LMFDB - Embedded newform 538.6.a.b.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,6,Mod(1,538)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("538.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	538.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$86.2864950594$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	538.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} +2.64462 q^{3} +16.0000 q^{4} -42.4071 q^{5} -10.5785 q^{6} -172.158 q^{7} -64.0000 q^{8} -236.006 q^{9} +169.629 q^{10} +648.081 q^{11} +42.3139 q^{12} -194.570 q^{13} +688.633 q^{14} -112.151 q^{15} +256.000 q^{16} -105.631 q^{17} +944.024 q^{18} +265.494 q^{19} -678.514 q^{20} -455.293 q^{21} -2592.32 q^{22} -3539.52 q^{23} -169.256 q^{24} -1326.63 q^{25} +778.279 q^{26} -1266.79 q^{27} -2754.53 q^{28} -4994.21 q^{29} +448.603 q^{30} -6587.50 q^{31} -1024.00 q^{32} +1713.93 q^{33} +422.523 q^{34} +7300.74 q^{35} -3776.10 q^{36} -4255.79 q^{37} -1061.98 q^{38} -514.563 q^{39} +2714.06 q^{40} -7107.79 q^{41} +1821.17 q^{42} +18226.1 q^{43} +10369.3 q^{44} +10008.3 q^{45} +14158.1 q^{46} -28805.4 q^{47} +677.023 q^{48} +12831.4 q^{49} +5306.54 q^{50} -279.353 q^{51} -3113.12 q^{52} +6660.77 q^{53} +5067.16 q^{54} -27483.3 q^{55} +11018.1 q^{56} +702.132 q^{57} +19976.8 q^{58} +12934.4 q^{59} -1794.41 q^{60} +11476.2 q^{61} +26350.0 q^{62} +40630.4 q^{63} +4096.00 q^{64} +8251.15 q^{65} -6855.72 q^{66} -27150.5 q^{67} -1690.09 q^{68} -9360.69 q^{69} -29202.9 q^{70} -38971.7 q^{71} +15104.4 q^{72} +61084.6 q^{73} +17023.2 q^{74} -3508.44 q^{75} +4247.91 q^{76} -111572. q^{77} +2058.25 q^{78} -44908.3 q^{79} -10856.2 q^{80} +53999.3 q^{81} +28431.1 q^{82} +107550. q^{83} -7284.69 q^{84} +4479.50 q^{85} -72904.4 q^{86} -13207.8 q^{87} -41477.2 q^{88} +31722.0 q^{89} -40033.4 q^{90} +33496.8 q^{91} -56632.3 q^{92} -17421.4 q^{93} +115222. q^{94} -11258.9 q^{95} -2708.09 q^{96} +73223.4 q^{97} -51325.7 q^{98} -152951. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 108 q^{2} + 33 q^{3} + 432 q^{4} + 139 q^{5} - 132 q^{6} - 25 q^{7} - 1728 q^{8} + 2346 q^{9} - 556 q^{10} + 1241 q^{11} + 528 q^{12} - 202 q^{13} + 100 q^{14} + 1786 q^{15} + 6912 q^{16} + 1550 q^{17}+ \cdots + 335378 q^{99}+O(q^{100}) $ 27 * q - 108 * q^2 + 33 * q^3 + 432 * q^4 + 139 * q^5 - 132 * q^6 - 25 * q^7 - 1728 * q^8 + 2346 * q^9 - 556 * q^10 + 1241 * q^11 + 528 * q^12 - 202 * q^13 + 100 * q^14 + 1786 * q^15 + 6912 * q^16 + 1550 * q^17 - 9384 * q^18 - 66 * q^19 + 2224 * q^20 - 1217 * q^21 - 4964 * q^22 + 10568 * q^23 - 2112 * q^24 + 15880 * q^25 + 808 * q^26 + 8256 * q^27 - 400 * q^28 - 2122 * q^29 - 7144 * q^30 - 1621 * q^31 - 27648 * q^32 + 18369 * q^33 - 6200 * q^34 + 23123 * q^35 + 37536 * q^36 + 8542 * q^37 + 264 * q^38 + 589 * q^39 - 8896 * q^40 + 10859 * q^41 + 4868 * q^42 + 23759 * q^43 + 19856 * q^44 + 28006 * q^45 - 42272 * q^46 + 38457 * q^47 + 8448 * q^48 + 57376 * q^49 - 63520 * q^50 + 58237 * q^51 - 3232 * q^52 + 73586 * q^53 - 33024 * q^54 + 25923 * q^55 + 1600 * q^56 + 52534 * q^57 + 8488 * q^58 + 68927 * q^59 + 28576 * q^60 - 76016 * q^61 + 6484 * q^62 - 20738 * q^63 + 110592 * q^64 + 74867 * q^65 - 73476 * q^66 + 94787 * q^67 + 24800 * q^68 - 90601 * q^69 - 92492 * q^70 + 15523 * q^71 - 150144 * q^72 - 96279 * q^73 - 34168 * q^74 - 123786 * q^75 - 1056 * q^76 - 199866 * q^77 - 2356 * q^78 - 246217 * q^79 + 35584 * q^80 - 54473 * q^81 - 43436 * q^82 - 56658 * q^83 - 19472 * q^84 - 120541 * q^85 - 95036 * q^86 - 352954 * q^87 - 79424 * q^88 - 83070 * q^89 - 112024 * q^90 - 132348 * q^91 + 169088 * q^92 + 40727 * q^93 - 153828 * q^94 - 72090 * q^95 - 33792 * q^96 - 36273 * q^97 - 229504 * q^98 + 335378 * 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
$3$	2.64462	0.169653	0.0848263	−	0.996396i	$-0.472966\pi$
$3$	2.64462	0.169653	0.0848263	+	0.996396i	$0.472966\pi$
$4$	16.0000	0.500000
$5$	−42.4071	−0.758602	−0.379301	−	0.925273i	$-0.623836\pi$
$5$	−42.4071	−0.758602	−0.379301	+	0.925273i	$0.623836\pi$
$6$	−10.5785	−0.119962
$7$	−172.158	−1.32795	−0.663976	−	0.747754i	$-0.731132\pi$
$7$	−172.158	−1.32795	−0.663976	+	0.747754i	$0.731132\pi$
$8$	−64.0000	−0.353553
$9$	−236.006	−0.971218
$10$	169.629	0.536413
$11$	648.081	1.61491	0.807454	−	0.589931i	$-0.200845\pi$
$11$	648.081	1.61491	0.807454	+	0.589931i	$0.200845\pi$
$12$	42.3139	0.0848263
$13$	−194.570	−0.319313	−0.159657	−	0.987173i	$-0.551039\pi$
$13$	−194.570	−0.319313	−0.159657	+	0.987173i	$0.551039\pi$
$14$	688.633	0.939004
$15$	−112.151	−0.128699
$16$	256.000	0.250000
$17$	−105.631	−0.0886478	−0.0443239	−	0.999017i	$-0.514113\pi$
$17$	−105.631	−0.0886478	−0.0443239	+	0.999017i	$0.514113\pi$
$18$	944.024	0.686755
$19$	265.494	0.168722	0.0843609	−	0.996435i	$-0.473115\pi$
$19$	265.494	0.168722	0.0843609	+	0.996435i	$0.473115\pi$
$20$	−678.514	−0.379301
$21$	−455.293	−0.225291
$22$	−2592.32	−1.14191
$23$	−3539.52	−1.39516	−0.697581	−	0.716506i	$-0.745740\pi$
$23$	−3539.52	−1.39516	−0.697581	+	0.716506i	$0.745740\pi$
$24$	−169.256	−0.0599812
$25$	−1326.63	−0.424523
$26$	778.279	0.225789
$27$	−1266.79	−0.334422
$28$	−2754.53	−0.663976
$29$	−4994.21	−1.10274	−0.551369	−	0.834262i	$-0.685894\pi$
$29$	−4994.21	−1.10274	−0.551369	+	0.834262i	$0.685894\pi$
$30$	448.603	0.0910038
$31$	−6587.50	−1.23116	−0.615582	−	0.788073i	$-0.711079\pi$
$31$	−6587.50	−1.23116	−0.615582	+	0.788073i	$0.711079\pi$
$32$	−1024.00	−0.176777
$33$	1713.93	0.273973
$34$	422.523	0.0626835
$35$	7300.74	1.00739
$36$	−3776.10	−0.485609
$37$	−4255.79	−0.511065	−0.255532	−	0.966801i	$-0.582251\pi$
$37$	−4255.79	−0.511065	−0.255532	+	0.966801i	$0.582251\pi$
$38$	−1061.98	−0.119304
$39$	−514.563	−0.0541723
$40$	2714.06	0.268206
$41$	−7107.79	−0.660351	−0.330175	−	0.943920i	$-0.607108\pi$
$41$	−7107.79	−0.660351	−0.330175	+	0.943920i	$0.607108\pi$
$42$	1821.17	0.159304
$43$	18226.1	1.50322	0.751610	−	0.659608i	$-0.229278\pi$
$43$	18226.1	1.50322	0.751610	+	0.659608i	$0.229278\pi$
$44$	10369.3	0.807454
$45$	10008.3	0.736768
$46$	14158.1	0.986529
$47$	−28805.4	−1.90208	−0.951040	−	0.309067i	$-0.899983\pi$
