LMFDB - Embedded newform 3174.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3174,2,Mod(1,3174)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3174, 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("3174.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3174 = 2 \cdot 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3174.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:	$25.3445176016$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.68251$ of defining polynomial
Character	$\chi$	$=$	3174.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.458044 q^{5} +1.00000 q^{6} -0.324635 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.458044 q^{5} +1.00000 q^{6} -0.324635 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.458044 q^{10} +2.48825 q^{11} -1.00000 q^{12} +1.35908 q^{13} +0.324635 q^{14} -0.458044 q^{15} +1.00000 q^{16} -2.69611 q^{17} -1.00000 q^{18} -2.89389 q^{19} +0.458044 q^{20} +0.324635 q^{21} -2.48825 q^{22} +1.00000 q^{24} -4.79020 q^{25} -1.35908 q^{26} -1.00000 q^{27} -0.324635 q^{28} -2.18119 q^{29} +0.458044 q^{30} +2.46297 q^{31} -1.00000 q^{32} -2.48825 q^{33} +2.69611 q^{34} -0.148697 q^{35} +1.00000 q^{36} +0.480474 q^{37} +2.89389 q^{38} -1.35908 q^{39} -0.458044 q^{40} +0.847432 q^{41} -0.324635 q^{42} -8.80752 q^{43} +2.48825 q^{44} +0.458044 q^{45} -2.71406 q^{47} -1.00000 q^{48} -6.89461 q^{49} +4.79020 q^{50} +2.69611 q^{51} +1.35908 q^{52} +8.72613 q^{53} +1.00000 q^{54} +1.13973 q^{55} +0.324635 q^{56} +2.89389 q^{57} +2.18119 q^{58} +10.8833 q^{59} -0.458044 q^{60} -12.0624 q^{61} -2.46297 q^{62} -0.324635 q^{63} +1.00000 q^{64} +0.622519 q^{65} +2.48825 q^{66} +10.5658 q^{67} -2.69611 q^{68} +0.148697 q^{70} -16.4068 q^{71} -1.00000 q^{72} +11.7624 q^{73} -0.480474 q^{74} +4.79020 q^{75} -2.89389 q^{76} -0.807771 q^{77} +1.35908 q^{78} -10.7275 q^{79} +0.458044 q^{80} +1.00000 q^{81} -0.847432 q^{82} +8.08342 q^{83} +0.324635 q^{84} -1.23494 q^{85} +8.80752 q^{86} +2.18119 q^{87} -2.48825 q^{88} -10.2983 q^{89} -0.458044 q^{90} -0.441205 q^{91} -2.46297 q^{93} +2.71406 q^{94} -1.32553 q^{95} +1.00000 q^{96} -0.224268 q^{97} +6.89461 q^{98} +2.48825 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} + q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 + q^7 - 5 * q^8 + 5 * q^9	$ 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} + q^{7} - 5 q^{8} + 5 q^{9} - q^{10} + 3 q^{11} - 5 q^{12} - 10 q^{13} - q^{14} - q^{15} + 5 q^{16} + 11 q^{17} - 5 q^{18} - 11 q^{19} + q^{20} - q^{21} - 3 q^{22} + 5 q^{24} + 6 q^{25} + 10 q^{26} - 5 q^{27} + q^{28} - 25 q^{29} + q^{30} - 4 q^{31} - 5 q^{32} - 3 q^{33} - 11 q^{34} + 9 q^{35} + 5 q^{36} - 22 q^{37} + 11 q^{38} + 10 q^{39} - q^{40} + 6 q^{41} + q^{42} + 6 q^{43} + 3 q^{44} + q^{45} + 4 q^{47} - 5 q^{48} - 4 q^{49} - 6 q^{50} - 11 q^{51} - 10 q^{52} - q^{53} + 5 q^{54} - 6 q^{55} - q^{56} + 11 q^{57} + 25 q^{58} + 6 q^{59} - q^{60} - 35 q^{61} + 4 q^{62} + q^{63} + 5 q^{64} - 2 q^{65} + 3 q^{66} - 5 q^{67} + 11 q^{68} - 9 q^{70} - 11 q^{71} - 5 q^{72} + 16 q^{73} + 22 q^{74} - 6 q^{75} - 11 q^{76} + 5 q^{77} - 10 q^{78} - 8 q^{79} + q^{80} + 5 q^{81} - 6 q^{82} + 26 q^{83} - q^{84} - 33 q^{85} - 6 q^{86} + 25 q^{87} - 3 q^{88} - q^{90} - 35 q^{91} + 4 q^{93} - 4 q^{94} + 11 q^{95} + 5 q^{96} - 19 q^{97} + 4 q^{98} + 3 q^{99}+O(q^{100}) $ 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 + q^7 - 5 * q^8 + 5 * q^9 - q^10 + 3 * q^11 - 5 * q^12 - 10 * q^13 - q^14 - q^15 + 5 * q^16 + 11 * q^17 - 5 * q^18 - 11 * q^19 + q^20 - q^21 - 3 * q^22 + 5 * q^24 + 6 * q^25 + 10 * q^26 - 5 * q^27 + q^28 - 25 * q^29 + q^30 - 4 * q^31 - 5 * q^32 - 3 * q^33 - 11 * q^34 + 9 * q^35 + 5 * q^36 - 22 * q^37 + 11 * q^38 + 10 * q^39 - q^40 + 6 * q^41 + q^42 + 6 * q^43 + 3 * q^44 + q^45 + 4 * q^47 - 5 * q^48 - 4 * q^49 - 6 * q^50 - 11 * q^51 - 10 * q^52 - q^53 + 5 * q^54 - 6 * q^55 - q^56 + 11 * q^57 + 25 * q^58 + 6 * q^59 - q^60 - 35 * q^61 + 4 * q^62 + q^63 + 5 * q^64 - 2 * q^65 + 3 * q^66 - 5 * q^67 + 11 * q^68 - 9 * q^70 - 11 * q^71 - 5 * q^72 + 16 * q^73 + 22 * q^74 - 6 * q^75 - 11 * q^76 + 5 * q^77 - 10 * q^78 - 8 * q^79 + q^80 + 5 * q^81 - 6 * q^82 + 26 * q^83 - q^84 - 33 * q^85 - 6 * q^86 + 25 * q^87 - 3 * q^88 - q^90 - 35 * q^91 + 4 * q^93 - 4 * q^94 + 11 * q^95 + 5 * q^96 - 19 * q^97 + 4 * q^98 + 3 * 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
$5$	0.458044	0.204844	0.102422	−	0.994741i	$-0.467341\pi$
$5$	0.458044	0.204844	0.102422	+	0.994741i	$0.467341\pi$
$6$	1.00000	0.408248
$7$	−0.324635	−0.122700	−0.0613502	−	0.998116i	$-0.519541\pi$
$7$	−0.324635	−0.122700	−0.0613502	+	0.998116i	$0.519541\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−0.458044	−0.144846
$11$	2.48825	0.750234	0.375117	−	0.926977i	$-0.377602\pi$
$11$	2.48825	0.750234	0.375117	+	0.926977i	$0.377602\pi$
$12$	−1.00000	−0.288675
$13$	1.35908	0.376941	0.188471	−	0.982079i	$-0.439647\pi$
$13$	1.35908	0.376941	0.188471	+	0.982079i	$0.439647\pi$
$14$	0.324635	0.0867623
$15$	−0.458044	−0.118267
$16$	1.00000	0.250000
$17$	−2.69611	−0.653902	−0.326951	−	0.945041i	$-0.606021\pi$
$17$	−2.69611	−0.653902	−0.326951	+	0.945041i	$0.606021\pi$
$18$	−1.00000	−0.235702
$19$	−2.89389	−0.663905	−0.331952	−	0.943296i	$-0.607707\pi$
$19$	−2.89389	−0.663905	−0.331952	+	0.943296i	$0.607707\pi$
$20$	0.458044	0.102422
$21$	0.324635	0.0708411
$22$	−2.48825	−0.530496
$23$	0	0
$23$	0	0
$24$	1.00000	0.204124
$25$	−4.79020	−0.958039
$26$	−1.35908	−0.266538
$27$	−1.00000	−0.192450
$28$	−0.324635	−0.0613502
$29$	−2.18119	−0.405036	−0.202518	−	0.979279i	$-0.564912\pi$
$29$	−2.18119	−0.405036	−0.202518	+	0.979279i	$0.564912\pi$
$30$	0.458044	0.0836271
$31$	2.46297	0.442363	0.221182	−	0.975233i	$-0.429009\pi$
$31$	2.46297	0.442363	0.221182	+	0.975233i	$0.429009\pi$
$32$	−1.00000	−0.176777
$33$	−2.48825	−0.433148
$34$	2.69611	0.462378
$35$	−0.148697	−0.0251344
$36$	1.00000	0.166667
$37$	0.480474	0.0789894	0.0394947	−	0.999220i	$-0.487425\pi$
$37$	0.480474	0.0789894	0.0394947	+	0.999220i	$0.487425\pi$
$38$	2.89389	0.469452
$39$	−1.35908	−0.217627
$40$	−0.458044	−0.0724232
