LMFDB - Embedded newform 350.8.a.v.1.3

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:	$4$
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} - 2473x^{2} - 31160x + 389808 $ x^4 - x^3 - 2473x^2 - 31160x + 389808
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-35.1621$ 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)

$f(q)$	$=$	$q-8.00000 q^{2} +8.12245 q^{3} +64.0000 q^{4} -64.9796 q^{6} +343.000 q^{7} -512.000 q^{8} -2121.03 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} +8.12245 q^{3} +64.0000 q^{4} -64.9796 q^{6} +343.000 q^{7} -512.000 q^{8} -2121.03 q^{9} -1568.25 q^{11} +519.837 q^{12} -6855.29 q^{13} -2744.00 q^{14} +4096.00 q^{16} -30329.5 q^{17} +16968.2 q^{18} -7158.12 q^{19} +2786.00 q^{21} +12546.0 q^{22} -12135.4 q^{23} -4158.70 q^{24} +54842.3 q^{26} -34991.7 q^{27} +21952.0 q^{28} +7299.31 q^{29} -4403.10 q^{31} -32768.0 q^{32} -12738.1 q^{33} +242636. q^{34} -135746. q^{36} +477796. q^{37} +57264.9 q^{38} -55681.8 q^{39} -462355. q^{41} -22288.0 q^{42} +177870. q^{43} -100368. q^{44} +97083.6 q^{46} +424536. q^{47} +33269.6 q^{48} +117649. q^{49} -246350. q^{51} -438738. q^{52} +685895. q^{53} +279934. q^{54} -175616. q^{56} -58141.5 q^{57} -58394.5 q^{58} -3438.30 q^{59} -302654. q^{61} +35224.8 q^{62} -727512. q^{63} +262144. q^{64} +101905. q^{66} +4.02722e6 q^{67} -1.94109e6 q^{68} -98569.6 q^{69} -2.50784e6 q^{71} +1.08597e6 q^{72} -5.59256e6 q^{73} -3.82236e6 q^{74} -458120. q^{76} -537911. q^{77} +445454. q^{78} +190724. q^{79} +4.35446e6 q^{81} +3.69884e6 q^{82} +4.96181e6 q^{83} +178304. q^{84} -1.42296e6 q^{86} +59288.3 q^{87} +802946. q^{88} -8.40090e6 q^{89} -2.35136e6 q^{91} -776668. q^{92} -35763.9 q^{93} -3.39628e6 q^{94} -266157. q^{96} -3.36217e6 q^{97} -941192. q^{98} +3.32631e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} - 42 q^{3} + 256 q^{4} + 336 q^{6} + 1372 q^{7} - 2048 q^{8} + 2210 q^{9}+O(q^{10}) $ 4 * q - 32 * q^2 - 42 * q^3 + 256 * q^4 + 336 * q^6 + 1372 * q^7 - 2048 * q^8 + 2210 * q^9	\( 4 q - 32 q^{2} - 42 q^{3} + 256 q^{4} + 336 q^{6} + 1372 q^{7} - 2048 q^{8} + 2210 q^{9} + 1324 q^{11} - 2688 q^{12} + 17458 q^{13} - 10976 q^{14} + 16384 q^{16} + 18158 q^{17} - 17680 q^{18} + 11984 q^{19} - 14406 q^{21} - 10592 q^{22} + 145300 q^{23} + 21504 q^{24} - 139664 q^{26} - 62244 q^{27} + 87808 q^{28} - 55578 q^{29} + 17206 q^{31} - 131072 q^{32} + 684320 q^{33} - 145264 q^{34} + 141440 q^{36} + 507898 q^{37} - 95872 q^{38} - 132344 q^{39} - 268660 q^{41} + 115248 q^{42} - 362460 q^{43} + 84736 q^{44} - 1162400 q^{46} + 855988 q^{47} - 172032 q^{48} + 470596 q^{49} - 1347956 q^{51} + 1117312 q^{52} - 1245360 q^{53} + 497952 q^{54} - 702464 q^{56} + 3348414 q^{57} + 444624 q^{58} - 1415834 q^{59} - 4333910 q^{61} - 137648 q^{62} + 758030 q^{63} + 1048576 q^{64} - 5474560 q^{66} + 2271660 q^{67} + 1162112 q^{68} - 10774750 q^{69} - 3370816 q^{71} - 1131520 q^{72} + 4604488 q^{73} - 4063184 q^{74} + 766976 q^{76} + 454132 q^{77} + 1058752 q^{78} - 9036996 q^{79} - 3280024 q^{81} + 2149280 q^{82} - 11603802 q^{83} - 921984 q^{84} + 2899680 q^{86} + 15868398 q^{87} - 677888 q^{88} + 307328 q^{89} + 5988094 q^{91} + 9299200 q^{92} - 30711032 q^{93} - 6847904 q^{94} + 1376256 q^{96} + 21766234 q^{97} - 3764768 q^{98} - 23711228 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 - 42 * q^3 + 256 * q^4 + 336 * q^6 + 1372 * q^7 - 2048 * q^8 + 2210 * q^9 + 1324 * q^11 - 2688 * q^12 + 17458 * q^13 - 10976 * q^14 + 16384 * q^16 + 18158 * q^17 - 17680 * q^18 + 11984 * q^19 - 14406 * q^21 - 10592 * q^22 + 145300 * q^23 + 21504 * q^24 - 139664 * q^26 - 62244 * q^27 + 87808 * q^28 - 55578 * q^29 + 17206 * q^31 - 131072 * q^32 + 684320 * q^33 - 145264 * q^34 + 141440 * q^36 + 507898 * q^37 - 95872 * q^38 - 132344 * q^39 - 268660 * q^41 + 115248 * q^42 - 362460 * q^43 + 84736 * q^44 - 1162400 * q^46 + 855988 * q^47 - 172032 * q^48 + 470596 * q^49 - 1347956 * q^51 + 1117312 * q^52 - 1245360 * q^53 + 497952 * q^54 - 702464 * q^56 + 3348414 * q^57 + 444624 * q^58 - 1415834 * q^59 - 4333910 * q^61 - 137648 * q^62 + 758030 * q^63 + 1048576 * q^64 - 5474560 * q^66 + 2271660 * q^67 + 1162112 * q^68 - 10774750 * q^69 - 3370816 * q^71 - 1131520 * q^72 + 4604488 * q^73 - 4063184 * q^74 + 766976 * q^76 + 454132 * q^77 + 1058752 * q^78 - 9036996 * q^79 - 3280024 * q^81 + 2149280 * q^82 - 11603802 * q^83 - 921984 * q^84 + 2899680 * q^86 + 15868398 * q^87 - 677888 * q^88 + 307328 * q^89 + 5988094 * q^91 + 9299200 * q^92 - 30711032 * q^93 - 6847904 * q^94 + 1376256 * q^96 + 21766234 * q^97 - 3764768 * q^98 - 23711228 * 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
$3$	8.12245	0.173685	0.0868426	−	0.996222i	$-0.472322\pi$
$3$	8.12245	0.173685	0.0868426	+	0.996222i	$0.472322\pi$
$4$	64.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−64.9796	−0.122814
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−512.000	−0.353553
$9$	−2121.03	−0.969833
$10$	0	0
$11$	−1568.25	−0.355256	−0.177628	−	0.984098i	$-0.556842\pi$
$11$	−1568.25	−0.355256	−0.177628	+	0.984098i	$0.556842\pi$
$12$	519.837	0.0868426
$13$	−6855.29	−0.865414	−0.432707	−	0.901535i	$-0.642442\pi$
$13$	−6855.29	−0.865414	−0.432707	+	0.901535i	$0.642442\pi$
$14$	−2744.00	−0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	−30329.5	−1.49725	−0.748625	−	0.662993i	$-0.769286\pi$
$17$	−30329.5	−1.49725	−0.748625	+	0.662993i	$0.769286\pi$
$18$	16968.2	0.685776
$19$	−7158.12	−0.239420	−0.119710	−	0.992809i	$-0.538197\pi$
$19$	−7158.12	−0.239420	−0.119710	+	0.992809i	$0.538197\pi$
$20$	0	0
$21$	2786.00	0.0656468
$22$	12546.0	0.251204
$23$	−12135.4	−0.207974	−0.103987	−	0.994579i	$-0.533160\pi$
$23$	−12135.4	−0.207974	−0.103987	+	0.994579i	$0.533160\pi$
$24$	−4158.70	−0.0614070
$25$	0	0
$26$	54842.3	0.611940
$27$	−34991.7	−0.342131
$28$	21952.0	0.188982
$29$	7299.31	0.0555762	0.0277881	−	0.999614i	$-0.491154\pi$
$29$	7299.31	0.0555762	0.0277881	+	0.999614i	$0.491154\pi$
$30$	0	0
$31$	−4403.10	−0.0265456	−0.0132728	−	0.999912i	$-0.504225\pi$
$31$	−4403.10	−0.0265456	−0.0132728	+	0.999912i	$0.504225\pi$