$47$	−28805.4	−1.90208	−0.951040	+	0.309067i	$0.899983\pi$
$48$	677.023	0.0424131
$49$	12831.4	0.763458
$50$	5306.54	0.300183
$51$	−279.353	−0.0150393
$52$	−3113.12	−0.159657
$53$	6660.77	0.325713	0.162856	−	0.986650i	$-0.447929\pi$
$53$	6660.77	0.325713	0.162856	+	0.986650i	$0.447929\pi$
$54$	5067.16	0.236472
$55$	−27483.3	−1.22507
$56$	11018.1	0.469502
$57$	702.132	0.0286241
$58$	19976.8	0.779753
$59$	12934.4	0.483746	0.241873	−	0.970308i	$-0.422238\pi$
$59$	12934.4	0.483746	0.241873	+	0.970308i	$0.422238\pi$
$60$	−1794.41	−0.0643494
$61$	11476.2	0.394888	0.197444	−	0.980314i	$-0.436736\pi$
$61$	11476.2	0.394888	0.197444	+	0.980314i	$0.436736\pi$
$62$	26350.0	0.870565
$63$	40630.4	1.28973
$64$	4096.00	0.125000
$65$	8251.15	0.242232
$66$	−6855.72	−0.193728
$67$	−27150.5	−0.738909	−0.369454	−	0.929249i	$-0.620455\pi$
$67$	−27150.5	−0.738909	−0.369454	+	0.929249i	$0.620455\pi$
$68$	−1690.09	−0.0443239
$69$	−9360.69	−0.236693
$70$	−29202.9	−0.712331
$71$	−38971.7	−0.917494	−0.458747	−	0.888567i	$-0.651702\pi$
$71$	−38971.7	−0.917494	−0.458747	+	0.888567i	$0.651702\pi$
$72$	15104.4	0.343377
$73$	61084.6	1.34161	0.670803	−	0.741636i	$-0.265950\pi$
$73$	61084.6	1.34161	0.670803	+	0.741636i	$0.265950\pi$
$74$	17023.2	0.361377
$75$	−3508.44	−0.0720214
$76$	4247.91	0.0843609
$77$	−111572.	−2.14452
$78$	2058.25	0.0383056
$79$	−44908.3	−0.809579	−0.404789	−	0.914410i	$-0.632655\pi$
$79$	−44908.3	−0.809579	−0.404789	+	0.914410i	$0.632655\pi$
$80$	−10856.2	−0.189651
$81$	53999.3	0.914482
$82$	28431.1	0.466938
$83$	107550.	1.71361	0.856807	−	0.515637i	$-0.172445\pi$
$83$	107550.	1.71361	0.856807	+	0.515637i	$0.172445\pi$
$84$	−7284.69	−0.112645
$85$	4479.50	0.0672484
$86$	−72904.4	−1.06294
$87$	−13207.8	−0.187082
$88$	−41477.2	−0.570956
$89$	31722.0	0.424508	0.212254	−	0.977215i	$-0.431920\pi$
$89$	31722.0	0.424508	0.212254	+	0.977215i	$0.431920\pi$
$90$	−40033.4	−0.520974
$91$	33496.8	0.424033
$92$	−56632.3	−0.697581
$93$	−17421.4	−0.208870
$94$	115222.	1.34497
$95$	−11258.9	−0.127993
$96$	−2708.09	−0.0299906
$97$	73223.4	0.790170	0.395085	−	0.918645i	$-0.370715\pi$
$97$	73223.4	0.790170	0.395085	+	0.918645i	$0.370715\pi$
$98$	−51325.7	−0.539846
$99$	−152951.	−1.56843
$100$	−21226.1	−0.212261
$101$	−130998.	−1.27779	−0.638897	−	0.769292i	$-0.720609\pi$
$101$	−130998.	−1.27779	−0.638897	+	0.769292i	$0.720609\pi$
$102$	1117.41	0.0106344
$103$	47905.4	0.444929	0.222465	−	0.974941i	$-0.428590\pi$
$103$	47905.4	0.444929	0.222465	+	0.974941i	$0.428590\pi$
$104$	12452.5	0.112894
$105$	19307.7	0.170906
$106$	−26643.1	−0.230314
$107$	−140665.	−1.18775	−0.593876	−	0.804557i	$-0.702403\pi$
$107$	−140665.	−1.18775	−0.593876	+	0.804557i	$0.702403\pi$
$108$	−20268.6	−0.167211
$109$	−190680.	−1.53723	−0.768616	−	0.639711i	$-0.779054\pi$
$109$	−190680.	−1.53723	−0.768616	+	0.639711i	$0.779054\pi$
$110$	109933.	0.866257
$111$	−11255.0	−0.0867034
$112$	−44072.5	−0.331988
$113$	140537.	1.03537	0.517685	−	0.855571i	$-0.326794\pi$
$113$	140537.	1.03537	0.517685	+	0.855571i	$0.326794\pi$
$114$	−2808.53	−0.0202403
$115$	150101.	1.05837
$116$	−79907.4	−0.551369
$117$	45919.6	0.310123
$118$	−51737.7	−0.342060
$119$	18185.2	0.117720
$120$	7177.66	0.0455019
$121$	258958.	1.60793
$122$	−45904.8	−0.279228
$123$	−18797.4	−0.112030
$124$	−105400.	−0.615582
$125$	188781.	1.08065
$126$	−162521.	−0.911978
$127$	207445.	1.14128	0.570641	−	0.821200i	$-0.306695\pi$
$127$	207445.	1.14128	0.570641	+	0.821200i	$0.306695\pi$
$128$	−16384.0	−0.0883883
$129$	48201.1	0.255025
$130$	−33004.6	−0.171284
$131$	35800.6	0.182269	0.0911343	−	0.995839i	$-0.470951\pi$
$131$	35800.6	0.182269	0.0911343	+	0.995839i	$0.470951\pi$
$132$	27422.9	0.136987
$133$	−45707.0	−0.224054
$134$	108602.	0.522487
$135$	53720.9	0.253693
$136$	6760.37	0.0313417
$137$	379065.	1.72549	0.862745	−	0.505640i	$-0.168744\pi$
$137$	379065.	1.72549	0.862745	+	0.505640i	$0.168744\pi$
$138$	37442.8	0.167367
$139$	−215908.	−0.947831	−0.473915	−	0.880570i	$-0.657160\pi$
$139$	−215908.	−0.947831	−0.473915	+	0.880570i	$0.657160\pi$
$140$	116812.	0.503694
$141$	−76179.3	−0.322693
$142$	155887.	0.648766
$143$	−126097.	−0.515661
$144$	−60417.5	−0.242805
$145$	211790.	0.836539
$146$	−244339.	−0.948658
$147$	33934.3	0.129523
$148$	−68092.6	−0.255532
$149$	234224.	0.864303	0.432151	−	0.901801i	$-0.357755\pi$
$149$	234224.	0.864303	0.432151	+	0.901801i	$0.357755\pi$
$150$	14033.8	0.0509268
$151$	437892.	1.56288	0.781439	−	0.623982i	$-0.214486\pi$
$151$	437892.	1.56288	0.781439	+	0.623982i	$0.214486\pi$
$152$	−16991.6	−0.0596521
$153$	24929.5	0.0860964
$154$	446290.	1.51641
$155$	279357.	0.933964
$156$	−8233.01	−0.0270862
$157$	321591.	1.04125	0.520624	−	0.853786i	$-0.325699\pi$
$157$	321591.	1.04125	0.520624	+	0.853786i	$0.325699\pi$
$158$	179633.	0.572459
$159$	17615.2	0.0552580
$160$	43424.9	0.134103
$161$	609357.	1.85271
$162$	−215997.	−0.646637
$163$	−167781.	−0.494624	−0.247312	−	0.968936i	$-0.579547\pi$
$163$	−167781.	−0.494624	−0.247312	+	0.968936i	$0.579547\pi$
$164$	−113725.	−0.330175
$165$	−72682.9	−0.207837
$166$	−430198.	−1.21171
$167$	−219262.	−0.608376	−0.304188	−	0.952612i	$-0.598385\pi$
$167$	−219262.	−0.608376	−0.304188	+	0.952612i	$0.598385\pi$
$168$	29138.8	0.0796522
$169$	−333436.	−0.898039
$170$	−17918.0	−0.0475518
$171$	−62658.2	−0.163866
$172$	291618.	0.751610
$173$	406899.	1.03364	0.516822	−	0.856093i	$-0.327115\pi$
$173$	406899.	1.03364	0.516822	+	0.856093i	$0.327115\pi$
$174$	52831.2	0.132287
$175$	228391.	0.563746
$176$	165909.	0.403727
$177$	34206.7	0.0820687
$178$	−126888.	−0.300172
$179$	−347306.	−0.810177	−0.405088	−	0.914278i	$-0.632759\pi$
$179$	−347306.	−0.810177	−0.405088	+	0.914278i	$0.632759\pi$
$180$	160133.	0.368384
$181$	−339461.	−0.770182	−0.385091	−	0.922879i	$-0.625830\pi$
$181$	−339461.	−0.770182	−0.385091	+	0.922879i	$0.625830\pi$
$182$	−133987.	−0.299836
$183$	30350.2	0.0669937
$184$	226529.	0.493264
$185$	180476.	0.387695
$186$	69685.7	0.147694
$187$	−68457.3	−0.143158
$188$	−460886.	−0.951040
$189$	218088.	0.444097
$190$	45035.4	0.0905045
$191$	−311944.	−0.618719	−0.309359	−	0.950945i	$-0.600115\pi$