$41$	0.847432	0.132347	0.0661733	−	0.997808i	$-0.478921\pi$
$41$	0.847432	0.132347	0.0661733	+	0.997808i	$0.478921\pi$
$42$	−0.324635	−0.0500922
$43$	−8.80752	−1.34313	−0.671567	−	0.740944i	$-0.734378\pi$
$43$	−8.80752	−1.34313	−0.671567	+	0.740944i	$0.734378\pi$
$44$	2.48825	0.375117
$45$	0.458044	0.0682812
$46$	0	0
$47$	−2.71406	−0.395886	−0.197943	−	0.980214i	$-0.563426\pi$
$47$	−2.71406	−0.395886	−0.197943	+	0.980214i	$0.563426\pi$
$48$	−1.00000	−0.144338
$49$	−6.89461	−0.984945
$50$	4.79020	0.677436
$51$	2.69611	0.377530
$52$	1.35908	0.188471
$53$	8.72613	1.19863	0.599313	−	0.800515i	$-0.295441\pi$
$53$	8.72613	1.19863	0.599313	+	0.800515i	$0.295441\pi$
$54$	1.00000	0.136083
$55$	1.13973	0.153681
$56$	0.324635	0.0433812
$57$	2.89389	0.383306
$58$	2.18119	0.286404
$59$	10.8833	1.41688	0.708441	−	0.705770i	$-0.249399\pi$
$59$	10.8833	1.41688	0.708441	+	0.705770i	$0.249399\pi$
$60$	−0.458044	−0.0591333
$61$	−12.0624	−1.54444	−0.772218	−	0.635357i	$-0.780853\pi$
$61$	−12.0624	−1.54444	−0.772218	+	0.635357i	$0.780853\pi$
$62$	−2.46297	−0.312798
$63$	−0.324635	−0.0409001
$64$	1.00000	0.125000
$65$	0.622519	0.0772140
$66$	2.48825	0.306282
$67$	10.5658	1.29082	0.645408	−	0.763838i	$-0.276687\pi$
$67$	10.5658	1.29082	0.645408	+	0.763838i	$0.276687\pi$
$68$	−2.69611	−0.326951
$69$	0	0
$70$	0.148697	0.0177727
$71$	−16.4068	−1.94713	−0.973564	−	0.228413i	$-0.926646\pi$
$71$	−16.4068	−1.94713	−0.973564	+	0.228413i	$0.926646\pi$
$72$	−1.00000	−0.117851
$73$	11.7624	1.37669	0.688345	−	0.725384i	$-0.258338\pi$
$73$	11.7624	1.37669	0.688345	+	0.725384i	$0.258338\pi$
$74$	−0.480474	−0.0558539
$75$	4.79020	0.553124
$76$	−2.89389	−0.331952
$77$	−0.807771	−0.0920541
$78$	1.35908	0.153886
$79$	−10.7275	−1.20694	−0.603469	−	0.797386i	$-0.706215\pi$
$79$	−10.7275	−1.20694	−0.603469	+	0.797386i	$0.706215\pi$
$80$	0.458044	0.0512109
$81$	1.00000	0.111111
$82$	−0.847432	−0.0935832
$83$	8.08342	0.887271	0.443635	−	0.896207i	$-0.353689\pi$
$83$	8.08342	0.887271	0.443635	+	0.896207i	$0.353689\pi$
$84$	0.324635	0.0354206
$85$	−1.23494	−0.133948
$86$	8.80752	0.949739
$87$	2.18119	0.233848
$88$	−2.48825	−0.265248
$89$	−10.2983	−1.09162	−0.545808	−	0.837910i	$-0.683777\pi$
$89$	−10.2983	−1.09162	−0.545808	+	0.837910i	$0.683777\pi$
$90$	−0.458044	−0.0482821
$91$	−0.441205	−0.0462508
$92$	0	0
$93$	−2.46297	−0.255399
$94$	2.71406	0.279933
$95$	−1.32553	−0.135997
$96$	1.00000	0.102062
$97$	−0.224268	−0.0227709	−0.0113855	−	0.999935i	$-0.503624\pi$
$97$	−0.224268	−0.0227709	−0.0113855	+	0.999935i	$0.503624\pi$
$98$	6.89461	0.696461
$99$	2.48825	0.250078
$100$	−4.79020	−0.479020
$101$	10.6957	1.06426	0.532129	−	0.846663i	$-0.321392\pi$
$101$	10.6957	1.06426	0.532129	+	0.846663i	$0.321392\pi$
$102$	−2.69611	−0.266954
$103$	2.02685	0.199712	0.0998559	−	0.995002i	$-0.468162\pi$
$103$	2.02685	0.199712	0.0998559	+	0.995002i	$0.468162\pi$
$104$	−1.35908	−0.133269
$105$	0.148697	0.0145114
$106$	−8.72613	−0.847556
$107$	2.16835	0.209622	0.104811	−	0.994492i	$-0.466576\pi$
$107$	2.16835	0.209622	0.104811	+	0.994492i	$0.466576\pi$
$108$	−1.00000	−0.0962250
$109$	−8.41887	−0.806382	−0.403191	−	0.915116i	$-0.632099\pi$
$109$	−8.41887	−0.806382	−0.403191	+	0.915116i	$0.632099\pi$
$110$	−1.13973	−0.108669
$111$	−0.480474	−0.0456046
$112$	−0.324635	−0.0306751
$113$	0.193888	0.0182394	0.00911970	−	0.999958i	$-0.497097\pi$
$113$	0.193888	0.0182394	0.00911970	+	0.999958i	$0.497097\pi$
$114$	−2.89389	−0.271038
$115$	0	0
$116$	−2.18119	−0.202518
$117$	1.35908	0.125647
$118$	−10.8833	−1.00189
$119$	0.875250	0.0802340
$120$	0.458044	0.0418135
$121$	−4.80864	−0.437149
$122$	12.0624	1.09208
$123$	−0.847432	−0.0764104
$124$	2.46297	0.221182
$125$	−4.48434	−0.401092
$126$	0.324635	0.0289208
$127$	12.9293	1.14729	0.573643	−	0.819106i	$-0.305530\pi$
$127$	12.9293	1.14729	0.573643	+	0.819106i	$0.305530\pi$
$128$	−1.00000	−0.0883883
$129$	8.80752	0.775459
$130$	−0.622519	−0.0545985
$131$	−5.89229	−0.514812	−0.257406	−	0.966303i	$-0.582868\pi$
$131$	−5.89229	−0.514812	−0.257406	+	0.966303i	$0.582868\pi$
$132$	−2.48825	−0.216574
$133$	0.939459	0.0814614
$134$	−10.5658	−0.912745
$135$	−0.458044	−0.0394222
$136$	2.69611	0.231189
$137$	−16.2119	−1.38508	−0.692538	−	0.721381i	$-0.743508\pi$
$137$	−16.2119	−1.38508	−0.692538	+	0.721381i	$0.743508\pi$
$138$	0	0
$139$	−19.6328	−1.66524	−0.832618	−	0.553848i	$-0.813159\pi$
$139$	−19.6328	−1.66524	−0.832618	+	0.553848i	$0.813159\pi$
$140$	−0.148697	−0.0125672
$141$	2.71406	0.228565
$142$	16.4068	1.37683
$143$	3.38173	0.282794
$144$	1.00000	0.0833333
$145$	−0.999080	−0.0829691
$146$	−11.7624	−0.973467
$147$	6.89461	0.568658
$148$	0.480474	0.0394947
$149$	3.17993	0.260510	0.130255	−	0.991481i	$-0.458420\pi$
$149$	3.17993	0.260510	0.130255	+	0.991481i	$0.458420\pi$
$150$	−4.79020	−0.391118
$151$	−1.27472	−0.103735	−0.0518676	−	0.998654i	$-0.516517\pi$
$151$	−1.27472	−0.103735	−0.0518676	+	0.998654i	$0.516517\pi$
$152$	2.89389	0.234726
$153$	−2.69611	−0.217967
$154$	0.807771	0.0650920
$155$	1.12815	0.0906153
$156$	−1.35908	−0.108814
$157$	−17.3357	−1.38354	−0.691769	−	0.722118i	$-0.743169\pi$
$157$	−17.3357	−1.38354	−0.691769	+	0.722118i	$0.743169\pi$
$158$	10.7275	0.853435
$159$	−8.72613	−0.692027
$160$	−0.458044	−0.0362116
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−5.27243	−0.412969	−0.206484	−	0.978450i	$-0.566202\pi$
$163$	−5.27243	−0.412969	−0.206484	+	0.978450i	$0.566202\pi$
$164$	0.847432	0.0661733
$165$	−1.13973	−0.0887276
$166$	−8.08342	−0.627395
$167$	−2.95796	−0.228894	−0.114447	−	0.993429i	$-0.536510\pi$
$167$	−2.95796	−0.228894	−0.114447	+	0.993429i	$0.536510\pi$
$168$	−0.324635	−0.0250461
$169$	−11.1529	−0.857915
$170$	1.23494	0.0947153
$171$	−2.89389	−0.221302
$172$	−8.80752	−0.671567
$173$	4.13279	0.314210	0.157105	−	0.987582i	$-0.449784\pi$
$173$	4.13279	0.314210	0.157105	+	0.987582i	$0.449784\pi$
$174$	−2.18119	−0.165355
$175$	1.55506	0.117552
$176$	2.48825	0.187559
$177$	−10.8833	−0.818038
$178$	10.2983	0.771889
$179$	10.1370	0.757677	0.378838	−	0.925463i	$-0.376324\pi$
$179$	10.1370	0.757677	0.378838	+	0.925463i	$0.376324\pi$
