LMFDB - Embedded newform 350.8.a.v.1.1

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			1.1
Root			$7.79308$ 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)

$f(q)$	$=$	$q-8.00000 q^{2} -75.8442 q^{3} +64.0000 q^{4} +606.753 q^{6} +343.000 q^{7} -512.000 q^{8} +3565.34 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} -75.8442 q^{3} +64.0000 q^{4} +606.753 q^{6} +343.000 q^{7} -512.000 q^{8} +3565.34 q^{9} -8257.03 q^{11} -4854.03 q^{12} +10951.6 q^{13} -2744.00 q^{14} +4096.00 q^{16} +5417.17 q^{17} -28522.7 q^{18} +16563.9 q^{19} -26014.6 q^{21} +66056.2 q^{22} +106775. q^{23} +38832.2 q^{24} -87612.5 q^{26} -104539. q^{27} +21952.0 q^{28} -213227. q^{29} +229691. q^{31} -32768.0 q^{32} +626248. q^{33} -43337.4 q^{34} +228182. q^{36} +507420. q^{37} -132511. q^{38} -830612. q^{39} -55304.2 q^{41} +208116. q^{42} -476551. q^{43} -528450. q^{44} -854200. q^{46} -1.17695e6 q^{47} -310658. q^{48} +117649. q^{49} -410861. q^{51} +700900. q^{52} -554739. q^{53} +836312. q^{54} -175616. q^{56} -1.25627e6 q^{57} +1.70582e6 q^{58} +295837. q^{59} -2.50601e6 q^{61} -1.83753e6 q^{62} +1.22291e6 q^{63} +262144. q^{64} -5.00998e6 q^{66} +450089. q^{67} +346699. q^{68} -8.09826e6 q^{69} -1.80575e6 q^{71} -1.82545e6 q^{72} +3.42850e6 q^{73} -4.05936e6 q^{74} +1.06009e6 q^{76} -2.83216e6 q^{77} +6.64490e6 q^{78} -1.23245e6 q^{79} +131277. q^{81} +442434. q^{82} -6.67291e6 q^{83} -1.66493e6 q^{84} +3.81241e6 q^{86} +1.61720e7 q^{87} +4.22760e6 q^{88} +6.27859e6 q^{89} +3.75639e6 q^{91} +6.83360e6 q^{92} -1.74207e7 q^{93} +9.41558e6 q^{94} +2.48526e6 q^{96} +1.82615e6 q^{97} -941192. q^{98} -2.94391e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−75.8442	−1.62180	−0.810901	−	0.585183i	$-0.801023\pi$
$3$	−75.8442	−1.62180	−0.810901	+	0.585183i	$0.801023\pi$
$4$	64.0000	0.500000
$5$	0	0
$5$	0	0
$6$	606.753	1.14679
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−512.000	−0.353553
$9$	3565.34	1.63024
$10$	0	0
$11$	−8257.03	−1.87046	−0.935232	−	0.354035i	$-0.884809\pi$
$11$	−8257.03	−1.87046	−0.935232	+	0.354035i	$0.884809\pi$
$12$	−4854.03	−0.810901
$13$	10951.6	1.38253	0.691265	−	0.722602i	$-0.257054\pi$
$13$	10951.6	1.38253	0.691265	+	0.722602i	$0.257054\pi$
$14$	−2744.00	−0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	5417.17	0.267425	0.133712	−	0.991020i	$-0.457310\pi$
$17$	5417.17	0.267425	0.133712	+	0.991020i	$0.457310\pi$
$18$	−28522.7	−1.15276
$19$	16563.9	0.554019	0.277009	−	0.960867i	$-0.410657\pi$
$19$	16563.9	0.554019	0.277009	+	0.960867i	$0.410657\pi$
$20$	0	0
$21$	−26014.6	−0.612984
$22$	66056.2	1.32262
$23$	106775.	1.82988	0.914939	−	0.403593i	$-0.132239\pi$
$23$	106775.	1.82988	0.914939	+	0.403593i	$0.132239\pi$
$24$	38832.2	0.573394
$25$	0	0
$26$	−87612.5	−0.977596
$27$	−104539.	−1.02213
$28$	21952.0	0.188982
$29$	−213227.	−1.62349	−0.811745	−	0.584012i	$-0.801482\pi$
$29$	−213227.	−1.62349	−0.811745	+	0.584012i	$0.801482\pi$
$30$	0	0
$31$	229691.	1.38477	0.692386	−	0.721527i	$-0.256560\pi$
$31$	229691.	1.38477	0.692386	+	0.721527i	$0.256560\pi$
$32$	−32768.0	−0.176777
$33$	626248.	3.03352
$34$	−43337.4	−0.189098
$35$	0	0
$36$	228182.	0.815121
$37$	507420.	1.64688	0.823439	−	0.567405i	$-0.192053\pi$
$37$	507420.	1.64688	0.823439	+	0.567405i	$0.192053\pi$
$38$	−132511.	−0.391750
$39$	−830612.	−2.24219
$40$	0	0
$41$	−55304.2	−0.125318	−0.0626592	−	0.998035i	$-0.519958\pi$
$41$	−55304.2	−0.125318	−0.0626592	+	0.998035i	$0.519958\pi$
$42$	208116.	0.433445
$43$	−476551.	−0.914050	−0.457025	−	0.889454i	$-0.651085\pi$
$43$	−476551.	−0.914050	−0.457025	+	0.889454i	$0.651085\pi$
$44$	−528450.	−0.935232
$45$	0	0
$46$	−854200.	−1.29392
$47$	−1.17695e6	−1.65354	−0.826770	−	0.562540i	$-0.809824\pi$
$47$	−1.17695e6	−1.65354	−0.826770	+	0.562540i	$0.809824\pi$
$48$	−310658.	−0.405451
$49$	117649.	0.142857
$50$	0	0
$51$	−410861.	−0.433710
$52$	700900.	0.691265
$53$	−554739.	−0.511827	−0.255914	−	0.966700i	$-0.582376\pi$
$53$	−554739.	−0.511827	−0.255914	+	0.966700i	$0.582376\pi$
$54$	836312.	0.722753
$55$	0	0
$56$	−175616.	−0.133631
$57$	−1.25627e6	−0.898509
$58$	1.70582e6	1.14798
$59$	295837.	0.187530	0.0937648	−	0.995594i	$-0.470110\pi$
$59$	295837.	0.187530	0.0937648	+	0.995594i	$0.470110\pi$
$60$	0	0
$61$	−2.50601e6	−1.41361	−0.706803	−	0.707410i	$-0.749863\pi$
$61$	−2.50601e6	−1.41361	−0.706803	+	0.707410i	$0.749863\pi$
$62$	−1.83753e6	−0.979182
$63$	1.22291e6	0.616174
$64$	262144.	0.125000
$65$	0	0
$66$	−5.00998e6	−2.14502
$67$	450089.	0.182825	0.0914127	−	0.995813i	$-0.470862\pi$
$67$	450089.	0.182825	0.0914127	+	0.995813i	$0.470862\pi$
$68$	346699.	0.133712
$69$	−8.09826e6	−2.96770
$70$	0	0
$71$	−1.80575e6	−0.598760	−0.299380	−	0.954134i	$-0.596780\pi$
$71$	−1.80575e6	−0.598760	−0.299380	+	0.954134i	$0.596780\pi$
$72$	−1.82545e6	−0.576378
$73$	3.42850e6	1.03151	0.515756	−	0.856736i	$-0.327511\pi$
$73$	3.42850e6	1.03151	0.515756	+	0.856736i	$0.327511\pi$
$74$	−4.05936e6	−1.16452
$75$	0	0
$76$	1.06009e6	0.277009
$77$	−2.83216e6	−0.706969
$78$	6.64490e6	1.58547
$79$	−1.23245e6	−0.281238	−0.140619	−	0.990064i	$-0.544909\pi$
$79$	−1.23245e6	−0.281238	−0.140619	+	0.990064i	$0.544909\pi$
$80$	0	0
$81$	131277.	0.0274468
$82$	442434.	0.0886135
$83$	−6.67291e6	−1.28098	−0.640489	−	0.767967i	$-0.721268\pi$
$83$	−6.67291e6	−1.28098	−0.640489	+	0.767967i	$0.721268\pi$
$84$	−1.66493e6	−0.306492
