LMFDB - Embedded newform 49.8.a.f.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

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

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

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

Embedding invariants

Embedding label			1.4
Root			$-2.05919$ of defining polynomial
Character	$\chi$	$=$	49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+19.0193 q^{2} +39.4615 q^{3} +233.735 q^{4} +350.005 q^{5} +750.531 q^{6} +2011.00 q^{8} -629.788 q^{9} +6656.85 q^{10} -6269.29 q^{11} +9223.52 q^{12} -4122.67 q^{13} +13811.7 q^{15} +8329.82 q^{16} +11142.7 q^{17} -11978.1 q^{18} +16879.6 q^{19} +81808.2 q^{20} -119238. q^{22} +34791.9 q^{23} +79357.1 q^{24} +44378.3 q^{25} -78410.4 q^{26} -111155. q^{27} +40847.7 q^{29} +262690. q^{30} -43911.2 q^{31} -98980.4 q^{32} -247396. q^{33} +211927. q^{34} -147203. q^{36} +77996.4 q^{37} +321038. q^{38} -162687. q^{39} +703859. q^{40} -41821.6 q^{41} +331256. q^{43} -1.46535e6 q^{44} -220429. q^{45} +661718. q^{46} +1.10085e6 q^{47} +328707. q^{48} +844045. q^{50} +439708. q^{51} -963611. q^{52} -1.56064e6 q^{53} -2.11409e6 q^{54} -2.19428e6 q^{55} +666093. q^{57} +776896. q^{58} -274335. q^{59} +3.22828e6 q^{60} -3.05087e6 q^{61} -835161. q^{62} -2.94876e6 q^{64} -1.44295e6 q^{65} -4.70530e6 q^{66} -645864. q^{67} +2.60444e6 q^{68} +1.37294e6 q^{69} +531485. q^{71} -1.26650e6 q^{72} +4.65746e6 q^{73} +1.48344e6 q^{74} +1.75124e6 q^{75} +3.94533e6 q^{76} -3.09419e6 q^{78} +2.35011e6 q^{79} +2.91548e6 q^{80} -3.00899e6 q^{81} -795418. q^{82} -107925. q^{83} +3.90000e6 q^{85} +6.30027e6 q^{86} +1.61191e6 q^{87} -1.26075e7 q^{88} +7.96573e6 q^{89} -4.19240e6 q^{90} +8.13206e6 q^{92} -1.73280e6 q^{93} +2.09375e7 q^{94} +5.90792e6 q^{95} -3.90592e6 q^{96} +1.44749e7 q^{97} +3.94832e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{2} + 28 q^{3} + 348 q^{4} + 252 q^{5} + 1022 q^{6} - 984 q^{8} + 2008 q^{9} + 4774 q^{10} - 3972 q^{11} - 5404 q^{12} - 1176 q^{13} - 16556 q^{15} + 57264 q^{16} + 56364 q^{17} - 35908 q^{18}+ \cdots + 15213976 q^{99}+O(q^{100}) $ 4 * q - 6 * q^2 + 28 * q^3 + 348 * q^4 + 252 * q^5 + 1022 * q^6 - 984 * q^8 + 2008 * q^9 + 4774 * q^10 - 3972 * q^11 - 5404 * q^12 - 1176 * q^13 - 16556 * q^15 + 57264 * q^16 + 56364 * q^17 - 35908 * q^18 + 41748 * q^19 + 162372 * q^20 - 152954 * q^22 + 131748 * q^23 + 389592 * q^24 - 68016 * q^25 + 113652 * q^26 + 189028 * q^27 + 34056 * q^29 + 605378 * q^30 + 401212 * q^31 - 453408 * q^32 + 24052 * q^33 + 82110 * q^34 - 360088 * q^36 - 5396 * q^37 + 31794 * q^38 + 364840 * q^39 - 443688 * q^40 + 410424 * q^41 + 46544 * q^43 - 1465836 * q^44 - 1209656 * q^45 + 1379202 * q^46 + 1470084 * q^47 - 2018800 * q^48 + 1395528 * q^50 - 1724796 * q^51 - 5269768 * q^52 - 642372 * q^53 - 4567318 * q^54 - 3063340 * q^55 + 3359068 * q^57 - 1972220 * q^58 + 752220 * q^59 + 2123324 * q^60 + 1325772 * q^61 - 3016314 * q^62 - 865856 * q^64 - 4868808 * q^65 - 6013294 * q^66 - 290916 * q^67 + 2453556 * q^68 + 4915260 * q^69 + 3377760 * q^71 + 4557744 * q^72 + 6706588 * q^73 + 3616878 * q^74 + 7223888 * q^75 + 2923004 * q^76 - 16288132 * q^78 + 3946244 * q^79 + 10695888 * q^80 - 11731292 * q^81 - 17563252 * q^82 + 9542064 * q^83 + 7006068 * q^85 + 23237832 * q^86 + 17836840 * q^87 - 8043336 * q^88 + 16165212 * q^89 + 3415748 * q^90 - 19442628 * q^92 + 12552852 * q^93 + 5720442 * q^94 - 3268452 * q^95 - 8448608 * q^96 + 1533112 * q^97 + 15213976 * q^99

Coefficient data

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

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

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

$2$	19.0193	1.68109	0.840543	−	0.541745i	$-0.182236\pi$
$2$	19.0193	1.68109	0.840543	+	0.541745i	$0.182236\pi$
$3$	39.4615	0.843819	0.421910	−	0.906638i	$-0.361360\pi$
$3$	39.4615	0.843819	0.421910	+	0.906638i	$0.361360\pi$
$4$	233.735	1.82605
$5$	350.005	1.25221	0.626107	−	0.779737i	$-0.284647\pi$
$5$	350.005	1.25221	0.626107	+	0.779737i	$0.284647\pi$
$6$	750.531	1.41853
$7$	0	0
$7$	0	0
$8$	2011.00	1.38866
$9$	−629.788	−0.287969
$10$	6656.85	2.10508
$11$	−6269.29	−1.42018	−0.710090	−	0.704111i	$-0.751346\pi$
$11$	−6269.29	−1.42018	−0.710090	+	0.704111i	$0.751346\pi$
$12$	9223.52	1.54086
$13$	−4122.67	−0.520448	−0.260224	−	0.965548i	$-0.583796\pi$
$13$	−4122.67	−0.520448	−0.260224	+	0.965548i	$0.583796\pi$
$14$	0	0
$15$	13811.7	1.05664
$16$	8329.82	0.508412
$17$	11142.7	0.550072	0.275036	−	0.961434i	$-0.411310\pi$
$17$	11142.7	0.550072	0.275036	+	0.961434i	$0.411310\pi$
$18$	−11978.1	−0.484101
$19$	16879.6	0.564577	0.282289	−	0.959330i	$-0.408906\pi$
$19$	16879.6	0.564577	0.282289	+	0.959330i	$0.408906\pi$
$20$	81808.2	2.28661
$21$	0	0
$22$	−119238.	−2.38745
$23$	34791.9	0.596253	0.298126	−	0.954526i	$-0.403638\pi$
$23$	34791.9	0.596253	0.298126	+	0.954526i	$0.403638\pi$
$24$	79357.1	1.17178
$25$	44378.3	0.568042
$26$	−78410.4	−0.874917
$27$	−111155.	−1.08681
$28$	0	0
$29$	40847.7	0.311010	0.155505	−	0.987835i	$-0.450299\pi$
$29$	40847.7	0.311010	0.155505	+	0.987835i	$0.450299\pi$
$30$	262690.	1.77631
$31$	−43911.2	−0.264734	−0.132367	−	0.991201i	$-0.542258\pi$
$31$	−43911.2	−0.264734	−0.132367	+	0.991201i	$0.542258\pi$
$32$	−98980.4	−0.533979
$33$	−247396.	−1.19838
$34$	211927.	0.924718
$35$	0	0
$36$	−147203.	−0.525846
$37$	77996.4	0.253144	0.126572	−	0.991957i	$-0.459602\pi$
$37$	77996.4	0.253144	0.126572	+	0.991957i	$0.459602\pi$
$38$	321038.	0.949103
$39$	−162687.	−0.439164
$40$	703859.	1.73890
$41$	−41821.6	−0.0947670	−0.0473835	−	0.998877i	$-0.515088\pi$
$41$	−41821.6	−0.0947670	−0.0473835	+	0.998877i	$0.515088\pi$
$42$	0	0
$43$	331256.	0.635366	0.317683	−	0.948197i	$-0.397095\pi$
$43$	331256.	0.635366	0.317683	+	0.948197i	$0.397095\pi$
$44$	−1.46535e6	−2.59332
$45$	−220429.	−0.360599
$46$	661718.	1.00235
$47$	1.10085e6	1.54663	0.773317	−	0.634019i	$-0.218596\pi$
$47$	1.10085e6	1.54663	0.773317	+	0.634019i	$0.218596\pi$
$48$	328707.	0.429008
