LMFDB - Embedded newform 245.6.a.k.1.2

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			1.2
Root			$-6.77005$ 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)

$f(q)$	$=$	$q-6.77005 q^{2} +23.4913 q^{3} +13.8335 q^{4} -25.0000 q^{5} -159.037 q^{6} +122.988 q^{8} +308.841 q^{9} +O(q^{10})$	\(q-6.77005 q^{2} +23.4913 q^{3} +13.8335 q^{4} -25.0000 q^{5} -159.037 q^{6} +122.988 q^{8} +308.841 q^{9} +169.251 q^{10} +679.557 q^{11} +324.968 q^{12} -685.019 q^{13} -587.283 q^{15} -1275.31 q^{16} -1101.75 q^{17} -2090.87 q^{18} +1531.25 q^{19} -345.838 q^{20} -4600.63 q^{22} +4196.42 q^{23} +2889.14 q^{24} +625.000 q^{25} +4637.61 q^{26} +1546.70 q^{27} -1805.73 q^{29} +3975.93 q^{30} +1367.40 q^{31} +4698.27 q^{32} +15963.7 q^{33} +7458.89 q^{34} +4272.37 q^{36} -7520.27 q^{37} -10366.7 q^{38} -16092.0 q^{39} -3074.70 q^{40} +4983.90 q^{41} +8205.89 q^{43} +9400.67 q^{44} -7721.04 q^{45} -28409.9 q^{46} -910.699 q^{47} -29958.6 q^{48} -4231.28 q^{50} -25881.5 q^{51} -9476.23 q^{52} +33703.7 q^{53} -10471.2 q^{54} -16988.9 q^{55} +35971.2 q^{57} +12224.9 q^{58} -8939.55 q^{59} -8124.19 q^{60} -2438.19 q^{61} -9257.37 q^{62} +9002.28 q^{64} +17125.5 q^{65} -108075. q^{66} +428.019 q^{67} -15241.1 q^{68} +98579.3 q^{69} +10802.1 q^{71} +37983.7 q^{72} +35325.7 q^{73} +50912.6 q^{74} +14682.1 q^{75} +21182.6 q^{76} +108943. q^{78} +99281.4 q^{79} +31882.7 q^{80} -38714.4 q^{81} -33741.2 q^{82} -17037.2 q^{83} +27543.7 q^{85} -55554.2 q^{86} -42418.9 q^{87} +83577.2 q^{88} +6485.74 q^{89} +52271.8 q^{90} +58051.2 q^{92} +32122.1 q^{93} +6165.47 q^{94} -38281.3 q^{95} +110369. q^{96} +103378. q^{97} +209875. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−6.77005	−1.19679	−0.598393	−	0.801203i	$-0.704194\pi$
$2$	−6.77005	−1.19679	−0.598393	+	0.801203i	$0.704194\pi$
$3$	23.4913	1.50697	0.753484	−	0.657466i	$-0.228372\pi$
$3$	23.4913	1.50697	0.753484	+	0.657466i	$0.228372\pi$
$4$	13.8335	0.432298
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	−159.037	−1.80352
$7$	0	0
$7$	0	0
$8$	122.988	0.679418
$9$	308.841	1.27095
$10$	169.251	0.535219
$11$	679.557	1.69334	0.846670	−	0.532119i	$-0.178604\pi$
$11$	679.557	1.69334	0.846670	+	0.532119i	$0.178604\pi$
$12$	324.968	0.651459
$13$	−685.019	−1.12420	−0.562100	−	0.827069i	$-0.690007\pi$
$13$	−685.019	−1.12420	−0.562100	+	0.827069i	$0.690007\pi$
$14$	0	0
$15$	−587.283	−0.673937
$16$	−1275.31	−1.24542
$17$	−1101.75	−0.924614	−0.462307	−	0.886720i	$-0.652978\pi$
$17$	−1101.75	−0.924614	−0.462307	+	0.886720i	$0.652978\pi$
$18$	−2090.87	−1.52106
$19$	1531.25	0.973113	0.486556	−	0.873649i	$-0.338253\pi$
$19$	1531.25	0.973113	0.486556	+	0.873649i	$0.338253\pi$
$20$	−345.838	−0.193329
$21$	0	0
$22$	−4600.63	−2.02657
$23$	4196.42	1.65409	0.827045	−	0.562136i	$-0.190020\pi$
$23$	4196.42	1.65409	0.827045	+	0.562136i	$0.190020\pi$
$24$	2889.14	1.02386
$25$	625.000	0.200000
$26$	4637.61	1.34543
$27$	1546.70	0.408316
$28$	0	0
$29$	−1805.73	−0.398710	−0.199355	−	0.979927i	$-0.563885\pi$
$29$	−1805.73	−0.398710	−0.199355	+	0.979927i	$0.563885\pi$
$30$	3975.93	0.806558
$31$	1367.40	0.255559	0.127780	−	0.991803i	$-0.459215\pi$
$31$	1367.40	0.255559	0.127780	+	0.991803i	$0.459215\pi$
$32$	4698.27	0.811079
$33$	15963.7	2.55181
$34$	7458.89	1.10657
$35$	0	0
$36$	4272.37	0.549430
$37$	−7520.27	−0.903086	−0.451543	−	0.892249i	$-0.649126\pi$
$37$	−7520.27	−0.903086	−0.451543	+	0.892249i	$0.649126\pi$
$38$	−10366.7	−1.16461
$39$	−16092.0	−1.69413
$40$	−3074.70	−0.303845
$41$	4983.90	0.463031	0.231515	−	0.972831i	$-0.425632\pi$
$41$	4983.90	0.463031	0.231515	+	0.972831i	$0.425632\pi$
$42$	0	0
$43$	8205.89	0.676791	0.338395	−	0.941004i	$-0.390116\pi$
$43$	8205.89	0.676791	0.338395	+	0.941004i	$0.390116\pi$
$44$	9400.67	0.732027
$45$	−7721.04	−0.568387
$46$	−28409.9	−1.97959
$47$	−910.699	−0.0601354	−0.0300677	−	0.999548i	$-0.509572\pi$
$47$	−910.699	−0.0601354	−0.0300677	+	0.999548i	$0.509572\pi$
$48$	−29958.6	−1.87680
$49$	0	0
$50$	−4231.28	−0.239357
$51$	−25881.5	−1.39336
$52$	−9476.23	−0.485990
$53$	33703.7	1.64812	0.824059	−	0.566505i	$-0.191705\pi$
$53$	33703.7	1.64812	0.824059	+	0.566505i	$0.191705\pi$
$54$	−10471.2	−0.488668
$55$	−16988.9	−0.757284
$56$	0	0
$57$	35971.2	1.46645
$58$	12224.9	0.477171
$59$	−8939.55	−0.334338	−0.167169	−	0.985928i	$-0.553463\pi$
$59$	−8939.55	−0.334338	−0.167169	+	0.985928i	$0.553463\pi$
$60$	−8124.19	−0.291341
$61$	−2438.19	−0.0838964	−0.0419482	−	0.999120i	$-0.513356\pi$
$61$	−2438.19	−0.0838964	−0.0419482	+	0.999120i	$0.513356\pi$
$62$	−9257.37	−0.305850
$63$	0	0
$64$	9002.28	0.274728
$65$	17125.5	0.502758
$66$	−108075.	−3.05397
$67$	428.019	0.0116487	0.00582433	−	0.999983i	$-0.498146\pi$
$67$	428.019	0.0116487	0.00582433	+	0.999983i	$0.498146\pi$
$68$	−15241.1	−0.399709
$69$	98579.3	2.49266
$70$	0	0
$71$	10802.1	0.254310	0.127155	−	0.991883i	$-0.459415\pi$
$71$	10802.1	0.254310	0.127155	+	0.991883i	$0.459415\pi$
$72$	37983.7	0.863508
$73$	35325.7	0.775861	0.387930	−	0.921689i	$-0.373190\pi$
$73$	35325.7	0.775861	0.387930	+	0.921689i	$0.373190\pi$
$74$	50912.6	1.08080
$75$	14682.1	0.301394
$76$	21182.6	0.420675
$77$	0	0
$78$	108943.	2.02752
$79$	99281.4	1.78978	0.894891	−	0.446284i	$-0.147253\pi$
$79$	99281.4	1.78978	0.894891	+	0.446284i	$0.147253\pi$
$80$	31882.7	0.556967
$81$	−38714.4	−0.655633
$82$	−33741.2	−0.554149
$83$	−17037.2	−0.271459	−0.135729	−	0.990746i	$-0.543338\pi$
