LMFDB - Embedded newform 637.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,6,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, 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("637.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	637.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:	$102.164493221$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.168897.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 100x + 256 $ x^3 - x^2 - 100*x + 256
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-10.6486$ of defining polynomial
Character	$\chi$	$=$	637.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.64858 q^{2} -10.2870 q^{3} +42.7979 q^{4} -92.1784 q^{5} +88.9676 q^{6} -93.3863 q^{8} -137.178 q^{9} +O(q^{10})$	\(q-8.64858 q^{2} -10.2870 q^{3} +42.7979 q^{4} -92.1784 q^{5} +88.9676 q^{6} -93.3863 q^{8} -137.178 q^{9} +797.212 q^{10} +571.711 q^{11} -440.260 q^{12} -169.000 q^{13} +948.236 q^{15} -561.873 q^{16} +855.571 q^{17} +1186.40 q^{18} +2355.90 q^{19} -3945.04 q^{20} -4944.49 q^{22} +2509.66 q^{23} +960.662 q^{24} +5371.86 q^{25} +1461.61 q^{26} +3910.88 q^{27} -5497.50 q^{29} -8200.89 q^{30} +144.924 q^{31} +7847.77 q^{32} -5881.17 q^{33} -7399.47 q^{34} -5870.95 q^{36} -515.975 q^{37} -20375.2 q^{38} +1738.50 q^{39} +8608.21 q^{40} +13928.9 q^{41} +7753.58 q^{43} +24468.0 q^{44} +12644.9 q^{45} -21705.0 q^{46} -8344.77 q^{47} +5779.97 q^{48} -46459.0 q^{50} -8801.22 q^{51} -7232.84 q^{52} -5976.21 q^{53} -33823.6 q^{54} -52699.5 q^{55} -24235.1 q^{57} +47545.6 q^{58} -2110.84 q^{59} +40582.5 q^{60} +17394.2 q^{61} -1253.38 q^{62} -49892.1 q^{64} +15578.2 q^{65} +50863.8 q^{66} -3121.05 q^{67} +36616.6 q^{68} -25816.8 q^{69} +43323.3 q^{71} +12810.6 q^{72} -6808.31 q^{73} +4462.45 q^{74} -55260.1 q^{75} +100828. q^{76} -15035.5 q^{78} -1280.80 q^{79} +51792.6 q^{80} -6896.72 q^{81} -120465. q^{82} -7148.44 q^{83} -78865.2 q^{85} -67057.5 q^{86} +56552.6 q^{87} -53390.0 q^{88} -107123. q^{89} -109360. q^{90} +107408. q^{92} -1490.82 q^{93} +72170.4 q^{94} -217163. q^{95} -80729.7 q^{96} -42456.2 q^{97} -78426.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 7 q^{2} - 8 q^{3} + 121 q^{4} - 56 q^{5} + 199 q^{6} + 327 q^{8} - 191 q^{9}+O(q^{10}) $ 3 * q + 7 * q^2 - 8 * q^3 + 121 * q^4 - 56 * q^5 + 199 * q^6 + 327 * q^8 - 191 * q^9	\( 3 q + 7 q^{2} - 8 q^{3} + 121 q^{4} - 56 q^{5} + 199 q^{6} + 327 q^{8} - 191 q^{9} + 1291 q^{10} + 556 q^{11} + 1091 q^{12} - 507 q^{13} + 1972 q^{15} + 2785 q^{16} - 908 q^{17} + 976 q^{18} - 148 q^{19} + 767 q^{20} - 3410 q^{22} + 3624 q^{23} + 13377 q^{24} + 2023 q^{25} - 