$49$	0	0
$50$	844045.	0.954928
$51$	439708.	0.464161
$52$	−963611.	−0.950364
$53$	−1.56064e6	−1.43992	−0.719960	−	0.694015i	$-0.755840\pi$
$53$	−1.56064e6	−1.43992	−0.719960	+	0.694015i	$0.755840\pi$
$54$	−2.11409e6	−1.82703
$55$	−2.19428e6	−1.77837
$56$	0	0
$57$	666093.	0.476401
$58$	776896.	0.522835
$59$	−274335.	−0.173900	−0.0869498	−	0.996213i	$-0.527712\pi$
$59$	−274335.	−0.173900	−0.0869498	+	0.996213i	$0.527712\pi$
$60$	3.22828e6	1.92948
$61$	−3.05087e6	−1.72095	−0.860476	−	0.509491i	$-0.829834\pi$
$61$	−3.05087e6	−1.72095	−0.860476	+	0.509491i	$0.829834\pi$
$62$	−835161.	−0.445040
$63$	0	0
$64$	−2.94876e6	−1.40608
$65$	−1.44295e6	−0.651712
$66$	−4.70530e6	−2.01457
$67$	−645864.	−0.262349	−0.131174	−	0.991359i	$-0.541875\pi$
$67$	−645864.	−0.262349	−0.131174	+	0.991359i	$0.541875\pi$
$68$	2.60444e6	1.00446
$69$	1.37294e6	0.503130
$70$	0	0
$71$	531485.	0.176233	0.0881165	−	0.996110i	$-0.471915\pi$
$71$	531485.	0.176233	0.0881165	+	0.996110i	$0.471915\pi$
$72$	−1.26650e6	−0.399892
$73$	4.65746e6	1.40126	0.700630	−	0.713525i	$-0.252902\pi$
$73$	4.65746e6	1.40126	0.700630	+	0.713525i	$0.252902\pi$
$74$	1.48344e6	0.425558
$75$	1.75124e6	0.479325
$76$	3.94533e6	1.03095
$77$	0	0
$78$	−3.09419e6	−0.738272
$79$	2.35011e6	0.536283	0.268141	−	0.963380i	$-0.413591\pi$
$79$	2.35011e6	0.536283	0.268141	+	0.963380i	$0.413591\pi$
$80$	2.91548e6	0.636641
$81$	−3.00899e6	−0.629105
$82$	−795418.	−0.159311
$83$	−107925.	−0.0207181	−0.0103591	−	0.999946i	$-0.503297\pi$
$83$	−107925.	−0.0207181	−0.0103591	+	0.999946i	$0.503297\pi$
$84$	0	0
$85$	3.90000e6	0.688808
$86$	6.30027e6	1.06811
$87$	1.61191e6	0.262437
$88$	−1.26075e7	−1.97215
$89$	7.96573e6	1.19774	0.598868	−	0.800848i	$-0.295618\pi$
$89$	7.96573e6	1.19774	0.598868	+	0.800848i	$0.295618\pi$
$90$	−4.19240e6	−0.606198
$91$	0	0
$92$	8.13206e6	1.08879
$93$	−1.73280e6	−0.223388
$94$	2.09375e7	2.60003
$95$	5.90792e6	0.706972
$96$	−3.90592e6	−0.450582
$97$	1.44749e7	1.61033	0.805167	−	0.593048i	$-0.202076\pi$
$97$	1.44749e7	1.61033	0.805167	+	0.593048i	$0.202076\pi$
$98$	0	0
$99$	3.94832e6	0.408968
$100$	1.03727e7	1.03727
$101$	−5.42470e6	−0.523903	−0.261951	−	0.965081i	$-0.584366\pi$
$101$	−5.42470e6	−0.523903	−0.261951	+	0.965081i	$0.584366\pi$
$102$	8.36295e6	0.780295
$103$	1.70839e7	1.54049	0.770243	−	0.637751i	$-0.220135\pi$
$103$	1.70839e7	1.54049	0.770243	+	0.637751i	$0.220135\pi$
$104$	−8.29069e6	−0.722726
$105$	0	0
$106$	−2.96824e7	−2.42063
$107$	−2.24913e7	−1.77489	−0.887447	−	0.460911i	$-0.847523\pi$
$107$	−2.24913e7	−1.77489	−0.887447	+	0.460911i	$0.847523\pi$
$108$	−2.59807e7	−1.98458
$109$	1.10912e6	0.0820322	0.0410161	−	0.999158i	$-0.486941\pi$
$109$	1.10912e6	0.0820322	0.0410161	+	0.999158i	$0.486941\pi$
$110$	−4.17337e7	−2.98960
$111$	3.07786e6	0.213608
$112$	0	0
$113$	−1.59055e7	−1.03698	−0.518492	−	0.855082i	$-0.673507\pi$
$113$	−1.59055e7	−1.03698	−0.518492	+	0.855082i	$0.673507\pi$
$114$	1.26686e7	0.800871
$115$	1.21773e7	0.746637
$116$	9.54752e6	0.567921
$117$	2.59641e6	0.149873
$118$	−5.21766e6	−0.292340
$119$	0	0
$120$	2.77754e7	1.46732
$121$	1.98168e7	1.01691
$122$	−5.80254e7	−2.89307
$123$	−1.65034e6	−0.0799662
$124$	−1.02636e7	−0.483418
$125$	−1.18115e7	−0.540904
$126$	0	0
$127$	2.24962e7	0.974532	0.487266	−	0.873254i	$-0.337994\pi$
$127$	2.24962e7	0.974532	0.487266	+	0.873254i	$0.337994\pi$
$128$	−4.34139e7	−1.82976
$129$	1.30719e7	0.536135
$130$	−2.74440e7	−1.09558
$131$	1.87818e7	0.729940	0.364970	−	0.931019i	$-0.381079\pi$
$131$	1.87818e7	0.729940	0.364970	+	0.931019i	$0.381079\pi$
$132$	−5.78249e7	−2.18830
$133$	0	0
$134$	−1.22839e7	−0.441031
$135$	−3.89047e7	−1.36092
$136$	2.24080e7	0.763865
$137$	3.93601e7	1.30778	0.653889	−	0.756590i	$-0.273136\pi$
$137$	3.93601e7	1.30778	0.653889	+	0.756590i	$0.273136\pi$
$138$	2.61124e7	0.845805
$139$	2.45180e7	0.774343	0.387172	−	0.922008i	$-0.373452\pi$
$139$	2.45180e7	0.774343	0.387172	+	0.922008i	$0.373452\pi$
$140$	0	0
$141$	4.34414e7	1.30508
$142$	1.01085e7	0.296263
$143$	2.58462e7	0.739130
$144$	−5.24602e6	−0.146407
$145$	1.42969e7	0.389452
$146$	8.85816e7	2.35564
$147$	0	0
$148$	1.82304e7	0.462255
$149$	4.33973e7	1.07476	0.537379	−	0.843341i	$-0.319414\pi$
$149$	4.33973e7	1.07476	0.537379	+	0.843341i	$0.319414\pi$
$150$	3.33073e7	0.805787
$151$	−3.36042e7	−0.794281	−0.397140	−	0.917758i	$-0.629997\pi$
$151$	−3.36042e7	−0.794281	−0.397140	+	0.917758i	$0.629997\pi$
$152$	3.39448e7	0.784008
$153$	−7.01754e6	−0.158404
$154$	0	0
$155$	−1.53691e7	−0.331504
$156$	−3.80256e7	−0.801935
$157$	−6.41848e7	−1.32368	−0.661841	−	0.749644i	$-0.730225\pi$
$157$	−6.41848e7	−1.32368	−0.661841	+	0.749644i	$0.730225\pi$
$158$	4.46975e7	0.901537
$159$	−6.15854e7	−1.21503
$160$	−3.46436e7	−0.668657
$161$	0	0
$162$	−5.72289e7	−1.05758
$163$	4.12358e7	0.745793	0.372897	−	0.927873i	$-0.378365\pi$
$163$	4.12358e7	0.745793	0.372897	+	0.927873i	$0.378365\pi$
$164$	−9.77515e6	−0.173049
$165$	−8.65896e7	−1.50062
$166$	−2.05267e6	−0.0348290
$167$	−5.94520e7	−0.987776	−0.493888	−	0.869526i	$-0.664425\pi$
$167$	−5.94520e7	−0.987776	−0.493888	+	0.869526i	$0.664425\pi$
$168$	0	0
$169$	−4.57521e7	−0.729134
$170$	7.41753e7	1.15795
$171$	−1.06305e7	−0.162581
$172$	7.74260e7	1.16021
$173$	−4.43816e7	−0.651691	−0.325845	−	0.945423i	$-0.605649\pi$
$173$	−4.43816e7	−0.651691	−0.325845	+	0.945423i	$0.605649\pi$
$174$	3.06575e7	0.441179
$175$	0	0
$176$	−5.22220e7	−0.722037
$177$	−1.08257e7	−0.146740
$178$	1.51503e8	2.01350
$179$	−6.36105e7	−0.828978	−0.414489	−	0.910054i	$-0.636040\pi$
$179$	−6.36105e7	−0.828978	−0.414489	+	0.910054i	$0.636040\pi$
$180$	−5.15218e7	−0.658472
$181$	−1.52914e6	−0.0191678	−0.00958389	−	0.999954i	$-0.503051\pi$
$181$	−1.52914e6	−0.0191678	−0.00958389	+	0.999954i	$0.503051\pi$
$182$	0	0
$183$	−1.20392e8	−1.45217
$184$	6.99664e7	0.827995
$185$	2.72991e7	0.316991
$186$	−3.29567e7	−0.375534
$187$	−6.98568e7	−0.781202
$188$	2.57308e8	2.82423
$189$	0	0
$190$	1.12365e8	1.18848
