LMFDB - Embedded newform 169.6.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,6,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, 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("169.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	169.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:	$27.1048655484$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.56155 q^{2} -1.63068 q^{3} -11.1922 q^{4} +103.462 q^{5} -7.43845 q^{6} -126.309 q^{7} -197.024 q^{8} -240.341 q^{9} +471.948 q^{10} +14.8296 q^{11} +18.2510 q^{12} -576.164 q^{14} -168.714 q^{15} -540.582 q^{16} -1051.12 q^{17} -1096.33 q^{18} +213.723 q^{19} -1157.97 q^{20} +205.969 q^{21} +67.6458 q^{22} -4231.16 q^{23} +321.283 q^{24} +7579.41 q^{25} +788.176 q^{27} +1413.68 q^{28} -504.955 q^{29} -769.597 q^{30} -4783.58 q^{31} +3838.86 q^{32} -24.1823 q^{33} -4794.75 q^{34} -13068.2 q^{35} +2689.95 q^{36} +4635.74 q^{37} +974.911 q^{38} -20384.5 q^{40} -7944.15 q^{41} +939.541 q^{42} -8516.41 q^{43} -165.976 q^{44} -24866.2 q^{45} -19300.7 q^{46} -24921.2 q^{47} +881.518 q^{48} -853.113 q^{49} +34573.9 q^{50} +1714.05 q^{51} -7808.46 q^{53} +3595.31 q^{54} +1534.30 q^{55} +24885.8 q^{56} -348.515 q^{57} -2303.38 q^{58} +37337.5 q^{59} +1888.29 q^{60} -18172.2 q^{61} -21820.5 q^{62} +30357.1 q^{63} +34809.8 q^{64} -110.309 q^{66} +34559.9 q^{67} +11764.4 q^{68} +6899.68 q^{69} -59611.1 q^{70} -41255.7 q^{71} +47352.8 q^{72} +1056.42 q^{73} +21146.2 q^{74} -12359.6 q^{75} -2392.04 q^{76} -1873.10 q^{77} -47719.3 q^{79} -55929.8 q^{80} +57117.6 q^{81} -36237.7 q^{82} +74799.0 q^{83} -2305.26 q^{84} -108751. q^{85} -38848.0 q^{86} +823.421 q^{87} -2921.78 q^{88} -9799.26 q^{89} -113428. q^{90} +47356.1 q^{92} +7800.50 q^{93} -113679. q^{94} +22112.3 q^{95} -6259.97 q^{96} +138432. q^{97} -3891.52 q^{98} -3564.15 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{2} - 28 q^{3} - 43 q^{4} + 42 q^{5} - 19 q^{6} + 36 q^{7} - 225 q^{8} + 212 q^{9} + 445 q^{10} + 376 q^{11} + 857 q^{12} - 505 q^{14} + 1452 q^{15} + 465 q^{16} - 2630 q^{17} - 898 q^{18} + 312 q^{19}+ \cdots + 159808 q^{99}+O(q^{100}) $ 2 * q + 5 * q^2 - 28 * q^3 - 43 * q^4 + 42 * q^5 - 19 * q^6 + 36 * q^7 - 225 * q^8 + 212 * q^9 + 445 * q^10 + 376 * q^11 + 857 * q^12 - 505 * q^14 + 1452 * q^15 + 465 * q^16 - 2630 * q^17 - 898 * q^18 + 312 * q^19 + 797 * q^20 - 4074 * q^21 + 226 * q^22 - 2624 * q^23 + 1059 * q^24 + 8232 * q^25 - 4732 * q^27 - 3749 * q^28 - 812 * q^29 - 59 * q^30 - 7720 * q^31 + 5175 * q^32 - 9548 * q^33 - 5487 * q^34 - 23044 * q^35 - 11698 * q^36 + 16858 * q^37 + 1018 * q^38 - 18665 * q^40 - 7840 * q^41 - 937 * q^42 + 2420 * q^43 - 11654 * q^44 - 52668 * q^45 - 18596 * q^46 - 9972 * q^47 - 25635 * q^48 + 8684 * q^49 + 34860 * q^50 + 43348 * q^51 - 43720 * q^53 + 1175 * q^54 - 20664 * q^55 + 20345 * q^56 - 2940 * q^57 - 2438 * q^58 + 38936 * q^59 - 49663 * q^60 + 1984 * q^61 - 23108 * q^62 + 103776 * q^63 + 3217 * q^64 - 4286 * q^66 + 69928 * q^67 + 61985 * q^68 - 35480 * q^69 - 63985 * q^70 - 67396 * q^71 + 34698 * q^72 - 74412 * q^73 + 26505 * q^74 - 29568 * q^75 - 5518 * q^76 + 56748 * q^77 - 55296 * q^79 - 117735 * q^80 + 92762 * q^81 - 36192 * q^82 + 75712 * q^83 + 133831 * q^84 - 11710 * q^85 - 34053 * q^86 + 8920 * q^87 - 13026 * q^88 - 116508 * q^89 - 125618 * q^90 - 3764 * q^92 + 85232 * q^93 - 107125 * q^94 + 16072 * q^95 - 41493 * q^96 + 34756 * q^97 + 290 * q^98 + 159808 * 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 $

$2$	4.56155	0.806376	0.403188	−	0.915117i	$-0.367902\pi$
$2$	4.56155	0.806376	0.403188	+	0.915117i	$0.367902\pi$
$3$	−1.63068	−0.104608	−0.0523042	−	0.998631i	$-0.516657\pi$
$3$	−1.63068	−0.104608	−0.0523042	+	0.998631i	$0.516657\pi$
$4$	−11.1922	−0.349757
$5$	103.462	1.85079	0.925393	−	0.379008i	$-0.123735\pi$
$5$	103.462	1.85079	0.925393	+	0.379008i	$0.123735\pi$
$6$	−7.43845	−0.0843537
$7$	−126.309	−0.974290	−0.487145	−	0.873321i	$-0.661962\pi$
$7$	−126.309	−0.974290	−0.487145	+	0.873321i	$0.661962\pi$
$8$	−197.024	−1.08841
$9$	−240.341	−0.989057
$10$	471.948	1.49243
$11$	14.8296	0.0369527	0.0184764	−	0.999829i	$-0.494118\pi$
$11$	14.8296	0.0369527	0.0184764	+	0.999829i	$0.494118\pi$
$12$	18.2510	0.0365875
$13$	0	0
$13$	0	0
$14$	−576.164	−0.785644
$15$	−168.714	−0.193608
$16$	−540.582	−0.527912
$17$	−1051.12	−0.882126	−0.441063	−	0.897476i	$-0.645398\pi$
$17$	−1051.12	−0.882126	−0.441063	+	0.897476i	$0.645398\pi$
$18$	−1096.33	−0.797552
$19$	213.723	0.135821	0.0679107	−	0.997691i	$-0.478367\pi$
$19$	213.723	0.135821	0.0679107	+	0.997691i	$0.478367\pi$
$20$	−1157.97	−0.647326
$21$	205.969	0.101919
$22$	67.6458	0.0297978
$23$	−4231.16	−1.66778	−0.833892	−	0.551928i	$-0.813892\pi$
$23$	−4231.16	−1.66778	−0.833892	+	0.551928i	$0.813892\pi$
$24$	321.283	0.113857
$25$	7579.41	2.42541
$26$	0	0
$27$	788.176	0.208072
$28$	1413.68	0.340765
$29$	−504.955	−0.111495	−0.0557477	−	0.998445i	$-0.517754\pi$
$29$	−504.955	−0.111495	−0.0557477	+	0.998445i	$0.517754\pi$
$30$	−769.597	−0.156121
$31$	−4783.58	−0.894022	−0.447011	−	0.894528i	$-0.647512\pi$
$31$	−4783.58	−0.894022	−0.447011	+	0.894528i	$0.647512\pi$
$32$	3838.86	0.662716
$33$	−24.1823	−0.00386557
$34$	−4794.75	−0.711325
$35$	−13068.2	−1.80320
$36$	2689.95	0.345930
$37$	4635.74	0.556692	0.278346	−	0.960481i	$-0.410214\pi$
$37$	4635.74	0.556692	0.278346	+	0.960481i	$0.410214\pi$
$38$	974.911	0.109523
$39$	0	0
$40$	−20384.5	−2.01442
$41$	−7944.15	−0.738054	−0.369027	−	0.929419i	$-0.620309\pi$
$41$	−7944.15	−0.738054	−0.369027	+	0.929419i	$0.620309\pi$
$42$	939.541	0.0821850
$43$	−8516.41	−0.702401	−0.351201	−	0.936300i	$-0.614226\pi$
$43$	−8516.41	−0.702401	−0.351201	+	0.936300i	$0.614226\pi$
$44$	−165.976	−0.0129245
$45$	−24866.2	−1.83053
$46$	−19300.7	−1.34486
$47$	−24921.2	−1.64560	−0.822801	−	0.568330i	$-0.807590\pi$
$47$	−24921.2	−1.64560	−0.822801	+	0.568330i	$0.807590\pi$
$48$	881.518	0.0552241
$49$	−853.113	−0.0507594
$50$	34573.9	1.95579
$51$	1714.05	0.0922777
$52$	0	0
