LMFDB - Embedded newform 62.6.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [62,6,Mod(1,62)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(62, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("62.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 62 = 2 \cdot 31 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	62.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.94379682840$
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} - 2x^{3} - 221x^{2} - 140x + 2412 $ x^4 - 2x^3 - 221x^2 - 140*x + 2412
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$15.8744$ of defining polynomial
Character	$\chi$	$=$	62.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} -15.2327 q^{3} +16.0000 q^{4} +4.15476 q^{5} +60.9309 q^{6} -174.927 q^{7} -64.0000 q^{8} -10.9643 q^{9} -16.6190 q^{10} -235.351 q^{11} -243.723 q^{12} +1029.68 q^{13} +699.709 q^{14} -63.2883 q^{15} +256.000 q^{16} +1455.25 q^{17} +43.8572 q^{18} +1049.81 q^{19} +66.4761 q^{20} +2664.62 q^{21} +941.406 q^{22} -59.4958 q^{23} +974.894 q^{24} -3107.74 q^{25} -4118.72 q^{26} +3868.57 q^{27} -2798.84 q^{28} +4193.48 q^{29} +253.153 q^{30} +961.000 q^{31} -1024.00 q^{32} +3585.04 q^{33} -5820.99 q^{34} -726.781 q^{35} -175.429 q^{36} +7798.04 q^{37} -4199.25 q^{38} -15684.8 q^{39} -265.905 q^{40} -4010.52 q^{41} -10658.5 q^{42} -1999.94 q^{43} -3765.62 q^{44} -45.5541 q^{45} +237.983 q^{46} +13311.7 q^{47} -3899.58 q^{48} +13792.6 q^{49} +12431.0 q^{50} -22167.4 q^{51} +16474.9 q^{52} -31868.1 q^{53} -15474.3 q^{54} -977.829 q^{55} +11195.3 q^{56} -15991.5 q^{57} -16773.9 q^{58} -35467.1 q^{59} -1012.61 q^{60} +45806.4 q^{61} -3844.00 q^{62} +1917.96 q^{63} +4096.00 q^{64} +4278.07 q^{65} -14340.2 q^{66} +55514.2 q^{67} +23284.0 q^{68} +906.283 q^{69} +2907.12 q^{70} -47049.1 q^{71} +701.716 q^{72} +79212.2 q^{73} -31192.2 q^{74} +47339.3 q^{75} +16797.0 q^{76} +41169.4 q^{77} +62739.2 q^{78} +814.146 q^{79} +1063.62 q^{80} -56264.5 q^{81} +16042.1 q^{82} +47660.7 q^{83} +42633.9 q^{84} +6046.20 q^{85} +7999.76 q^{86} -63878.1 q^{87} +15062.5 q^{88} -46071.5 q^{89} +182.216 q^{90} -180119. q^{91} -951.933 q^{92} -14638.6 q^{93} -53246.8 q^{94} +4361.72 q^{95} +15598.3 q^{96} +54266.6 q^{97} -55170.3 q^{98} +2580.47 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{2} + 10 q^{3} + 64 q^{4} + 2 q^{5} - 40 q^{6} + 146 q^{7} - 256 q^{8} - 36 q^{9} - 8 q^{10} - 270 q^{11} + 160 q^{12} + 1038 q^{13} - 584 q^{14} + 1864 q^{15} + 1024 q^{16} + 2020 q^{17} + 144 q^{18}+ \cdots - 324354 q^{99}+O(q^{100}) $ 4 * q - 16 * q^2 + 10 * q^3 + 64 * q^4 + 2 * q^5 - 40 * q^6 + 146 * q^7 - 256 * q^8 - 36 * q^9 - 8 * q^10 - 270 * q^11 + 160 * q^12 + 1038 * q^13 - 584 * q^14 + 1864 * q^15 + 1024 * q^16 + 2020 * q^17 + 144 * q^18 + 6214 * q^19 + 32 * q^20 + 7256 * q^21 + 1080 * q^22 + 8292 * q^23 - 640 * q^24 + 9974 * q^25 - 4152 * q^26 + 5032 * q^27 + 2336 * q^28 + 1894 * q^29 - 7456 * q^30 + 3844 * q^31 - 4096 * q^32 - 11280 * q^33 - 8080 * q^34 - 9146 * q^35 - 576 * q^36 - 5470 * q^37 - 24856 * q^38 - 1628 * q^39 - 128 * q^40 - 16878 * q^41 - 29024 * q^42 + 1530 * q^43 - 4320 * q^44 + 5642 * q^45 - 33168 * q^46 - 25460 * q^47 + 2560 * q^48 + 21630 * q^49 - 39896 * q^50 - 40372 * q^51 + 16608 * q^52 - 7790 * q^53 - 20128 * q^54 + 5760 * q^55 - 9344 * q^56 + 15036 * q^57 - 7576 * q^58 - 35802 * q^59 + 29824 * q^60 + 48366 * q^61 - 15376 * q^62 + 53634 * q^63 + 16384 * q^64 + 12096 * q^65 + 45120 * q^66 + 54696 * q^67 + 32320 * q^68 + 110760 * q^69 + 36584 * q^70 - 35342 * q^71 + 2304 * q^72 + 59336 * q^73 + 21880 * q^74 - 24510 * q^75 + 99424 * q^76 - 86976 * q^77 + 6512 * q^78 - 3652 * q^79 + 512 * q^80 - 73452 * q^81 + 67512 * q^82 - 80074 * q^83 + 116096 * q^84 + 81860 * q^85 - 6120 * q^86 + 112364 * q^87 + 17280 * q^88 - 12060 * q^89 - 22568 * q^90 - 74952 * q^91 + 132672 * q^92 + 9610 * q^93 + 101840 * q^94 - 286138 * q^95 - 10240 * q^96 + 232670 * q^97 - 86520 * q^98 - 324354 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	−15.2327	−0.977179	−0.488590	−	0.872514i	$-0.662488\pi$
$3$	−15.2327	−0.977179	−0.488590	+	0.872514i	$0.662488\pi$
$4$	16.0000	0.500000
$5$	4.15476	0.0743226	0.0371613	−	0.999309i	$-0.488168\pi$
$5$	4.15476	0.0743226	0.0371613	+	0.999309i	$0.488168\pi$
$6$	60.9309	0.690970
$7$	−174.927	−1.34931	−0.674656	−	0.738132i	$-0.735708\pi$
$7$	−174.927	−1.34931	−0.674656	+	0.738132i	$0.735708\pi$
$8$	−64.0000	−0.353553
$9$	−10.9643	−0.0451206
$10$	−16.6190	−0.0525540
$11$	−235.351	−0.586456	−0.293228	−	0.956043i	$-0.594729\pi$
$11$	−235.351	−0.586456	−0.293228	+	0.956043i	$0.594729\pi$
$12$	−243.723	−0.488590
$13$	1029.68	1.68983	0.844916	−	0.534899i	$-0.179650\pi$
$13$	1029.68	1.68983	0.844916	+	0.534899i	$0.179650\pi$
$14$	699.709	0.954108
$15$	−63.2883	−0.0726265
$16$	256.000	0.250000
$17$	1455.25	1.22128	0.610639	−	0.791909i	$-0.290913\pi$
$17$	1455.25	1.22128	0.610639	+	0.791909i	$0.290913\pi$
$18$	43.8572	0.0319051
$19$	1049.81	0.667157	0.333579	−	0.942722i	$-0.391744\pi$
$19$	1049.81	0.667157	0.333579	+	0.942722i	$0.391744\pi$
$20$	66.4761	0.0371613
$21$	2664.62	1.31852
$22$	941.406	0.414687
$23$	−59.4958	−0.0234513	−0.0117256	−	0.999931i	$-0.503732\pi$
$23$	−59.4958	−0.0234513	−0.0117256	+	0.999931i	$0.503732\pi$
$24$	974.894	0.345485
$25$	−3107.74	−0.994476
$26$	−4118.72	−1.19489
$27$	3868.57	1.02127
$28$	−2798.84	−0.674656
$29$	4193.48	0.925933	0.462966	−	0.886376i	$-0.346785\pi$
$29$	4193.48	0.925933	0.462966	+	0.886376i	$0.346785\pi$
$30$	253.153	0.0513547
$31$	961.000	0.179605
$31$	961.000	0.179605
$32$	−1024.00	−0.176777
$33$	3585.04	0.573072
$34$	−5820.99	−0.863574
$35$	−726.781	−0.100284
$36$	−175.429	−0.0225603
$37$	7798.04	0.936443	0.468221	−	0.883611i	$-0.344895\pi$
$37$	7798.04	0.936443	0.468221	+	0.883611i	$0.344895\pi$
$38$	−4199.25	−0.471751
$39$	−15684.8	−1.65127
$40$	−265.905	−0.0262770
$41$	−4010.52	−0.372599	−0.186299	−	0.982493i	$-0.559649\pi$
$41$	−4010.52	−0.372599	−0.186299	+	0.982493i	$0.559649\pi$
$42$	−10658.5	−0.932335
$43$	−1999.94	−0.164947	−0.0824737	−	0.996593i	$-0.526282\pi$
$43$	−1999.94	−0.164947	−0.0824737	+	0.996593i	$0.526282\pi$
$44$	−3765.62	−0.293228