1183 q^{26} + 4276 q^{27} - 8758 q^{29} + 1717 q^{30} + 2608 q^{31} + 34871 q^{32} + 1828 q^{33} - 15989 q^{34} - 4678 q^{36} - 20632 q^{37} - 33786 q^{38} + 1352 q^{39} + 43509 q^{40} + 10998 q^{41} + 2032 q^{43} + 49790 q^{44} + 13918 q^{45} - 3408 q^{46} - 34260 q^{47} + 76223 q^{48} - 65038 q^{50} + 13468 q^{51} - 20449 q^{52} - 12570 q^{53} - 55595 q^{54} - 35312 q^{55} + 1716 q^{57} - 19126 q^{58} - 63948 q^{59} + 110465 q^{60} + 12754 q^{61} + 40340 q^{62} + 117393 q^{64} + 9464 q^{65} + 112354 q^{66} + 56132 q^{67} + 49175 q^{68} + 20112 q^{69} + 77580 q^{71} + 28794 q^{72} + 43026 q^{73} - 131519 q^{74} - 23548 q^{75} + 99638 q^{76} - 33631 q^{78} - 61872 q^{79} + 264683 q^{80} - 108221 q^{81} - 142260 q^{82} - 98092 q^{83} - 55226 q^{85} - 69897 q^{86} - 139072 q^{87} + 158010 q^{88} - 33694 q^{89} - 101258 q^{90} + 300864 q^{92} + 105392 q^{93} - 72843 q^{94} - 196560 q^{95} + 243641 q^{96} - 76334 q^{97} - 60332 q^{99}+O(q^{100}) \) 3 * q + 7 * q^2 - 8 * q^3 + 121 * q^4 - 56 * q^5 + 199 * q^6 + 327 * q^8 - 191 * q^9 + 1291 * q^10 + 556 * q^11 + 1091 * q^12 - 507 * q^13 + 1972 * q^15 + 2785 * q^16 - 908 * q^17 + 976 * q^18 - 148 * q^19 + 767 * q^20 - 3410 * q^22 + 3624 * q^23 + 13377 * q^24 + 2023 * q^25 - 1183 * q^26 + 4276 * q^27 - 8758 * q^29 + 1717 * q^30 + 2608 * q^31 + 34871 * q^32 + 1828 * q^33 - 15989 * q^34 - 4678 * q^36 - 20632 * q^37 - 33786 * q^38 + 1352 * q^39 + 43509 * q^40 + 10998 * q^41 + 2032 * q^43 + 49790 * q^44 + 13918 * q^45 - 3408 * q^46 - 34260 * q^47 + 76223 * q^48 - 65038 * q^50 + 13468 * q^51 - 20449 * q^52 - 12570 * q^53 - 55595 * q^54 - 35312 * q^55 + 1716 * q^57 - 19126 * q^58 - 63948 * q^59 + 110465 * q^60 + 12754 * q^61 + 40340 * q^62 + 117393 * q^64 + 9464 * q^65 + 112354 * q^66 + 56132 * q^67 + 49175 * q^68 + 20112 * q^69 + 77580 * q^71 + 28794 * q^72 + 43026 * q^73 - 131519 * q^74 - 23548 * q^75 + 99638 * q^76 - 33631 * q^78 - 61872 * q^79 + 264683 * q^80 - 108221 * q^81 - 142260 * q^82 - 98092 * q^83 - 55226 * q^85 - 69897 * q^86 - 139072 * q^87 + 158010 * q^88 - 33694 * q^89 - 101258 * q^90 + 300864 * q^92 + 105392 * q^93 - 72843 * q^94 - 196560 * q^95 + 243641 * q^96 - 76334 * q^97 - 60332 * 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 $

$2$	−8.64858	−1.52887	−0.764433	−	0.644703i	$-0.776981\pi$
$2$	−8.64858	−1.52887	−0.764433	+	0.644703i	$0.776981\pi$
$3$	−10.2870	−0.659909	−0.329954	−	0.943997i	$-0.607033\pi$
$3$	−10.2870	−0.659909	−0.329954	+	0.943997i	$0.607033\pi$
$4$	42.7979	1.33743
$5$	−92.1784	−1.64894	−0.824469	−	0.565907i	$-0.808526\pi$
$5$	−92.1784	−1.64894	−0.824469	+	0.565907i	$0.808526\pi$
