LMFDB - Embedded newform 245.6.a.j.1.3

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 225x^{6} + 537x^{5} + 15166x^{4} - 25016x^{3} - 317696x^{2} + 463952x + 1012704 $ x^8 - 3x^7 - 225x^6 + 537x^5 + 15166x^4 - 25016x^3 - 317696x^2 + 463952*x + 1012704
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{5}\cdot 7^{3} $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-6.63433$ 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)

$f(q)$	$=$	$q-6.63433 q^{2} +30.5785 q^{3} +12.0144 q^{4} +25.0000 q^{5} -202.868 q^{6} +132.591 q^{8} +692.045 q^{9} +O(q^{10})$	\(q-6.63433 q^{2} +30.5785 q^{3} +12.0144 q^{4} +25.0000 q^{5} -202.868 q^{6} +132.591 q^{8} +692.045 q^{9} -165.858 q^{10} +85.8033 q^{11} +367.382 q^{12} -324.642 q^{13} +764.463 q^{15} -1264.11 q^{16} +1828.83 q^{17} -4591.26 q^{18} -794.161 q^{19} +300.360 q^{20} -569.248 q^{22} +325.914 q^{23} +4054.44 q^{24} +625.000 q^{25} +2153.78 q^{26} +13731.1 q^{27} +4174.34 q^{29} -5071.70 q^{30} -686.406 q^{31} +4143.64 q^{32} +2623.74 q^{33} -12133.1 q^{34} +8314.51 q^{36} -2975.62 q^{37} +5268.73 q^{38} -9927.06 q^{39} +3314.78 q^{40} +10136.9 q^{41} -8438.98 q^{43} +1030.88 q^{44} +17301.1 q^{45} -2162.22 q^{46} -11892.4 q^{47} -38654.8 q^{48} -4146.46 q^{50} +55922.8 q^{51} -3900.37 q^{52} -31336.9 q^{53} -91096.9 q^{54} +2145.08 q^{55} -24284.3 q^{57} -27694.0 q^{58} +1042.10 q^{59} +9184.56 q^{60} -2001.38 q^{61} +4553.85 q^{62} +12961.4 q^{64} -8116.04 q^{65} -17406.8 q^{66} +20518.7 q^{67} +21972.3 q^{68} +9965.96 q^{69} +61262.0 q^{71} +91759.1 q^{72} +17727.0 q^{73} +19741.2 q^{74} +19111.6 q^{75} -9541.37 q^{76} +65859.4 q^{78} -30261.9 q^{79} -31602.9 q^{80} +251711. q^{81} -67251.8 q^{82} +41359.5 q^{83} +45720.7 q^{85} +55987.0 q^{86} +127645. q^{87} +11376.8 q^{88} +115229. q^{89} -114781. q^{90} +3915.66 q^{92} -20989.3 q^{93} +78898.4 q^{94} -19854.0 q^{95} +126706. q^{96} +18962.8 q^{97} +59379.8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{2} - 2 q^{3} + 203 q^{4} + 200 q^{5} - 56 q^{6} + 249 q^{8} + 1218 q^{9}+O(q^{10}) $ 8 * q + 3 * q^2 - 2 * q^3 + 203 * q^4 + 200 * q^5 - 56 * q^6 + 249 * q^8 + 1218 * q^9	\( 8 q + 3 q^{2} - 2 q^{3} + 203 q^{4} + 200 q^{5} - 56 q^{6} + 249 q^{8} + 1218 q^{9} + 75 q^{10} + 120 q^{11} - 884 q^{12} + 1994 q^{13} - 50 q^{15} + 6451 q^{16} + 1856 q^{17} + 2013 q^{18} - 1828 q^{19} + 5075 q^{20} + 2199 q^{22} + 4822 q^{23} - 2008 q^{24} + 5000 q^{25} - 7457 q^{26} + 7798 q^{27} + 10502 q^{29} - 1400 q^{30} + 5148 q^{31} + 12061 q^{32} - 2872 q^{33} - 40682 q^{34} + 40949 q^{36} + 7810 q^{37} + 18567 q^{38} - 2572 q^{39} + 6225 q^{40} - 18192 q^{41} + 63190 q^{43} - 4765 q^{44} + 30450 q^{45} + 6567 q^{46} + 11816 q^{47} - 107336 q^{48} + 1875 q^{50} + 28788 q^{51} + 167255 q^{52} - 39902 q^{53} - 126138 q^{54} + 3000 q^{55} + 38748 q^{57} + 67552 q^{58} + 50752 q^{59} - 22100 q^{60} - 2146 q^{61} + 101572 q^{62} + 270039 q^{64} + 49850 q^{65} + 164358 q^{66} + 50498 q^{67} - 125090 q^{68} - 118822 q^{69} + 183976 q^{71} + 522343 q^{72} + 54436 q^{73} + 228885 q^{74} - 1250 q^{75} - 65789 q^{76} + 391806 q^{78} + 51040 q^{79} + 161275 q^{80} + 124612 q^{81} - 195887 q^{82} + 60438 q^{83} + 46400 q^{85} + 260996 q^{86} + 409482 q^{87} + 205001 q^{88} - 96678 q^{89} + 50325 q^{90} + 174679 q^{92} - 51428 q^{93} - 139395 q^{94} - 45700 q^{95} - 587116 q^{96} - 195312 q^{97} + 397244 q^{99}+O(q^{100}) \) 8 * q + 3 * q^2 - 2 * q^3 + 203 * q^4 + 200 * q^5 - 56 * q^6 + 249 * q^8 + 1218 * q^9 + 75 * q^10 + 120 * q^11 - 884 * q^12 + 1994 * q^13 - 50 * q^15 + 6451 * q^16 + 1856 * q^17 + 2013 * q^18 - 1828 * q^19 + 5075 * q^20 + 2199 * q^22 + 4822 * q^23 - 2008 * q^24 + 5000 * q^25 - 7457 * q^26 + 7798 * q^27 + 10502 * q^29 - 1400 * q^30 + 5148 * q^31 + 12061 * q^32 - 2872 * q^33 - 40682 * q^34 + 40949 * q^36 + 7810 * q^37 + 18567 * q^38 - 2572 * q^39 + 6225 * q^40 - 18192 * q^41 + 63190 * q^43 - 4765 * q^44 + 30450 * q^45 + 6567 * q^46 + 11816 * q^47 - 107336 * q^48 + 1875 * q^50 + 28788 * q^51 + 167255 * q^52 - 39902 * q^53 - 126138 * q^54 + 3000 * q^55 + 38748 * q^57 + 67552 * q^58 + 50752 * q^59 - 22100 * q^60 - 2146 * q^61 + 101572 * q^62 + 270039 * q^64 + 49850 * q^65 + 164358 * q^66 + 50498 * q^67 - 125090 * q^68 - 118822 * q^69 + 183976 * q^71 + 522343 * q^72 + 54436 * q^73 + 228885 * q^74 - 1250 * q^75 - 65789 * q^76 + 391806 * q^78 + 51040 * q^79 + 161275 * q^80 + 124612 * q^81 - 195887 * q^82 + 60438 * q^83 + 46400 * q^85 + 260996 * q^86 + 409482 * q^87 + 205001 * q^88 - 96678 * q^89 + 50325 * q^90 + 174679 * q^92 - 51428 * q^93 - 139395 * q^94 - 45700 * q^95 - 587116 * q^96 - 195312 * q^97 + 397244 * 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 $

$2$	−6.63433	−1.17280	−0.586398	−	0.810023i	$-0.699454\pi$
$2$	−6.63433	−1.17280	−0.586398	+	0.810023i	$0.699454\pi$
$3$	30.5785	1.96161	0.980806	−	0.194985i	$-0.0624659\pi$
$3$	30.5785	1.96161	0.980806	+	0.194985i	$0.0624659\pi$
$4$	12.0144	0.375450
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	−202.868	−2.30057
$7$	0	0
$7$	0	0
$8$	132.591	0.732470
$9$	692.045	2.84792
$10$	−165.858	−0.524490
$11$	85.8033	0.213807	0.106904	−	0.994269i	$-0.465906\pi$
$11$	85.8033	0.213807	0.106904	+	0.994269i	$0.465906\pi$
$12$	367.382	0.736487
$13$	−324.642	−0.532777	−0.266389	−	0.963866i	$-0.585830\pi$
$13$	−324.642	−0.532777	−0.266389	+	0.963866i	$0.585830\pi$
$14$	0	0
$15$	764.463	0.877260
$16$	−1264.11	−1.23449
$17$	1828.83	1.53480	0.767398	−	0.641171i	$-0.221551\pi$
$17$	1828.83	1.53480	0.767398	+	0.641171i	$0.221551\pi$
$18$	−4591.26	−3.34003
$19$	−794.161	−0.504690	−0.252345	−	0.967637i	$-0.581202\pi$
$19$	−794.161	−0.504690	−0.252345	+	0.967637i	$0.581202\pi$
$20$	300.360	0.167906
$21$	0	0
$22$	−569.248	−0.250752
$23$	325.914	0.128465	0.0642323	−	0.997935i	$-0.479540\pi$
$23$	325.914	0.128465	0.0642323	+	0.997935i	$0.479540\pi$
$24$	4054.44	1.43682
$25$	625.000	0.200000
$26$	2153.78	0.624839
$27$	13731.1	3.62491
$28$	0	0
$29$	4174.34	0.921708	0.460854	−	0.887476i	$-0.347543\pi$
$29$	4174.34	0.921708	0.460854	+	0.887476i	$0.347543\pi$
