	0.138801
$592$	7.42838	0.305304
$593$	−24.0300	−0.986792	−0.493396	−	0.869805i	$-0.664245\pi$
$593$	−24.0300	−0.986792	−0.493396	+	0.869805i	$0.664245\pi$
$594$	−1.00000	−0.0410305
$595$	−13.3654	−0.547929
$596$	−9.08077	−0.371963
$597$	−17.9859	−0.736114
$598$	−4.29396	−0.175593
$599$	47.8941	1.95690	0.978449	−	0.206487i	$-0.0662032\pi$
$599$	47.8941	1.95690	0.978449	+	0.206487i	$0.0662032\pi$
$600$	−1.28044	−0.0522736
$601$	23.3171	0.951126	0.475563	−	0.879682i	$-0.342244\pi$
$601$	23.3171	0.951126	0.475563	+	0.879682i	$0.342244\pi$
$602$	−10.6055	−0.432250
$603$	10.1876	0.414870
$604$	12.9000	0.524895
$605$	−1.92862	−0.0784094
$606$	8.58883	0.348898
$607$	−39.5147	−1.60385	−0.801925	−	0.597424i	$-0.796191\pi$
$607$	−39.5147	−1.60385	−0.801925	+	0.597424i	$0.796191\pi$
$608$	7.77911	0.315484
$609$	−9.99484	−0.405011
$610$	−1.92862	−0.0780874
$611$	−2.06936	−0.0837174
$612$	−3.06469	−0.123883
$613$	40.9813	1.65522	0.827610	−	0.561304i	$-0.189700\pi$
$613$	40.9813	1.65522	0.827610	+	0.561304i	$0.189700\pi$
$614$	18.4307	0.743801
$615$	−0.792208	−0.0319449
$616$	2.26126	0.0911086
$617$	−13.5410	−0.545140	−0.272570	−	0.962136i	$-0.587874\pi$
$617$	−13.5410	−0.545140	−0.272570	+	0.962136i	$0.587874\pi$
$618$	8.13013	0.327042
$619$	28.0970	1.12932	0.564658	−	0.825325i	$-0.309008\pi$
$619$	28.0970	1.12932	0.564658	+	0.825325i	$0.309008\pi$
$620$	1.90717	0.0765936
$621$	4.91749	0.197332
$622$	21.0613	0.844480
$623$	39.6983	1.59048
$624$	−0.873201	−0.0349560
$625$	−16.9583	−0.678332
$626$	−15.9547	−0.637677
$627$	−7.77911	−0.310668
$628$	9.80139	0.391118
$629$	−22.7657	−0.907728
$630$	4.36110	0.173750
$631$	28.6837	1.14188	0.570940	−	0.820992i	$-0.306579\pi$
$631$	28.6837	1.14188	0.570940	+	0.820992i	$0.306579\pi$
$632$	5.89360	0.234435
$633$	−18.3538	−0.729498
$634$	−27.5581	−1.09447
$635$	3.68570	0.146262
$636$	−2.81425	−0.111592
$637$	1.64749	0.0652760
$638$	−4.42004	−0.174991
$639$	8.05201	0.318532
$640$	−1.92862	−0.0762353
$641$	−1.37921	−0.0544756	−0.0272378	−	0.999629i	$-0.508671\pi$
$641$	−1.37921	−0.0544756	−0.0272378	+	0.999629i	$0.508671\pi$
$642$	−14.9345	−0.589417
$643$	−22.1406	−0.873142	−0.436571	−	0.899670i	$-0.643807\pi$
$643$	−22.1406	−0.873142	−0.436571	+	0.899670i	$0.643807\pi$
$644$	−11.1197	−0.438178
$645$	−9.04543	−0.356164
$646$	−23.8406	−0.937995
$647$	4.35297	0.171133	0.0855665	−	0.996332i	$-0.472730\pi$
$647$	4.35297	0.171133	0.0855665	+	0.996332i	$0.472730\pi$
$648$	1.00000	0.0392837
$649$	−1.62420	−0.0637553
$650$	1.11808	0.0438546
$651$	2.23610	0.0876398
