LMFDB - Embedded newform 5070.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.12311 q^{11} +1.00000 q^{12} -0.561553 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.12311 q^{17} -1.00000 q^{18} -0.561553 q^{19} -1.00000 q^{20} +0.561553 q^{21} +4.12311 q^{22} +4.68466 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +0.561553 q^{28} +2.43845 q^{29} +1.00000 q^{30} +6.68466 q^{31} -1.00000 q^{32} -4.12311 q^{33} +3.12311 q^{34} -0.561553 q^{35} +1.00000 q^{36} +4.12311 q^{37} +0.561553 q^{38} +1.00000 q^{40} -12.2462 q^{41} -0.561553 q^{42} +0.438447 q^{43} -4.12311 q^{44} -1.00000 q^{45} -4.68466 q^{46} -7.00000 q^{47} +1.00000 q^{48} -6.68466 q^{49} -1.00000 q^{50} -3.12311 q^{51} +8.56155 q^{53} -1.00000 q^{54} +4.12311 q^{55} -0.561553 q^{56} -0.561553 q^{57} -2.43845 q^{58} -6.43845 q^{59} -1.00000 q^{60} +6.00000 q^{61} -6.68466 q^{62} +0.561553 q^{63} +1.00000 q^{64} +4.12311 q^{66} +2.24621 q^{67} -3.12311 q^{68} +4.68466 q^{69} +0.561553 q^{70} +13.1231 q^{71} -1.00000 q^{72} -9.36932 q^{73} -4.12311 q^{74} +1.00000 q^{75} -0.561553 q^{76} -2.31534 q^{77} +11.5616 q^{79} -1.00000 q^{80} +1.00000 q^{81} +12.2462 q^{82} +7.12311 q^{83} +0.561553 q^{84} +3.12311 q^{85} -0.438447 q^{86} +2.43845 q^{87} +4.12311 q^{88} +18.8078 q^{89} +1.00000 q^{90} +4.68466 q^{92} +6.68466 q^{93} +7.00000 q^{94} +0.561553 q^{95} -1.00000 q^{96} +7.12311 q^{97} +6.68466 q^{98} -4.12311 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 3 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 3 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 3 q^{14} - 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 3 q^{19} - 2 q^{20} - 3 q^{21} - 3 q^{23} - 2 q^{24} + 2 q^{25} + 2 q^{27} - 3 q^{28} + 9 q^{29} + 2 q^{30} + q^{31} - 2 q^{32} - 2 q^{34} + 3 q^{35} + 2 q^{36} - 3 q^{38} + 2 q^{40} - 8 q^{41} + 3 q^{42} + 5 q^{43} - 2 q^{45} + 3 q^{46} - 14 q^{47} + 2 q^{48} - q^{49} - 2 q^{50} + 2 q^{51} + 13 q^{53} - 2 q^{54} + 3 q^{56} + 3 q^{57} - 9 q^{58} - 17 q^{59} - 2 q^{60} + 12 q^{61} - q^{62} - 3 q^{63} + 2 q^{64} - 12 q^{67} + 2 q^{68} - 3 q^{69} - 3 q^{70} + 18 q^{71} - 2 q^{72} + 6 q^{73} + 2 q^{75} + 3 q^{76} - 17 q^{77} + 19 q^{79} - 2 q^{80} + 2 q^{81} + 8 q^{82} + 6 q^{83} - 3 q^{84} - 2 q^{85} - 5 q^{86} + 9 q^{87} + 17 q^{89} + 2 q^{90} - 3 q^{92} + q^{93} + 14 q^{94} - 3 q^{95} - 2 q^{96} + 6 q^{97} + q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 3 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 3 * q^14 - 2 * q^15 + 2 * q^16 + 2 * q^17 - 2 * q^18 + 3 * q^19 - 2 * q^20 - 3 * q^21 - 3 * q^23 - 2 * q^24 + 2 * q^25 + 2 * q^27 - 3 * q^28 + 9 * q^29 + 2 * q^30 + q^31 - 2 * q^32 - 2 * q^34 + 3 * q^35 + 2 * q^36 - 3 * q^38 + 2 * q^40 - 8 * q^41 + 3 * q^42 + 5 * q^43 - 2 * q^45 + 3 * q^46 - 14 * q^47 + 2 * q^48 - q^49 - 2 * q^50 + 2 * q^51 + 13 * q^53 - 2 * q^54 + 3 * q^56 + 3 * q^57 - 9 * q^58 - 17 * q^59 - 2 * q^60 + 12 * q^61 - q^62 - 3 * q^63 + 2 * q^64 - 12 * q^67 + 2 * q^68 - 3 * q^69 - 3 * q^70 + 18 * q^71 - 2 * q^72 + 6 * q^73 + 2 * q^75 + 3 * q^76 - 17 * q^77 + 19 * q^79 - 2 * q^80 + 2 * q^81 + 8 * q^82 + 6 * q^83 - 3 * q^84 - 2 * q^85 - 5 * q^86 + 9 * q^87 + 17 * q^89 + 2 * q^90 - 3 * q^92 + q^93 + 14 * q^94 - 3 * q^95 - 2 * q^96 + 6 * q^97 + q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	0.561553	0.212247	0.106124	−	0.994353i	$-0.466156\pi$
$7$	0.561553	0.212247	0.106124	+	0.994353i	$0.466156\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−4.12311	−1.24316	−0.621582	−	0.783349i	$-0.713510\pi$
$11$	−4.12311	−1.24316	−0.621582	+	0.783349i	$0.713510\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−0.561553	−0.150081
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−3.12311	−0.757464	−0.378732	−	0.925506i	$-0.623640\pi$
$17$	−3.12311	−0.757464	−0.378732	+	0.925506i	$0.623640\pi$
$18$	−1.00000	−0.235702
$19$	−0.561553	−0.128829	−0.0644145	−	0.997923i	$-0.520518\pi$
$19$	−0.561553	−0.128829	−0.0644145	+	0.997923i	$0.520518\pi$
$20$	−1.00000	−0.223607
$21$	0.561553	0.122541
$22$	4.12311	0.879049
$23$	4.68466	0.976819	0.488409	−	0.872615i	$-0.337577\pi$
$23$	4.68466	0.976819	0.488409	+	0.872615i	$0.337577\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0.561553	0.106124
$29$	2.43845	0.452808	0.226404	−	0.974033i	$-0.427303\pi$
$29$	2.43845	0.452808	0.226404	+	0.974033i	$0.427303\pi$
$30$	1.00000	0.182574
$31$	6.68466	1.20060	0.600300	−	0.799775i	$-0.295048\pi$
$31$	6.68466	1.20060	0.600300	+	0.799775i	$0.295048\pi$
$32$	−1.00000	−0.176777
$33$	−4.12311	−0.717741
$34$	3.12311	0.535608
$35$	−0.561553	−0.0949197
$36$	1.00000	0.166667
$37$	4.12311	0.677834	0.338917	−	0.940816i	$-0.389939\pi$
$37$	4.12311	0.677834	0.338917	+	0.940816i	$0.389939\pi$
$38$	0.561553	0.0910959
$39$	0	0
$40$	1.00000	0.158114
$41$	−12.2462	−1.91254	−0.956268	−	0.292490i	$-0.905516\pi$
$41$	−12.2462	−1.91254	−0.956268	+	0.292490i	$0.905516\pi$
$42$	−0.561553	−0.0866495
$43$	0.438447	0.0668626	0.0334313	−	0.999441i	$-0.489357\pi$
$43$	0.438447	0.0668626	0.0334313	+	0.999441i	$0.489357\pi$
$44$	−4.12311	−0.621582
$45$	−1.00000	−0.149071
$46$	−4.68466	−0.690715
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	1.00000	0.144338
$49$	−6.68466	−0.954951
$50$	−1.00000	−0.141421
$51$	−3.12311	−0.437322
$52$	0	0
$53$	8.56155	1.17602	0.588010	−	0.808854i	$-0.299912\pi$
$53$	8.56155	1.17602	0.588010	+	0.808854i	$0.299912\pi$
$54$	−1.00000	−0.136083
$55$	4.12311	0.555959
$56$	−0.561553	−0.0750407
$57$	−0.561553	−0.0743795
$58$	−2.43845	−0.320184
$59$	−6.43845	−0.838214	−0.419107	−	0.907937i	$-0.637657\pi$
$59$	−6.43845	−0.838214	−0.419107	+	0.907937i	$0.637657\pi$
$60$	−1.00000	−0.129099
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−6.68466	−0.848952