$83$	−17037.2	−0.271459	−0.135729	+	0.990746i	$0.543338\pi$
$84$	0	0
$85$	27543.7	0.413500
$86$	−55554.2	−0.809974
$87$	−42418.9	−0.600844
$88$	83577.2	1.15049
$89$	6485.74	0.0867930	0.0433965	−	0.999058i	$-0.486182\pi$
$89$	6485.74	0.0867930	0.0433965	+	0.999058i	$0.486182\pi$
$90$	52271.8	0.680238
$91$	0	0
$92$	58051.2	0.715059
$93$	32122.1	0.385120
$94$	6165.47	0.0719692
$95$	−38281.3	−0.435189
$96$	110369.	1.22227
$97$	103378.	1.11558	0.557789	−	0.829983i	$-0.311650\pi$
$97$	103378.	1.11558	0.557789	+	0.829983i	$0.311650\pi$
$98$	0	0
$99$	209875.	2.15215
$100$	8645.96	0.0864596
$101$	127206.	1.24081	0.620403	−	0.784283i	$-0.286969\pi$
$101$	127206.	1.24081	0.620403	+	0.784283i	$0.286969\pi$
$102$	175219.	1.66756
$103$	28754.8	0.267065	0.133532	−	0.991044i	$-0.457368\pi$
$103$	28754.8	0.267065	0.133532	+	0.991044i	$0.457368\pi$
$104$	−84248.9	−0.763803
$105$	0	0
$106$	−228176.	−1.97244
$107$	131234.	1.10812	0.554060	−	0.832477i	$-0.313078\pi$
$107$	131234.	1.10812	0.554060	+	0.832477i	$0.313078\pi$
$108$	21396.3	0.176514
$109$	72992.6	0.588454	0.294227	−	0.955736i	$-0.404938\pi$
$109$	72992.6	0.588454	0.294227	+	0.955736i	$0.404938\pi$
$110$	115016.	0.906308
$111$	−176661.	−1.36092
$112$	0	0
$113$	−143536.	−1.05746	−0.528731	−	0.848790i	$-0.677332\pi$
$113$	−143536.	−1.05746	−0.528731	+	0.848790i	$0.677332\pi$
$114$	−243526.	−1.75503
$115$	−104910.	−0.739731
$116$	−24979.6	−0.172362
$117$	−211562.	−1.42881
$118$	60521.2	0.400131
$119$	0	0
$120$	−72228.6	−0.457885
$121$	300746.	1.86740
$122$	16506.7	0.100406
$123$	117078.	0.697772
$124$	18916.0	0.110478
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	16619.1	0.0914317	0.0457159	−	0.998954i	$-0.485443\pi$
$127$	16619.1	0.0914317	0.0457159	+	0.998954i	$0.485443\pi$
$128$	−211291.	−1.13987
$129$	192767.	1.01990
$130$	−115940.	−0.601694
$131$	284181.	1.44683	0.723413	−	0.690415i	$-0.242572\pi$
$131$	284181.	1.44683	0.723413	+	0.690415i	$0.242572\pi$
$132$	220834.	1.10314
$133$	0	0
$134$	−2897.71	−0.0139410
$135$	−38667.5	−0.182605
$136$	−135502.	−0.628200
$137$	92118.7	0.419321	0.209660	−	0.977774i	$-0.432764\pi$
$137$	92118.7	0.419321	0.209660	+	0.977774i	$0.432764\pi$
$138$	−667386.	−2.98318
$139$	333882.	1.46574	0.732869	−	0.680369i	$-0.238181\pi$
$139$	333882.	1.46574	0.732869	+	0.680369i	$0.238181\pi$
$140$	0	0
$141$	−21393.5	−0.0906221
$142$	−73131.0	−0.304355
$143$	−465509.	−1.90365
$144$	−393867.	−1.58286
$145$	45143.2	0.178309
$146$	−239157.	−0.928539
$147$	0	0
$148$	−104032.	−0.390402
$149$	118435.	0.437034	0.218517	−	0.975833i	$-0.429878\pi$
$149$	118435.	0.437034	0.218517	+	0.975833i	$0.429878\pi$
$150$	−99398.3	−0.360704
$151$	−85529.1	−0.305261	−0.152630	−	0.988283i	$-0.548774\pi$
$151$	−85529.1	−0.305261	−0.152630	+	0.988283i	$0.548774\pi$
$152$	188326.	0.661151
$153$	−340266.	−1.17514
$154$	0	0
$155$	−34185.0	−0.114290
$156$	−222609.	−0.732371
$157$	376610.	1.21939	0.609694	−	0.792637i	$-0.291292\pi$
$157$	376610.	1.21939	0.609694	+	0.792637i	$0.291292\pi$
$158$	−672140.	−2.14199
$159$	791744.	2.48366
$160$	−117457.	−0.362726
$161$	0	0
$162$	262099.	0.784652
$163$	−48668.2	−0.143475	−0.0717375	−	0.997424i	$-0.522854\pi$
$163$	−48668.2	−0.143475	−0.0717375	+	0.997424i	$0.522854\pi$
$164$	68945.0	0.200167
$165$	−399092.	−1.14120
$166$	115343.	0.324878
$167$	−681113.	−1.88985	−0.944927	−	0.327282i	$-0.893867\pi$
$167$	−681113.	−1.88985	−0.944927	+	0.327282i	$0.893867\pi$
$168$	0	0
$169$	97957.4	0.263828
$170$	−186472.	−0.494871
$171$	472915.	1.23678
$172$	113516.	0.292575
$173$	−672704.	−1.70887	−0.854435	−	0.519558i	$-0.826097\pi$
$173$	−672704.	−1.70887	−0.854435	+	0.519558i	$0.826097\pi$
$174$	287178.	0.719081
$175$	0	0
$176$	−866643.	−2.10891
$177$	−210002.	−0.503837
$178$	−43908.8	−0.103873
$179$	−2628.33	−0.00613123	−0.00306562	−	0.999995i	$-0.500976\pi$
$179$	−2628.33	−0.00613123	−0.00306562	+	0.999995i	$0.500976\pi$
$180$	−106809.	−0.245713
$181$	−412847.	−0.936682	−0.468341	−	0.883548i	$-0.655148\pi$
$181$	−412847.	−0.936682	−0.468341	+	0.883548i	$0.655148\pi$
$182$	0	0
$183$	−57276.3	−0.126429
$184$	516108.	1.12382
$185$	188007.	0.403872
$186$	−217468.	−0.460906
$187$	−748701.	−1.56569
$188$	−12598.2	−0.0259964
$189$	0	0
$190$	259167.	0.520829
$191$	−597848.	−1.18579	−0.592895	−	0.805280i	$-0.702015\pi$
$191$	−597848.	−1.18579	−0.592895	+	0.805280i	$0.702015\pi$
$192$	211475.	0.414006
$193$	720637.	1.39259	0.696295	−	0.717756i	$-0.254831\pi$
$193$	720637.	1.39259	0.696295	+	0.717756i	$0.254831\pi$
$194$	−699876.	−1.33511
$195$	402299.	0.757640
$196$	0	0
$197$	−1.04486e6	−1.91820	−0.959100	−	0.283069i	$-0.908647\pi$
$197$	−1.04486e6	−1.91820	−0.959100	+	0.283069i	$0.908647\pi$
$198$	−1.42087e6	−2.57567
$199$	−223645.	−0.400338	−0.200169	−	0.979761i	$-0.564149\pi$
$199$	−223645.	−0.400338	−0.200169	+	0.979761i	$0.564149\pi$
$200$	76867.4	0.135884
$201$	10054.7	0.0175542
$202$	−861191.	−1.48498
$203$	0	0
$204$	−358033.	−0.602348
$205$	−124598.	−0.207074
$206$	−194671.	−0.319620
$207$	1.29603e6	2.10227
$208$	873609.	1.40010
$209$	1.04057e6	1.64781
$210$	0	0
$211$	223498.	0.345595	0.172798	−	0.984957i	$-0.444719\pi$
$211$	223498.	0.345595	0.172798	+	0.984957i	$0.444719\pi$
$212$	466241.	0.712478
$213$	253756.	0.383237
$214$	−888460.	−1.32618
$215$	−205147.	−0.302670
$216$	190225.	0.277418
$217$	0	0
$218$	−494163.	−0.704254
$219$	829847.	1.16920