$180$	0.458044	0.0341406
$181$	−26.6380	−1.97998	−0.989992	−	0.141121i	$-0.954929\pi$
$181$	−26.6380	−1.97998	−0.989992	+	0.141121i	$0.954929\pi$
$182$	0.441205	0.0327043
$183$	12.0624	0.891681
$184$	0	0
$185$	0.220078	0.0161805
$186$	2.46297	0.180594
$187$	−6.70857	−0.490579
$188$	−2.71406	−0.197943
$189$	0.324635	0.0236137
$190$	1.32553	0.0961642
$191$	23.3924	1.69261	0.846306	−	0.532698i	$-0.178822\pi$
$191$	23.3924	1.69261	0.846306	+	0.532698i	$0.178822\pi$
$192$	−1.00000	−0.0721688
$193$	6.01660	0.433085	0.216542	−	0.976273i	$-0.430522\pi$
$193$	6.01660	0.433085	0.216542	+	0.976273i	$0.430522\pi$
$194$	0.224268	0.0161015
$195$	−0.622519	−0.0445795
$196$	−6.89461	−0.492472
$197$	15.4640	1.10176	0.550881	−	0.834584i	$-0.314292\pi$
$197$	15.4640	1.10176	0.550881	+	0.834584i	$0.314292\pi$
$198$	−2.48825	−0.176832
$199$	−11.7110	−0.830171	−0.415086	−	0.909782i	$-0.636248\pi$
$199$	−11.7110	−0.830171	−0.415086	+	0.909782i	$0.636248\pi$
$200$	4.79020	0.338718
$201$	−10.5658	−0.745253
$202$	−10.6957	−0.752544
$203$	0.708089	0.0496981
$204$	2.69611	0.188765
$205$	0.388162	0.0271104
$206$	−2.02685	−0.141218
$207$	0	0
$208$	1.35908	0.0942353
$209$	−7.20072	−0.498084
$210$	−0.148697	−0.0102611
$211$	10.6750	0.734898	0.367449	−	0.930044i	$-0.380231\pi$
$211$	10.6750	0.734898	0.367449	+	0.930044i	$0.380231\pi$
$212$	8.72613	0.599313
$213$	16.4068	1.12418
$214$	−2.16835	−0.148225
$215$	−4.03423	−0.275132
$216$	1.00000	0.0680414
$217$	−0.799567	−0.0542782
$218$	8.41887	0.570198
$219$	−11.7624	−0.794832
$220$	1.13973	0.0768404
$221$	−3.66422	−0.246482
$222$	0.480474	0.0322473
$223$	7.57171	0.507039	0.253519	−	0.967330i	$-0.418412\pi$
$223$	7.57171	0.507039	0.253519	+	0.967330i	$0.418412\pi$
$224$	0.324635	0.0216906
$225$	−4.79020	−0.319346
$226$	−0.193888	−0.0128972
$227$	−12.6447	−0.839256	−0.419628	−	0.907696i	$-0.637839\pi$
$227$	−12.6447	−0.839256	−0.419628	+	0.907696i	$0.637839\pi$
$228$	2.89389	0.191653
$229$	−8.39479	−0.554743	−0.277372	−	0.960763i	$-0.589463\pi$
$229$	−8.39479	−0.554743	−0.277372	+	0.960763i	$0.589463\pi$
$230$	0	0
$231$	0.807771	0.0531474
$232$	2.18119	0.143202
$233$	26.2816	1.72177	0.860883	−	0.508803i	$-0.169912\pi$
$233$	26.2816	1.72177	0.860883	+	0.508803i	$0.169912\pi$
$234$	−1.35908	−0.0888459
$235$	−1.24316	−0.0810947
$236$	10.8833	0.708441
$237$	10.7275	0.696826
$238$	−0.875250	−0.0567340
$239$	7.85374	0.508016	0.254008	−	0.967202i	$-0.418251\pi$
$239$	7.85374	0.508016	0.254008	+	0.967202i	$0.418251\pi$
$240$	−0.458044	−0.0295666
$241$	−26.5043	−1.70729	−0.853644	−	0.520856i	$-0.825613\pi$
$241$	−26.5043	−1.70729	−0.853644	+	0.520856i	$0.825613\pi$
$242$	4.80864	0.309111
$243$	−1.00000	−0.0641500
$244$	−12.0624	−0.772218
$245$	−3.15804	−0.201760
$246$	0.847432	0.0540303
$247$	−3.93303	−0.250253
$248$	−2.46297	−0.156399
$249$	−8.08342	−0.512266
$250$	4.48434	0.283615
$251$	−18.0533	−1.13951	−0.569757	−	0.821813i	$-0.692963\pi$
$251$	−18.0533	−1.13951	−0.569757	+	0.821813i	$0.692963\pi$
$252$	−0.324635	−0.0204501
$253$	0	0
$254$	−12.9293	−0.811253
$255$	1.23494	0.0773347
$256$	1.00000	0.0625000
$257$	−22.8558	−1.42570	−0.712852	−	0.701314i	$-0.752597\pi$
$257$	−22.8558	−1.42570	−0.712852	+	0.701314i	$0.752597\pi$
$258$	−8.80752	−0.548332
$259$	−0.155979	−0.00969203
$260$	0.622519	0.0386070
$261$	−2.18119	−0.135012
$262$	5.89229	0.364027
$263$	−18.6195	−1.14813	−0.574063	−	0.818811i	$-0.694633\pi$
$263$	−18.6195	−1.14813	−0.574063	+	0.818811i	$0.694633\pi$
$264$	2.48825	0.153141
$265$	3.99695	0.245531
$266$	−0.939459	−0.0576019
$267$	10.2983	0.630245
$268$	10.5658	0.645408
$269$	0.788737	0.0480901	0.0240451	−	0.999711i	$-0.492345\pi$
$269$	0.788737	0.0480901	0.0240451	+	0.999711i	$0.492345\pi$
$270$	0.458044	0.0278757
$271$	29.3202	1.78108	0.890539	−	0.454907i	$-0.150327\pi$
$271$	29.3202	1.78108	0.890539	+	0.454907i	$0.150327\pi$
$272$	−2.69611	−0.163475
$273$	0.441205	0.0267029
$274$	16.2119	0.979397
$275$	−11.9192	−0.718754
$276$	0	0
$277$	−25.4356	−1.52828	−0.764138	−	0.645053i	$-0.776835\pi$
$277$	−25.4356	−1.52828	−0.764138	+	0.645053i	$0.776835\pi$
$278$	19.6328	1.17750
$279$	2.46297	0.147454
$280$	0.148697	0.00888636
$281$	−23.5820	−1.40679	−0.703394	−	0.710801i	$-0.748333\pi$
$281$	−23.5820	−1.40679	−0.703394	+	0.710801i	$0.748333\pi$
$282$	−2.71406	−0.161620
$283$	−26.6302	−1.58300	−0.791500	−	0.611169i	$-0.790699\pi$
$283$	−26.6302	−1.58300	−0.791500	+	0.611169i	$0.790699\pi$
$284$	−16.4068	−0.973564
$285$	1.32553	0.0785177
$286$	−3.38173	−0.199966
$287$	−0.275106	−0.0162390
$288$	−1.00000	−0.0589256
$289$	−9.73101	−0.572413
$290$	0.999080	0.0586680
$291$	0.224268	0.0131468
$292$	11.7624	0.688345
$293$	−6.59236	−0.385130	−0.192565	−	0.981284i	$-0.561681\pi$
$293$	−6.59236	−0.385130	−0.192565	+	0.981284i	$0.561681\pi$
$294$	−6.89461	−0.402102
$295$	4.98503	0.290239
$296$	−0.480474	−0.0279270
$297$	−2.48825	−0.144383
$298$	−3.17993	−0.184208
$299$	0	0
$300$	4.79020	0.276562
$301$	2.85923	0.164803
$302$	1.27472	0.0733518
$303$	−10.6957	−0.614449
$304$	−2.89389	−0.165976
$305$	−5.52513	−0.316368
$306$	2.69611	0.154126
$307$	22.2953	1.27246	0.636231	−	0.771499i	$-0.280493\pi$
$307$	22.2953	1.27246	0.636231	+	0.771499i	$0.280493\pi$
$308$	−0.807771	−0.0460270
$309$	−2.02685	−0.115304
$310$	−1.12815	−0.0640747
$311$	−12.9065	−0.731859	−0.365930	−	0.930643i	$-0.619249\pi$
$311$	−12.9065	−0.731859	−0.365930	+	0.930643i	$0.619249\pi$
$312$	1.35908	0.0769428
$313$	−19.2227	−1.08653	−0.543266	−	0.839560i	$-0.682813\pi$
$313$	−19.2227	−1.08653	−0.543266	+	0.839560i	$0.682813\pi$
$314$	17.3357	0.978310
$315$	−0.148697	−0.00837814
$316$	−10.7275	−0.603469
$317$	−10.7412	−0.603285	−0.301642	−	0.953421i	$-0.597535\pi$
$317$	−10.7412	−0.603285	−0.301642	+	0.953421i	$0.597535\pi$
$318$	8.72613	0.489337
$319$	−5.42733	−0.303872
$320$	0.458044	0.0256055
$321$	−2.16835	−0.121025
$322$	0	0
$323$	7.80224	0.434128
$324$	1.00000	0.0555556
$325$	−6.51026	−0.361124
$326$	5.27243	0.292013
$327$	8.41887	0.465565
$328$	−0.847432	−0.0467916
$329$	0.881077	0.0485753
$330$	1.13973	0.0627399