$652$	−0.451975	−0.0177007
$653$	−15.1101	−0.591302	−0.295651	−	0.955296i	$-0.595537\pi$
$653$	−15.1101	−0.591302	−0.295651	+	0.955296i	$0.595537\pi$
$654$	−6.73374	−0.263310
$655$	−14.9842	−0.585482
$656$	0.410765	0.0160377
$657$	1.69404	0.0660908
$658$	−5.35885	−0.208910
$659$	−9.45104	−0.368160	−0.184080	−	0.982911i	$-0.558931\pi$
$659$	−9.45104	−0.368160	−0.184080	+	0.982911i	$0.558931\pi$
$660$	1.92862	0.0750713
$661$	−10.3530	−0.402687	−0.201343	−	0.979521i	$-0.564531\pi$
$661$	−10.3530	−0.402687	−0.201343	+	0.979521i	$0.564531\pi$
$662$	8.33039	0.323770
$663$	2.67609	0.103931
$664$	13.8812	0.538697
$665$	33.9254	1.31557
$666$	7.42838	0.287844
$667$	21.7355	0.841603
$668$	15.2873	0.591482
$669$	−19.7968	−0.765387
$670$	−19.6479	−0.759065
$671$	−1.00000	−0.0386046
$672$	−2.26126	−0.0872298
$673$	−5.51622	−0.212635	−0.106317	−	0.994332i	$-0.533906\pi$
$673$	−5.51622	−0.212635	−0.106317	+	0.994332i	$0.533906\pi$
$674$	−5.68929	−0.219143
$675$	−1.28044	−0.0492840
$676$	−12.2375	−0.470674
$677$	2.35644	0.0905652	0.0452826	−	0.998974i	$-0.485581\pi$
$677$	2.35644	0.0905652	0.0452826	+	0.998974i	$0.485581\pi$
$678$	4.31004	0.165526
$679$	−37.1512	−1.42573
$680$	5.91062	0.226662
$681$	−25.4316	−0.974542
$682$	0.988877	0.0378661
$683$	20.2829	0.776105	0.388052	−	0.921637i	$-0.373148\pi$
$683$	20.2829	0.776105	0.388052	+	0.921637i	$0.373148\pi$
$684$	7.77911	0.297442
$685$	26.8924	1.02750
$686$	20.0952	0.767237
$687$	24.6913	0.942032
$688$	4.69011	0.178809
$689$	2.45741	0.0936199
$690$	−9.48396	−0.361048
$691$	−26.8635	−1.02194	−0.510968	−	0.859600i	$-0.670713\pi$
$691$	−26.8635	−1.02194	−0.510968	+	0.859600i	$0.670713\pi$
$692$	18.0507	0.686186
$693$	2.26126	0.0858980
$694$	−25.6170	−0.972410
$695$	12.1179	0.459658
$696$	4.42004	0.167541
$697$	−1.25887	−0.0476830
$698$	6.52564	0.246999
$699$	−15.7394	−0.595318
$700$	2.89539	0.109436
$701$	0.0667184	0.00251992	0.00125996	−	0.999999i	$-0.499599\pi$
$701$	0.0667184	0.00251992	0.00125996	+	0.999999i	$0.499599\pi$
$702$	−0.873201	−0.0329568
$703$	57.7861	2.17945
$704$	−1.00000	−0.0376889
$705$	−4.57055	−0.172137
$706$	−28.3757	−1.06793
$707$	−19.4215	−0.730422
$708$	1.62420	0.0610410
$709$	10.1936	0.382828	0.191414	−	0.981509i	$-0.438693\pi$
$709$	10.1936	0.382828	0.191414	+	0.981509i	$0.438693\pi$
$710$	−15.5292	−0.582802
$711$	5.89360	0.221027
$712$	−17.5559	−0.657934
$713$	−4.86280	−0.182113
$714$	6.93005	0.259351
$715$	−1.68407	−0.0629807
$716$	−1.82203	−0.0680926
$717$	10.0399	0.374947
$718$	−33.2373	−1.24040
