LMFDB - Embedded newform 4010.2.a.j.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4010,2,Mod(1,4010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4010, 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("4010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4010 = 2 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4010.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:	$32.0200112105$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 16 x^{10} + 30 x^{9} + 93 x^{8} - 162 x^{7} - 238 x^{6} + 391 x^{5} + 240 x^{4} + \cdots - 8 $ x^12 - 2x^11 - 16x^10 + 30x^9 + 93x^8 - 162x^7 - 238x^6 + 391x^5 + 240x^4 - 408x^3 - 42x^2 + 120*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.660223$ of defining polynomial
Character	$\chi$	$=$	4010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.660223 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.660223 q^{6} +0.752610 q^{7} +1.00000 q^{8} -2.56411 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.660223 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.660223 q^{6} +0.752610 q^{7} +1.00000 q^{8} -2.56411 q^{9} -1.00000 q^{10} -1.80356 q^{11} +0.660223 q^{12} -2.15624 q^{13} +0.752610 q^{14} -0.660223 q^{15} +1.00000 q^{16} -6.44743 q^{17} -2.56411 q^{18} +6.40123 q^{19} -1.00000 q^{20} +0.496891 q^{21} -1.80356 q^{22} +0.238594 q^{23} +0.660223 q^{24} +1.00000 q^{25} -2.15624 q^{26} -3.67355 q^{27} +0.752610 q^{28} +8.14441 q^{29} -0.660223 q^{30} +0.727741 q^{31} +1.00000 q^{32} -1.19075 q^{33} -6.44743 q^{34} -0.752610 q^{35} -2.56411 q^{36} -10.5005 q^{37} +6.40123 q^{38} -1.42360 q^{39} -1.00000 q^{40} +4.78911 q^{41} +0.496891 q^{42} -2.24046 q^{43} -1.80356 q^{44} +2.56411 q^{45} +0.238594 q^{46} -5.98009 q^{47} +0.660223 q^{48} -6.43358 q^{49} +1.00000 q^{50} -4.25675 q^{51} -2.15624 q^{52} -3.19964 q^{53} -3.67355 q^{54} +1.80356 q^{55} +0.752610 q^{56} +4.22624 q^{57} +8.14441 q^{58} -7.95642 q^{59} -0.660223 q^{60} -7.38653 q^{61} +0.727741 q^{62} -1.92977 q^{63} +1.00000 q^{64} +2.15624 q^{65} -1.19075 q^{66} -15.7664 q^{67} -6.44743 q^{68} +0.157525 q^{69} -0.752610 q^{70} -8.98492 q^{71} -2.56411 q^{72} -4.89859 q^{73} -10.5005 q^{74} +0.660223 q^{75} +6.40123 q^{76} -1.35738 q^{77} -1.42360 q^{78} -1.31309 q^{79} -1.00000 q^{80} +5.26695 q^{81} +4.78911 q^{82} -16.4875 q^{83} +0.496891 q^{84} +6.44743 q^{85} -2.24046 q^{86} +5.37713 q^{87} -1.80356 q^{88} +9.45590 q^{89} +2.56411 q^{90} -1.62281 q^{91} +0.238594 q^{92} +0.480472 q^{93} -5.98009 q^{94} -6.40123 q^{95} +0.660223 q^{96} +6.49591 q^{97} -6.43358 q^{98} +4.62452 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} - 2 q^{3} + 12 q^{4} - 12 q^{5} - 2 q^{6} - 9 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 - 2 * q^3 + 12 * q^4 - 12 * q^5 - 2 * q^6 - 9 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} - 2 q^{3} + 12 q^{4} - 12 q^{5} - 2 q^{6} - 9 q^{7} + 12 q^{8} - 12 q^{10} + q^{11} - 2 q^{12} - 6 q^{13} - 9 q^{14} + 2 q^{15} + 12 q^{16} - 11 q^{17} - 13 q^{19} - 12 q^{20} - 14 q^{21} + q^{22} - 21 q^{23} - 2 q^{24} + 12 q^{25} - 6 q^{26} - 2 q^{27} - 9 q^{28} - 10 q^{29} + 2 q^{30} - 11 q^{31} + 12 q^{32} - 22 q^{33} - 11 q^{34} + 9 q^{35} - 29 q^{37} - 13 q^{38} - 2 q^{39} - 12 q^{40} - q^{41} - 14 q^{42} - 23 q^{43} + q^{44} - 21 q^{46} - 17 q^{47} - 2 q^{48} - 3 q^{49} + 12 q^{50} - 19 q^{51} - 6 q^{52} - 47 q^{53} - 2 q^{54} - q^{55} - 9 q^{56} - 11 q^{57} - 10 q^{58} + 14 q^{59} + 2 q^{60} - 22 q^{61} - 11 q^{62} - 28 q^{63} + 12 q^{64} + 6 q^{65} - 22 q^{66} - 28 q^{67} - 11 q^{68} - q^{69} + 9 q^{70} - 18 q^{71} - 2 q^{73} - 29 q^{74} - 2 q^{75} - 13 q^{76} - 11 q^{77} - 2 q^{78} - 39 q^{79} - 12 q^{80} - 44 q^{81} - q^{82} - 5 q^{83} - 14 q^{84} + 11 q^{85} - 23 q^{86} - 6 q^{87} + q^{88} - 8 q^{89} - 12 q^{91} - 21 q^{92} - 30 q^{93} - 17 q^{94} + 13 q^{95} - 2 q^{96} - 32 q^{97} - 3 q^{98} - 13 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 - 2 * q^3 + 12 * q^4 - 12 * q^5 - 2 * q^6 - 9 * q^7 + 12 * q^8 - 12 * q^10 + q^11 - 2 * q^12 - 6 * q^13 - 9 * q^14 + 2 * q^15 + 12 * q^16 - 11 * q^17 - 13 * q^19 - 12 * q^20 - 14 * q^21 + q^22 - 21 * q^23 - 2 * q^24 + 12 * q^25 - 6 * q^26 - 2 * q^27 - 9 * q^28 - 10 * q^29 + 2 * q^30 - 11 * q^31 + 12 * q^32 - 22 * q^33 - 11 * q^34 + 9 * q^35 - 29 * q^37 - 13 * q^38 - 2 * q^39 - 12 * q^40 - q^41 - 14 * q^42 - 23 * q^43 + q^44 - 21 * q^46 - 17 * q^47 - 2 * q^48 - 3 * q^49 + 12 * q^50 - 19 * q^51 - 6 * q^52 - 47 * q^53 - 2 * q^54 - q^55 - 9 * q^56 - 11 * q^57 - 10 * q^58 + 14 * q^59 + 2 * q^60 - 22 * q^61 - 11 * q^62 - 28 * q^63 + 12 * q^64 + 6 * q^65 - 22 * q^66 - 28 * q^67 - 11 * q^68 - q^69 + 9 * q^70 - 18 * q^71 - 2 * q^73 - 29 * q^74 - 2 * q^75 - 13 * q^76 - 11 * q^77 - 2 * q^78 - 39 * q^79 - 12 * q^80 - 44 * q^81 - q^82 - 5 * q^83 - 14 * q^84 + 11 * q^85 - 23 * q^86 - 6 * q^87 + q^88 - 8 * q^89 - 12 * q^91 - 21 * q^92 - 30 * q^93 - 17 * q^94 + 13 * q^95 - 2 * q^96 - 32 * q^97 - 3 * q^98 - 13 * 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
$3$	0.660223	0.381180	0.190590	−	0.981670i	$-0.438960\pi$
$3$	0.660223	0.381180	0.190590	+	0.981670i	$0.438960\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.660223	0.269535
$7$	0.752610	0.284460	0.142230	−	0.989834i	$-0.454573\pi$
$7$	0.752610	0.284460	0.142230	+	0.989834i	$0.454573\pi$
$8$	1.00000	0.353553
$9$	−2.56411	−0.854702
$10$	−1.00000	−0.316228
$11$	−1.80356	−0.543794	−0.271897	−	0.962326i	$-0.587651\pi$
$11$	−1.80356	−0.543794	−0.271897	+	0.962326i	$0.587651\pi$
$12$	0.660223	0.190590
$13$	−2.15624	−0.598034	−0.299017	−	0.954248i	$-0.596659\pi$
$13$	−2.15624	−0.598034	−0.299017	+	0.954248i	$0.596659\pi$
$14$	0.752610	0.201143
$15$	−0.660223	−0.170469
$16$	1.00000	0.250000
$17$	−6.44743	−1.56373	−0.781866	−	0.623447i	$-0.785732\pi$
$17$	−6.44743	−1.56373	−0.781866	+	0.623447i	$0.785732\pi$
$18$	−2.56411	−0.604365
$19$	6.40123	1.46854	0.734271	−	0.678856i	$-0.237524\pi$
$19$	6.40123	1.46854	0.734271	+	0.678856i	$0.237524\pi$
$20$	−1.00000	−0.223607
$21$	0.496891	0.108430
$22$	−1.80356	−0.384520
$23$	0.238594	0.0497503	0.0248751	−	0.999691i	$-0.492081\pi$
$23$	0.238594	0.0497503	0.0248751	+	0.999691i	$0.492081\pi$
$24$	0.660223	0.134768
$25$	1.00000	0.200000
$26$	−2.15624	−0.422874
