LMFDB - Embedded newform 245.6.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,6,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("245.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	245.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:	$39.2940358542$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 109x^{4} + 41x^{3} + 2208x^{2} - 3204x + 560 $ x^6 - x^5 - 109x^4 + 41x^3 + 2208x^2 - 3204x + 560
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.35048$ of defining polynomial
Character	$\chi$	$=$	245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.350479 q^{2} -21.5910 q^{3} -31.8772 q^{4} -25.0000 q^{5} -7.56720 q^{6} -22.3876 q^{8} +223.173 q^{9} +O(q^{10})$	\(q+0.350479 q^{2} -21.5910 q^{3} -31.8772 q^{4} -25.0000 q^{5} -7.56720 q^{6} -22.3876 q^{8} +223.173 q^{9} -8.76197 q^{10} -193.196 q^{11} +688.261 q^{12} +691.199 q^{13} +539.776 q^{15} +1012.22 q^{16} +649.991 q^{17} +78.2173 q^{18} +696.551 q^{19} +796.929 q^{20} -67.7112 q^{22} -2979.44 q^{23} +483.371 q^{24} +625.000 q^{25} +242.251 q^{26} +428.092 q^{27} +6777.77 q^{29} +189.180 q^{30} +151.853 q^{31} +1071.17 q^{32} +4171.30 q^{33} +227.808 q^{34} -7114.11 q^{36} -11014.0 q^{37} +244.126 q^{38} -14923.7 q^{39} +559.690 q^{40} +7366.94 q^{41} -19245.8 q^{43} +6158.54 q^{44} -5579.32 q^{45} -1044.23 q^{46} +12794.2 q^{47} -21854.9 q^{48} +219.049 q^{50} -14034.0 q^{51} -22033.5 q^{52} +9071.01 q^{53} +150.037 q^{54} +4829.90 q^{55} -15039.3 q^{57} +2375.46 q^{58} -36912.7 q^{59} -17206.5 q^{60} -23419.7 q^{61} +53.2212 q^{62} -32015.7 q^{64} -17280.0 q^{65} +1461.95 q^{66} +64419.4 q^{67} -20719.9 q^{68} +64329.2 q^{69} -83135.9 q^{71} -4996.30 q^{72} +28914.7 q^{73} -3860.16 q^{74} -13494.4 q^{75} -22204.1 q^{76} -5230.44 q^{78} +1690.97 q^{79} -25305.6 q^{80} -63473.9 q^{81} +2581.96 q^{82} +80334.4 q^{83} -16249.8 q^{85} -6745.25 q^{86} -146339. q^{87} +4325.20 q^{88} +118850. q^{89} -1955.43 q^{90} +94976.2 q^{92} -3278.66 q^{93} +4484.08 q^{94} -17413.8 q^{95} -23127.6 q^{96} -147605. q^{97} -43116.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9}+O(q^{10}) $ 6 * q - 5 * q^2 + 20 * q^3 + 31 * q^4 - 150 * q^5 + 96 * q^6 - 135 * q^8 + 378 * q^9	\( 6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9} + 125 q^{10} - 924 q^{11} + 370 q^{12} - 150 q^{13} - 500 q^{15} + 435 q^{16} + 1540 q^{17} + 195 q^{18} + 92 q^{19} - 775 q^{20} - 6855 q^{22} - 3920 q^{23} - 7200 q^{24} + 3750 q^{25} + 2635 q^{26} + 2060 q^{27} + 1264 q^{29} - 2400 q^{30} - 7160 q^{31} + 9105 q^{32} + 4460 q^{33} + 2166 q^{34} - 26375 q^{36} - 14170 q^{37} - 46215 q^{38} - 15376 q^{39} + 3375 q^{40} + 4098 q^{41} - 24460 q^{43} - 27873 q^{44} - 9450 q^{45} + 6815 q^{46} + 42940 q^{47} + 11610 q^{48} - 3125 q^{50} - 42008 q^{51} - 36115 q^{52} - 2450 q^{53} + 19566 q^{54} + 23100 q^{55} - 97100 q^{57} - 36110 q^{58} - 64600 q^{59} - 9250 q^{60} + 73620 q^{61} + 111440 q^{62} - 157997 q^{64} + 3750 q^{65} - 139138 q^{66} - 142620 q^{67} + 124330 q^{68} + 17344 q^{69} - 154256 q^{71} - 117495 q^{72} - 5120 q^{73} + 2785 q^{74} + 12500 q^{75} + 7775 q^{76} - 214090 q^{78} - 222504 q^{79} - 10875 q^{80} - 43986 q^{81} + 31665 q^{82} + 179580 q^{83} - 38500 q^{85} - 207160 q^{86} - 209300 q^{87} - 45145 q^{88} + 41648 q^{89} - 4875 q^{90} - 292185 q^{92} - 198520 q^{93} - 333699 q^{94} - 2300 q^{95} + 61824 q^{96} - 73980 q^{97} - 190772 q^{99}+O(q^{100}) \) 6 * q - 5 * q^2 + 20 * q^3 + 31 * q^4 - 150 * q^5 + 96 * q^6 - 135 * q^8 + 378 * q^9 + 125 * q^10 - 924 * q^11 + 370 * q^12 - 150 * q^13 - 500 * q^15 + 435 * q^16 + 1540 * q^17 + 195 * q^18 + 92 * q^19 - 775 * q^20 - 6855 * q^22 - 3920 * q^23 - 7200 * q^24 + 3750 * q^25 + 2635 * q^26 + 2060 * q^27 + 1264 * q^29 - 2400 * q^30 - 7160 * q^31 + 9105 * q^32 + 4460 * q^33 + 2166 * q^34 - 26375 * q^36 - 14170 * q^37 - 46215 * q^38 - 15376 * q^39 + 3375 * q^40 + 4098 * q^41 - 24460 * q^43 - 27873 * q^44 - 9450 * q^45 + 6815 * q^46 + 42940 * q^47 + 11610 * q^48 - 3125 * q^50 - 42008 * q^51 - 36115 * q^52 - 2450 * q^53 + 19566 * q^54 + 23100 * q^55 - 97100 * q^57 - 36110 * q^58 - 64600 * q^59 - 9250 * q^60 + 73620 * q^61 + 111440 * q^62 - 157997 * q^64 + 3750 * q^65 - 139138 * q^66 - 142620 * q^67 + 124330 * q^68 + 17344 * q^69 - 154256 * q^71 - 117495 * q^72 - 5120 * q^73 + 2785 * q^74 + 12500 * q^75 + 7775 * q^76 - 214090 * q^78 - 222504 * q^79 - 10875 * q^80 - 43986 * q^81 + 31665 * q^82 + 179580 * q^83 - 38500 * q^85 - 207160 * q^86 - 209300 * q^87 - 45145 * q^88 + 41648 * q^89 - 4875 * q^90 - 292185 * q^92 - 198520 * q^93 - 333699 * q^94 - 2300 * q^95 + 61824 * q^96 - 73980 * q^97 - 190772 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0.350479	0.0619565	0.0309782	−	0.999520i	$-0.490138\pi$
$2$	0.350479	0.0619565	0.0309782	+	0.999520i	$0.490138\pi$
$3$	−21.5910	−1.38507	−0.692533	−	0.721387i	$-0.743505\pi$
$3$	−21.5910	−1.38507	−0.692533	+	0.721387i	$0.743505\pi$
$4$	−31.8772	−0.996161
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	−7.56720	−0.0858138
$7$	0	0
$7$	0	0
$8$	−22.3876	−0.123675
$9$	223.173	0.918406
$10$	−8.76197	−0.0277078
$11$	−193.196	−0.481412	−0.240706	−	0.970598i	$-0.577379\pi$
$11$	−193.196	−0.481412	−0.240706	+	0.970598i	$0.577379\pi$
$12$	688.261	1.37975
$13$	691.199	1.13434	0.567172	−	0.823599i	$-0.308037\pi$
$13$	691.199	1.13434	0.567172	+	0.823599i	$0.308037\pi$
$14$	0	0
$15$	539.776	0.619420
$16$	1012.22	0.988499
$17$	649.991	0.545488	0.272744	−	0.962087i	$-0.412069\pi$
$17$	649.991	0.545488	0.272744	+	0.962087i	$0.412069\pi$
$18$	78.2173	0.0569012
$19$	696.551	0.442659	0.221329	−	0.975199i	$-0.428960\pi$
$19$	696.551	0.442659	0.221329	+	0.975199i	$0.428960\pi$
$20$	796.929	0.445497
$21$	0	0
$22$	−67.7112	−0.0298266
$23$	−2979.44	−1.17440	−0.587199	−	0.809443i	$-0.699769\pi$
$23$	−2979.44	−1.17440	−0.587199	+	0.809443i	$0.699769\pi$
$24$	483.371	0.171298
$25$	625.000	0.200000
$26$	242.251	0.0702800
$27$	428.092	0.113013
$28$	0	0
$29$	6777.77	1.49655	0.748276	−	0.663388i	$-0.230882\pi$
$29$	6777.77	1.49655	0.748276	+	0.663388i	$0.230882\pi$
$30$	189.180	0.0383771