$45$	−45.5541	−0.00335348
$46$	237.983	0.0165826
$47$	13311.7	0.879000	0.439500	−	0.898243i	$-0.355156\pi$
$47$	13311.7	0.879000	0.439500	+	0.898243i	$0.355156\pi$
$48$	−3899.58	−0.244295
$49$	13792.6	0.820645
$50$	12431.0	0.703201
$51$	−22167.4	−1.19341
$52$	16474.9	0.844916
$53$	−31868.1	−1.55835	−0.779177	−	0.626804i	$-0.784363\pi$
$53$	−31868.1	−1.55835	−0.779177	+	0.626804i	$0.784363\pi$
$54$	−15474.3	−0.722147
$55$	−977.829	−0.0435869
$56$	11195.3	0.477054
$57$	−15991.5	−0.651932
$58$	−16773.9	−0.654733
$59$	−35467.1	−1.32646	−0.663231	−	0.748414i	$-0.730815\pi$
$59$	−35467.1	−1.32646	−0.663231	+	0.748414i	$0.730815\pi$
$60$	−1012.61	−0.0363132
$61$	45806.4	1.57617	0.788083	−	0.615569i	$-0.211074\pi$
$61$	45806.4	1.57617	0.788083	+	0.615569i	$0.211074\pi$
$62$	−3844.00	−0.127000
$63$	1917.96	0.0608818
$64$	4096.00	0.125000
$65$	4278.07	0.125593
$66$	−14340.2	−0.405223
$67$	55514.2	1.51083	0.755417	−	0.655244i	$-0.227434\pi$
$67$	55514.2	1.51083	0.755417	+	0.655244i	$0.227434\pi$
$68$	23284.0	0.610639
$69$	906.283	0.0229161
$70$	2907.12	0.0709118
$71$	−47049.1	−1.10766	−0.553829	−	0.832630i	$-0.686834\pi$
$71$	−47049.1	−1.10766	−0.553829	+	0.832630i	$0.686834\pi$
$72$	701.716	0.0159525
$73$	79212.2	1.73974	0.869871	−	0.493280i	$-0.164202\pi$
$73$	79212.2	1.73974	0.869871	+	0.493280i	$0.164202\pi$
$74$	−31192.2	−0.662165
$75$	47339.3	0.971782
$76$	16797.0	0.333579
$77$	41169.4	0.791312
$78$	62739.2	1.16762
$79$	814.146	0.0146769	0.00733845	−	0.999973i	$-0.497664\pi$
$79$	814.146	0.0146769	0.00733845	+	0.999973i	$0.497664\pi$
$80$	1063.62	0.0185806
$81$	−56264.5	−0.952844
$82$	16042.1	0.263467
$83$	47660.7	0.759390	0.379695	−	0.925112i	$-0.376029\pi$
$83$	47660.7	0.759390	0.379695	+	0.925112i	$0.376029\pi$
$84$	42633.9	0.659260
$85$	6046.20	0.0907686
$86$	7999.76	0.116635
$87$	−63878.1	−0.904802
$88$	15062.5	0.207343
$89$	−46071.5	−0.616535	−0.308267	−	0.951300i	$-0.599749\pi$
$89$	−46071.5	−0.616535	−0.308267	+	0.951300i	$0.599749\pi$
$90$	182.216	0.00237127
$91$	−180119.	−2.28011
$92$	−951.933	−0.0117256
$93$	−14638.6	−0.175507
$94$	−53246.8	−0.621547
$95$	4361.72	0.0495848
$96$	15598.3	0.172743
$97$	54266.6	0.585603	0.292802	−	0.956173i	$-0.405412\pi$
$97$	54266.6	0.585603	0.292802	+	0.956173i	$0.405412\pi$
$98$	−55170.3	−0.580283
$99$	2580.47	0.0264612
$100$	−49723.8	−0.497238
$101$	82647.7	0.806171	0.403086	−	0.915162i	$-0.367938\pi$
$101$	82647.7	0.806171	0.403086	+	0.915162i	$0.367938\pi$
$102$	88669.5	0.843867
$103$	84118.3	0.781263	0.390631	−	0.920547i	$-0.372257\pi$
$103$	84118.3	0.781263	0.390631	+	0.920547i	$0.372257\pi$
$104$	−65899.4	−0.597446
$105$	11070.8	0.0979958
$106$	127472.	1.10192
$107$	−108238.	−0.913942	−0.456971	−	0.889482i	$-0.651066\pi$
$107$	−108238.	−0.913942	−0.456971	+	0.889482i	$0.651066\pi$
$108$	61897.1	0.510635
$109$	−76250.3	−0.614717	−0.307359	−	0.951594i	$-0.599445\pi$
$109$	−76250.3	−0.614717	−0.307359	+	0.951594i	$0.599445\pi$
$110$	3911.31	0.0308206
$111$	−118785.	−0.915072
$112$	−44781.4	−0.337328
$113$	113394.	0.835400	0.417700	−	0.908585i	$-0.362836\pi$
$113$	113394.	0.835400	0.417700	+	0.908585i	$0.362836\pi$
$114$	63966.1	0.460986
$115$	−247.191	−0.00174296
$116$	67095.7	0.462966
$117$	−11289.7	−0.0762462
$118$	141868.	0.937951
$119$	−254563.	−1.64789
$120$	4050.45	0.0256773
$121$	−105661.	−0.656070
$122$	−183226.	−1.11452
$123$	61091.1	0.364096
$124$	15376.0	0.0898027
$125$	−25895.5	−0.148235
$126$	−7671.83	−0.0430499
$127$	127074.	0.699110	0.349555	−	0.936916i	$-0.386333\pi$
$127$	127074.	0.699110	0.349555	+	0.936916i	$0.386333\pi$
$128$	−16384.0	−0.0883883
$129$	30464.5	0.161183
$130$	−17112.3	−0.0888074
$131$	9469.80	0.0482128	0.0241064	−	0.999709i	$-0.492326\pi$
$131$	9469.80	0.0482128	0.0241064	+	0.999709i	$0.492326\pi$
$132$	57360.7	0.286536
$133$	−183641.	−0.900204
$134$	−222057.	−1.06832
$135$	16073.0	0.0759034
$136$	−93135.9	−0.431787
$137$	66791.2	0.304031	0.152016	−	0.988378i	$-0.451424\pi$
$137$	66791.2	0.304031	0.152016	+	0.988378i	$0.451424\pi$
$138$	−3625.13	−0.0162041
$139$	264672.	1.16190	0.580952	−	0.813938i	$-0.302680\pi$
$139$	264672.	1.16190	0.580952	+	0.813938i	$0.302680\pi$
$140$	−11628.5	−0.0501422
$141$	−202773.	−0.858941
$142$	188197.	0.783233
$143$	−242336.	−0.991011
$144$	−2806.86	−0.0112802
$145$	17422.9	0.0688177
$146$	−316849.	−1.23018
$147$	−210098.	−0.801917
$148$	124769.	0.468221
$149$	272372.	1.00507	0.502535	−	0.864557i	$-0.332401\pi$
$149$	272372.	1.00507	0.502535	+	0.864557i	$0.332401\pi$
$150$	−189357.	−0.687153
$151$	223300.	0.796978	0.398489	−	0.917173i	$-0.369535\pi$
$151$	223300.	0.796978	0.398489	+	0.917173i	$0.369535\pi$
$152$	−67188.1	−0.235876
$153$	−15955.8	−0.0551048
$154$	−164678.	−0.559542
$155$	3992.72	0.0133487
$156$	−250957.	−0.825634
$157$	−7971.22	−0.0258093	−0.0129046	−	0.999917i	$-0.504108\pi$
$157$	−7971.22	−0.0258093	−0.0129046	+	0.999917i	$0.504108\pi$
$158$	−3256.58	−0.0103781
$159$	485437.	1.52279
$160$	−4254.47	−0.0131385
$161$	10407.4	0.0316431
$162$	225058.	0.673762
$163$	−192360.	−0.567083	−0.283541	−	0.958960i	$-0.591509\pi$
$163$	−192360.	−0.567083	−0.283541	+	0.958960i	$0.591509\pi$
$164$	−64168.3	−0.186299
$165$	14895.0	0.0425922
$166$	−190643.	−0.536970
$167$	23682.7	0.0657114	0.0328557	−	0.999460i	$-0.489540\pi$
$167$	23682.7	0.0657114	0.0328557	+	0.999460i	$0.489540\pi$
$168$	−170536.	−0.466167
$169$	688946.	1.85553
$170$	−24184.8	−0.0641831
$171$	−11510.5	−0.0301025
$172$	−31999.0	−0.0824737
$173$	712817.	1.81077	0.905385	−	0.424593i	$-0.139583\pi$
$173$	712817.	1.81077	0.905385	+	0.424593i	$0.139583\pi$
$174$	255512.	0.639792
$175$	543628.	1.34186
$176$	−60250.0	−0.146614
$177$	540260.	1.29619
$178$	184286.	0.435956
$179$	−591690.	−1.38026	−0.690131	−	0.723684i	$-0.742447\pi$
$179$	−591690.	−1.38026	−0.690131	+	0.723684i	$0.742447\pi$
$180$	−728.865	−0.00167674
$181$	649491.	1.47359	0.736794	−	0.676117i	$-0.236339\pi$
$181$	649491.	1.47359	0.736794	+	0.676117i	$0.236339\pi$
$182$	720476.	1.61228
$183$	−697756.	−1.54020
$184$	3807.73	0.00829129
$185$	32399.0	0.0695988
$186$	58554.6	0.124102
$187$	−342495.	−0.716226
$188$	212987.	0.439500
$189$	−676718.	−1.37801
