LMFDB - Embedded newform 538.6.a.c.1.7

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:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	\(q-4.00000 q^{2} -18.8532 q^{3} +16.0000 q^{4} -69.9105 q^{5} +75.4127 q^{6} +251.245 q^{7} -64.0000 q^{8} +112.442 q^{9} +279.642 q^{10} -554.401 q^{11} -301.651 q^{12} -295.362 q^{13} -1004.98 q^{14} +1318.03 q^{15} +256.000 q^{16} +1041.08 q^{17} -449.767 q^{18} +698.194 q^{19} -1118.57 q^{20} -4736.77 q^{21} +2217.60 q^{22} -3561.09 q^{23} +1206.60 q^{24} +1762.48 q^{25} +1181.45 q^{26} +2461.43 q^{27} +4019.92 q^{28} -5777.94 q^{29} -5272.14 q^{30} +851.123 q^{31} -1024.00 q^{32} +10452.2 q^{33} -4164.32 q^{34} -17564.7 q^{35} +1799.07 q^{36} +11407.6 q^{37} -2792.78 q^{38} +5568.51 q^{39} +4474.27 q^{40} -2368.35 q^{41} +18947.1 q^{42} -4542.00 q^{43} -8870.42 q^{44} -7860.87 q^{45} +14244.3 q^{46} +15678.2 q^{47} -4826.41 q^{48} +46317.2 q^{49} -7049.92 q^{50} -19627.7 q^{51} -4725.79 q^{52} +29228.9 q^{53} -9845.74 q^{54} +38758.5 q^{55} -16079.7 q^{56} -13163.2 q^{57} +23111.8 q^{58} +7562.69 q^{59} +21088.6 q^{60} -40717.5 q^{61} -3404.49 q^{62} +28250.5 q^{63} +4096.00 q^{64} +20648.9 q^{65} -41808.9 q^{66} +65923.1 q^{67} +16657.3 q^{68} +67137.7 q^{69} +70258.7 q^{70} +81488.4 q^{71} -7196.28 q^{72} +13007.3 q^{73} -45630.3 q^{74} -33228.3 q^{75} +11171.1 q^{76} -139291. q^{77} -22274.0 q^{78} +49650.7 q^{79} -17897.1 q^{80} -73729.2 q^{81} +9473.38 q^{82} +27957.3 q^{83} -75788.3 q^{84} -72782.5 q^{85} +18168.0 q^{86} +108932. q^{87} +35481.7 q^{88} -112434. q^{89} +31443.5 q^{90} -74208.3 q^{91} -56977.4 q^{92} -16046.4 q^{93} -62712.6 q^{94} -48811.1 q^{95} +19305.6 q^{96} -4913.93 q^{97} -185269. q^{98} -62337.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 120 q^{2} - 30 q^{3} + 480 q^{4} - 136 q^{5} + 120 q^{6} - 123 q^{7} - 1920 q^{8} + 2670 q^{9} + 544 q^{10} - 1058 q^{11} - 480 q^{12} - 371 q^{13} + 492 q^{14} - 1364 q^{15} + 7680 q^{16} - 1918 q^{17}+ \cdots - 78063 q^{99}+O(q^{100}) $ 30 * q - 120 * q^2 - 30 * q^3 + 480 * q^4 - 136 * q^5 + 120 * q^6 - 123 * q^7 - 1920 * q^8 + 2670 * q^9 + 544 * q^10 - 1058 * q^11 - 480 * q^12 - 371 * q^13 + 492 * q^14 - 1364 * q^15 + 7680 * q^16 - 1918 * q^17 - 10680 * q^18 - 3315 * q^19 - 2176 * q^20 - 1217 * q^21 + 4232 * q^22 - 9534 * q^23 + 1920 * q^24 + 22130 * q^25 + 1484 * q^26 - 12156 * q^27 - 1968 * q^28 - 8009 * q^29 + 5456 * q^30 - 3543 * q^31 - 30720 * q^32 - 3411 * q^33 + 7672 * q^34 - 11177 * q^35 + 42720 * q^36 - 6517 * q^37 + 13260 * q^38 - 2453 * q^39 + 8704 * q^40 - 5951 * q^41 + 4868 * q^42 - 18768 * q^43 - 16928 * q^44 - 22619 * q^45 + 38136 * q^46 - 41067 * q^47 - 7680 * q^48 + 107797 * q^49 - 88520 * q^50 - 4187 * q^51 - 5936 * q^52 - 64055 * q^53 + 48624 * q^54 + 31973 * q^55 + 7872 * q^56 - 25442 * q^57 + 32036 * q^58 - 66832 * q^59 - 21824 * q^60 + 46777 * q^61 + 14172 * q^62 - 115994 * q^63 + 122880 * q^64 - 68783 * q^65 + 13644 * q^66 - 107218 * q^67 - 30688 * q^68 + 228589 * q^69 + 44708 * q^70 + 4733 * q^71 - 170880 * q^72 + 150385 * q^73 + 26068 * q^74 + 133325 * q^75 - 53040 * q^76 + 292020 * q^77 + 9812 * q^78 + 31827 * q^79 - 34816 * q^80 + 425214 * q^81 + 23804 * q^82 + 107301 * q^83 - 19472 * q^84 + 338625 * q^85 + 75072 * q^86 + 288484 * q^87 + 67712 * q^88 - 99690 * q^89 + 90476 * q^90 - 109140 * q^91 - 152544 * q^92 - 114545 * q^93 + 164268 * q^94 - 155668 * q^95 + 30720 * q^96 - 79959 * q^97 - 431188 * q^98 - 78063 * 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
$3$	−18.8532	−1.20943	−0.604716	−	0.796442i	$-0.706713\pi$
$3$	−18.8532	−1.20943	−0.604716	+	0.796442i	$0.706713\pi$
$4$	16.0000	0.500000
$5$	−69.9105	−1.25060	−0.625299	−	0.780386i	$-0.715023\pi$
$5$	−69.9105	−1.25060	−0.625299	+	0.780386i	$0.715023\pi$
$6$	75.4127	0.855197
$7$	251.245	1.93800	0.968998	−	0.247070i	$-0.0794676\pi$
$7$	251.245	1.93800	0.968998	+	0.247070i	$0.0794676\pi$
$8$	−64.0000	−0.353553
$9$	112.442	0.462724
$10$	279.642	0.884306
$11$	−554.401	−1.38147	−0.690736	−	0.723107i	$-0.742713\pi$
$11$	−554.401	−1.38147	−0.690736	+	0.723107i	$0.742713\pi$
$12$	−301.651	−0.604716
$13$	−295.362	−0.484726	−0.242363	−	0.970186i	$-0.577922\pi$
$13$	−295.362	−0.484726	−0.242363	+	0.970186i	$0.577922\pi$
$14$	−1004.98	−1.37037
$15$	1318.03	1.51251
$16$	256.000	0.250000
$17$	1041.08	0.873699	0.436850	−	0.899535i	$-0.356094\pi$
$17$	1041.08	0.873699	0.436850	+	0.899535i	$0.356094\pi$
$18$	−449.767	−0.327195
$19$	698.194	0.443703	0.221851	−	0.975080i	$-0.428790\pi$
$19$	698.194	0.443703	0.221851	+	0.975080i	$0.428790\pi$
$20$	−1118.57	−0.625299
$21$	−4736.77	−2.34387
$22$	2217.60	0.976849
$23$	−3561.09	−1.40366	−0.701832	−	0.712343i	$-0.747634\pi$
$23$	−3561.09	−1.40366	−0.701832	+	0.712343i	$0.747634\pi$
$24$	1206.60	0.427598
$25$	1762.48	0.563993
$26$	1181.45	0.342753
$27$	2461.43	0.649799
$28$	4019.92	0.968998
$29$	−5777.94	−1.27579	−0.637893	−	0.770125i	$-0.720194\pi$
$29$	−5777.94	−1.27579	−0.637893	+	0.770125i	$0.720194\pi$
$30$	−5272.14	−1.06951
$31$	851.123	0.159070	0.0795349	−	0.996832i	$-0.474656\pi$
$31$	851.123	0.159070	0.0795349	+	0.996832i	$0.474656\pi$
$32$	−1024.00	−0.176777
$33$	10452.2	1.67080
$34$	−4164.32	−0.617799
$35$	−17564.7	−2.42365
$36$	1799.07	0.231362
$37$	11407.6	1.36990	0.684950	−	0.728590i	$-0.259824\pi$
$37$	11407.6	1.36990	0.684950	+	0.728590i	$0.259824\pi$
$38$	−2792.78	−0.313745
$39$	5568.51	0.586243
$40$	4474.27	0.442153
$41$	−2368.35	−0.220032	−0.110016	−	0.993930i	$-0.535090\pi$
$41$	−2368.35	−0.220032	−0.110016	+	0.993930i	$0.535090\pi$
$42$	18947.1	1.65737
$43$	−4542.00	−0.374607	−0.187303	−	0.982302i	$-0.559975\pi$
$43$	−4542.00	−0.374607	−0.187303	+	0.982302i	$0.559975\pi$
$44$	−8870.42	−0.690736
$45$	−7860.87	−0.578681
$46$	14244.3	0.992540
$47$	15678.2	1.03526	0.517631	−	0.855604i	$-0.326814\pi$
$47$	15678.2	1.03526	0.517631	+	0.855604i	$0.326814\pi$
$48$	−4826.41	−0.302358
$49$	46317.2	2.75583
$50$	−7049.92	−0.398804
$51$	−19627.7	−1.05668
$52$	−4725.79	−0.242363
$53$	29228.9	1.42930	0.714649	−	0.699483i	$-0.246587\pi$