$32$	−32768.0	−0.176777
$33$	−12738.1	−0.0617028
$34$	242636.	1.05872
$35$	0	0
$36$	−135746.	−0.484917
$37$	477796.	1.55073	0.775365	−	0.631514i	$-0.217566\pi$
$37$	477796.	1.55073	0.775365	+	0.631514i	$0.217566\pi$
$38$	57264.9	0.169296
$39$	−55681.8	−0.150310
$40$	0	0
$41$	−462355.	−1.04769	−0.523845	−	0.851814i	$-0.675503\pi$
$41$	−462355.	−1.04769	−0.523845	+	0.851814i	$0.675503\pi$
$42$	−22288.0	−0.0464193
$43$	177870.	0.341164	0.170582	−	0.985343i	$-0.445435\pi$
$43$	177870.	0.341164	0.170582	+	0.985343i	$0.445435\pi$
$44$	−100368.	−0.177628
$45$	0	0
$46$	97083.6	0.147060
$47$	424536.	0.596447	0.298223	−	0.954496i	$-0.403606\pi$
$47$	424536.	0.596447	0.298223	+	0.954496i	$0.403606\pi$
$48$	33269.6	0.0434213
$49$	117649.	0.142857
$50$	0	0
$51$	−246350.	−0.260050
$52$	−438738.	−0.432707
$53$	685895.	0.632837	0.316419	−	0.948620i	$-0.397520\pi$
$53$	685895.	0.632837	0.316419	+	0.948620i	$0.397520\pi$
$54$	279934.	0.241923
$55$	0	0
$56$	−175616.	−0.133631
$57$	−58141.5	−0.0415838
$58$	−58394.5	−0.0392983
$59$	−3438.30	−0.00217953	−0.00108976	−	0.999999i	$-0.500347\pi$
$59$	−3438.30	−0.00217953	−0.00108976	+	0.999999i	$0.500347\pi$
$60$	0	0
$61$	−302654.	−0.170723	−0.0853614	−	0.996350i	$-0.527204\pi$
$61$	−302654.	−0.170723	−0.0853614	+	0.996350i	$0.527204\pi$
$62$	35224.8	0.0187706
$63$	−727512.	−0.366563
$64$	262144.	0.125000
$65$	0	0
$66$	101905.	0.0436304
$67$	4.02722e6	1.63585	0.817924	−	0.575326i	$-0.195125\pi$
$67$	4.02722e6	1.63585	0.817924	+	0.575326i	$0.195125\pi$
$68$	−1.94109e6	−0.748625
$69$	−98569.6	−0.0361219
$70$	0	0
$71$	−2.50784e6	−0.831565	−0.415783	−	0.909464i	$-0.636492\pi$
$71$	−2.50784e6	−0.831565	−0.415783	+	0.909464i	$0.636492\pi$
$72$	1.08597e6	0.342888
$73$	−5.59256e6	−1.68260	−0.841299	−	0.540570i	$-0.818209\pi$
$73$	−5.59256e6	−1.68260	−0.841299	+	0.540570i	$0.818209\pi$
$74$	−3.82236e6	−1.09653
$75$	0	0
$76$	−458120.	−0.119710
$77$	−537911.	−0.134274
$78$	445454.	0.106285
$79$	190724.	0.0435221	0.0217611	−	0.999763i	$-0.493073\pi$
$79$	190724.	0.0435221	0.0217611	+	0.999763i	$0.493073\pi$
$80$	0	0
$81$	4.35446e6	0.910410
$82$	3.69884e6	0.740828
$83$	4.96181e6	0.952504	0.476252	−	0.879309i	$-0.341995\pi$
$83$	4.96181e6	0.952504	0.476252	+	0.879309i	$0.341995\pi$
$84$	178304.	0.0328234
$85$	0	0
$86$	−1.42296e6	−0.241239
$87$	59288.3	0.00965277
$88$	802946.	0.125602
$89$	−8.40090e6	−1.26317	−0.631583	−	0.775308i	$-0.717595\pi$
$89$	−8.40090e6	−1.26317	−0.631583	+	0.775308i	$0.717595\pi$
$90$	0	0
$91$	−2.35136e6	−0.327096
$92$	−776668.	−0.103987
$93$	−35763.9	−0.00461058
$94$	−3.39628e6	−0.421752
$95$	0	0
$96$	−266157.	−0.0307035
$97$	−3.36217e6	−0.374040	−0.187020	−	0.982356i	$-0.559883\pi$
$97$	−3.36217e6	−0.374040	−0.187020	+	0.982356i	$0.559883\pi$
$98$	−941192.	−0.101015
$99$	3.32631e6	0.344539
$100$	0	0
$101$	2.66199e6	0.257087	0.128544	−	0.991704i	$-0.458970\pi$
$101$	2.66199e6	0.257087	0.128544	+	0.991704i	$0.458970\pi$
$102$	1.97080e6	0.183883
$103$	249972.	0.0225404	0.0112702	−	0.999936i	$-0.496413\pi$
$103$	249972.	0.0225404	0.0112702	+	0.999936i	$0.496413\pi$
$104$	3.50991e6	0.305970
$105$	0	0
$106$	−5.48716e6	−0.447483
$107$	1.70478e7	1.34532	0.672658	−	0.739954i	$-0.265153\pi$
$107$	1.70478e7	1.34532	0.672658	+	0.739954i	$0.265153\pi$
$108$	−2.23947e6	−0.171065
$109$	−6.63923e6	−0.491049	−0.245524	−	0.969390i	$-0.578960\pi$
$109$	−6.63923e6	−0.491049	−0.245524	+	0.969390i	$0.578960\pi$
$110$	0	0
$111$	3.88087e6	0.269339
$112$	1.40493e6	0.0944911
$113$	−6.31215e6	−0.411531	−0.205766	−	0.978601i	$-0.565968\pi$
$113$	−6.31215e6	−0.411531	−0.205766	+	0.978601i	$0.565968\pi$
$114$	465132.	0.0294042
$115$	0	0
$116$	467156.	0.0277881
$117$	1.45402e7	0.839307
$118$	27506.4	0.00154116
$119$	−1.04030e7	−0.565908
$120$	0	0
$121$	−1.70278e7	−0.873793
$122$	2.42123e6	0.120719
$123$	−3.75546e6	−0.181968
$124$	−281798.	−0.0132728
$125$	0	0
$126$	5.82009e6	0.259199
$127$	4.27455e7	1.85173	0.925865	−	0.377854i	$-0.123338\pi$
$127$	4.27455e7	1.85173	0.925865	+	0.377854i	$0.123338\pi$
$128$	−2.09715e6	−0.0883883
$129$	1.44474e6	0.0592552
$130$	0	0
$131$	2.54855e6	0.0990475	0.0495237	−	0.998773i	$-0.484230\pi$
$131$	2.54855e6	0.0990475	0.0495237	+	0.998773i	$0.484230\pi$
$132$	−815236.	−0.0308514
$133$	−2.45523e6	−0.0904924
$134$	−3.22177e7	−1.15672
$135$	0	0
$136$	1.55287e7	0.529358
$137$	−2.95134e7	−0.980611	−0.490306	−	0.871551i	$-0.663115\pi$
$137$	−2.95134e7	−0.980611	−0.490306	+	0.871551i	$0.663115\pi$
$138$	788557.	0.0255421
$139$	−1.48720e7	−0.469698	−0.234849	−	0.972032i	$-0.575460\pi$
$139$	−1.48720e7	−0.469698	−0.234849	+	0.972032i	$0.575460\pi$
$140$	0	0
$141$	3.44827e6	0.103594
$142$	2.00627e7	0.588005
$143$	1.07508e7	0.307444
$144$	−8.68772e6	−0.242458
$145$	0	0
$146$	4.47405e7	1.18978
$147$	955599.	0.0248122
$148$	3.05789e7	0.775365
$149$	−2.74618e7	−0.680106	−0.340053	−	0.940406i	$-0.610445\pi$
$149$	−2.74618e7	−0.680106	−0.340053	+	0.940406i	$0.610445\pi$
$150$	0	0
$151$	4.17984e7	0.987962	0.493981	−	0.869473i	$-0.335541\pi$
$151$	4.17984e7	0.987962	0.493981	+	0.869473i	$0.335541\pi$
$152$	3.66496e6	0.0846479
$153$	6.43297e7	1.45208
$154$	4.30329e6	0.0949462
$155$	0	0
$156$	−3.56363e6	−0.0751548
$157$	7.73011e7	1.59418	0.797089	−	0.603862i	$-0.206372\pi$
$157$	7.73011e7	1.59418	0.797089	+	0.603862i	$0.206372\pi$
$158$	−1.52579e6	−0.0307748
$159$	5.57115e6	0.109914
$160$	0	0
$161$	−4.16246e6	−0.0786066
$162$	−3.48357e7	−0.643757
$163$	−5.24095e6	−0.0947879	−0.0473940	−	0.998876i	$-0.515092\pi$
$163$	−5.24095e6	−0.0947879	−0.0473940	+	0.998876i	$0.515092\pi$
$164$	−2.95907e7	−0.523845
$165$	0	0
$166$	−3.96945e7	−0.673522
$167$	7.89943e7	1.31247	0.656233	−	0.754558i	$-0.272149\pi$
$167$	7.89943e7	1.31247	0.656233	+	0.754558i	$0.272149\pi$
$168$	−1.42643e6	−0.0232097
$169$	−1.57535e7	−0.251059
$170$	0	0
$171$	1.51826e7	0.232198
$172$	1.13837e7	0.170582
$173$	6.36119e7	0.934064	0.467032	−	0.884240i	$-0.345323\pi$
$173$	6.36119e7	0.934064	0.467032	+	0.884240i	$0.345323\pi$
$174$	−474307.	−0.00682554
$175$	0	0
$176$	−6.42356e6	−0.0888141
$177$	−27927.4	−0.000378551 0
$178$	6.72072e7	0.893194