$30$	−5071.70	−1.02885
$31$	−686.406	−0.128285	−0.0641427	−	0.997941i	$-0.520431\pi$
$31$	−686.406	−0.128285	−0.0641427	+	0.997941i	$0.520431\pi$
$32$	4143.64	0.715332
$33$	2623.74	0.419407
$34$	−12133.1	−1.80000
$35$	0	0
$36$	8314.51	1.06925
$37$	−2975.62	−0.357332	−0.178666	−	0.983910i	$-0.557178\pi$
$37$	−2975.62	−0.357332	−0.178666	+	0.983910i	$0.557178\pi$
$38$	5268.73	0.591898
$39$	−9927.06	−1.04510
$40$	3314.78	0.327570
$41$	10136.9	0.941774	0.470887	−	0.882193i	$-0.343934\pi$
$41$	10136.9	0.941774	0.470887	+	0.882193i	$0.343934\pi$
$42$	0	0
$43$	−8438.98	−0.696015	−0.348008	−	0.937492i	$-0.613142\pi$
$43$	−8438.98	−0.696015	−0.348008	+	0.937492i	$0.613142\pi$
$44$	1030.88	0.0802739
$45$	17301.1	1.27363
$46$	−2162.22	−0.150663
$47$	−11892.4	−0.785283	−0.392642	−	0.919692i	$-0.628439\pi$
$47$	−11892.4	−0.785283	−0.392642	+	0.919692i	$0.628439\pi$
$48$	−38654.8	−2.42159
$49$	0	0
$50$	−4146.46	−0.234559
$51$	55922.8	3.01067
$52$	−3900.37	−0.200031
$53$	−31336.9	−1.53238	−0.766189	−	0.642615i	$-0.777849\pi$
$53$	−31336.9	−1.53238	−0.766189	+	0.642615i	$0.777849\pi$
$54$	−91096.9	−4.25128
$55$	2145.08	0.0956175
$56$	0	0
$57$	−24284.3	−0.990006
$58$	−27694.0	−1.08098
$59$	1042.10	0.0389743	0.0194871	−	0.999810i	$-0.493797\pi$
$59$	1042.10	0.0389743	0.0194871	+	0.999810i	$0.493797\pi$
$60$	9184.56	0.329367
$61$	−2001.38	−0.0688659	−0.0344330	−	0.999407i	$-0.510963\pi$
$61$	−2001.38	−0.0688659	−0.0344330	+	0.999407i	$0.510963\pi$
$62$	4553.85	0.150452
$63$	0	0
$64$	12961.4	0.395549
$65$	−8116.04	−0.238265
$66$	−17406.8	−0.491879
$67$	20518.7	0.558423	0.279211	−	0.960230i	$-0.409927\pi$
$67$	20518.7	0.558423	0.279211	+	0.960230i	$0.409927\pi$
$68$	21972.3	0.576239
$69$	9965.96	0.251998
$70$	0	0
$71$	61262.0	1.44227	0.721133	−	0.692796i	$-0.243621\pi$
$71$	61262.0	1.44227	0.721133	+	0.692796i	$0.243621\pi$
$72$	91759.1	2.08602
$73$	17727.0	0.389339	0.194670	−	0.980869i	$-0.437637\pi$
$73$	17727.0	0.389339	0.194670	+	0.980869i	$0.437637\pi$
$74$	19741.2	0.419078
$75$	19111.6	0.392322
$76$	−9541.37	−0.189486
$77$	0	0
$78$	65859.4	1.22569
$79$	−30261.9	−0.545542	−0.272771	−	0.962079i	$-0.587940\pi$
$79$	−30261.9	−0.545542	−0.272771	+	0.962079i	$0.587940\pi$
$80$	−31602.9	−0.552079
$81$	251711.	4.26274
$82$	−67251.8	−1.10451
$83$	41359.5	0.658992	0.329496	−	0.944157i	$-0.393121\pi$
$83$	41359.5	0.658992	0.329496	+	0.944157i	$0.393121\pi$
$84$	0	0
$85$	45720.7	0.686382
$86$	55987.0	0.816283
$87$	127645.	1.80803
$88$	11376.8	0.156607
$89$	115229.	1.54201	0.771006	−	0.636828i	$-0.219754\pi$
$89$	115229.	1.54201	0.771006	+	0.636828i	$0.219754\pi$
$90$	−114781.	−1.49371
$91$	0	0
$92$	3915.66	0.0482320
$93$	−20989.3	−0.251646
$94$	78898.4	0.920977
$95$	−19854.0	−0.225704
$96$	126706.	1.40320
$97$	18962.8	0.204632	0.102316	−	0.994752i	$-0.467375\pi$
$97$	18962.8	0.204632	0.102316	+	0.994752i	$0.467375\pi$
$98$	0	0
$99$	59379.8	0.608907
$100$	7509.00	0.0750900
$101$	−121043.	−1.18069	−0.590346	−	0.807151i	$-0.701009\pi$
$101$	−121043.	−1.18069	−0.590346	+	0.807151i	$0.701009\pi$
$102$	−371011.	−3.53091
$103$	93113.2	0.864805	0.432402	−	0.901681i	$-0.357666\pi$
$103$	93113.2	0.864805	0.432402	+	0.901681i	$0.357666\pi$
$104$	−43044.6	−0.390243
$105$	0	0
$106$	207899.	1.79717
$107$	−21586.9	−0.182277	−0.0911384	−	0.995838i	$-0.529051\pi$
$107$	−21586.9	−0.182277	−0.0911384	+	0.995838i	$0.529051\pi$
$108$	164971.	1.36097
$109$	−21898.4	−0.176541	−0.0882705	−	0.996097i	$-0.528134\pi$
$109$	−21898.4	−0.176541	−0.0882705	+	0.996097i	$0.528134\pi$
$110$	−14231.2	−0.112140
$111$	−90989.9	−0.700948
$112$	0	0
$113$	−66325.7	−0.488636	−0.244318	−	0.969695i	$-0.578564\pi$
$113$	−66325.7	−0.488636	−0.244318	+	0.969695i	$0.578564\pi$
$114$	161110.	1.16107
$115$	8147.84	0.0574511
$116$	50152.2	0.346055
$117$	−224667.	−1.51731
$118$	−6913.62	−0.0457089
$119$	0	0
$120$	101361.	0.642566
$121$	−153689.	−0.954286
$122$	13277.8	0.0807657
$123$	309972.	1.84740
$124$	−8246.76	−0.0481647
$125$	15625.0	0.0894427
$126$	0	0
$127$	224823.	1.23689	0.618446	−	0.785827i	$-0.287763\pi$
$127$	224823.	1.23689	0.618446	+	0.785827i	$0.287763\pi$
$128$	−218587.	−1.17923
$129$	−258051.	−1.36531
$130$	53844.5	0.279437
$131$	310838.	1.58254	0.791271	−	0.611466i	$-0.209420\pi$
$131$	310838.	1.58254	0.791271	+	0.611466i	$0.209420\pi$
$132$	31522.6	0.157466
$133$	0	0
$134$	−136128.	−0.654916
$135$	343278.	1.62111
$136$	242486.	1.12419
$137$	−17065.4	−0.0776811	−0.0388405	−	0.999245i	$-0.512366\pi$
$137$	−17065.4	−0.0776811	−0.0388405	+	0.999245i	$0.512366\pi$
$138$	−66117.5	−0.295542
$139$	−123482.	−0.542082	−0.271041	−	0.962568i	$-0.587368\pi$
$139$	−123482.	−0.542082	−0.271041	+	0.962568i	$0.587368\pi$
$140$	0	0
$141$	−363653.	−1.54042
$142$	−406433.	−1.69148
$143$	−27855.3	−0.113912
$144$	−874825.	−3.51572
$145$	104359.	0.412200
$146$	−117607.	−0.456616
$147$	0	0
$148$	−35750.2	−0.134160
$149$	32706.3	0.120689	0.0603443	−	0.998178i	$-0.480780\pi$
$149$	32706.3	0.120689	0.0603443	+	0.998178i	$0.480780\pi$
$150$	−126793.	−0.460114
$151$	449087.	1.60283	0.801417	−	0.598107i	$-0.204080\pi$
$151$	449087.	1.60283	0.801417	+	0.598107i	$0.204080\pi$
$152$	−105299.	−0.369670
$153$	1.26563e6	4.37098
$154$	0	0
$155$	−17160.2	−0.0573709
$156$	−119268.	−0.392384
$157$	−31696.8	−0.102628	−0.0513140	−	0.998683i	$-0.516341\pi$
$157$	−31696.8	−0.102628	−0.0513140	+	0.998683i	$0.516341\pi$
$158$	200767.	0.639809
$159$	−958235.	−3.00593
$160$	103591.	0.319906
$161$	0	0
$162$	−1.66993e6	−4.99933
$163$	501206.	1.47757	0.738784	−	0.673942i	$-0.235400\pi$
$163$	501206.	1.47757	0.738784	+	0.673942i	$0.235400\pi$
$164$	121789.	0.353589
$165$	65593.4	0.187564
$166$	−274393.	−0.772863
$167$	295459.	0.819798	0.409899	−	0.912131i	$-0.365564\pi$
$167$	295459.	0.819798	0.409899	+	0.912131i	$0.365564\pi$
$168$	0	0
$169$	−265901.	−0.716148
$170$	−303326.	−0.804985
$171$	−549595.	−1.43732
$172$	−101389.	−0.261319
$173$	55664.7	0.141405	0.0707025	−	0.997497i	$-0.477476\pi$
$173$	55664.7	0.141405	0.0707025	+	0.997497i	$0.477476\pi$
$174$	−846841.	−2.12045
$175$	0	0
$176$	−108465.	−0.263942
