LMFDB - Embedded newform 49.8.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [49,8,Mod(1,49)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("49.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 49 = 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	49.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-6,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$15.3068662487$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 102x^{2} + 240x + 900 $ x^4 - x^3 - 102x^2 + 240x + 900
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.14290$ of defining polynomial
Character	$\chi$	$=$	49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-7.38077 q^{2} +74.2434 q^{3} -73.5243 q^{4} -290.819 q^{5} -547.974 q^{6} +1487.40 q^{8} +3325.09 q^{9} +2146.47 q^{10} +3192.88 q^{11} -5458.70 q^{12} +5186.70 q^{13} -21591.4 q^{15} -1567.07 q^{16} -2449.11 q^{17} -24541.7 q^{18} +32318.2 q^{19} +21382.2 q^{20} -23565.9 q^{22} +62941.4 q^{23} +110430. q^{24} +6450.60 q^{25} -38281.8 q^{26} +84495.7 q^{27} +141752. q^{29} +159361. q^{30} +282376. q^{31} -178821. q^{32} +237051. q^{33} +18076.3 q^{34} -244475. q^{36} -131368. q^{37} -238533. q^{38} +385078. q^{39} -432565. q^{40} -65254.0 q^{41} -257876. q^{43} -234755. q^{44} -966999. q^{45} -464556. q^{46} +127457. q^{47} -116345. q^{48} -47610.4 q^{50} -181830. q^{51} -381348. q^{52} +790646. q^{53} -623643. q^{54} -928551. q^{55} +2.39942e6 q^{57} -1.04624e6 q^{58} +1.21435e6 q^{59} +1.58749e6 q^{60} +762084. q^{61} -2.08415e6 q^{62} +1.52042e6 q^{64} -1.50839e6 q^{65} -1.74962e6 q^{66} -3.44286e6 q^{67} +180069. q^{68} +4.67299e6 q^{69} -3.99576e6 q^{71} +4.94575e6 q^{72} -769338. q^{73} +969594. q^{74} +478915. q^{75} -2.37617e6 q^{76} -2.84217e6 q^{78} -5.83380e6 q^{79} +455734. q^{80} -998721. q^{81} +481625. q^{82} +1.39747e6 q^{83} +712248. q^{85} +1.90332e6 q^{86} +1.05241e7 q^{87} +4.74911e6 q^{88} +1.61702e6 q^{89} +7.13719e6 q^{90} -4.62772e6 q^{92} +2.09645e7 q^{93} -940733. q^{94} -9.39875e6 q^{95} -1.32763e7 q^{96} -5.68544e6 q^{97} +1.06166e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{2} + 28 q^{3} + 348 q^{4} + 252 q^{5} + 1022 q^{6} - 984 q^{8} + 2008 q^{9} + 4774 q^{10} - 3972 q^{11} - 5404 q^{12} - 1176 q^{13} - 16556 q^{15} + 57264 q^{16} + 56364 q^{17} - 35908 q^{18}+ \cdots + 15213976 q^{99}+O(q^{100}) $ 4 * q - 6 * q^2 + 28 * q^3 + 348 * q^4 + 252 * q^5 + 1022 * q^6 - 984 * q^8 + 2008 * q^9 + 4774 * q^10 - 3972 * q^11 - 5404 * q^12 - 1176 * q^13 - 16556 * q^15 + 57264 * q^16 + 56364 * q^17 - 35908 * q^18 + 41748 * q^19 + 162372 * q^20 - 152954 * q^22 + 131748 * q^23 + 389592 * q^24 - 68016 * q^25 + 113652 * q^26 + 189028 * q^27 + 34056 * q^29 + 605378 * q^30 + 401212 * q^31 - 453408 * q^32 + 24052 * q^33 + 82110 * q^34 - 360088 * q^36 - 5396 * q^37 + 31794 * q^38 + 364840 * q^39 - 443688 * q^40 + 410424 * q^41 + 46544 * q^43 - 1465836 * q^44 - 1209656 * q^45 + 1379202 * q^46 + 1470084 * q^47 - 2018800 * q^48 + 1395528 * q^50 - 1724796 * q^51 - 5269768 * q^52 - 642372 * q^53 - 4567318 * q^54 - 3063340 * q^55 + 3359068 * q^57 - 1972220 * q^58 + 752220 * q^59 + 2123324 * q^60 + 1325772 * q^61 - 3016314 * q^62 - 865856 * q^64 - 4868808 * q^65 - 6013294 * q^66 - 290916 * q^67 + 2453556 * q^68 + 4915260 * q^69 + 3377760 * q^71 + 4557744 * q^72 + 6706588 * q^73 + 3616878 * q^74 + 7223888 * q^75 + 2923004 * q^76 - 16288132 * q^78 + 3946244 * q^79 + 10695888 * q^80 - 11731292 * q^81 - 17563252 * q^82 + 9542064 * q^83 + 7006068 * q^85 + 23237832 * q^86 + 17836840 * q^87 - 8043336 * q^88 + 16165212 * q^89 + 3415748 * q^90 - 19442628 * q^92 + 12552852 * q^93 + 5720442 * q^94 - 3268452 * q^95 - 8448608 * q^96 + 1533112 * q^97 + 15213976 * q^99

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 $

$2$	−7.38077	−0.652374	−0.326187	−	0.945305i	$-0.605764\pi$
$2$	−7.38077	−0.652374	−0.326187	+	0.945305i	$0.605764\pi$
$3$	74.2434	1.58757	0.793787	−	0.608196i	$-0.208107\pi$
$3$	74.2434	1.58757	0.793787	+	0.608196i	$0.208107\pi$
$4$	−73.5243	−0.574408
$5$	−290.819	−1.04047	−0.520233	−	0.854025i	$-0.674155\pi$
$5$	−290.819	−1.04047	−0.520233	+	0.854025i	$0.674155\pi$
$6$	−547.974	−1.03569
$7$	0	0
$7$	0	0
$8$	1487.40	1.02710
$9$	3325.09	1.52039
$10$	2146.47	0.678772
$11$	3192.88	0.723284	0.361642	−	0.932317i	$-0.382216\pi$
$11$	3192.88	0.723284	0.361642	+	0.932317i	$0.382216\pi$
$12$	−5458.70	−0.911915
$13$	5186.70	0.654771	0.327385	−	0.944891i	$-0.393832\pi$
$13$	5186.70	0.654771	0.327385	+	0.944891i	$0.393832\pi$
$14$	0	0
$15$	−21591.4	−1.65181
$16$	−1567.07	−0.0956464
$17$	−2449.11	−0.120903	−0.0604515	−	0.998171i	$-0.519254\pi$
$17$	−2449.11	−0.120903	−0.0604515	+	0.998171i	$0.519254\pi$
$18$	−24541.7	−0.991861
$19$	32318.2	1.08096	0.540480	−	0.841357i	$-0.318243\pi$
$19$	32318.2	1.08096	0.540480	+	0.841357i	$0.318243\pi$
$20$	21382.2	0.597652
$21$	0	0
$22$	−23565.9	−0.471851
$23$	62941.4	1.07867	0.539335	−	0.842091i	$-0.318675\pi$
$23$	62941.4	1.07867	0.539335	+	0.842091i	$0.318675\pi$
$24$	110430.	1.63060
$25$	6450.60	0.0825677
$26$	−38281.8	−0.427155
$27$	84495.7	0.826154
$28$	0	0
$29$	141752.	1.07928	0.539642	−	0.841895i	$-0.318560\pi$
$29$	141752.	1.07928	0.539642	+	0.841895i	$0.318560\pi$
$30$	159361.	1.07760
$31$	282376.	1.70240	0.851200	−	0.524842i	$-0.175876\pi$
$31$	282376.	1.70240	0.851200	+	0.524842i	$0.175876\pi$
$32$	−178821.	−0.964706
$33$	237051.	1.14827
$34$	18076.3	0.0788740
$35$	0	0
$36$	−244475.	−0.873324
$37$	−131368.	−0.426366	−0.213183	−	0.977012i	$-0.568383\pi$
$37$	−131368.	−0.426366	−0.213183	+	0.977012i	$0.568383\pi$
$38$	−238533.	−0.705190
$39$	385078.	1.03950
$40$	−432565.	−1.06866
$41$	−65254.0	−0.147864	−0.0739322	−	0.997263i	$-0.523555\pi$
$41$	−65254.0	−0.147864	−0.0739322	+	0.997263i	$0.523555\pi$
$42$	0	0
$43$	−257876.	−0.494620	−0.247310	−	0.968936i	$-0.579547\pi$
$43$	−257876.	−0.494620	−0.247310	+	0.968936i	$0.579547\pi$
$44$	−234755.	−0.415460
$45$	−966999.	−1.58191
$46$	−464556.	−0.703697