$191$	−311944.	−0.618719	−0.309359	+	0.950945i	$0.600115\pi$
$192$	10832.4	0.0212066
$193$	80846.4	0.156231	0.0781155	−	0.996944i	$-0.475110\pi$
$193$	80846.4	0.156231	0.0781155	+	0.996944i	$0.475110\pi$
$194$	−292894.	−0.558735
$195$	21821.2	0.0410952
$196$	205303.	0.381729
$197$	−960730.	−1.76375	−0.881873	−	0.471488i	$-0.843717\pi$
$197$	−960730.	−1.76375	−0.881873	+	0.471488i	$0.843717\pi$
$198$	611804.	1.10905
$199$	362141.	0.648254	0.324127	−	0.946014i	$-0.394930\pi$
$199$	362141.	0.648254	0.324127	+	0.946014i	$0.394930\pi$
$200$	84904.6	0.150091
$201$	−71802.8	−0.125358
$202$	523992.	0.903537
$203$	859794.	1.46438
$204$	−4469.65	−0.00751967
$205$	301421.	0.500943
$206$	−191621.	−0.314613
$207$	835348.	1.35501
$208$	−49809.9	−0.0798283
$209$	172062.	0.272470
$210$	−77230.7	−0.120849
$211$	779268.	1.20498	0.602491	−	0.798126i	$-0.294175\pi$
$211$	779268.	1.20498	0.602491	+	0.798126i	$0.294175\pi$
$212$	106572.	0.162856
$213$	−103065.	−0.155655
$214$	562659.	0.839867
$215$	−772917.	−1.14035
$216$	81074.5	0.118236
$217$	1.13409e6	1.63493
$218$	762721.	1.08699
$219$	161546.	0.227607
$220$	−439732.	−0.612536
$221$	20552.6	0.0283064
$222$	45019.8	0.0613086
$223$	−350070.	−0.471404	−0.235702	−	0.971825i	$-0.575739\pi$
$223$	−350070.	−0.471404	−0.235702	+	0.971825i	$0.575739\pi$
$224$	176290.	0.234751
$225$	313094.	0.412304
$226$	−562150.	−0.732118
$227$	1.19359e6	1.53741	0.768704	−	0.639605i	$-0.220902\pi$
$227$	1.19359e6	1.53741	0.768704	+	0.639605i	$0.220902\pi$
$228$	11234.1	0.0143120
$229$	−123666.	−0.155834	−0.0779168	−	0.996960i	$-0.524827\pi$
$229$	−123666.	−0.155834	−0.0779168	+	0.996960i	$0.524827\pi$
$230$	−600404.	−0.748383
$231$	−295067.	−0.363823
$232$	319630.	0.389876
$233$	1.05599e6	1.27429	0.637146	−	0.770743i	$-0.280115\pi$
$233$	1.05599e6	1.27429	0.637146	+	0.770743i	$0.280115\pi$
$234$	−183679.	−0.219290
$235$	1.22155e6	1.44292
$236$	206951.	0.241873
$237$	−118766.	−0.137347
$238$	−72740.8	−0.0832407
$239$	812322.	0.919884	0.459942	−	0.887949i	$-0.347870\pi$
$239$	812322.	0.919884	0.459942	+	0.887949i	$0.347870\pi$
$240$	−28710.6	−0.0321747
$241$	−1.31322e6	−1.45644	−0.728221	−	0.685342i	$-0.759653\pi$
$241$	−1.31322e6	−1.45644	−0.728221	+	0.685342i	$0.759653\pi$
$242$	−1.03583e6	−1.13698
$243$	450638.	0.489566
$244$	183619.	0.197444
$245$	−544144.	−0.579160
$246$	75189.6	0.0792173
$247$	−51657.1	−0.0538751
$248$	421600.	0.435282
$249$	284428.	0.290719
$250$	−755124.	−0.764132
$251$	−344173.	−0.344820	−0.172410	−	0.985025i	$-0.555155\pi$
$251$	−344173.	−0.344820	−0.172410	+	0.985025i	$0.555155\pi$
$252$	650086.	0.644866
$253$	−2.29390e6	−2.25306
$254$	−829779.	−0.807008
$255$	11846.6	0.0114089
$256$	65536.0	0.0625000
$257$	285569.	0.269698	0.134849	−	0.990866i	$-0.456945\pi$
$257$	285569.	0.269698	0.134849	+	0.990866i	$0.456945\pi$
$258$	−192805.	−0.180330
$259$	732669.	0.678669
$260$	132018.	0.121116
$261$	1.17866e6	1.07100
$262$	−143202.	−0.128883
$263$	873850.	0.779018	0.389509	−	0.921023i	$-0.372645\pi$
$263$	873850.	0.779018	0.389509	+	0.921023i	$0.372645\pi$
$264$	−109692.	−0.0968642
$265$	−282464.	−0.247086
$266$	182828.	0.158430
$267$	83892.7	0.0720188
$268$	−434408.	−0.369454
$269$	72361.0	0.0609711
$269$	72361.0	0.0609711
$270$	−214884.	−0.179388
$271$	−872993.	−0.722084	−0.361042	−	0.932550i	$-0.617579\pi$
$271$	−872993.	−0.722084	−0.361042	+	0.932550i	$0.617579\pi$
$272$	−27041.5	−0.0221620
$273$	88586.3	0.0719383
$274$	−1.51626e6	−1.22011
$275$	−859767.	−0.685565
$276$	−149771.	−0.118346
$277$	1.85667e6	1.45390	0.726952	−	0.686689i	$-0.240936\pi$
$277$	1.85667e6	1.45390	0.726952	+	0.686689i	$0.240936\pi$
$278$	863630.	0.670218
$279$	1.55469e6	1.19573
$280$	−467247.	−0.356165
$281$	625660.	0.472686	0.236343	−	0.971670i	$-0.424051\pi$
$281$	625660.	0.472686	0.236343	+	0.971670i	$0.424051\pi$
$282$	304717.	0.228178
$283$	−217469.	−0.161410	−0.0807052	−	0.996738i	$-0.525717\pi$
$283$	−217469.	−0.161410	−0.0807052	+	0.996738i	$0.525717\pi$
$284$	−623547.	−0.458747
$285$	−29775.4	−0.0217143
$286$	504388.	0.364628
$287$	1.22366e6	0.876914
$288$	241670.	0.171689
$289$	−1.40870e6	−0.992142
$290$	−847161.	−0.591522
$291$	193648.	0.134054
$292$	977354.	0.670803
$293$	2.18405e6	1.48625	0.743127	−	0.669150i	$-0.233342\pi$
$293$	2.18405e6	1.48625	0.743127	+	0.669150i	$0.233342\pi$
$294$	−135737.	−0.0915863
$295$	−548512.	−0.366971
$296$	272371.	0.180689
$297$	−820983.	−0.540061
$298$	−936896.	−0.611154
$299$	688684.	0.445494
$300$	−56135.1	−0.0360107
$301$	−3.13777e6	−1.99620
$302$	−1.75157e6	−1.10512
$303$	−346440.	−0.216781
$304$	67966.5	0.0421804
$305$	−486673.	−0.299563
$306$	−99718.0	−0.0608793
$307$	−1.17552e6	−0.711845	−0.355922	−	0.934516i	$-0.615833\pi$
$307$	−1.17552e6	−0.711845	−0.355922	+	0.934516i	$0.615833\pi$
$308$	−1.78516e6	−1.07226
$309$	126692.	0.0754834
$310$	−1.11743e6	−0.660412
$311$	187420.	0.109879	0.0549396	−	0.998490i	$-0.482503\pi$
$311$	187420.	0.109879	0.0549396	+	0.998490i	$0.482503\pi$
$312$	32932.1	0.0191528
$313$	−685459.	−0.395477	−0.197738	−	0.980255i	$-0.563360\pi$
$313$	−685459.	−0.395477	−0.197738	+	0.980255i	$0.563360\pi$
$314$	−1.28636e6	−0.736274
$315$	−1.72302e6	−0.978393
$316$	−718533.	−0.404789
$317$	−655449.	−0.366345	−0.183173	−	0.983081i	$-0.558637\pi$
$317$	−655449.	−0.366345	−0.183173	+	0.983081i	$0.558637\pi$
$318$	−70460.9	−0.0390733
$319$	−3.23666e6	−1.78082
$320$	−173700.	−0.0948253
$321$	−372005.	−0.201505
$322$	−2.43743e6	−1.31006
$323$	−28044.4	−0.0149568
$324$	863988.	0.457241
$325$	258123.	0.135556
$326$	671126.	0.349752
$327$	−504277.	−0.260795
$328$	454898.	0.233469
$329$	4.95908e6	2.52587
$330$	290731.	0.146963
$331$	−3.45469e6	−1.73316	−0.866580	−	0.499038i	$-0.833687\pi$
$331$	−3.45469e6	−1.73316	−0.866580	+	0.499038i	$0.833687\pi$
$332$	1.72079e6	0.856807
$333$	1.00439e6	0.496355
$334$	877048.	0.430187
$335$	1.15138e6	0.560538
$336$	−116555.	−0.0563226
$337$	−641336.	−0.307617	−0.153809	−	0.988101i	$-0.549154\pi$
$337$	−641336.	−0.307617	−0.153809	+	0.988101i	$0.549154\pi$
$338$	1.33374e6	0.635009
$339$	371668.	0.175653