$177$	31865.8	0.0764524
$178$	−764469.	−1.80846
$179$	−508702.	−1.18667	−0.593336	−	0.804955i	$-0.702189\pi$
$179$	−508702.	−1.18667	−0.593336	+	0.804955i	$0.702189\pi$
$180$	207863.	0.478184
$181$	−725751.	−1.64661	−0.823306	−	0.567598i	$-0.807873\pi$
$181$	−725751.	−1.64661	−0.823306	+	0.567598i	$0.807873\pi$
$182$	0	0
$183$	−61199.1	−0.135088
$184$	43213.3	0.0940964
$185$	−74390.4	−0.159804
$186$	139250.	0.295129
$187$	156920.	0.328150
$188$	−142880.	−0.294834
$189$	0	0
$190$	131718.	0.264705
$191$	−590207.	−1.17063	−0.585317	−	0.810805i	$-0.699030\pi$
$191$	−590207.	−1.17063	−0.585317	+	0.810805i	$0.699030\pi$
$192$	396339.	0.775914
$193$	−394221.	−0.761810	−0.380905	−	0.924614i	$-0.624387\pi$
$193$	−394221.	−0.761810	−0.380905	+	0.924614i	$0.624387\pi$
$194$	−125805.	−0.239991
$195$	−248176.	−0.467384
$196$	0	0
$197$	252511.	0.463569	0.231785	−	0.972767i	$-0.425544\pi$
$197$	252511.	0.463569	0.231785	+	0.972767i	$0.425544\pi$
$198$	−393945.	−0.714123
$199$	−852191.	−1.52547	−0.762736	−	0.646710i	$-0.776145\pi$
$199$	−852191.	−1.52547	−0.762736	+	0.646710i	$0.776145\pi$
$200$	82869.5	0.146494
$201$	627431.	1.09541
$202$	803040.	1.38471
$203$	0	0
$204$	671879.	1.13036
$205$	253423.	0.421174
$206$	−617744.	−1.01424
$207$	225547.	0.365857
$208$	410384.	0.657707
$209$	−68141.7	−0.107906
$210$	0	0
$211$	452398.	0.699544	0.349772	−	0.936835i	$-0.386259\pi$
$211$	452398.	0.699544	0.349772	+	0.936835i	$0.386259\pi$
$212$	−376494.	−0.575331
$213$	1.87330e6	2.82917
$214$	143215.	0.213773
$215$	−210974.	−0.311267
$216$	1.82063e6	2.65514
$217$	0	0
$218$	145281.	0.207046
$219$	542065.	0.763733
$220$	25771.9	0.0358996
$221$	−593714.	−0.817705
$222$	603657.	0.822069
$223$	−812327.	−1.09388	−0.546939	−	0.837173i	$-0.684207\pi$
$223$	−812327.	−1.09388	−0.546939	+	0.837173i	$0.684207\pi$
$224$	0	0
$225$	432528.	0.569585
$226$	440027.	0.573071
$227$	−1.26725e6	−1.63229	−0.816145	−	0.577847i	$-0.803893\pi$
$227$	−1.26725e6	−1.63229	−0.816145	+	0.577847i	$0.803893\pi$
$228$	−291761.	−0.371698
$229$	96862.3	0.122058	0.0610290	−	0.998136i	$-0.480562\pi$
$229$	96862.3	0.122058	0.0610290	+	0.998136i	$0.480562\pi$
$230$	−54055.5	−0.0673784
$231$	0	0
$232$	553481.	0.675123
$233$	−1.30610e6	−1.57611	−0.788054	−	0.615606i	$-0.788911\pi$
$233$	−1.30610e6	−1.57611	−0.788054	+	0.615606i	$0.788911\pi$
$234$	1.49051e6	1.77949
$235$	−297311.	−0.351189
$236$	12520.2	0.0146329
$237$	−925363.	−1.07014
$238$	0	0
$239$	−506413.	−0.573469	−0.286734	−	0.958010i	$-0.592570\pi$
$239$	−506413.	−0.573469	−0.286734	+	0.958010i	$0.592570\pi$
$240$	−966369.	−1.08297
$241$	818155.	0.907388	0.453694	−	0.891158i	$-0.350106\pi$
$241$	818155.	0.907388	0.453694	+	0.891158i	$0.350106\pi$
$242$	1.01962e6	1.11918
$243$	4.36027e6	4.73694
$244$	−24045.3	−0.0258557
$245$	0	0
$246$	−2.05646e6	−2.16662
$247$	257818.	0.268887
$248$	−91011.4	−0.0939651
$249$	1.26471e6	1.29269
$250$	−103661.	−0.104898
$251$	599979.	0.601107	0.300554	−	0.953765i	$-0.402829\pi$
$251$	599979.	0.601107	0.300554	+	0.953765i	$0.402829\pi$
$252$	0	0
$253$	27964.5	0.0274666
$254$	−1.49155e6	−1.45062
$255$	1.39807e6	1.34641
$256$	1.03541e6	0.987447
$257$	−1.43571e6	−1.35592	−0.677962	−	0.735097i	$-0.737137\pi$
$257$	−1.43571e6	−1.35592	−0.677962	+	0.735097i	$0.737137\pi$
$258$	1.71200e6	1.60123
$259$	0	0
$260$	−97509.3	−0.0894567
$261$	2.88884e6	2.62495
$262$	−2.06220e6	−1.85600
$263$	377192.	0.336258	0.168129	−	0.985765i	$-0.446227\pi$
$263$	377192.	0.336258	0.168129	+	0.985765i	$0.446227\pi$
$264$	347885.	0.307203
$265$	−783422.	−0.685300
$266$	0	0
$267$	3.52354e6	3.02483
$268$	246520.	0.209660
$269$	−943813.	−0.795253	−0.397627	−	0.917547i	$-0.630166\pi$
$269$	−943813.	−0.795253	−0.397627	+	0.917547i	$0.630166\pi$
$270$	−2.27742e6	−1.90123
$271$	−1.84724e6	−1.52792	−0.763959	−	0.645265i	$-0.776747\pi$
$271$	−1.84724e6	−1.52792	−0.763959	+	0.645265i	$0.776747\pi$
$272$	−2.31185e6	−1.89469
$273$	0	0
$274$	113218.	0.0911040
$275$	53627.1	0.0427615
$276$	119735.	0.0946125
$277$	1.72957e6	1.35438	0.677189	−	0.735809i	$-0.263198\pi$
$277$	1.72957e6	1.35438	0.677189	+	0.735809i	$0.263198\pi$
$278$	819218.	0.635752
$279$	−475024.	−0.365347
$280$	0	0
$281$	−1.35071e6	−1.02046	−0.510231	−	0.860038i	$-0.670440\pi$
$281$	−1.35071e6	−1.02046	−0.510231	+	0.860038i	$0.670440\pi$
$282$	2.41260e6	1.80660
$283$	−1.16652e6	−0.865819	−0.432910	−	0.901437i	$-0.642513\pi$
$283$	−1.16652e6	−0.865819	−0.432910	+	0.901437i	$0.642513\pi$
$284$	736027.	0.541499
$285$	−607107.	−0.442744
$286$	184802.	0.133595
$287$	0	0
$288$	2.86759e6	2.03721
$289$	1.92476e6	1.35560
$290$	−692350.	−0.483427
$291$	579854.	0.401408
$292$	212979.	0.146177
$293$	−1.21134e6	−0.824321	−0.412160	−	0.911111i	$-0.635226\pi$
$293$	−1.21134e6	−0.824321	−0.412160	+	0.911111i	$0.635226\pi$
$294$	0	0
$295$	26052.4	0.0174298
$296$	−394540.	−0.261735
$297$	1.17818e6	0.775032
$298$	−216985.	−0.141543
$299$	−105805.	−0.0684430
$300$	229614.	0.147297
$301$	0	0
$302$	−2.97939e6	−1.87980
$303$	−3.70132e6	−2.31606
$304$	1.00391e6	0.623033
$305$	−50034.4	−0.0307978
$306$	−8.39663e6	−5.12627
$307$	−1.38256e6	−0.837220	−0.418610	−	0.908166i	$-0.637483\pi$
$307$	−1.38256e6	−0.837220	−0.418610	+	0.908166i	$0.637483\pi$
$308$	0	0
$309$	2.84726e6	1.69641
$310$	113846.	0.0672844
$311$	−2.88612e6	−1.69205	−0.846027	−	0.533140i	$-0.821012\pi$
$311$	−2.88612e6	−1.69205	−0.846027	+	0.533140i	$0.821012\pi$
$312$	−1.31624e6	−0.765506
$313$	−2.16417e6	−1.24862	−0.624309	−	0.781177i	$-0.714619\pi$
$313$	−2.16417e6	−1.24862	−0.624309	+	0.781177i	$0.714619\pi$
$314$	210287.	0.120362
$315$	0	0
$316$	−363578.	−0.204824
$317$	1.55130e6	0.867056	0.433528	−	0.901140i	$-0.357268\pi$
$317$	1.55130e6	0.867056	0.433528	+	0.901140i	$0.357268\pi$
$318$	6.35725e6	3.52534
$319$	358173.	0.197068
$320$	324034.	0.176895
$321$	−660096.	−0.357556
$322$	0	0
$323$	−1.45238e6	−0.774596
$324$	3.02415e6	1.60045
$325$	−202901.	−0.106555
$326$	−3.32517e6	−1.73289
$327$	−669619.	−0.346305
$328$	1.34407e6	0.689821
$329$	0	0
$330$	−435169.	−0.219975
$331$	−1.98219e6	−0.994431	−0.497216	−	0.867627i	$-0.665644\pi$