$47$	127457.	0.179070	0.0895349	−	0.995984i	$-0.471462\pi$
$47$	127457.	0.179070	0.0895349	+	0.995984i	$0.471462\pi$
$48$	−116345.	−0.151846
$49$	0	0
$50$	−47610.4	−0.0538650
$51$	−181830.	−0.191942
$52$	−381348.	−0.376106
$53$	790646.	0.729485	0.364743	−	0.931108i	$-0.381157\pi$
$53$	790646.	0.729485	0.364743	+	0.931108i	$0.381157\pi$
$54$	−623643.	−0.538961
$55$	−928551.	−0.752552
$56$	0	0
$57$	2.39942e6	1.71610
$58$	−1.04624e6	−0.704096
$59$	1.21435e6	0.769769	0.384884	−	0.922965i	$-0.374241\pi$
$59$	1.21435e6	0.769769	0.384884	+	0.922965i	$0.374241\pi$
$60$	1.58749e6	0.948816
$61$	762084.	0.429881	0.214941	−	0.976627i	$-0.431044\pi$
$61$	762084.	0.429881	0.214941	+	0.976627i	$0.431044\pi$
$62$	−2.08415e6	−1.11060
$63$	0	0
$64$	1.52042e6	0.724995
$65$	−1.50839e6	−0.681266
$66$	−1.74962e6	−0.749099
$67$	−3.44286e6	−1.39848	−0.699241	−	0.714886i	$-0.746479\pi$
$67$	−3.44286e6	−1.39848	−0.699241	+	0.714886i	$0.746479\pi$
$68$	180069.	0.0694477
$69$	4.67299e6	1.71247
$70$	0	0
$71$	−3.99576e6	−1.32494	−0.662469	−	0.749089i	$-0.730491\pi$
$71$	−3.99576e6	−1.32494	−0.662469	+	0.749089i	$0.730491\pi$
$72$	4.94575e6	1.56159
$73$	−769338.	−0.231466	−0.115733	−	0.993280i	$-0.536922\pi$
$73$	−769338.	−0.231466	−0.115733	+	0.993280i	$0.536922\pi$
$74$	969594.	0.278150
$75$	478915.	0.131082
$76$	−2.37617e6	−0.620913
$77$	0	0
$78$	−2.84217e6	−0.678140
$79$	−5.83380e6	−1.33124	−0.665620	−	0.746291i	$-0.731833\pi$
$79$	−5.83380e6	−1.33124	−0.665620	+	0.746291i	$0.731833\pi$
$80$	455734.	0.0995168
$81$	−998721.	−0.208808
$82$	481625.	0.0964629
$83$	1.39747e6	0.268267	0.134134	−	0.990963i	$-0.457175\pi$
$83$	1.39747e6	0.268267	0.134134	+	0.990963i	$0.457175\pi$
$84$	0	0
$85$	712248.	0.125795
$86$	1.90332e6	0.322677
$87$	1.05241e7	1.71344
$88$	4.74911e6	0.742887
$89$	1.61702e6	0.243136	0.121568	−	0.992583i	$-0.461208\pi$
$89$	1.61702e6	0.243136	0.121568	+	0.992583i	$0.461208\pi$
$90$	7.13719e6	1.03200
$91$	0	0
$92$	−4.62772e6	−0.619598
$93$	2.09645e7	2.70268
$94$	−940733.	−0.116820
$95$	−9.39875e6	−1.12470
$96$	−1.32763e7	−1.53154
$97$	−5.68544e6	−0.632504	−0.316252	−	0.948675i	$-0.602424\pi$
$97$	−5.68544e6	−0.632504	−0.316252	+	0.948675i	$0.602424\pi$
$98$	0	0
$99$	1.06166e7	1.09967
$100$	−474276.	−0.0474276
$101$	1.00880e7	0.974277	0.487138	−	0.873325i	$-0.338041\pi$
$101$	1.00880e7	0.974277	0.487138	+	0.873325i	$0.338041\pi$
$102$	1.34205e6	0.125218
$103$	9.33755e6	0.841981	0.420991	−	0.907065i	$-0.361683\pi$
$103$	9.33755e6	0.841981	0.420991	+	0.907065i	$0.361683\pi$
$104$	7.71471e6	0.672517
$105$	0	0
$106$	−5.83557e6	−0.475897
$107$	−1.44049e7	−1.13676	−0.568379	−	0.822767i	$-0.692429\pi$
$107$	−1.44049e7	−1.13676	−0.568379	+	0.822767i	$0.692429\pi$
$108$	−6.21248e6	−0.474550
$109$	8.55998e6	0.633111	0.316556	−	0.948574i	$-0.397474\pi$
$109$	8.55998e6	0.633111	0.316556	+	0.948574i	$0.397474\pi$
$110$	6.85342e6	0.490945
$111$	−9.75319e6	−0.676887
$112$	0	0
$113$	1.15607e7	0.753722	0.376861	−	0.926270i	$-0.377003\pi$
$113$	1.15607e7	0.753722	0.376861	+	0.926270i	$0.377003\pi$
$114$	−1.77095e7	−1.11954
$115$	−1.83045e7	−1.12232
$116$	−1.04222e7	−0.619950
$117$	1.72462e7	0.995506
$118$	−8.96280e6	−0.502177
$119$	0	0
$120$	−3.21151e7	−1.69658
$121$	−9.29266e6	−0.476860
$122$	−5.62477e6	−0.280443
$123$	−4.84468e6	−0.234746
$124$	−2.07615e7	−0.977873
$125$	2.08443e7	0.954556
$126$	0	0
$127$	2.65100e7	1.14841	0.574204	−	0.818712i	$-0.305312\pi$
$127$	2.65100e7	1.14841	0.574204	+	0.818712i	$0.305312\pi$
$128$	1.16673e7	0.491738
$129$	−1.91456e7	−0.785245
$130$	1.11331e7	0.444440
$131$	4.54574e6	0.176667	0.0883333	−	0.996091i	$-0.471846\pi$
$131$	4.54574e6	0.176667	0.0883333	+	0.996091i	$0.471846\pi$
$132$	−1.74290e7	−0.659574
$133$	0	0
$134$	2.54109e7	0.912334
$135$	−2.45729e7	−0.859585
$136$	−3.64282e6	−0.124180
$137$	4.93543e7	1.63984	0.819922	−	0.572475i	$-0.194017\pi$
$137$	4.93543e7	1.63984	0.819922	+	0.572475i	$0.194017\pi$
$138$	−3.44902e7	−1.11717
$139$	−4.41231e7	−1.39352	−0.696761	−	0.717303i	$-0.745376\pi$
$139$	−4.41231e7	−1.39352	−0.696761	+	0.717303i	$0.745376\pi$
$140$	0	0
$141$	9.46288e6	0.284287
$142$	2.94918e7	0.864355
$143$	1.65605e7	0.473585
$144$	−5.21065e6	−0.145420
$145$	−4.12241e7	−1.12296
$146$	5.67830e6	0.151002
$147$	0	0
$148$	9.65872e6	0.244908
$149$	2.22735e7	0.551616	0.275808	−	0.961213i	$-0.411055\pi$
$149$	2.22735e7	0.551616	0.275808	+	0.961213i	$0.411055\pi$
$150$	−3.53476e6	−0.0855146
$151$	−6.95161e7	−1.64311	−0.821554	−	0.570130i	$-0.806893\pi$
$151$	−6.95161e7	−1.64311	−0.821554	+	0.570130i	$0.806893\pi$
$152$	4.80702e7	1.11026
$153$	−8.14351e6	−0.183820
$154$	0	0
$155$	−8.21202e7	−1.77129
$156$	−2.83126e7	−0.597096
$157$	−1.22922e7	−0.253501	−0.126751	−	0.991935i	$-0.540455\pi$
$157$	−1.22922e7	−0.253501	−0.126751	+	0.991935i	$0.540455\pi$
$158$	4.30579e7	0.868466
$159$	5.87003e7	1.15811
$160$	5.20047e7	1.00374
$161$	0	0
$162$	7.37133e6	0.136221
$163$	−1.76180e7	−0.318640	−0.159320	−	0.987227i	$-0.550930\pi$
$163$	−1.76180e7	−0.318640	−0.159320	+	0.987227i	$0.550930\pi$
$164$	4.79776e6	0.0849346
$165$	−6.89388e7	−1.19473
$166$	−1.03144e7	−0.175011
$167$	1.13558e8	1.88673	0.943367	−	0.331751i	$-0.107639\pi$
$167$	1.13558e8	1.88673	0.943367	+	0.331751i	$0.107639\pi$
$168$	0	0
$169$	−3.58467e7	−0.571275
$170$	−5.25693e6	−0.0820656
$171$	1.07461e8	1.64348
$172$	1.89602e7	0.284114
$173$	−2.16280e7	−0.317582	−0.158791	−	0.987312i	$-0.550760\pi$
$173$	−2.16280e7	−0.317582	−0.158791	+	0.987312i	$0.550760\pi$
$174$	−7.76763e7	−1.11780
$175$	0	0
$176$	−5.00348e6	−0.0691795
$177$	9.01572e7	1.22206
$178$	−1.19348e7	−0.158616
$179$	1.08286e8	1.41120	0.705599	−	0.708611i	$-0.250678\pi$
$179$	1.08286e8	1.41120	0.705599	+	0.708611i	$0.250678\pi$
$180$	7.10979e7	0.908663
$181$	6.56731e7	0.823214	0.411607	−	0.911361i	$-0.364968\pi$
$181$	6.56731e7	0.823214	0.411607	+	0.911361i	$0.364968\pi$
$182$	0	0
$183$	5.65798e7	0.682468
$184$	9.36193e7	1.10791
$185$	3.82042e7	0.443619
$186$	−1.54734e8	−1.76316
$187$	−7.81973e6	−0.0874472