$53$	−7808.46	−0.381835	−0.190917	−	0.981606i	$-0.561146\pi$
$53$	−7808.46	−0.381835	−0.190917	+	0.981606i	$0.561146\pi$
$54$	3595.31	0.167784
$55$	1534.30	0.0683916
$56$	24885.8	1.06043
$57$	−348.515	−0.0142081
$58$	−2303.38	−0.0899073
$59$	37337.5	1.39642	0.698209	−	0.715894i	$-0.253980\pi$
$59$	37337.5	1.39642	0.698209	+	0.715894i	$0.253980\pi$
$60$	1888.29	0.0677157
$61$	−18172.2	−0.625292	−0.312646	−	0.949870i	$-0.601215\pi$
$61$	−18172.2	−0.625292	−0.312646	+	0.949870i	$0.601215\pi$
$62$	−21820.5	−0.720918
$63$	30357.1	0.963628
$64$	34809.8	1.06231
$65$	0	0
$66$	−110.309	−0.00311710
$67$	34559.9	0.940559	0.470279	−	0.882518i	$-0.344153\pi$
$67$	34559.9	0.940559	0.470279	+	0.882518i	$0.344153\pi$
$68$	11764.4	0.308530
$69$	6899.68	0.174464
$70$	−59611.1	−1.45406
$71$	−41255.7	−0.971265	−0.485632	−	0.874163i	$-0.661411\pi$
$71$	−41255.7	−0.971265	−0.485632	+	0.874163i	$0.661411\pi$
$72$	47352.8	1.07650
$73$	1056.42	0.0232022	0.0116011	−	0.999933i	$-0.496307\pi$
$73$	1056.42	0.0232022	0.0116011	+	0.999933i	$0.496307\pi$
$74$	21146.2	0.448903
$75$	−12359.6	−0.253718
$76$	−2392.04	−0.0475045
$77$	−1873.10	−0.0360027
$78$	0	0
$79$	−47719.3	−0.860253	−0.430126	−	0.902769i	$-0.641531\pi$
$79$	−47719.3	−0.860253	−0.430126	+	0.902769i	$0.641531\pi$
$80$	−55929.8	−0.977053
$81$	57117.6	0.967291
$82$	−36237.7	−0.595149
$83$	74799.0	1.19179	0.595896	−	0.803061i	$-0.296797\pi$
$83$	74799.0	1.19179	0.595896	+	0.803061i	$0.296797\pi$
$84$	−2305.26	−0.0356469
$85$	−108751.	−1.63263
$86$	−38848.0	−0.566400
$87$	823.421	0.0116634
$88$	−2921.78	−0.0402198
$89$	−9799.26	−0.131135	−0.0655675	−	0.997848i	$-0.520886\pi$
$89$	−9799.26	−0.131135	−0.0655675	+	0.997848i	$0.520886\pi$
$90$	−113428.	−1.47610
$91$	0	0
$92$	47356.1	0.583320
$93$	7800.50	0.0935222
$94$	−113679.	−1.32697
$95$	22112.3	0.251376
$96$	−6259.97	−0.0693257
$97$	138432.	1.49385	0.746927	−	0.664906i	$-0.231528\pi$
$97$	138432.	1.49385	0.746927	+	0.664906i	$0.231528\pi$
$98$	−3891.52	−0.0409312
$99$	−3564.15	−0.0365484
$100$	−84830.5	−0.848305
$101$	139151.	1.35733	0.678663	−	0.734450i	$-0.262560\pi$
$101$	139151.	1.35733	0.678663	+	0.734450i	$0.262560\pi$
$102$	7818.71	0.0744106
$103$	98512.2	0.914950	0.457475	−	0.889223i	$-0.348754\pi$
$103$	98512.2	0.914950	0.457475	+	0.889223i	$0.348754\pi$
$104$	0	0
$105$	21310.0	0.188630
$106$	−35618.7	−0.307902
$107$	−26848.4	−0.226704	−0.113352	−	0.993555i	$-0.536159\pi$
$107$	−26848.4	−0.226704	−0.113352	+	0.993555i	$0.536159\pi$
$108$	−8821.45	−0.0727747
$109$	63220.9	0.509676	0.254838	−	0.966984i	$-0.417978\pi$
$109$	63220.9	0.509676	0.254838	+	0.966984i	$0.417978\pi$
$110$	6998.78	0.0551494
$111$	−7559.43	−0.0582346
$112$	68280.2	0.514340
$113$	114434.	0.843058	0.421529	−	0.906815i	$-0.361494\pi$
$113$	114434.	0.843058	0.421529	+	0.906815i	$0.361494\pi$
$114$	−1589.77	−0.0114570
$115$	−437765.	−3.08671
$116$	5651.57	0.0389964
$117$	0	0
$118$	170317.	1.12604
$119$	132766.	0.859446
$120$	33240.6	0.210725
$121$	−160831.	−0.998634
$122$	−82893.4	−0.504221
$123$	12954.4	0.0772066
$124$	53538.9	0.312691
$125$	460863.	2.63813
$126$	138476.	0.777047
$127$	−248871.	−1.36919	−0.684596	−	0.728922i	$-0.740022\pi$
$127$	−248871.	−1.36919	−0.684596	+	0.728922i	$0.740022\pi$
$128$	35943.2	0.193906
$129$	13887.6	0.0734770
$130$	0	0
$131$	102963.	0.524205	0.262102	−	0.965040i	$-0.415584\pi$
$131$	102963.	0.524205	0.262102	+	0.965040i	$0.415584\pi$
$132$	270.654	0.00135201
$133$	−26995.1	−0.132329
$134$	157647.	0.758444
$135$	81546.3	0.385097
$136$	207096.	0.960117
$137$	36037.4	0.164041	0.0820204	−	0.996631i	$-0.473863\pi$
$137$	36037.4	0.164041	0.0820204	+	0.996631i	$0.473863\pi$
$138$	31473.3	0.140684
$139$	152655.	0.670151	0.335076	−	0.942191i	$-0.391238\pi$
$139$	152655.	0.670151	0.335076	+	0.942191i	$0.391238\pi$
$140$	146262.	0.630683
$141$	40638.6	0.172144
$142$	−188190.	−0.783205
$143$	0	0
$144$	129924.	0.522136
$145$	−52243.7	−0.206354
$146$	4818.92	0.0187097
$147$	1391.16	0.00530986
$148$	−51884.3	−0.194707
$149$	−72547.1	−0.267704	−0.133852	−	0.991001i	$-0.542735\pi$
$149$	−72547.1	−0.267704	−0.133852	+	0.991001i	$0.542735\pi$
$150$	−56379.0	−0.204592
$151$	−489021.	−1.74536	−0.872681	−	0.488291i	$-0.837621\pi$
$151$	−489021.	−1.74536	−0.872681	+	0.488291i	$0.837621\pi$
$152$	−42108.6	−0.147830
$153$	252627.	0.872473
$154$	−8544.26	−0.0290317
$155$	−494919.	−1.65464
$156$	0	0
$157$	89467.9	0.289680	0.144840	−	0.989455i	$-0.453733\pi$
$157$	89467.9	0.289680	0.144840	+	0.989455i	$0.453733\pi$
$158$	−217674.	−0.693687
$159$	12733.1	0.0399431
$160$	397177.	1.22655
$161$	534432.	1.62490
$162$	260545.	0.780000
$163$	225668.	0.665275	0.332637	−	0.943055i	$-0.392061\pi$
$163$	225668.	0.665275	0.332637	+	0.943055i	$0.392061\pi$
$164$	88912.8	0.258140
$165$	−2501.95	−0.00715434
$166$	341200.	0.961033
$167$	−209528.	−0.581367	−0.290683	−	0.956819i	$-0.593883\pi$
$167$	−209528.	−0.581367	−0.290683	+	0.956819i	$0.593883\pi$
$168$	−40580.9	−0.110930
$169$	0	0
$170$	−496074.	−1.31651
$171$	−51366.5	−0.134335
$172$	95317.6	0.245670
$173$	−465184.	−1.18171	−0.590853	−	0.806779i	$-0.701209\pi$
$173$	−465184.	−1.18171	−0.590853	+	0.806779i	$0.701209\pi$
$174$	3756.08	0.00940506
$175$	−957345.	−2.36305
$176$	−8016.60	−0.0195078
$177$	−60885.7	−0.146077
$178$	−44699.9	−0.105744
$179$	−472573.	−1.10239	−0.551197	−	0.834375i	$-0.685829\pi$
$179$	−472573.	−1.10239	−0.551197	+	0.834375i	$0.685829\pi$
$180$	278308.	0.640243
$181$	−74099.4	−0.168120	−0.0840598	−	0.996461i	$-0.526789\pi$
$181$	−74099.4	−0.168120	−0.0840598	+	0.996461i	$0.526789\pi$
$182$	0	0
$183$	29633.1	0.0654108
$184$	833638.	1.81524
$185$	479624.	1.03032
$186$	35582.4	0.0754141
$187$	−15587.7	−0.0325970
$188$	278924.	0.575561
$189$	−99553.5	−0.202722
$190$	100866.	0.202704
$191$	−224128.	−0.444543	−0.222271	−	0.974985i	$-0.571347\pi$
$191$	−224128.	−0.444543	−0.222271	+	0.974985i	$0.571347\pi$
$192$	−56763.8	−0.111127
$193$	662265.	1.27979	0.639895	−	0.768463i	$-0.278978\pi$
