LMFDB - Embedded newform 115.2.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,2,Mod(1,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, 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("115.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	115.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:	$0.918279623245$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.32973$ of defining polynomial
Character	$\chi$	$=$	115.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.32973 q^{2} +1.56155 q^{3} -0.231826 q^{4} +1.00000 q^{5} +2.07644 q^{6} -3.50407 q^{7} -2.96772 q^{8} -0.561553 q^{9} +O(q^{10})$	\(q+1.32973 q^{2} +1.56155 q^{3} -0.231826 q^{4} +1.00000 q^{5} +2.07644 q^{6} -3.50407 q^{7} -2.96772 q^{8} -0.561553 q^{9} +1.32973 q^{10} -0.659454 q^{11} -0.362008 q^{12} +5.91023 q^{13} -4.65945 q^{14} +1.56155 q^{15} -3.48261 q^{16} +0.844614 q^{17} -0.746712 q^{18} +0.659454 q^{19} -0.231826 q^{20} -5.47179 q^{21} -0.876894 q^{22} -1.00000 q^{23} -4.63425 q^{24} +1.00000 q^{25} +7.85900 q^{26} -5.56155 q^{27} +0.812333 q^{28} +1.59383 q^{29} +2.07644 q^{30} +6.75485 q^{31} +1.30452 q^{32} -1.02977 q^{33} +1.12311 q^{34} -3.50407 q^{35} +0.130182 q^{36} -11.7503 q^{37} +0.876894 q^{38} +9.22914 q^{39} -2.96772 q^{40} +6.40617 q^{41} -7.27598 q^{42} -9.47179 q^{43} +0.152878 q^{44} -0.561553 q^{45} -1.32973 q^{46} +6.88046 q^{47} -5.43827 q^{48} +5.27849 q^{49} +1.32973 q^{50} +1.31891 q^{51} -1.37014 q^{52} +6.64881 q^{53} -7.39535 q^{54} -0.659454 q^{55} +10.3991 q^{56} +1.02977 q^{57} +2.11936 q^{58} -4.97586 q^{59} -0.362008 q^{60} +5.78256 q^{61} +8.98210 q^{62} +1.96772 q^{63} +8.69987 q^{64} +5.91023 q^{65} -1.36932 q^{66} +8.31640 q^{67} -0.195803 q^{68} -1.56155 q^{69} -4.65945 q^{70} -3.63174 q^{71} +1.66653 q^{72} -13.9102 q^{73} -15.6247 q^{74} +1.56155 q^{75} -0.152878 q^{76} +2.31077 q^{77} +12.2722 q^{78} +1.02977 q^{79} -3.48261 q^{80} -7.00000 q^{81} +8.51845 q^{82} -8.27849 q^{83} +1.26850 q^{84} +0.844614 q^{85} -12.5949 q^{86} +2.48886 q^{87} +1.95708 q^{88} +17.6030 q^{89} -0.746712 q^{90} -20.7099 q^{91} +0.231826 q^{92} +10.5481 q^{93} +9.14914 q^{94} +0.659454 q^{95} +2.03708 q^{96} -8.34868 q^{97} +7.01895 q^{98} +0.370318 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 9 * q^8 + 6 * q^9	\( 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9} + 2 q^{10} + 4 q^{11} - 19 q^{12} - 12 q^{14} - 2 q^{15} + 8 q^{16} - q^{17} + 3 q^{18} - 4 q^{19} + 4 q^{20} + 10 q^{21} - 20 q^{22} - 4 q^{23} - 30 q^{24} + 4 q^{25} - q^{26} - 14 q^{27} - 22 q^{28} + 19 q^{29} - q^{30} - q^{31} + 20 q^{32} - 2 q^{33} - 12 q^{34} - 3 q^{35} + 23 q^{36} - 3 q^{37} + 20 q^{38} + 9 q^{40} + 13 q^{41} + 6 q^{42} - 6 q^{43} - 18 q^{44} + 6 q^{45} - 2 q^{46} + 6 q^{47} - 21 q^{48} + 9 q^{49} + 2 q^{50} - 8 q^{51} - q^{52} + 19 q^{53} - 7 q^{54} + 4 q^{55} - 10 q^{56} + 2 q^{57} + 21 q^{58} + 23 q^{59} - 19 q^{60} - 13 q^{62} - 13 q^{63} + 27 q^{64} + 44 q^{66} - 3 q^{67} - 4 q^{68} + 2 q^{69} - 12 q^{70} - 3 q^{71} + 39 q^{72} - 32 q^{73} - 12 q^{74} - 2 q^{75} + 18 q^{76} + 18 q^{77} + 43 q^{78} + 2 q^{79} + 8 q^{80} - 28 q^{81} - 5 q^{82} - 21 q^{83} + 28 q^{84} - q^{85} - 2 q^{86} - 18 q^{87} - 14 q^{88} + 3 q^{90} - 40 q^{91} - 4 q^{92} - 8 q^{93} + 47 q^{94} - 4 q^{95} - 61 q^{96} - 18 q^{97} + 16 q^{98} + 6 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 9 * q^8 + 6 * q^9 + 2 * q^10 + 4 * q^11 - 19 * q^12 - 12 * q^14 - 2 * q^15 + 8 * q^16 - q^17 + 3 * q^18 - 4 * q^19 + 4 * q^20 + 10 * q^21 - 20 * q^22 - 4 * q^23 - 30 * q^24 + 4 * q^25 - q^26 - 14 * q^27 - 22 * q^28 + 19 * q^29 - q^30 - q^31 + 20 * q^32 - 2 * q^33 - 12 * q^34 - 3 * q^35 + 23 * q^36 - 3 * q^37 + 20 * q^38 + 9 * q^40 + 13 * q^41 + 6 * q^42 - 6 * q^43 - 18 * q^44 + 6 * q^45 - 2 * q^46 + 6 * q^47 - 21 * q^48 + 9 * q^49 + 2 * q^50 - 8 * q^51 - q^52 + 19 * q^53 - 7 * q^54 + 4 * q^55 - 10 * q^56 + 2 * q^57 + 21 * q^58 + 23 * q^59 - 19 * q^60 - 13 * q^62 - 13 * q^63 + 27 * q^64 + 44 * q^66 - 3 * q^67 - 4 * q^68 + 2 * q^69 - 12 * q^70 - 3 * q^71 + 39 * q^72 - 32 * q^73 - 12 * q^74 - 2 * q^75 + 18 * q^76 + 18 * q^77 + 43 * q^78 + 2 * q^79 + 8 * q^80 - 28 * q^81 - 5 * q^82 - 21 * q^83 + 28 * q^84 - q^85 - 2 * q^86 - 18 * q^87 - 14 * q^88 + 3 * q^90 - 40 * q^91 - 4 * q^92 - 8 * q^93 + 47 * q^94 - 4 * q^95 - 61 * q^96 - 18 * q^97 + 16 * q^98 + 6 * 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 $

$2$	1.32973	0.940259	0.470130	−	0.882597i	$-0.344207\pi$
$2$	1.32973	0.940259	0.470130	+	0.882597i	$0.344207\pi$
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	−0.231826	−0.115913
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.07644	0.847703
$7$	−3.50407	−1.32441	−0.662207	−	0.749321i	$-0.730380\pi$
$7$	−3.50407	−1.32441	−0.662207	+	0.749321i	$0.730380\pi$
$8$	−2.96772	−1.04925
$9$	−0.561553	−0.187184
$10$	1.32973	0.420497
$11$	−0.659454	−0.198833	−0.0994165	−	0.995046i	$-0.531698\pi$
$11$	−0.659454	−0.198833	−0.0994165	+	0.995046i	$0.531698\pi$
$12$	−0.362008	−0.104503
$13$	5.91023	1.63920	0.819602	−	0.572933i	$-0.194195\pi$
$13$	5.91023	1.63920	0.819602	+	0.572933i	$0.194195\pi$
$14$	−4.65945	−1.24529
$15$	1.56155	0.403191
$16$	−3.48261	−0.870651
$17$	0.844614	0.204849	0.102424	−	0.994741i	$-0.467340\pi$
$17$	0.844614	0.204849	0.102424	+	0.994741i	$0.467340\pi$
$18$	−0.746712	−0.176002
$19$	0.659454	0.151289	0.0756446	−	0.997135i	$-0.475899\pi$
$19$	0.659454	0.151289	0.0756446	+	0.997135i	$0.475899\pi$
$20$	−0.231826	−0.0518378
$21$	−5.47179	−1.19404
$22$	−0.876894	−0.186955
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−4.63425	−0.945962
$25$	1.00000	0.200000
$26$	7.85900	1.54128
$27$	−5.56155	−1.07032
$28$	0.812333	0.153516
$29$	1.59383	0.295967	0.147984	−	0.988990i	$-0.452722\pi$
$29$	1.59383	0.295967	0.147984	+	0.988990i	$0.452722\pi$
$30$	2.07644	0.379104
$31$	6.75485	1.21321	0.606603	−	0.795005i	$-0.292532\pi$
$31$	6.75485	1.21321	0.606603	+	0.795005i	$0.292532\pi$
$32$	1.30452	0.230609
$33$	−1.02977	−0.179260
$34$	1.12311	0.192611
$35$	−3.50407	−0.592296
$36$	0.130182	0.0216971
$37$	−11.7503	−1.93173	−0.965867	−	0.259038i	$-0.916594\pi$
$37$	−11.7503	−1.93173	−0.965867	+	0.259038i	$0.916594\pi$