$191$	−5.50290e7	−0.571446	−0.285723	−	0.958312i	$-0.592234\pi$
$191$	−5.50290e7	−0.571446	−0.285723	+	0.958312i	$0.592234\pi$
$192$	−1.16362e8	−1.18648
$193$	1.35920e8	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$193$	1.35920e8	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$194$	2.75304e8	2.70711
$195$	−5.69412e7	−0.549927
$196$	0	0
$197$	1.66347e8	1.55018	0.775091	−	0.631850i	$-0.217704\pi$
$197$	1.66347e8	1.55018	0.775091	+	0.631850i	$0.217704\pi$
$198$	7.50944e7	0.687510
$199$	−6.39211e7	−0.574987	−0.287494	−	0.957783i	$-0.592822\pi$
$199$	−6.39211e7	−0.574987	−0.287494	+	0.957783i	$0.592822\pi$
$200$	8.92447e7	0.788819
$201$	−2.54868e7	−0.221375
$202$	−1.03174e8	−0.880726
$203$	0	0
$204$	1.02775e8	0.847582
$205$	−1.46378e7	−0.118669
$206$	3.24925e8	2.58969
$207$	−2.19115e7	−0.171702
$208$	−3.43411e7	−0.264602
$209$	−1.05823e8	−0.801802
$210$	0	0
$211$	−6.44393e7	−0.472239	−0.236120	−	0.971724i	$-0.575876\pi$
$211$	−6.44393e7	−0.472239	−0.236120	+	0.971724i	$0.575876\pi$
$212$	−3.64777e8	−2.62937
$213$	2.09732e7	0.148709
$214$	−4.27770e8	−2.98375
$215$	1.15941e8	0.795615
$216$	−2.23532e8	−1.50922
$217$	0	0
$218$	2.10946e7	0.137903
$219$	1.83790e8	1.18241
$220$	−5.12879e8	−3.24740
$221$	−4.59377e7	−0.286284
$222$	5.85387e7	0.359094
$223$	9.27372e7	0.559998	0.279999	−	0.960000i	$-0.409666\pi$
$223$	9.27372e7	0.559998	0.279999	+	0.960000i	$0.409666\pi$
$224$	0	0
$225$	−2.79489e7	−0.163578
$226$	−3.02512e8	−1.74326
$227$	−3.22533e7	−0.183014	−0.0915070	−	0.995804i	$-0.529168\pi$
$227$	−3.22533e7	−0.183014	−0.0915070	+	0.995804i	$0.529168\pi$
$228$	1.55689e8	0.869933
$229$	−2.85949e8	−1.57349	−0.786746	−	0.617277i	$-0.788236\pi$
$229$	−2.85949e8	−1.57349	−0.786746	+	0.617277i	$0.788236\pi$
$230$	2.31604e8	1.25516
$231$	0	0
$232$	8.21447e7	0.431889
$233$	6.43667e7	0.333362	0.166681	−	0.986011i	$-0.446695\pi$
$233$	6.43667e7	0.333362	0.166681	+	0.986011i	$0.446695\pi$
$234$	4.93819e7	0.251949
$235$	3.85304e8	1.93672
$236$	−6.41215e7	−0.317550
$237$	9.27390e7	0.452526
$238$	0	0
$239$	−7.70718e7	−0.365177	−0.182588	−	0.983189i	$-0.558448\pi$
$239$	−7.70718e7	−0.365177	−0.182588	+	0.983189i	$0.558448\pi$
$240$	1.15049e8	0.537210
$241$	−2.31733e8	−1.06642	−0.533210	−	0.845983i	$-0.679014\pi$
$241$	−2.31733e8	−1.06642	−0.533210	+	0.845983i	$0.679014\pi$
$242$	3.76901e8	1.70952
$243$	1.24356e8	0.555962
$244$	−7.13093e8	−3.14255
$245$	0	0
$246$	−3.13884e7	−0.134430
$247$	−6.95888e7	−0.293833
$248$	−8.83054e7	−0.367626
$249$	−4.25890e6	−0.0174824
$250$	−2.24647e8	−0.909306
$251$	4.88898e8	1.95146	0.975731	−	0.218975i	$-0.0702712\pi$
$251$	4.88898e8	1.95146	0.975731	+	0.218975i	$0.0702712\pi$
$252$	0	0
$253$	−2.18120e8	−0.846787
$254$	4.27862e8	1.63827
$255$	1.53900e8	0.581230
$256$	−4.48261e8	−1.66990
$257$	3.84669e7	0.141358	0.0706791	−	0.997499i	$-0.477483\pi$
$257$	3.84669e7	0.141358	0.0706791	+	0.997499i	$0.477483\pi$
$258$	2.48618e8	0.901288
$259$	0	0
$260$	−3.37268e8	−1.19006
$261$	−2.57254e7	−0.0895613
$262$	3.57217e8	1.22709
$263$	2.47568e8	0.839171	0.419585	−	0.907716i	$-0.362176\pi$
$263$	2.47568e8	0.839171	0.419585	+	0.907716i	$0.362176\pi$
$264$	−4.97512e8	−1.66414
$265$	−5.46233e8	−1.80309
$266$	0	0
$267$	3.14340e8	1.01067
$268$	−1.50961e8	−0.479062
$269$	−2.48276e7	−0.0777682	−0.0388841	−	0.999244i	$-0.512380\pi$
$269$	−2.48276e7	−0.0777682	−0.0388841	+	0.999244i	$0.512380\pi$
$270$	−7.39941e8	−2.28783
$271$	1.83516e8	0.560121	0.280061	−	0.959982i	$-0.409645\pi$
$271$	1.83516e8	0.560121	0.280061	+	0.959982i	$0.409645\pi$
$272$	9.28167e7	0.279663
$273$	0	0
$274$	7.48602e8	2.19849
$275$	−2.78220e8	−0.806723
$276$	3.20904e8	0.918741
$277$	−3.78154e8	−1.06903	−0.534515	−	0.845159i	$-0.679506\pi$
$277$	−3.78154e8	−1.06903	−0.534515	+	0.845159i	$0.679506\pi$
$278$	4.66316e8	1.30174
$279$	2.76547e7	0.0762351
$280$	0	0
$281$	−6.03667e8	−1.62302	−0.811512	−	0.584335i	$-0.801355\pi$
$281$	−6.03667e8	−1.62302	−0.811512	+	0.584335i	$0.801355\pi$
$282$	8.26226e8	2.19395
$283$	−1.55837e8	−0.408713	−0.204356	−	0.978897i	$-0.565510\pi$
$283$	−1.55837e8	−0.408713	−0.204356	+	0.978897i	$0.565510\pi$
$284$	1.24227e8	0.321810
$285$	2.33136e8	0.596557
$286$	4.91577e8	1.24254
$287$	0	0
$288$	6.23367e7	0.153769
$289$	−2.86179e8	−0.697421
$290$	2.71917e8	0.654702
$291$	5.71204e8	1.35883
$292$	1.08861e9	2.55877
$293$	4.10409e8	0.953191	0.476596	−	0.879123i	$-0.341871\pi$
$293$	4.10409e8	0.953191	0.476596	+	0.879123i	$0.341871\pi$
$294$	0	0
$295$	−9.60185e7	−0.217760
$296$	1.56851e8	0.351532
$297$	6.96861e8	1.54347
$298$	8.25387e8	1.80676
$299$	−1.43435e8	−0.310318
$300$	4.09324e8	0.875272
$301$	0	0
$302$	−6.39129e8	−1.33525
$303$	−2.14067e8	−0.442079
$304$	1.40604e8	0.287038
$305$	−1.06782e9	−2.15500
$306$	−1.33469e8	−0.266290
$307$	6.12316e8	1.20779	0.603895	−	0.797064i	$-0.293615\pi$
$307$	6.12316e8	1.20779	0.603895	+	0.797064i	$0.293615\pi$
$308$	0	0
$309$	6.74158e8	1.29989
$310$	−2.92310e8	−0.557286
$311$	8.14865e7	0.153612	0.0768058	−	0.997046i	$-0.475528\pi$
$311$	8.14865e7	0.153612	0.0768058	+	0.997046i	$0.475528\pi$
$312$	−3.27163e8	−0.609851
$313$	−4.66510e8	−0.859916	−0.429958	−	0.902849i	$-0.641472\pi$
$313$	−4.66510e8	−0.859916	−0.429958	+	0.902849i	$0.641472\pi$
$314$	−1.22075e9	−2.22522
$315$	0	0
$316$	5.49302e8	0.979279
$317$	4.37865e8	0.772027	0.386014	−	0.922493i	$-0.373852\pi$
$317$	4.37865e8	0.772027	0.386014	+	0.922493i	$0.373852\pi$
$318$	−1.17131e9	−2.04257
$319$	−2.56086e8	−0.441691
$320$	−1.03208e9	−1.76071
$321$	−8.87543e8	−1.49769
$322$	0	0
$323$	1.88084e8	0.310558
$324$	−7.03305e8	−1.14878
$325$	−1.82957e8	−0.295636
$326$	7.84278e8	1.25374
$327$	4.37674e7	0.0692203
$328$	−8.41032e7	−0.131599
$329$	0	0
$330$	−1.64688e9	−2.52268
$331$	1.18359e9	1.79392	0.896960	−	0.442112i	$-0.145771\pi$
$331$	1.18359e9	1.79392	0.896960	+	0.442112i	$0.145771\pi$
$332$	−2.52259e7	−0.0378324
$333$	−4.91212e7	−0.0728977
$334$	−1.13074e9	−1.66054
$335$	−2.26055e8	−0.328517
$336$	0	0
$337$	2.24486e8	0.319510	0.159755	−	0.987157i	$-0.448930\pi$