$190$	−17446.9	−0.0350618
$191$	662573.	1.31417	0.657083	−	0.753818i	$-0.271790\pi$
$191$	662573.	1.31417	0.657083	+	0.753818i	$0.271790\pi$
$192$	−62393.2	−0.122147
$193$	944863.	1.82589	0.912947	−	0.408078i	$-0.133801\pi$
$193$	944863.	1.82589	0.912947	+	0.408078i	$0.133801\pi$
$194$	−217066.	−0.414084
$195$	−65166.6	−0.122727
$196$	220681.	0.410322
$197$	−443922.	−0.814969	−0.407485	−	0.913212i	$-0.633594\pi$
$197$	−443922.	−0.814969	−0.407485	+	0.913212i	$0.633594\pi$
$198$	−10321.9	−0.0187109
$199$	−24623.0	−0.0440765	−0.0220383	−	0.999757i	$-0.507016\pi$
$199$	−24623.0	−0.0440765	−0.0220383	+	0.999757i	$0.507016\pi$
$200$	198895.	0.351600
$201$	−845631.	−1.47636
$202$	−330591.	−0.570049
$203$	−733554.	−1.24937
$204$	−354678.	−0.596704
$205$	−16662.7	−0.0276925
$206$	−336473.	−0.552436
$207$	652.331	0.00105814
$208$	263598.	0.422458
$209$	−247075.	−0.391258
$210$	−44283.4	−0.0692935
$211$	−389799.	−0.602747	−0.301373	−	0.953506i	$-0.597445\pi$
$211$	−389799.	−0.602747	−0.301373	+	0.953506i	$0.597445\pi$
$212$	−509889.	−0.779177
$213$	716686.	1.08238
$214$	432950.	0.646255
$215$	−8309.27	−0.0122593
$216$	−247588.	−0.361074
$217$	−168105.	−0.242344
$218$	305001.	0.434671
$219$	−1.20662e6	−1.70004
$220$	−15645.3	−0.0217935
$221$	1.49844e6	2.06375
$222$	475141.	0.647054
$223$	−467711.	−0.629819	−0.314910	−	0.949122i	$-0.601974\pi$
$223$	−467711.	−0.629819	−0.314910	+	0.949122i	$0.601974\pi$
$224$	179126.	0.238527
$225$	34074.2	0.0448714
$226$	−453577.	−0.590717
$227$	−898140.	−1.15686	−0.578428	−	0.815733i	$-0.696334\pi$
$227$	−898140.	−1.15686	−0.578428	+	0.815733i	$0.696334\pi$
$228$	−255864.	−0.325966
$229$	−630277.	−0.794223	−0.397112	−	0.917770i	$-0.629987\pi$
$229$	−630277.	−0.794223	−0.397112	+	0.917770i	$0.629987\pi$
$230$	988.763	0.00123246
$231$	−627122.	−0.773254
$232$	−268383.	−0.327367
$233$	−473923.	−0.571898	−0.285949	−	0.958245i	$-0.592309\pi$
$233$	−473923.	−0.571898	−0.285949	+	0.958245i	$0.592309\pi$
$234$	45158.9	0.0539142
$235$	55306.9	0.0653296
$236$	−567473.	−0.663231
$237$	−12401.7	−0.0143420
$238$	1.01825e6	1.16523
$239$	123418.	0.139761	0.0698803	−	0.997555i	$-0.477738\pi$
$239$	123418.	0.139761	0.0698803	+	0.997555i	$0.477738\pi$
$240$	−16201.8	−0.0181566
$241$	−1.80023e6	−1.99657	−0.998287	−	0.0584992i	$-0.981368\pi$
$241$	−1.80023e6	−1.99657	−0.998287	+	0.0584992i	$0.981368\pi$
$242$	422643.	0.463911
$243$	−83001.1	−0.0901713
$244$	732903.	0.788083
$245$	57304.8	0.0609924
$246$	−244365.	−0.257454
$247$	1.08097e6	1.12738
$248$	−61504.0	−0.0635001
$249$	−726002.	−0.742061
$250$	103582.	0.104818
$251$	885581.	0.887246	0.443623	−	0.896213i	$-0.353693\pi$
$251$	885581.	0.887246	0.443623	+	0.896213i	$0.353693\pi$
$252$	30687.3	0.0304409
$253$	14002.4	0.0137531
$254$	−508294.	−0.494346
$255$	−92100.1	−0.0886972
$256$	65536.0	0.0625000
$257$	786523.	0.742812	0.371406	−	0.928471i	$-0.378876\pi$
$257$	786523.	0.742812	0.371406	+	0.928471i	$0.378876\pi$
$258$	−121858.	−0.113974
$259$	−1.36409e6	−1.26355
$260$	68449.1	0.0627963
$261$	−45978.6	−0.0417787
$262$	−37879.2	−0.0340916
$263$	1.48145e6	1.32068	0.660341	−	0.750966i	$-0.270412\pi$
$263$	1.48145e6	1.32068	0.660341	+	0.750966i	$0.270412\pi$
$264$	−229443.	−0.202612
$265$	−132404.	−0.115821
$266$	734564.	0.636540
$267$	701794.	0.602465
$268$	888227.	0.755417
$269$	−1.16750e6	−0.983727	−0.491863	−	0.870672i	$-0.663684\pi$
$269$	−1.16750e6	−0.983727	−0.491863	+	0.870672i	$0.663684\pi$
$270$	−64291.8	−0.0536718
$271$	1.20110e6	0.993477	0.496738	−	0.867900i	$-0.334531\pi$
$271$	1.20110e6	0.993477	0.496738	+	0.867900i	$0.334531\pi$
$272$	372543.	0.305320
$273$	2.74370e6	2.22808
$274$	−267165.	−0.214982
$275$	731411.	0.583216
$276$	14500.5	0.0114581
$277$	−1.84842e6	−1.44745	−0.723723	−	0.690091i	$-0.757571\pi$
$277$	−1.84842e6	−1.44745	−0.723723	+	0.690091i	$0.757571\pi$
$278$	−1.05869e6	−0.821590
$279$	−10536.7	−0.00810390
$280$	46514.0	0.0354559
$281$	982107.	0.741981	0.370991	−	0.928637i	$-0.379018\pi$
$281$	982107.	0.741981	0.370991	+	0.928637i	$0.379018\pi$
$282$	811094.	0.607363
$283$	−76041.8	−0.0564399	−0.0282200	−	0.999602i	$-0.508984\pi$
$283$	−76041.8	−0.0564399	−0.0282200	+	0.999602i	$0.508984\pi$
$284$	−752786.	−0.553829
$285$	−66440.9	−0.0484533
$286$	969346.	0.700751
$287$	701550.	0.502752
$288$	11227.5	0.00797627
$289$	697890.	0.491521
$290$	−69691.6	−0.0486615
$291$	−826628.	−0.572239
$292$	1.26739e6	0.869871
$293$	−2.66036e6	−1.81039	−0.905194	−	0.424998i	$-0.860275\pi$
$293$	−2.66036e6	−1.81039	−0.905194	+	0.424998i	$0.860275\pi$
$294$	840393.	0.567041
$295$	−147357.	−0.0985861
$296$	−499075.	−0.331082
$297$	−910473.	−0.598930
$298$	−1.08949e6	−0.710692
$299$	−61261.6	−0.0396287
$300$	757429.	0.485891
$301$	349844.	0.222566
$302$	−893200.	−0.563549
$303$	−1.25895e6	−0.787774
$304$	268752.	0.166789
$305$	190315.	0.117145
$306$	63823.2	0.0389650
$307$	951120.	0.575956	0.287978	−	0.957637i	$-0.407017\pi$
$307$	951120.	0.575956	0.287978	+	0.957637i	$0.407017\pi$
$308$	658710.	0.395656
$309$	−1.28135e6	−0.763434
$310$	−15970.9	−0.00943898
$311$	699861.	0.410309	0.205154	−	0.978730i	$-0.434230\pi$
$311$	699861.	0.410309	0.205154	+	0.978730i	$0.434230\pi$
$312$	1.00383e6	0.583812
$313$	3.03373e6	1.75031	0.875156	−	0.483841i	$-0.160759\pi$
$313$	3.03373e6	1.75031	0.875156	+	0.483841i	$0.160759\pi$
$314$	31884.9	0.0182499
$315$	7968.65	0.00452489
$316$	13026.3	0.00733845
$317$	2.04461e6	1.14278	0.571390	−	0.820679i	$-0.306404\pi$
$317$	2.04461e6	1.14278	0.571390	+	0.820679i	$0.306404\pi$
$318$	−1.94175e6	−1.07678
$319$	−986941.	−0.543019
$320$	17017.9	0.00929032
$321$	1.64875e6	0.893085
$322$	−41629.8	−0.0223751
$323$	1.52774e6	0.814785
$324$	−900231.	−0.476422
$325$	−3.19997e6	−1.68050
$326$	769441.	0.400988
$327$	1.16150e6	0.600689
$328$	256673.	0.131733
$329$	−2.32858e6	−1.18605
$330$	−59579.9	−0.0301172
$331$	3.17478e6	1.59273	0.796367	−	0.604814i	$-0.206753\pi$
$331$	3.17478e6	1.59273	0.796367	+	0.604814i	$0.206753\pi$
$332$	762571.	0.379695
$333$	−85500.1	−0.0422529
$334$	−94730.9	−0.0464649
$335$	230648.	0.112289
$336$	682142.	0.329630
$337$	288974.	0.138607	0.0693033	−	0.997596i	$-0.477922\pi$