$188$	−9.37121e6	−0.102859
$189$	0	0
$190$	6.93699e7	0.733726
$191$	−1.44600e7	−0.150159	−0.0750795	−	0.997178i	$-0.523921\pi$
$191$	−1.44600e7	−0.150159	−0.0750795	+	0.997178i	$0.523921\pi$
$192$	1.12882e8	1.15098
$193$	−6.23712e7	−0.624501	−0.312251	−	0.950000i	$-0.601083\pi$
$193$	−6.23712e7	−0.624501	−0.312251	+	0.950000i	$0.601083\pi$
$194$	4.19629e7	0.412629
$195$	−1.11988e8	−1.08156
$196$	0	0
$197$	−1.35525e7	−0.126296	−0.0631478	−	0.998004i	$-0.520114\pi$
$197$	−1.35525e7	−0.126296	−0.0631478	+	0.998004i	$0.520114\pi$
$198$	−7.83588e7	−0.717397
$199$	−2.97247e7	−0.267381	−0.133691	−	0.991023i	$-0.542683\pi$
$199$	−2.97247e7	−0.267381	−0.133691	+	0.991023i	$0.542683\pi$
$200$	9.59464e6	0.0848055
$201$	−2.55610e8	−2.22019
$202$	−7.44575e7	−0.635593
$203$	0	0
$204$	1.33690e7	0.110253
$205$	1.89771e7	0.153848
$206$	−6.89182e7	−0.549287
$207$	2.09286e8	1.64000
$208$	−8.12793e6	−0.0626265
$209$	1.03188e8	0.781841
$210$	0	0
$211$	−1.45624e8	−1.06719	−0.533597	−	0.845739i	$-0.679160\pi$
$211$	−1.45624e8	−1.06719	−0.533597	+	0.845739i	$0.679160\pi$
$212$	−5.81317e7	−0.419022
$213$	−2.96659e8	−2.10344
$214$	1.06319e8	0.741591
$215$	7.49952e7	0.514635
$216$	1.25679e8	0.848545
$217$	0	0
$218$	−6.31792e7	−0.413025
$219$	−5.71183e7	−0.367469
$220$	6.82710e7	0.432272
$221$	−1.27028e7	−0.0791638
$222$	7.19860e7	0.441583
$223$	−2.01327e8	−1.21573	−0.607863	−	0.794042i	$-0.707973\pi$
$223$	−2.01327e8	−1.21573	−0.607863	+	0.794042i	$0.707973\pi$
$224$	0	0
$225$	2.14488e7	0.125535
$226$	−8.53271e7	−0.491708
$227$	−3.07709e8	−1.74602	−0.873010	−	0.487703i	$-0.837835\pi$
$227$	−3.07709e8	−1.74602	−0.873010	+	0.487703i	$0.837835\pi$
$228$	−1.76415e8	−0.985744
$229$	1.85267e8	1.01947	0.509734	−	0.860332i	$-0.329744\pi$
$229$	1.85267e8	1.01947	0.509734	+	0.860332i	$0.329744\pi$
$230$	1.35102e8	0.732172
$231$	0	0
$232$	2.10842e8	1.10854
$233$	−2.49102e8	−1.29012	−0.645062	−	0.764130i	$-0.723169\pi$
$233$	−2.49102e8	−1.29012	−0.645062	+	0.764130i	$0.723169\pi$
$234$	−1.27290e8	−0.649442
$235$	−3.70670e7	−0.186316
$236$	−8.92839e7	−0.442162
$237$	−4.33121e8	−2.11344
$238$	0	0
$239$	3.36485e6	0.0159431	0.00797156	−	0.999968i	$-0.497463\pi$
$239$	3.36485e6	0.0159431	0.00797156	+	0.999968i	$0.497463\pi$
$240$	3.38352e7	0.157990
$241$	2.18549e8	1.00575	0.502873	−	0.864360i	$-0.332276\pi$
$241$	2.18549e8	1.00575	0.502873	+	0.864360i	$0.332276\pi$
$242$	6.85870e7	0.311091
$243$	−2.58941e8	−1.15765
$244$	−5.60317e7	−0.246928
$245$	0	0
$246$	3.57575e7	0.153142
$247$	1.67625e8	0.707781
$248$	4.20007e8	1.74854
$249$	1.03753e8	0.425894
$250$	−1.53847e8	−0.622728
$251$	3.10664e8	1.24003	0.620017	−	0.784588i	$-0.287126\pi$
$251$	3.10664e8	1.24003	0.620017	+	0.784588i	$0.287126\pi$
$252$	0	0
$253$	2.00965e8	0.780185
$254$	−1.95664e8	−0.749191
$255$	5.28797e7	0.199709
$256$	−2.80728e8	−1.04579
$257$	2.96149e8	1.08829	0.544144	−	0.838992i	$-0.316854\pi$
$257$	2.96149e8	1.08829	0.544144	+	0.838992i	$0.316854\pi$
$258$	1.41309e8	0.512273
$259$	0	0
$260$	1.10903e8	0.391325
$261$	4.71337e8	1.64093
$262$	−3.35510e7	−0.115253
$263$	2.59447e8	0.879433	0.439717	−	0.898137i	$-0.355079\pi$
$263$	2.59447e8	0.879433	0.439717	+	0.898137i	$0.355079\pi$
$264$	3.52590e8	1.17939
$265$	−2.29935e8	−0.759004
$266$	0	0
$267$	1.20053e8	0.385997
$268$	2.53134e8	0.803300
$269$	1.46457e8	0.458751	0.229376	−	0.973338i	$-0.426332\pi$
$269$	1.46457e8	0.458751	0.229376	+	0.973338i	$0.426332\pi$
$270$	1.81367e8	0.560771
$271$	−2.51571e8	−0.767836	−0.383918	−	0.923367i	$-0.625425\pi$
$271$	−2.51571e8	−0.767836	−0.383918	+	0.923367i	$0.625425\pi$
$272$	3.83793e6	0.0115639
$273$	0	0
$274$	−3.64272e8	−1.06979
$275$	2.05960e7	0.0597198
$276$	−3.43578e8	−0.983657
$277$	−4.65305e8	−1.31540	−0.657701	−	0.753279i	$-0.728471\pi$
$277$	−4.65305e8	−1.31540	−0.657701	+	0.753279i	$0.728471\pi$
$278$	3.25662e8	0.909097
$279$	9.38924e8	2.58831
$280$	0	0
$281$	−5.58490e8	−1.50156	−0.750781	−	0.660551i	$-0.770323\pi$
$281$	−5.58490e8	−1.50156	−0.750781	+	0.660551i	$0.770323\pi$
$282$	−6.98433e7	−0.185461
$283$	7.52795e7	0.197435	0.0987176	−	0.995115i	$-0.468526\pi$
$283$	7.52795e7	0.197435	0.0987176	+	0.995115i	$0.468526\pi$
$284$	2.93786e8	0.761056
$285$	−6.97795e8	−1.78555
$286$	−1.22229e8	−0.308955
$287$	0	0
$288$	−5.94597e8	−1.46673
$289$	−4.04341e8	−0.985382
$290$	3.04265e8	0.732588
$291$	−4.22107e8	−1.00415
$292$	5.65650e7	0.132956
$293$	−5.90696e8	−1.37192	−0.685958	−	0.727641i	$-0.740617\pi$
$293$	−5.90696e8	−1.37192	−0.685958	+	0.727641i	$0.740617\pi$
$294$	0	0
$295$	−3.53155e8	−0.800918
$296$	−1.95397e8	−0.437922
$297$	2.69785e8	0.597544
$298$	−1.64396e8	−0.359860
$299$	3.26458e8	0.706282
$300$	−3.52119e7	−0.0752947
$301$	0	0
$302$	5.13082e8	1.07192
$303$	7.48971e8	1.54674
$304$	−5.06449e7	−0.103390
$305$	−2.21629e8	−0.447277
$306$	6.01054e7	0.119919
$307$	6.49771e8	1.28167	0.640835	−	0.767679i	$-0.278588\pi$
$307$	6.49771e8	1.28167	0.640835	+	0.767679i	$0.278588\pi$
$308$	0	0
$309$	6.93252e8	1.33671
$310$	6.06110e8	1.15554
$311$	4.48581e7	0.0845629	0.0422815	−	0.999106i	$-0.486537\pi$
$311$	4.48581e7	0.0845629	0.0422815	+	0.999106i	$0.486537\pi$
$312$	5.72767e8	1.06767
$313$	5.72019e6	0.0105440	0.00527200	−	0.999986i	$-0.498322\pi$
$313$	5.72019e6	0.0105440	0.00527200	+	0.999986i	$0.498322\pi$
$314$	9.07257e7	0.165378
$315$	0	0
$316$	4.28926e8	0.764676
$317$	−2.79676e8	−0.493115	−0.246558	−	0.969128i	$-0.579299\pi$
$317$	−2.79676e8	−0.493115	−0.246558	+	0.969128i	$0.579299\pi$
$318$	−4.33253e8	−0.755521
$319$	4.52597e8	0.780629
$320$	−4.42168e8	−0.754332
$321$	−1.06947e9	−1.80469
$322$	0	0
$323$	−7.91509e7	−0.130691
$324$	7.34303e7	0.119941
$325$	3.34573e7	0.0540629
$326$	1.30034e8	0.207872
$327$	6.35523e8	1.00511
$328$	−9.70591e7	−0.151872
$329$	0	0
$330$	5.08821e8	0.779411
$331$	−8.77705e8	−1.33030	−0.665152	−	0.746708i	$-0.731633\pi$
$331$	−8.77705e8	−1.33030	−0.665152	+	0.746708i	$0.731633\pi$
$332$	−1.02748e8	−0.154095
$333$	−4.36809e8	−0.648242
$334$	−8.38146e8	−1.23086
$335$	1.00125e9	1.45507