$31$	151.853	0.0283804	0.0141902	−	0.999899i	$-0.495483\pi$
$31$	151.853	0.0283804	0.0141902	+	0.999899i	$0.495483\pi$
$32$	1071.17	0.184919
$33$	4171.30	0.666787
$34$	227.808	0.0337965
$35$	0	0
$36$	−7114.11	−0.914881
$37$	−11014.0	−1.32263	−0.661317	−	0.750107i	$-0.730002\pi$
$37$	−11014.0	−1.32263	−0.661317	+	0.750107i	$0.730002\pi$
$38$	244.126	0.0274256
$39$	−14923.7	−1.57114
$40$	559.690	0.0553092
$41$	7366.94	0.684427	0.342214	−	0.939622i	$-0.388823\pi$
$41$	7366.94	0.684427	0.342214	+	0.939622i	$0.388823\pi$
$42$	0	0
$43$	−19245.8	−1.58732	−0.793662	−	0.608360i	$-0.791828\pi$
$43$	−19245.8	−1.58732	−0.793662	+	0.608360i	$0.791828\pi$
$44$	6158.54	0.479564
$45$	−5579.32	−0.410724
$46$	−1044.23	−0.0727616
$47$	12794.2	0.844826	0.422413	−	0.906404i	$-0.361183\pi$
$47$	12794.2	0.844826	0.422413	+	0.906404i	$0.361183\pi$
$48$	−21854.9	−1.36914
$49$	0	0
$50$	219.049	0.0123913
$51$	−14034.0	−0.755536
$52$	−22033.5	−1.12999
$53$	9071.01	0.443574	0.221787	−	0.975095i	$-0.428811\pi$
$53$	9071.01	0.443574	0.221787	+	0.975095i	$0.428811\pi$
$54$	150.037	0.00700189
$55$	4829.90	0.215294
$56$	0	0
$57$	−15039.3	−0.613111
$58$	2375.46	0.0927211
$59$	−36912.7	−1.38053	−0.690265	−	0.723557i	$-0.742506\pi$
$59$	−36912.7	−1.38053	−0.690265	+	0.723557i	$0.742506\pi$
$60$	−17206.5	−0.617042
$61$	−23419.7	−0.805856	−0.402928	−	0.915232i	$-0.632008\pi$
$61$	−23419.7	−0.805856	−0.402928	+	0.915232i	$0.632008\pi$
$62$	53.2212	0.00175835
$63$	0	0
$64$	−32015.7	−0.977042
$65$	−17280.0	−0.507294
$66$	1461.95	0.0413118
$67$	64419.4	1.75319	0.876596	−	0.481227i	$-0.159809\pi$
$67$	64419.4	1.75319	0.876596	+	0.481227i	$0.159809\pi$
$68$	−20719.9	−0.543394
$69$	64329.2	1.62662
$70$	0	0
$71$	−83135.9	−1.95723	−0.978617	−	0.205692i	$-0.934055\pi$
$71$	−83135.9	−1.95723	−0.978617	+	0.205692i	$0.934055\pi$
$72$	−4996.30	−0.113584
$73$	28914.7	0.635056	0.317528	−	0.948249i	$-0.397147\pi$
$73$	28914.7	0.635056	0.317528	+	0.948249i	$0.397147\pi$
$74$	−3860.16	−0.0819457
$75$	−13494.4	−0.277013
$76$	−22204.1	−0.440959
$77$	0	0
$78$	−5230.44	−0.0973424
$79$	1690.97	0.0304838	0.0152419	−	0.999884i	$-0.495148\pi$
$79$	1690.97	0.0304838	0.0152419	+	0.999884i	$0.495148\pi$
$80$	−25305.6	−0.442070
$81$	−63473.9	−1.07494
$82$	2581.96	0.0424047
$83$	80334.4	1.27999	0.639995	−	0.768379i	$-0.278936\pi$
$83$	80334.4	1.27999	0.639995	+	0.768379i	$0.278936\pi$
$84$	0	0
$85$	−16249.8	−0.243949
$86$	−6745.25	−0.0983450
$87$	−146339.	−2.07282
$88$	4325.20	0.0595387
$89$	118850.	1.59047	0.795235	−	0.606301i	$-0.207347\pi$
$89$	118850.	1.59047	0.795235	+	0.606301i	$0.207347\pi$
$90$	−1955.43	−0.0254470
$91$	0	0
$92$	94976.2	1.16989
$93$	−3278.66	−0.0393087
$94$	4484.08	0.0523424
$95$	−17413.8	−0.197963
$96$	−23127.6	−0.256125
$97$	−147605.	−1.59284	−0.796418	−	0.604747i	$-0.793274\pi$
$97$	−147605.	−1.59284	−0.796418	+	0.604747i	$0.793274\pi$
$98$	0	0
$99$	−43116.1	−0.442131
$100$	−19923.2	−0.199232
$101$	8253.31	0.0805053	0.0402527	−	0.999190i	$-0.487184\pi$
$101$	8253.31	0.0805053	0.0402527	+	0.999190i	$0.487184\pi$
$102$	−4918.61	−0.0468104
$103$	185892.	1.72651	0.863253	−	0.504772i	$-0.168423\pi$
$103$	185892.	1.72651	0.863253	+	0.504772i	$0.168423\pi$
$104$	−15474.3	−0.140290
$105$	0	0
$106$	3179.20	0.0274823
$107$	−98555.4	−0.832187	−0.416093	−	0.909322i	$-0.636601\pi$
$107$	−98555.4	−0.832187	−0.416093	+	0.909322i	$0.636601\pi$
$108$	−13646.4	−0.112579
$109$	116887.	0.942319	0.471160	−	0.882048i	$-0.343836\pi$
$109$	116887.	0.942319	0.471160	+	0.882048i	$0.343836\pi$
$110$	1692.78	0.0133389
$111$	237803.	1.83193
$112$	0	0
$113$	−26879.3	−0.198026	−0.0990130	−	0.995086i	$-0.531569\pi$
$113$	−26879.3	−0.198026	−0.0990130	+	0.995086i	$0.531569\pi$
$114$	−5270.94	−0.0379862
$115$	74486.1	0.525207
$116$	−216056.	−1.49081
$117$	154257.	1.04179
$118$	−12937.1	−0.0855328
$119$	0	0
$120$	−12084.3	−0.0766069
$121$	−123726.	−0.768243
$122$	−8208.12	−0.0499280
$123$	−159060.	−0.947977
$124$	−4840.63	−0.0282715
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	102250.	0.562540	0.281270	−	0.959629i	$-0.409244\pi$
$127$	102250.	0.562540	0.281270	+	0.959629i	$0.409244\pi$
$128$	−45498.1	−0.245453
$129$	415537.	2.19855
$130$	−6056.27	−0.0314302
$131$	−354998.	−1.80737	−0.903686	−	0.428196i	$-0.859149\pi$
$131$	−354998.	−1.80737	−0.903686	+	0.428196i	$0.859149\pi$
$132$	−132969.	−0.664227
$133$	0	0
$134$	22577.6	0.108622
$135$	−10702.3	−0.0505409
$136$	−14551.7	−0.0674633
$137$	228246.	1.03897	0.519483	−	0.854481i	$-0.326124\pi$
$137$	228246.	1.03897	0.519483	+	0.854481i	$0.326124\pi$
$138$	22546.0	0.100780
$139$	−216438.	−0.950160	−0.475080	−	0.879942i	$-0.657581\pi$
$139$	−216438.	−0.950160	−0.475080	+	0.879942i	$0.657581\pi$
$140$	0	0
$141$	−276239.	−1.17014
$142$	−29137.4	−0.121263
$143$	−133537.	−0.546087
$144$	225900.	0.907843
$145$	−169444.	−0.669278
$146$	10134.0	0.0393459
$147$	0	0
$148$	351094.	1.31756
$149$	220224.	0.812642	0.406321	−	0.913730i	$-0.366812\pi$
$149$	220224.	0.812642	0.406321	+	0.913730i	$0.366812\pi$
$150$	−4729.50	−0.0171628
$151$	−123531.	−0.440895	−0.220447	−	0.975399i	$-0.570752\pi$
$151$	−123531.	−0.440895	−0.220447	+	0.975399i	$0.570752\pi$
$152$	−15594.1	−0.0547459
$153$	145060.	0.500979
$154$	0	0
$155$	−3796.32	−0.0126921
$156$	475725.	1.56511
$157$	−249357.	−0.807368	−0.403684	−	0.914898i	$-0.632271\pi$
$157$	−249357.	−0.807368	−0.403684	+	0.914898i	$0.632271\pi$
$158$	592.650	0.00188867
$159$	−195853.	−0.614379
$160$	−26779.1	−0.0826983
$161$	0	0
$162$	−22246.3	−0.0665993
$163$	150583.	0.443923	0.221961	−	0.975055i	$-0.428754\pi$
$163$	150583.	0.443923	0.221961	+	0.975055i	$0.428754\pi$
$164$	−234837.	−0.681800
$165$	−104283.	−0.298196
$166$	28155.5	0.0793037
$167$	241046.	0.668820	0.334410	−	0.942428i	$-0.391463\pi$
$167$	241046.	0.668820	0.334410	+	0.942428i	$0.391463\pi$
$168$	0	0
$169$	106463.	0.286737
$170$	−5695.20	−0.0151143
$171$	155451.	0.406540
$172$	613502.	1.58123
$173$	−310571.	−0.788942	−0.394471	−	0.918908i	$-0.629072\pi$
$173$	−310571.	−0.788942	−0.394471	+	0.918908i	$0.629072\pi$
$174$	−51288.7	−0.128425