$27$	−3.67355	−0.706975
$28$	0.752610	0.142230
$29$	8.14441	1.51238	0.756190	−	0.654352i	$-0.227059\pi$
$29$	8.14441	1.51238	0.756190	+	0.654352i	$0.227059\pi$
$30$	−0.660223	−0.120540
$31$	0.727741	0.130706	0.0653531	−	0.997862i	$-0.479183\pi$
$31$	0.727741	0.130706	0.0653531	+	0.997862i	$0.479183\pi$
$32$	1.00000	0.176777
$33$	−1.19075	−0.207283
$34$	−6.44743	−1.10573
$35$	−0.752610	−0.127214
$36$	−2.56411	−0.427351
$37$	−10.5005	−1.72626	−0.863132	−	0.504978i	$-0.831501\pi$
$37$	−10.5005	−1.72626	−0.863132	+	0.504978i	$0.831501\pi$
$38$	6.40123	1.03842
$39$	−1.42360	−0.227959
$40$	−1.00000	−0.158114
$41$	4.78911	0.747933	0.373966	−	0.927442i	$-0.377998\pi$
$41$	4.78911	0.747933	0.373966	+	0.927442i	$0.377998\pi$
$42$	0.496891	0.0766719
$43$	−2.24046	−0.341667	−0.170834	−	0.985300i	$-0.554646\pi$
$43$	−2.24046	−0.341667	−0.170834	+	0.985300i	$0.554646\pi$
$44$	−1.80356	−0.271897
$45$	2.56411	0.382234
$46$	0.238594	0.0351788
$47$	−5.98009	−0.872285	−0.436143	−	0.899878i	$-0.643656\pi$
$47$	−5.98009	−0.872285	−0.436143	+	0.899878i	$0.643656\pi$
$48$	0.660223	0.0952950
$49$	−6.43358	−0.919083
$50$	1.00000	0.141421
$51$	−4.25675	−0.596064
$52$	−2.15624	−0.299017
$53$	−3.19964	−0.439504	−0.219752	−	0.975556i	$-0.570525\pi$
$53$	−3.19964	−0.439504	−0.219752	+	0.975556i	$0.570525\pi$
$54$	−3.67355	−0.499907
$55$	1.80356	0.243192
$56$	0.752610	0.100572
$57$	4.22624	0.559779
$58$	8.14441	1.06941
$59$	−7.95642	−1.03584	−0.517919	−	0.855430i	$-0.673293\pi$
$59$	−7.95642	−1.03584	−0.517919	+	0.855430i	$0.673293\pi$
$60$	−0.660223	−0.0852345
$61$	−7.38653	−0.945748	−0.472874	−	0.881130i	$-0.656783\pi$
$61$	−7.38653	−0.945748	−0.472874	+	0.881130i	$0.656783\pi$
$62$	0.727741	0.0924232
$63$	−1.92977	−0.243128
$64$	1.00000	0.125000
$65$	2.15624	0.267449
$66$	−1.19075	−0.146571
$67$	−15.7664	−1.92617	−0.963086	−	0.269194i	$-0.913243\pi$
$67$	−15.7664	−1.92617	−0.963086	+	0.269194i	$0.913243\pi$
$68$	−6.44743	−0.781866
$69$	0.157525	0.0189638
$70$	−0.752610	−0.0899541
$71$	−8.98492	−1.06631	−0.533157	−	0.846017i	$-0.678994\pi$
$71$	−8.98492	−1.06631	−0.533157	+	0.846017i	$0.678994\pi$
$72$	−2.56411	−0.302183
$73$	−4.89859	−0.573336	−0.286668	−	0.958030i	$-0.592548\pi$
$73$	−4.89859	−0.573336	−0.286668	+	0.958030i	$0.592548\pi$
$74$	−10.5005	−1.22065
$75$	0.660223	0.0762360
$76$	6.40123	0.734271
$77$	−1.35738	−0.154687
$78$	−1.42360	−0.161191
$79$	−1.31309	−0.147734	−0.0738670	−	0.997268i	$-0.523534\pi$
$79$	−1.31309	−0.147734	−0.0738670	+	0.997268i	$0.523534\pi$
$80$	−1.00000	−0.111803
$81$	5.26695	0.585217
$82$	4.78911	0.528868
$83$	−16.4875	−1.80973	−0.904867	−	0.425694i	$-0.860030\pi$
$83$	−16.4875	−1.80973	−0.904867	+	0.425694i	$0.860030\pi$
$84$	0.496891	0.0542152
$85$	6.44743	0.699322
$86$	−2.24046	−0.241595
$87$	5.37713	0.576489
$88$	−1.80356	−0.192260
$89$	9.45590	1.00232	0.501162	−	0.865354i	$-0.332906\pi$
$89$	9.45590	1.00232	0.501162	+	0.865354i	$0.332906\pi$
$90$	2.56411	0.270280
$91$	−1.62281	−0.170117
$92$	0.238594	0.0248751
$93$	0.480472	0.0498226
$94$	−5.98009	−0.616799
$95$	−6.40123	−0.656752
$96$	0.660223	0.0673838
$97$	6.49591	0.659560	0.329780	−	0.944058i	$-0.393025\pi$
$97$	6.49591	0.659560	0.329780	+	0.944058i	$0.393025\pi$
$98$	−6.43358	−0.649890
$99$	4.62452	0.464781
$100$	1.00000	0.100000
$101$	7.64420	0.760626	0.380313	−	0.924858i	$-0.375816\pi$
$101$	7.64420	0.760626	0.380313	+	0.924858i	$0.375816\pi$
$102$	−4.25675	−0.421481
$103$	−12.5858	−1.24012	−0.620059	−	0.784555i	$-0.712891\pi$
$103$	−12.5858	−1.24012	−0.620059	+	0.784555i	$0.712891\pi$
$104$	−2.15624	−0.211437
$105$	−0.496891	−0.0484916
$106$	−3.19964	−0.310777
$107$	5.32288	0.514582	0.257291	−	0.966334i	$-0.417170\pi$
$107$	5.32288	0.514582	0.257291	+	0.966334i	$0.417170\pi$
$108$	−3.67355	−0.353488
$109$	−1.27565	−0.122185	−0.0610927	−	0.998132i	$-0.519459\pi$
$109$	−1.27565	−0.122185	−0.0610927	+	0.998132i	$0.519459\pi$
$110$	1.80356	0.171963
$111$	−6.93265	−0.658018
$112$	0.752610	0.0711150
$113$	−13.8771	−1.30545	−0.652724	−	0.757596i	$-0.726374\pi$
$113$	−13.8771	−1.30545	−0.652724	+	0.757596i	$0.726374\pi$
$114$	4.22624	0.395824
$115$	−0.238594	−0.0222490
$116$	8.14441	0.756190
$117$	5.52883	0.511141
$118$	−7.95642	−0.732447
$119$	−4.85240	−0.444819
$120$	−0.660223	−0.0602699
$121$	−7.74717	−0.704288
$122$	−7.38653	−0.668745
$123$	3.16188	0.285097
$124$	0.727741	0.0653531
$125$	−1.00000	−0.0894427
$126$	−1.92977	−0.171918
$127$	−1.14246	−0.101377	−0.0506883	−	0.998715i	$-0.516142\pi$
$127$	−1.14246	−0.101377	−0.0506883	+	0.998715i	$0.516142\pi$
$128$	1.00000	0.0883883
$129$	−1.47920	−0.130237
$130$	2.15624	0.189115
$131$	9.70605	0.848021	0.424011	−	0.905657i	$-0.360622\pi$
$131$	9.70605	0.848021	0.424011	+	0.905657i	$0.360622\pi$
$132$	−1.19075	−0.103642
$133$	4.81763	0.417741
$134$	−15.7664	−1.36201
$135$	3.67355	0.316169
$136$	−6.44743	−0.552863
$137$	23.0076	1.96567	0.982837	−	0.184475i	$-0.0590585\pi$
$137$	23.0076	1.96567	0.982837	+	0.184475i	$0.0590585\pi$
$138$	0.157525	0.0134094
$139$	−1.70038	−0.144224	−0.0721122	−	0.997397i	$-0.522974\pi$
$139$	−1.70038	−0.144224	−0.0721122	+	0.997397i	$0.522974\pi$
$140$	−0.752610	−0.0636072
$141$	−3.94819	−0.332498
$142$	−8.98492	−0.753997
$143$	3.88891	0.325207
$144$	−2.56411	−0.213675
$145$	−8.14441	−0.676357
$146$	−4.89859	−0.405410
$147$	−4.24760	−0.350336
$148$	−10.5005	−0.863132
$149$	−1.77860	−0.145709	−0.0728544	−	0.997343i	$-0.523211\pi$
$149$	−1.77860	−0.145709	−0.0728544	+	0.997343i	$0.523211\pi$
$150$	0.660223	0.0539070
$151$	12.9178	1.05124	0.525618	−	0.850721i	$-0.323834\pi$
$151$	12.9178	1.05124	0.525618	+	0.850721i	$0.323834\pi$
$152$	6.40123	0.519208
$153$	16.5319	1.33652
$154$	−1.35738	−0.109381
$155$	−0.727741	−0.0584536
$156$	−1.42360	−0.113979
$157$	13.5958	1.08506	0.542531	−	0.840036i	$-0.317466\pi$
$157$	13.5958	1.08506	0.542531	+	0.840036i	$0.317466\pi$
$158$	−1.31309	−0.104464
$159$	−2.11248	−0.167530
$160$	−1.00000	−0.0790569
$161$	0.179568	0.0141520
$162$	5.26695	0.413811
$163$	10.7103	0.838899	0.419449	−	0.907779i	$-0.362223\pi$
$163$	10.7103	0.838899	0.419449	+	0.907779i	$0.362223\pi$
$164$	4.78911	0.373966
$165$	1.19075	0.0927000
$166$	−16.4875	−1.27968
$167$	−3.79004	−0.293282	−0.146641	−	0.989190i	$-0.546846\pi$