$85$	0	0
$86$	3.81241e6	0.646331
$87$	1.61720e7	2.63298
$88$	4.22760e6	0.661309
$89$	6.27859e6	0.944054	0.472027	−	0.881584i	$-0.343522\pi$
$89$	6.27859e6	0.944054	0.472027	+	0.881584i	$0.343522\pi$
$90$	0	0
$91$	3.75639e6	0.522547
$92$	6.83360e6	0.914939
$93$	−1.74207e7	−2.24583
$94$	9.41558e6	1.16923
$95$	0	0
$96$	2.48526e6	0.286697
$97$	1.82615e6	0.203159	0.101579	−	0.994827i	$-0.467610\pi$
$97$	1.82615e6	0.203159	0.101579	+	0.994827i	$0.467610\pi$
$98$	−941192.	−0.101015
$99$	−2.94391e7	−3.04931
$100$	0	0
$101$	1.46479e7	1.41466	0.707329	−	0.706885i	$-0.249900\pi$
$101$	1.46479e7	1.41466	0.707329	+	0.706885i	$0.249900\pi$
$102$	3.28689e6	0.306679
$103$	−8.29071e6	−0.747587	−0.373793	−	0.927512i	$-0.621943\pi$
$103$	−8.29071e6	−0.747587	−0.373793	+	0.927512i	$0.621943\pi$
$104$	−5.60720e6	−0.488798
$105$	0	0
$106$	4.43791e6	0.361917
$107$	−1.09502e7	−0.864132	−0.432066	−	0.901842i	$-0.642215\pi$
$107$	−1.09502e7	−0.864132	−0.432066	+	0.901842i	$0.642215\pi$
$108$	−6.69050e6	−0.511064
$109$	−4.59122e6	−0.339574	−0.169787	−	0.985481i	$-0.554308\pi$
$109$	−4.59122e6	−0.339574	−0.169787	+	0.985481i	$0.554308\pi$
$110$	0	0
$111$	−3.84848e7	−2.67091
$112$	1.40493e6	0.0944911
$113$	−3.04095e6	−0.198260	−0.0991298	−	0.995075i	$-0.531606\pi$
$113$	−3.04095e6	−0.198260	−0.0991298	+	0.995075i	$0.531606\pi$
$114$	1.00502e7	0.635342
$115$	0	0
$116$	−1.36465e7	−0.811745
$117$	3.90460e7	2.25386
$118$	−2.36669e6	−0.132603
$119$	1.85809e6	0.101077
$120$	0	0
$121$	4.86914e7	2.49864
$122$	2.00481e7	0.999570
$123$	4.19450e6	0.203242
$124$	1.47002e7	0.692386
$125$	0	0
$126$	−9.78329e6	−0.435700
$127$	−1.19194e7	−0.516348	−0.258174	−	0.966098i	$-0.583121\pi$
$127$	−1.19194e7	−0.516348	−0.258174	+	0.966098i	$0.583121\pi$
$128$	−2.09715e6	−0.0883883
$129$	3.61436e7	1.48241
$130$	0	0
$131$	1.77697e7	0.690605	0.345303	−	0.938491i	$-0.387776\pi$
$131$	1.77697e7	0.690605	0.345303	+	0.938491i	$0.387776\pi$
$132$	4.00798e7	1.51676
$133$	5.68141e6	0.209399
$134$	−3.60071e6	−0.129277
$135$	0	0
$136$	−2.77359e6	−0.0945489
$137$	4.13076e7	1.37249	0.686244	−	0.727372i	$-0.259258\pi$
$137$	4.13076e7	1.37249	0.686244	+	0.727372i	$0.259258\pi$
$138$	6.47861e7	2.09848
$139$	−542170.	−0.0171231	−0.00856157	−	0.999963i	$-0.502725\pi$
$139$	−542170.	−0.0171231	−0.00856157	+	0.999963i	$0.502725\pi$
$140$	0	0
$141$	8.92646e7	2.68171
$142$	1.44460e7	0.423388
$143$	−9.04274e7	−2.58597
$144$	1.46036e7	0.407560
$145$	0	0
$146$	−2.74280e7	−0.729389
$147$	−8.92299e6	−0.231686
$148$	3.24749e7	0.823439
$149$	−4.57677e7	−1.13346	−0.566731	−	0.823903i	$-0.691792\pi$
$149$	−4.57677e7	−1.13346	−0.566731	+	0.823903i	$0.691792\pi$
$150$	0	0
$151$	−4.07482e7	−0.963140	−0.481570	−	0.876408i	$-0.659933\pi$
$151$	−4.07482e7	−0.963140	−0.481570	+	0.876408i	$0.659933\pi$
$152$	−8.48071e6	−0.195875
$153$	1.93141e7	0.435967
$154$	2.26573e7	0.499903
$155$	0	0
$156$	−5.31592e7	−1.12109
$157$	8.09425e7	1.66928	0.834638	−	0.550799i	$-0.185677\pi$
$157$	8.09425e7	1.66928	0.834638	+	0.550799i	$0.185677\pi$
$158$	9.85960e6	0.198865
$159$	4.20737e7	0.830082
$160$	0	0
$161$	3.66238e7	0.691629
$162$	−1.05022e6	−0.0194079
$163$	5.82935e7	1.05430	0.527150	−	0.849773i	$-0.323261\pi$
$163$	5.82935e7	1.05430	0.527150	+	0.849773i	$0.323261\pi$
$164$	−3.53947e6	−0.0626592
$165$	0	0
$166$	5.33833e7	0.905789
$167$	4.80177e7	0.797799	0.398899	−	0.916995i	$-0.369392\pi$
$167$	4.80177e7	0.797799	0.398899	+	0.916995i	$0.369392\pi$
$168$	1.33195e7	0.216722
$169$	5.71882e7	0.911387
$170$	0	0
$171$	5.90558e7	0.903185
$172$	−3.04993e7	−0.457025
$173$	−7.10435e7	−1.04319	−0.521595	−	0.853193i	$-0.674663\pi$
$173$	−7.10435e7	−1.04319	−0.521595	+	0.853193i	$0.674663\pi$
$174$	−1.29376e8	−1.86180
$175$	0	0
$176$	−3.38208e7	−0.467616
$177$	−2.24375e7	−0.304136
$178$	−5.02287e7	−0.667547
$179$	−7.63389e7	−0.994857	−0.497428	−	0.867505i	$-0.665722\pi$
$179$	−7.63389e7	−0.994857	−0.497428	+	0.867505i	$0.665722\pi$
$180$	0	0
$181$	8.03682e7	1.00742	0.503708	−	0.863874i	$-0.331969\pi$
$181$	8.03682e7	1.00742	0.503708	+	0.863874i	$0.331969\pi$
$182$	−3.00511e7	−0.369497
$183$	1.90066e8	2.29259
$184$	−5.46688e7	−0.646959
$185$	0	0
$186$	1.39366e8	1.58804
$187$	−4.47297e7	−0.500208
$188$	−7.53246e7	−0.826770
$189$	−3.58569e7	−0.386328
$190$	0	0
$191$	5.03211e7	0.522556	0.261278	−	0.965264i	$-0.415856\pi$
$191$	5.03211e7	0.522556	0.261278	+	0.965264i	$0.415856\pi$
$192$	−1.98821e7	−0.202725
$193$	4.14979e7	0.415504	0.207752	−	0.978182i	$-0.433385\pi$
$193$	4.14979e7	0.415504	0.207752	+	0.978182i	$0.433385\pi$
$194$	−1.46092e7	−0.143655
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	−4.77397e7	−0.444885	−0.222443	−	0.974946i	$-0.571403\pi$
$197$	−4.77397e7	−0.444885	−0.222443	+	0.974946i	$0.571403\pi$
$198$	2.35513e8	2.15619
$199$	−1.17280e8	−1.05496	−0.527481	−	0.849567i	$-0.676864\pi$
$199$	−1.17280e8	−1.05496	−0.527481	+	0.849567i	$0.676864\pi$
$200$	0	0
$201$	−3.41366e7	−0.296506
$202$	−1.17183e8	−1.00031
$203$	−7.31369e7	−0.613622
$204$	−2.62951e7	−0.216855
$205$	0	0
$206$	6.63257e7	0.528624
$207$	3.80689e8	2.98314
$208$	4.48576e7	0.345632
$209$	−1.36768e8	−1.03627
$210$	0	0
$211$	−5.50053e7	−0.403103	−0.201552	−	0.979478i	$-0.564598\pi$
$211$	−5.50053e7	−0.403103	−0.201552	+	0.979478i	$0.564598\pi$
$212$	−3.55033e7	−0.255914
$213$	1.36956e8	0.971071
$214$	8.76018e7	0.611033
$215$	0	0
$216$	5.35240e7	0.361377
