LMFDB - Embedded newform 338.2.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,2,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	338.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:	$2.69894358832$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.69202 q^{3} +1.00000 q^{4} -2.49396 q^{5} +2.69202 q^{6} +1.60388 q^{7} +1.00000 q^{8} +4.24698 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.69202 q^{3} +1.00000 q^{4} -2.49396 q^{5} +2.69202 q^{6} +1.60388 q^{7} +1.00000 q^{8} +4.24698 q^{9} -2.49396 q^{10} -2.04892 q^{11} +2.69202 q^{12} +1.60388 q^{14} -6.71379 q^{15} +1.00000 q^{16} -4.54288 q^{17} +4.24698 q^{18} +4.85086 q^{19} -2.49396 q^{20} +4.31767 q^{21} -2.04892 q^{22} +2.71379 q^{23} +2.69202 q^{24} +1.21983 q^{25} +3.35690 q^{27} +1.60388 q^{28} -9.20775 q^{29} -6.71379 q^{30} -5.10992 q^{31} +1.00000 q^{32} -5.51573 q^{33} -4.54288 q^{34} -4.00000 q^{35} +4.24698 q^{36} -7.60388 q^{37} +4.85086 q^{38} -2.49396 q^{40} +3.46681 q^{41} +4.31767 q^{42} +11.3448 q^{43} -2.04892 q^{44} -10.5918 q^{45} +2.71379 q^{46} -0.219833 q^{47} +2.69202 q^{48} -4.42758 q^{49} +1.21983 q^{50} -12.2295 q^{51} -2.71379 q^{53} +3.35690 q^{54} +5.10992 q^{55} +1.60388 q^{56} +13.0586 q^{57} -9.20775 q^{58} +4.07606 q^{59} -6.71379 q^{60} +10.4155 q^{61} -5.10992 q^{62} +6.81163 q^{63} +1.00000 q^{64} -5.51573 q^{66} +12.0761 q^{67} -4.54288 q^{68} +7.30559 q^{69} -4.00000 q^{70} -1.28621 q^{71} +4.24698 q^{72} -3.62565 q^{73} -7.60388 q^{74} +3.28382 q^{75} +4.85086 q^{76} -3.28621 q^{77} -5.32975 q^{79} -2.49396 q^{80} -3.70410 q^{81} +3.46681 q^{82} -4.85086 q^{83} +4.31767 q^{84} +11.3297 q^{85} +11.3448 q^{86} -24.7875 q^{87} -2.04892 q^{88} +16.5700 q^{89} -10.5918 q^{90} +2.71379 q^{92} -13.7560 q^{93} -0.219833 q^{94} -12.0978 q^{95} +2.69202 q^{96} -4.64071 q^{97} -4.42758 q^{98} -8.70171 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} + 3 q^{6} - 4 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 2 * q^5 + 3 * q^6 - 4 * q^7 + 3 * q^8 + 8 * q^9	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} + 3 q^{6} - 4 q^{7} + 3 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} + 3 q^{12} - 4 q^{14} - 12 q^{15} + 3 q^{16} + 5 q^{17} + 8 q^{18} + q^{19} + 2 q^{20} - 4 q^{21} + 3 q^{22} + 3 q^{24} + 5 q^{25} + 6 q^{27} - 4 q^{28} - 10 q^{29} - 12 q^{30} - 16 q^{31} + 3 q^{32} - 4 q^{33} + 5 q^{34} - 12 q^{35} + 8 q^{36} - 14 q^{37} + q^{38} + 2 q^{40} + 7 q^{41} - 4 q^{42} + 11 q^{43} + 3 q^{44} - 4 q^{45} - 2 q^{47} + 3 q^{48} + 3 q^{49} + 5 q^{50} - 16 q^{51} + 6 q^{54} + 16 q^{55} - 4 q^{56} + 8 q^{57} - 10 q^{58} - 3 q^{59} - 12 q^{60} - 4 q^{61} - 16 q^{62} - 6 q^{63} + 3 q^{64} - 4 q^{66} + 21 q^{67} + 5 q^{68} - 14 q^{69} - 12 q^{70} - 12 q^{71} + 8 q^{72} + q^{73} - 14 q^{74} - 23 q^{75} + q^{76} - 18 q^{77} - 18 q^{79} + 2 q^{80} - 25 q^{81} + 7 q^{82} - q^{83} - 4 q^{84} + 36 q^{85} + 11 q^{86} - 10 q^{87} + 3 q^{88} + 25 q^{89} - 4 q^{90} - 2 q^{93} - 2 q^{94} - 18 q^{95} + 3 q^{96} + 23 q^{97} + 3 q^{98} + q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 2 * q^5 + 3 * q^6 - 4 * q^7 + 3 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 + 3 * q^12 - 4 * q^14 - 12 * q^15 + 3 * q^16 + 5 * q^17 + 8 * q^18 + q^19 + 2 * q^20 - 4 * q^21 + 3 * q^22 + 3 * q^24 + 5 * q^25 + 6 * q^27 - 4 * q^28 - 10 * q^29 - 12 * q^30 - 16 * q^31 + 3 * q^32 - 4 * q^33 + 5 * q^34 - 12 * q^35 + 8 * q^36 - 14 * q^37 + q^38 + 2 * q^40 + 7 * q^41 - 4 * q^42 + 11 * q^43 + 3 * q^44 - 4 * q^45 - 2 * q^47 + 3 * q^48 + 3 * q^49 + 5 * q^50 - 16 * q^51 + 6 * q^54 + 16 * q^55 - 4 * q^56 + 8 * q^57 - 10 * q^58 - 3 * q^59 - 12 * q^60 - 4 * q^61 - 16 * q^62 - 6 * q^63 + 3 * q^64 - 4 * q^66 + 21 * q^67 + 5 * q^68 - 14 * q^69 - 12 * q^70 - 12 * q^71 + 8 * q^72 + q^73 - 14 * q^74 - 23 * q^75 + q^76 - 18 * q^77 - 18 * q^79 + 2 * q^80 - 25 * q^81 + 7 * q^82 - q^83 - 4 * q^84 + 36 * q^85 + 11 * q^86 - 10 * q^87 + 3 * q^88 + 25 * q^89 - 4 * q^90 - 2 * q^93 - 2 * q^94 - 18 * q^95 + 3 * q^96 + 23 * q^97 + 3 * q^98 + 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
$3$	2.69202	1.55424	0.777120	−	0.629353i	$-0.216680\pi$
$3$	2.69202	1.55424	0.777120	+	0.629353i	$0.216680\pi$
$4$	1.00000	0.500000
$5$	−2.49396	−1.11533	−0.557666	−	0.830065i	$-0.688303\pi$
$5$	−2.49396	−1.11533	−0.557666	+	0.830065i	$0.688303\pi$
$6$	2.69202	1.09901
$7$	1.60388	0.606208	0.303104	−	0.952957i	$-0.401977\pi$
$7$	1.60388	0.606208	0.303104	+	0.952957i	$0.401977\pi$
$8$	1.00000	0.353553
$9$	4.24698	1.41566
$10$	−2.49396	−0.788659
$11$	−2.04892	−0.617772	−0.308886	−	0.951099i	$-0.599956\pi$
$11$	−2.04892	−0.617772	−0.308886	+	0.951099i	$0.599956\pi$
$12$	2.69202	0.777120
$13$	0	0
$13$	0	0
$14$	1.60388	0.428654
$15$	−6.71379	−1.73349
$16$	1.00000	0.250000
$17$	−4.54288	−1.10181	−0.550905	−	0.834568i	$-0.685717\pi$
$17$	−4.54288	−1.10181	−0.550905	+	0.834568i	$0.685717\pi$
$18$	4.24698	1.00102
$19$	4.85086	1.11286	0.556431	−	0.830894i	$-0.312170\pi$
$19$	4.85086	1.11286	0.556431	+	0.830894i	$0.312170\pi$
$20$	−2.49396	−0.557666
$21$	4.31767	0.942192
$22$	−2.04892	−0.436831
$23$	2.71379	0.565865	0.282932	−	0.959140i	$-0.408693\pi$
$23$	2.71379	0.565865	0.282932	+	0.959140i	$0.408693\pi$
$24$	2.69202	0.549507
$25$	1.21983	0.243967
$26$	0	0
$27$	3.35690	0.646035
$28$	1.60388	0.303104
$29$	−9.20775	−1.70984	−0.854918	−	0.518763i	$-0.826393\pi$
$29$	−9.20775	−1.70984	−0.854918	+	0.518763i	$0.826393\pi$
$30$	−6.71379	−1.22577
$31$	−5.10992	−0.917768	−0.458884	−	0.888496i	$-0.651751\pi$
$31$	−5.10992	−0.917768	−0.458884	+	0.888496i	$0.651751\pi$
$32$	1.00000	0.176777
$33$	−5.51573	−0.960165
$34$	−4.54288	−0.779097
$35$	−4.00000	−0.676123
$36$	4.24698	0.707830
$37$	−7.60388	−1.25007	−0.625035	−	0.780597i	$-0.714915\pi$
$37$	−7.60388	−1.25007	−0.625035	+	0.780597i	$0.714915\pi$
$38$	4.85086	0.786913
$39$	0	0
$40$	−2.49396	−0.394330
$41$	3.46681	0.541425	0.270713	−	0.962660i	$-0.412741\pi$
$41$	3.46681	0.541425	0.270713	+	0.962660i	$0.412741\pi$
$42$	4.31767	0.666231
$43$	11.3448	1.73007	0.865034	−	0.501713i	$-0.167297\pi$