$719$	7.48537	0.279157	0.139579	−	0.990211i	$-0.455425\pi$
$719$	7.48537	0.279157	0.139579	+	0.990211i	$0.455425\pi$
$720$	−1.92862	−0.0718753
$721$	−18.3843	−0.684667
$722$	41.5145	1.54501
$723$	2.32941	0.0866318
$724$	17.3994	0.646643
$725$	−5.65958	−0.210191
$726$	1.00000	0.0371135
$727$	−18.1436	−0.672908	−0.336454	−	0.941700i	$-0.609228\pi$
$727$	−18.1436	−0.672908	−0.336454	+	0.941700i	$0.609228\pi$
$728$	1.97453	0.0731810
$729$	1.00000	0.0370370
$730$	−3.26715	−0.120923
$731$	−14.3738	−0.531633
$732$	1.00000	0.0369611
$733$	−16.5833	−0.612517	−0.306259	−	0.951948i	$-0.599077\pi$
$733$	−16.5833	−0.612517	−0.306259	+	0.951948i	$0.599077\pi$
$734$	0.752222	0.0277650
$735$	3.63877	0.134218
$736$	4.91749	0.181261
$737$	−10.1876	−0.375264
$738$	0.410765	0.0151205
$739$	9.21743	0.339069	0.169534	−	0.985524i	$-0.445774\pi$
$739$	9.21743	0.339069	0.169534	+	0.985524i	$0.445774\pi$
$740$	−14.3265	−0.526652
$741$	−6.79272	−0.249537
$742$	6.36375	0.233621
$743$	37.2079	1.36503	0.682513	−	0.730874i	$-0.260887\pi$
$743$	37.2079	1.36503	0.682513	+	0.730874i	$0.260887\pi$
$744$	−0.988877	−0.0362540
$745$	17.5133	0.641639
$746$	14.4260	0.528174
$747$	13.8812	0.507888
$748$	3.06469	0.112056
$749$	33.7707	1.23395
$750$	12.1126	0.442288
$751$	22.2661	0.812500	0.406250	−	0.913762i	$-0.366836\pi$
$751$	22.2661	0.812500	0.406250	+	0.913762i	$0.366836\pi$
$752$	2.36986	0.0864198
$753$	14.0398	0.511637
$754$	−3.85958	−0.140558
$755$	−24.8792	−0.905448
$756$	−2.26126	−0.0822410
$757$	23.6339	0.858989	0.429495	−	0.903069i	$-0.358692\pi$
$757$	23.6339	0.858989	0.429495	+	0.903069i	$0.358692\pi$
$758$	4.02822	0.146311
$759$	−4.91749	−0.178494
$760$	−15.0029	−0.544213
$761$	26.9054	0.975320	0.487660	−	0.873033i	$-0.337850\pi$
$761$	26.9054	0.975320	0.487660	+	0.873033i	$0.337850\pi$
$762$	−1.91106	−0.0692303
$763$	15.2267	0.551244
$764$	−23.8388	−0.862456
$765$	5.91062	0.213699
$766$	−36.8520	−1.33152
$767$	−1.41825	−0.0512100
$768$	1.00000	0.0360844
$769$	32.7482	1.18093	0.590465	−	0.807063i	$-0.298944\pi$
$769$	32.7482	1.18093	0.590465	+	0.807063i	$0.298944\pi$
$770$	−4.36110	−0.157163
$771$	6.37262	0.229504
$772$	−6.05697	−0.217995
$773$	−48.8029	−1.75532	−0.877659	−	0.479286i	$-0.840896\pi$
$773$	−48.8029	−1.75532	−0.877659	+	0.479286i	$0.840896\pi$
$774$	4.69011	0.168583
$775$	1.26619	0.0454830
$776$	16.4295	0.589784
$777$	−16.7975	−0.602605
$778$	−24.1552	−0.866007
$779$	3.19538	0.114486
$780$	1.68407	0.0602994
$781$	−8.05201	−0.288123
$782$	−15.0706	−0.538924
$783$	4.42004	0.157959
$784$	−1.88673	−0.0673831