$63$	0.561553	0.0707490
$64$	1.00000	0.125000
$65$	0	0
$66$	4.12311	0.507519
$67$	2.24621	0.274418	0.137209	−	0.990542i	$-0.456187\pi$
$67$	2.24621	0.274418	0.137209	+	0.990542i	$0.456187\pi$
$68$	−3.12311	−0.378732
$69$	4.68466	0.563967
$70$	0.561553	0.0671184
$71$	13.1231	1.55743	0.778713	−	0.627380i	$-0.215873\pi$
$71$	13.1231	1.55743	0.778713	+	0.627380i	$0.215873\pi$
$72$	−1.00000	−0.117851
$73$	−9.36932	−1.09660	−0.548298	−	0.836283i	$-0.684724\pi$
$73$	−9.36932	−1.09660	−0.548298	+	0.836283i	$0.684724\pi$
$74$	−4.12311	−0.479301
$75$	1.00000	0.115470
$76$	−0.561553	−0.0644145
$77$	−2.31534	−0.263858
$78$	0	0
$79$	11.5616	1.30078	0.650388	−	0.759602i	$-0.274606\pi$
$79$	11.5616	1.30078	0.650388	+	0.759602i	$0.274606\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	12.2462	1.35237
$83$	7.12311	0.781862	0.390931	−	0.920420i	$-0.372153\pi$
$83$	7.12311	0.781862	0.390931	+	0.920420i	$0.372153\pi$
$84$	0.561553	0.0612704
$85$	3.12311	0.338748
$86$	−0.438447	−0.0472790
$87$	2.43845	0.261429
$88$	4.12311	0.439525
$89$	18.8078	1.99362	0.996810	−	0.0798174i	$-0.0254337\pi$
$89$	18.8078	1.99362	0.996810	+	0.0798174i	$0.0254337\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	4.68466	0.488409
$93$	6.68466	0.693167
$94$	7.00000	0.721995
$95$	0.561553	0.0576141
$96$	−1.00000	−0.102062
$97$	7.12311	0.723242	0.361621	−	0.932325i	$-0.382223\pi$
$97$	7.12311	0.723242	0.361621	+	0.932325i	$0.382223\pi$
$98$	6.68466	0.675252
$99$	−4.12311	−0.414388
$100$	1.00000	0.100000
$101$	−8.87689	−0.883284	−0.441642	−	0.897191i	$-0.645604\pi$
$101$	−8.87689	−0.883284	−0.441642	+	0.897191i	$0.645604\pi$
$102$	3.12311	0.309234
$103$	4.56155	0.449463	0.224732	−	0.974421i	$-0.427849\pi$
$103$	4.56155	0.449463	0.224732	+	0.974421i	$0.427849\pi$
$104$	0	0
$105$	−0.561553	−0.0548019
$106$	−8.56155	−0.831572
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	1.00000	0.0962250
$109$	3.75379	0.359548	0.179774	−	0.983708i	$-0.442463\pi$
$109$	3.75379	0.359548	0.179774	+	0.983708i	$0.442463\pi$
$110$	−4.12311	−0.393123
$111$	4.12311	0.391348
$112$	0.561553	0.0530618
$113$	8.68466	0.816984	0.408492	−	0.912762i	$-0.366055\pi$
$113$	8.68466	0.816984	0.408492	+	0.912762i	$0.366055\pi$
$114$	0.561553	0.0525942
$115$	−4.68466	−0.436847
$116$	2.43845	0.226404
$117$	0	0
$118$	6.43845	0.592707
$119$	−1.75379	−0.160770
$120$	1.00000	0.0912871
$121$	6.00000	0.545455
$122$	−6.00000	−0.543214
$123$	−12.2462	−1.10420
$124$	6.68466	0.600300
$125$	−1.00000	−0.0894427
$126$	−0.561553	−0.0500271
$127$	−8.56155	−0.759715	−0.379857	−	0.925045i	$-0.624027\pi$
$127$	−8.56155	−0.759715	−0.379857	+	0.925045i	$0.624027\pi$
$128$	−1.00000	−0.0883883
$129$	0.438447	0.0386031
$130$	0	0
$131$	−2.12311	−0.185497	−0.0927483	−	0.995690i	$-0.529565\pi$
$131$	−2.12311	−0.185497	−0.0927483	+	0.995690i	$0.529565\pi$
$132$	−4.12311	−0.358870
$133$	−0.315342	−0.0273436
$134$	−2.24621	−0.194043
$135$	−1.00000	−0.0860663
$136$	3.12311	0.267804
$137$	−11.8078	−1.00881	−0.504403	−	0.863469i	$-0.668287\pi$
$137$	−11.8078	−1.00881	−0.504403	+	0.863469i	$0.668287\pi$
$138$	−4.68466	−0.398785
$139$	12.5616	1.06546	0.532729	−	0.846286i	$-0.321167\pi$
$139$	12.5616	1.06546	0.532729	+	0.846286i	$0.321167\pi$
$140$	−0.561553	−0.0474599
$141$	−7.00000	−0.589506
$142$	−13.1231	−1.10127
$143$	0	0
$144$	1.00000	0.0833333
$145$	−2.43845	−0.202502
$146$	9.36932	0.775410
$147$	−6.68466	−0.551341
$148$	4.12311	0.338917
$149$	2.19224	0.179595	0.0897975	−	0.995960i	$-0.471378\pi$
$149$	2.19224	0.179595	0.0897975	+	0.995960i	$0.471378\pi$
$150$	−1.00000	−0.0816497
$151$	15.3693	1.25074	0.625369	−	0.780329i	$-0.284949\pi$
$151$	15.3693	1.25074	0.625369	+	0.780329i	$0.284949\pi$
$152$	0.561553	0.0455479
$153$	−3.12311	−0.252488
$154$	2.31534	0.186576
$155$	−6.68466	−0.536925
$156$	0	0
$157$	13.8769	1.10750	0.553748	−	0.832684i	$-0.313197\pi$
$157$	13.8769	1.10750	0.553748	+	0.832684i	$0.313197\pi$
$158$	−11.5616	−0.919788
$159$	8.56155	0.678975
$160$	1.00000	0.0790569
$161$	2.63068	0.207327
$162$	−1.00000	−0.0785674
$163$	14.4384	1.13091	0.565453	−	0.824780i	$-0.308701\pi$
$163$	14.4384	1.13091	0.565453	+	0.824780i	$0.308701\pi$
$164$	−12.2462	−0.956268
$165$	4.12311	0.320983
$166$	−7.12311	−0.552860
$167$	−4.36932	−0.338108	−0.169054	−	0.985607i	$-0.554071\pi$
$167$	−4.36932	−0.338108	−0.169054	+	0.985607i	$0.554071\pi$
$168$	−0.561553	−0.0433247
$169$	0	0
$170$	−3.12311	−0.239531
$171$	−0.561553	−0.0429430
$172$	0.438447	0.0334313
$173$	20.1771	1.53404	0.767018	−	0.641626i	$-0.221740\pi$
$173$	20.1771	1.53404	0.767018	+	0.641626i	$0.221740\pi$
$174$	−2.43845	−0.184858
$175$	0.561553	0.0424494
$176$	−4.12311	−0.310791
$177$	−6.43845	−0.483943
$178$	−18.8078	−1.40970
$179$	−12.6847	−0.948096	−0.474048	−	0.880499i	$-0.657208\pi$
$179$	−12.6847	−0.948096	−0.474048	+	0.880499i	$0.657208\pi$
$180$	−1.00000	−0.0745356
$181$	2.87689	0.213838	0.106919	−	0.994268i	$-0.465901\pi$
$181$	2.87689	0.213838	0.106919	+	0.994268i	$0.465901\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	−4.68466	−0.345358
$185$	−4.12311	−0.303137
$186$	−6.68466	−0.490143
$187$	12.8769	0.941652
$188$	−7.00000	−0.510527
$189$	0.561553	0.0408470
$190$	−0.561553	−0.0407393
$191$	10.8769	0.787024	0.393512	−	0.919319i	$-0.371260\pi$
$191$	10.8769	0.787024	0.393512	+	0.919319i	$0.371260\pi$
$192$	1.00000	0.0721688
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	−7.12311	−0.511409
$195$	0	0
$196$	−6.68466	−0.477476
$197$	−12.8078	−0.912515	−0.456258	−	0.889848i	$-0.650810\pi$
$197$	−12.8078	−0.912515	−0.456258	+	0.889848i	$0.650810\pi$
$198$	4.12311	0.293016
$199$	2.87689	0.203938	0.101969	−	0.994788i	$-0.467486\pi$
$199$	2.87689	0.203938	0.101969	+	0.994788i	$0.467486\pi$
$200$	−1.00000	−0.0707107
