LMFDB - Embedded newform 350.8.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,8,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("350.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	350.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:	$109.334758919$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1969}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 492 $ x^2 - x - 492
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-21.6867$ of defining polynomial
Character	$\chi$	$=$	350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{2} +9.37342 q^{3} +64.0000 q^{4} -74.9873 q^{6} -343.000 q^{7} -512.000 q^{8} -2099.14 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} +9.37342 q^{3} +64.0000 q^{4} -74.9873 q^{6} -343.000 q^{7} -512.000 q^{8} -2099.14 q^{9} +3881.05 q^{11} +599.899 q^{12} +11585.6 q^{13} +2744.00 q^{14} +4096.00 q^{16} +15242.4 q^{17} +16793.1 q^{18} -32461.7 q^{19} -3215.08 q^{21} -31048.4 q^{22} -56146.1 q^{23} -4799.19 q^{24} -92684.6 q^{26} -40175.8 q^{27} -21952.0 q^{28} -26442.0 q^{29} -45448.2 q^{31} -32768.0 q^{32} +36378.7 q^{33} -121939. q^{34} -134345. q^{36} +555245. q^{37} +259694. q^{38} +108596. q^{39} +306385. q^{41} +25720.7 q^{42} -780755. q^{43} +248387. q^{44} +449169. q^{46} +531243. q^{47} +38393.5 q^{48} +117649. q^{49} +142873. q^{51} +741477. q^{52} +363593. q^{53} +321406. q^{54} +175616. q^{56} -304277. q^{57} +211536. q^{58} -2.14095e6 q^{59} +888824. q^{61} +363586. q^{62} +720005. q^{63} +262144. q^{64} -291030. q^{66} -4.34541e6 q^{67} +975513. q^{68} -526281. q^{69} +663207. q^{71} +1.07476e6 q^{72} -1.34202e6 q^{73} -4.44196e6 q^{74} -2.07755e6 q^{76} -1.33120e6 q^{77} -868771. q^{78} +7.05834e6 q^{79} +4.21423e6 q^{81} -2.45108e6 q^{82} -6.60874e6 q^{83} -205765. q^{84} +6.24604e6 q^{86} -247851. q^{87} -1.98710e6 q^{88} -5.32998e6 q^{89} -3.97385e6 q^{91} -3.59335e6 q^{92} -426005. q^{93} -4.24994e6 q^{94} -307148. q^{96} +2.09810e6 q^{97} -941192. q^{98} -8.14686e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{2} - 70 q^{3} + 128 q^{4} + 560 q^{6} - 686 q^{7} - 1024 q^{8} + 2014 q^{9}+O(q^{10}) $ 2 * q - 16 * q^2 - 70 * q^3 + 128 * q^4 + 560 * q^6 - 686 * q^7 - 1024 * q^8 + 2014 * q^9	\( 2 q - 16 q^{2} - 70 q^{3} + 128 q^{4} + 560 q^{6} - 686 q^{7} - 1024 q^{8} + 2014 q^{9} - 3420 q^{11} - 4480 q^{12} + 6398 q^{13} + 5488 q^{14} + 8192 q^{16} + 38472 q^{17} - 16112 q^{18} - 43358 q^{19} + 24010 q^{21} + 27360 q^{22} - 89928 q^{23} + 35840 q^{24} - 51184 q^{26} - 193060 q^{27} - 43904 q^{28} + 159576 q^{29} - 143612 q^{31} - 65536 q^{32} + 615888 q^{33} - 307776 q^{34} + 128896 q^{36} + 271832 q^{37} + 346864 q^{38} + 520352 q^{39} + 64848 q^{41} - 192080 q^{42} - 1527964 q^{43} - 218880 q^{44} + 719424 q^{46} - 485436 q^{47} - 286720 q^{48} + 235298 q^{49} - 1700940 q^{51} + 409472 q^{52} + 145716 q^{53} + 1544480 q^{54} + 351232 q^{56} + 560596 q^{57} - 1276608 q^{58} - 4183662 q^{59} - 280658 q^{61} + 1148896 q^{62} - 690802 q^{63} + 524288 q^{64} - 4927104 q^{66} - 5671648 q^{67} + 2462208 q^{68} + 2155104 q^{69} - 619272 q^{71} - 1031168 q^{72} - 3939628 q^{73} - 2174656 q^{74} - 2774912 q^{76} + 1173060 q^{77} - 4162816 q^{78} + 4656616 q^{79} + 7353742 q^{81} - 518784 q^{82} - 1235850 q^{83} + 1536640 q^{84} + 12223712 q^{86} - 15012732 q^{87} + 1751040 q^{88} - 17241420 q^{89} - 2194514 q^{91} - 5755392 q^{92} + 7365592 q^{93} + 3883488 q^{94} + 2293760 q^{96} + 740936 q^{97} - 1882384 q^{98} - 38177100 q^{99}+O(q^{100}) \) 2 * q - 16 * q^2 - 70 * q^3 + 128 * q^4 + 560 * q^6 - 686 * q^7 - 1024 * q^8 + 2014 * q^9 - 3420 * q^11 - 4480 * q^12 + 6398 * q^13 + 5488 * q^14 + 8192 * q^16 + 38472 * q^17 - 16112 * q^18 - 43358 * q^19 + 24010 * q^21 + 27360 * q^22 - 89928 * q^23 + 35840 * q^24 - 51184 * q^26 - 193060 * q^27 - 43904 * q^28 + 159576 * q^29 - 143612 * q^31 - 65536 * q^32 + 615888 * q^33 - 307776 * q^34 + 128896 * q^36 + 271832 * q^37 + 346864 * q^38 + 520352 * q^39 + 64848 * q^41 - 192080 * q^42 - 1527964 * q^43 - 218880 * q^44 + 719424 * q^46 - 485436 * q^47 - 286720 * q^48 + 235298 * q^49 - 1700940 * q^51 + 409472 * q^52 + 145716 * q^53 + 1544480 * q^54 + 351232 * q^56 + 560596 * q^57 - 1276608 * q^58 - 4183662 * q^59 - 280658 * q^61 + 1148896 * q^62 - 690802 * q^63 + 524288 * q^64 - 4927104 * q^66 - 5671648 * q^67 + 2462208 * q^68 + 2155104 * q^69 - 619272 * q^71 - 1031168 * q^72 - 3939628 * q^73 - 2174656 * q^74 - 2774912 * q^76 + 1173060 * q^77 - 4162816 * q^78 + 4656616 * q^79 + 7353742 * q^81 - 518784 * q^82 - 1235850 * q^83 + 1536640 * q^84 + 12223712 * q^86 - 15012732 * q^87 + 1751040 * q^88 - 17241420 * q^89 - 2194514 * q^91 - 5755392 * q^92 + 7365592 * q^93 + 3883488 * q^94 + 2293760 * q^96 + 740936 * q^97 - 1882384 * q^98 - 38177100 * 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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	9.37342	0.200435	0.100217	−	0.994966i	$-0.468046\pi$
$3$	9.37342	0.200435	0.100217	+	0.994966i	$0.468046\pi$
$4$	64.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−74.9873	−0.141729
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−512.000	−0.353553
$9$	−2099.14	−0.959826
$10$	0	0
$11$	3881.05	0.879174	0.439587	−	0.898200i	$-0.355125\pi$
$11$	3881.05	0.879174	0.439587	+	0.898200i	$0.355125\pi$
$12$	599.899	0.100217
$13$	11585.6	1.46257	0.731284	−	0.682073i	$-0.238922\pi$
$13$	11585.6	1.46257	0.731284	+	0.682073i	$0.238922\pi$
$14$	2744.00	0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	15242.4	0.752457	0.376229	−	0.926527i	$-0.377221\pi$
$17$	15242.4	0.752457	0.376229	+	0.926527i	$0.377221\pi$
$18$	16793.1	0.678699
$19$	−32461.7	−1.08576	−0.542880	−	0.839810i	$-0.682666\pi$
$19$	−32461.7	−1.08576	−0.542880	+	0.839810i	$0.682666\pi$
$20$	0	0
$21$	−3215.08	−0.0757573
$22$	−31048.4	−0.621670
$23$	−56146.1	−0.962215	−0.481108	−	0.876662i	$-0.659765\pi$
$23$	−56146.1	−0.962215	−0.481108	+	0.876662i	$0.659765\pi$
$24$	−4799.19	−0.0708645
$25$	0	0
$26$	−92684.6	−1.03419
$27$	−40175.8	−0.392818
$28$	−21952.0	−0.188982
$29$	−26442.0	−0.201326	−0.100663	−	0.994921i	$-0.532096\pi$
$29$	−26442.0	−0.201326	−0.100663	+	0.994921i	$0.532096\pi$