$337$	288974.	0.138607	0.0693033	+	0.997596i	$0.477922\pi$
$338$	−2.75578e6	−1.31206
$339$	−1.72730e6	−0.816336
$340$	96739.3	0.0453843
$341$	−226173.	−0.105331
$342$	46041.9	0.0212857
$343$	527306.	0.242007
$344$	127996.	0.0583177
$345$	3765.39	0.00170319
$346$	−2.85127e6	−1.28041
$347$	3.73614e6	1.66571	0.832856	−	0.553490i	$-0.186704\pi$
$347$	3.73614e6	1.66571	0.832856	+	0.553490i	$0.186704\pi$
$348$	−1.02205e6	−0.452401
$349$	−3.49319e6	−1.53518	−0.767588	−	0.640943i	$-0.778543\pi$
$349$	−3.49319e6	−1.53518	−0.767588	+	0.640943i	$0.778543\pi$
$350$	−2.17451e6	−0.948838
$351$	3.98338e6	1.72577
$352$	241000.	0.103672
$353$	−2.86521e6	−1.22383	−0.611914	−	0.790925i	$-0.709600\pi$
$353$	−2.86521e6	−1.22383	−0.611914	+	0.790925i	$0.709600\pi$
$354$	−2.16104e6	−0.916546
$355$	−195478.	−0.0823240
$356$	−737144.	−0.308267
$357$	3.87768e6	1.61028
$358$	2.36676e6	0.975993
$359$	−4.21678e6	−1.72681	−0.863406	−	0.504510i	$-0.831673\pi$
$359$	−4.21678e6	−1.72681	−0.863406	+	0.504510i	$0.831673\pi$
$360$	2915.46	0.00118563
$361$	−1.37399e6	−0.554901
$362$	−2.59796e6	−1.04198
$363$	1.60950e6	0.641098
$364$	−2.88190e6	−1.14006
$365$	329107.	0.129302
$366$	2.79102e6	1.08908
$367$	−2.80326e6	−1.08642	−0.543211	−	0.839596i	$-0.682792\pi$
$367$	−2.80326e6	−1.08642	−0.543211	+	0.839596i	$0.682792\pi$
$368$	−15230.9	−0.00586282
$369$	43972.6	0.0168119
$370$	−129596.	−0.0492138
$371$	5.57460e6	2.10271
$372$	−234218.	−0.0877533
$373$	−3.03545e6	−1.12967	−0.564834	−	0.825204i	$-0.691060\pi$
$373$	−3.03545e6	−1.12967	−0.564834	+	0.825204i	$0.691060\pi$
$374$	1.36998e6	0.506448
$375$	394459.	0.144852
$376$	−851949.	−0.310774
$377$	4.31794e6	1.56467
$378$	2.70687e6	0.974402
$379$	−1.35861e6	−0.485845	−0.242922	−	0.970046i	$-0.578106\pi$
$379$	−1.35861e6	−0.485845	−0.242922	+	0.970046i	$0.578106\pi$
$380$	69787.6	0.0247924
$381$	−1.93567e6	−0.683156
$382$	−2.65029e6	−0.929256
$383$	−4.81834e6	−1.67842	−0.839209	−	0.543809i	$-0.816982\pi$
$383$	−4.81834e6	−1.67842	−0.839209	+	0.543809i	$0.816982\pi$
$384$	249573.	0.0863713
$385$	171049.	0.0588124
$386$	−3.77945e6	−1.29110
$387$	21928.0	0.00744253
$388$	868266.	0.292802
$389$	−2.84397e6	−0.952909	−0.476455	−	0.879199i	$-0.658078\pi$
$389$	−2.84397e6	−0.952909	−0.476455	+	0.879199i	$0.658078\pi$
$390$	260666.	0.0867808
$391$	−86581.2	−0.0286406
$392$	−882725.	−0.290142
$393$	−144251.	−0.0471126
$394$	1.77569e6	0.576270
$395$	3382.58	0.00109083
$396$	41287.5	0.0132306
$397$	−4.23047e6	−1.34714	−0.673569	−	0.739124i	$-0.735240\pi$
$397$	−4.23047e6	−1.34714	−0.673569	+	0.739124i	$0.735240\pi$
$398$	98491.8	0.0311668
$399$	2.79735e6	0.879660
$400$	−795581.	−0.248619
$401$	938762.	0.291538	0.145769	−	0.989319i	$-0.453434\pi$
$401$	938762.	0.291538	0.145769	+	0.989319i	$0.453434\pi$
$402$	3.38253e6	1.04394
$403$	989521.	0.303503
$404$	1.32236e6	0.403086
$405$	−233765.	−0.0708178
$406$	2.93422e6	0.883440
$407$	−1.83528e6	−0.549182
$408$	1.41871e6	0.421933
$409$	1.18216e6	0.349437	0.174718	−	0.984618i	$-0.444098\pi$
$409$	1.18216e6	0.349437	0.174718	+	0.984618i	$0.444098\pi$
$410$	66651.0	0.0195815
$411$	−1.01741e6	−0.297093
$412$	1.34589e6	0.390631
$413$	6.20416e6	1.78981
$414$	−2609.32	−0.000748216 0
$415$	198019.	0.0564399
$416$	−1.05439e6	−0.298723
$417$	−4.03167e6	−1.13539
$418$	988301.	0.276661
$419$	5.55949e6	1.54703	0.773517	−	0.633775i	$-0.218495\pi$
$419$	5.55949e6	1.54703	0.773517	+	0.633775i	$0.218495\pi$
$420$	177134.	0.0489979
$421$	−539989.	−0.148484	−0.0742420	−	0.997240i	$-0.523654\pi$
$421$	−539989.	−0.148484	−0.0742420	+	0.997240i	$0.523654\pi$
$422$	1.55920e6	0.426206
$423$	−145954.	−0.0396610
$424$	2.03956e6	0.550961
$425$	−4.52253e6	−1.21453
$426$	−2.86675e6	−0.765359
$427$	−8.01279e6	−2.12674
$428$	−1.73180e6	−0.456971
$429$	3.69144e6	0.968396
$430$	33237.1	0.00866865
$431$	4.82032e6	1.24992	0.624961	−	0.780656i	$-0.285115\pi$
$431$	4.82032e6	1.24992	0.624961	+	0.780656i	$0.285115\pi$
$432$	990353.	0.255318
$433$	684866.	0.175544	0.0877720	−	0.996141i	$-0.472025\pi$
$433$	684866.	0.175544	0.0877720	+	0.996141i	$0.472025\pi$
$434$	672421.	0.171363
$435$	−265398.	−0.0672473
$436$	−1.22001e6	−0.307359
$437$	−62459.5	−0.0156457
$438$	4.82647e6	1.20211
$439$	3.94888e6	0.977940	0.488970	−	0.872301i	$-0.337373\pi$
$439$	3.94888e6	0.977940	0.488970	+	0.872301i	$0.337373\pi$
$440$	62581.0	0.0154103
$441$	−151226.	−0.0370280
$442$	−5.99375e6	−1.45930
$443$	3.73154e6	0.903397	0.451699	−	0.892171i	$-0.350818\pi$
$443$	3.73154e6	0.903397	0.451699	+	0.892171i	$0.350818\pi$
$444$	−1.90057e6	−0.457536
$445$	−191416.	−0.0458224
$446$	1.87085e6	0.445349
$447$	−4.14896e6	−0.982133
$448$	−716502.	−0.168664
$449$	−5.80405e6	−1.35867	−0.679337	−	0.733826i	$-0.737733\pi$
$449$	−5.80405e6	−1.35867	−0.679337	+	0.733826i	$0.737733\pi$
$450$	−136297.	−0.0317289
$451$	943882.	0.218513
$452$	1.81431e6	0.417700
$453$	−3.40147e6	−0.778790
$454$	3.59256e6	0.818021
$455$	−748351.	−0.169464
$456$	1.02346e6	0.230493
$457$	1.86959e6	0.418750	0.209375	−	0.977835i	$-0.432857\pi$
$457$	1.86959e6	0.418750	0.209375	+	0.977835i	$0.432857\pi$
$458$	2.52111e6	0.561600
$459$	5.62972e6	1.24726
$460$	−3955.05	−0.000871481 0
$461$	3.30270e6	0.723797	0.361898	−	0.932218i	$-0.382129\pi$
$461$	3.30270e6	0.723797	0.361898	+	0.932218i	$0.382129\pi$
$462$	2.50849e6	0.546773
$463$	1.42250e6	0.308389	0.154195	−	0.988041i	$-0.450722\pi$
$463$	1.42250e6	0.308389	0.154195	+	0.988041i	$0.450722\pi$
$464$	1.07353e6	0.231483
$465$	−60820.0	−0.0130441
$466$	1.89569e6	0.404393
$467$	−6.32630e6	−1.34232	−0.671162	−	0.741311i	$-0.734205\pi$
$467$	−6.32630e6	−1.34232	−0.671162	+	0.741311i	$0.734205\pi$
$468$	−180635.	−0.0381231
$469$	−9.71094e6	−2.03859
$470$	−221228.	−0.0461950
$471$	121423.	0.0252203
$472$	2.26989e6	0.468975
$473$	470689.	0.0967344
$474$	49606.6	0.0101413
$475$	−3.26255e6	−0.663472
$476$	−4.07300e6	−0.823943
$477$	349411.	0.0703139
$478$	−493673.	−0.0988257
$479$	−7.24181e6	−1.44214	−0.721072	−	0.692860i	$-0.756350\pi$
$479$	−7.24181e6	−1.44214	−0.721072	+	0.692860i	$0.756350\pi$
$480$	64807.2	0.0128387
$481$	8.02948e6	1.58243
$482$	7.20092e6	1.41179
$483$	−158534.	−0.0309210
$484$	−1.69057e6	−0.328035