$201$	2.24621	0.158436
$202$	8.87689	0.624576
$203$	1.36932	0.0961072
$204$	−3.12311	−0.218661
$205$	12.2462	0.855312
$206$	−4.56155	−0.317818
$207$	4.68466	0.325606
$208$	0	0
$209$	2.31534	0.160156
$210$	0.561553	0.0387508
$211$	−21.9309	−1.50978	−0.754892	−	0.655850i	$-0.772311\pi$
$211$	−21.9309	−1.50978	−0.754892	+	0.655850i	$0.772311\pi$
$212$	8.56155	0.588010
$213$	13.1231	0.899180
$214$	−2.00000	−0.136717
$215$	−0.438447	−0.0299018
$216$	−1.00000	−0.0680414
$217$	3.75379	0.254824
$218$	−3.75379	−0.254239
$219$	−9.36932	−0.633120
$220$	4.12311	0.277980
$221$	0	0
$222$	−4.12311	−0.276725
$223$	−27.3002	−1.82816	−0.914078	−	0.405539i	$-0.867084\pi$
$223$	−27.3002	−1.82816	−0.914078	+	0.405539i	$0.867084\pi$
$224$	−0.561553	−0.0375203
$225$	1.00000	0.0666667
$226$	−8.68466	−0.577695
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	−0.561553	−0.0371897
$229$	24.2462	1.60223	0.801117	−	0.598507i	$-0.204239\pi$
$229$	24.2462	1.60223	0.801117	+	0.598507i	$0.204239\pi$
$230$	4.68466	0.308897
$231$	−2.31534	−0.152338
$232$	−2.43845	−0.160092
$233$	−5.31534	−0.348220	−0.174110	−	0.984726i	$-0.555705\pi$
$233$	−5.31534	−0.348220	−0.174110	+	0.984726i	$0.555705\pi$
$234$	0	0
$235$	7.00000	0.456630
$236$	−6.43845	−0.419107
$237$	11.5616	0.751004
$238$	1.75379	0.113681
$239$	11.3693	0.735420	0.367710	−	0.929941i	$-0.380142\pi$
$239$	11.3693	0.735420	0.367710	+	0.929941i	$0.380142\pi$
$240$	−1.00000	−0.0645497
$241$	28.1231	1.81157	0.905784	−	0.423739i	$-0.139283\pi$
$241$	28.1231	1.81157	0.905784	+	0.423739i	$0.139283\pi$
$242$	−6.00000	−0.385695
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	6.68466	0.427067
$246$	12.2462	0.780790
$247$	0	0
$248$	−6.68466	−0.424476
$249$	7.12311	0.451408
$250$	1.00000	0.0632456
$251$	−8.12311	−0.512726	−0.256363	−	0.966581i	$-0.582524\pi$
$251$	−8.12311	−0.512726	−0.256363	+	0.966581i	$0.582524\pi$
$252$	0.561553	0.0353745
$253$	−19.3153	−1.21435
$254$	8.56155	0.537200
$255$	3.12311	0.195576
$256$	1.00000	0.0625000
$257$	−5.56155	−0.346920	−0.173460	−	0.984841i	$-0.555495\pi$
$257$	−5.56155	−0.346920	−0.173460	+	0.984841i	$0.555495\pi$
$258$	−0.438447	−0.0272965
$259$	2.31534	0.143868
$260$	0	0
$261$	2.43845	0.150936
$262$	2.12311	0.131166
$263$	−13.0000	−0.801614	−0.400807	−	0.916162i	$-0.631270\pi$
$263$	−13.0000	−0.801614	−0.400807	+	0.916162i	$0.631270\pi$
$264$	4.12311	0.253760
$265$	−8.56155	−0.525932
$266$	0.315342	0.0193348
$267$	18.8078	1.15102
$268$	2.24621	0.137209
$269$	21.3693	1.30291	0.651455	−	0.758687i	$-0.274159\pi$
$269$	21.3693	1.30291	0.651455	+	0.758687i	$0.274159\pi$
$270$	1.00000	0.0608581
$271$	13.1771	0.800451	0.400225	−	0.916417i	$-0.368932\pi$
$271$	13.1771	0.800451	0.400225	+	0.916417i	$0.368932\pi$
$272$	−3.12311	−0.189366
$273$	0	0
$274$	11.8078	0.713333
$275$	−4.12311	−0.248633
$276$	4.68466	0.281983
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	−12.5616	−0.753392
$279$	6.68466	0.400200
$280$	0.561553	0.0335592
$281$	−16.2462	−0.969168	−0.484584	−	0.874745i	$-0.661029\pi$
$281$	−16.2462	−0.969168	−0.484584	+	0.874745i	$0.661029\pi$
$282$	7.00000	0.416844
$283$	15.5616	0.925038	0.462519	−	0.886609i	$-0.346946\pi$
$283$	15.5616	0.925038	0.462519	+	0.886609i	$0.346946\pi$
$284$	13.1231	0.778713
$285$	0.561553	0.0332635
$286$	0	0
$287$	−6.87689	−0.405930
$288$	−1.00000	−0.0589256
$289$	−7.24621	−0.426248
$290$	2.43845	0.143191
$291$	7.12311	0.417564
$292$	−9.36932	−0.548298
$293$	−3.93087	−0.229644	−0.114822	−	0.993386i	$-0.536630\pi$
$293$	−3.93087	−0.229644	−0.114822	+	0.993386i	$0.536630\pi$
$294$	6.68466	0.389857
$295$	6.43845	0.374861
$296$	−4.12311	−0.239651
$297$	−4.12311	−0.239247
$298$	−2.19224	−0.126993
$299$	0	0
$300$	1.00000	0.0577350
$301$	0.246211	0.0141914
$302$	−15.3693	−0.884405
$303$	−8.87689	−0.509964
$304$	−0.561553	−0.0322073
$305$	−6.00000	−0.343559
$306$	3.12311	0.178536
$307$	−25.6155	−1.46196	−0.730978	−	0.682401i	$-0.760936\pi$
$307$	−25.6155	−1.46196	−0.730978	+	0.682401i	$0.760936\pi$
$308$	−2.31534	−0.131929
$309$	4.56155	0.259498
$310$	6.68466	0.379663
$311$	30.7386	1.74303	0.871514	−	0.490371i	$-0.163139\pi$
$311$	30.7386	1.74303	0.871514	+	0.490371i	$0.163139\pi$
$312$	0	0
$313$	31.3693	1.77310	0.886549	−	0.462634i	$-0.153096\pi$
$313$	31.3693	1.77310	0.886549	+	0.462634i	$0.153096\pi$
$314$	−13.8769	−0.783118
$315$	−0.561553	−0.0316399
$316$	11.5616	0.650388
$317$	24.8078	1.39334	0.696671	−	0.717390i	$-0.254664\pi$
$317$	24.8078	1.39334	0.696671	+	0.717390i	$0.254664\pi$
$318$	−8.56155	−0.480108
$319$	−10.0540	−0.562915
$320$	−1.00000	−0.0559017
$321$	2.00000	0.111629
$322$	−2.63068	−0.146602
$323$	1.75379	0.0975834
$324$	1.00000	0.0555556
$325$	0	0
$326$	−14.4384	−0.799672
$327$	3.75379	0.207585
$328$	12.2462	0.676184
$329$	−3.93087	−0.216716
$330$	−4.12311	−0.226969
$331$	−30.7386	−1.68955	−0.844774	−	0.535123i	$-0.820265\pi$
$331$	−30.7386	−1.68955	−0.844774	+	0.535123i	$0.820265\pi$
$332$	7.12311	0.390931
$333$	4.12311	0.225945
$334$	4.36932	0.239078
$335$	−2.24621	−0.122724
$336$	0.561553	0.0306352
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	0	0
$339$	8.68466	0.471686
$340$	3.12311	0.169374
$341$	−27.5616	−1.49254
$342$	0.561553	0.0303653
$343$	−7.68466	−0.414933
$344$	−0.438447	−0.0236395
$345$	−4.68466	−0.252214
$346$	−20.1771	−1.08473
$347$	0.876894	0.0470742	0.0235371	−	0.999723i	$-0.492507\pi$
$347$	0.876894	0.0470742	0.0235371	+	0.999723i	$0.492507\pi$
$348$	2.43845	0.130714
$349$	−8.49242	−0.454589	−0.227294	−	0.973826i	$-0.572988\pi$
$349$	−8.49242	−0.454589	−0.227294	+	0.973826i	$0.572988\pi$
$350$	−0.561553	−0.0300163
$351$	0	0
$352$	4.12311	0.219762
$353$	25.8617	1.37648	0.688241	−	0.725482i	$-0.258383\pi$
$353$	25.8617	1.37648	0.688241	+	0.725482i	$0.258383\pi$