$331$	−1.98219e6	−0.994431	−0.497216	+	0.867627i	$0.665644\pi$
$332$	496910.	0.247419
$333$	−2.05926e6	−1.01766
$334$	−1.96018e6	−0.961455
$335$	512968.	0.249734
$336$	0	0
$337$	727886.	0.349131	0.174565	−	0.984646i	$-0.444148\pi$
$337$	727886.	0.349131	0.174565	+	0.984646i	$0.444148\pi$
$338$	1.76407e6	0.839896
$339$	−2.02814e6	−0.958515
$340$	549307.	0.257702
$341$	−58895.9	−0.0274283
$342$	3.64620e6	1.68568
$343$	0	0
$344$	−1.11893e6	−0.509810
$345$	249149.	0.112697
$346$	−369298.	−0.165839
$347$	3.66140e6	1.63239	0.816194	−	0.577777i	$-0.196080\pi$
$347$	3.66140e6	1.63239	0.816194	+	0.577777i	$0.196080\pi$
$348$	1.53358e6	0.678826
$349$	2.41958e6	1.06335	0.531674	−	0.846949i	$-0.321563\pi$
$349$	2.41958e6	1.06335	0.531674	+	0.846949i	$0.321563\pi$
$350$	0	0
$351$	−4.45770e6	−1.93127
$352$	355538.	0.152943
$353$	1.27768e6	0.545741	0.272871	−	0.962051i	$-0.412027\pi$
$353$	1.27768e6	0.545741	0.272871	+	0.962051i	$0.412027\pi$
$354$	−211408.	−0.0896631
$355$	1.53155e6	0.645001
$356$	1.38441e6	0.578948
$357$	0	0
$358$	3.37490e6	1.39172
$359$	−1.98231e6	−0.811775	−0.405887	−	0.913923i	$-0.633037\pi$
$359$	−1.98231e6	−0.811775	−0.405887	+	0.913923i	$0.633037\pi$
$360$	2.29398e6	0.932895
$361$	−1.84541e6	−0.745288
$362$	4.81487e6	1.93114
$363$	−4.69957e6	−1.87194
$364$	0	0
$365$	443175.	0.174118
$366$	406016.	0.158431
$367$	2.82732e6	1.09575	0.547874	−	0.836561i	$-0.315437\pi$
$367$	2.82732e6	1.09575	0.547874	+	0.836561i	$0.315437\pi$
$368$	−411992.	−0.158588
$369$	7.01522e6	2.68210
$370$	493531.	0.187417
$371$	0	0
$372$	−252173.	−0.0944805
$373$	410264.	0.152683	0.0763416	−	0.997082i	$-0.475676\pi$
$373$	410264.	0.152683	0.0763416	+	0.997082i	$0.475676\pi$
$374$	−1.04106e6	−0.384853
$375$	477789.	0.175452
$376$	−1.57683e6	−0.575196
$377$	−1.35517e6	−0.491065
$378$	0	0
$379$	494501.	0.176835	0.0884177	−	0.996083i	$-0.471819\pi$
$379$	494501.	0.176835	0.0884177	+	0.996083i	$0.471819\pi$
$380$	−238534.	−0.0847406
$381$	6.87476e6	2.42630
$382$	3.91563e6	1.37291
$383$	−2.57122e6	−0.895659	−0.447829	−	0.894119i	$-0.647803\pi$
$383$	−2.57122e6	−0.895659	−0.447829	+	0.894119i	$0.647803\pi$
$384$	−6.68405e6	−2.31319
$385$	0	0
$386$	2.61539e6	0.893447
$387$	−5.84015e6	−1.98220
$388$	227826.	0.0768289
$389$	−3.05022e6	−1.02201	−0.511007	−	0.859576i	$-0.670728\pi$
$389$	−3.05022e6	−1.02201	−0.511007	+	0.859576i	$0.670728\pi$
$390$	1.64649e6	0.548146
$391$	596040.	0.197167
$392$	0	0
$393$	9.50495e6	3.10433
$394$	−1.67524e6	−0.543672
$395$	−756547.	−0.243974
$396$	713412.	0.228614
$397$	3.47152e6	1.10546	0.552730	−	0.833360i	$-0.313586\pi$
$397$	3.47152e6	1.10546	0.552730	+	0.833360i	$0.313586\pi$
$398$	5.65372e6	1.78907
$399$	0	0
$400$	−790072.	−0.246897
$401$	140919.	0.0437630	0.0218815	−	0.999761i	$-0.493034\pi$
$401$	140919.	0.0437630	0.0218815	+	0.999761i	$0.493034\pi$
$402$	−4.16259e6	−1.28469
$403$	222836.	0.0683475
$404$	−1.45426e6	−0.443290
$405$	6.29277e6	1.90636
$406$	0	0
$407$	−255318.	−0.0764003
$408$	7.41488e6	2.20523
$409$	699346.	0.206721	0.103360	−	0.994644i	$-0.467041\pi$
$409$	699346.	0.206721	0.103360	+	0.994644i	$0.467041\pi$
$410$	−1.68130e6	−0.493951
$411$	−521835.	−0.152380
$412$	1.11870e6	0.324691
$413$	0	0
$414$	−1.49635e6	−0.429076
$415$	1.03399e6	0.294710
$416$	−1.34520e6	−0.381113
$417$	−3.77588e6	−1.06335
$418$	452075.	0.126552
$419$	−228295.	−0.0635274	−0.0317637	−	0.999495i	$-0.510112\pi$
$419$	−228295.	−0.0635274	−0.0317637	+	0.999495i	$0.510112\pi$
$420$	0	0
$421$	−2.57951e6	−0.709302	−0.354651	−	0.934999i	$-0.615400\pi$
$421$	−2.57951e6	−0.709302	−0.354651	+	0.934999i	$0.615400\pi$
$422$	−3.00136e6	−0.820422
$423$	−8.23011e6	−2.23643
$424$	−4.15499e6	−1.12242
$425$	1.14302e6	0.306959
$426$	−1.24281e7	−3.31804
$427$	0	0
$428$	−259354.	−0.0684358
$429$	−851775.	−0.223451
$430$	1.39967e6	0.365053
$431$	−4.36730e6	−1.13245	−0.566226	−	0.824250i	$-0.691597\pi$
$431$	−4.36730e6	−1.13245	−0.566226	+	0.824250i	$0.691597\pi$
$432$	−1.73577e7	−4.47490
$433$	−1.40410e6	−0.359896	−0.179948	−	0.983676i	$-0.557593\pi$
$433$	−1.40410e6	−0.359896	−0.179948	+	0.983676i	$0.557593\pi$
$434$	0	0
$435$	3.19113e6	0.808577
$436$	−263096.	−0.0662823
$437$	−258828.	−0.0648347
$438$	−3.59624e6	−0.895703
$439$	−4.81275e6	−1.19188	−0.595940	−	0.803029i	$-0.703220\pi$
$439$	−4.81275e6	−1.19188	−0.595940	+	0.803029i	$0.703220\pi$
$440$	284419.	0.0700369
$441$	0	0
$442$	3.93890e6	0.959001
$443$	−3.16968e6	−0.767372	−0.383686	−	0.923464i	$-0.625346\pi$
$443$	−3.16968e6	−0.767372	−0.383686	+	0.923464i	$0.625346\pi$
$444$	−1.09319e6	−0.263171
$445$	2.88073e6	0.689609
$446$	5.38925e6	1.28289
$447$	1.00011e6	0.236744
$448$	0	0
$449$	−6.35197e6	−1.48694	−0.743469	−	0.668771i	$-0.766821\pi$
$449$	−6.35197e6	−1.48694	−0.743469	+	0.668771i	$0.766821\pi$
$450$	−2.86954e6	−0.668006
$451$	869783.	0.201358
$452$	−796864.	−0.183459
$453$	1.37324e7	3.14414
$454$	8.40736e6	1.91434
$455$	0	0
$456$	−3.21988e6	−0.725149
$457$	7.85181e6	1.75865	0.879325	−	0.476223i	$-0.157994\pi$
$457$	7.85181e6	1.75865	0.879325	+	0.476223i	$0.157994\pi$
$458$	−642617.	−0.143149
$459$	2.51119e7	5.56349
$460$	97891.4	0.0215700
$461$	−1.98815e6	−0.435710	−0.217855	−	0.975981i	$-0.569906\pi$
$461$	−1.98815e6	−0.435710	−0.217855	+	0.975981i	$0.569906\pi$
$462$	0	0
$463$	415445.	0.0900661	0.0450330	−	0.998985i	$-0.485661\pi$
$463$	415445.	0.0900661	0.0450330	+	0.998985i	$0.485661\pi$
$464$	−5.27685e6	−1.13784
$465$	−524732.	−0.112540
$466$	8.66509e6	1.84845
$467$	8.00336e6	1.69817	0.849083	−	0.528259i	$-0.177155\pi$
$467$	8.00336e6	1.69817	0.849083	+	0.528259i	$0.177155\pi$
$468$	−2.69924e6	−0.569674
$469$	0	0
$470$	1.97246e6	0.411873
$471$	−969241.	−0.201317
$472$	138173.	0.0285475
$473$	−724092.	−0.148813
$474$	6.13917e6	1.25506
$475$	−496351.	−0.100938
$476$	0	0
$477$	−2.16865e7	−4.36409
$478$	3.35971e6	0.672562
$479$	71170.0	0.0141729	0.00708644	−	0.999975i	$-0.497744\pi$
$479$	71170.0	0.0141729	0.00708644	+	0.999975i	$0.497744\pi$
$480$	3.16766e6	0.627532
$481$	966009.	0.190379
$482$	−5.42791e6	−1.06418
$483$	0	0
$484$	−1.84648e6	−0.358287