$485$	225465.	0.0435235
$486$	332004.	0.0637607
$487$	4.99256e6	0.953896	0.476948	−	0.878931i	$-0.341743\pi$
$487$	4.99256e6	0.953896	0.476948	+	0.878931i	$0.341743\pi$
$488$	−2.93161e6	−0.557259
$489$	2.93017e6	0.554141
$490$	−229219.	−0.0431282
$491$	4.15456e6	0.777716	0.388858	−	0.921298i	$-0.372870\pi$
$491$	4.15456e6	0.777716	0.388858	+	0.921298i	$0.372870\pi$
$492$	977458.	0.182048
$493$	6.10255e6	1.13082
$494$	−4.32388e6	−0.797180
$495$	10721.2	0.00196667
$496$	246016.	0.0449013
$497$	8.23018e6	1.49458
$498$	2.90401e6	0.524716
$499$	8.96959e6	1.61258	0.806290	−	0.591520i	$-0.201472\pi$
$499$	8.96959e6	1.61258	0.806290	+	0.591520i	$0.201472\pi$
$500$	−414328.	−0.0741173
$501$	−360752.	−0.0642118
$502$	−3.54233e6	−0.627378
$503$	3.70803e6	0.653467	0.326734	−	0.945116i	$-0.394052\pi$
$503$	3.70803e6	0.653467	0.326734	+	0.945116i	$0.394052\pi$
$504$	−122749.	−0.0215250
$505$	343381.	0.0599167
$506$	−56009.7	−0.00972494
$507$	−1.04945e7	−1.81319
$508$	2.03318e6	0.349555
$509$	5.11101e6	0.874405	0.437202	−	0.899363i	$-0.355969\pi$
$509$	5.11101e6	0.874405	0.437202	+	0.899363i	$0.355969\pi$
$510$	368401.	0.0627184
$511$	−1.38564e7	−2.34746
$512$	−262144.	−0.0441942
$513$	4.06127e6	0.681348
$514$	−3.14609e6	−0.525247
$515$	349491.	0.0580655
$516$	487432.	0.0805916
$517$	−3.13293e6	−0.515495
$518$	5.45636e6	0.893468
$519$	−1.08581e7	−1.76945
$520$	−273796.	−0.0444037
$521$	6.62277e6	1.06892	0.534460	−	0.845194i	$-0.320515\pi$
$521$	6.62277e6	1.06892	0.534460	+	0.845194i	$0.320515\pi$
$522$	183914.	0.0295420
$523$	−5.90383e6	−0.943799	−0.471899	−	0.881652i	$-0.656431\pi$
$523$	−5.90383e6	−0.943799	−0.471899	+	0.881652i	$0.656431\pi$
$524$	151517.	0.0241064
$525$	−8.28094e6	−1.31124
$526$	−5.92580e6	−0.933863
$527$	1.39849e6	0.219348
$528$	917771.	0.143268
$529$	−6.43280e6	−0.999450
$530$	529617.	0.0818977
$531$	388872.	0.0598508
$532$	−2.93826e6	−0.450102
$533$	−4.12955e6	−0.629629
$534$	−2.80718e6	−0.426007
$535$	−449701.	−0.0679265
$536$	−3.55291e6	−0.534160
$537$	9.01304e6	1.34876
$538$	4.66998e6	0.695600
$539$	−3.24610e6	−0.481272
$540$	257167.	0.0379517
$541$	−4.62041e6	−0.678715	−0.339357	−	0.940658i	$-0.610210\pi$
$541$	−4.62041e6	−0.678715	−0.339357	+	0.940658i	$0.610210\pi$
$542$	−4.80442e6	−0.702494
$543$	−9.89351e6	−1.43996
$544$	−1.49017e6	−0.215894
$545$	−316802.	−0.0456874
$546$	−1.09748e7	−1.57549
$547$	7.61421e6	1.08807	0.544035	−	0.839063i	$-0.316896\pi$
$547$	7.61421e6	1.08807	0.544035	+	0.839063i	$0.316896\pi$
$548$	1.06866e6	0.152016
$549$	−502236.	−0.0711175
$550$	−2.92564e6	−0.412396
$551$	4.40237e6	0.617743
$552$	−58002.1	−0.00810207
$553$	−142416.	−0.0198037
$554$	7.39370e6	1.02350
$555$	−493525.	−0.0680105
$556$	4.23474e6	0.580952
$557$	−8.83369e6	−1.20644	−0.603218	−	0.797577i	$-0.706115\pi$
$557$	−8.83369e6	−1.20644	−0.603218	+	0.797577i	$0.706115\pi$
$558$	42146.8	0.00573032
$559$	−2.05930e6	−0.278733
$560$	−186056.	−0.0250711
$561$	5.21713e6	0.699881
$562$	−3.92843e6	−0.524660
$563$	9.26342e6	1.23169	0.615843	−	0.787869i	$-0.288815\pi$
$563$	9.26342e6	1.23169	0.615843	+	0.787869i	$0.288815\pi$
$564$	−3.24437e6	−0.429470
$565$	471125.	0.0620891
$566$	304167.	0.0399091
$567$	9.84219e6	1.28568
$568$	3.01115e6	0.391616
$569$	1.18171e6	0.153013	0.0765067	−	0.997069i	$-0.475623\pi$
$569$	1.18171e6	0.153013	0.0765067	+	0.997069i	$0.475623\pi$
$570$	265764.	0.0342616
$571$	−1.38342e7	−1.77568	−0.887840	−	0.460151i	$-0.847795\pi$
$571$	−1.38342e7	−1.77568	−0.887840	+	0.460151i	$0.847795\pi$
$572$	−3.87738e6	−0.495506
$573$	−1.00928e7	−1.28418
$574$	−2.80620e6	−0.355499
$575$	184897.	0.0233218
$576$	−44909.8	−0.00564008
$577$	−7.78469e6	−0.973423	−0.486712	−	0.873563i	$-0.661804\pi$
$577$	−7.78469e6	−0.973423	−0.486712	+	0.873563i	$0.661804\pi$
$578$	−2.79156e6	−0.347558
$579$	−1.43928e7	−1.78423
$580$	278766.	0.0344089
$581$	−8.33716e6	−1.02466
$582$	3.30651e6	0.404634
$583$	7.50020e6	0.913906
$584$	−5.06958e6	−0.615092
$585$	−46906.1	−0.00566682
$586$	1.06414e7	1.28014
$587$	691004.	0.0827723	0.0413862	−	0.999143i	$-0.486823\pi$
$587$	691004.	0.0827723	0.0413862	+	0.999143i	$0.486823\pi$
$588$	−3.36157e6	−0.400958
$589$	1.00887e6	0.119825
$590$	589428.	0.0697109
$591$	6.76214e6	0.796371
$592$	1.99630e6	0.234111
$593$	1.66505e7	1.94442	0.972209	−	0.234115i	$-0.0752191\pi$
$593$	1.66505e7	1.94442	0.972209	+	0.234115i	$0.0752191\pi$
$594$	3.64189e6	0.423507
$595$	−1.05765e6	−0.122475
$596$	4.35795e6	0.502535
$597$	375075.	0.0430707
$598$	245046.	0.0280218
$599$	5.35814e6	0.610165	0.305082	−	0.952326i	$-0.401316\pi$
$599$	5.35814e6	0.610165	0.305082	+	0.952326i	$0.401316\pi$
$600$	−3.02971e6	−0.343577
$601$	326874.	0.0369143	0.0184571	−	0.999830i	$-0.494125\pi$
$601$	326874.	0.0369143	0.0184571	+	0.999830i	$0.494125\pi$
$602$	−1.39938e6	−0.157378
$603$	−608674.	−0.0681698
$604$	3.57280e6	0.398489
$605$	−438995.	−0.0487608
$606$	5.03579e6	0.557040
$607$	1.72684e7	1.90230	0.951152	−	0.308724i	$-0.0999018\pi$
$607$	1.72684e7	1.90230	0.951152	+	0.308724i	$0.0999018\pi$
$608$	−1.07501e6	−0.117938
$609$	1.11740e7	1.22086
$610$	−761258.	−0.0828338
$611$	1.37068e7	1.48536
$612$	−255293.	−0.0275524
$613$	−8.78767e6	−0.944545	−0.472273	−	0.881453i	$-0.656566\pi$
$613$	−8.78767e6	−0.944545	−0.472273	+	0.881453i	$0.656566\pi$
$614$	−3.80448e6	−0.407263
$615$	253819.	0.0270605
$616$	−2.63484e6	−0.279771
$617$	3.55288e6	0.375723	0.187861	−	0.982196i	$-0.439844\pi$
$617$	3.55288e6	0.375723	0.187861	+	0.982196i	$0.439844\pi$
$618$	5.12540e6	0.539829
$619$	1.34718e7	1.41319	0.706595	−	0.707619i	$-0.250231\pi$
$619$	1.34718e7	1.41319	0.706595	+	0.707619i	$0.250231\pi$
$620$	63883.6	0.00667437
$621$	−230164.	−0.0239501
$622$	−2.79944e6	−0.290132
$623$	8.05917e6	0.831898
$624$	−4.01531e6	−0.412817
$625$	9.60409e6	0.983459
$626$	−1.21349e7	−1.23766
$627$	3.76363e6	0.382329
$628$	−127540.	−0.0129046
$629$	1.13481e7	1.14366
$630$	−31874.6	−0.00319958
$631$	−1.19893e6	−0.119872	−0.0599362	−	0.998202i	$-0.519090\pi$
$631$	−1.19893e6	−0.119872	−0.0599362	+	0.998202i	$0.519090\pi$
$632$	−52105.3	−0.00518907
$633$	5.93770e6	0.588992
$634$	−8.17844e6	−0.808067
$635$	527960.	0.0519597
$636$	7.76700e6	0.761396