$179$	5.82926e7	0.759675	0.379837	−	0.925053i	$-0.375980\pi$
$179$	5.82926e7	0.759675	0.379837	+	0.925053i	$0.375980\pi$
$180$	0	0
$181$	1.51100e8	1.89404	0.947022	−	0.321168i	$-0.104075\pi$
$181$	1.51100e8	1.89404	0.947022	+	0.321168i	$0.104075\pi$
$182$	1.88109e7	0.231292
$183$	−2.45829e6	−0.0296520
$184$	6.21335e6	0.0735298
$185$	0	0
$186$	286112.	0.00326017
$187$	4.75644e7	0.531908
$188$	2.71703e7	0.298223
$189$	−1.20022e7	−0.129313
$190$	0	0
$191$	1.00664e8	1.04534	0.522670	−	0.852535i	$-0.324936\pi$
$191$	1.00664e8	1.04534	0.522670	+	0.852535i	$0.324936\pi$
$192$	2.12925e6	0.0217107
$193$	7.58322e7	0.759282	0.379641	−	0.925134i	$-0.376048\pi$
$193$	7.58322e7	0.759282	0.379641	+	0.925134i	$0.376048\pi$
$194$	2.68973e7	0.264486
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	8.92225e7	0.831462	0.415731	−	0.909488i	$-0.363526\pi$
$197$	8.92225e7	0.831462	0.415731	+	0.909488i	$0.363526\pi$
$198$	−2.66104e7	−0.243626
$199$	−1.18034e7	−0.106175	−0.0530873	−	0.998590i	$-0.516906\pi$
$199$	−1.18034e7	−0.106175	−0.0530873	+	0.998590i	$0.516906\pi$
$200$	0	0
$201$	3.27109e7	0.284123
$202$	−2.12959e7	−0.181788
$203$	2.50366e6	0.0210058
$204$	−1.57664e7	−0.130025
$205$	0	0
$206$	−1.99978e6	−0.0159384
$207$	2.57396e7	0.201700
$208$	−2.80793e7	−0.216354
$209$	1.12257e7	0.0850556
$210$	0	0
$211$	1.90523e8	1.39624	0.698119	−	0.715982i	$-0.254021\pi$
$211$	1.90523e8	1.39624	0.698119	+	0.715982i	$0.254021\pi$
$212$	4.38973e7	0.316419
$213$	−2.03698e7	−0.144431
$214$	−1.36382e8	−0.951282
$215$	0	0
$216$	1.79158e7	0.120962
$217$	−1.51026e6	−0.0100333
$218$	5.31138e7	0.347224
$219$	−4.54253e7	−0.292242
$220$	0	0
$221$	2.07918e8	1.29574
$222$	−3.10470e7	−0.190451
$223$	−5.48678e7	−0.331322	−0.165661	−	0.986183i	$-0.552976\pi$
$223$	−5.48678e7	−0.331322	−0.165661	+	0.986183i	$0.552976\pi$
$224$	−1.12394e7	−0.0668153
$225$	0	0
$226$	5.04972e7	0.290997
$227$	5.97953e7	0.339294	0.169647	−	0.985505i	$-0.445737\pi$
$227$	5.97953e7	0.339294	0.169647	+	0.985505i	$0.445737\pi$
$228$	−3.72106e6	−0.0207919
$229$	−1.72334e8	−0.948302	−0.474151	−	0.880444i	$-0.657245\pi$
$229$	−1.72334e8	−0.948302	−0.474151	+	0.880444i	$0.657245\pi$
$230$	0	0
$231$	−4.36916e6	−0.0233215
$232$	−3.73725e6	−0.0196492
$233$	−2.30822e8	−1.19545	−0.597725	−	0.801702i	$-0.703928\pi$
$233$	−2.30822e8	−1.19545	−0.597725	+	0.801702i	$0.703928\pi$
$234$	−1.16322e8	−0.593480
$235$	0	0
$236$	−220051.	−0.00108976
$237$	1.54915e6	0.00755915
$238$	8.32242e7	0.400157
$239$	1.12257e8	0.531887	0.265943	−	0.963989i	$-0.414317\pi$
$239$	1.12257e8	0.531887	0.265943	+	0.963989i	$0.414317\pi$
$240$	0	0
$241$	−7.28605e7	−0.335299	−0.167650	−	0.985847i	$-0.553618\pi$
$241$	−7.28605e7	−0.335299	−0.167650	+	0.985847i	$0.553618\pi$
$242$	1.36222e8	0.617865
$243$	1.11896e8	0.500256
$244$	−1.93698e7	−0.0853614
$245$	0	0
$246$	3.00437e7	0.128671
$247$	4.90710e7	0.207198
$248$	2.25439e6	0.00938528
$249$	4.03021e7	0.165436
$250$	0	0
$251$	−2.62256e8	−1.04681	−0.523404	−	0.852085i	$-0.675338\pi$
$251$	−2.62256e8	−1.04681	−0.523404	+	0.852085i	$0.675338\pi$
$252$	−4.65608e7	−0.183281
$253$	1.90315e7	0.0738839
$254$	−3.41964e8	−1.30937
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	2.87621e8	1.05695	0.528475	−	0.848949i	$-0.322764\pi$
$257$	2.87621e8	1.05695	0.528475	+	0.848949i	$0.322764\pi$
$258$	−1.15579e7	−0.0418997
$259$	1.63884e8	0.586121
$260$	0	0
$261$	−1.54820e7	−0.0538997
$262$	−2.03884e7	−0.0700371
$263$	8.02212e7	0.271922	0.135961	−	0.990714i	$-0.456588\pi$
$263$	8.02212e7	0.271922	0.135961	+	0.990714i	$0.456588\pi$
$264$	6.52189e6	0.0218152
$265$	0	0
$266$	1.96419e7	0.0639878
$267$	−6.82359e7	−0.219393
$268$	2.57742e8	0.817924
$269$	−5.50125e7	−0.172317	−0.0861585	−	0.996281i	$-0.527459\pi$
$269$	−5.50125e7	−0.172317	−0.0861585	+	0.996281i	$0.527459\pi$
$270$	0	0
$271$	3.07796e8	0.939442	0.469721	−	0.882815i	$-0.344354\pi$
$271$	3.07796e8	0.939442	0.469721	+	0.882815i	$0.344354\pi$
$272$	−1.24230e8	−0.374313
$273$	−1.90988e7	−0.0568117
$274$	2.36107e8	0.693397
$275$	0	0
$276$	−6.30845e6	−0.0180610
$277$	3.21773e8	0.909641	0.454820	−	0.890583i	$-0.349703\pi$
$277$	3.21773e8	0.909641	0.454820	+	0.890583i	$0.349703\pi$
$278$	1.18976e8	0.332126
$279$	9.33908e6	0.0257448
$280$	0	0
$281$	−8.09521e7	−0.217649	−0.108824	−	0.994061i	$-0.534709\pi$
$281$	−8.09521e7	−0.217649	−0.108824	+	0.994061i	$0.534709\pi$
$282$	−2.75862e7	−0.0732520
$283$	−3.07805e8	−0.807278	−0.403639	−	0.914918i	$-0.632255\pi$
$283$	−3.07805e8	−0.807278	−0.403639	+	0.914918i	$0.632255\pi$
$284$	−1.60502e8	−0.415783
$285$	0	0
$286$	−8.60066e7	−0.217396
$287$	−1.58588e8	−0.395989
$288$	6.95018e7	0.171444
$289$	5.09542e8	1.24176
$290$	0	0
$291$	−2.73090e7	−0.0649652
$292$	−3.57924e8	−0.841299
$293$	−6.33399e7	−0.147109	−0.0735547	−	0.997291i	$-0.523434\pi$
$293$	−6.33399e7	−0.147109	−0.0735547	+	0.997291i	$0.523434\pi$
$294$	−7.64479e6	−0.0175449
$295$	0	0
$296$	−2.44631e8	−0.548266
$297$	5.48759e7	0.121544
$298$	2.19694e8	0.480908
$299$	8.31920e7	0.179983
$300$	0	0
$301$	6.10094e7	0.128948
$302$	−3.34387e8	−0.698595
$303$	2.16219e7	0.0446523
$304$	−2.93197e7	−0.0598551
$305$	0	0
$306$	−5.14638e8	−1.02678
$307$	8.77945e8	1.73174	0.865871	−	0.500267i	$-0.166765\pi$
$307$	8.77945e8	1.73174	0.865871	+	0.500267i	$0.166765\pi$
$308$	−3.44263e7	−0.0671371
$309$	2.03039e6	0.00391493
$310$	0	0
$311$	−9.28974e8	−1.75123	−0.875613	−	0.483014i	$-0.839542\pi$
$311$	−9.28974e8	−1.75123	−0.875613	+	0.483014i	$0.839542\pi$
$312$	2.85091e7	0.0531425
$313$	−7.40062e8	−1.36415	−0.682076	−	0.731281i	$-0.738923\pi$
$313$	−7.40062e8	−1.36415	−0.682076	+	0.731281i	$0.738923\pi$
$314$	−6.18409e8	−1.12725
$315$	0	0
$316$	1.22063e7	0.0217611
$317$	−2.49190e8	−0.439362	−0.219681	−	0.975572i	$-0.570502\pi$
$317$	−2.49190e8	−0.439362	−0.219681	+	0.975572i	$0.570502\pi$
$318$	−4.45692e7	−0.0777213
$319$	−1.14472e7	−0.0197438
$320$	0	0
$321$	1.38470e8	0.233661
$322$	3.32997e7	0.0555833
$323$	2.17102e8	0.358472
$324$	2.78686e8	0.455205
$325$	0	0
$326$	4.19276e7	0.0670252
$327$	−5.39268e7	−0.0852879
$328$	2.36726e8	0.370414
$329$	1.45616e8	0.225436