$485$	474070.	0.0915141
$486$	−2.89275e7	−5.55546
$487$	−4.58222e6	−0.875494	−0.437747	−	0.899098i	$-0.644223\pi$
$487$	−4.58222e6	−0.875494	−0.437747	+	0.899098i	$0.644223\pi$
$488$	−265365.	−0.0504422
$489$	1.53261e7	2.89842
$490$	0	0
$491$	5.97569e6	1.11863	0.559313	−	0.828957i	$-0.311065\pi$
$491$	5.97569e6	1.11863	0.559313	+	0.828957i	$0.311065\pi$
$492$	3.72413e6	0.693605
$493$	7.63416e6	1.41463
$494$	−1.71045e6	−0.315350
$495$	1.48449e6	0.272311
$496$	867696.	0.158367
$497$	0	0
$498$	−8.39053e6	−1.51606
$499$	5.06213e6	0.910085	0.455043	−	0.890470i	$-0.349624\pi$
$499$	5.06213e6	0.910085	0.455043	+	0.890470i	$0.349624\pi$
$500$	187725.	0.0335813
$501$	9.03471e6	1.60813
$502$	−3.98046e6	−0.704976
$503$	−1.54242e6	−0.271820	−0.135910	−	0.990721i	$-0.543396\pi$
$503$	−1.54242e6	−0.271820	−0.135910	+	0.990721i	$0.543396\pi$
$504$	0	0
$505$	−3.02608e6	−0.528021
$506$	−185526.	−0.0322128
$507$	−8.13085e6	−1.40481
$508$	2.70112e6	0.464391
$509$	−4.21778e6	−0.721588	−0.360794	−	0.932645i	$-0.617494\pi$
$509$	−4.21778e6	−0.721588	−0.360794	+	0.932645i	$0.617494\pi$
$510$	−9.27527e6	−1.57907
$511$	0	0
$512$	125493.	0.0211566
$513$	−1.09047e7	−1.82945
$514$	9.52501e6	1.59022
$515$	2.32783e6	0.386752
$516$	−3.10033e6	−0.512606
$517$	−1.02041e6	−0.167899
$518$	0	0
$519$	1.70214e6	0.277382
$520$	−1.07612e6	−0.174522
$521$	3.23776e6	0.522577	0.261289	−	0.965261i	$-0.415853\pi$
$521$	3.23776e6	0.522577	0.261289	+	0.965261i	$0.415853\pi$
$522$	−1.91655e7	−3.07853
$523$	−8.24760e6	−1.31848	−0.659240	−	0.751933i	$-0.729122\pi$
$523$	−8.24760e6	−1.31848	−0.659240	+	0.751933i	$0.729122\pi$
$524$	3.73453e6	0.594165
$525$	0	0
$526$	−2.50242e6	−0.394363
$527$	−1.25532e6	−0.196892
$528$	−3.31671e6	−0.517753
$529$	−6.33012e6	−0.983497
$530$	5.19748e6	0.803717
$531$	721178.	0.110996
$532$	0	0
$533$	−3.29087e6	−0.501756
$534$	−2.33763e7	−3.54751
$535$	−539673.	−0.0815166
$536$	2.72060e6	0.409028
$537$	−1.55553e7	−2.32779
$538$	6.26157e6	0.932669
$539$	0	0
$540$	4.12428e6	0.608645
$541$	1.30093e7	1.91099	0.955497	−	0.295002i	$-0.0953203\pi$
$541$	1.30093e7	1.91099	0.955497	+	0.295002i	$0.0953203\pi$
$542$	1.22552e7	1.79193
$543$	−2.21924e7	−3.23001
$544$	7.57801e6	1.09789
$545$	−547459.	−0.0789515
$546$	0	0
$547$	1.06475e7	1.52152	0.760762	−	0.649031i	$-0.224825\pi$
$547$	1.06475e7	1.52152	0.760762	+	0.649031i	$0.224825\pi$
$548$	−205031.	−0.0291653
$549$	−1.38504e6	−0.196125
$550$	−355780.	−0.0501504
$551$	−3.31510e6	−0.465177
$552$	1.32140e6	0.184581
$553$	0	0
$554$	−1.14746e7	−1.58841
$555$	−2.27475e6	−0.313473
$556$	−1.48356e6	−0.203525
$557$	3.80607e6	0.519803	0.259902	−	0.965635i	$-0.416310\pi$
$557$	3.80607e6	0.519803	0.259902	+	0.965635i	$0.416310\pi$
$558$	3.15147e6	0.428477
$559$	2.73964e6	0.370821
$560$	0	0
$561$	4.79837e6	0.643704
$562$	8.96107e6	1.19679
$563$	7.30416e6	0.971179	0.485589	−	0.874187i	$-0.338605\pi$
$563$	7.30416e6	0.971179	0.485589	+	0.874187i	$0.338605\pi$
$564$	−4.36907e6	−0.578351
$565$	−1.65814e6	−0.218525
$566$	7.73911e6	1.01543
$567$	0	0
$568$	8.12281e6	1.05642
$569$	2.90873e6	0.376637	0.188319	−	0.982108i	$-0.439696\pi$
$569$	2.90873e6	0.376637	0.188319	+	0.982108i	$0.439696\pi$
$570$	4.02775e6	0.519248
$571$	2.69786e6	0.346281	0.173141	−	0.984897i	$-0.444608\pi$
$571$	2.69786e6	0.346281	0.173141	+	0.984897i	$0.444608\pi$
$572$	−334665.	−0.0427681
$573$	−1.80477e7	−2.29633
$574$	0	0
$575$	203696.	0.0256929
$576$	8.96985e6	1.12649
$577$	−1.42203e7	−1.77815	−0.889077	−	0.457758i	$-0.848652\pi$
$577$	−1.42203e7	−1.77815	−0.889077	+	0.457758i	$0.848652\pi$
$578$	−1.27695e7	−1.58984
$579$	−1.20547e7	−1.49438
$580$	1.25381e6	0.154761
$581$	0	0
$582$	−3.84694e6	−0.470770
$583$	−2.68881e6	−0.327634
$584$	2.35044e6	0.285179
$585$	−5.61667e6	−0.678561
$586$	8.03642e6	0.966760
$587$	115775.	0.0138682	0.00693411	−	0.999976i	$-0.497793\pi$
$587$	115775.	0.0138682	0.00693411	+	0.999976i	$0.497793\pi$
$588$	0	0
$589$	545117.	0.0647443
$590$	−172840.	−0.0204416
$591$	7.72140e6	0.909343
$592$	3.76152e6	0.441122
$593$	1.11369e7	1.30055	0.650276	−	0.759698i	$-0.274653\pi$
$593$	1.11369e7	1.30055	0.650276	+	0.759698i	$0.274653\pi$
$594$	−7.81642e6	−0.908954
$595$	0	0
$596$	392947.	0.0453125
$597$	−2.60587e7	−2.99239
$598$	701947.	0.0802696
$599$	−891104.	−0.101475	−0.0507377	−	0.998712i	$-0.516157\pi$
$599$	−891104.	−0.101475	−0.0507377	+	0.998712i	$0.516157\pi$
$600$	2.53403e6	0.287364
$601$	3.38207e6	0.381941	0.190971	−	0.981596i	$-0.438836\pi$
$601$	3.38207e6	0.381941	0.190971	+	0.981596i	$0.438836\pi$
$602$	0	0
$603$	1.41999e7	1.59034
$604$	5.39551e6	0.601784
$605$	−3.84222e6	−0.426770
$606$	2.45558e7	2.71626
$607$	6.48824e6	0.714752	0.357376	−	0.933961i	$-0.383671\pi$
$607$	6.48824e6	0.714752	0.357376	+	0.933961i	$0.383671\pi$
$608$	−3.29072e6	−0.361021
$609$	0	0
$610$	331945.	0.0361195
$611$	3.86078e6	0.418381
$612$	1.52058e7	1.64108
$613$	−4.49484e6	−0.483128	−0.241564	−	0.970385i	$-0.577660\pi$
$613$	−4.49484e6	−0.483128	−0.241564	+	0.970385i	$0.577660\pi$
$614$	9.17240e6	0.981888
$615$	7.74931e6	0.826181
$616$	0	0
$617$	−3.99996e6	−0.423003	−0.211501	−	0.977378i	$-0.567835\pi$
$617$	−3.99996e6	−0.423003	−0.211501	+	0.977378i	$0.567835\pi$
$618$	−1.88897e7	−1.98954
$619$	−1.26318e7	−1.32507	−0.662533	−	0.749033i	$-0.730518\pi$
$619$	−1.26318e7	−1.32507	−0.662533	+	0.749033i	$0.730518\pi$
$620$	−206169.	−0.0215399
$621$	4.47517e6	0.465672
$622$	1.91475e7	1.98443
$623$	0	0
$624$	1.25489e7	1.29017
$625$	390625.	0.0400000
$626$	1.43578e7	1.46437
$627$	−2.08367e6	−0.211670
$628$	−380818.	−0.0385317
$629$	−5.44189e6	−0.548432
$630$	0	0
$631$	−3.30584e6	−0.330528	−0.165264	−	0.986249i	$-0.552848\pi$
$631$	−3.30584e6	−0.330528	−0.165264	+	0.986249i	$0.552848\pi$
$632$	−4.01246e6	−0.399593
$633$	1.38337e7	1.37223
$634$	−1.02918e7	−1.01688
$635$	5.62058e6	0.553155
$636$	−1.15126e7	−1.12858
$637$	0	0
$638$	−2.37624e6	−0.231120
$639$	4.23961e7	4.10746
$640$	−5.46466e6	−0.527368
$641$	−1.55783e7	−1.49753	−0.748766	−	0.662835i	$-0.769353\pi$
$641$	−1.55783e7	−1.49753	−0.748766	+	0.662835i	$0.769353\pi$