$354$	6.43845	0.342200
$355$	−13.1231	−0.696502
$356$	18.8078	0.996810
$357$	−1.75379	−0.0928203
$358$	12.6847	0.670405
$359$	−13.1231	−0.692611	−0.346306	−	0.938122i	$-0.612564\pi$
$359$	−13.1231	−0.692611	−0.346306	+	0.938122i	$0.612564\pi$
$360$	1.00000	0.0527046
$361$	−18.6847	−0.983403
$362$	−2.87689	−0.151206
$363$	6.00000	0.314918
$364$	0	0
$365$	9.36932	0.490412
$366$	−6.00000	−0.313625
$367$	12.4924	0.652099	0.326050	−	0.945353i	$-0.394282\pi$
$367$	12.4924	0.652099	0.326050	+	0.945353i	$0.394282\pi$
$368$	4.68466	0.244205
$369$	−12.2462	−0.637512
$370$	4.12311	0.214350
$371$	4.80776	0.249607
$372$	6.68466	0.346583
$373$	−25.8078	−1.33628	−0.668138	−	0.744038i	$-0.732908\pi$
$373$	−25.8078	−1.33628	−0.668138	+	0.744038i	$0.732908\pi$
$374$	−12.8769	−0.665848
$375$	−1.00000	−0.0516398
$376$	7.00000	0.360997
$377$	0	0
$378$	−0.561553	−0.0288832
$379$	17.6847	0.908400	0.454200	−	0.890900i	$-0.349925\pi$
$379$	17.6847	0.908400	0.454200	+	0.890900i	$0.349925\pi$
$380$	0.561553	0.0288071
$381$	−8.56155	−0.438622
$382$	−10.8769	−0.556510
$383$	−0.192236	−0.00982280	−0.00491140	−	0.999988i	$-0.501563\pi$
$383$	−0.192236	−0.00982280	−0.00491140	+	0.999988i	$0.501563\pi$
$384$	−1.00000	−0.0510310
$385$	2.31534	0.118001
$386$	0	0
$387$	0.438447	0.0222875
$388$	7.12311	0.361621
$389$	−18.0540	−0.915373	−0.457686	−	0.889114i	$-0.651322\pi$
$389$	−18.0540	−0.915373	−0.457686	+	0.889114i	$0.651322\pi$
$390$	0	0
$391$	−14.6307	−0.739905
$392$	6.68466	0.337626
$393$	−2.12311	−0.107097
$394$	12.8078	0.645246
$395$	−11.5616	−0.581725
$396$	−4.12311	−0.207194
$397$	20.1231	1.00995	0.504975	−	0.863134i	$-0.331502\pi$
$397$	20.1231	1.00995	0.504975	+	0.863134i	$0.331502\pi$
$398$	−2.87689	−0.144206
$399$	−0.315342	−0.0157868
$400$	1.00000	0.0500000
$401$	−0.315342	−0.0157474	−0.00787370	−	0.999969i	$-0.502506\pi$
$401$	−0.315342	−0.0157474	−0.00787370	+	0.999969i	$0.502506\pi$
$402$	−2.24621	−0.112031
$403$	0	0
$404$	−8.87689	−0.441642
$405$	−1.00000	−0.0496904
$406$	−1.36932	−0.0679581
$407$	−17.0000	−0.842659
$408$	3.12311	0.154617
$409$	18.8078	0.929984	0.464992	−	0.885315i	$-0.346057\pi$
$409$	18.8078	0.929984	0.464992	+	0.885315i	$0.346057\pi$
$410$	−12.2462	−0.604797
$411$	−11.8078	−0.582434
$412$	4.56155	0.224732
$413$	−3.61553	−0.177909
$414$	−4.68466	−0.230238
$415$	−7.12311	−0.349660
$416$	0	0
$417$	12.5616	0.615142
$418$	−2.31534	−0.113247
$419$	32.4924	1.58736	0.793679	−	0.608336i	$-0.208163\pi$
$419$	32.4924	1.58736	0.793679	+	0.608336i	$0.208163\pi$
$420$	−0.561553	−0.0274010
$421$	32.4924	1.58358	0.791792	−	0.610791i	$-0.209148\pi$
$421$	32.4924	1.58358	0.791792	+	0.610791i	$0.209148\pi$
$422$	21.9309	1.06758
$423$	−7.00000	−0.340352
$424$	−8.56155	−0.415786
$425$	−3.12311	−0.151493
$426$	−13.1231	−0.635817
$427$	3.36932	0.163053
$428$	2.00000	0.0966736
$429$	0	0
$430$	0.438447	0.0211438
$431$	22.7386	1.09528	0.547641	−	0.836714i	$-0.315526\pi$
$431$	22.7386	1.09528	0.547641	+	0.836714i	$0.315526\pi$
$432$	1.00000	0.0481125
$433$	−18.7386	−0.900521	−0.450261	−	0.892897i	$-0.648669\pi$
$433$	−18.7386	−0.900521	−0.450261	+	0.892897i	$0.648669\pi$
$434$	−3.75379	−0.180188
$435$	−2.43845	−0.116915
$436$	3.75379	0.179774
$437$	−2.63068	−0.125843
$438$	9.36932	0.447683
$439$	25.1231	1.19906	0.599530	−	0.800352i	$-0.295354\pi$
$439$	25.1231	1.19906	0.599530	+	0.800352i	$0.295354\pi$
$440$	−4.12311	−0.196561
$441$	−6.68466	−0.318317
$442$	0	0
$443$	−13.1231	−0.623498	−0.311749	−	0.950165i	$-0.600915\pi$
$443$	−13.1231	−0.623498	−0.311749	+	0.950165i	$0.600915\pi$
$444$	4.12311	0.195674
$445$	−18.8078	−0.891574
$446$	27.3002	1.29270
$447$	2.19224	0.103689
$448$	0.561553	0.0265309
$449$	20.1771	0.952215	0.476108	−	0.879387i	$-0.342047\pi$
$449$	20.1771	0.952215	0.476108	+	0.879387i	$0.342047\pi$
$450$	−1.00000	−0.0471405
$451$	50.4924	2.37760
$452$	8.68466	0.408492
$453$	15.3693	0.722113
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0.561553	0.0262971
$457$	−20.2462	−0.947078	−0.473539	−	0.880773i	$-0.657024\pi$
$457$	−20.2462	−0.947078	−0.473539	+	0.880773i	$0.657024\pi$
$458$	−24.2462	−1.13295
$459$	−3.12311	−0.145774
$460$	−4.68466	−0.218423
$461$	30.0540	1.39975	0.699877	−	0.714264i	$-0.253238\pi$
$461$	30.0540	1.39975	0.699877	+	0.714264i	$0.253238\pi$
$462$	2.31534	0.107719
$463$	7.61553	0.353924	0.176962	−	0.984218i	$-0.443373\pi$
$463$	7.61553	0.353924	0.176962	+	0.984218i	$0.443373\pi$
$464$	2.43845	0.113202
$465$	−6.68466	−0.309994
$466$	5.31534	0.246228
$467$	−17.8617	−0.826543	−0.413271	−	0.910608i	$-0.635614\pi$
$467$	−17.8617	−0.826543	−0.413271	+	0.910608i	$0.635614\pi$
$468$	0	0
$469$	1.26137	0.0582445
$470$	−7.00000	−0.322886
$471$	13.8769	0.639414
$472$	6.43845	0.296354
$473$	−1.80776	−0.0831211
$474$	−11.5616	−0.531040
$475$	−0.561553	−0.0257658
$476$	−1.75379	−0.0803848
$477$	8.56155	0.392007
$478$	−11.3693	−0.520020
$479$	38.2462	1.74751	0.873757	−	0.486363i	$-0.161677\pi$
$479$	38.2462	1.74751	0.873757	+	0.486363i	$0.161677\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−28.1231	−1.28097
$483$	2.63068	0.119700
$484$	6.00000	0.272727
$485$	−7.12311	−0.323444
$486$	−1.00000	−0.0453609
$487$	−13.0540	−0.591532	−0.295766	−	0.955260i	$-0.595575\pi$
$487$	−13.0540	−0.591532	−0.295766	+	0.955260i	$0.595575\pi$
$488$	−6.00000	−0.271607
$489$	14.4384	0.652929
$490$	−6.68466	−0.301982
$491$	−17.4384	−0.786986	−0.393493	−	0.919328i	$-0.628733\pi$
$491$	−17.4384	−0.786986	−0.393493	+	0.919328i	$0.628733\pi$
$492$	−12.2462	−0.552102
$493$	−7.61553	−0.342986
$494$	0	0
$495$	4.12311	0.185320
$496$	6.68466	0.300150
$497$	7.36932	0.330559
$498$	−7.12311	−0.319194
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−1.00000	−0.0447214
$501$	−4.36932	−0.195207