$217$	7.87841e7	0.523395
$218$	3.67297e7	0.240115
$219$	−2.60032e8	−1.67291
$220$	0	0
$221$	5.93265e7	0.369722
$222$	3.07879e8	1.88862
$223$	1.98605e8	1.19928	0.599642	−	0.800268i	$-0.295310\pi$
$223$	1.98605e8	1.19928	0.599642	+	0.800268i	$0.295310\pi$
$224$	−1.12394e7	−0.0668153
$225$	0	0
$226$	2.43276e7	0.140191
$227$	−1.00012e8	−0.567494	−0.283747	−	0.958899i	$-0.591578\pi$
$227$	−1.00012e8	−0.567494	−0.283747	+	0.958899i	$0.591578\pi$
$228$	−8.04015e7	−0.449254
$229$	−1.10029e7	−0.0605458	−0.0302729	−	0.999542i	$-0.509638\pi$
$229$	−1.10029e7	−0.0605458	−0.0302729	+	0.999542i	$0.509638\pi$
$230$	0	0
$231$	2.14803e8	1.14656
$232$	1.09172e8	0.573990
$233$	3.54238e8	1.83463	0.917316	−	0.398159i	$-0.130351\pi$
$233$	3.54238e8	1.83463	0.917316	+	0.398159i	$0.130351\pi$
$234$	−3.12368e8	−1.59372
$235$	0	0
$236$	1.89335e7	0.0937648
$237$	9.34741e7	0.456113
$238$	−1.48647e7	−0.0714722
$239$	−7.10274e7	−0.336537	−0.168269	−	0.985741i	$-0.553818\pi$
$239$	−7.10274e7	−0.336537	−0.168269	+	0.985741i	$0.553818\pi$
$240$	0	0
$241$	3.55847e7	0.163758	0.0818792	−	0.996642i	$-0.473908\pi$
$241$	3.55847e7	0.163758	0.0818792	+	0.996642i	$0.473908\pi$
$242$	−3.89531e8	−1.76680
$243$	2.18670e8	0.977614
$244$	−1.60385e8	−0.706803
$245$	0	0
$246$	−3.35560e7	−0.143714
$247$	1.81400e8	0.765947
$248$	−1.17602e8	−0.489591
$249$	5.06101e8	2.07749
$250$	0	0
$251$	1.03520e8	0.413205	0.206603	−	0.978425i	$-0.433759\pi$
$251$	1.03520e8	0.413205	0.206603	+	0.978425i	$0.433759\pi$
$252$	7.82663e7	0.308087
$253$	−8.81644e8	−3.42272
$254$	9.53554e7	0.365113
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−3.70739e7	−0.136239	−0.0681197	−	0.997677i	$-0.521700\pi$
$257$	−3.70739e7	−0.136239	−0.0681197	+	0.997677i	$0.521700\pi$
$258$	−2.89149e8	−1.04822
$259$	1.74045e8	0.622461
$260$	0	0
$261$	−7.60227e8	−2.64668
$262$	−1.42157e8	−0.488332
$263$	3.95495e8	1.34059	0.670294	−	0.742095i	$-0.266168\pi$
$263$	3.95495e8	1.34059	0.670294	+	0.742095i	$0.266168\pi$
$264$	−3.20639e8	−1.07251
$265$	0	0
$266$	−4.54513e7	−0.148068
$267$	−4.76194e8	−1.53107
$268$	2.88057e7	0.0914127
$269$	−1.06489e7	−0.0333557	−0.0166779	−	0.999861i	$-0.505309\pi$
$269$	−1.06489e7	−0.0333557	−0.0166779	+	0.999861i	$0.505309\pi$
$270$	0	0
$271$	−3.41595e7	−0.104260	−0.0521301	−	0.998640i	$-0.516601\pi$
$271$	−3.41595e7	−0.104260	−0.0521301	+	0.998640i	$0.516601\pi$
$272$	2.21887e7	0.0668562
$273$	−2.84900e8	−0.847468
$274$	−3.30461e8	−0.970495
$275$	0	0
$276$	−5.18289e8	−1.48385
$277$	2.22289e8	0.628403	0.314202	−	0.949356i	$-0.398263\pi$
$277$	2.22289e8	0.628403	0.314202	+	0.949356i	$0.398263\pi$
$278$	4.33736e6	0.0121079
$279$	8.18927e8	2.25751
$280$	0	0
$281$	7.11258e8	1.91229	0.956147	−	0.292886i	$-0.0946158\pi$
$281$	7.11258e8	1.91229	0.956147	+	0.292886i	$0.0946158\pi$
$282$	−7.14117e8	−1.89626
$283$	−3.53874e8	−0.928102	−0.464051	−	0.885809i	$-0.653605\pi$
$283$	−3.53874e8	−0.928102	−0.464051	+	0.885809i	$0.653605\pi$
$284$	−1.15568e8	−0.299380
$285$	0	0
$286$	7.23419e8	1.82856
$287$	−1.89694e7	−0.0473659
$288$	−1.16829e8	−0.288189
$289$	−3.80993e8	−0.928484
$290$	0	0
$291$	−1.38503e8	−0.329484
$292$	2.19424e8	0.515756
$293$	6.55354e8	1.52209	0.761043	−	0.648701i	$-0.224687\pi$
$293$	6.55354e8	1.52209	0.761043	+	0.648701i	$0.224687\pi$
$294$	7.13839e7	0.163827
$295$	0	0
$296$	−2.59799e8	−0.582259
$297$	8.63182e8	1.91185
$298$	3.66142e8	0.801479
$299$	1.16935e9	2.52986
$300$	0	0
$301$	−1.63457e8	−0.345478
$302$	3.25986e8	0.681043
$303$	−1.11096e9	−2.29429
$304$	6.78457e7	0.138505
$305$	0	0
$306$	−1.54512e8	−0.308275
$307$	−1.14979e8	−0.226795	−0.113398	−	0.993550i	$-0.536173\pi$
$307$	−1.14979e8	−0.226795	−0.113398	+	0.993550i	$0.536173\pi$
$308$	−1.81258e8	−0.353485
$309$	6.28802e8	1.21244
$310$	0	0
$311$	2.26226e7	0.0426462	0.0213231	−	0.999773i	$-0.493212\pi$
$311$	2.26226e7	0.0426462	0.0213231	+	0.999773i	$0.493212\pi$
$312$	4.25273e8	0.792733
$313$	−6.05498e8	−1.11611	−0.558056	−	0.829803i	$-0.688453\pi$
$313$	−6.05498e8	−1.11611	−0.558056	+	0.829803i	$0.688453\pi$
$314$	−6.47540e8	−1.18036
$315$	0	0
$316$	−7.88768e7	−0.140619
$317$	1.08278e9	1.90912	0.954561	−	0.298016i	$-0.0963247\pi$
$317$	1.08278e9	1.90912	0.954561	+	0.298016i	$0.0963247\pi$
$318$	−3.36590e8	−0.586957
$319$	1.76062e9	3.03668
$320$	0	0
$321$	8.30511e8	1.40145
$322$	−2.92991e8	−0.489055
$323$	8.97294e7	0.148158
$324$	8.40176e6	0.0137234
$325$	0	0
$326$	−4.66348e8	−0.745502
$327$	3.48217e8	0.550723
$328$	2.83158e7	0.0443068
$329$	−4.03693e8	−0.624979
$330$	0	0
$331$	2.04797e7	0.0310402	0.0155201	−	0.999880i	$-0.495060\pi$
$331$	2.04797e7	0.0310402	0.0155201	+	0.999880i	$0.495060\pi$
$332$	−4.27066e8	−0.640489
$333$	1.80912e9	2.68481
$334$	−3.84141e8	−0.564129
$335$	0	0
$336$	−1.06556e8	−0.153246
$337$	8.42450e8	1.19906	0.599528	−	0.800354i	$-0.295355\pi$
$337$	8.42450e8	1.19906	0.599528	+	0.800354i	$0.295355\pi$
$338$	−4.57506e8	−0.644448
$339$	2.30638e8	0.321538
$340$	0	0
$341$	−1.89657e9	−2.59017
$342$	−4.72447e8	−0.638648
$343$	4.03536e7	0.0539949
$344$	2.43994e8	0.323165
$345$	0	0
$346$	5.68348e8	0.737646
$347$	6.03778e8	0.775754	0.387877	−	0.921711i	$-0.373209\pi$
$347$	6.03778e8	0.775754	0.387877	+	0.921711i	$0.373209\pi$
$348$	1.03501e9	1.31649
$349$	−1.49549e9	−1.88319	−0.941595	−	0.336746i	$-0.890674\pi$
$349$	−1.49549e9	−1.88319	−0.941595	+	0.336746i	$0.890674\pi$
$350$	0	0