$53$	29228.9	1.42930	0.714649	+	0.699483i	$0.246587\pi$
$54$	−9845.74	−0.459477
$55$	38758.5	1.72767
$56$	−16079.7	−0.685185
$57$	−13163.2	−0.536628
$58$	23111.8	0.902117
$59$	7562.69	0.282844	0.141422	−	0.989949i	$-0.454833\pi$
$59$	7562.69	0.282844	0.141422	+	0.989949i	$0.454833\pi$
$60$	21088.6	0.756256
$61$	−40717.5	−1.40106	−0.700530	−	0.713623i	$-0.747053\pi$
$61$	−40717.5	−1.40106	−0.700530	+	0.713623i	$0.747053\pi$
$62$	−3404.49	−0.112479
$63$	28250.5	0.896756
$64$	4096.00	0.125000
$65$	20648.9	0.606197
$66$	−41808.9	−1.18143
$67$	65923.1	1.79412	0.897058	−	0.441913i	$-0.145700\pi$
$67$	65923.1	1.79412	0.897058	+	0.441913i	$0.145700\pi$
$68$	16657.3	0.436850
$69$	67137.7	1.69763
$70$	70258.7	1.71378
$71$	81488.4	1.91845	0.959224	−	0.282648i	$-0.0912130\pi$
$71$	81488.4	1.91845	0.959224	+	0.282648i	$0.0912130\pi$
$72$	−7196.28	−0.163597
$73$	13007.3	0.285679	0.142840	−	0.989746i	$-0.454377\pi$
$73$	13007.3	0.285679	0.142840	+	0.989746i	$0.454377\pi$
$74$	−45630.3	−0.968666
$75$	−33228.3	−0.682111
$76$	11171.1	0.221851
$77$	−139291.	−2.67729
$78$	−22274.0	−0.414536
$79$	49650.7	0.895071	0.447535	−	0.894266i	$-0.352302\pi$
$79$	49650.7	0.895071	0.447535	+	0.894266i	$0.352302\pi$
$80$	−17897.1	−0.312649
$81$	−73729.2	−1.24861
$82$	9473.38	0.155586
$83$	27957.3	0.445451	0.222725	−	0.974881i	$-0.428505\pi$
$83$	27957.3	0.445451	0.222725	+	0.974881i	$0.428505\pi$
$84$	−75788.3	−1.17194
$85$	−72782.5	−1.09265
$86$	18168.0	0.264887
$87$	108932.	1.54298
$88$	35481.7	0.488424
$89$	−112434.	−1.50461	−0.752305	−	0.658816i	$-0.771058\pi$
$89$	−112434.	−1.50461	−0.752305	+	0.658816i	$0.771058\pi$
$90$	31443.5	0.409189
$91$	−74208.3	−0.939397
$92$	−56977.4	−0.701832
$93$	−16046.4	−0.192384
$94$	−62712.6	−0.732041
$95$	−48811.1	−0.554894
$96$	19305.6	0.213799
$97$	−4913.93	−0.0530273	−0.0265136	−	0.999648i	$-0.508441\pi$
$97$	−4913.93	−0.0530273	−0.0265136	+	0.999648i	$0.508441\pi$
$98$	−185269.	−1.94866
$99$	−62337.9	−0.639240
$100$	28199.7	0.281997
$101$	−129473.	−1.26292	−0.631462	−	0.775407i	$-0.717545\pi$
$101$	−129473.	−1.26292	−0.631462	+	0.775407i	$0.717545\pi$
$102$	78510.7	0.747185
$103$	−168318.	−1.56329	−0.781644	−	0.623725i	$-0.785618\pi$
$103$	−168318.	−1.56329	−0.781644	+	0.623725i	$0.785618\pi$
$104$	18903.2	0.171377
$105$	331150.	2.93124
$106$	−116916.	−1.01067
$107$	−108423.	−0.915508	−0.457754	−	0.889079i	$-0.651346\pi$
$107$	−108423.	−0.915508	−0.457754	+	0.889079i	$0.651346\pi$
$108$	39383.0	0.324899
$109$	−197910.	−1.59551	−0.797757	−	0.602979i	$-0.793980\pi$
$109$	−197910.	−1.59551	−0.797757	+	0.602979i	$0.793980\pi$
$110$	−155034.	−1.22164
$111$	−215069.	−1.65680
$112$	64318.8	0.484499
$113$	246225.	1.81400	0.906999	−	0.421133i	$-0.138368\pi$
$113$	246225.	1.81400	0.906999	+	0.421133i	$0.138368\pi$
$114$	52652.7	0.379453
$115$	248957.	1.75542
$116$	−92447.0	−0.637893
$117$	−33211.0	−0.224294
$118$	−30250.8	−0.200001
$119$	261567.	1.69323
$120$	−84354.2	−0.534753
$121$	146309.	0.908467
$122$	162870.	0.990700
$123$	44650.8	0.266113
$124$	13618.0	0.0795349
$125$	95254.5	0.545269
$126$	−113002.	−0.634102
$127$	258489.	1.42211	0.711056	−	0.703136i	$-0.248217\pi$
$127$	258489.	1.42211	0.711056	+	0.703136i	$0.248217\pi$
$128$	−16384.0	−0.0883883
$129$	85631.0	0.453061
$130$	−82595.7	−0.428646
$131$	133258.	0.678448	0.339224	−	0.940706i	$-0.389836\pi$
$131$	133258.	0.678448	0.339224	+	0.940706i	$0.389836\pi$
$132$	167235.	0.835398
$133$	175418.	0.859894
$134$	−263692.	−1.26863
$135$	−172080.	−0.812637
$136$	−66629.2	−0.308899
$137$	−317756.	−1.44642	−0.723208	−	0.690631i	$-0.757333\pi$
$137$	−317756.	−1.44642	−0.723208	+	0.690631i	$0.757333\pi$
$138$	−268551.	−1.20041
$139$	−92680.8	−0.406867	−0.203434	−	0.979089i	$-0.565210\pi$
$139$	−92680.8	−0.406867	−0.203434	+	0.979089i	$0.565210\pi$
$140$	−281035.	−1.21183
$141$	−295583.	−1.25208
$142$	−325954.	−1.35655
$143$	163749.	0.669636
$144$	28785.1	0.115681
$145$	403939.	1.59550
$146$	−52029.1	−0.202006
$147$	−873225.	−3.33298
$148$	182521.	0.684950
$149$	16076.4	0.0593231	0.0296615	−	0.999560i	$-0.490557\pi$
$149$	16076.4	0.0593231	0.0296615	+	0.999560i	$0.490557\pi$
$150$	132913.	0.482325
$151$	255779.	0.912900	0.456450	−	0.889749i	$-0.349121\pi$
$151$	255779.	0.912900	0.456450	+	0.889749i	$0.349121\pi$
$152$	−44684.4	−0.156873
$153$	117061.	0.404281
$154$	557162.	1.89313
$155$	−59502.4	−0.198932
$156$	89096.2	0.293121
$157$	−336742.	−1.09030	−0.545152	−	0.838337i	$-0.683528\pi$
$157$	−336742.	−1.09030	−0.545152	+	0.838337i	$0.683528\pi$
$158$	−198603.	−0.632911
$159$	−551057.	−1.72864
$160$	71588.4	0.221076
$161$	−894706.	−2.72029
$162$	294917.	0.882901
$163$	124878.	0.368143	0.184072	−	0.982913i	$-0.441072\pi$
$163$	124878.	0.368143	0.184072	+	0.982913i	$0.441072\pi$
$164$	−37893.5	−0.110016
$165$	−730720.	−2.08949
$166$	−111829.	−0.314981
$167$	−418305.	−1.16065	−0.580325	−	0.814385i	$-0.697075\pi$
$167$	−418305.	−1.16065	−0.580325	+	0.814385i	$0.697075\pi$
$168$	303153.	0.828684
$169$	−284054.	−0.765041
$170$	291130.	0.772617
$171$	78506.2	0.205312
$172$	−72672.0	−0.187303
$173$	603777.	1.53377	0.766887	−	0.641782i	$-0.221804\pi$
$173$	603777.	1.53377	0.766887	+	0.641782i	$0.221804\pi$
$174$	−435730.	−1.09105
$175$	442815.	1.09302
$176$	−141927.	−0.345368
$177$	−142581.	−0.342080
$178$	449737.	1.06392
$179$	−764371.	−1.78308	−0.891542	−	0.452938i	$-0.850376\pi$
$179$	−764371.	−1.78308	−0.891542	+	0.452938i	$0.850376\pi$
$180$	−125774.	−0.289340
$181$	−443900.	−1.00714	−0.503569	−	0.863955i	$-0.667980\pi$
$181$	−443900.	−1.00714	−0.503569	+	0.863955i	$0.667980\pi$
$182$	296833.	0.664254
$183$	767655.	1.69449
$184$	227910.	0.496270
$185$	−797510.	−1.71319
$186$	64185.4	0.136036
$187$	−577176.	−1.20699
$188$	250851.	0.517631
$189$	618424.	1.25931
$190$	195244.	0.392369
$191$	−229314.	−0.454828	−0.227414	−	0.973798i	$-0.573027\pi$
$191$	−229314.	−0.454828	−0.227414	+	0.973798i	$0.573027\pi$
$192$	−77222.6	−0.151179
$193$	423980.	0.819316	0.409658	−	0.912239i	$-0.365648\pi$
$193$	423980.	0.819316	0.409658	+	0.912239i	$0.365648\pi$
$194$	19655.7	0.0374959
$195$	−389297.	−0.733154