$175$	0	0
$176$	−195558.	−0.475875
$177$	796983.	1.91212
$178$	41654.6	0.0985400
$179$	8295.97	0.0193524	0.00967619	−	0.999953i	$-0.496920\pi$
$179$	8295.97	0.0193524	0.00967619	+	0.999953i	$0.496920\pi$
$180$	177853.	0.409147
$181$	−353858.	−0.802847	−0.401423	−	0.915893i	$-0.631484\pi$
$181$	−353858.	−0.802847	−0.401423	+	0.915893i	$0.631484\pi$
$182$	0	0
$183$	505656.	1.11616
$184$	66702.5	0.145244
$185$	275349.	0.591500
$186$	−1149.10	−0.00243543
$187$	−125576.	−0.262604
$188$	−407841.	−0.841583
$189$	0	0
$190$	−6103.16	−0.0122651
$191$	−397824.	−0.789055	−0.394527	−	0.918884i	$-0.629092\pi$
$191$	−397824.	−0.789055	−0.394527	+	0.918884i	$0.629092\pi$
$192$	691252.	1.35327
$193$	−13959.9	−0.0269768	−0.0134884	−	0.999909i	$-0.504294\pi$
$193$	−13959.9	−0.0269768	−0.0134884	+	0.999909i	$0.504294\pi$
$194$	−51732.3	−0.0986865
$195$	373093.	0.702636
$196$	0	0
$197$	286397.	0.525779	0.262890	−	0.964826i	$-0.415325\pi$
$197$	286397.	0.525779	0.262890	+	0.964826i	$0.415325\pi$
$198$	−15111.3	−0.0273929
$199$	−723932.	−1.29588	−0.647941	−	0.761691i	$-0.724370\pi$
$199$	−723932.	−1.29588	−0.647941	+	0.761691i	$0.724370\pi$
$200$	−13992.2	−0.0247350
$201$	−1.39088e6	−2.42829
$202$	2892.61	0.00498783
$203$	0	0
$204$	447363.	0.752636
$205$	−184173.	−0.306085
$206$	65151.3	0.106968
$207$	−664930.	−1.07857
$208$	699648.	1.12130
$209$	−134571.	−0.213101
$210$	0	0
$211$	518960.	0.802468	0.401234	−	0.915976i	$-0.368581\pi$
$211$	518960.	0.802468	0.401234	+	0.915976i	$0.368581\pi$
$212$	−289158.	−0.441871
$213$	1.79499e6	2.71090
$214$	−34541.6	−0.0515594
$215$	481146.	0.709872
$216$	−9583.96	−0.0139769
$217$	0	0
$218$	40966.2	0.0583828
$219$	−624299.	−0.879594
$220$	−153964.	−0.214467
$221$	449273.	0.618771
$222$	83344.9	0.113500
$223$	−287841.	−0.387606	−0.193803	−	0.981040i	$-0.562082\pi$
$223$	−287841.	−0.387606	−0.193803	+	0.981040i	$0.562082\pi$
$224$	0	0
$225$	139483.	0.183681
$226$	−9420.64	−0.0122690
$227$	493689.	0.635900	0.317950	−	0.948107i	$-0.397006\pi$
$227$	493689.	0.635900	0.317950	+	0.948107i	$0.397006\pi$
$228$	479409.	0.610758
$229$	361035.	0.454947	0.227473	−	0.973784i	$-0.426954\pi$
$229$	361035.	0.454947	0.227473	+	0.973784i	$0.426954\pi$
$230$	26105.8	0.0325400
$231$	0	0
$232$	−151738.	−0.185086
$233$	−37489.0	−0.0452391	−0.0226195	−	0.999744i	$-0.507201\pi$
$233$	−37489.0	−0.0452391	−0.0226195	+	0.999744i	$0.507201\pi$
$234$	54063.7	0.0645456
$235$	−319854.	−0.377817
$236$	1.17667e6	1.37523
$237$	−36509.8	−0.0422220
$238$	0	0
$239$	488378.	0.553046	0.276523	−	0.961007i	$-0.410818\pi$
$239$	488378.	0.553046	0.276523	+	0.961007i	$0.410818\pi$
$240$	546373.	0.612296
$241$	−286185.	−0.317398	−0.158699	−	0.987327i	$-0.550730\pi$
$241$	−286185.	−0.317398	−0.158699	+	0.987327i	$0.550730\pi$
$242$	−43363.4	−0.0475976
$243$	1.26644e6	1.37584
$244$	746555.	0.802763
$245$	0	0
$246$	−55747.1	−0.0587333
$247$	481456.	0.502127
$248$	−3399.62	−0.00350995
$249$	−1.73450e6	−1.77287
$250$	−5476.23	−0.00554156
$251$	−979810.	−0.981652	−0.490826	−	0.871258i	$-0.663305\pi$
$251$	−979810.	−0.981652	−0.490826	+	0.871258i	$0.663305\pi$
$252$	0	0
$253$	575617.	0.565369
$254$	35836.4	0.0348530
$255$	350849.	0.337886
$256$	1.00856e6	0.961835
$257$	−1.06258e6	−1.00353	−0.501764	−	0.865004i	$-0.667316\pi$
$257$	−1.06258e6	−1.00353	−0.501764	+	0.865004i	$0.667316\pi$
$258$	145637.	0.136214
$259$	0	0
$260$	550837.	0.505347
$261$	1.51261e6	1.37444
$262$	−124419.	−0.111978
$263$	−1.21334e6	−1.08166	−0.540832	−	0.841130i	$-0.681891\pi$
$263$	−1.21334e6	−1.08166	−0.540832	+	0.841130i	$0.681891\pi$
$264$	−93385.5	−0.0824650
$265$	−226775.	−0.198372
$266$	0	0
$267$	−2.56610e6	−2.20291
$268$	−2.05351e6	−1.74646
$269$	−835870.	−0.704301	−0.352150	−	0.935943i	$-0.614549\pi$
$269$	−835870.	−0.704301	−0.352150	+	0.935943i	$0.614549\pi$
$270$	−3750.93	−0.00313134
$271$	−577614.	−0.477765	−0.238883	−	0.971048i	$-0.576781\pi$
$271$	−577614.	−0.477765	−0.238883	+	0.971048i	$0.576781\pi$
$272$	657935.	0.539214
$273$	0	0
$274$	79995.4	0.0643707
$275$	−120748.	−0.0962823
$276$	−2.05063e6	−1.62037
$277$	−1.85825e6	−1.45514	−0.727569	−	0.686035i	$-0.759350\pi$
$277$	−1.85825e6	−1.45514	−0.727569	+	0.686035i	$0.759350\pi$
$278$	−75857.0	−0.0588686
$279$	33889.4	0.0260647
$280$	0	0
$281$	−2.12343e6	−1.60425	−0.802125	−	0.597157i	$-0.796297\pi$
$281$	−2.12343e6	−1.60425	−0.802125	+	0.597157i	$0.796297\pi$
$282$	−96816.0	−0.0724977
$283$	−1.19467e6	−0.886708	−0.443354	−	0.896347i	$-0.646212\pi$
$283$	−1.19467e6	−0.886708	−0.443354	+	0.896347i	$0.646212\pi$
$284$	2.65014e6	1.94972
$285$	375981.	0.274192
$286$	−46801.9	−0.0338336
$287$	0	0
$288$	239055.	0.169831
$289$	−997369.	−0.702443
$290$	−59386.6	−0.0414661
$291$	3.18694e6	2.20618
$292$	−921720.	−0.632618
$293$	1.33436e6	0.908037	0.454018	−	0.890992i	$-0.349990\pi$
$293$	1.33436e6	0.908037	0.454018	+	0.890992i	$0.349990\pi$
$294$	0	0
$295$	922818.	0.617392
$296$	246576.	0.163577
$297$	−82705.8	−0.0544058
$298$	77183.9	0.0503484
$299$	−2.05939e6	−1.33217
$300$	430163.	0.275950
$301$	0	0
$302$	−43295.1	−0.0273163
$303$	−178197.	−0.111505
$304$	705065.	0.437568
$305$	585494.	0.360390
$306$	50840.5	0.0310389
$307$	−225396.	−0.136490	−0.0682450	−	0.997669i	$-0.521740\pi$
$307$	−225396.	−0.136490	−0.0682450	+	0.997669i	$0.521740\pi$
$308$	0	0
$309$	−4.01360e6	−2.39132
$310$	−1330.53	−0.000786358 0
$311$	−2.12161e6	−1.24384	−0.621919	−	0.783082i	$-0.713647\pi$
$311$	−2.12161e6	−1.24384	−0.621919	+	0.783082i	$0.713647\pi$
$312$	334106.	0.194311
$313$	886070.	0.511219	0.255610	−	0.966780i	$-0.417724\pi$
$313$	886070.	0.511219	0.255610	+	0.966780i	$0.417724\pi$
$314$	−87394.2	−0.0500217
$315$	0	0
$316$	−53903.4	−0.0303667
$317$	37425.0	0.0209177	0.0104588	−	0.999945i	$-0.496671\pi$
$317$	37425.0	0.0209177	0.0104588	+	0.999945i	$0.496671\pi$
$318$	−68642.2	−0.0380648
$319$	−1.30944e6	−0.720458
$320$	800393.	0.436946
$321$	2.12791e6	1.15263
$322$	0	0
$323$	452752.	0.241465
$324$	2.02337e6	1.07081
$325$	432000.	0.226869
$326$	52776.2	0.0275039
$327$	−2.52370e6	−1.30517
$328$	−164928.	−0.0846467
$329$	0	0
$330$	−36548.8	−0.0184752