$502$	8.12311	0.362552
$503$	−43.0540	−1.91968	−0.959841	−	0.280545i	$-0.909485\pi$
$503$	−43.0540	−1.91968	−0.959841	+	0.280545i	$0.909485\pi$
$504$	−0.561553	−0.0250136
$505$	8.87689	0.395017
$506$	19.3153	0.858672
$507$	0	0
$508$	−8.56155	−0.379857
$509$	−40.5464	−1.79719	−0.898594	−	0.438782i	$-0.855410\pi$
$509$	−40.5464	−1.79719	−0.898594	+	0.438782i	$0.855410\pi$
$510$	−3.12311	−0.138293
$511$	−5.26137	−0.232749
$512$	−1.00000	−0.0441942
$513$	−0.561553	−0.0247932
$514$	5.56155	0.245310
$515$	−4.56155	−0.201006
$516$	0.438447	0.0193016
$517$	28.8617	1.26934
$518$	−2.31534	−0.101730
$519$	20.1771	0.885676
$520$	0	0
$521$	−6.31534	−0.276680	−0.138340	−	0.990385i	$-0.544177\pi$
$521$	−6.31534	−0.276680	−0.138340	+	0.990385i	$0.544177\pi$
$522$	−2.43845	−0.106728
$523$	−29.8078	−1.30340	−0.651701	−	0.758476i	$-0.725944\pi$
$523$	−29.8078	−1.30340	−0.651701	+	0.758476i	$0.725944\pi$
$524$	−2.12311	−0.0927483
$525$	0.561553	0.0245082
$526$	13.0000	0.566827
$527$	−20.8769	−0.909412
$528$	−4.12311	−0.179435
$529$	−1.05398	−0.0458250
$530$	8.56155	0.371890
$531$	−6.43845	−0.279405
$532$	−0.315342	−0.0136718
$533$	0	0
$534$	−18.8078	−0.813892
$535$	−2.00000	−0.0864675
$536$	−2.24621	−0.0970215
$537$	−12.6847	−0.547383
$538$	−21.3693	−0.921297
$539$	27.5616	1.18716
$540$	−1.00000	−0.0430331
$541$	27.3693	1.17670	0.588349	−	0.808607i	$-0.299778\pi$
$541$	27.3693	1.17670	0.588349	+	0.808607i	$0.299778\pi$
$542$	−13.1771	−0.566004
$543$	2.87689	0.123459
$544$	3.12311	0.133902
$545$	−3.75379	−0.160795
$546$	0	0
$547$	−5.61553	−0.240103	−0.120051	−	0.992768i	$-0.538306\pi$
$547$	−5.61553	−0.240103	−0.120051	+	0.992768i	$0.538306\pi$
$548$	−11.8078	−0.504403
$549$	6.00000	0.256074
$550$	4.12311	0.175810
$551$	−1.36932	−0.0583349
$552$	−4.68466	−0.199392
$553$	6.49242	0.276086
$554$	1.00000	0.0424859
$555$	−4.12311	−0.175016
$556$	12.5616	0.532729
$557$	8.56155	0.362765	0.181382	−	0.983413i	$-0.441943\pi$
$557$	8.56155	0.362765	0.181382	+	0.983413i	$0.441943\pi$
$558$	−6.68466	−0.282984
$559$	0	0
$560$	−0.561553	−0.0237299
$561$	12.8769	0.543663
$562$	16.2462	0.685305
$563$	−32.9848	−1.39015	−0.695073	−	0.718939i	$-0.744628\pi$
$563$	−32.9848	−1.39015	−0.695073	+	0.718939i	$0.744628\pi$
$564$	−7.00000	−0.294753
$565$	−8.68466	−0.365366
$566$	−15.5616	−0.654101
$567$	0.561553	0.0235830
$568$	−13.1231	−0.550633
$569$	26.1771	1.09740	0.548700	−	0.836019i	$-0.315123\pi$
$569$	26.1771	1.09740	0.548700	+	0.836019i	$0.315123\pi$
$570$	−0.561553	−0.0235209
$571$	23.6847	0.991172	0.495586	−	0.868559i	$-0.334953\pi$
$571$	23.6847	0.991172	0.495586	+	0.868559i	$0.334953\pi$
$572$	0	0
$573$	10.8769	0.454389
$574$	6.87689	0.287036
$575$	4.68466	0.195364
$576$	1.00000	0.0416667
$577$	40.7386	1.69597	0.847986	−	0.530019i	$-0.177815\pi$
$577$	40.7386	1.69597	0.847986	+	0.530019i	$0.177815\pi$
$578$	7.24621	0.301403
$579$	0	0
$580$	−2.43845	−0.101251
$581$	4.00000	0.165948
$582$	−7.12311	−0.295262
$583$	−35.3002	−1.46198
$584$	9.36932	0.387705
$585$	0	0
$586$	3.93087	0.162383
$587$	40.2462	1.66114	0.830569	−	0.556915i	$-0.188015\pi$
$587$	40.2462	1.66114	0.830569	+	0.556915i	$0.188015\pi$
$588$	−6.68466	−0.275671
$589$	−3.75379	−0.154672
$590$	−6.43845	−0.265067
$591$	−12.8078	−0.526841
$592$	4.12311	0.169459
$593$	−33.1771	−1.36242	−0.681210	−	0.732088i	$-0.738546\pi$
$593$	−33.1771	−1.36242	−0.681210	+	0.732088i	$0.738546\pi$
$594$	4.12311	0.169173
$595$	1.75379	0.0718983
$596$	2.19224	0.0897975
$597$	2.87689	0.117743
$598$	0	0
$599$	−14.0000	−0.572024	−0.286012	−	0.958226i	$-0.592330\pi$
$599$	−14.0000	−0.572024	−0.286012	+	0.958226i	$0.592330\pi$
$600$	−1.00000	−0.0408248
$601$	29.9848	1.22311	0.611554	−	0.791203i	$-0.290545\pi$
$601$	29.9848	1.22311	0.611554	+	0.791203i	$0.290545\pi$
$602$	−0.246211	−0.0100348
$603$	2.24621	0.0914728
$604$	15.3693	0.625369
$605$	−6.00000	−0.243935
$606$	8.87689	0.360599
$607$	32.4233	1.31602	0.658010	−	0.753009i	$-0.271398\pi$
$607$	32.4233	1.31602	0.658010	+	0.753009i	$0.271398\pi$
$608$	0.561553	0.0227740
$609$	1.36932	0.0554875
$610$	6.00000	0.242933
$611$	0	0
$612$	−3.12311	−0.126244
$613$	−19.8769	−0.802820	−0.401410	−	0.915898i	$-0.631480\pi$
$613$	−19.8769	−0.802820	−0.401410	+	0.915898i	$0.631480\pi$
$614$	25.6155	1.03376
$615$	12.2462	0.493815
$616$	2.31534	0.0932878
$617$	29.3153	1.18019	0.590096	−	0.807333i	$-0.299090\pi$
$617$	29.3153	1.18019	0.590096	+	0.807333i	$0.299090\pi$
$618$	−4.56155	−0.183493
$619$	20.5616	0.826439	0.413219	−	0.910632i	$-0.364404\pi$
$619$	20.5616	0.826439	0.413219	+	0.910632i	$0.364404\pi$
$620$	−6.68466	−0.268462
$621$	4.68466	0.187989
$622$	−30.7386	−1.23251
$623$	10.5616	0.423140
$624$	0	0
$625$	1.00000	0.0400000
$626$	−31.3693	−1.25377
$627$	2.31534	0.0924658
$628$	13.8769	0.553748
$629$	−12.8769	−0.513435
$630$	0.561553	0.0223728
$631$	13.7538	0.547530	0.273765	−	0.961797i	$-0.411731\pi$
$631$	13.7538	0.547530	0.273765	+	0.961797i	$0.411731\pi$
$632$	−11.5616	−0.459894
$633$	−21.9309	−0.871674
$634$	−24.8078	−0.985242
$635$	8.56155	0.339755
$636$	8.56155	0.339488
$637$	0	0
$638$	10.0540	0.398041
$639$	13.1231	0.519142
$640$	1.00000	0.0395285
$641$	−11.4384	−0.451792	−0.225896	−	0.974151i	$-0.572531\pi$
$641$	−11.4384	−0.451792	−0.225896	+	0.974151i	$0.572531\pi$
$642$	−2.00000	−0.0789337
$643$	−21.7538	−0.857886	−0.428943	−	0.903332i	$-0.641114\pi$
$643$	−21.7538	−0.857886	−0.428943	+	0.903332i	$0.641114\pi$
$644$	2.63068	0.103663
$645$	−0.438447	−0.0172638
$646$	−1.75379	−0.0690019
$647$	35.0540	1.37811	0.689057	−	0.724707i	$-0.258025\pi$
$647$	35.0540	1.37811	0.689057	+	0.724707i	$0.258025\pi$
$648$	−1.00000	−0.0392837
$649$	26.5464	1.04204
$650$	0	0
$651$	3.75379	0.147123
$652$	14.4384	0.565453