$336$	0	0
$337$	1.05320e8	0.149902	0.0749508	−	0.997187i	$-0.476120\pi$
$337$	1.05320e8	0.149902	0.0749508	+	0.997187i	$0.476120\pi$
$338$	2.64576e8	0.372685
$339$	8.58309e8	1.19659
$340$	−5.23675e7	−0.0722580
$341$	9.01593e8	1.23132
$342$	−7.93144e8	−1.07216
$343$	0	0
$344$	−3.83566e8	−0.508025
$345$	−1.35899e9	−1.78176
$346$	1.59631e8	0.207182
$347$	3.59025e8	0.461287	0.230644	−	0.973038i	$-0.425917\pi$
$347$	3.59025e8	0.461287	0.230644	+	0.973038i	$0.425917\pi$
$348$	−7.73780e8	−0.984216
$349$	−1.99439e8	−0.251142	−0.125571	−	0.992085i	$-0.540076\pi$
$349$	−1.99439e8	−0.251142	−0.125571	+	0.992085i	$0.540076\pi$
$350$	0	0
$351$	4.38254e8	0.540942
$352$	−5.70956e8	−0.697756
$353$	−1.64620e9	−1.99192	−0.995958	−	0.0898223i	$-0.971370\pi$
$353$	−1.64620e9	−1.99192	−0.995958	+	0.0898223i	$0.971370\pi$
$354$	−6.65429e8	−0.797243
$355$	1.16204e9	1.37855
$356$	−1.18890e8	−0.139660
$357$	0	0
$358$	−7.99236e8	−0.920629
$359$	2.41422e7	0.0275388	0.0137694	−	0.999905i	$-0.495617\pi$
$359$	2.41422e7	0.0275388	0.0137694	+	0.999905i	$0.495617\pi$
$360$	−1.43832e9	−1.62479
$361$	1.50595e8	0.168475
$362$	−4.84718e8	−0.537043
$363$	−6.89919e8	−0.757051
$364$	0	0
$365$	2.23738e8	0.240832
$366$	−4.17602e8	−0.445224
$367$	−1.55386e9	−1.64089	−0.820447	−	0.571723i	$-0.806275\pi$
$367$	−1.55386e9	−1.64089	−0.820447	+	0.571723i	$0.806275\pi$
$368$	−9.86336e7	−0.103171
$369$	−2.16975e8	−0.224811
$370$	−2.81976e8	−0.289405
$371$	0	0
$372$	−1.54140e9	−1.55244
$373$	1.52310e9	1.51967	0.759833	−	0.650118i	$-0.225281\pi$
$373$	1.52310e9	1.51967	0.759833	+	0.650118i	$0.225281\pi$
$374$	5.77156e7	0.0570483
$375$	1.54755e9	1.51543
$376$	1.89581e8	0.183923
$377$	7.35224e8	0.706684
$378$	0	0
$379$	9.32142e8	0.879518	0.439759	−	0.898116i	$-0.355064\pi$
$379$	9.32142e8	0.879518	0.439759	+	0.898116i	$0.355064\pi$
$380$	6.91036e8	0.646038
$381$	1.96819e9	1.82318
$382$	1.06726e8	0.0979598
$383$	−1.75474e9	−1.59594	−0.797970	−	0.602697i	$-0.794093\pi$
$383$	−1.75474e9	−1.59594	−0.797970	+	0.602697i	$0.794093\pi$
$384$	8.66217e8	0.780670
$385$	0	0
$386$	4.60347e8	0.407408
$387$	−8.57461e8	−0.752014
$388$	4.18018e8	0.363316
$389$	−9.20669e8	−0.793012	−0.396506	−	0.918032i	$-0.629777\pi$
$389$	−9.20669e8	−0.793012	−0.396506	+	0.918032i	$0.629777\pi$
$390$	8.26558e8	0.705581
$391$	−1.54151e8	−0.130415
$392$	0	0
$393$	3.37491e8	0.280471
$394$	1.00028e8	0.0823919
$395$	1.69658e9	1.38511
$396$	−7.80580e8	−0.631661
$397$	−9.35556e8	−0.750418	−0.375209	−	0.926940i	$-0.622429\pi$
$397$	−9.35556e8	−0.750418	−0.375209	+	0.926940i	$0.622429\pi$
$398$	2.19391e8	0.174433
$399$	0	0
$400$	−1.01085e7	−0.00789730
$401$	1.94892e9	1.50934	0.754671	−	0.656103i	$-0.227796\pi$
$401$	1.94892e9	1.50934	0.754671	+	0.656103i	$0.227796\pi$
$402$	1.88659e9	1.44840
$403$	1.46460e9	1.11468
$404$	−7.41716e8	−0.559633
$405$	2.90447e8	0.217257
$406$	0	0
$407$	−4.19442e8	−0.308384
$408$	−2.70455e8	−0.197145
$409$	−3.00520e8	−0.217191	−0.108596	−	0.994086i	$-0.534635\pi$
$409$	−3.00520e8	−0.217191	−0.108596	+	0.994086i	$0.534635\pi$
$410$	−1.40066e8	−0.100366
$411$	3.66423e9	2.60337
$412$	−6.86536e8	−0.483641
$413$	0	0
$414$	−1.54469e9	−1.06989
$415$	−4.06409e8	−0.279123
$416$	−9.27493e8	−0.631661
$417$	−3.27585e9	−2.21232
$418$	−7.61609e8	−0.510053
$419$	−7.41204e8	−0.492253	−0.246127	−	0.969238i	$-0.579158\pi$
$419$	−7.41204e8	−0.492253	−0.246127	+	0.969238i	$0.579158\pi$
$420$	0	0
$421$	2.25657e9	1.47388	0.736940	−	0.675958i	$-0.236270\pi$
$421$	2.25657e9	1.47388	0.736940	+	0.675958i	$0.236270\pi$
$422$	1.07481e9	0.696210
$423$	4.23807e8	0.272256
$424$	1.17601e9	0.749256
$425$	−1.57982e7	−0.00998268
$426$	2.18957e9	1.37223
$427$	0	0
$428$	1.05911e9	0.652964
$429$	1.22951e9	0.751851
$430$	−5.53522e8	−0.335734
$431$	2.19006e7	0.0131760	0.00658802	−	0.999978i	$-0.497903\pi$
$431$	2.19006e7	0.0131760	0.00658802	+	0.999978i	$0.497903\pi$
$432$	−1.32411e8	−0.0790187
$433$	1.42535e9	0.843749	0.421875	−	0.906654i	$-0.361372\pi$
$433$	1.42535e9	0.843749	0.421875	+	0.906654i	$0.361372\pi$
$434$	0	0
$435$	−3.06062e9	−1.78278
$436$	−6.29367e8	−0.363665
$437$	2.03415e9	1.16600
$438$	4.21577e8	0.239727
$439$	2.51236e9	1.41728	0.708640	−	0.705571i	$-0.249309\pi$
$439$	2.51236e9	1.41728	0.708640	+	0.705571i	$0.249309\pi$
$440$	−1.38113e9	−0.772948
$441$	0	0
$442$	9.37564e7	0.0516444
$443$	−1.47011e9	−0.803408	−0.401704	−	0.915770i	$-0.631582\pi$
$443$	−1.47011e9	−0.803408	−0.401704	+	0.915770i	$0.631582\pi$
$444$	7.17096e8	0.388810
$445$	−4.70259e8	−0.252975
$446$	1.48595e9	0.793108
$447$	1.65366e9	0.875731
$448$	0	0
$449$	−2.83997e9	−1.48064	−0.740322	−	0.672252i	$-0.765327\pi$
$449$	−2.83997e9	−1.48064	−0.740322	+	0.672252i	$0.765327\pi$
$450$	−1.58309e8	−0.0818957
$451$	−2.08349e8	−0.106948
$452$	−8.49995e8	−0.432944
$453$	−5.16112e9	−2.60855
$454$	2.27113e9	1.13906
$455$	0	0
$456$	3.56890e9	1.76261
$457$	−4.12022e8	−0.201936	−0.100968	−	0.994890i	$-0.532194\pi$
$457$	−4.12022e8	−0.201936	−0.100968	+	0.994890i	$0.532194\pi$
$458$	−1.36741e9	−0.665075
$459$	−2.06939e8	−0.0998846
$460$	1.34583e9	0.644670
$461$	−4.14604e9	−1.97097	−0.985484	−	0.169766i	$-0.945699\pi$
$461$	−4.14604e9	−1.97097	−0.985484	+	0.169766i	$0.945699\pi$
$462$	0	0
$463$	−2.49603e9	−1.16873	−0.584367	−	0.811490i	$-0.698657\pi$
$463$	−2.49603e9	−1.16873	−0.584367	+	0.811490i	$0.698657\pi$
$464$	−2.22135e8	−0.103230
$465$	−6.09689e9	−2.81205
$466$	1.83856e9	0.841644
$467$	−2.73423e9	−1.24230	−0.621148	−	0.783693i	$-0.713334\pi$
$467$	−2.73423e9	−1.24230	−0.621148	+	0.783693i	$0.713334\pi$
$468$	−1.26802e9	−0.571827
$469$	0	0
$470$	2.73583e8	0.121548
$471$	−9.12614e8	−0.402452
$472$	1.80622e9	0.790632
$473$	−8.23369e8	−0.357751
$474$	3.19677e9	1.37875
$475$	2.08472e8	0.0892524
$476$	0	0
$477$	2.62897e9	1.10910
$478$	−2.48352e7	−0.0104009
$479$	1.21613e9	0.505598	0.252799	−	0.967519i	$-0.418649\pi$
$479$	1.21613e9	0.505598	0.252799	+	0.967519i	$0.418649\pi$
$480$	3.86100e9	1.59351
$481$	−6.81365e8	−0.279172
$482$	−1.61306e9	−0.656122