$38$	0.876894	0.142251
$39$	9.22914	1.47785
$40$	−2.96772	−0.469238
$41$	6.40617	1.00048	0.500238	−	0.865888i	$-0.333246\pi$
$41$	6.40617	1.00048	0.500238	+	0.865888i	$0.333246\pi$
$42$	−7.27598	−1.12271
$43$	−9.47179	−1.44443	−0.722217	−	0.691667i	$-0.756877\pi$
$43$	−9.47179	−1.44443	−0.722217	+	0.691667i	$0.756877\pi$
$44$	0.152878	0.0230473
$45$	−0.561553	−0.0837114
$46$	−1.32973	−0.196058
$47$	6.88046	1.00362	0.501809	−	0.864978i	$-0.332668\pi$
$47$	6.88046	1.00362	0.501809	+	0.864978i	$0.332668\pi$
$48$	−5.43827	−0.784947
$49$	5.27849	0.754070
$50$	1.32973	0.188052
$51$	1.31891	0.184684
$52$	−1.37014	−0.190005
$53$	6.64881	0.913284	0.456642	−	0.889650i	$-0.349052\pi$
$53$	6.64881	0.913284	0.456642	+	0.889650i	$0.349052\pi$
$54$	−7.39535	−1.00638
$55$	−0.659454	−0.0889208
$56$	10.3991	1.38964
$57$	1.02977	0.136397
$58$	2.11936	0.278286
$59$	−4.97586	−0.647801	−0.323901	−	0.946091i	$-0.604994\pi$
$59$	−4.97586	−0.647801	−0.323901	+	0.946091i	$0.604994\pi$
$60$	−0.362008	−0.0467350
$61$	5.78256	0.740381	0.370190	−	0.928956i	$-0.379292\pi$
$61$	5.78256	0.740381	0.370190	+	0.928956i	$0.379292\pi$
$62$	8.98210	1.14073
$63$	1.96772	0.247909
$64$	8.69987	1.08748
$65$	5.91023	0.733074
$66$	−1.36932	−0.168551
$67$	8.31640	1.01601	0.508005	−	0.861354i	$-0.330383\pi$
$67$	8.31640	1.01601	0.508005	+	0.861354i	$0.330383\pi$
$68$	−0.195803	−0.0237446
$69$	−1.56155	−0.187989
$70$	−4.65945	−0.556911
$71$	−3.63174	−0.431009	−0.215504	−	0.976503i	$-0.569140\pi$
$71$	−3.63174	−0.431009	−0.215504	+	0.976503i	$0.569140\pi$
$72$	1.66653	0.196403
$73$	−13.9102	−1.62807	−0.814035	−	0.580816i	$-0.802734\pi$
$73$	−13.9102	−1.62807	−0.814035	+	0.580816i	$0.802734\pi$
$74$	−15.6247	−1.81633
$75$	1.56155	0.180313
$76$	−0.152878	−0.0175364
$77$	2.31077	0.263337
$78$	12.2722	1.38956
$79$	1.02977	0.115858	0.0579292	−	0.998321i	$-0.481550\pi$
$79$	1.02977	0.115858	0.0579292	+	0.998321i	$0.481550\pi$
$80$	−3.48261	−0.389367
$81$	−7.00000	−0.777778
$82$	8.51845	0.940706
$83$	−8.27849	−0.908683	−0.454341	−	0.890828i	$-0.650125\pi$
$83$	−8.27849	−0.908683	−0.454341	+	0.890828i	$0.650125\pi$
$84$	1.26850	0.138405
$85$	0.844614	0.0916112
$86$	−12.5949	−1.35814
$87$	2.48886	0.266833
$88$	1.95708	0.208625
$89$	17.6030	1.86592	0.932959	−	0.359984i	$-0.117218\pi$
$89$	17.6030	1.86592	0.932959	+	0.359984i	$0.117218\pi$
$90$	−0.746712	−0.0787104
$91$	−20.7099	−2.17098
$92$	0.231826	0.0241695
$93$	10.5481	1.09378
$94$	9.14914	0.943661
$95$	0.659454	0.0676586
$96$	2.03708	0.207909
$97$	−8.34868	−0.847680	−0.423840	−	0.905737i	$-0.639318\pi$
$97$	−8.34868	−0.847680	−0.423840	+	0.905737i	$0.639318\pi$
$98$	7.01895	0.709021
$99$	0.370318	0.0372184
$100$	−0.231826	−0.0231826
$101$	11.5974	1.15398	0.576992	−	0.816750i	$-0.304226\pi$
$101$	11.5974	1.15398	0.576992	+	0.816750i	$0.304226\pi$
$102$	1.75379	0.173651
$103$	−18.9436	−1.86657	−0.933283	−	0.359142i	$-0.883069\pi$
$103$	−18.9436	−1.86657	−0.933283	+	0.359142i	$0.883069\pi$
$104$	−17.5399	−1.71993
$105$	−5.47179	−0.533992
$106$	8.84110	0.858724
$107$	−1.30826	−0.126475	−0.0632374	−	0.997999i	$-0.520143\pi$
$107$	−1.30826	−0.126475	−0.0632374	+	0.997999i	$0.520143\pi$
$108$	1.28931	0.124064
$109$	0.578621	0.0554218	0.0277109	−	0.999616i	$-0.491178\pi$
$109$	0.578621	0.0554218	0.0277109	+	0.999616i	$0.491178\pi$
$110$	−0.876894	−0.0836086
$111$	−18.3487	−1.74158
$112$	12.2033	1.15310
$113$	−4.62717	−0.435288	−0.217644	−	0.976028i	$-0.569837\pi$
$113$	−4.62717	−0.435288	−0.217644	+	0.976028i	$0.569837\pi$
$114$	1.36932	0.128248
$115$	−1.00000	−0.0932505
$116$	−0.369491	−0.0343064
$117$	−3.31891	−0.306833
$118$	−6.61653	−0.609101
$119$	−2.95958	−0.271305
$120$	−4.63425	−0.423047
$121$	−10.5651	−0.960465
$122$	7.68923	0.696150
$123$	10.0036	0.901991
$124$	−1.56595	−0.140626
$125$	1.00000	0.0894427
$126$	2.61653	0.233099
$127$	−0.634250	−0.0562806	−0.0281403	−	0.999604i	$-0.508959\pi$
$127$	−0.634250	−0.0562806	−0.0281403	+	0.999604i	$0.508959\pi$
$128$	8.95941	0.791907
$129$	−14.7907	−1.30225
$130$	7.85900	0.689280
$131$	−14.7780	−1.29116	−0.645580	−	0.763693i	$-0.723384\pi$
$131$	−14.7780	−1.29116	−0.645580	+	0.763693i	$0.723384\pi$
$132$	0.238728	0.0207786
$133$	−2.31077	−0.200369
$134$	11.0585	0.955313
$135$	−5.56155	−0.478662
$136$	−2.50658	−0.214937
$137$	1.51471	0.129411	0.0647053	−	0.997904i	$-0.479389\pi$
$137$	1.51471	0.129411	0.0647053	+	0.997904i	$0.479389\pi$
$138$	−2.07644	−0.176758
$139$	−10.3200	−0.875328	−0.437664	−	0.899139i	$-0.644194\pi$
$139$	−10.3200	−0.875328	−0.437664	+	0.899139i	$0.644194\pi$
$140$	0.812333	0.0686547
$141$	10.7442	0.904825
$142$	−4.82923	−0.405260
$143$	−3.89753	−0.325928
$144$	1.95567	0.162972
$145$	1.59383	0.132361
$146$	−18.4968	−1.53081
$147$	8.24264	0.679842
$148$	2.72402	0.223913
$149$	13.0081	1.06567	0.532834	−	0.846220i	$-0.321127\pi$
$149$	13.0081	1.06567	0.532834	+	0.846220i	$0.321127\pi$
$150$	2.07644	0.169541
$151$	11.0333	0.897880	0.448940	−	0.893562i	$-0.351802\pi$
$151$	11.0333	0.897880	0.448940	+	0.893562i	$0.351802\pi$
$152$	−1.95708	−0.158740
$153$	−0.474295	−0.0383445
$154$	3.07270	0.247605
$155$	6.75485	0.542562
$156$	−2.13955	−0.171301
$157$	−1.61904	−0.129213	−0.0646066	−	0.997911i	$-0.520579\pi$
$157$	−1.61904	−0.129213	−0.0646066	+	0.997911i	$0.520579\pi$
$158$	1.36932	0.108937
$159$	10.3825	0.823383
$160$	1.30452	0.103132
$161$	3.50407	0.276159
$162$	−9.30809	−0.731313
$163$	−2.55342	−0.199999	−0.0999995	−	0.994987i	$-0.531884\pi$
$163$	−2.55342	−0.199999	−0.0999995	+	0.994987i	$0.531884\pi$
$164$	−1.48511	−0.115968
$165$	−1.02977	−0.0801677
$166$	−11.0081	−0.854397
$167$	14.2462	1.10240	0.551202	−	0.834372i	$-0.314169\pi$
$167$	14.2462	1.10240	0.551202	+	0.834372i	$0.314169\pi$
$168$	16.2387	1.25285
$169$	21.9309	1.68699
$170$	1.12311	0.0861383
$171$	−0.370318	−0.0283190
$172$	2.19580	0.167428
$173$	20.5733	1.56416	0.782078	−	0.623181i	$-0.214160\pi$
$173$	20.5733	1.56416	0.782078	+	0.623181i	$0.214160\pi$
$174$	3.30950	0.250892
$175$	−3.50407	−0.264883
$176$	2.29662	0.173114
$177$	−7.77006	−0.584034
$178$	23.4072	1.75445
$179$	−17.6712	−1.32080	−0.660402	−	0.750912i	$-0.729614\pi$
$179$	−17.6712	−1.32080	−0.660402	+	0.750912i	$0.729614\pi$