$220$	−235017.	−0.327372
$221$	754719.	1.03945
$222$	1.19600e6	1.62873
$223$	−1.03126e6	−1.38870	−0.694348	−	0.719640i	$-0.744307\pi$
$223$	−1.03126e6	−1.38870	−0.694348	+	0.719640i	$0.744307\pi$
$224$	0	0
$225$	193026.	0.254190
$226$	971745.	1.26556
$227$	800607.	1.03123	0.515614	−	0.856821i	$-0.327564\pi$
$227$	800607.	1.03123	0.515614	+	0.856821i	$0.327564\pi$
$228$	497608.	0.633943
$229$	32444.4	0.0408838	0.0204419	−	0.999791i	$-0.493493\pi$
$229$	32444.4	0.0408838	0.0204419	+	0.999791i	$0.493493\pi$
$230$	710248.	0.885300
$231$	0	0
$232$	−222083.	−0.270891
$233$	244506.	0.295053	0.147527	−	0.989058i	$-0.452869\pi$
$233$	244506.	0.295053	0.147527	+	0.989058i	$0.452869\pi$
$234$	1.43229e6	1.70998
$235$	22767.5	0.0268934
$236$	−123666.	−0.144534
$237$	2.33225e6	2.69714
$238$	0	0
$239$	−15420.3	−0.0174622	−0.00873108	−	0.999962i	$-0.502779\pi$
$239$	−15420.3	−0.0174622	−0.00873108	+	0.999962i	$0.502779\pi$
$240$	748965.	0.839332
$241$	−100864.	−0.111865	−0.0559326	−	0.998435i	$-0.517813\pi$
$241$	−100864.	−0.111865	−0.0559326	+	0.998435i	$0.517813\pi$
$242$	−2.03607e6	−2.23488
$243$	−1.28530e6	−1.39633
$244$	−33728.8	−0.0362682
$245$	0	0
$246$	−792626.	−0.835085
$247$	−1.04894e6	−1.09397
$248$	168174.	0.173632
$249$	−400227.	−0.409080
$250$	105782.	0.107044
$251$	−884113.	−0.885775	−0.442888	−	0.896577i	$-0.646046\pi$
$251$	−884113.	−0.885775	−0.442888	+	0.896577i	$0.646046\pi$
$252$	0	0
$253$	2.85170e6	2.80093
$254$	−112512.	−0.109424
$255$	647038.	0.623131
$256$	1.14237e6	1.08945
$257$	680976.	0.643130	0.321565	−	0.946887i	$-0.395791\pi$
$257$	680976.	0.643130	0.321565	+	0.946887i	$0.395791\pi$
$258$	−1.30504e6	−1.22060
$259$	0	0
$260$	236906.	0.217341
$261$	−557684.	−0.506742
$262$	−1.92392e6	−1.73154
$263$	982168.	0.875581	0.437791	−	0.899077i	$-0.355761\pi$
$263$	982168.	0.875581	0.437791	+	0.899077i	$0.355761\pi$
$264$	1.96334e6	1.73374
$265$	−842593.	−0.737060
$266$	0	0
$267$	152359.	0.130794
$268$	5921.02	0.00503569
$269$	−629600.	−0.530498	−0.265249	−	0.964180i	$-0.585454\pi$
$269$	−629600.	−0.530498	−0.265249	+	0.964180i	$0.585454\pi$
$270$	261781.	0.218539
$271$	−961792.	−0.795533	−0.397766	−	0.917487i	$-0.630215\pi$
$271$	−961792.	−0.795533	−0.397766	+	0.917487i	$0.630215\pi$
$272$	1.40507e6	1.15153
$273$	0	0
$274$	−623648.	−0.501837
$275$	424723.	0.338668
$276$	1.36370e6	1.07757
$277$	450504.	0.352777	0.176388	−	0.984321i	$-0.443559\pi$
$277$	450504.	0.352777	0.176388	+	0.984321i	$0.443559\pi$
$278$	−2.26040e6	−1.75418
$279$	422310.	0.324804
$280$	0	0
$281$	865431.	0.653832	0.326916	−	0.945053i	$-0.393990\pi$
$281$	865431.	0.653832	0.326916	+	0.945053i	$0.393990\pi$
$282$	144835.	0.108455
$283$	−1.36525e6	−1.01332	−0.506659	−	0.862147i	$-0.669120\pi$
$283$	−1.36525e6	−1.01332	−0.506659	+	0.862147i	$0.669120\pi$
$284$	149432.	0.109938
$285$	−899279.	−0.655816
$286$	3.15152e6	2.27827
$287$	0	0
$288$	1.45102e6	1.03084
$289$	−206005.	−0.145089
$290$	−305622.	−0.213397
$291$	2.42849e6	1.68114
$292$	488679.	0.335403
$293$	−2.34303e6	−1.59445	−0.797223	−	0.603685i	$-0.793698\pi$
$293$	−2.34303e6	−1.59445	−0.797223	+	0.603685i	$0.793698\pi$
$294$	0	0
$295$	223489.	0.149520
$296$	−924902.	−0.613573
$297$	1.05107e6	0.691418
$298$	−801812.	−0.523036
$299$	−2.87462e6	−1.85953
$300$	203105.	0.130292
$301$	0	0
$302$	579036.	0.365332
$303$	2.98823e6	1.86986
$304$	−1.95282e6	−1.21193
$305$	60954.8	0.0375196
$306$	2.30362e6	1.40639
$307$	−2.65973e6	−1.61061	−0.805307	−	0.592858i	$-0.798000\pi$
$307$	−2.65973e6	−1.61061	−0.805307	+	0.592858i	$0.798000\pi$
$308$	0	0
$309$	675487.	0.402458
$310$	231434.	0.136780
$311$	−397996.	−0.233334	−0.116667	−	0.993171i	$-0.537221\pi$
$311$	−397996.	−0.233334	−0.116667	+	0.993171i	$0.537221\pi$
$312$	−1.97912e6	−1.15103
$313$	−395132.	−0.227972	−0.113986	−	0.993482i	$-0.536362\pi$
$313$	−395132.	−0.227972	−0.113986	+	0.993482i	$0.536362\pi$
$314$	−2.54966e6	−1.45935
$315$	0	0
$316$	1.37341e6	0.773719
$317$	−978276.	−0.546781	−0.273390	−	0.961903i	$-0.588145\pi$
$317$	−978276.	−0.546781	−0.273390	+	0.961903i	$0.588145\pi$
$318$	−5.36015e6	−2.97241
$319$	−1.22709e6	−0.675152
$320$	−225057.	−0.122862
$321$	3.08286e6	1.66990
$322$	0	0
$323$	−1.68706e6	−0.899754
$324$	−535558.	−0.283429
$325$	−428137.	−0.224840
$326$	329486.	0.171709
$327$	1.71469e6	0.886781
$328$	612959.	0.314592
$329$	0	0
$330$	2.70187e6	1.36578
$331$	3.42465e6	1.71809	0.859046	−	0.511899i	$-0.171058\pi$
$331$	3.42465e6	1.71809	0.859046	+	0.511899i	$0.171058\pi$
$332$	−235685.	−0.117351
$333$	−2.32257e6	−1.14778
$334$	4.61117e6	2.26175
$335$	−10700.5	−0.00520944
$336$	0	0
$337$	−2.15859e6	−1.03537	−0.517686	−	0.855571i	$-0.673206\pi$
$337$	−2.15859e6	−1.03537	−0.517686	+	0.855571i	$0.673206\pi$
$338$	−663176.	−0.315746
$339$	−3.37185e6	−1.59356
$340$	381027.	0.178755
$341$	929227.	0.432749
$342$	−3.20165e6	−1.48016
$343$	0	0
$344$	1.00922e6	0.459824
$345$	−2.46448e6	−1.11475
$346$	4.55424e6	2.04515
$347$	−3.57389e6	−1.59338	−0.796688	−	0.604391i	$-0.793416\pi$
$347$	−3.57389e6	−1.59338	−0.796688	+	0.604391i	$0.793416\pi$
$348$	−586803.	−0.259743
$349$	2.67898e6	1.17735	0.588675	−	0.808370i	$-0.299650\pi$
$349$	2.67898e6	1.17735	0.588675	+	0.808370i	$0.299650\pi$
$350$	0	0
$351$	−1.05952e6	−0.459030
$352$	3.19274e6	1.37343
$353$	1.17456e6	0.501695	0.250848	−	0.968027i	$-0.419291\pi$
$353$	1.17456e6	0.501695	0.250848	+	0.968027i	$0.419291\pi$