$483$	0	0
$484$	6.83236e8	0.273913
$485$	1.65343e9	0.658098
$486$	1.91118e9	0.755222
$487$	−2.38138e9	−0.934280	−0.467140	−	0.884183i	$-0.654716\pi$
$487$	−2.38138e9	−0.934280	−0.467140	+	0.884183i	$0.654716\pi$
$488$	1.13353e9	0.441532
$489$	−1.30802e9	−0.505865
$490$	0	0
$491$	4.90402e9	1.86968	0.934839	−	0.355072i	$-0.115544\pi$
$491$	4.90402e9	1.86968	0.934839	+	0.355072i	$0.115544\pi$
$492$	3.56202e8	0.134840
$493$	−3.47166e8	−0.130489
$494$	−1.23720e9	−0.461738
$495$	−3.08751e9	−1.14417
$496$	−4.42503e8	−0.162828
$497$	0	0
$498$	−7.65774e8	−0.277842
$499$	−2.90707e9	−1.04738	−0.523690	−	0.851909i	$-0.675445\pi$
$499$	−2.90707e9	−1.04738	−0.523690	+	0.851909i	$0.675445\pi$
$500$	−1.53256e9	−0.548305
$501$	8.43095e9	2.99533
$502$	−2.29294e9	−0.808966
$503$	−7.90853e7	−0.0277082	−0.0138541	−	0.999904i	$-0.504410\pi$
$503$	−7.90853e7	−0.0277082	−0.0138541	+	0.999904i	$0.504410\pi$
$504$	0	0
$505$	−2.93379e9	−1.01370
$506$	−1.48327e9	−0.508972
$507$	−2.66138e9	−0.906941
$508$	−1.94913e9	−0.659655
$509$	−3.39847e9	−1.14228	−0.571138	−	0.820854i	$-0.693498\pi$
$509$	−3.39847e9	−1.14228	−0.571138	+	0.820854i	$0.693498\pi$
$510$	−3.90293e8	−0.130285
$511$	0	0
$512$	5.78577e8	0.190509
$513$	2.73075e9	0.893040
$514$	−2.18581e9	−0.709971
$515$	−2.71553e9	−0.876052
$516$	1.40767e9	0.451052
$517$	4.06957e8	0.129518
$518$	0	0
$519$	−1.60574e9	−0.504184
$520$	−2.24358e9	−0.699730
$521$	−3.59315e9	−1.11312	−0.556561	−	0.830807i	$-0.687880\pi$
$521$	−3.59315e9	−1.11312	−0.556561	+	0.830807i	$0.687880\pi$
$522$	−3.47883e9	−1.07050
$523$	2.19278e9	0.670253	0.335126	−	0.942173i	$-0.391221\pi$
$523$	2.19278e9	0.670253	0.335126	+	0.942173i	$0.391221\pi$
$524$	−3.34222e8	−0.101479
$525$	0	0
$526$	−1.91492e9	−0.573719
$527$	−6.91570e8	−0.205825
$528$	−3.71475e8	−0.109828
$529$	5.56794e8	0.163531
$530$	1.69709e9	0.495154
$531$	4.03781e9	1.17035
$532$	0	0
$533$	−3.38453e8	−0.0968173
$534$	−8.86083e8	−0.251814
$535$	4.18923e9	1.18276
$536$	−5.12092e9	−1.43639
$537$	8.03955e9	2.24038
$538$	−1.08097e9	−0.299277
$539$	0	0
$540$	1.80671e9	0.493753
$541$	−4.78889e8	−0.130030	−0.0650152	−	0.997884i	$-0.520710\pi$
$541$	−4.78889e8	−0.130030	−0.0650152	+	0.997884i	$0.520710\pi$
$542$	1.85679e9	0.500916
$543$	4.87580e9	1.30691
$544$	4.37954e8	0.116636
$545$	−2.48940e9	−0.658730
$546$	0	0
$547$	−1.52985e9	−0.399664	−0.199832	−	0.979830i	$-0.564040\pi$
$547$	−1.52985e9	−0.399664	−0.199832	+	0.979830i	$0.564040\pi$
$548$	−3.62874e9	−0.941941
$549$	2.53400e9	0.653587
$550$	−1.52014e8	−0.0389597
$551$	4.58117e9	1.16666
$552$	6.95062e9	1.75888
$553$	0	0
$554$	3.43431e9	0.858134
$555$	2.83641e9	0.704277
$556$	3.24412e9	0.800451
$557$	4.91768e8	0.120578	0.0602889	−	0.998181i	$-0.480798\pi$
$557$	4.91768e8	0.120578	0.0602889	+	0.998181i	$0.480798\pi$
$558$	−6.92998e9	−1.68854
$559$	−1.33753e9	−0.323863
$560$	0	0
$561$	−5.80564e8	−0.138829
$562$	4.12209e9	0.979580
$563$	−1.12449e9	−0.265569	−0.132785	−	0.991145i	$-0.542392\pi$
$563$	−1.12449e9	−0.265569	−0.132785	+	0.991145i	$0.542392\pi$
$564$	−6.95751e8	−0.163297
$565$	−3.36208e9	−0.784221
$566$	−5.55621e8	−0.128801
$567$	0	0
$568$	−5.94331e9	−1.36085
$569$	3.07826e9	0.700506	0.350253	−	0.936655i	$-0.386096\pi$
$569$	3.07826e9	0.700506	0.350253	+	0.936655i	$0.386096\pi$
$570$	5.15026e9	1.16484
$571$	9.19642e8	0.206725	0.103362	−	0.994644i	$-0.467040\pi$
$571$	9.19642e8	0.206725	0.103362	+	0.994644i	$0.467040\pi$
$572$	−1.21760e9	−0.272031
$573$	−1.07356e9	−0.238388
$574$	0	0
$575$	4.06010e8	0.0890633
$576$	5.05555e9	1.10227
$577$	4.86320e9	1.05392	0.526959	−	0.849891i	$-0.323332\pi$
$577$	4.86320e9	1.05392	0.526959	+	0.849891i	$0.323332\pi$
$578$	2.98434e9	0.642838
$579$	−4.63065e9	−0.991442
$580$	3.03097e9	0.645036
$581$	0	0
$582$	3.11547e9	0.655078
$583$	2.52444e9	0.527625
$584$	−1.14432e9	−0.237739
$585$	−5.01553e9	−1.03579
$586$	4.35979e9	0.895002
$587$	1.16578e9	0.237894	0.118947	−	0.992901i	$-0.462048\pi$
$587$	1.16578e9	0.237894	0.118947	+	0.992901i	$0.462048\pi$
$588$	0	0
$589$	9.12588e9	1.84023
$590$	2.60655e9	0.522498
$591$	−1.00619e9	−0.200504
$592$	2.05862e8	0.0407804
$593$	−3.35508e9	−0.660711	−0.330356	−	0.943857i	$-0.607169\pi$
$593$	−3.35508e9	−0.660711	−0.330356	+	0.943857i	$0.607169\pi$
$594$	−1.99122e9	−0.389822
$595$	0	0
$596$	−1.63765e9	−0.316853
$597$	−2.20686e9	−0.424488
$598$	−2.40951e9	−0.460760
$599$	−2.32145e9	−0.441332	−0.220666	−	0.975349i	$-0.570823\pi$
$599$	−2.32145e9	−0.441332	−0.220666	+	0.975349i	$0.570823\pi$
$600$	7.12339e8	0.134635
$601$	4.38113e9	0.823238	0.411619	−	0.911356i	$-0.364964\pi$
$601$	4.38113e9	0.823238	0.411619	+	0.911356i	$0.364964\pi$
$602$	0	0
$603$	−1.14478e10	−2.12624
$604$	5.11112e9	0.943815
$605$	2.70248e9	0.496157
$606$	−5.52798e9	−1.00905
$607$	4.32861e9	0.785577	0.392788	−	0.919629i	$-0.371511\pi$
$607$	4.32861e9	0.785577	0.392788	+	0.919629i	$0.371511\pi$
$608$	−5.77919e9	−1.04281
$609$	0	0
$610$	1.63579e9	0.291792
$611$	6.61083e8	0.117250
$612$	5.98746e8	0.105588
$613$	1.90325e9	0.333721	0.166860	−	0.985981i	$-0.446637\pi$
$613$	1.90325e9	0.333721	0.166860	+	0.985981i	$0.446637\pi$
$614$	−4.79581e9	−0.836128
$615$	1.40893e9	0.244245
$616$	0	0
$617$	−8.11496e9	−1.39088	−0.695438	−	0.718586i	$-0.744790\pi$
$617$	−8.11496e9	−1.39088	−0.695438	+	0.718586i	$0.744790\pi$
$618$	−5.11673e9	−0.872033
$619$	−4.60343e9	−0.780125	−0.390062	−	0.920788i	$-0.627547\pi$
$619$	−4.60343e9	−0.780125	−0.390062	+	0.920788i	$0.627547\pi$
$620$	6.03783e9	1.01744
$621$	5.31827e9	0.891149
$622$	−3.31087e8	−0.0551666
$623$	0	0
$624$	−6.03445e8	−0.0994241
$625$	−6.56586e9	−1.07575
$626$	−4.22194e7	−0.00687863
$627$	7.66106e9	1.24123
$628$	9.03774e8	0.145613
$629$	3.21734e8	0.0515490
$630$	0	0
$631$	3.73470e9	0.591770	0.295885	−	0.955224i	$-0.404385\pi$
$631$	3.73470e9	0.591770	0.295885	+	0.955224i	$0.404385\pi$
$632$	−8.67721e9	−1.36732
$633$	−1.08116e10	−1.69425
$634$	2.06423e9	0.321695
$635$	−7.70960e9	−1.19488
$636$	−4.31590e9	−0.665229
$637$	0	0
$638$	−3.34051e9	−0.509262