$180$	0.130182	0.00970322
$181$	−9.55262	−0.710041	−0.355020	−	0.934859i	$-0.615526\pi$
$181$	−9.55262	−0.710041	−0.355020	+	0.934859i	$0.615526\pi$
$182$	−27.5385	−2.04129
$183$	9.02977	0.667500
$184$	2.96772	0.218783
$185$	−11.7503	−0.863898
$186$	14.0260	1.02844
$187$	−0.556984	−0.0407307
$188$	−1.59507	−0.116332
$189$	19.4881	1.41755
$190$	0.876894	0.0636166
$191$	−4.87689	−0.352880	−0.176440	−	0.984311i	$-0.556458\pi$
$191$	−4.87689	−0.352880	−0.176440	+	0.984311i	$0.556458\pi$
$192$	13.5853	0.980435
$193$	13.5903	0.978253	0.489126	−	0.872213i	$-0.337316\pi$
$193$	13.5903	0.978253	0.489126	+	0.872213i	$0.337316\pi$
$194$	−11.1015	−0.797039
$195$	9.22914	0.660913
$196$	−1.22369	−0.0874064
$197$	9.46822	0.674583	0.337291	−	0.941400i	$-0.390489\pi$
$197$	9.46822	0.674583	0.337291	+	0.941400i	$0.390489\pi$
$198$	0.492423	0.0349949
$199$	13.9784	0.990900	0.495450	−	0.868636i	$-0.335003\pi$
$199$	13.9784	0.990900	0.495450	+	0.868636i	$0.335003\pi$
$200$	−2.96772	−0.209849
$201$	12.9865	0.915997
$202$	15.4214	1.08504
$203$	−5.58490	−0.391983
$204$	−0.305757	−0.0214073
$205$	6.40617	0.447426
$206$	−25.1898	−1.75506
$207$	0.561553	0.0390306
$208$	−20.5830	−1.42718
$209$	−0.434880	−0.0300813
$210$	−7.27598	−0.502091
$211$	2.58926	0.178252	0.0891262	−	0.996020i	$-0.471593\pi$
$211$	2.58926	0.178252	0.0891262	+	0.996020i	$0.471593\pi$
$212$	−1.54136	−0.105861
$213$	−5.67116	−0.388581
$214$	−1.73964	−0.118919
$215$	−9.47179	−0.645971
$216$	16.5051	1.12303
$217$	−23.6694	−1.60679
$218$	0.769408	0.0521109
$219$	−21.7216	−1.46781
$220$	0.152878	0.0103071
$221$	4.99186	0.335789
$222$	−24.3987	−1.63754
$223$	−26.1816	−1.75325	−0.876626	−	0.481172i	$-0.840211\pi$
$223$	−26.1816	−1.75325	−0.876626	+	0.481172i	$0.840211\pi$
$224$	−4.57114	−0.305422
$225$	−0.561553	−0.0374369
$226$	−6.15288	−0.409283
$227$	16.7312	1.11049	0.555243	−	0.831688i	$-0.312625\pi$
$227$	16.7312	1.11049	0.555243	+	0.831688i	$0.312625\pi$
$228$	−0.238728	−0.0158101
$229$	−6.65945	−0.440069	−0.220035	−	0.975492i	$-0.570617\pi$
$229$	−6.65945	−0.440069	−0.220035	+	0.975492i	$0.570617\pi$
$230$	−1.32973	−0.0876796
$231$	3.60839	0.237415
$232$	−4.73005	−0.310543
$233$	−25.3604	−1.66141	−0.830707	−	0.556710i	$-0.812064\pi$
$233$	−25.3604	−1.66141	−0.830707	+	0.556710i	$0.812064\pi$
$234$	−4.41324	−0.288503
$235$	6.88046	0.448832
$236$	1.15353	0.0750885
$237$	1.60804	0.104454
$238$	−3.93544	−0.255097
$239$	−10.4349	−0.674980	−0.337490	−	0.941329i	$-0.609578\pi$
$239$	−10.4349	−0.674980	−0.337490	+	0.941329i	$0.609578\pi$
$240$	−5.43827	−0.351039
$241$	11.3189	0.729115	0.364558	−	0.931181i	$-0.381220\pi$
$241$	11.3189	0.729115	0.364558	+	0.931181i	$0.381220\pi$
$242$	−14.0487	−0.903086
$243$	5.75379	0.369106
$244$	−1.34055	−0.0858196
$245$	5.27849	0.337230
$246$	13.3020	0.848106
$247$	3.89753	0.247994
$248$	−20.0465	−1.27295
$249$	−12.9273	−0.819235
$250$	1.32973	0.0840993
$251$	20.1979	1.27488	0.637441	−	0.770499i	$-0.279993\pi$
$251$	20.1979	1.27488	0.637441	+	0.770499i	$0.279993\pi$
$252$	−0.456168	−0.0287359
$253$	0.659454	0.0414595
$254$	−0.843380	−0.0529184
$255$	1.31891	0.0825933
$256$	−5.48617	−0.342886
$257$	−8.40054	−0.524011	−0.262006	−	0.965066i	$-0.584384\pi$
$257$	−8.40054	−0.524011	−0.262006	+	0.965066i	$0.584384\pi$
$258$	−19.6676	−1.22445
$259$	41.1738	2.55841
$260$	−1.37014	−0.0849727
$261$	−0.895022	−0.0554005
$262$	−19.6507	−1.21402
$263$	28.6434	1.76623	0.883115	−	0.469156i	$-0.155442\pi$
$263$	28.6434	1.76623	0.883115	+	0.469156i	$0.155442\pi$
$264$	3.05608	0.188089
$265$	6.64881	0.408433
$266$	−3.07270	−0.188399
$267$	27.4881	1.68224
$268$	−1.92795	−0.117769
$269$	−26.1117	−1.59206	−0.796028	−	0.605260i	$-0.793069\pi$
$269$	−26.1117	−1.59206	−0.796028	+	0.605260i	$0.793069\pi$
$270$	−7.39535	−0.450067
$271$	13.9032	0.844557	0.422278	−	0.906466i	$-0.361230\pi$
$271$	13.9032	0.844557	0.422278	+	0.906466i	$0.361230\pi$
$272$	−2.94146	−0.178352
$273$	−32.3395	−1.95728
$274$	2.01415	0.121679
$275$	−0.659454	−0.0397666
$276$	0.362008	0.0217903
$277$	27.6712	1.66260	0.831299	−	0.555825i	$-0.187598\pi$
$277$	27.6712	1.66260	0.831299	+	0.555825i	$0.187598\pi$
$278$	−13.7227	−0.823035
$279$	−3.79320	−0.227093
$280$	10.3991	0.621464
$281$	15.6513	0.933679	0.466840	−	0.884342i	$-0.345393\pi$
$281$	15.6513	0.933679	0.466840	+	0.884342i	$0.345393\pi$
$282$	14.2869	0.850770
$283$	23.6641	1.40668	0.703342	−	0.710852i	$-0.251690\pi$
$283$	23.6641	1.40668	0.703342	+	0.710852i	$0.251690\pi$
$284$	0.841931	0.0499594
$285$	1.02977	0.0609985
$286$	−5.18265	−0.306457
$287$	−22.4476	−1.32504
$288$	−0.732559	−0.0431664
$289$	−16.2866	−0.958037
$290$	2.11936	0.124453
$291$	−13.0369	−0.764237
$292$	3.22475	0.188714
$293$	−24.0507	−1.40506	−0.702528	−	0.711657i	$-0.747945\pi$
$293$	−24.0507	−1.40506	−0.702528	+	0.711657i	$0.747945\pi$
$294$	10.9605	0.639227
$295$	−4.97586	−0.289705
$296$	34.8715	2.02687
$297$	3.66759	0.212815
$298$	17.2973	1.00200
$299$	−5.91023	−0.341798
$300$	−0.362008	−0.0209005
$301$	33.1898	1.91303
$302$	14.6713	0.844240
$303$	18.1100	1.04039
$304$	−2.29662	−0.131720
$305$	5.78256	0.331108
$306$	−0.630683	−0.0360538
$307$	−0.370318	−0.0211352	−0.0105676	−	0.999944i	$-0.503364\pi$
$307$	−0.370318	−0.0211352	−0.0105676	+	0.999944i	$0.503364\pi$
$308$	−0.535696	−0.0305241
$309$	−29.5814	−1.68283
$310$	8.98210	0.510149
$311$	4.35225	0.246793	0.123397	−	0.992357i	$-0.460621\pi$
$311$	4.35225	0.246793	0.123397	+	0.992357i	$0.460621\pi$
$312$	−27.3895	−1.55063
$313$	−6.82834	−0.385961	−0.192980	−	0.981203i	$-0.561815\pi$
$313$	−6.82834	−0.385961	−0.192980	+	0.981203i	$0.561815\pi$
$314$	−2.15288	−0.121494
$315$	1.96772	0.110868
$316$	−0.238728	−0.0134295
$317$	−26.1979	−1.47142	−0.735711	−	0.677296i	$-0.763152\pi$
$317$	−26.1979	−1.47142	−0.735711	+	0.677296i	$0.763152\pi$
$318$	13.8059	0.774194
$319$	−1.05106	−0.0588481
$320$	8.69987	0.486338
$321$	−2.04292	−0.114025
$322$	4.65945	0.259661
$323$	0.556984	0.0309914
$324$	1.62278	0.0901544
$325$	5.91023	0.327841
$326$	−3.39535	−0.188051
$327$	0.903547	0.0499663
$328$	−19.0117	−1.04975
$329$	−24.1096	−1.32921
$330$	−1.36932	−0.0753784
$331$	−12.6307	−0.694248	−0.347124	−	0.937819i	$-0.612842\pi$