$43$	11.3448	1.73007	0.865034	+	0.501713i	$0.167297\pi$
$44$	−2.04892	−0.308886
$45$	−10.5918	−1.57893
$46$	2.71379	0.400127
$47$	−0.219833	−0.0320659	−0.0160329	−	0.999871i	$-0.505104\pi$
$47$	−0.219833	−0.0320659	−0.0160329	+	0.999871i	$0.505104\pi$
$48$	2.69202	0.388560
$49$	−4.42758	−0.632512
$50$	1.21983	0.172510
$51$	−12.2295	−1.71248
$52$	0	0
$53$	−2.71379	−0.372768	−0.186384	−	0.982477i	$-0.559677\pi$
$53$	−2.71379	−0.372768	−0.186384	+	0.982477i	$0.559677\pi$
$54$	3.35690	0.456816
$55$	5.10992	0.689021
$56$	1.60388	0.214327
$57$	13.0586	1.72965
$58$	−9.20775	−1.20904
$59$	4.07606	0.530658	0.265329	−	0.964158i	$-0.414519\pi$
$59$	4.07606	0.530658	0.265329	+	0.964158i	$0.414519\pi$
$60$	−6.71379	−0.866747
$61$	10.4155	1.33357	0.666784	−	0.745251i	$-0.267670\pi$
$61$	10.4155	1.33357	0.666784	+	0.745251i	$0.267670\pi$
$62$	−5.10992	−0.648960
$63$	6.81163	0.858184
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.51573	−0.678939
$67$	12.0761	1.47533	0.737663	−	0.675169i	$-0.235929\pi$
$67$	12.0761	1.47533	0.737663	+	0.675169i	$0.235929\pi$
$68$	−4.54288	−0.550905
$69$	7.30559	0.879489
$70$	−4.00000	−0.478091
$71$	−1.28621	−0.152645	−0.0763224	−	0.997083i	$-0.524318\pi$
$71$	−1.28621	−0.152645	−0.0763224	+	0.997083i	$0.524318\pi$
$72$	4.24698	0.500511
$73$	−3.62565	−0.424350	−0.212175	−	0.977232i	$-0.568055\pi$
$73$	−3.62565	−0.424350	−0.212175	+	0.977232i	$0.568055\pi$
$74$	−7.60388	−0.883933
$75$	3.28382	0.379182
$76$	4.85086	0.556431
$77$	−3.28621	−0.374498
$78$	0	0
$79$	−5.32975	−0.599644	−0.299822	−	0.953995i	$-0.596927\pi$
$79$	−5.32975	−0.599644	−0.299822	+	0.953995i	$0.596927\pi$
$80$	−2.49396	−0.278833
$81$	−3.70410	−0.411567
$82$	3.46681	0.382845
$83$	−4.85086	−0.532451	−0.266225	−	0.963911i	$-0.585777\pi$
$83$	−4.85086	−0.532451	−0.266225	+	0.963911i	$0.585777\pi$
$84$	4.31767	0.471096
$85$	11.3297	1.22888
$86$	11.3448	1.22334
$87$	−24.7875	−2.65750
$88$	−2.04892	−0.218415
$89$	16.5700	1.75642	0.878209	−	0.478276i	$-0.158738\pi$
$89$	16.5700	1.75642	0.878209	+	0.478276i	$0.158738\pi$
$90$	−10.5918	−1.11647
$91$	0	0
$92$	2.71379	0.282932
$93$	−13.7560	−1.42643
$94$	−0.219833	−0.0226740
$95$	−12.0978	−1.24121
$96$	2.69202	0.274753
$97$	−4.64071	−0.471193	−0.235596	−	0.971851i	$-0.575704\pi$
$97$	−4.64071	−0.471193	−0.235596	+	0.971851i	$0.575704\pi$
$98$	−4.42758	−0.447253
$99$	−8.70171	−0.874555
$100$	1.21983	0.121983
$101$	7.42758	0.739072	0.369536	−	0.929216i	$-0.379517\pi$
$101$	7.42758	0.739072	0.369536	+	0.929216i	$0.379517\pi$
$102$	−12.2295	−1.21090
$103$	0.518122	0.0510521	0.0255261	−	0.999674i	$-0.491874\pi$
$103$	0.518122	0.0510521	0.0255261	+	0.999674i	$0.491874\pi$
$104$	0	0
$105$	−10.7681	−1.05086
$106$	−2.71379	−0.263587
$107$	3.51035	0.339359	0.169679	−	0.985499i	$-0.445727\pi$
$107$	3.51035	0.339359	0.169679	+	0.985499i	$0.445727\pi$
$108$	3.35690	0.323017
$109$	3.38404	0.324133	0.162066	−	0.986780i	$-0.448184\pi$
$109$	3.38404	0.324133	0.162066	+	0.986780i	$0.448184\pi$
$110$	5.10992	0.487211
$111$	−20.4698	−1.94291
$112$	1.60388	0.151552
$113$	−5.44935	−0.512632	−0.256316	−	0.966593i	$-0.582509\pi$
$113$	−5.44935	−0.512632	−0.256316	+	0.966593i	$0.582509\pi$
$114$	13.0586	1.22305
$115$	−6.76809	−0.631127
$116$	−9.20775	−0.854918
$117$	0	0
$118$	4.07606	0.375232
$119$	−7.28621	−0.667926
$120$	−6.71379	−0.612883
$121$	−6.80194	−0.618358
$122$	10.4155	0.942975
$123$	9.33273	0.841504
$124$	−5.10992	−0.458884
$125$	9.42758	0.843229
$126$	6.81163	0.606828
$127$	−6.19567	−0.549777	−0.274888	−	0.961476i	$-0.588641\pi$
$127$	−6.19567	−0.549777	−0.274888	+	0.961476i	$0.588641\pi$
$128$	1.00000	0.0883883
$129$	30.5405	2.68894
$130$	0	0
$131$	−4.13706	−0.361457	−0.180728	−	0.983533i	$-0.557846\pi$
$131$	−4.13706	−0.361457	−0.180728	+	0.983533i	$0.557846\pi$
$132$	−5.51573	−0.480083
$133$	7.78017	0.674626
$134$	12.0761	1.04321
$135$	−8.37196	−0.720544
$136$	−4.54288	−0.389548
$137$	19.3817	1.65589	0.827943	−	0.560812i	$-0.189511\pi$
$137$	19.3817	1.65589	0.827943	+	0.560812i	$0.189511\pi$
$138$	7.30559	0.621893
$139$	18.5864	1.57648	0.788240	−	0.615368i	$-0.210993\pi$
$139$	18.5864	1.57648	0.788240	+	0.615368i	$0.210993\pi$
$140$	−4.00000	−0.338062
$141$	−0.591794	−0.0498380
$142$	−1.28621	−0.107936
$143$	0	0
$144$	4.24698	0.353915
$145$	22.9638	1.90704
$146$	−3.62565	−0.300061
$147$	−11.9191	−0.983075
$148$	−7.60388	−0.625035
$149$	3.65817	0.299689	0.149844	−	0.988710i	$-0.452123\pi$
$149$	3.65817	0.299689	0.149844	+	0.988710i	$0.452123\pi$
$150$	3.28382	0.268122
$151$	−14.5918	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$151$	−14.5918	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$152$	4.85086	0.393456
$153$	−19.2935	−1.55979
$154$	−3.28621	−0.264810
$155$	12.7439	1.02362
$156$	0	0
$157$	21.6039	1.72418	0.862088	−	0.506758i	$-0.169156\pi$
$157$	21.6039	1.72418	0.862088	+	0.506758i	$0.169156\pi$
$158$	−5.32975	−0.424012
$159$	−7.30559	−0.579371
$160$	−2.49396	−0.197165
$161$	4.35258	0.343032
$162$	−3.70410	−0.291022
$163$	−13.6093	−1.06596	−0.532979	−	0.846128i	$-0.678928\pi$
$163$	−13.6093	−1.06596	−0.532979	+	0.846128i	$0.678928\pi$
$164$	3.46681	0.270713
$165$	13.7560	1.07090
$166$	−4.85086	−0.376499
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	4.31767	0.333115
$169$	0	0
$170$	11.3297	0.868952
$171$	20.6015	1.57543
$172$	11.3448	0.865034
$173$	−4.21983	−0.320828	−0.160414	−	0.987050i	$-0.551283\pi$
$173$	−4.21983	−0.320828	−0.160414	+	0.987050i	$0.551283\pi$
$174$	−24.7875	−1.87913
$175$	1.95646	0.147894
$176$	−2.04892	−0.154443
$177$	10.9729	0.824770
$178$	16.5700	1.24198
$179$	15.8291	1.18312	0.591561	−	0.806260i	$-0.298512\pi$
$179$	15.8291	1.18312	0.591561	+	0.806260i	$0.298512\pi$
$180$	−10.5918	−0.789466
$181$	−15.3056	−1.13766	−0.568828	−	0.822457i	$-0.692603\pi$
$181$	−15.3056	−1.13766	−0.568828	+	0.822457i	$0.692603\pi$
$182$	0	0
$183$	28.0388	2.07268
$184$	2.71379	0.200063
$185$	18.9638	1.39424
$186$	−13.7560	−1.00864
$187$	9.30798	0.680667
$188$	−0.219833	−0.0160329
$189$	5.38404	0.391631
$190$	−12.0978	−0.877669