$340$	71672.0	0.0336242
$341$	−4.26923e6	−1.98822
$342$	250633.	0.115870
$343$	684427.	0.314117
$344$	−1.16647e6	−0.531468
$345$	396960.	0.179556
$346$	−1.62759e6	−0.730897
$347$	4.16069e6	1.85499	0.927496	−	0.373834i	$-0.121957\pi$
$347$	4.16069e6	1.85499	0.927496	+	0.373834i	$0.121957\pi$
$348$	−211325.	−0.0935411
$349$	−361045.	−0.158671	−0.0793356	−	0.996848i	$-0.525280\pi$
$349$	−361045.	−0.158671	−0.0793356	+	0.996848i	$0.525280\pi$
$350$	−913563.	−0.398629
$351$	246479.	0.106785
$352$	−663635.	−0.285478
$353$	−1.41983e6	−0.606458	−0.303229	−	0.952918i	$-0.598065\pi$
$353$	−1.41983e6	−0.606458	−0.303229	+	0.952918i	$0.598065\pi$
$354$	−136827.	−0.0580313
$355$	1.65268e6	0.696013
$356$	507552.	0.212254
$357$	48093.0	0.0199715
$358$	1.38922e6	0.572881
$359$	−2.60383e6	−1.06629	−0.533146	−	0.846023i	$-0.678990\pi$
$359$	−2.60383e6	−1.06629	−0.533146	+	0.846023i	$0.678990\pi$
$360$	−640534.	−0.260487
$361$	−2.40561e6	−0.971533
$362$	1.35784e6	0.544601
$363$	684847.	0.272789
$364$	535948.	0.212016
$365$	−2.59043e6	−1.01774
$366$	−121401.	−0.0473717
$367$	−3.14883e6	−1.22035	−0.610174	−	0.792267i	$-0.708901\pi$
$367$	−3.14883e6	−1.22035	−0.610174	+	0.792267i	$0.708901\pi$
$368$	−906117.	−0.348791
$369$	1.67748e6	0.641345
$370$	−721904.	−0.274142
$371$	−1.14671e6	−0.432531
$372$	−278743.	−0.104435
$373$	646752.	0.240694	0.120347	−	0.992732i	$-0.461599\pi$
$373$	646752.	0.240694	0.120347	+	0.992732i	$0.461599\pi$
$374$	273829.	0.101228
$375$	499255.	0.183334
$376$	1.84354e6	0.672487
$377$	971723.	0.352119
$378$	−872353.	−0.314024
$379$	−3.85752e6	−1.37946	−0.689731	−	0.724066i	$-0.742271\pi$
$379$	−3.85752e6	−1.37946	−0.689731	+	0.724066i	$0.742271\pi$
$380$	−180142.	−0.0639963
$381$	548613.	0.193621
$382$	1.24778e6	0.437500
$383$	4.00290e6	1.39437	0.697184	−	0.716892i	$-0.254436\pi$
$383$	4.00290e6	1.39437	0.697184	+	0.716892i	$0.254436\pi$
$384$	−43329.5	−0.0149953
$385$	4.73147e6	1.62684
$386$	−323385.	−0.110472
$387$	−4.30147e6	−1.45995
$388$	1.17157e6	0.395085
$389$	−380075.	−0.127349	−0.0636744	−	0.997971i	$-0.520282\pi$
$389$	−380075.	−0.127349	−0.0636744	+	0.997971i	$0.520282\pi$
$390$	−87284.7	−0.0290587
$391$	373882.	0.123678
$392$	−821212.	−0.269923
$393$	94679.0	0.0309223
$394$	3.84292e6	1.24716
$395$	1.90443e6	0.614148
$396$	−2.44722e6	−0.784214
$397$	6.02621e6	1.91897	0.959485	−	0.281761i	$-0.0909186\pi$
$397$	6.02621e6	1.91897	0.959485	+	0.281761i	$0.0909186\pi$
$398$	−1.44856e6	−0.458384
$399$	−120878.	−0.0380114
$400$	−339618.	−0.106131
$401$	−486904.	−0.151211	−0.0756053	−	0.997138i	$-0.524089\pi$
$401$	−486904.	−0.151211	−0.0756053	+	0.997138i	$0.524089\pi$
$402$	287211.	0.0886413
$403$	1.28173e6	0.393127
$404$	−2.09597e6	−0.638897
$405$	−2.28996e6	−0.693728
$406$	−3.43918e6	−1.03547
$407$	−2.75810e6	−0.825322
$408$	17878.6	0.00531721
$409$	−3.72734e6	−1.10177	−0.550884	−	0.834582i	$-0.685710\pi$
$409$	−3.72734e6	−1.10177	−0.550884	+	0.834582i	$0.685710\pi$
$410$	−1.20568e6	−0.354221
$411$	1.00248e6	0.292734
$412$	766486.	0.222465
$413$	−2.22677e6	−0.642391
$414$	−3.34139e6	−0.958135
$415$	−4.56087e6	−1.29995
$416$	199239.	0.0564471
$417$	−570994.	−0.160802
$418$	−688247.	−0.192665
$419$	992231.	0.276107	0.138054	−	0.990425i	$-0.455915\pi$
$419$	992231.	0.276107	0.138054	+	0.990425i	$0.455915\pi$
$420$	308923.	0.0854529
$421$	−4.13355e6	−1.13663	−0.568313	−	0.822812i	$-0.692404\pi$
$421$	−4.13355e6	−1.13663	−0.568313	+	0.822812i	$0.692404\pi$
$422$	−3.11707e6	−0.852051
$423$	6.79824e6	1.84733
$424$	−426289.	−0.115157
$425$	140133.	0.0376330
$426$	412261.	0.110065
$427$	−1.97572e6	−0.524392
$428$	−2.25064e6	−0.593876
$429$	−333479.	−0.0874833
$430$	3.09167e6	0.806346
$431$	6.79852e6	1.76287	0.881437	−	0.472301i	$-0.156576\pi$
$431$	6.79852e6	1.76287	0.881437	+	0.472301i	$0.156576\pi$
$432$	−324298.	−0.0836055
$433$	−1.58309e6	−0.405774	−0.202887	−	0.979202i	$-0.565032\pi$
$433$	−1.58309e6	−0.405774	−0.202887	+	0.979202i	$0.565032\pi$
$434$	−4.53636e6	−1.15607
$435$	560105.	0.141921
$436$	−3.05088e6	−0.768616
$437$	−939722.	−0.235394
$438$	−646183.	−0.160942
$439$	6.99769e6	1.73298	0.866489	−	0.499195i	$-0.166371\pi$
$439$	6.99769e6	1.73298	0.866489	+	0.499195i	$0.166371\pi$
$440$	1.75893e6	0.433129
$441$	−3.02829e6	−0.741484
$442$	−82210.2	−0.0200157
$443$	−1.76557e6	−0.427440	−0.213720	−	0.976895i	$-0.568558\pi$
$443$	−1.76557e6	−0.427440	−0.213720	+	0.976895i	$0.568558\pi$
$444$	−180079.	−0.0433517
$445$	−1.34524e6	−0.322032
$446$	1.40028e6	0.333333
$447$	619434.	0.146631
$448$	−705160.	−0.165994
$449$	94486.2	0.0221183	0.0110592	−	0.999939i	$-0.496480\pi$
$449$	94486.2	0.0221183	0.0110592	+	0.999939i	$0.496480\pi$
$450$	−1.25237e6	−0.291543
$451$	−4.60642e6	−1.06641
$452$	2.24860e6	0.517685
$453$	1.15806e6	0.265146
$454$	−4.77435e6	−1.08711
$455$	−1.42050e6	−0.321672
$456$	−44936.4	−0.0101201
$457$	1.19691e6	0.268084	0.134042	−	0.990976i	$-0.457204\pi$
$457$	1.19691e6	0.268084	0.134042	+	0.990976i	$0.457204\pi$
$458$	494664.	0.110191
$459$	133812.	0.0296458
$460$	2.40161e6	0.529187
$461$	3.95013e6	0.865683	0.432842	−	0.901470i	$-0.357511\pi$
$461$	3.95013e6	0.865683	0.432842	+	0.901470i	$0.357511\pi$
$462$	1.18027e6	0.257262
$463$	−8.41496e6	−1.82431	−0.912157	−	0.409840i	$-0.865584\pi$
$463$	−8.41496e6	−1.82431	−0.912157	+	0.409840i	$0.865584\pi$
$464$	−1.27852e6	−0.275684
$465$	738793.	0.158449
$466$	−4.22395e6	−0.901061
$467$	−8.52363e6	−1.80856	−0.904279	−	0.426941i	$-0.859591\pi$
$467$	−8.52363e6	−1.80856	−0.904279	+	0.426941i	$0.859591\pi$
$468$	734714.	0.155061
$469$	4.67418e6	0.981236
$470$	−4.88622e6	−1.02030
$471$	850486.	0.176650
$472$	−827803.	−0.171030
$473$	1.18120e7	2.42756
$474$	475062.	0.0971191
$475$	−352214.	−0.0716262
$476$	290963.	0.0588600
$477$	−1.57198e6	−0.316338
$478$	−3.24929e6	−0.650456
$479$	−796614.	−0.158639	−0.0793193	−	0.996849i	$-0.525275\pi$
$479$	−796614.	−0.158639	−0.0793193	+	0.996849i	$0.525275\pi$
$480$	114842.	0.0227509
$481$	828048.	0.163190
$482$	5.25286e6	1.02986
$483$	1.61152e6	0.314317
$484$	4.14333e6	0.803964
$485$	−3.10520e6	−0.599425