$785$	−18.9031	−0.674682
$786$	7.76942	0.277126
$787$	13.5759	0.483929	0.241965	−	0.970285i	$-0.422208\pi$
$787$	13.5759	0.483929	0.241965	+	0.970285i	$0.422208\pi$
$788$	3.37431	0.120205
$789$	12.5964	0.448444
$790$	−11.3665	−0.404402
$791$	−9.74609	−0.346531
$792$	−1.00000	−0.0355335
$793$	−0.873201	−0.0310083
$794$	−5.63136	−0.199849
$795$	5.42762	0.192498
$796$	−17.9859	−0.637493
$797$	−11.8630	−0.420209	−0.210105	−	0.977679i	$-0.567380\pi$
$797$	−11.8630	−0.420209	−0.210105	+	0.977679i	$0.567380\pi$
$798$	−17.5905	−0.622699
$799$	−7.26288	−0.256942
$800$	−1.28044	−0.0452702
$801$	−17.5559	−0.620306
$802$	20.6288	0.728430
$803$	−1.69404	−0.0597813
$804$	10.1876	0.359288
$805$	21.4457	0.755860
$806$	0.863488	0.0304151
$807$	−6.49463	−0.228622
$808$	8.58883	0.302154
$809$	26.4415	0.929633	0.464816	−	0.885407i	$-0.346120\pi$
$809$	26.4415	0.929633	0.464816	+	0.885407i	$0.346120\pi$
$810$	−1.92862	−0.0677647
$811$	−38.9219	−1.36673	−0.683366	−	0.730076i	$-0.739485\pi$
$811$	−38.9219	−1.36673	−0.683366	+	0.730076i	$0.739485\pi$
$812$	−9.99484	−0.350750
$813$	−0.898941	−0.0315272
$814$	−7.42838	−0.260364
$815$	0.871686	0.0305338
$816$	−3.06469	−0.107286
$817$	36.4849	1.27644
$818$	15.5884	0.545035
$819$	1.97453	0.0689957
$820$	−0.792208	−0.0276651
$821$	−47.6989	−1.66470	−0.832351	−	0.554249i	$-0.813006\pi$
$821$	−47.6989	−1.66470	−0.832351	+	0.554249i	$0.813006\pi$
$822$	−13.9439	−0.486348
$823$	−4.59217	−0.160073	−0.0800364	−	0.996792i	$-0.525504\pi$
$823$	−4.59217	−0.160073	−0.0800364	+	0.996792i	$0.525504\pi$
$824$	8.13013	0.283227
$825$	1.28044	0.0445791
$826$	−3.67272	−0.127790
$827$	−43.9899	−1.52968	−0.764839	−	0.644222i	$-0.777181\pi$
$827$	−43.9899	−1.52968	−0.764839	+	0.644222i	$0.777181\pi$
$828$	4.91749	0.170895
$829$	−1.07509	−0.0373395	−0.0186697	−	0.999826i	$-0.505943\pi$
$829$	−1.07509	−0.0373395	−0.0186697	+	0.999826i	$0.505943\pi$
$830$	−26.7716	−0.929256
$831$	1.14137	0.0395937
$832$	−0.873201	−0.0302728
$833$	5.78223	0.200343
$834$	−6.28321	−0.217570
$835$	−29.4833	−1.02031
$836$	−7.77911	−0.269046
$837$	−0.988877	−0.0341806
$838$	−9.23481	−0.319011
$839$	53.1719	1.83570	0.917849	−	0.396929i	$-0.129924\pi$
$839$	53.1719	1.83570	0.917849	+	0.396929i	$0.129924\pi$
$840$	4.36110	0.150472
$841$	−9.46323	−0.326318
$842$	9.90944	0.341502
$843$	3.14846	0.108439
$844$	−18.3538	−0.631764
$845$	23.6015	0.811916
$846$	2.36986	0.0814774
$847$	−2.26126	−0.0776976
$848$	−2.81425	−0.0966419
$849$	31.5485	1.08274
$850$	3.92414	0.134597
$851$	36.5290	1.25220