$642$	4.37930e6	0.419341
$643$	504385.	0.0481100	0.0240550	−	0.999711i	$-0.492342\pi$
$643$	504385.	0.0481100	0.0240550	+	0.999711i	$0.492342\pi$
$644$	0	0
$645$	−6.45128e6	−0.610586
$646$	9.63560e6	0.908443
$647$	3.69440e6	0.346963	0.173481	−	0.984837i	$-0.444498\pi$
$647$	3.69440e6	0.346963	0.173481	+	0.984837i	$0.444498\pi$
$648$	3.33746e7	3.12233
$649$	89415.4	0.00833298
$650$	1.34611e6	0.124968
$651$	0	0
$652$	6.02169e6	0.554753
$653$	1.16742e7	1.07138	0.535689	−	0.844415i	$-0.320052\pi$
$653$	1.16742e7	1.07138	0.535689	+	0.844415i	$0.320052\pi$
$654$	4.44248e6	0.406145
$655$	7.77094e6	0.707734
$656$	−1.28142e7	−1.16261
$657$	1.22679e7	1.10881
$658$	0	0
$659$	−2.88804e6	−0.259054	−0.129527	−	0.991576i	$-0.541346\pi$
$659$	−2.88804e6	−0.259054	−0.129527	+	0.991576i	$0.541346\pi$
$660$	788066.	0.0704211
$661$	−1.69341e7	−1.50750	−0.753752	−	0.657159i	$-0.771758\pi$
$661$	−1.69341e7	−1.50750	−0.753752	+	0.657159i	$0.771758\pi$
$662$	1.31505e7	1.16627
$663$	−1.81549e7	−1.60402
$664$	5.48391e6	0.482692
$665$	0	0
$666$	1.36618e7	1.19350
$667$	1.36048e6	0.118407
$668$	3.54977e6	0.307793
$669$	−2.48397e7	−2.14576
$670$	−3.40320e6	−0.292887
$671$	−171725.	−0.0147240
$672$	0	0
$673$	−1.63004e7	−1.38727	−0.693633	−	0.720329i	$-0.743991\pi$
$673$	−1.63004e7	−1.38727	−0.693633	+	0.720329i	$0.743991\pi$
$674$	−4.82904e6	−0.409459
$675$	8.58196e6	0.724982
$676$	−3.19464e6	−0.268878
$677$	−1.45832e7	−1.22287	−0.611435	−	0.791295i	$-0.709407\pi$
$677$	−1.45832e7	−1.22287	−0.611435	+	0.791295i	$0.709407\pi$
$678$	1.34554e7	1.12414
$679$	0	0
$680$	6.06216e6	0.502754
$681$	−3.87506e7	−3.20192
$682$	390735.	0.0321678
$683$	1.07632e7	0.882859	0.441429	−	0.897296i	$-0.354472\pi$
$683$	1.07632e7	0.882859	0.441429	+	0.897296i	$0.354472\pi$
$684$	−6.60306e6	−0.539641
$685$	−426635.	−0.0347400
$686$	0	0
$687$	2.96190e6	0.239430
$688$	1.06678e7	0.859222
$689$	1.01733e7	0.816416
$690$	−1.65294e6	−0.132170
$691$	−1.31160e7	−1.04497	−0.522487	−	0.852647i	$-0.674996\pi$
$691$	−1.31160e7	−1.04497	−0.522487	+	0.852647i	$0.674996\pi$
$692$	668778.	0.0530905
$693$	0	0
$694$	−2.42910e7	−1.91446
$695$	−3.08704e6	−0.242426
$696$	1.69246e7	1.32433
$697$	1.85387e7	1.44543
$698$	−1.60523e7	−1.24709
$699$	−3.99385e7	−3.09171
$700$	0	0
$701$	1.60167e7	1.23106	0.615528	−	0.788115i	$-0.288943\pi$
$701$	1.60167e7	1.23106	0.615528	+	0.788115i	$0.288943\pi$
$702$	2.95739e7	2.26498
$703$	2.36312e6	0.180342
$704$	1.11213e6	0.0845713
$705$	−9.09133e6	−0.688897
$706$	−8.47659e6	−0.640043
$707$	0	0
$708$	382848.	0.0287040
$709$	29218.7	0.00218296	0.00109148	−	0.999999i	$-0.499653\pi$
$709$	29218.7	0.00218296	0.00109148	+	0.999999i	$0.499653\pi$
$710$	−1.01608e7	−0.756455
$711$	−2.09426e7	−1.55366
$712$	1.52784e7	1.12948
$713$	−223709.	−0.0164801
$714$	0	0
$715$	−696383.	−0.0509429
$716$	−6.11174e6	−0.445536
$717$	−1.54853e7	−1.12492
$718$	1.31513e7	0.952046
$719$	−1.93974e7	−1.39933	−0.699666	−	0.714470i	$-0.746668\pi$
$719$	−1.93974e7	−1.39933	−0.699666	+	0.714470i	$0.746668\pi$
$720$	−2.18706e7	−1.57228
$721$	0	0
$722$	1.22430e7	0.874071
$723$	2.50180e7	1.77994
$724$	−8.71946e6	−0.618220
$725$	2.60897e6	0.184342
$726$	3.11785e7	2.19540
$727$	−1.41427e7	−0.992424	−0.496212	−	0.868201i	$-0.665276\pi$
$727$	−1.41427e7	−0.992424	−0.496212	+	0.868201i	$0.665276\pi$
$728$	0	0
$729$	7.21649e7	5.02930
$730$	−2.94017e6	−0.204205
$731$	−1.54334e7	−1.06824
$732$	−735271.	−0.0507189
$733$	−1.76749e7	−1.21506	−0.607528	−	0.794299i	$-0.707839\pi$
$733$	−1.76749e7	−1.21506	−0.607528	+	0.794299i	$0.707839\pi$
$734$	−1.87574e7	−1.28509
$735$	0	0
$736$	1.35047e6	0.0918947
$737$	1.76057e6	0.119395
$738$	−4.65413e7	−3.14556
$739$	1.38906e7	0.935642	0.467821	−	0.883823i	$-0.345039\pi$
$739$	1.38906e7	0.935642	0.467821	+	0.883823i	$0.345039\pi$
$740$	−893756.	−0.0599984
$741$	7.88368e6	0.527453
$742$	0	0
$743$	−6.42472e6	−0.426955	−0.213477	−	0.976948i	$-0.568479\pi$
$743$	−6.42472e6	−0.426955	−0.213477	+	0.976948i	$0.568479\pi$
$744$	−2.78299e6	−0.184323
$745$	817658.	0.0539736
$746$	−2.72183e6	−0.179066
$747$	2.86227e7	1.87676
$748$	1.88529e6	0.123204
$749$	0	0
$750$	−3.16981e6	−0.205769
$751$	2.11552e7	1.36873	0.684364	−	0.729140i	$-0.260080\pi$
$751$	2.11552e7	1.36873	0.684364	+	0.729140i	$0.260080\pi$
$752$	1.50334e7	0.969422
$753$	1.83465e7	1.17914
$754$	8.99062e6	0.575919
$755$	1.12272e7	0.716809
$756$	0	0
$757$	−7.71120e6	−0.489083	−0.244541	−	0.969639i	$-0.578637\pi$
$757$	−7.71120e6	−0.489083	−0.244541	+	0.969639i	$0.578637\pi$
$758$	−3.28068e6	−0.207392
$759$	855112.	0.0538789
$760$	−2.63247e6	−0.165321
$761$	−4.01663e6	−0.251420	−0.125710	−	0.992067i	$-0.540121\pi$
$761$	−4.01663e6	−0.251420	−0.125710	+	0.992067i	$0.540121\pi$
$762$	−4.56094e7	−2.84556
$763$	0	0
$764$	−7.09098e6	−0.439514
$765$	3.16408e7	1.95476
$766$	1.70583e7	1.05043
$767$	−338308.	−0.0207646
$768$	3.16614e7	1.93699
$769$	1.66044e7	1.01253	0.506265	−	0.862378i	$-0.331026\pi$
$769$	1.66044e7	1.01253	0.506265	+	0.862378i	$0.331026\pi$
$770$	0	0
$771$	−4.39020e7	−2.65980
$772$	−4.73633e6	−0.286021
$773$	1.25920e7	0.757957	0.378979	−	0.925405i	$-0.376275\pi$
$773$	1.25920e7	0.757957	0.378979	+	0.925405i	$0.376275\pi$
$774$	3.87455e7	2.32471
$775$	−429004.	−0.0256571
$776$	2.51430e6	0.149887
$777$	0	0
$778$	2.02362e7	1.19861
$779$	−8.05036e6	−0.475304
$780$	−2.98169e6	−0.175479
$781$	5.25649e6	0.308367
$782$	−3.95433e6	−0.231236
$783$	5.73185e7	3.34111
$784$	0	0
$785$	−792420.	−0.0458967
$786$	−6.30590e7	−3.64075
$787$	2.68566e6	0.154566	0.0772829	−	0.997009i	$-0.475376\pi$
$787$	2.68566e6	0.154566	0.0772829	+	0.997009i	$0.475376\pi$
$788$	3.03377e6	0.174047
$789$	1.15340e7	0.659609
$790$	5.01919e6	0.286131
$791$	0	0
$792$	7.87324e6	0.446006
$793$	649731.	0.0366902
$794$	−2.30312e7	−1.29648
$795$	−2.39559e7	−1.34429
$796$	−1.02386e7	−0.572739
$797$	−3.05957e7	−1.70614	−0.853070	−	0.521796i	$-0.825262\pi$
$797$	−3.05957e7	−1.70614	−0.853070	+	0.521796i	$0.825262\pi$
$798$	0	0
$799$	−2.17492e7	−1.20525
$800$	2.58978e6	0.143066
$801$	7.97439e7	4.39153
$802$	−934901.	−0.0513251
$803$	1.52104e6	0.0832436