$191$	3.60388	0.260767	0.130384	−	0.991464i	$-0.458379\pi$
$191$	3.60388	0.260767	0.130384	+	0.991464i	$0.458379\pi$
$192$	2.69202	0.194280
$193$	11.2228	0.807836	0.403918	−	0.914795i	$-0.367648\pi$
$193$	11.2228	0.807836	0.403918	+	0.914795i	$0.367648\pi$
$194$	−4.64071	−0.333184
$195$	0	0
$196$	−4.42758	−0.316256
$197$	−23.1051	−1.64617	−0.823086	−	0.567916i	$-0.807750\pi$
$197$	−23.1051	−1.64617	−0.823086	+	0.567916i	$0.807750\pi$
$198$	−8.70171	−0.618404
$199$	21.2620	1.50723	0.753613	−	0.657318i	$-0.228309\pi$
$199$	21.2620	1.50723	0.753613	+	0.657318i	$0.228309\pi$
$200$	1.21983	0.0862552
$201$	32.5090	2.29301
$202$	7.42758	0.522603
$203$	−14.7681	−1.03652
$204$	−12.2295	−0.856238
$205$	−8.64609	−0.603869
$206$	0.518122	0.0360993
$207$	11.5254	0.801072
$208$	0	0
$209$	−9.93900	−0.687495
$210$	−10.7681	−0.743069
$211$	1.70709	0.117521	0.0587604	−	0.998272i	$-0.481285\pi$
$211$	1.70709	0.117521	0.0587604	+	0.998272i	$0.481285\pi$
$212$	−2.71379	−0.186384
$213$	−3.46250	−0.237247
$214$	3.51035	0.239963
$215$	−28.2935	−1.92960
$216$	3.35690	0.228408
$217$	−8.19567	−0.556358
$218$	3.38404	0.229196
$219$	−9.76032	−0.659541
$220$	5.10992	0.344510
$221$	0	0
$222$	−20.4698	−1.37384
$223$	−6.21983	−0.416511	−0.208255	−	0.978074i	$-0.566779\pi$
$223$	−6.21983	−0.416511	−0.208255	+	0.978074i	$0.566779\pi$
$224$	1.60388	0.107163
$225$	5.18060	0.345374
$226$	−5.44935	−0.362486
$227$	0.955395	0.0634118	0.0317059	−	0.999497i	$-0.489906\pi$
$227$	0.955395	0.0634118	0.0317059	+	0.999497i	$0.489906\pi$
$228$	13.0586	0.864827
$229$	22.4155	1.48126	0.740629	−	0.671914i	$-0.234528\pi$
$229$	22.4155	1.48126	0.740629	+	0.671914i	$0.234528\pi$
$230$	−6.76809	−0.446274
$231$	−8.84654	−0.582060
$232$	−9.20775	−0.604518
$233$	2.99031	0.195902	0.0979509	−	0.995191i	$-0.468771\pi$
$233$	2.99031	0.195902	0.0979509	+	0.995191i	$0.468771\pi$
$234$	0	0
$235$	0.548253	0.0357641
$236$	4.07606	0.265329
$237$	−14.3478	−0.931990
$238$	−7.28621	−0.472295
$239$	11.1293	0.719894	0.359947	−	0.932973i	$-0.382795\pi$
$239$	11.1293	0.719894	0.359947	+	0.932973i	$0.382795\pi$
$240$	−6.71379	−0.433373
$241$	−5.20775	−0.335461	−0.167730	−	0.985833i	$-0.553644\pi$
$241$	−5.20775	−0.335461	−0.167730	+	0.985833i	$0.553644\pi$
$242$	−6.80194	−0.437245
$243$	−20.0422	−1.28571
$244$	10.4155	0.666784
$245$	11.0422	0.705461
$246$	9.33273	0.595033
$247$	0	0
$248$	−5.10992	−0.324480
$249$	−13.0586	−0.827556
$250$	9.42758	0.596253
$251$	−22.6950	−1.43250	−0.716248	−	0.697846i	$-0.754142\pi$
$251$	−22.6950	−1.43250	−0.716248	+	0.697846i	$0.754142\pi$
$252$	6.81163	0.429092
$253$	−5.56033	−0.349575
$254$	−6.19567	−0.388751
$255$	30.4999	1.90998
$256$	1.00000	0.0625000
$257$	10.4306	0.650641	0.325320	−	0.945604i	$-0.394528\pi$
$257$	10.4306	0.650641	0.325320	+	0.945604i	$0.394528\pi$
$258$	30.5405	1.90137
$259$	−12.1957	−0.757802
$260$	0	0
$261$	−39.1051	−2.42055
$262$	−4.13706	−0.255589
$263$	−7.10992	−0.438416	−0.219208	−	0.975678i	$-0.570347\pi$
$263$	−7.10992	−0.438416	−0.219208	+	0.975678i	$0.570347\pi$
$264$	−5.51573	−0.339470
$265$	6.76809	0.415760
$266$	7.78017	0.477033
$267$	44.6069	2.72990
$268$	12.0761	0.737663
$269$	2.02416	0.123415	0.0617077	−	0.998094i	$-0.480345\pi$
$269$	2.02416	0.123415	0.0617077	+	0.998094i	$0.480345\pi$
$270$	−8.37196	−0.509501
$271$	15.9651	0.969810	0.484905	−	0.874567i	$-0.338854\pi$
$271$	15.9651	0.969810	0.484905	+	0.874567i	$0.338854\pi$
$272$	−4.54288	−0.275452
$273$	0	0
$274$	19.3817	1.17089
$275$	−2.49934	−0.150716
$276$	7.30559	0.439745
$277$	13.7017	0.823256	0.411628	−	0.911352i	$-0.364960\pi$
$277$	13.7017	0.823256	0.411628	+	0.911352i	$0.364960\pi$
$278$	18.5864	1.11474
$279$	−21.7017	−1.29925
$280$	−4.00000	−0.239046
$281$	−15.2024	−0.906898	−0.453449	−	0.891282i	$-0.649807\pi$
$281$	−15.2024	−0.906898	−0.453449	+	0.891282i	$0.649807\pi$
$282$	−0.591794	−0.0352408
$283$	20.6069	1.22495	0.612475	−	0.790490i	$-0.290174\pi$
$283$	20.6069	1.22495	0.612475	+	0.790490i	$0.290174\pi$
$284$	−1.28621	−0.0763224
$285$	−32.5676	−1.92914
$286$	0	0
$287$	5.56033	0.328216
$288$	4.24698	0.250256
$289$	3.63773	0.213984
$290$	22.9638	1.34848
$291$	−12.4929	−0.732346
$292$	−3.62565	−0.212175
$293$	−2.17629	−0.127140	−0.0635702	−	0.997977i	$-0.520249\pi$
$293$	−2.17629	−0.127140	−0.0635702	+	0.997977i	$0.520249\pi$
$294$	−11.9191	−0.695139
$295$	−10.1655	−0.591861
$296$	−7.60388	−0.441966
$297$	−6.87800	−0.399102
$298$	3.65817	0.211912
$299$	0	0
$300$	3.28382	0.189591
$301$	18.1957	1.04878
$302$	−14.5918	−0.839663
$303$	19.9952	1.14870
$304$	4.85086	0.278216
$305$	−25.9758	−1.48737
$306$	−19.2935	−1.10294
$307$	−17.0127	−0.970965	−0.485482	−	0.874247i	$-0.661356\pi$
$307$	−17.0127	−0.970965	−0.485482	+	0.874247i	$0.661356\pi$
$308$	−3.28621	−0.187249
$309$	1.39480	0.0793472
$310$	12.7439	0.723806
$311$	−4.71379	−0.267295	−0.133647	−	0.991029i	$-0.542669\pi$
$311$	−4.71379	−0.267295	−0.133647	+	0.991029i	$0.542669\pi$
$312$	0	0
$313$	1.67696	0.0947872	0.0473936	−	0.998876i	$-0.484909\pi$
$313$	1.67696	0.0947872	0.0473936	+	0.998876i	$0.484909\pi$
$314$	21.6039	1.21918
$315$	−16.9879	−0.957161
$316$	−5.32975	−0.299822
$317$	−29.1400	−1.63667	−0.818334	−	0.574743i	$-0.805102\pi$
$317$	−29.1400	−1.63667	−0.818334	+	0.574743i	$0.805102\pi$
$318$	−7.30559	−0.409677
$319$	18.8659	1.05629
$320$	−2.49396	−0.139417
$321$	9.44994	0.527444
$322$	4.35258	0.242560
$323$	−22.0368	−1.22616
$324$	−3.70410	−0.205784
$325$	0	0
$326$	−13.6093	−0.753747
$327$	9.10992	0.503780
$328$	3.46681	0.191423
$329$	−0.352584	−0.0194386
$330$	13.7560	0.757243
$331$	−8.74392	−0.480609	−0.240305	−	0.970698i	$-0.577247\pi$
$331$	−8.74392	−0.480609	−0.240305	+	0.970698i	$0.577247\pi$
$332$	−4.85086	−0.266225
$333$	−32.2935	−1.76967
$334$	−14.0000	−0.766046
$335$	−30.1172	−1.64548
$336$	4.31767	0.235548
$337$	−3.10560	−0.169173	−0.0845865	−	0.996416i	$-0.526957\pi$
$337$	−3.10560	−0.169173	−0.0845865	+	0.996416i	$0.526957\pi$
$338$	0	0
$339$	−14.6698	−0.796753
$340$	11.3297	0.614442
$341$	10.4698	0.566971
$342$	20.6015	1.11400