$331$	−17.5358	−0.963855	−0.481928	−	0.876211i	$-0.660063\pi$
$331$	−17.5358	−0.963855	−0.481928	+	0.876211i	$0.660063\pi$
$332$	8.08342	0.443635
$333$	0.480474	0.0263298
$334$	2.95796	0.161852
$335$	4.83960	0.264416
$336$	0.324635	0.0177103
$337$	−13.2185	−0.720058	−0.360029	−	0.932941i	$-0.617233\pi$
$337$	−13.2185	−0.720058	−0.360029	+	0.932941i	$0.617233\pi$
$338$	11.1529	0.606638
$339$	−0.193888	−0.0105305
$340$	−1.23494	−0.0669738
$341$	6.12848	0.331876
$342$	2.89389	0.156484
$343$	4.51067	0.243554
$344$	8.80752	0.474869
$345$	0	0
$346$	−4.13279	−0.222180
$347$	−15.9720	−0.857422	−0.428711	−	0.903442i	$-0.641032\pi$
$347$	−15.9720	−0.857422	−0.428711	+	0.903442i	$0.641032\pi$
$348$	2.18119	0.116924
$349$	2.86564	0.153394	0.0766970	−	0.997054i	$-0.475563\pi$
$349$	2.86564	0.153394	0.0766970	+	0.997054i	$0.475563\pi$
$350$	−1.55506	−0.0831217
$351$	−1.35908	−0.0725423
$352$	−2.48825	−0.132624
$353$	8.05226	0.428578	0.214289	−	0.976770i	$-0.431257\pi$
$353$	8.05226	0.428578	0.214289	+	0.976770i	$0.431257\pi$
$354$	10.8833	0.578440
$355$	−7.51504	−0.398857
$356$	−10.2983	−0.545808
$357$	−0.875250	−0.0463231
$358$	−10.1370	−0.535758
$359$	7.86143	0.414910	0.207455	−	0.978245i	$-0.433482\pi$
$359$	7.86143	0.414910	0.207455	+	0.978245i	$0.433482\pi$
$360$	−0.458044	−0.0241411
$361$	−10.6254	−0.559230
$362$	26.6380	1.40006
$363$	4.80864	0.252388
$364$	−0.441205	−0.0231254
$365$	5.38772	0.282006
$366$	−12.0624	−0.630514
$367$	13.9011	0.725631	0.362816	−	0.931861i	$-0.381816\pi$
$367$	13.9011	0.725631	0.362816	+	0.931861i	$0.381816\pi$
$368$	0	0
$369$	0.847432	0.0441156
$370$	−0.220078	−0.0114413
$371$	−2.83280	−0.147072
$372$	−2.46297	−0.127699
$373$	−9.66207	−0.500283	−0.250141	−	0.968209i	$-0.580477\pi$
$373$	−9.66207	−0.500283	−0.250141	+	0.968209i	$0.580477\pi$
$374$	6.70857	0.346892
$375$	4.48434	0.231571
$376$	2.71406	0.139967
$377$	−2.96441	−0.152675
$378$	−0.324635	−0.0166974
$379$	−19.9331	−1.02389	−0.511947	−	0.859017i	$-0.671076\pi$
$379$	−19.9331	−1.02389	−0.511947	+	0.859017i	$0.671076\pi$
$380$	−1.32553	−0.0679984
$381$	−12.9293	−0.662386
$382$	−23.3924	−1.19686
$383$	27.9735	1.42938	0.714688	−	0.699443i	$-0.246568\pi$
$383$	27.9735	1.42938	0.714688	+	0.699443i	$0.246568\pi$
$384$	1.00000	0.0510310
$385$	−0.369995	−0.0188567
$386$	−6.01660	−0.306237
$387$	−8.80752	−0.447711
$388$	−0.224268	−0.0113855
$389$	−26.4792	−1.34255	−0.671276	−	0.741208i	$-0.734253\pi$
$389$	−26.4792	−1.34255	−0.671276	+	0.741208i	$0.734253\pi$
$390$	0.622519	0.0315225
$391$	0	0
$392$	6.89461	0.348231
$393$	5.89229	0.297227
$394$	−15.4640	−0.779063
$395$	−4.91367	−0.247234
$396$	2.48825	0.125039
$397$	−16.4478	−0.825492	−0.412746	−	0.910846i	$-0.635430\pi$
$397$	−16.4478	−0.825492	−0.412746	+	0.910846i	$0.635430\pi$
$398$	11.7110	0.587020
$399$	−0.939459	−0.0470318
$400$	−4.79020	−0.239510
$401$	20.2422	1.01084	0.505422	−	0.862872i	$-0.331337\pi$
$401$	20.2422	1.01084	0.505422	+	0.862872i	$0.331337\pi$
$402$	10.5658	0.526974
$403$	3.34738	0.166745
$404$	10.6957	0.532129
$405$	0.458044	0.0227604
$406$	−0.708089	−0.0351419
$407$	1.19554	0.0592606
$408$	−2.69611	−0.133477
$409$	17.4324	0.861978	0.430989	−	0.902357i	$-0.358165\pi$
$409$	17.4324	0.861978	0.430989	+	0.902357i	$0.358165\pi$
$410$	−0.388162	−0.0191699
$411$	16.2119	0.799674
$412$	2.02685	0.0998559
$413$	−3.53309	−0.173852
$414$	0	0
$415$	3.70257	0.181752
$416$	−1.35908	−0.0666344
$417$	19.6328	0.961424
$418$	7.20072	0.352199
$419$	−16.4612	−0.804184	−0.402092	−	0.915599i	$-0.631717\pi$
$419$	−16.4612	−0.804184	−0.402092	+	0.915599i	$0.631717\pi$
$420$	0.148697	0.00725568
$421$	−30.9773	−1.50974	−0.754870	−	0.655875i	$-0.772300\pi$
$421$	−30.9773	−1.50974	−0.754870	+	0.655875i	$0.772300\pi$
$422$	−10.6750	−0.519651
$423$	−2.71406	−0.131962
$424$	−8.72613	−0.423778
$425$	12.9149	0.626463
$426$	−16.4068	−0.794912
$427$	3.91589	0.189503
$428$	2.16835	0.104811
$429$	−3.38173	−0.163271
$430$	4.03423	0.194548
$431$	−18.3159	−0.882245	−0.441123	−	0.897447i	$-0.645420\pi$
$431$	−18.3159	−0.882245	−0.441123	+	0.897447i	$0.645420\pi$
$432$	−1.00000	−0.0481125
$433$	15.1498	0.728054	0.364027	−	0.931388i	$-0.381402\pi$
$433$	15.1498	0.728054	0.364027	+	0.931388i	$0.381402\pi$
$434$	0.799567	0.0383805
$435$	0.999080	0.0479022
$436$	−8.41887	−0.403191
$437$	0	0
$438$	11.7624	0.562031
$439$	5.83176	0.278335	0.139167	−	0.990269i	$-0.455557\pi$
$439$	5.83176	0.278335	0.139167	+	0.990269i	$0.455557\pi$
$440$	−1.13973	−0.0543343
$441$	−6.89461	−0.328315
$442$	3.66422	0.174289
$443$	−17.3465	−0.824155	−0.412077	−	0.911149i	$-0.635197\pi$
$443$	−17.3465	−0.824155	−0.412077	+	0.911149i	$0.635197\pi$
$444$	−0.480474	−0.0228023
$445$	−4.71707	−0.223611
$446$	−7.57171	−0.358531
$447$	−3.17993	−0.150405
$448$	−0.324635	−0.0153376
$449$	−18.5051	−0.873307	−0.436654	−	0.899630i	$-0.643836\pi$
$449$	−18.5051	−0.873307	−0.436654	+	0.899630i	$0.643836\pi$
$450$	4.79020	0.225812
$451$	2.10862	0.0992910
$452$	0.193888	0.00911970
$453$	1.27472	0.0598915
$454$	12.6447	0.593443
$455$	−0.202091	−0.00947419
$456$	−2.89389	−0.135519
$457$	−5.76183	−0.269527	−0.134764	−	0.990878i	$-0.543027\pi$
$457$	−5.76183	−0.269527	−0.134764	+	0.990878i	$0.543027\pi$
$458$	8.39479	0.392263
$459$	2.69611	0.125843
$460$	0	0
$461$	22.5329	1.04946	0.524730	−	0.851269i	$-0.324166\pi$
$461$	22.5329	1.04946	0.524730	+	0.851269i	$0.324166\pi$
$462$	−0.807771	−0.0375809
$463$	−29.9997	−1.39420	−0.697102	−	0.716972i	$-0.745527\pi$
$463$	−29.9997	−1.39420	−0.697102	+	0.716972i	$0.745527\pi$
$464$	−2.18119	−0.101259
$465$	−1.12815	−0.0523168
$466$	−26.2816	−1.21747
$467$	−11.6433	−0.538788	−0.269394	−	0.963030i	$-0.586823\pi$
$467$	−11.6433	−0.538788	−0.269394	+	0.963030i	$0.586823\pi$
$468$	1.35908	0.0628235
$469$	−3.43002	−0.158384
$470$	1.24316	0.0573426
$471$	17.3357	0.798787
$472$	−10.8833	−0.500944
$473$	−21.9153	−1.00766
$474$	−10.7275	−0.492731
$475$	13.8623	0.636047
$476$	0.875250	0.0401170
$477$	8.72613	0.399542
$478$	−7.85374	−0.359222
$479$	13.3394	0.609493	0.304746	−	0.952434i	$-0.401428\pi$
$479$	13.3394	0.609493	0.304746	+	0.952434i	$0.401428\pi$