$330$	0	0
$331$	−3.40070e8	−0.515431	−0.257715	−	0.966221i	$-0.582970\pi$
$331$	−3.40070e8	−0.515431	−0.257715	+	0.966221i	$0.582970\pi$
$332$	3.17556e8	0.476252
$333$	−1.01342e9	−1.50395
$334$	−6.31955e8	−0.928054
$335$	0	0
$336$	1.14115e7	0.0164117
$337$	−5.86495e8	−0.834756	−0.417378	−	0.908733i	$-0.637051\pi$
$337$	−5.86495e8	−0.834756	−0.417378	+	0.908733i	$0.637051\pi$
$338$	1.26028e8	0.177525
$339$	−5.12702e7	−0.0714769
$340$	0	0
$341$	6.90517e6	0.00943048
$342$	−1.21460e8	−0.164189
$343$	4.03536e7	0.0539949
$344$	−9.10695e7	−0.120620
$345$	0	0
$346$	−5.08895e8	−0.660483
$347$	−9.58314e8	−1.23127	−0.615637	−	0.788030i	$-0.711101\pi$
$347$	−9.58314e8	−1.23127	−0.615637	+	0.788030i	$0.711101\pi$
$348$	3.79445e6	0.00482638
$349$	6.52289e8	0.821393	0.410697	−	0.911772i	$-0.365286\pi$
$349$	6.52289e8	0.821393	0.410697	+	0.911772i	$0.365286\pi$
$350$	0	0
$351$	2.39878e8	0.296085
$352$	5.13885e7	0.0628010
$353$	−8.14776e7	−0.0985887	−0.0492943	−	0.998784i	$-0.515697\pi$
$353$	−8.14776e7	−0.0985887	−0.0492943	+	0.998784i	$0.515697\pi$
$354$	223420.	0.000267676 0
$355$	0	0
$356$	−5.37657e8	−0.631583
$357$	−8.44981e7	−0.0982898
$358$	−4.66341e8	−0.537171
$359$	1.13536e9	1.29510	0.647552	−	0.762022i	$-0.275793\pi$
$359$	1.13536e9	1.29510	0.647552	+	0.762022i	$0.275793\pi$
$360$	0	0
$361$	−8.42633e8	−0.942678
$362$	−1.20880e9	−1.33929
$363$	−1.38307e8	−0.151765
$364$	−1.50487e8	−0.163548
$365$	0	0
$366$	1.96663e7	0.0209671
$367$	5.55248e8	0.586349	0.293174	−	0.956059i	$-0.405288\pi$
$367$	5.55248e8	0.586349	0.293174	+	0.956059i	$0.405288\pi$
$368$	−4.97068e7	−0.0519934
$369$	9.80668e8	1.01608
$370$	0	0
$371$	2.35262e8	0.239190
$372$	−2.28889e6	−0.00230529
$373$	−1.08270e8	−0.108026	−0.0540130	−	0.998540i	$-0.517201\pi$
$373$	−1.08270e8	−0.108026	−0.0540130	+	0.998540i	$0.517201\pi$
$374$	−3.80515e8	−0.376116
$375$	0	0
$376$	−2.17362e8	−0.210876
$377$	−5.00389e7	−0.0480964
$378$	9.60173e7	0.0914383
$379$	1.02345e8	0.0965668	0.0482834	−	0.998834i	$-0.484625\pi$
$379$	1.02345e8	0.0965668	0.0482834	+	0.998834i	$0.484625\pi$
$380$	0	0
$381$	3.47199e8	0.321618
$382$	−8.05312e8	−0.739166
$383$	−8.11322e8	−0.737900	−0.368950	−	0.929449i	$-0.620283\pi$
$383$	−8.11322e8	−0.737900	−0.368950	+	0.929449i	$0.620283\pi$
$384$	−1.70340e7	−0.0153518
$385$	0	0
$386$	−6.06657e8	−0.536893
$387$	−3.77267e8	−0.330872
$388$	−2.15179e8	−0.187020
$389$	1.07770e9	0.928266	0.464133	−	0.885765i	$-0.346366\pi$
$389$	1.07770e9	0.928266	0.464133	+	0.885765i	$0.346366\pi$
$390$	0	0
$391$	3.68062e8	0.311389
$392$	−6.02363e7	−0.0505076
$393$	2.07005e7	0.0172031
$394$	−7.13780e8	−0.587932
$395$	0	0
$396$	2.12884e8	0.172270
$397$	−4.78761e8	−0.384018	−0.192009	−	0.981393i	$-0.561500\pi$
$397$	−4.78761e8	−0.384018	−0.192009	+	0.981393i	$0.561500\pi$
$398$	9.44270e7	0.0750767
$399$	−1.99425e7	−0.0157172
$400$	0	0
$401$	−1.57281e9	−1.21807	−0.609034	−	0.793144i	$-0.708443\pi$
$401$	−1.57281e9	−1.21807	−0.609034	+	0.793144i	$0.708443\pi$
$402$	−2.61687e8	−0.200905
$403$	3.01845e7	0.0229729
$404$	1.70367e8	0.128544
$405$	0	0
$406$	−2.00293e7	−0.0148534
$407$	−7.49304e8	−0.550906
$408$	1.26131e8	0.0919417
$409$	1.70492e9	1.23217	0.616086	−	0.787679i	$-0.288717\pi$
$409$	1.70492e9	1.23217	0.616086	+	0.787679i	$0.288717\pi$
$410$	0	0
$411$	−2.39721e8	−0.170318
$412$	1.59982e7	0.0112702
$413$	−1.17934e6	−0.000823783 0
$414$	−2.05917e8	−0.142623
$415$	0	0
$416$	2.24634e8	0.152985
$417$	−1.20797e8	−0.0815796
$418$	−8.98059e7	−0.0601434
$419$	−5.32172e8	−0.353429	−0.176715	−	0.984262i	$-0.556547\pi$
$419$	−5.32172e8	−0.353429	−0.176715	+	0.984262i	$0.556547\pi$
$420$	0	0
$421$	1.33991e8	0.0875162	0.0437581	−	0.999042i	$-0.486067\pi$
$421$	1.33991e8	0.0875162	0.0437581	+	0.999042i	$0.486067\pi$
$422$	−1.52419e9	−0.987289
$423$	−9.00451e8	−0.578454
$424$	−3.51178e8	−0.223742
$425$	0	0
$426$	1.62959e8	0.102128
$427$	−1.03810e8	−0.0645271
$428$	1.09106e9	0.672658
$429$	8.73231e7	0.0533984
$430$	0	0
$431$	1.66271e9	1.00034	0.500169	−	0.865928i	$-0.333271\pi$
$431$	1.66271e9	1.00034	0.500169	+	0.865928i	$0.333271\pi$
$432$	−1.43326e8	−0.0855327
$433$	1.23313e9	0.729965	0.364983	−	0.931014i	$-0.381075\pi$
$433$	1.23313e9	0.729965	0.364983	+	0.931014i	$0.381075\pi$
$434$	1.20821e7	0.00709460
$435$	0	0
$436$	−4.24911e8	−0.245524
$437$	8.68670e7	0.0497931
$438$	3.63402e8	0.206647
$439$	−1.77669e8	−0.100227	−0.0501137	−	0.998744i	$-0.515958\pi$
$439$	−1.77669e8	−0.100227	−0.0501137	+	0.998744i	$0.515958\pi$
$440$	0	0
$441$	−2.49537e8	−0.138548
$442$	−1.66334e9	−0.916228
$443$	−2.79169e9	−1.52565	−0.762825	−	0.646605i	$-0.776188\pi$
$443$	−2.79169e9	−1.52565	−0.762825	+	0.646605i	$0.776188\pi$
$444$	2.48376e8	0.134669
$445$	0	0
$446$	4.38942e8	0.234280
$447$	−2.23057e8	−0.118124
$448$	8.99154e7	0.0472456
$449$	2.76253e9	1.44028	0.720138	−	0.693831i	$-0.244079\pi$
$449$	2.76253e9	1.44028	0.720138	+	0.693831i	$0.244079\pi$
$450$	0	0
$451$	7.25090e8	0.372198
$452$	−4.03978e8	−0.205766
$453$	3.39506e8	0.171594
$454$	−4.78363e8	−0.239917
$455$	0	0
$456$	2.97684e7	0.0147021
$457$	−7.69684e8	−0.377230	−0.188615	−	0.982051i	$-0.560400\pi$
$457$	−7.69684e8	−0.377230	−0.188615	+	0.982051i	$0.560400\pi$
$458$	1.37867e9	0.670551
$459$	1.06128e9	0.512256
$460$	0	0
$461$	3.88680e9	1.84773	0.923865	−	0.382719i	$-0.125012\pi$
$461$	3.88680e9	1.84773	0.923865	+	0.382719i	$0.125012\pi$
$462$	3.49532e7	0.0164908
$463$	−1.84763e9	−0.865132	−0.432566	−	0.901602i	$-0.642392\pi$
$463$	−1.84763e9	−0.865132	−0.432566	+	0.901602i	$0.642392\pi$
$464$	2.98980e7	0.0138941
$465$	0	0
$466$	1.84657e9	0.845310
$467$	−2.13137e9	−0.968389	−0.484194	−	0.874960i	$-0.660887\pi$
$467$	−2.13137e9	−0.968389	−0.484194	+	0.874960i	$0.660887\pi$
$468$	9.30575e8	0.419654
$469$	1.38133e9	0.618293
$470$	0	0
$471$	6.27875e8	0.276885
$472$	1.76041e6	0.000770579 0
$473$	−2.78945e8	−0.121201
$474$	−1.23932e7	−0.00534513
$475$	0	0
$476$	−6.65794e8	−0.282954
$477$	−1.45480e9	−0.613747
$478$	−8.98052e8	−0.376101
$479$	1.12045e9	0.465819	0.232910	−	0.972498i	$-0.425175\pi$
$479$	1.12045e9	0.465819	0.232910	+	0.972498i	$0.425175\pi$