$354$	1.42172e6	0.602985
$355$	−270053.	−0.113731
$356$	89720.8	0.0375204
$357$	0	0
$358$	17793.9	0.00733777
$359$	−3.11282e6	−1.27473	−0.637365	−	0.770562i	$-0.719976\pi$
$359$	−3.11282e6	−1.27473	−0.637365	+	0.770562i	$0.719976\pi$
$360$	−949593.	−0.386173
$361$	−131361.	−0.0530514
$362$	2.79499e6	1.12101
$363$	7.06492e6	2.81411
$364$	0	0
$365$	−883143.	−0.346975
$366$	387763.	0.151309
$367$	−2.13590e6	−0.827780	−0.413890	−	0.910327i	$-0.635830\pi$
$367$	−2.13590e6	−0.827780	−0.413890	+	0.910327i	$0.635830\pi$
$368$	−5.35172e6	−2.06003
$369$	1.53924e6	0.588490
$370$	−1.27281e6	−0.483349
$371$	0	0
$372$	444361.	0.166487
$373$	−2.58520e6	−0.962104	−0.481052	−	0.876692i	$-0.659745\pi$
$373$	−2.58520e6	−0.962104	−0.481052	+	0.876692i	$0.659745\pi$
$374$	5.06874e6	1.87379
$375$	−367052.	−0.134787
$376$	−112005.	−0.0408571
$377$	1.23696e6	0.448230
$378$	0	0
$379$	2.67172e6	0.955416	0.477708	−	0.878519i	$-0.341468\pi$
$379$	2.67172e6	0.955416	0.477708	+	0.878519i	$0.341468\pi$
$380$	−529566.	−0.188131
$381$	390403.	0.137785
$382$	4.04746e6	1.41914
$383$	417098.	0.145292	0.0726458	−	0.997358i	$-0.476856\pi$
$383$	417098.	0.145292	0.0726458	+	0.997358i	$0.476856\pi$
$384$	−4.96349e6	−1.71775
$385$	0	0
$386$	−4.87875e6	−1.66663
$387$	2.53432e6	0.860169
$388$	1.43009e6	0.482262
$389$	1.44459e6	0.484030	0.242015	−	0.970273i	$-0.422192\pi$
$389$	1.44459e6	0.484030	0.242015	+	0.970273i	$0.422192\pi$
$390$	−2.72359e6	−0.906733
$391$	−4.62340e6	−1.52939
$392$	0	0
$393$	6.67578e6	2.18032
$394$	7.07377e6	2.29567
$395$	−2.48204e6	−0.800415
$396$	2.90332e6	0.930371
$397$	2.38444e6	0.759294	0.379647	−	0.925131i	$-0.376045\pi$
$397$	2.38444e6	0.759294	0.379647	+	0.925131i	$0.376045\pi$
$398$	1.51409e6	0.479119
$399$	0	0
$400$	−797067.	−0.249083
$401$	4.68007e6	1.45342	0.726711	−	0.686944i	$-0.241048\pi$
$401$	4.68007e6	1.45342	0.726711	+	0.686944i	$0.241048\pi$
$402$	−68071.0	−0.0210086
$403$	−936696.	−0.287300
$404$	1.75971e6	0.536398
$405$	967861.	0.293208
$406$	0	0
$407$	−5.11045e6	−1.52923
$408$	−3.18311e6	−0.946677
$409$	2.10911e6	0.623435	0.311718	−	0.950175i	$-0.399096\pi$
$409$	2.10911e6	0.623435	0.311718	+	0.950175i	$0.399096\pi$
$410$	843531.	0.247823
$411$	2.16399e6	0.631903
$412$	397780.	0.115452
$413$	0	0
$414$	−8.77416e6	−2.51597
$415$	425931.	0.121400
$416$	−3.21840e6	−0.911816
$417$	7.84333e6	2.20882
$418$	−7.04473e6	−1.97208
$419$	919295.	0.255811	0.127906	−	0.991786i	$-0.459175\pi$
$419$	919295.	0.255811	0.127906	+	0.991786i	$0.459175\pi$
$420$	0	0
$421$	−498570.	−0.137095	−0.0685473	−	0.997648i	$-0.521836\pi$
$421$	−498570.	−0.137095	−0.0685473	+	0.997648i	$0.521836\pi$
$422$	−1.51309e6	−0.413604
$423$	−281262.	−0.0764292
$424$	4.14515e6	1.11976
$425$	−688593.	−0.184923
$426$	−1.71794e6	−0.458653
$427$	0	0
$428$	1.81543e6	0.479038
$429$	−1.09354e7	−2.86874
$430$	1.38886e6	0.362231
$431$	2.61474e6	0.678008	0.339004	−	0.940785i	$-0.389910\pi$
$431$	2.61474e6	0.678008	0.339004	+	0.940785i	$0.389910\pi$
$432$	−1.97252e6	−0.508524
$433$	3.41998e6	0.876605	0.438303	−	0.898827i	$-0.355580\pi$
$433$	3.41998e6	0.876605	0.438303	+	0.898827i	$0.355580\pi$
$434$	0	0
$435$	1.06047e6	0.268705
$436$	1.00975e6	0.254387
$437$	6.42578e6	1.60962
$438$	−5.61810e6	−1.39928
$439$	−3.10856e6	−0.769837	−0.384918	−	0.922951i	$-0.625770\pi$
$439$	−3.10856e6	−0.769837	−0.384918	+	0.922951i	$0.625770\pi$
$440$	−2.08943e6	−0.514513
$441$	0	0
$442$	−5.10948e6	−1.24400
$443$	−5.22719e6	−1.26549	−0.632746	−	0.774360i	$-0.718072\pi$
$443$	−5.22719e6	−1.26549	−0.632746	+	0.774360i	$0.718072\pi$
$444$	−2.44385e6	−0.588324
$445$	−162144.	−0.0388150
$446$	6.98169e6	1.66197
$447$	2.78220e6	0.658596
$448$	0	0
$449$	1.86850e6	0.437398	0.218699	−	0.975792i	$-0.429819\pi$
$449$	1.86850e6	0.437398	0.218699	+	0.975792i	$0.429819\pi$
$450$	−1.30679e6	−0.304212
$451$	3.38684e6	0.784068
$452$	−1.98561e6	−0.457138
$453$	−2.00919e6	−0.460019
$454$	−5.42015e6	−1.23416
$455$	0	0
$456$	4.42401e6	0.996333
$457$	944245.	0.211492	0.105746	−	0.994393i	$-0.466277\pi$
$457$	944245.	0.211492	0.105746	+	0.994393i	$0.466277\pi$
$458$	−219650.	−0.0489292
$459$	−1.70408e6	−0.377535
$460$	−1.45128e6	−0.319784
$461$	6.15997e6	1.34998	0.674988	−	0.737829i	$-0.264149\pi$
$461$	6.15997e6	1.34998	0.674988	+	0.737829i	$0.264149\pi$
$462$	0	0
$463$	3.76264e6	0.815719	0.407860	−	0.913045i	$-0.366275\pi$
$463$	3.76264e6	0.815719	0.407860	+	0.913045i	$0.366275\pi$
$464$	2.30286e6	0.496560
$465$	−803051.	−0.172231
$466$	−1.65532e6	−0.353116
$467$	−8.00623e6	−1.69878	−0.849388	−	0.527769i	$-0.823029\pi$
$467$	−8.00623e6	−1.69878	−0.849388	+	0.527769i	$0.823029\pi$
$468$	−2.92665e6	−0.617670
$469$	0	0
$470$	−154137.	−0.0321856
$471$	8.84705e6	1.83758
$472$	−1.09946e6	−0.227155
$473$	5.57636e6	1.14604
$474$	−1.57894e7	−3.22791
$475$	957034.	0.194623
$476$	0	0
$477$	1.04091e7	2.09468
$478$	104396.	0.0208985
$479$	5.62866e6	1.12090	0.560450	−	0.828189i	$-0.310628\pi$
$479$	5.62866e6	1.12090	0.560450	+	0.828189i	$0.310628\pi$
$480$	−2.75921e6	−0.546616
$481$	5.15152e6	1.01525
$482$	682856.	0.133879
$483$	0	0
$484$	4.16038e6	0.807272
$485$	−2.58446e6	−0.498902
$486$	8.70155e6	1.67111
$487$	−4.69075e6	−0.896231	−0.448116	−	0.893976i	$-0.647905\pi$
$487$	−4.69075e6	−0.896231	−0.448116	+	0.893976i	$0.647905\pi$
$488$	−299868.	−0.0570007
$489$	−1.14328e6	−0.216212
$490$	0	0
$491$	−4.98230e6	−0.932667	−0.466333	−	0.884609i	$-0.654425\pi$