$804$	7.53821e6	0.411271
$805$	0	0
$806$	−1.47837e6	−0.0801577
$807$	−2.88604e7	−1.55998
$808$	−1.60492e7	−0.864821
$809$	−379974.	−0.0204119	−0.0102059	−	0.999948i	$-0.503249\pi$
$809$	−379974.	−0.0204119	−0.0102059	+	0.999948i	$0.503249\pi$
$810$	−4.17483e7	−2.23577
$811$	2.19008e7	1.16925	0.584625	−	0.811304i	$-0.301242\pi$
$811$	2.19008e7	1.16925	0.584625	+	0.811304i	$0.301242\pi$
$812$	0	0
$813$	−5.64858e7	−2.99718
$814$	1.69386e6	0.0896019
$815$	1.25302e7	0.660788
$816$	−7.06929e7	−3.71664
$817$	6.70191e6	0.351272
$818$	−4.63969e6	−0.242441
$819$	0	0
$820$	3.04473e6	0.158130
$821$	1.44749e7	0.749475	0.374737	−	0.927131i	$-0.377733\pi$
$821$	1.44749e7	0.749475	0.374737	+	0.927131i	$0.377733\pi$
$822$	3.46203e6	0.178711
$823$	7.63351e6	0.392848	0.196424	−	0.980519i	$-0.437067\pi$
$823$	7.63351e6	0.392848	0.196424	+	0.980519i	$0.437067\pi$
$824$	1.23460e7	0.633443
$825$	1.63984e6	0.0838814
$826$	0	0
$827$	2.35052e7	1.19509	0.597545	−	0.801835i	$-0.296143\pi$
$827$	2.35052e7	1.19509	0.597545	+	0.801835i	$0.296143\pi$
$828$	2.70981e6	0.137361
$829$	49933.2	0.00252350	0.00126175	−	0.999999i	$-0.499598\pi$
$829$	49933.2	0.00252350	0.00126175	+	0.999999i	$0.499598\pi$
$830$	−6.85982e6	−0.345635
$831$	5.28878e7	2.65676
$832$	−4.20780e6	−0.210740
$833$	0	0
$834$	2.50505e7	1.24710
$835$	7.38649e6	0.366625
$836$	−818681.	−0.0405134
$837$	−9.42513e6	−0.465022
$838$	1.51458e6	0.0745047
$839$	1.73936e7	0.853071	0.426536	−	0.904471i	$-0.359734\pi$
$839$	1.73936e7	0.853071	0.426536	+	0.904471i	$0.359734\pi$
$840$	0	0
$841$	−3.08600e6	−0.150455
$842$	1.71133e7	0.831867
$843$	−4.13027e7	−2.00175
$844$	5.43529e6	0.262644
$845$	−6.64752e6	−0.320271
$846$	5.46013e7	2.62287
$847$	0	0
$848$	3.96134e7	1.89170
$849$	−3.56705e7	−1.69840
$850$	−7.58316e6	−0.360000
$851$	−969794.	−0.0459045
$852$	2.25066e7	1.06221
$853$	1.10470e6	0.0519843	0.0259922	−	0.999662i	$-0.491726\pi$
$853$	1.10470e6	0.0519843	0.0259922	+	0.999662i	$0.491726\pi$
$854$	0	0
$855$	−1.37399e7	−0.642788
$856$	−2.86224e6	−0.133512
$857$	−3.57155e7	−1.66114	−0.830568	−	0.556917i	$-0.811984\pi$
$857$	−3.57155e7	−1.66114	−0.830568	+	0.556917i	$0.811984\pi$
$858$	5.65096e6	0.262062
$859$	−2.29998e7	−1.06351	−0.531754	−	0.846899i	$-0.678467\pi$
$859$	−2.29998e7	−1.06351	−0.531754	+	0.846899i	$0.678467\pi$
$860$	−2.53473e6	−0.116865
$861$	0	0
$862$	2.89741e7	1.32813
$863$	2.20963e6	0.100993	0.0504966	−	0.998724i	$-0.483920\pi$
$863$	2.20963e6	0.100993	0.0504966	+	0.998724i	$0.483920\pi$
$864$	5.68969e7	2.59301
$865$	1.39162e6	0.0632382
$866$	9.31525e6	0.422085
$867$	5.88562e7	2.65916
$868$	0	0
$869$	−2.59657e6	−0.116641
$870$	−2.11710e7	−0.948296
$871$	−6.66123e6	−0.297515
$872$	−2.90353e6	−0.129311
$873$	1.31231e7	0.582775
$874$	1.71715e6	0.0760379
$875$	0	0
$876$	6.51259e6	0.286743
$877$	−2.30761e7	−1.01313	−0.506563	−	0.862203i	$-0.669084\pi$
$877$	−2.30761e7	−1.01313	−0.506563	+	0.862203i	$0.669084\pi$
$878$	3.19294e7	1.39783
$879$	−3.70409e7	−1.61700
$880$	−2.71163e6	−0.118039
$881$	−3.23900e7	−1.40596	−0.702978	−	0.711212i	$-0.748147\pi$
$881$	−3.23900e7	−1.40596	−0.702978	+	0.711212i	$0.748147\pi$
$882$	0	0
$883$	1.16517e7	0.502906	0.251453	−	0.967870i	$-0.419092\pi$
$883$	1.16517e7	0.502906	0.251453	+	0.967870i	$0.419092\pi$
$884$	−7.13311e6	−0.307007
$885$	796644.	0.0341906
$886$	2.10287e7	0.899971
$887$	6.86404e6	0.292935	0.146467	−	0.989215i	$-0.453210\pi$
$887$	6.86404e6	0.292935	0.146467	+	0.989215i	$0.453210\pi$
$888$	−1.20645e7	−0.513423
$889$	0	0
$890$	−1.91117e7	−0.808770
$891$	2.15976e7	0.911405
$892$	−9.75961e6	−0.410696
$893$	9.44451e6	0.396325
$894$	−6.63507e6	−0.277653
$895$	−1.27175e7	−0.530696
$896$	0	0
$897$	−3.23536e6	−0.134259
$898$	4.21411e7	1.74387
$899$	−2.86530e6	−0.118242
$900$	5.19657e6	0.213850
$901$	−5.73097e7	−2.35189
$902$	−5.77043e6	−0.236152
$903$	0	0
$904$	−8.79421e6	−0.357911
$905$	−1.81438e7	−0.736387
$906$	−9.11055e7	−3.68743
$907$	−7.63555e6	−0.308193	−0.154096	−	0.988056i	$-0.549247\pi$
$907$	−7.63555e6	−0.308193	−0.154096	+	0.988056i	$0.549247\pi$
$908$	−1.52252e7	−0.612843
$909$	−8.37673e7	−3.36252
$910$	0	0
$911$	7.18966e6	0.287020	0.143510	−	0.989649i	$-0.454161\pi$
$911$	7.18966e6	0.287020	0.143510	+	0.989649i	$0.454161\pi$
$912$	3.06981e7	1.22215
$913$	3.54878e6	0.140897
$914$	−5.20915e7	−2.06254
$915$	−1.52998e6	−0.0604133
$916$	1.16374e6	0.0458266
$917$	0	0
$918$	−1.66601e8	−6.52484
$919$	−4.78811e7	−1.87014	−0.935072	−	0.354458i	$-0.884665\pi$
$919$	−4.78811e7	−1.87014	−0.935072	+	0.354458i	$0.884665\pi$
$920$	1.08033e6	0.0420812
$921$	−4.22768e7	−1.64230
$922$	1.31901e7	0.510999
$923$	−1.98882e7	−0.768407
$924$	0	0
$925$	−1.85976e6	−0.0714665
$926$	−2.75620e6	−0.105629
$927$	6.44385e7	2.46290
$928$	1.72970e7	0.659327
$929$	9.19961e6	0.349728	0.174864	−	0.984593i	$-0.444051\pi$
$929$	9.19961e6	0.349728	0.174864	+	0.984593i	$0.444051\pi$
$930$	3.48125e6	0.131986
$931$	0	0
$932$	−1.56920e7	−0.591750
$933$	−8.82534e7	−3.31915
$934$	−5.30970e7	−1.99160
$935$	3.92299e6	0.146753
$936$	−2.97888e7	−1.11138
$937$	3.23616e7	1.20415	0.602076	−	0.798439i	$-0.294341\pi$
$937$	3.23616e7	1.20415	0.602076	+	0.798439i	$0.294341\pi$
$938$	0	0
$939$	−6.61770e7	−2.44930
$940$	−3.57201e6	−0.131854
$941$	1.60957e7	0.592564	0.296282	−	0.955101i	$-0.404253\pi$
$941$	1.60957e7	0.592564	0.296282	+	0.955101i	$0.404253\pi$
$942$	6.43027e6	0.236103
$943$	3.30377e6	0.120985
$944$	−1.31733e6	−0.0481132
$945$	0	0
$946$	4.80387e6	0.174527
$947$	2.95005e7	1.06894	0.534471	−	0.845187i	$-0.320511\pi$
$947$	2.95005e7	1.06894	0.534471	+	0.845187i	$0.320511\pi$
$948$	−1.11177e7	−0.401785
$949$	−5.75492e6	−0.207431
$950$	3.29296e6	0.118380
$951$	4.74364e7	1.70083
$952$	0	0
$953$	3.08601e7	1.10069	0.550345	−	0.834938i	$-0.314496\pi$
$953$	3.08601e7	1.10069	0.550345	+	0.834938i	$0.314496\pi$
$954$	1.43876e8	5.11819
$955$	−1.47552e7	−0.523523
$956$	−6.08424e6	−0.215309
$957$	1.09524e7	0.386571
$958$	−472165.	−0.0166219
$959$	0	0
$960$	9.90847e6	0.346999
$961$	−2.81580e7	−0.983543
$962$	−6.40882e6	−0.223275
$963$	−1.49391e7	−0.519110
$964$	9.82964e6	0.340679