$337$	2.24486e8	0.319510	0.159755	+	0.987157i	$0.448930\pi$
$338$	−8.70174e8	−1.22574
$339$	−6.27655e8	−0.875028
$340$	9.11565e8	1.25780
$341$	2.75292e8	0.375970
$342$	−2.02186e8	−0.273312
$343$	0	0
$344$	6.66156e8	0.882310
$345$	4.80536e8	0.630027
$346$	−8.44108e8	−1.09555
$347$	5.66996e8	0.728496	0.364248	−	0.931302i	$-0.381326\pi$
$347$	5.66996e8	0.728496	0.364248	+	0.931302i	$0.381326\pi$
$348$	3.76760e8	0.479223
$349$	−1.28642e9	−1.61992	−0.809958	−	0.586487i	$-0.800510\pi$
$349$	−1.28642e9	−1.61992	−0.809958	+	0.586487i	$0.800510\pi$
$350$	0	0
$351$	4.58255e8	0.565629
$352$	6.20537e8	0.758347
$353$	−3.12019e8	−0.377546	−0.188773	−	0.982021i	$-0.560451\pi$
$353$	−3.12019e8	−0.377546	−0.188773	+	0.982021i	$0.560451\pi$
$354$	−2.05897e8	−0.246682
$355$	1.86022e8	0.220682
$356$	1.86187e9	2.18713
$357$	0	0
$358$	−1.20983e9	−1.39358
$359$	−1.44457e9	−1.64781	−0.823906	−	0.566726i	$-0.808210\pi$
$359$	−1.44457e9	−1.64781	−0.823906	+	0.566726i	$0.808210\pi$
$360$	−4.43282e8	−0.500750
$361$	−6.08952e8	−0.681253
$362$	−2.90832e7	−0.0322227
$363$	7.82000e8	0.858091
$364$	0	0
$365$	1.63013e9	1.75468
$366$	−2.28977e9	−2.44123
$367$	1.28857e9	1.36075	0.680373	−	0.732866i	$-0.261818\pi$
$367$	1.28857e9	1.36075	0.680373	+	0.732866i	$0.261818\pi$
$368$	2.89810e8	0.303142
$369$	2.63387e7	0.0272899
$370$	5.19210e8	0.532890
$371$	0	0
$372$	−4.05016e8	−0.407917
$373$	−8.95722e8	−0.893701	−0.446850	−	0.894609i	$-0.647454\pi$
$373$	−8.95722e8	−0.893701	−0.446850	+	0.894609i	$0.647454\pi$
$374$	−1.32863e9	−1.31327
$375$	−4.66100e8	−0.456425
$376$	2.21382e9	2.14775
$377$	−1.68402e8	−0.161865
$378$	0	0
$379$	−1.52727e9	−1.44105	−0.720527	−	0.693427i	$-0.756100\pi$
$379$	−1.52727e9	−1.44105	−0.720527	+	0.693427i	$0.756100\pi$
$380$	1.38089e9	1.29097
$381$	8.87734e8	0.822329
$382$	−1.04661e9	−0.960650
$383$	1.75124e9	1.59276	0.796380	−	0.604796i	$-0.206745\pi$
$383$	1.75124e9	1.59276	0.796380	+	0.604796i	$0.206745\pi$
$384$	−1.71318e9	−1.54398
$385$	0	0
$386$	2.58510e9	2.28782
$387$	−2.08621e8	−0.182966
$388$	3.38330e9	2.94055
$389$	−6.20733e8	−0.534664	−0.267332	−	0.963604i	$-0.586142\pi$
$389$	−6.20733e8	−0.534664	−0.267332	+	0.963604i	$0.586142\pi$
$390$	−1.08298e9	−0.924475
$391$	3.87676e8	0.327982
$392$	0	0
$393$	7.41157e8	0.615937
$394$	3.16380e9	2.60599
$395$	8.22550e8	0.671541
$396$	9.22859e8	0.746796
$397$	−1.59069e9	−1.27590	−0.637952	−	0.770076i	$-0.720218\pi$
$397$	−1.59069e9	−1.27590	−0.637952	+	0.770076i	$0.720218\pi$
$398$	−1.21574e9	−0.966604
$399$	0	0
$400$	3.69663e8	0.288799
$401$	4.98917e8	0.386388	0.193194	−	0.981161i	$-0.438115\pi$
$401$	4.98917e8	0.386388	0.193194	+	0.981161i	$0.438115\pi$
$402$	−4.84741e8	−0.372151
$403$	1.81031e8	0.137780
$404$	−1.26794e9	−0.956673
$405$	−1.05316e9	−0.787775
$406$	0	0
$407$	−4.88981e8	−0.359511
$408$	8.84253e8	0.644564
$409$	−2.37156e9	−1.71397	−0.856984	−	0.515343i	$-0.827664\pi$
$409$	−2.37156e9	−1.71397	−0.856984	+	0.515343i	$0.827664\pi$
$410$	−2.78400e8	−0.199492
$411$	1.55321e9	1.10353
$412$	3.99311e9	2.81301
$413$	0	0
$414$	−4.16742e8	−0.288646
$415$	−3.77744e7	−0.0259435
$416$	4.08064e8	0.277908
$417$	9.67519e8	0.653406
$418$	−2.01268e9	−1.34790
$419$	8.82367e8	0.586003	0.293002	−	0.956112i	$-0.405346\pi$
$419$	8.82367e8	0.586003	0.293002	+	0.956112i	$0.405346\pi$
$420$	0	0
$421$	−1.22486e9	−0.800019	−0.400010	−	0.916511i	$-0.630993\pi$
$421$	−1.22486e9	−0.800019	−0.400010	+	0.916511i	$0.630993\pi$
$422$	−1.22559e9	−0.793875
$423$	−6.93305e8	−0.445382
$424$	−3.13845e9	−1.99956
$425$	4.94494e8	0.312464
$426$	3.98897e8	0.249992
$427$	0	0
$428$	−5.25700e9	−3.24105
$429$	1.01993e9	0.623692
$430$	2.20512e9	1.33750
$431$	6.42740e8	0.386692	0.193346	−	0.981131i	$-0.438066\pi$
$431$	6.42740e8	0.386692	0.193346	+	0.981131i	$0.438066\pi$
$432$	−9.25899e8	−0.552548
$433$	2.02652e9	1.19962	0.599811	−	0.800142i	$-0.295243\pi$
$433$	2.02652e9	1.19962	0.599811	+	0.800142i	$0.295243\pi$
$434$	0	0
$435$	5.64177e8	0.328627
$436$	2.59239e8	0.149795
$437$	5.87271e8	0.336631
$438$	3.49557e9	1.98773
$439$	−1.28650e8	−0.0725746	−0.0362873	−	0.999341i	$-0.511553\pi$
$439$	−1.28650e8	−0.0725746	−0.0362873	+	0.999341i	$0.511553\pi$
$440$	−4.41269e9	−2.46956
$441$	0	0
$442$	−8.73704e8	−0.481267
$443$	8.55957e7	0.0467777	0.0233889	−	0.999726i	$-0.492554\pi$
$443$	8.55957e7	0.0467777	0.0233889	+	0.999726i	$0.492554\pi$
$444$	7.19401e8	0.390059
$445$	2.78804e9	1.49982
$446$	1.76380e9	0.941406
$447$	1.71252e9	0.906902
$448$	0	0
$449$	4.64175e8	0.242002	0.121001	−	0.992652i	$-0.461390\pi$
$449$	4.64175e8	0.242002	0.121001	+	0.992652i	$0.461390\pi$
$450$	−5.31569e8	−0.274989
$451$	2.62191e8	0.134586
$452$	−3.71766e9	−1.89359
$453$	−1.32607e9	−0.670229
$454$	−6.13437e8	−0.307662
$455$	0	0
$456$	1.33951e9	0.661561
$457$	−5.45972e8	−0.267586	−0.133793	−	0.991009i	$-0.542716\pi$
$457$	−5.45972e8	−0.267586	−0.133793	+	0.991009i	$0.542716\pi$
$458$	−5.43856e9	−2.64518
$459$	−1.23856e9	−0.597825
$460$	2.84626e9	1.36340
$461$	−1.36175e9	−0.647355	−0.323678	−	0.946167i	$-0.604919\pi$
$461$	−1.36175e9	−0.647355	−0.323678	+	0.946167i	$0.604919\pi$
$462$	0	0
$463$	−1.76640e9	−0.827097	−0.413549	−	0.910482i	$-0.635711\pi$
$463$	−1.76640e9	−0.827097	−0.413549	+	0.910482i	$0.635711\pi$
$464$	3.40254e8	0.158121
$465$	−6.06489e8	−0.279729
$466$	1.22421e9	0.560409
$467$	−1.71412e9	−0.778811	−0.389405	−	0.921066i	$-0.627319\pi$
$467$	−1.71412e9	−0.778811	−0.389405	+	0.921066i	$0.627319\pi$
$468$	6.06870e8	0.273675
$469$	0	0
$470$	7.32823e9	3.25579
$471$	−2.53283e9	−1.11695
$472$	−5.51687e8	−0.241488
$473$	−2.07674e9	−0.902335
$474$	1.76383e9	0.760735
$475$	7.49086e8	0.320704
$476$	0	0
$477$	9.82875e8	0.414652
$478$	−1.46585e9	−0.613894
$479$	3.59504e9	1.49462	0.747308	−	0.664477i	$-0.231346\pi$
$479$	3.59504e9	1.49462	0.747308	+	0.664477i	$0.231346\pi$
$480$	−1.36709e9	−0.564226
$481$	−3.21553e8	−0.131748
$482$	−4.40740e9	−1.79274
$483$	0	0
$484$	4.63186e9	1.85694
$485$	5.06630e9	2.01648