$480$	0.458044	0.0209068
$481$	0.653003	0.0297744
$482$	26.5043	1.20724
$483$	0	0
$484$	−4.80864	−0.218574
$485$	−0.102725	−0.00466448
$486$	1.00000	0.0453609
$487$	−13.5203	−0.612663	−0.306331	−	0.951925i	$-0.599102\pi$
$487$	−13.5203	−0.612663	−0.306331	+	0.951925i	$0.599102\pi$
$488$	12.0624	0.546041
$489$	5.27243	0.238428
$490$	3.15804	0.142666
$491$	40.9468	1.84790	0.923952	−	0.382508i	$-0.124940\pi$
$491$	40.9468	1.84790	0.923952	+	0.382508i	$0.124940\pi$
$492$	−0.847432	−0.0382052
$493$	5.88071	0.264854
$494$	3.93303	0.176956
$495$	1.13973	0.0512269
$496$	2.46297	0.110591
$497$	5.32622	0.238913
$498$	8.08342	0.362227
$499$	21.3933	0.957697	0.478848	−	0.877898i	$-0.341054\pi$
$499$	21.3933	0.957697	0.478848	+	0.877898i	$0.341054\pi$
$500$	−4.48434	−0.200546
$501$	2.95796	0.132152
$502$	18.0533	0.805759
$503$	21.7245	0.968648	0.484324	−	0.874889i	$-0.339066\pi$
$503$	21.7245	0.968648	0.484324	+	0.874889i	$0.339066\pi$
$504$	0.324635	0.0144604
$505$	4.89909	0.218006
$506$	0	0
$507$	11.1529	0.495318
$508$	12.9293	0.573643
$509$	31.3484	1.38949	0.694747	−	0.719255i	$-0.255517\pi$
$509$	31.3484	1.38949	0.694747	+	0.719255i	$0.255517\pi$
$510$	−1.23494	−0.0546839
$511$	−3.81850	−0.168920
$512$	−1.00000	−0.0441942
$513$	2.89389	0.127769
$514$	22.8558	1.00813
$515$	0.928389	0.0409097
$516$	8.80752	0.387729
$517$	−6.75324	−0.297007
$518$	0.155979	0.00685330
$519$	−4.13279	−0.181409
$520$	−0.622519	−0.0272993
$521$	−9.13076	−0.400026	−0.200013	−	0.979793i	$-0.564098\pi$
$521$	−9.13076	−0.400026	−0.200013	+	0.979793i	$0.564098\pi$
$522$	2.18119	0.0954679
$523$	17.2377	0.753753	0.376877	−	0.926263i	$-0.376998\pi$
$523$	17.2377	0.753753	0.376877	+	0.926263i	$0.376998\pi$
$524$	−5.89229	−0.257406
$525$	−1.55506	−0.0678686
$526$	18.6195	0.811847
$527$	−6.64044	−0.289262
$528$	−2.48825	−0.108287
$529$	0	0
$530$	−3.99695	−0.173617
$531$	10.8833	0.472294
$532$	0.939459	0.0407307
$533$	1.15173	0.0498869
$534$	−10.2983	−0.445650
$535$	0.993199	0.0429397
$536$	−10.5658	−0.456373
$537$	−10.1370	−0.437445
$538$	−0.788737	−0.0340048
$539$	−17.1555	−0.738939
$540$	−0.458044	−0.0197111
$541$	42.7863	1.83953	0.919764	−	0.392471i	$-0.128380\pi$
$541$	42.7863	1.83953	0.919764	+	0.392471i	$0.128380\pi$
$542$	−29.3202	−1.25941
$543$	26.6380	1.14314
$544$	2.69611	0.115595
$545$	−3.85622	−0.165182
$546$	−0.441205	−0.0188818
$547$	17.9526	0.767599	0.383799	−	0.923416i	$-0.374615\pi$
$547$	17.9526	0.767599	0.383799	+	0.923416i	$0.374615\pi$
$548$	−16.2119	−0.692538
$549$	−12.0624	−0.514812
$550$	11.9192	0.508236
$551$	6.31212	0.268905
$552$	0	0
$553$	3.48252	0.148092
$554$	25.4356	1.08065
$555$	−0.220078	−0.00934181
$556$	−19.6328	−0.832618
$557$	14.1138	0.598023	0.299011	−	0.954250i	$-0.403343\pi$
$557$	14.1138	0.598023	0.299011	+	0.954250i	$0.403343\pi$
$558$	−2.46297	−0.104266
$559$	−11.9701	−0.506282
$560$	−0.148697	−0.00628360
$561$	6.70857	0.283236
$562$	23.5820	0.994749
$563$	22.2335	0.937029	0.468514	−	0.883456i	$-0.344789\pi$
$563$	22.2335	0.937029	0.468514	+	0.883456i	$0.344789\pi$
$564$	2.71406	0.114282
$565$	0.0888091	0.00373623
$566$	26.6302	1.11935
$567$	−0.324635	−0.0136334
$568$	16.4068	0.688414
$569$	−13.1629	−0.551817	−0.275909	−	0.961184i	$-0.588979\pi$
$569$	−13.1629	−0.551817	−0.275909	+	0.961184i	$0.588979\pi$
$570$	−1.32553	−0.0555204
$571$	20.9313	0.875947	0.437973	−	0.898988i	$-0.355696\pi$
$571$	20.9313	0.875947	0.437973	+	0.898988i	$0.355696\pi$
$572$	3.38173	0.141397
$573$	−23.3924	−0.977230
$574$	0.275106	0.0114827
$575$	0	0
$576$	1.00000	0.0416667
$577$	−1.03276	−0.0429945	−0.0214972	−	0.999769i	$-0.506843\pi$
$577$	−1.03276	−0.0429945	−0.0214972	+	0.999769i	$0.506843\pi$
$578$	9.73101	0.404757
$579$	−6.01660	−0.250041
$580$	−0.999080	−0.0414845
$581$	−2.62416	−0.108868
$582$	−0.224268	−0.00929619
$583$	21.7127	0.899250
$584$	−11.7624	−0.486733
$585$	0.622519	0.0257380
$586$	6.59236	0.272328
$587$	−4.68509	−0.193374	−0.0966871	−	0.995315i	$-0.530825\pi$
$587$	−4.68509	−0.193374	−0.0966871	+	0.995315i	$0.530825\pi$
$588$	6.89461	0.284329
$589$	−7.12759	−0.293687
$590$	−4.98503	−0.205230
$591$	−15.4640	−0.636102
$592$	0.480474	0.0197474
$593$	−31.3586	−1.28774	−0.643871	−	0.765134i	$-0.722673\pi$
$593$	−31.3586	−1.28774	−0.643871	+	0.765134i	$0.722673\pi$
$594$	2.48825	0.102094
$595$	0.400903	0.0164354
$596$	3.17993	0.130255
$597$	11.7110	0.479300
$598$	0	0
$599$	27.2585	1.11375	0.556875	−	0.830596i	$-0.312000\pi$
$599$	27.2585	1.11375	0.556875	+	0.830596i	$0.312000\pi$
$600$	−4.79020	−0.195559
$601$	17.2603	0.704061	0.352031	−	0.935988i	$-0.385491\pi$
$601$	17.2603	0.704061	0.352031	+	0.935988i	$0.385491\pi$
$602$	−2.85923	−0.116533
$603$	10.5658	0.430272
$604$	−1.27472	−0.0518676
$605$	−2.20257	−0.0895471
$606$	10.6957	0.434481
$607$	−45.6597	−1.85327	−0.926634	−	0.375964i	$-0.877312\pi$
$607$	−45.6597	−1.85327	−0.926634	+	0.375964i	$0.877312\pi$
$608$	2.89389	0.117363
$609$	−0.708089	−0.0286932
$610$	5.52513	0.223706
$611$	−3.68862	−0.149226
$612$	−2.69611	−0.108984
$613$	−2.01747	−0.0814847	−0.0407424	−	0.999170i	$-0.512972\pi$
$613$	−2.01747	−0.0814847	−0.0407424	+	0.999170i	$0.512972\pi$
$614$	−22.2953	−0.899766
$615$	−0.388162	−0.0156522
$616$	0.807771	0.0325460
$617$	37.4980	1.50961	0.754807	−	0.655947i	$-0.227731\pi$
$617$	37.4980	1.50961	0.754807	+	0.655947i	$0.227731\pi$
$618$	2.02685	0.0815320
$619$	26.2989	1.05704	0.528520	−	0.848921i	$-0.322747\pi$
$619$	26.2989	1.05704	0.528520	+	0.848921i	$0.322747\pi$
$620$	1.12815	0.0453077
$621$	0	0
$622$	12.9065	0.517503
$623$	3.34318	0.133942
$624$	−1.35908	−0.0544068
$625$	21.8969	0.875878
$626$	19.2227	0.768295
$627$	7.20072	0.287569
$628$	−17.3357	−0.691769
$629$	−1.29541	−0.0516513
$630$	0.148697	0.00592424
$631$	20.4519	0.814176	0.407088	−	0.913389i	$-0.366544\pi$
$631$	20.4519	0.814176	0.407088	+	0.913389i	$0.366544\pi$
$632$	10.7275	0.426717
$633$	−10.6750	−0.424294
$634$	10.7412	0.426587
$635$	5.92217	0.235014
$636$	−8.72613	−0.346013
$637$	−9.37033	−0.371266
$638$	5.42733	0.214870
$639$	−16.4068	−0.649043
$640$	−0.458044	−0.0181058