$486$	−1.80255e6	−0.346176
$487$	5.22787e6	0.998855	0.499427	−	0.866356i	$-0.333544\pi$
$487$	5.22787e6	0.998855	0.499427	+	0.866356i	$0.333544\pi$
$488$	−734477.	−0.139614
$489$	−443719.	−0.0839142
$490$	2.17658e6	0.409528
$491$	1.46114e6	0.273520	0.136760	−	0.990604i	$-0.456331\pi$
$491$	1.46114e6	0.273520	0.136760	+	0.990604i	$0.456331\pi$
$492$	−300758.	−0.0560151
$493$	527542.	0.0977552
$494$	206629.	0.0380954
$495$	6.48622e6	1.18981
$496$	−1.68640e6	−0.307791
$497$	6.70929e6	1.21839
$498$	−1.13771e6	−0.205569
$499$	3.87785e6	0.697172	0.348586	−	0.937277i	$-0.386662\pi$
$499$	3.87785e6	0.697172	0.348586	+	0.937277i	$0.386662\pi$
$500$	3.02050e6	0.540323
$501$	−579865.	−0.103213
$502$	1.37669e6	0.243824
$503$	−6.93030e6	−1.22133	−0.610664	−	0.791890i	$-0.709097\pi$
$503$	−6.93030e6	−1.22133	−0.610664	+	0.791890i	$0.709097\pi$
$504$	−2.60034e6	−0.455989
$505$	5.55525e6	0.969338
$506$	9.17559e6	1.59315
$507$	−881811.	−0.152355
$508$	3.31911e6	0.570641
$509$	5.73900e6	0.981843	0.490922	−	0.871204i	$-0.336660\pi$
$509$	5.73900e6	0.981843	0.490922	+	0.871204i	$0.336660\pi$
$510$	−47386.3	−0.00806729
$511$	−1.05162e7	−1.78159
$512$	−262144.	−0.0441942
$513$	−336325.	−0.0564243
$514$	−1.14227e6	−0.190705
$515$	−2.03153e6	−0.337524
$516$	771218.	0.127513
$517$	−1.86682e7	−3.07168
$518$	−2.93068e6	−0.479892
$519$	1.07609e6	0.175360
$520$	−528074.	−0.0856418
$521$	414941.	0.0669718	0.0334859	−	0.999439i	$-0.489339\pi$
$521$	414941.	0.0669718	0.0334859	+	0.999439i	$0.489339\pi$
$522$	−4.71466e6	−0.757310
$523$	−9.70473e6	−1.55142	−0.775709	−	0.631090i	$-0.782608\pi$
$523$	−9.70473e6	−1.55142	−0.775709	+	0.631090i	$0.782608\pi$
$524$	572809.	0.0911343
$525$	604007.	0.0956410
$526$	−3.49540e6	−0.550849
$527$	695842.	0.109140
$528$	438766.	0.0684933
$529$	6.09186e6	0.946478
$530$	1.12986e6	0.174716
$531$	−3.05260e6	−0.469823
$532$	−731312.	−0.112027
$533$	1.38296e6	0.210859
$534$	−335571.	−0.0509250
$535$	5.96519e6	0.901031
$536$	1.73763e6	0.261244
$537$	−918493.	−0.137449
$538$	−289444.	−0.0431131
$539$	8.31581e6	1.23291
$540$	859535.	0.126847
$541$	7.52073e6	1.10476	0.552379	−	0.833593i	$-0.313720\pi$
$541$	7.52073e6	1.10476	0.552379	+	0.833593i	$0.313720\pi$
$542$	3.49197e6	0.510590
$543$	−897745.	−0.130663
$544$	108166.	0.0156709
$545$	8.08620e6	1.16615
$546$	−354345.	−0.0508680
$547$	1.15446e6	0.164972	0.0824861	−	0.996592i	$-0.473714\pi$
$547$	1.15446e6	0.164972	0.0824861	+	0.996592i	$0.473714\pi$
$548$	6.06504e6	0.862745
$549$	−2.70845e6	−0.383522
$550$	3.43907e6	0.484768
$551$	−1.32593e6	−0.186056
$552$	599084.	0.0836836
$553$	7.73133e6	1.07508
$554$	−7.42669e6	−1.02807
$555$	477290.	0.0657734
$556$	−3.45452e6	−0.473915
$557$	3.97240e6	0.542518	0.271259	−	0.962506i	$-0.412560\pi$
$557$	3.97240e6	0.542518	0.271259	+	0.962506i	$0.412560\pi$
$558$	−6.21875e6	−0.845508
$559$	−3.54625e6	−0.479998
$560$	1.86899e6	0.251847
$561$	−181044.	−0.0242871
$562$	−2.50264e6	−0.334239
$563$	−5.83753e6	−0.776173	−0.388086	−	0.921623i	$-0.626864\pi$
$563$	−5.83753e6	−0.776173	−0.388086	+	0.921623i	$0.626864\pi$
$564$	−1.21887e6	−0.161346
$565$	−5.95979e6	−0.785434
$566$	869877.	0.114134
$567$	−9.29642e6	−1.21439
$568$	2.49419e6	0.324383
$569$	1.16912e7	1.51383	0.756915	−	0.653513i	$-0.226706\pi$
$569$	1.16912e7	1.51383	0.756915	+	0.653513i	$0.226706\pi$
$570$	119102.	0.0153543
$571$	8.42334e6	1.08117	0.540585	−	0.841289i	$-0.318203\pi$
$571$	8.42334e6	1.08117	0.540585	+	0.841289i	$0.318203\pi$
$572$	−2.01755e6	−0.257831
$573$	−824974.	−0.104967
$574$	−4.89465e6	−0.620072
$575$	4.69565e6	0.592278
$576$	−966680.	−0.121402
$577$	4.55890e6	0.570060	0.285030	−	0.958519i	$-0.407996\pi$
$577$	4.55890e6	0.570060	0.285030	+	0.958519i	$0.407996\pi$
$578$	5.63480e6	0.701550
$579$	213808.	0.0265050
$580$	3.38864e6	0.418269
$581$	−1.85155e7	−2.27560
$582$	−774593.	−0.0947908
$583$	4.31672e6	0.525996
$584$	−3.90942e6	−0.474329
$585$	−1.94732e6	−0.235260
$586$	−8.73619e6	−1.05094
$587$	−8.64175e6	−1.03516	−0.517579	−	0.855636i	$-0.673166\pi$
$587$	−8.64175e6	−1.03516	−0.517579	+	0.855636i	$0.673166\pi$
$588$	542948.	0.0647613
$589$	−1.74894e6	−0.207724
$590$	2.19405e6	0.259487
$591$	−2.54077e6	−0.299224
$592$	−1.08948e6	−0.127766
$593$	−2.18979e6	−0.255721	−0.127860	−	0.991792i	$-0.540811\pi$
$593$	−2.18979e6	−0.255721	−0.127860	+	0.991792i	$0.540811\pi$
$594$	3.28393e6	0.381881
$595$	−771182.	−0.0893027
$596$	3.74759e6	0.432151
$597$	957726.	0.109978
$598$	−2.75473e6	−0.315012
$599$	−1.37049e7	−1.56067	−0.780333	−	0.625365i	$-0.784950\pi$
$599$	−1.37049e7	−1.56067	−0.780333	+	0.625365i	$0.784950\pi$
$600$	224540.	0.0254634
$601$	−6.69658e6	−0.756253	−0.378126	−	0.925754i	$-0.623432\pi$
$601$	−6.69658e6	−0.756253	−0.378126	+	0.925754i	$0.623432\pi$
$602$	1.25511e7	1.41153
$603$	6.40768e6	0.717642
$604$	7.00628e6	0.781439
$605$	−1.09817e7	−1.21978
$606$	1.38576e6	0.153287
$607$	1.77118e7	1.95115	0.975574	−	0.219669i	$-0.0704979\pi$
$607$	1.77118e7	1.95115	0.975574	+	0.219669i	$0.0704979\pi$
$608$	−271866.	−0.0298261
$609$	2.27383e6	0.248436
$610$	1.94669e6	0.211823
$611$	5.60466e6	0.607360
$612$	398872.	0.0430482
$613$	−1.54369e7	−1.65924	−0.829619	−	0.558330i	$-0.811442\pi$
$613$	−1.54369e7	−1.65924	−0.829619	+	0.558330i	$0.811442\pi$
$614$	4.70209e6	0.503350
$615$	797144.	0.0849863
$616$	7.14064e6	0.758203
$617$	−8.28216e6	−0.875851	−0.437926	−	0.899011i	$-0.644287\pi$
$617$	−8.28216e6	−0.875851	−0.437926	+	0.899011i	$0.644287\pi$
$618$	−506766.	−0.0533748
$619$	9.17280e6	0.962223	0.481111	−	0.876660i	$-0.340233\pi$
$619$	9.17280e6	0.962223	0.481111	+	0.876660i	$0.340233\pi$
$620$	4.46971e6	0.466982
$621$	4.48383e6	0.466573
$622$	−749681.	−0.0776963
$623$	−5.46120e6	−0.563726
$624$	−131728.	−0.0135431
$625$	−3.85994e6	−0.395258
$626$	2.74184e6	0.279644
$627$	455038.	0.0462252
$628$	5.14545e6	0.520624
$629$	449542.	0.0453048
$630$	6.89207e6	0.691828
$631$	8.75086e6	0.874938	0.437469	−	0.899234i	$-0.355875\pi$
$631$	8.75086e6	0.874938	0.437469	+	0.899234i	$0.355875\pi$
$632$	2.87413e6	0.286229
$633$	2.06087e6	0.204428
$634$	2.62180e6	0.259045
$635$	−8.79714e6	−0.865779