$193$	662265.	1.27979	0.639895	+	0.768463i	$0.278978\pi$
$194$	631467.	1.20461
$195$	0	0
$196$	9548.24	0.0177535
$197$	−666464.	−1.22352	−0.611760	−	0.791043i	$-0.709538\pi$
$197$	−666464.	−1.22352	−0.611760	+	0.791043i	$0.709538\pi$
$198$	−16258.1	−0.0294717
$199$	645720.	1.15588	0.577938	−	0.816081i	$-0.303858\pi$
$199$	645720.	1.15588	0.577938	+	0.816081i	$0.303858\pi$
$200$	−1.49332e6	−2.63985
$201$	−56356.3	−0.0983903
$202$	634746.	1.09451
$203$	63780.1	0.108629
$204$	−19184.0	−0.0322748
$205$	−821919.	−1.36598
$206$	449369.	0.737794
$207$	1.01692e6	1.64953
$208$	0	0
$209$	3169.43	0.00501897
$210$	97206.9	0.152107
$211$	−868021.	−1.34222	−0.671110	−	0.741357i	$-0.734182\pi$
$211$	−868021.	−1.34222	−0.671110	+	0.741357i	$0.734182\pi$
$212$	87394.1	0.133550
$213$	67274.9	0.101602
$214$	−122470.	−0.182808
$215$	−881125.	−1.29999
$216$	−155289.	−0.226468
$217$	604207.	0.871037
$218$	288385.	0.410991
$219$	−1722.69	−0.00242715
$220$	−17172.2	−0.0239205
$221$	0	0
$222$	−34482.7	−0.0469590
$223$	58342.9	0.0785644	0.0392822	−	0.999228i	$-0.487493\pi$
$223$	58342.9	0.0785644	0.0392822	+	0.999228i	$0.487493\pi$
$224$	−484882.	−0.645678
$225$	−1.82164e6	−2.39887
$226$	521995.	0.679822
$227$	−111768.	−0.143964	−0.0719820	−	0.997406i	$-0.522932\pi$
$227$	−111768.	−0.143964	−0.0719820	+	0.997406i	$0.522932\pi$
$228$	3900.67	0.00496937
$229$	−1.14984e6	−1.44893	−0.724467	−	0.689309i	$-0.757914\pi$
$229$	−1.14984e6	−1.44893	−0.724467	+	0.689309i	$0.757914\pi$
$230$	−1.99689e6	−2.48905
$231$	3054.44	0.00376618
$232$	99488.0	0.121353
$233$	630470.	0.760807	0.380404	−	0.924821i	$-0.375785\pi$
$233$	630470.	0.760807	0.380404	+	0.924821i	$0.375785\pi$
$234$	0	0
$235$	−2.57840e6	−3.04566
$236$	−417891.	−0.488408
$237$	77815.0	0.0899897
$238$	605618.	0.693037
$239$	165628.	0.187559	0.0937797	−	0.995593i	$-0.470105\pi$
$239$	165628.	0.187559	0.0937797	+	0.995593i	$0.470105\pi$
$240$	91203.8	0.102208
$241$	−690968.	−0.766329	−0.383165	−	0.923680i	$-0.625166\pi$
$241$	−690968.	−0.766329	−0.383165	+	0.923680i	$0.625166\pi$
$242$	−733639.	−0.805275
$243$	−284667.	−0.309259
$244$	203387.	0.218700
$245$	−88264.9	−0.0939448
$246$	59092.1	0.0622575
$247$	0	0
$248$	942478.	0.973065
$249$	−121974.	−0.124671
$250$	2.10225e6	2.12733
$251$	−386887.	−0.387615	−0.193807	−	0.981040i	$-0.562084\pi$
$251$	−386887.	−0.387615	−0.193807	+	0.981040i	$0.562084\pi$
$252$	−339764.	−0.337036
$253$	−62746.2	−0.0616292
$254$	−1.13524e6	−1.10408
$255$	177339.	0.170786
$256$	−949957.	−0.905950
$257$	258260.	0.243907	0.121953	−	0.992536i	$-0.461084\pi$
$257$	258260.	0.243907	0.121953	+	0.992536i	$0.461084\pi$
$258$	63348.8	0.0592501
$259$	−585535.	−0.542379
$260$	0	0
$261$	121361.	0.110275
$262$	469669.	0.422706
$263$	−1.19053e6	−1.06133	−0.530665	−	0.847582i	$-0.678058\pi$
$263$	−1.19053e6	−1.06133	−0.530665	+	0.847582i	$0.678058\pi$
$264$	4764.49	0.00420733
$265$	−807879.	−0.706695
$266$	−123140.	−0.106707
$267$	15979.5	0.0137178
$268$	−386803.	−0.328967
$269$	850968.	0.717022	0.358511	−	0.933525i	$-0.383285\pi$
$269$	850968.	0.717022	0.358511	+	0.933525i	$0.383285\pi$
$270$	371978.	0.310533
$271$	40926.1	0.0338514	0.0169257	−	0.999857i	$-0.494612\pi$
$271$	40926.1	0.0338514	0.0169257	+	0.999857i	$0.494612\pi$
$272$	568218.	0.465685
$273$	0	0
$274$	164387.	0.132279
$275$	112399.	0.0896256
$276$	−77222.8	−0.0610201
$277$	1.00054e6	0.783490	0.391745	−	0.920074i	$-0.371872\pi$
$277$	1.00054e6	0.783490	0.391745	+	0.920074i	$0.371872\pi$
$278$	696342.	0.540394
$279$	1.14969e6	0.884239
$280$	2.57474e6	1.96263
$281$	1.73596e6	1.31151	0.655757	−	0.754972i	$-0.272350\pi$
$281$	1.73596e6	1.31151	0.655757	+	0.754972i	$0.272350\pi$
$282$	185375.	0.138813
$283$	−1.27363e6	−0.945319	−0.472660	−	0.881245i	$-0.656706\pi$
$283$	−1.27363e6	−0.945319	−0.472660	+	0.881245i	$0.656706\pi$
$284$	461743.	0.339707
$285$	−36058.1	−0.0262961
$286$	0	0
$287$	1.00342e6	0.719078
$288$	−922636.	−0.655464
$289$	−315001.	−0.221854
$290$	−238312.	−0.166399
$291$	−225739.	−0.156270
$292$	−11823.7	−0.00811515
$293$	2043.70	0.00139075	0.000695374	−	1.00000i	$-0.499779\pi$
$293$	2043.70	0.00139075	0.000695374		1.00000i	$0.499779\pi$
$294$	6345.84	0.00428174
$295$	3.86302e6	2.58447
$296$	−913351.	−0.605910
$297$	11688.3	0.00768883
$298$	−330927.	−0.215870
$299$	0	0
$300$	138332.	0.0887398
$301$	1.07570e6	0.684342
$302$	−2.23070e6	−1.40742
$303$	−226912.	−0.141988
$304$	−115535.	−0.0717018
$305$	−1.88013e6	−1.15728
$306$	1.15237e6	0.703541
$307$	401308.	0.243014	0.121507	−	0.992591i	$-0.461227\pi$
$307$	401308.	0.243014	0.121507	+	0.992591i	$0.461227\pi$
$308$	20964.2	0.0125922
$309$	−160642.	−0.0957114
$310$	−2.25760e6	−1.33427
$311$	−1.92628e6	−1.12933	−0.564663	−	0.825322i	$-0.690994\pi$
$311$	−1.92628e6	−1.12933	−0.564663	+	0.825322i	$0.690994\pi$
$312$	0	0
$313$	1.64519e6	0.949196	0.474598	−	0.880203i	$-0.342593\pi$
$313$	1.64519e6	0.949196	0.474598	+	0.880203i	$0.342593\pi$
$314$	408113.	0.233591
$315$	3.14081e6	1.78347
$316$	534085.	0.300880
$317$	1.99476e6	1.11492	0.557459	−	0.830205i	$-0.311777\pi$
$317$	1.99476e6	1.11492	0.557459	+	0.830205i	$0.311777\pi$
$318$	58082.8	0.0322092
$319$	−7488.26	−0.00412006
$320$	3.60150e6	1.96611
$321$	43781.2	0.0237151
$322$	2.43784e6	1.31028
$323$	−224649.	−0.119812
$324$	−639273.	−0.338317
$325$	0	0
$326$	1.02940e6	0.536462
$327$	−103093.	−0.0533164
$328$	1.56519e6	0.803306
$329$	3.14777e6	1.60329
$330$	−11412.8	−0.00576909
$331$	675924.	0.339100	0.169550	−	0.985522i	$-0.445769\pi$
$331$	675924.	0.339100	0.169550	+	0.985522i	$0.445769\pi$
$332$	−837168.	−0.416838
$333$	−1.11416e6	−0.550600
$334$	−955772.	−0.468800
$335$	3.57564e6	1.74077
$336$	−111343.	−0.0538042
$337$	−2.13552e6	−1.02430	−0.512152	−	0.858895i	$-0.671152\pi$
$337$	−2.13552e6	−1.02430	−0.512152	+	0.858895i	$0.671152\pi$
$338$	0	0
$339$	−186605.	−0.0881909
$340$	1.21717e6	0.571023
$341$	−70938.3	−0.0330366
$342$	−234311.	−0.108325
$343$	2.23063e6	1.02374
$344$	1.67793e6	0.764502
$345$	713855.	0.322896
$346$	−2.12196e6	−0.952899