$639$	−1.32863e10	−2.01442
$640$	−3.39306e9	−0.511636
$641$	−7.28967e9	−1.09321	−0.546607	−	0.837390i	$-0.684081\pi$
$641$	−7.28967e9	−1.09321	−0.546607	+	0.837390i	$0.684081\pi$
$642$	7.89352e9	1.17733
$643$	4.01680e9	0.595856	0.297928	−	0.954588i	$-0.403704\pi$
$643$	4.01680e9	0.595856	0.297928	+	0.954588i	$0.403704\pi$
$644$	0	0
$645$	5.56791e9	0.817020
$646$	5.84194e8	0.0852597
$647$	3.54285e9	0.514267	0.257133	−	0.966376i	$-0.417222\pi$
$647$	3.54285e9	0.514267	0.257133	+	0.966376i	$0.417222\pi$
$648$	−1.48550e9	−0.214467
$649$	3.87727e9	0.556761
$650$	−2.46941e8	−0.0352692
$651$	0	0
$652$	1.29535e9	0.183030
$653$	1.11981e10	1.57379	0.786897	−	0.617084i	$-0.211686\pi$
$653$	1.11981e10	1.57379	0.786897	+	0.617084i	$0.211686\pi$
$654$	−4.69064e9	−0.655708
$655$	−1.32199e9	−0.183816
$656$	1.02258e8	0.0141427
$657$	−2.55812e9	−0.351918
$658$	0	0
$659$	8.68249e9	1.18180	0.590902	−	0.806743i	$-0.298772\pi$
$659$	8.68249e9	1.18180	0.590902	+	0.806743i	$0.298772\pi$
$660$	5.06868e9	0.686263
$661$	1.03500e10	1.39391	0.696955	−	0.717115i	$-0.254538\pi$
$661$	1.03500e10	1.39391	0.696955	+	0.717115i	$0.254538\pi$
$662$	6.47814e9	0.867855
$663$	−9.43100e8	−0.125678
$664$	2.07860e9	0.275538
$665$	0	0
$666$	3.22399e9	0.422896
$667$	8.92206e9	1.16419
$668$	−8.34928e9	−1.08376
$669$	−1.49472e10	−1.93005
$670$	−7.38998e9	−0.949251
$671$	2.43325e9	0.310926
$672$	0	0
$673$	1.94227e9	0.245617	0.122808	−	0.992430i	$-0.460810\pi$
$673$	1.94227e9	0.245617	0.122808	+	0.992430i	$0.460810\pi$
$674$	−7.77342e8	−0.0977918
$675$	5.45047e8	0.0682136
$676$	2.63560e9	0.328145
$677$	5.39351e9	0.668053	0.334027	−	0.942564i	$-0.391592\pi$
$677$	5.39351e9	0.668053	0.334027	+	0.942564i	$0.391592\pi$
$678$	−6.33498e9	−0.780623
$679$	0	0
$680$	1.05940e9	0.129205
$681$	−2.28453e10	−2.77193
$682$	−6.65445e9	−0.803279
$683$	1.11516e10	1.33926	0.669628	−	0.742697i	$-0.266454\pi$
$683$	1.11516e10	1.33926	0.669628	+	0.742697i	$0.266454\pi$
$684$	−7.90099e9	−0.944029
$685$	−1.43531e10	−1.70620
$686$	0	0
$687$	1.37549e10	1.61848
$688$	4.04110e8	0.0473086
$689$	4.10084e9	0.477646
$690$	1.00304e10	1.16238
$691$	−4.28327e9	−0.493858	−0.246929	−	0.969034i	$-0.579421\pi$
$691$	−4.28327e9	−0.493858	−0.246929	+	0.969034i	$0.579421\pi$
$692$	1.59019e9	0.182422
$693$	0	0
$694$	−2.64988e9	−0.300932
$695$	1.28318e10	1.44991
$696$	1.56537e10	1.75988
$697$	1.59814e8	0.0178773
$698$	1.47201e9	0.163839
$699$	−1.84942e10	−2.04817
$700$	0	0
$701$	−9.26658e8	−0.101603	−0.0508015	−	0.998709i	$-0.516178\pi$
$701$	−9.26658e8	−0.101603	−0.0508015	+	0.998709i	$0.516178\pi$
$702$	−3.23465e9	−0.352896
$703$	−4.24557e9	−0.460885
$704$	4.85454e9	0.524377
$705$	−2.75198e9	−0.295790
$706$	1.21502e10	1.29947
$707$	0	0
$708$	−6.62874e9	−0.701964
$709$	−5.58852e9	−0.588891	−0.294446	−	0.955668i	$-0.595135\pi$
$709$	−5.58852e9	−0.588891	−0.294446	+	0.955668i	$0.595135\pi$
$710$	−8.57677e9	−0.899331
$711$	−1.93979e10	−2.02400
$712$	2.40516e9	0.249726
$713$	1.77731e10	1.83633
$714$	0	0
$715$	−4.81611e9	−0.492749
$716$	−7.96168e9	−0.810604
$717$	2.49818e8	0.0253109
$718$	−1.78188e8	−0.0179656
$719$	1.32009e10	1.32450	0.662252	−	0.749281i	$-0.269601\pi$
$719$	1.32009e10	1.32450	0.662252	+	0.749281i	$0.269601\pi$
$720$	1.51536e9	0.151304
$721$	0	0
$722$	−1.11151e9	−0.109909
$723$	1.62258e10	1.59669
$724$	−4.82857e9	−0.472861
$725$	9.14384e8	0.0891139
$726$	5.09213e9	0.493880
$727$	−7.32749e9	−0.707269	−0.353635	−	0.935384i	$-0.615054\pi$
$727$	−7.32749e9	−0.707269	−0.353635	+	0.935384i	$0.615054\pi$
$728$	0	0
$729$	−1.70404e10	−1.62905
$730$	−1.65136e9	−0.157113
$731$	6.31567e8	0.0598011
$732$	−4.15999e9	−0.392016
$733$	−1.39234e10	−1.30581	−0.652907	−	0.757438i	$-0.726451\pi$
$733$	−1.39234e10	−1.30581	−0.652907	+	0.757438i	$0.726451\pi$
$734$	1.14687e10	1.07048
$735$	0	0
$736$	−1.12553e10	−1.04060
$737$	−1.09926e10	−1.01150
$738$	1.60145e9	0.146661
$739$	2.95420e9	0.269268	0.134634	−	0.990895i	$-0.457014\pi$
$739$	2.95420e9	0.269268	0.134634	+	0.990895i	$0.457014\pi$
$740$	−2.80894e9	−0.254818
$741$	1.24450e10	1.12365
$742$	0	0
$743$	−5.39743e9	−0.482754	−0.241377	−	0.970431i	$-0.577599\pi$
$743$	−5.39743e9	−0.482754	−0.241377	+	0.970431i	$0.577599\pi$
$744$	3.11827e10	2.77593
$745$	−6.47756e9	−0.573937
$746$	−1.12417e10	−0.991390
$747$	4.64670e9	0.407871
$748$	5.74940e8	0.0502304
$749$	0	0
$750$	−1.14221e10	−0.988625
$751$	1.12321e9	0.0967661	0.0483830	−	0.998829i	$-0.484593\pi$
$751$	1.12321e9	0.0967661	0.0483830	+	0.998829i	$0.484593\pi$
$752$	−1.99735e8	−0.0171274
$753$	2.30648e10	1.96864
$754$	−5.42652e9	−0.461022
$755$	2.02166e10	1.70960
$756$	0	0
$757$	9.72000e9	0.814387	0.407193	−	0.913342i	$-0.366507\pi$
$757$	9.72000e9	0.814387	0.407193	+	0.913342i	$0.366507\pi$
$758$	−6.87992e9	−0.573775
$759$	1.49203e10	1.23860
$760$	−1.39797e10	−1.15518
$761$	1.91127e10	1.57208	0.786041	−	0.618174i	$-0.212127\pi$
$761$	1.91127e10	1.57208	0.786041	+	0.618174i	$0.212127\pi$
$762$	−1.45268e10	−1.18940
$763$	0	0
$764$	1.06316e9	0.0862526
$765$	2.36829e9	0.191258
$766$	1.29513e10	1.04115
$767$	6.29845e9	0.504022
$768$	−2.08422e10	−1.66027
$769$	−1.69816e10	−1.34660	−0.673298	−	0.739371i	$-0.735123\pi$
$769$	−1.69816e10	−1.34660	−0.673298	+	0.739371i	$0.735123\pi$
$770$	0	0
$771$	2.19871e10	1.72774
$772$	4.58580e9	0.358719
$773$	1.63789e10	1.27543	0.637713	−	0.770274i	$-0.279880\pi$
$773$	1.63789e10	1.27543	0.637713	+	0.770274i	$0.279880\pi$
$774$	6.32872e9	0.490594
$775$	1.82149e9	0.140563
$776$	−8.45655e9	−0.649646
$777$	0	0
$778$	6.79524e9	0.517341
$779$	−2.10889e9	−0.159836
$780$	8.23384e9	0.621257
$781$	−1.27580e10	−0.958306
$782$	1.13775e9	0.0850791
$783$	1.19774e10	0.891655
$784$	0	0
$785$	3.57480e9	0.263759
$786$	−2.49094e9	−0.182972
$787$	−1.35516e10	−0.991016	−0.495508	−	0.868603i	$-0.665018\pi$
$787$	−1.35516e10	−0.991016	−0.495508	+	0.868603i	$0.665018\pi$
$788$	9.96439e8	0.0725453
$789$	1.92622e10	1.39616
$790$	−1.25220e10	−0.903609
$791$	0	0
$792$	1.57912e10	1.12948
$793$	3.95270e9	0.281474