$852$	8.05201	0.275857
$853$	−2.56311	−0.0877592	−0.0438796	−	0.999037i	$-0.513972\pi$
$853$	−2.56311	−0.0877592	−0.0438796	+	0.999037i	$0.513972\pi$
$854$	−2.26126	−0.0773786
$855$	−15.0029	−0.513089
$856$	−14.9345	−0.510450
$857$	−47.4104	−1.61951	−0.809754	−	0.586770i	$-0.800399\pi$
$857$	−47.4104	−1.61951	−0.809754	+	0.586770i	$0.800399\pi$
$858$	0.873201	0.0298106
$859$	36.0966	1.23160	0.615800	−	0.787903i	$-0.288833\pi$
$859$	36.0966	1.23160	0.615800	+	0.787903i	$0.288833\pi$
$860$	−9.04543	−0.308447
$861$	−0.928844	−0.0316549
$862$	−17.7159	−0.603404
$863$	−31.6842	−1.07854	−0.539271	−	0.842132i	$-0.681300\pi$
$863$	−31.6842	−1.07854	−0.539271	+	0.842132i	$0.681300\pi$
$864$	1.00000	0.0340207
$865$	−34.8129	−1.18368
$866$	20.8748	0.709356
$867$	−7.60766	−0.258370
$868$	2.23610	0.0758983
$869$	−5.89360	−0.199927
$870$	−8.52457	−0.289010
$871$	−8.89579	−0.301422
$872$	−6.73374	−0.228033
$873$	16.4295	0.556054
$874$	38.2537	1.29395
$875$	−27.3896	−0.925937
$876$	1.69404	0.0572363
$877$	55.2199	1.86464	0.932321	−	0.361632i	$-0.117780\pi$
$877$	55.2199	1.86464	0.932321	+	0.361632i	$0.117780\pi$
$878$	−5.42028	−0.182926
$879$	−5.45862	−0.184115
$880$	1.92862	0.0650137
$881$	−42.2415	−1.42315	−0.711576	−	0.702610i	$-0.752018\pi$
$881$	−42.2415	−1.42315	−0.711576	+	0.702610i	$0.752018\pi$
$882$	−1.88673	−0.0635294
$883$	5.64098	0.189834	0.0949171	−	0.995485i	$-0.469741\pi$
$883$	5.64098	0.189834	0.0949171	+	0.995485i	$0.469741\pi$
$884$	2.67609	0.0900067
$885$	−3.13245	−0.105296
$886$	9.40677	0.316026
$887$	20.1615	0.676958	0.338479	−	0.940974i	$-0.390088\pi$
$887$	20.1615	0.676958	0.338479	+	0.940974i	$0.390088\pi$
$888$	7.42838	0.249280
$889$	4.32139	0.144935
$890$	33.8585	1.13494
$891$	−1.00000	−0.0335013
$892$	−19.7968	−0.662845
$893$	18.4354	0.616916
$894$	−9.08077	−0.303706
$895$	3.51400	0.117460
$896$	−2.26126	−0.0755432
$897$	−4.29396	−0.143371
$898$	−8.39029	−0.279988
$899$	−4.37088	−0.145777
$900$	−1.28044	−0.0426812
$901$	8.62483	0.287335
$902$	−0.410765	−0.0136770
$903$	−10.6055	−0.352930
$904$	4.31004	0.143350
$905$	−33.5568	−1.11546
$906$	12.9000	0.428575
$907$	52.6786	1.74916	0.874582	−	0.484878i	$-0.161136\pi$
$907$	52.6786	1.74916	0.874582	+	0.484878i	$0.161136\pi$
$908$	−25.4316	−0.843978
$909$	8.58883	0.284874
$910$	−3.80811	−0.126238
$911$	−0.216308	−0.00716660	−0.00358330	−	0.999994i	$-0.501141\pi$
$911$	−0.216308	−0.00716660	−0.00358330	+	0.999994i	$0.501141\pi$
$912$	7.77911	0.257592
$913$	−13.8812	−0.459402
$914$	−40.6310	−1.34395
$915$	−1.92862	−0.0637581