$331$	−12.6307	−0.694248	−0.347124	+	0.937819i	$0.612842\pi$
$332$	1.91917	0.105328
$333$	6.59840	0.361590
$334$	18.9436	1.03655
$335$	8.31640	0.454374
$336$	19.0561	1.03959
$337$	−3.65132	−0.198900	−0.0994500	−	0.995043i	$-0.531708\pi$
$337$	−3.65132	−0.198900	−0.0994500	+	0.995043i	$0.531708\pi$
$338$	29.1621	1.58621
$339$	−7.22558	−0.392439
$340$	−0.195803	−0.0106189
$341$	−4.45451	−0.241225
$342$	−0.492423	−0.0266272
$343$	6.03228	0.325713
$344$	28.1096	1.51557
$345$	−1.56155	−0.0840712
$346$	27.3568	1.47071
$347$	25.0031	1.34224	0.671119	−	0.741350i	$-0.265814\pi$
$347$	25.0031	1.34224	0.671119	+	0.741350i	$0.265814\pi$
$348$	−0.576980	−0.0309294
$349$	0.580682	0.0310832	0.0155416	−	0.999879i	$-0.495053\pi$
$349$	0.580682	0.0310832	0.0155416	+	0.999879i	$0.495053\pi$
$350$	−4.65945	−0.249058
$351$	−32.8701	−1.75448
$352$	−0.860273	−0.0458527
$353$	15.6499	0.832959	0.416479	−	0.909145i	$-0.363264\pi$
$353$	15.6499	0.832959	0.416479	+	0.909145i	$0.363264\pi$
$354$	−10.3321	−0.549143
$355$	−3.63174	−0.192753
$356$	−4.08083	−0.216284
$357$	−4.62155	−0.244598
$358$	−23.4978	−1.24190
$359$	26.2196	1.38382	0.691908	−	0.721986i	$-0.256771\pi$
$359$	26.2196	1.38382	0.691908	+	0.721986i	$0.256771\pi$
$360$	1.66653	0.0878339
$361$	−18.5651	−0.977112
$362$	−12.7024	−0.667622
$363$	−16.4980	−0.865920
$364$	4.80108	0.251645
$365$	−13.9102	−0.728095
$366$	12.0071	0.627623
$367$	−20.9113	−1.09156	−0.545780	−	0.837928i	$-0.683767\pi$
$367$	−20.9113	−1.09156	−0.545780	+	0.837928i	$0.683767\pi$
$368$	3.48261	0.181543
$369$	−3.59740	−0.187273
$370$	−15.6247	−0.812288
$371$	−23.2979	−1.20957
$372$	−2.44531	−0.126783
$373$	−16.3220	−0.845123	−0.422561	−	0.906334i	$-0.638869\pi$
$373$	−16.3220	−0.845123	−0.422561	+	0.906334i	$0.638869\pi$
$374$	−0.740637	−0.0382974
$375$	1.56155	0.0806382
$376$	−20.4193	−1.05304
$377$	9.41993	0.485151
$378$	25.9138	1.33286
$379$	18.9003	0.970843	0.485422	−	0.874280i	$-0.338666\pi$
$379$	18.9003	0.970843	0.485422	+	0.874280i	$0.338666\pi$
$380$	−0.152878	−0.00784250
$381$	−0.990415	−0.0507405
$382$	−6.48494	−0.331798
$383$	−4.06105	−0.207510	−0.103755	−	0.994603i	$-0.533086\pi$
$383$	−4.06105	−0.207510	−0.103755	+	0.994603i	$0.533086\pi$
$384$	13.9906	0.713954
$385$	2.31077	0.117768
$386$	18.0714	0.919811
$387$	5.31891	0.270375
$388$	1.93544	0.0982570
$389$	11.5310	0.584644	0.292322	−	0.956320i	$-0.405572\pi$
$389$	11.5310	0.584644	0.292322	+	0.956320i	$0.405572\pi$
$390$	12.2722	0.621429
$391$	−0.844614	−0.0427139
$392$	−15.6651	−0.791206
$393$	−23.0766	−1.16406
$394$	12.5901	0.634283
$395$	1.02977	0.0518135
$396$	−0.0858493	−0.00431409
$397$	−14.0252	−0.703905	−0.351952	−	0.936018i	$-0.614482\pi$
$397$	−14.0252	−0.703905	−0.351952	+	0.936018i	$0.614482\pi$
$398$	18.5874	0.931703
$399$	−3.60839	−0.180646
$400$	−3.48261	−0.174130
$401$	15.5814	0.778098	0.389049	−	0.921217i	$-0.372804\pi$
$401$	15.5814	0.778098	0.389049	+	0.921217i	$0.372804\pi$
$402$	17.2685	0.861275
$403$	39.9227	1.98869
$404$	−2.68857	−0.133762
$405$	−7.00000	−0.347833
$406$	−7.42639	−0.368566
$407$	7.74877	0.384092
$408$	−3.91415	−0.193779
$409$	−1.64424	−0.0813025	−0.0406513	−	0.999173i	$-0.512943\pi$
$409$	−1.64424	−0.0813025	−0.0406513	+	0.999173i	$0.512943\pi$
$410$	8.51845	0.420696
$411$	2.36530	0.116672
$412$	4.39161	0.216359
$413$	17.4357	0.857956
$414$	0.746712	0.0366989
$415$	−8.27849	−0.406375
$416$	7.71004	0.378016
$417$	−16.1152	−0.789164
$418$	−0.578272	−0.0282842
$419$	8.70986	0.425505	0.212752	−	0.977106i	$-0.431757\pi$
$419$	8.70986	0.425505	0.212752	+	0.977106i	$0.431757\pi$
$420$	1.26850	0.0618965
$421$	17.0028	0.828664	0.414332	−	0.910126i	$-0.364015\pi$
$421$	17.0028	0.828664	0.414332	+	0.910126i	$0.364015\pi$
$422$	3.44302	0.167603
$423$	−3.86374	−0.187862
$424$	−19.7318	−0.958261
$425$	0.844614	0.0409698
$426$	−7.54109	−0.365367
$427$	−20.2625	−0.980570
$428$	0.303289	0.0146600
$429$	−6.08620	−0.293844
$430$	−12.5949	−0.607380
$431$	−4.38282	−0.211113	−0.105556	−	0.994413i	$-0.533662\pi$
$431$	−4.38282	−0.211113	−0.105556	+	0.994413i	$0.533662\pi$
$432$	19.3687	0.931877
$433$	22.4764	1.08015	0.540074	−	0.841618i	$-0.318396\pi$
$433$	22.4764	1.08015	0.540074	+	0.841618i	$0.318396\pi$
$434$	−31.4739	−1.51080
$435$	2.48886	0.119331
$436$	−0.134139	−0.00642410
$437$	−0.659454	−0.0315460
$438$	−28.8838	−1.38012
$439$	−12.6580	−0.604134	−0.302067	−	0.953287i	$-0.597677\pi$
$439$	−12.6580	−0.604134	−0.302067	+	0.953287i	$0.597677\pi$
$440$	1.95708	0.0932999
$441$	−2.96415	−0.141150
$442$	6.63782	0.315729
$443$	12.7351	0.605061	0.302531	−	0.953140i	$-0.402169\pi$
$443$	12.7351	0.605061	0.302531	+	0.953140i	$0.402169\pi$
$444$	4.25369	0.201871
$445$	17.6030	0.834464
$446$	−34.8145	−1.64851
$447$	20.3129	0.960767
$448$	−30.4849	−1.44028
$449$	−6.41887	−0.302925	−0.151463	−	0.988463i	$-0.548398\pi$
$449$	−6.41887	−0.302925	−0.151463	+	0.988463i	$0.548398\pi$
$450$	−0.746712	−0.0352003
$451$	−4.22457	−0.198927
$452$	1.07270	0.0504554
$453$	17.2291	0.809496
$454$	22.2479	1.04414
$455$	−20.7099	−0.970893
$456$	−3.05608	−0.143114
$457$	−23.4824	−1.09846	−0.549231	−	0.835671i	$-0.685079\pi$
$457$	−23.4824	−1.09846	−0.549231	+	0.835671i	$0.685079\pi$
$458$	−8.85526	−0.413779
$459$	−4.69736	−0.219254
$460$	0.231826	0.0108089
$461$	21.4162	0.997450	0.498725	−	0.866760i	$-0.333802\pi$
$461$	21.4162	0.997450	0.498725	+	0.866760i	$0.333802\pi$
$462$	4.79818	0.223232
$463$	14.7782	0.686801	0.343401	−	0.939189i	$-0.388421\pi$
$463$	14.7782	0.686801	0.343401	+	0.939189i	$0.388421\pi$
$464$	−5.55069	−0.257684
$465$	10.5481	0.489154
$466$	−33.7224	−1.56216
$467$	−38.6509	−1.78855	−0.894276	−	0.447516i	$-0.852309\pi$
$467$	−38.6509	−1.78855	−0.894276	+	0.447516i	$0.852309\pi$
$468$	0.769408	0.0355659
$469$	−29.1412	−1.34562
$470$	9.14914	0.422018
$471$	−2.52821	−0.116494
$472$	14.7669	0.679704
$473$	6.24621	0.287201
$474$	2.13826	0.0982136
$475$	0.659454	0.0302578
$476$	0.686107	0.0314477
$477$	−3.73366	−0.170952
$478$	−13.8756	−0.634656
$479$	7.87088	0.359630	0.179815	−	0.983700i	$-0.442450\pi$
$479$	7.87088	0.359630	0.179815	+	0.983700i	$0.442450\pi$
$480$	2.03708	0.0929796
$481$	−69.4469	−3.16651
$482$	15.0511	0.685557
$483$	5.47179	0.248975