$196$	741075.	1.37791
$197$	303119.	0.556477	0.278239	−	0.960512i	$-0.410249\pi$
$197$	303119.	0.556477	0.278239	+	0.960512i	$0.410249\pi$
$198$	249351.	0.452011
$199$	352737.	0.631421	0.315710	−	0.948856i	$-0.397757\pi$
$199$	352737.	0.631421	0.315710	+	0.948856i	$0.397757\pi$
$200$	−112799.	−0.199402
$201$	−1.24286e6	−2.16986
$202$	517894.	0.893023
$203$	−1.45168e6	−2.47247
$204$	−314043.	−0.528340
$205$	165572.	0.275171
$206$	673274.	1.10541
$207$	−400415.	−0.649508
$208$	−75612.7	−0.121182
$209$	−387080.	−0.612963
$210$	−1.32460e6	−2.07270
$211$	−1.17010e6	−1.80933	−0.904663	−	0.426128i	$-0.859877\pi$
$211$	−1.17010e6	−1.80933	−0.904663	+	0.426128i	$0.859877\pi$
$212$	467662.	0.714649
$213$	−1.53631e6	−2.32023
$214$	433692.	0.647362
$215$	317533.	0.468482
$216$	−157532.	−0.229739
$217$	213841.	0.308277
$218$	791639.	1.12820
$219$	−245228.	−0.345510
$220$	620135.	0.863833
$221$	−307496.	−0.423505
$222$	860276.	1.17154
$223$	267505.	0.360221	0.180111	−	0.983646i	$-0.442354\pi$
$223$	267505.	0.360221	0.180111	+	0.983646i	$0.442354\pi$
$224$	−257275.	−0.342592
$225$	198176.	0.260973
$226$	−984901.	−1.28269
$227$	−795470.	−1.02461	−0.512305	−	0.858803i	$-0.671208\pi$
$227$	−795470.	−1.02461	−0.512305	+	0.858803i	$0.671208\pi$
$228$	−210611.	−0.268314
$229$	254230.	0.320360	0.160180	−	0.987088i	$-0.448793\pi$
$229$	254230.	0.320360	0.160180	+	0.987088i	$0.448793\pi$
$230$	−995830.	−1.24127
$231$	2.62607e6	3.23800
$232$	369788.	0.451059
$233$	−659664.	−0.796036	−0.398018	−	0.917377i	$-0.630302\pi$
$233$	−659664.	−0.796036	−0.398018	+	0.917377i	$0.630302\pi$
$234$	132844.	0.158600
$235$	−1.09607e6	−1.29470
$236$	121003.	0.141422
$237$	−936072.	−1.08253
$238$	−1.04627e6	−1.19729
$239$	29221.6	0.0330909	0.0165455	−	0.999863i	$-0.494733\pi$
$239$	29221.6	0.0330909	0.0165455	+	0.999863i	$0.494733\pi$
$240$	337417.	0.378128
$241$	681088.	0.755372	0.377686	−	0.925934i	$-0.376720\pi$
$241$	681088.	0.755372	0.377686	+	0.925934i	$0.376720\pi$
$242$	−585238.	−0.642383
$243$	791900.	0.860309
$244$	−651481.	−0.700530
$245$	−3.23806e6	−3.44643
$246$	−178603.	−0.188171
$247$	−206220.	−0.215074
$248$	−54471.9	−0.0562397
$249$	−527083.	−0.538742
$250$	−381018.	−0.385563
$251$	1.33876e6	1.34127	0.670636	−	0.741786i	$-0.266021\pi$
$251$	1.33876e6	1.34127	0.670636	+	0.741786i	$0.266021\pi$
$252$	452008.	0.448378
$253$	1.97427e6	1.93912
$254$	−1.03396e6	−1.00558
$255$	1.37218e6	1.32148
$256$	65536.0	0.0625000
$257$	806338.	0.761525	0.380763	−	0.924673i	$-0.375662\pi$
$257$	806338.	0.761525	0.380763	+	0.924673i	$0.375662\pi$
$258$	−342524.	−0.320363
$259$	2.86610e6	2.65486
$260$	330383.	0.303099
$261$	−649682.	−0.590336
$262$	−533034.	−0.479735
$263$	−1.16728e6	−1.04060	−0.520301	−	0.853983i	$-0.674180\pi$
$263$	−1.16728e6	−1.04060	−0.520301	+	0.853983i	$0.674180\pi$
$264$	−668942.	−0.590716
$265$	−2.04341e6	−1.78748
$266$	−701672.	−0.608037
$267$	2.11974e6	1.81972
$268$	1.05477e6	0.897058
$269$	−72361.0	−0.0609711
$269$	−72361.0	−0.0609711
$270$	688321.	0.574621
$271$	−118480.	−0.0979991	−0.0489996	−	0.998799i	$-0.515603\pi$
$271$	−118480.	−0.0979991	−0.0489996	+	0.998799i	$0.515603\pi$
$272$	266517.	0.218425
$273$	1.39906e6	1.13614
$274$	1.27103e6	1.02277
$275$	−977121.	−0.779142
$276$	1.07420e6	0.848817
$277$	1.28672e6	1.00759	0.503795	−	0.863823i	$-0.331937\pi$
$277$	1.28672e6	1.00759	0.503795	+	0.863823i	$0.331937\pi$
$278$	370723.	0.287699
$279$	95701.8	0.0736054
$280$	1.12414e6	0.856890
$281$	160733.	0.121434	0.0607169	−	0.998155i	$-0.480661\pi$
$281$	160733.	0.121434	0.0607169	+	0.998155i	$0.480661\pi$
$282$	1.18233e6	0.885353
$283$	−2.23047e6	−1.65551	−0.827754	−	0.561092i	$-0.810381\pi$
$283$	−2.23047e6	−1.65551	−0.827754	+	0.561092i	$0.810381\pi$
$284$	1.30381e6	0.959224
$285$	920244.	0.671106
$286$	−654996.	−0.473504
$287$	−595036.	−0.426421
$288$	−115140.	−0.0817987
$289$	−336008.	−0.236649
$290$	−1.61575e6	−1.12819
$291$	92643.0	0.0641328
$292$	208116.	0.142840
$293$	411529.	0.280047	0.140024	−	0.990148i	$-0.455282\pi$
$293$	411529.	0.280047	0.140024	+	0.990148i	$0.455282\pi$
$294$	3.49290e6	2.35677
$295$	−528712.	−0.353723
$296$	−730085.	−0.484333
$297$	−1.36462e6	−0.897679
$298$	−64305.7	−0.0419477
$299$	1.05181e6	0.680392
$300$	−531653.	−0.341056
$301$	−1.14116e6	−0.725987
$302$	−1.02312e6	−0.645518
$303$	2.44099e6	1.52742
$304$	178738.	0.110926
$305$	2.84658e6	1.75216
$306$	−468244.	−0.285870
$307$	−565267.	−0.342300	−0.171150	−	0.985245i	$-0.554748\pi$
$307$	−565267.	−0.342300	−0.171150	+	0.985245i	$0.554748\pi$
$308$	−2.22865e6	−1.33864
$309$	3.17334e6	1.89069
$310$	238010.	0.140666
$311$	−2.52349e6	−1.47945	−0.739727	−	0.672907i	$-0.765045\pi$
$311$	−2.52349e6	−1.47945	−0.739727	+	0.672907i	$0.765045\pi$
$312$	−356385.	−0.207268
$313$	−2.11561e6	−1.22060	−0.610302	−	0.792169i	$-0.708952\pi$
$313$	−2.11561e6	−1.22060	−0.610302	+	0.792169i	$0.708952\pi$
$314$	1.34697e6	0.770962
$315$	−1.97500e6	−1.12148
$316$	794411.	0.447535
$317$	−743761.	−0.415705	−0.207853	−	0.978160i	$-0.566647\pi$
$317$	−743761.	−0.415705	−0.207853	+	0.978160i	$0.566647\pi$
$318$	2.20423e6	1.22233
$319$	3.20330e6	1.76246
$320$	−286353.	−0.156325
$321$	2.04412e6	1.10724
$322$	3.57882e6	1.92354
$323$	726876.	0.387663
$324$	−1.17967e6	−0.624305
$325$	−520570.	−0.273382
$326$	−499512.	−0.260316
$327$	3.73122e6	1.92967
$328$	151574.	0.0777930
$329$	3.93906e6	2.00633
$330$	2.92288e6	1.47749
$331$	2.29328e6	1.15050	0.575251	−	0.817977i	$-0.304904\pi$
$331$	2.29328e6	1.15050	0.575251	+	0.817977i	$0.304904\pi$
$332$	447317.	0.222725
$333$	1.28269e6	0.633885
$334$	1.67322e6	0.820704
$335$	−4.60872e6	−2.24372
$336$	−1.21261e6	−0.585968
$337$	−1.72298e6	−0.826427	−0.413213	−	0.910634i	$-0.635594\pi$
$337$	−1.72298e6	−0.826427	−0.413213	+	0.910634i	$0.635594\pi$
$338$	1.13622e6	0.540965
$339$	−4.64213e6	−2.19390
$340$	−1.16452e6	−0.546323
$341$	−471863.	−0.219751
$342$	−314025.	−0.145177
$343$	7.41429e6	3.40278
$344$	290688.	0.132444
$345$	−4.69363e6	−2.12306
$346$	−2.41511e6	−1.08454
$347$	2.22978e6	0.994120	0.497060	−	0.867716i	$-0.334413\pi$