$331$	−1.60549e6	−0.805448	−0.402724	−	0.915321i	$-0.631937\pi$
$331$	−1.60549e6	−0.805448	−0.402724	+	0.915321i	$0.631937\pi$
$332$	−2.56083e6	−1.27508
$333$	−2.45802e6	−1.21471
$334$	84481.6	0.0414377
$335$	−1.61048e6	−0.784051
$336$	0	0
$337$	−2.83244e6	−1.35858	−0.679291	−	0.733869i	$-0.737713\pi$
$337$	−2.83244e6	−1.35858	−0.679291	+	0.733869i	$0.737713\pi$
$338$	37313.2	0.0177652
$339$	580352.	0.274279
$340$	517997.	0.243013
$341$	−29337.4	−0.0136627
$342$	54482.3	0.0251878
$343$	0	0
$344$	430868.	0.196312
$345$	−1.60823e6	−0.727446
$346$	−108848.	−0.0488801
$347$	−609249.	−0.271626	−0.135813	−	0.990735i	$-0.543365\pi$
$347$	−609249.	−0.271626	−0.135813	+	0.990735i	$0.543365\pi$
$348$	4.66487e6	2.06487
$349$	−1.12578e6	−0.494755	−0.247377	−	0.968919i	$-0.579569\pi$
$349$	−1.12578e6	−0.494755	−0.247377	+	0.968919i	$0.579569\pi$
$350$	0	0
$351$	295897.	0.128196
$352$	−206945.	−0.0890222
$353$	1.05039e6	0.448657	0.224328	−	0.974514i	$-0.427981\pi$
$353$	1.05039e6	0.448657	0.224328	+	0.974514i	$0.427981\pi$
$354$	279326.	0.118469
$355$	2.07840e6	0.875301
$356$	−3.78861e6	−1.58437
$357$	0	0
$358$	2907.56	0.00119901
$359$	−2.18022e6	−0.892821	−0.446410	−	0.894828i	$-0.647298\pi$
$359$	−2.18022e6	−0.892821	−0.446410	+	0.894828i	$0.647298\pi$
$360$	124907.	0.0507963
$361$	−1.99092e6	−0.804053
$362$	−124020.	−0.0497416
$363$	2.67138e6	1.06407
$364$	0	0
$365$	−722868.	−0.284006
$366$	177222.	0.0691536
$367$	−866531.	−0.335830	−0.167915	−	0.985802i	$-0.553703\pi$
$367$	−866531.	−0.335830	−0.167915	+	0.985802i	$0.553703\pi$
$368$	−3.01586e6	−1.16089
$369$	1.64410e6	0.628582
$370$	96504.1	0.0366472
$371$	0	0
$372$	104514.	0.0391578
$373$	−715676.	−0.266345	−0.133172	−	0.991093i	$-0.542516\pi$
$373$	−715676.	−0.266345	−0.133172	+	0.991093i	$0.542516\pi$
$374$	−44011.6	−0.0162700
$375$	337360.	0.123884
$376$	−286430.	−0.104484
$377$	4.68479e6	1.69761
$378$	0	0
$379$	−2.97948e6	−1.06547	−0.532737	−	0.846281i	$-0.678837\pi$
$379$	−2.97948e6	−1.06547	−0.532737	+	0.846281i	$0.678837\pi$
$380$	555102.	0.197203
$381$	−2.20768e6	−0.779155
$382$	−139429.	−0.0488871
$383$	2.06400e6	0.718974	0.359487	−	0.933150i	$-0.382952\pi$
$383$	2.06400e6	0.718974	0.359487	+	0.933150i	$0.382952\pi$
$384$	982352.	0.339969
$385$	0	0
$386$	−4892.66	−0.00167139
$387$	−4.29514e6	−1.45781
$388$	4.70522e6	1.58672
$389$	−1.80356e6	−0.604305	−0.302152	−	0.953260i	$-0.597705\pi$
$389$	−1.80356e6	−0.604305	−0.302152	+	0.953260i	$0.597705\pi$
$390$	130761.	0.0435328
$391$	−1.93661e6	−0.640620
$392$	0	0
$393$	7.66477e6	2.50333
$394$	100376.	0.0325754
$395$	−42274.3	−0.0136327
$396$	1.37442e6	0.440434
$397$	892591.	0.284234	0.142117	−	0.989850i	$-0.454609\pi$
$397$	892591.	0.284234	0.142117	+	0.989850i	$0.454609\pi$
$398$	−253723.	−0.0802883
$399$	0	0
$400$	632639.	0.197700
$401$	5.41514e6	1.68170	0.840850	−	0.541268i	$-0.182056\pi$
$401$	5.41514e6	1.68170	0.840850	+	0.541268i	$0.182056\pi$
$402$	−487474.	−0.150448
$403$	104961.	0.0321931
$404$	−263092.	−0.0801963
$405$	1.58685e6	0.480726
$406$	0	0
$407$	2.12786e6	0.636731
$408$	314187.	0.0934410
$409$	5.84591e6	1.72800	0.864000	−	0.503492i	$-0.167952\pi$
$409$	5.84591e6	1.72800	0.864000	+	0.503492i	$0.167952\pi$
$410$	−64548.9	−0.0189640
$411$	−4.92807e6	−1.43904
$412$	−5.92571e6	−1.71988
$413$	0	0
$414$	−233044.	−0.0668247
$415$	−2.00836e6	−0.572429
$416$	740389.	0.209762
$417$	4.67312e6	1.31603
$418$	−47164.3	−0.0132030
$419$	783896.	0.218134	0.109067	−	0.994034i	$-0.465214\pi$
$419$	783896.	0.218134	0.109067	+	0.994034i	$0.465214\pi$
$420$	0	0
$421$	−3.82391e6	−1.05148	−0.525741	−	0.850644i	$-0.676212\pi$
$421$	−3.82391e6	−1.05148	−0.525741	+	0.850644i	$0.676212\pi$
$422$	181884.	0.0497181
$423$	2.85531e6	0.775893
$424$	−203078.	−0.0548591
$425$	406244.	0.109098
$426$	629106.	0.167958
$427$	0	0
$428$	3.14167e6	0.828993
$429$	2.88320e6	0.756366
$430$	168631.	0.0439812
$431$	5.17122e6	1.34091	0.670456	−	0.741949i	$-0.266099\pi$
$431$	5.17122e6	1.34091	0.670456	+	0.741949i	$0.266099\pi$
$432$	433325.	0.111713
$433$	3.31682e6	0.850162	0.425081	−	0.905155i	$-0.360246\pi$
$433$	3.31682e6	0.850162	0.425081	+	0.905155i	$0.360246\pi$
$434$	0	0
$435$	3.65848e6	0.926994
$436$	−3.72601e6	−0.938702
$437$	−2.07533e6	−0.519858
$438$	−218804.	−0.0544966
$439$	−2.52987e6	−0.626523	−0.313262	−	0.949667i	$-0.601422\pi$
$439$	−2.52987e6	−0.626523	−0.313262	+	0.949667i	$0.601422\pi$
$440$	−108130.	−0.0266265
$441$	0	0
$442$	157461.	0.0383369
$443$	85543.0	0.0207098	0.0103549	−	0.999946i	$-0.496704\pi$
$443$	85543.0	0.0207098	0.0103549	+	0.999946i	$0.496704\pi$
$444$	−7.58048e6	−1.82490
$445$	−2.97126e6	−0.711280
$446$	−100882.	−0.0240147
$447$	−4.75487e6	−1.12556
$448$	0	0
$449$	4.79130e6	1.12160	0.560800	−	0.827951i	$-0.310494\pi$
$449$	4.79130e6	1.12160	0.560800	+	0.827951i	$0.310494\pi$
$450$	48885.8	0.0113802
$451$	−1.42326e6	−0.329491
$452$	856837.	0.197266
$453$	2.66717e6	0.610668
$454$	173028.	0.0393981
$455$	0	0
$456$	336693.	0.0758266
$457$	2.85563e6	0.639605	0.319802	−	0.947484i	$-0.396383\pi$
$457$	2.85563e6	0.639605	0.319802	+	0.947484i	$0.396383\pi$
$458$	126535.	0.0281869
$459$	278256.	0.0616472
$460$	−2.37440e6	−0.523191
$461$	6.56708e6	1.43920	0.719598	−	0.694391i	$-0.244326\pi$
$461$	6.56708e6	1.43920	0.719598	+	0.694391i	$0.244326\pi$
$462$	0	0
$463$	−5.14996e6	−1.11648	−0.558241	−	0.829679i	$-0.688523\pi$
$463$	−5.14996e6	−1.11648	−0.558241	+	0.829679i	$0.688523\pi$
$464$	6.86061e6	1.47934
$465$	81966.4	0.0175794
$466$	−13139.1	−0.00280285
$467$	−4.25721e6	−0.903302	−0.451651	−	0.892195i	$-0.649165\pi$
$467$	−4.25721e6	−0.903302	−0.451651	+	0.892195i	$0.649165\pi$
$468$	−4.91727e6	−1.03779
$469$	0	0
$470$	−112102.	−0.0234082
$471$	5.38386e6	1.11826
$472$	826387.	0.170737
$473$	3.71822e6	0.764156
$474$	−12795.9	−0.00261593
$475$	435344.	0.0885317
$476$	0	0
$477$	2.02440e6	0.407381
$478$	171166.	0.0342648
$479$	3.95142e6	0.786891	0.393446	−	0.919348i	$-0.371283\pi$
$479$	3.95142e6	0.786891	0.393446	+	0.919348i	$0.371283\pi$
$480$	578189.	0.114543
$481$	−7.61285e6	−1.50032
$482$	−100302.	−0.0196649