$30$	0	0
$31$	−45448.2	−0.274000	−0.137000	−	0.990571i	$-0.543746\pi$
$31$	−45448.2	−0.274000	−0.137000	+	0.990571i	$0.543746\pi$
$32$	−32768.0	−0.176777
$33$	36378.7	0.176217
$34$	−121939.	−0.532068
$35$	0	0
$36$	−134345.	−0.479913
$37$	555245.	1.80210	0.901049	−	0.433717i	$-0.142798\pi$
$37$	555245.	1.80210	0.901049	+	0.433717i	$0.142798\pi$
$38$	259694.	0.767749
$39$	108596.	0.293150
$40$	0	0
$41$	306385.	0.694264	0.347132	−	0.937816i	$-0.387156\pi$
$41$	306385.	0.694264	0.347132	+	0.937816i	$0.387156\pi$
$42$	25720.7	0.0535685
$43$	−780755.	−1.49753	−0.748765	−	0.662836i	$-0.769353\pi$
$43$	−780755.	−1.49753	−0.748765	+	0.662836i	$0.769353\pi$
$44$	248387.	0.439587
$45$	0	0
$46$	449169.	0.680389
$47$	531243.	0.746364	0.373182	−	0.927758i	$-0.378267\pi$
$47$	531243.	0.746364	0.373182	+	0.927758i	$0.378267\pi$
$48$	38393.5	0.0501087
$49$	117649.	0.142857
$50$	0	0
$51$	142873.	0.150819
$52$	741477.	0.731284
$53$	363593.	0.335467	0.167733	−	0.985832i	$-0.446355\pi$
$53$	363593.	0.335467	0.167733	+	0.985832i	$0.446355\pi$
$54$	321406.	0.277764
$55$	0	0
$56$	175616.	0.133631
$57$	−304277.	−0.217624
$58$	211536.	0.142359
$59$	−2.14095e6	−1.35714	−0.678570	−	0.734535i	$-0.737400\pi$
$59$	−2.14095e6	−1.35714	−0.678570	+	0.734535i	$0.737400\pi$
$60$	0	0
$61$	888824.	0.501373	0.250687	−	0.968068i	$-0.419344\pi$
$61$	888824.	0.501373	0.250687	+	0.968068i	$0.419344\pi$
$62$	363586.	0.193747
$63$	720005.	0.362780
$64$	262144.	0.125000
$65$	0	0
$66$	−291030.	−0.124604
$67$	−4.34541e6	−1.76510	−0.882549	−	0.470221i	$-0.844174\pi$
$67$	−4.34541e6	−1.76510	−0.882549	+	0.470221i	$0.844174\pi$
$68$	975513.	0.376229
$69$	−526281.	−0.192862
$70$	0	0
$71$	663207.	0.219910	0.109955	−	0.993937i	$-0.464929\pi$
$71$	663207.	0.219910	0.109955	+	0.993937i	$0.464929\pi$
$72$	1.07476e6	0.339350
$73$	−1.34202e6	−0.403765	−0.201882	−	0.979410i	$-0.564706\pi$
$73$	−1.34202e6	−0.403765	−0.201882	+	0.979410i	$0.564706\pi$
$74$	−4.44196e6	−1.27428
$75$	0	0
$76$	−2.07755e6	−0.542880
$77$	−1.33120e6	−0.332297
$78$	−868771.	−0.207288
$79$	7.05834e6	1.61067	0.805337	−	0.592818i	$-0.201984\pi$
$79$	7.05834e6	1.61067	0.805337	+	0.592818i	$0.201984\pi$
$80$	0	0
$81$	4.21423e6	0.881091
$82$	−2.45108e6	−0.490919
$83$	−6.60874e6	−1.26866	−0.634330	−	0.773063i	$-0.718724\pi$
$83$	−6.60874e6	−1.26866	−0.634330	+	0.773063i	$0.718724\pi$
$84$	−205765.	−0.0378786
$85$	0	0
$86$	6.24604e6	1.05891
$87$	−247851.	−0.0403528
$88$	−1.98710e6	−0.310835
$89$	−5.32998e6	−0.801420	−0.400710	−	0.916205i	$-0.631237\pi$
$89$	−5.32998e6	−0.801420	−0.400710	+	0.916205i	$0.631237\pi$
$90$	0	0
$91$	−3.97385e6	−0.552799
$92$	−3.59335e6	−0.481108
$93$	−426005.	−0.0549192
$94$	−4.24994e6	−0.527759
$95$	0	0
$96$	−307148.	−0.0354322
$97$	2.09810e6	0.233413	0.116707	−	0.993166i	$-0.462766\pi$
$97$	2.09810e6	0.233413	0.116707	+	0.993166i	$0.462766\pi$
$98$	−941192.	−0.101015
$99$	−8.14686e6	−0.843854
$100$	0	0
$101$	1.95538e7	1.88845	0.944227	−	0.329294i	$-0.106811\pi$
$101$	1.95538e7	1.88845	0.944227	+	0.329294i	$0.106811\pi$
$102$	−1.14299e6	−0.106645
$103$	8.95093e6	0.807119	0.403560	−	0.914953i	$-0.367773\pi$
$103$	8.95093e6	0.807119	0.403560	+	0.914953i	$0.367773\pi$
$104$	−5.93181e6	−0.517096
$105$	0	0
$106$	−2.90874e6	−0.237211
$107$	1.39509e7	1.10093	0.550466	−	0.834858i	$-0.314450\pi$
$107$	1.39509e7	1.10093	0.550466	+	0.834858i	$0.314450\pi$
$108$	−2.57125e6	−0.196409
$109$	5.84295e6	0.432155	0.216077	−	0.976376i	$-0.430674\pi$
$109$	5.84295e6	0.432155	0.216077	+	0.976376i	$0.430674\pi$
$110$	0	0
$111$	5.20454e6	0.361203
$112$	−1.40493e6	−0.0944911
$113$	1.20951e7	0.788562	0.394281	−	0.918990i	$-0.370994\pi$
$113$	1.20951e7	0.788562	0.394281	+	0.918990i	$0.370994\pi$
$114$	2.43422e6	0.153884
$115$	0	0
$116$	−1.69229e6	−0.100663
$117$	−2.43197e7	−1.40381
$118$	1.71276e7	0.959644
$119$	−5.22814e6	−0.284402
$120$	0	0
$121$	−4.42462e6	−0.227053
$122$	−7.11059e6	−0.354524
$123$	2.87188e6	0.139155
$124$	−2.90868e6	−0.137000
$125$	0	0
$126$	−5.76004e6	−0.256524
$127$	325760.	0.0141119	0.00705594	−	0.999975i	$-0.497754\pi$
$127$	325760.	0.0141119	0.00705594	+	0.999975i	$0.497754\pi$
$128$	−2.09715e6	−0.0883883
$129$	−7.31834e6	−0.300157
$130$	0	0
$131$	6.38049e6	0.247973	0.123986	−	0.992284i	$-0.460432\pi$
$131$	6.38049e6	0.247973	0.123986	+	0.992284i	$0.460432\pi$
$132$	2.32824e6	0.0881086
$133$	1.11344e7	0.410379
$134$	3.47633e7	1.24811
$135$	0	0
$136$	−7.80411e6	−0.266034
$137$	−4.62324e6	−0.153612	−0.0768059	−	0.997046i	$-0.524472\pi$
$137$	−4.62324e6	−0.153612	−0.0768059	+	0.997046i	$0.524472\pi$
$138$	4.21025e6	0.136374
$139$	−3.55394e7	−1.12243	−0.561214	−	0.827671i	$-0.689665\pi$
$139$	−3.55394e7	−1.12243	−0.561214	+	0.827671i	$0.689665\pi$
$140$	0	0
$141$	4.97956e6	0.149597
$142$	−5.30565e6	−0.155500
$143$	4.49642e7	1.28585
$144$	−8.59807e6	−0.239956
$145$	0	0
$146$	1.07362e7	0.285505
$147$	1.10277e6	0.0286336
$148$	3.55357e7	0.901049
$149$	1.01708e7	0.251885	0.125943	−	0.992038i	$-0.459804\pi$
$149$	1.01708e7	0.251885	0.125943	+	0.992038i	$0.459804\pi$
$150$	0	0
$151$	2.45272e7	0.579735	0.289867	−	0.957067i	$-0.406389\pi$
$151$	2.45272e7	0.579735	0.289867	+	0.957067i	$0.406389\pi$
$152$	1.66204e7	0.383874
$153$	−3.19959e7	−0.722228
$154$	1.06496e7	0.234969
$155$	0	0
$156$	6.95017e6	0.146575
$157$	1.27379e7	0.262693	0.131346	−	0.991337i	$-0.458070\pi$
$157$	1.27379e7	0.262693	0.131346	+	0.991337i	$0.458070\pi$
$158$	−5.64667e7	−1.13892
$159$	3.40810e6	0.0672393
$160$	0	0
$161$	1.92581e7	0.363683
$162$	−3.37139e7	−0.623026
$163$	8.51531e7	1.54008	0.770041	−	0.637995i	$-0.220236\pi$
$163$	8.51531e7	1.54008	0.770041	+	0.637995i	$0.220236\pi$
$164$	1.96087e7	0.347132
$165$	0	0
$166$	5.28699e7	0.897078
$167$	3.74260e6	0.0621822	0.0310911	−	0.999517i	$-0.490102\pi$
$167$	3.74260e6	0.0621822	0.0310911	+	0.999517i	$0.490102\pi$
$168$	1.64612e6	0.0267842
$169$	7.14770e7	1.13910
$170$	0	0
$171$	6.81417e7	1.04214
$172$	−4.99683e7	−0.748765
$173$	7.26865e7	1.06731	0.533657	−	0.845701i	$-0.320817\pi$
$173$	7.26865e7	1.06731	0.533657	+	0.845701i	$0.320817\pi$
$174$	1.98281e6	0.0285338
$175$	0	0
$176$	1.58968e7	0.219794
$177$	−2.00680e7	−0.272018