$347$	−2.57257e6	−1.14695	−0.573473	−	0.819225i	$-0.694404\pi$
$347$	−2.57257e6	−1.14695	−0.573473	+	0.819225i	$0.694404\pi$
$348$	−9215.92	−0.00407935
$349$	−2.02363e6	−0.889339	−0.444670	−	0.895695i	$-0.646679\pi$
$349$	−2.02363e6	−0.889339	−0.444670	+	0.895695i	$0.646679\pi$
$350$	−4.36698e6	−1.90551
$351$	0	0
$352$	56928.7	0.0244892
$353$	−2.04810e6	−0.874810	−0.437405	−	0.899265i	$-0.644102\pi$
$353$	−2.04810e6	−0.874810	−0.437405	+	0.899265i	$0.644102\pi$
$354$	−277733.	−0.117793
$355$	−4.26840e6	−1.79760
$356$	109676.	0.0458654
$357$	−216499.	−0.0899053
$358$	−2.15567e6	−0.888944
$359$	1.59901e6	0.654808	0.327404	−	0.944885i	$-0.393826\pi$
$359$	1.59901e6	0.654808	0.327404	+	0.944885i	$0.393826\pi$
$360$	4.89922e6	1.99238
$361$	−2.43042e6	−0.981553
$362$	−338008.	−0.135568
$363$	262265.	0.104466
$364$	0	0
$365$	109299.	0.0429424
$366$	135173.	0.0527457
$367$	−3.86389e6	−1.49747	−0.748737	−	0.662867i	$-0.769339\pi$
$367$	−3.86389e6	−1.49747	−0.748737	+	0.662867i	$0.769339\pi$
$368$	2.28729e6	0.880444
$369$	1.90930e6	0.729977
$370$	2.18783e6	0.830824
$371$	986276.	0.372018
$372$	−87305.0	−0.0327101
$373$	−1.56702e6	−0.583179	−0.291589	−	0.956544i	$-0.594184\pi$
$373$	−1.56702e6	−0.583179	−0.291589	+	0.956544i	$0.594184\pi$
$374$	−71104.0	−0.0262854
$375$	−751521.	−0.275971
$376$	4.91007e6	1.79109
$377$	0	0
$378$	−454118.	−0.163471
$379$	−3.19239e6	−1.14161	−0.570805	−	0.821086i	$-0.693369\pi$
$379$	−3.19239e6	−1.14161	−0.570805	+	0.821086i	$0.693369\pi$
$380$	−247486.	−0.0879208
$381$	405829.	0.143229
$382$	−1.02237e6	−0.358469
$383$	−400432.	−0.139486	−0.0697432	−	0.997565i	$-0.522218\pi$
$383$	−400432.	−0.139486	−0.0697432	+	0.997565i	$0.522218\pi$
$384$	−58611.9	−0.0202842
$385$	−193795.	−0.0666333
$386$	3.02096e6	1.03199
$387$	2.04684e6	0.694715
$388$	−1.54937e6	−0.522487
$389$	413440.	0.138528	0.0692642	−	0.997598i	$-0.477935\pi$
$389$	413440.	0.138528	0.0692642	+	0.997598i	$0.477935\pi$
$390$	0	0
$391$	4.44746e6	1.47120
$392$	168083.	0.0552471
$393$	−167899.	−0.0548362
$394$	−3.04011e6	−0.986618
$395$	−4.93714e6	−1.59214
$396$	39890.8	0.0127831
$397$	102926.	0.0327753	0.0163877	−	0.999866i	$-0.494783\pi$
$397$	102926.	0.0327753	0.0163877	+	0.999866i	$0.494783\pi$
$398$	2.94548e6	0.932071
$399$	44020.5	0.0138428
$400$	−4.09729e6	−1.28040
$401$	−2.23365e6	−0.693671	−0.346836	−	0.937926i	$-0.612744\pi$
$401$	−2.23365e6	−0.693671	−0.346836	+	0.937926i	$0.612744\pi$
$402$	−257072.	−0.0793396
$403$	0	0
$404$	−1.55741e6	−0.474734
$405$	5.90950e6	1.79025
$406$	290937.	0.0875958
$407$	68746.0	0.0205713
$408$	−337708.	−0.100436
$409$	4.46150e6	1.31878	0.659390	−	0.751801i	$-0.270815\pi$
$409$	4.46150e6	1.31878	0.659390	+	0.751801i	$0.270815\pi$
$410$	−3.74923e6	−1.10149
$411$	−58765.6	−0.0171600
$412$	−1.10257e6	−0.320010
$413$	−4.71606e6	−1.36052
$414$	4.63874e6	1.33014
$415$	7.73887e6	2.20575
$416$	0	0
$417$	−248931.	−0.0701034
$418$	14457.5	0.00404718
$419$	4.22792e6	1.17650	0.588250	−	0.808679i	$-0.299817\pi$
$419$	4.22792e6	1.17650	0.588250	+	0.808679i	$0.299817\pi$
$420$	−238507.	−0.0659748
$421$	−4.11791e6	−1.13233	−0.566163	−	0.824293i	$-0.691573\pi$
$421$	−4.11791e6	−1.13233	−0.566163	+	0.824293i	$0.691573\pi$
$422$	−3.95952e6	−1.08233
$423$	5.98959e6	1.62759
$424$	1.53845e6	0.415594
$425$	−7.96688e6	−2.13952
$426$	306878.	0.0819298
$427$	2.29531e6	0.609216
$428$	300493.	0.0792912
$429$	0	0
$430$	−4.01930e6	−1.04828
$431$	−1.15324e6	−0.299038	−0.149519	−	0.988759i	$-0.547773\pi$
$431$	−1.15324e6	−0.299038	−0.149519	+	0.988759i	$0.547773\pi$
$432$	−426074.	−0.109844
$433$	−33734.3	−0.00864673	−0.00432337	−	0.999991i	$-0.501376\pi$
$433$	−33734.3	−0.00864673	−0.00432337	+	0.999991i	$0.501376\pi$
$434$	2.75612e6	0.702383
$435$	85192.9	0.0215864
$436$	−707583.	−0.178263
$437$	−904298.	−0.226521
$438$	−7858.13	−0.00195719
$439$	7.48363e6	1.85332	0.926661	−	0.375898i	$-0.122666\pi$
$439$	7.48363e6	1.85332	0.926661	+	0.375898i	$0.122666\pi$
$440$	−302293.	−0.0744383
$441$	205038.	0.0502039
$442$	0	0
$443$	−3.28028e6	−0.794148	−0.397074	−	0.917786i	$-0.629974\pi$
$443$	−3.28028e6	−0.794148	−0.397074	+	0.917786i	$0.629974\pi$
$444$	84606.9	0.0203680
$445$	−1.01385e6	−0.242703
$446$	266134.	0.0633525
$447$	118301.	0.0280040
$448$	−4.39678e6	−1.03500
$449$	−7.95356e6	−1.86185	−0.930927	−	0.365205i	$-0.880999\pi$
$449$	−7.95356e6	−1.86185	−0.930927	+	0.365205i	$0.880999\pi$
$450$	−8.30951e6	−1.93439
$451$	−117808.	−0.0272731
$452$	−1.28077e6	−0.294866
$453$	797439.	0.182579
$454$	−509837.	−0.116089
$455$	0	0
$456$	68665.8	0.0154642
$457$	3.35187e6	0.750753	0.375377	−	0.926872i	$-0.377513\pi$
$457$	3.35187e6	0.750753	0.375377	+	0.926872i	$0.377513\pi$
$458$	−5.24506e6	−1.16839
$459$	−828468.	−0.183546
$460$	4.89956e6	1.07960
$461$	−4.68627e6	−1.02701	−0.513505	−	0.858086i	$-0.671653\pi$
$461$	−4.68627e6	−1.02701	−0.513505	+	0.858086i	$0.671653\pi$
$462$	13933.0	0.00303696
$463$	−6.64697e6	−1.44102	−0.720512	−	0.693442i	$-0.756093\pi$
$463$	−6.64697e6	−1.44102	−0.720512	+	0.693442i	$0.756093\pi$
$464$	272969.	0.0588599
$465$	807056.	0.173090
$466$	2.87592e6	0.613497
$467$	−3.14141e6	−0.666549	−0.333275	−	0.942830i	$-0.608154\pi$
$467$	−3.14141e6	−0.666549	−0.333275	+	0.942830i	$0.608154\pi$
$468$	0	0
$469$	−4.36522e6	−0.916377
$470$	−1.17615e7	−2.45594
$471$	−145894.	−0.0303029
$472$	−7.35638e6	−1.51988
$473$	−126295.	−0.0259556
$474$	354957.	0.0725655
$475$	1.61990e6	0.329423
$476$	−1.48595e6	−0.300598
$477$	1.87669e6	0.377656
$478$	755521.	0.151243
$479$	6.68286e6	1.33083	0.665416	−	0.746473i	$-0.268254\pi$
$479$	6.68286e6	1.33083	0.665416	+	0.746473i	$0.268254\pi$
$480$	−647670.	−0.128307
$481$	0	0
$482$	−3.15189e6	−0.617950
$483$	−871489.	−0.169979
$484$	1.80006e6	0.349280
$485$	1.43225e7	2.76481
$486$	−1.29853e6	−0.249379
$487$	4.06478e6	0.776631	0.388316	−	0.921526i	$-0.373057\pi$
$487$	4.06478e6	0.776631	0.388316	+	0.921526i	$0.373057\pi$
$488$	3.58035e6	0.680575
$489$	−367993.	−0.0695933
$490$	−402625.	−0.0757549