$480$	0	0
$481$	−3.27543e9	−1.34202
$482$	5.82884e8	0.237092
$483$	−3.38094e7	−0.0136528
$484$	−1.08978e9	−0.436896
$485$	0	0
$486$	−8.95167e8	−0.353734
$487$	3.36201e9	1.31901	0.659505	−	0.751700i	$-0.270766\pi$
$487$	3.36201e9	1.31901	0.659505	+	0.751700i	$0.270766\pi$
$488$	1.54959e8	0.0603596
$489$	−4.25693e7	−0.0164633
$490$	0	0
$491$	−3.38603e9	−1.29094	−0.645470	−	0.763786i	$-0.723338\pi$
$491$	−3.38603e9	−1.29094	−0.645470	+	0.763786i	$0.723338\pi$
$492$	−2.40350e8	−0.0909841
$493$	−2.21385e8	−0.0832115
$494$	−3.92568e8	−0.146511
$495$	0	0
$496$	−1.80351e7	−0.00663640
$497$	−8.60190e8	−0.314302
$498$	−3.22417e8	−0.116981
$499$	1.78302e9	0.642398	0.321199	−	0.947012i	$-0.395914\pi$
$499$	1.78302e9	0.642398	0.321199	+	0.947012i	$0.395914\pi$
$500$	0	0
$501$	6.41628e8	0.227956
$502$	2.09805e9	0.740205
$503$	1.96251e9	0.687581	0.343790	−	0.939046i	$-0.388289\pi$
$503$	1.96251e9	0.687581	0.343790	+	0.939046i	$0.388289\pi$
$504$	3.72486e8	0.129599
$505$	0	0
$506$	−1.52252e8	−0.0522438
$507$	−1.27957e8	−0.0436052
$508$	2.73571e9	0.925865
$509$	−1.81259e9	−0.609238	−0.304619	−	0.952474i	$-0.598529\pi$
$509$	−1.81259e9	−0.609238	−0.304619	+	0.952474i	$0.598529\pi$
$510$	0	0
$511$	−1.91825e9	−0.635962
$512$	−1.34218e8	−0.0441942
$513$	2.50475e8	0.0819132
$514$	−2.30097e9	−0.747376
$515$	0	0
$516$	9.24635e7	0.0296276
$517$	−6.65779e8	−0.211891
$518$	−1.31107e9	−0.414450
$519$	5.16684e8	0.162233
$520$	0	0
$521$	1.41631e9	0.438758	0.219379	−	0.975640i	$-0.429597\pi$
$521$	1.41631e9	0.438758	0.219379	+	0.975640i	$0.429597\pi$
$522$	1.23856e8	0.0381128
$523$	3.18021e9	0.972075	0.486037	−	0.873938i	$-0.338442\pi$
$523$	3.18021e9	0.972075	0.486037	+	0.873938i	$0.338442\pi$
$524$	1.63107e8	0.0495237
$525$	0	0
$526$	−6.41770e8	−0.192278
$527$	1.33544e8	0.0397454
$528$	−5.21751e7	−0.0154257
$529$	−3.25756e9	−0.956747
$530$	0	0
$531$	7.29273e6	0.00211378
$532$	−1.57135e8	−0.0452462
$533$	3.16958e9	0.906685
$534$	5.45887e8	0.155135
$535$	0	0
$536$	−2.06193e9	−0.578360
$537$	4.73479e8	0.131944
$538$	4.40100e8	0.121847
$539$	−1.84503e8	−0.0507509
$540$	0	0
$541$	2.09280e9	0.568247	0.284124	−	0.958788i	$-0.408297\pi$
$541$	2.09280e9	0.568247	0.284124	+	0.958788i	$0.408297\pi$
$542$	−2.46237e9	−0.664286
$543$	1.22731e9	0.328968
$544$	9.93838e8	0.264679
$545$	0	0
$546$	1.52791e8	0.0401719
$547$	2.89817e9	0.757127	0.378563	−	0.925575i	$-0.376418\pi$
$547$	2.89817e9	0.757127	0.378563	+	0.925575i	$0.376418\pi$
$548$	−1.88886e9	−0.490306
$549$	6.41936e8	0.165573
$550$	0	0
$551$	−5.22493e7	−0.0133061
$552$	5.04676e7	0.0127710
$553$	6.54183e7	0.0164498
$554$	−2.57418e9	−0.643213
$555$	0	0
$556$	−9.51810e8	−0.234849
$557$	4.62851e9	1.13488	0.567438	−	0.823416i	$-0.307935\pi$
$557$	4.62851e9	1.13488	0.567438	+	0.823416i	$0.307935\pi$
$558$	−7.47126e7	−0.0182043
$559$	−1.21935e9	−0.295248
$560$	0	0
$561$	3.86340e8	0.0923845
$562$	6.47617e8	0.153901
$563$	2.76931e9	0.654021	0.327011	−	0.945021i	$-0.393959\pi$
$563$	2.76931e9	0.654021	0.327011	+	0.945021i	$0.393959\pi$
$564$	2.20689e8	0.0517970
$565$	0	0
$566$	2.46244e9	0.570832
$567$	1.49358e9	0.344103
$568$	1.28402e9	0.294003
$569$	−5.09524e9	−1.15950	−0.579751	−	0.814794i	$-0.696850\pi$
$569$	−5.09524e9	−1.15950	−0.579751	+	0.814794i	$0.696850\pi$
$570$	0	0
$571$	8.03829e9	1.80691	0.903456	−	0.428680i	$-0.141021\pi$
$571$	8.03829e9	1.80691	0.903456	+	0.428680i	$0.141021\pi$
$572$	6.88053e8	0.153722
$573$	8.17638e8	0.181560
$574$	1.26870e9	0.280007
$575$	0	0
$576$	−5.56014e8	−0.121229
$577$	2.23461e9	0.484268	0.242134	−	0.970243i	$-0.422153\pi$
$577$	2.23461e9	0.484268	0.242134	+	0.970243i	$0.422153\pi$
$578$	−4.07634e9	−0.878057
$579$	6.15943e8	0.131876
$580$	0	0
$581$	1.70190e9	0.360013
$582$	2.18472e8	0.0459374
$583$	−1.07566e9	−0.224819
$584$	2.86339e9	0.594888
$585$	0	0
$586$	5.06719e8	0.104022
$587$	7.07231e9	1.44320	0.721602	−	0.692308i	$-0.243406\pi$
$587$	7.07231e9	1.44320	0.721602	+	0.692308i	$0.243406\pi$
$588$	6.11583e7	0.0124061
$589$	3.15179e7	0.00635556
$590$	0	0
$591$	7.24705e8	0.144413
$592$	1.95705e9	0.387682
$593$	−1.86473e9	−0.367219	−0.183610	−	0.982999i	$-0.558778\pi$
$593$	−1.86473e9	−0.367219	−0.183610	+	0.982999i	$0.558778\pi$
$594$	−4.39007e8	−0.0859447
$595$	0	0
$596$	−1.75755e9	−0.340053
$597$	−9.58724e7	−0.0184409
$598$	−6.65536e8	−0.127267
$599$	−2.05712e9	−0.391080	−0.195540	−	0.980696i	$-0.562646\pi$
$599$	−2.05712e9	−0.391080	−0.195540	+	0.980696i	$0.562646\pi$
$600$	0	0
$601$	3.69684e8	0.0694657	0.0347329	−	0.999397i	$-0.488942\pi$
$601$	3.69684e8	0.0694657	0.0347329	+	0.999397i	$0.488942\pi$
$602$	−4.88075e8	−0.0911799
$603$	−8.54183e9	−1.58650
$604$	2.67510e9	0.493981
$605$	0	0
$606$	−1.72975e8	−0.0315739
$607$	−3.13002e9	−0.568050	−0.284025	−	0.958817i	$-0.591670\pi$
$607$	−3.13002e9	−0.568050	−0.284025	+	0.958817i	$0.591670\pi$
$608$	2.34557e8	0.0423240
$609$	2.03359e7	0.00364840
$610$	0	0
$611$	−2.91031e9	−0.516173
$612$	4.11710e9	0.726042
$613$	−6.41139e9	−1.12419	−0.562096	−	0.827072i	$-0.690005\pi$
$613$	−6.41139e9	−1.12419	−0.562096	+	0.827072i	$0.690005\pi$
$614$	−7.02356e9	−1.22453
$615$	0	0
$616$	2.75410e8	0.0474731
$617$	−1.24237e9	−0.212938	−0.106469	−	0.994316i	$-0.533954\pi$
$617$	−1.24237e9	−0.212938	−0.106469	+	0.994316i	$0.533954\pi$
$618$	−1.62431e7	−0.00276827
$619$	−7.46312e9	−1.26475	−0.632373	−	0.774664i	$-0.717919\pi$
$619$	−7.46312e9	−1.26475	−0.632373	+	0.774664i	$0.717919\pi$
$620$	0	0
$621$	4.24640e8	0.0711542
$622$	7.43179e9	1.23830
$623$	−2.88151e9	−0.477432
$624$	−2.28072e8	−0.0375774
$625$	0	0
$626$	5.92049e9	0.964601
$627$	9.11806e7	0.0147729
$628$	4.94727e9	0.797089
$629$	−1.44913e10	−2.32183
$630$	0	0
$631$	−5.15836e9	−0.817351	−0.408676	−	0.912680i	$-0.634009\pi$
$631$	−5.15836e9	−0.817351	−0.408676	+	0.912680i	$0.634009\pi$
$632$	−9.76506e7	−0.0153874
$633$	1.54752e9	0.242506
$634$	1.99352e9	0.310676
$635$	0	0
$636$	3.56554e8	0.0549572
$637$	−8.06518e8	−0.123631
$638$	9.15774e7	0.0139610
$639$	5.31920e9	0.806480
$640$	0	0
$641$	−4.57861e9	−0.686642	−0.343321	−	0.939218i	$-0.611552\pi$
$641$	−4.57861e9	−0.686642	−0.343321	+	0.939218i	$0.611552\pi$