$178$	4.26398e7	0.566690
$179$	1.02445e8	1.33507	0.667536	−	0.744578i	$-0.267349\pi$
$179$	1.02445e8	1.33507	0.667536	+	0.744578i	$0.267349\pi$
$180$	0	0
$181$	4.79161e6	0.0600630	0.0300315	−	0.999549i	$-0.490439\pi$
$181$	4.79161e6	0.0600630	0.0300315	+	0.999549i	$0.490439\pi$
$182$	3.17908e7	0.390888
$183$	8.33131e6	0.100493
$184$	2.87468e7	0.340194
$185$	0	0
$186$	3.40804e6	0.0388337
$187$	5.91565e7	0.661541
$188$	3.39996e7	0.373182
$189$	1.37803e7	0.148471
$190$	0	0
$191$	2.60173e7	0.270175	0.135087	−	0.990834i	$-0.456868\pi$
$191$	2.60173e7	0.270175	0.135087	+	0.990834i	$0.456868\pi$
$192$	2.45718e6	0.0250544
$193$	7.58326e7	0.759286	0.379643	−	0.925133i	$-0.376047\pi$
$193$	7.58326e7	0.759286	0.379643	+	0.925133i	$0.376047\pi$
$194$	−1.67848e7	−0.165048
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	−1.19192e8	−1.11075	−0.555374	−	0.831601i	$-0.687425\pi$
$197$	−1.19192e8	−1.11075	−0.555374	+	0.831601i	$0.687425\pi$
$198$	6.51749e7	0.596695
$199$	6.72662e7	0.605078	0.302539	−	0.953137i	$-0.402166\pi$
$199$	6.72662e7	0.605078	0.302539	+	0.953137i	$0.402166\pi$
$200$	0	0
$201$	−4.07313e7	−0.353787
$202$	−1.56430e8	−1.33534
$203$	9.06959e6	0.0760942
$204$	9.14389e6	0.0754094
$205$	0	0
$206$	−7.16074e7	−0.570720
$207$	1.17858e8	0.923559
$208$	4.74545e7	0.365642
$209$	−1.25986e8	−0.954573
$210$	0	0
$211$	2.32421e8	1.70328	0.851641	−	0.524126i	$-0.175608\pi$
$211$	2.32421e8	1.70328	0.851641	+	0.524126i	$0.175608\pi$
$212$	2.32699e7	0.167733
$213$	6.21651e6	0.0440776
$214$	−1.11608e8	−0.778476
$215$	0	0
$216$	2.05700e7	0.138882
$217$	1.55887e7	0.103562
$218$	−4.67436e7	−0.305580
$219$	−1.25793e7	−0.0809286
$220$	0	0
$221$	1.76592e8	1.10052
$222$	−4.16363e7	−0.255409
$223$	2.53319e8	1.52968	0.764840	−	0.644221i	$-0.222818\pi$
$223$	2.53319e8	1.52968	0.764840	+	0.644221i	$0.222818\pi$
$224$	1.12394e7	0.0668153
$225$	0	0
$226$	−9.67610e7	−0.557598
$227$	−4.50124e7	−0.255412	−0.127706	−	0.991812i	$-0.540761\pi$
$227$	−4.50124e7	−0.255412	−0.127706	+	0.991812i	$0.540761\pi$
$228$	−1.94738e7	−0.108812
$229$	1.86973e8	1.02886	0.514429	−	0.857533i	$-0.328004\pi$
$229$	1.86973e8	1.02886	0.514429	+	0.857533i	$0.328004\pi$
$230$	0	0
$231$	−1.24779e7	−0.0666039
$232$	1.35383e7	0.0711796
$233$	2.57588e8	1.33408	0.667038	−	0.745024i	$-0.267562\pi$
$233$	2.57588e8	1.33408	0.667038	+	0.745024i	$0.267562\pi$
$234$	1.94558e8	0.992644
$235$	0	0
$236$	−1.37021e8	−0.678570
$237$	6.61607e7	0.322835
$238$	4.18251e7	0.201103
$239$	−3.15071e8	−1.49285	−0.746425	−	0.665469i	$-0.768231\pi$
$239$	−3.15071e8	−1.49285	−0.746425	+	0.665469i	$0.768231\pi$
$240$	0	0
$241$	8.39900e7	0.386516	0.193258	−	0.981148i	$-0.438095\pi$
$241$	8.39900e7	0.386516	0.193258	+	0.981148i	$0.438095\pi$
$242$	3.53970e7	0.160551
$243$	1.27366e8	0.569419
$244$	5.68847e7	0.250687
$245$	0	0
$246$	−2.29750e7	−0.0983973
$247$	−3.76088e8	−1.58800
$248$	2.32695e7	0.0968737
$249$	−6.19464e7	−0.254284
$250$	0	0
$251$	−5.33130e6	−0.0212802	−0.0106401	−	0.999943i	$-0.503387\pi$
$251$	−5.33130e6	−0.0212802	−0.0106401	+	0.999943i	$0.503387\pi$
$252$	4.60803e7	0.181390
$253$	−2.17906e8	−0.845955
$254$	−2.60608e6	−0.00997861
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	2.09942e8	0.771495	0.385747	−	0.922604i	$-0.373944\pi$
$257$	2.09942e8	0.771495	0.385747	+	0.922604i	$0.373944\pi$
$258$	5.85467e7	0.212243
$259$	−1.90449e8	−0.681129
$260$	0	0
$261$	5.55053e7	0.193238
$262$	−5.10439e7	−0.175343
$263$	2.50805e8	0.850143	0.425071	−	0.905160i	$-0.360249\pi$
$263$	2.50805e8	0.850143	0.425071	+	0.905160i	$0.360249\pi$
$264$	−1.86259e7	−0.0623022
$265$	0	0
$266$	−8.90750e7	−0.290182
$267$	−4.99601e7	−0.160633
$268$	−2.78106e8	−0.882549
$269$	3.39197e8	1.06248	0.531238	−	0.847223i	$-0.321727\pi$
$269$	3.39197e8	1.06248	0.531238	+	0.847223i	$0.321727\pi$
$270$	0	0
$271$	7.59574e7	0.231834	0.115917	−	0.993259i	$-0.463019\pi$
$271$	7.59574e7	0.231834	0.115917	+	0.993259i	$0.463019\pi$
$272$	6.24328e7	0.188114
$273$	−3.72486e7	−0.110800
$274$	3.69859e7	0.108620
$275$	0	0
$276$	−3.36820e7	−0.0964308
$277$	5.00883e7	0.141598	0.0707990	−	0.997491i	$-0.477445\pi$
$277$	5.00883e7	0.141598	0.0707990	+	0.997491i	$0.477445\pi$
$278$	2.84315e8	0.793676
$279$	9.54021e7	0.262992
$280$	0	0
$281$	3.41917e8	0.919283	0.459641	−	0.888105i	$-0.347978\pi$
$281$	3.41917e8	0.919283	0.459641	+	0.888105i	$0.347978\pi$
$282$	−3.98365e7	−0.105781
$283$	−3.69077e8	−0.967977	−0.483988	−	0.875074i	$-0.660812\pi$
$283$	−3.69077e8	−0.967977	−0.483988	+	0.875074i	$0.660812\pi$
$284$	4.24452e7	0.109955
$285$	0	0
$286$	−3.59714e8	−0.909234
$287$	−1.05090e8	−0.262407
$288$	6.87846e7	0.169675
$289$	−1.78008e8	−0.433808
$290$	0	0
$291$	1.96664e7	0.0467842
$292$	−8.58892e7	−0.201882
$293$	−5.92780e8	−1.37676	−0.688378	−	0.725352i	$-0.741677\pi$
$293$	−5.92780e8	−1.37676	−0.688378	+	0.725352i	$0.741677\pi$
$294$	−8.82218e6	−0.0202470
$295$	0	0
$296$	−2.84285e8	−0.637138
$297$	−1.55924e8	−0.345355
$298$	−8.13663e7	−0.178110
$299$	−6.50485e8	−1.40730
$300$	0	0
$301$	2.67799e8	0.566013
$302$	−1.96218e8	−0.409934
$303$	1.83286e8	0.378512
$304$	−1.32963e8	−0.271440
$305$	0	0
$306$	2.55967e8	0.510692
$307$	9.43846e8	1.86173	0.930865	−	0.365363i	$-0.119055\pi$
$307$	9.43846e8	1.86173	0.930865	+	0.365363i	$0.119055\pi$
$308$	−8.51968e7	−0.166148
$309$	8.39008e7	0.161775
$310$	0	0
$311$	−1.64065e8	−0.309282	−0.154641	−	0.987971i	$-0.549422\pi$
$311$	−1.64065e8	−0.309282	−0.154641	+	0.987971i	$0.549422\pi$
$312$	−5.56014e7	−0.103644
$313$	−3.46012e8	−0.637801	−0.318901	−	0.947788i	$-0.603314\pi$
$313$	−3.46012e8	−0.637801	−0.318901	+	0.947788i	$0.603314\pi$
$314$	−1.01903e8	−0.185752
$315$	0	0
$316$	4.51734e8	0.805337
$317$	9.53240e8	1.68072	0.840359	−	0.542030i	$-0.182344\pi$
$317$	9.53240e8	1.68072	0.840359	+	0.542030i	$0.182344\pi$
$318$	−2.72648e7	−0.0475453
$319$	−1.02623e8	−0.177001
$320$	0	0
$321$	1.30768e8	0.220665
$322$	−1.54065e8	−0.257163
$323$	−4.94795e8	−0.816989
$324$	2.69711e8	0.440546
$325$	0	0
$326$	−6.81225e8	−1.08900
$327$	5.47684e7	0.0866189
$328$	−1.56869e8	−0.245459
$329$	−1.82216e8	−0.282099
$330$	0	0
$331$	2.24263e8	0.339906	0.169953	−	0.985452i	$-0.445638\pi$