$483$	0	0
$484$	3.94404e6	0.765294
$485$	3.69012e6	0.712338
$486$	443861.	0.0852425
$487$	1.46528e6	0.279962	0.139981	−	0.990154i	$-0.455296\pi$
$487$	1.46528e6	0.279962	0.139981	+	0.990154i	$0.455296\pi$
$488$	524312.	0.0996644
$489$	−3.25125e6	−0.614862
$490$	0	0
$491$	−9.72476e6	−1.82044	−0.910218	−	0.414130i	$-0.864086\pi$
$491$	−9.72476e6	−1.82044	−0.910218	+	0.414130i	$0.864086\pi$
$492$	5.07038e6	0.944338
$493$	4.40549e6	0.816350
$494$	168740.	0.0311101
$495$	1.07790e6	0.197727
$496$	153709.	0.0280540
$497$	0	0
$498$	−607907.	−0.109841
$499$	−3.38445e6	−0.608467	−0.304233	−	0.952598i	$-0.598400\pi$
$499$	−3.38445e6	−0.608467	−0.304233	+	0.952598i	$0.598400\pi$
$500$	498081.	0.0890994
$501$	−5.20444e6	−0.926359
$502$	−343403.	−0.0608197
$503$	−4.17285e6	−0.735381	−0.367691	−	0.929948i	$-0.619851\pi$
$503$	−4.17285e6	−0.735381	−0.367691	+	0.929948i	$0.619851\pi$
$504$	0	0
$505$	−206333.	−0.0360031
$506$	201741.	0.0350283
$507$	−2.29866e6	−0.397150
$508$	−3.25944e6	−0.560381
$509$	2.26525e6	0.387545	0.193772	−	0.981046i	$-0.437928\pi$
$509$	2.26525e6	0.387545	0.193772	+	0.981046i	$0.437928\pi$
$510$	122965.	0.0209342
$511$	0	0
$512$	1.80942e6	0.305045
$513$	298188.	0.0500262
$514$	−372413.	−0.0621751
$515$	−4.64730e6	−0.772117
$516$	−1.32461e7	−2.19011
$517$	−2.47178e6	−0.406709
$518$	0	0
$519$	6.70554e6	1.09274
$520$	386857.	0.0627397
$521$	4.56954e6	0.737527	0.368764	−	0.929523i	$-0.379781\pi$
$521$	4.56954e6	0.737527	0.368764	+	0.929523i	$0.379781\pi$
$522$	530139.	0.0851556
$523$	6.98635e6	1.11685	0.558427	−	0.829554i	$-0.311405\pi$
$523$	6.98635e6	1.11685	0.558427	+	0.829554i	$0.311405\pi$
$524$	1.13163e7	1.80043
$525$	0	0
$526$	−425250.	−0.0670162
$527$	98702.9	0.0154812
$528$	4.22229e6	0.659118
$529$	2.44073e6	0.379211
$530$	−79480.0	−0.0122905
$531$	−8.23791e6	−1.26789
$532$	0	0
$533$	5.09202e6	0.776376
$534$	−899365.	−0.136484
$535$	2.46388e6	0.372165
$536$	−1.44219e6	−0.216826
$537$	−179118.	−0.0268043
$538$	−292955.	−0.0436360
$539$	0	0
$540$	341159.	0.0503469
$541$	−3.26412e6	−0.479482	−0.239741	−	0.970837i	$-0.577063\pi$
$541$	−3.26412e6	−0.479482	−0.239741	+	0.970837i	$0.577063\pi$
$542$	−202442.	−0.0296007
$543$	7.64016e6	1.11200
$544$	696248.	0.100871
$545$	−2.92216e6	−0.421418
$546$	0	0
$547$	2.63534e6	0.376589	0.188295	−	0.982113i	$-0.439704\pi$
$547$	2.63534e6	0.376589	0.188295	+	0.982113i	$0.439704\pi$
$548$	−7.27583e6	−1.03498
$549$	−5.22665e6	−0.740103
$550$	−42319.5	−0.00596532
$551$	4.72106e6	0.662462
$552$	−1.44018e6	−0.201172
$553$	0	0
$554$	−651276.	−0.0901552
$555$	−5.94507e6	−0.819266
$556$	6.89944e6	0.946513
$557$	1.32907e7	1.81514	0.907568	−	0.419906i	$-0.137937\pi$
$557$	1.32907e7	1.81514	0.907568	+	0.419906i	$0.137937\pi$
$558$	11877.5	0.00161488
$559$	−1.33027e7	−1.80057
$560$	0	0
$561$	2.71131e6	0.363724
$562$	−744217.	−0.0993937
$563$	−1.87464e6	−0.249257	−0.124629	−	0.992203i	$-0.539774\pi$
$563$	−1.87464e6	−0.249257	−0.124629	+	0.992203i	$0.539774\pi$
$564$	8.80572e6	1.16565
$565$	671983.	0.0885599
$566$	−418705.	−0.0549373
$567$	0	0
$568$	1.86121e6	0.242061
$569$	−2.33433e6	−0.302260	−0.151130	−	0.988514i	$-0.548291\pi$
$569$	−2.33433e6	−0.302260	−0.151130	+	0.988514i	$0.548291\pi$
$570$	131774.	0.0169880
$571$	−1.51285e7	−1.94180	−0.970900	−	0.239484i	$-0.923022\pi$
$571$	−1.51285e7	−1.94180	−0.970900	+	0.239484i	$0.923022\pi$
$572$	4.25678e6	0.543990
$573$	8.58942e6	1.09289
$574$	0	0
$575$	−1.86215e6	−0.234880
$576$	−7.14503e6	−0.897321
$577$	9.50067e6	1.18800	0.593998	−	0.804467i	$-0.297549\pi$
$577$	9.50067e6	1.18800	0.593998	+	0.804467i	$0.297549\pi$
$578$	−349557.	−0.0435209
$579$	301409.	0.0373646
$580$	5.40140e6	0.666709
$581$	0	0
$582$	1.11695e6	0.136687
$583$	−1.75248e6	−0.213542
$584$	−647331.	−0.0785407
$585$	−3.85642e6	−0.465902
$586$	467664.	0.0562588
$587$	8.74269e6	1.04725	0.523624	−	0.851949i	$-0.324580\pi$
$587$	8.74269e6	1.04725	0.523624	+	0.851949i	$0.324580\pi$
$588$	0	0
$589$	105773.	0.0125628
$590$	323428.	0.0382514
$591$	−6.18361e6	−0.728239
$592$	−1.11486e7	−1.30742
$593$	−5.07761e6	−0.592957	−0.296478	−	0.955040i	$-0.595812\pi$
$593$	−5.07761e6	−0.592957	−0.296478	+	0.955040i	$0.595812\pi$
$594$	−28986.6	−0.00337079
$595$	0	0
$596$	−7.02012e6	−0.809522
$597$	1.56304e7	1.79488
$598$	−721772.	−0.0825367
$599$	−5.23339e6	−0.595958	−0.297979	−	0.954572i	$-0.596313\pi$
$599$	−5.23339e6	−0.595958	−0.297979	+	0.954572i	$0.596313\pi$
$600$	302107.	0.0342596
$601$	6.88552e6	0.777590	0.388795	−	0.921324i	$-0.372891\pi$
$601$	6.88552e6	0.777590	0.388795	+	0.921324i	$0.372891\pi$
$602$	0	0
$603$	1.43766e7	1.61014
$604$	3.93783e6	0.439202
$605$	3.09316e6	0.343569
$606$	−62454.4	−0.00690847
$607$	1.63103e7	1.79676	0.898378	−	0.439224i	$-0.144747\pi$
$607$	1.63103e7	1.79676	0.898378	+	0.439224i	$0.144747\pi$
$608$	746122.	0.0818560
$609$	0	0
$610$	205203.	0.0223285
$611$	8.84331e6	0.958323
$612$	−4.62411e6	−0.499056
$613$	−1.15378e7	−1.24014	−0.620071	−	0.784546i	$-0.712896\pi$
$613$	−1.15378e7	−1.24014	−0.620071	+	0.784546i	$0.712896\pi$
$614$	−78996.6	−0.00845644
$615$	3.97650e6	0.423948
$616$	0	0
$617$	−4.55233e6	−0.481416	−0.240708	−	0.970598i	$-0.577380\pi$
$617$	−4.55233e6	−0.481416	−0.240708	+	0.970598i	$0.577380\pi$
$618$	−1.40668e6	−0.148158
$619$	−8.26487e6	−0.866981	−0.433491	−	0.901158i	$-0.642718\pi$
$619$	−8.26487e6	−0.866981	−0.433491	+	0.901158i	$0.642718\pi$
$620$	121016.	0.0126434
$621$	−1.27548e6	−0.132722
$622$	−743578.	−0.0770638
$623$	0	0
$624$	−1.51061e7	−1.55307
$625$	390625.	0.0400000
$626$	310549.	0.0316734
$627$	2.90553e6	0.295159
$628$	7.94878e6	0.804269
$629$	−7.15898e6	−0.721480
$630$	0	0
$631$	6.24165e6	0.624059	0.312030	−	0.950072i	$-0.398991\pi$
$631$	6.24165e6	0.624059	0.312030	+	0.950072i	$0.398991\pi$
$632$	−37856.8	−0.00377008
$633$	−1.12049e7	−1.11147
$634$	13116.7	0.00129599
$635$	−2.55625e6	−0.251576
$636$	6.24322e6	0.612021
$637$	0	0
$638$	−458931.	−0.0446370
$639$	−1.85537e7	−1.79754
$640$	1.13745e6	0.109770
$641$	1.26929e7	1.22016	0.610078	−	0.792342i	$-0.291138\pi$
$641$	1.26929e7	1.22016	0.610078	+	0.792342i	$0.291138\pi$