$491$	−2.10434e6	−0.393923	−0.196962	−	0.980411i	$-0.563107\pi$
$491$	−2.10434e6	−0.393923	−0.196962	+	0.980411i	$0.563107\pi$
$492$	−144989.	−0.0270036
$493$	530768.	0.0983530
$494$	0	0
$495$	−368755.	−0.0676432
$496$	2.58592e6	0.471966
$497$	5.21095e6	0.946293
$498$	−556389.	−0.100532
$499$	−5.96715e6	−1.07279	−0.536396	−	0.843966i	$-0.680215\pi$
$499$	−5.96715e6	−1.07279	−0.536396	+	0.843966i	$0.680215\pi$
$500$	−5.15808e6	−0.922706
$501$	341673.	0.0608158
$502$	−1.76481e6	−0.312563
$503$	−1.00144e7	−1.76483	−0.882417	−	0.470467i	$-0.844085\pi$
$503$	−1.00144e7	−1.76483	−0.882417	+	0.470467i	$0.844085\pi$
$504$	−5.98108e6	−1.04882
$505$	1.43969e7	2.51212
$506$	−286220.	−0.0496963
$507$	0	0
$508$	2.78542e6	0.478885
$509$	8.47321e6	1.44962	0.724809	−	0.688950i	$-0.241928\pi$
$509$	8.47321e6	1.44962	0.724809	+	0.688950i	$0.241928\pi$
$510$	808940.	0.137718
$511$	−133435.	−0.0226057
$512$	−5.48346e6	−0.924442
$513$	168452.	0.0282606
$514$	1.17807e6	0.196681
$515$	1.01923e7	1.69338
$516$	−155433.	−0.0256991
$517$	−369571.	−0.0608095
$518$	−2.67095e6	−0.437362
$519$	758567.	0.123616
$520$	0	0
$521$	−197614.	−0.0318951	−0.0159476	−	0.999873i	$-0.505076\pi$
$521$	−197614.	−0.0318951	−0.0159476	+	0.999873i	$0.505076\pi$
$522$	553596.	0.0889235
$523$	8.27263e6	1.32248	0.661240	−	0.750174i	$-0.270030\pi$
$523$	8.27263e6	1.32248	0.661240	+	0.750174i	$0.270030\pi$
$524$	−1.15238e6	−0.183345
$525$	1.56113e6	0.247195
$526$	−5.43066e6	−0.855831
$527$	5.02812e6	0.788640
$528$	13072.5	0.00204068
$529$	1.14664e7	1.78150
$530$	−3.68518e6	−0.569862
$531$	−8.97374e6	−1.38114
$532$	302136.	0.0462832
$533$	0	0
$534$	72891.3	0.0110617
$535$	−2.77779e6	−0.419580
$536$	−6.80913e6	−1.02372
$537$	770618.	0.115320
$538$	3.88174e6	0.578190
$539$	−12651.3	−0.00187570
$540$	−912686.	−0.134690
$541$	−363216.	−0.0533546	−0.0266773	−	0.999644i	$-0.508493\pi$
$541$	−363216.	−0.0533546	−0.0266773	+	0.999644i	$0.508493\pi$
$542$	186686.	0.0272970
$543$	120833.	0.0175867
$544$	−4.03511e6	−0.584599
$545$	6.54097e6	0.943302
$546$	0	0
$547$	−620452.	−0.0886624	−0.0443312	−	0.999017i	$-0.514116\pi$
$547$	−620452.	−0.0886624	−0.0443312	+	0.999017i	$0.514116\pi$
$548$	−403339.	−0.0573745
$549$	4.36752e6	0.618449
$550$	512715.	0.0722719
$551$	−107921.	−0.0151435
$552$	−1.35940e6	−0.189889
$553$	6.02736e6	0.838136
$554$	4.56400e6	0.631788
$555$	−782114.	−0.107780
$556$	−1.70855e6	−0.234390
$557$	−3.89737e6	−0.532272	−0.266136	−	0.963935i	$-0.585747\pi$
$557$	−3.89737e6	−0.532272	−0.266136	+	0.963935i	$0.585747\pi$
$558$	5.24437e6	0.713029
$559$	0	0
$560$	7.06442e6	0.951933
$561$	25418.5	0.00340992
$562$	7.91865e6	1.05757
$563$	510725.	0.0679073	0.0339536	−	0.999423i	$-0.489190\pi$
$563$	510725.	0.0679073	0.0339536	+	0.999423i	$0.489190\pi$
$564$	−454837.	−0.0602085
$565$	1.18395e7	1.56032
$566$	−5.80975e6	−0.762283
$567$	−7.21445e6	−0.942422
$568$	8.12834e6	1.05714
$569$	9.75625e6	1.26329	0.631644	−	0.775259i	$-0.282381\pi$
$569$	9.75625e6	1.26329	0.631644	+	0.775259i	$0.282381\pi$
$570$	−164481.	−0.0212045
$571$	−1.41952e7	−1.82201	−0.911006	−	0.412393i	$-0.864693\pi$
$571$	−1.41952e7	−1.82201	−0.911006	+	0.412393i	$0.864693\pi$
$572$	0	0
$573$	365482.	0.0465029
$574$	4.57713e6	0.579847
$575$	−3.20697e7	−4.04506
$576$	−8.36622e6	−1.05069
$577$	1.16423e6	0.145579	0.0727896	−	0.997347i	$-0.476810\pi$
$577$	1.16423e6	0.145579	0.0727896	+	0.997347i	$0.476810\pi$
$578$	−1.43689e6	−0.178898
$579$	−1.07994e6	−0.133877
$580$	584723.	0.0721740
$581$	−9.44777e6	−1.16115
$582$	−1.02972e6	−0.126012
$583$	−115796.	−0.0141098
$584$	−208140.	−0.0252536
$585$	0	0
$586$	9322.45	0.00112147
$587$	−6.58038e6	−0.788234	−0.394117	−	0.919060i	$-0.628950\pi$
$587$	−6.58038e6	−0.788234	−0.394117	+	0.919060i	$0.628950\pi$
$588$	−15570.2	−0.00185716
$589$	−1.02236e6	−0.121427
$590$	1.76214e7	2.08406
$591$	1.08679e6	0.127991
$592$	−2.50600e6	−0.293885
$593$	1.91423e6	0.223541	0.111771	−	0.993734i	$-0.464348\pi$
$593$	1.91423e6	0.223541	0.111771	+	0.993734i	$0.464348\pi$
$594$	53316.8	0.00620009
$595$	1.37362e7	1.59065
$596$	811964.	0.0936313
$597$	−1.05296e6	−0.120914
$598$	0	0
$599$	−2.33678e6	−0.266104	−0.133052	−	0.991109i	$-0.542478\pi$
$599$	−2.33678e6	−0.266104	−0.133052	+	0.991109i	$0.542478\pi$
$600$	2.43514e6	0.276150
$601$	1.04273e7	1.17757	0.588786	−	0.808289i	$-0.299606\pi$
$601$	1.04273e7	1.17757	0.588786	+	0.808289i	$0.299606\pi$
$602$	4.90684e6	0.551837
$603$	−8.30617e6	−0.930266
$604$	5.47324e6	0.610453
$605$	−1.66399e7	−1.84826
$606$	−1.03507e6	−0.114495
$607$	−120274.	−0.0132495	−0.00662474	−	0.999978i	$-0.502109\pi$
$607$	−120274.	−0.0132495	−0.00662474	+	0.999978i	$0.502109\pi$
$608$	820455.	0.0900111
$609$	−104005.	−0.0113635
$610$	−8.57633e6	−0.933205
$611$	0	0
$612$	−2.82747e6	−0.305154
$613$	−1.34576e7	−1.44649	−0.723245	−	0.690592i	$-0.757350\pi$
$613$	−1.34576e7	−1.44649	−0.723245	+	0.690592i	$0.757350\pi$
$614$	1.83059e6	0.195961
$615$	1.34029e6	0.142893
$616$	369046.	0.0391858
$617$	6.84879e6	0.724270	0.362135	−	0.932126i	$-0.382048\pi$
$617$	6.84879e6	0.724270	0.362135	+	0.932126i	$0.382048\pi$
$618$	−732778.	−0.0771794
$619$	5.40663e6	0.567153	0.283577	−	0.958950i	$-0.408479\pi$
$619$	5.40663e6	0.567153	0.283577	+	0.958950i	$0.408479\pi$
$620$	5.53925e6	0.578724
$621$	−3.33490e6	−0.347019
$622$	−8.78684e6	−0.910661
$623$	1.23773e6	0.127763
$624$	0	0
$625$	2.39962e7	2.45721
$626$	7.50463e6	0.765409
$627$	−5168.33	−0.000525027 0
$628$	−1.00135e6	−0.101318
$629$	−4.87273e6	−0.491072
$630$	1.43270e7	1.43815
$631$	9.62552e6	0.962390	0.481195	−	0.876614i	$-0.340203\pi$
$631$	9.62552e6	0.962390	0.481195	+	0.876614i	$0.340203\pi$
$632$	9.40183e6	0.936310
$633$	1.41547e6	0.140408
$634$	9.09920e6	0.899043
$635$	−2.57487e7	−2.53408
$636$	−142512.	−0.0139704
$637$	0	0
$638$	−34158.1	−0.00332232
$639$	9.91542e6	0.960636
$640$	3.71876e6	0.358879
$641$	−1.58752e7	−1.52607	−0.763037	−	0.646355i	$-0.776293\pi$
$641$	−1.58752e7	−1.52607	−0.763037	+	0.646355i	$0.776293\pi$