$484$	2.44927	0.111330
$485$	−8.34868	−0.379094
$486$	7.65097	0.347055
$487$	−16.0702	−0.728212	−0.364106	−	0.931357i	$-0.618625\pi$
$487$	−16.0702	−0.728212	−0.364106	+	0.931357i	$0.618625\pi$
$488$	−17.1610	−0.776843
$489$	−3.98730	−0.180312
$490$	7.01895	0.317084
$491$	−39.2473	−1.77120	−0.885602	−	0.464444i	$-0.846254\pi$
$491$	−39.2473	−1.77120	−0.885602	+	0.464444i	$0.846254\pi$
$492$	−2.31908	−0.104552
$493$	1.34617	0.0606286
$494$	5.18265	0.233179
$495$	0.370318	0.0166446
$496$	−23.5245	−1.05628
$497$	12.7259	0.570833
$498$	−17.1898	−0.770293
$499$	−4.99392	−0.223559	−0.111779	−	0.993733i	$-0.535655\pi$
$499$	−4.99392	−0.223559	−0.111779	+	0.993733i	$0.535655\pi$
$500$	−0.231826	−0.0103676
$501$	22.2462	0.993887
$502$	26.8577	1.19872
$503$	5.62281	0.250709	0.125354	−	0.992112i	$-0.459993\pi$
$503$	5.62281	0.250709	0.125354	+	0.992112i	$0.459993\pi$
$504$	−5.83964	−0.260118
$505$	11.5974	0.516078
$506$	0.876894	0.0389827
$507$	34.2462	1.52093
$508$	0.147035	0.00652364
$509$	34.4619	1.52749	0.763747	−	0.645515i	$-0.223357\pi$
$509$	34.4619	1.52749	0.763747	+	0.645515i	$0.223357\pi$
$510$	1.75379	0.0776591
$511$	48.7424	2.15624
$512$	−25.2139	−1.11431
$513$	−3.66759	−0.161928
$514$	−11.1704	−0.492706
$515$	−18.9436	−0.834754
$516$	3.42886	0.150947
$517$	−4.53735	−0.199552
$518$	54.7499	2.40557
$519$	32.1262	1.41018
$520$	−17.5399	−0.769176
$521$	−17.1231	−0.750177	−0.375088	−	0.926989i	$-0.622388\pi$
$521$	−17.1231	−0.750177	−0.375088	+	0.926989i	$0.622388\pi$
$522$	−1.19013	−0.0520908
$523$	15.4881	0.677246	0.338623	−	0.940922i	$-0.390039\pi$
$523$	15.4881	0.677246	0.338623	+	0.940922i	$0.390039\pi$
$524$	3.42592	0.149662
$525$	−5.47179	−0.238808
$526$	38.0880	1.66071
$527$	5.70524	0.248524
$528$	3.58629	0.156073
$529$	1.00000	0.0434783
$530$	8.84110	0.384033
$531$	2.79421	0.121258
$532$	0.535696	0.0232254
$533$	37.8619	1.63998
$534$	36.5516	1.58174
$535$	−1.30826	−0.0565612
$536$	−24.6807	−1.06605
$537$	−27.5944	−1.19079
$538$	−34.7214	−1.49695
$539$	−3.48092	−0.149934
$540$	1.28931	0.0554831
$541$	−28.9634	−1.24523	−0.622617	−	0.782527i	$-0.713931\pi$
$541$	−28.9634	−1.24523	−0.622617	+	0.782527i	$0.713931\pi$
$542$	18.4874	0.794102
$543$	−14.9169	−0.640146
$544$	1.10182	0.0472401
$545$	0.578621	0.0247854
$546$	−43.0028	−1.84035
$547$	−7.57782	−0.324004	−0.162002	−	0.986790i	$-0.551795\pi$
$547$	−7.57782	−0.324004	−0.162002	+	0.986790i	$0.551795\pi$
$548$	−0.351149	−0.0150003
$549$	−3.24721	−0.138588
$550$	−0.876894	−0.0373909
$551$	1.05106	0.0447767
$552$	4.63425	0.197247
$553$	−3.60839	−0.153445
$554$	36.7951	1.56327
$555$	−18.3487	−0.778858
$556$	2.39243	0.101462
$557$	37.9248	1.60693	0.803463	−	0.595355i	$-0.202989\pi$
$557$	37.9248	1.60693	0.803463	+	0.595355i	$0.202989\pi$
$558$	−5.04393	−0.213526
$559$	−55.9805	−2.36772
$560$	12.2033	0.515683
$561$	−0.869760	−0.0367213
$562$	20.8120	0.877901
$563$	1.24907	0.0526420	0.0263210	−	0.999654i	$-0.491621\pi$
$563$	1.24907	0.0526420	0.0263210	+	0.999654i	$0.491621\pi$
$564$	−2.49078	−0.104881
$565$	−4.62717	−0.194667
$566$	31.4668	1.32265
$567$	24.5285	1.03010
$568$	10.7780	0.452234
$569$	−15.4197	−0.646429	−0.323214	−	0.946326i	$-0.604763\pi$
$569$	−15.4197	−0.646429	−0.323214	+	0.946326i	$0.604763\pi$
$570$	1.36932	0.0573544
$571$	8.49242	0.355397	0.177698	−	0.984085i	$-0.443135\pi$
$571$	8.49242	0.355397	0.177698	+	0.984085i	$0.443135\pi$
$572$	0.903547	0.0377792
$573$	−7.61553	−0.318143
$574$	−29.8492	−1.24588
$575$	−1.00000	−0.0417029
$576$	−4.88544	−0.203560
$577$	17.1987	0.715992	0.357996	−	0.933723i	$-0.383460\pi$
$577$	17.1987	0.715992	0.357996	+	0.933723i	$0.383460\pi$
$578$	−21.6568	−0.900803
$579$	21.2220	0.881957
$580$	−0.369491	−0.0153423
$581$	29.0084	1.20347
$582$	−17.3355	−0.718581
$583$	−4.38459	−0.181591
$584$	41.2817	1.70825
$585$	−3.31891	−0.137220
$586$	−31.9808	−1.32112
$587$	14.5860	0.602027	0.301014	−	0.953620i	$-0.402675\pi$
$587$	14.5860	0.602027	0.301014	+	0.953620i	$0.402675\pi$
$588$	−1.91086	−0.0788024
$589$	4.45451	0.183545
$590$	−6.61653	−0.272398
$591$	14.7851	0.608179
$592$	40.9216	1.68187
$593$	4.38659	0.180136	0.0900678	−	0.995936i	$-0.471292\pi$
$593$	4.38659	0.180136	0.0900678	+	0.995936i	$0.471292\pi$
$594$	4.87689	0.200101
$595$	−2.95958	−0.121331
$596$	−3.01562	−0.123525
$597$	21.8280	0.893359
$598$	−7.85900	−0.321378
$599$	29.7863	1.21704	0.608518	−	0.793540i	$-0.291764\pi$
$599$	29.7863	1.21704	0.608518	+	0.793540i	$0.291764\pi$
$600$	−4.63425	−0.189192
$601$	14.6828	0.598924	0.299462	−	0.954108i	$-0.403193\pi$
$601$	14.6828	0.598924	0.299462	+	0.954108i	$0.403193\pi$
$602$	44.1334	1.79874
$603$	−4.67010	−0.190181
$604$	−2.55781	−0.104076
$605$	−10.5651	−0.429533
$606$	24.0813	0.978236
$607$	22.3220	0.906023	0.453012	−	0.891505i	$-0.350350\pi$
$607$	22.3220	0.906023	0.453012	+	0.891505i	$0.350350\pi$
$608$	0.860273	0.0348887
$609$	−8.72112	−0.353398
$610$	7.68923	0.311328
$611$	40.6651	1.64514
$612$	0.109954	0.00444462
$613$	−5.19892	−0.209983	−0.104991	−	0.994473i	$-0.533481\pi$
$613$	−5.19892	−0.209983	−0.104991	+	0.994473i	$0.533481\pi$
$614$	−0.492423	−0.0198726
$615$	10.0036	0.403383
$616$	−6.85772	−0.276306
$617$	−33.4720	−1.34753	−0.673767	−	0.738944i	$-0.735325\pi$
$617$	−33.4720	−1.34753	−0.673767	+	0.738944i	$0.735325\pi$
$618$	−39.3352	−1.58229
$619$	−0.826486	−0.0332193	−0.0166096	−	0.999862i	$-0.505287\pi$
$619$	−0.826486	−0.0332193	−0.0166096	+	0.999862i	$0.505287\pi$
$620$	−1.56595	−0.0628899
$621$	5.56155	0.223177
$622$	5.78730	0.232050
$623$	−61.6822	−2.47125
$624$	−32.1415	−1.28669
$625$	1.00000	0.0400000
$626$	−9.07983	−0.362903
$627$	−0.679088	−0.0271202
$628$	0.375334	0.0149775
$629$	−9.92445	−0.395714
$630$	2.61653	0.104245
$631$	−32.6453	−1.29959	−0.649794	−	0.760110i	$-0.725145\pi$
$631$	−32.6453	−1.29959	−0.649794	+	0.760110i	$0.725145\pi$
$632$	−3.05608	−0.121564
$633$	4.04327	0.160706
$634$	−34.8361	−1.38352
$635$	−0.634250	−0.0251695
$636$	−2.40692	−0.0954407
$637$	31.1971	1.23608
$638$	−1.39762	−0.0553325
$639$	2.03942	0.0806780
$640$	8.95941	0.354152
$641$	−14.6707	−0.579458	−0.289729	−	0.957109i	$-0.593565\pi$
$641$	−14.6707	−0.579458	−0.289729	+	0.957109i	$0.593565\pi$
$642$	−2.71653	−0.107213