$347$	2.22978e6	0.994120	0.497060	+	0.867716i	$0.334413\pi$
$348$	1.74292e6	0.771488
$349$	826665.	0.363301	0.181650	−	0.983363i	$-0.441856\pi$
$349$	826665.	0.363301	0.181650	+	0.983363i	$0.441856\pi$
$350$	−1.77126e6	−0.772880
$351$	−727014.	−0.314974
$352$	567707.	0.244212
$353$	2.30753e6	0.985623	0.492812	−	0.870136i	$-0.335969\pi$
$353$	2.30753e6	0.985623	0.492812	+	0.870136i	$0.335969\pi$
$354$	570323.	0.241887
$355$	−5.69689e6	−2.39920
$356$	−1.79895e6	−0.752305
$357$	−4.93136e6	−2.04784
$358$	3.05748e6	1.26083
$359$	−1.10481e6	−0.452429	−0.226215	−	0.974077i	$-0.572635\pi$
$359$	−1.10481e6	−0.452429	−0.226215	+	0.974077i	$0.572635\pi$
$360$	503095.	0.204595
$361$	−1.98862e6	−0.803128
$362$	1.77560e6	0.712154
$363$	−2.75840e6	−1.09873
$364$	−1.18733e6	−0.469698
$365$	−909345.	−0.357270
$366$	−3.07062e6	−1.19818
$367$	−1.02020e6	−0.395383	−0.197692	−	0.980264i	$-0.563344\pi$
$367$	−1.02020e6	−0.395383	−0.197692	+	0.980264i	$0.563344\pi$
$368$	−911638.	−0.350916
$369$	−266301.	−0.101814
$370$	3.19004e6	1.21141
$371$	7.34362e6	2.76997
$372$	−256742.	−0.0961920
$373$	−4.50224e6	−1.67555	−0.837774	−	0.546017i	$-0.816143\pi$
$373$	−4.50224e6	−1.67555	−0.837774	+	0.546017i	$0.816143\pi$
$374$	2.30870e6	0.853472
$375$	−1.79585e6	−0.659465
$376$	−1.00340e6	−0.366021
$377$	1.70658e6	0.618407
$378$	−2.47370e6	−0.890465
$379$	−4.18405e6	−1.49623	−0.748115	−	0.663569i	$-0.769041\pi$
$379$	−4.18405e6	−1.49623	−0.748115	+	0.663569i	$0.769041\pi$
$380$	−780978.	−0.277447
$381$	−4.87334e6	−1.71995
$382$	917256.	0.321612
$383$	2.78375e6	0.969692	0.484846	−	0.874600i	$-0.338876\pi$
$383$	2.78375e6	0.969692	0.484846	+	0.874600i	$0.338876\pi$
$384$	308890.	0.106900
$385$	9.73788e6	3.34821
$386$	−1.69592e6	−0.579344
$387$	−510711.	−0.173339
$388$	−78622.8	−0.0265136
$389$	2.10399e6	0.704969	0.352484	−	0.935818i	$-0.385337\pi$
$389$	2.10399e6	0.704969	0.352484	+	0.935818i	$0.385337\pi$
$390$	1.55719e6	0.518418
$391$	−3.70738e6	−1.22638
$392$	−2.96430e6	−0.974332
$393$	−2.51234e6	−0.820536
$394$	−1.21248e6	−0.393489
$395$	−3.47110e6	−1.11937
$396$	−997406.	−0.319620
$397$	−3.33402e6	−1.06168	−0.530838	−	0.847473i	$-0.678123\pi$
$397$	−3.33402e6	−1.06168	−0.530838	+	0.847473i	$0.678123\pi$
$398$	−1.41095e6	−0.446482
$399$	−3.30718e6	−1.03998
$400$	451195.	0.140998
$401$	2.15526e6	0.669327	0.334663	−	0.942338i	$-0.391377\pi$
$401$	2.15526e6	0.669327	0.334663	+	0.942338i	$0.391377\pi$
$402$	4.97144e6	1.53432
$403$	−251389.	−0.0771053
$404$	−2.07158e6	−0.631462
$405$	5.15445e6	1.56151
$406$	5.80672e6	1.74830
$407$	−6.32437e6	−1.89248
$408$	1.25617e6	0.373593
$409$	3.01358e6	0.890788	0.445394	−	0.895335i	$-0.353064\pi$
$409$	3.01358e6	0.890788	0.445394	+	0.895335i	$0.353064\pi$
$410$	−662289.	−0.194575
$411$	5.99071e6	1.74934
$412$	−2.69310e6	−0.781644
$413$	1.90009e6	0.548150
$414$	1.60166e6	0.459272
$415$	−1.95451e6	−0.557080
$416$	302451.	0.0856883
$417$	1.74733e6	0.492078
$418$	1.54832e6	0.433431
$419$	−2.28129e6	−0.634814	−0.317407	−	0.948289i	$-0.602812\pi$
$419$	−2.28129e6	−0.634814	−0.317407	+	0.948289i	$0.602812\pi$
$420$	5.29840e6	1.46562
$421$	−1.88462e6	−0.518224	−0.259112	−	0.965847i	$-0.583430\pi$
$421$	−1.88462e6	−0.518224	−0.259112	+	0.965847i	$0.583430\pi$
$422$	4.68040e6	1.27939
$423$	1.76288e6	0.479040
$424$	−1.87065e6	−0.505333
$425$	1.83488e6	0.492761
$426$	6.14526e6	1.64065
$427$	−1.02301e7	−2.71525
$428$	−1.73477e6	−0.457754
$429$	−3.08719e6	−0.809878
$430$	−1.27013e6	−0.331267
$431$	2.51782e6	0.652877	0.326439	−	0.945218i	$-0.394151\pi$
$431$	2.51782e6	0.652877	0.326439	+	0.945218i	$0.394151\pi$
$432$	630127.	0.162450
$433$	2.60939e6	0.668834	0.334417	−	0.942425i	$-0.391461\pi$
$433$	2.60939e6	0.668834	0.334417	+	0.942425i	$0.391461\pi$
$434$	−855362.	−0.217985
$435$	−7.61552e6	−1.92964
$436$	−3.16656e6	−0.797757
$437$	−2.48633e6	−0.622809
$438$	980913.	0.244312
$439$	−96190.9	−0.0238217	−0.0119108	−	0.999929i	$-0.503791\pi$
$439$	−96190.9	−0.0238217	−0.0119108	+	0.999929i	$0.503791\pi$
$440$	−2.48054e6	−0.610822
$441$	5.20799e6	1.27519
$442$	1.22998e6	0.299463
$443$	714512.	0.172982	0.0864908	−	0.996253i	$-0.472435\pi$
$443$	714512.	0.172982	0.0864908	+	0.996253i	$0.472435\pi$
$444$	−3.44110e6	−0.828400
$445$	7.86034e6	1.88166
$446$	−1.07002e6	−0.254715
$447$	−303091.	−0.0717471
$448$	1.02910e6	0.242249
$449$	480835.	0.112559	0.0562796	−	0.998415i	$-0.482076\pi$
$449$	480835.	0.112559	0.0562796	+	0.998415i	$0.482076\pi$
$450$	−792706.	−0.184536
$451$	1.31301e6	0.303968
$452$	3.93961e6	0.906999
$453$	−4.82225e6	−1.10409
$454$	3.18188e6	0.724509
$455$	5.18794e6	1.17481
$456$	842443.	0.189727
$457$	−5.83406e6	−1.30671	−0.653357	−	0.757050i	$-0.726640\pi$
$457$	−5.83406e6	−1.30671	−0.653357	+	0.757050i	$0.726640\pi$
$458$	−1.01692e6	−0.226528
$459$	2.56255e6	0.567729
$460$	3.98332e6	0.877709
$461$	−2.54633e6	−0.558036	−0.279018	−	0.960286i	$-0.590009\pi$
$461$	−2.54633e6	−0.558036	−0.279018	+	0.960286i	$0.590009\pi$
$462$	−1.05043e7	−2.28961
$463$	2.21247e6	0.479650	0.239825	−	0.970816i	$-0.422910\pi$
$463$	2.21247e6	0.479650	0.239825	+	0.970816i	$0.422910\pi$
$464$	−1.47915e6	−0.318947
$465$	1.12181e6	0.240595
$466$	2.63866e6	0.562883
$467$	−3.86803e6	−0.820725	−0.410362	−	0.911923i	$-0.634598\pi$
$467$	−3.86803e6	−0.820725	−0.410362	+	0.911923i	$0.634598\pi$
$468$	−531377.	−0.112147
$469$	1.65629e7	3.47699
$470$	4.38427e6	0.915489
$471$	6.34865e6	1.31865
$472$	−484012.	−0.100000
$473$	2.51809e6	0.517509
$474$	3.74429e6	0.765462
$475$	1.23055e6	0.250246
$476$	4.18506e6	0.846613
$477$	3.28655e6	0.661370
$478$	−116886.	−0.0233988
$479$	871432.	0.173538	0.0867691	−	0.996228i	$-0.472346\pi$
$479$	871432.	0.173538	0.0867691	+	0.996228i	$0.472346\pi$
$480$	−1.34967e6	−0.267377
$481$	−3.36937e6	−0.664027
$482$	−2.72435e6	−0.534129
$483$	1.68680e7	3.29001
$484$	2.34095e6	0.454233
$485$	343535.	0.0663158
$486$	−3.16760e6	−0.608331
$487$	−8.60687e6	−1.64446	−0.822229	−	0.569157i	$-0.807270\pi$
$487$	−8.60687e6	−1.64446	−0.822229	+	0.569157i	$0.807270\pi$
$488$	2.60592e6	0.495350
$489$	−2.35434e6	−0.445244