$331$	2.24263e8	0.339906	0.169953	+	0.985452i	$0.445638\pi$
$332$	−4.22959e8	−0.634330
$333$	−1.16554e9	−1.72970
$334$	−2.99408e7	−0.0439694
$335$	0	0
$336$	−1.31690e7	−0.0189393
$337$	−3.56750e8	−0.507761	−0.253880	−	0.967236i	$-0.581707\pi$
$337$	−3.56750e8	−0.507761	−0.253880	+	0.967236i	$0.581707\pi$
$338$	−5.71816e8	−0.805468
$339$	1.13373e8	0.158055
$340$	0	0
$341$	−1.76387e8	−0.240894
$342$	−5.45134e8	−0.736905
$343$	−4.03536e7	−0.0539949
$344$	3.99747e8	0.529456
$345$	0	0
$346$	−5.81492e8	−0.754705
$347$	−1.61839e7	−0.0207936	−0.0103968	−	0.999946i	$-0.503309\pi$
$347$	−1.61839e7	−0.0207936	−0.0103968	+	0.999946i	$0.503309\pi$
$348$	−1.58625e7	−0.0201764
$349$	2.07780e8	0.261647	0.130823	−	0.991406i	$-0.458238\pi$
$349$	2.07780e8	0.261647	0.130823	+	0.991406i	$0.458238\pi$
$350$	0	0
$351$	−4.65459e8	−0.574522
$352$	−1.27174e8	−0.155417
$353$	1.31624e9	1.59266	0.796328	−	0.604865i	$-0.206773\pi$
$353$	1.31624e9	1.59266	0.796328	+	0.604865i	$0.206773\pi$
$354$	1.60544e8	0.192346
$355$	0	0
$356$	−3.41119e8	−0.400710
$357$	−4.90055e7	−0.0570041
$358$	−8.19559e8	−0.944038
$359$	−6.19271e8	−0.706400	−0.353200	−	0.935548i	$-0.614906\pi$
$359$	−6.19271e8	−0.706400	−0.353200	+	0.935548i	$0.614906\pi$
$360$	0	0
$361$	1.59893e8	0.178877
$362$	−3.83329e7	−0.0424709
$363$	−4.14738e7	−0.0455093
$364$	−2.54327e8	−0.276399
$365$	0	0
$366$	−6.66505e7	−0.0710591
$367$	−2.78628e8	−0.294235	−0.147117	−	0.989119i	$-0.547000\pi$
$367$	−2.78628e8	−0.294235	−0.147117	+	0.989119i	$0.547000\pi$
$368$	−2.29974e8	−0.240554
$369$	−6.43146e8	−0.666373
$370$	0	0
$371$	−1.24712e8	−0.126795
$372$	−2.72643e7	−0.0274596
$373$	1.65056e8	0.164684	0.0823420	−	0.996604i	$-0.473760\pi$
$373$	1.65056e8	0.164684	0.0823420	+	0.996604i	$0.473760\pi$
$374$	−4.73252e8	−0.467780
$375$	0	0
$376$	−2.71996e8	−0.263880
$377$	−3.06345e8	−0.294453
$378$	−1.10242e8	−0.104985
$379$	4.29433e8	0.405190	0.202595	−	0.979263i	$-0.435063\pi$
$379$	4.29433e8	0.405190	0.202595	+	0.979263i	$0.435063\pi$
$380$	0	0
$381$	3.05348e6	0.00282851
$382$	−2.08138e8	−0.191043
$383$	−1.34378e9	−1.22217	−0.611084	−	0.791566i	$-0.709266\pi$
$383$	−1.34378e9	−1.22217	−0.611084	+	0.791566i	$0.709266\pi$
$384$	−1.96575e7	−0.0177161
$385$	0	0
$386$	−6.06661e8	−0.536897
$387$	1.63891e9	1.43737
$388$	1.34279e8	0.116707
$389$	−1.07871e8	−0.0929139	−0.0464569	−	0.998920i	$-0.514793\pi$
$389$	−1.07871e8	−0.0929139	−0.0464569	+	0.998920i	$0.514793\pi$
$390$	0	0
$391$	−8.55801e8	−0.724026
$392$	−6.02363e7	−0.0505076
$393$	5.98070e7	0.0497024
$394$	9.53537e8	0.785418
$395$	0	0
$396$	−5.21399e8	−0.421927
$397$	1.86525e9	1.49613	0.748066	−	0.663625i	$-0.230983\pi$
$397$	1.86525e9	1.49613	0.748066	+	0.663625i	$0.230983\pi$
$398$	−5.38130e8	−0.427855
$399$	1.04367e8	0.0822543
$400$	0	0
$401$	9.94724e6	0.00770367	0.00385183	−	0.999993i	$-0.498774\pi$
$401$	9.94724e6	0.00770367	0.00385183	+	0.999993i	$0.498774\pi$
$402$	3.25850e8	0.250165
$403$	−5.26543e8	−0.400744
$404$	1.25144e9	0.944227
$405$	0	0
$406$	−7.25567e7	−0.0538067
$407$	2.15493e9	1.58436
$408$	−7.31511e7	−0.0533225
$409$	−8.76376e8	−0.633372	−0.316686	−	0.948530i	$-0.602570\pi$
$409$	−8.76376e8	−0.633372	−0.316686	+	0.948530i	$0.602570\pi$
$410$	0	0
$411$	−4.33356e7	−0.0307892
$412$	5.72859e8	0.403560
$413$	7.34347e8	0.512951
$414$	−9.42868e8	−0.653055
$415$	0	0
$416$	−3.79636e8	−0.258548
$417$	−3.33126e8	−0.224974
$418$	1.00789e9	0.674985
$419$	−2.01605e9	−1.33891	−0.669457	−	0.742851i	$-0.733473\pi$
$419$	−2.01605e9	−1.33891	−0.669457	+	0.742851i	$0.733473\pi$
$420$	0	0
$421$	5.69546e8	0.371999	0.186000	−	0.982550i	$-0.440448\pi$
$421$	5.69546e8	0.371999	0.186000	+	0.982550i	$0.440448\pi$
$422$	−1.85937e9	−1.20440
$423$	−1.11515e9	−0.716380
$424$	−1.86159e8	−0.118605
$425$	0	0
$426$	−4.97321e7	−0.0311676
$427$	−3.04867e8	−0.189501
$428$	8.92860e8	0.550466
$429$	4.21468e8	0.257730
$430$	0	0
$431$	−7.71940e8	−0.464422	−0.232211	−	0.972665i	$-0.574596\pi$
$431$	−7.71940e8	−0.464422	−0.232211	+	0.972665i	$0.574596\pi$
$432$	−1.64560e8	−0.0982044
$433$	−1.24002e9	−0.734043	−0.367021	−	0.930213i	$-0.619622\pi$
$433$	−1.24002e9	−0.734043	−0.367021	+	0.930213i	$0.619622\pi$
$434$	−1.24710e8	−0.0732296
$435$	0	0
$436$	3.73949e8	0.216077
$437$	1.82260e9	1.04474
$438$	1.00634e8	0.0572252
$439$	1.90621e9	1.07534	0.537669	−	0.843156i	$-0.319305\pi$
$439$	1.90621e9	1.07534	0.537669	+	0.843156i	$0.319305\pi$
$440$	0	0
$441$	−2.46962e8	−0.137118
$442$	−1.41274e9	−0.778185
$443$	5.04738e8	0.275837	0.137919	−	0.990444i	$-0.455959\pi$
$443$	5.04738e8	0.275837	0.137919	+	0.990444i	$0.455959\pi$
$444$	3.33091e8	0.180602
$445$	0	0
$446$	−2.02655e9	−1.08165
$447$	9.53351e7	0.0504866
$448$	−8.99154e7	−0.0472456
$449$	−2.02721e9	−1.05691	−0.528453	−	0.848962i	$-0.677228\pi$
$449$	−2.02721e9	−1.05691	−0.528453	+	0.848962i	$0.677228\pi$
$450$	0	0
$451$	1.18910e9	0.610379
$452$	7.74088e8	0.394281
$453$	2.29904e8	0.116199
$454$	3.60100e8	0.180604
$455$	0	0
$456$	1.55790e8	0.0769419
$457$	−2.03969e9	−0.999672	−0.499836	−	0.866120i	$-0.666606\pi$
$457$	−2.03969e9	−0.999672	−0.499836	+	0.866120i	$0.666606\pi$
$458$	−1.49579e9	−0.727512
$459$	−6.12375e8	−0.295579
$460$	0	0
$461$	−3.41390e9	−1.62292	−0.811461	−	0.584406i	$-0.801327\pi$
$461$	−3.41390e9	−1.62292	−0.811461	+	0.584406i	$0.801327\pi$
$462$	9.98231e7	0.0470960
$463$	1.97893e9	0.926608	0.463304	−	0.886199i	$-0.346664\pi$
$463$	1.97893e9	0.926608	0.463304	+	0.886199i	$0.346664\pi$
$464$	−1.08306e8	−0.0503316
$465$	0	0
$466$	−2.06071e9	−0.943334
$467$	−1.86422e9	−0.847007	−0.423504	−	0.905894i	$-0.639200\pi$
$467$	−1.86422e9	−0.847007	−0.423504	+	0.905894i	$0.639200\pi$
$468$	−1.55646e9	−0.701905
$469$	1.49047e9	0.667144
$470$	0	0
$471$	1.19397e8	0.0526528
$472$	1.09617e9	0.479822
$473$	−3.03015e9	−1.31659
$474$	−5.29286e8	−0.228279
$475$	0	0
$476$	−3.34601e8	−0.142201
$477$	−7.63231e8	−0.321990
$478$	2.52057e9	1.05560
$479$	3.71828e9	1.54585	0.772926	−	0.634496i	$-0.218792\pi$
$479$	3.71828e9	1.54585	0.772926	+	0.634496i	$0.218792\pi$
$480$	0	0
$481$	6.43283e9	2.63569
$482$	−6.71920e8	−0.273308
$483$	1.80514e8	0.0728948
$484$	−2.83176e8	−0.113526
$485$	0	0
$486$	−1.01893e9	−0.402640