$653$	−44.5616	−1.74383	−0.871914	−	0.489659i	$-0.837121\pi$
$653$	−44.5616	−1.74383	−0.871914	+	0.489659i	$0.837121\pi$
$654$	−3.75379	−0.146785
$655$	2.12311	0.0829566
$656$	−12.2462	−0.478134
$657$	−9.36932	−0.365532
$658$	3.93087	0.153241
$659$	−6.05398	−0.235829	−0.117915	−	0.993024i	$-0.537621\pi$
$659$	−6.05398	−0.235829	−0.117915	+	0.993024i	$0.537621\pi$
$660$	4.12311	0.160492
$661$	−29.6155	−1.15191	−0.575955	−	0.817481i	$-0.695370\pi$
$661$	−29.6155	−1.15191	−0.575955	+	0.817481i	$0.695370\pi$
$662$	30.7386	1.19469
$663$	0	0
$664$	−7.12311	−0.276430
$665$	0.315342	0.0122284
$666$	−4.12311	−0.159767
$667$	11.4233	0.442312
$668$	−4.36932	−0.169054
$669$	−27.3002	−1.05549
$670$	2.24621	0.0867787
$671$	−24.7386	−0.955024
$672$	−0.561553	−0.0216624
$673$	25.7538	0.992736	0.496368	−	0.868112i	$-0.334667\pi$
$673$	25.7538	0.992736	0.496368	+	0.868112i	$0.334667\pi$
$674$	6.00000	0.231111
$675$	1.00000	0.0384900
$676$	0	0
$677$	−37.1231	−1.42676	−0.713378	−	0.700779i	$-0.752836\pi$
$677$	−37.1231	−1.42676	−0.713378	+	0.700779i	$0.752836\pi$
$678$	−8.68466	−0.333532
$679$	4.00000	0.153506
$680$	−3.12311	−0.119766
$681$	20.0000	0.766402
$682$	27.5616	1.05539
$683$	−17.1231	−0.655197	−0.327599	−	0.944817i	$-0.606239\pi$
$683$	−17.1231	−0.655197	−0.327599	+	0.944817i	$0.606239\pi$
$684$	−0.561553	−0.0214715
$685$	11.8078	0.451151
$686$	7.68466	0.293402
$687$	24.2462	0.925051
$688$	0.438447	0.0167156
$689$	0	0
$690$	4.68466	0.178342
$691$	20.5616	0.782198	0.391099	−	0.920349i	$-0.372095\pi$
$691$	20.5616	0.782198	0.391099	+	0.920349i	$0.372095\pi$
$692$	20.1771	0.767018
$693$	−2.31534	−0.0879526
$694$	−0.876894	−0.0332865
$695$	−12.5616	−0.476487
$696$	−2.43845	−0.0924291
$697$	38.2462	1.44868
$698$	8.49242	0.321443
$699$	−5.31534	−0.201045
$700$	0.561553	0.0212247
$701$	−36.3002	−1.37104	−0.685520	−	0.728054i	$-0.740425\pi$
$701$	−36.3002	−1.37104	−0.685520	+	0.728054i	$0.740425\pi$
$702$	0	0
$703$	−2.31534	−0.0873248
$704$	−4.12311	−0.155395
$705$	7.00000	0.263635
$706$	−25.8617	−0.973319
$707$	−4.98485	−0.187474
$708$	−6.43845	−0.241972
$709$	−34.2462	−1.28614	−0.643072	−	0.765806i	$-0.722340\pi$
$709$	−34.2462	−1.28614	−0.643072	+	0.765806i	$0.722340\pi$
$710$	13.1231	0.492501
$711$	11.5616	0.433592
$712$	−18.8078	−0.704851
$713$	31.3153	1.17277
$714$	1.75379	0.0656339
$715$	0	0
$716$	−12.6847	−0.474048
$717$	11.3693	0.424595
$718$	13.1231	0.489750
$719$	44.9848	1.67765	0.838826	−	0.544400i	$-0.183243\pi$
$719$	44.9848	1.67765	0.838826	+	0.544400i	$0.183243\pi$
$720$	−1.00000	−0.0372678
$721$	2.56155	0.0953972
$722$	18.6847	0.695371
$723$	28.1231	1.04591
$724$	2.87689	0.106919
$725$	2.43845	0.0905617
$726$	−6.00000	−0.222681
$727$	38.4233	1.42504	0.712521	−	0.701651i	$-0.247554\pi$
$727$	38.4233	1.42504	0.712521	+	0.701651i	$0.247554\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−9.36932	−0.346774
$731$	−1.36932	−0.0506460
$732$	6.00000	0.221766
$733$	−20.0691	−0.741270	−0.370635	−	0.928779i	$-0.620860\pi$
$733$	−20.0691	−0.741270	−0.370635	+	0.928779i	$0.620860\pi$
$734$	−12.4924	−0.461104
$735$	6.68466	0.246567
$736$	−4.68466	−0.172679
$737$	−9.26137	−0.341147
$738$	12.2462	0.450789
$739$	27.5464	1.01331	0.506655	−	0.862149i	$-0.330882\pi$
$739$	27.5464	1.01331	0.506655	+	0.862149i	$0.330882\pi$
$740$	−4.12311	−0.151568
$741$	0	0
$742$	−4.80776	−0.176499
$743$	−13.1771	−0.483420	−0.241710	−	0.970349i	$-0.577708\pi$
$743$	−13.1771	−0.483420	−0.241710	+	0.970349i	$0.577708\pi$
$744$	−6.68466	−0.245071
$745$	−2.19224	−0.0803173
$746$	25.8078	0.944889
$747$	7.12311	0.260621
$748$	12.8769	0.470826
$749$	1.12311	0.0410374
$750$	1.00000	0.0365148
$751$	−23.8078	−0.868758	−0.434379	−	0.900730i	$-0.643032\pi$
$751$	−23.8078	−0.868758	−0.434379	+	0.900730i	$0.643032\pi$
$752$	−7.00000	−0.255264
$753$	−8.12311	−0.296022
$754$	0	0
$755$	−15.3693	−0.559347
$756$	0.561553	0.0204235
$757$	−28.4233	−1.03306	−0.516531	−	0.856268i	$-0.672777\pi$
$757$	−28.4233	−1.03306	−0.516531	+	0.856268i	$0.672777\pi$
$758$	−17.6847	−0.642336
$759$	−19.3153	−0.701102
$760$	−0.561553	−0.0203697
$761$	17.9309	0.649994	0.324997	−	0.945715i	$-0.394637\pi$
$761$	17.9309	0.649994	0.324997	+	0.945715i	$0.394637\pi$
$762$	8.56155	0.310152
$763$	2.10795	0.0763129
$764$	10.8769	0.393512
$765$	3.12311	0.112916
$766$	0.192236	0.00694577
$767$	0	0
$768$	1.00000	0.0360844
$769$	33.3153	1.20138	0.600691	−	0.799481i	$-0.294892\pi$
$769$	33.3153	1.20138	0.600691	+	0.799481i	$0.294892\pi$
$770$	−2.31534	−0.0834391
$771$	−5.56155	−0.200294
$772$	0	0
$773$	−24.1771	−0.869589	−0.434795	−	0.900530i	$-0.643179\pi$
$773$	−24.1771	−0.869589	−0.434795	+	0.900530i	$0.643179\pi$
$774$	−0.438447	−0.0157597
$775$	6.68466	0.240120
$776$	−7.12311	−0.255705
$777$	2.31534	0.0830624
$778$	18.0540	0.647266
$779$	6.87689	0.246390
$780$	0	0
$781$	−54.1080	−1.93613
$782$	14.6307	0.523192
$783$	2.43845	0.0871430
$784$	−6.68466	−0.238738
$785$	−13.8769	−0.495288
$786$	2.12311	0.0757287
$787$	−31.3153	−1.11627	−0.558136	−	0.829750i	$-0.688483\pi$
$787$	−31.3153	−1.11627	−0.558136	+	0.829750i	$0.688483\pi$
$788$	−12.8078	−0.456258
$789$	−13.0000	−0.462812
$790$	11.5616	0.411342
$791$	4.87689	0.173402
$792$	4.12311	0.146508
$793$	0	0
$794$	−20.1231	−0.714142
$795$	−8.56155	−0.303647
$796$	2.87689	0.101969
$797$	13.1231	0.464844	0.232422	−	0.972615i	$-0.425335\pi$
$797$	13.1231	0.464844	0.232422	+	0.972615i	$0.425335\pi$
$798$	0.315342	0.0111630
$799$	21.8617	0.773413
$800$	−1.00000	−0.0353553
$801$	18.8078	0.664540
$802$	0.315342	0.0111351
$803$	38.6307	1.36325
$804$	2.24621	0.0792178
$805$	−2.63068	−0.0927194
$806$	0	0
$807$	21.3693	0.752236
$808$	8.87689	0.312288
$809$	−14.4924	−0.509526	−0.254763	−	0.967003i	$-0.581998\pi$
$809$	−14.4924	−0.509526	−0.254763	+	0.967003i	$0.581998\pi$