$343$	−18.3284	−0.989642
$344$	11.3448	0.611671
$345$	−18.2198	−0.980923
$346$	−4.21983	−0.226860
$347$	17.9758	0.964993	0.482497	−	0.875898i	$-0.339730\pi$
$347$	17.9758	0.964993	0.482497	+	0.875898i	$0.339730\pi$
$348$	−24.7875	−1.32875
$349$	−15.1642	−0.811722	−0.405861	−	0.913935i	$-0.633028\pi$
$349$	−15.1642	−0.811722	−0.405861	+	0.913935i	$0.633028\pi$
$350$	1.95646	0.104577
$351$	0	0
$352$	−2.04892	−0.109208
$353$	3.30021	0.175652	0.0878262	−	0.996136i	$-0.472008\pi$
$353$	3.30021	0.175652	0.0878262	+	0.996136i	$0.472008\pi$
$354$	10.9729	0.583201
$355$	3.20775	0.170250
$356$	16.5700	0.878209
$357$	−19.6146	−1.03812
$358$	15.8291	0.836593
$359$	−18.8901	−0.996980	−0.498490	−	0.866895i	$-0.666112\pi$
$359$	−18.8901	−0.996980	−0.498490	+	0.866895i	$0.666112\pi$
$360$	−10.5918	−0.558237
$361$	4.53079	0.238463
$362$	−15.3056	−0.804444
$363$	−18.3110	−0.961076
$364$	0	0
$365$	9.04221	0.473291
$366$	28.0388	1.46561
$367$	−0.195669	−0.0102139	−0.00510693	−	0.999987i	$-0.501626\pi$
$367$	−0.195669	−0.0102139	−0.00510693	+	0.999987i	$0.501626\pi$
$368$	2.71379	0.141466
$369$	14.7235	0.766474
$370$	18.9638	0.985879
$371$	−4.35258	−0.225975
$372$	−13.7560	−0.713216
$373$	−10.7681	−0.557550	−0.278775	−	0.960356i	$-0.589928\pi$
$373$	−10.7681	−0.557550	−0.278775	+	0.960356i	$0.589928\pi$
$374$	9.30798	0.481304
$375$	25.3793	1.31058
$376$	−0.219833	−0.0113370
$377$	0	0
$378$	5.38404	0.276925
$379$	−27.9627	−1.43635	−0.718173	−	0.695864i	$-0.755022\pi$
$379$	−27.9627	−1.43635	−0.718173	+	0.695864i	$0.755022\pi$
$380$	−12.0978	−0.620606
$381$	−16.6789	−0.854485
$382$	3.60388	0.184390
$383$	−32.8310	−1.67759	−0.838793	−	0.544451i	$-0.816738\pi$
$383$	−32.8310	−1.67759	−0.838793	+	0.544451i	$0.816738\pi$
$384$	2.69202	0.137377
$385$	8.19567	0.417690
$386$	11.2228	0.571226
$387$	48.1812	2.44919
$388$	−4.64071	−0.235596
$389$	8.90946	0.451728	0.225864	−	0.974159i	$-0.427480\pi$
$389$	8.90946	0.451728	0.225864	+	0.974159i	$0.427480\pi$
$390$	0	0
$391$	−12.3284	−0.623475
$392$	−4.42758	−0.223627
$393$	−11.1371	−0.561791
$394$	−23.1051	−1.16402
$395$	13.2922	0.668802
$396$	−8.70171	−0.437277
$397$	−11.7888	−0.591662	−0.295831	−	0.955240i	$-0.595597\pi$
$397$	−11.7888	−0.591662	−0.295831	+	0.955240i	$0.595597\pi$
$398$	21.2620	1.06577
$399$	20.9444	1.04853
$400$	1.21983	0.0609916
$401$	−3.21313	−0.160456	−0.0802280	−	0.996777i	$-0.525565\pi$
$401$	−3.21313	−0.160456	−0.0802280	+	0.996777i	$0.525565\pi$
$402$	32.5090	1.62140
$403$	0	0
$404$	7.42758	0.369536
$405$	9.23788	0.459034
$406$	−14.7681	−0.732928
$407$	15.5797	0.772258
$408$	−12.2295	−0.605452
$409$	−12.0218	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$409$	−12.0218	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$410$	−8.64609	−0.427000
$411$	52.1758	2.57364
$412$	0.518122	0.0255261
$413$	6.53750	0.321689
$414$	11.5254	0.566443
$415$	12.0978	0.593859
$416$	0	0
$417$	50.0350	2.45023
$418$	−9.93900	−0.486132
$419$	−3.56033	−0.173934	−0.0869669	−	0.996211i	$-0.527717\pi$
$419$	−3.56033	−0.173934	−0.0869669	+	0.996211i	$0.527717\pi$
$420$	−10.7681	−0.525429
$421$	1.28621	0.0626860	0.0313430	−	0.999509i	$-0.490022\pi$
$421$	1.28621	0.0626860	0.0313430	+	0.999509i	$0.490022\pi$
$422$	1.70709	0.0830997
$423$	−0.933624	−0.0453944
$424$	−2.71379	−0.131793
$425$	−5.54155	−0.268805
$426$	−3.46250	−0.167759
$427$	16.7052	0.808420
$428$	3.51035	0.169679
$429$	0	0
$430$	−28.2935	−1.36443
$431$	−26.9879	−1.29996	−0.649981	−	0.759950i	$-0.725223\pi$
$431$	−26.9879	−1.29996	−0.649981	+	0.759950i	$0.725223\pi$
$432$	3.35690	0.161509
$433$	−16.0170	−0.769727	−0.384864	−	0.922973i	$-0.625752\pi$
$433$	−16.0170	−0.769727	−0.384864	+	0.922973i	$0.625752\pi$
$434$	−8.19567	−0.393405
$435$	61.8189	2.96399
$436$	3.38404	0.162066
$437$	13.1642	0.629730
$438$	−9.76032	−0.466366
$439$	−37.1400	−1.77260	−0.886299	−	0.463114i	$-0.846732\pi$
$439$	−37.1400	−1.77260	−0.886299	+	0.463114i	$0.846732\pi$
$440$	5.10992	0.243606
$441$	−18.8039	−0.895422
$442$	0	0
$443$	−41.2083	−1.95787	−0.978934	−	0.204178i	$-0.934548\pi$
$443$	−41.2083	−1.95787	−0.978934	+	0.204178i	$0.934548\pi$
$444$	−20.4698	−0.971454
$445$	−41.3250	−1.95899
$446$	−6.21983	−0.294518
$447$	9.84787	0.465788
$448$	1.60388	0.0757760
$449$	10.1263	0.477890	0.238945	−	0.971033i	$-0.423198\pi$
$449$	10.1263	0.477890	0.238945	+	0.971033i	$0.423198\pi$
$450$	5.18060	0.244216
$451$	−7.10321	−0.334477
$452$	−5.44935	−0.256316
$453$	−39.2814	−1.84560
$454$	0.955395	0.0448389
$455$	0	0
$456$	13.0586	0.611525
$457$	−21.7560	−1.01770	−0.508851	−	0.860854i	$-0.669930\pi$
$457$	−21.7560	−1.01770	−0.508851	+	0.860854i	$0.669930\pi$
$458$	22.4155	1.04741
$459$	−15.2500	−0.711807
$460$	−6.76809	−0.315564
$461$	38.5676	1.79627	0.898137	−	0.439716i	$-0.144921\pi$
$461$	38.5676	1.79627	0.898137	+	0.439716i	$0.144921\pi$
$462$	−8.84654	−0.411578
$463$	33.0073	1.53398	0.766990	−	0.641660i	$-0.221754\pi$
$463$	33.0073	1.53398	0.766990	+	0.641660i	$0.221754\pi$
$464$	−9.20775	−0.427459
$465$	34.3069	1.59095
$466$	2.99031	0.138523
$467$	6.53989	0.302630	0.151315	−	0.988486i	$-0.451649\pi$
$467$	6.53989	0.302630	0.151315	+	0.988486i	$0.451649\pi$
$468$	0	0
$469$	19.3685	0.894354
$470$	0.548253	0.0252890
$471$	58.1581	2.67978
$472$	4.07606	0.187616
$473$	−23.2446	−1.06879
$474$	−14.3478	−0.659016
$475$	5.91723	0.271501
$476$	−7.28621	−0.333963
$477$	−11.5254	−0.527713
$478$	11.1293	0.509042
$479$	−8.30691	−0.379553	−0.189776	−	0.981827i	$-0.560776\pi$
$479$	−8.30691	−0.379553	−0.189776	+	0.981827i	$0.560776\pi$
$480$	−6.71379	−0.306441
$481$	0	0
$482$	−5.20775	−0.237207
$483$	11.7172	0.533153
$484$	−6.80194	−0.309179
$485$	11.5737	0.525537
$486$	−20.0422	−0.909133
$487$	31.6534	1.43435	0.717176	−	0.696892i	$-0.245434\pi$
$487$	31.6534	1.43435	0.717176	+	0.696892i	$0.245434\pi$
$488$	10.4155	0.471488
$489$	−36.6364	−1.65676
$490$	11.0422	0.498836
$491$	−36.6631	−1.65458	−0.827291	−	0.561774i	$-0.810119\pi$
$491$	−36.6631	−1.65458	−0.827291	+	0.561774i	$0.810119\pi$
$492$	9.33273	0.420752
$493$	41.8297	1.88391
$494$	0	0