$487$	2.16341e9	0.848767	0.424383	−	0.905483i	$-0.360491\pi$
$487$	2.16341e9	0.848767	0.424383	+	0.905483i	$0.360491\pi$
$488$	−4.55078e8	−0.177262
$489$	7.98175e8	0.308686
$490$	0	0
$491$	2.37608e8	0.0905890	0.0452945	−	0.998974i	$-0.485577\pi$
$491$	2.37608e8	0.0905890	0.0452945	+	0.998974i	$0.485577\pi$
$492$	1.83800e8	0.0695774
$493$	−4.03039e8	−0.151489
$494$	3.00870e9	1.12288
$495$	0	0
$496$	−1.86156e8	−0.0685000
$497$	−2.27480e8	−0.0831181
$498$	4.95571e8	0.179806
$499$	2.65166e9	0.955358	0.477679	−	0.878535i	$-0.341478\pi$
$499$	2.65166e9	0.955358	0.477679	+	0.878535i	$0.341478\pi$
$500$	0	0
$501$	3.50810e7	0.0124635
$502$	4.26504e7	0.0150474
$503$	−2.15865e9	−0.756300	−0.378150	−	0.925744i	$-0.623440\pi$
$503$	−2.15865e9	−0.756300	−0.378150	+	0.925744i	$0.623440\pi$
$504$	−3.68642e8	−0.128262
$505$	0	0
$506$	1.74325e9	0.598180
$507$	6.69984e8	0.228316
$508$	2.08486e7	0.00705594
$509$	2.47831e9	0.832998	0.416499	−	0.909136i	$-0.363257\pi$
$509$	2.47831e9	0.832998	0.416499	+	0.909136i	$0.363257\pi$
$510$	0	0
$511$	4.60312e8	0.152609
$512$	−1.34218e8	−0.0441942
$513$	1.30418e9	0.426506
$514$	−1.67953e9	−0.545529
$515$	0	0
$516$	−4.68374e8	−0.150079
$517$	2.06178e9	0.656184
$518$	1.52359e9	0.481631
$519$	6.81321e8	0.213927
$520$	0	0
$521$	−3.99735e9	−1.23834	−0.619170	−	0.785257i	$-0.712531\pi$
$521$	−3.99735e9	−1.23834	−0.619170	+	0.785257i	$0.712531\pi$
$522$	−4.44043e8	−0.136640
$523$	3.93132e9	1.20166	0.600831	−	0.799376i	$-0.294837\pi$
$523$	3.93132e9	1.20166	0.600831	+	0.799376i	$0.294837\pi$
$524$	4.08351e8	0.123986
$525$	0	0
$526$	−2.00644e9	−0.601142
$527$	−6.92739e8	−0.206173
$528$	1.49007e8	0.0440543
$529$	−2.52441e8	−0.0741421
$530$	0	0
$531$	4.49416e9	1.30262
$532$	7.12600e8	0.205190
$533$	3.54965e9	1.01541
$534$	3.99681e8	0.113584
$535$	0	0
$536$	2.22485e9	0.624056
$537$	9.60258e8	0.267595
$538$	−2.71358e9	−0.751284
$539$	4.56602e8	0.125596
$540$	0	0
$541$	−2.74766e9	−0.746057	−0.373028	−	0.927820i	$-0.621681\pi$
$541$	−2.74766e9	−0.746057	−0.373028	+	0.927820i	$0.621681\pi$
$542$	−6.07659e8	−0.163932
$543$	4.49138e7	0.0120387
$544$	−4.99463e8	−0.133017
$545$	0	0
$546$	2.97989e8	0.0783475
$547$	1.40581e9	0.367258	0.183629	−	0.982996i	$-0.441216\pi$
$547$	1.40581e9	0.367258	0.183629	+	0.982996i	$0.441216\pi$
$548$	−2.95887e8	−0.0768059
$549$	−1.86576e9	−0.481231
$550$	0	0
$551$	8.58352e8	0.218592
$552$	2.69456e8	0.0681869
$553$	−2.42101e9	−0.608777
$554$	−4.00707e8	−0.100125
$555$	0	0
$556$	−2.27452e9	−0.561214
$557$	−9.87805e8	−0.242202	−0.121101	−	0.992640i	$-0.538643\pi$
$557$	−9.87805e8	−0.242202	−0.121101	+	0.992640i	$0.538643\pi$
$558$	−7.63217e8	−0.185964
$559$	−9.04550e9	−2.19024
$560$	0	0
$561$	5.54498e8	0.132596
$562$	−2.73534e9	−0.650031
$563$	−1.88115e9	−0.444268	−0.222134	−	0.975016i	$-0.571302\pi$
$563$	−1.88115e9	−0.444268	−0.222134	+	0.975016i	$0.571302\pi$
$564$	3.18692e8	0.0747987
$565$	0	0
$566$	2.95262e9	0.684463
$567$	−1.44548e9	−0.333021
$568$	−3.39562e8	−0.0777499
$569$	−3.88531e9	−0.884164	−0.442082	−	0.896975i	$-0.645760\pi$
$569$	−3.88531e9	−0.884164	−0.442082	+	0.896975i	$0.645760\pi$
$570$	0	0
$571$	2.44900e9	0.550505	0.275253	−	0.961372i	$-0.411238\pi$
$571$	2.44900e9	0.550505	0.275253	+	0.961372i	$0.411238\pi$
$572$	2.87771e9	0.642926
$573$	2.43871e8	0.0541525
$574$	8.40722e8	0.185550
$575$	0	0
$576$	−5.50277e8	−0.119978
$577$	−2.88218e9	−0.624605	−0.312303	−	0.949983i	$-0.601100\pi$
$577$	−2.88218e9	−0.624605	−0.312303	+	0.949983i	$0.601100\pi$
$578$	1.42407e9	0.306748
$579$	7.10811e8	0.152188
$580$	0	0
$581$	2.26680e9	0.479508
$582$	−1.57331e8	−0.0330814
$583$	1.41112e9	0.294934
$584$	6.87114e8	0.142752
$585$	0	0
$586$	4.74224e9	0.973513
$587$	−4.72250e9	−0.963692	−0.481846	−	0.876256i	$-0.660034\pi$
$587$	−4.72250e9	−0.963692	−0.481846	+	0.876256i	$0.660034\pi$
$588$	7.05775e7	0.0143168
$589$	1.47533e9	0.297499
$590$	0	0
$591$	−1.11724e9	−0.222633
$592$	2.27428e9	0.450525
$593$	−1.23167e8	−0.0242550	−0.0121275	−	0.999926i	$-0.503860\pi$
$593$	−1.23167e8	−0.0242550	−0.0121275	+	0.999926i	$0.503860\pi$
$594$	1.24739e9	0.244203
$595$	0	0
$596$	6.50931e8	0.125943
$597$	6.30514e8	0.121279
$598$	5.20388e9	0.995115
$599$	1.35497e9	0.257594	0.128797	−	0.991671i	$-0.458888\pi$
$599$	1.35497e9	0.257594	0.128797	+	0.991671i	$0.458888\pi$
$600$	0	0
$601$	−5.25107e9	−0.986704	−0.493352	−	0.869830i	$-0.664228\pi$
$601$	−5.25107e9	−0.986704	−0.493352	+	0.869830i	$0.664228\pi$
$602$	−2.14239e9	−0.400231
$603$	9.12161e9	1.69419
$604$	1.56974e9	0.289867
$605$	0	0
$606$	−1.46629e9	−0.267649
$607$	−4.99012e9	−0.905630	−0.452815	−	0.891605i	$-0.649580\pi$
$607$	−4.99012e9	−0.905630	−0.452815	+	0.891605i	$0.649580\pi$
$608$	1.06371e9	0.191937
$609$	8.50130e7	0.0152519
$610$	0	0
$611$	6.15476e9	1.09161
$612$	−2.04774e9	−0.361114
$613$	2.46028e9	0.431393	0.215696	−	0.976460i	$-0.430798\pi$
$613$	2.46028e9	0.431393	0.215696	+	0.976460i	$0.430798\pi$
$614$	−7.55077e9	−1.31644
$615$	0	0
$616$	6.81575e8	0.117485
$617$	1.93223e9	0.331178	0.165589	−	0.986195i	$-0.447047\pi$
$617$	1.93223e9	0.331178	0.165589	+	0.986195i	$0.447047\pi$
$618$	−6.71206e8	−0.114392
$619$	−8.13699e9	−1.37894	−0.689472	−	0.724313i	$-0.742157\pi$
$619$	−8.13699e9	−1.37894	−0.689472	+	0.724313i	$0.742157\pi$
$620$	0	0
$621$	2.25571e9	0.377975
$622$	1.31252e9	0.218695
$623$	1.82818e9	0.302908
$624$	4.44811e8	0.0732874
$625$	0	0
$626$	2.76809e9	0.450994
$627$	−1.18092e9	−0.191330
$628$	8.15223e8	0.131346
$629$	8.46326e9	1.35600
$630$	0	0
$631$	8.83737e9	1.40030	0.700149	−	0.713997i	$-0.253117\pi$
$631$	8.83737e9	1.40030	0.700149	+	0.713997i	$0.253117\pi$
$632$	−3.61387e9	−0.569459
$633$	2.17858e9	0.341397
$634$	−7.62592e9	−1.18845
$635$	0	0
$636$	2.18119e8	0.0336196
$637$	1.36303e9	0.208938
$638$	8.20981e8	0.125159
$639$	−1.39216e9	−0.211075
$640$	0	0
$641$	−1.25905e10	−1.88817	−0.944086	−	0.329700i	$-0.893052\pi$
$641$	−1.25905e10	−1.88817	−0.944086	+	0.329700i	$0.893052\pi$
$642$	−1.04614e9	−0.156034
$643$	−6.47578e9	−0.960625	−0.480312	−	0.877098i	$-0.659477\pi$
$643$	−6.47578e9	−0.960625	−0.480312	+	0.877098i	$0.659477\pi$
$644$	1.23252e9	0.181842
$645$	0	0
$646$	3.95836e9	0.577698