$6$	88.9676	1.00891
$7$	0	0
$7$	0	0
$8$	−93.3863	−0.515892
$9$	−137.178	−0.564520
$10$	797.212	2.52101
$11$	571.711	1.42461	0.712304	−	0.701871i	$-0.247652\pi$
$11$	571.711	1.42461	0.712304	+	0.701871i	$0.247652\pi$
$12$	−440.260	−0.882585
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	0	0
$15$	948.236	1.08815
$16$	−561.873	−0.548704
$17$	855.571	0.718015	0.359008	−	0.933335i	$-0.383115\pi$
$17$	855.571	0.718015	0.359008	+	0.933335i	$0.383115\pi$
$18$	1186.40	0.863076
$19$	2355.90	1.49718	0.748589	−	0.663034i	$-0.230732\pi$
$19$	2355.90	1.49718	0.748589	+	0.663034i	$0.230732\pi$
$20$	−3945.04	−2.20535
$21$	0	0
$22$	−4944.49	−2.17804
$23$	2509.66	0.989227	0.494613	−	0.869113i	$-0.335310\pi$
$23$	2509.66	0.989227	0.494613	+	0.869113i	$0.335310\pi$
$24$	960.662	0.340441
$25$	5371.86	1.71900
$26$	1461.61	0.424031
$27$	3910.88	1.03244
$28$	0	0
$29$	−5497.50	−1.21387	−0.606933	−	0.794753i	$-0.707600\pi$
$29$	−5497.50	−1.21387	−0.606933	+	0.794753i	$0.707600\pi$
$30$	−8200.89	−1.66363
$31$	144.924	0.0270854	0.0135427	−	0.999908i	$-0.495689\pi$
$31$	144.924	0.0270854	0.0135427	+	0.999908i	$0.495689\pi$
$32$	7847.77	1.35479
$33$	−5881.17	−0.940111
$34$	−7399.47	−1.09775
$35$	0	0
$36$	−5870.95	−0.755009
$37$	−515.975	−0.0619618	−0.0309809	−	0.999520i	$-0.509863\pi$
$37$	−515.975	−0.0619618	−0.0309809	+	0.999520i	$0.509863\pi$
$38$	−20375.2	−2.28899
$39$	1738.50	0.183026
$40$	8608.21	0.850673
$41$	13928.9	1.29407	0.647033	−	0.762462i	$-0.276009\pi$
$41$	13928.9	1.29407	0.647033	+	0.762462i	$0.276009\pi$
$42$	0	0
$43$	7753.58	0.639486	0.319743	−	0.947504i	$-0.396403\pi$
$43$	7753.58	0.639486	0.319743	+	0.947504i	$0.396403\pi$
$44$	24468.0	1.90532
$45$	12644.9	0.930859
$46$	−21705.0	−1.51240
$47$	−8344.77	−0.551023	−0.275512	−	0.961298i	$-0.588847\pi$
$47$	−8344.77	−0.551023	−0.275512	+	0.961298i	$0.588847\pi$
$48$	5779.97	0.362095
$49$	0	0
$50$	−46459.0	−2.62812
$51$	−8801.22	−0.473825
$52$	−7232.84	−0.370937
$53$	−5976.21	−0.292238	−0.146119	−	0.989267i	$-0.546678\pi$
$53$	−5976.21	−0.292238	−0.146119	+	0.989267i	$0.546678\pi$
$54$	−33823.6	−1.57846
$55$	−52699.5	−2.34909
$56$	0	0
$57$	−24235.1	−0.988001
$58$	47545.6	1.85584
$59$	−2110.84	−0.0789451	−0.0394726	−	0.999221i	$-0.512568\pi$
$59$	−2110.84	−0.0789451	−0.0394726	+	0.999221i	$0.512568\pi$
$60$	40582.5	1.45533
$61$	17394.2	0.598522	0.299261	−	0.954171i	$-0.403260\pi$
$61$	17394.2	0.598522	0.299261	+	0.954171i	$0.403260\pi$
$62$	−1253.38	−0.0414099
$63$	0	0
$64$	−49892.1	−1.52259
$65$	15578.2	0.457333
$66$	50863.8	1.43730
$67$	−3121.05	−0.0849402	−0.0424701	−	0.999098i	$-0.513523\pi$