$810$	1.00000	0.0351364
$811$	−47.5464	−1.66958	−0.834790	−	0.550569i	$-0.814411\pi$
$811$	−47.5464	−1.66958	−0.834790	+	0.550569i	$0.814411\pi$
$812$	1.36932	0.0480536
$813$	13.1771	0.462140
$814$	17.0000	0.595850
$815$	−14.4384	−0.505757
$816$	−3.12311	−0.109331
$817$	−0.246211	−0.00861384
$818$	−18.8078	−0.657598
$819$	0	0
$820$	12.2462	0.427656
$821$	8.43845	0.294504	0.147252	−	0.989099i	$-0.452957\pi$
$821$	8.43845	0.294504	0.147252	+	0.989099i	$0.452957\pi$
$822$	11.8078	0.411843
$823$	5.05398	0.176171	0.0880853	−	0.996113i	$-0.471925\pi$
$823$	5.05398	0.176171	0.0880853	+	0.996113i	$0.471925\pi$
$824$	−4.56155	−0.158909
$825$	−4.12311	−0.143548
$826$	3.61553	0.125800
$827$	28.2462	0.982217	0.491109	−	0.871098i	$-0.336592\pi$
$827$	28.2462	0.982217	0.491109	+	0.871098i	$0.336592\pi$
$828$	4.68466	0.162803
$829$	12.3845	0.430130	0.215065	−	0.976600i	$-0.431004\pi$
$829$	12.3845	0.430130	0.215065	+	0.976600i	$0.431004\pi$
$830$	7.12311	0.247247
$831$	−1.00000	−0.0346896
$832$	0	0
$833$	20.8769	0.723342
$834$	−12.5616	−0.434971
$835$	4.36932	0.151206
$836$	2.31534	0.0800778
$837$	6.68466	0.231056
$838$	−32.4924	−1.12243
$839$	−36.7386	−1.26836	−0.634179	−	0.773186i	$-0.718662\pi$
$839$	−36.7386	−1.26836	−0.634179	+	0.773186i	$0.718662\pi$
$840$	0.561553	0.0193754
$841$	−23.0540	−0.794965
$842$	−32.4924	−1.11976
$843$	−16.2462	−0.559549
$844$	−21.9309	−0.754892
$845$	0	0
$846$	7.00000	0.240665
$847$	3.36932	0.115771
$848$	8.56155	0.294005
$849$	15.5616	0.534071
$850$	3.12311	0.107122
$851$	19.3153	0.662121
$852$	13.1231	0.449590
$853$	−2.30019	−0.0787569	−0.0393784	−	0.999224i	$-0.512538\pi$
$853$	−2.30019	−0.0787569	−0.0393784	+	0.999224i	$0.512538\pi$
$854$	−3.36932	−0.115296
$855$	0.561553	0.0192047
$856$	−2.00000	−0.0683586
$857$	−6.19224	−0.211523	−0.105761	−	0.994392i	$-0.533728\pi$
$857$	−6.19224	−0.211523	−0.105761	+	0.994392i	$0.533728\pi$
$858$	0	0
$859$	39.7926	1.35771	0.678853	−	0.734274i	$-0.262477\pi$
$859$	39.7926	1.35771	0.678853	+	0.734274i	$0.262477\pi$
$860$	−0.438447	−0.0149509
$861$	−6.87689	−0.234364
$862$	−22.7386	−0.774481
$863$	−2.43845	−0.0830057	−0.0415029	−	0.999138i	$-0.513215\pi$
$863$	−2.43845	−0.0830057	−0.0415029	+	0.999138i	$0.513215\pi$
$864$	−1.00000	−0.0340207
$865$	−20.1771	−0.686041
$866$	18.7386	0.636765
$867$	−7.24621	−0.246094
$868$	3.75379	0.127412
$869$	−47.6695	−1.61708
$870$	2.43845	0.0826711
$871$	0	0
$872$	−3.75379	−0.127119
$873$	7.12311	0.241081
$874$	2.63068	0.0889842
$875$	−0.561553	−0.0189839
$876$	−9.36932	−0.316560
$877$	−8.43845	−0.284946	−0.142473	−	0.989799i	$-0.545505\pi$
$877$	−8.43845	−0.284946	−0.142473	+	0.989799i	$0.545505\pi$
$878$	−25.1231	−0.847864
$879$	−3.93087	−0.132585
$880$	4.12311	0.138990
$881$	−24.5616	−0.827500	−0.413750	−	0.910391i	$-0.635781\pi$
$881$	−24.5616	−0.827500	−0.413750	+	0.910391i	$0.635781\pi$
$882$	6.68466	0.225084
$883$	−19.1771	−0.645360	−0.322680	−	0.946508i	$-0.604584\pi$
$883$	−19.1771	−0.645360	−0.322680	+	0.946508i	$0.604584\pi$
$884$	0	0
$885$	6.43845	0.216426
$886$	13.1231	0.440879
$887$	−33.1080	−1.11166	−0.555828	−	0.831297i	$-0.687599\pi$
$887$	−33.1080	−1.11166	−0.555828	+	0.831297i	$0.687599\pi$
$888$	−4.12311	−0.138362
$889$	−4.80776	−0.161247
$890$	18.8078	0.630438
$891$	−4.12311	−0.138129
$892$	−27.3002	−0.914078
$893$	3.93087	0.131542
$894$	−2.19224	−0.0733193
$895$	12.6847	0.424001
$896$	−0.561553	−0.0187602
$897$	0	0
$898$	−20.1771	−0.673318
$899$	16.3002	0.543642
$900$	1.00000	0.0333333
$901$	−26.7386	−0.890793
$902$	−50.4924	−1.68121
$903$	0.246211	0.00819340
$904$	−8.68466	−0.288847
$905$	−2.87689	−0.0956312
$906$	−15.3693	−0.510611
$907$	−53.9157	−1.79024	−0.895121	−	0.445823i	$-0.852911\pi$
$907$	−53.9157	−1.79024	−0.895121	+	0.445823i	$0.852911\pi$
$908$	20.0000	0.663723
$909$	−8.87689	−0.294428
$910$	0	0
$911$	−18.6307	−0.617262	−0.308631	−	0.951182i	$-0.599871\pi$
$911$	−18.6307	−0.617262	−0.308631	+	0.951182i	$0.599871\pi$
$912$	−0.561553	−0.0185949
$913$	−29.3693	−0.971983
$914$	20.2462	0.669685
$915$	−6.00000	−0.198354
$916$	24.2462	0.801117
$917$	−1.19224	−0.0393711
$918$	3.12311	0.103078
$919$	16.6307	0.548596	0.274298	−	0.961645i	$-0.411555\pi$
$919$	16.6307	0.548596	0.274298	+	0.961645i	$0.411555\pi$
$920$	4.68466	0.154449
$921$	−25.6155	−0.844060
$922$	−30.0540	−0.989775
$923$	0	0
$924$	−2.31534	−0.0761691
$925$	4.12311	0.135567
$926$	−7.61553	−0.250262
$927$	4.56155	0.149821
$928$	−2.43845	−0.0800460
$929$	25.1231	0.824262	0.412131	−	0.911125i	$-0.364785\pi$
$929$	25.1231	0.824262	0.412131	+	0.911125i	$0.364785\pi$
$930$	6.68466	0.219199
$931$	3.75379	0.123025
$932$	−5.31534	−0.174110
$933$	30.7386	1.00634
$934$	17.8617	0.584454
$935$	−12.8769	−0.421119
$936$	0	0
$937$	53.2311	1.73898	0.869491	−	0.493948i	$-0.164447\pi$
$937$	53.2311	1.73898	0.869491	+	0.493948i	$0.164447\pi$
$938$	−1.26137	−0.0411851
$939$	31.3693	1.02370
$940$	7.00000	0.228315
$941$	25.3693	0.827016	0.413508	−	0.910500i	$-0.364303\pi$
$941$	25.3693	0.827016	0.413508	+	0.910500i	$0.364303\pi$
$942$	−13.8769	−0.452134
$943$	−57.3693	−1.86820
$944$	−6.43845	−0.209554
$945$	−0.561553	−0.0182673
$946$	1.80776	0.0587755
$947$	43.8617	1.42532	0.712658	−	0.701512i	$-0.247491\pi$
$947$	43.8617	1.42532	0.712658	+	0.701512i	$0.247491\pi$
$948$	11.5616	0.375502
$949$	0	0
$950$	0.561553	0.0182192
$951$	24.8078	0.804447
$952$	1.75379	0.0568406
$953$	51.1771	1.65779	0.828894	−	0.559406i	$-0.188971\pi$
$953$	51.1771	1.65779	0.828894	+	0.559406i	$0.188971\pi$
$954$	−8.56155	−0.277191
$955$	−10.8769	−0.351968
$956$	11.3693	0.367710
$957$	−10.0540	−0.324999
$958$	−38.2462	−1.23568
$959$	−6.63068	−0.214116
$960$	−1.00000	−0.0322749
$961$	13.6847	0.441441
$962$	0	0
$963$	2.00000	0.0644491