$167$	−3.79004	−0.293282	−0.146641	+	0.989190i	$0.546846\pi$
$168$	0.496891	0.0383360
$169$	−8.35062	−0.642355
$170$	6.44743	0.494495
$171$	−16.4134	−1.25517
$172$	−2.24046	−0.170834
$173$	18.0799	1.37459	0.687294	−	0.726379i	$-0.258798\pi$
$173$	18.0799	1.37459	0.687294	+	0.726379i	$0.258798\pi$
$174$	5.37713	0.407639
$175$	0.752610	0.0568920
$176$	−1.80356	−0.135948
$177$	−5.25301	−0.394841
$178$	9.45590	0.708750
$179$	20.5143	1.53331	0.766655	−	0.642059i	$-0.221920\pi$
$179$	20.5143	1.53331	0.766655	+	0.642059i	$0.221920\pi$
$180$	2.56411	0.191117
$181$	−8.20378	−0.609782	−0.304891	−	0.952387i	$-0.598620\pi$
$181$	−8.20378	−0.609782	−0.304891	+	0.952387i	$0.598620\pi$
$182$	−1.62281	−0.120291
$183$	−4.87676	−0.360500
$184$	0.238594	0.0175894
$185$	10.5005	0.772009
$186$	0.480472	0.0352299
$187$	11.6283	0.850348
$188$	−5.98009	−0.436143
$189$	−2.76475	−0.201106
$190$	−6.40123	−0.464394
$191$	−5.40522	−0.391108	−0.195554	−	0.980693i	$-0.562650\pi$
$191$	−5.40522	−0.391108	−0.195554	+	0.980693i	$0.562650\pi$
$192$	0.660223	0.0476475
$193$	−16.7157	−1.20322	−0.601610	−	0.798790i	$-0.705474\pi$
$193$	−16.7157	−1.20322	−0.601610	+	0.798790i	$0.705474\pi$
$194$	6.49591	0.466379
$195$	1.42360	0.101946
$196$	−6.43358	−0.459541
$197$	20.9588	1.49325	0.746626	−	0.665244i	$-0.231673\pi$
$197$	20.9588	1.49325	0.746626	+	0.665244i	$0.231673\pi$
$198$	4.62452	0.328650
$199$	1.57105	0.111369	0.0556843	−	0.998448i	$-0.482266\pi$
$199$	1.57105	0.111369	0.0556843	+	0.998448i	$0.482266\pi$
$200$	1.00000	0.0707107
$201$	−10.4093	−0.734219
$202$	7.64420	0.537844
$203$	6.12957	0.430211
$204$	−4.25675	−0.298032
$205$	−4.78911	−0.334486
$206$	−12.5858	−0.876895
$207$	−0.611780	−0.0425217
$208$	−2.15624	−0.149508
$209$	−11.5450	−0.798584
$210$	−0.496891	−0.0342887
$211$	−11.9345	−0.821607	−0.410803	−	0.911724i	$-0.634752\pi$
$211$	−11.9345	−0.821607	−0.410803	+	0.911724i	$0.634752\pi$
$212$	−3.19964	−0.219752
$213$	−5.93205	−0.406458
$214$	5.32288	0.363865
$215$	2.24046	0.152798
$216$	−3.67355	−0.249954
$217$	0.547705	0.0371807
$218$	−1.27565	−0.0863981
$219$	−3.23416	−0.218544
$220$	1.80356	0.121596
$221$	13.9022	0.935165
$222$	−6.93265	−0.465289
$223$	−9.36978	−0.627447	−0.313723	−	0.949514i	$-0.601576\pi$
$223$	−9.36978	−0.627447	−0.313723	+	0.949514i	$0.601576\pi$
$224$	0.752610	0.0502859
$225$	−2.56411	−0.170940
$226$	−13.8771	−0.923091
$227$	−10.0378	−0.666234	−0.333117	−	0.942885i	$-0.608100\pi$
$227$	−10.0378	−0.666234	−0.333117	+	0.942885i	$0.608100\pi$
$228$	4.22624	0.279890
$229$	2.10973	0.139415	0.0697074	−	0.997567i	$-0.477793\pi$
$229$	2.10973	0.139415	0.0697074	+	0.997567i	$0.477793\pi$
$230$	−0.238594	−0.0157324
$231$	−0.896172	−0.0589638
$232$	8.14441	0.534707
$233$	8.24084	0.539875	0.269938	−	0.962878i	$-0.412997\pi$
$233$	8.24084	0.539875	0.269938	+	0.962878i	$0.412997\pi$
$234$	5.52883	0.361431
$235$	5.98009	0.390098
$236$	−7.95642	−0.517919
$237$	−0.866932	−0.0563133
$238$	−4.85240	−0.314535
$239$	1.40125	0.0906392	0.0453196	−	0.998973i	$-0.485569\pi$
$239$	1.40125	0.0906392	0.0453196	+	0.998973i	$0.485569\pi$
$240$	−0.660223	−0.0426172
$241$	−26.6416	−1.71614	−0.858070	−	0.513534i	$-0.828336\pi$
$241$	−26.6416	−1.71614	−0.858070	+	0.513534i	$0.828336\pi$
$242$	−7.74717	−0.498007
$243$	14.4980	0.930048
$244$	−7.38653	−0.472874
$245$	6.43358	0.411026
$246$	3.16188	0.201594
$247$	−13.8026	−0.878238
$248$	0.727741	0.0462116
$249$	−10.8854	−0.689835
$250$	−1.00000	−0.0632456
$251$	7.95803	0.502306	0.251153	−	0.967947i	$-0.419190\pi$
$251$	7.95803	0.502306	0.251153	+	0.967947i	$0.419190\pi$
$252$	−1.92977	−0.121564
$253$	−0.430319	−0.0270539
$254$	−1.14246	−0.0716841
$255$	4.25675	0.266568
$256$	1.00000	0.0625000
$257$	21.6887	1.35290	0.676452	−	0.736487i	$-0.263517\pi$
$257$	21.6887	1.35290	0.676452	+	0.736487i	$0.263517\pi$
$258$	−1.47920	−0.0920913
$259$	−7.90275	−0.491053
$260$	2.15624	0.133724
$261$	−20.8831	−1.29263
$262$	9.70605	0.599642
$263$	20.2278	1.24730	0.623649	−	0.781704i	$-0.285649\pi$
$263$	20.2278	1.24730	0.623649	+	0.781704i	$0.285649\pi$
$264$	−1.19075	−0.0732857
$265$	3.19964	0.196552
$266$	4.81763	0.295388
$267$	6.24301	0.382066
$268$	−15.7664	−0.963086
$269$	13.7099	0.835910	0.417955	−	0.908468i	$-0.362747\pi$
$269$	13.7099	0.835910	0.417955	+	0.908468i	$0.362747\pi$
$270$	3.67355	0.223565
$271$	17.5606	1.06673	0.533364	−	0.845886i	$-0.320928\pi$
$271$	17.5606	1.06673	0.533364	+	0.845886i	$0.320928\pi$
$272$	−6.44743	−0.390933
$273$	−1.07142	−0.0648451
$274$	23.0076	1.38994
$275$	−1.80356	−0.108759
$276$	0.157525	0.00948191
$277$	−6.40653	−0.384931	−0.192466	−	0.981304i	$-0.561648\pi$
$277$	−6.40653	−0.384931	−0.192466	+	0.981304i	$0.561648\pi$
$278$	−1.70038	−0.101982
$279$	−1.86600	−0.111715
$280$	−0.752610	−0.0449771
$281$	−2.19176	−0.130749	−0.0653747	−	0.997861i	$-0.520824\pi$
$281$	−2.19176	−0.130749	−0.0653747	+	0.997861i	$0.520824\pi$
$282$	−3.94819	−0.235111
$283$	−7.49892	−0.445765	−0.222882	−	0.974845i	$-0.571547\pi$
$283$	−7.49892	−0.445765	−0.222882	+	0.974845i	$0.571547\pi$
$284$	−8.98492	−0.533157
$285$	−4.22624	−0.250341
$286$	3.88891	0.229956
$287$	3.60433	0.212757
$288$	−2.56411	−0.151091
$289$	24.5694	1.44526
$290$	−8.14441	−0.478256
$291$	4.28875	0.251411
$292$	−4.89859	−0.286668
$293$	−28.0355	−1.63785	−0.818926	−	0.573899i	$-0.805430\pi$
$293$	−28.0355	−1.63785	−0.818926	+	0.573899i	$0.805430\pi$
$294$	−4.24760	−0.247725
$295$	7.95642	0.463240
$296$	−10.5005	−0.610327
$297$	6.62547	0.384449
$298$	−1.77860	−0.103032
$299$	−0.514466	−0.0297524
$300$	0.660223	0.0381180
$301$	−1.68619	−0.0971906
$302$	12.9178	0.743335
$303$	5.04688	0.289936
$304$	6.40123	0.367136
$305$	7.38653	0.422951
$306$	16.5319	0.945065
$307$	11.8771	0.677864	0.338932	−	0.940811i	$-0.389934\pi$
$307$	11.8771	0.677864	0.338932	+	0.940811i	$0.389934\pi$
$308$	−1.35738	−0.0773437
$309$	−8.30945	−0.472708
$310$	−0.727741	−0.0413329
$311$	23.3821	1.32588	0.662939	−	0.748673i	$-0.269309\pi$
$311$	23.3821	1.32588	0.662939	+	0.748673i	$0.269309\pi$
$312$	−1.42360	−0.0805956
$313$	−14.2402	−0.804906	−0.402453	−	0.915441i	$-0.631842\pi$
$313$	−14.2402	−0.804906	−0.402453	+	0.915441i	$0.631842\pi$
$314$	13.5958	0.767255
$315$	1.92977	0.108730
$316$	−1.31309	−0.0738670