$642$	745788.	0.0714131
$643$	−5.69724e6	−0.543421	−0.271711	−	0.962379i	$-0.587589\pi$
$643$	−5.69724e6	−0.543421	−0.271711	+	0.962379i	$0.587589\pi$
$644$	0	0
$645$	−1.03884e7	−0.983220
$646$	158680.	0.0149603
$647$	4.03296e6	0.378759	0.189379	−	0.981904i	$-0.439352\pi$
$647$	4.03296e6	0.378759	0.189379	+	0.981904i	$0.439352\pi$
$648$	1.42103e6	0.132943
$649$	7.13139e6	0.664603
$650$	151407.	0.0140560
$651$	0	0
$652$	−4.80017e6	−0.442219
$653$	2.11914e7	1.94481	0.972403	−	0.233309i	$-0.0749554\pi$
$653$	2.11914e7	1.94481	0.972403	+	0.233309i	$0.0749554\pi$
$654$	−884504.	−0.0808640
$655$	8.87495e6	0.808281
$656$	7.45698e6	0.676556
$657$	6.45298e6	0.583239
$658$	0	0
$659$	−459929.	−0.0412551	−0.0206275	−	0.999787i	$-0.506566\pi$
$659$	−459929.	−0.0412551	−0.0206275	+	0.999787i	$0.506566\pi$
$660$	3.32423e6	0.297051
$661$	−1.38029e7	−1.22876	−0.614380	−	0.789010i	$-0.710594\pi$
$661$	−1.38029e7	−1.22876	−0.614380	+	0.789010i	$0.710594\pi$
$662$	−562690.	−0.0499027
$663$	−9.70027e6	−0.857038
$664$	−1.79849e6	−0.158303
$665$	0	0
$666$	−861483.	−0.0752595
$667$	−2.01940e7	−1.75755
$668$	−7.68387e6	−0.666253
$669$	6.21478e6	0.536859
$670$	−564441.	−0.0485771
$671$	4.52460e6	0.387949
$672$	0	0
$673$	−6.76968e6	−0.576143	−0.288072	−	0.957609i	$-0.593014\pi$
$673$	−6.76968e6	−0.576143	−0.288072	+	0.957609i	$0.593014\pi$
$674$	−992710.	−0.0841730
$675$	267558.	0.0226026
$676$	−3.39375e6	−0.285636
$677$	−1.50919e7	−1.26553	−0.632766	−	0.774343i	$-0.718080\pi$
$677$	−1.50919e7	−1.26553	−0.632766	+	0.774343i	$0.718080\pi$
$678$	203401.	0.0169934
$679$	0	0
$680$	363793.	0.0301705
$681$	−1.06593e7	−0.880763
$682$	−10282.1	−0.000846490 0
$683$	−8.27256e6	−0.678560	−0.339280	−	0.940685i	$-0.610183\pi$
$683$	−8.27256e6	−0.678560	−0.339280	+	0.940685i	$0.610183\pi$
$684$	−4.95534e6	−0.404980
$685$	−5.70615e6	−0.464640
$686$	0	0
$687$	−7.79511e6	−0.630131
$688$	−1.94811e7	−1.56907
$689$	6.26988e6	0.503166
$690$	−563651.	−0.0450700
$691$	971323.	0.0773871	0.0386935	−	0.999251i	$-0.487680\pi$
$691$	971323.	0.0773871	0.0386935	+	0.999251i	$0.487680\pi$
$692$	9.90011e6	0.785914
$693$	0	0
$694$	−213529.	−0.0168290
$695$	5.41096e6	0.424925
$696$	3.27618e6	0.256357
$697$	4.78844e6	0.373347
$698$	−394562.	−0.0306533
$699$	809425.	0.0626591
$700$	0	0
$701$	413147.	0.0317548	0.0158774	−	0.999874i	$-0.494946\pi$
$701$	413147.	0.0317548	0.0158774	+	0.999874i	$0.494946\pi$
$702$	103706.	0.00794255
$703$	−7.67179e6	−0.585475
$704$	6.18531e6	0.470359
$705$	6.90598e6	0.523302
$706$	368140.	0.0277972
$707$	0	0
$708$	−2.54056e7	−1.90478
$709$	4.66361e6	0.348423	0.174212	−	0.984708i	$-0.444262\pi$
$709$	4.66361e6	0.348423	0.174212	+	0.984708i	$0.444262\pi$
$710$	728434.	0.0542306
$711$	377379.	0.0279965
$712$	−2.66077e6	−0.196702
$713$	−452436.	−0.0333299
$714$	0	0
$715$	3.33843e6	0.244217
$716$	−264452.	−0.0192781
$717$	−1.05446e7	−0.766005
$718$	−764121.	−0.0553160
$719$	−1.68074e7	−1.21249	−0.606244	−	0.795279i	$-0.707325\pi$
$719$	−1.68074e7	−1.21249	−0.606244	+	0.795279i	$0.707325\pi$
$720$	−5.64751e6	−0.406000
$721$	0	0
$722$	−697774.	−0.0498163
$723$	6.17903e6	0.439617
$724$	1.12800e7	0.799765
$725$	4.23611e6	0.299310
$726$	936261.	0.0659258
$727$	−5.44669e6	−0.382205	−0.191103	−	0.981570i	$-0.561206\pi$
$727$	−5.44669e6	−0.382205	−0.191103	+	0.981570i	$0.561206\pi$
$728$	0	0
$729$	−1.19196e7	−0.830698
$730$	−253350.	−0.0175960
$731$	−1.25096e7	−0.865865
$732$	−1.61189e7	−1.11188
$733$	201931.	0.0138817	0.00694085	−	0.999976i	$-0.497791\pi$
$733$	201931.	0.0138817	0.00694085	+	0.999976i	$0.497791\pi$
$734$	−303701.	−0.0208068
$735$	0	0
$736$	−3.19148e6	−0.217169
$737$	−1.24456e7	−0.844007
$738$	576222.	0.0389448
$739$	−1.42050e7	−0.956823	−0.478411	−	0.878136i	$-0.658787\pi$
$739$	−1.42050e7	−0.956823	−0.478411	+	0.878136i	$0.658787\pi$
$740$	−8.77735e6	−0.589229
$741$	−1.03951e7	−0.695479
$742$	0	0
$743$	−2.23566e7	−1.48571	−0.742854	−	0.669454i	$-0.766528\pi$
$743$	−2.23566e7	−1.48571	−0.742854	+	0.669454i	$0.766528\pi$
$744$	73401.3	0.00486151
$745$	−5.50560e6	−0.363424
$746$	−250829.	−0.0165018
$747$	1.79285e7	1.17555
$748$	4.00300e6	0.261596
$749$	0	0
$750$	118238.	0.00767542
$751$	−501912.	−0.0324734	−0.0162367	−	0.999868i	$-0.505169\pi$
$751$	−501912.	−0.0324734	−0.0162367	+	0.999868i	$0.505169\pi$
$752$	1.29505e7	0.835109
$753$	2.11551e7	1.35965
$754$	1.64192e6	0.105178
$755$	3.08828e6	0.197174
$756$	0	0
$757$	−6.27081e6	−0.397726	−0.198863	−	0.980027i	$-0.563725\pi$
$757$	−6.27081e6	−0.397726	−0.198863	+	0.980027i	$0.563725\pi$
$758$	−1.04425e6	−0.0660130
$759$	−1.24282e7	−0.783073
$760$	389853.	0.0244831
$761$	1.80496e7	1.12981	0.564906	−	0.825155i	$-0.308912\pi$
$761$	1.80496e7	1.12981	0.564906	+	0.825155i	$0.308912\pi$
$762$	−773746.	−0.0482737
$763$	0	0
$764$	1.26815e7	0.786026
$765$	−3.62650e6	−0.224045
$766$	723389.	0.0445451
$767$	−2.55140e7	−1.56600
$768$	−2.17758e7	−1.33220
$769$	−1.15070e7	−0.701689	−0.350844	−	0.936434i	$-0.614105\pi$
$769$	−1.15070e7	−0.701689	−0.350844	+	0.936434i	$0.614105\pi$
$770$	0	0
$771$	2.29422e7	1.38995
$772$	445003.	0.0268732
$773$	−1.81370e7	−1.09174	−0.545868	−	0.837871i	$-0.683800\pi$
$773$	−1.81370e7	−1.09174	−0.545868	+	0.837871i	$0.683800\pi$
$774$	−1.50536e6	−0.0903206
$775$	94908.0	0.00567608
$776$	3.30452e6	0.196994
$777$	0	0
$778$	−632109.	−0.0374406
$779$	5.13145e6	0.302968
$780$	−1.18931e7	−0.699938
$781$	1.60615e7	0.942235
$782$	−678741.	−0.0396905
$783$	2.90151e6	0.169130
$784$	0	0
$785$	6.23391e6	0.361066
$786$	2.68634e6	0.155097
$787$	−2.33931e6	−0.134633	−0.0673164	−	0.997732i	$-0.521444\pi$
$787$	−2.33931e6	−0.134633	−0.0673164	+	0.997732i	$0.521444\pi$
$788$	−9.12953e6	−0.523761
$789$	2.61972e7	1.49818
$790$	−14816.2	−0.000844637 0
$791$	0	0
$792$	965266.	0.0546807
$793$	−1.61877e7	−0.914118
$794$	312834.	0.0176101
$795$	4.89631e6	0.274759
$796$	2.30769e7	1.29091
$797$	−3.74432e6	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$797$	−3.74432e6	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$798$	0	0
$799$	8.31608e6	0.460842
$800$	669479.	0.0369838