$351$	−1.14487e9	−1.41312
$352$	2.70566e8	0.330654
$353$	−7.60354e8	−0.920036	−0.460018	−	0.887910i	$-0.652157\pi$
$353$	−7.60354e8	−0.920036	−0.460018	+	0.887910i	$0.652157\pi$
$354$	1.79500e8	0.215057
$355$	0	0
$356$	4.01830e8	0.472027
$357$	−1.40925e8	−0.163927
$358$	6.10711e8	0.703470
$359$	6.63167e8	0.756471	0.378235	−	0.925709i	$-0.376531\pi$
$359$	6.63167e8	0.756471	0.378235	+	0.925709i	$0.376531\pi$
$360$	0	0
$361$	−6.19510e8	−0.693063
$362$	−6.42945e8	−0.712351
$363$	−3.69296e9	−4.05229
$364$	2.40409e8	0.261273
$365$	0	0
$366$	−1.52053e9	−1.62111
$367$	−1.16137e9	−1.22642	−0.613210	−	0.789920i	$-0.710122\pi$
$367$	−1.16137e9	−1.22642	−0.613210	+	0.789920i	$0.710122\pi$
$368$	4.37350e8	0.457469
$369$	−1.97178e8	−0.204299
$370$	0	0
$371$	−1.90276e8	−0.193453
$372$	−1.11493e9	−1.12291
$373$	−1.14417e8	−0.114159	−0.0570796	−	0.998370i	$-0.518179\pi$
$373$	−1.14417e8	−0.114159	−0.0570796	+	0.998370i	$0.518179\pi$
$374$	3.57838e8	0.353701
$375$	0	0
$376$	6.02597e8	0.584615
$377$	−2.33517e9	−2.24452
$378$	2.86855e8	0.273175
$379$	−7.42466e8	−0.700550	−0.350275	−	0.936647i	$-0.613912\pi$
$379$	−7.42466e8	−0.700550	−0.350275	+	0.936647i	$0.613912\pi$
$380$	0	0
$381$	9.04019e8	0.837413
$382$	−4.02569e8	−0.369503
$383$	−5.68496e8	−0.517049	−0.258525	−	0.966005i	$-0.583236\pi$
$383$	−5.68496e8	−0.517049	−0.258525	+	0.966005i	$0.583236\pi$
$384$	1.59057e8	0.143348
$385$	0	0
$386$	−3.31983e8	−0.293806
$387$	−1.69907e9	−1.49012
$388$	1.16874e8	0.101579
$389$	−3.92756e8	−0.338298	−0.169149	−	0.985590i	$-0.554102\pi$
$389$	−3.92756e8	−0.338298	−0.169149	+	0.985590i	$0.554102\pi$
$390$	0	0
$391$	5.78418e8	0.489354
$392$	−6.02363e7	−0.0505076
$393$	−1.34773e9	−1.12002
$394$	3.81918e8	0.314581
$395$	0	0
$396$	−1.88410e9	−1.52465
$397$	−1.34076e9	−1.07543	−0.537716	−	0.843126i	$-0.680713\pi$
$397$	−1.34076e9	−1.07543	−0.537716	+	0.843126i	$0.680713\pi$
$398$	9.38238e8	0.745971
$399$	−4.30902e8	−0.339604
$400$	0	0
$401$	2.19846e9	1.70260	0.851301	−	0.524678i	$-0.175814\pi$
$401$	2.19846e9	1.70260	0.851301	+	0.524678i	$0.175814\pi$
$402$	2.73093e8	0.209662
$403$	2.51548e9	1.91449
$404$	9.37467e8	0.707329
$405$	0	0
$406$	5.85095e8	0.433896
$407$	−4.18978e9	−3.08043
$408$	2.10361e8	0.153340
$409$	−1.72102e9	−1.24381	−0.621905	−	0.783093i	$-0.713641\pi$
$409$	−1.72102e9	−1.24381	−0.621905	+	0.783093i	$0.713641\pi$
$410$	0	0
$411$	−3.13294e9	−2.22590
$412$	−5.30606e8	−0.373793
$413$	1.01472e8	0.0708795
$414$	−3.04551e9	−2.10940
$415$	0	0
$416$	−3.58861e8	−0.244399
$417$	4.11204e7	0.0277703
$418$	1.09415e9	0.732755
$419$	−4.67442e8	−0.310440	−0.155220	−	0.987880i	$-0.549609\pi$
$419$	−4.67442e8	−0.310440	−0.155220	+	0.987880i	$0.549609\pi$
$420$	0	0
$421$	2.04183e9	1.33362	0.666811	−	0.745227i	$-0.267659\pi$
$421$	2.04183e9	1.33362	0.666811	+	0.745227i	$0.267659\pi$
$422$	4.40043e8	0.285037
$423$	−4.19622e9	−2.69567
$424$	2.84027e8	0.180958
$425$	0	0
$426$	−1.09564e9	−0.686651
$427$	−8.59562e8	−0.534293
$428$	−7.00815e8	−0.432066
$429$	6.85839e9	4.19393
$430$	0	0
$431$	1.59016e9	0.956690	0.478345	−	0.878172i	$-0.341237\pi$
$431$	1.59016e9	0.956690	0.478345	+	0.878172i	$0.341237\pi$
$432$	−4.28192e8	−0.255532
$433$	−1.18592e9	−0.702016	−0.351008	−	0.936372i	$-0.614161\pi$
$433$	−1.18592e9	−0.702016	−0.351008	+	0.936372i	$0.614161\pi$
$434$	−6.30272e8	−0.370096
$435$	0	0
$436$	−2.93838e8	−0.169787
$437$	1.76861e9	1.01379
$438$	2.08025e9	1.18292
$439$	1.57260e9	0.887143	0.443571	−	0.896239i	$-0.353711\pi$
$439$	1.57260e9	0.887143	0.443571	+	0.896239i	$0.353711\pi$
$440$	0	0
$441$	4.19459e8	0.232892
$442$	−4.74612e8	−0.261433
$443$	8.09762e8	0.442532	0.221266	−	0.975214i	$-0.428981\pi$
$443$	8.09762e8	0.442532	0.221266	+	0.975214i	$0.428981\pi$
$444$	−2.46303e9	−1.33545
$445$	0	0
$446$	−1.58884e9	−0.848022
$447$	3.47121e9	1.83825
$448$	8.99154e7	0.0472456
$449$	−1.24344e7	−0.00648281	−0.00324140	−	0.999995i	$-0.501032\pi$
$449$	−1.24344e7	−0.00648281	−0.00324140	+	0.999995i	$0.501032\pi$
$450$	0	0
$451$	4.56649e8	0.234404
$452$	−1.94621e8	−0.0991298
$453$	3.09051e9	1.56202
$454$	8.00096e8	0.401279
$455$	0	0
$456$	6.43212e8	0.317671
$457$	3.21385e8	0.157514	0.0787571	−	0.996894i	$-0.474905\pi$
$457$	3.21385e8	0.157514	0.0787571	+	0.996894i	$0.474905\pi$
$458$	8.80235e7	0.0428124
$459$	−5.66306e8	−0.273342
$460$	0	0
$461$	1.33225e9	0.633332	0.316666	−	0.948537i	$-0.397437\pi$
$461$	1.33225e9	0.633332	0.316666	+	0.948537i	$0.397437\pi$
$462$	−1.71842e9	−0.810743
$463$	1.48661e9	0.696088	0.348044	−	0.937478i	$-0.386846\pi$
$463$	1.48661e9	0.696088	0.348044	+	0.937478i	$0.386846\pi$
$464$	−8.73379e8	−0.405873
$465$	0	0
$466$	−2.83390e9	−1.29728
$467$	−2.66164e9	−1.20932	−0.604659	−	0.796485i	$-0.706691\pi$
$467$	−2.66164e9	−1.20932	−0.604659	+	0.796485i	$0.706691\pi$
$468$	2.49895e9	1.12693
$469$	1.54380e8	0.0691015
$470$	0	0
$471$	−6.13902e9	−2.70723
$472$	−1.51468e8	−0.0663017
$473$	3.93490e9	1.70970
$474$	−7.47793e8	−0.322520
$475$	0	0
$476$	1.18918e8	0.0505385
$477$	−1.97783e9	−0.834402
$478$	5.68219e8	0.237968
$479$	2.68138e9	1.11477	0.557383	−	0.830256i	$-0.311806\pi$
$479$	2.68138e9	1.11477	0.557383	+	0.830256i	$0.311806\pi$
$480$	0	0
$481$	5.55704e9	2.27686
$482$	−2.84678e8	−0.115795
$483$	−2.77770e9	−1.12168
$484$	3.11625e9	1.24932
$485$	0	0
$486$	−1.74936e9	−0.691278
$487$	−5.31276e8	−0.208434	−0.104217	−	0.994555i	$-0.533234\pi$