$636$	281843.	0.0276290
$637$	−2.49661e6	−0.243782
$638$	1.29466e7	1.25923
$639$	9.19755e6	0.891087
$640$	694799.	0.0670516
$641$	1.35977e6	0.130714	0.0653568	−	0.997862i	$-0.479181\pi$
$641$	1.35977e6	0.130714	0.0653568	+	0.997862i	$0.479181\pi$
$642$	1.48802e6	0.142486
$643$	9.04005e6	0.862270	0.431135	−	0.902287i	$-0.358113\pi$
$643$	9.04005e6	0.862270	0.431135	+	0.902287i	$0.358113\pi$
$644$	9.74972e6	0.926355
$645$	−2.04407e6	−0.193463
$646$	112177.	0.0105761
$647$	4.61515e6	0.433436	0.216718	−	0.976234i	$-0.430465\pi$
$647$	4.61515e6	0.433436	0.216718	+	0.976234i	$0.430465\pi$
$648$	−3.45595e6	−0.323318
$649$	8.38256e6	0.781205
$650$	−1.03249e6	−0.0958524
$651$	2.99924e6	0.277370
$652$	−2.68450e6	−0.247312
$653$	−7.50726e6	−0.688967	−0.344484	−	0.938792i	$-0.611946\pi$
$653$	−7.50726e6	−0.688967	−0.344484	+	0.938792i	$0.611946\pi$
$654$	2.01711e6	0.184410
$655$	−1.51820e6	−0.138269
$656$	−1.81959e6	−0.165088
$657$	−1.44163e7	−1.30299
$658$	−1.98363e7	−1.78606
$659$	2.48414e6	0.222824	0.111412	−	0.993774i	$-0.464463\pi$
$659$	2.48414e6	0.222824	0.111412	+	0.993774i	$0.464463\pi$
$660$	−1.16293e6	−0.103918
$661$	1.72264e7	1.53352	0.766760	−	0.641933i	$-0.221867\pi$
$661$	1.72264e7	1.53352	0.766760	+	0.641933i	$0.221867\pi$
$662$	1.38187e7	1.22553
$663$	54353.7	0.00480226
$664$	−6.88317e6	−0.605854
$665$	1.93830e6	0.169968
$666$	−4.01757e6	−0.350976
$667$	1.76771e7	1.53850
$668$	−3.50819e6	−0.304188
$669$	−925804.	−0.0799749
$670$	−4.60550e6	−0.396360
$671$	7.43751e6	0.637707
$672$	466220.	0.0398261
$673$	1.10827e7	0.943210	0.471605	−	0.881810i	$-0.343675\pi$
$673$	1.10827e7	0.943210	0.471605	+	0.881810i	$0.343675\pi$
$674$	2.56534e6	0.217518
$675$	1.68057e6	0.141970
$676$	−5.33497e6	−0.449020
$677$	1.76385e7	1.47907	0.739536	−	0.673117i	$-0.235045\pi$
$677$	1.76385e7	1.47907	0.739536	+	0.673117i	$0.235045\pi$
$678$	−1.48667e6	−0.124206
$679$	−1.26060e7	−1.04931
$680$	−286688.	−0.0237759
$681$	3.15658e6	0.260825
$682$	1.70769e7	1.40588
$683$	1.33312e7	1.09350	0.546748	−	0.837297i	$-0.315866\pi$
$683$	1.33312e7	1.09350	0.546748	+	0.837297i	$0.315866\pi$
$684$	−1.00253e6	−0.0819328
$685$	−1.60751e7	−1.30896
$686$	−2.73771e6	−0.222114
$687$	−327050.	−0.0264376
$688$	4.66588e6	0.375805
$689$	−1.29598e6	−0.104004
$690$	−1.58784e6	−0.126965
$691$	−4.45922e6	−0.355274	−0.177637	−	0.984096i	$-0.556845\pi$
$691$	−4.45922e6	−0.355274	−0.177637	+	0.984096i	$0.556845\pi$
$692$	6.51038e6	0.516822
$693$	2.63318e7	2.08280
$694$	−1.66428e7	−1.31168
$695$	9.15602e6	0.719026
$696$	845299.	0.0661435
$697$	750801.	0.0585387
$698$	1.44418e6	0.112197
$699$	2.79269e6	0.216187
$700$	3.65425e6	0.281873
$701$	−1.51937e7	−1.16780	−0.583899	−	0.811826i	$-0.698474\pi$
$701$	−1.51937e7	−1.16780	−0.583899	+	0.811826i	$0.698474\pi$
$702$	−985916.	−0.0755087
$703$	−1.12989e6	−0.0862277
$704$	2.65454e6	0.201863
$705$	3.23055e6	0.244795
$706$	5.67934e6	0.428831
$707$	2.25524e7	1.69685
$708$	547307.	0.0410344
$709$	−1.07823e7	−0.805558	−0.402779	−	0.915297i	$-0.631956\pi$
$709$	−1.07823e7	−0.805558	−0.402779	+	0.915297i	$0.631956\pi$
$710$	−6.61071e6	−0.492155
$711$	1.05986e7	0.786277
$712$	−2.03021e6	−0.150086
$713$	2.33166e7	1.71767
$714$	−192372.	−0.0141220
$715$	5.34741e6	0.391182
$716$	−5.55690e6	−0.405088
$717$	2.14828e6	0.156061
$718$	1.04153e7	0.753982
$719$	6.55088e6	0.472582	0.236291	−	0.971682i	$-0.424068\pi$
$719$	6.55088e6	0.472582	0.236291	+	0.971682i	$0.424068\pi$
$720$	2.56214e6	0.184192
$721$	−8.24730e6	−0.590845
$722$	9.62245e6	0.686978
$723$	−3.47296e6	−0.247089
$724$	−5.43137e6	−0.385091
$725$	6.62549e6	0.468137
$726$	−2.73939e6	−0.192891
$727$	−1.70595e7	−1.19710	−0.598550	−	0.801086i	$-0.704256\pi$
$727$	−1.70595e7	−1.19710	−0.598550	+	0.801086i	$0.704256\pi$
$728$	−2.14379e6	−0.149918
$729$	−1.19301e7	−0.831426
$730$	1.03617e7	0.719654
$731$	−1.92524e6	−0.133257
$732$	485603.	0.0334968
$733$	−1.46612e7	−1.00788	−0.503939	−	0.863739i	$-0.668117\pi$
$733$	−1.46612e7	−1.00788	−0.503939	+	0.863739i	$0.668117\pi$
$734$	1.25953e7	0.862917
$735$	−1.43906e6	−0.0982561
$736$	3.62447e6	0.246632
$737$	−1.75957e7	−1.19327
$738$	−6.70992e6	−0.453499
$739$	−1.91597e7	−1.29056	−0.645279	−	0.763947i	$-0.723259\pi$
$739$	−1.91597e7	−1.29056	−0.645279	+	0.763947i	$0.723259\pi$
$740$	2.88761e6	0.193847
$741$	−136614.	−0.00914005
$742$	4.58682e6	0.305846
$743$	−2.19055e7	−1.45573	−0.727865	−	0.685721i	$-0.759487\pi$
$743$	−2.19055e7	−1.45573	−0.727865	+	0.685721i	$0.759487\pi$
$744$	1.11497e6	0.0738468
$745$	−9.93278e6	−0.655662
$746$	−2.58701e6	−0.170196
$747$	−2.53823e7	−1.66429
$748$	−1.09532e6	−0.0715790
$749$	2.42166e7	1.57728
$750$	−1.99702e6	−0.129637
$751$	1.18458e7	0.766413	0.383207	−	0.923663i	$-0.374820\pi$
$751$	1.18458e7	0.766413	0.383207	+	0.923663i	$0.374820\pi$
$752$	−7.37418e6	−0.475520
$753$	−910206.	−0.0584995
$754$	−3.88689e6	−0.248985
$755$	−1.85698e7	−1.18560
$756$	3.48941e6	0.222048
$757$	−1.46228e7	−0.927451	−0.463726	−	0.885979i	$-0.653488\pi$
$757$	−1.46228e7	−0.927451	−0.463726	+	0.885979i	$0.653488\pi$
$758$	1.54301e7	0.975427
$759$	−6.06649e6	−0.382237
$760$	720566.	0.0452522
$761$	−448992.	−0.0281045	−0.0140523	−	0.999901i	$-0.504473\pi$
$761$	−448992.	−0.0281045	−0.0140523	+	0.999901i	$0.504473\pi$
$762$	−2.19445e6	−0.136911
$763$	3.28271e7	2.04137
$764$	−4.99111e6	−0.309359
$765$	−1.05719e6	−0.0653129
$766$	−1.60116e7	−0.985967
$767$	−2.51665e6	−0.154466
$768$	173318.	0.0106033
$769$	−2.95734e7	−1.80337	−0.901686	−	0.432391i	$-0.857670\pi$
$769$	−2.95734e7	−1.80337	−0.901686	+	0.432391i	$0.857670\pi$
$770$	−1.89259e7	−1.15035
$771$	755221.	0.0457550
$772$	1.29354e6	0.0781155
$773$	−1.14951e7	−0.691933	−0.345967	−	0.938247i	$-0.612449\pi$
$773$	−1.14951e7	−0.691933	−0.345967	+	0.938247i	$0.612449\pi$
$774$	1.72059e7	1.03234
$775$	8.73920e6	0.522658
$776$	−4.68630e6	−0.279367
$777$	1.93763e6	0.115138
$778$	1.52030e6	0.0900492
$779$	−1.88708e6	−0.111416
$780$	349139.	0.0205476
$781$	−2.52568e7	−1.48167
$782$	−1.49553e6	−0.0874536
$783$	6.32662e6	0.368780
$784$	3.28485e6	0.190864
$785$	−1.36377e7	−0.789893