$647$	−8.69520e9	−1.26216	−0.631081	−	0.775717i	$-0.717388\pi$
$647$	−8.69520e9	−1.26216	−0.631081	+	0.775717i	$0.717388\pi$
$648$	−2.15769e9	−0.311513
$649$	−8.30914e9	−1.19316
$650$	0	0
$651$	1.46120e8	0.0207575
$652$	5.44980e9	0.770041
$653$	3.87898e9	0.545156	0.272578	−	0.962134i	$-0.412124\pi$
$653$	3.87898e9	0.545156	0.272578	+	0.962134i	$0.412124\pi$
$654$	−4.38147e8	−0.0612488
$655$	0	0
$656$	1.25495e9	0.173566
$657$	2.81708e9	0.387544
$658$	1.45773e9	0.199474
$659$	−1.30333e9	−0.177400	−0.0887000	−	0.996058i	$-0.528271\pi$
$659$	−1.30333e9	−0.177400	−0.0887000	+	0.996058i	$0.528271\pi$
$660$	0	0
$661$	6.85888e9	0.923736	0.461868	−	0.886949i	$-0.347179\pi$
$661$	6.85888e9	0.923736	0.461868	+	0.886949i	$0.347179\pi$
$662$	−1.79410e9	−0.240350
$663$	1.65527e9	0.220583
$664$	3.38367e9	0.448539
$665$	0	0
$666$	9.32429e9	1.22308
$667$	1.48461e9	0.193719
$668$	2.39527e8	0.0310911
$669$	2.37446e9	0.306601
$670$	0	0
$671$	3.44957e9	0.440794
$672$	1.05352e8	0.0133921
$673$	6.44953e9	0.815596	0.407798	−	0.913072i	$-0.366297\pi$
$673$	6.44953e9	0.815596	0.407798	+	0.913072i	$0.366297\pi$
$674$	2.85400e9	0.359041
$675$	0	0
$676$	4.57453e9	0.569552
$677$	−3.83027e9	−0.474426	−0.237213	−	0.971458i	$-0.576234\pi$
$677$	−3.83027e9	−0.474426	−0.237213	+	0.971458i	$0.576234\pi$
$678$	−9.06981e8	−0.111762
$679$	−7.19649e8	−0.0882219
$680$	0	0
$681$	−4.21920e8	−0.0511936
$682$	1.41109e9	0.170338
$683$	1.97836e9	0.237592	0.118796	−	0.992919i	$-0.462097\pi$
$683$	1.97836e9	0.237592	0.118796	+	0.992919i	$0.462097\pi$
$684$	4.36107e9	0.521071
$685$	0	0
$686$	3.22829e8	0.0381802
$687$	1.75258e9	0.206219
$688$	−3.19797e9	−0.374382
$689$	4.21243e9	0.490643
$690$	0	0
$691$	1.00255e10	1.15593	0.577967	−	0.816060i	$-0.303846\pi$
$691$	1.00255e10	1.15593	0.577967	+	0.816060i	$0.303846\pi$
$692$	4.65193e9	0.533657
$693$	2.79437e9	0.318947
$694$	1.29471e8	0.0147033
$695$	0	0
$696$	1.26900e8	0.0142669
$697$	4.67005e9	0.522404
$698$	−1.66224e9	−0.185012
$699$	2.41448e9	0.267395
$700$	0	0
$701$	1.21616e9	0.133345	0.0666727	−	0.997775i	$-0.478762\pi$
$701$	1.21616e9	0.133345	0.0666727	+	0.997775i	$0.478762\pi$
$702$	3.72367e9	0.406249
$703$	−1.80242e10	−1.95665
$704$	1.01739e9	0.109897
$705$	0	0
$706$	−1.05299e10	−1.12618
$707$	−6.70696e9	−0.713769
$708$	−1.28435e9	−0.136009
$709$	1.04706e10	1.10334	0.551671	−	0.834062i	$-0.313990\pi$
$709$	1.04706e10	1.10334	0.551671	+	0.834062i	$0.313990\pi$
$710$	0	0
$711$	−1.48164e10	−1.54597
$712$	2.72895e9	0.283345
$713$	2.55174e9	0.263647
$714$	3.92044e8	0.0403080
$715$	0	0
$716$	6.55647e9	0.667536
$717$	−2.95330e9	−0.299219
$718$	4.95417e9	0.499500
$719$	−2.99586e9	−0.300587	−0.150294	−	0.988641i	$-0.548022\pi$
$719$	−2.99586e9	−0.300587	−0.150294	+	0.988641i	$0.548022\pi$
$720$	0	0
$721$	−3.07017e9	−0.305062
$722$	−1.27914e9	−0.126485
$723$	7.87273e8	0.0774714
$724$	3.06663e8	0.0300315
$725$	0	0
$726$	3.31790e8	0.0321800
$727$	6.52469e9	0.629781	0.314890	−	0.949128i	$-0.398032\pi$
$727$	6.52469e9	0.629781	0.314890	+	0.949128i	$0.398032\pi$
$728$	2.03461e9	0.195444
$729$	−8.02267e9	−0.766960
$730$	0	0
$731$	−1.19006e10	−1.12683
$732$	5.33204e8	0.0502464
$733$	1.19856e10	1.12408	0.562040	−	0.827110i	$-0.310017\pi$
$733$	1.19856e10	1.12408	0.562040	+	0.827110i	$0.310017\pi$
$734$	2.22903e9	0.208056
$735$	0	0
$736$	1.83980e9	0.170097
$737$	−1.68647e10	−1.55183
$738$	5.14517e9	0.471197
$739$	1.81843e10	1.65746	0.828728	−	0.559651i	$-0.189065\pi$
$739$	1.81843e10	1.65746	0.828728	+	0.559651i	$0.189065\pi$
$740$	0	0
$741$	−3.52523e9	−0.318290
$742$	9.97698e8	0.0896573
$743$	−6.10111e9	−0.545692	−0.272846	−	0.962058i	$-0.587965\pi$
$743$	−6.10111e9	−0.545692	−0.272846	+	0.962058i	$0.587965\pi$
$744$	2.18114e8	0.0194169
$745$	0	0
$746$	−1.32045e9	−0.116449
$747$	1.38727e10	1.21769
$748$	3.78602e9	0.330771
$749$	−4.78517e9	−0.416113
$750$	0	0
$751$	−3.69384e9	−0.318228	−0.159114	−	0.987260i	$-0.550864\pi$
$751$	−3.69384e9	−0.318228	−0.159114	+	0.987260i	$0.550864\pi$
$752$	2.17597e9	0.186591
$753$	−4.99725e7	−0.00426529
$754$	2.45076e9	0.208210
$755$	0	0
$756$	8.81938e8	0.0742356
$757$	−1.35894e10	−1.13858	−0.569292	−	0.822135i	$-0.692783\pi$
$757$	−1.35894e10	−1.13858	−0.569292	+	0.822135i	$0.692783\pi$
$758$	−3.43546e9	−0.286512
$759$	−2.04252e9	−0.169559
$760$	0	0
$761$	1.79994e10	1.48051	0.740255	−	0.672326i	$-0.234705\pi$
$761$	1.79994e10	1.48051	0.740255	+	0.672326i	$0.234705\pi$
$762$	−2.44279e7	−0.00200006
$763$	−2.00413e9	−0.163339
$764$	1.66511e9	0.135087
$765$	0	0
$766$	1.07502e10	0.864204
$767$	−2.48042e10	−1.98491
$768$	1.57260e8	0.0125272
$769$	1.41340e10	1.12079	0.560394	−	0.828226i	$-0.310650\pi$
$769$	1.41340e10	1.12079	0.560394	+	0.828226i	$0.310650\pi$
$770$	0	0
$771$	1.96787e9	0.154634
$772$	4.85329e9	0.379643
$773$	7.39484e9	0.575838	0.287919	−	0.957655i	$-0.407037\pi$
$773$	7.39484e9	0.575838	0.287919	+	0.957655i	$0.407037\pi$
$774$	−1.31113e10	−1.01637
$775$	0	0
$776$	−1.07423e9	−0.0825241
$777$	−1.78516e9	−0.136522
$778$	8.62967e8	0.0657000
$779$	−9.94581e9	−0.753805
$780$	0	0
$781$	2.57394e9	0.193339
$782$	6.84641e9	0.511964
$783$	1.06233e9	0.0790845
$784$	4.81890e8	0.0357143
$785$	0	0
$786$	−4.78456e8	−0.0351449
$787$	9.49146e9	0.694099	0.347050	−	0.937847i	$-0.387184\pi$
$787$	9.49146e9	0.694099	0.347050	+	0.937847i	$0.387184\pi$
$788$	−7.62829e9	−0.555374
$789$	2.35090e9	0.170398
$790$	0	0
$791$	−4.14863e9	−0.298049
$792$	4.17119e9	0.298347
$793$	1.02975e10	0.733292
$794$	−1.49220e10	−1.05792
$795$	0	0
$796$	4.30504e9	0.302539
$797$	8.83350e9	0.618057	0.309029	−	0.951053i	$-0.399996\pi$
$797$	8.83350e9	0.618057	0.309029	+	0.951053i	$0.399996\pi$
$798$	−8.34937e8	−0.0581626
$799$	8.09742e9	0.561607
$800$	0	0
$801$	1.11884e10	0.769224
$802$	−7.95779e7	−0.00544731
$803$	−5.20844e9	−0.354980
$804$	−2.60680e9	−0.176894
$805$	0	0
$806$	4.21235e9	0.283368
$807$	3.17943e9	0.212957
$808$	−1.00116e10	−0.667670
$809$	6.15622e9	0.408784	0.204392	−	0.978889i	$-0.434478\pi$
$809$	6.15622e9	0.408784	0.204392	+	0.978889i	$0.434478\pi$
$810$	0	0
$811$	−1.53945e10	−1.01343	−0.506715	−	0.862114i	$-0.669140\pi$
$811$	−1.53945e10	−1.01343	−0.506715	+	0.862114i	$0.669140\pi$
$812$	5.80454e8	0.0380471