$637$	1.42019e7	1.38675
$638$	3.94777e6	0.383972
$639$	515861.	0.0499782
$640$	−68071.6	−0.00656925
$641$	−1.61440e6	−0.155191	−0.0775953	−	0.996985i	$-0.524724\pi$
$641$	−1.61440e6	−0.155191	−0.0775953	+	0.996985i	$0.524724\pi$
$642$	−6.59501e6	−0.631507
$643$	5.51394e6	0.525938	0.262969	−	0.964804i	$-0.415298\pi$
$643$	5.51394e6	0.525938	0.262969	+	0.964804i	$0.415298\pi$
$644$	166519.	0.0158216
$645$	126573.	0.0119796
$646$	−6.11096e6	−0.576140
$647$	−1.67629e7	−1.57430	−0.787150	−	0.616761i	$-0.788444\pi$
$647$	−1.67629e7	−1.57430	−0.787150	+	0.616761i	$0.788444\pi$
$648$	3.60093e6	0.336881
$649$	8.34722e6	0.777912
$650$	1.27999e7	1.18829
$651$	2.56070e6	0.236813
$652$	−3.07776e6	−0.283541
$653$	−7.92672e6	−0.727462	−0.363731	−	0.931504i	$-0.618497\pi$
$653$	−7.92672e6	−0.727462	−0.363731	+	0.931504i	$0.618497\pi$
$654$	−4.64600e6	−0.424751
$655$	39344.7	0.00358330
$656$	−1.02669e6	−0.0931496
$657$	−868507.	−0.0784982
$658$	9.31432e6	0.838661
$659$	−1.79313e7	−1.60841	−0.804206	−	0.594351i	$-0.797409\pi$
$659$	−1.79313e7	−1.60841	−0.804206	+	0.594351i	$0.797409\pi$
$660$	238320.	0.0212961
$661$	1.63759e6	0.145781	0.0728904	−	0.997340i	$-0.476778\pi$
$661$	1.63759e6	0.145781	0.0728904	+	0.997340i	$0.476778\pi$
$662$	−1.26991e7	−1.12623
$663$	−2.28253e7	−2.01666
$664$	−3.05028e6	−0.268485
$665$	−762984.	−0.0669055
$666$	342001.	0.0298773
$667$	−249494.	−0.0217143
$668$	378924.	0.0328557
$669$	7.12452e6	0.615446
$670$	−922592.	−0.0794004
$671$	−1.07806e7	−0.924351
$672$	−2.72857e6	−0.233084
$673$	6.68369e6	0.568825	0.284413	−	0.958702i	$-0.408201\pi$
$673$	6.68369e6	0.568825	0.284413	+	0.958702i	$0.408201\pi$
$674$	−1.15590e6	−0.0980097
$675$	−1.20225e7	−1.01563
$676$	1.10231e7	0.927765
$677$	435721.	0.0365373	0.0182687	−	0.999833i	$-0.494185\pi$
$677$	435721.	0.0365373	0.0182687	+	0.999833i	$0.494185\pi$
$678$	6.90920e6	0.577236
$679$	−9.49271e6	−0.790162
$680$	−386957.	−0.0320915
$681$	1.36811e7	1.13046
$682$	904691.	0.0744799
$683$	−7.88577e6	−0.646834	−0.323417	−	0.946257i	$-0.604832\pi$
$683$	−7.88577e6	−0.646834	−0.323417	+	0.946257i	$0.604832\pi$
$684$	−184168.	−0.0150513
$685$	277501.	0.0225964
$686$	−2.10922e6	−0.171125
$687$	9.60083e6	0.776098
$688$	−511984.	−0.0412369
$689$	−3.28139e7	−2.63336
$690$	−15061.6	−0.00120433
$691$	−2.42470e6	−0.193180	−0.0965900	−	0.995324i	$-0.530794\pi$
$691$	−2.42470e6	−0.193180	−0.0965900	+	0.995324i	$0.530794\pi$
$692$	1.14051e7	0.905385
$693$	−451394.	−0.0357045
$694$	−1.49446e7	−1.17784
$695$	1.09965e6	0.0863557
$696$	4.08820e6	0.319896
$697$	−5.83630e6	−0.455047
$698$	1.39727e7	1.08553
$699$	7.21914e6	0.558847
$700$	8.69805e6	0.670930
$701$	−1.34755e7	−1.03574	−0.517871	−	0.855459i	$-0.673275\pi$
$701$	−1.34755e7	−1.03574	−0.517871	+	0.855459i	$0.673275\pi$
$702$	−1.59335e7	−1.22031
$703$	8.18649e6	0.624754
$704$	−964000.	−0.0733070
$705$	−842475.	−0.0638387
$706$	1.14608e7	0.865376
$707$	−1.44573e7	−1.08778
$708$	8.64415e6	0.648096
$709$	1.62749e7	1.21591	0.607956	−	0.793971i	$-0.291990\pi$
$709$	1.62749e7	1.21591	0.607956	+	0.793971i	$0.291990\pi$
$710$	781911.	0.0582119
$711$	−8926.55	−0.000662231 0
$712$	2.94858e6	0.217978
$713$	−57175.5	−0.00421198
$714$	−1.55107e7	−1.13864
$715$	−1.00685e6	−0.0736545
$716$	−9.46704e6	−0.690131
$717$	−1.88000e6	−0.136571
$718$	1.68671e7	1.22104
$719$	−5.81254e6	−0.419319	−0.209659	−	0.977775i	$-0.567235\pi$
$719$	−5.81254e6	−0.419319	−0.209659	+	0.977775i	$0.567235\pi$
$720$	−11661.8	−0.000838370 0
$721$	−1.47146e7	−1.05417
$722$	5.49596e6	0.392374
$723$	2.74224e7	1.95101
$724$	1.03918e7	0.736794
$725$	−1.30322e7	−0.920818
$726$	−6.43800e6	−0.453325
$727$	−6.21696e6	−0.436256	−0.218128	−	0.975920i	$-0.569995\pi$
$727$	−6.21696e6	−0.436256	−0.218128	+	0.975920i	$0.569995\pi$
$728$	1.15276e7	0.806141
$729$	1.49366e7	1.04096
$730$	−1.31643e6	−0.0914304
$731$	−2.91041e6	−0.201447
$732$	−1.11641e7	−0.770098
$733$	1.81331e7	1.24656	0.623279	−	0.781999i	$-0.285800\pi$
$733$	1.81331e7	1.24656	0.623279	+	0.781999i	$0.285800\pi$
$734$	1.12131e7	0.768217
$735$	−872908.	−0.0596005
$736$	60923.7	0.00414564
$737$	−1.30653e7	−0.886037
$738$	−175890.	−0.0118878
$739$	−2.73725e7	−1.84375	−0.921876	−	0.387485i	$-0.873344\pi$
$739$	−2.73725e7	−1.84375	−0.921876	+	0.387485i	$0.873344\pi$
$740$	518384.	0.0347994
$741$	−1.64661e7	−1.10166
$742$	−2.22984e7	−1.48684
$743$	−2.90995e7	−1.93381	−0.966904	−	0.255142i	$-0.917878\pi$
$743$	−2.90995e7	−1.93381	−0.966904	+	0.255142i	$0.917878\pi$
$744$	936873.	0.0620509
$745$	1.13164e6	0.0746994
$746$	1.21418e7	0.798796
$747$	−522567.	−0.0342642
$748$	−5.47992e6	−0.358113
$749$	1.89337e7	1.23319
$750$	−1.57784e6	−0.102426
$751$	4.40114e6	0.284751	0.142376	−	0.989813i	$-0.454526\pi$
$751$	4.40114e6	0.284751	0.142376	+	0.989813i	$0.454526\pi$
$752$	3.40780e6	0.219750
$753$	−1.34898e7	−0.866999
$754$	−1.72717e7	−1.10639
$755$	927758.	0.0592335
$756$	−1.08275e7	−0.689006
$757$	6.88173e6	0.436474	0.218237	−	0.975896i	$-0.429969\pi$
$757$	6.88173e6	0.436474	0.218237	+	0.975896i	$0.429969\pi$
$758$	5.43445e6	0.343544
$759$	−213295.	−0.0134393
$760$	−279150.	−0.0175309
$761$	1.81095e7	1.13356	0.566781	−	0.823869i	$-0.308189\pi$
$761$	1.81095e7	1.13356	0.566781	+	0.823869i	$0.308189\pi$
$762$	7.74270e6	0.483064
$763$	1.33383e7	0.829446
$764$	1.06012e7	0.657083
$765$	−66292.5	−0.00409553
$766$	1.92733e7	1.18682
$767$	−3.65197e7	−2.24150
$768$	−998291.	−0.0610737
$769$	−1.21359e7	−0.740044	−0.370022	−	0.929023i	$-0.620650\pi$
$769$	−1.21359e7	−0.740044	−0.370022	+	0.929023i	$0.620650\pi$
$770$	−684196.	−0.0415866
$771$	−1.19809e7	−0.725860
$772$	1.51178e7	0.912947
$773$	−6.85275e6	−0.412493	−0.206246	−	0.978500i	$-0.566125\pi$
$773$	−6.85275e6	−0.412493	−0.206246	+	0.978500i	$0.566125\pi$
$774$	−87711.8	−0.00526266
$775$	−2.98654e6	−0.178613
$776$	−3.47306e6	−0.207042
$777$	2.07788e7	1.23472
$778$	1.13759e7	0.673808
$779$	−4.21030e6	−0.248582
$780$	−1.04267e6	−0.0613633
$781$	1.10731e7	0.649593
$782$	346325.	0.0202519
$783$	1.62228e7	0.945628
$784$	3.53090e6	0.205161
$785$	−33118.5	−0.00191821
$786$	577003.	0.0333136
$787$	−9.20801e6	−0.529943	−0.264971	−	0.964256i	$-0.585363\pi$