$801$	2.65242e7	1.46070
$802$	1.89789e6	0.104192
$803$	−5.58621e6	−0.305723
$804$	4.43373e7	2.41896
$805$	0	0
$806$	36786.4	0.00199457
$807$	1.80473e7	0.975502
$808$	−184772.	−0.00995651
$809$	−6.10119e6	−0.327750	−0.163875	−	0.986481i	$-0.552399\pi$
$809$	−6.10119e6	−0.327750	−0.163875	+	0.986481i	$0.552399\pi$
$810$	556157.	0.0297841
$811$	−3.52610e7	−1.88253	−0.941267	−	0.337662i	$-0.890364\pi$
$811$	−3.52610e7	−1.88253	−0.941267	+	0.337662i	$0.890364\pi$
$812$	0	0
$813$	1.24713e7	0.661736
$814$	745768.	0.0394496
$815$	−3.76458e6	−0.198528
$816$	−1.42055e7	−0.746846
$817$	−1.34057e7	−0.702642
$818$	2.04887e6	0.107061
$819$	0	0
$820$	5.87093e6	0.304910
$821$	−6.08908e6	−0.315278	−0.157639	−	0.987497i	$-0.550388\pi$
$821$	−6.08908e6	−0.315278	−0.157639	+	0.987497i	$0.550388\pi$
$822$	−1.72718e6	−0.0891577
$823$	2.98213e7	1.53471	0.767357	−	0.641220i	$-0.221571\pi$
$823$	2.98213e7	1.53471	0.767357	+	0.641220i	$0.221571\pi$
$824$	−4.16168e6	−0.213526
$825$	2.60706e6	0.133357
$826$	0	0
$827$	1.57103e7	0.798768	0.399384	−	0.916784i	$-0.369224\pi$
$827$	1.57103e7	0.798768	0.399384	+	0.916784i	$0.369224\pi$
$828$	2.11961e7	1.07443
$829$	2.28370e7	1.15413	0.577063	−	0.816699i	$-0.304199\pi$
$829$	2.28370e7	1.15413	0.577063	+	0.816699i	$0.304199\pi$
$830$	−703888.	−0.0354657
$831$	4.01215e7	2.01546
$832$	−2.21292e7	−1.10830
$833$	0	0
$834$	1.63783e6	0.0815369
$835$	−6.02616e6	−0.299105
$836$	4.28974e6	0.212283
$837$	65007.0	0.00320735
$838$	274739.	0.0135148
$839$	3.24143e7	1.58976	0.794881	−	0.606765i	$-0.207533\pi$
$839$	3.24143e7	1.58976	0.794881	+	0.606765i	$0.207533\pi$
$840$	0	0
$841$	2.54270e7	1.23967
$842$	−1.34020e6	−0.0651462
$843$	4.58470e7	2.22199
$844$	−1.65430e7	−0.799388
$845$	−2.66159e6	−0.128233
$846$	1.00072e6	0.0480716
$847$	0	0
$848$	9.18189e6	0.438472
$849$	2.57941e7	1.22815
$850$	142380.	0.00675930
$851$	3.28155e7	1.55330
$852$	−5.72192e7	−2.70049
$853$	−7.88571e6	−0.371081	−0.185540	−	0.982637i	$-0.559404\pi$
$853$	−7.88571e6	−0.371081	−0.185540	+	0.982637i	$0.559404\pi$
$854$	0	0
$855$	−3.88628e6	−0.181810
$856$	2.20642e6	0.102921
$857$	−6.51644e6	−0.303081	−0.151540	−	0.988451i	$-0.548423\pi$
$857$	−6.51644e6	−0.303081	−0.151540	+	0.988451i	$0.548423\pi$
$858$	1.01050e6	0.0468618
$859$	−1.98740e6	−0.0918973	−0.0459487	−	0.998944i	$-0.514631\pi$
$859$	−1.98740e6	−0.0918973	−0.0459487	+	0.998944i	$0.514631\pi$
$860$	−1.53376e7	−0.707148
$861$	0	0
$862$	1.81240e6	0.0830782
$863$	2.24957e7	1.02819	0.514093	−	0.857734i	$-0.328129\pi$
$863$	2.24957e7	1.02819	0.514093	+	0.857734i	$0.328129\pi$
$864$	458558.	0.0208983
$865$	7.76427e6	0.352826
$866$	1.16247e6	0.0526731
$867$	2.15342e7	0.972930
$868$	0	0
$869$	−326689.	−0.0146752
$870$	1.28222e6	0.0574333
$871$	4.45266e7	1.98872
$872$	−2.61681e6	−0.116542
$873$	−3.29413e7	−1.46287
$874$	−727361.	−0.0322086
$875$	0	0
$876$	1.99009e7	0.876218
$877$	−2.15240e7	−0.944983	−0.472492	−	0.881335i	$-0.656645\pi$
$877$	−2.15240e7	−0.944983	−0.472492	+	0.881335i	$0.656645\pi$
$878$	−886666.	−0.0388172
$879$	−2.88102e7	−1.25769
$880$	4.88894e6	0.212818
$881$	3.32106e7	1.44157	0.720786	−	0.693157i	$-0.243781\pi$
$881$	3.32106e7	1.44157	0.720786	+	0.693157i	$0.243781\pi$
$882$	0	0
$883$	−4.37588e7	−1.88870	−0.944352	−	0.328936i	$-0.893310\pi$
$883$	−4.37588e7	−1.88870	−0.944352	+	0.328936i	$0.893310\pi$
$884$	−1.43216e7	−0.616395
$885$	−1.99246e7	−0.855128
$886$	29981.0	0.00128311
$887$	4.04382e7	1.72577	0.862886	−	0.505399i	$-0.168655\pi$
$887$	4.04382e7	1.72577	0.862886	+	0.505399i	$0.168655\pi$
$888$	−5.32384e6	−0.226565
$889$	0	0
$890$	−1.04136e6	−0.0440684
$891$	1.22629e7	0.517487
$892$	9.17555e6	0.386118
$893$	8.91178e6	0.373969
$894$	−1.66648e6	−0.0697359
$895$	−207399.	−0.00865465
$896$	0	0
$897$	4.44643e7	1.84515
$898$	1.67925e6	0.0694904
$899$	1.02922e6	0.0424727
$900$	−4.44632e6	−0.182976
$901$	5.89607e6	0.241964
$902$	−498824.	−0.0204141
$903$	0	0
$904$	601763.	0.0244909
$905$	8.84645e6	0.359044
$906$	934786.	0.0378348
$907$	−9.38936e6	−0.378981	−0.189491	−	0.981883i	$-0.560684\pi$
$907$	−9.38936e6	−0.378981	−0.189491	+	0.981883i	$0.560684\pi$
$908$	−1.57374e7	−0.633459
$909$	1.84191e6	0.0739366
$910$	0	0
$911$	3.19004e7	1.27350	0.636752	−	0.771068i	$-0.280277\pi$
$911$	3.19004e7	1.27350	0.636752	+	0.771068i	$0.280277\pi$
$912$	−1.52231e7	−0.606060
$913$	−1.55203e7	−0.616202
$914$	1.00084e6	0.0396277
$915$	−1.26414e7	−0.499163
$916$	−1.15088e7	−0.453200
$917$	0	0
$918$	97522.9	0.00381944
$919$	−1.38425e7	−0.540662	−0.270331	−	0.962767i	$-0.587133\pi$
$919$	−1.38425e7	−0.540662	−0.270331	+	0.962767i	$0.587133\pi$
$920$	−1.66756e6	−0.0649550
$921$	4.86654e6	0.189048
$922$	2.30162e6	0.0891675
$923$	−5.74635e7	−2.22018
$924$	0	0
$925$	−6.88373e6	−0.264527
$926$	−1.80495e6	−0.0691733
$927$	4.14860e7	1.58563
$928$	7.26011e6	0.276741
$929$	−1.41459e7	−0.537764	−0.268882	−	0.963173i	$-0.586654\pi$
$929$	−1.41459e7	−0.537764	−0.268882	+	0.963173i	$0.586654\pi$
$930$	28727.5	0.00108916
$931$	0	0
$932$	1.19504e6	0.0450654
$933$	4.58077e7	1.72280
$934$	−1.49206e6	−0.0559654
$935$	3.13939e6	0.117440
$936$	−3.45344e6	−0.128843
$937$	1.57848e7	0.587341	0.293670	−	0.955907i	$-0.405123\pi$
$937$	1.57848e7	0.587341	0.293670	+	0.955907i	$0.405123\pi$
$938$	0	0
$939$	−1.91312e7	−0.708072
$940$	1.01960e7	0.376367
$941$	−2.49437e7	−0.918305	−0.459153	−	0.888357i	$-0.651847\pi$
$941$	−2.49437e7	−0.918305	−0.459153	+	0.888357i	$0.651847\pi$
$942$	1.88693e6	0.0692833
$943$	−2.19494e7	−0.803790
$944$	−3.73639e7	−1.36465
$945$	0	0
$946$	1.30316e6	0.0473444
$947$	324473.	0.0117572	0.00587859	−	0.999983i	$-0.498129\pi$
$947$	324473.	0.0117572	0.00587859	+	0.999983i	$0.498129\pi$
$948$	1.16383e6	0.0420599
$949$	1.99858e7	0.720372
$950$	152579.	0.00548512
$951$	−808044.	−0.0289724
$952$	0	0
$953$	−1.45659e6	−0.0519522	−0.0259761	−	0.999663i	$-0.508269\pi$
$953$	−1.45659e6	−0.0519522	−0.0259761	+	0.999663i	$0.508269\pi$
$954$	709510.	0.0252399
$955$	9.94559e6	0.352876
$956$	−1.55681e7	−0.550923
$957$	2.82721e7	0.997881
$958$	1.38489e6	0.0487530
$959$	0	0
$960$	−1.72813e7	−0.605199