$490$	1.29522e7	2.43699
$491$	−5.64835e6	−1.05735	−0.528674	−	0.848825i	$-0.677311\pi$
$491$	−5.64835e6	−1.05735	−0.528674	+	0.848825i	$0.677311\pi$
$492$	714413.	0.133057
$493$	−6.01530e6	−1.11465
$494$	824880.	0.152081
$495$	4.35807e6	0.799432
$496$	217887.	0.0397675
$497$	2.04736e7	3.71794
$498$	2.10833e6	0.380948
$499$	4909.17	0.000882586 0	0.000441293	−	1.00000i	$-0.499860\pi$
$499$	4909.17	0.000882586 0	0.000441293		1.00000i	$0.499860\pi$
$500$	1.52407e6	0.272634
$501$	7.88637e6	1.40373
$502$	−5.35502e6	−0.948423
$503$	−3.87946e6	−0.683678	−0.341839	−	0.939759i	$-0.611050\pi$
$503$	−3.87946e6	−0.683678	−0.341839	+	0.939759i	$0.611050\pi$
$504$	−1.80803e6	−0.317051
$505$	9.05156e6	1.57941
$506$	−7.89708e6	−1.37117
$507$	5.35532e6	0.925264
$508$	4.13583e6	0.711056
$509$	−192712.	−0.0329696	−0.0164848	−	0.999864i	$-0.505248\pi$
$509$	−192712.	−0.0329696	−0.0164848	+	0.999864i	$0.505248\pi$
$510$	−5.48872e6	−0.934428
$511$	3.26801e6	0.553645
$512$	−262144.	−0.0441942
$513$	1.71856e6	0.288318
$514$	−3.22535e6	−0.538480
$515$	1.17672e7	1.95504
$516$	1.37010e6	0.226531
$517$	−8.69199e6	−1.43019
$518$	−1.14644e7	−1.87727
$519$	−1.13831e7	−1.85499
$520$	−1.32153e6	−0.214323
$521$	3.17552e6	0.512531	0.256266	−	0.966606i	$-0.417508\pi$
$521$	3.17552e6	0.512531	0.256266	+	0.966606i	$0.417508\pi$
$522$	2.59873e6	0.417431
$523$	−1.13172e7	−1.80919	−0.904597	−	0.426268i	$-0.859828\pi$
$523$	−1.13172e7	−1.80919	−0.904597	+	0.426268i	$0.859828\pi$
$524$	2.13214e6	0.339224
$525$	−8.34846e6	−1.32193
$526$	4.66911e6	0.735816
$527$	886087.	0.138979
$528$	2.67577e6	0.417699
$529$	6.24499e6	0.970270
$530$	8.17363e6	1.26394
$531$	850363.	0.130878
$532$	2.80669e6	0.429947
$533$	699519.	0.106655
$534$	−8.47897e6	−1.28674
$535$	7.57991e6	1.14493
$536$	−4.21908e6	−0.634316
$537$	1.44108e7	2.15652
$538$	289444.	0.0431131
$539$	−2.56783e7	−3.80710
$540$	−2.75328e6	−0.406318
$541$	−8.74242e6	−1.28422	−0.642108	−	0.766614i	$-0.721940\pi$
$541$	−8.74242e6	−1.28422	−0.642108	+	0.766614i	$0.721940\pi$
$542$	473920.	0.0692959
$543$	8.36892e6	1.21806
$544$	−1.06607e6	−0.154450
$545$	1.38360e7	1.99535
$546$	−5.59625e6	−0.803369
$547$	−1.04425e6	−0.149223	−0.0746116	−	0.997213i	$-0.523772\pi$
$547$	−1.04425e6	−0.149223	−0.0746116	+	0.997213i	$0.523772\pi$
$548$	−5.08410e6	−0.723208
$549$	−4.57835e6	−0.648304
$550$	3.90848e6	0.550936
$551$	−4.03412e6	−0.566070
$552$	−4.29682e6	−0.600204
$553$	1.24745e7	1.73464
$554$	−5.14687e6	−0.712473
$555$	1.50356e7	2.07199
$556$	−1.48289e6	−0.203434
$557$	−225408.	−0.0307844	−0.0153922	−	0.999882i	$-0.504900\pi$
$557$	−225408.	−0.0307844	−0.0153922	+	0.999882i	$0.504900\pi$
$558$	−382807.	−0.0520469
$559$	1.34153e6	0.181582
$560$	−4.49656e6	−0.605913
$561$	1.08816e7	1.45977
$562$	−642933.	−0.0858667
$563$	−1.49819e6	−0.199203	−0.0996015	−	0.995027i	$-0.531757\pi$
$563$	−1.49819e6	−0.199203	−0.0996015	+	0.995027i	$0.531757\pi$
$564$	−4.72933e6	−0.626039
$565$	−1.72137e7	−2.26858
$566$	8.92190e6	1.17062
$567$	−1.85241e7	−2.41980
$568$	−5.21526e6	−0.678273
$569$	−5.08603e6	−0.658564	−0.329282	−	0.944232i	$-0.606807\pi$
$569$	−5.08603e6	−0.658564	−0.329282	+	0.944232i	$0.606807\pi$
$570$	−3.68098e6	−0.474543
$571$	−5.84412e6	−0.750117	−0.375058	−	0.927001i	$-0.622377\pi$
$571$	−5.84412e6	−0.750117	−0.375058	+	0.927001i	$0.622377\pi$
$572$	2.61998e6	0.334818
$573$	4.32329e6	0.550083
$574$	2.38014e6	0.301525
$575$	−6.27634e6	−0.791657
$576$	460562.	0.0578404
$577$	8.07540e6	1.00977	0.504887	−	0.863185i	$-0.331534\pi$
$577$	8.07540e6	1.00977	0.504887	+	0.863185i	$0.331534\pi$
$578$	1.34403e6	0.167336
$579$	−7.99336e6	−0.990907
$580$	6.46302e6	0.797748
$581$	7.02414e6	0.863282
$582$	−370572.	−0.0453488
$583$	−1.62045e7	−1.97454
$584$	−832465.	−0.101003
$585$	2.32180e6	0.280502
$586$	−1.64612e6	−0.198023
$587$	−1.39140e7	−1.66670	−0.833348	−	0.552749i	$-0.813579\pi$
$587$	−1.39140e7	−1.66670	−0.833348	+	0.552749i	$0.813579\pi$
$588$	−1.39716e7	−1.66649
$589$	594249.	0.0705798
$590$	2.11485e6	0.250120
$591$	−5.71475e6	−0.673021
$592$	2.92034e6	0.342475
$593$	1.25773e7	1.46876	0.734379	−	0.678739i	$-0.237473\pi$
$593$	1.25773e7	1.46876	0.734379	+	0.678739i	$0.237473\pi$
$594$	5.45849e6	0.634755
$595$	−1.82863e7	−2.11754
$596$	257223.	0.0296615
$597$	−6.65022e6	−0.763660
$598$	−4.20724e6	−0.481110
$599$	−1.29484e7	−1.47452	−0.737258	−	0.675611i	$-0.763880\pi$
$599$	−1.29484e7	−1.47452	−0.737258	+	0.675611i	$0.763880\pi$
$600$	2.12661e6	0.241163
$601$	−1.13130e7	−1.27759	−0.638793	−	0.769378i	$-0.720566\pi$
$601$	−1.13130e7	−1.27759	−0.638793	+	0.769378i	$0.720566\pi$
$602$	4.56462e6	0.513350
$603$	7.41251e6	0.830180
$604$	4.09247e6	0.456450
$605$	−1.02286e7	−1.13613
$606$	−9.76394e6	−1.08005
$607$	7.81963e6	0.861419	0.430709	−	0.902491i	$-0.358263\pi$
$607$	7.81963e6	0.861419	0.430709	+	0.902491i	$0.358263\pi$
$608$	−714951.	−0.0784363
$609$	2.73688e7	2.99028
$610$	−1.13863e7	−1.23897
$611$	−4.63073e6	−0.501819
$612$	1.87298e6	0.202141
$613$	7.28729e6	0.783276	0.391638	−	0.920119i	$-0.371909\pi$
$613$	7.28729e6	0.783276	0.391638	+	0.920119i	$0.371909\pi$
$614$	2.26107e6	0.242043
$615$	−3.12156e6	−0.332801
$616$	8.91460e6	0.946564
$617$	−5.46615e6	−0.578054	−0.289027	−	0.957321i	$-0.593332\pi$
$617$	−5.46615e6	−0.578054	−0.289027	+	0.957321i	$0.593332\pi$
$618$	−1.26933e7	−1.33692
$619$	1.40249e7	1.47120	0.735600	−	0.677416i	$-0.236900\pi$
$619$	1.40249e7	1.47120	0.735600	+	0.677416i	$0.236900\pi$
$620$	−952039.	−0.0994662
$621$	−8.76538e6	−0.912099
$622$	1.00940e7	1.04613
$623$	−2.82486e7	−2.91593
$624$	1.42554e6	0.146561
$625$	−1.21670e7	−1.24590
$626$	8.46244e6	0.863098
$627$	7.29768e6	0.741337
$628$	−5.38787e6	−0.545152
$629$	1.18762e7	1.19688
$630$	7.90002e6	0.793007
$631$	−1.53355e7	−1.53329	−0.766645	−	0.642071i	$-0.778075\pi$
$631$	−1.53355e7	−1.53329	−0.766645	+	0.642071i	$0.778075\pi$
$632$	−3.17764e6	−0.316455
$633$	2.20601e7	2.18825
$634$	2.97505e6	0.293948
$635$	−1.80711e7	−1.77849
$636$	−8.81691e6	−0.864318
$637$	−1.36803e7	−1.33582
$638$	−1.28132e7	−1.24625
$639$	9.16270e6	0.887711