$317$	−5.86145	−0.329212	−0.164606	−	0.986359i	$-0.552635\pi$
$317$	−5.86145	−0.329212	−0.164606	+	0.986359i	$0.552635\pi$
$318$	−2.11248	−0.118462
$319$	−14.6889	−0.822423
$320$	−1.00000	−0.0559017
$321$	3.51429	0.196149
$322$	0.179568	0.0100069
$323$	−41.2715	−2.29641
$324$	5.26695	0.292608
$325$	−2.15624	−0.119607
$326$	10.7103	0.593191
$327$	−0.842216	−0.0465746
$328$	4.78911	0.264434
$329$	−4.50067	−0.248130
$330$	1.19075	0.0655488
$331$	−31.2399	−1.71710	−0.858550	−	0.512730i	$-0.828634\pi$
$331$	−31.2399	−1.71710	−0.858550	+	0.512730i	$0.828634\pi$
$332$	−16.4875	−0.904867
$333$	26.9243	1.47544
$334$	−3.79004	−0.207382
$335$	15.7664	0.861410
$336$	0.496891	0.0271076
$337$	11.5269	0.627909	0.313955	−	0.949438i	$-0.398346\pi$
$337$	11.5269	0.627909	0.313955	+	0.949438i	$0.398346\pi$
$338$	−8.35062	−0.454214
$339$	−9.16198	−0.497611
$340$	6.44743	0.349661
$341$	−1.31252	−0.0710772
$342$	−16.4134	−0.887536
$343$	−10.1102	−0.545902
$344$	−2.24046	−0.120798
$345$	−0.157525	−0.00848088
$346$	18.0799	0.971981
$347$	−24.0345	−1.29024	−0.645121	−	0.764081i	$-0.723193\pi$
$347$	−24.0345	−1.29024	−0.645121	+	0.764081i	$0.723193\pi$
$348$	5.37713	0.288245
$349$	−31.2170	−1.67101	−0.835505	−	0.549483i	$-0.814825\pi$
$349$	−31.2170	−1.67101	−0.835505	+	0.549483i	$0.814825\pi$
$350$	0.752610	0.0402287
$351$	7.92107	0.422795
$352$	−1.80356	−0.0961301
$353$	27.7468	1.47682	0.738408	−	0.674355i	$-0.235578\pi$
$353$	27.7468	1.47682	0.738408	+	0.674355i	$0.235578\pi$
$354$	−5.25301	−0.279194
$355$	8.98492	0.476870
$356$	9.45590	0.501162
$357$	−3.20367	−0.169556
$358$	20.5143	1.08421
$359$	8.54510	0.450993	0.225497	−	0.974244i	$-0.427600\pi$
$359$	8.54510	0.450993	0.225497	+	0.974244i	$0.427600\pi$
$360$	2.56411	0.135140
$361$	21.9757	1.15662
$362$	−8.20378	−0.431181
$363$	−5.11486	−0.268461
$364$	−1.62281	−0.0850583
$365$	4.89859	0.256404
$366$	−4.87676	−0.254912
$367$	15.5378	0.811067	0.405534	−	0.914080i	$-0.367086\pi$
$367$	15.5378	0.811067	0.405534	+	0.914080i	$0.367086\pi$
$368$	0.238594	0.0124376
$369$	−12.2798	−0.639259
$370$	10.5005	0.545893
$371$	−2.40808	−0.125021
$372$	0.480472	0.0249113
$373$	−29.3423	−1.51929	−0.759643	−	0.650340i	$-0.774626\pi$
$373$	−29.3423	−1.51929	−0.759643	+	0.650340i	$0.774626\pi$
$374$	11.6283	0.601287
$375$	−0.660223	−0.0340938
$376$	−5.98009	−0.308399
$377$	−17.5613	−0.904454
$378$	−2.76475	−0.142204
$379$	3.29449	0.169227	0.0846133	−	0.996414i	$-0.473034\pi$
$379$	3.29449	0.169227	0.0846133	+	0.996414i	$0.473034\pi$
$380$	−6.40123	−0.328376
$381$	−0.754277	−0.0386428
$382$	−5.40522	−0.276555
$383$	−6.97352	−0.356330	−0.178165	−	0.984001i	$-0.557016\pi$
$383$	−6.97352	−0.356330	−0.178165	+	0.984001i	$0.557016\pi$
$384$	0.660223	0.0336919
$385$	1.35738	0.0691784
$386$	−16.7157	−0.850805
$387$	5.74478	0.292023
$388$	6.49591	0.329780
$389$	12.4424	0.630853	0.315426	−	0.948950i	$-0.397852\pi$
$389$	12.4424	0.630853	0.315426	+	0.948950i	$0.397852\pi$
$390$	1.42360	0.0720869
$391$	−1.53832	−0.0777961
$392$	−6.43358	−0.324945
$393$	6.40816	0.323249
$394$	20.9588	1.05589
$395$	1.31309	0.0660687
$396$	4.62452	0.232391
$397$	20.5807	1.03291	0.516457	−	0.856313i	$-0.327251\pi$
$397$	20.5807	1.03291	0.516457	+	0.856313i	$0.327251\pi$
$398$	1.57105	0.0787496
$399$	3.18071	0.159235
$400$	1.00000	0.0500000
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	−10.4093	−0.519171
$403$	−1.56919	−0.0781667
$404$	7.64420	0.380313
$405$	−5.26695	−0.261717
$406$	6.12957	0.304205
$407$	18.9382	0.938732
$408$	−4.25675	−0.210740
$409$	22.4858	1.11185	0.555925	−	0.831232i	$-0.312364\pi$
$409$	22.4858	1.11185	0.555925	+	0.831232i	$0.312364\pi$
$410$	−4.78911	−0.236517
$411$	15.1902	0.749276
$412$	−12.5858	−0.620059
$413$	−5.98808	−0.294654
$414$	−0.611780	−0.0300673
$415$	16.4875	0.809338
$416$	−2.15624	−0.105718
$417$	−1.12263	−0.0549755
$418$	−11.5450	−0.564684
$419$	25.3845	1.24012	0.620058	−	0.784556i	$-0.287109\pi$
$419$	25.3845	1.24012	0.620058	+	0.784556i	$0.287109\pi$
$420$	−0.496891	−0.0242458
$421$	9.03773	0.440472	0.220236	−	0.975447i	$-0.429317\pi$
$421$	9.03773	0.440472	0.220236	+	0.975447i	$0.429317\pi$
$422$	−11.9345	−0.580964
$423$	15.3336	0.745544
$424$	−3.19964	−0.155388
$425$	−6.44743	−0.312746
$426$	−5.93205	−0.287409
$427$	−5.55917	−0.269027
$428$	5.32288	0.257291
$429$	2.56755	0.123962
$430$	2.24046	0.108045
$431$	−27.8264	−1.34035	−0.670174	−	0.742204i	$-0.733781\pi$
$431$	−27.8264	−1.34035	−0.670174	+	0.742204i	$0.733781\pi$
$432$	−3.67355	−0.176744
$433$	−1.38610	−0.0666119	−0.0333059	−	0.999445i	$-0.510604\pi$
$433$	−1.38610	−0.0666119	−0.0333059	+	0.999445i	$0.510604\pi$
$434$	0.547705	0.0262907
$435$	−5.37713	−0.257814
$436$	−1.27565	−0.0610927
$437$	1.52729	0.0730604
$438$	−3.23416	−0.154534
$439$	−2.44065	−0.116486	−0.0582430	−	0.998302i	$-0.518550\pi$
$439$	−2.44065	−0.116486	−0.0582430	+	0.998302i	$0.518550\pi$
$440$	1.80356	0.0859813
$441$	16.4964	0.785541
$442$	13.9022	0.661261
$443$	2.34524	0.111426	0.0557128	−	0.998447i	$-0.482257\pi$
$443$	2.34524	0.111426	0.0557128	+	0.998447i	$0.482257\pi$
$444$	−6.93265	−0.329009
$445$	−9.45590	−0.448253
$446$	−9.36978	−0.443672
$447$	−1.17428	−0.0555413
$448$	0.752610	0.0355575
$449$	−27.8979	−1.31658	−0.658292	−	0.752763i	$-0.728721\pi$
$449$	−27.8979	−1.31658	−0.658292	+	0.752763i	$0.728721\pi$
$450$	−2.56411	−0.120873
$451$	−8.63744	−0.406721
$452$	−13.8771	−0.652724
$453$	8.52863	0.400710
$454$	−10.0378	−0.471099
$455$	1.62281	0.0760785
$456$	4.22624	0.197912
$457$	8.35174	0.390678	0.195339	−	0.980736i	$-0.437419\pi$
$457$	8.35174	0.390678	0.195339	+	0.980736i	$0.437419\pi$
$458$	2.10973	0.0985812
$459$	23.6850	1.10552
$460$	−0.238594	−0.0111245
$461$	−11.8433	−0.551596	−0.275798	−	0.961216i	$-0.588942\pi$
$461$	−11.8433	−0.551596	−0.275798	+	0.961216i	$0.588942\pi$
$462$	−0.896172	−0.0416937
$463$	38.4470	1.78678	0.893392	−	0.449278i	$-0.148319\pi$
$463$	38.4470	1.78678	0.893392	+	0.449278i	$0.148319\pi$
$464$	8.14441	0.378095
$465$	−0.480472	−0.0222813
$466$	8.24084	0.381750
$467$	−7.55036	−0.349389	−0.174695	−	0.984623i	$-0.555894\pi$
$467$	−7.55036	−0.349389	−0.174695	+	0.984623i	$0.555894\pi$
$468$	5.52883	0.255570
$469$	−11.8659	−0.547919
$470$	5.98009	0.275841