$794$	6.90512e9	0.489553
$795$	−1.70711e10	−1.20497
$796$	2.18548e9	0.153586
$797$	−5.21891e7	−0.00365153	−0.00182577	−	0.999998i	$-0.500581\pi$
$797$	−5.21891e7	−0.00365153	−0.00182577	+	0.999998i	$0.500581\pi$
$798$	0	0
$799$	−3.12157e8	−0.0216501
$800$	−1.15351e9	−0.0796535
$801$	5.37673e9	0.369662
$802$	−1.43845e10	−0.984655
$803$	−2.45641e9	−0.167416
$804$	1.87935e10	1.27530
$805$	0	0
$806$	−1.08099e10	−0.727189
$807$	1.08735e10	0.728301
$808$	1.50050e10	1.00068
$809$	1.36046e10	0.903369	0.451685	−	0.892178i	$-0.350823\pi$
$809$	1.36046e10	0.903369	0.451685	+	0.892178i	$0.350823\pi$
$810$	−2.14372e9	−0.141733
$811$	−3.10614e9	−0.204479	−0.102239	−	0.994760i	$-0.532601\pi$
$811$	−3.10614e9	−0.204479	−0.102239	+	0.994760i	$0.532601\pi$
$812$	0	0
$813$	−1.86775e10	−1.21900
$814$	3.09580e9	0.201181
$815$	5.12365e9	0.331534
$816$	2.84941e8	0.0183586
$817$	−8.33410e9	−0.534665
$818$	2.21807e9	0.141690
$819$	0	0
$820$	−1.39528e9	−0.0883715
$821$	1.05773e10	0.667075	0.333537	−	0.942737i	$-0.391758\pi$
$821$	1.05773e10	0.667075	0.333537	+	0.942737i	$0.391758\pi$
$822$	−2.70448e10	−1.69837
$823$	1.02811e10	0.642894	0.321447	−	0.946928i	$-0.395831\pi$
$823$	1.02811e10	0.642894	0.321447	+	0.946928i	$0.395831\pi$
$824$	1.38887e10	0.864801
$825$	1.52912e9	0.0948096
$826$	0	0
$827$	1.76194e9	0.108323	0.0541617	−	0.998532i	$-0.482751\pi$
$827$	1.76194e9	0.108323	0.0541617	+	0.998532i	$0.482751\pi$
$828$	−1.53876e10	−0.942029
$829$	−1.85017e10	−1.12790	−0.563951	−	0.825808i	$-0.690719\pi$
$829$	−1.85017e10	−1.12790	−0.563951	+	0.825808i	$0.690719\pi$
$830$	2.99961e9	0.182092
$831$	−3.45458e10	−2.08830
$832$	7.88599e9	0.474706
$833$	0	0
$834$	2.41783e10	1.44326
$835$	−3.30249e10	−1.96308
$836$	−7.58685e9	−0.449096
$837$	2.38595e10	1.40644
$838$	5.47065e9	0.321133
$839$	−1.91428e10	−1.11902	−0.559512	−	0.828822i	$-0.689012\pi$
$839$	−1.91428e10	−1.11902	−0.559512	+	0.828822i	$0.689012\pi$
$840$	0	0
$841$	2.84370e9	0.164854
$842$	−1.66552e10	−0.961521
$843$	−4.14642e10	−2.38384
$844$	1.07069e10	0.613005
$845$	1.04249e10	0.594392
$846$	−3.12802e9	−0.177613
$847$	0	0
$848$	−1.23900e9	−0.0697726
$849$	5.58901e9	0.313443
$850$	1.16603e8	0.00651244
$851$	−8.26847e9	−0.459909
$852$	2.18117e10	1.20823
$853$	7.28376e9	0.401822	0.200911	−	0.979609i	$-0.435610\pi$
$853$	7.28376e9	0.401822	0.200911	+	0.979609i	$0.435610\pi$
$854$	0	0
$855$	−3.12517e10	−1.70998
$856$	−2.14260e10	−1.16757
$857$	2.77327e10	1.50508	0.752539	−	0.658547i	$-0.228829\pi$
$857$	2.77327e10	1.50508	0.752539	+	0.658547i	$0.228829\pi$
$858$	−9.07473e9	−0.490488
$859$	1.98597e9	0.106905	0.0534524	−	0.998570i	$-0.482977\pi$
$859$	1.98597e9	0.106905	0.0534524	+	0.998570i	$0.482977\pi$
$860$	−5.51397e9	−0.295611
$861$	0	0
$862$	−1.61643e8	−0.00859570
$863$	−9.69107e9	−0.513256	−0.256628	−	0.966510i	$-0.582612\pi$
$863$	−9.69107e9	−0.513256	−0.256628	+	0.966510i	$0.582612\pi$
$864$	−1.51096e10	−0.796996
$865$	6.28984e9	0.330433
$866$	−1.05202e10	−0.550440
$867$	−3.00196e10	−1.56437
$868$	0	0
$869$	−1.86266e10	−0.962865
$870$	2.25897e10	1.16304
$871$	−1.78571e10	−0.915686
$872$	1.27322e10	0.650271
$873$	−1.89046e10	−0.961651
$874$	−1.50136e10	−0.760668
$875$	0	0
$876$	4.19958e9	0.211077
$877$	8.06479e9	0.403733	0.201867	−	0.979413i	$-0.435299\pi$
$877$	8.06479e9	0.403733	0.201867	+	0.979413i	$0.435299\pi$
$878$	−1.85431e10	−0.924596
$879$	−4.38553e10	−2.17802
$880$	1.45511e9	0.0719789
$881$	3.72296e10	1.83431	0.917156	−	0.398529i	$-0.130479\pi$
$881$	3.72296e10	1.83431	0.917156	+	0.398529i	$0.130479\pi$
$882$	0	0
$883$	1.71532e10	0.838458	0.419229	−	0.907880i	$-0.362300\pi$
$883$	1.71532e10	0.838458	0.419229	+	0.907880i	$0.362300\pi$
$884$	9.33965e8	0.0454724
$885$	−2.62194e10	−1.27152
$886$	1.08505e10	0.524122
$887$	−2.19309e10	−1.05517	−0.527586	−	0.849502i	$-0.676903\pi$
$887$	−2.19309e10	−1.05517	−0.527586	+	0.849502i	$0.676903\pi$
$888$	−1.45069e10	−0.695233
$889$	0	0
$890$	3.47088e9	0.165034
$891$	−3.18880e9	−0.151027
$892$	1.48024e10	0.698323
$893$	4.11920e9	0.193567
$894$	−1.22053e10	−0.571304
$895$	−3.14917e10	−1.46830
$896$	0	0
$897$	2.42374e10	1.12127
$898$	2.09611e10	0.965934
$899$	4.00273e10	1.83737
$900$	−1.57701e9	−0.0721083
$901$	−1.93638e9	−0.0881970
$902$	1.53777e9	0.0697700
$903$	0	0
$904$	1.71955e10	0.774150
$905$	−1.90990e10	−0.856525
$906$	3.80930e10	1.70175
$907$	−2.50595e9	−0.111519	−0.0557593	−	0.998444i	$-0.517758\pi$
$907$	−2.50595e9	−0.111519	−0.0557593	+	0.998444i	$0.517758\pi$
$908$	2.26241e10	1.00293
$909$	3.35437e10	1.48128
$910$	0	0
$911$	−8.68527e9	−0.380600	−0.190300	−	0.981726i	$-0.560946\pi$
$911$	−8.68527e9	−0.380600	−0.190300	+	0.981726i	$0.560946\pi$
$912$	−3.76005e9	−0.164139
$913$	4.46195e9	0.194033
$914$	3.04104e9	0.131738
$915$	−1.64545e10	−0.710084
$916$	−1.36216e10	−0.585591
$917$	0	0
$918$	1.52737e9	0.0651621
$919$	6.68328e9	0.284044	0.142022	−	0.989864i	$-0.454640\pi$
$919$	6.68328e9	0.284044	0.142022	+	0.989864i	$0.454640\pi$
$920$	−2.72262e10	−1.15274
$921$	4.82413e10	2.03474
$922$	3.06009e10	1.28581
$923$	−2.07248e10	−0.867531
$924$	0	0
$925$	−8.47400e8	−0.0352040
$926$	1.84226e10	0.762451
$927$	3.10482e10	1.28014
$928$	−2.53483e10	−1.04119
$929$	−1.21779e10	−0.498330	−0.249165	−	0.968461i	$-0.580156\pi$
$929$	−1.21779e10	−0.498330	−0.249165	+	0.968461i	$0.580156\pi$
$930$	4.49997e10	1.83451
$931$	0	0
$932$	1.83151e10	0.741059
$933$	3.33042e9	0.134250
$934$	2.01807e10	0.810442
$935$	2.27412e9	0.0909858
$936$	2.56521e10	1.02249
$937$	4.28141e10	1.70019	0.850097	−	0.526626i	$-0.176543\pi$
$937$	4.28141e10	1.70019	0.850097	+	0.526626i	$0.176543\pi$
$938$	0	0
$939$	4.24687e8	0.0167394
$940$	2.72533e9	0.107021
$941$	−3.94834e10	−1.54473	−0.772363	−	0.635181i	$-0.780925\pi$
$941$	−3.94834e10	−1.54473	−0.772363	+	0.635181i	$0.780925\pi$
$942$	6.73579e9	0.262549
$943$	−4.10718e9	−0.159497
$944$	−1.90297e9	−0.0736256
$945$	0	0
$946$	6.07709e9	0.233387
$947$	1.11103e10	0.425108	0.212554	−	0.977149i	$-0.431822\pi$