$916$	24.6913	0.815824
$917$	−17.5686	−0.580167
$918$	−3.06469	−0.101150
$919$	35.6867	1.17719	0.588597	−	0.808426i	$-0.299680\pi$
$919$	35.6867	1.17719	0.588597	+	0.808426i	$0.299680\pi$
$920$	−9.48396	−0.312677
$921$	18.4307	0.607311
$922$	19.9901	0.658340
$923$	−7.03102	−0.231429
$924$	2.26126	0.0743898
$925$	−9.51156	−0.312738
$926$	1.22887	0.0403833
$927$	8.13013	0.267029
$928$	4.42004	0.145095
$929$	−10.0922	−0.331114	−0.165557	−	0.986200i	$-0.552942\pi$
$929$	−10.0922	−0.331114	−0.165557	+	0.986200i	$0.552942\pi$
$930$	1.90717	0.0625384
$931$	−14.6770	−0.481021
$932$	−15.7394	−0.515560
$933$	21.0613	0.689515
$934$	−37.0365	−1.21187
$935$	−5.91062	−0.193298
$936$	−0.873201	−0.0285415
$937$	23.5555	0.769524	0.384762	−	0.923016i	$-0.374284\pi$
$937$	23.5555	0.769524	0.384762	+	0.923016i	$0.374284\pi$
$938$	−23.0367	−0.752174
$939$	−15.9547	−0.520661
$940$	−4.57055	−0.149075
$941$	30.6754	0.999989	0.499995	−	0.866029i	$-0.333335\pi$
$941$	30.6754	0.999989	0.499995	+	0.866029i	$0.333335\pi$
$942$	9.80139	0.319347
$943$	2.01993	0.0657781
$944$	1.62420	0.0528631
$945$	4.36110	0.141866
$946$	−4.69011	−0.152489
$947$	−41.6853	−1.35459	−0.677295	−	0.735712i	$-0.736848\pi$
$947$	−41.6853	−1.35459	−0.677295	+	0.735712i	$0.736848\pi$
$948$	5.89360	0.191415
$949$	−1.47924	−0.0480181
$950$	−9.96065	−0.323166
$951$	−27.5581	−0.893634
$952$	6.93005	0.224604
$953$	−0.164222	−0.00531966	−0.00265983	−	0.999996i	$-0.500847\pi$
$953$	−0.164222	−0.00531966	−0.00265983	+	0.999996i	$0.500847\pi$
$954$	−2.81425	−0.0911149
$955$	45.9759	1.48774
$956$	10.0399	0.324714
$957$	−4.42004	−0.142880
$958$	21.5676	0.696817
$959$	31.5306	1.01818
$960$	−1.92862	−0.0622459
$961$	−30.0221	−0.968456
$962$	−6.48647	−0.209132
$963$	−14.9345	−0.481257
$964$	2.32941	0.0750253
$965$	11.6816	0.376043
$966$	−11.1197	−0.357771
$967$	−57.7842	−1.85821	−0.929107	−	0.369812i	$-0.879422\pi$
$967$	−57.7842	−1.85821	−0.929107	+	0.369812i	$0.879422\pi$
$968$	1.00000	0.0321412
$969$	−23.8406	−0.765870
$970$	−31.6862	−1.01738
$971$	−47.9513	−1.53883	−0.769415	−	0.638750i	$-0.779452\pi$
$971$	−47.9513	−1.53883	−0.769415	+	0.638750i	$0.779452\pi$
$972$	1.00000	0.0320750
$973$	14.2079	0.455486
$974$	−32.0318	−1.02637
$975$	1.11808	0.0358071
$976$	1.00000	0.0320092
$977$	−25.3670	−0.811563	−0.405781	−	0.913970i	$-0.633001\pi$
$977$	−25.3670	−0.811563	−0.405781	+	0.913970i	$0.633001\pi$
$978$	−0.451975	−0.0144526
$979$	17.5559	0.561088
$980$	3.63877	0.116236
$981$	−6.73374	−0.214992
$982$	31.8127	1.01518