$643$	27.1504	1.07071	0.535353	−	0.844628i	$-0.320179\pi$
$643$	27.1504	1.07071	0.535353	+	0.844628i	$0.320179\pi$
$644$	−0.812333	−0.0320104
$645$	−14.7907	−0.582383
$646$	0.740637	0.0291400
$647$	17.2671	0.678838	0.339419	−	0.940635i	$-0.389770\pi$
$647$	17.2671	0.678838	0.339419	+	0.940635i	$0.389770\pi$
$648$	20.7740	0.816081
$649$	3.28135	0.128804
$650$	7.85900	0.308255
$651$	−36.9611	−1.44862
$652$	0.591948	0.0231825
$653$	25.0837	0.981603	0.490801	−	0.871271i	$-0.336704\pi$
$653$	25.0837	0.981603	0.490801	+	0.871271i	$0.336704\pi$
$654$	1.20147	0.0469813
$655$	−14.7780	−0.577424
$656$	−22.3102	−0.871065
$657$	7.81133	0.304749
$658$	−32.0592	−1.24980
$659$	16.8195	0.655193	0.327597	−	0.944818i	$-0.393761\pi$
$659$	16.8195	0.655193	0.327597	+	0.944818i	$0.393761\pi$
$660$	0.238728	0.00929246
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−16.7954	−0.652773
$663$	7.79506	0.302735
$664$	24.5682	0.953433
$665$	−2.31077	−0.0896079
$666$	8.77407	0.339988
$667$	−1.59383	−0.0617135
$668$	−3.30264	−0.127783
$669$	−40.8840	−1.58067
$670$	11.0585	0.427229
$671$	−3.81333	−0.147212
$672$	−7.13807	−0.275357
$673$	−31.8365	−1.22721	−0.613604	−	0.789614i	$-0.710281\pi$
$673$	−31.8365	−1.22721	−0.613604	+	0.789614i	$0.710281\pi$
$674$	−4.85526	−0.187018
$675$	−5.56155	−0.214064
$676$	−5.08414	−0.195544
$677$	−46.5522	−1.78915	−0.894574	−	0.446920i	$-0.852521\pi$
$677$	−46.5522	−1.78915	−0.894574	+	0.446920i	$0.852521\pi$
$678$	−9.60804	−0.368995
$679$	29.2543	1.12268
$680$	−2.50658	−0.0961228
$681$	26.1266	1.00117
$682$	−5.92329	−0.226814
$683$	−42.5214	−1.62704	−0.813518	−	0.581540i	$-0.802451\pi$
$683$	−42.5214	−1.62704	−0.813518	+	0.581540i	$0.802451\pi$
$684$	0.0858493	0.00328253
$685$	1.51471	0.0578742
$686$	8.02129	0.306254
$687$	−10.3991	−0.396750
$688$	32.9865	1.25760
$689$	39.2960	1.49706
$690$	−2.07644	−0.0790487
$691$	21.5488	0.819757	0.409878	−	0.912140i	$-0.365571\pi$
$691$	21.5488	0.819757	0.409878	+	0.912140i	$0.365571\pi$
$692$	−4.76941	−0.181306
$693$	−1.29762	−0.0492925
$694$	33.2473	1.26205
$695$	−10.3200	−0.391459
$696$	−7.38622	−0.279974
$697$	5.41074	0.204946
$698$	0.772148	0.0292262
$699$	−39.6016	−1.49787
$700$	0.812333	0.0307033
$701$	0.251576	0.00950191	0.00475096	−	0.999989i	$-0.498488\pi$
$701$	0.251576	0.00950191	0.00475096	+	0.999989i	$0.498488\pi$
$702$	−43.7082	−1.64966
$703$	−7.74877	−0.292250
$704$	−5.73717	−0.216228
$705$	10.7442	0.404650
$706$	20.8101	0.783197
$707$	−40.6381	−1.52835
$708$	1.80130	0.0676970
$709$	−38.3254	−1.43934	−0.719670	−	0.694316i	$-0.755707\pi$
$709$	−38.3254	−1.43934	−0.719670	+	0.694316i	$0.755707\pi$
$710$	−4.82923	−0.181238
$711$	−0.578272	−0.0216869
$712$	−52.2408	−1.95781
$713$	−6.75485	−0.252971
$714$	−6.14539	−0.229986
$715$	−3.89753	−0.145759
$716$	4.09663	0.153098
$717$	−16.2947	−0.608537
$718$	34.8649	1.30114
$719$	−10.8055	−0.402976	−0.201488	−	0.979491i	$-0.564578\pi$
$719$	−10.8055	−0.402976	−0.201488	+	0.979491i	$0.564578\pi$
$720$	1.95567	0.0728834
$721$	66.3796	2.47210
$722$	−24.6865	−0.918738
$723$	17.6751	0.657343
$724$	2.21454	0.0823028
$725$	1.59383	0.0591935
$726$	−21.9378	−0.814189
$727$	13.3421	0.494829	0.247415	−	0.968910i	$-0.420419\pi$
$727$	13.3421	0.494829	0.247415	+	0.968910i	$0.420419\pi$
$728$	61.4611	2.27790
$729$	29.9848	1.11055
$730$	−18.4968	−0.684598
$731$	−8.00000	−0.295891
$732$	−2.09333	−0.0773718
$733$	−18.1422	−0.670099	−0.335049	−	0.942201i	$-0.608753\pi$
$733$	−18.1422	−0.670099	−0.335049	+	0.942201i	$0.608753\pi$
$734$	−27.8063	−1.02635
$735$	8.24264	0.304035
$736$	−1.30452	−0.0480854
$737$	−5.48429	−0.202016
$738$	−4.78356	−0.176085
$739$	−5.06562	−0.186342	−0.0931709	−	0.995650i	$-0.529700\pi$
$739$	−5.06562	−0.186342	−0.0931709	+	0.995650i	$0.529700\pi$
$740$	2.72402	0.100137
$741$	6.08620	0.223582
$742$	−30.9798	−1.13731
$743$	−18.9436	−0.694972	−0.347486	−	0.937685i	$-0.612965\pi$
$743$	−18.9436	−0.694972	−0.347486	+	0.937685i	$0.612965\pi$
$744$	−31.3037	−1.14765
$745$	13.0081	0.476581
$746$	−21.7038	−0.794634
$747$	4.64881	0.170091
$748$	0.129123	0.00472121
$749$	4.58425	0.167505
$750$	2.07644	0.0758208
$751$	−32.6666	−1.19202	−0.596010	−	0.802977i	$-0.703248\pi$
$751$	−32.6666	−1.19202	−0.596010	+	0.802977i	$0.703248\pi$
$752$	−23.9619	−0.873802
$753$	31.5401	1.14939
$754$	12.5259	0.456168
$755$	11.0333	0.401544
$756$	−4.51783	−0.164312
$757$	12.0128	0.436611	0.218306	−	0.975880i	$-0.429947\pi$
$757$	12.0128	0.436611	0.218306	+	0.975880i	$0.429947\pi$
$758$	25.1322	0.912844
$759$	1.02977	0.0373784
$760$	−1.95708	−0.0709906
$761$	−20.7928	−0.753737	−0.376868	−	0.926267i	$-0.622999\pi$
$761$	−20.7928	−0.753737	−0.376868	+	0.926267i	$0.622999\pi$
$762$	−1.31698	−0.0477092
$763$	−2.02753	−0.0734014
$764$	1.13059	0.0409033
$765$	−0.474295	−0.0171482
$766$	−5.40009	−0.195113
$767$	−29.4085	−1.06188
$768$	−8.56695	−0.309133
$769$	24.6491	0.888868	0.444434	−	0.895812i	$-0.353405\pi$
$769$	24.6491	0.888868	0.444434	+	0.895812i	$0.353405\pi$
$770$	3.07270	0.110732
$771$	−13.1179	−0.472429
$772$	−3.15059	−0.113392
$773$	−7.44638	−0.267828	−0.133914	−	0.990993i	$-0.542755\pi$
$773$	−7.44638	−0.267828	−0.133914	+	0.990993i	$0.542755\pi$
$774$	7.07270	0.254223
$775$	6.75485	0.242641
$776$	24.7765	0.889426
$777$	64.2950	2.30657
$778$	15.3331	0.549717
$779$	4.22457	0.151361
$780$	−2.13955	−0.0766082
$781$	2.39497	0.0856987
$782$	−1.12311	−0.0401622
$783$	−8.86419	−0.316780
$784$	−18.3829	−0.656532
$785$	−1.61904	−0.0577859
$786$	−30.6856	−1.09452
$787$	4.72463	0.168415	0.0842074	−	0.996448i	$-0.473164\pi$
$787$	4.72463	0.168415	0.0842074	+	0.996448i	$0.473164\pi$
$788$	−2.19498	−0.0781928
$789$	44.7283	1.59237
$790$	1.36932	0.0487181
$791$	16.2139	0.576501
$792$	−1.09900	−0.0390513
$793$	34.1763	1.21364
$794$	−18.6497	−0.661853
$795$	10.3825	0.368228
$796$	−3.24054	−0.114858
$797$	−0.325538	−0.0115312	−0.00576558	−	0.999983i	$-0.501835\pi$
$797$	−0.325538	−0.0115312	−0.00576558	+	0.999983i	$0.501835\pi$
$798$	−4.79818	−0.169854
$799$	5.81133	0.205590
$800$	1.30452	0.0461219
$801$	−9.88503	−0.349270
$802$	20.7190	0.731613
$803$	9.17316	0.323714
$804$	−3.01060	−0.106176
$805$	3.50407	0.123502
$806$	53.0863	1.86989