$491$	−4.98230e6	−0.932667	−0.466333	+	0.884609i	$0.654425\pi$
$492$	1.61961e6	0.301646
$493$	1.98946e6	0.368653
$494$	7.10135e6	1.30925
$495$	−5.24688e6	−0.962472
$496$	−1.74386e6	−0.318278
$497$	0	0
$498$	2.70956e6	0.489581
$499$	3.71500e6	0.667894	0.333947	−	0.942592i	$-0.391619\pi$
$499$	3.71500e6	0.667894	0.333947	+	0.942592i	$0.391619\pi$
$500$	−216149.	−0.0386659
$501$	−1.60002e7	−2.84795
$502$	5.98549e6	1.06008
$503$	−3.81219e6	−0.671822	−0.335911	−	0.941894i	$-0.609044\pi$
$503$	−3.81219e6	−0.671822	−0.335911	+	0.941894i	$0.609044\pi$
$504$	0	0
$505$	−3.18015e6	−0.554906
$506$	−1.93062e7	−3.35212
$507$	2.30115e6	0.397580
$508$	229900.	0.0395257
$509$	−9.04579e6	−1.54758	−0.773788	−	0.633445i	$-0.781640\pi$
$509$	−9.04579e6	−1.54758	−0.773788	+	0.633445i	$0.781640\pi$
$510$	−4.38048e6	−0.745755
$511$	0	0
$512$	−972628.	−0.163973
$513$	2.36839e6	0.397338
$514$	−4.61024e6	−0.769690
$515$	−718869.	−0.119435
$516$	2.66665e6	0.440901
$517$	−618872.	−0.101830
$518$	0	0
$519$	−1.58027e7	−2.57521
$520$	2.10622e6	0.341583
$521$	−2.71943e6	−0.438918	−0.219459	−	0.975622i	$-0.570429\pi$
$521$	−2.71943e6	−0.438918	−0.219459	+	0.975622i	$0.570429\pi$
$522$	3.77554e6	0.606462
$523$	8.92836e6	1.42731	0.713654	−	0.700498i	$-0.247039\pi$
$523$	8.92836e6	1.42731	0.713654	+	0.700498i	$0.247039\pi$
$524$	3.93123e6	0.625460
$525$	0	0
$526$	−6.64933e6	−1.04788
$527$	−1.50653e6	−0.236294
$528$	−2.03586e7	−3.17806
$529$	1.11736e7	1.73601
$530$	5.70439e6	0.882104
$531$	−2.76090e6	−0.424928
$532$	0	0
$533$	−3.41407e6	−0.520540
$534$	−1.03147e6	−0.156533
$535$	−3.28085e6	−0.495566
$536$	52641.1	0.00791432
$537$	−61743.0	−0.00923957
$538$	4.26242e6	0.634893
$539$	0	0
$540$	−534908.	−0.0789396
$541$	−8.98350e6	−1.31963	−0.659815	−	0.751428i	$-0.729366\pi$
$541$	−8.98350e6	−1.31963	−0.659815	+	0.751428i	$0.729366\pi$
$542$	6.51138e6	0.952083
$543$	−9.69831e6	−1.41155
$544$	−5.17632e6	−0.749935
$545$	−1.82481e6	−0.263165
$546$	0	0
$547$	1.31609e7	1.88068	0.940342	−	0.340230i	$-0.110505\pi$
$547$	1.31609e7	1.88068	0.940342	+	0.340230i	$0.110505\pi$
$548$	1.27433e6	0.181271
$549$	−753014.	−0.106628
$550$	−2.87539e6	−0.405313
$551$	−2.76503e6	−0.387990
$552$	1.21241e7	1.69356
$553$	0	0
$554$	−3.04994e6	−0.422198
$555$	4.41652e6	0.608623
$556$	4.61877e6	0.633636
$557$	−4.07803e6	−0.556945	−0.278472	−	0.960444i	$-0.589828\pi$
$557$	−4.07803e6	−0.556945	−0.278472	+	0.960444i	$0.589828\pi$
$558$	−2.85906e6	−0.388721
$559$	−5.62118e6	−0.760849
$560$	0	0
$561$	−1.75880e7	−2.35944
$562$	−5.85901e6	−0.782498
$563$	8.10182e6	1.07724	0.538619	−	0.842549i	$-0.318946\pi$
$563$	8.10182e6	1.07724	0.538619	+	0.842549i	$0.318946\pi$
$564$	−295948.	−0.0391757
$565$	3.58840e6	0.472911
$566$	9.24280e6	1.21273
$567$	0	0
$568$	1.32853e6	0.172783
$569$	−5.45110e6	−0.705836	−0.352918	−	0.935654i	$-0.614810\pi$
$569$	−5.45110e6	−0.705836	−0.352918	+	0.935654i	$0.614810\pi$
$570$	6.08816e6	0.784872
$571$	5.26286e6	0.675510	0.337755	−	0.941234i	$-0.390332\pi$
$571$	5.26286e6	0.675510	0.337755	+	0.941234i	$0.390332\pi$
$572$	−6.43963e6	−0.822945
$573$	−1.40442e7	−1.78695
$574$	0	0
$575$	2.62276e6	0.330818
$576$	2.78028e6	0.349166
$577$	6.84574e6	0.856014	0.428007	−	0.903775i	$-0.359216\pi$
$577$	6.84574e6	0.856014	0.428007	+	0.903775i	$0.359216\pi$
$578$	1.39467e6	0.173640
$579$	1.69287e7	2.09859
$580$	624490.	0.0770824
$581$	0	0
$582$	−1.64410e7	−2.01197
$583$	2.29036e7	2.79082
$584$	4.34463e6	0.527134
$585$	5.28905e6	0.638981
$586$	1.58625e7	1.90821
$587$	−7.05308e6	−0.844857	−0.422429	−	0.906396i	$-0.638822\pi$
$587$	−7.05308e6	−0.844857	−0.422429	+	0.906396i	$0.638822\pi$
$588$	0	0
$589$	2.09384e6	0.248688
$590$	−1.51303e6	−0.178944
$591$	−2.45452e7	−2.89066
$592$	9.59065e6	1.12472
$593$	−8.79571e6	−1.02715	−0.513576	−	0.858044i	$-0.671679\pi$
$593$	−8.79571e6	−1.02715	−0.513576	+	0.858044i	$0.671679\pi$
$594$	−7.11580e6	−0.827480
$595$	0	0
$596$	1.63838e6	0.188929
$597$	−5.25371e6	−0.603296
$598$	1.94613e7	2.22546
$599$	−1.35162e7	−1.53918	−0.769588	−	0.638541i	$-0.779538\pi$
$599$	−1.35162e7	−1.53918	−0.769588	+	0.638541i	$0.779538\pi$
$600$	1.80572e6	0.204772
$601$	−8.57754e6	−0.968672	−0.484336	−	0.874882i	$-0.660939\pi$
$601$	−8.57754e6	−0.968672	−0.484336	+	0.874882i	$0.660939\pi$
$602$	0	0
$603$	132190.	0.0148049
$604$	−1.18317e6	−0.131964
$605$	−7.51866e6	−0.835126
$606$	−2.02305e7	−2.23782
$607$	9.64688e6	1.06271	0.531356	−	0.847149i	$-0.321683\pi$
$607$	9.64688e6	1.06271	0.531356	+	0.847149i	$0.321683\pi$
$608$	7.19425e6	0.789272
$609$	0	0
$610$	−412667.	−0.0449029
$611$	623846.	0.0676043
$612$	−4.70708e6	−0.508011
$613$	3.74910e6	0.402973	0.201486	−	0.979491i	$-0.435423\pi$
$613$	3.74910e6	0.402973	0.201486	+	0.979491i	$0.435423\pi$
$614$	1.80065e7	1.92756
$615$	−2.92696e6	−0.312053
$616$	0	0
$617$	5.92799e6	0.626895	0.313447	−	0.949606i	$-0.398516\pi$
$617$	5.92799e6	0.626895	0.313447	+	0.949606i	$0.398516\pi$
$618$	−4.57308e6	−0.481657
$619$	1.44089e6	0.151149	0.0755744	−	0.997140i	$-0.475921\pi$
$619$	1.44089e6	0.151149	0.0755744	+	0.997140i	$0.475921\pi$
$620$	−472900.	−0.0494072
$621$	6.49060e6	0.675392
$622$	2.69445e6	0.279251
$623$	0	0
$624$	2.05222e7	2.10990
$625$	390625.	0.0400000
$626$	2.67506e6	0.272834
$627$	2.44444e7	2.48320
$628$	5.20984e6	0.527139
$629$	8.28545e6	0.835006
$630$	0	0
$631$	−3.20551e6	−0.320497	−0.160249	−	0.987077i	$-0.551230\pi$