$813$	7.11980e8	0.0464677
$814$	−1.72395e10	−1.12031
$815$	0	0
$816$	5.85209e8	0.0377047
$817$	2.53447e10	1.62596
$818$	7.01101e9	0.447862
$819$	8.34167e9	0.530590
$820$	0	0
$821$	−2.74907e10	−1.73374	−0.866871	−	0.498532i	$-0.833872\pi$
$821$	−2.74907e10	−1.73374	−0.866871	+	0.498532i	$0.833872\pi$
$822$	3.46684e8	0.0217712
$823$	−1.61300e9	−0.100864	−0.0504318	−	0.998728i	$-0.516060\pi$
$823$	−1.61300e9	−0.100864	−0.0504318	+	0.998728i	$0.516060\pi$
$824$	−4.58288e9	−0.285360
$825$	0	0
$826$	−5.87477e9	−0.362711
$827$	−9.03950e9	−0.555745	−0.277872	−	0.960618i	$-0.589629\pi$
$827$	−9.03950e9	−0.555745	−0.277872	+	0.960618i	$0.589629\pi$
$828$	7.54294e9	0.461779
$829$	2.46326e10	1.50165	0.750826	−	0.660501i	$-0.229656\pi$
$829$	2.46326e10	1.50165	0.750826	+	0.660501i	$0.229656\pi$
$830$	0	0
$831$	4.69499e8	0.0283812
$832$	3.03709e9	0.182821
$833$	1.79325e9	0.107494
$834$	2.66500e9	0.159080
$835$	0	0
$836$	−8.06308e9	−0.477286
$837$	1.82592e9	0.107632
$838$	1.61284e10	0.946755
$839$	−6.84347e9	−0.400046	−0.200023	−	0.979791i	$-0.564102\pi$
$839$	−6.84347e9	−0.400046	−0.200023	+	0.979791i	$0.564102\pi$
$840$	0	0
$841$	−1.65507e10	−0.959468
$842$	−4.55637e9	−0.263043
$843$	3.20493e9	0.184256
$844$	1.48749e10	0.851641
$845$	0	0
$846$	8.92123e9	0.506557
$847$	1.51764e9	0.0858179
$848$	1.48928e9	0.0838667
$849$	−3.45952e9	−0.194016
$850$	0	0
$851$	−3.11748e10	−1.73401
$852$	3.97857e8	0.0220388
$853$	−3.54072e10	−1.95330	−0.976652	−	0.214826i	$-0.931081\pi$
$853$	−3.54072e10	−1.95330	−0.976652	+	0.214826i	$0.931081\pi$
$854$	2.43893e9	0.133998
$855$	0	0
$856$	−7.14288e9	−0.389238
$857$	1.87299e10	1.01649	0.508245	−	0.861213i	$-0.330294\pi$
$857$	1.87299e10	1.01649	0.508245	+	0.861213i	$0.330294\pi$
$858$	−3.37175e9	−0.182242
$859$	−5.38272e9	−0.289752	−0.144876	−	0.989450i	$-0.546278\pi$
$859$	−5.38272e9	−0.289752	−0.144876	+	0.989450i	$0.546278\pi$
$860$	0	0
$861$	−9.85054e8	−0.0525956
$862$	6.17552e9	0.328396
$863$	−7.29444e9	−0.386326	−0.193163	−	0.981167i	$-0.561875\pi$
$863$	−7.29444e9	−0.386326	−0.193163	+	0.981167i	$0.561875\pi$
$864$	1.31648e9	0.0694410
$865$	0	0
$866$	9.92016e9	0.519046
$867$	−1.66854e9	−0.0869503
$868$	9.97679e8	0.0517811
$869$	2.73938e10	1.41606
$870$	0	0
$871$	−5.03440e10	−2.58157
$872$	−2.99159e9	−0.152790
$873$	−4.40421e9	−0.224036
$874$	−1.45808e10	−0.738740
$875$	0	0
$876$	−8.05075e8	−0.0404643
$877$	−3.05623e10	−1.52998	−0.764992	−	0.644039i	$-0.777257\pi$
$877$	−3.05623e10	−1.52998	−0.764992	+	0.644039i	$0.777257\pi$
$878$	−1.52497e10	−0.760379
$879$	−5.55637e9	−0.275950
$880$	0	0
$881$	−3.10736e10	−1.53100	−0.765500	−	0.643436i	$-0.777508\pi$
$881$	−3.10736e10	−1.53100	−0.765500	+	0.643436i	$0.777508\pi$
$882$	1.97569e9	0.0969571
$883$	−9.10432e9	−0.445026	−0.222513	−	0.974930i	$-0.571426\pi$
$883$	−9.10432e9	−0.445026	−0.222513	+	0.974930i	$0.571426\pi$
$884$	1.13019e10	0.550260
$885$	0	0
$886$	−4.03790e9	−0.195046
$887$	5.03417e8	0.0242212	0.0121106	−	0.999927i	$-0.496145\pi$
$887$	5.03417e8	0.0242212	0.0121106	+	0.999927i	$0.496145\pi$
$888$	−2.66472e9	−0.127705
$889$	−1.11736e8	−0.00533379
$890$	0	0
$891$	1.63557e10	0.774633
$892$	1.62124e10	0.764840
$893$	−1.72451e10	−0.810373
$894$	−7.62680e8	−0.0356994
$895$	0	0
$896$	7.19323e8	0.0334077
$897$	−6.09727e9	−0.282073
$898$	1.62177e10	0.747346
$899$	1.20174e9	0.0551634
$900$	0	0
$901$	5.54202e9	0.252424
$902$	−9.51278e9	−0.431603
$903$	2.51019e9	0.113449
$904$	−6.19271e9	−0.278799
$905$	0	0
$906$	−1.83923e9	−0.0821652
$907$	−3.55392e10	−1.58155	−0.790774	−	0.612108i	$-0.790322\pi$
$907$	−3.55392e10	−1.58155	−0.790774	+	0.612108i	$0.790322\pi$
$908$	−2.88080e9	−0.127706
$909$	−4.10462e10	−1.81259
$910$	0	0
$911$	3.60841e10	1.58125	0.790627	−	0.612299i	$-0.209755\pi$
$911$	3.60841e10	1.58125	0.790627	+	0.612299i	$0.209755\pi$
$912$	−1.24632e9	−0.0544061
$913$	−2.56488e10	−1.11537
$914$	1.63175e10	0.706875
$915$	0	0
$916$	1.19663e10	0.514429
$917$	−2.18851e9	−0.0937250
$918$	4.89900e9	0.209006
$919$	−7.06975e9	−0.300469	−0.150235	−	0.988650i	$-0.548003\pi$
$919$	−7.06975e9	−0.300469	−0.150235	+	0.988650i	$0.548003\pi$
$920$	0	0
$921$	8.84706e9	0.373156
$922$	2.73112e10	1.14758
$923$	7.68363e9	0.321633
$924$	−7.98585e8	−0.0333019
$925$	0	0
$926$	−1.58314e10	−0.655211
$927$	−1.87892e10	−0.774694
$928$	8.66450e8	0.0355898
$929$	−1.55102e10	−0.634691	−0.317345	−	0.948310i	$-0.602791\pi$
$929$	−1.55102e10	−0.634691	−0.317345	+	0.948310i	$0.602791\pi$
$930$	0	0
$931$	−3.81909e9	−0.155109
$932$	1.64856e10	0.667038
$933$	−1.53785e9	−0.0619910
$934$	1.49137e10	0.598925
$935$	0	0
$936$	1.24517e10	0.496322
$937$	3.61814e10	1.43680	0.718400	−	0.695630i	$-0.244875\pi$
$937$	3.61814e10	1.43680	0.718400	+	0.695630i	$0.244875\pi$
$938$	−1.19238e10	−0.471742
$939$	−3.24331e9	−0.127838
$940$	0	0
$941$	1.86518e10	0.729720	0.364860	−	0.931062i	$-0.381117\pi$
$941$	1.86518e10	0.729720	0.364860	+	0.931062i	$0.381117\pi$
$942$	−9.55178e8	−0.0372311
$943$	−1.72023e10	−0.668031
$944$	−8.76934e9	−0.339285
$945$	0	0
$946$	2.42412e10	0.930969
$947$	−8.55630e9	−0.327387	−0.163693	−	0.986511i	$-0.552341\pi$
$947$	−8.55630e9	−0.327387	−0.163693	+	0.986511i	$0.552341\pi$
$948$	4.23429e9	0.161418
$949$	−1.55481e10	−0.590533
$950$	0	0
$951$	8.93512e9	0.336875
$952$	2.67681e9	0.100551
$953$	5.92820e8	0.0221870	0.0110935	−	0.999938i	$-0.496469\pi$
$953$	5.92820e8	0.0221870	0.0110935	+	0.999938i	$0.496469\pi$
$954$	6.10585e9	0.227681
$955$	0	0
$956$	−2.01646e10	−0.746425
$957$	−9.61924e8	−0.0354772
$958$	−2.97462e10	−1.09308
$959$	1.58577e9	0.0580598
$960$	0	0
$961$	−2.54471e10	−0.924924
$962$	−5.14626e10	−1.86371
$963$	−2.92850e10	−1.05670
$964$	5.37536e9	0.193258
$965$	0	0
$966$	−1.44411e9	−0.0515444
$967$	5.25393e10	1.86849	0.934247	−	0.356626i	$-0.116073\pi$
$967$	5.25393e10	1.86849	0.934247	+	0.356626i	$0.116073\pi$
$968$	2.26541e9	0.0802753
$969$	−4.63792e9	−0.163753
$970$	0	0
$971$	−9.16576e9	−0.321293	−0.160647	−	0.987012i	$-0.551358\pi$
$971$	−9.16576e9	−0.321293	−0.160647	+	0.987012i	$0.551358\pi$
$972$	8.15143e9	0.284710
$973$	1.21900e10	0.424238
$974$	−1.73073e10	−0.600169
$975$	0	0
$976$	3.64062e9	0.125343