$67$	−3121.05	−0.0849402	−0.0424701	+	0.999098i	$0.513523\pi$
$68$	36616.6	0.960298
$69$	−25816.8	−0.652800
$70$	0	0
$71$	43323.3	1.01994	0.509971	−	0.860191i	$-0.329656\pi$
$71$	43323.3	1.01994	0.509971	+	0.860191i	$0.329656\pi$
$72$	12810.6	0.291231
$73$	−6808.31	−0.149531	−0.0747657	−	0.997201i	$-0.523821\pi$
$73$	−6808.31	−0.149531	−0.0747657	+	0.997201i	$0.523821\pi$
$74$	4462.45	0.0947314
$75$	−55260.1	−1.13438
$76$	100828.	2.00238
$77$	0	0
$78$	−15035.5	−0.279822
$79$	−1280.80	−0.0230895	−0.0115447	−	0.999933i	$-0.503675\pi$
$79$	−1280.80	−0.0230895	−0.0115447	+	0.999933i	$0.503675\pi$
$80$	51792.6	0.904779
$81$	−6896.72	−0.116797
$82$	−120465.	−1.97846
$83$	−7148.44	−0.113898	−0.0569490	−	0.998377i	$-0.518137\pi$
$83$	−7148.44	−0.113898	−0.0569490	+	0.998377i	$0.518137\pi$
$84$	0	0
$85$	−78865.2	−1.18396
$86$	−67057.5	−0.977690
$87$	56552.6	0.801041
$88$	−53390.0	−0.734943
$89$	−107123.	−1.43353	−0.716765	−	0.697315i	$-0.754378\pi$
$89$	−107123.	−1.43353	−0.716765	+	0.697315i	$0.754378\pi$
$90$	−109360.	−1.42316
$91$	0	0
$92$	107408.	1.32303
$93$	−1490.82	−0.0178739
$94$	72170.4	0.842441
$95$	−217163.	−2.46875
$96$	−80729.7	−0.894036
$97$	−42456.2	−0.458155	−0.229077	−	0.973408i	$-0.573571\pi$
$97$	−42456.2	−0.458155	−0.229077	+	0.973408i	$0.573571\pi$
$98$	0	0
$99$	−78426.5	−0.804220
$100$	229904.	2.29904
$101$	87947.3	0.857866	0.428933	−	0.903336i	$-0.358890\pi$
$101$	87947.3	0.857866	0.428933	+	0.903336i	$0.358890\pi$
$102$	76118.1	0.724415
$103$	−96885.8	−0.899844	−0.449922	−	0.893068i	$-0.648548\pi$
$103$	−96885.8	−0.899844	−0.449922	+	0.893068i	$0.648548\pi$
$104$	15782.3	0.143083
$105$	0	0
$106$	51685.7	0.446792
$107$	106509.	0.899346	0.449673	−	0.893193i	$-0.351540\pi$
$107$	106509.	0.899346	0.449673	+	0.893193i	$0.351540\pi$
$108$	167377.	1.38082
$109$	−142520.	−1.14897	−0.574485	−	0.818515i	$-0.694798\pi$
$109$	−142520.	−1.14897	−0.574485	+	0.818515i	$0.694798\pi$
$110$	455775.	3.59144
$111$	5307.81	0.0408892
$112$	0	0
$113$	−209540.	−1.54373	−0.771864	−	0.635787i	$-0.780676\pi$
$113$	−209540.	−1.54373	−0.771864	+	0.635787i	$0.780676\pi$
$114$	209599.	1.51052
$115$	−231337.	−1.63117
$116$	−235282.	−1.62346
$117$	23183.2	0.156570
$118$	18255.8	0.120697
$119$	0	0
$120$	−88552.3	−0.561367
$121$	165803.	1.02951
$122$	−150435.	−0.915061
$123$	−143286.	−0.853966
$124$	6202.42	0.0362249
$125$	−207112.	−1.18558
$126$	0	0
$127$	−54468.2	−0.299663	−0.149832	−	0.988712i	$-0.547873\pi$
$127$	−54468.2	−0.299663	−0.149832	+	0.988712i	$0.547873\pi$
$128$	180367.	0.973043
$129$	−79760.8	−0.422003
$130$	−134729.	−0.699201
$131$	−258252.	−1.31482	−0.657408	−	0.753535i	$-0.728347\pi$