$642$	199710.	0.0191233
$643$	−1.57235e7	−1.49976	−0.749880	−	0.661574i	$-0.769889\pi$
$643$	−1.57235e7	−1.49976	−0.749880	+	0.661574i	$0.769889\pi$
$644$	−5.98149e6	−0.568322
$645$	1.43684e6	0.135990
$646$	−1.02475e6	−0.0966132
$647$	−1.58173e7	−1.48549	−0.742747	−	0.669572i	$-0.766477\pi$
$647$	−1.58173e7	−1.48549	−0.742747	+	0.669572i	$0.766477\pi$
$648$	−1.12535e7	−1.05281
$649$	553699.	0.0516015
$650$	0	0
$651$	−985270.	−0.0911178
$652$	−2.52573e6	−0.232685
$653$	5.31229e6	0.487527	0.243763	−	0.969835i	$-0.421618\pi$
$653$	5.31229e6	0.487527	0.243763	+	0.969835i	$0.421618\pi$
$654$	−470265.	−0.0429931
$655$	1.06527e7	0.970192
$656$	4.29447e6	0.389628
$657$	−253901.	−0.0229483
$658$	1.43587e7	1.29286
$659$	9.90554e6	0.888514	0.444257	−	0.895899i	$-0.353468\pi$
$659$	9.90554e6	0.888514	0.444257	+	0.895899i	$0.353468\pi$
$660$	28002.5	0.00250228
$661$	1.29988e7	1.15717	0.578587	−	0.815621i	$-0.303604\pi$
$661$	1.29988e7	1.15717	0.578587	+	0.815621i	$0.303604\pi$
$662$	3.08326e6	0.273442
$663$	0	0
$664$	−1.47372e7	−1.29716
$665$	−2.79297e6	−0.244913
$666$	−5.08229e6	−0.443991
$667$	2.13654e6	0.185950
$668$	2.34508e6	0.203337
$669$	−95138.8	−0.00821850
$670$	1.63105e7	1.40372
$671$	−269486.	−0.0231062
$672$	790688.	0.0675433
$673$	−1.32503e7	−1.12769	−0.563844	−	0.825881i	$-0.690678\pi$
$673$	−1.32503e7	−1.12769	−0.563844	+	0.825881i	$0.690678\pi$
$674$	−9.74129e6	−0.825975
$675$	5.97391e6	0.504660
$676$	0	0
$677$	2.23310e7	1.87257	0.936284	−	0.351245i	$-0.114242\pi$
$677$	2.23310e7	1.87257	0.936284	+	0.351245i	$0.114242\pi$
$678$	−851208.	−0.0711151
$679$	−1.74852e7	−1.45545
$680$	2.14266e7	1.77697
$681$	182259.	0.0150598
$682$	−323589.	−0.0266399
$683$	1.40049e7	1.14876	0.574380	−	0.818588i	$-0.305243\pi$
$683$	1.40049e7	1.14876	0.574380	+	0.818588i	$0.305243\pi$
$684$	574906.	0.0469847
$685$	3.72851e6	0.303605
$686$	1.01751e7	0.825523
$687$	1.87502e6	0.151571
$688$	4.60382e6	0.370806
$689$	0	0
$690$	3.25629e6	0.260376
$691$	5.24817e6	0.418132	0.209066	−	0.977902i	$-0.432958\pi$
$691$	5.24817e6	0.418132	0.209066	+	0.977902i	$0.432958\pi$
$692$	5.20645e6	0.413310
$693$	450183.	0.0356087
$694$	−1.17349e7	−0.924870
$695$	1.57940e7	1.24031
$696$	−162233.	−0.0126945
$697$	8.35027e6	0.651056
$698$	−9.23090e6	−0.717142
$699$	−1.02810e6	−0.0795868
$700$	1.07148e7	0.826495
$701$	−2.13994e7	−1.64477	−0.822386	−	0.568930i	$-0.807358\pi$
$701$	−2.13994e7	−1.64477	−0.822386	+	0.568930i	$0.807358\pi$
$702$	0	0
$703$	990767.	0.0756107
$704$	516214.	0.0392553
$705$	4.20456e6	0.318601
$706$	−9.34250e6	−0.705426
$707$	−1.75760e7	−1.32243
$708$	681447.	0.0510915
$709$	−4.35333e6	−0.325242	−0.162621	−	0.986689i	$-0.551995\pi$
$709$	−4.35333e6	−0.325242	−0.162621	+	0.986689i	$0.551995\pi$
$710$	−1.94705e7	−1.44954
$711$	1.14689e7	0.850839
$712$	1.93069e6	0.142729
$713$	2.02401e7	1.49104
$714$	−987571.	−0.0724975
$715$	0	0
$716$	5.28915e6	0.385570
$717$	−270087.	−0.0196203
$718$	7.29395e6	0.528021
$719$	2.21389e7	1.59710	0.798552	−	0.601926i	$-0.205600\pi$
$719$	2.21389e7	1.59710	0.798552	+	0.601926i	$0.205600\pi$
$720$	1.34422e7	0.966361
$721$	−1.24430e7	−0.891426
$722$	−1.10865e7	−0.791501
$723$	1.12675e6	0.0801645
$724$	829338.	0.0588010
$725$	−3.82726e6	−0.270422
$726$	1.19633e6	0.0842385
$727$	−4.83218e6	−0.339084	−0.169542	−	0.985523i	$-0.554229\pi$
$727$	−4.83218e6	−0.339084	−0.169542	+	0.985523i	$0.554229\pi$
$728$	0	0
$729$	−1.34154e7	−0.934940
$730$	498575.	0.0346277
$731$	8.95178e6	0.619606
$732$	−331661.	−0.0228779
$733$	2.47827e7	1.70368	0.851840	−	0.523803i	$-0.175487\pi$
$733$	2.47827e7	1.70368	0.851840	+	0.523803i	$0.175487\pi$
$734$	−1.76253e7	−1.20753
$735$	143932.	0.00982741
$736$	−1.62428e7	−1.10527
$737$	512509.	0.0347562
$738$	8.70939e6	0.588636
$739$	−7.09289e6	−0.477762	−0.238881	−	0.971049i	$-0.576781\pi$
$739$	−7.09289e6	−0.477762	−0.238881	+	0.971049i	$0.576781\pi$
$740$	−5.36806e6	−0.360361
$741$	0	0
$742$	4.49895e6	0.299986
$743$	−1.95117e7	−1.29665	−0.648327	−	0.761362i	$-0.724531\pi$
$743$	−1.95117e7	−1.29665	−0.648327	+	0.761362i	$0.724531\pi$
$744$	−1.53688e6	−0.101791
$745$	−7.50588e6	−0.495462
$746$	−7.14803e6	−0.470262
$747$	−1.79773e7	−1.17875
$748$	174461.	0.0114010
$749$	3.39118e6	0.220875
$750$	−3.42810e6	−0.222536
$751$	1.66103e7	1.07468	0.537339	−	0.843366i	$-0.319430\pi$
$751$	1.66103e7	1.07468	0.537339	+	0.843366i	$0.319430\pi$
$752$	1.34720e7	0.868733
$753$	630891.	0.0405477
$754$	0	0
$755$	−5.05952e7	−3.23029
$756$	1.11423e6	0.0709037
$757$	1.22902e6	0.0779508	0.0389754	−	0.999240i	$-0.487591\pi$
$757$	1.22902e6	0.0779508	0.0389754	+	0.999240i	$0.487591\pi$
$758$	−1.45622e7	−0.920567
$759$	102319.	0.00644693
$760$	−4.35664e6	−0.273601
$761$	−1.37482e7	−0.860569	−0.430284	−	0.902693i	$-0.641587\pi$
$761$	−1.37482e7	−0.860569	−0.430284	+	0.902693i	$0.641587\pi$
$762$	1.85121e6	0.115496
$763$	−7.98535e6	−0.496572
$764$	2.50850e6	0.155482
$765$	2.61374e7	1.61476
$766$	−1.82659e6	−0.112478
$767$	0	0
$768$	1.54908e6	0.0947699
$769$	1.01549e7	0.619240	0.309620	−	0.950860i	$-0.399798\pi$
$769$	1.01549e7	0.619240	0.309620	+	0.950860i	$0.399798\pi$
$770$	−884007.	−0.0537315
$771$	−421140.	−0.0255147
$772$	−7.41222e6	−0.447616
$773$	−1.41511e7	−0.851807	−0.425903	−	0.904769i	$-0.640044\pi$
$773$	−1.41511e7	−0.851807	−0.425903	+	0.904769i	$0.640044\pi$
$774$	9.33677e6	0.560202
$775$	−3.62567e7	−2.16837
$776$	−2.72745e7	−1.62593
$777$	954821.	0.0567374
$778$	1.88593e6	0.111706
$779$	−1.69785e6	−0.100243
$780$	0	0
$781$	−611803.	−0.0358909
$782$	2.02873e7	1.18634
$783$	−397993.	−0.0231991
$784$	461178.	0.0267965
$785$	9.25654e6	0.536136
$786$	−765882.	−0.0442186
$787$	1.03095e6	0.0593338	0.0296669	−	0.999560i	$-0.490555\pi$
$787$	1.03095e6	0.0593338	0.0296669	+	0.999560i	$0.490555\pi$
$788$	7.45923e6	0.427935
$789$	1.94137e6	0.111024
$790$	−2.25210e7	−1.28387
$791$	−1.44540e7	−0.821383
$792$	702222.	0.0397797
$793$	0	0
$794$	469500.	0.0264292
$795$	1.31740e6	0.0739262
$796$	−7.22705e6	−0.404276