$786$	−378716.	−0.0218654
$787$	9.20477e6	0.529756	0.264878	−	0.964282i	$-0.414668\pi$
$787$	9.20477e6	0.529756	0.264878	+	0.964282i	$0.414668\pi$
$788$	−1.53717e7	−0.881873
$789$	2.31100e6	0.132162
$790$	−7.61773e6	−0.434268
$791$	−2.41947e7	−1.37492
$792$	9.78887e6	0.554523
$793$	−2.23292e6	−0.126093
$794$	−2.41048e7	−1.35692
$795$	−747011.	−0.0419188
$796$	5.79425e6	0.324127
$797$	−1.73402e7	−0.966958	−0.483479	−	0.875356i	$-0.660627\pi$
$797$	−1.73402e7	−0.966958	−0.483479	+	0.875356i	$0.660627\pi$
$798$	483511.	0.0268781
$799$	3.04273e6	0.168615
$800$	1.35847e6	0.0750457
$801$	−7.48658e6	−0.412289
$802$	1.94762e6	0.106922
$803$	3.95878e7	2.16657
$804$	−1.14884e6	−0.0626789
$805$	−2.58411e7	−1.40547
$806$	−5.12691e6	−0.277983
$807$	191367.	0.0103439
$808$	8.38387e6	0.451769
$809$	2.72210e6	0.146229	0.0731145	−	0.997324i	$-0.476706\pi$
$809$	2.72210e6	0.146229	0.0731145	+	0.997324i	$0.476706\pi$
$810$	9.15982e6	0.490540
$811$	−3.04561e7	−1.62600	−0.813002	−	0.582261i	$-0.802168\pi$
$811$	−3.04561e7	−1.62600	−0.813002	+	0.582261i	$0.802168\pi$
$812$	1.37567e7	0.732191
$813$	−2.30874e6	−0.122503
$814$	1.10324e7	0.583591
$815$	7.11513e6	0.375223
$816$	−71514.5	−0.00375983
$817$	4.83892e6	0.253626
$818$	1.49094e7	0.779068
$819$	−7.90544e6	−0.411828
$820$	4.82273e6	0.250472
$821$	3.66630e6	0.189832	0.0949160	−	0.995485i	$-0.469742\pi$
$821$	3.66630e6	0.189832	0.0949160	+	0.995485i	$0.469742\pi$
$822$	−4.00993e6	−0.206994
$823$	2.04062e7	1.05018	0.525089	−	0.851047i	$-0.324032\pi$
$823$	2.04062e7	1.05018	0.525089	+	0.851047i	$0.324032\pi$
$824$	−3.06594e6	−0.157306
$825$	−2.27376e6	−0.116308
$826$	8.90707e6	0.454239
$827$	−5.43452e6	−0.276311	−0.138155	−	0.990411i	$-0.544117\pi$
$827$	−5.43452e6	−0.276311	−0.138155	+	0.990411i	$0.544117\pi$
$828$	1.33656e7	0.677503
$829$	3.02973e6	0.153115	0.0765574	−	0.997065i	$-0.475607\pi$
$829$	3.02973e6	0.153115	0.0765574	+	0.997065i	$0.475607\pi$
$830$	1.82435e7	0.919205
$831$	4.91019e6	0.246658
$832$	−796958.	−0.0399142
$833$	−1.35539e6	−0.0676789
$834$	2.28397e6	0.113704
$835$	9.29828e6	0.461515
$836$	2.75299e6	0.136235
$837$	8.34497e6	0.411729
$838$	−3.96892e6	−0.195237
$839$	3.30101e7	1.61898	0.809491	−	0.587133i	$-0.199743\pi$
$839$	3.30101e7	1.61898	0.809491	+	0.587133i	$0.199743\pi$
$840$	−1.23569e6	−0.0604243
$841$	4.43100e6	0.216029
$842$	1.65342e7	0.803716
$843$	1.65463e6	0.0801924
$844$	1.24683e7	0.602491
$845$	1.41401e7	0.681254
$846$	−2.71930e7	−1.30626
$847$	−4.45818e7	−2.13525
$848$	1.70516e6	0.0814282
$849$	−575124.	−0.0273837
$850$	−560533.	−0.0266106
$851$	1.50635e7	0.713018
$852$	−1.64905e6	−0.0778276
$853$	−2.47097e7	−1.16277	−0.581386	−	0.813628i	$-0.697489\pi$
$853$	−2.47097e7	−1.16277	−0.581386	+	0.813628i	$0.697489\pi$
$854$	7.90288e6	0.370801
$855$	2.65716e6	0.124309
$856$	9.00254e6	0.419934
$857$	−3.36607e7	−1.56557	−0.782784	−	0.622294i	$-0.786201\pi$
$857$	−3.36607e7	−1.56557	−0.782784	+	0.622294i	$0.786201\pi$
$858$	1.33392e6	0.0618600
$859$	4.07908e6	0.188616	0.0943081	−	0.995543i	$-0.469936\pi$
$859$	4.07908e6	0.188616	0.0943081	+	0.995543i	$0.469936\pi$
$860$	−1.23667e7	−0.570173
$861$	3.23613e6	0.148771
$862$	−2.71941e7	−1.24654
$863$	4.14774e7	1.89576	0.947881	−	0.318623i	$-0.103220\pi$
$863$	4.14774e7	1.89576	0.947881	+	0.318623i	$0.103220\pi$
$864$	1.29719e6	0.0591180
$865$	−1.72554e7	−0.784124
$866$	6.33234e6	0.286926
$867$	−3.72548e6	−0.168319
$868$	1.81455e7	0.817464
$869$	−2.91042e7	−1.30739
$870$	−2.24042e6	−0.100353
$871$	5.28267e6	0.235943
$872$	1.22035e7	0.543493
$873$	−1.72812e7	−0.767428
$874$	3.75889e6	0.166449
$875$	−3.25002e7	−1.43505
$876$	2.58473e6	0.113803
$877$	−1.27324e7	−0.558999	−0.279499	−	0.960146i	$-0.590168\pi$
$877$	−1.27324e7	−0.558999	−0.279499	+	0.960146i	$0.590168\pi$
$878$	−2.79908e7	−1.22540
$879$	5.77598e6	0.252147
$880$	−7.03572e6	−0.306268
$881$	2.09875e6	0.0911005	0.0455503	−	0.998962i	$-0.485496\pi$
$881$	2.09875e6	0.0911005	0.0455503	+	0.998962i	$0.485496\pi$
$882$	1.21132e7	0.524308
$883$	2.75482e7	1.18903	0.594514	−	0.804085i	$-0.297345\pi$
$883$	2.75482e7	1.18903	0.594514	+	0.804085i	$0.297345\pi$
$884$	328841.	0.0141532
$885$	−1.45061e6	−0.0622575
$886$	7.06227e6	0.302246
$887$	2.32724e7	0.993187	0.496594	−	0.867983i	$-0.334584\pi$
$887$	2.32724e7	0.993187	0.496594	+	0.867983i	$0.334584\pi$
$888$	720317.	0.0306543
$889$	−3.57133e7	−1.51557
$890$	5.38096e6	0.227711
$891$	3.49959e7	1.47680
$892$	−5.60113e6	−0.235702
$893$	−7.64766e6	−0.320922
$894$	−2.47774e6	−0.103684
$895$	1.47283e7	0.614602
$896$	2.82064e6	0.117376
$897$	1.82131e6	0.0755792
$898$	−377945.	−0.0156400
$899$	3.28994e7	1.35765
$900$	5.00950e6	0.206152
$901$	−703582.	−0.0288737
$902$	1.84257e7	0.754063
$903$	−8.29822e6	−0.338661
$904$	−8.99439e6	−0.366059
$905$	1.43956e7	0.584261
$906$	−4.63224e6	−0.187487
$907$	−7.37611e6	−0.297721	−0.148860	−	0.988858i	$-0.547561\pi$
$907$	−7.37611e6	−0.297721	−0.148860	+	0.988858i	$0.547561\pi$
$908$	1.90974e7	0.768704
$909$	3.09163e7	1.24102
$910$	5.68201e6	0.227457
$911$	6.41549e6	0.256114	0.128057	−	0.991767i	$-0.459126\pi$
$911$	6.41549e6	0.256114	0.128057	+	0.991767i	$0.459126\pi$
$912$	179746.	0.00715602
$913$	6.97008e7	2.76733
$914$	−4.78763e6	−0.189564
$915$	−1.28707e6	−0.0508216
$916$	−1.97865e6	−0.0779168
$917$	−6.16336e6	−0.242044
$918$	−535248.	−0.0209627
$919$	−2.03466e7	−0.794701	−0.397350	−	0.917667i	$-0.630070\pi$
$919$	−2.03466e7	−0.794701	−0.397350	+	0.917667i	$0.630070\pi$
$920$	−9.60646e6	−0.374191
$921$	−3.10881e6	−0.120766
$922$	−1.58005e7	−0.612131
$923$	7.58271e6	0.292968
$924$	−4.72107e6	−0.181912
$925$	5.64587e6	0.216959
$926$	3.36599e7	1.28999
$927$	−1.13060e7	−0.432123
$928$	5.11407e6	0.194938
$929$	1.39985e7	0.532159	0.266079	−	0.963951i	$-0.414272\pi$
$929$	1.39985e7	0.532159	0.266079	+	0.963951i	$0.414272\pi$
$930$	−2.95517e6	−0.112041
$931$	3.40667e6	0.128812
$932$	1.68958e7	0.637146
$933$	495655.	0.0186413
$934$	3.40945e7	1.27884
$935$	2.90308e6	0.108600
$936$	−2.93886e6	−0.109645
$937$	−1.96842e7	−0.732435	−0.366217	−	0.930529i	$-0.619347\pi$