$642$	−1.10776e9	−0.165224
$643$	3.25934e9	0.483494	0.241747	−	0.970339i	$-0.422280\pi$
$643$	3.25934e9	0.483494	0.241747	+	0.970339i	$0.422280\pi$
$644$	−2.66397e8	−0.0393033
$645$	0	0
$646$	−1.73682e9	−0.253478
$647$	9.75723e9	1.41632	0.708160	−	0.706052i	$-0.249526\pi$
$647$	9.75723e9	1.41632	0.708160	+	0.706052i	$0.249526\pi$
$648$	−2.22949e9	−0.321879
$649$	5.39213e6	0.000774290 0
$650$	0	0
$651$	−1.22670e7	−0.00174263
$652$	−3.35421e8	−0.0473940
$653$	4.87557e9	0.685219	0.342610	−	0.939478i	$-0.388689\pi$
$653$	4.87557e9	0.685219	0.342610	+	0.939478i	$0.388689\pi$
$654$	4.31415e8	0.0603077
$655$	0	0
$656$	−1.89381e9	−0.261922
$657$	1.18620e10	1.63184
$658$	−1.16493e9	−0.159407
$659$	−5.19574e9	−0.707210	−0.353605	−	0.935395i	$-0.615044\pi$
$659$	−5.19574e9	−0.707210	−0.353605	+	0.935395i	$0.615044\pi$
$660$	0	0
$661$	9.49831e9	1.27921	0.639604	−	0.768704i	$-0.279098\pi$
$661$	9.49831e9	1.27921	0.639604	+	0.768704i	$0.279098\pi$
$662$	2.72056e9	0.364464
$663$	1.68880e9	0.225051
$664$	−2.54045e9	−0.336761
$665$	0	0
$666$	8.10733e9	1.06345
$667$	−8.85804e7	−0.0115584
$668$	5.05564e9	0.656233
$669$	−4.45661e8	−0.0575458
$670$	0	0
$671$	4.74637e8	0.0606503
$672$	−9.12917e7	−0.0116048
$673$	4.20205e9	0.531383	0.265692	−	0.964058i	$-0.414400\pi$
$673$	4.20205e9	0.531383	0.265692	+	0.964058i	$0.414400\pi$
$674$	4.69196e9	0.590261
$675$	0	0
$676$	−1.00823e9	−0.125529
$677$	−1.34115e10	−1.66119	−0.830593	−	0.556879i	$-0.811999\pi$
$677$	−1.34115e10	−1.66119	−0.830593	+	0.556879i	$0.811999\pi$
$678$	4.10161e8	0.0505418
$679$	−1.15322e9	−0.141374
$680$	0	0
$681$	4.85685e8	0.0589304
$682$	−5.52414e7	−0.00666836
$683$	6.53462e9	0.784780	0.392390	−	0.919799i	$-0.371648\pi$
$683$	6.53462e9	0.784780	0.392390	+	0.919799i	$0.371648\pi$
$684$	9.71683e8	0.116099
$685$	0	0
$686$	−3.22829e8	−0.0381802
$687$	−1.39977e9	−0.164706
$688$	7.28556e8	0.0852910
$689$	−4.70201e9	−0.547666
$690$	0	0
$691$	−5.38588e9	−0.620989	−0.310494	−	0.950575i	$-0.600495\pi$
$691$	−5.38588e9	−0.620989	−0.310494	+	0.950575i	$0.600495\pi$
$692$	4.07116e9	0.467032
$693$	1.14092e9	0.130224
$694$	7.66652e9	0.870642
$695$	0	0
$696$	−3.03556e7	−0.00341277
$697$	1.40230e10	1.56865
$698$	−5.21831e9	−0.580813
$699$	−1.87484e9	−0.207632
$700$	0	0
$701$	−7.53668e9	−0.826356	−0.413178	−	0.910650i	$-0.635581\pi$
$701$	−7.53668e9	−0.826356	−0.413178	+	0.910650i	$0.635581\pi$
$702$	−1.91903e9	−0.209364
$703$	−3.42012e9	−0.371276
$704$	−4.11108e8	−0.0444070
$705$	0	0
$706$	6.51821e8	0.0697127
$707$	9.13061e8	0.0971699
$708$	−1.78736e6	−0.000189276 0
$709$	1.49758e10	1.57807	0.789036	−	0.614347i	$-0.210580\pi$
$709$	1.49758e10	1.57807	0.789036	+	0.614347i	$0.210580\pi$
$710$	0	0
$711$	−4.04530e8	−0.0422092
$712$	4.30126e9	0.446597
$713$	5.34335e7	0.00552078
$714$	6.75985e8	0.0695014
$715$	0	0
$716$	3.73073e9	0.379837
$717$	9.11799e8	0.0923808
$718$	−9.08291e9	−0.915776
$719$	−1.71309e10	−1.71882	−0.859409	−	0.511289i	$-0.829168\pi$
$719$	−1.71309e10	−1.71882	−0.859409	+	0.511289i	$0.829168\pi$
$720$	0	0
$721$	8.57404e7	0.00851946
$722$	6.74106e9	0.666574
$723$	−5.91806e8	−0.0582365
$724$	9.67042e9	0.947022
$725$	0	0
$726$	1.10646e9	0.107314
$727$	1.59520e10	1.53973	0.769865	−	0.638207i	$-0.220324\pi$
$727$	1.59520e10	1.53973	0.769865	+	0.638207i	$0.220324\pi$
$728$	1.20390e9	0.115646
$729$	−8.61434e9	−0.823523
$730$	0	0
$731$	−5.39472e9	−0.510808
$732$	−1.57331e8	−0.0148260
$733$	1.08379e10	1.01644	0.508220	−	0.861227i	$-0.330304\pi$
$733$	1.08379e10	1.01644	0.508220	+	0.861227i	$0.330304\pi$
$734$	−4.44199e9	−0.414611
$735$	0	0
$736$	3.97654e8	0.0367649
$737$	−6.31569e9	−0.581145
$738$	−7.84534e9	−0.718480
$739$	1.83957e9	0.167672	0.0838359	−	0.996480i	$-0.473283\pi$
$739$	1.83957e9	0.167672	0.0838359	+	0.996480i	$0.473283\pi$
$740$	0	0
$741$	3.98577e8	0.0359872
$742$	−1.88210e9	−0.169133
$743$	1.50293e10	1.34424	0.672122	−	0.740441i	$-0.265383\pi$
$743$	1.50293e10	1.34424	0.672122	+	0.740441i	$0.265383\pi$
$744$	1.83111e7	0.00163008
$745$	0	0
$746$	8.66163e8	0.0763860
$747$	−1.05241e10	−0.923770
$748$	3.04412e9	0.265954
$749$	5.84738e9	0.508481
$750$	0	0
$751$	1.58853e10	1.36853	0.684265	−	0.729233i	$-0.260123\pi$
$751$	1.58853e10	1.36853	0.684265	+	0.729233i	$0.260123\pi$
$752$	1.73890e9	0.149112
$753$	−2.13016e9	−0.181815
$754$	4.00311e8	0.0340093
$755$	0	0
$756$	−7.68139e8	−0.0646567
$757$	−2.07844e10	−1.74141	−0.870707	−	0.491803i	$-0.836338\pi$
$757$	−2.07844e10	−1.74141	−0.870707	+	0.491803i	$0.836338\pi$
$758$	−8.18757e8	−0.0682830
$759$	1.54582e8	0.0128325
$760$	0	0
$761$	5.61591e9	0.461928	0.230964	−	0.972962i	$-0.425812\pi$
$761$	5.61591e9	0.461928	0.230964	+	0.972962i	$0.425812\pi$
$762$	−2.77759e9	−0.227418
$763$	−2.27725e9	−0.185599
$764$	6.44249e9	0.522670
$765$	0	0
$766$	6.49058e9	0.521774
$767$	2.35705e7	0.00188619
$768$	1.36272e8	0.0108553
$769$	−1.64625e9	−0.130543	−0.0652716	−	0.997868i	$-0.520791\pi$
$769$	−1.64625e9	−0.130543	−0.0652716	+	0.997868i	$0.520791\pi$
$770$	0	0
$771$	2.33619e9	0.183577
$772$	4.85326e9	0.379641
$773$	3.77661e9	0.294086	0.147043	−	0.989130i	$-0.453024\pi$
$773$	3.77661e9	0.294086	0.147043	+	0.989130i	$0.453024\pi$
$774$	3.01814e9	0.233962
$775$	0	0
$776$	1.72143e9	0.132243
$777$	1.33114e9	0.101801
$778$	−8.62157e9	−0.656383
$779$	3.30960e9	0.250838
$780$	0	0
$781$	3.93293e9	0.295419
$782$	−2.94450e9	−0.220185
$783$	−2.55416e8	−0.0190143
$784$	4.81890e8	0.0357143
$785$	0	0
$786$	−1.65604e8	−0.0121644
$787$	−1.64925e10	−1.20607	−0.603037	−	0.797713i	$-0.706043\pi$
$787$	−1.64925e10	−1.20607	−0.603037	+	0.797713i	$0.706043\pi$
$788$	5.71024e9	0.415731
$789$	6.51593e8	0.0472288
$790$	0	0
$791$	−2.16507e9	−0.155544
$792$	−1.70307e9	−0.121813
$793$	2.07478e9	0.147746
$794$	3.83009e9	0.271542
$795$	0	0
$796$	−7.55416e8	−0.0530873
$797$	−2.04858e10	−1.43334	−0.716670	−	0.697413i	$-0.754335\pi$
$797$	−2.04858e10	−1.43334	−0.716670	+	0.697413i	$0.754335\pi$
$798$	1.59540e8	0.0111137
$799$	−1.28760e10	−0.893030
$800$	0	0
$801$	1.78185e10	1.22506
$802$	1.25825e10	0.861304
$803$	8.77055e9	0.597754
$804$	2.09350e9	0.142061