$797$	−1.43337e7	−0.799303	−0.399651	−	0.916667i	$-0.630869\pi$
$797$	−1.43337e7	−0.799303	−0.399651	+	0.916667i	$0.630869\pi$
$798$	200802.	0.0111625
$799$	2.61952e7	1.45163
$800$	2.90963e7	1.60736
$801$	2.35516e6	0.129700
$802$	−1.01889e7	−0.559360
$803$	15666.3	0.000857386 0
$804$	630753.	0.0344127
$805$	5.52935e7	3.00735
$806$	0	0
$807$	−1.38766e6	−0.0750065
$808$	−2.74161e7	−1.47733
$809$	1.55020e7	0.832751	0.416376	−	0.909193i	$-0.363300\pi$
$809$	1.55020e7	0.832751	0.416376	+	0.909193i	$0.363300\pi$
$810$	2.69565e7	1.44361
$811$	−2.45861e7	−1.31261	−0.656307	−	0.754494i	$-0.727882\pi$
$811$	−2.45861e7	−1.31261	−0.656307	+	0.754494i	$0.727882\pi$
$812$	−713842.	−0.0379938
$813$	−66737.5	−0.00354114
$814$	313589.	0.0165882
$815$	2.33481e7	1.23128
$816$	−926583.	−0.0487146
$817$	−1.82016e6	−0.0954011
$818$	2.03513e7	1.06343
$819$	0	0
$820$	9.19911e6	0.477761
$821$	3.33396e6	0.172624	0.0863122	−	0.996268i	$-0.472492\pi$
$821$	3.33396e6	0.172624	0.0863122	+	0.996268i	$0.472492\pi$
$822$	−268062.	−0.0138375
$823$	3.08787e6	0.158913	0.0794564	−	0.996838i	$-0.474682\pi$
$823$	3.08787e6	0.158913	0.0794564	+	0.996838i	$0.474682\pi$
$824$	−1.94092e7	−0.995842
$825$	−183288.	−0.00937559
$826$	−2.15125e7	−1.09709
$827$	−6.89555e6	−0.350595	−0.175297	−	0.984516i	$-0.556089\pi$
$827$	−6.89555e6	−0.350595	−0.175297	+	0.984516i	$0.556089\pi$
$828$	−1.13816e7	−0.576936
$829$	−2.00007e7	−1.01078	−0.505391	−	0.862890i	$-0.668652\pi$
$829$	−2.00007e7	−1.01078	−0.505391	+	0.862890i	$0.668652\pi$
$830$	3.53012e7	1.77867
$831$	−1.63156e6	−0.0819596
$832$	0	0
$833$	896725.	0.0447762
$834$	−1.13551e6	−0.0565297
$835$	−2.16782e7	−1.07599
$836$	−35473.0	−0.00175542
$837$	−3.77030e6	−0.186021
$838$	1.92859e7	0.948701
$839$	−5.23988e6	−0.256990	−0.128495	−	0.991710i	$-0.541015\pi$
$839$	−5.23988e6	−0.256990	−0.128495	+	0.991710i	$0.541015\pi$
$840$	−4.19858e6	−0.205307
$841$	−2.02562e7	−0.987569
$842$	−1.87841e7	−0.913081
$843$	−2.83079e6	−0.137195
$844$	9.71509e6	0.469452
$845$	0	0
$846$	2.73218e7	1.31245
$847$	2.03144e7	0.972959
$848$	4.22111e6	0.201575
$849$	2.07689e6	0.0988883
$850$	−3.63413e7	−1.72526
$851$	−1.96146e7	−0.928442
$852$	−752957.	−0.0355362
$853$	1.32853e7	0.625170	0.312585	−	0.949890i	$-0.398805\pi$
$853$	1.32853e7	0.625170	0.312585	+	0.949890i	$0.398805\pi$
$854$	1.04702e7	0.491257
$855$	−5.31449e6	−0.248626
$856$	5.28976e6	0.246747
$857$	−8.34473e6	−0.388115	−0.194057	−	0.980990i	$-0.562165\pi$
$857$	−8.34473e6	−0.388115	−0.194057	+	0.980990i	$0.562165\pi$
$858$	0	0
$859$	−4.07521e7	−1.88437	−0.942187	−	0.335088i	$-0.891234\pi$
$859$	−4.07521e7	−1.88437	−0.942187	+	0.335088i	$0.891234\pi$
$860$	9.86176e6	0.454683
$861$	−1.63625e6	−0.0752216
$862$	−5.26056e6	−0.241137
$863$	1.73853e7	0.794614	0.397307	−	0.917686i	$-0.369945\pi$
$863$	1.73853e7	0.794614	0.397307	+	0.917686i	$0.369945\pi$
$864$	3.02570e6	0.137893
$865$	−4.81289e7	−2.18708
$866$	−153881.	−0.00697252
$867$	513667.	0.0232078
$868$	−6.76243e6	−0.304652
$869$	−707656.	−0.0317887
$870$	388612.	0.0174068
$871$	0	0
$872$	−1.24560e7	−0.554738
$873$	−3.32710e7	−1.47751
$874$	−4.12500e6	−0.182661
$875$	−5.82109e7	−2.57030
$876$	19280.7	0.000848912 0
$877$	−2.58279e6	−0.113394	−0.0566971	−	0.998391i	$-0.518057\pi$
$877$	−2.58279e6	−0.113394	−0.0566971	+	0.998391i	$0.518057\pi$
$878$	3.41370e7	1.49447
$879$	−3332.63	−0.000145484 0
$880$	−829414.	−0.0361048
$881$	−1.66814e7	−0.724090	−0.362045	−	0.932161i	$-0.617921\pi$
$881$	−1.66814e7	−0.724090	−0.362045	+	0.932161i	$0.617921\pi$
$882$	935291.	0.0404833
$883$	2.36384e7	1.02027	0.510137	−	0.860093i	$-0.329595\pi$
$883$	2.36384e7	1.02027	0.510137	+	0.860093i	$0.329595\pi$
$884$	0	0
$885$	−6.29936e6	−0.270358
$886$	−1.49632e7	−0.640382
$887$	−5.25660e6	−0.224334	−0.112167	−	0.993689i	$-0.535779\pi$
$887$	−5.25660e6	−0.224334	−0.112167	+	0.993689i	$0.535779\pi$
$888$	1.48939e6	0.0633833
$889$	3.14345e7	1.33399
$890$	−4.62474e6	−0.195710
$891$	847029.	0.0357441
$892$	−652988.	−0.0274785
$893$	−5.32625e6	−0.223508
$894$	539638.	0.0225818
$895$	−4.88935e7	−2.04030
$896$	−4.53994e6	−0.188921
$897$	0	0
$898$	−3.62806e7	−1.50135
$899$	2.41549e6	0.0996795
$900$	2.03882e7	0.839022
$901$	8.20763e6	0.336826
$902$	−537389.	−0.0219924
$903$	−1.75412e6	−0.0715879
$904$	−2.25461e7	−0.917595
$905$	−7.66648e6	−0.311153
$906$	3.63756e6	0.147228
$907$	3.22789e7	1.30287	0.651435	−	0.758705i	$-0.274167\pi$
$907$	3.22789e7	1.30287	0.651435	+	0.758705i	$0.274167\pi$
$908$	1.25094e6	0.0503525
$909$	−3.34438e7	−1.34247
$910$	0	0
$911$	4.20975e7	1.68058	0.840292	−	0.542134i	$-0.182383\pi$
$911$	4.20975e7	1.68058	0.840292	+	0.542134i	$0.182383\pi$
$912$	188401.	0.00750061
$913$	1.10924e6	0.0440400
$914$	1.52897e7	0.605389
$915$	3.06590e6	0.121061
$916$	1.28693e7	0.506775
$917$	−1.30051e7	−0.510728
$918$	−3.77910e6	−0.148007
$919$	−2.19460e7	−0.857168	−0.428584	−	0.903502i	$-0.640987\pi$
$919$	−2.19460e7	−0.857168	−0.428584	+	0.903502i	$0.640987\pi$
$920$	8.62500e7	3.35961
$921$	−654407.	−0.0254213
$922$	−2.13767e7	−0.828157
$923$	0	0
$924$	−34186.0	−0.00131725
$925$	3.51362e7	1.35021
$926$	−3.03205e7	−1.16201
$927$	−2.36765e7	−0.904937
$928$	−1.93845e6	−0.0738899
$929$	1.35921e7	0.516710	0.258355	−	0.966050i	$-0.416820\pi$
$929$	1.35921e7	0.516710	0.258355	+	0.966050i	$0.416820\pi$
$930$	3.68143e6	0.139575
$931$	−182330.	−0.00689421
$932$	−7.05637e6	−0.266098
$933$	3.14116e6	0.118137
$934$	−1.43297e7	−0.537489
$935$	−1.61273e6	−0.0603300
$936$	0	0
$937$	3.01018e7	1.12007	0.560033	−	0.828470i	$-0.310788\pi$
$937$	3.01018e7	1.12007	0.560033	+	0.828470i	$0.310788\pi$
$938$	−1.99122e7	−0.738944
$939$	−2.68279e6	−0.0992938
$940$	2.88581e7	1.06524
$941$	1.06275e7	0.391252	0.195626	−	0.980679i	$-0.437326\pi$
$941$	1.06275e7	0.391252	0.195626	+	0.980679i	$0.437326\pi$
$942$	−665503.	−0.0244356
$943$	3.36130e7	1.23091
$944$	−2.01840e7	−0.737187
$945$	−1.03000e7	−0.375196
$946$	−576099.	−0.0209300
$947$	2.41709e7	0.875826	0.437913	−	0.899017i	$-0.355718\pi$