$631$	−3.20551e6	−0.320497	−0.160249	+	0.987077i	$0.551230\pi$
$632$	1.22104e7	1.21601
$633$	5.25026e6	0.520801
$634$	6.62298e6	0.654380
$635$	−415476.	−0.0408895
$636$	1.09526e7	1.07368
$637$	0	0
$638$	8.30749e6	0.808012
$639$	3.33615e6	0.323216
$640$	5.28226e6	0.509765
$641$	−2.62918e6	−0.252741	−0.126370	−	0.991983i	$-0.540333\pi$
$641$	−2.62918e6	−0.252741	−0.126370	+	0.991983i	$0.540333\pi$
$642$	−2.08711e7	−1.99852
$643$	−7.74923e6	−0.739148	−0.369574	−	0.929201i	$-0.620496\pi$
$643$	−7.74923e6	−0.739148	−0.369574	+	0.929201i	$0.620496\pi$
$644$	0	0
$645$	−4.81917e6	−0.456114
$646$	1.14215e7	1.07681
$647$	−1.53834e7	−1.44475	−0.722373	−	0.691504i	$-0.756949\pi$
$647$	−1.53834e7	−1.44475	−0.722373	+	0.691504i	$0.756949\pi$
$648$	−4.76141e6	−0.445449
$649$	−6.07493e6	−0.566148
$650$	2.89850e6	0.269086
$651$	0	0
$652$	−673253.	−0.0620240
$653$	829826.	0.0761560	0.0380780	−	0.999275i	$-0.487876\pi$
$653$	829826.	0.0761560	0.0380780	+	0.999275i	$0.487876\pi$
$654$	−1.16085e7	−1.06129
$655$	−7.10452e6	−0.647041
$656$	−6.35600e6	−0.576666
$657$	1.09100e7	0.986082
$658$	0	0
$659$	6.26272e6	0.561758	0.280879	−	0.959743i	$-0.409374\pi$
$659$	6.26272e6	0.561758	0.280879	+	0.959743i	$0.409374\pi$
$660$	−5.52085e6	−0.493340
$661$	1.13887e7	1.01384	0.506919	−	0.861994i	$-0.330784\pi$
$661$	1.13887e7	1.01384	0.506919	+	0.861994i	$0.330784\pi$
$662$	−2.31850e7	−2.05619
$663$	1.77293e7	1.56642
$664$	−2.09537e6	−0.184434
$665$	0	0
$666$	1.57239e7	1.37365
$667$	−7.57759e6	−0.659502
$668$	−9.42220e6	−0.816980
$669$	−2.42257e7	−2.09272
$670$	72442.7	0.00623459
$671$	−1.65689e6	−0.142065
$672$	0	0
$673$	−1.46408e7	−1.24603	−0.623014	−	0.782211i	$-0.714092\pi$
$673$	−1.46408e7	−1.24603	−0.623014	+	0.782211i	$0.714092\pi$
$674$	1.46138e7	1.23912
$675$	966688.	0.0816633
$676$	1.35510e6	0.114052
$677$	7.51341e6	0.630036	0.315018	−	0.949086i	$-0.397989\pi$
$677$	7.51341e6	0.630036	0.315018	+	0.949086i	$0.397989\pi$
$678$	2.28275e7	1.90715
$679$	0	0
$680$	3.38754e6	0.280939
$681$	1.88073e7	1.55403
$682$	−6.29091e6	−0.517908
$683$	1.89830e7	1.55709	0.778546	−	0.627588i	$-0.215958\pi$
$683$	1.89830e7	1.55709	0.778546	+	0.627588i	$0.215958\pi$
$684$	6.54208e6	0.534657
$685$	−2.30297e6	−0.187526
$686$	0	0
$687$	762162.	0.0616106
$688$	−1.04650e7	−0.842886
$689$	−2.30877e7	−1.85281
$690$	1.66847e7	1.33412
$691$	−1.18823e7	−0.946685	−0.473343	−	0.880878i	$-0.656953\pi$
$691$	−1.18823e7	−0.946685	−0.473343	+	0.880878i	$0.656953\pi$
$692$	−9.30588e6	−0.738741
$693$	0	0
$694$	2.41954e7	1.90693
$695$	−8.34706e6	−0.655498
$696$	−5.21701e6	−0.408224
$697$	−5.49101e6	−0.428125
$698$	−1.81368e7	−1.40904
$699$	5.74378e6	0.444636
$700$	0	0
$701$	1.64049e7	1.26090	0.630448	−	0.776231i	$-0.282871\pi$
$701$	1.64049e7	1.26090	0.630448	+	0.776231i	$0.282871\pi$
$702$	7.17299e6	0.549361
$703$	−1.15154e7	−0.878805
$704$	6.11756e6	0.465207
$705$	534838.	0.0405274
$706$	−7.95186e6	−0.600422
$707$	0	0
$708$	−2.90507e6	−0.217807
$709$	−1.30955e7	−0.978375	−0.489188	−	0.872179i	$-0.662707\pi$
$709$	−1.30955e7	−0.978375	−0.489188	+	0.872179i	$0.662707\pi$
$710$	1.82827e6	0.136112
$711$	3.06622e7	2.27473
$712$	797668.	0.0589688
$713$	5.73819e6	0.422718
$714$	0	0
$715$	1.16377e7	0.851340
$716$	−36359.1	−0.00265052
$717$	−362243.	−0.0263149
$718$	2.10740e7	1.52558
$719$	−2.27950e6	−0.164444	−0.0822220	−	0.996614i	$-0.526202\pi$
$719$	−2.27950e6	−0.164444	−0.0822220	+	0.996614i	$0.526202\pi$
$720$	9.84669e6	0.707879
$721$	0	0
$722$	889317.	0.0634912
$723$	−2.36943e6	−0.168577
$724$	−5.71113e6	−0.404926
$725$	−1.12858e6	−0.0797420
$726$	−4.78299e7	−3.36789
$727$	−2.02817e7	−1.42321	−0.711603	−	0.702582i	$-0.752031\pi$
$727$	−2.02817e7	−1.42321	−0.711603	+	0.702582i	$0.752031\pi$
$728$	0	0
$729$	−2.07858e7	−1.44860
$730$	5.97892e6	0.415255
$731$	−9.04083e6	−0.625770
$732$	−792333.	−0.0546550
$733$	−2.80673e7	−1.92948	−0.964740	−	0.263205i	$-0.915220\pi$
$733$	−2.80673e7	−1.92948	−0.964740	+	0.263205i	$0.915220\pi$
$734$	1.44601e7	0.990676
$735$	0	0
$736$	1.97159e7	1.34160
$737$	290863.	0.0197251
$738$	−1.04207e7	−0.704297
$739$	1.74967e7	1.17854	0.589271	−	0.807936i	$-0.299415\pi$
$739$	1.74967e7	1.17854	0.589271	+	0.807936i	$0.299415\pi$
$740$	2.60080e6	0.174593
$741$	−2.46409e7	−1.64858
$742$	0	0
$743$	1.17150e7	0.778523	0.389261	−	0.921127i	$-0.372730\pi$
$743$	1.17150e7	0.778523	0.389261	+	0.921127i	$0.372730\pi$
$744$	3.95062e6	0.261657
$745$	−2.96088e6	−0.195448
$746$	1.75019e7	1.15143
$747$	−5.26181e6	−0.345011
$748$	−1.03572e7	−0.676842
$749$	0	0
$750$	2.48496e6	0.161312
$751$	4.06898e6	0.263261	0.131630	−	0.991299i	$-0.457979\pi$
$751$	4.06898e6	0.263261	0.131630	+	0.991299i	$0.457979\pi$
$752$	1.16142e6	0.0748936
$753$	−2.07690e7	−1.33483
$754$	−8.37426e6	−0.536436
$755$	2.13823e6	0.136517
$756$	0	0
$757$	8.45414e6	0.536204	0.268102	−	0.963391i	$-0.413604\pi$
$757$	8.45414e6	0.536204	0.268102	+	0.963391i	$0.413604\pi$
$758$	−1.80876e7	−1.14343
$759$	6.69902e7	4.22092
$760$	−4.70814e6	−0.295676
$761$	−678347.	−0.0424610	−0.0212305	−	0.999775i	$-0.506758\pi$
$761$	−678347.	−0.0424610	−0.0212305	+	0.999775i	$0.506758\pi$
$762$	−2.64305e6	−0.164899
$763$	0	0
$764$	−8.27035e6	−0.512614
$765$	8.50665e6	0.525539
$766$	−2.82377e6	−0.173883
$767$	6.12376e6	0.375863
$768$	2.68359e7	1.64177
$769$	2.33463e7	1.42365	0.711824	−	0.702358i	$-0.247869\pi$