$471$	8.97626	0.413604
$472$	−7.95642	−0.366224
$473$	4.04081	0.185796
$474$	−0.866932	−0.0398195
$475$	6.40123	0.293709
$476$	−4.85240	−0.222409
$477$	8.20421	0.375645
$478$	1.40125	0.0640916
$479$	−12.6211	−0.576671	−0.288336	−	0.957529i	$-0.593102\pi$
$479$	−12.6211	−0.576671	−0.288336	+	0.957529i	$0.593102\pi$
$480$	−0.660223	−0.0301349
$481$	22.6415	1.03236
$482$	−26.6416	−1.21349
$483$	0.118555	0.00539445
$484$	−7.74717	−0.352144
$485$	−6.49591	−0.294964
$486$	14.4980	0.657644
$487$	30.0375	1.36113	0.680565	−	0.732688i	$-0.261734\pi$
$487$	30.0375	1.36113	0.680565	+	0.732688i	$0.261734\pi$
$488$	−7.38653	−0.334372
$489$	7.07122	0.319772
$490$	6.43358	0.290639
$491$	1.72676	0.0779277	0.0389639	−	0.999241i	$-0.487594\pi$
$491$	1.72676	0.0779277	0.0389639	+	0.999241i	$0.487594\pi$
$492$	3.16188	0.142549
$493$	−52.5106	−2.36496
$494$	−13.8026	−0.621008
$495$	−4.62452	−0.207857
$496$	0.727741	0.0326765
$497$	−6.76214	−0.303323
$498$	−10.8854	−0.487787
$499$	21.0407	0.941909	0.470955	−	0.882157i	$-0.343910\pi$
$499$	21.0407	0.941909	0.470955	+	0.882157i	$0.343910\pi$
$500$	−1.00000	−0.0447214
$501$	−2.50227	−0.111793
$502$	7.95803	0.355184
$503$	−42.0950	−1.87692	−0.938462	−	0.345383i	$-0.887749\pi$
$503$	−42.0950	−1.87692	−0.938462	+	0.345383i	$0.887749\pi$
$504$	−1.92977	−0.0859588
$505$	−7.64420	−0.340162
$506$	−0.430319	−0.0191300
$507$	−5.51328	−0.244853
$508$	−1.14246	−0.0506883
$509$	−2.44923	−0.108560	−0.0542801	−	0.998526i	$-0.517286\pi$
$509$	−2.44923	−0.108560	−0.0542801	+	0.998526i	$0.517286\pi$
$510$	4.25675	0.188492
$511$	−3.68673	−0.163091
$512$	1.00000	0.0441942
$513$	−23.5153	−1.03822
$514$	21.6887	0.956648
$515$	12.5858	0.554597
$516$	−1.47920	−0.0651184
$517$	10.7854	0.474343
$518$	−7.90275	−0.347227
$519$	11.9368	0.523966
$520$	2.15624	0.0945575
$521$	29.3417	1.28548	0.642741	−	0.766083i	$-0.277797\pi$
$521$	29.3417	1.28548	0.642741	+	0.766083i	$0.277797\pi$
$522$	−20.8831	−0.914030
$523$	13.8014	0.603492	0.301746	−	0.953388i	$-0.402431\pi$
$523$	13.8014	0.603492	0.301746	+	0.953388i	$0.402431\pi$
$524$	9.70605	0.424011
$525$	0.496891	0.0216861
$526$	20.2278	0.881973
$527$	−4.69206	−0.204389
$528$	−1.19075	−0.0518208
$529$	−22.9431	−0.997525
$530$	3.19964	0.138983
$531$	20.4011	0.885332
$532$	4.81763	0.208871
$533$	−10.3265	−0.447289
$534$	6.24301	0.270161
$535$	−5.32288	−0.230128
$536$	−15.7664	−0.681005
$537$	13.5440	0.584468
$538$	13.7099	0.591078
$539$	11.6033	0.499791
$540$	3.67355	0.158085
$541$	25.8442	1.11113	0.555564	−	0.831474i	$-0.312502\pi$
$541$	25.8442	1.11113	0.555564	+	0.831474i	$0.312502\pi$
$542$	17.5606	0.754291
$543$	−5.41633	−0.232437
$544$	−6.44743	−0.276431
$545$	1.27565	0.0546430
$546$	−1.07142	−0.0458524
$547$	−35.6914	−1.52605	−0.763027	−	0.646366i	$-0.776288\pi$
$547$	−35.6914	−1.52605	−0.763027	+	0.646366i	$0.776288\pi$
$548$	23.0076	0.982837
$549$	18.9398	0.808332
$550$	−1.80356	−0.0769041
$551$	52.1343	2.22099
$552$	0.157525	0.00670472
$553$	−0.988244	−0.0420244
$554$	−6.40653	−0.272188
$555$	6.93265	0.294275
$556$	−1.70038	−0.0721122
$557$	−9.32290	−0.395024	−0.197512	−	0.980300i	$-0.563286\pi$
$557$	−9.32290	−0.395024	−0.197512	+	0.980300i	$0.563286\pi$
$558$	−1.86600	−0.0789943
$559$	4.83098	0.204329
$560$	−0.752610	−0.0318036
$561$	7.67730	0.324136
$562$	−2.19176	−0.0924537
$563$	23.0396	0.971003	0.485501	−	0.874236i	$-0.338637\pi$
$563$	23.0396	0.971003	0.485501	+	0.874236i	$0.338637\pi$
$564$	−3.94819	−0.166249
$565$	13.8771	0.583814
$566$	−7.49892	−0.315203
$567$	3.96396	0.166471
$568$	−8.98492	−0.376999
$569$	−17.6166	−0.738526	−0.369263	−	0.929325i	$-0.620390\pi$
$569$	−17.6166	−0.738526	−0.369263	+	0.929325i	$0.620390\pi$
$570$	−4.22624	−0.177018
$571$	5.22973	0.218858	0.109429	−	0.993995i	$-0.465098\pi$
$571$	5.22973	0.218858	0.109429	+	0.993995i	$0.465098\pi$
$572$	3.88891	0.162604
$573$	−3.56865	−0.149083
$574$	3.60433	0.150442
$575$	0.238594	0.00995006
$576$	−2.56411	−0.106838
$577$	13.4073	0.558153	0.279077	−	0.960269i	$-0.409972\pi$
$577$	13.4073	0.558153	0.279077	+	0.960269i	$0.409972\pi$
$578$	24.5694	1.02195
$579$	−11.0361	−0.458644
$580$	−8.14441	−0.338178
$581$	−12.4086	−0.514797
$582$	4.28875	0.177775
$583$	5.77074	0.239000
$584$	−4.89859	−0.202705
$585$	−5.52883	−0.228589
$586$	−28.0355	−1.15814
$587$	−22.5554	−0.930961	−0.465480	−	0.885058i	$-0.654118\pi$
$587$	−22.5554	−0.930961	−0.465480	+	0.885058i	$0.654118\pi$
$588$	−4.24760	−0.175168
$589$	4.65844	0.191948
$590$	7.95642	0.327560
$591$	13.8375	0.569198
$592$	−10.5005	−0.431566
$593$	−8.35193	−0.342973	−0.171486	−	0.985186i	$-0.554857\pi$
$593$	−8.35193	−0.342973	−0.171486	+	0.985186i	$0.554857\pi$
$594$	6.62547	0.271846
$595$	4.85240	0.198929
$596$	−1.77860	−0.0728544
$597$	1.03724	0.0424515
$598$	−0.514466	−0.0210381
$599$	−3.54168	−0.144709	−0.0723545	−	0.997379i	$-0.523051\pi$
$599$	−3.54168	−0.144709	−0.0723545	+	0.997379i	$0.523051\pi$
$600$	0.660223	0.0269535
$601$	−32.7122	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$601$	−32.7122	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$602$	−1.68619	−0.0687241
$603$	40.4267	1.64630
$604$	12.9178	0.525618
$605$	7.74717	0.314967
$606$	5.04688	0.205016
$607$	21.5938	0.876465	0.438232	−	0.898862i	$-0.355605\pi$
$607$	21.5938	0.876465	0.438232	+	0.898862i	$0.355605\pi$
$608$	6.40123	0.259604
$609$	4.04688	0.163988
$610$	7.38653	0.299072
$611$	12.8945	0.521656
$612$	16.5319	0.668262
$613$	−23.1957	−0.936864	−0.468432	−	0.883500i	$-0.655181\pi$
$613$	−23.1957	−0.936864	−0.468432	+	0.883500i	$0.655181\pi$
$614$	11.8771	0.479322
$615$	−3.16188	−0.127499
$616$	−1.35738	−0.0546903
$617$	31.4470	1.26601	0.633005	−	0.774148i	$-0.281821\pi$
$617$	31.4470	1.26601	0.633005	+	0.774148i	$0.281821\pi$
$618$	−8.30945	−0.334255
$619$	16.0213	0.643951	0.321976	−	0.946748i	$-0.395653\pi$
$619$	16.0213	0.643951	0.321976	+	0.946748i	$0.395653\pi$
$620$	−0.727741	−0.0292268
$621$	−0.876487	−0.0351722
$622$	23.3821	0.937538
$623$	7.11661	0.285121
$624$	−1.42360	−0.0569897
$625$	1.00000	0.0400000
$626$	−14.2402	−0.569154
$627$	−7.62228	−0.304405
$628$	13.5958	0.542531
$629$	67.7010	2.69942
$630$	1.92977	0.0768839
$631$	0.682676	0.0271769	0.0135885	−	0.999908i	$-0.495675\pi$
$631$	0.682676	0.0271769	0.0135885	+	0.999908i	$0.495675\pi$