$495$	21.7017	0.975419
$496$	−5.10992	−0.229442
$497$	−2.06292	−0.0925345
$498$	−13.0586	−0.585170
$499$	29.8920	1.33815	0.669075	−	0.743195i	$-0.266691\pi$
$499$	29.8920	1.33815	0.669075	+	0.743195i	$0.266691\pi$
$500$	9.42758	0.421614
$501$	−37.6883	−1.68379
$502$	−22.6950	−1.01293
$503$	15.5905	0.695145	0.347572	−	0.937653i	$-0.387006\pi$
$503$	15.5905	0.695145	0.347572	+	0.937653i	$0.387006\pi$
$504$	6.81163	0.303414
$505$	−18.5241	−0.824311
$506$	−5.56033	−0.247187
$507$	0	0
$508$	−6.19567	−0.274888
$509$	17.1642	0.760790	0.380395	−	0.924824i	$-0.375788\pi$
$509$	17.1642	0.760790	0.380395	+	0.924824i	$0.375788\pi$
$510$	30.4999	1.35056
$511$	−5.81508	−0.257244
$512$	1.00000	0.0441942
$513$	16.2838	0.718948
$514$	10.4306	0.460073
$515$	−1.29218	−0.0569401
$516$	30.5405	1.34447
$517$	0.450419	0.0198094
$518$	−12.1957	−0.535847
$519$	−11.3599	−0.498643
$520$	0	0
$521$	6.15452	0.269634	0.134817	−	0.990870i	$-0.456955\pi$
$521$	6.15452	0.269634	0.134817	+	0.990870i	$0.456955\pi$
$522$	−39.1051	−1.71159
$523$	6.93900	0.303421	0.151711	−	0.988425i	$-0.451522\pi$
$523$	6.93900	0.303421	0.151711	+	0.988425i	$0.451522\pi$
$524$	−4.13706	−0.180728
$525$	5.26683	0.229863
$526$	−7.10992	−0.310007
$527$	23.2137	1.01121
$528$	−5.51573	−0.240041
$529$	−15.6353	−0.679797
$530$	6.76809	0.293987
$531$	17.3110	0.751232
$532$	7.78017	0.337313
$533$	0	0
$534$	44.6069	1.93033
$535$	−8.75468	−0.378498
$536$	12.0761	0.521607
$537$	42.6122	1.83885
$538$	2.02416	0.0872679
$539$	9.07175	0.390748
$540$	−8.37196	−0.360272
$541$	0.426256	0.0183262	0.00916308	−	0.999958i	$-0.497083\pi$
$541$	0.426256	0.0183262	0.00916308	+	0.999958i	$0.497083\pi$
$542$	15.9651	0.685759
$543$	−41.2030	−1.76819
$544$	−4.54288	−0.194774
$545$	−8.43967	−0.361516
$546$	0	0
$547$	3.72348	0.159205	0.0796023	−	0.996827i	$-0.474635\pi$
$547$	3.72348	0.159205	0.0796023	+	0.996827i	$0.474635\pi$
$548$	19.3817	0.827943
$549$	44.2344	1.88788
$550$	−2.49934	−0.106572
$551$	−44.6655	−1.90281
$552$	7.30559	0.310946
$553$	−8.54825	−0.363509
$554$	13.7017	0.582130
$555$	51.0508	2.16699
$556$	18.5864	0.788240
$557$	42.7961	1.81333	0.906664	−	0.421853i	$-0.138620\pi$
$557$	42.7961	1.81333	0.906664	+	0.421853i	$0.138620\pi$
$558$	−21.7017	−0.918707
$559$	0	0
$560$	−4.00000	−0.169031
$561$	25.0573	1.05792
$562$	−15.2024	−0.641273
$563$	2.92500	0.123274	0.0616370	−	0.998099i	$-0.480368\pi$
$563$	2.92500	0.123274	0.0616370	+	0.998099i	$0.480368\pi$
$564$	−0.591794	−0.0249190
$565$	13.5905	0.571755
$566$	20.6069	0.866171
$567$	−5.94092	−0.249495
$568$	−1.28621	−0.0539681
$569$	−26.5894	−1.11469	−0.557343	−	0.830282i	$-0.688179\pi$
$569$	−26.5894	−1.11469	−0.557343	+	0.830282i	$0.688179\pi$
$570$	−32.5676	−1.36411
$571$	−2.34422	−0.0981027	−0.0490513	−	0.998796i	$-0.515620\pi$
$571$	−2.34422	−0.0981027	−0.0490513	+	0.998796i	$0.515620\pi$
$572$	0	0
$573$	9.70171	0.405295
$574$	5.56033	0.232084
$575$	3.31037	0.138052
$576$	4.24698	0.176957
$577$	−40.4064	−1.68214	−0.841070	−	0.540926i	$-0.818074\pi$
$577$	−40.4064	−1.68214	−0.841070	+	0.540926i	$0.818074\pi$
$578$	3.63773	0.151310
$579$	30.2121	1.25557
$580$	22.9638	0.953518
$581$	−7.78017	−0.322776
$582$	−12.4929	−0.517847
$583$	5.56033	0.230286
$584$	−3.62565	−0.150030
$585$	0	0
$586$	−2.17629	−0.0899018
$587$	43.2180	1.78380	0.891900	−	0.452234i	$-0.149373\pi$
$587$	43.2180	1.78380	0.891900	+	0.452234i	$0.149373\pi$
$588$	−11.9191	−0.491537
$589$	−24.7875	−1.02135
$590$	−10.1655	−0.418509
$591$	−62.1995	−2.55855
$592$	−7.60388	−0.312517
$593$	7.01746	0.288172	0.144086	−	0.989565i	$-0.453976\pi$
$593$	7.01746	0.288172	0.144086	+	0.989565i	$0.453976\pi$
$594$	−6.87800	−0.282208
$595$	18.1715	0.744959
$596$	3.65817	0.149844
$597$	57.2379	2.34259
$598$	0	0
$599$	14.9772	0.611950	0.305975	−	0.952039i	$-0.401018\pi$
$599$	14.9772	0.611950	0.305975	+	0.952039i	$0.401018\pi$
$600$	3.28382	0.134061
$601$	−37.2771	−1.52057	−0.760283	−	0.649593i	$-0.774940\pi$
$601$	−37.2771	−1.52057	−0.760283	+	0.649593i	$0.774940\pi$
$602$	18.1957	0.741600
$603$	51.2868	2.08856
$604$	−14.5918	−0.593732
$605$	16.9638	0.689675
$606$	19.9952	0.812250
$607$	0.803003	0.0325929	0.0162964	−	0.999867i	$-0.494812\pi$
$607$	0.803003	0.0325929	0.0162964	+	0.999867i	$0.494812\pi$
$608$	4.85086	0.196728
$609$	−39.7560	−1.61099
$610$	−25.9758	−1.05173
$611$	0	0
$612$	−19.2935	−0.779894
$613$	−4.87071	−0.196726	−0.0983630	−	0.995151i	$-0.531361\pi$
$613$	−4.87071	−0.196726	−0.0983630	+	0.995151i	$0.531361\pi$
$614$	−17.0127	−0.686576
$615$	−23.2755	−0.938557
$616$	−3.28621	−0.132405
$617$	−3.34290	−0.134580	−0.0672899	−	0.997733i	$-0.521435\pi$
$617$	−3.34290	−0.134580	−0.0672899	+	0.997733i	$0.521435\pi$
$618$	1.39480	0.0561069
$619$	34.1715	1.37347	0.686734	−	0.726908i	$-0.259044\pi$
$619$	34.1715	1.37347	0.686734	+	0.726908i	$0.259044\pi$
$620$	12.7439	0.511808
$621$	9.10992	0.365568
$622$	−4.71379	−0.189006
$623$	26.5763	1.06476
$624$	0	0
$625$	−29.6112	−1.18445
$626$	1.67696	0.0670246
$627$	−26.7560	−1.06853
$628$	21.6039	0.862088
$629$	34.5435	1.37734
$630$	−16.9879	−0.676815
$631$	−39.8297	−1.58559	−0.792797	−	0.609486i	$-0.791376\pi$
$631$	−39.8297	−1.58559	−0.792797	+	0.609486i	$0.791376\pi$
$632$	−5.32975	−0.212006
$633$	4.59551	0.182655
$634$	−29.1400	−1.15730
$635$	15.4517	0.613184
$636$	−7.30559	−0.289685
$637$	0	0
$638$	18.8659	0.746909
$639$	−5.46250	−0.216093
$640$	−2.49396	−0.0985824
$641$	−13.4910	−0.532861	−0.266431	−	0.963854i	$-0.585844\pi$
$641$	−13.4910	−0.532861	−0.266431	+	0.963854i	$0.585844\pi$
$642$	9.44994	0.372960
$643$	16.6638	0.657156	0.328578	−	0.944477i	$-0.393431\pi$
$643$	16.6638	0.657156	0.328578	+	0.944477i	$0.393431\pi$
$644$	4.35258	0.171516
$645$	−76.1667	−2.99906
$646$	−22.0368	−0.867028
$647$	48.7332	1.91590	0.957949	−	0.286938i	$-0.0926372\pi$
$647$	48.7332	1.91590	0.957949	+	0.286938i	$0.0926372\pi$
$648$	−3.70410	−0.145511
$649$	−8.35152	−0.327826
$650$	0	0
$651$	−22.0629	−0.864714
$652$	−13.6093	−0.532979
$653$	47.0702	1.84200	0.921000	−	0.389563i	$-0.127374\pi$