$487$	−5.31276e8	−0.208434	−0.104217	+	0.994555i	$0.533234\pi$
$488$	1.28308e9	0.499785
$489$	−4.42123e9	−1.70986
$490$	0	0
$491$	1.93216e9	0.736642	0.368321	−	0.929699i	$-0.379933\pi$
$491$	1.93216e9	0.736642	0.368321	+	0.929699i	$0.379933\pi$
$492$	2.68448e8	0.101621
$493$	−1.15509e9	−0.434161
$494$	−1.45120e9	−0.541606
$495$	0	0
$496$	9.40815e8	0.346193
$497$	−6.19372e8	−0.226310
$498$	−4.04881e9	−1.46901
$499$	2.51357e9	0.905606	0.452803	−	0.891611i	$-0.350424\pi$
$499$	2.51357e9	0.905606	0.452803	+	0.891611i	$0.350424\pi$
$500$	0	0
$501$	−3.64186e9	−1.29387
$502$	−8.28160e8	−0.292180
$503$	−5.42778e9	−1.90167	−0.950833	−	0.309703i	$-0.899770\pi$
$503$	−5.42778e9	−1.90167	−0.950833	+	0.309703i	$0.899770\pi$
$504$	−6.26131e8	−0.217850
$505$	0	0
$506$	7.05315e9	2.42023
$507$	−4.33739e9	−1.47809
$508$	−7.62843e8	−0.258174
$509$	−1.75614e9	−0.590264	−0.295132	−	0.955456i	$-0.595364\pi$
$509$	−1.75614e9	−0.590264	−0.295132	+	0.955456i	$0.595364\pi$
$510$	0	0
$511$	1.17598e9	0.389875
$512$	−1.34218e8	−0.0441942
$513$	−1.73157e9	−0.566278
$514$	2.96591e8	0.0963357
$515$	0	0
$516$	2.31319e9	0.741204
$517$	9.71809e9	3.09289
$518$	−1.39236e9	−0.440147
$519$	5.38824e9	1.69185
$520$	0	0
$521$	2.22607e9	0.689616	0.344808	−	0.938673i	$-0.387944\pi$
$521$	2.22607e9	0.689616	0.344808	+	0.938673i	$0.387944\pi$
$522$	6.08182e9	1.87149
$523$	5.04206e9	1.54118	0.770588	−	0.637334i	$-0.219963\pi$
$523$	5.04206e9	1.54118	0.770588	+	0.637334i	$0.219963\pi$
$524$	1.13726e9	0.345303
$525$	0	0
$526$	−3.16396e9	−0.947939
$527$	1.24428e9	0.370322
$528$	2.56511e9	0.758381
$529$	7.99607e9	2.34845
$530$	0	0
$531$	1.05476e9	0.305719
$532$	3.63610e8	0.104700
$533$	−6.05668e8	−0.173256
$534$	3.80955e9	1.08263
$535$	0	0
$536$	−2.30445e8	−0.0646385
$537$	5.78986e9	1.61346
$538$	8.51909e7	0.0235861
$539$	−9.71431e8	−0.267209
$540$	0	0
$541$	3.63487e9	0.986957	0.493478	−	0.869758i	$-0.335725\pi$
$541$	3.63487e9	0.986957	0.493478	+	0.869758i	$0.335725\pi$
$542$	2.73276e8	0.0737231
$543$	−6.09546e9	−1.63383
$544$	−1.77510e8	−0.0472744
$545$	0	0
$546$	2.27920e9	0.599250
$547$	4.37535e9	1.14303	0.571514	−	0.820592i	$-0.306356\pi$
$547$	4.37535e9	1.14303	0.571514	+	0.820592i	$0.306356\pi$
$548$	2.64369e9	0.686244
$549$	−8.93478e9	−2.30452
$550$	0	0
$551$	−3.53187e9	−0.899444
$552$	4.14631e9	1.04924
$553$	−4.22730e8	−0.106298
$554$	−1.77831e9	−0.444348
$555$	0	0
$556$	−3.46989e7	−0.00856157
$557$	5.29674e9	1.29872	0.649359	−	0.760482i	$-0.275037\pi$
$557$	5.29674e9	1.29872	0.649359	+	0.760482i	$0.275037\pi$
$558$	−6.55141e9	−1.59630
$559$	−5.21898e9	−1.26370
$560$	0	0
$561$	3.39249e9	0.811239
$562$	−5.69006e9	−1.35220
$563$	2.95189e9	0.697140	0.348570	−	0.937283i	$-0.386667\pi$
$563$	2.95189e9	0.697140	0.348570	+	0.937283i	$0.386667\pi$
$564$	5.71294e9	1.34086
$565$	0	0
$566$	2.83099e9	0.656267
$567$	4.50282e7	0.0103739
$568$	9.24543e8	0.211694
$569$	8.84764e8	0.201342	0.100671	−	0.994920i	$-0.467901\pi$
$569$	8.84764e8	0.201342	0.100671	+	0.994920i	$0.467901\pi$
$570$	0	0
$571$	−2.87842e9	−0.647035	−0.323518	−	0.946222i	$-0.604865\pi$
$571$	−2.87842e9	−0.647035	−0.323518	+	0.946222i	$0.604865\pi$
$572$	−5.78735e9	−1.29299
$573$	−3.81656e9	−0.847483
$574$	1.51755e8	0.0334928
$575$	0	0
$576$	9.34632e8	0.203780
$577$	−3.13356e9	−0.679082	−0.339541	−	0.940591i	$-0.610272\pi$
$577$	−3.13356e9	−0.679082	−0.339541	+	0.940591i	$0.610272\pi$
$578$	3.04794e9	0.656537
$579$	−3.14737e9	−0.673865
$580$	0	0
$581$	−2.28881e9	−0.484165
$582$	1.10802e9	0.232980
$583$	4.58050e9	0.957355
$584$	−1.75539e9	−0.364694
$585$	0	0
$586$	−5.24283e9	−1.07628
$587$	−4.99566e9	−1.01944	−0.509718	−	0.860342i	$-0.670250\pi$
$587$	−4.99566e9	−1.01944	−0.509718	+	0.860342i	$0.670250\pi$
$588$	−5.71071e8	−0.115843
$589$	3.80458e9	0.767190
$590$	0	0
$591$	3.62078e9	0.721516
$592$	2.07839e9	0.411719
$593$	7.24014e9	1.42579	0.712895	−	0.701270i	$-0.247383\pi$
$593$	7.24014e9	1.42579	0.712895	+	0.701270i	$0.247383\pi$
$594$	−6.90545e9	−1.35188
$595$	0	0
$596$	−2.92913e9	−0.566731
$597$	8.89499e9	1.71094
$598$	−9.35482e9	−1.78888
$599$	2.73076e9	0.519146	0.259573	−	0.965723i	$-0.416418\pi$
$599$	2.73076e9	0.519146	0.259573	+	0.965723i	$0.416418\pi$
$600$	0	0
$601$	−2.98434e9	−0.560774	−0.280387	−	0.959887i	$-0.590463\pi$
$601$	−2.98434e9	−0.560774	−0.280387	+	0.959887i	$0.590463\pi$
$602$	1.30766e9	0.244290
$603$	1.60472e9	0.298049
$604$	−2.60789e9	−0.481570
$605$	0	0
$606$	8.88768e9	1.62231
$607$	5.69754e9	1.03401	0.517007	−	0.855981i	$-0.327046\pi$
$607$	5.69754e9	1.03401	0.517007	+	0.855981i	$0.327046\pi$
$608$	−5.42765e8	−0.0979376
$609$	5.54701e9	0.995173
$610$	0	0
$611$	−1.28894e10	−2.28607
$612$	1.23610e9	0.217983
$613$	4.49004e9	0.787296	0.393648	−	0.919261i	$-0.371213\pi$
$613$	4.49004e9	0.787296	0.393648	+	0.919261i	$0.371213\pi$
$614$	9.19832e8	0.160368
$615$	0	0
$616$	1.45007e9	0.249951
$617$	−9.29800e9	−1.59364	−0.796822	−	0.604214i	$-0.793487\pi$
$617$	−9.29800e9	−1.59364	−0.796822	+	0.604214i	$0.793487\pi$
$618$	−5.03042e9	−0.857323
$619$	−1.25016e9	−0.211859	−0.105930	−	0.994374i	$-0.533782\pi$
$619$	−1.25016e9	−0.211859	−0.105930	+	0.994374i	$0.533782\pi$
$620$	0	0
$621$	−1.11621e10	−1.87037
$622$	−1.80981e8	−0.0301554
$623$	2.15356e9	0.356819
$624$	−3.40219e9	−0.560547
$625$	0	0
$626$	4.84399e9	0.789210
$627$	1.03731e10	1.68063
$628$	5.18032e9	0.834638
$629$	2.74878e9	0.440416