$983$	−2.80933	−0.0896037	−0.0448018	−	0.998996i	$-0.514266\pi$
$983$	−2.80933	−0.0896037	−0.0448018	+	0.998996i	$0.514266\pi$
$984$	0.410765	0.0130947
$985$	−6.50776	−0.207355
$986$	−13.5461	−0.431395
$987$	−5.35885	−0.170574
$988$	−6.79272	−0.216105
$989$	23.0636	0.733380
$990$	1.92862	0.0612955
$991$	−8.62422	−0.273957	−0.136979	−	0.990574i	$-0.543739\pi$
$991$	−8.62422	−0.273957	−0.136979	+	0.990574i	$0.543739\pi$
$992$	−0.988877	−0.0313969
$993$	8.33039	0.264357
$994$	−18.2076	−0.577511
$995$	34.6879	1.09968
$996$	13.8812	0.439844
$997$	−16.8507	−0.533668	−0.266834	−	0.963743i	$-0.585978\pi$
$997$	−16.8507	−0.533668	−0.266834	+	0.963743i	$0.585978\pi$
$998$	10.1537	0.321410
$999$	7.42838	0.235023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4026.2.a.bb.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4026.2.a.bb.1.3	✓	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:	\( N \)	\(=\)	\( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4026.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.92862 q^{5} +1.00000 q^{6} -2.26126 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.92862 q^{5} +1.00000 q^{6} -2.26126 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.92862 q^{10} -1.00000 q^{11} +1.00000 q^{12} -0.873201 q^{13} -2.26126 q^{14} -1.92862 q^{15} +1.00000 q^{16} -3.06469 q^{17} +1.00000 q^{18} +7.77911 q^{19} -1.92862 q^{20} -2.26126 q^{21} -1.00000 q^{22} +4.91749 q^{23} +1.00000 q^{24} -1.28044 q^{25} -0.873201 q^{26} +1.00000 q^{27} -2.26126 q^{28} +4.42004 q^{29} -1.92862 q^{30} -0.988877 q^{31} +1.00000 q^{32} -1.00000 q^{33} -3.06469 q^{34} +4.36110 q^{35} +1.00000 q^{36} +7.42838 q^{37} +7.77911 q^{38} -0.873201 q^{39} -1.92862 q^{40} +0.410765 q^{41} -2.26126 q^{42} +4.69011 q^{43} -1.00000 q^{44} -1.92862 q^{45} +4.91749 q^{46} +2.36986 q^{47} +1.00000 q^{48} -1.88673 q^{49} -1.28044 q^{50} -3.06469 q^{51} -0.873201 q^{52} -2.81425 q^{53} +1.00000 q^{54} +1.92862 q^{55} -2.26126 q^{56} +7.77911 q^{57} +4.42004 q^{58} +1.62420 q^{59} -1.92862 q^{60} +1.00000 q^{61} -0.988877 q^{62} -2.26126 q^{63} +1.00000 q^{64} +1.68407 q^{65} -1.00000 q^{66} +10.1876 q^{67} -3.06469 q^{68} +4.91749 q^{69} +4.36110 q^{70} +8.05201 q^{71} +1.00000 q^{72} +1.69404 q^{73} +7.42838 q^{74} -1.28044 q^{75} +7.77911 q^{76} +2.26126 q^{77} -0.873201 q^{78} +5.89360 q^{79} -1.92862 q^{80} +1.00000 q^{81} +0.410765 q^{82} +13.8812 q^{83} -2.26126 q^{84} +5.91062 q^{85} +4.69011 q^{86} +4.42004 q^{87} -1.00000 q^{88} -17.5559 q^{89} -1.92862 q^{90} +1.97453 q^{91} +4.91749 