$653$	47.0702	1.84200	0.921000	+	0.389563i	$0.127374\pi$
$654$	9.10992	0.356226
$655$	10.3177	0.403145
$656$	3.46681	0.135356
$657$	−15.3980	−0.600735
$658$	−0.352584	−0.0137452
$659$	−22.2524	−0.866829	−0.433414	−	0.901195i	$-0.642691\pi$
$659$	−22.2524	−0.866829	−0.433414	+	0.901195i	$0.642691\pi$
$660$	13.7560	0.535452
$661$	2.57242	0.100055	0.0500277	−	0.998748i	$-0.484069\pi$
$661$	2.57242	0.100055	0.0500277	+	0.998748i	$0.484069\pi$
$662$	−8.74392	−0.339842
$663$	0	0
$664$	−4.85086	−0.188250
$665$	−19.4034	−0.752432
$666$	−32.2935	−1.25135
$667$	−24.9879	−0.967536
$668$	−14.0000	−0.541676
$669$	−16.7439	−0.647357
$670$	−30.1172	−1.16353
$671$	−21.3405	−0.823841
$672$	4.31767	0.166558
$673$	24.1691	0.931651	0.465825	−	0.884877i	$-0.345757\pi$
$673$	24.1691	0.931651	0.465825	+	0.884877i	$0.345757\pi$
$674$	−3.10560	−0.119623
$675$	4.09485	0.157611
$676$	0	0
$677$	−4.91425	−0.188870	−0.0944349	−	0.995531i	$-0.530104\pi$
$677$	−4.91425	−0.188870	−0.0944349	+	0.995531i	$0.530104\pi$
$678$	−14.6698	−0.563389
$679$	−7.44312	−0.285641
$680$	11.3297	0.434476
$681$	2.57194	0.0985571
$682$	10.4698	0.400909
$683$	1.96556	0.0752100	0.0376050	−	0.999293i	$-0.488027\pi$
$683$	1.96556	0.0752100	0.0376050	+	0.999293i	$0.488027\pi$
$684$	20.6015	0.787717
$685$	−48.3370	−1.84686
$686$	−18.3284	−0.699782
$687$	60.3430	2.30223
$688$	11.3448	0.432517
$689$	0	0
$690$	−18.2198	−0.693617
$691$	−25.5077	−0.970359	−0.485179	−	0.874415i	$-0.661246\pi$
$691$	−25.5077	−0.970359	−0.485179	+	0.874415i	$0.661246\pi$
$692$	−4.21983	−0.160414
$693$	−13.9565	−0.530162
$694$	17.9758	0.682353
$695$	−46.3538	−1.75830
$696$	−24.7875	−0.939566
$697$	−15.7493	−0.596547
$698$	−15.1642	−0.573974
$699$	8.04998	0.304478
$700$	1.95646	0.0739472
$701$	−41.8491	−1.58062	−0.790308	−	0.612709i	$-0.790080\pi$
$701$	−41.8491	−1.58062	−0.790308	+	0.612709i	$0.790080\pi$
$702$	0	0
$703$	−36.8853	−1.39116
$704$	−2.04892	−0.0772215
$705$	1.47591	0.0555860
$706$	3.30021	0.124205
$707$	11.9129	0.448031
$708$	10.9729	0.412385
$709$	−33.3056	−1.25082	−0.625409	−	0.780297i	$-0.715068\pi$
$709$	−33.3056	−1.25082	−0.625409	+	0.780297i	$0.715068\pi$
$710$	3.20775	0.120385
$711$	−22.6353	−0.848891
$712$	16.5700	0.620988
$713$	−13.8672	−0.519333
$714$	−19.6146	−0.734059
$715$	0	0
$716$	15.8291	0.591561
$717$	29.9603	1.11889
$718$	−18.8901	−0.704972
$719$	6.37196	0.237634	0.118817	−	0.992916i	$-0.462090\pi$
$719$	6.37196	0.237634	0.118817	+	0.992916i	$0.462090\pi$
$720$	−10.5918	−0.394733
$721$	0.831004	0.0309482
$722$	4.53079	0.168619
$723$	−14.0194	−0.521386
$724$	−15.3056	−0.568828
$725$	−11.2319	−0.417143
$726$	−18.3110	−0.679584
$727$	23.7995	0.882676	0.441338	−	0.897341i	$-0.354504\pi$
$727$	23.7995	0.882676	0.441338	+	0.897341i	$0.354504\pi$
$728$	0	0
$729$	−42.8418	−1.58673
$730$	9.04221	0.334667
$731$	−51.5381	−1.90621
$732$	28.0388	1.03634
$733$	27.4142	1.01257	0.506283	−	0.862368i	$-0.331019\pi$
$733$	27.4142	1.01257	0.506283	+	0.862368i	$0.331019\pi$
$734$	−0.195669	−0.00722229
$735$	29.7259	1.09646
$736$	2.71379	0.100032
$737$	−24.7429	−0.911415
$738$	14.7235	0.541979
$739$	−35.3274	−1.29954	−0.649769	−	0.760132i	$-0.725134\pi$
$739$	−35.3274	−1.29954	−0.649769	+	0.760132i	$0.725134\pi$
$740$	18.9638	0.697122
$741$	0	0
$742$	−4.35258	−0.159788
$743$	23.9758	0.879588	0.439794	−	0.898099i	$-0.355051\pi$
$743$	23.9758	0.879588	0.439794	+	0.898099i	$0.355051\pi$
$744$	−13.7560	−0.504320
$745$	−9.12333	−0.334253
$746$	−10.7681	−0.394248
$747$	−20.6015	−0.753769
$748$	9.30798	0.340333
$749$	5.63017	0.205722
$750$	25.3793	0.926719
$751$	−24.8659	−0.907370	−0.453685	−	0.891162i	$-0.649891\pi$
$751$	−24.8659	−0.907370	−0.453685	+	0.891162i	$0.649891\pi$
$752$	−0.219833	−0.00801647
$753$	−61.0954	−2.22644
$754$	0	0
$755$	36.3913	1.32442
$756$	5.38404	0.195816
$757$	4.93362	0.179316	0.0896578	−	0.995973i	$-0.471423\pi$
$757$	4.93362	0.179316	0.0896578	+	0.995973i	$0.471423\pi$
$758$	−27.9627	−1.01565
$759$	−14.9685	−0.543324
$760$	−12.0978	−0.438835
$761$	15.0067	0.543993	0.271996	−	0.962298i	$-0.412316\pi$
$761$	15.0067	0.543993	0.271996	+	0.962298i	$0.412316\pi$
$762$	−16.6789	−0.604212
$763$	5.42758	0.196492
$764$	3.60388	0.130384
$765$	48.1172	1.73968
$766$	−32.8310	−1.18623
$767$	0	0
$768$	2.69202	0.0971400
$769$	−22.2640	−0.802859	−0.401430	−	0.915890i	$-0.631487\pi$
$769$	−22.2640	−0.802859	−0.401430	+	0.915890i	$0.631487\pi$
$770$	8.19567	0.295351
$771$	28.0793	1.01125
$772$	11.2228	0.403918
$773$	11.7366	0.422137	0.211069	−	0.977471i	$-0.432306\pi$
$773$	11.7366	0.422137	0.211069	+	0.977471i	$0.432306\pi$
$774$	48.1812	1.73184
$775$	−6.23324	−0.223905
$776$	−4.64071	−0.166592
$777$	−32.8310	−1.17781
$778$	8.90946	0.319420
$779$	16.8170	0.602532
$780$	0	0
$781$	2.63533	0.0942997
$782$	−12.3284	−0.440863
$783$	−30.9095	−1.10461
$784$	−4.42758	−0.158128
$785$	−53.8792	−1.92303
$786$	−11.1371	−0.397246
$787$	14.9498	0.532901	0.266451	−	0.963849i	$-0.414149\pi$
$787$	14.9498	0.532901	0.266451	+	0.963849i	$0.414149\pi$
$788$	−23.1051	−0.823086
$789$	−19.1400	−0.681404
$790$	13.2922	0.472914
$791$	−8.74008	−0.310762
$792$	−8.70171	−0.309202
$793$	0	0
$794$	−11.7888	−0.418369
$795$	18.2198	0.646191
$796$	21.2620	0.753613
$797$	35.1728	1.24589	0.622943	−	0.782267i	$-0.285937\pi$
$797$	35.1728	1.24589	0.622943	+	0.782267i	$0.285937\pi$
$798$	20.9444	0.741423
$799$	0.998672	0.0353305
$800$	1.21983	0.0431276
$801$	70.3726	2.48649
$802$	−3.21313	−0.113459
$803$	7.42865	0.262151
$804$	32.5090	1.14650
$805$	−10.8552	−0.382594
$806$	0	0
$807$	5.44909	0.191817
$808$	7.42758	0.261301
$809$	42.9487	1.51000	0.754998	−	0.655727i	$-0.227638\pi$
$809$	42.9487	1.51000	0.754998	+	0.655727i	$0.227638\pi$
$810$	9.23788	0.324586
$811$	20.4644	0.718603	0.359301	−	0.933222i	$-0.383015\pi$
$811$	20.4644	0.718603	0.359301	+	0.933222i	$0.383015\pi$
$812$	−14.7681	−0.518258
$813$	42.9783	1.50732
$814$	15.5797	0.546069
$815$	33.9409	1.18890
$816$	−12.2295	−0.428119
$817$	55.0320	1.92533
$818$	−12.0218	−0.420331
$819$	0	0
$820$	−8.64609	−0.301934
$821$	36.9987	1.29126	0.645631	−	0.763649i	$-0.276594\pi$