$641$	28.4557	1.12393	0.561965	−	0.827161i	$-0.310045\pi$
$641$	28.4557	1.12393	0.561965	+	0.827161i	$0.310045\pi$
$642$	2.16835	0.0855778
$643$	35.1379	1.38570	0.692851	−	0.721080i	$-0.256354\pi$
$643$	35.1379	1.38570	0.692851	+	0.721080i	$0.256354\pi$
$644$	0	0
$645$	4.03423	0.158848
$646$	−7.80224	−0.306975
$647$	−12.2486	−0.481544	−0.240772	−	0.970582i	$-0.577401\pi$
$647$	−12.2486	−0.481544	−0.240772	+	0.970582i	$0.577401\pi$
$648$	−1.00000	−0.0392837
$649$	27.0803	1.06299
$650$	6.51026	0.255353
$651$	0.799567	0.0313375
$652$	−5.27243	−0.206484
$653$	−23.9493	−0.937210	−0.468605	−	0.883408i	$-0.655243\pi$
$653$	−23.9493	−0.937210	−0.468605	+	0.883408i	$0.655243\pi$
$654$	−8.41887	−0.329204
$655$	−2.69893	−0.105456
$656$	0.847432	0.0330867
$657$	11.7624	0.458897
$658$	−0.881077	−0.0343480
$659$	−9.37070	−0.365031	−0.182515	−	0.983203i	$-0.558424\pi$
$659$	−9.37070	−0.365031	−0.182515	+	0.983203i	$0.558424\pi$
$660$	−1.13973	−0.0443638
$661$	20.5838	0.800615	0.400308	−	0.916381i	$-0.368903\pi$
$661$	20.5838	0.800615	0.400308	+	0.916381i	$0.368903\pi$
$662$	17.5358	0.681549
$663$	3.66422	0.142307
$664$	−8.08342	−0.313698
$665$	0.430314	0.0166869
$666$	−0.480474	−0.0186180
$667$	0	0
$668$	−2.95796	−0.114447
$669$	−7.57171	−0.292739
$670$	−4.83960	−0.186970
$671$	−30.0143	−1.15869
$672$	−0.324635	−0.0125231
$673$	−5.04670	−0.194536	−0.0972681	−	0.995258i	$-0.531010\pi$
$673$	−5.04670	−0.194536	−0.0972681	+	0.995258i	$0.531010\pi$
$674$	13.2185	0.509158
$675$	4.79020	0.184375
$676$	−11.1529	−0.428958
$677$	35.8747	1.37878	0.689389	−	0.724391i	$-0.257879\pi$
$677$	35.8747	1.37878	0.689389	+	0.724391i	$0.257879\pi$
$678$	0.193888	0.00744621
$679$	0.0728051	0.00279400
$680$	1.23494	0.0473576
$681$	12.6447	0.484545
$682$	−6.12848	−0.234672
$683$	22.1365	0.847029	0.423514	−	0.905889i	$-0.360796\pi$
$683$	22.1365	0.847029	0.423514	+	0.905889i	$0.360796\pi$
$684$	−2.89389	−0.110651
$685$	−7.42577	−0.283724
$686$	−4.51067	−0.172218
$687$	8.39479	0.320281
$688$	−8.80752	−0.335783
$689$	11.8595	0.451811
$690$	0	0
$691$	19.2304	0.731560	0.365780	−	0.930701i	$-0.380802\pi$
$691$	19.2304	0.731560	0.365780	+	0.930701i	$0.380802\pi$
$692$	4.13279	0.157105
$693$	−0.807771	−0.0306847
$694$	15.9720	0.606289
$695$	−8.99271	−0.341113
$696$	−2.18119	−0.0826777
$697$	−2.28477	−0.0865417
$698$	−2.86564	−0.108466
$699$	−26.2816	−0.994062
$700$	1.55506	0.0587759
$701$	−23.7269	−0.896154	−0.448077	−	0.893995i	$-0.647891\pi$
$701$	−23.7269	−0.896154	−0.448077	+	0.893995i	$0.647891\pi$
$702$	1.35908	0.0512952
$703$	−1.39044	−0.0524415
$704$	2.48825	0.0937793
$705$	1.24316	0.0468200
$706$	−8.05226	−0.303051
$707$	−3.47218	−0.130585
$708$	−10.8833	−0.409019
$709$	−21.0057	−0.788885	−0.394443	−	0.918921i	$-0.629062\pi$
$709$	−21.0057	−0.788885	−0.394443	+	0.918921i	$0.629062\pi$
$710$	7.51504	0.282034
$711$	−10.7275	−0.402313
$712$	10.2983	0.385945
$713$	0	0
$714$	0.875250	0.0327554
$715$	1.54898	0.0579286
$716$	10.1370	0.378838
$717$	−7.85374	−0.293303
$718$	−7.86143	−0.293386
$719$	12.3412	0.460251	0.230125	−	0.973161i	$-0.426086\pi$
$719$	12.3412	0.460251	0.230125	+	0.973161i	$0.426086\pi$
$720$	0.458044	0.0170703
$721$	−0.657987	−0.0245047
$722$	10.6254	0.395436
$723$	26.5043	0.985704
$724$	−26.6380	−0.989992
$725$	10.4483	0.388040
$726$	−4.80864	−0.178465
$727$	11.8297	0.438739	0.219369	−	0.975642i	$-0.429600\pi$
$727$	11.8297	0.438739	0.219369	+	0.975642i	$0.429600\pi$
$728$	0.441205	0.0163521
$729$	1.00000	0.0370370
$730$	−5.38772	−0.199408
$731$	23.7460	0.878277
$732$	12.0624	0.445840
$733$	45.7853	1.69112	0.845559	−	0.533882i	$-0.179267\pi$
$733$	45.7853	1.69112	0.845559	+	0.533882i	$0.179267\pi$
$734$	−13.9011	−0.513099
$735$	3.15804	0.116486
$736$	0	0
$737$	26.2903	0.968415
$738$	−0.847432	−0.0311944
$739$	−8.85715	−0.325816	−0.162908	−	0.986641i	$-0.552087\pi$
$739$	−8.85715	−0.325816	−0.162908	+	0.986641i	$0.552087\pi$
$740$	0.220078	0.00809024
$741$	3.93303	0.144484
$742$	2.83280	0.103996
$743$	−35.3025	−1.29512	−0.647562	−	0.762012i	$-0.724211\pi$
$743$	−35.3025	−1.29512	−0.647562	+	0.762012i	$0.724211\pi$
$744$	2.46297	0.0902970
$745$	1.45655	0.0533638
$746$	9.66207	0.353753
$747$	8.08342	0.295757
$748$	−6.70857	−0.245290
$749$	−0.703920	−0.0257207
$750$	−4.48434	−0.163745
$751$	17.7586	0.648021	0.324011	−	0.946053i	$-0.394969\pi$
$751$	17.7586	0.648021	0.324011	+	0.946053i	$0.394969\pi$
$752$	−2.71406	−0.0989714
$753$	18.0533	0.657899
$754$	2.96441	0.107957
$755$	−0.583878	−0.0212495
$756$	0.324635	0.0118069
$757$	8.03022	0.291863	0.145932	−	0.989295i	$-0.453382\pi$
$757$	8.03022	0.291863	0.145932	+	0.989295i	$0.453382\pi$
$758$	19.9331	0.724002
$759$	0	0
$760$	1.32553	0.0480821
$761$	20.8102	0.754370	0.377185	−	0.926138i	$-0.376892\pi$
$761$	20.8102	0.754370	0.377185	+	0.926138i	$0.376892\pi$
$762$	12.9293	0.468377
$763$	2.73306	0.0989434
$764$	23.3924	0.846306
$765$	−1.23494	−0.0446492
$766$	−27.9735	−1.01072
$767$	14.7913	0.534081
$768$	−1.00000	−0.0360844
$769$	29.3239	1.05745	0.528724	−	0.848794i	$-0.322671\pi$
$769$	29.3239	1.05745	0.528724	+	0.848794i	$0.322671\pi$
$770$	0.369995	0.0133337
$771$	22.8558	0.823131
$772$	6.01660	0.216542
$773$	−41.7014	−1.49990	−0.749948	−	0.661497i	$-0.769921\pi$
$773$	−41.7014	−1.49990	−0.749948	+	0.661497i	$0.769921\pi$
$774$	8.80752	0.316580
$775$	−11.7981	−0.423801
$776$	0.224268	0.00805074
$777$	0.155979	0.00559570
$778$	26.4792	0.949327
$779$	−2.45238	−0.0878656
$780$	−0.622519	−0.0222898
$781$	−40.8241	−1.46080
$782$	0	0
$783$	2.18119	0.0779492
$784$	−6.89461	−0.246236
$785$	−7.94052	−0.283409
$786$	−5.89229	−0.210171
$787$	44.1710	1.57452	0.787262	−	0.616618i	$-0.211498\pi$
$787$	44.1710	1.57452	0.787262	+	0.616618i	$0.211498\pi$
$788$	15.4640	0.550881
$789$	18.6195	0.662870
$790$	4.91367	0.174821
$791$	−0.0629427	−0.00223798
$792$	−2.48825	−0.0884159
$793$	−16.3938	−0.582162
$794$	16.4478	0.583711
$795$	−3.99695	−0.141757
$796$	−11.7110	−0.415086
$797$	−30.9863	−1.09759	−0.548796	−	0.835957i	$-0.684913\pi$
$797$	−30.9863	−1.09759	−0.548796	+	0.835957i	$0.684913\pi$
$798$	0.939459	0.0332565