$807$	−40.7747	−1.43534
$808$	−34.4178	−1.21081
$809$	−16.0435	−0.564061	−0.282030	−	0.959405i	$-0.591008\pi$
$809$	−16.0435	−0.564061	−0.282030	+	0.959405i	$0.591008\pi$
$810$	−9.30809	−0.327053
$811$	18.2442	0.640639	0.320319	−	0.947310i	$-0.396210\pi$
$811$	18.2442	0.640639	0.320319	+	0.947310i	$0.396210\pi$
$812$	1.29472	0.0454359
$813$	21.7105	0.761421
$814$	10.3038	0.361146
$815$	−2.55342	−0.0894423
$816$	−4.59324	−0.160796
$817$	−6.24621	−0.218527
$818$	−2.18639	−0.0764454
$819$	11.6297	0.406374
$820$	−1.48511	−0.0518624
$821$	36.0163	1.25698	0.628488	−	0.777819i	$-0.283674\pi$
$821$	36.0163	1.25698	0.628488	+	0.777819i	$0.283674\pi$
$822$	3.14521	0.109702
$823$	−13.5111	−0.470969	−0.235484	−	0.971878i	$-0.575668\pi$
$823$	−13.5111	−0.470969	−0.235484	+	0.971878i	$0.575668\pi$
$824$	56.2192	1.95849
$825$	−1.02977	−0.0358521
$826$	23.1848	0.806701
$827$	6.29476	0.218890	0.109445	−	0.993993i	$-0.465093\pi$
$827$	6.29476	0.218890	0.109445	+	0.993993i	$0.465093\pi$
$828$	−0.130182	−0.00452415
$829$	−45.9194	−1.59485	−0.797424	−	0.603420i	$-0.793804\pi$
$829$	−45.9194	−1.59485	−0.797424	+	0.603420i	$0.793804\pi$
$830$	−11.0081	−0.382098
$831$	43.2100	1.49894
$832$	51.4183	1.78261
$833$	4.45829	0.154470
$834$	−21.4288	−0.742018
$835$	14.2462	0.493010
$836$	0.100816	0.00348681
$837$	−37.5674	−1.29852
$838$	11.5817	0.400085
$839$	−22.6829	−0.783099	−0.391550	−	0.920157i	$-0.628061\pi$
$839$	−22.6829	−0.783099	−0.391550	+	0.920157i	$0.628061\pi$
$840$	16.2387	0.560289
$841$	−26.4597	−0.912403
$842$	22.6090	0.779159
$843$	24.4404	0.841771
$844$	−0.600258	−0.0206617
$845$	21.9309	0.754445
$846$	−5.13772	−0.176639
$847$	37.0209	1.27205
$848$	−23.1552	−0.795152
$849$	36.9527	1.26821
$850$	1.12311	0.0385222
$851$	11.7503	0.402794
$852$	1.31472	0.0450416
$853$	−29.2919	−1.00294	−0.501468	−	0.865176i	$-0.667206\pi$
$853$	−29.2919	−1.00294	−0.501468	+	0.865176i	$0.667206\pi$
$854$	−26.9436	−0.921990
$855$	−0.370318	−0.0126646
$856$	3.88256	0.132703
$857$	−39.5853	−1.35221	−0.676104	−	0.736806i	$-0.736333\pi$
$857$	−39.5853	−1.35221	−0.676104	+	0.736806i	$0.736333\pi$
$858$	−8.09298	−0.276290
$859$	44.6853	1.52464	0.762321	−	0.647199i	$-0.224060\pi$
$859$	44.6853	1.52464	0.762321	+	0.647199i	$0.224060\pi$
$860$	2.19580	0.0748763
$861$	−35.0532	−1.19461
$862$	−5.82795	−0.198501
$863$	−26.3810	−0.898020	−0.449010	−	0.893527i	$-0.648223\pi$
$863$	−26.3810	−0.898020	−0.449010	+	0.893527i	$0.648223\pi$
$864$	−7.25517	−0.246826
$865$	20.5733	0.699512
$866$	29.8875	1.01562
$867$	−25.4324	−0.863731
$868$	5.48718	0.186247
$869$	−0.679088	−0.0230365
$870$	3.30950	0.112203
$871$	49.1519	1.66545
$872$	−1.71718	−0.0581512
$873$	4.68823	0.158672
$874$	−0.876894	−0.0296614
$875$	−3.50407	−0.118459
$876$	5.03562	0.170138
$877$	−3.08897	−0.104307	−0.0521535	−	0.998639i	$-0.516609\pi$
$877$	−3.08897	−0.104307	−0.0521535	+	0.998639i	$0.516609\pi$
$878$	−16.8317	−0.568042
$879$	−37.5564	−1.26675
$880$	2.29662	0.0774190
$881$	−56.2213	−1.89414	−0.947072	−	0.321020i	$-0.895974\pi$
$881$	−56.2213	−1.89414	−0.947072	+	0.321020i	$0.895974\pi$
$882$	−3.94151	−0.132718
$883$	36.2950	1.22142	0.610712	−	0.791852i	$-0.290883\pi$
$883$	36.2950	1.22142	0.610712	+	0.791852i	$0.290883\pi$
$884$	−1.15724	−0.0389223
$885$	−7.77006	−0.261188
$886$	16.9342	0.568914
$887$	14.1185	0.474054	0.237027	−	0.971503i	$-0.423827\pi$
$887$	14.1185	0.474054	0.237027	+	0.971503i	$0.423827\pi$
$888$	54.4537	1.82735
$889$	2.22246	0.0745388
$890$	23.4072	0.784612
$891$	4.61618	0.154648
$892$	6.06958	0.203224
$893$	4.53735	0.151837
$894$	27.0106	0.903370
$895$	−17.6712	−0.590682
$896$	−31.3944	−1.04881
$897$	−9.22914	−0.308152
$898$	−8.53535	−0.284828
$899$	10.7661	0.359070
$900$	0.130182	0.00433941
$901$	5.61568	0.187085
$902$	−5.61753	−0.187043
$903$	51.8276	1.72471
$904$	13.7322	0.456725
$905$	−9.55262	−0.317540
$906$	22.9101	0.761136
$907$	−34.8321	−1.15658	−0.578291	−	0.815831i	$-0.696280\pi$
$907$	−34.8321	−1.15658	−0.578291	+	0.815831i	$0.696280\pi$
$908$	−3.87871	−0.128719
$909$	−6.51255	−0.216008
$910$	−27.5385	−0.912891
$911$	−38.8161	−1.28603	−0.643017	−	0.765852i	$-0.722318\pi$
$911$	−38.8161	−1.28603	−0.643017	+	0.765852i	$0.722318\pi$
$912$	−3.58629	−0.118754
$913$	5.45929	0.180676
$914$	−31.2252	−1.03284
$915$	9.02977	0.298515
$916$	1.54383	0.0510097
$917$	51.7831	1.71003
$918$	−6.24621	−0.206156
$919$	−44.6020	−1.47129	−0.735643	−	0.677370i	$-0.763120\pi$
$919$	−44.6020	−1.47129	−0.735643	+	0.677370i	$0.763120\pi$
$920$	2.96772	0.0978428
$921$	−0.578272	−0.0190547
$922$	28.4776	0.937861
$923$	−21.4644	−0.706511
$924$	−0.836518	−0.0275194
$925$	−11.7503	−0.386347
$926$	19.6510	0.645771
$927$	10.6378	0.349392
$928$	2.07919	0.0682528
$929$	22.9632	0.753397	0.376698	−	0.926336i	$-0.377059\pi$
$929$	22.9632	0.753397	0.376698	+	0.926336i	$0.377059\pi$
$930$	14.0260	0.459932
$931$	3.48092	0.114083
$932$	5.87919	0.192579
$933$	6.79627	0.222500
$934$	−51.3952	−1.68170
$935$	−0.556984	−0.0182153
$936$	9.84959	0.321944
$937$	−7.96421	−0.260179	−0.130090	−	0.991502i	$-0.541527\pi$
$937$	−7.96421	−0.260179	−0.130090	+	0.991502i	$0.541527\pi$
$938$	−38.7499	−1.26523
$939$	−10.6628	−0.347968
$940$	−1.59507	−0.0520254
$941$	−1.95330	−0.0636759	−0.0318379	−	0.999493i	$-0.510136\pi$
$941$	−1.95330	−0.0636759	−0.0318379	+	0.999493i	$0.510136\pi$
$942$	−3.36183	−0.109534
$943$	−6.40617	−0.208613
$944$	17.3289	0.564009
$945$	19.4881	0.633947
$946$	8.30576	0.270043
$947$	−7.75736	−0.252080	−0.126040	−	0.992025i	$-0.540227\pi$
$947$	−7.75736	−0.252080	−0.126040	+	0.992025i	$0.540227\pi$
$948$	−0.372786	−0.0121075
$949$	−82.2127	−2.66874
$950$	0.876894	0.0284502
$951$	−40.9094	−1.32658
$952$	8.78321	0.284666
$953$	−52.3363	−1.69534	−0.847669	−	0.530525i	$-0.821995\pi$
$953$	−52.3363	−1.69534	−0.847669	+	0.530525i	$0.821995\pi$
$954$	−4.96475	−0.160740
$955$	−4.87689	−0.157813
$956$	2.41909	0.0782388
$957$	−1.64129	−0.0530553
$958$	10.4661	0.338145
$959$	−5.30765	−0.171393
$960$	13.5853	0.438464
$961$	14.6280	0.471870
$962$	−92.3454	−2.97734
$963$	0.734660	0.0236741
$964$	−2.62401	−0.0845138
$965$	13.5903	0.437488
$966$	7.27598	0.234101
$967$	−8.71007	−0.280097	−0.140048	−	0.990145i	$-0.544726\pi$