$769$	2.33463e7	1.42365	0.711824	+	0.702358i	$0.247869\pi$
$770$	0	0
$771$	1.59970e7	0.969177
$772$	9.96896e6	0.602014
$773$	−7.64877e6	−0.460408	−0.230204	−	0.973142i	$-0.573939\pi$
$773$	−7.64877e6	−0.460408	−0.230204	+	0.973142i	$0.573939\pi$
$774$	−1.71574e7	−1.02944
$775$	854626.	0.0511119
$776$	1.27143e7	0.757945
$777$	0	0
$778$	−9.77998e6	−0.579280
$779$	7.63162e6	0.450581
$780$	5.56522e6	0.327526
$781$	7.34066e6	0.430633
$782$	3.13006e7	1.83036
$783$	−2.79292e6	−0.162800
$784$	0	0
$785$	−9.41524e6	−0.545327
$786$	−4.51953e7	−2.60938
$787$	2.26164e7	1.30163	0.650814	−	0.759237i	$-0.274428\pi$
$787$	2.26164e7	1.30163	0.650814	+	0.759237i	$0.274428\pi$
$788$	−1.44541e7	−0.829233
$789$	2.30724e7	1.31947
$790$	1.68035e7	0.957926
$791$	0	0
$792$	2.58121e7	1.46221
$793$	1.67021e6	0.0943164
$794$	−1.61428e7	−0.908713
$795$	−1.97936e7	−1.11073
$796$	−3.09380e6	−0.173065
$797$	7.01242e6	0.391041	0.195521	−	0.980700i	$-0.437360\pi$
$797$	7.01242e6	0.391041	0.195521	+	0.980700i	$0.437360\pi$
$798$	0	0
$799$	1.00336e6	0.0556020
$800$	2.93642e6	0.162216
$801$	2.00307e6	0.110310
$802$	−3.16843e7	−1.73943
$803$	2.40058e7	1.31380
$804$	139092.	0.00758863
$805$	0	0
$806$	6.34147e6	0.343837
$807$	−1.47901e7	−0.799444
$808$	1.56448e7	0.843027
$809$	−1.56622e7	−0.841357	−0.420679	−	0.907210i	$-0.638208\pi$
$809$	−1.56622e7	−0.841357	−0.420679	+	0.907210i	$0.638208\pi$
$810$	−6.55247e6	−0.350907
$811$	−95904.4	−0.00512019	−0.00256010	−	0.999997i	$-0.500815\pi$
$811$	−95904.4	−0.00512019	−0.00256010	+	0.999997i	$0.500815\pi$
$812$	0	0
$813$	−2.25938e7	−1.19884
$814$	3.45980e7	1.83016
$815$	1.21671e6	0.0641640
$816$	3.30069e7	1.73532
$817$	1.25653e7	0.658594
$818$	−1.42788e7	−0.746119
$819$	0	0
$820$	−1.72362e6	−0.0895175
$821$	1.44185e7	0.746558	0.373279	−	0.927719i	$-0.378233\pi$
$821$	1.44185e7	0.746558	0.373279	+	0.927719i	$0.378233\pi$
$822$	−1.46503e7	−0.756253
$823$	−1.68179e6	−0.0865509	−0.0432754	−	0.999063i	$-0.513779\pi$
$823$	−1.68179e6	−0.0865509	−0.0432754	+	0.999063i	$0.513779\pi$
$824$	3.53649e6	0.181449
$825$	9.97730e6	0.510362
$826$	0	0
$827$	−2.03174e7	−1.03301	−0.516506	−	0.856284i	$-0.672767\pi$
$827$	−2.03174e7	−1.03301	−0.516506	+	0.856284i	$0.672767\pi$
$828$	1.79286e7	0.908806
$829$	−2.22764e7	−1.12579	−0.562897	−	0.826527i	$-0.690313\pi$
$829$	−2.22764e7	−1.12579	−0.562897	+	0.826527i	$0.690313\pi$
$830$	−2.88357e6	−0.145290
$831$	1.05829e7	0.531623
$832$	−6.16673e6	−0.308849
$833$	0	0
$834$	−5.30997e7	−2.64349
$835$	1.70278e7	0.845168
$836$	1.43948e7	0.712345
$837$	2.11496e6	0.104349
$838$	−6.22367e6	−0.306151
$839$	1.12927e7	0.553850	0.276925	−	0.960892i	$-0.410685\pi$
$839$	1.12927e7	0.553850	0.276925	+	0.960892i	$0.410685\pi$
$840$	0	0
$841$	−1.72505e7	−0.841030
$842$	3.37534e6	0.164073
$843$	2.03301e7	0.985305
$844$	3.09177e6	0.149400
$845$	−2.44894e6	−0.117987
$846$	1.90415e6	0.0914695
$847$	0	0
$848$	−4.29826e7	−2.05259
$849$	−3.20715e7	−1.52704
$850$	4.66181e6	0.221313
$851$	−3.15582e7	−1.49378
$852$	3.51035e6	0.165673
$853$	−2.19428e7	−1.03257	−0.516285	−	0.856417i	$-0.672685\pi$
$853$	−2.19428e7	−1.03257	−0.516285	+	0.856417i	$0.672685\pi$
$854$	0	0
$855$	−1.18229e7	−0.553105
$856$	1.61402e7	0.752877
$857$	−1.75433e7	−0.815942	−0.407971	−	0.912995i	$-0.633764\pi$
$857$	−1.75433e7	−0.815942	−0.407971	+	0.912995i	$0.633764\pi$
$858$	7.40332e7	3.43327
$859$	6.13059e6	0.283478	0.141739	−	0.989904i	$-0.454731\pi$
$859$	6.13059e6	0.283478	0.141739	+	0.989904i	$0.454731\pi$
$860$	−2.83791e6	−0.130844
$861$	0	0
$862$	−1.77019e7	−0.811430
$863$	−2.70291e7	−1.23539	−0.617695	−	0.786418i	$-0.711933\pi$
$863$	−2.70291e7	−1.23539	−0.617695	+	0.786418i	$0.711933\pi$
$864$	7.26682e6	0.331177
$865$	1.68176e7	0.764230
$866$	−2.31534e7	−1.04911
$867$	−4.83934e6	−0.218644
$868$	0	0
$869$	6.74673e7	3.03071
$870$	−7.17945e6	−0.321583
$871$	−293201.	−0.0130954
$872$	8.97720e6	0.399806
$873$	3.19275e7	1.41785
$874$	−4.35028e7	−1.92637
$875$	0	0
$876$	1.14797e7	0.505441
$877$	1.65213e7	0.725345	0.362672	−	0.931917i	$-0.381864\pi$
$877$	1.65213e7	0.725345	0.362672	+	0.931917i	$0.381864\pi$
$878$	2.10451e7	0.921330
$879$	−5.50409e7	−2.40278
$880$	2.16661e7	0.943134
$881$	1.18753e6	0.0515472	0.0257736	−	0.999668i	$-0.491795\pi$
$881$	1.18753e6	0.0515472	0.0257736	+	0.999668i	$0.491795\pi$
$882$	0	0
$883$	−3.06022e7	−1.32084	−0.660420	−	0.750896i	$-0.729622\pi$
$883$	−3.06022e7	−1.32084	−0.660420	+	0.750896i	$0.729622\pi$
$884$	1.04404e7	0.449353
$885$	5.25004e6	0.225323
$886$	3.53883e7	1.51452
$887$	−1.06403e7	−0.454092	−0.227046	−	0.973884i	$-0.572907\pi$
$887$	−1.06403e7	−0.454092	−0.227046	+	0.973884i	$0.572907\pi$
$888$	−2.17272e7	−0.924635
$889$	0	0
$890$	1.09772e6	0.0464533
$891$	−2.63087e7	−1.11021
$892$	−1.42660e7	−0.600330
$893$	−1.39451e6	−0.0585185
$894$	−1.88356e7	−0.788199
$895$	65708.3	0.00274197
$896$	0	0
$897$	−6.75286e7	−2.80225
$898$	−1.26498e7	−0.523472
$899$	−2.46916e6	−0.101894
$900$	2.67023e6	0.109886
$901$	−3.71330e7	−1.52387
$902$	−2.29291e7	−0.938362
$903$	0	0
$904$	−1.76532e7	−0.718458
$905$	1.03212e7	0.418897
$906$	1.36023e7	0.550544
$907$	−3.32206e7	−1.34088	−0.670439	−	0.741964i	$-0.733894\pi$
$907$	−3.32206e7	−1.34088	−0.670439	+	0.741964i	$0.733894\pi$
$908$	1.10752e7	0.445798
$909$	3.92865e7	1.57701
$910$	0	0
$911$	2.32493e7	0.928140	0.464070	−	0.885798i	$-0.346389\pi$
$911$	2.32493e7	0.928140	0.464070	+	0.885798i	$0.346389\pi$