$630$	0	0
$631$	3.88978e9	0.616342	0.308171	−	0.951331i	$-0.400283\pi$
$631$	3.88978e9	0.616342	0.308171	+	0.951331i	$0.400283\pi$
$632$	6.31014e8	0.0994327
$633$	4.17184e9	0.653754
$634$	−8.66226e9	−1.34995
$635$	0	0
$636$	2.69272e9	0.415041
$637$	1.28844e9	0.197504
$638$	−1.40850e10	−2.14726
$639$	−6.43811e9	−0.976124
$640$	0	0
$641$	5.83217e9	0.874636	0.437318	−	0.899307i	$-0.355928\pi$
$641$	5.83217e9	0.874636	0.437318	+	0.899307i	$0.355928\pi$
$642$	−6.64409e9	−0.990975
$643$	7.36497e9	1.09253	0.546264	−	0.837613i	$-0.316050\pi$
$643$	7.36497e9	1.09253	0.546264	+	0.837613i	$0.316050\pi$
$644$	2.34392e9	0.345814
$645$	0	0
$646$	−7.17835e8	−0.104764
$647$	1.83394e9	0.266208	0.133104	−	0.991102i	$-0.457506\pi$
$647$	1.83394e9	0.266208	0.133104	+	0.991102i	$0.457506\pi$
$648$	−6.72140e7	−0.00970393
$649$	−2.44273e9	−0.350767
$650$	0	0
$651$	−5.97531e9	−0.848842
$652$	3.73079e9	0.527150
$653$	−6.95469e9	−0.977421	−0.488710	−	0.872446i	$-0.662533\pi$
$653$	−6.95469e9	−0.977421	−0.488710	+	0.872446i	$0.662533\pi$
$654$	−2.78574e9	−0.389420
$655$	0	0
$656$	−2.26526e8	−0.0313296
$657$	1.22238e10	1.68161
$658$	3.22954e9	0.441927
$659$	−8.59395e9	−1.16975	−0.584876	−	0.811122i	$-0.698857\pi$
$659$	−8.59395e9	−1.16975	−0.584876	+	0.811122i	$0.698857\pi$
$660$	0	0
$661$	1.30760e10	1.76104	0.880521	−	0.474007i	$-0.157193\pi$
$661$	1.30760e10	1.76104	0.880521	+	0.474007i	$0.157193\pi$
$662$	−1.63837e8	−0.0219488
$663$	−4.49957e9	−0.599617
$664$	3.41653e9	0.452894
$665$	0	0
$666$	−1.44730e10	−1.89845
$667$	−2.27673e10	−2.97079
$668$	3.07313e9	0.398899
$669$	−1.50630e10	−1.94500
$670$	0	0
$671$	2.06922e10	2.64410
$672$	8.52445e8	0.108361
$673$	1.23354e10	1.55991	0.779956	−	0.625835i	$-0.215242\pi$
$673$	1.23354e10	1.55991	0.779956	+	0.625835i	$0.215242\pi$
$674$	−6.73960e9	−0.847860
$675$	0	0
$676$	3.66005e9	0.455694
$677$	6.80108e9	0.842398	0.421199	−	0.906968i	$-0.361609\pi$
$677$	6.80108e9	0.842398	0.421199	+	0.906968i	$0.361609\pi$
$678$	−1.84510e9	−0.227361
$679$	6.26370e8	0.0767869
$680$	0	0
$681$	7.58532e9	0.920363
$682$	1.51725e10	1.83152
$683$	5.65060e9	0.678612	0.339306	−	0.940676i	$-0.389808\pi$
$683$	5.65060e9	0.678612	0.339306	+	0.940676i	$0.389808\pi$
$684$	3.77957e9	0.451592
$685$	0	0
$686$	−3.22829e8	−0.0381802
$687$	8.34509e8	0.0981934
$688$	−1.95195e9	−0.228513
$689$	−6.07526e9	−0.707616
$690$	0	0
$691$	−8.70075e9	−1.00319	−0.501595	−	0.865102i	$-0.667253\pi$
$691$	−8.70075e9	−1.00319	−0.501595	+	0.865102i	$0.667253\pi$
$692$	−4.54678e9	−0.521595
$693$	−1.00976e10	−1.15253
$694$	−4.83022e9	−0.548541
$695$	0	0
$696$	−8.28009e9	−0.930899
$697$	−2.99593e8	−0.0335132
$698$	1.19639e10	1.33162
$699$	−2.68669e10	−2.97541
$700$	0	0
$701$	−1.00909e10	−1.10641	−0.553203	−	0.833046i	$-0.686595\pi$
$701$	−1.00909e10	−1.10641	−0.553203	+	0.833046i	$0.686595\pi$
$702$	9.15892e9	0.999228
$703$	8.40484e9	0.912401
$704$	−2.16453e9	−0.233808
$705$	0	0
$706$	6.08284e9	0.650563
$707$	5.02424e9	0.534690
$708$	−1.43600e9	−0.152068
$709$	−1.07656e10	−1.13442	−0.567212	−	0.823572i	$-0.691978\pi$
$709$	−1.07656e10	−1.13442	−0.567212	+	0.823572i	$0.691978\pi$
$710$	0	0
$711$	−4.39410e9	−0.458486
$712$	−3.21464e9	−0.333774
$713$	2.45253e10	2.53396
$714$	1.12740e9	0.115914
$715$	0	0
$716$	−4.88569e9	−0.497428
$717$	5.38701e9	0.545797
$718$	−5.30533e9	−0.534906
$719$	−1.56066e10	−1.56588	−0.782939	−	0.622098i	$-0.786280\pi$
$719$	−1.56066e10	−1.56588	−0.782939	+	0.622098i	$0.786280\pi$
$720$	0	0
$721$	−2.84371e9	−0.282561
$722$	4.95608e9	0.490070
$723$	−2.69889e9	−0.265584
$724$	5.14356e9	0.503708
$725$	0	0
$726$	2.95436e10	2.86540
$727$	4.05190e9	0.391100	0.195550	−	0.980694i	$-0.437351\pi$
$727$	4.05190e9	0.391100	0.195550	+	0.980694i	$0.437351\pi$
$728$	−1.92327e9	−0.184748
$729$	−1.68720e10	−1.61294
$730$	0	0
$731$	−2.58156e9	−0.244439
$732$	1.21642e10	1.14629
$733$	−1.28047e10	−1.20090	−0.600449	−	0.799663i	$-0.705012\pi$
$733$	−1.28047e10	−1.20090	−0.600449	+	0.799663i	$0.705012\pi$
$734$	9.29095e9	0.867210
$735$	0	0
$736$	−3.49880e9	−0.323480
$737$	−3.71640e9	−0.341968
$738$	1.57743e9	0.144461
$739$	1.99840e10	1.82149	0.910744	−	0.412972i	$-0.135509\pi$
$739$	1.99840e10	1.82149	0.910744	+	0.412972i	$0.135509\pi$
$740$	0	0
$741$	−1.37582e10	−1.24221
$742$	1.52220e9	0.136792
$743$	8.37488e9	0.749062	0.374531	−	0.927214i	$-0.377804\pi$
$743$	8.37488e9	0.749062	0.374531	+	0.927214i	$0.377804\pi$
$744$	8.91942e9	0.794019
$745$	0	0
$746$	9.15339e8	0.0807228
$747$	−2.37912e10	−2.08831
$748$	−2.86270e9	−0.250104
$749$	−3.75593e9	−0.326611
$750$	0	0
$751$	−2.84507e9	−0.245106	−0.122553	−	0.992462i	$-0.539108\pi$
$751$	−2.84507e9	−0.245106	−0.122553	+	0.992462i	$0.539108\pi$
$752$	−4.82078e9	−0.413385
$753$	−7.85139e9	−0.670137
$754$	1.86814e10	1.58712
$755$	0	0
$756$	−2.29484e9	−0.193164
$757$	−1.29223e10	−1.08269	−0.541345	−	0.840800i	$-0.682085\pi$
$757$	−1.29223e10	−1.08269	−0.541345	+	0.840800i	$0.682085\pi$
$758$	5.93972e9	0.495364
$759$	6.68676e10	5.55098
$760$	0	0
$761$	4.85655e9	0.399468	0.199734	−	0.979850i	$-0.435992\pi$
$761$	4.85655e9	0.399468	0.199734	+	0.979850i	$0.435992\pi$
$762$	−7.23215e9	−0.592141
$763$	−1.57479e9	−0.128347
$764$	3.22055e9	0.261278
$765$	0	0
$766$	4.54797e9	0.365609
$767$	3.23987e9	0.259265
$768$	−1.27245e9	−0.101363
$769$	8.40558e9	0.666539	0.333269	−	0.942832i	$-0.391848\pi$