$131$	−258252.	−1.31482	−0.657408	+	0.753535i	$0.728347\pi$
$132$	−251702.	−1.25734
$133$	0	0
$134$	26992.6	0.129862
$135$	−360499.	−1.70243
$136$	−79898.6	−0.370418
$137$	293754.	1.33716	0.668580	−	0.743641i	$-0.266903\pi$
$137$	293754.	1.33716	0.668580	+	0.743641i	$0.266903\pi$
$138$	223279.	0.998044
$139$	−180400.	−0.791955	−0.395977	−	0.918260i	$-0.629594\pi$
$139$	−180400.	−0.791955	−0.395977	+	0.918260i	$0.629594\pi$
$140$	0	0
$141$	85842.4	0.363625
$142$	−374685.	−1.55936
$143$	−96619.2	−0.395115
$144$	77076.9	0.309755
$145$	506751.	2.00159
$146$	58882.2	0.228614
$147$	0	0
$148$	−22082.6	−0.0828698
$149$	516009.	1.90411	0.952054	−	0.305930i	$-0.0989675\pi$
$149$	516009.	1.90411	0.952054	+	0.305930i	$0.0989675\pi$
$150$	477922.	1.73432
$151$	−333770.	−1.19126	−0.595628	−	0.803260i	$-0.703097\pi$
$151$	−333770.	−1.19126	−0.595628	+	0.803260i	$0.703097\pi$
$152$	−220009.	−0.772381
$153$	−117366.	−0.405334
$154$	0	0
$155$	−13358.8	−0.0446621
$156$	74404.0	0.244785
$157$	265860.	0.860804	0.430402	−	0.902637i	$-0.358372\pi$
$157$	265860.	0.860804	0.430402	+	0.902637i	$0.358372\pi$
$158$	11077.1	0.0353007
$159$	61477.0	0.192850
$160$	−723395.	−2.23396
$161$	0	0
$162$	59646.8	0.178566
$163$	−231417.	−0.682223	−0.341112	−	0.940023i	$-0.610803\pi$
$163$	−231417.	−0.682223	−0.341112	+	0.940023i	$0.610803\pi$
$164$	596127.	1.73073
$165$	542117.	1.55018
$166$	61823.8	0.174135
$167$	656226.	1.82080	0.910400	−	0.413730i	$-0.135774\pi$
$167$	656226.	1.82080	0.910400	+	0.413730i	$0.135774\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	682071.	1.81012
$171$	−323179.	−0.845187
$172$	331837.	0.855271
$173$	−45111.6	−0.114597	−0.0572985	−	0.998357i	$-0.518249\pi$
$173$	−45111.6	−0.114597	−0.0572985	+	0.998357i	$0.518249\pi$
$174$	−489100.	−1.22468
$175$	0	0
$176$	−321229.	−0.781688
$177$	21714.1	0.0520966
$178$	926460.	2.19168
$179$	521846.	1.21733	0.608667	−	0.793426i	$-0.291705\pi$
$179$	521846.	1.21733	0.608667	+	0.793426i	$0.291705\pi$
$180$	541175.	1.24496
$181$	122780.	0.278569	0.139284	−	0.990252i	$-0.455520\pi$
$181$	122780.	0.278569	0.139284	+	0.990252i	$0.455520\pi$
$182$	0	0
$183$	−178934.	−0.394970
$184$	−234368.	−0.510334
$185$	47561.7	0.102171
$186$	12893.5	0.0273268
$187$	489140.	1.02289
$188$	−357139.	−0.736957
$189$	0	0
$190$	1.87815e6	3.77440
$191$	350208.	0.694613	0.347306	−	0.937752i	$-0.387096\pi$
$191$	350208.	0.694613	0.347306	+	0.937752i	$0.387096\pi$
$192$	513238.	1.00477
$193$	876316.	1.69343	0.846716	−	0.532046i	$-0.178577\pi$
$193$	876316.	1.69343	0.846716	+	0.532046i	$0.178577\pi$
$194$	367186.	0.700457