$912$	−4.58742e7	−1.82634
$913$	−1.15778e7	−0.459672
$914$	−6.39258e6	−0.253111
$915$	1.43191e6	0.0565408
$916$	448821.	0.0176740
$917$	0	0
$918$	1.15367e7	0.451829
$919$	1.94409e7	0.759323	0.379662	−	0.925125i	$-0.376040\pi$
$919$	1.94409e7	0.759323	0.379662	+	0.925125i	$0.376040\pi$
$920$	−1.29027e7	−0.502587
$921$	−6.24805e7	−2.42714
$922$	−4.17033e7	−1.61563
$923$	−7.39966e6	−0.285896
$924$	0	0
$925$	−4.70017e6	−0.180617
$926$	−2.54733e7	−0.976242
$927$	8.88067e6	0.339427
$928$	−8.48380e6	−0.323386
$929$	1.38184e7	0.525314	0.262657	−	0.964889i	$-0.415401\pi$
$929$	1.38184e7	0.525314	0.262657	+	0.964889i	$0.415401\pi$
$930$	5.43670e6	0.206124
$931$	0	0
$932$	3.38239e6	0.127551
$933$	−9.34945e6	−0.351627
$934$	5.42026e7	2.03307
$935$	1.87175e7	0.700196
$936$	−2.60196e7	−0.970757
$937$	−1.20217e6	−0.0447318	−0.0223659	−	0.999750i	$-0.507120\pi$
$937$	−1.20217e6	−0.0447318	−0.0223659	+	0.999750i	$0.507120\pi$
$938$	0	0
$939$	−9.28216e6	−0.343546
$940$	314955.	0.0116259
$941$	−2.62045e7	−0.964720	−0.482360	−	0.875973i	$-0.660220\pi$
$941$	−2.62045e7	−0.964720	−0.482360	+	0.875973i	$0.660220\pi$
$942$	−5.98949e7	−2.19919
$943$	2.09145e7	0.765894
$944$	1.14007e7	0.416390
$945$	0	0
$946$	−3.77523e7	−1.37156
$947$	−2.44193e7	−0.884826	−0.442413	−	0.896812i	$-0.645877\pi$
$947$	−2.44193e7	−0.884826	−0.442413	+	0.896812i	$0.645877\pi$
$948$	3.22633e7	1.16597
$949$	−2.41988e7	−0.872223
$950$	−6.47916e6	−0.232922
$951$	−2.29810e7	−0.823981
$952$	0	0
$953$	1.40817e7	0.502254	0.251127	−	0.967954i	$-0.419199\pi$
$953$	1.40817e7	0.502254	0.251127	+	0.967954i	$0.419199\pi$
$954$	−7.04701e7	−2.50688
$955$	1.49462e7	0.530301
$956$	−213317.	−0.00754885
$957$	−2.88261e7	−1.01743
$958$	−3.81063e7	−1.34148
$959$	0	0
$960$	−5.28688e6	−0.185149
$961$	−2.67594e7	−0.934689
$962$	−3.48761e7	−1.21504
$963$	4.05305e7	1.40837
$964$	−1.39531e6	−0.0483590
$965$	−1.80159e7	−0.622785
$966$	0	0
$967$	6.39355e6	0.219875	0.109938	−	0.993939i	$-0.464935\pi$
$967$	6.39355e6	0.219875	0.109938	+	0.993939i	$0.464935\pi$
$968$	3.69881e7	1.26874
$969$	−3.96312e7	−1.35590
$970$	1.74969e7	0.597079
$971$	1.69327e6	0.0576340	0.0288170	−	0.999585i	$-0.490826\pi$
$971$	1.69327e6	0.0576340	0.0288170	+	0.999585i	$0.490826\pi$
$972$	−1.77803e7	−0.603632
$973$	0	0
$974$	3.17566e7	1.07260
$975$	−1.00575e7	−0.338827
$976$	3.10944e6	0.104486
$977$	6.74819e6	0.226178	0.113089	−	0.993585i	$-0.463925\pi$
$977$	6.74819e6	0.226178	0.113089	+	0.993585i	$0.463925\pi$
$978$	7.74006e6	0.258760
$979$	4.40743e6	0.146970
$980$	0	0
$981$	2.25431e7	0.747897
$982$	3.37304e7	1.11620
$983$	1.31858e7	0.435234	0.217617	−	0.976034i	$-0.430172\pi$
$983$	1.31858e7	0.435234	0.217617	+	0.976034i	$0.430172\pi$
$984$	1.43992e7	0.474079
$985$	2.61216e7	0.857845
$986$	−1.34687e7	−0.441199
$987$	0	0
$988$	−1.45105e7	−0.472923
$989$	3.44353e7	1.11947
$990$	3.55216e7	1.15187
$991$	−2.42501e7	−0.784386	−0.392193	−	0.919883i	$-0.628284\pi$
$991$	−2.42501e7	−0.784386	−0.392193	+	0.919883i	$0.628284\pi$
$992$	6.42443e6	0.207279
$993$	8.04495e7	2.58911
$994$	0	0
$995$	5.59113e6	0.179036
$996$	−5.53655e6	−0.176844
$997$	−3.74969e7	−1.19470	−0.597348	−	0.801982i	$-0.703779\pi$
$997$	−3.74969e7	−1.19470	−0.597348	+	0.801982i	$0.703779\pi$
$998$	−2.51507e7	−0.799326
$999$	−1.16316e7	−0.368745

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.6.a.k.1.2		8
7.3	odd	6		35.6.e.b.16.7	yes	16
7.5	odd	6		35.6.e.b.11.7	✓	16
7.6	odd	2		245.6.a.j.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.6.e.b.11.7	✓	16	7.5	odd	6
35.6.e.b.16.7	yes	16	7.3	odd	6
245.6.a.j.1.2		8	7.6	odd	2
245.6.a.k.1.2		8	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)

\(f(q)\)	\(=\)	\(q-6.77005 q^{2} +23.4913 q^{3} +13.8335 q^{4} -25.0000 q^{5} -159.037 q^{6} +122.988 q^{8} +308.841 q^{9} +O(q^{10})\)	\(q-6.77005 q^{2} +23.4913 q^{3} +13.8335 q^{4} -25.0000 q^{5} -159.037 q^{6} +122.988 q^{8} +308.841 q^{9} +169.251 q^{10} +679.557 q^{11} +324.968 q^{12} -685.019 q^{13} -587.283 q^{15} -1275.31 q^{16} -1101.75 q^{17} -2090.87 q^{18} +1531.25 q^{19} -345.838 q^{20} -4600.63 q^{22} +4196.42 q^{23} +2889.14 q^{24} +625.000 q^{25} +4637.61 q^{26} +1546.70 q^{27} -1805.73 q^{29} +3975.93 q^{30} +1367.40 q^{31} +4698.27 q^{32} +15963.7 q^{33} +7458.89 q^{34} +4272.37 q^{36} -7520.27 q^{37} -10366.7 q^{38} -16092.0 q^{39} -3074.70 q^{40} +4983.90 q^{41} +8205.89 q^{43} +9400.67 q^{44} -7721.04 q^{45} -28409.9 q^{46} -910.699 q^{47} -29958.6 q^{48} -4231.28 q^{50} -25881.5 q^{51} -9476.23 q^{52} +33703.7 q^{53} -10471.2 q^{54} -16988.9 q^{55} +35971.2 q^{57} +12224.9 q^{58} -8939.55 q^{59} -8124.19 q^{60} -2438.19 q^{61} -9257.37 q^{62} +9002.28 q^{64} +17125.5 q^{65} -108075. q^{66} +428.019 q^{67} -15241.1 q^{68} +98579.3 q^{69} +10802.1 q^{71} +37983.7 q^{72} +35325.7 q^{73} +50912.6 q^{74} +14682.1 q^{75} +21182.6 q^{76} +108943. q^{78} +99281.4 q^{79} +31882.7 q^{80} -38714.4 q^{81} -33741.2 q^{82} -17037.2 q^{83} +27543.7 q^{85} -55554.2 q^{86} -42418.9 q^{87} +83577.2 q^{88} +6485.74 q^{89} +52271.8 q^{90} +58051.2 q^{92} +32122.1 q^{93} +6165.47 q^{94} -38281.3 q^{95} +110369. q^{96} +103378. q^{97} +209875. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} + 2 q^{3} + 203 q^{4} - 200 q^{5} + 56 q^{6} + 249 q^{8} + 1218 q^{9}+O(q^{10}) \) 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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