$769$	8.40558e9	0.666539	0.333269	+	0.942832i	$0.391848\pi$
$770$	0	0
$771$	2.81184e9	0.220953
$772$	2.65586e9	0.207752
$773$	2.34233e10	1.82398	0.911988	−	0.410218i	$-0.134547\pi$
$773$	2.34233e10	1.82398	0.911988	+	0.410218i	$0.134547\pi$
$774$	1.35925e10	1.05368
$775$	0	0
$776$	−9.34990e8	−0.0718275
$777$	−1.32003e10	−1.00951
$778$	3.14205e9	0.239213
$779$	−9.16053e8	−0.0694288
$780$	0	0
$781$	1.49101e10	1.11996
$782$	−4.62735e9	−0.346026
$783$	2.22906e10	1.65941
$784$	4.81890e8	0.0357143
$785$	0	0
$786$	1.07818e10	0.791977
$787$	6.62950e9	0.484807	0.242404	−	0.970176i	$-0.422064\pi$
$787$	6.62950e9	0.484807	0.242404	+	0.970176i	$0.422064\pi$
$788$	−3.05534e9	−0.222443
$789$	−2.99960e10	−2.17417
$790$	0	0
$791$	−1.04304e9	−0.0749351
$792$	1.50728e10	1.07809
$793$	−2.74447e10	−1.95435
$794$	1.07260e10	0.760445
$795$	0	0
$796$	−7.50590e9	−0.527481
$797$	−1.28007e10	−0.895635	−0.447817	−	0.894125i	$-0.647799\pi$
$797$	−1.28007e10	−0.895635	−0.447817	+	0.894125i	$0.647799\pi$
$798$	3.44722e9	0.240137
$799$	−6.37573e9	−0.442197
$800$	0	0
$801$	2.23853e10	1.53904
$802$	−1.75877e10	−1.20392
$803$	−2.83092e10	−1.92941
$804$	−2.18474e9	−0.148253
$805$	0	0
$806$	−2.01238e10	−1.35375
$807$	8.07654e8	0.0540964
$808$	−7.49974e9	−0.500157
$809$	−2.65270e9	−0.176144	−0.0880720	−	0.996114i	$-0.528071\pi$
$809$	−2.65270e9	−0.176144	−0.0880720	+	0.996114i	$0.528071\pi$
$810$	0	0
$811$	−2.24586e10	−1.47846	−0.739232	−	0.673451i	$-0.764811\pi$
$811$	−2.24586e10	−1.47846	−0.739232	+	0.673451i	$0.764811\pi$
$812$	−4.68076e9	−0.306811
$813$	2.59080e9	0.169089
$814$	3.35182e10	2.17819
$815$	0	0
$816$	−1.68289e9	−0.108427
$817$	−7.89354e9	−0.506401
$818$	1.37681e10	0.879506
$819$	1.33928e10	0.851878
$820$	0	0
$821$	−1.51120e10	−0.953064	−0.476532	−	0.879157i	$-0.658106\pi$
$821$	−1.51120e10	−0.953064	−0.476532	+	0.879157i	$0.658106\pi$
$822$	2.50635e10	1.57395
$823$	−2.52223e10	−1.57719	−0.788596	−	0.614911i	$-0.789192\pi$
$823$	−2.52223e10	−1.57719	−0.788596	+	0.614911i	$0.789192\pi$
$824$	4.24485e9	0.264312
$825$	0	0
$826$	−8.11775e8	−0.0501194
$827$	−1.26920e9	−0.0780298	−0.0390149	−	0.999239i	$-0.512422\pi$
$827$	−1.26920e9	−0.0780298	−0.0390149	+	0.999239i	$0.512422\pi$
$828$	2.43641e10	1.49157
$829$	1.89290e10	1.15395	0.576974	−	0.816762i	$-0.304233\pi$
$829$	1.89290e10	1.15395	0.576974	+	0.816762i	$0.304233\pi$
$830$	0	0
$831$	−1.68593e10	−1.01915
$832$	2.87089e9	0.172816
$833$	6.37325e8	0.0382035
$834$	−3.28963e8	−0.0196366
$835$	0	0
$836$	−8.75318e9	−0.518136
$837$	−2.40117e10	−1.41541
$838$	3.73953e9	0.219515
$839$	−1.70646e10	−0.997539	−0.498769	−	0.866735i	$-0.666215\pi$
$839$	−1.70646e10	−0.997539	−0.498769	+	0.866735i	$0.666215\pi$
$840$	0	0
$841$	2.82160e10	1.63572
$842$	−1.63346e10	−0.943012
$843$	−5.39447e10	−3.10136
$844$	−3.52034e9	−0.201552
$845$	0	0
$846$	3.35697e10	1.90613
$847$	1.67011e10	0.944396
$848$	−2.27221e9	−0.127957
$849$	2.68392e10	1.50520
$850$	0	0
$851$	5.41797e10	3.01358
$852$	8.76515e9	0.485535
$853$	1.00513e10	0.554498	0.277249	−	0.960798i	$-0.410577\pi$
$853$	1.00513e10	0.554498	0.277249	+	0.960798i	$0.410577\pi$
$854$	6.87649e9	0.377802
$855$	0	0
$856$	5.60652e9	0.305517
$857$	−1.81010e9	−0.0982358	−0.0491179	−	0.998793i	$-0.515641\pi$
$857$	−1.81010e9	−0.0982358	−0.0491179	+	0.998793i	$0.515641\pi$
$858$	−5.48671e10	−2.96556
$859$	−2.56644e8	−0.0138151	−0.00690756	−	0.999976i	$-0.502199\pi$
$859$	−2.56644e8	−0.0138151	−0.00690756	+	0.999976i	$0.502199\pi$
$860$	0	0
$861$	1.43872e9	0.0768181
$862$	−1.27213e10	−0.676482
$863$	−4.72638e9	−0.250317	−0.125159	−	0.992137i	$-0.539944\pi$
$863$	−4.72638e9	−0.250317	−0.125159	+	0.992137i	$0.539944\pi$
$864$	3.42553e9	0.180688
$865$	0	0
$866$	9.48734e9	0.496400
$867$	2.88961e10	1.50582
$868$	5.04218e9	0.261697
$869$	1.01764e10	0.526046
$870$	0	0
$871$	4.92917e9	0.252761
$872$	2.35070e9	0.120058
$873$	6.51085e9	0.331198
$874$	−1.41489e10	−0.716855
$875$	0	0
$876$	−1.66420e10	−0.836454
$877$	9.14540e9	0.457830	0.228915	−	0.973446i	$-0.426482\pi$
$877$	9.14540e9	0.457830	0.228915	+	0.973446i	$0.426482\pi$
$878$	−1.25808e10	−0.627305
$879$	−4.97048e10	−2.46852
$880$	0	0
$881$	−2.57156e10	−1.26701	−0.633505	−	0.773738i	$-0.718384\pi$
$881$	−2.57156e10	−1.26701	−0.633505	+	0.773738i	$0.718384\pi$
$882$	−3.35567e9	−0.164679
$883$	1.52139e10	0.743669	0.371834	−	0.928299i	$-0.378729\pi$
$883$	1.52139e10	0.743669	0.371834	+	0.928299i	$0.378729\pi$
$884$	3.79690e9	0.184861
$885$	0	0
$886$	−6.47810e9	−0.312917
$887$	8.86899e9	0.426719	0.213359	−	0.976974i	$-0.431559\pi$
$887$	8.86899e9	0.426719	0.213359	+	0.976974i	$0.431559\pi$
$888$	1.97042e10	0.944309
$889$	−4.08836e9	−0.195161
$890$	0	0
$891$	−1.08396e9	−0.0513383
$892$	1.27107e10	0.599642
$893$	−1.94948e10	−0.916092
$894$	−2.77697e10	−1.29984
$895$	0	0
$896$	−7.19323e8	−0.0334077
$897$	−8.86886e10	−4.10293
$898$	9.94753e7	0.00458404
$899$	−4.89764e10	−2.24816
$900$	0	0
$901$	−3.00512e9	−0.136875
$902$	−3.65319e9	−0.165748
$903$	1.23973e10	0.560298
$904$	1.55696e9	0.0700953
$905$	0	0
$906$	−2.47241e10	−1.10452
$907$	−2.03617e9	−0.0906125	−0.0453063	−	0.998973i	$-0.514426\pi$
$907$	−2.03617e9	−0.0906125	−0.0453063	+	0.998973i	$0.514426\pi$
$908$	−6.40076e9	−0.283747
$909$	5.22248e10	2.30623
$910$	0	0
$911$	−1.29154e10	−0.565972	−0.282986	−	0.959124i	$-0.591325\pi$
$911$	−1.29154e10	−0.565972	−0.282986	+	0.959124i	$0.591325\pi$