$805$	0	0
$806$	−2.41476e8	−0.0162443
$807$	−4.46836e8	−0.0299289
$808$	−1.36294e9	−0.0908941
$809$	3.65098e9	0.242432	0.121216	−	0.992626i	$-0.461321\pi$
$809$	3.65098e9	0.242432	0.121216	+	0.992626i	$0.461321\pi$
$810$	0	0
$811$	−1.79825e10	−1.18380	−0.591898	−	0.806013i	$-0.701621\pi$
$811$	−1.79825e10	−1.18380	−0.591898	+	0.806013i	$0.701621\pi$
$812$	1.60235e8	0.0105029
$813$	2.50006e9	0.163167
$814$	5.99444e9	0.389550
$815$	0	0
$816$	−1.00905e9	−0.0650126
$817$	−1.27322e9	−0.0816817
$818$	−1.36393e10	−0.871278
$819$	4.98730e9	0.317228
$820$	0	0
$821$	−2.45590e9	−0.154885	−0.0774427	−	0.996997i	$-0.524675\pi$
$821$	−2.45590e9	−0.154885	−0.0774427	+	0.996997i	$0.524675\pi$
$822$	1.91777e9	0.120433
$823$	1.71346e10	1.07146	0.535728	−	0.844390i	$-0.320037\pi$
$823$	1.71346e10	1.07146	0.535728	+	0.844390i	$0.320037\pi$
$824$	−1.27986e8	−0.00796922
$825$	0	0
$826$	9.43470e6	0.000582503 0
$827$	−1.70537e10	−1.04846	−0.524228	−	0.851578i	$-0.675646\pi$
$827$	−1.70537e10	−1.04846	−0.524228	+	0.851578i	$0.675646\pi$
$828$	1.64733e9	0.100850
$829$	2.05756e10	1.25433	0.627163	−	0.778888i	$-0.284216\pi$
$829$	2.05756e10	1.25433	0.627163	+	0.778888i	$0.284216\pi$
$830$	0	0
$831$	2.61358e9	0.157991
$832$	−1.79707e9	−0.108177
$833$	−3.56824e9	−0.213893
$834$	9.66379e8	0.0576855
$835$	0	0
$836$	7.18447e8	0.0425278
$837$	1.54072e8	0.00908207
$838$	4.25737e9	0.249912
$839$	−1.27697e10	−0.746472	−0.373236	−	0.927736i	$-0.621752\pi$
$839$	−1.27697e10	−0.746472	−0.373236	+	0.927736i	$0.621752\pi$
$840$	0	0
$841$	−1.71966e10	−0.996911
$842$	−1.07193e9	−0.0618833
$843$	−6.57530e8	−0.0378024
$844$	1.21935e10	0.698119
$845$	0	0
$846$	7.20361e9	0.409029
$847$	−5.84052e9	−0.330263
$848$	2.80943e9	0.158209
$849$	−2.50013e9	−0.140212
$850$	0	0
$851$	−5.79826e9	−0.322511
$852$	−1.30367e9	−0.0722153
$853$	1.21052e10	0.667807	0.333904	−	0.942607i	$-0.391634\pi$
$853$	1.21052e10	0.667807	0.333904	+	0.942607i	$0.391634\pi$
$854$	8.30481e8	0.0456276
$855$	0	0
$856$	−8.72845e9	−0.475641
$857$	−5.38657e8	−0.0292334	−0.0146167	−	0.999893i	$-0.504653\pi$
$857$	−5.38657e8	−0.0292334	−0.0146167	+	0.999893i	$0.504653\pi$
$858$	−6.98585e8	−0.0377584
$859$	−1.80931e10	−0.973948	−0.486974	−	0.873416i	$-0.661899\pi$
$859$	−1.80931e10	−0.973948	−0.486974	+	0.873416i	$0.661899\pi$
$860$	0	0
$861$	−1.28812e9	−0.0687775
$862$	−1.33017e10	−0.707345
$863$	−3.56217e10	−1.88659	−0.943293	−	0.331962i	$-0.892289\pi$
$863$	−3.56217e10	−1.88659	−0.943293	+	0.331962i	$0.892289\pi$
$864$	1.14661e9	0.0604808
$865$	0	0
$866$	−9.86506e9	−0.516163
$867$	4.13873e9	0.215675
$868$	−9.66568e7	−0.00501664
$869$	−2.99103e8	−0.0154615
$870$	0	0
$871$	−2.76077e10	−1.41569
$872$	3.39928e9	0.173612
$873$	7.13124e9	0.362757
$874$	−6.94936e8	−0.0352091
$875$	0	0
$876$	−2.90722e9	−0.146121
$877$	−1.13282e10	−0.567102	−0.283551	−	0.958957i	$-0.591513\pi$
$877$	−1.13282e10	−0.567102	−0.283551	+	0.958957i	$0.591513\pi$
$878$	1.42135e9	0.0708715
$879$	−5.14475e8	−0.0255507
$880$	0	0
$881$	3.29464e10	1.62327	0.811637	−	0.584162i	$-0.198577\pi$
$881$	3.29464e10	1.62327	0.811637	+	0.584162i	$0.198577\pi$
$882$	1.99629e9	0.0979680
$883$	−1.80365e10	−0.881639	−0.440820	−	0.897596i	$-0.645312\pi$
$883$	−1.80365e10	−0.881639	−0.440820	+	0.897596i	$0.645312\pi$
$884$	1.33067e10	0.647871
$885$	0	0
$886$	2.23336e10	1.07880
$887$	−3.25522e10	−1.56620	−0.783101	−	0.621894i	$-0.786363\pi$
$887$	−3.25522e10	−1.56620	−0.783101	+	0.621894i	$0.786363\pi$
$888$	−1.98701e9	−0.0952257
$889$	1.46617e10	0.699888
$890$	0	0
$891$	−6.82890e9	−0.323429
$892$	−3.51154e9	−0.165661
$893$	−3.03888e9	−0.142802
$894$	1.78446e9	0.0835266
$895$	0	0
$896$	−7.19323e8	−0.0334077
$897$	6.75723e8	0.0312604
$898$	−2.21003e10	−1.01843
$899$	−3.21396e7	−0.00147530
$900$	0	0
$901$	−2.08029e10	−0.947516
$902$	−5.80072e9	−0.263184
$903$	4.95546e8	0.0223963
$904$	3.23182e9	0.145498
$905$	0	0
$906$	−2.71605e9	−0.121336
$907$	2.41647e10	1.07536	0.537682	−	0.843148i	$-0.319300\pi$
$907$	2.41647e10	1.07536	0.537682	+	0.843148i	$0.319300\pi$
$908$	3.82690e9	0.169647
$909$	−5.64614e9	−0.249332
$910$	0	0
$911$	−1.97815e10	−0.866854	−0.433427	−	0.901189i	$-0.642696\pi$
$911$	−1.97815e10	−0.866854	−0.433427	+	0.901189i	$0.642696\pi$
$912$	−2.38148e8	−0.0103959
$913$	−7.78137e9	−0.338383
$914$	6.15747e9	0.266742
$915$	0	0
$916$	−1.10294e10	−0.474151
$917$	8.74152e8	0.0374364
$918$	−8.49027e9	−0.362220
$919$	2.71038e10	1.15193	0.575965	−	0.817474i	$-0.304626\pi$
$919$	2.71038e10	1.15193	0.575965	+	0.817474i	$0.304626\pi$
$920$	0	0
$921$	7.13107e9	0.300778
$922$	−3.10944e10	−1.30654
$923$	1.71920e10	0.719648
$924$	−2.79626e8	−0.0116607
$925$	0	0
$926$	1.47811e10	0.611741
$927$	−5.30197e8	−0.0218604
$928$	−2.39184e8	−0.00982458
$929$	−3.62682e10	−1.48413	−0.742063	−	0.670331i	$-0.766152\pi$
$929$	−3.62682e10	−1.48413	−0.742063	+	0.670331i	$0.766152\pi$
$930$	0	0
$931$	−8.42145e8	−0.0342029
$932$	−1.47726e10	−0.597725
$933$	−7.54555e9	−0.304162
$934$	1.70510e10	0.684754
$935$	0	0
$936$	−7.44460e9	−0.296740
$937$	−4.10598e10	−1.63053	−0.815265	−	0.579089i	$-0.803408\pi$
$937$	−4.10598e10	−1.63053	−0.815265	+	0.579089i	$0.803408\pi$
$938$	−1.10507e10	−0.437199
$939$	−6.01112e9	−0.236933
$940$	0	0
$941$	1.35312e10	0.529388	0.264694	−	0.964332i	$-0.414729\pi$
$941$	1.35312e10	0.529388	0.264694	+	0.964332i	$0.414729\pi$
$942$	−5.02300e9	−0.195787
$943$	5.61089e9	0.217892
$944$	−1.40833e7	−0.000544881 0
$945$	0	0
$946$	2.23156e9	0.0857018
$947$	−6.83192e9	−0.261407	−0.130704	−	0.991421i	$-0.541724\pi$
$947$	−6.83192e9	−0.261407	−0.130704	+	0.991421i	$0.541724\pi$
$948$	9.91453e7	0.00377958
$949$	3.83386e10	1.45614
$950$	0	0
$951$	−2.02403e9	−0.0763108
$952$	5.32635e9	0.200079
$953$	−1.42589e8	−0.00533654	−0.00266827	−	0.999996i	$-0.500849\pi$
$953$	−1.42589e8	−0.00533654	−0.00266827	+	0.999996i	$0.500849\pi$
$954$	1.16384e10	0.433984
$955$	0	0
$956$	7.18442e9	0.265943
$957$	−9.29791e7	−0.00342921
$958$	−8.96358e9	−0.329384
$959$	−1.01231e10	−0.370636
$960$	0	0
$961$	−2.74932e10	−0.999295
$962$	2.62034e10	0.948954
$963$	−3.61587e10	−1.30473