q^{92} -0.988877 q^{93} +2.36986 q^{94} -15.0029 q^{95} +1.00000 q^{96} +16.4295 q^{97} -1.88673 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} + 13 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 5 * q^5 + 8 * q^6 + 13 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} + 13 q^{7} + 8 q^{8} + 8 q^{9} + 5 q^{10} - 8 q^{11} + 8 q^{12} + 10 q^{13} + 13 q^{14} + 5 q^{15} + 8 q^{16} + 4 q^{17} + 8 q^{18} + 11 q^{19} + 5 q^{20} + 13 q^{21} - 8 q^{22} + 2 q^{23} + 8 q^{24} + 23 q^{25} + 10 q^{26} + 8 q^{27} + 13 q^{28} + 10 q^{29} + 5 q^{30} + 9 q^{31} + 8 q^{32} - 8 q^{33} + 4 q^{34} - 3 q^{35} + 8 q^{36} + 9 q^{37} + 11 q^{38} + 10 q^{39} + 5 q^{40} + 3 q^{41} + 13 q^{42} + 16 q^{43} - 8 q^{44} + 5 q^{45} + 2 q^{46} - 16 q^{47} + 8 q^{48} + 17 q^{49} + 23 q^{50} + 4 q^{51} + 10 q^{52} + 7 q^{53} + 8 q^{54} - 5 q^{55} + 13 q^{56} + 11 q^{57} + 10 q^{58} - 14 q^{59} + 5 q^{60} + 8 q^{61} + 9 q^{62} + 13 q^{63} + 8 q^{64} + 22 q^{65} - 8 q^{66} + 8 q^{67} + 4 q^{68} + 2 q^{69} - 3 q^{70} + 11 q^{71} + 8 q^{72} + 14 q^{73} + 9 q^{74} + 23 q^{75} + 11 q^{76} - 13 q^{77} + 10 q^{78} + 22 q^{79} + 5 q^{80} + 8 q^{81} + 3 q^{82} - 16 q^{83} + 13 q^{84} + 3 q^{85} + 16 q^{86} + 10 q^{87} - 8 q^{88} + q^{89} + 5 q^{90} + 15 q^{91} + 2 q^{92} + 9 q^{93} - 16 q^{94} - 9 q^{95} + 8 q^{96} + 24 q^{97} + 17 q^{98} - 8 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 5 * q^5 + 8 * q^6 + 13 * q^7 + 8 * q^8 + 8 * q^9 + 5 * q^10 - 8 * q^11 + 8 * q^12 + 10 * q^13 + 13 * q^14 + 5 * q^15 + 8 * q^16 + 4 * q^17 + 8 * q^18 + 11 * q^19 + 5 * q^20 + 13 * q^21 - 8 * q^22 + 2 * q^23 + 8 * q^24 + 23 * q^25 + 10 * q^26 + 8 * q^27 + 13 * q^28 + 10 * q^29 + 5 * q^30 + 9 * q^31 + 8 * q^32 - 8 * q^33 + 4 * q^34 - 3 * q^35 + 8 * q^36 + 9 * q^37 + 11 * q^38 + 10 * q^39 + 5 * q^40 + 3 * q^41 + 13 * q^42 + 16 * q^43 - 8 * q^44 + 5 * q^45 + 2 * q^46 - 16 * q^47 + 8 * q^48 + 17 * q^49 + 23 * q^50 + 4 * q^51 + 10 * q^52 + 7 * q^53 + 8 * q^54 - 5 * q^55 + 13 * q^56 + 11 * q^57 + 10 * q^58 - 14 * q^59 + 5 * q^60 + 8 * q^61 + 9 * q^62 + 13 * q^63 + 8 * q^64 + 22 * q^65 - 8 * q^66 + 8 * q^67 + 4 * q^68 + 2 * q^69 - 3 * q^70 + 11 * q^71 + 8 * q^72 + 14 * q^73 + 9 * q^74 + 23 * q^75 + 11 * q^76 - 13 * q^77 + 10 * q^78 + 22 * q^79 + 5 * q^80 + 8 * q^81 + 3 * q^82 - 16 * q^83 + 13 * q^84 + 3 * q^85 + 16 * q^86 + 10 * q^87 - 8 * q^88 + q^89 + 5 * q^90 + 15 * q^91 + 2 * q^92 + 9 * q^93 - 16 * q^94 - 9 * q^95 + 8 * q^96 + 24 * q^97 + 17 * q^98 - 8 * q^99

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