$799$	7.31738	0.258870
$800$	4.79020	0.169359
$801$	−10.2983	−0.363872
$802$	−20.2422	−0.714775
$803$	29.2678	1.03284
$804$	−10.5658	−0.372627
$805$	0	0
$806$	−3.34738	−0.117906
$807$	−0.788737	−0.0277648
$808$	−10.6957	−0.376272
$809$	−25.4796	−0.895817	−0.447908	−	0.894080i	$-0.647831\pi$
$809$	−25.4796	−0.895817	−0.447908	+	0.894080i	$0.647831\pi$
$810$	−0.458044	−0.0160940
$811$	−21.6232	−0.759293	−0.379646	−	0.925132i	$-0.623954\pi$
$811$	−21.6232	−0.759293	−0.379646	+	0.925132i	$0.623954\pi$
$812$	0.708089	0.0248491
$813$	−29.3202	−1.02831
$814$	−1.19554	−0.0419035
$815$	−2.41501	−0.0845941
$816$	2.69611	0.0943826
$817$	25.4880	0.891713
$818$	−17.4324	−0.609511
$819$	−0.441205	−0.0154169
$820$	0.388162	0.0135552
$821$	−47.8261	−1.66914	−0.834572	−	0.550899i	$-0.814285\pi$
$821$	−47.8261	−1.66914	−0.834572	+	0.550899i	$0.814285\pi$
$822$	−16.2119	−0.565455
$823$	−21.3120	−0.742889	−0.371445	−	0.928455i	$-0.621137\pi$
$823$	−21.3120	−0.742889	−0.371445	+	0.928455i	$0.621137\pi$
$824$	−2.02685	−0.0706088
$825$	11.9192	0.414973
$826$	3.53309	0.122932
$827$	8.52696	0.296512	0.148256	−	0.988949i	$-0.452634\pi$
$827$	8.52696	0.296512	0.148256	+	0.988949i	$0.452634\pi$
$828$	0	0
$829$	−28.0337	−0.973651	−0.486826	−	0.873499i	$-0.661845\pi$
$829$	−28.0337	−0.973651	−0.486826	+	0.873499i	$0.661845\pi$
$830$	−3.70257	−0.128518
$831$	25.4356	0.882350
$832$	1.35908	0.0471176
$833$	18.5886	0.644057
$834$	−19.6328	−0.679830
$835$	−1.35488	−0.0468875
$836$	−7.20072	−0.249042
$837$	−2.46297	−0.0851328
$838$	16.4612	0.568644
$839$	39.4372	1.36152	0.680761	−	0.732505i	$-0.261649\pi$
$839$	39.4372	1.36152	0.680761	+	0.732505i	$0.261649\pi$
$840$	−0.148697	−0.00513054
$841$	−24.2424	−0.835946
$842$	30.9773	1.06755
$843$	23.5820	0.812209
$844$	10.6750	0.367449
$845$	−5.10852	−0.175739
$846$	2.71406	0.0933112
$847$	1.56105	0.0536383
$848$	8.72613	0.299656
$849$	26.6302	0.913945
$850$	−12.9149	−0.442976
$851$	0	0
$852$	16.4068	0.562088
$853$	46.1009	1.57846	0.789232	−	0.614095i	$-0.210479\pi$
$853$	46.1009	1.57846	0.789232	+	0.614095i	$0.210479\pi$
$854$	−3.91589	−0.133999
$855$	−1.32553	−0.0453322
$856$	−2.16835	−0.0741125
$857$	−12.5505	−0.428718	−0.214359	−	0.976755i	$-0.568766\pi$
$857$	−12.5505	−0.428718	−0.214359	+	0.976755i	$0.568766\pi$
$858$	3.38173	0.115450
$859$	−4.90320	−0.167295	−0.0836476	−	0.996495i	$-0.526657\pi$
$859$	−4.90320	−0.167295	−0.0836476	+	0.996495i	$0.526657\pi$
$860$	−4.03423	−0.137566
$861$	0.275106	0.00937559
$862$	18.3159	0.623842
$863$	−39.7277	−1.35235	−0.676173	−	0.736743i	$-0.736363\pi$
$863$	−39.7277	−1.35235	−0.676173	+	0.736743i	$0.736363\pi$
$864$	1.00000	0.0340207
$865$	1.89300	0.0643640
$866$	−15.1498	−0.514812
$867$	9.73101	0.330483
$868$	−0.799567	−0.0271391
$869$	−26.6927	−0.905487
$870$	−0.999080	−0.0338720
$871$	14.3598	0.486562
$872$	8.41887	0.285099
$873$	−0.224268	−0.00759031
$874$	0	0
$875$	1.45577	0.0492142
$876$	−11.7624	−0.397416
$877$	2.18187	0.0736764	0.0368382	−	0.999321i	$-0.488271\pi$
$877$	2.18187	0.0736764	0.0368382	+	0.999321i	$0.488271\pi$
$878$	−5.83176	−0.196812
$879$	6.59236	0.222355
$880$	1.13973	0.0384202
$881$	−50.7112	−1.70850	−0.854251	−	0.519861i	$-0.825984\pi$
$881$	−50.7112	−1.70850	−0.854251	+	0.519861i	$0.825984\pi$
$882$	6.89461	0.232154
$883$	−38.2922	−1.28863	−0.644317	−	0.764759i	$-0.722858\pi$
$883$	−38.2922	−1.28863	−0.644317	+	0.764759i	$0.722858\pi$
$884$	−3.66422	−0.123241
$885$	−4.98503	−0.167570
$886$	17.3465	0.582765
$887$	21.5525	0.723661	0.361831	−	0.932244i	$-0.382152\pi$
$887$	21.5525	0.723661	0.361831	+	0.932244i	$0.382152\pi$
$888$	0.480474	0.0161236
$889$	−4.19729	−0.140772
$890$	4.71707	0.158117
$891$	2.48825	0.0833594
$892$	7.57171	0.253519
$893$	7.85419	0.262830
$894$	3.17993	0.106353
$895$	4.64321	0.155205
$896$	0.324635	0.0108453
$897$	0	0
$898$	18.5051	0.617522
$899$	−5.37221	−0.179173
$900$	−4.79020	−0.159673
$901$	−23.5266	−0.783783
$902$	−2.10862	−0.0702093
$903$	−2.85923	−0.0951491
$904$	−0.193888	−0.00644860
$905$	−12.2014	−0.405587
$906$	−1.27472	−0.0423497
$907$	28.7244	0.953778	0.476889	−	0.878964i	$-0.341764\pi$
$907$	28.7244	0.953778	0.476889	+	0.878964i	$0.341764\pi$
$908$	−12.6447	−0.419628
$909$	10.6957	0.354752
$910$	0.202091	0.00669926
$911$	−3.63697	−0.120498	−0.0602491	−	0.998183i	$-0.519190\pi$
$911$	−3.63697	−0.120498	−0.0602491	+	0.998183i	$0.519190\pi$
$912$	2.89389	0.0958264
$913$	20.1135	0.665661
$914$	5.76183	0.190584
$915$	5.52513	0.182655
$916$	−8.39479	−0.277372
$917$	1.91284	0.0631676
$918$	−2.69611	−0.0889847
$919$	14.2577	0.470318	0.235159	−	0.971957i	$-0.424439\pi$
$919$	14.2577	0.470318	0.235159	+	0.971957i	$0.424439\pi$
$920$	0	0
$921$	−22.2953	−0.734656
$922$	−22.5329	−0.742081
$923$	−22.2982	−0.733953
$924$	0.807771	0.0265737
$925$	−2.30156	−0.0756749
$926$	29.9997	0.985850
$927$	2.02685	0.0665706
$928$	2.18119	0.0716009
$929$	52.1683	1.71159	0.855794	−	0.517317i	$-0.173069\pi$
$929$	52.1683	1.71159	0.855794	+	0.517317i	$0.173069\pi$
$930$	1.12815	0.0369936
$931$	19.9523	0.653910
$932$	26.2816	0.860883
$933$	12.9065	0.422539
$934$	11.6433	0.380981
$935$	−3.07282	−0.100492
$936$	−1.35908	−0.0444229
$937$	−36.8202	−1.20286	−0.601432	−	0.798924i	$-0.705403\pi$
$937$	−36.8202	−1.20286	−0.601432	+	0.798924i	$0.705403\pi$
$938$	3.43002	0.111994
$939$	19.2227	0.627310
$940$	−1.24316	−0.0405473
$941$	−16.4890	−0.537527	−0.268764	−	0.963206i	$-0.586615\pi$
$941$	−16.4890	−0.537527	−0.268764	+	0.963206i	$0.586615\pi$
$942$	−17.3357	−0.564827
$943$	0	0
$944$	10.8833	0.354221
$945$	0.148697	0.00483712
$946$	21.9153	0.712527
$947$	48.5457	1.57752	0.788762	−	0.614699i	$-0.210723\pi$
$947$	48.5457	1.57752	0.788762	+	0.614699i	$0.210723\pi$
$948$	10.7275	0.348413
$949$	15.9861	0.518931
$950$	−13.8623	−0.449753
$951$	10.7412	0.348307
$952$	−0.875250	−0.0283670
$953$	56.7652	1.83880	0.919402	−	0.393318i	$-0.128673\pi$
$953$	56.7652	1.83880	0.919402	+	0.393318i	$0.128673\pi$
$954$	−8.72613	−0.282519
$955$	10.7147	0.346721
$956$	7.85374	0.254008
$957$	5.42733	0.175441
$958$	−13.3394	−0.430976
$959$	5.26295	0.169949