$632$	−1.31309	−0.0522319
$633$	−7.87945	−0.313180
$634$	−5.86145	−0.232788
$635$	1.14246	0.0453370
$636$	−2.11248	−0.0837652
$637$	13.8724	0.549643
$638$	−14.6889	−0.581541
$639$	23.0383	0.911380
$640$	−1.00000	−0.0395285
$641$	−16.4359	−0.649179	−0.324590	−	0.945855i	$-0.605226\pi$
$641$	−16.4359	−0.649179	−0.324590	+	0.945855i	$0.605226\pi$
$642$	3.51429	0.138698
$643$	19.2811	0.760373	0.380187	−	0.924910i	$-0.375860\pi$
$643$	19.2811	0.760373	0.380187	+	0.924910i	$0.375860\pi$
$644$	0.179568	0.00707598
$645$	1.47920	0.0582436
$646$	−41.2715	−1.62381
$647$	−32.1131	−1.26250	−0.631248	−	0.775581i	$-0.717457\pi$
$647$	−32.1131	−1.26250	−0.631248	+	0.775581i	$0.717457\pi$
$648$	5.26695	0.206905
$649$	14.3499	0.563282
$650$	−2.15624	−0.0845748
$651$	0.361608	0.0141725
$652$	10.7103	0.419449
$653$	−27.9495	−1.09375	−0.546874	−	0.837215i	$-0.684182\pi$
$653$	−27.9495	−1.09375	−0.546874	+	0.837215i	$0.684182\pi$
$654$	−0.842216	−0.0329332
$655$	−9.70605	−0.379247
$656$	4.78911	0.186983
$657$	12.5605	0.490032
$658$	−4.50067	−0.175454
$659$	−4.72812	−0.184181	−0.0920907	−	0.995751i	$-0.529355\pi$
$659$	−4.72812	−0.184181	−0.0920907	+	0.995751i	$0.529355\pi$
$660$	1.19075	0.0463500
$661$	−2.57846	−0.100291	−0.0501453	−	0.998742i	$-0.515968\pi$
$661$	−2.57846	−0.100291	−0.0501453	+	0.998742i	$0.515968\pi$
$662$	−31.2399	−1.21417
$663$	9.17857	0.356466
$664$	−16.4875	−0.639838
$665$	−4.81763	−0.186820
$666$	26.9243	1.04329
$667$	1.94321	0.0752413
$668$	−3.79004	−0.146641
$669$	−6.18615	−0.239170
$670$	15.7664	0.609109
$671$	13.3220	0.514292
$672$	0.496891	0.0191680
$673$	−35.7544	−1.37823	−0.689115	−	0.724652i	$-0.742001\pi$
$673$	−35.7544	−1.37823	−0.689115	+	0.724652i	$0.742001\pi$
$674$	11.5269	0.443999
$675$	−3.67355	−0.141395
$676$	−8.35062	−0.321178
$677$	−46.8057	−1.79889	−0.899445	−	0.437034i	$-0.856029\pi$
$677$	−46.8057	−1.79889	−0.899445	+	0.437034i	$0.856029\pi$
$678$	−9.16198	−0.351864
$679$	4.88889	0.187618
$680$	6.44743	0.247248
$681$	−6.62721	−0.253955
$682$	−1.31252	−0.0502592
$683$	−15.4146	−0.589822	−0.294911	−	0.955525i	$-0.595290\pi$
$683$	−15.4146	−0.589822	−0.294911	+	0.955525i	$0.595290\pi$
$684$	−16.4134	−0.627583
$685$	−23.0076	−0.879076
$686$	−10.1102	−0.386011
$687$	1.39289	0.0531422
$688$	−2.24046	−0.0854168
$689$	6.89920	0.262839
$690$	−0.157525	−0.00599689
$691$	9.38580	0.357053	0.178526	−	0.983935i	$-0.442867\pi$
$691$	9.38580	0.357053	0.178526	+	0.983935i	$0.442867\pi$
$692$	18.0799	0.687294
$693$	3.48046	0.132212
$694$	−24.0345	−0.912338
$695$	1.70038	0.0644991
$696$	5.37713	0.203820
$697$	−30.8774	−1.16957
$698$	−31.2170	−1.18158
$699$	5.44080	0.205790
$700$	0.752610	0.0284460
$701$	6.60518	0.249474	0.124737	−	0.992190i	$-0.460191\pi$
$701$	6.60518	0.249474	0.124737	+	0.992190i	$0.460191\pi$
$702$	7.92107	0.298961
$703$	−67.2158	−2.53509
$704$	−1.80356	−0.0679742
$705$	3.94819	0.148698
$706$	27.7468	1.04427
$707$	5.75310	0.216368
$708$	−5.25301	−0.197420
$709$	13.2952	0.499311	0.249656	−	0.968335i	$-0.419683\pi$
$709$	13.2952	0.499311	0.249656	+	0.968335i	$0.419683\pi$
$710$	8.98492	0.337198
$711$	3.36690	0.126269
$712$	9.45590	0.354375
$713$	0.173635	0.00650267
$714$	−3.20367	−0.119894
$715$	−3.88891	−0.145437
$716$	20.5143	0.766655
$717$	0.925137	0.0345499
$718$	8.54510	0.318900
$719$	4.72153	0.176083	0.0880416	−	0.996117i	$-0.471939\pi$
$719$	4.72153	0.176083	0.0880416	+	0.996117i	$0.471939\pi$
$720$	2.56411	0.0955586
$721$	−9.47221	−0.352764
$722$	21.9757	0.817852
$723$	−17.5894	−0.654158
$724$	−8.20378	−0.304891
$725$	8.14441	0.302476
$726$	−5.11486	−0.189830
$727$	19.1904	0.711734	0.355867	−	0.934537i	$-0.384186\pi$
$727$	19.1904	0.711734	0.355867	+	0.934537i	$0.384186\pi$
$728$	−1.62281	−0.0601453
$729$	−6.22892	−0.230701
$730$	4.89859	0.181305
$731$	14.4452	0.534276
$732$	−4.87676	−0.180250
$733$	−14.8139	−0.547165	−0.273583	−	0.961848i	$-0.588209\pi$
$733$	−14.8139	−0.547165	−0.273583	+	0.961848i	$0.588209\pi$
$734$	15.5378	0.573511
$735$	4.24760	0.156675
$736$	0.238594	0.00879469
$737$	28.4356	1.04744
$738$	−12.2798	−0.452025
$739$	21.6610	0.796813	0.398406	−	0.917209i	$-0.369563\pi$
$739$	21.6610	0.796813	0.398406	+	0.917209i	$0.369563\pi$
$740$	10.5005	0.386004
$741$	−9.11280	−0.334767
$742$	−2.40808	−0.0884035
$743$	5.19283	0.190506	0.0952532	−	0.995453i	$-0.469634\pi$
$743$	5.19283	0.190506	0.0952532	+	0.995453i	$0.469634\pi$
$744$	0.480472	0.0176150
$745$	1.77860	0.0651630
$746$	−29.3423	−1.07430
$747$	42.2756	1.54678
$748$	11.6283	0.425174
$749$	4.00605	0.146378
$750$	−0.660223	−0.0241080
$751$	−40.8624	−1.49109	−0.745544	−	0.666456i	$-0.767810\pi$
$751$	−40.8624	−1.49109	−0.745544	+	0.666456i	$0.767810\pi$
$752$	−5.98009	−0.218071
$753$	5.25408	0.191469
$754$	−17.5613	−0.639546
$755$	−12.9178	−0.470127
$756$	−2.76475	−0.100553
$757$	1.17490	0.0427026	0.0213513	−	0.999772i	$-0.493203\pi$
$757$	1.17490	0.0427026	0.0213513	+	0.999772i	$0.493203\pi$
$758$	3.29449	0.119661
$759$	−0.284106	−0.0103124
$760$	−6.40123	−0.232197
$761$	−13.6837	−0.496033	−0.248016	−	0.968756i	$-0.579779\pi$
$761$	−13.6837	−0.496033	−0.248016	+	0.968756i	$0.579779\pi$
$762$	−0.754277	−0.0273246
$763$	−0.960069	−0.0347568
$764$	−5.40522	−0.195554
$765$	−16.5319	−0.597712
$766$	−6.97352	−0.251964
$767$	17.1560	0.619466
$768$	0.660223	0.0238238
$769$	20.7869	0.749596	0.374798	−	0.927106i	$-0.377712\pi$
$769$	20.7869	0.749596	0.374798	+	0.927106i	$0.377712\pi$
$770$	1.35738	0.0489165
$771$	14.3194	0.515700
$772$	−16.7157	−0.601610
$773$	24.1629	0.869079	0.434539	−	0.900653i	$-0.356911\pi$
$773$	24.1629	0.869079	0.434539	+	0.900653i	$0.356911\pi$
$774$	5.74478	0.206492
$775$	0.727741	0.0261412
$776$	6.49591	0.233190
$777$	−5.21758	−0.187180
$778$	12.4424	0.446080
$779$	30.6562	1.09837
$780$	1.42360	0.0509731
$781$	16.2048	0.579855
$782$	−1.53832	−0.0550102
$783$	−29.9189	−1.06922
$784$	−6.43358	−0.229771
$785$	−13.5958	−0.485255
$786$	6.40816	0.228571
$787$	−11.4355	−0.407630	−0.203815	−	0.979009i	$-0.565334\pi$
$787$	−11.4355	−0.407630	−0.203815	+	0.979009i	$0.565334\pi$
$788$	20.9588	0.746626
$789$	13.3549	0.475445
$790$	1.31309	0.0467176
$791$	−10.4440	−0.371347
$792$	4.62452	0.164325
$793$	15.9271	0.565589
$794$	20.5807	0.730381