$486$	2.36517e9	0.934620
$487$	2.10947e9	0.827603	0.413802	−	0.910367i	$-0.364201\pi$
$487$	2.10947e9	0.827603	0.413802	+	0.910367i	$0.364201\pi$
$488$	−6.13529e9	−2.38982
$489$	1.62723e9	0.629315
$490$	0	0
$491$	−1.92961e9	−0.735671	−0.367835	−	0.929891i	$-0.619901\pi$
$491$	−1.92961e9	−0.735671	−0.367835	+	0.929891i	$0.619901\pi$
$492$	−3.85742e8	−0.146022
$493$	4.55154e8	0.171078
$494$	−1.32353e9	−0.493958
$495$	1.38193e9	0.512116
$496$	−3.65772e8	−0.134594
$497$	0	0
$498$	−8.10014e7	−0.0293893
$499$	−3.82144e9	−1.37681	−0.688407	−	0.725325i	$-0.741690\pi$
$499$	−3.82144e9	−1.37681	−0.688407	+	0.725325i	$0.741690\pi$
$500$	−2.76076e9	−0.987719
$501$	−2.34606e9	−0.833505
$502$	9.29850e9	3.28057
$503$	1.26027e8	0.0441547	0.0220773	−	0.999756i	$-0.492972\pi$
$503$	1.26027e8	0.0441547	0.0220773	+	0.999756i	$0.492972\pi$
$504$	0	0
$505$	−1.89867e9	−0.656039
$506$	−4.14850e9	−1.42352
$507$	−1.80545e9	−0.615258
$508$	5.25814e9	1.77955
$509$	2.05967e8	0.0692284	0.0346142	−	0.999401i	$-0.488980\pi$
$509$	2.05967e8	0.0692284	0.0346142	+	0.999401i	$0.488980\pi$
$510$	2.92707e9	0.977097
$511$	0	0
$512$	−2.96865e9	−0.977494
$513$	−1.87624e9	−0.613590
$514$	7.31614e8	0.237635
$515$	5.97946e9	1.92902
$516$	3.05535e9	0.979009
$517$	−6.90157e9	−2.19650
$518$	0	0
$519$	−1.75137e9	−0.549909
$520$	−2.90178e9	−0.905009
$521$	−2.53117e9	−0.784131	−0.392066	−	0.919937i	$-0.628239\pi$
$521$	−2.53117e9	−0.784131	−0.392066	+	0.919937i	$0.628239\pi$
$522$	−4.89280e8	−0.150560
$523$	−2.84816e9	−0.870578	−0.435289	−	0.900291i	$-0.643354\pi$
$523$	−2.84816e9	−0.870578	−0.435289	+	0.900291i	$0.643354\pi$
$524$	4.38995e9	1.33291
$525$	0	0
$526$	4.70858e9	1.41072
$527$	−4.89290e8	−0.145623
$528$	−2.06076e9	−0.609268
$529$	−2.19435e9	−0.644482
$530$	−1.03890e10	−3.03115
$531$	1.72773e8	0.0500777
$532$	0	0
$533$	1.72417e8	0.0493212
$534$	5.97853e9	1.69903
$535$	−7.87208e9	−2.22255
$536$	−1.29883e9	−0.364314
$537$	−2.51017e9	−0.699508
$538$	−4.72205e8	−0.130735
$539$	0	0
$540$	−9.09337e9	−2.48512
$541$	5.96667e9	1.62010	0.810049	−	0.586362i	$-0.199440\pi$
$541$	5.96667e9	1.62010	0.810049	+	0.586362i	$0.199440\pi$
$542$	3.49035e9	0.941612
$543$	−6.03422e7	−0.0161741
$544$	−1.10291e9	−0.293727
$545$	3.88196e8	0.102722
$546$	0	0
$547$	4.54821e9	1.18819	0.594093	−	0.804396i	$-0.297511\pi$
$547$	4.54821e9	1.18819	0.594093	+	0.804396i	$0.297511\pi$
$548$	9.19981e9	2.38807
$549$	1.92140e9	0.495581
$550$	−5.29156e9	−1.35617
$551$	6.89491e8	0.175589
$552$	2.76098e9	0.698678
$553$	0	0
$554$	−7.19224e9	−1.79713
$555$	1.07726e9	0.267483
$556$	5.73071e9	1.41399
$557$	−4.18975e9	−1.02729	−0.513647	−	0.858002i	$-0.671706\pi$
$557$	−4.18975e9	−1.02729	−0.513647	+	0.858002i	$0.671706\pi$
$558$	5.25975e8	0.128158
$559$	−1.36566e9	−0.330675
$560$	0	0
$561$	−2.75666e9	−0.659193
$562$	−1.14813e10	−2.72844
$563$	1.69918e9	0.401291	0.200646	−	0.979664i	$-0.435696\pi$
$563$	1.69918e9	0.401291	0.200646	+	0.979664i	$0.435696\pi$
$564$	1.01538e10	2.38314
$565$	−5.56699e9	−1.29853
$566$	−2.96392e9	−0.687082
$567$	0	0
$568$	1.06882e9	0.244728
$569$	−4.48579e8	−0.102081	−0.0510406	−	0.998697i	$-0.516254\pi$
$569$	−4.48579e8	−0.102081	−0.0510406	+	0.998697i	$0.516254\pi$
$570$	4.43408e9	1.00286
$571$	−3.09015e8	−0.0694630	−0.0347315	−	0.999397i	$-0.511058\pi$
$571$	−3.09015e8	−0.0694630	−0.0347315	+	0.999397i	$0.511058\pi$
$572$	6.04115e9	1.34969
$573$	−2.17153e9	−0.482197
$574$	0	0
$575$	1.54400e9	0.338697
$576$	1.85709e9	0.404906
$577$	−5.69700e9	−1.23461	−0.617307	−	0.786723i	$-0.711776\pi$
$577$	−5.69700e9	−1.23461	−0.617307	+	0.786723i	$0.711776\pi$
$578$	−5.44293e9	−1.17242
$579$	5.36359e9	1.14837
$580$	3.34168e9	0.711159
$581$	0	0
$582$	1.08639e10	2.28431
$583$	9.78413e9	2.04495
$584$	9.36614e9	1.94588
$585$	9.08755e8	0.187673
$586$	7.80570e9	1.60240
$587$	−1.74145e8	−0.0355368	−0.0177684	−	0.999842i	$-0.505656\pi$
$587$	−1.74145e8	−0.0355368	−0.0177684	+	0.999842i	$0.505656\pi$
$588$	0	0
$589$	−7.41202e8	−0.149463
$590$	−1.82621e9	−0.366073
$591$	6.56430e9	1.30807
$592$	6.49695e8	0.128702
$593$	3.83650e8	0.0755517	0.0377758	−	0.999286i	$-0.487973\pi$
$593$	3.83650e8	0.0755517	0.0377758	+	0.999286i	$0.487973\pi$
$594$	1.32538e10	2.59471
$595$	0	0
$596$	1.01434e10	1.96256
$597$	−2.52242e9	−0.485186
$598$	−2.72805e9	−0.521672
$599$	−1.26883e9	−0.241217	−0.120609	−	0.992700i	$-0.538485\pi$
$599$	−1.26883e9	−0.241217	−0.120609	+	0.992700i	$0.538485\pi$
$600$	3.52173e9	0.665621
$601$	2.12270e9	0.398867	0.199434	−	0.979911i	$-0.436090\pi$
$601$	2.12270e9	0.398867	0.199434	+	0.979911i	$0.436090\pi$
$602$	0	0
$603$	4.06757e8	0.0755483
$604$	−7.85446e9	−1.45040
$605$	6.93596e9	1.27339
$606$	−4.07141e9	−0.743174
$607$	−3.47687e9	−0.630999	−0.315499	−	0.948926i	$-0.602172\pi$
$607$	−3.47687e9	−0.630999	−0.315499	+	0.948926i	$0.602172\pi$
$608$	−1.67075e9	−0.301473
$609$	0	0
$610$	−2.03092e10	−3.62274
$611$	−4.53846e9	−0.804942
$612$	−1.64024e9	−0.289253
$613$	4.57439e9	0.802087	0.401043	−	0.916059i	$-0.368648\pi$
$613$	4.57439e9	0.802087	0.401043	+	0.916059i	$0.368648\pi$
$614$	1.16458e10	2.03040
$615$	−5.77628e8	−0.100135
$616$	0	0
$617$	4.07855e9	0.699049	0.349525	−	0.936927i	$-0.386343\pi$
$617$	4.07855e9	0.699049	0.349525	+	0.936927i	$0.386343\pi$
$618$	1.28220e10	2.18523
$619$	4.85260e8	0.0822351	0.0411176	−	0.999154i	$-0.486908\pi$
$619$	4.85260e8	0.0822351	0.0411176	+	0.999154i	$0.486908\pi$
$620$	−3.59230e9	−0.605343
$621$	−3.86728e9	−0.648015
$622$	1.54982e9	0.258234
$623$	0	0
$624$	−1.35515e9	−0.223276
$625$	−7.60114e9	−1.24537
$626$	−8.87271e9	−1.44559
$627$	−4.17593e9	−0.676576
$628$	−1.50022e10	−2.41711
$629$	8.69091e8	0.139248
$630$	0	0
$631$	5.74787e9	0.910760	0.455380	−	0.890297i	$-0.349503\pi$
$631$	5.74787e9	0.910760	0.455380	+	0.890297i	$0.349503\pi$
$632$	4.72607e9	0.744716
$633$	−2.54287e9	−0.398485
$634$	8.32789e9	1.29784
$635$	7.87378e9	1.22032
$636$	−1.43946e10	−2.21871
$637$	0	0
$638$	−4.87058e9	−0.742521
$639$	−3.34723e8	−0.0507496
$640$	−1.51951e10	−2.29125