$640$	1.14541e6	0.110538
$641$	−617357.	−0.0593460	−0.0296730	−	0.999560i	$-0.509447\pi$
$641$	−617357.	−0.0593460	−0.0296730	+	0.999560i	$0.509447\pi$
$642$	−8.17647e6	−0.782939
$643$	−2.25889e6	−0.215460	−0.107730	−	0.994180i	$-0.534358\pi$
$643$	−2.25889e6	−0.215460	−0.107730	+	0.994180i	$0.534358\pi$
$644$	−1.43153e7	−1.36015
$645$	−5.98651e6	−0.566597
$646$	−2.90751e6	−0.274119
$647$	−1.78782e7	−1.67905	−0.839524	−	0.543322i	$-0.817166\pi$
$647$	−1.78782e7	−1.67905	−0.839524	+	0.543322i	$0.817166\pi$
$648$	4.71867e6	0.441450
$649$	−4.19276e6	−0.390741
$650$	2.08228e6	0.193310
$651$	−4.03157e6	−0.372839
$652$	1.99805e6	0.184072
$653$	−318239.	−0.0292059	−0.0146030	−	0.999893i	$-0.504648\pi$
$653$	−318239.	−0.0292059	−0.0146030	+	0.999893i	$0.504648\pi$
$654$	−1.49249e7	−1.36448
$655$	−9.31617e6	−0.848465
$656$	−606296.	−0.0550079
$657$	1.46256e6	0.132191
$658$	−1.57563e7	−1.41869
$659$	−1.51263e6	−0.135681	−0.0678403	−	0.997696i	$-0.521611\pi$
$659$	−1.51263e6	−0.135681	−0.0678403	+	0.997696i	$0.521611\pi$
$660$	−1.16915e7	−1.04475
$661$	4.40951e6	0.392542	0.196271	−	0.980550i	$-0.437117\pi$
$661$	4.40951e6	0.392542	0.196271	+	0.980550i	$0.437117\pi$
$662$	−9.17313e6	−0.813528
$663$	5.79727e6	0.512200
$664$	−1.78927e6	−0.157491
$665$	−1.22636e7	−1.07538
$666$	−5.13076e6	−0.448225
$667$	2.05757e7	1.79077
$668$	−6.69287e6	−0.580325
$669$	−5.04331e6	−0.435663
$670$	1.84349e7	1.58655
$671$	2.25738e7	1.93553
$672$	4.85045e6	0.414342
$673$	1.65886e7	1.41180	0.705899	−	0.708312i	$-0.250543\pi$
$673$	1.65886e7	1.41180	0.705899	+	0.708312i	$0.250543\pi$
$674$	6.89190e6	0.584372
$675$	4.33823e6	0.366482
$676$	−4.54487e6	−0.382520
$677$	−1.15300e7	−0.966848	−0.483424	−	0.875386i	$-0.660607\pi$
$677$	−1.15300e7	−0.966848	−0.483424	+	0.875386i	$0.660607\pi$
$678$	1.85685e7	1.55132
$679$	−1.23460e6	−0.102767
$680$	4.65808e6	0.386309
$681$	1.49971e7	1.23920
$682$	1.88745e6	0.155387
$683$	1.79284e7	1.47059	0.735293	−	0.677750i	$-0.237045\pi$
$683$	1.79284e7	1.47059	0.735293	+	0.677750i	$0.237045\pi$
$684$	1.25610e6	0.102656
$685$	2.22145e7	1.80888
$686$	−2.96572e7	−2.40613
$687$	−4.79304e6	−0.387453
$688$	−1.16275e6	−0.0936517
$689$	−8.63311e6	−0.692818
$690$	1.87745e7	1.50123
$691$	1.20893e7	0.963177	0.481588	−	0.876397i	$-0.340060\pi$
$691$	1.20893e7	0.963177	0.481588	+	0.876397i	$0.340060\pi$
$692$	9.66044e6	0.766887
$693$	−1.56621e7	−1.23884
$694$	−8.91913e6	−0.702949
$695$	6.47937e6	0.508827
$696$	−6.97168e6	−0.545524
$697$	−2.46564e6	−0.192242
$698$	−3.30666e6	−0.256892
$699$	1.24368e7	0.962751
$700$	7.08503e6	0.546508
$701$	7.92968e6	0.609482	0.304741	−	0.952435i	$-0.401430\pi$
$701$	7.92968e6	0.609482	0.304741	+	0.952435i	$0.401430\pi$
$702$	2.90806e6	0.222721
$703$	7.96471e6	0.607829
$704$	−2.27083e6	−0.172684
$705$	2.06644e7	1.56585
$706$	−9.23013e6	−0.696941
$707$	−3.25296e7	−2.44754
$708$	−2.28129e6	−0.171040
$709$	−1.32327e7	−0.988628	−0.494314	−	0.869284i	$-0.664581\pi$
$709$	−1.32327e7	−0.988628	−0.494314	+	0.869284i	$0.664581\pi$
$710$	2.27876e7	1.69649
$711$	5.58281e6	0.414170
$712$	7.19579e6	0.531960
$713$	−3.03092e6	−0.223281
$714$	1.97254e7	1.44804
$715$	−1.14478e7	−0.837445
$716$	−1.22299e7	−0.891542
$717$	−550919.	−0.0400212
$718$	4.41923e6	0.319916
$719$	2.01622e7	1.45450	0.727252	−	0.686370i	$-0.240797\pi$
$719$	2.01622e7	1.45450	0.727252	+	0.686370i	$0.240797\pi$
$720$	−2.01238e6	−0.144670
$721$	−4.22892e7	−3.02964
$722$	7.95450e6	0.567897
$723$	−1.28407e7	−0.913570
$724$	−7.10240e6	−0.503569
$725$	−1.01835e7	−0.719535
$726$	1.10336e7	0.776918
$727$	−6.81712e6	−0.478371	−0.239186	−	0.970974i	$-0.576880\pi$
$727$	−6.81712e6	−0.478371	−0.239186	+	0.970974i	$0.576880\pi$
$728$	4.74933e6	0.332127
$729$	2.98637e6	0.208125
$730$	3.63738e6	0.252628
$731$	−4.72859e6	−0.327294
$732$	1.22825e7	0.847243
$733$	1.69084e6	0.116237	0.0581183	−	0.998310i	$-0.481490\pi$
$733$	1.69084e6	0.116237	0.0581183	+	0.998310i	$0.481490\pi$
$734$	4.08078e6	0.279578
$735$	6.10476e7	4.16822
$736$	3.64655e6	0.248135
$737$	−3.65478e7	−2.47852
$738$	1.06520e6	0.0719933
$739$	2.37957e7	1.60283	0.801414	−	0.598110i	$-0.204081\pi$
$739$	2.37957e7	1.60283	0.801414	+	0.598110i	$0.204081\pi$
$740$	−1.27602e7	−0.856597
$741$	3.88790e6	0.260118
$742$	−2.93745e7	−1.95867
$743$	346125.	0.0230017	0.0115009	−	0.999934i	$-0.496339\pi$
$743$	346125.	0.0230017	0.0115009	+	0.999934i	$0.496339\pi$
$744$	1.02697e6	0.0680180
$745$	−1.12391e6	−0.0741892
$746$	1.80090e7	1.18479
$747$	3.14357e6	0.206121
$748$	−9.23482e6	−0.603496
$749$	−2.72408e7	−1.77425
$750$	7.18339e6	0.466312
$751$	−5.97385e6	−0.386505	−0.193252	−	0.981149i	$-0.561904\pi$
$751$	−5.97385e6	−0.386505	−0.193252	+	0.981149i	$0.561904\pi$
$752$	4.01361e6	0.258816
$753$	−2.52398e7	−1.62218
$754$	−6.82634e6	−0.437280
$755$	−1.78817e7	−1.14167
$756$	9.89478e6	0.629654
$757$	−1.78157e7	−1.12996	−0.564979	−	0.825106i	$-0.691116\pi$
$757$	−1.78157e7	−1.12996	−0.564979	+	0.825106i	$0.691116\pi$
$758$	1.67362e7	1.05799
$759$	−3.72212e7	−2.34523
$760$	3.12391e6	0.196185
$761$	6.61514e6	0.414073	0.207037	−	0.978333i	$-0.433618\pi$
$761$	6.61514e6	0.414073	0.207037	+	0.978333i	$0.433618\pi$
$762$	1.94934e7	1.21619
$763$	−4.97239e7	−3.09210
$764$	−3.66902e6	−0.227414
$765$	−8.18379e6	−0.505593
$766$	−1.11350e7	−0.685676
$767$	−2.23373e6	−0.137102
$768$	−1.23556e6	−0.0755894
$769$	−321645.	−0.0196138	−0.00980688	−	0.999952i	$-0.503122\pi$
$769$	−321645.	−0.0196138	−0.00980688	+	0.999952i	$0.503122\pi$
$770$	−3.89515e7	−2.36754
$771$	−1.52020e7	−0.921012
$772$	6.78367e6	0.409658
$773$	3.10097e7	1.86659	0.933295	−	0.359111i	$-0.116920\pi$
$773$	3.10097e7	1.86659	0.933295	+	0.359111i	$0.116920\pi$
$774$	2.04284e6	0.122569
$775$	1.50009e6	0.0897144
$776$	314491.	0.0187480
$777$	−5.40351e7	−3.21087
$778$	−8.41596e6	−0.498488
$779$	−1.65357e6	−0.0976287
$780$	−6.22876e6	−0.366577
$781$	−4.51772e7	−2.65028
$782$	1.48295e7	0.867181
$783$	−1.42220e7	−0.829005
$784$	1.18572e7	0.688957
$785$	2.35418e7	1.36353
$786$	1.00494e7	0.580207
$787$	6.33232e6	0.364440	0.182220	−	0.983258i	$-0.441672\pi$