$795$	2.11248	0.0749219
$796$	1.57105	0.0556843
$797$	4.44494	0.157448	0.0787239	−	0.996896i	$-0.474915\pi$
$797$	4.44494	0.157448	0.0787239	+	0.996896i	$0.474915\pi$
$798$	3.18071	0.112596
$799$	38.5562	1.36402
$800$	1.00000	0.0353553
$801$	−24.2459	−0.856688
$802$	1.00000	0.0353112
$803$	8.83490	0.311777
$804$	−10.4093	−0.367109
$805$	−0.179568	−0.00632895
$806$	−1.56919	−0.0552722
$807$	9.05162	0.318632
$808$	7.64420	0.268922
$809$	22.1370	0.778295	0.389148	−	0.921175i	$-0.372770\pi$
$809$	22.1370	0.778295	0.389148	+	0.921175i	$0.372770\pi$
$810$	−5.26695	−0.185062
$811$	−43.2094	−1.51729	−0.758644	−	0.651505i	$-0.774138\pi$
$811$	−43.2094	−1.51729	−0.758644	+	0.651505i	$0.774138\pi$
$812$	6.12957	0.215106
$813$	11.5939	0.406616
$814$	18.9382	0.663784
$815$	−10.7103	−0.375167
$816$	−4.25675	−0.149016
$817$	−14.3417	−0.501753
$818$	22.4858	0.786197
$819$	4.16105	0.145399
$820$	−4.78911	−0.167243
$821$	6.42501	0.224234	0.112117	−	0.993695i	$-0.464237\pi$
$821$	6.42501	0.224234	0.112117	+	0.993695i	$0.464237\pi$
$822$	15.1902	0.529818
$823$	−38.9013	−1.35601	−0.678007	−	0.735055i	$-0.737156\pi$
$823$	−38.9013	−1.35601	−0.678007	+	0.735055i	$0.737156\pi$
$824$	−12.5858	−0.438448
$825$	−1.19075	−0.0414567
$826$	−5.98808	−0.208352
$827$	−42.4758	−1.47703	−0.738515	−	0.674238i	$-0.764472\pi$
$827$	−42.4758	−1.47703	−0.738515	+	0.674238i	$0.764472\pi$
$828$	−0.611780	−0.0212608
$829$	2.95115	0.102498	0.0512489	−	0.998686i	$-0.483680\pi$
$829$	2.95115	0.102498	0.0512489	+	0.998686i	$0.483680\pi$
$830$	16.4875	0.572288
$831$	−4.22974	−0.146728
$832$	−2.15624	−0.0747542
$833$	41.4801	1.43720
$834$	−1.12263	−0.0388735
$835$	3.79004	0.131160
$836$	−11.5450	−0.399292
$837$	−2.67340	−0.0924061
$838$	25.3845	0.876894
$839$	10.9433	0.377805	0.188902	−	0.981996i	$-0.439507\pi$
$839$	10.9433	0.377805	0.188902	+	0.981996i	$0.439507\pi$
$840$	−0.496891	−0.0171444
$841$	37.3315	1.28729
$842$	9.03773	0.311461
$843$	−1.44705	−0.0498391
$844$	−11.9345	−0.410803
$845$	8.35062	0.287270
$846$	15.3336	0.527179
$847$	−5.83060	−0.200342
$848$	−3.19964	−0.109876
$849$	−4.95096	−0.169917
$850$	−6.44743	−0.221145
$851$	−2.50535	−0.0858822
$852$	−5.93205	−0.203229
$853$	−15.2454	−0.521992	−0.260996	−	0.965340i	$-0.584051\pi$
$853$	−15.2454	−0.521992	−0.260996	+	0.965340i	$0.584051\pi$
$854$	−5.55917	−0.190231
$855$	16.4134	0.561327
$856$	5.32288	0.181932
$857$	52.1369	1.78096	0.890481	−	0.455020i	$-0.150368\pi$
$857$	52.1369	1.78096	0.890481	+	0.455020i	$0.150368\pi$
$858$	2.56755	0.0876547
$859$	−7.22227	−0.246421	−0.123210	−	0.992381i	$-0.539319\pi$
$859$	−7.22227	−0.246421	−0.123210	+	0.992381i	$0.539319\pi$
$860$	2.24046	0.0763991
$861$	2.37966	0.0810987
$862$	−27.8264	−0.947769
$863$	19.2752	0.656137	0.328068	−	0.944654i	$-0.393602\pi$
$863$	19.2752	0.656137	0.328068	+	0.944654i	$0.393602\pi$
$864$	−3.67355	−0.124977
$865$	−18.0799	−0.614735
$866$	−1.38610	−0.0471017
$867$	16.2213	0.550904
$868$	0.547705	0.0185903
$869$	2.36823	0.0803368
$870$	−5.37713	−0.182302
$871$	33.9962	1.15192
$872$	−1.27565	−0.0431990
$873$	−16.6562	−0.563727
$874$	1.52729	0.0516615
$875$	−0.752610	−0.0254429
$876$	−3.23416	−0.109272
$877$	−32.7868	−1.10713	−0.553565	−	0.832806i	$-0.686733\pi$
$877$	−32.7868	−1.10713	−0.553565	+	0.832806i	$0.686733\pi$
$878$	−2.44065	−0.0823681
$879$	−18.5097	−0.624317
$880$	1.80356	0.0607980
$881$	−9.26471	−0.312136	−0.156068	−	0.987746i	$-0.549882\pi$
$881$	−9.26471	−0.312136	−0.156068	+	0.987746i	$0.549882\pi$
$882$	16.4964	0.555462
$883$	−15.2948	−0.514710	−0.257355	−	0.966317i	$-0.582851\pi$
$883$	−15.2948	−0.514710	−0.257355	+	0.966317i	$0.582851\pi$
$884$	13.9022	0.467582
$885$	5.25301	0.176578
$886$	2.34524	0.0787898
$887$	17.1709	0.576543	0.288272	−	0.957549i	$-0.406919\pi$
$887$	17.1709	0.576543	0.288272	+	0.957549i	$0.406919\pi$
$888$	−6.93265	−0.232644
$889$	−0.859824	−0.0288376
$890$	−9.45590	−0.316963
$891$	−9.49926	−0.318237
$892$	−9.36978	−0.313723
$893$	−38.2799	−1.28099
$894$	−1.17428	−0.0392736
$895$	−20.5143	−0.685717
$896$	0.752610	0.0251429
$897$	−0.339663	−0.0113410
$898$	−27.8979	−0.930966
$899$	5.92703	0.197677
$900$	−2.56411	−0.0854702
$901$	20.6295	0.687267
$902$	−8.63744	−0.287595
$903$	−1.11326	−0.0370471
$904$	−13.8771	−0.461545
$905$	8.20378	0.272703
$906$	8.52863	0.283345
$907$	23.2378	0.771600	0.385800	−	0.922582i	$-0.373926\pi$
$907$	23.2378	0.771600	0.385800	+	0.922582i	$0.373926\pi$
$908$	−10.0378	−0.333117
$909$	−19.6005	−0.650109
$910$	1.62281	0.0537956
$911$	−26.9686	−0.893511	−0.446755	−	0.894656i	$-0.647421\pi$
$911$	−26.9686	−0.893511	−0.446755	+	0.894656i	$0.647421\pi$
$912$	4.22624	0.139945
$913$	29.7361	0.984122
$914$	8.35174	0.276251
$915$	4.87676	0.161221
$916$	2.10973	0.0697074
$917$	7.30487	0.241228
$918$	23.6850	0.781721
$919$	−1.10679	−0.0365097	−0.0182549	−	0.999833i	$-0.505811\pi$
$919$	−1.10679	−0.0365097	−0.0182549	+	0.999833i	$0.505811\pi$
$920$	−0.238594	−0.00786621
$921$	7.84156	0.258388
$922$	−11.8433	−0.390038
$923$	19.3737	0.637692
$924$	−0.896172	−0.0294819
$925$	−10.5005	−0.345253
$926$	38.4470	1.26345
$927$	32.2714	1.05993
$928$	8.14441	0.267353
$929$	−5.20578	−0.170796	−0.0853980	−	0.996347i	$-0.527216\pi$
$929$	−5.20578	−0.170796	−0.0853980	+	0.996347i	$0.527216\pi$
$930$	−0.480472	−0.0157553
$931$	−41.1828	−1.34971
$932$	8.24084	0.269938
$933$	15.4374	0.505399
$934$	−7.55036	−0.247055
$935$	−11.6283	−0.380287
$936$	5.52883	0.180715
$937$	−0.792750	−0.0258980	−0.0129490	−	0.999916i	$-0.504122\pi$
$937$	−0.792750	−0.0258980	−0.0129490	+	0.999916i	$0.504122\pi$
$938$	−11.8659	−0.387437
$939$	−9.40174	−0.306814
$940$	5.98009	0.195049
$941$	−38.3149	−1.24903	−0.624515	−	0.781013i	$-0.714703\pi$
$941$	−38.3149	−1.24903	−0.624515	+	0.781013i	$0.714703\pi$
$942$	8.97626	0.292462
$943$	1.14265	0.0372099
$944$	−7.95642	−0.258959
$945$	2.76475	0.0899374
$946$	4.04081	0.131378
$947$	−51.3635	−1.66909	−0.834545	−	0.550939i	$-0.814270\pi$
$947$	−51.3635	−1.66909	−0.834545	+	0.550939i	$0.814270\pi$
$948$	−0.866932	−0.0281566
$949$	10.5625	0.342875
$950$	6.40123	0.207683
$951$	−3.86986	−0.125489
$952$	−4.85240	−0.157267
$953$	−30.9742	−1.00335	−0.501676	−	0.865056i	$-0.667283\pi$
$953$	−30.9742	−1.00335	−0.501676	+	0.865056i	$0.667283\pi$