$937$	−1.96842e7	−0.732435	−0.366217	+	0.930529i	$0.619347\pi$
$938$	−1.86967e7	−0.693838
$939$	−1.81278e6	−0.0670936
$940$	1.95449e7	0.721461
$941$	−4.11760e7	−1.51590	−0.757949	−	0.652314i	$-0.773798\pi$
$941$	−4.11760e7	−1.51590	−0.757949	+	0.652314i	$0.773798\pi$
$942$	−3.40194e6	−0.124911
$943$	2.51581e7	0.921296
$944$	3.31121e6	0.120936
$945$	−9.24850e6	−0.336893
$946$	−4.72480e7	−1.71655
$947$	−4.16384e7	−1.50876	−0.754378	−	0.656440i	$-0.772061\pi$
$947$	−4.16384e7	−1.50876	−0.754378	+	0.656440i	$0.772061\pi$
$948$	−1.90025e6	−0.0686735
$949$	−1.18852e7	−0.428392
$950$	1.40885e6	0.0506474
$951$	−1.73341e6	−0.0621514
$952$	−1.16385e6	−0.0416203
$953$	2.35585e7	0.840263	0.420131	−	0.907463i	$-0.361984\pi$
$953$	2.35585e7	0.840263	0.420131	+	0.907463i	$0.361984\pi$
$954$	6.28793e6	0.223685
$955$	1.32287e7	0.469361
$956$	1.29971e7	0.459942
$957$	−8.55973e6	−0.302120
$958$	3.18646e6	0.112174
$959$	−6.52591e7	−2.29137
$960$	−459370.	−0.0160873
$961$	1.47660e7	0.515766
$962$	−3.31219e6	−0.115393
$963$	3.31977e7	1.15357
$964$	−2.10115e7	−0.728221
$965$	−3.42846e6	−0.118517
$966$	−6.44608e6	−0.222256
$967$	4.89280e7	1.68264	0.841319	−	0.540538i	$-0.181779\pi$
$967$	4.89280e7	1.68264	0.841319	+	0.540538i	$0.181779\pi$
$968$	−1.65733e7	−0.568488
$969$	−74166.7	−0.00253746
$970$	1.24208e7	0.423857
$971$	2.26979e7	0.772568	0.386284	−	0.922380i	$-0.373758\pi$
$971$	2.26979e7	0.772568	0.386284	+	0.922380i	$0.373758\pi$
$972$	7.21020e6	0.244783
$973$	3.71702e7	1.25867
$974$	−2.09115e7	−0.706297
$975$	682637.	0.0229974
$976$	2.93791e6	0.0987219
$977$	−2.80773e7	−0.941065	−0.470533	−	0.882383i	$-0.655938\pi$
$977$	−2.80773e7	−0.941065	−0.470533	+	0.882383i	$0.655938\pi$
$978$	1.77487e6	0.0593363
$979$	2.05584e7	0.685541
$980$	−8.70631e6	−0.289580
$981$	4.50017e7	1.49299
$982$	−5.84457e6	−0.193408
$983$	−1.53300e7	−0.506010	−0.253005	−	0.967465i	$-0.581419\pi$
$983$	−1.53300e7	−0.506010	−0.253005	+	0.967465i	$0.581419\pi$
$984$	1.20303e6	0.0396087
$985$	4.07418e7	1.33798
$986$	−2.11017e6	−0.0691234
$987$	1.31149e7	0.428521
$988$	−826514.	−0.0269375
$989$	−6.45116e7	−2.09724
$990$	−2.59449e7	−0.841324
$991$	4.10889e7	1.32905	0.664523	−	0.747267i	$-0.268635\pi$
$991$	4.10889e7	1.32905	0.664523	+	0.747267i	$0.268635\pi$
$992$	6.74560e6	0.217641
$993$	−9.13634e6	−0.294035
$994$	−2.68372e7	−0.861531
$995$	−1.53574e7	−0.491766
$996$	4.55084e6	0.145360
$997$	−5.63505e7	−1.79539	−0.897697	−	0.440613i	$-0.854761\pi$
$997$	−5.63505e7	−1.79539	−0.897697	+	0.440613i	$0.854761\pi$
$998$	−1.55114e7	−0.492975
$999$	5.39119e6	0.170911

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.6.a.b.1.14	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.6.a.b.1.14	✓	27	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +2.64462 q^{3} +16.0000 q^{4} -42.4071 q^{5} -10.5785 q^{6} -172.158 q^{7} -64.0000 q^{8} -236.006 q^{9} +169.629 q^{10} +648.081 q^{11} +42.3139 q^{12} -194.570 q^{13} +688.633 q^{14} -112.151 q^{15} +256.000 q^{16} -105.631 q^{17} +944.024 q^{18} +265.494 q^{19} -678.514 q^{20} -455.293 q^{21} -2592.32 q^{22} -3539.52 q^{23} -169.256 q^{24} -1326.63 q^{25} +778.279 q^{26} -1266.79 q^{27} -2754.53 q^{28} -4994.21 q^{29} +448.603 q^{30} -6587.50 q^{31} -1024.00 q^{32} +1713.93 q^{33} +422.523 q^{34} +7300.74 q^{35} -3776.10 q^{36} -4255.79 q^{37} -1061.98 q^{38} -514.563 q^{39} +2714.06 q^{40} -7107.79 q^{41} +1821.17 q^{42} +18226.1 q^{43} +10369.3 q^{44} +10008.3 q^{45} +14158.1 q^{46} -28805.4 q^{47} +677.023 q^{48} +12831.4 q^{49} +5306.54 q^{50} -279.353 q^{51} -3113.12 q^{52} +6660.77 q^{53} +5067.16 q^{54} -27483.3 q^{55} +11018.1 q^{56} +702.132 q^{57} +19976.8 q^{58} +12934.4 q^{59} -1794.41 q^{60} +11476.2 q^{61} +26350.0 q^{62} +40630.4 q^{63} +4096.00 q^{64} +8251.15 q^{65} -6855.72 q^{66} -27150.5 q^{67} -1690.09 q^{68} -9360.69 q^{69} -29202.9 q^{70} -38971.7 q^{71} +15104.4 q^{72} +61084.6 q^{73} +17023.2 q^{74} -3508.44 q^{75} +4247.91 q^{76} -111572. q^{77} +2058.25 q^{78} -44908.3 q^{79} -10856.2 q^{80} +53999.3 q^{81} +28431.1 q^{82} +107550. q^{83} -7284.69 q^{84} +4479.50 q^{85} -72904.4 q^{86} -13207.8 q^{87} -41477.2 q^{88} +31722.0 q^{89} -40033.4 q^{90} +33496.8 q^{91} -56632.3 q^{92} -17421.4 q^{93} +115222. q^{94} -11258.9 q^{95} -2708.09 q^{96} +73223.4 q^{97} -51325.7 q^{98} -152951. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 108 q^{2} + 33 q^{3} + 432 q^{4} + 139 q^{5} - 132 q^{6} - 25 q^{7} - 1728 q^{8} + 2346 q^{9} - 556 q^{10} + 1241 q^{11} + 528 q^{12} - 202 q^{13} + 100 q^{14} + 1786 q^{15} + 6912 q^{16} + 1550 q^{17}+ \cdots + 335378 q^{99}+O(q^{100}) \) 27 * q - 108 * q^2 + 33 * q^3 + 432 * q^4 + 139 * q^5 - 132 * q^6 - 25 * q^7 - 1728 * q^8 + 2346 * q^9 - 556 * q^10 + 1241 * q^11 + 528 * q^12 - 202 * q^13 + 100 * q^14 + 1786 * q^15 + 6912 * q^16 + 1550 * q^17 - 9384 * q^18 - 66 * q^19 + 2224 * q^20 - 1217 * q^21 - 4964 * q^22 + 10568 * q^23 - 2112 * q^24 + 15880 * q^25 + 808 * q^26 + 8256 * q^27 - 400 * q^28 - 2122 * q^29 - 7144 * q^30 - 1621 * q^31 - 27648 * q^32 + 18369 * q^33 - 6200 * q^34 + 23123 * q^35 + 37536 * q^36 + 8542 * q^37 + 264 * q^38 + 589 * q^39 - 8896 * q^40 + 10859 * q^41 + 4868 * q^42 + 23759 * q^43 + 19856 * q^44 + 28006 * q^45 - 42272 * q^46 + 38457 * q^47 + 8448 * q^48 + 57376 * q^49 - 63520 * q^50 + 58237 * q^51 - 3232 * q^52 + 73586 * q^53 - 33024 * q^54 + 25923 * q^55 + 1600 * q^56 + 52534 * q^57 + 8488 * q^58 + 68927 * q^59 + 28576 * q^60 - 76016 * q^61 + 6484 * q^62 - 20738 * q^63 + 110592 * q^64 + 74867 * q^65 - 73476 * q^66 + 94787 * q^67 + 24800 * q^68 - 90601 * q^69 - 92492 * q^70 + 15523 * q^71 - 150144 * q^72 - 96279 * q^73 - 34168 * q^74 - 123786 * q^75 - 1056 * q^76 - 199866 * q^77 - 2356 * q^78 - 246217 * q^79 + 35584 * q^80 - 54473 * q^81 - 43436 * q^82 - 56658 * q^83 - 19472 * q^84 - 120541 * q^85 - 95036 * q^86 - 352954 * q^87 - 79424 * q^88 - 83070 * q^89 - 112024 * q^90 - 132348 * q^91 + 169088 * q^92 + 40727 * q^93 - 153828 * q^94 - 72090 * q^95 - 33792 * q^96 - 36273 * q^97 - 229504 * q^98 + 335378 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more