$821$	36.9987	1.29126	0.645631	+	0.763649i	$0.276594\pi$
$822$	52.1758	1.81984
$823$	9.27545	0.323322	0.161661	−	0.986846i	$-0.448315\pi$
$823$	9.27545	0.323322	0.161661	+	0.986846i	$0.448315\pi$
$824$	0.518122	0.0180496
$825$	−6.72827	−0.234248
$826$	6.53750	0.227469
$827$	−6.05669	−0.210612	−0.105306	−	0.994440i	$-0.533582\pi$
$827$	−6.05669	−0.210612	−0.105306	+	0.994440i	$0.533582\pi$
$828$	11.5254	0.400536
$829$	9.94092	0.345262	0.172631	−	0.984987i	$-0.444773\pi$
$829$	9.94092	0.345262	0.172631	+	0.984987i	$0.444773\pi$
$830$	12.0978	0.419922
$831$	36.8853	1.27954
$832$	0	0
$833$	20.1140	0.696908
$834$	50.0350	1.73257
$835$	34.9154	1.20830
$836$	−9.93900	−0.343748
$837$	−17.1535	−0.592910
$838$	−3.56033	−0.122990
$839$	−34.4698	−1.19003	−0.595015	−	0.803715i	$-0.702854\pi$
$839$	−34.4698	−1.19003	−0.595015	+	0.803715i	$0.702854\pi$
$840$	−10.7681	−0.371534
$841$	55.7827	1.92354
$842$	1.28621	0.0443257
$843$	−40.9251	−1.40954
$844$	1.70709	0.0587604
$845$	0	0
$846$	−0.933624	−0.0320987
$847$	−10.9095	−0.374854
$848$	−2.71379	−0.0931920
$849$	55.4741	1.90387
$850$	−5.54155	−0.190074
$851$	−20.6353	−0.707370
$852$	−3.46250	−0.118623
$853$	−22.1521	−0.758474	−0.379237	−	0.925299i	$-0.623814\pi$
$853$	−22.1521	−0.758474	−0.379237	+	0.925299i	$0.623814\pi$
$854$	16.7052	0.571639
$855$	−51.3793	−1.75713
$856$	3.51035	0.119981
$857$	−30.5187	−1.04250	−0.521250	−	0.853404i	$-0.674534\pi$
$857$	−30.5187	−1.04250	−0.521250	+	0.853404i	$0.674534\pi$
$858$	0	0
$859$	−20.9071	−0.713340	−0.356670	−	0.934230i	$-0.616088\pi$
$859$	−20.9071	−0.713340	−0.356670	+	0.934230i	$0.616088\pi$
$860$	−28.2935	−0.964800
$861$	14.9685	0.510127
$862$	−26.9879	−0.919212
$863$	55.6969	1.89595	0.947973	−	0.318352i	$-0.103129\pi$
$863$	55.6969	1.89595	0.947973	+	0.318352i	$0.103129\pi$
$864$	3.35690	0.114204
$865$	10.5241	0.357830
$866$	−16.0170	−0.544279
$867$	9.79284	0.332582
$868$	−8.19567	−0.278179
$869$	10.9202	0.370443
$870$	61.8189	2.09586
$871$	0	0
$872$	3.38404	0.114598
$873$	−19.7090	−0.667049
$874$	13.1642	0.445286
$875$	15.1207	0.511172
$876$	−9.76032	−0.329771
$877$	32.7922	1.10732	0.553658	−	0.832744i	$-0.313232\pi$
$877$	32.7922	1.10732	0.553658	+	0.832744i	$0.313232\pi$
$878$	−37.1400	−1.25342
$879$	−5.85862	−0.197607
$880$	5.10992	0.172255
$881$	22.4101	0.755016	0.377508	−	0.926006i	$-0.376781\pi$
$881$	22.4101	0.755016	0.377508	+	0.926006i	$0.376781\pi$
$882$	−18.8039	−0.633159
$883$	−38.9670	−1.31134	−0.655672	−	0.755046i	$-0.727615\pi$
$883$	−38.9670	−1.31134	−0.655672	+	0.755046i	$0.727615\pi$
$884$	0	0
$885$	−27.3658	−0.919893
$886$	−41.2083	−1.38442
$887$	−34.8745	−1.17097	−0.585486	−	0.810682i	$-0.699096\pi$
$887$	−34.8745	−1.17097	−0.585486	+	0.810682i	$0.699096\pi$
$888$	−20.4698	−0.686921
$889$	−9.93708	−0.333279
$890$	−41.3250	−1.38522
$891$	7.58940	0.254254
$892$	−6.21983	−0.208255
$893$	−1.06638	−0.0356849
$894$	9.84787	0.329362
$895$	−39.4771	−1.31957
$896$	1.60388	0.0535817
$897$	0	0
$898$	10.1263	0.337919
$899$	47.0508	1.56923
$900$	5.18060	0.172687
$901$	12.3284	0.410719
$902$	−7.10321	−0.236511
$903$	48.9831	1.63006
$904$	−5.44935	−0.181243
$905$	38.1715	1.26886
$906$	−39.2814	−1.30504
$907$	15.1317	0.502439	0.251220	−	0.967930i	$-0.419168\pi$
$907$	15.1317	0.502439	0.251220	+	0.967930i	$0.419168\pi$
$908$	0.955395	0.0317059
$909$	31.5448	1.04627
$910$	0	0
$911$	23.0810	0.764707	0.382353	−	0.924016i	$-0.375114\pi$
$911$	23.0810	0.764707	0.382353	+	0.924016i	$0.375114\pi$
$912$	13.0586	0.432414
$913$	9.93900	0.328933
$914$	−21.7560	−0.719625
$915$	−69.9275	−2.31173
$916$	22.4155	0.740629
$917$	−6.63533	−0.219118
$918$	−15.2500	−0.503324
$919$	−39.9275	−1.31709	−0.658544	−	0.752543i	$-0.728827\pi$
$919$	−39.9275	−1.31709	−0.658544	+	0.752543i	$0.728827\pi$
$920$	−6.76809	−0.223137
$921$	−45.7985	−1.50911
$922$	38.5676	1.27016
$923$	0	0
$924$	−8.84654	−0.291030
$925$	−9.27545	−0.304975
$926$	33.0073	1.08469
$927$	2.20046	0.0722724
$928$	−9.20775	−0.302259
$929$	45.1771	1.48221	0.741107	−	0.671387i	$-0.234301\pi$
$929$	45.1771	1.48221	0.741107	+	0.671387i	$0.234301\pi$
$930$	34.3069	1.12497
$931$	−21.4776	−0.703899
$932$	2.99031	0.0979509
$933$	−12.6896	−0.415440
$934$	6.53989	0.213992
$935$	−23.2137	−0.759170
$936$	0	0
$937$	29.0901	0.950331	0.475165	−	0.879896i	$-0.342388\pi$
$937$	29.0901	0.950331	0.475165	+	0.879896i	$0.342388\pi$
$938$	19.3685	0.632404
$939$	4.51440	0.147322
$940$	0.548253	0.0178821
$941$	26.1220	0.851553	0.425776	−	0.904828i	$-0.360001\pi$
$941$	26.1220	0.851553	0.425776	+	0.904828i	$0.360001\pi$
$942$	58.1581	1.89489
$943$	9.40821	0.306373
$944$	4.07606	0.132665
$945$	−13.4276	−0.436799
$946$	−23.2446	−0.755747
$947$	−42.6698	−1.38658	−0.693291	−	0.720658i	$-0.743840\pi$
$947$	−42.6698	−1.38658	−0.693291	+	0.720658i	$0.743840\pi$
$948$	−14.3478	−0.465995
$949$	0	0
$950$	5.91723	0.191980
$951$	−78.4456	−2.54377
$952$	−7.28621	−0.236147
$953$	−29.9038	−0.968680	−0.484340	−	0.874880i	$-0.660940\pi$
$953$	−29.9038	−0.968680	−0.484340	+	0.874880i	$0.660940\pi$
$954$	−11.5254	−0.373149
$955$	−8.98792	−0.290842
$956$	11.1293	0.359947
$957$	50.7875	1.64173
$958$	−8.30691	−0.268384
$959$	31.0858	1.00381
$960$	−6.71379	−0.216687
$961$	−4.88876	−0.157702
$962$	0	0
$963$	14.9084	0.480416
$964$	−5.20775	−0.167730
$965$	−27.9892	−0.901006
$966$	11.7172	0.376996
$967$	5.16900	0.166224	0.0831119	−	0.996540i	$-0.473514\pi$
$967$	5.16900	0.166224	0.0831119	+	0.996540i	$0.473514\pi$
$968$	−6.80194	−0.218623
$969$	−59.3236	−1.90575
$970$	11.5737	0.371611
$971$	35.9715	1.15438	0.577191	−	0.816609i	$-0.304149\pi$
$971$	35.9715	1.15438	0.577191	+	0.816609i	$0.304149\pi$
$972$	−20.0422	−0.642854
$973$	29.8103	0.955674
$974$	31.6534	1.01424
$975$	0	0
$976$	10.4155	0.333392
$977$	−34.2435	−1.09555	−0.547774	−	0.836627i	$-0.684525\pi$
$977$	−34.2435	−1.09555	−0.547774	+	0.836627i	$0.684525\pi$
$978$	−36.6364	−1.17150
$979$	−33.9506	−1.08507
$980$	11.0422	0.352731
$981$	14.3720	0.458861
$982$	−36.6631	−1.16997
$983$	54.8939	1.75084	0.875422	−	0.483359i	$-0.160584\pi$