$977$	2.01603e10	0.691618	0.345809	−	0.938305i	$-0.387604\pi$
$977$	2.01603e10	0.691618	0.345809	+	0.938305i	$0.387604\pi$
$978$	−6.38540e9	−0.218274
$979$	−2.06859e10	−0.704588
$980$	0	0
$981$	−1.22652e10	−0.414793
$982$	−1.90086e9	−0.0640561
$983$	1.18989e9	0.0399549	0.0199775	−	0.999800i	$-0.493641\pi$
$983$	1.18989e9	0.0399549	0.0199775	+	0.999800i	$0.493641\pi$
$984$	−1.47040e9	−0.0491986
$985$	0	0
$986$	3.22431e9	0.107119
$987$	−1.70799e9	−0.0565425
$988$	−2.40696e10	−0.793999
$989$	4.38364e10	1.44095
$990$	0	0
$991$	2.09986e10	0.685382	0.342691	−	0.939448i	$-0.388662\pi$
$991$	2.09986e10	0.685382	0.342691	+	0.939448i	$0.388662\pi$
$992$	1.48925e9	0.0484368
$993$	2.10211e9	0.0681291
$994$	1.81984e9	0.0587734
$995$	0	0
$996$	−3.96457e9	−0.127142
$997$	−3.21657e10	−1.02792	−0.513961	−	0.857814i	$-0.671822\pi$
$997$	−3.21657e10	−1.02792	−0.513961	+	0.857814i	$0.671822\pi$
$998$	−2.12133e10	−0.675540
$999$	−2.23074e10	−0.707896

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.8.a.j.1.2		2
5.2	odd	4		350.8.c.k.99.1		4
5.3	odd	4		350.8.c.k.99.4		4
5.4	even	2		14.8.a.c.1.1	✓	2
15.14	odd	2		126.8.a.i.1.1		2
20.19	odd	2		112.8.a.g.1.2		2
35.4	even	6		98.8.c.g.79.2		4
35.9	even	6		98.8.c.g.67.2		4
35.19	odd	6		98.8.c.k.67.1		4
35.24	odd	6		98.8.c.k.79.1		4
35.34	odd	2		98.8.a.g.1.2		2
40.19	odd	2		448.8.a.s.1.1		2
40.29	even	2		448.8.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.8.a.c.1.1	✓	2	5.4	even	2
98.8.a.g.1.2		2	35.34	odd	2
98.8.c.g.67.2		4	35.9	even	6
98.8.c.g.79.2		4	35.4	even	6
98.8.c.k.67.1		4	35.19	odd	6
98.8.c.k.79.1		4	35.24	odd	6
112.8.a.g.1.2		2	20.19	odd	2
126.8.a.i.1.1		2	15.14	odd	2
350.8.a.j.1.2		2	1.1	even	1	trivial
350.8.c.k.99.1		4	5.2	odd	4
350.8.c.k.99.4		4	5.3	odd	4
448.8.a.l.1.2		2	40.29	even	2
448.8.a.s.1.1		2	40.19	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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +9.37342 q^{3} +64.0000 q^{4} -74.9873 q^{6} -343.000 q^{7} -512.000 q^{8} -2099.14 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} +9.37342 q^{3} +64.0000 q^{4} -74.9873 q^{6} -343.000 q^{7} -512.000 q^{8} -2099.14 q^{9} +3881.05 q^{11} +599.899 q^{12} +11585.6 q^{13} +2744.00 q^{14} +4096.00 q^{16} +15242.4 q^{17} +16793.1 q^{18} -32461.7 q^{19} -3215.08 q^{21} -31048.4 q^{22} -56146.1 q^{23} -4799.19 q^{24} -92684.6 q^{26} -40175.8 q^{27} -21952.0 q^{28} -26442.0 q^{29} -45448.2 q^{31} -32768.0 q^{32} +36378.7 q^{33} -121939. q^{34} -134345. q^{36} +555245. q^{37} +259694. q^{38} +108596. q^{39} +306385. q^{41} +25720.7 q^{42} -780755. q^{43} +248387. q^{44} +449169. q^{46} +531243. q^{47} +38393.5 q^{48} +117649. q^{49} +142873. q^{51} +741477. q^{52} +363593. q^{53} +321406. q^{54} +175616. q^{56} -304277. q^{57} +211536. q^{58} -2.14095e6 q^{59} +888824. q^{61} +363586. q^{62} +720005. q^{63} +262144. q^{64} -291030. q^{66} -4.34541e6 q^{67} +975513. q^{68} -526281. q^{69} +663207. q^{71} +1.07476e6 q^{72} -1.34202e6 q^{73} -4.44196e6 q^{74} -2.07755e6 q^{76} -1.33120e6 q^{77} -868771. q^{78} +7.05834e6 q^{79} +4.21423e6 q^{81} -2.45108e6 q^{82} -6.60874e6 q^{83} -205765. q^{84} +6.24604e6 q^{86} -247851. q^{87} -1.98710e6 q^{88} -5.32998e6 q^{89} -3.97385e6 q^{91} -3.59335e6 q^{92} -426005. q^{93} -4.24994e6 q^{94} -307148. q^{96} +2.09810e6 q^{97} -941192. q^{98} -8.14686e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{2} - 70 q^{3} + 128 q^{4} + 560 q^{6} - 686 q^{7} - 1024 q^{8} + 2014 q^{9}+O(q^{10}) \) 2 * q - 16 * q^2 - 70 * q^3 + 128 * q^4 + 560 * q^6 - 686 * q^7 - 1024 * q^8 + 2014 * q^9	\( 2 q - 16 q^{2} - 70 q^{3} + 128 q^{4} + 560 q^{6} - 686 q^{7} - 1024 q^{8} + 2014 q^{9} - 3420 q^{11} - 4480 q^{12} + 6398 q^{13} + 5488 q^{14} + 8192 q^{16} + 38472 q^{17} - 16112 q^{18} - 43358 q^{19} + 24010 q^{21} + 27360 q^{22} - 89928 q^{23} + 35840 q^{24} - 51184 q^{26} - 193060 q^{27} - 43904 q^{28} + 159576 q^{29} - 143612 q^{31} - 65536 q^{32} + 615888 q^{33} - 307776 q^{34} + 128896 q^{36} + 271832 q^{37} + 346864 q^{38} + 520352 q^{39} + 64848 q^{41} - 192080 q^{42} - 1527964 q^{43} - 218880 q^{44} + 719424 q^{46} - 485436 q^{47} - 286720 q^{48} + 235298 q^{49} - 1700940 q^{51} + 409472 q^{52} + 145716 q^{53} + 1544480 q^{54} + 351232 q^{56} + 560596 q^{57} - 1276608 q^{58} - 4183662 q^{59} - 280658 q^{61} + 1148896 q^{62} - 690802 q^{63} + 524288 q^{64} - 4927104 q^{66} - 5671648 q^{67} + 2462208 q^{68} + 2155104 q^{69} - 619272 q^{71} - 1031168 q^{72} - 3939628 q^{73} - 2174656 q^{74} - 2774912 q^{76} + 1173060 q^{77} - 4162816 q^{78} + 4656616 q^{79} + 7353742 q^{81} - 518784 q^{82} - 1235850 q^{83} + 1536640 q^{84} + 12223712 q^{86} - 15012732 q^{87} + 1751040 q^{88} - 17241420 q^{89} - 2194514 q^{91} - 5755392 q^{92} + 7365592 q^{93} + 3883488 q^{94} + 2293760 q^{96} + 740936 q^{97} - 1882384 q^{98} - 38177100 q^{99}+O(q^{100}) \) 2 * q - 16 * q^2 - 70 * q^3 + 128 * q^4 + 560 * q^6 - 686 * q^7 - 1024 * q^8 + 2014 * q^9 - 3420 * q^11 - 4480 * q^12 + 6398 * q^13 + 5488 * q^14 + 8192 * q^16 + 38472 * q^17 - 16112 * q^18 - 43358 * q^19 + 24010 * q^21 + 27360 * q^22 - 89928 * q^23 + 35840 * q^24 - 51184 * q^26 - 193060 * q^27 - 43904 * q^28 + 159576 * q^29 - 143612 * q^31 - 65536 * q^32 + 615888 * q^33 - 307776 * q^34 + 128896 * q^36 + 271832 * q^37 + 346864 * q^38 + 520352 * q^39 + 64848 * q^41 - 192080 * q^42 - 1527964 * q^43 - 218880 * q^44 + 719424 * q^46 - 485436 * q^47 - 286720 * q^48 + 235298 * q^49 - 1700940 * q^51 + 409472 * q^52 + 145716 * q^53 + 1544480 * q^54 + 351232 * q^56 + 560596 * q^57 - 1276608 * q^58 - 4183662 * q^59 - 280658 * q^61 + 1148896 * q^62 - 690802 * q^63 + 524288 * q^64 - 4927104 * q^66 - 5671648 * q^67 + 2462208 * q^68 + 2155104 * q^69 - 619272 * q^71 - 1031168 * q^72 - 3939628 * q^73 - 2174656 * q^74 - 2774912 * q^76 + 1173060 * q^77 - 4162816 * q^78 + 4656616 * q^79 + 7353742 * q^81 - 518784 * q^82 - 1235850 * q^83 + 1536640 * q^84 + 12223712 * q^86 - 15012732 * q^87 + 1751040 * q^88 - 17241420 * q^89 - 2194514 * q^91 - 5755392 * q^92 + 7365592 * q^93 + 3883488 * q^94 + 2293760 * q^96 + 740936 * q^97 - 1882384 * q^98 - 38177100 * q^99

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