$947$	1.11103e10	0.425108	0.212554	+	0.977149i	$0.431822\pi$
$948$	3.18449e10	1.21398
$949$	−3.99032e9	−0.151557
$950$	−1.53868e9	−0.0582259
$951$	−2.07641e10	−0.782856
$952$	0	0
$953$	−3.33302e10	−1.24742	−0.623711	−	0.781655i	$-0.714376\pi$
$953$	−3.33302e10	−1.24742	−0.623711	+	0.781655i	$0.714376\pi$
$954$	−1.94038e10	−0.723548
$955$	4.20524e9	0.156235
$956$	−2.47398e8	−0.00915787
$957$	3.36024e10	1.23930
$958$	−8.97597e9	−0.329839
$959$	0	0
$960$	−3.28281e10	−1.19756
$961$	5.22234e10	1.89816
$962$	5.02899e9	0.182124
$963$	−4.78977e10	−1.72831
$964$	−1.60686e10	−0.577709
$965$	1.81387e10	0.649772
$966$	0	0
$967$	−5.24781e10	−1.86632	−0.933158	−	0.359466i	$-0.882959\pi$
$967$	−5.24781e10	−1.86632	−0.933158	+	0.359466i	$0.882959\pi$
$968$	−1.38219e10	−0.489785
$969$	−5.87644e9	−0.207482
$970$	−1.22036e10	−0.429326
$971$	3.48968e10	1.22326	0.611630	−	0.791144i	$-0.290514\pi$
$971$	3.48968e10	1.22326	0.611630	+	0.791144i	$0.290514\pi$
$972$	1.90384e10	0.664965
$973$	0	0
$974$	1.75764e10	0.609499
$975$	2.48399e9	0.0858288
$976$	−1.19424e9	−0.0411166
$977$	−5.64445e10	−1.93638	−0.968191	−	0.250214i	$-0.919499\pi$
$977$	−5.64445e10	−1.93638	−0.968191	+	0.250214i	$0.919499\pi$
$978$	9.65421e9	0.330013
$979$	5.16295e9	0.175857
$980$	0	0
$981$	2.84627e10	0.962575
$982$	−3.61954e10	−1.21973
$983$	−8.41243e7	−0.00282478	−0.00141239	−	0.999999i	$-0.500450\pi$
$983$	−8.41243e7	−0.00282478	−0.00141239	+	0.999999i	$0.500450\pi$
$984$	−7.20600e9	−0.241108
$985$	3.94133e9	0.131406
$986$	2.56235e9	0.0851274
$987$	0	0
$988$	−1.23245e10	−0.406556
$989$	−1.62311e10	−0.533532
$990$	2.27882e10	0.746427
$991$	1.29923e10	0.424062	0.212031	−	0.977263i	$-0.431992\pi$
$991$	1.29923e10	0.424062	0.212031	+	0.977263i	$0.431992\pi$
$992$	−5.04948e10	−1.64231
$993$	−6.51639e10	−2.11195
$994$	0	0
$995$	8.64449e9	0.278201
$996$	−7.62834e9	−0.244637
$997$	−2.22679e8	−0.00711617	−0.00355808	−	0.999994i	$-0.501133\pi$
$997$	−2.22679e8	−0.00711617	−0.00355808	+	0.999994i	$0.501133\pi$
$998$	2.14564e10	0.683283
$999$	−1.11000e10	−0.352244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.8.a.f.1.2		4
3.2	odd	2		441.8.a.s.1.3		4
7.2	even	3		7.8.c.a.4.3	yes	8
7.3	odd	6		49.8.c.g.30.3		8
7.4	even	3		7.8.c.a.2.3	✓	8
7.5	odd	6		49.8.c.g.18.3		8
7.6	odd	2		49.8.a.e.1.2		4
21.2	odd	6		63.8.e.b.46.2		8
21.11	odd	6		63.8.e.b.37.2		8
21.20	even	2		441.8.a.t.1.3		4
28.11	odd	6		112.8.i.c.65.4		8
28.23	odd	6		112.8.i.c.81.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.8.c.a.2.3	✓	8	7.4	even	3
7.8.c.a.4.3	yes	8	7.2	even	3
49.8.a.e.1.2		4	7.6	odd	2
49.8.a.f.1.2		4	1.1	even	1	trivial
49.8.c.g.18.3		8	7.5	odd	6
49.8.c.g.30.3		8	7.3	odd	6
63.8.e.b.37.2		8	21.11	odd	6
63.8.e.b.46.2		8	21.2	odd	6
112.8.i.c.65.4		8	28.11	odd	6
112.8.i.c.81.4		8	28.23	odd	6
441.8.a.s.1.3		4	3.2	odd	2
441.8.a.t.1.3		4	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)

\(f(q)\)	\(=\)	\(q-7.38077 q^{2} +74.2434 q^{3} -73.5243 q^{4} -290.819 q^{5} -547.974 q^{6} +1487.40 q^{8} +3325.09 q^{9} +2146.47 q^{10} +3192.88 q^{11} -5458.70 q^{12} +5186.70 q^{13} -21591.4 q^{15} -1567.07 q^{16} -2449.11 q^{17} -24541.7 q^{18} +32318.2 q^{19} +21382.2 q^{20} -23565.9 q^{22} +62941.4 q^{23} +110430. q^{24} +6450.60 q^{25} -38281.8 q^{26} +84495.7 q^{27} +141752. q^{29} +159361. q^{30} +282376. q^{31} -178821. q^{32} +237051. q^{33} +18076.3 q^{34} -244475. q^{36} -131368. q^{37} -238533. q^{38} +385078. q^{39} -432565. q^{40} -65254.0 q^{41} -257876. q^{43} -234755. q^{44} -966999. q^{45} -464556. q^{46} +127457. q^{47} -116345. q^{48} -47610.4 q^{50} -181830. q^{51} -381348. q^{52} +790646. q^{53} -623643. q^{54} -928551. q^{55} +2.39942e6 q^{57} -1.04624e6 q^{58} +1.21435e6 q^{59} +1.58749e6 q^{60} +762084. q^{61} -2.08415e6 q^{62} +1.52042e6 q^{64} -1.50839e6 q^{65} -1.74962e6 q^{66} -3.44286e6 q^{67} +180069. q^{68} +4.67299e6 q^{69} -3.99576e6 q^{71} +4.94575e6 q^{72} -769338. q^{73} +969594. q^{74} +478915. q^{75} -2.37617e6 q^{76} -2.84217e6 q^{78} -5.83380e6 q^{79} +455734. q^{80} -998721. q^{81} +481625. q^{82} +1.39747e6 q^{83} +712248. q^{85} +1.90332e6 q^{86} +1.05241e7 q^{87} +4.74911e6 q^{88} +1.61702e6 q^{89} +7.13719e6 q^{90} -4.62772e6 q^{92} +2.09645e7 q^{93} -940733. q^{94} -9.39875e6 q^{95} -1.32763e7 q^{96} -5.68544e6 q^{97} +1.06166e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{2} + 28 q^{3} + 348 q^{4} + 252 q^{5} + 1022 q^{6} - 984 q^{8} + 2008 q^{9} + 4774 q^{10} - 3972 q^{11} - 5404 q^{12} - 1176 q^{13} - 16556 q^{15} + 57264 q^{16} + 56364 q^{17} - 35908 q^{18}+ \cdots + 15213976 q^{99}+O(q^{100}) \) 4 * q - 6 * q^2 + 28 * q^3 + 348 * q^4 + 252 * q^5 + 1022 * q^6 - 984 * q^8 + 2008 * q^9 + 4774 * q^10 - 3972 * q^11 - 5404 * q^12 - 1176 * q^13 - 16556 * q^15 + 57264 * q^16 + 56364 * q^17 - 35908 * q^18 + 41748 * q^19 + 162372 * q^20 - 152954 * q^22 + 131748 * q^23 + 389592 * q^24 - 68016 * q^25 + 113652 * q^26 + 189028 * q^27 + 34056 * q^29 + 605378 * q^30 + 401212 * q^31 - 453408 * q^32 + 24052 * q^33 + 82110 * q^34 - 360088 * q^36 - 5396 * q^37 + 31794 * q^38 + 364840 * q^39 - 443688 * q^40 + 410424 * q^41 + 46544 * q^43 - 1465836 * q^44 - 1209656 * q^45 + 1379202 * q^46 + 1470084 * q^47 - 2018800 * q^48 + 1395528 * q^50 - 1724796 * q^51 - 5269768 * q^52 - 642372 * q^53 - 4567318 * q^54 - 3063340 * q^55 + 3359068 * q^57 - 1972220 * q^58 + 752220 * q^59 + 2123324 * q^60 + 1325772 * q^61 - 3016314 * q^62 - 865856 * q^64 - 4868808 * q^65 - 6013294 * q^66 - 290916 * q^67 + 2453556 * q^68 + 4915260 * q^69 + 3377760 * q^71 + 4557744 * q^72 + 6706588 * q^73 + 3616878 * q^74 + 7223888 * q^75 + 2923004 * q^76 - 16288132 * q^78 + 3946244 * q^79 + 10695888 * q^80 - 11731292 * q^81 - 17563252 * q^82 + 9542064 * q^83 + 7006068 * q^85 + 23237832 * q^86 + 17836840 * q^87 - 8043336 * q^88 + 16165212 * q^89 + 3415748 * q^90 - 19442628 * q^92 + 12552852 * q^93 + 5720442 * q^94 - 3268452 * q^95 - 8448608 * q^96 + 1533112 * q^97 + 15213976 * q^99

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