$787$	−9.20801e6	−0.529943	−0.264971	+	0.964256i	$0.585363\pi$
$788$	−7.10275e6	−0.407485
$789$	−2.25665e7	−1.29054
$790$	−13530.3	−0.000771330 0
$791$	−1.98357e7	−1.12722
$792$	−165150.	−0.00935546
$793$	4.71659e7	2.66345
$794$	1.69219e7	0.952570
$795$	2.01688e6	0.113178
$796$	−393967.	−0.0220383
$797$	−1.24590e7	−0.694762	−0.347381	−	0.937724i	$-0.612929\pi$
$797$	−1.24590e7	−0.694762	−0.347381	+	0.937724i	$0.612929\pi$
$798$	−1.11894e7	−0.622014
$799$	1.93718e7	1.07350
$800$	3.18232e6	0.175800
$801$	505142.	0.0278184
$802$	−3.75505e6	−0.206148
$803$	−1.86427e7	−1.02028
$804$	−1.35301e7	−0.738178
$805$	43240.4	0.00235180
$806$	−3.95809e6	−0.214609
$807$	1.77841e7	0.961277
$808$	−5.28945e6	−0.285025
$809$	1.39177e7	0.747645	0.373823	−	0.927500i	$-0.378047\pi$
$809$	1.39177e7	0.747645	0.373823	+	0.927500i	$0.378047\pi$
$810$	935061.	0.0500757
$811$	2.23695e7	1.19427	0.597137	−	0.802140i	$-0.296305\pi$
$811$	2.23695e7	1.19427	0.597137	+	0.802140i	$0.296305\pi$
$812$	−1.17369e7	−0.624686
$813$	−1.82961e7	−0.970805
$814$	7.34112e6	0.388330
$815$	−799211.	−0.0421471
$816$	−5.67485e6	−0.298352
$817$	−2.09956e6	−0.110046
$818$	−4.72865e6	−0.247089
$819$	1.97488e6	0.102880
$820$	−266604.	−0.0138462
$821$	5.80930e6	0.300792	0.150396	−	0.988626i	$-0.451945\pi$
$821$	5.80930e6	0.300792	0.150396	+	0.988626i	$0.451945\pi$
$822$	4.06965e6	0.210076
$823$	−2.13414e7	−1.09830	−0.549152	−	0.835722i	$-0.685049\pi$
$823$	−2.13414e7	−1.09830	−0.549152	+	0.835722i	$0.685049\pi$
$824$	−5.38357e6	−0.276218
$825$	−1.11414e7	−0.569907
$826$	−2.48166e7	−1.26559
$827$	2.57909e7	1.31130	0.655651	−	0.755064i	$-0.272394\pi$
$827$	2.57909e7	1.31130	0.655651	+	0.755064i	$0.272394\pi$
$828$	10437.3	0.000529069 0
$829$	−2.36243e7	−1.19391	−0.596956	−	0.802274i	$-0.703623\pi$
$829$	−2.36243e7	−1.19391	−0.596956	+	0.802274i	$0.703623\pi$
$830$	−792075.	−0.0399090
$831$	2.81565e7	1.41441
$832$	4.21756e6	0.211229
$833$	2.00716e7	1.00224
$834$	1.61267e7	0.802841
$835$	98396.0	0.00488384
$836$	−3.95320e6	−0.195629
$837$	3.71769e6	0.183426
$838$	−2.22380e7	−1.09392
$839$	2.79824e7	1.37240	0.686198	−	0.727415i	$-0.259278\pi$
$839$	2.79824e7	1.37240	0.686198	+	0.727415i	$0.259278\pi$
$840$	−708534.	−0.0346468
$841$	−2.92588e6	−0.142648
$842$	2.15996e6	0.104994
$843$	−1.49602e7	−0.725049
$844$	−6.23679e6	−0.301373
$845$	2.86240e6	0.137908
$846$	583815.	0.0280446
$847$	1.84829e7	0.885243
$848$	−8.15823e6	−0.389589
$849$	1.15832e6	0.0551519
$850$	1.80901e7	0.858804
$851$	−463951.	−0.0219608
$852$	1.14670e7	0.541190
$853$	−2.04470e7	−0.962182	−0.481091	−	0.876671i	$-0.659759\pi$
$853$	−2.04470e7	−0.962182	−0.481091	+	0.876671i	$0.659759\pi$
$854$	3.20512e7	1.50383
$855$	−47823.3	−0.00223730
$856$	6.92721e6	0.323127
$857$	3.08472e7	1.43471	0.717355	−	0.696708i	$-0.245352\pi$
$857$	3.08472e7	1.43471	0.717355	+	0.696708i	$0.245352\pi$
$858$	−1.47658e7	−0.684759
$859$	−1.90536e7	−0.881038	−0.440519	−	0.897743i	$-0.645206\pi$
$859$	−1.90536e7	−0.881038	−0.440519	+	0.897743i	$0.645206\pi$
$860$	−132948.	−0.00612966
$861$	−1.06865e7	−0.491279
$862$	−1.92813e7	−0.883828
$863$	−1.60269e7	−0.732525	−0.366263	−	0.930512i	$-0.619363\pi$
$863$	−1.60269e7	−0.732525	−0.366263	+	0.930512i	$0.619363\pi$
$864$	−3.96141e6	−0.180537
$865$	2.96158e6	0.134581
$866$	−2.73946e6	−0.124128
$867$	−1.06308e7	−0.480304
$868$	−2.68968e6	−0.121172
$869$	−191610.	−0.00860736
$870$	1.06159e6	0.0475510
$871$	5.71618e7	2.55305
$872$	4.88002e6	0.217335
$873$	−594996.	−0.0264228
$874$	249838.	0.0110632
$875$	4.52983e6	0.200015
$876$	−1.93059e7	−0.850020
$877$	6.46053e6	0.283641	0.141821	−	0.989892i	$-0.454704\pi$
$877$	6.46053e6	0.283641	0.141821	+	0.989892i	$0.454704\pi$
$878$	−1.57955e7	−0.691508
$879$	4.05245e7	1.76907
$880$	−250324.	−0.0108967
$881$	4.43941e7	1.92702	0.963508	−	0.267679i	$-0.0862567\pi$
$881$	4.43941e7	1.92702	0.963508	+	0.267679i	$0.0862567\pi$
$882$	604904.	0.0261827
$883$	3.31650e7	1.43146	0.715729	−	0.698378i	$-0.246095\pi$
$883$	3.31650e7	1.43146	0.715729	+	0.698378i	$0.246095\pi$
$884$	2.39750e7	1.03188
$885$	2.24465e6	0.0963363
$886$	−1.49262e7	−0.638798
$887$	−3.31641e7	−1.41533	−0.707667	−	0.706546i	$-0.750252\pi$
$887$	−3.31641e7	−1.41533	−0.707667	+	0.706546i	$0.750252\pi$
$888$	7.60226e6	0.323527
$889$	−2.22286e7	−0.943318
$890$	765664.	0.0324014
$891$	1.32419e7	0.558800
$892$	−7.48338e6	−0.314910
$893$	1.39748e7	0.586431
$894$	1.65958e7	0.694473
$895$	−2.45833e6	−0.102585
$896$	2.86601e6	0.119264
$897$	933181.	0.0387244
$898$	2.32162e7	0.960728
$899$	4.02993e6	0.166302
$900$	545187.	0.0224357
$901$	−4.63759e7	−1.90318
$902$	−3.77553e6	−0.154512
$903$	−5.32908e6	−0.217487
$904$	−7.25723e6	−0.295359
$905$	2.69848e6	0.109521
$906$	1.36059e7	0.550688
$907$	2.70053e7	1.09001	0.545005	−	0.838433i	$-0.316528\pi$
$907$	2.70053e7	1.09001	0.545005	+	0.838433i	$0.316528\pi$
$908$	−1.43702e7	−0.578428
$909$	−906175.	−0.0363749
$910$	2.99340e6	0.119829
$911$	1.82765e7	0.729621	0.364811	−	0.931082i	$-0.381134\pi$
$911$	1.82765e7	0.729621	0.364811	+	0.931082i	$0.381134\pi$
$912$	−4.09383e6	−0.162983
$913$	−1.12170e7	−0.445349
$914$	−7.47835e6	−0.296101
$915$	−2.89901e6	−0.114471
$916$	−1.00844e7	−0.397112
$917$	−1.65653e6	−0.0650542
$918$	−2.25189e7	−0.881943
$919$	2.05975e7	0.804498	0.402249	−	0.915530i	$-0.368229\pi$
$919$	2.05975e7	0.804498	0.402249	+	0.915530i	$0.368229\pi$
$920$	15820.2	0.000616230 0
$921$	−1.44881e7	−0.562812
$922$	−1.32108e7	−0.511801
$923$	−4.84455e7	−1.87176
$924$	−1.00340e7	−0.386627
$925$	−2.42343e7	−0.931270
$926$	−5.68999e6	−0.218064
$927$	−922299.	−0.0352511
$928$	−4.29412e6	−0.163683
$929$	−2.79827e7	−1.06378	−0.531888	−	0.846815i	$-0.678517\pi$
$929$	−2.79827e7	−1.06378	−0.531888	+	0.846815i	$0.678517\pi$
$930$	243280.	0.00922357
$931$	1.44796e7	0.547499
$932$	−7.58277e6	−0.285949
$933$	−1.06608e7	−0.400945
$934$	2.53052e7	0.949166
$935$	−1.42298e6	−0.0532317
$936$	722542.	0.0269571
$937$	−1.18863e7	−0.442281	−0.221141	−	0.975242i	$-0.570978\pi$
$937$	−1.18863e7	−0.442281	−0.221141	+	0.975242i	$0.570978\pi$
$938$	3.88438e7	1.44150
$939$	−4.62119e7	−1.71037
$940$	884911.	0.0326648
$941$	−2.99706e7	−1.10337	−0.551686	−	0.834052i	$-0.686015\pi$