$641$	−5.67745e9	−0.851432	−0.425716	−	0.904857i	$-0.639978\pi$
$641$	−5.67745e9	−0.851432	−0.425716	+	0.904857i	$0.639978\pi$
$642$	−1.68805e10	−2.51774
$643$	−9.43725e9	−1.39993	−0.699966	−	0.714176i	$-0.746802\pi$
$643$	−9.43725e9	−1.39993	−0.699966	+	0.714176i	$0.746802\pi$
$644$	0	0
$645$	4.57522e9	0.671356
$646$	3.57723e9	0.522075
$647$	−6.61857e8	−0.0960725	−0.0480362	−	0.998846i	$-0.515296\pi$
$647$	−6.61857e8	−0.0960725	−0.0480362	+	0.998846i	$0.515296\pi$
$648$	−6.05108e9	−0.873615
$649$	1.71988e9	0.246969
$650$	−3.47972e9	−0.496990
$651$	0	0
$652$	9.63824e9	1.36186
$653$	−2.13437e9	−0.299967	−0.149983	−	0.988689i	$-0.547922\pi$
$653$	−2.13437e9	−0.299967	−0.149983	+	0.988689i	$0.547922\pi$
$654$	8.32426e8	0.116365
$655$	6.57371e9	0.914041
$656$	−3.48366e8	−0.0481806
$657$	−2.93321e9	−0.403519
$658$	0	0
$659$	−9.15512e9	−1.24613	−0.623067	−	0.782168i	$-0.714114\pi$
$659$	−9.15512e9	−1.24613	−0.623067	+	0.782168i	$0.714114\pi$
$660$	−2.02390e10	−2.74022
$661$	2.81453e9	0.379054	0.189527	−	0.981876i	$-0.439305\pi$
$661$	2.81453e9	0.379054	0.189527	+	0.981876i	$0.439305\pi$
$662$	2.25111e10	3.01573
$663$	−1.81277e9	−0.241572
$664$	−2.17038e8	−0.0287705
$665$	0	0
$666$	−9.34251e8	−0.122547
$667$	1.42117e9	0.185441
$668$	−1.38960e10	−1.80373
$669$	3.65955e9	0.472537
$670$	−4.29942e9	−0.552266
$671$	1.91268e10	2.44406
$672$	0	0
$673$	−1.09650e10	−1.38661	−0.693307	−	0.720643i	$-0.743847\pi$
$673$	−1.09650e10	−1.38661	−0.693307	+	0.720643i	$0.743847\pi$
$674$	4.26956e9	0.537123
$675$	−4.93286e9	−0.617356
$676$	−1.06938e10	−1.33144
$677$	1.03601e10	1.28322	0.641612	−	0.767029i	$-0.278266\pi$
$677$	1.03601e10	1.28322	0.641612	+	0.767029i	$0.278266\pi$
$678$	−1.19376e10	−1.47100
$679$	0	0
$680$	7.84290e9	0.956523
$681$	−1.27277e9	−0.154431
$682$	5.23587e9	0.632038
$683$	−8.47880e9	−1.01827	−0.509134	−	0.860687i	$-0.670034\pi$
$683$	−8.47880e9	−1.01827	−0.509134	+	0.860687i	$0.670034\pi$
$684$	−2.48472e9	−0.296881
$685$	1.37762e10	1.63762
$686$	0	0
$687$	−1.12840e10	−1.32774
$688$	2.75930e9	0.323028
$689$	6.43402e9	0.749403
$690$	9.13946e9	1.05913
$691$	−9.95648e8	−0.114798	−0.0573988	−	0.998351i	$-0.518281\pi$
$691$	−9.95648e8	−0.114798	−0.0573988	+	0.998351i	$0.518281\pi$
$692$	−1.03735e10	−1.19002
$693$	0	0
$694$	1.07839e10	1.22466
$695$	8.58142e9	0.969644
$696$	3.24156e9	0.364436
$697$	−4.66006e8	−0.0521287
$698$	−2.44668e10	−2.72322
$699$	2.54001e9	0.281297
$700$	0	0
$701$	1.18853e10	1.30316	0.651579	−	0.758581i	$-0.274107\pi$
$701$	1.18853e10	1.30316	0.651579	+	0.758581i	$0.274107\pi$
$702$	8.71569e9	0.950872
$703$	1.31654e9	0.142920
$704$	1.84866e10	1.99688
$705$	1.52047e10	1.63424
$706$	−5.93439e9	−0.634687
$707$	0	0
$708$	−2.53033e9	−0.267955
$709$	−1.38325e10	−1.45760	−0.728802	−	0.684725i	$-0.759922\pi$
$709$	−1.38325e10	−1.45760	−0.728802	+	0.684725i	$0.759922\pi$
$710$	3.53802e9	0.370985
$711$	−1.48007e9	−0.154433
$712$	1.60191e10	1.66325
$713$	−1.52775e9	−0.157848
$714$	0	0
$715$	9.04629e9	0.925549
$716$	−1.48680e10	−1.51376
$717$	−3.04137e9	−0.308143
$718$	−2.74747e10	−2.77011
$719$	−5.48605e9	−0.550438	−0.275219	−	0.961382i	$-0.588750\pi$
$719$	−5.48605e9	−0.550438	−0.275219	+	0.961382i	$0.588750\pi$
$720$	−1.83613e9	−0.183333
$721$	0	0
$722$	−1.15819e10	−1.14524
$723$	−9.14453e9	−0.899865
$724$	−3.57413e8	−0.0350014
$725$	1.81275e9	0.176667
$726$	1.48731e10	1.44253
$727$	4.80453e9	0.463746	0.231873	−	0.972746i	$-0.425515\pi$
$727$	4.80453e9	0.463746	0.231873	+	0.972746i	$0.425515\pi$
$728$	0	0
$729$	1.14879e10	1.09824
$730$	3.10040e10	2.94977
$731$	3.69109e9	0.349497
$732$	−2.81397e10	−2.65174
$733$	9.96386e9	0.934467	0.467233	−	0.884134i	$-0.345251\pi$
$733$	9.96386e9	0.934467	0.467233	+	0.884134i	$0.345251\pi$
$734$	2.45078e10	2.28753
$735$	0	0
$736$	−3.44372e9	−0.318387
$737$	4.04910e9	0.372583
$738$	5.00945e8	0.0458767
$739$	6.89414e9	0.628383	0.314192	−	0.949360i	$-0.398267\pi$
$739$	6.89414e9	0.628383	0.314192	+	0.949360i	$0.398267\pi$
$740$	6.38074e9	0.578842
$741$	−2.74608e9	−0.247942
$742$	0	0
$743$	7.75055e9	0.693221	0.346610	−	0.938009i	$-0.387333\pi$
$743$	7.75055e9	0.693221	0.346610	+	0.938009i	$0.387333\pi$
$744$	−3.48467e9	−0.310210
$745$	1.51893e10	1.34583
$746$	−1.70360e10	−1.50239
$747$	6.79701e7	0.00596617
$748$	−1.63279e10	−1.42651
$749$	0	0
$750$	−8.86491e9	−0.767290
$751$	8.59013e9	0.740048	0.370024	−	0.929022i	$-0.379349\pi$
$751$	8.59013e9	0.740048	0.370024	+	0.929022i	$0.379349\pi$
$752$	9.16992e9	0.786327
$753$	1.92926e10	1.64668
$754$	−3.20289e9	−0.272108
$755$	−1.17616e10	−0.994610
$756$	0	0
$757$	−3.04455e9	−0.255087	−0.127543	−	0.991833i	$-0.540709\pi$
$757$	−3.04455e9	−0.255087	−0.127543	+	0.991833i	$0.540709\pi$
$758$	−2.90477e10	−2.42253
$759$	−8.60736e9	−0.714535
$760$	1.18808e10	0.981746
$761$	−2.88950e9	−0.237671	−0.118835	−	0.992914i	$-0.537916\pi$
$761$	−2.88950e9	−0.237671	−0.118835	+	0.992914i	$0.537916\pi$
$762$	1.68841e10	1.38241
$763$	0	0
$764$	−1.28622e10	−1.04349
$765$	−2.45617e9	−0.198355
$766$	3.33074e10	2.67757
$767$	1.13099e9	0.0905057
$768$	−1.76891e10	−1.40910
$769$	−3.71789e9	−0.294818	−0.147409	−	0.989076i	$-0.547093\pi$
$769$	−3.71789e9	−0.294818	−0.147409	+	0.989076i	$0.547093\pi$
$770$	0	0
$771$	1.51796e9	0.119281
$772$	3.17691e10	2.48510
$773$	−4.56149e9	−0.355205	−0.177602	−	0.984102i	$-0.556834\pi$
$773$	−4.56149e9	−0.355205	−0.177602	+	0.984102i	$0.556834\pi$
$774$	−3.96783e9	−0.307581
$775$	−1.94870e9	−0.150380
$776$	2.91091e10	2.23621
$777$	0	0
$778$	−1.18059e10	−0.898817
$779$	−7.05930e8	−0.0535033
$780$	−1.33091e10	−1.00420
$781$	−3.33203e9	−0.250283
$782$	7.37333e9	0.551366
$783$	−4.54042e9	−0.338010
$784$	0	0
$785$	−2.24650e10	−1.65753
$786$	1.40963e10	1.03544
$787$	6.51750e9	0.476617	0.238308	−	0.971190i	$-0.423407\pi$
$787$	6.51750e9	0.476617	0.238308	+	0.971190i	$0.423407\pi$
$788$	3.88810e10	2.83071
$789$	9.76943e9	0.708108
$790$	1.56443e10	1.12892
$791$	0	0
$792$	7.94007e9	0.567919
$793$	1.25777e10	0.895665