$961$	−2.86061e7	−0.999195
$962$	−2.66814e6	−0.0929547
$963$	−2.19949e7	−0.764286
$964$	9.12276e6	0.316180
$965$	348998.	0.0120644
$966$	0	0
$967$	−2.94281e7	−1.01204	−0.506018	−	0.862523i	$-0.668883\pi$
$967$	−2.94281e7	−1.01204	−0.506018	+	0.862523i	$0.668883\pi$
$968$	2.76993e6	0.0950126
$969$	−9.77538e6	−0.334445
$970$	1.29331e6	0.0441340
$971$	−1.18934e7	−0.404816	−0.202408	−	0.979301i	$-0.564877\pi$
$971$	−1.18934e7	−0.404816	−0.202408	+	0.979301i	$0.564877\pi$
$972$	−4.03705e7	−1.37056
$973$	0	0
$974$	513550.	0.0173454
$975$	−9.32732e6	−0.314228
$976$	−2.37060e7	−0.796588
$977$	2.96089e7	0.992399	0.496200	−	0.868208i	$-0.334728\pi$
$977$	2.96089e7	0.992399	0.496200	+	0.868208i	$0.334728\pi$
$978$	−1.13949e6	−0.0380947
$979$	−2.29614e7	−0.765671
$980$	0	0
$981$	2.60859e7	0.865432
$982$	−3.40832e6	−0.112788
$983$	−4.24526e7	−1.40126	−0.700632	−	0.713523i	$-0.747099\pi$
$983$	−4.24526e7	−1.40126	−0.700632	+	0.713523i	$0.747099\pi$
$984$	3.56097e6	0.117241
$985$	−7.15993e6	−0.235136
$986$	1.54403e6	0.0505782
$987$	0	0
$988$	−1.53474e7	−0.500200
$989$	5.73418e7	1.86415
$990$	377782.	0.0122505
$991$	−3.56197e7	−1.15214	−0.576071	−	0.817400i	$-0.695415\pi$
$991$	−3.56197e7	−1.15214	−0.576071	+	0.817400i	$0.695415\pi$
$992$	162659.	0.00524808
$993$	3.46642e7	1.11560
$994$	0	0
$995$	1.80983e7	0.579536
$996$	5.52910e7	1.76606
$997$	−3.62311e7	−1.15437	−0.577184	−	0.816614i	$-0.695848\pi$
$997$	−3.62311e7	−1.15437	−0.577184	+	0.816614i	$0.695848\pi$
$998$	−1.18618e6	−0.0376985
$999$	−4.71500e6	−0.149475

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.6.a.i.1.4		6
7.2	even	3		35.6.e.a.11.3	✓	12
7.4	even	3		35.6.e.a.16.3	yes	12
7.6	odd	2		245.6.a.h.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.6.e.a.11.3	✓	12	7.2	even	3
35.6.e.a.16.3	yes	12	7.4	even	3
245.6.a.h.1.4		6	7.6	odd	2
245.6.a.i.1.4		6	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:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

\(f(q)\)	\(=\)	\(q+0.350479 q^{2} -21.5910 q^{3} -31.8772 q^{4} -25.0000 q^{5} -7.56720 q^{6} -22.3876 q^{8} +223.173 q^{9} +O(q^{10})\)	\(q+0.350479 q^{2} -21.5910 q^{3} -31.8772 q^{4} -25.0000 q^{5} -7.56720 q^{6} -22.3876 q^{8} +223.173 q^{9} -8.76197 q^{10} -193.196 q^{11} +688.261 q^{12} +691.199 q^{13} +539.776 q^{15} +1012.22 q^{16} +649.991 q^{17} +78.2173 q^{18} +696.551 q^{19} +796.929 q^{20} -67.7112 q^{22} -2979.44 q^{23} +483.371 q^{24} +625.000 q^{25} +242.251 q^{26} +428.092 q^{27} +6777.77 q^{29} +189.180 q^{30} +151.853 q^{31} +1071.17 q^{32} +4171.30 q^{33} +227.808 q^{34} -7114.11 q^{36} -11014.0 q^{37} +244.126 q^{38} -14923.7 q^{39} +559.690 q^{40} +7366.94 q^{41} -19245.8 q^{43} +6158.54 q^{44} -5579.32 q^{45} -1044.23 q^{46} +12794.2 q^{47} -21854.9 q^{48} +219.049 q^{50} -14034.0 q^{51} -22033.5 q^{52} +9071.01 q^{53} +150.037 q^{54} +4829.90 q^{55} -15039.3 q^{57} +2375.46 q^{58} -36912.7 q^{59} -17206.5 q^{60} -23419.7 q^{61} +53.2212 q^{62} -32015.7 q^{64} -17280.0 q^{65} +1461.95 q^{66} +64419.4 q^{67} -20719.9 q^{68} +64329.2 q^{69} -83135.9 q^{71} -4996.30 q^{72} +28914.7 q^{73} -3860.16 q^{74} -13494.4 q^{75} -22204.1 q^{76} -5230.44 q^{78} +1690.97 q^{79} -25305.6 q^{80} -63473.9 q^{81} +2581.96 q^{82} +80334.4 q^{83} -16249.8 q^{85} -6745.25 q^{86} -146339. q^{87} +4325.20 q^{88} +118850. q^{89} -1955.43 q^{90} +94976.2 q^{92} -3278.66 q^{93} +4484.08 q^{94} -17413.8 q^{95} -23127.6 q^{96} -147605. q^{97} -43116.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9}+O(q^{10}) \) 6 * q - 5 * q^2 + 20 * q^3 + 31 * q^4 - 150 * q^5 + 96 * q^6 - 135 * q^8 + 378 * q^9	\( 6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9} + 125 q^{10} - 924 q^{11} + 370 q^{12} - 150 q^{13} - 500 q^{15} + 435 q^{16} + 1540 q^{17} + 195 q^{18} + 92 q^{19} - 775 q^{20} - 6855 q^{22} - 3920 q^{23} - 7200 q^{24} + 3750 q^{25} + 2635 q^{26} + 2060 q^{27} + 1264 q^{29} - 2400 q^{30} - 7160 q^{31} + 9105 q^{32} + 4460 q^{33} + 2166 q^{34} - 26375 q^{36} - 14170 q^{37} - 46215 q^{38} - 15376 q^{39} + 3375 q^{40} + 4098 q^{41} - 24460 q^{43} - 27873 q^{44} - 9450 q^{45} + 6815 q^{46} + 42940 q^{47} + 11610 q^{48} - 3125 q^{50} - 42008 q^{51} - 36115 q^{52} - 2450 q^{53} + 19566 q^{54} + 23100 q^{55} - 97100 q^{57} - 36110 q^{58} - 64600 q^{59} - 9250 q^{60} + 73620 q^{61} + 111440 q^{62} - 157997 q^{64} + 3750 q^{65} - 139138 q^{66} - 142620 q^{67} + 124330 q^{68} + 17344 q^{69} - 154256 q^{71} - 117495 q^{72} - 5120 q^{73} + 2785 q^{74} + 12500 q^{75} + 7775 q^{76} - 214090 q^{78} - 222504 q^{79} - 10875 q^{80} - 43986 q^{81} + 31665 q^{82} + 179580 q^{83} - 38500 q^{85} - 207160 q^{86} - 209300 q^{87} - 45145 q^{88} + 41648 q^{89} - 4875 q^{90} - 292185 q^{92} - 198520 q^{93} - 333699 q^{94} - 2300 q^{95} + 61824 q^{96} - 73980 q^{97} - 190772 q^{99}+O(q^{100}) \) 6 * q - 5 * q^2 + 20 * q^3 + 31 * q^4 - 150 * q^5 + 96 * q^6 - 135 * q^8 + 378 * q^9 + 125 * q^10 - 924 * q^11 + 370 * q^12 - 150 * q^13 - 500 * q^15 + 435 * q^16 + 1540 * q^17 + 195 * q^18 + 92 * q^19 - 775 * q^20 - 6855 * q^22 - 3920 * q^23 - 7200 * q^24 + 3750 * q^25 + 2635 * q^26 + 2060 * q^27 + 1264 * q^29 - 2400 * q^30 - 7160 * q^31 + 9105 * q^32 + 4460 * q^33 + 2166 * q^34 - 26375 * q^36 - 14170 * q^37 - 46215 * q^38 - 15376 * q^39 + 3375 * q^40 + 4098 * q^41 - 24460 * q^43 - 27873 * q^44 - 9450 * q^45 + 6815 * q^46 + 42940 * q^47 + 11610 * q^48 - 3125 * q^50 - 42008 * q^51 - 36115 * q^52 - 2450 * q^53 + 19566 * q^54 + 23100 * q^55 - 97100 * q^57 - 36110 * q^58 - 64600 * q^59 - 9250 * q^60 + 73620 * q^61 + 111440 * q^62 - 157997 * q^64 + 3750 * q^65 - 139138 * q^66 - 142620 * q^67 + 124330 * q^68 + 17344 * q^69 - 154256 * q^71 - 117495 * q^72 - 5120 * q^73 + 2785 * q^74 + 12500 * q^75 + 7775 * q^76 - 214090 * q^78 - 222504 * q^79 - 10875 * q^80 - 43986 * q^81 + 31665 * q^82 + 179580 * q^83 - 38500 * q^85 - 207160 * q^86 - 209300 * q^87 - 45145 * q^88 + 41648 * q^89 - 4875 * q^90 - 292185 * q^92 - 198520 * q^93 - 333699 * q^94 - 2300 * q^95 + 61824 * q^96 - 73980 * q^97 - 190772 * q^99

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