$787$	6.33232e6	0.364440	0.182220	+	0.983258i	$0.441672\pi$
$788$	4.84990e6	0.278239
$789$	2.20069e7	1.25854
$790$	1.38844e7	0.791516
$791$	6.18630e7	3.51552
$792$	3.98962e6	0.226005
$793$	1.20264e7	0.679131
$794$	1.33361e7	0.750718
$795$	3.85247e7	2.16183
$796$	5.64380e6	0.315710
$797$	−7.53655e6	−0.420268	−0.210134	−	0.977673i	$-0.567390\pi$
$797$	−7.53655e6	−0.420268	−0.210134	+	0.977673i	$0.567390\pi$
$798$	1.32287e7	0.735379
$799$	1.63222e7	0.904508
$800$	−1.80478e6	−0.0997009
$801$	−1.26423e7	−0.696218
$802$	−8.62103e6	−0.473285
$803$	−7.21124e6	−0.394658
$804$	−1.98857e7	−1.08493
$805$	6.25494e7	3.40199
$806$	1.00556e6	0.0545217
$807$	1.36423e6	0.0737403
$808$	8.28630e6	0.446511
$809$	2.35347e7	1.26426	0.632131	−	0.774862i	$-0.282181\pi$
$809$	2.35347e7	1.26426	0.632131	+	0.774862i	$0.282181\pi$
$810$	−2.06178e7	−1.10415
$811$	5.91709e6	0.315905	0.157952	−	0.987447i	$-0.449511\pi$
$811$	5.91709e6	0.315905	0.157952	+	0.987447i	$0.449511\pi$
$812$	−2.32269e7	−1.23623
$813$	2.23373e6	0.118523
$814$	2.52975e7	1.33819
$815$	−8.73028e6	−0.460399
$816$	−5.02468e6	−0.264170
$817$	−3.17120e6	−0.166214
$818$	−1.20543e7	−0.629882
$819$	−8.34412e6	−0.434681
$820$	2.64916e6	0.137586
$821$	−5.82707e6	−0.301712	−0.150856	−	0.988556i	$-0.548203\pi$
$821$	−5.82707e6	−0.301712	−0.150856	+	0.988556i	$0.548203\pi$
$822$	−2.39629e7	−1.23697
$823$	2.32044e7	1.19418	0.597091	−	0.802173i	$-0.296323\pi$
$823$	2.32044e7	1.19418	0.597091	+	0.802173i	$0.296323\pi$
$824$	1.07724e7	0.552705
$825$	1.84218e7	0.942318
$826$	−7.60036e6	−0.387600
$827$	−1.71686e7	−0.872913	−0.436457	−	0.899725i	$-0.643767\pi$
$827$	−1.71686e7	−0.872913	−0.436457	+	0.899725i	$0.643767\pi$
$828$	−6.40664e6	−0.324754
$829$	1.90451e7	0.962490	0.481245	−	0.876586i	$-0.340185\pi$
$829$	1.90451e7	0.962490	0.481245	+	0.876586i	$0.340185\pi$
$830$	7.81803e6	0.393915
$831$	−2.42587e7	−1.21861
$832$	−1.20980e6	−0.0605908
$833$	4.82199e7	2.40776
$834$	−6.98931e6	−0.347952
$835$	2.92439e7	1.45151
$836$	−6.19327e6	−0.306482
$837$	2.09498e6	0.103363
$838$	9.12518e6	0.448881
$839$	2.13728e7	1.04823	0.524115	−	0.851648i	$-0.324396\pi$
$839$	2.13728e7	1.04823	0.524115	+	0.851648i	$0.324396\pi$
$840$	−2.11936e7	−1.03635
$841$	1.28734e7	0.627631
$842$	7.53846e6	0.366440
$843$	−3.03033e6	−0.146866
$844$	−1.87216e7	−0.904663
$845$	1.98584e7	0.956758
$846$	−7.05152e6	−0.338733
$847$	3.67596e7	1.76060
$848$	7.48260e6	0.357324
$849$	4.20515e7	2.00222
$850$	−7.33953e6	−0.348434
$851$	−4.06234e7	−1.92288
$852$	−2.45810e7	−1.16011
$853$	−3.37158e7	−1.58658	−0.793289	−	0.608846i	$-0.791633\pi$
$853$	−3.37158e7	−1.58658	−0.793289	+	0.608846i	$0.791633\pi$
$854$	4.09204e7	1.91997
$855$	−5.48841e6	−0.256762
$856$	6.93907e6	0.323681
$857$	−8.27844e6	−0.385032	−0.192516	−	0.981294i	$-0.561665\pi$
$857$	−8.27844e6	−0.385032	−0.192516	+	0.981294i	$0.561665\pi$
$858$	1.23487e7	0.572670
$859$	−9.17440e6	−0.424224	−0.212112	−	0.977245i	$-0.568034\pi$
$859$	−9.17440e6	−0.424224	−0.212112	+	0.977245i	$0.568034\pi$
$860$	5.08053e6	0.234241
$861$	1.12183e7	0.515726
$862$	−1.00713e7	−0.461654
$863$	−5.97748e6	−0.273207	−0.136603	−	0.990626i	$-0.543619\pi$
$863$	−5.97748e6	−0.273207	−0.136603	+	0.990626i	$0.543619\pi$
$864$	−2.52051e6	−0.114869
$865$	−4.22104e7	−1.91813
$866$	−1.04375e7	−0.472937
$867$	6.33482e6	0.286211
$868$	3.42145e6	0.154138
$869$	−2.75264e7	−1.23652
$870$	3.04621e7	1.36446
$871$	−1.94712e7	−0.869655
$872$	1.26662e7	0.564100
$873$	−552531.	−0.0245370
$874$	9.94532e6	0.440393
$875$	2.39322e7	1.05673
$876$	−3.92365e6	−0.172755
$877$	−1.47797e7	−0.648885	−0.324442	−	0.945905i	$-0.605177\pi$
$877$	−1.47797e7	−0.648885	−0.324442	+	0.945905i	$0.605177\pi$
$878$	384763.	0.0168445
$879$	−7.75863e6	−0.338698
$880$	9.92217e6	0.431916
$881$	−3.67113e6	−0.159353	−0.0796764	−	0.996821i	$-0.525389\pi$
$881$	−3.67113e6	−0.159353	−0.0796764	+	0.996821i	$0.525389\pi$
$882$	−2.08319e7	−0.901692
$883$	2.03832e7	0.879772	0.439886	−	0.898054i	$-0.355019\pi$
$883$	2.03832e7	0.879772	0.439886	+	0.898054i	$0.355019\pi$
$884$	−4.91993e6	−0.211752
$885$	9.96789e6	0.427804
$886$	−2.85805e6	−0.122316
$887$	−3.63778e7	−1.55248	−0.776242	−	0.630435i	$-0.782877\pi$
$887$	−3.63778e7	−1.55248	−0.776242	+	0.630435i	$0.782877\pi$
$888$	1.37644e7	0.585768
$889$	6.49443e7	2.75605
$890$	−3.14413e7	−1.33053
$891$	4.08755e7	1.72492
$892$	4.28008e6	0.180111
$893$	1.09464e7	0.459349
$894$	1.21237e6	0.0507329
$895$	5.34376e7	2.22992
$896$	−4.11640e6	−0.171296
$897$	−1.98299e7	−0.822887
$898$	−1.92334e6	−0.0795913
$899$	−4.91774e6	−0.202939
$900$	3.17082e6	0.130487
$901$	3.04296e7	1.24878
$902$	−5.25205e6	−0.214938
$903$	2.15144e7	0.878031
$904$	−1.57584e7	−0.641345
$905$	3.10333e7	1.25952
$906$	1.92890e7	0.780709
$907$	−1.80791e7	−0.729726	−0.364863	−	0.931061i	$-0.618884\pi$
$907$	−1.80791e7	−0.729726	−0.364863	+	0.931061i	$0.618884\pi$
$908$	−1.27275e7	−0.512305
$909$	−1.45582e7	−0.584385
$910$	−2.07518e7	−0.830714
$911$	−2.82325e7	−1.12708	−0.563538	−	0.826090i	$-0.690560\pi$
$911$	−2.82325e7	−1.12708	−0.563538	+	0.826090i	$0.690560\pi$
$912$	−3.36977e6	−0.134157
$913$	−1.54995e7	−0.615378
$914$	2.33363e7	0.923987
$915$	−5.36671e7	−2.11912
$916$	4.06768e6	0.160180
$917$	3.34806e7	1.31483
$918$	−1.02502e7	−0.401445
$919$	−6.71231e6	−0.262170	−0.131085	−	0.991371i	$-0.541846\pi$
$919$	−6.71231e6	−0.262170	−0.131085	+	0.991371i	$0.541846\pi$
$920$	−1.59333e7	−0.620634
$921$	1.06571e7	0.413989
$922$	1.01853e7	0.394591
$923$	−2.40686e7	−0.929921
$924$	4.20171e7	1.61900
$925$	2.01056e7	0.772615
$926$	−8.84986e6	−0.339163
$927$	−1.89260e7	−0.723370
$928$	5.91661e6	0.225529
$929$	3.88845e7	1.47821	0.739106	−	0.673589i	$-0.235248\pi$
$929$	3.88845e7	1.47821	0.739106	+	0.673589i	$0.235248\pi$
$930$	−4.48724e6	−0.170126
$931$	3.23384e7	1.22277
$932$	−1.05546e7	−0.398018
$933$	4.75758e7	1.78930
$934$	1.54721e7	0.580340
$935$	4.03507e7	1.50946
$936$	2.12551e6	0.0793000
$937$	−1.84069e7	−0.684906	−0.342453	−	0.939535i	$-0.611258\pi$
$937$	−1.84069e7	−0.684906	−0.342453	+	0.939535i	$0.611258\pi$
$938$	−6.62515e7	−2.45860
$939$	3.98860e7	1.47624