$794$	−3.02538e10	−2.14491
$795$	−2.15552e10	−1.52148
$796$	−1.49406e10	−1.04996
$797$	5.00841e9	0.350425	0.175213	−	0.984531i	$-0.443939\pi$
$797$	5.00841e9	0.350425	0.175213	+	0.984531i	$0.443939\pi$
$798$	0	0
$799$	1.22665e10	0.850760
$800$	−4.39258e9	−0.303323
$801$	−5.01672e9	−0.344910
$802$	9.48906e9	0.649551
$803$	−2.91989e10	−1.99004
$804$	−5.95714e9	−0.404242
$805$	0	0
$806$	3.44310e9	0.231620
$807$	−9.79736e8	−0.0656223
$808$	−1.09091e10	−0.727525
$809$	8.98130e8	0.0596375	0.0298187	−	0.999555i	$-0.490507\pi$
$809$	8.98130e8	0.0596375	0.0298187	+	0.999555i	$0.490507\pi$
$810$	−2.00304e10	−1.32432
$811$	−1.37233e10	−0.903413	−0.451707	−	0.892167i	$-0.649185\pi$
$811$	−1.37233e10	−0.903413	−0.451707	+	0.892167i	$0.649185\pi$
$812$	0	0
$813$	7.24183e9	0.472641
$814$	−9.30009e9	−0.604369
$815$	1.44327e10	0.933893
$816$	3.66269e9	0.235985
$817$	5.59145e9	0.358713
$818$	−4.51055e10	−2.88133
$819$	0	0
$820$	−3.42135e9	−0.216695
$821$	1.45744e10	0.919159	0.459580	−	0.888137i	$-0.348000\pi$
$821$	1.45744e10	0.919159	0.459580	+	0.888137i	$0.348000\pi$
$822$	2.95410e10	1.85513
$823$	−2.55933e10	−1.60039	−0.800196	−	0.599738i	$-0.795271\pi$
$823$	−2.55933e10	−1.60039	−0.800196	+	0.599738i	$0.795271\pi$
$824$	3.43558e10	2.13922
$825$	−1.09790e10	−0.680728
$826$	0	0
$827$	5.33790e9	0.328172	0.164086	−	0.986446i	$-0.447533\pi$
$827$	5.33790e9	0.328172	0.164086	+	0.986446i	$0.447533\pi$
$828$	−5.12148e9	−0.313537
$829$	−1.88092e10	−1.14664	−0.573322	−	0.819330i	$-0.694345\pi$
$829$	−1.88092e10	−1.14664	−0.573322	+	0.819330i	$0.694345\pi$
$830$	−7.18444e8	−0.0436133
$831$	−1.49225e10	−0.902068
$832$	1.21568e10	0.731789
$833$	0	0
$834$	1.84015e10	1.09843
$835$	−2.08085e10	−1.23691
$836$	−2.47344e10	−1.46413
$837$	4.88094e9	0.287716
$838$	1.67820e10	0.985122
$839$	−1.81462e10	−1.06077	−0.530383	−	0.847758i	$-0.677952\pi$
$839$	−1.81462e10	−1.06077	−0.530383	+	0.847758i	$0.677952\pi$
$840$	0	0
$841$	−1.55813e10	−0.903273
$842$	−2.32961e10	−1.34490
$843$	−2.38216e10	−1.36954
$844$	−1.50617e10	−0.862333
$845$	−1.60134e10	−0.913033
$846$	−1.31862e10	−0.748726
$847$	0	0
$848$	−1.29999e10	−0.732072
$849$	−6.14957e9	−0.344880
$850$	9.40495e9	0.525279
$851$	2.71364e9	0.150938
$852$	4.90217e9	0.271550
$853$	2.76121e10	1.52327	0.761637	−	0.648004i	$-0.224396\pi$
$853$	2.76121e10	1.52327	0.761637	+	0.648004i	$0.224396\pi$
$854$	0	0
$855$	−3.72074e9	−0.203586
$856$	−4.52301e10	−2.46473
$857$	−2.08004e9	−0.112885	−0.0564427	−	0.998406i	$-0.517976\pi$
$857$	−2.08004e9	−0.112885	−0.0564427	+	0.998406i	$0.517976\pi$
$858$	1.93984e10	1.04848
$859$	1.20480e10	0.648545	0.324272	−	0.945964i	$-0.394881\pi$
$859$	1.20480e10	0.648545	0.324272	+	0.945964i	$0.394881\pi$
$860$	2.70995e10	1.45283
$861$	0	0
$862$	1.22245e10	0.650062
$863$	2.45958e10	1.30264	0.651319	−	0.758804i	$-0.274216\pi$
$863$	2.45958e10	1.30264	0.651319	+	0.758804i	$0.274216\pi$
$864$	1.10021e10	0.580336
$865$	−1.55338e10	−0.816057
$866$	3.85431e10	2.01667
$867$	−1.12931e10	−0.588497
$868$	0	0
$869$	−1.47335e10	−0.761618
$870$	1.07303e10	0.552450
$871$	2.66268e9	0.136539
$872$	2.23043e9	0.113915
$873$	−9.11615e9	−0.463726
$874$	1.11695e10	0.565905
$875$	0	0
$876$	4.29581e10	2.15914
$877$	−1.35458e10	−0.678118	−0.339059	−	0.940765i	$-0.610109\pi$
$877$	−1.35458e10	−0.678118	−0.339059	+	0.940765i	$0.610109\pi$
$878$	−2.44684e9	−0.122004
$879$	1.61954e10	0.804321
$880$	−1.82779e10	−0.904145
$881$	2.21441e10	1.09105	0.545523	−	0.838096i	$-0.316331\pi$
$881$	2.21441e10	1.09105	0.545523	+	0.838096i	$0.316331\pi$
$882$	0	0
$883$	2.02852e10	0.991556	0.495778	−	0.868449i	$-0.334883\pi$
$883$	2.02852e10	0.991556	0.495778	+	0.868449i	$0.334883\pi$
$884$	−1.07372e10	−0.522769
$885$	−3.78903e9	−0.183750
$886$	1.62797e9	0.0786374
$887$	3.17249e10	1.52640	0.763198	−	0.646164i	$-0.223628\pi$
$887$	3.17249e10	1.52640	0.763198	+	0.646164i	$0.223628\pi$
$888$	6.18957e9	0.296630
$889$	0	0
$890$	5.30267e10	2.52133
$891$	1.88642e10	0.893443
$892$	2.16759e10	1.02259
$893$	1.85819e10	0.873194
$894$	3.25710e10	1.52458
$895$	−2.22640e10	−1.03806
$896$	0	0
$897$	−5.66018e9	−0.261853
$898$	8.82829e9	0.406827
$899$	−1.79367e9	−0.0823350
$900$	−6.53263e9	−0.298703
$901$	−1.73898e10	−0.792060
$902$	4.98670e9	0.226251
$903$	0	0
$904$	−3.19859e10	−1.44002
$905$	−5.35206e8	−0.0240022
$906$	−2.52210e10	−1.12671
$907$	2.65662e8	0.0118223	0.00591117	−	0.999983i	$-0.498118\pi$
$907$	2.65662e8	0.0118223	0.00591117	+	0.999983i	$0.498118\pi$
$908$	−7.53872e9	−0.334193
$909$	3.41641e9	0.150868
$910$	0	0
$911$	−3.13936e10	−1.37571	−0.687854	−	0.725849i	$-0.741447\pi$
$911$	−3.13936e10	−1.37571	−0.687854	+	0.725849i	$0.741447\pi$
$912$	5.54843e9	0.242208
$913$	6.76615e8	0.0294235
$914$	−1.03840e10	−0.449836
$915$	−4.21377e10	−1.81843
$916$	−6.68362e10	−2.87328
$917$	0	0
$918$	−2.35567e10	−1.00500
$919$	3.35660e9	0.142658	0.0713288	−	0.997453i	$-0.477276\pi$
$919$	3.35660e9	0.142658	0.0713288	+	0.997453i	$0.477276\pi$
$920$	2.44886e10	1.03683
$921$	2.41629e10	1.01916
$922$	−2.58995e10	−1.08826
$923$	−2.19114e9	−0.0917200
$924$	0	0
$925$	3.46135e9	0.143797
$926$	−3.35958e10	−1.39042
$927$	−1.07593e10	−0.443612
$928$	−4.04313e9	−0.166073
$929$	−3.09152e10	−1.26508	−0.632539	−	0.774528i	$-0.717987\pi$
$929$	−3.09152e10	−1.26508	−0.632539	+	0.774528i	$0.717987\pi$
$930$	−1.15350e10	−0.470249
$931$	0	0
$932$	1.50447e10	0.608735
$933$	3.21558e9	0.129621
$934$	−3.26014e10	−1.30925
$935$	−2.44502e10	−0.978232
$936$	5.22138e9	0.208123
$937$	−5.91421e9	−0.234860	−0.117430	−	0.993081i	$-0.537466\pi$
$937$	−5.91421e9	−0.234860	−0.117430	+	0.993081i	$0.537466\pi$
$938$	0	0
$939$	−1.84092e10	−0.725614
$940$	9.00590e10	3.53655
$941$	5.01276e10	1.96116	0.980580	−	0.196121i	$-0.0628344\pi$
$941$	5.01276e10	1.96116	0.980580	+	0.196121i	$0.0628344\pi$
$942$	−4.81727e10	−1.87769
$943$	−1.45505e9	−0.0565051
$944$	−2.28516e9	−0.0884126
$945$	0	0
$946$	−3.94982e10	−1.51690
$947$	−6.47196e9	−0.247635	−0.123817	−	0.992305i	$-0.539514\pi$