$947$	2.41709e7	0.875826	0.437913	+	0.899017i	$0.355718\pi$
$948$	−870924.	−0.0314745
$949$	0	0
$950$	7.38925e6	0.265639
$951$	−3.25282e6	−0.116630
$952$	−2.61580e7	−0.935432
$953$	4.27043e7	1.52314	0.761569	−	0.648084i	$-0.224430\pi$
$953$	4.27043e7	1.52314	0.761569	+	0.648084i	$0.224430\pi$
$954$	8.56063e6	0.304533
$955$	−2.31888e7	−0.822754
$956$	−1.85375e6	−0.0656003
$957$	12211.0	0.000430993 0
$958$	3.04842e7	1.07315
$959$	−4.55184e6	−0.159823
$960$	−5.87290e6	−0.205672
$961$	−5.74655e6	−0.200724
$962$	0	0
$963$	6.45276e6	0.224223
$964$	7.73348e6	0.268029
$965$	6.85193e7	2.36862
$966$	−3.97535e6	−0.137067
$967$	4.42692e7	1.52242	0.761211	−	0.648504i	$-0.224605\pi$
$967$	4.42692e7	1.52242	0.761211	+	0.648504i	$0.224605\pi$
$968$	3.16875e7	1.08693
$969$	366332.	0.0125333
$970$	6.53329e7	2.22947
$971$	3.88962e7	1.32391	0.661956	−	0.749542i	$-0.269726\pi$
$971$	3.88962e7	1.32391	0.661956	+	0.749542i	$0.269726\pi$
$972$	3.18606e6	0.108166
$973$	−1.92816e7	−0.652922
$974$	1.85417e7	0.626257
$975$	0	0
$976$	9.82357e6	0.330099
$977$	2.71611e7	0.910354	0.455177	−	0.890401i	$-0.349576\pi$
$977$	2.71611e7	0.910354	0.455177	+	0.890401i	$0.349576\pi$
$978$	−1.67862e6	−0.0561184
$979$	−145319.	−0.00484580
$980$	987881.	0.0328579
$981$	−1.51946e7	−0.504099
$982$	−9.59904e6	−0.317650
$983$	1.98048e7	0.653714	0.326857	−	0.945074i	$-0.394011\pi$
$983$	1.98048e7	0.653714	0.326857	+	0.945074i	$0.394011\pi$
$984$	−2.55232e6	−0.0840326
$985$	−6.89538e7	−2.26448
$986$	2.42113e6	0.0793096
$987$	−5.13301e6	−0.167718
$988$	0	0
$989$	3.60343e7	1.17145
$990$	−1.68209e6	−0.0545459
$991$	−1.44104e7	−0.466115	−0.233057	−	0.972463i	$-0.574873\pi$
$991$	−1.44104e7	−0.466115	−0.233057	+	0.972463i	$0.574873\pi$
$992$	−1.83635e7	−0.592483
$993$	−1.10222e6	−0.0354727
$994$	2.37700e7	0.763068
$995$	6.68075e7	2.13928
$996$	1.36516e6	0.0436048
$997$	3.16635e7	1.00884	0.504418	−	0.863459i	$-0.331707\pi$
$997$	3.16635e7	1.00884	0.504418	+	0.863459i	$0.331707\pi$
$998$	−2.72195e7	−0.865074
$999$	3.65378e6	0.115832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.6.a.a.1.2		2
13.5	odd	4		169.6.b.a.168.1		4
13.8	odd	4		169.6.b.a.168.4		4
13.12	even	2		13.6.a.a.1.1	✓	2
39.38	odd	2		117.6.a.c.1.2		2
52.51	odd	2		208.6.a.h.1.1		2
65.12	odd	4		325.6.b.b.274.1		4
65.38	odd	4		325.6.b.b.274.4		4
65.64	even	2		325.6.a.b.1.2		2
91.90	odd	2		637.6.a.a.1.1		2
104.51	odd	2		832.6.a.i.1.2		2
104.77	even	2		832.6.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.a.1.1	✓	2	13.12	even	2
117.6.a.c.1.2		2	39.38	odd	2
169.6.a.a.1.2		2	1.1	even	1	trivial
169.6.b.a.168.1		4	13.5	odd	4
169.6.b.a.168.4		4	13.8	odd	4
208.6.a.h.1.1		2	52.51	odd	2
325.6.a.b.1.2		2	65.64	even	2
325.6.b.b.274.1		4	65.12	odd	4
325.6.b.b.274.4		4	65.38	odd	4
637.6.a.a.1.1		2	91.90	odd	2
832.6.a.i.1.2		2	104.51	odd	2
832.6.a.p.1.1		2	104.77	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q+4.56155 q^{2} -1.63068 q^{3} -11.1922 q^{4} +103.462 q^{5} -7.43845 q^{6} -126.309 q^{7} -197.024 q^{8} -240.341 q^{9} +471.948 q^{10} +14.8296 q^{11} +18.2510 q^{12} -576.164 q^{14} -168.714 q^{15} -540.582 q^{16} -1051.12 q^{17} -1096.33 q^{18} +213.723 q^{19} -1157.97 q^{20} +205.969 q^{21} +67.6458 q^{22} -4231.16 q^{23} +321.283 q^{24} +7579.41 q^{25} +788.176 q^{27} +1413.68 q^{28} -504.955 q^{29} -769.597 q^{30} -4783.58 q^{31} +3838.86 q^{32} -24.1823 q^{33} -4794.75 q^{34} -13068.2 q^{35} +2689.95 q^{36} +4635.74 q^{37} +974.911 q^{38} -20384.5 q^{40} -7944.15 q^{41} +939.541 q^{42} -8516.41 q^{43} -165.976 q^{44} -24866.2 q^{45} -19300.7 q^{46} -24921.2 q^{47} +881.518 q^{48} -853.113 q^{49} +34573.9 q^{50} +1714.05 q^{51} -7808.46 q^{53} +3595.31 q^{54} +1534.30 q^{55} +24885.8 q^{56} -348.515 q^{57} -2303.38 q^{58} +37337.5 q^{59} +1888.29 q^{60} -18172.2 q^{61} -21820.5 q^{62} +30357.1 q^{63} +34809.8 q^{64} -110.309 q^{66} +34559.9 q^{67} +11764.4 q^{68} +6899.68 q^{69} -59611.1 q^{70} -41255.7 q^{71} +47352.8 q^{72} +1056.42 q^{73} +21146.2 q^{74} -12359.6 q^{75} -2392.04 q^{76} -1873.10 q^{77} -47719.3 q^{79} -55929.8 q^{80} +57117.6 q^{81} -36237.7 q^{82} +74799.0 q^{83} -2305.26 q^{84} -108751. q^{85} -38848.0 q^{86} +823.421 q^{87} -2921.78 q^{88} -9799.26 q^{89} -113428. q^{90} +47356.1 q^{92} +7800.50 q^{93} -113679. q^{94} +22112.3 q^{95} -6259.97 q^{96} +138432. q^{97} -3891.52 q^{98} -3564.15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{2} - 28 q^{3} - 43 q^{4} + 42 q^{5} - 19 q^{6} + 36 q^{7} - 225 q^{8} + 212 q^{9} + 445 q^{10} + 376 q^{11} + 857 q^{12} - 505 q^{14} + 1452 q^{15} + 465 q^{16} - 2630 q^{17} - 898 q^{18} + 312 q^{19}+ \cdots + 159808 q^{99}+O(q^{100}) \) 2 * q + 5 * q^2 - 28 * q^3 - 43 * q^4 + 42 * q^5 - 19 * q^6 + 36 * q^7 - 225 * q^8 + 212 * q^9 + 445 * q^10 + 376 * q^11 + 857 * q^12 - 505 * q^14 + 1452 * q^15 + 465 * q^16 - 2630 * q^17 - 898 * q^18 + 312 * q^19 + 797 * q^20 - 4074 * q^21 + 226 * q^22 - 2624 * q^23 + 1059 * q^24 + 8232 * q^25 - 4732 * q^27 - 3749 * q^28 - 812 * q^29 - 59 * q^30 - 7720 * q^31 + 5175 * q^32 - 9548 * q^33 - 5487 * q^34 - 23044 * q^35 - 11698 * q^36 + 16858 * q^37 + 1018 * q^38 - 18665 * q^40 - 7840 * q^41 - 937 * q^42 + 2420 * q^43 - 11654 * q^44 - 52668 * q^45 - 18596 * q^46 - 9972 * q^47 - 25635 * q^48 + 8684 * q^49 + 34860 * q^50 + 43348 * q^51 - 43720 * q^53 + 1175 * q^54 - 20664 * q^55 + 20345 * q^56 - 2940 * q^57 - 2438 * q^58 + 38936 * q^59 - 49663 * q^60 + 1984 * q^61 - 23108 * q^62 + 103776 * q^63 + 3217 * q^64 - 4286 * q^66 + 69928 * q^67 + 61985 * q^68 - 35480 * q^69 - 63985 * q^70 - 67396 * q^71 + 34698 * q^72 - 74412 * q^73 + 26505 * q^74 - 29568 * q^75 - 5518 * q^76 + 56748 * q^77 - 55296 * q^79 - 117735 * q^80 + 92762 * q^81 - 36192 * q^82 + 75712 * q^83 + 133831 * q^84 - 11710 * q^85 - 34053 * q^86 + 8920 * q^87 - 13026 * q^88 - 116508 * q^89 - 125618 * q^90 - 3764 * q^92 + 85232 * q^93 - 107125 * q^94 + 16072 * q^95 - 41493 * q^96 + 34756 * q^97 + 290 * q^98 + 159808 * q^99

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