$967$	−8.71007	−0.280097	−0.140048	+	0.990145i	$0.544726\pi$
$968$	31.3543	1.00777
$969$	0.869760	0.0279407
$970$	−11.1015	−0.356447
$971$	−42.5016	−1.36394	−0.681970	−	0.731380i	$-0.738877\pi$
$971$	−42.5016	−1.36394	−0.681970	+	0.731380i	$0.738877\pi$
$972$	−1.33388	−0.0427841
$973$	36.1619	1.15930
$974$	−21.3690	−0.684708
$975$	9.22914	0.295569
$976$	−20.1384	−0.644614
$977$	32.3185	1.03396	0.516981	−	0.855997i	$-0.327056\pi$
$977$	32.3185	1.03396	0.516981	+	0.855997i	$0.327056\pi$
$978$	−5.30202	−0.169540
$979$	−11.6084	−0.371006
$980$	−1.22369	−0.0390893
$981$	−0.324926	−0.0103741
$982$	−52.1882	−1.66539
$983$	−41.8095	−1.33352	−0.666758	−	0.745275i	$-0.732318\pi$
$983$	−41.8095	−1.33352	−0.666758	+	0.745275i	$0.732318\pi$
$984$	−29.6878	−0.946412
$985$	9.46822	0.301683
$986$	1.79004	0.0570066
$987$	−37.6484	−1.19836
$988$	−0.903547	−0.0287457
$989$	9.47179	0.301185
$990$	0.492423	0.0156502
$991$	40.4189	1.28395	0.641974	−	0.766727i	$-0.278116\pi$
$991$	40.4189	1.28395	0.641974	+	0.766727i	$0.278116\pi$
$992$	8.81186	0.279777
$993$	−19.7236	−0.625909
$994$	16.9219	0.536731
$995$	13.9784	0.443144
$996$	2.99688	0.0949598
$997$	−20.7599	−0.657473	−0.328737	−	0.944422i	$-0.606623\pi$
$997$	−20.7599	−0.657473	−0.328737	+	0.944422i	$0.606623\pi$
$998$	−6.64056	−0.210203
$999$	65.3498	2.06758

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.2.a.c.1.3	✓	4
3.2	odd	2		1035.2.a.o.1.2		4
4.3	odd	2		1840.2.a.u.1.2		4
5.2	odd	4		575.2.b.e.24.6		8
5.3	odd	4		575.2.b.e.24.3		8
5.4	even	2		575.2.a.h.1.2		4
7.6	odd	2		5635.2.a.v.1.3		4
8.3	odd	2		7360.2.a.cg.1.4		4
8.5	even	2		7360.2.a.cj.1.1		4
15.14	odd	2		5175.2.a.bx.1.3		4
20.19	odd	2		9200.2.a.cl.1.3		4
23.22	odd	2		2645.2.a.m.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.a.c.1.3	✓	4	1.1	even	1	trivial
575.2.a.h.1.2		4	5.4	even	2
575.2.b.e.24.3		8	5.3	odd	4
575.2.b.e.24.6		8	5.2	odd	4
1035.2.a.o.1.2		4	3.2	odd	2
1840.2.a.u.1.2		4	4.3	odd	2
2645.2.a.m.1.3		4	23.22	odd	2
5175.2.a.bx.1.3		4	15.14	odd	2
5635.2.a.v.1.3		4	7.6	odd	2
7360.2.a.cg.1.4		4	8.3	odd	2
7360.2.a.cj.1.1		4	8.5	even	2
9200.2.a.cl.1.3		4	20.19	odd	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:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	115.a (trivial)

\(f(q)\)	\(=\)	\(q+1.32973 q^{2} +1.56155 q^{3} -0.231826 q^{4} +1.00000 q^{5} +2.07644 q^{6} -3.50407 q^{7} -2.96772 q^{8} -0.561553 q^{9} +O(q^{10})\)	\(q+1.32973 q^{2} +1.56155 q^{3} -0.231826 q^{4} +1.00000 q^{5} +2.07644 q^{6} -3.50407 q^{7} -2.96772 q^{8} -0.561553 q^{9} +1.32973 q^{10} -0.659454 q^{11} -0.362008 q^{12} +5.91023 q^{13} -4.65945 q^{14} +1.56155 q^{15} -3.48261 q^{16} +0.844614 q^{17} -0.746712 q^{18} +0.659454 q^{19} -0.231826 q^{20} -5.47179 q^{21} -0.876894 q^{22} -1.00000 q^{23} -4.63425 q^{24} +1.00000 q^{25} +7.85900 q^{26} -5.56155 q^{27} +0.812333 q^{28} +1.59383 q^{29} +2.07644 q^{30} +6.75485 q^{31} +1.30452 q^{32} -1.02977 q^{33} +1.12311 q^{34} -3.50407 q^{35} +0.130182 q^{36} -11.7503 q^{37} +0.876894 q^{38} +9.22914 q^{39} -2.96772 q^{40} +6.40617 q^{41} -7.27598 q^{42} -9.47179 q^{43} +0.152878 q^{44} -0.561553 q^{45} -1.32973 q^{46} +6.88046 q^{47} -5.43827 q^{48} +5.27849 q^{49} +1.32973 q^{50} +1.31891 q^{51} -1.37014 q^{52} +6.64881 q^{53} -7.39535 q^{54} -0.659454 q^{55} +10.3991 q^{56} +1.02977 q^{57} +2.11936 q^{58} -4.97586 q^{59} -0.362008 q^{60} +5.78256 q^{61} +8.98210 q^{62} +1.96772 q^{63} +8.69987 q^{64} +5.91023 q^{65} -1.36932 q^{66} +8.31640 q^{67} -0.195803 q^{68} -1.56155 q^{69} -4.65945 q^{70} -3.63174 q^{71} +1.66653 q^{72} -13.9102 q^{73} -15.6247 q^{74} +1.56155 q^{75} -0.152878 q^{76} +2.31077 q^{77} +12.2722 q^{78} +1.02977 q^{79} -3.48261 q^{80} -7.00000 q^{81} +8.51845 q^{82} -8.27849 q^{83} +1.26850 q^{84} +0.844614 q^{85} -12.5949 q^{86} +2.48886 q^{87} +1.95708 q^{88} +17.6030 q^{89} -0.746712 q^{90} -20.7099 q^{91} +0.231826 q^{92} +10.5481 q^{93} +9.14914 q^{94} +0.659454 q^{95} +2.03708 q^{96} -8.34868 q^{97} +7.01895 q^{98} +0.370318 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 9 * q^8 + 6 * q^9	\( 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9} + 2 q^{10} + 4 q^{11} - 19 q^{12} - 12 q^{14} - 2 q^{15} + 8 q^{16} - q^{17} + 3 q^{18} - 4 q^{19} + 4 q^{20} + 10 q^{21} - 20 q^{22} - 4 q^{23} - 30 q^{24} + 4 q^{25} - q^{26} - 14 q^{27} - 22 q^{28} + 19 q^{29} - q^{30} - q^{31} + 20 q^{32} - 2 q^{33} - 12 q^{34} - 3 q^{35} + 23 q^{36} - 3 q^{37} + 20 q^{38} + 9 q^{40} + 13 q^{41} + 6 q^{42} - 6 q^{43} - 18 q^{44} + 6 q^{45} - 2 q^{46} + 6 q^{47} - 21 q^{48} + 9 q^{49} + 2 q^{50} - 8 q^{51} - q^{52} + 19 q^{53} - 7 q^{54} + 4 q^{55} - 10 q^{56} + 2 q^{57} + 21 q^{58} + 23 q^{59} - 19 q^{60} - 13 q^{62} - 13 q^{63} + 27 q^{64} + 44 q^{66} - 3 q^{67} - 4 q^{68} + 2 q^{69} - 12 q^{70} - 3 q^{71} + 39 q^{72} - 32 q^{73} - 12 q^{74} - 2 q^{75} + 18 q^{76} + 18 q^{77} + 43 q^{78} + 2 q^{79} + 8 q^{80} - 28 q^{81} - 5 q^{82} - 21 q^{83} + 28 q^{84} - q^{85} - 2 q^{86} - 18 q^{87} - 14 q^{88} + 3 q^{90} - 40 q^{91} - 4 q^{92} - 8 q^{93} + 47 q^{94} - 4 q^{95} - 61 q^{96} - 18 q^{97} + 16 q^{98} + 6 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 9 * q^8 + 6 * q^9 + 2 * q^10 + 4 * q^11 - 19 * q^12 - 12 * q^14 - 2 * q^15 + 8 * q^16 - q^17 + 3 * q^18 - 4 * q^19 + 4 * q^20 + 10 * q^21 - 20 * q^22 - 4 * q^23 - 30 * q^24 + 4 * q^25 - q^26 - 14 * q^27 - 22 * q^28 + 19 * q^29 - q^30 - q^31 + 20 * q^32 - 2 * q^33 - 12 * q^34 - 3 * q^35 + 23 * q^36 - 3 * q^37 + 20 * q^38 + 9 * q^40 + 13 * q^41 + 6 * q^42 - 6 * q^43 - 18 * q^44 + 6 * q^45 - 2 * q^46 + 6 * q^47 - 21 * q^48 + 9 * q^49 + 2 * q^50 - 8 * q^51 - q^52 + 19 * q^53 - 7 * q^54 + 4 * q^55 - 10 * q^56 + 2 * q^57 + 21 * q^58 + 23 * q^59 - 19 * q^60 - 13 * q^62 - 13 * q^63 + 27 * q^64 + 44 * q^66 - 3 * q^67 - 4 * q^68 + 2 * q^69 - 12 * q^70 - 3 * q^71 + 39 * q^72 - 32 * q^73 - 12 * q^74 - 2 * q^75 + 18 * q^76 + 18 * q^77 + 43 * q^78 + 2 * q^79 + 8 * q^80 - 28 * q^81 - 5 * q^82 - 21 * q^83 + 28 * q^84 - q^85 - 2 * q^86 - 18 * q^87 - 14 * q^88 + 3 * q^90 - 40 * q^91 - 4 * q^92 - 8 * q^93 + 47 * q^94 - 4 * q^95 - 61 * q^96 - 18 * q^97 + 16 * q^98 + 6 * q^99

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