$947$	−6.47196e9	−0.247635	−0.123817	+	0.992305i	$0.539514\pi$
$948$	2.16763e10	0.826335
$949$	−1.92012e10	−0.729282
$950$	1.42471e10	0.539130
$951$	1.72788e10	0.651452
$952$	0	0
$953$	1.92800e9	0.0721575	0.0360788	−	0.999349i	$-0.488513\pi$
$953$	1.92800e9	0.0721575	0.0360788	+	0.999349i	$0.488513\pi$
$954$	1.86936e10	0.697066
$955$	−1.92604e10	−0.715573
$956$	−1.80144e10	−0.666831
$957$	−1.01055e10	−0.372707
$958$	6.83753e10	2.51258
$959$	0	0
$960$	−4.07274e10	−1.48572
$961$	−2.55844e10	−0.929916
$962$	−6.11573e9	−0.221480
$963$	1.41648e10	0.511114
$964$	−5.41640e10	−1.94734
$965$	4.75725e10	1.70416
$966$	0	0
$967$	9.68312e9	0.344368	0.172184	−	0.985065i	$-0.444918\pi$
$967$	9.68312e9	0.344368	0.172184	+	0.985065i	$0.444918\pi$
$968$	3.98515e10	1.41215
$969$	7.42208e9	0.262055
$970$	9.63576e10	3.38988
$971$	4.24418e10	1.48774	0.743870	−	0.668325i	$-0.232988\pi$
$971$	4.24418e10	1.48774	0.743870	+	0.668325i	$0.232988\pi$
$972$	2.90663e10	1.01522
$973$	0	0
$974$	4.01207e10	1.39127
$975$	−7.21977e9	−0.249464
$976$	−2.54132e10	−0.874952
$977$	4.37670e10	1.50147	0.750734	−	0.660605i	$-0.229700\pi$
$977$	4.37670e10	1.50147	0.750734	+	0.660605i	$0.229700\pi$
$978$	3.09488e10	1.05793
$979$	−4.99395e10	−1.70100
$980$	0	0
$981$	−6.98508e8	−0.0236227
$982$	−3.66998e10	−1.23673
$983$	1.14491e9	0.0384445	0.0192223	−	0.999815i	$-0.493881\pi$
$983$	1.14491e9	0.0384445	0.0192223	+	0.999815i	$0.493881\pi$
$984$	−3.31884e9	−0.111046
$985$	5.82221e10	1.94116
$986$	8.65672e9	0.287597
$987$	0	0
$988$	−1.62653e10	−0.536554
$989$	1.15250e10	0.378839
$990$	2.62834e10	0.860911
$991$	1.50163e10	0.490122	0.245061	−	0.969508i	$-0.421192\pi$
$991$	1.50163e10	0.490122	0.245061	+	0.969508i	$0.421192\pi$
$992$	4.34635e9	0.141362
$993$	4.67062e10	1.51374
$994$	0	0
$995$	−2.23727e10	−0.720008
$996$	−9.95453e8	−0.0319237
$997$	−5.34366e10	−1.70768	−0.853839	−	0.520538i	$-0.825732\pi$
$997$	−5.34366e10	−1.70768	−0.853839	+	0.520538i	$0.825732\pi$
$998$	−7.26812e10	−2.31454
$999$	−8.66967e9	−0.275121

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.8.a.f.1.4		4
3.2	odd	2		441.8.a.s.1.1		4
7.2	even	3		7.8.c.a.4.1	yes	8
7.3	odd	6		49.8.c.g.30.1		8
7.4	even	3		7.8.c.a.2.1	✓	8
7.5	odd	6		49.8.c.g.18.1		8
7.6	odd	2		49.8.a.e.1.4		4
21.2	odd	6		63.8.e.b.46.4		8
21.11	odd	6		63.8.e.b.37.4		8
21.20	even	2		441.8.a.t.1.1		4
28.11	odd	6		112.8.i.c.65.3		8
28.23	odd	6		112.8.i.c.81.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.8.c.a.2.1	✓	8	7.4	even	3
7.8.c.a.4.1	yes	8	7.2	even	3
49.8.a.e.1.4		4	7.6	odd	2
49.8.a.f.1.4		4	1.1	even	1	trivial
49.8.c.g.18.1		8	7.5	odd	6
49.8.c.g.30.1		8	7.3	odd	6
63.8.e.b.37.4		8	21.11	odd	6
63.8.e.b.46.4		8	21.2	odd	6
112.8.i.c.65.3		8	28.11	odd	6
112.8.i.c.81.3		8	28.23	odd	6
441.8.a.s.1.1		4	3.2	odd	2
441.8.a.t.1.1		4	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+19.0193 q^{2} +39.4615 q^{3} +233.735 q^{4} +350.005 q^{5} +750.531 q^{6} +2011.00 q^{8} -629.788 q^{9} +6656.85 q^{10} -6269.29 q^{11} +9223.52 q^{12} -4122.67 q^{13} +13811.7 q^{15} +8329.82 q^{16} +11142.7 q^{17} -11978.1 q^{18} +16879.6 q^{19} +81808.2 q^{20} -119238. q^{22} +34791.9 q^{23} +79357.1 q^{24} +44378.3 q^{25} -78410.4 q^{26} -111155. q^{27} +40847.7 q^{29} +262690. q^{30} -43911.2 q^{31} -98980.4 q^{32} -247396. q^{33} +211927. q^{34} -147203. q^{36} +77996.4 q^{37} +321038. q^{38} -162687. q^{39} +703859. q^{40} -41821.6 q^{41} +331256. q^{43} -1.46535e6 q^{44} -220429. q^{45} +661718. q^{46} +1.10085e6 q^{47} +328707. q^{48} +844045. q^{50} +439708. q^{51} -963611. q^{52} -1.56064e6 q^{53} -2.11409e6 q^{54} -2.19428e6 q^{55} +666093. q^{57} +776896. q^{58} -274335. q^{59} +3.22828e6 q^{60} -3.05087e6 q^{61} -835161. q^{62} -2.94876e6 q^{64} -1.44295e6 q^{65} -4.70530e6 q^{66} -645864. q^{67} +2.60444e6 q^{68} +1.37294e6 q^{69} +531485. q^{71} -1.26650e6 q^{72} +4.65746e6 q^{73} +1.48344e6 q^{74} +1.75124e6 q^{75} +3.94533e6 q^{76} -3.09419e6 q^{78} +2.35011e6 q^{79} +2.91548e6 q^{80} -3.00899e6 q^{81} -795418. q^{82} -107925. q^{83} +3.90000e6 q^{85} +6.30027e6 q^{86} +1.61191e6 q^{87} -1.26075e7 q^{88} +7.96573e6 q^{89} -4.19240e6 q^{90} +8.13206e6 q^{92} -1.73280e6 q^{93} +2.09375e7 q^{94} +5.90792e6 q^{95} -3.90592e6 q^{96} +1.44749e7 q^{97} +3.94832e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{2} + 28 q^{3} + 348 q^{4} + 252 q^{5} + 1022 q^{6} - 984 q^{8} + 2008 q^{9} + 4774 q^{10} - 3972 q^{11} - 5404 q^{12} - 1176 q^{13} - 16556 q^{15} + 57264 q^{16} + 56364 q^{17} - 35908 q^{18}+ \cdots + 15213976 q^{99}+O(q^{100}) \) 4 * q - 6 * q^2 + 28 * q^3 + 348 * q^4 + 252 * q^5 + 1022 * q^6 - 984 * q^8 + 2008 * q^9 + 4774 * q^10 - 3972 * q^11 - 5404 * q^12 - 1176 * q^13 - 16556 * q^15 + 57264 * q^16 + 56364 * q^17 - 35908 * q^18 + 41748 * q^19 + 162372 * q^20 - 152954 * q^22 + 131748 * q^23 + 389592 * q^24 - 68016 * q^25 + 113652 * q^26 + 189028 * q^27 + 34056 * q^29 + 605378 * q^30 + 401212 * q^31 - 453408 * q^32 + 24052 * q^33 + 82110 * q^34 - 360088 * q^36 - 5396 * q^37 + 31794 * q^38 + 364840 * q^39 - 443688 * q^40 + 410424 * q^41 + 46544 * q^43 - 1465836 * q^44 - 1209656 * q^45 + 1379202 * q^46 + 1470084 * q^47 - 2018800 * q^48 + 1395528 * q^50 - 1724796 * q^51 - 5269768 * q^52 - 642372 * q^53 - 4567318 * q^54 - 3063340 * q^55 + 3359068 * q^57 - 1972220 * q^58 + 752220 * q^59 + 2123324 * q^60 + 1325772 * q^61 - 3016314 * q^62 - 865856 * q^64 - 4868808 * q^65 - 6013294 * q^66 - 290916 * q^67 + 2453556 * q^68 + 4915260 * q^69 + 3377760 * q^71 + 4557744 * q^72 + 6706588 * q^73 + 3616878 * q^74 + 7223888 * q^75 + 2923004 * q^76 - 16288132 * q^78 + 3946244 * q^79 + 10695888 * q^80 - 11731292 * q^81 - 17563252 * q^82 + 9542064 * q^83 + 7006068 * q^85 + 23237832 * q^86 + 17836840 * q^87 - 8043336 * q^88 + 16165212 * q^89 + 3415748 * q^90 - 19442628 * q^92 + 12552852 * q^93 + 5720442 * q^94 - 3268452 * q^95 - 8448608 * q^96 + 1533112 * q^97 + 15213976 * q^99

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