$941$	−2.99706e7	−1.10337	−0.551686	+	0.834052i	$0.686015\pi$
$942$	−485694.	−0.0178334
$943$	238609.	0.00873792
$944$	−9.07957e6	−0.331616
$945$	−2.81160e6	−0.102417
$946$	−1.88275e6	−0.0684015
$947$	2.88301e6	0.104465	0.0522325	−	0.998635i	$-0.483366\pi$
$947$	2.88301e6	0.104465	0.0522325	+	0.998635i	$0.483366\pi$
$948$	−198427.	−0.00717099
$949$	8.15631e7	2.93987
$950$	1.30502e7	0.469146
$951$	−3.11450e7	−1.11670
$952$	1.62920e7	0.582616
$953$	−1.21859e7	−0.434637	−0.217319	−	0.976101i	$-0.569731\pi$
$953$	−1.21859e7	−0.434637	−0.217319	+	0.976101i	$0.569731\pi$
$954$	−1.39765e6	−0.0497194
$955$	2.75283e6	0.0976722
$956$	1.97469e6	0.0698803
$957$	1.50338e7	0.530626
$958$	2.89672e7	1.01975
$959$	−1.16836e7	−0.410233
$960$	−259229.	−0.00907831
$961$	923521.	0.0322581
$962$	−3.21179e7	−1.11895
$963$	1.18675e6	0.0412376
$964$	−2.88037e7	−0.998287
$965$	3.92568e6	0.135705
$966$	634135.	0.0218645
$967$	4.36736e7	1.50194	0.750970	−	0.660337i	$-0.229586\pi$
$967$	4.36736e7	1.50194	0.750970	+	0.660337i	$0.229586\pi$
$968$	6.76228e6	0.231956
$969$	−2.32716e7	−0.796191
$970$	−901859.	−0.0307758
$971$	2.72990e6	0.0929178	0.0464589	−	0.998920i	$-0.485206\pi$
$971$	2.72990e6	0.0929178	0.0464589	+	0.998920i	$0.485206\pi$
$972$	−1.32802e6	−0.0450856
$973$	−4.62983e7	−1.56777
$974$	−1.99702e7	−0.674506
$975$	4.87443e7	1.64215
$976$	1.17264e7	0.394041
$977$	−5.61565e6	−0.188219	−0.0941096	−	0.995562i	$-0.530000\pi$
$977$	−5.61565e6	−0.188219	−0.0941096	+	0.995562i	$0.530000\pi$
$978$	−1.17207e7	−0.391837
$979$	1.08430e7	0.361570
$980$	916877.	0.0304962
$981$	836032.	0.0277364
$982$	−1.66182e7	−0.549928
$983$	7.21208e6	0.238055	0.119027	−	0.992891i	$-0.462022\pi$
$983$	7.21208e6	0.238055	0.119027	+	0.992891i	$0.462022\pi$
$984$	−3.90983e6	−0.128727
$985$	−1.84439e6	−0.0605706
$986$	−2.44102e7	−0.799612
$987$	3.54706e7	1.15898
$988$	1.72955e7	0.563692
$989$	118988.	0.00386823
$990$	−42884.9	−0.00139064
$991$	−2.05082e7	−0.663351	−0.331675	−	0.943394i	$-0.607614\pi$
$991$	−2.05082e7	−0.663351	−0.331675	+	0.943394i	$0.607614\pi$
$992$	−984064.	−0.0317500
$993$	−4.83605e7	−1.55639
$994$	−3.29207e7	−1.05683
$995$	−102302.	−0.00327588
$996$	−1.16160e7	−0.371030
$997$	−3.56424e7	−1.13561	−0.567805	−	0.823163i	$-0.692207\pi$
$997$	−3.56424e7	−1.13561	−0.567805	+	0.823163i	$0.692207\pi$
$998$	−3.58784e7	−1.14027
$999$	3.01672e7	0.956361

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	62.6.a.c.1.1	✓	4
3.2	odd	2		558.6.a.k.1.3		4
4.3	odd	2		496.6.a.d.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
62.6.a.c.1.1	✓	4	1.1	even	1	trivial
496.6.a.d.1.4		4	4.3	odd	2
558.6.a.k.1.3		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 62 = 2 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	62.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} -15.2327 q^{3} +16.0000 q^{4} +4.15476 q^{5} +60.9309 q^{6} -174.927 q^{7} -64.0000 q^{8} -10.9643 q^{9} -16.6190 q^{10} -235.351 q^{11} -243.723 q^{12} +1029.68 q^{13} +699.709 q^{14} -63.2883 q^{15} +256.000 q^{16} +1455.25 q^{17} +43.8572 q^{18} +1049.81 q^{19} +66.4761 q^{20} +2664.62 q^{21} +941.406 q^{22} -59.4958 q^{23} +974.894 q^{24} -3107.74 q^{25} -4118.72 q^{26} +3868.57 q^{27} -2798.84 q^{28} +4193.48 q^{29} +253.153 q^{30} +961.000 q^{31} -1024.00 q^{32} +3585.04 q^{33} -5820.99 q^{34} -726.781 q^{35} -175.429 q^{36} +7798.04 q^{37} -4199.25 q^{38} -15684.8 q^{39} -265.905 q^{40} -4010.52 q^{41} -10658.5 q^{42} -1999.94 q^{43} -3765.62 q^{44} -45.5541 q^{45} +237.983 q^{46} +13311.7 q^{47} -3899.58 q^{48} +13792.6 q^{49} +12431.0 q^{50} -22167.4 q^{51} +16474.9 q^{52} -31868.1 q^{53} -15474.3 q^{54} -977.829 q^{55} +11195.3 q^{56} -15991.5 q^{57} -16773.9 q^{58} -35467.1 q^{59} -1012.61 q^{60} +45806.4 q^{61} -3844.00 q^{62} +1917.96 q^{63} +4096.00 q^{64} +4278.07 q^{65} -14340.2 q^{66} +55514.2 q^{67} +23284.0 q^{68} +906.283 q^{69} +2907.12 q^{70} -47049.1 q^{71} +701.716 q^{72} +79212.2 q^{73} -31192.2 q^{74} +47339.3 q^{75} +16797.0 q^{76} +41169.4 q^{77} +62739.2 q^{78} +814.146 q^{79} +1063.62 q^{80} -56264.5 q^{81} +16042.1 q^{82} +47660.7 q^{83} +42633.9 q^{84} +6046.20 q^{85} +7999.76 q^{86} -63878.1 q^{87} +15062.5 q^{88} -46071.5 q^{89} +182.216 q^{90} -180119. q^{91} -951.933 q^{92} -14638.6 q^{93} -53246.8 q^{94} +4361.72 q^{95} +15598.3 q^{96} +54266.6 q^{97} -55170.3 q^{98} +2580.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{2} + 10 q^{3} + 64 q^{4} + 2 q^{5} - 40 q^{6} + 146 q^{7} - 256 q^{8} - 36 q^{9} - 8 q^{10} - 270 q^{11} + 160 q^{12} + 1038 q^{13} - 584 q^{14} + 1864 q^{15} + 1024 q^{16} + 2020 q^{17} + 144 q^{18}+ \cdots - 324354 q^{99}+O(q^{100}) \) 4 * q - 16 * q^2 + 10 * q^3 + 64 * q^4 + 2 * q^5 - 40 * q^6 + 146 * q^7 - 256 * q^8 - 36 * q^9 - 8 * q^10 - 270 * q^11 + 160 * q^12 + 1038 * q^13 - 584 * q^14 + 1864 * q^15 + 1024 * q^16 + 2020 * q^17 + 144 * q^18 + 6214 * q^19 + 32 * q^20 + 7256 * q^21 + 1080 * q^22 + 8292 * q^23 - 640 * q^24 + 9974 * q^25 - 4152 * q^26 + 5032 * q^27 + 2336 * q^28 + 1894 * q^29 - 7456 * q^30 + 3844 * q^31 - 4096 * q^32 - 11280 * q^33 - 8080 * q^34 - 9146 * q^35 - 576 * q^36 - 5470 * q^37 - 24856 * q^38 - 1628 * q^39 - 128 * q^40 - 16878 * q^41 - 29024 * q^42 + 1530 * q^43 - 4320 * q^44 + 5642 * q^45 - 33168 * q^46 - 25460 * q^47 + 2560 * q^48 + 21630 * q^49 - 39896 * q^50 - 40372 * q^51 + 16608 * q^52 - 7790 * q^53 - 20128 * q^54 + 5760 * q^55 - 9344 * q^56 + 15036 * q^57 - 7576 * q^58 - 35802 * q^59 + 29824 * q^60 + 48366 * q^61 - 15376 * q^62 + 53634 * q^63 + 16384 * q^64 + 12096 * q^65 + 45120 * q^66 + 54696 * q^67 + 32320 * q^68 + 110760 * q^69 + 36584 * q^70 - 35342 * q^71 + 2304 * q^72 + 59336 * q^73 + 21880 * q^74 - 24510 * q^75 + 99424 * q^76 - 86976 * q^77 + 6512 * q^78 - 3652 * q^79 + 512 * q^80 - 73452 * q^81 + 67512 * q^82 - 80074 * q^83 + 116096 * q^84 + 81860 * q^85 - 6120 * q^86 + 112364 * q^87 + 17280 * q^88 - 12060 * q^89 - 22568 * q^90 - 74952 * q^91 + 132672 * q^92 + 9610 * q^93 + 101840 * q^94 - 286138 * q^95 - 10240 * q^96 + 232670 * q^97 - 86520 * q^98 - 324354 * q^99

Introduction

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