$912$	−5.14570e9	−0.224627
$913$	5.50984e10	2.39603
$914$	−2.57108e9	−0.111379
$915$	0	0
$916$	−7.04188e8	−0.0302729
$917$	6.09500e9	0.261024
$918$	4.53045e9	0.193282
$919$	1.40035e10	0.595156	0.297578	−	0.954697i	$-0.403821\pi$
$919$	1.40035e10	0.595156	0.297578	+	0.954697i	$0.403821\pi$
$920$	0	0
$921$	8.72048e9	0.367817
$922$	−1.06580e10	−0.447833
$923$	−1.97758e10	−0.827804
$924$	1.37474e10	0.573282
$925$	0	0
$926$	−1.18929e10	−0.492208
$927$	−2.95592e10	−1.21875
$928$	6.98703e9	0.286995
$929$	−2.88001e10	−1.17852	−0.589262	−	0.807942i	$-0.700581\pi$
$929$	−2.88001e10	−1.17852	−0.589262	+	0.807942i	$0.700581\pi$
$930$	0	0
$931$	1.94872e9	0.0791455
$932$	2.26712e10	0.917316
$933$	−1.71579e9	−0.0691637
$934$	2.12931e10	0.855117
$935$	0	0
$936$	−1.99916e10	−0.796859
$937$	3.28754e10	1.30552	0.652759	−	0.757566i	$-0.273612\pi$
$937$	3.28754e10	1.30552	0.652759	+	0.757566i	$0.273612\pi$
$938$	−1.23504e9	−0.0488621
$939$	4.59235e10	1.81011
$940$	0	0
$941$	7.89628e9	0.308929	0.154465	−	0.987998i	$-0.450635\pi$
$941$	7.89628e9	0.308929	0.154465	+	0.987998i	$0.450635\pi$
$942$	4.91122e10	1.91430
$943$	−5.90511e9	−0.229317
$944$	1.21175e9	0.0468824
$945$	0	0
$946$	−3.14792e10	−1.20894
$947$	1.26500e10	0.484021	0.242010	−	0.970274i	$-0.422193\pi$
$947$	1.26500e10	0.484021	0.242010	+	0.970274i	$0.422193\pi$
$948$	5.98234e9	0.228056
$949$	3.75474e10	1.42610
$950$	0	0
$951$	−8.21227e10	−3.09622
$952$	−9.51342e8	−0.0357361
$953$	9.76921e9	0.365624	0.182812	−	0.983148i	$-0.441480\pi$
$953$	9.76921e9	0.365624	0.182812	+	0.983148i	$0.441480\pi$
$954$	1.58227e10	0.590011
$955$	0	0
$956$	−4.54575e9	−0.168269
$957$	−1.33533e11	−4.92489
$958$	−2.14510e10	−0.788258
$959$	1.41685e10	0.518751
$960$	0	0
$961$	2.52454e10	0.917593
$962$	−4.44563e10	−1.60998
$963$	−3.90413e10	−1.40874
$964$	2.27742e9	0.0818792
$965$	0	0
$966$	2.22216e10	0.793151
$967$	−6.79022e9	−0.241486	−0.120743	−	0.992684i	$-0.538528\pi$
$967$	−6.79022e9	−0.241486	−0.120743	+	0.992684i	$0.538528\pi$
$968$	−2.49300e10	−0.883401
$969$	−6.80545e9	−0.240283
$970$	0	0
$971$	8.84052e9	0.309892	0.154946	−	0.987923i	$-0.450480\pi$
$971$	8.84052e9	0.309892	0.154946	+	0.987923i	$0.450480\pi$
$972$	1.39949e10	0.488807
$973$	−1.85964e8	−0.00647194
$974$	4.25020e9	0.147385
$975$	0	0
$976$	−1.02646e10	−0.353402
$977$	−3.47398e10	−1.19178	−0.595890	−	0.803066i	$-0.703201\pi$
$977$	−3.47398e10	−1.19178	−0.595890	+	0.803066i	$0.703201\pi$
$978$	3.53698e10	1.20906
$979$	−5.18425e10	−1.76582
$980$	0	0
$981$	−1.63692e10	−0.553588
$982$	−1.54572e10	−0.520885
$983$	5.94367e9	0.199580	0.0997901	−	0.995009i	$-0.468183\pi$
$983$	5.94367e9	0.199580	0.0997901	+	0.995009i	$0.468183\pi$
$984$	−2.14759e9	−0.0718568
$985$	0	0
$986$	9.24071e9	0.306998
$987$	3.06178e10	1.01359
$988$	1.16096e10	0.382974
$989$	−5.08837e10	−1.67260
$990$	0	0
$991$	−4.60247e10	−1.50222	−0.751109	−	0.660178i	$-0.770481\pi$
$991$	−4.60247e10	−1.50222	−0.751109	+	0.660178i	$0.770481\pi$
$992$	−7.52652e9	−0.244795
$993$	−1.55326e9	−0.0503411
$994$	4.95497e9	0.160025
$995$	0	0
$996$	3.23905e10	1.03875
$997$	4.83457e10	1.54499	0.772493	−	0.635023i	$-0.219009\pi$
$997$	4.83457e10	1.54499	0.772493	+	0.635023i	$0.219009\pi$
$998$	−2.01086e10	−0.640360
$999$	−5.30452e10	−1.68332

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.8.a.v.1.1	✓	4
5.2	odd	4		350.8.c.o.99.4		8
5.3	odd	4		350.8.c.o.99.5		8
5.4	even	2		350.8.a.y.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.8.a.v.1.1	✓	4	1.1	even	1	trivial
350.8.a.y.1.4	yes	4	5.4	even	2
350.8.c.o.99.4		8	5.2	odd	4
350.8.c.o.99.5		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)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -75.8442 q^{3} +64.0000 q^{4} +606.753 q^{6} +343.000 q^{7} -512.000 q^{8} +3565.34 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} -75.8442 q^{3} +64.0000 q^{4} +606.753 q^{6} +343.000 q^{7} -512.000 q^{8} +3565.34 q^{9} -8257.03 q^{11} -4854.03 q^{12} +10951.6 q^{13} -2744.00 q^{14} +4096.00 q^{16} +5417.17 q^{17} -28522.7 q^{18} +16563.9 q^{19} -26014.6 q^{21} +66056.2 q^{22} +106775. q^{23} +38832.2 q^{24} -87612.5 q^{26} -104539. q^{27} +21952.0 q^{28} -213227. q^{29} +229691. q^{31} -32768.0 q^{32} +626248. q^{33} -43337.4 q^{34} +228182. q^{36} +507420. q^{37} -132511. q^{38} -830612. q^{39} -55304.2 q^{41} +208116. q^{42} -476551. q^{43} -528450. q^{44} -854200. q^{46} -1.17695e6 q^{47} -310658. q^{48} +117649. q^{49} -410861. q^{51} +700900. q^{52} -554739. q^{53} +836312. q^{54} -175616. q^{56} -1.25627e6 q^{57} +1.70582e6 q^{58} +295837. q^{59} -2.50601e6 q^{61} -1.83753e6 q^{62} +1.22291e6 q^{63} +262144. q^{64} -5.00998e6 q^{66} +450089. q^{67} +346699. q^{68} -8.09826e6 q^{69} -1.80575e6 q^{71} -1.82545e6 q^{72} +3.42850e6 q^{73} -4.05936e6 q^{74} +1.06009e6 q^{76} -2.83216e6 q^{77} +6.64490e6 q^{78} -1.23245e6 q^{79} +131277. q^{81} +442434. q^{82} -6.67291e6 q^{83} -1.66493e6 q^{84} +3.81241e6 q^{86} +1.61720e7 q^{87} +4.22760e6 q^{88} +6.27859e6 q^{89} +3.75639e6 q^{91} +6.83360e6 q^{92} -1.74207e7 q^{93} +9.41558e6 q^{94} +2.48526e6 q^{96} +1.82615e6 q^{97} -941192. q^{98} -2.94391e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} - 42 q^{3} + 256 q^{4} + 336 q^{6} + 1372 q^{7} - 2048 q^{8} + 2210 q^{9}+O(q^{10}) \) 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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