$964$	28.1231	0.905784
$965$	0	0
$966$	−2.63068	−0.0846408
$967$	−15.6847	−0.504385	−0.252192	−	0.967677i	$-0.581152\pi$
$967$	−15.6847	−0.504385	−0.252192	+	0.967677i	$0.581152\pi$
$968$	−6.00000	−0.192847
$969$	1.75379	0.0563398
$970$	7.12311	0.228709
$971$	−4.31534	−0.138486	−0.0692430	−	0.997600i	$-0.522058\pi$
$971$	−4.31534	−0.138486	−0.0692430	+	0.997600i	$0.522058\pi$
$972$	1.00000	0.0320750
$973$	7.05398	0.226140
$974$	13.0540	0.418276
$975$	0	0
$976$	6.00000	0.192055
$977$	48.3002	1.54526	0.772630	−	0.634857i	$-0.218941\pi$
$977$	48.3002	1.54526	0.772630	+	0.634857i	$0.218941\pi$
$978$	−14.4384	−0.461691
$979$	−77.5464	−2.47839
$980$	6.68466	0.213534
$981$	3.75379	0.119849
$982$	17.4384	0.556483
$983$	−32.1231	−1.02457	−0.512284	−	0.858816i	$-0.671200\pi$
$983$	−32.1231	−1.02457	−0.512284	+	0.858816i	$0.671200\pi$
$984$	12.2462	0.390395
$985$	12.8078	0.408089
$986$	7.61553	0.242528
$987$	−3.93087	−0.125121
$988$	0	0
$989$	2.05398	0.0653126
$990$	−4.12311	−0.131041
$991$	49.4233	1.56998	0.784991	−	0.619507i	$-0.212667\pi$
$991$	49.4233	1.56998	0.784991	+	0.619507i	$0.212667\pi$
$992$	−6.68466	−0.212238
$993$	−30.7386	−0.975461
$994$	−7.36932	−0.233741
$995$	−2.87689	−0.0912037
$996$	7.12311	0.225704
$997$	−37.5464	−1.18911	−0.594553	−	0.804056i	$-0.702671\pi$
$997$	−37.5464	−1.18911	−0.594553	+	0.804056i	$0.702671\pi$
$998$	4.00000	0.126618
$999$	4.12311	0.130449

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bb.1.2		2
13.4	even	6		390.2.i.g.211.2	yes	4
13.5	odd	4		5070.2.b.r.1351.4		4
13.8	odd	4		5070.2.b.r.1351.1		4
13.10	even	6		390.2.i.g.61.2	✓	4
13.12	even	2		5070.2.a.bi.1.1		2
39.17	odd	6		1170.2.i.o.991.2		4
39.23	odd	6		1170.2.i.o.451.2		4
65.4	even	6		1950.2.i.bi.601.1		4
65.17	odd	12		1950.2.z.n.1849.4		8
65.23	odd	12		1950.2.z.n.1699.4		8
65.43	odd	12		1950.2.z.n.1849.1		8
65.49	even	6		1950.2.i.bi.451.1		4
65.62	odd	12		1950.2.z.n.1699.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.g.61.2	✓	4	13.10	even	6
390.2.i.g.211.2	yes	4	13.4	even	6
1170.2.i.o.451.2		4	39.23	odd	6
1170.2.i.o.991.2		4	39.17	odd	6
1950.2.i.bi.451.1		4	65.49	even	6
1950.2.i.bi.601.1		4	65.4	even	6
1950.2.z.n.1699.1		8	65.62	odd	12
1950.2.z.n.1699.4		8	65.23	odd	12
1950.2.z.n.1849.1		8	65.43	odd	12
1950.2.z.n.1849.4		8	65.17	odd	12
5070.2.a.bb.1.2		2	1.1	even	1	trivial
5070.2.a.bi.1.1		2	13.12	even	2
5070.2.b.r.1351.1		4	13.8	odd	4
5070.2.b.r.1351.4		4	13.5	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.12311 q^{11} +1.00000 q^{12} -0.561553 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.12311 q^{17} -1.00000 q^{18} -0.561553 q^{19} -1.00000 q^{20} +0.561553 q^{21} +4.12311 q^{22} +4.68466 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +0.561553 q^{28} +2.43845 q^{29} +1.00000 q^{30} +6.68466 q^{31} -1.00000 q^{32} -4.12311 q^{33} +3.12311 q^{34} -0.561553 q^{35} +1.00000 q^{36} +4.12311 q^{37} +0.561553 q^{38} +1.00000 q^{40} -12.2462 q^{41} -0.561553 q^{42} +0.438447 q^{43} -4.12311 q^{44} -1.00000 q^{45} -4.68466 q^{46} -7.00000 q^{47} +1.00000 q^{48} -6.68466 q^{49} -1.00000 q^{50} -3.12311 q^{51} +8.56155 q^{53} -1.00000 q^{54} +4.12311 q^{55} -0.561553 q^{56} -0.561553 q^{57} -2.43845 q^{58} -6.43845 q^{59} -1.00000 q^{60} +6.00000 q^{61} -6.68466 q^{62} +0.561553 q^{63} +1.00000 q^{64} +4.12311 q^{66} +2.24621 q^{67} -3.12311 q^{68} +4.68466 q^{69} +0.561553 q^{70} +13.1231 q^{71} -1.00000 q^{72} -9.36932 q^{73} -4.12311 q^{74} +1.00000 q^{75} -0.561553 q^{76} -2.31534 q^{77} +11.5616 q^{79} -1.00000 q^{80} +1.00000 q^{81} +12.2462 q^{82} +7.12311 q^{83} +0.561553 q^{84} +3.12311 q^{85} -0.438447 q^{86} +2.43845 q^{87} +4.12311 q^{88} +18.8078 q^{89} +1.00000 q^{90} +4.68466 q^{92} +6.68466 q^{93} +7.00000 q^{94} +0.561553 q^{95} -1.00000 q^{96} +7.12311 q^{97} +6.68466 q^{98} -4.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 3 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 3 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 3 q^{14} - 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 3 q^{19} - 2 q^{20} - 3 q^{21} - 3 q^{23} - 2 q^{24} + 2 q^{25} + 2 q^{27} - 3 q^{28} + 9 q^{29} + 2 q^{30} + q^{31} - 2 q^{32} - 2 q^{34} + 3 q^{35} + 2 q^{36} - 3 q^{38} + 2 q^{40} - 8 q^{41} + 3 q^{42} + 5 q^{43} - 2 q^{45} + 3 q^{46} - 14 q^{47} + 2 q^{48} - q^{49} - 2 q^{50} + 2 q^{51} + 13 q^{53} - 2 q^{54} + 3 q^{56} + 3 q^{57} - 9 q^{58} - 17 q^{59} - 2 q^{60} + 12 q^{61} - q^{62} - 3 q^{63} + 2 q^{64} - 12 q^{67} + 2 q^{68} - 3 q^{69} - 3 q^{70} + 18 q^{71} - 2 q^{72} + 6 q^{73} + 2 q^{75} + 3 q^{76} - 17 q^{77} + 19 q^{79} - 2 q^{80} + 2 q^{81} + 8 q^{82} + 6 q^{83} - 3 q^{84} - 2 q^{85} - 5 q^{86} + 9 q^{87} + 17 q^{89} + 2 q^{90} - 3 q^{92} + q^{93} + 14 q^{94} - 3 q^{95} - 2 q^{96} + 6 q^{97} + q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 3 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 3 * q^14 - 2 * q^15 + 2 * q^16 + 2 * q^17 - 2 * q^18 + 3 * q^19 - 2 * q^20 - 3 * q^21 - 3 * q^23 - 2 * q^24 + 2 * q^25 + 2 * q^27 - 3 * q^28 + 9 * q^29 + 2 * q^30 + q^31 - 2 * q^32 - 2 * q^34 + 3 * q^35 + 2 * q^36 - 3 * q^38 + 2 * q^40 - 8 * q^41 + 3 * q^42 + 5 * q^43 - 2 * q^45 + 3 * q^46 - 14 * q^47 + 2 * q^48 - q^49 - 2 * q^50 + 2 * q^51 + 13 * q^53 - 2 * q^54 + 3 * q^56 + 3 * q^57 - 9 * q^58 - 17 * q^59 - 2 * q^60 + 12 * q^61 - q^62 - 3 * q^63 + 2 * q^64 - 12 * q^67 + 2 * q^68 - 3 * q^69 - 3 * q^70 + 18 * q^71 - 2 * q^72 + 6 * q^73 + 2 * q^75 + 3 * q^76 - 17 * q^77 + 19 * q^79 - 2 * q^80 + 2 * q^81 + 8 * q^82 + 6 * q^83 - 3 * q^84 - 2 * q^85 - 5 * q^86 + 9 * q^87 + 17 * q^89 + 2 * q^90 - 3 * q^92 + q^93 + 14 * q^94 - 3 * q^95 - 2 * q^96 + 6 * q^97 + q^98

Introduction

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