$964$	−4.66307e9	−0.167650
$965$	0	0
$966$	2.70475e8	0.00965400
$967$	5.48898e9	0.195209	0.0976043	−	0.995225i	$-0.468882\pi$
$967$	5.48898e9	0.195209	0.0976043	+	0.995225i	$0.468882\pi$
$968$	8.71821e9	0.308932
$969$	1.76340e9	0.0622614
$970$	0	0
$971$	−5.22931e10	−1.83306	−0.916531	−	0.399964i	$-0.869023\pi$
$971$	−5.22931e10	−1.83306	−0.916531	+	0.399964i	$0.869023\pi$
$972$	7.16134e9	0.250128
$973$	−5.10111e9	−0.177529
$974$	−2.68961e10	−0.932681
$975$	0	0
$976$	−1.23967e9	−0.0426807
$977$	−4.06965e10	−1.39613	−0.698065	−	0.716034i	$-0.745955\pi$
$977$	−4.06965e10	−1.39613	−0.698065	+	0.716034i	$0.745955\pi$
$978$	3.40555e8	0.0116413
$979$	1.31747e10	0.448748
$980$	0	0
$981$	1.40820e10	0.476236
$982$	2.70883e10	0.912832
$983$	−2.31376e10	−0.776930	−0.388465	−	0.921464i	$-0.626995\pi$
$983$	−2.31376e10	−0.776930	−0.388465	+	0.921464i	$0.626995\pi$
$984$	1.92280e9	0.0643355
$985$	0	0
$986$	1.77108e9	0.0588394
$987$	1.18276e9	0.0391548
$988$	3.14054e9	0.103599
$989$	−2.15853e9	−0.0709531
$990$	0	0
$991$	−4.28251e10	−1.39779	−0.698893	−	0.715226i	$-0.746324\pi$
$991$	−4.28251e10	−1.39779	−0.698893	+	0.715226i	$0.746324\pi$
$992$	1.44281e8	0.00469264
$993$	−2.76220e9	−0.0895227
$994$	6.88152e9	0.222245
$995$	0	0
$996$	2.57933e9	0.0827179
$997$	1.31943e10	0.421652	0.210826	−	0.977524i	$-0.432385\pi$
$997$	1.31943e10	0.421652	0.210826	+	0.977524i	$0.432385\pi$
$998$	−1.42642e10	−0.454244
$999$	−1.67189e10	−0.530553

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.8.a.v.1.3	✓	4
5.2	odd	4		350.8.c.o.99.2		8
5.3	odd	4		350.8.c.o.99.7		8
5.4	even	2		350.8.a.y.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.8.a.v.1.3	✓	4	1.1	even	1	trivial
350.8.a.y.1.2	yes	4	5.4	even	2
350.8.c.o.99.2		8	5.2	odd	4
350.8.c.o.99.7		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +8.12245 q^{3} +64.0000 q^{4} -64.9796 q^{6} +343.000 q^{7} -512.000 q^{8} -2121.03 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} +8.12245 q^{3} +64.0000 q^{4} -64.9796 q^{6} +343.000 q^{7} -512.000 q^{8} -2121.03 q^{9} -1568.25 q^{11} +519.837 q^{12} -6855.29 q^{13} -2744.00 q^{14} +4096.00 q^{16} -30329.5 q^{17} +16968.2 q^{18} -7158.12 q^{19} +2786.00 q^{21} +12546.0 q^{22} -12135.4 q^{23} -4158.70 q^{24} +54842.3 q^{26} -34991.7 q^{27} +21952.0 q^{28} +7299.31 q^{29} -4403.10 q^{31} -32768.0 q^{32} -12738.1 q^{33} +242636. q^{34} -135746. q^{36} +477796. q^{37} +57264.9 q^{38} -55681.8 q^{39} -462355. q^{41} -22288.0 q^{42} +177870. q^{43} -100368. q^{44} +97083.6 q^{46} +424536. q^{47} +33269.6 q^{48} +117649. q^{49} -246350. q^{51} -438738. q^{52} +685895. q^{53} +279934. q^{54} -175616. q^{56} -58141.5 q^{57} -58394.5 q^{58} -3438.30 q^{59} -302654. q^{61} +35224.8 q^{62} -727512. q^{63} +262144. q^{64} +101905. q^{66} +4.02722e6 q^{67} -1.94109e6 q^{68} -98569.6 q^{69} -2.50784e6 q^{71} +1.08597e6 q^{72} -5.59256e6 q^{73} -3.82236e6 q^{74} -458120. q^{76} -537911. q^{77} +445454. q^{78} +190724. q^{79} +4.35446e6 q^{81} +3.69884e6 q^{82} +4.96181e6 q^{83} +178304. q^{84} -1.42296e6 q^{86} +59288.3 q^{87} +802946. q^{88} -8.40090e6 q^{89} -2.35136e6 q^{91} -776668. q^{92} -35763.9 q^{93} -3.39628e6 q^{94} -266157. q^{96} -3.36217e6 q^{97} -941192. q^{98} +3.32631e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} - 42 q^{3} + 256 q^{4} + 336 q^{6} + 1372 q^{7} - 2048 q^{8} + 2210 q^{9}+O(q^{10}) \) 4 * q - 32 * q^2 - 42 * q^3 + 256 * q^4 + 336 * q^6 + 1372 * q^7 - 2048 * q^8 + 2210 * q^9	\( 4 q - 32 q^{2} - 42 q^{3} + 256 q^{4} + 336 q^{6} + 1372 q^{7} - 2048 q^{8} + 2210 q^{9} + 1324 q^{11} - 2688 q^{12} + 17458 q^{13} - 10976 q^{14} + 16384 q^{16} + 18158 q^{17} - 17680 q^{18} + 11984 q^{19} - 14406 q^{21} - 10592 q^{22} + 145300 q^{23} + 21504 q^{24} - 139664 q^{26} - 62244 q^{27} + 87808 q^{28} - 55578 q^{29} + 17206 q^{31} - 131072 q^{32} + 684320 q^{33} - 145264 q^{34} + 141440 q^{36} + 507898 q^{37} - 95872 q^{38} - 132344 q^{39} - 268660 q^{41} + 115248 q^{42} - 362460 q^{43} + 84736 q^{44} - 1162400 q^{46} + 855988 q^{47} - 172032 q^{48} + 470596 q^{49} - 1347956 q^{51} + 1117312 q^{52} - 1245360 q^{53} + 497952 q^{54} - 702464 q^{56} + 3348414 q^{57} + 444624 q^{58} - 1415834 q^{59} - 4333910 q^{61} - 137648 q^{62} + 758030 q^{63} + 1048576 q^{64} - 5474560 q^{66} + 2271660 q^{67} + 1162112 q^{68} - 10774750 q^{69} - 3370816 q^{71} - 1131520 q^{72} + 4604488 q^{73} - 4063184 q^{74} + 766976 q^{76} + 454132 q^{77} + 1058752 q^{78} - 9036996 q^{79} - 3280024 q^{81} + 2149280 q^{82} - 11603802 q^{83} - 921984 q^{84} + 2899680 q^{86} + 15868398 q^{87} - 677888 q^{88} + 307328 q^{89} + 5988094 q^{91} + 9299200 q^{92} - 30711032 q^{93} - 6847904 q^{94} + 1376256 q^{96} + 21766234 q^{97} - 3764768 q^{98} - 23711228 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 - 42 * q^3 + 256 * q^4 + 336 * q^6 + 1372 * q^7 - 2048 * q^8 + 2210 * q^9 + 1324 * q^11 - 2688 * q^12 + 17458 * q^13 - 10976 * q^14 + 16384 * q^16 + 18158 * q^17 - 17680 * q^18 + 11984 * q^19 - 14406 * q^21 - 10592 * q^22 + 145300 * q^23 + 21504 * q^24 - 139664 * q^26 - 62244 * q^27 + 87808 * q^28 - 55578 * q^29 + 17206 * q^31 - 131072 * q^32 + 684320 * q^33 - 145264 * q^34 + 141440 * q^36 + 507898 * q^37 - 95872 * q^38 - 132344 * q^39 - 268660 * q^41 + 115248 * q^42 - 362460 * q^43 + 84736 * q^44 - 1162400 * q^46 + 855988 * q^47 - 172032 * q^48 + 470596 * q^49 - 1347956 * q^51 + 1117312 * q^52 - 1245360 * q^53 + 497952 * q^54 - 702464 * q^56 + 3348414 * q^57 + 444624 * q^58 - 1415834 * q^59 - 4333910 * q^61 - 137648 * q^62 + 758030 * q^63 + 1048576 * q^64 - 5474560 * q^66 + 2271660 * q^67 + 1162112 * q^68 - 10774750 * q^69 - 3370816 * q^71 - 1131520 * q^72 + 4604488 * q^73 - 4063184 * q^74 + 766976 * q^76 + 454132 * q^77 + 1058752 * q^78 - 9036996 * q^79 - 3280024 * q^81 + 2149280 * q^82 - 11603802 * q^83 - 921984 * q^84 + 2899680 * q^86 + 15868398 * q^87 - 677888 * q^88 + 307328 * q^89 + 5988094 * q^91 + 9299200 * q^92 - 30711032 * q^93 - 6847904 * q^94 + 1376256 * q^96 + 21766234 * q^97 - 3764768 * q^98 - 23711228 * q^99

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