$954$	8.20421	0.265621
$955$	5.40522	0.174909
$956$	1.40125	0.0453196
$957$	−9.69798	−0.313491
$958$	−12.6211	−0.407768
$959$	17.3158	0.559155
$960$	−0.660223	−0.0213086
$961$	−30.4704	−0.982916
$962$	22.6415	0.729992
$963$	−13.6484	−0.439814
$964$	−26.6416	−0.858070
$965$	16.7157	0.538097
$966$	0.118555	0.00381445
$967$	−10.5899	−0.340550	−0.170275	−	0.985397i	$-0.554466\pi$
$967$	−10.5899	−0.340550	−0.170275	+	0.985397i	$0.554466\pi$
$968$	−7.74717	−0.249004
$969$	−27.2484	−0.875345
$970$	−6.49591	−0.208571
$971$	16.8333	0.540205	0.270103	−	0.962832i	$-0.412942\pi$
$971$	16.8333	0.540205	0.270103	+	0.962832i	$0.412942\pi$
$972$	14.4980	0.465024
$973$	−1.27972	−0.0410261
$974$	30.0375	0.962464
$975$	−1.42360	−0.0455917
$976$	−7.38653	−0.236437
$977$	−7.17353	−0.229502	−0.114751	−	0.993394i	$-0.536607\pi$
$977$	−7.17353	−0.229502	−0.114751	+	0.993394i	$0.536607\pi$
$978$	7.07122	0.226113
$979$	−17.0543	−0.545057
$980$	6.43358	0.205513
$981$	3.27091	0.104432
$982$	1.72676	0.0551032
$983$	−34.8313	−1.11095	−0.555473	−	0.831534i	$-0.687463\pi$
$983$	−34.8313	−1.11095	−0.555473	+	0.831534i	$0.687463\pi$
$984$	3.16188	0.100797
$985$	−20.9588	−0.667802
$986$	−52.5106	−1.67228
$987$	−2.97145	−0.0945823
$988$	−13.8026	−0.439119
$989$	−0.534561	−0.0169980
$990$	−4.62452	−0.146977
$991$	−24.3488	−0.773465	−0.386733	−	0.922192i	$-0.626396\pi$
$991$	−24.3488	−0.773465	−0.386733	+	0.922192i	$0.626396\pi$
$992$	0.727741	0.0231058
$993$	−20.6253	−0.654524
$994$	−6.76214	−0.214482
$995$	−1.57105	−0.0498056
$996$	−10.8854	−0.344917
$997$	8.62217	0.273067	0.136534	−	0.990635i	$-0.456404\pi$
$997$	8.62217	0.273067	0.136534	+	0.990635i	$0.456404\pi$
$998$	21.0407	0.666030
$999$	38.5740	1.22043

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.j.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.j.1.8	✓	12	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:	\( N \)	\(=\)	\( 4010 = 2 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4010.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.660223 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.660223 q^{6} +0.752610 q^{7} +1.00000 q^{8} -2.56411 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.660223 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.660223 q^{6} +0.752610 q^{7} +1.00000 q^{8} -2.56411 q^{9} -1.00000 q^{10} -1.80356 q^{11} +0.660223 q^{12} -2.15624 q^{13} +0.752610 q^{14} -0.660223 q^{15} +1.00000 q^{16} -6.44743 q^{17} -2.56411 q^{18} +6.40123 q^{19} -1.00000 q^{20} +0.496891 q^{21} -1.80356 q^{22} +0.238594 q^{23} +0.660223 q^{24} +1.00000 q^{25} -2.15624 q^{26} -3.67355 q^{27} +0.752610 q^{28} +8.14441 q^{29} -0.660223 q^{30} +0.727741 q^{31} +1.00000 q^{32} -1.19075 q^{33} -6.44743 q^{34} -0.752610 q^{35} -2.56411 q^{36} -10.5005 q^{37} +6.40123 q^{38} -1.42360 q^{39} -1.00000 q^{40} +4.78911 q^{41} +0.496891 q^{42} -2.24046 q^{43} -1.80356 q^{44} +2.56411 q^{45} +0.238594 q^{46} -5.98009 q^{47} +0.660223 q^{48} -6.43358 q^{49} +1.00000 q^{50} -4.25675 q^{51} -2.15624 q^{52} -3.19964 q^{53} -3.67355 q^{54} +1.80356 q^{55} +0.752610 q^{56} +4.22624 q^{57} +8.14441 q^{58} -7.95642 q^{59} -0.660223 q^{60} -7.38653 q^{61} +0.727741 q^{62} -1.92977 q^{63} +1.00000 q^{64} +2.15624 q^{65} -1.19075 q^{66} -15.7664 q^{67} -6.44743 q^{68} +0.157525 q^{69} -0.752610 q^{70} -8.98492 q^{71} -2.56411 q^{72} -4.89859 q^{73} -10.5005 q^{74} +0.660223 q^{75} +6.40123 q^{76} -1.35738 q^{77} -1.42360 q^{78} -1.31309 q^{79} -1.00000 q^{80} +5.26695 q^{81} +4.78911 q^{82} -16.4875 q^{83} +0.496891 q^{84} +6.44743 q^{85} -2.24046 q^{86} +5.37713 q^{87} -1.80356 q^{88} +9.45590 q^{89} +2.56411 q^{90} -1.62281 q^{91} +0.238594 q^{92} +0.480472 q^{93} -5.98009 q^{94} -6.40123 q^{95} +0.660223 q^{96} +6.49591 q^{97} -6.43358 q^{98} +4.62452 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} - 2 q^{3} + 12 q^{4} - 12 q^{5} - 2 q^{6} - 9 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 - 2 * q^3 + 12 * q^4 - 12 * q^5 - 2 * q^6 - 9 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} - 2 q^{3} + 12 q^{4} - 12 q^{5} - 2 q^{6} - 9 q^{7} + 12 q^{8} - 12 q^{10} + q^{11} - 2 q^{12} - 6 q^{13} - 9 q^{14} + 2 q^{15} + 12 q^{16} - 11 q^{17} - 13 q^{19} - 12 q^{20} - 14 q^{21} + q^{22} - 21 q^{23} - 2 q^{24} + 12 q^{25} - 6 q^{26} - 2 q^{27} - 9 q^{28} - 10 q^{29} + 2 q^{30} - 11 q^{31} + 12 q^{32} - 22 q^{33} - 11 q^{34} + 9 q^{35} - 29 q^{37} - 13 q^{38} - 2 q^{39} - 12 q^{40} - q^{41} - 14 q^{42} - 23 q^{43} + q^{44} - 21 q^{46} - 17 q^{47} - 2 q^{48} - 3 q^{49} + 12 q^{50} - 19 q^{51} - 6 q^{52} - 47 q^{53} - 2 q^{54} - q^{55} - 9 q^{56} - 11 q^{57} - 10 q^{58} + 14 q^{59} + 2 q^{60} - 22 q^{61} - 11 q^{62} - 28 q^{63} + 12 q^{64} + 6 q^{65} - 22 q^{66} - 28 q^{67} - 11 q^{68} - q^{69} + 9 q^{70} - 18 q^{71} - 2 q^{73} - 29 q^{74} - 2 q^{75} - 13 q^{76} - 11 q^{77} - 2 q^{78} - 39 q^{79} - 12 q^{80} - 44 q^{81} - q^{82} - 5 q^{83} - 14 q^{84} + 11 q^{85} - 23 q^{86} - 6 q^{87} + q^{88} - 8 q^{89} - 12 q^{91} - 21 q^{92} - 30 q^{93} - 17 q^{94} + 13 q^{95} - 2 q^{96} - 32 q^{97} - 3 q^{98} - 13 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 - 2 * q^3 + 12 * q^4 - 12 * q^5 - 2 * q^6 - 9 * q^7 + 12 * q^8 - 12 * q^10 + q^11 - 2 * q^12 - 6 * q^13 - 9 * q^14 + 2 * q^15 + 12 * q^16 - 11 * q^17 - 13 * q^19 - 12 * q^20 - 14 * q^21 + q^22 - 21 * q^23 - 2 * q^24 + 12 * q^25 - 6 * q^26 - 2 * q^27 - 9 * q^28 - 10 * q^29 + 2 * q^30 - 11 * q^31 + 12 * q^32 - 22 * q^33 - 11 * q^34 + 9 * q^35 - 29 * q^37 - 13 * q^38 - 2 * q^39 - 12 * q^40 - q^41 - 14 * q^42 - 23 * q^43 + q^44 - 21 * q^46 - 17 * q^47 - 2 * q^48 - 3 * q^49 + 12 * q^50 - 19 * q^51 - 6 * q^52 - 47 * q^53 - 2 * q^54 - q^55 - 9 * q^56 - 11 * q^57 - 10 * q^58 + 14 * q^59 + 2 * q^60 - 22 * q^61 - 11 * q^62 - 28 * q^63 + 12 * q^64 + 6 * q^65 - 22 * q^66 - 28 * q^67 - 11 * q^68 - q^69 + 9 * q^70 - 18 * q^71 - 2 * q^73 - 29 * q^74 - 2 * q^75 - 13 * q^76 - 11 * q^77 - 2 * q^78 - 39 * q^79 - 12 * q^80 - 44 * q^81 - q^82 - 5 * q^83 - 14 * q^84 + 11 * q^85 - 23 * q^86 - 6 * q^87 + q^88 - 8 * q^89 - 12 * q^91 - 21 * q^92 - 30 * q^93 - 17 * q^94 + 13 * q^95 - 2 * q^96 - 32 * q^97 - 3 * q^98 - 13 * q^99

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