$983$	54.8939	1.75084	0.875422	+	0.483359i	$0.160584\pi$
$984$	9.33273	0.297517
$985$	57.6233	1.83603
$986$	41.8297	1.33213
$987$	−0.949164	−0.0302122
$988$	0	0
$989$	30.7875	0.978984
$990$	21.7017	0.689726
$991$	−16.6655	−0.529396	−0.264698	−	0.964331i	$-0.585272\pi$
$991$	−16.6655	−0.529396	−0.264698	+	0.964331i	$0.585272\pi$
$992$	−5.10992	−0.162240
$993$	−23.5388	−0.746982
$994$	−2.06292	−0.0654318
$995$	−53.0267	−1.68106
$996$	−13.0586	−0.413778
$997$	16.3961	0.519270	0.259635	−	0.965707i	$-0.416398\pi$
$997$	16.3961	0.519270	0.259635	+	0.965707i	$0.416398\pi$
$998$	29.8920	0.946215
$999$	−25.5254	−0.807588

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.2.a.h.1.3	yes	3
3.2	odd	2		3042.2.a.z.1.3		3
4.3	odd	2		2704.2.a.w.1.1		3
5.4	even	2		8450.2.a.bn.1.1		3
13.2	odd	12		338.2.e.e.147.4		12
13.3	even	3		338.2.c.h.191.1		6
13.4	even	6		338.2.c.i.315.1		6
13.5	odd	4		338.2.b.d.337.3		6
13.6	odd	12		338.2.e.e.23.1		12
13.7	odd	12		338.2.e.e.23.4		12
13.8	odd	4		338.2.b.d.337.6		6
13.9	even	3		338.2.c.h.315.1		6
13.10	even	6		338.2.c.i.191.1		6
13.11	odd	12		338.2.e.e.147.1		12
13.12	even	2		338.2.a.g.1.3	✓	3
39.5	even	4		3042.2.b.n.1351.4		6
39.8	even	4		3042.2.b.n.1351.3		6
39.38	odd	2		3042.2.a.bi.1.1		3
52.31	even	4		2704.2.f.m.337.2		6
52.47	even	4		2704.2.f.m.337.1		6
52.51	odd	2		2704.2.a.v.1.1		3
65.64	even	2		8450.2.a.bx.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.2.a.g.1.3	✓	3	13.12	even	2
338.2.a.h.1.3	yes	3	1.1	even	1	trivial
338.2.b.d.337.3		6	13.5	odd	4
338.2.b.d.337.6		6	13.8	odd	4
338.2.c.h.191.1		6	13.3	even	3
338.2.c.h.315.1		6	13.9	even	3
338.2.c.i.191.1		6	13.10	even	6
338.2.c.i.315.1		6	13.4	even	6
338.2.e.e.23.1		12	13.6	odd	12
338.2.e.e.23.4		12	13.7	odd	12
338.2.e.e.147.1		12	13.11	odd	12
338.2.e.e.147.4		12	13.2	odd	12
2704.2.a.v.1.1		3	52.51	odd	2
2704.2.a.w.1.1		3	4.3	odd	2
2704.2.f.m.337.1		6	52.47	even	4
2704.2.f.m.337.2		6	52.31	even	4
3042.2.a.z.1.3		3	3.2	odd	2
3042.2.a.bi.1.1		3	39.38	odd	2
3042.2.b.n.1351.3		6	39.8	even	4
3042.2.b.n.1351.4		6	39.5	even	4
8450.2.a.bn.1.1		3	5.4	even	2
8450.2.a.bx.1.1		3	65.64	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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.69202 q^{3} +1.00000 q^{4} -2.49396 q^{5} +2.69202 q^{6} +1.60388 q^{7} +1.00000 q^{8} +4.24698 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.69202 q^{3} +1.00000 q^{4} -2.49396 q^{5} +2.69202 q^{6} +1.60388 q^{7} +1.00000 q^{8} +4.24698 q^{9} -2.49396 q^{10} -2.04892 q^{11} +2.69202 q^{12} +1.60388 q^{14} -6.71379 q^{15} +1.00000 q^{16} -4.54288 q^{17} +4.24698 q^{18} +4.85086 q^{19} -2.49396 q^{20} +4.31767 q^{21} -2.04892 q^{22} +2.71379 q^{23} +2.69202 q^{24} +1.21983 q^{25} +3.35690 q^{27} +1.60388 q^{28} -9.20775 q^{29} -6.71379 q^{30} -5.10992 q^{31} +1.00000 q^{32} -5.51573 q^{33} -4.54288 q^{34} -4.00000 q^{35} +4.24698 q^{36} -7.60388 q^{37} +4.85086 q^{38} -2.49396 q^{40} +3.46681 q^{41} +4.31767 q^{42} +11.3448 q^{43} -2.04892 q^{44} -10.5918 q^{45} +2.71379 q^{46} -0.219833 q^{47} +2.69202 q^{48} -4.42758 q^{49} +1.21983 q^{50} -12.2295 q^{51} -2.71379 q^{53} +3.35690 q^{54} +5.10992 q^{55} +1.60388 q^{56} +13.0586 q^{57} -9.20775 q^{58} +4.07606 q^{59} -6.71379 q^{60} +10.4155 q^{61} -5.10992 q^{62} +6.81163 q^{63} +1.00000 q^{64} -5.51573 q^{66} +12.0761 q^{67} -4.54288 q^{68} +7.30559 q^{69} -4.00000 q^{70} -1.28621 q^{71} +4.24698 q^{72} -3.62565 q^{73} -7.60388 q^{74} +3.28382 q^{75} +4.85086 q^{76} -3.28621 q^{77} -5.32975 q^{79} -2.49396 q^{80} -3.70410 q^{81} +3.46681 q^{82} -4.85086 q^{83} +4.31767 q^{84} +11.3297 q^{85} +11.3448 q^{86} -24.7875 q^{87} -2.04892 q^{88} +16.5700 q^{89} -10.5918 q^{90} +2.71379 q^{92} -13.7560 q^{93} -0.219833 q^{94} -12.0978 q^{95} +2.69202 q^{96} -4.64071 q^{97} -4.42758 q^{98} -8.70171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} + 3 q^{6} - 4 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 2 * q^5 + 3 * q^6 - 4 * q^7 + 3 * q^8 + 8 * q^9	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} + 3 q^{6} - 4 q^{7} + 3 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} + 3 q^{12} - 4 q^{14} - 12 q^{15} + 3 q^{16} + 5 q^{17} + 8 q^{18} + q^{19} + 2 q^{20} - 4 q^{21} + 3 q^{22} + 3 q^{24} + 5 q^{25} + 6 q^{27} - 4 q^{28} - 10 q^{29} - 12 q^{30} - 16 q^{31} + 3 q^{32} - 4 q^{33} + 5 q^{34} - 12 q^{35} + 8 q^{36} - 14 q^{37} + q^{38} + 2 q^{40} + 7 q^{41} - 4 q^{42} + 11 q^{43} + 3 q^{44} - 4 q^{45} - 2 q^{47} + 3 q^{48} + 3 q^{49} + 5 q^{50} - 16 q^{51} + 6 q^{54} + 16 q^{55} - 4 q^{56} + 8 q^{57} - 10 q^{58} - 3 q^{59} - 12 q^{60} - 4 q^{61} - 16 q^{62} - 6 q^{63} + 3 q^{64} - 4 q^{66} + 21 q^{67} + 5 q^{68} - 14 q^{69} - 12 q^{70} - 12 q^{71} + 8 q^{72} + q^{73} - 14 q^{74} - 23 q^{75} + q^{76} - 18 q^{77} - 18 q^{79} + 2 q^{80} - 25 q^{81} + 7 q^{82} - q^{83} - 4 q^{84} + 36 q^{85} + 11 q^{86} - 10 q^{87} + 3 q^{88} + 25 q^{89} - 4 q^{90} - 2 q^{93} - 2 q^{94} - 18 q^{95} + 3 q^{96} + 23 q^{97} + 3 q^{98} + q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 2 * q^5 + 3 * q^6 - 4 * q^7 + 3 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 + 3 * q^12 - 4 * q^14 - 12 * q^15 + 3 * q^16 + 5 * q^17 + 8 * q^18 + q^19 + 2 * q^20 - 4 * q^21 + 3 * q^22 + 3 * q^24 + 5 * q^25 + 6 * q^27 - 4 * q^28 - 10 * q^29 - 12 * q^30 - 16 * q^31 + 3 * q^32 - 4 * q^33 + 5 * q^34 - 12 * q^35 + 8 * q^36 - 14 * q^37 + q^38 + 2 * q^40 + 7 * q^41 - 4 * q^42 + 11 * q^43 + 3 * q^44 - 4 * q^45 - 2 * q^47 + 3 * q^48 + 3 * q^49 + 5 * q^50 - 16 * q^51 + 6 * q^54 + 16 * q^55 - 4 * q^56 + 8 * q^57 - 10 * q^58 - 3 * q^59 - 12 * q^60 - 4 * q^61 - 16 * q^62 - 6 * q^63 + 3 * q^64 - 4 * q^66 + 21 * q^67 + 5 * q^68 - 14 * q^69 - 12 * q^70 - 12 * q^71 + 8 * q^72 + q^73 - 14 * q^74 - 23 * q^75 + q^76 - 18 * q^77 - 18 * q^79 + 2 * q^80 - 25 * q^81 + 7 * q^82 - q^83 - 4 * q^84 + 36 * q^85 + 11 * q^86 - 10 * q^87 + 3 * q^88 + 25 * q^89 - 4 * q^90 - 2 * q^93 - 2 * q^94 - 18 * q^95 + 3 * q^96 + 23 * q^97 + 3 * q^98 + q^99

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