$960$	−0.458044	−0.0147833
$961$	−24.9338	−0.804315
$962$	−0.653003	−0.0210536
$963$	2.16835	0.0698740
$964$	−26.5043	−0.853644
$965$	2.75587	0.0887146
$966$	0	0
$967$	26.3683	0.847948	0.423974	−	0.905674i	$-0.360635\pi$
$967$	26.3683	0.847948	0.423974	+	0.905674i	$0.360635\pi$
$968$	4.80864	0.154555
$969$	−7.80224	−0.250644
$970$	0.102725	0.00329829
$971$	5.85306	0.187834	0.0939169	−	0.995580i	$-0.470061\pi$
$971$	5.85306	0.187834	0.0939169	+	0.995580i	$0.470061\pi$
$972$	−1.00000	−0.0320750
$973$	6.37350	0.204325
$974$	13.5203	0.433218
$975$	6.51026	0.208495
$976$	−12.0624	−0.386109
$977$	−31.7051	−1.01434	−0.507169	−	0.861847i	$-0.669308\pi$
$977$	−31.7051	−1.01434	−0.507169	+	0.861847i	$0.669308\pi$
$978$	−5.27243	−0.168594
$979$	−25.6247	−0.818968
$980$	−3.15804	−0.100880
$981$	−8.41887	−0.268794
$982$	−40.9468	−1.30667
$983$	25.9223	0.826794	0.413397	−	0.910551i	$-0.364342\pi$
$983$	25.9223	0.826794	0.413397	+	0.910551i	$0.364342\pi$
$984$	0.847432	0.0270152
$985$	7.08318	0.225689
$986$	−5.88071	−0.187280
$987$	−0.881077	−0.0280450
$988$	−3.93303	−0.125127
$989$	0	0
$990$	−1.13973	−0.0362229
$991$	−39.4166	−1.25211	−0.626054	−	0.779779i	$-0.715331\pi$
$991$	−39.4166	−1.25211	−0.626054	+	0.779779i	$0.715331\pi$
$992$	−2.46297	−0.0781995
$993$	17.5358	0.556482
$994$	−5.32622	−0.168937
$995$	−5.36416	−0.170055
$996$	−8.08342	−0.256133
$997$	−29.7606	−0.942526	−0.471263	−	0.881993i	$-0.656202\pi$
$997$	−29.7606	−0.942526	−0.471263	+	0.881993i	$0.656202\pi$
$998$	−21.3933	−0.677194
$999$	−0.480474	−0.0152015

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3174.2.a.x.1.3		5
3.2	odd	2		9522.2.a.bx.1.3		5
23.3	even	11		138.2.e.d.55.1	✓	10
23.8	even	11		138.2.e.d.133.1	yes	10
23.22	odd	2		3174.2.a.w.1.3		5
69.8	odd	22		414.2.i.a.271.1		10
69.26	odd	22		414.2.i.a.55.1		10
69.68	even	2		9522.2.a.by.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.d.55.1	✓	10	23.3	even	11
138.2.e.d.133.1	yes	10	23.8	even	11
414.2.i.a.55.1		10	69.26	odd	22
414.2.i.a.271.1		10	69.8	odd	22
3174.2.a.w.1.3		5	23.22	odd	2
3174.2.a.x.1.3		5	1.1	even	1	trivial
9522.2.a.bx.1.3		5	3.2	odd	2
9522.2.a.by.1.3		5	69.68	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3174 = 2 \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3174.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.458044 q^{5} +1.00000 q^{6} -0.324635 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.458044 q^{5} +1.00000 q^{6} -0.324635 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.458044 q^{10} +2.48825 q^{11} -1.00000 q^{12} +1.35908 q^{13} +0.324635 q^{14} -0.458044 q^{15} +1.00000 q^{16} -2.69611 q^{17} -1.00000 q^{18} -2.89389 q^{19} +0.458044 q^{20} +0.324635 q^{21} -2.48825 q^{22} +1.00000 q^{24} -4.79020 q^{25} -1.35908 q^{26} -1.00000 q^{27} -0.324635 q^{28} -2.18119 q^{29} +0.458044 q^{30} +2.46297 q^{31} -1.00000 q^{32} -2.48825 q^{33} +2.69611 q^{34} -0.148697 q^{35} +1.00000 q^{36} +0.480474 q^{37} +2.89389 q^{38} -1.35908 q^{39} -0.458044 q^{40} +0.847432 q^{41} -0.324635 q^{42} -8.80752 q^{43} +2.48825 q^{44} +0.458044 q^{45} -2.71406 q^{47} -1.00000 q^{48} -6.89461 q^{49} +4.79020 q^{50} +2.69611 q^{51} +1.35908 q^{52} +8.72613 q^{53} +1.00000 q^{54} +1.13973 q^{55} +0.324635 q^{56} +2.89389 q^{57} +2.18119 q^{58} +10.8833 q^{59} -0.458044 q^{60} -12.0624 q^{61} -2.46297 q^{62} -0.324635 q^{63} +1.00000 q^{64} +0.622519 q^{65} +2.48825 q^{66} +10.5658 q^{67} -2.69611 q^{68} +0.148697 q^{70} -16.4068 q^{71} -1.00000 q^{72} +11.7624 q^{73} -0.480474 q^{74} +4.79020 q^{75} -2.89389 q^{76} -0.807771 q^{77} +1.35908 q^{78} -10.7275 q^{79} +0.458044 q^{80} +1.00000 q^{81} -0.847432 q^{82} +8.08342 q^{83} +0.324635 q^{84} -1.23494 q^{85} +8.80752 q^{86} +2.18119 q^{87} -2.48825 q^{88} -10.2983 q^{89} -0.458044 q^{90} -0.441205 q^{91} -2.46297 q^{93} +2.71406 q^{94} -1.32553 q^{95} +1.00000 q^{96} -0.224268 q^{97} +6.89461 q^{98} +2.48825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} + q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 + q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} + q^{7} - 5 q^{8} + 5 q^{9} - q^{10} + 3 q^{11} - 5 q^{12} - 10 q^{13} - q^{14} - q^{15} + 5 q^{16} + 11 q^{17} - 5 q^{18} - 11 q^{19} + q^{20} - q^{21} - 3 q^{22} + 5 q^{24} + 6 q^{25} + 10 q^{26} - 5 q^{27} + q^{28} - 25 q^{29} + q^{30} - 4 q^{31} - 5 q^{32} - 3 q^{33} - 11 q^{34} + 9 q^{35} + 5 q^{36} - 22 q^{37} + 11 q^{38} + 10 q^{39} - q^{40} + 6 q^{41} + q^{42} + 6 q^{43} + 3 q^{44} + q^{45} + 4 q^{47} - 5 q^{48} - 4 q^{49} - 6 q^{50} - 11 q^{51} - 10 q^{52} - q^{53} + 5 q^{54} - 6 q^{55} - q^{56} + 11 q^{57} + 25 q^{58} + 6 q^{59} - q^{60} - 35 q^{61} + 4 q^{62} + q^{63} + 5 q^{64} - 2 q^{65} + 3 q^{66} - 5 q^{67} + 11 q^{68} - 9 q^{70} - 11 q^{71} - 5 q^{72} + 16 q^{73} + 22 q^{74} - 6 q^{75} - 11 q^{76} + 5 q^{77} - 10 q^{78} - 8 q^{79} + q^{80} + 5 q^{81} - 6 q^{82} + 26 q^{83} - q^{84} - 33 q^{85} - 6 q^{86} + 25 q^{87} - 3 q^{88} - q^{90} - 35 q^{91} + 4 q^{93} - 4 q^{94} + 11 q^{95} + 5 q^{96} - 19 q^{97} + 4 q^{98} + 3 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 + q^7 - 5 * q^8 + 5 * q^9 - q^10 + 3 * q^11 - 5 * q^12 - 10 * q^13 - q^14 - q^15 + 5 * q^16 + 11 * q^17 - 5 * q^18 - 11 * q^19 + q^20 - q^21 - 3 * q^22 + 5 * q^24 + 6 * q^25 + 10 * q^26 - 5 * q^27 + q^28 - 25 * q^29 + q^30 - 4 * q^31 - 5 * q^32 - 3 * q^33 - 11 * q^34 + 9 * q^35 + 5 * q^36 - 22 * q^37 + 11 * q^38 + 10 * q^39 - q^40 + 6 * q^41 + q^42 + 6 * q^43 + 3 * q^44 + q^45 + 4 * q^47 - 5 * q^48 - 4 * q^49 - 6 * q^50 - 11 * q^51 - 10 * q^52 - q^53 + 5 * q^54 - 6 * q^55 - q^56 + 11 * q^57 + 25 * q^58 + 6 * q^59 - q^60 - 35 * q^61 + 4 * q^62 + q^63 + 5 * q^64 - 2 * q^65 + 3 * q^66 - 5 * q^67 + 11 * q^68 - 9 * q^70 - 11 * q^71 - 5 * q^72 + 16 * q^73 + 22 * q^74 - 6 * q^75 - 11 * q^76 + 5 * q^77 - 10 * q^78 - 8 * q^79 + q^80 + 5 * q^81 - 6 * q^82 + 26 * q^83 - q^84 - 33 * q^85 - 6 * q^86 + 25 * q^87 - 3 * q^88 - q^90 - 35 * q^91 + 4 * q^93 - 4 * q^94 + 11 * q^95 + 5 * q^96 - 19 * q^97 + 4 * q^98 + 3 * q^99

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