$940$	−1.75371e7	−0.647348
$941$	−3.29379e7	−1.21261	−0.606307	−	0.795231i	$-0.707350\pi$
$941$	−3.29379e7	−1.21261	−0.606307	+	0.795231i	$0.707350\pi$
$942$	−2.53946e7	−0.932425
$943$	8.43388e6	0.308851
$944$	1.93605e6	0.0707109
$945$	−4.32343e7	−1.57489
$946$	−1.00724e7	−0.365934
$947$	−2.32660e7	−0.843038	−0.421519	−	0.906820i	$-0.638503\pi$
$947$	−2.32660e7	−0.843038	−0.421519	+	0.906820i	$0.638503\pi$
$948$	−1.49772e7	−0.541263
$949$	−3.84185e6	−0.138476
$950$	−4.92221e6	−0.176950
$951$	1.40223e7	0.502767
$952$	−1.67403e7	−0.598646
$953$	2.06940e7	0.738095	0.369047	−	0.929411i	$-0.379684\pi$
$953$	2.06940e7	0.738095	0.369047	+	0.929411i	$0.379684\pi$
$954$	−1.31462e7	−0.467659
$955$	1.60315e7	0.568807
$956$	467545.	0.0165455
$957$	−6.03923e7	−2.13158
$958$	−3.48573e6	−0.122710
$959$	−7.98348e7	−2.80315
$960$	5.39867e6	0.189064
$961$	−2.79047e7	−0.974697
$962$	1.34775e7	0.469538
$963$	−1.21913e7	−0.423627
$964$	1.08974e7	0.377686
$965$	−2.96406e7	−1.02463
$966$	−6.74722e7	−2.32639
$967$	2.38531e7	0.820311	0.410156	−	0.912016i	$-0.365474\pi$
$967$	2.38531e7	0.820311	0.410156	+	0.912016i	$0.365474\pi$
$968$	−9.36381e6	−0.321192
$969$	−1.37039e7	−0.468852
$970$	−1.37414e6	−0.0468923
$971$	4.65256e7	1.58360	0.791798	−	0.610783i	$-0.209145\pi$
$971$	4.65256e7	1.58360	0.791798	+	0.610783i	$0.209145\pi$
$972$	1.26704e7	0.430155
$973$	−2.32856e7	−0.788507
$974$	3.44275e7	1.16281
$975$	9.81438e6	0.330637
$976$	−1.04237e7	−0.350265
$977$	−3.63182e7	−1.21727	−0.608637	−	0.793449i	$-0.708283\pi$
$977$	−3.63182e7	−1.21727	−0.608637	+	0.793449i	$0.708283\pi$
$978$	9.41737e6	0.314835
$979$	6.23337e7	2.07858
$980$	−5.18089e7	−1.72321
$981$	−2.22533e7	−0.738282
$982$	2.25934e7	0.747658
$983$	3.93404e6	0.129854	0.0649270	−	0.997890i	$-0.479319\pi$
$983$	3.93404e6	0.129854	0.0649270	+	0.997890i	$0.479319\pi$
$984$	−2.85765e6	−0.0940853
$985$	−2.11912e7	−0.695929
$986$	2.40612e7	0.788179
$987$	−7.42638e7	−2.42652
$988$	−3.29952e6	−0.107537
$989$	1.61744e7	0.525822
$990$	−1.74323e7	−0.565284
$991$	1.59324e7	0.515344	0.257672	−	0.966232i	$-0.417045\pi$
$991$	1.59324e7	0.515344	0.257672	+	0.966232i	$0.417045\pi$
$992$	−871550.	−0.0281198
$993$	−4.32356e7	−1.39145
$994$	−8.18943e7	−2.62898
$995$	−2.46600e7	−0.789653
$996$	−8.43333e6	−0.269371
$997$	2.15385e7	0.686244	0.343122	−	0.939291i	$-0.388516\pi$
$997$	2.15385e7	0.686244	0.343122	+	0.939291i	$0.388516\pi$
$998$	−19636.7	−0.000624083 0
$999$	2.80790e7	0.890160

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.6.a.c.1.7	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.6.a.c.1.7	✓	30	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)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} -18.8532 q^{3} +16.0000 q^{4} -69.9105 q^{5} +75.4127 q^{6} +251.245 q^{7} -64.0000 q^{8} +112.442 q^{9} +279.642 q^{10} -554.401 q^{11} -301.651 q^{12} -295.362 q^{13} -1004.98 q^{14} +1318.03 q^{15} +256.000 q^{16} +1041.08 q^{17} -449.767 q^{18} +698.194 q^{19} -1118.57 q^{20} -4736.77 q^{21} +2217.60 q^{22} -3561.09 q^{23} +1206.60 q^{24} +1762.48 q^{25} +1181.45 q^{26} +2461.43 q^{27} +4019.92 q^{28} -5777.94 q^{29} -5272.14 q^{30} +851.123 q^{31} -1024.00 q^{32} +10452.2 q^{33} -4164.32 q^{34} -17564.7 q^{35} +1799.07 q^{36} +11407.6 q^{37} -2792.78 q^{38} +5568.51 q^{39} +4474.27 q^{40} -2368.35 q^{41} +18947.1 q^{42} -4542.00 q^{43} -8870.42 q^{44} -7860.87 q^{45} +14244.3 q^{46} +15678.2 q^{47} -4826.41 q^{48} +46317.2 q^{49} -7049.92 q^{50} -19627.7 q^{51} -4725.79 q^{52} +29228.9 q^{53} -9845.74 q^{54} +38758.5 q^{55} -16079.7 q^{56} -13163.2 q^{57} +23111.8 q^{58} +7562.69 q^{59} +21088.6 q^{60} -40717.5 q^{61} -3404.49 q^{62} +28250.5 q^{63} +4096.00 q^{64} +20648.9 q^{65} -41808.9 q^{66} +65923.1 q^{67} +16657.3 q^{68} +67137.7 q^{69} +70258.7 q^{70} +81488.4 q^{71} -7196.28 q^{72} +13007.3 q^{73} -45630.3 q^{74} -33228.3 q^{75} +11171.1 q^{76} -139291. q^{77} -22274.0 q^{78} +49650.7 q^{79} -17897.1 q^{80} -73729.2 q^{81} +9473.38 q^{82} +27957.3 q^{83} -75788.3 q^{84} -72782.5 q^{85} +18168.0 q^{86} +108932. q^{87} +35481.7 q^{88} -112434. q^{89} +31443.5 q^{90} -74208.3 q^{91} -56977.4 q^{92} -16046.4 q^{93} -62712.6 q^{94} -48811.1 q^{95} +19305.6 q^{96} -4913.93 q^{97} -185269. q^{98} -62337.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 120 q^{2} - 30 q^{3} + 480 q^{4} - 136 q^{5} + 120 q^{6} - 123 q^{7} - 1920 q^{8} + 2670 q^{9} + 544 q^{10} - 1058 q^{11} - 480 q^{12} - 371 q^{13} + 492 q^{14} - 1364 q^{15} + 7680 q^{16} - 1918 q^{17}+ \cdots - 78063 q^{99}+O(q^{100}) \) 30 * q - 120 * q^2 - 30 * q^3 + 480 * q^4 - 136 * q^5 + 120 * q^6 - 123 * q^7 - 1920 * q^8 + 2670 * q^9 + 544 * q^10 - 1058 * q^11 - 480 * q^12 - 371 * q^13 + 492 * q^14 - 1364 * q^15 + 7680 * q^16 - 1918 * q^17 - 10680 * q^18 - 3315 * q^19 - 2176 * q^20 - 1217 * q^21 + 4232 * q^22 - 9534 * q^23 + 1920 * q^24 + 22130 * q^25 + 1484 * q^26 - 12156 * q^27 - 1968 * q^28 - 8009 * q^29 + 5456 * q^30 - 3543 * q^31 - 30720 * q^32 - 3411 * q^33 + 7672 * q^34 - 11177 * q^35 + 42720 * q^36 - 6517 * q^37 + 13260 * q^38 - 2453 * q^39 + 8704 * q^40 - 5951 * q^41 + 4868 * q^42 - 18768 * q^43 - 16928 * q^44 - 22619 * q^45 + 38136 * q^46 - 41067 * q^47 - 7680 * q^48 + 107797 * q^49 - 88520 * q^50 - 4187 * q^51 - 5936 * q^52 - 64055 * q^53 + 48624 * q^54 + 31973 * q^55 + 7872 * q^56 - 25442 * q^57 + 32036 * q^58 - 66832 * q^59 - 21824 * q^60 + 46777 * q^61 + 14172 * q^62 - 115994 * q^63 + 122880 * q^64 - 68783 * q^65 + 13644 * q^66 - 107218 * q^67 - 30688 * q^68 + 228589 * q^69 + 44708 * q^70 + 4733 * q^71 - 170880 * q^72 + 150385 * q^73 + 26068 * q^74 + 133325 * q^75 - 53040 * q^76 + 292020 * q^77 + 9812 * q^78 + 31827 * q^79 - 34816 * q^80 + 425214 * q^81 + 23804 * q^82 + 107301 * q^83 - 19472 * q^84 + 338625 * q^85 + 75072 * q^86 + 288484 * q^87 + 67712 * q^88 - 99690 * q^89 + 90476 * q^90 - 109140 * q^91 - 152544 * q^92 - 114545 * q^93 + 164268 * q^94 - 155668 * q^95 + 30720 * q^96 - 79959 * q^97 - 431188 * q^98 - 78063 * q^99

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