$965$	−9.85553e6	−0.340692
$966$	0	0
$967$	−5.58630e6	−0.192114	−0.0960568	−	0.995376i	$-0.530623\pi$
$967$	−5.58630e6	−0.192114	−0.0960568	+	0.995376i	$0.530623\pi$
$968$	−2.03778e7	−0.698986
$969$	−4.44118e7	−1.51946
$970$	−3.14514e6	−0.107327
$971$	3.25113e7	1.10659	0.553294	−	0.832986i	$-0.313371\pi$
$971$	3.25113e7	1.10659	0.553294	+	0.832986i	$0.313371\pi$
$972$	5.23860e7	1.77848
$973$	0	0
$974$	3.04000e7	1.02678
$975$	−6.20441e6	−0.209021
$976$	2.52997e6	0.0850141
$977$	3.67465e7	1.23163	0.615814	−	0.787892i	$-0.288827\pi$
$977$	3.67465e7	1.23163	0.615814	+	0.787892i	$0.288827\pi$
$978$	−1.01679e8	−3.39925
$979$	9.88705e6	0.329693
$980$	0	0
$981$	−1.51547e7	−0.502775
$982$	−3.96447e7	−1.31192
$983$	−2.06566e7	−0.681827	−0.340914	−	0.940095i	$-0.610736\pi$
$983$	−2.06566e7	−0.681827	−0.340914	+	0.940095i	$0.610736\pi$
$984$	4.10996e7	1.35316
$985$	6.31277e6	0.207314
$986$	−5.06476e7	−1.65908
$987$	0	0
$988$	3.09753e6	0.100954
$989$	−2.75038e6	−0.0894132
$990$	−9.84863e6	−0.319366
$991$	4.26712e7	1.38023	0.690115	−	0.723700i	$-0.257560\pi$
$991$	4.26712e7	1.38023	0.690115	+	0.723700i	$0.257560\pi$
$992$	−2.84422e6	−0.0917666
$993$	−6.06124e7	−1.95069
$994$	0	0
$995$	−2.13048e7	−0.682212
$996$	1.51948e7	0.485339
$997$	3.07826e6	0.0980770	0.0490385	−	0.998797i	$-0.484384\pi$
$997$	3.07826e6	0.0980770	0.0490385	+	0.998797i	$0.484384\pi$
$998$	−3.35839e7	−1.06734
$999$	−4.08586e7	−1.29530

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.6.a.j.1.3		8
7.2	even	3		35.6.e.b.11.6	✓	16
7.4	even	3		35.6.e.b.16.6	yes	16
7.6	odd	2		245.6.a.k.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.6.e.b.11.6	✓	16	7.2	even	3
35.6.e.b.16.6	yes	16	7.4	even	3
245.6.a.j.1.3		8	1.1	even	1	trivial
245.6.a.k.1.3		8	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

\(f(q)\)	\(=\)	\(q-6.63433 q^{2} +30.5785 q^{3} +12.0144 q^{4} +25.0000 q^{5} -202.868 q^{6} +132.591 q^{8} +692.045 q^{9} +O(q^{10})\)	\(q-6.63433 q^{2} +30.5785 q^{3} +12.0144 q^{4} +25.0000 q^{5} -202.868 q^{6} +132.591 q^{8} +692.045 q^{9} -165.858 q^{10} +85.8033 q^{11} +367.382 q^{12} -324.642 q^{13} +764.463 q^{15} -1264.11 q^{16} +1828.83 q^{17} -4591.26 q^{18} -794.161 q^{19} +300.360 q^{20} -569.248 q^{22} +325.914 q^{23} +4054.44 q^{24} +625.000 q^{25} +2153.78 q^{26} +13731.1 q^{27} +4174.34 q^{29} -5071.70 q^{30} -686.406 q^{31} +4143.64 q^{32} +2623.74 q^{33} -12133.1 q^{34} +8314.51 q^{36} -2975.62 q^{37} +5268.73 q^{38} -9927.06 q^{39} +3314.78 q^{40} +10136.9 q^{41} -8438.98 q^{43} +1030.88 q^{44} +17301.1 q^{45} -2162.22 q^{46} -11892.4 q^{47} -38654.8 q^{48} -4146.46 q^{50} +55922.8 q^{51} -3900.37 q^{52} -31336.9 q^{53} -91096.9 q^{54} +2145.08 q^{55} -24284.3 q^{57} -27694.0 q^{58} +1042.10 q^{59} +9184.56 q^{60} -2001.38 q^{61} +4553.85 q^{62} +12961.4 q^{64} -8116.04 q^{65} -17406.8 q^{66} +20518.7 q^{67} +21972.3 q^{68} +9965.96 q^{69} +61262.0 q^{71} +91759.1 q^{72} +17727.0 q^{73} +19741.2 q^{74} +19111.6 q^{75} -9541.37 q^{76} +65859.4 q^{78} -30261.9 q^{79} -31602.9 q^{80} +251711. q^{81} -67251.8 q^{82} +41359.5 q^{83} +45720.7 q^{85} +55987.0 q^{86} +127645. q^{87} +11376.8 q^{88} +115229. q^{89} -114781. q^{90} +3915.66 q^{92} -20989.3 q^{93} +78898.4 q^{94} -19854.0 q^{95} +126706. q^{96} +18962.8 q^{97} +59379.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} - 2 q^{3} + 203 q^{4} + 200 q^{5} - 56 q^{6} + 249 q^{8} + 1218 q^{9}+O(q^{10}) \) 8 * q + 3 * q^2 - 2 * q^3 + 203 * q^4 + 200 * q^5 - 56 * q^6 + 249 * q^8 + 1218 * q^9	\( 8 q + 3 q^{2} - 2 q^{3} + 203 q^{4} + 200 q^{5} - 56 q^{6} + 249 q^{8} + 1218 q^{9} + 75 q^{10} + 120 q^{11} - 884 q^{12} + 1994 q^{13} - 50 q^{15} + 6451 q^{16} + 1856 q^{17} + 2013 q^{18} - 1828 q^{19} + 5075 q^{20} + 2199 q^{22} + 4822 q^{23} - 2008 q^{24} + 5000 q^{25} - 7457 q^{26} + 7798 q^{27} + 10502 q^{29} - 1400 q^{30} + 5148 q^{31} + 12061 q^{32} - 2872 q^{33} - 40682 q^{34} + 40949 q^{36} + 7810 q^{37} + 18567 q^{38} - 2572 q^{39} + 6225 q^{40} - 18192 q^{41} + 63190 q^{43} - 4765 q^{44} + 30450 q^{45} + 6567 q^{46} + 11816 q^{47} - 107336 q^{48} + 1875 q^{50} + 28788 q^{51} + 167255 q^{52} - 39902 q^{53} - 126138 q^{54} + 3000 q^{55} + 38748 q^{57} + 67552 q^{58} + 50752 q^{59} - 22100 q^{60} - 2146 q^{61} + 101572 q^{62} + 270039 q^{64} + 49850 q^{65} + 164358 q^{66} + 50498 q^{67} - 125090 q^{68} - 118822 q^{69} + 183976 q^{71} + 522343 q^{72} + 54436 q^{73} + 228885 q^{74} - 1250 q^{75} - 65789 q^{76} + 391806 q^{78} + 51040 q^{79} + 161275 q^{80} + 124612 q^{81} - 195887 q^{82} + 60438 q^{83} + 46400 q^{85} + 260996 q^{86} + 409482 q^{87} + 205001 q^{88} - 96678 q^{89} + 50325 q^{90} + 174679 q^{92} - 51428 q^{93} - 139395 q^{94} - 45700 q^{95} - 587116 q^{96} - 195312 q^{97} + 397244 q^{99}+O(q^{100}) \) 8 * q + 3 * q^2 - 2 * q^3 + 203 * q^4 + 200 * q^5 - 56 * q^6 + 249 * q^8 + 1218 * q^9 + 75 * q^10 + 120 * q^11 - 884 * q^12 + 1994 * q^13 - 50 * q^15 + 6451 * q^16 + 1856 * q^17 + 2013 * q^18 - 1828 * q^19 + 5075 * q^20 + 2199 * q^22 + 4822 * q^23 - 2008 * q^24 + 5000 * q^25 - 7457 * q^26 + 7798 * q^27 + 10502 * q^29 - 1400 * q^30 + 5148 * q^31 + 12061 * q^32 - 2872 * q^33 - 40682 * q^34 + 40949 * q^36 + 7810 * q^37 + 18567 * q^38 - 2572 * q^39 + 6225 * q^40 - 18192 * q^41 + 63190 * q^43 - 4765 * q^44 + 30450 * q^45 + 6567 * q^46 + 11816 * q^47 - 107336 * q^48 + 1875 * q^50 + 28788 * q^51 + 167255 * q^52 - 39902 * q^53 - 126138 * q^54 + 3000 * q^55 + 38748 * q^57 + 67552 * q^58 + 50752 * q^59 - 22100 * q^60 - 2146 * q^61 + 101572 * q^62 + 270039 * q^64 + 49850 * q^65 + 164358 * q^66 + 50498 * q^67 - 125090 * q^68 - 118822 * q^69 + 183976 * q^71 + 522343 * q^72 + 54436 * q^73 + 228885 * q^74 - 1250 * q^75 - 65789 * q^76 + 391806 * q^78 + 51040 * q^79 + 161275 * q^80 + 124612 * q^81 - 195887 * q^82 + 60438 * q^83 + 46400 * q^85 + 260996 * q^86 + 409482 * q^87 + 205001 * q^88 - 96678 * q^89 + 50325 * q^90 + 174679 * q^92 - 51428 * q^93 - 139395 * q^94 - 45700 * q^95 - 587116 * q^96 - 195312 * q^97 + 397244 * q^99

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