$195$	−160252.	−0.301798
$196$	0	0
$197$	306582.	0.562835	0.281418	−	0.959585i	$-0.409195\pi$
$197$	306582.	0.562835	0.281418	+	0.959585i	$0.409195\pi$
$198$	678277.	1.22954
$199$	327849.	0.586869	0.293434	−	0.955979i	$-0.405202\pi$
$199$	327849.	0.586869	0.293434	+	0.955979i	$0.405202\pi$
$200$	−501659.	−0.886815
$201$	32106.1	0.0560528
$202$	−760619.	−1.31156
$203$	0	0
$204$	−376674.	−0.633709
$205$	−1.28394e6	−2.13384
$206$	837924.	1.37574
$207$	−344272.	−0.558439
$208$	94956.6	0.152183
$209$	1.34690e6	2.13289
$210$	0	0
$211$	−509483.	−0.787814	−0.393907	−	0.919150i	$-0.628877\pi$
$211$	−509483.	−0.787814	−0.393907	+	0.919150i	$0.628877\pi$
$212$	−255769.	−0.390849
$213$	−445665.	−0.673069
$214$	−921151.	−1.37498
$215$	−714713.	−1.05447
$216$	−365223.	−0.532627
$217$	0	0
$218$	1.23259e6	1.75662
$219$	70036.8	0.0986771
$220$	−2.25543e6	−3.14175
$221$	−144591.	−0.199142
$222$	−45905.0	−0.0625141
$223$	15061.4	0.0202816	0.0101408	−	0.999949i	$-0.496772\pi$
$223$	15061.4	0.0202816	0.0101408	+	0.999949i	$0.496772\pi$
$224$	0	0
$225$	−736904.	−0.970408
$226$	1.81222e6	2.36016
$227$	−273713.	−0.352558	−0.176279	−	0.984340i	$-0.556406\pi$
$227$	−273713.	−0.352558	−0.176279	+	0.984340i	$0.556406\pi$
$228$	−1.03721e6	−1.32139
$229$	−743056.	−0.936339	−0.468169	−	0.883639i	$-0.655086\pi$
$229$	−743056.	−0.936339	−0.468169	+	0.883639i	$0.655086\pi$
$230$	2.00073e6	2.49385
$231$	0	0
$232$	513392.	0.626223
$233$	1.12122e6	1.35301	0.676506	−	0.736437i	$-0.263493\pi$
$233$	1.12122e6	1.35301	0.676506	+	0.736437i	$0.263493\pi$
$234$	−200501.	−0.239374
$235$	769208.	0.908603
$236$	−90339.5	−0.105584
$237$	13175.6	0.0152369
$238$	0	0
$239$	279838.	0.316892	0.158446	−	0.987368i	$-0.449352\pi$
$239$	279838.	0.316892	0.158446	+	0.987368i	$0.449352\pi$
$240$	−532788.	−0.597072
$241$	−816889.	−0.905983	−0.452992	−	0.891515i	$-0.649643\pi$
$241$	−816889.	−0.905983	−0.452992	+	0.891515i	$0.649643\pi$
$242$	−1.43396e6	−1.57398
$243$	−879398.	−0.955366
$244$	744436.	0.800484
$245$	0	0
$246$	1.23922e6	1.30560
$247$	−398148.	−0.415242
$248$	−13533.9	−0.0139731
$249$	73535.7	0.0751623
$250$	1.79123e6	1.81259
$251$	766697.	0.768139	0.384069	−	0.923304i	$-0.374522\pi$
$251$	766697.	0.768139	0.384069	+	0.923304i	$0.374522\pi$
$252$	0	0
$253$	1.43480e6	1.40926
$254$	471072.	0.458145
$255$	811283.	0.781307
$256$	36629.4	0.0349325
$257$	1.82413e6	1.72276	0.861379	−	0.507963i	$-0.169601\pi$
$257$	1.82413e6	1.72276	0.861379	+	0.507963i	$0.169601\pi$
$258$	689817.	0.645186
$259$	0	0
$260$	666712.	0.611653
$261$	754139.	0.685252
$262$	2.23351e6	2.01018
$263$	−294257.	−0.262323	−0.131162	−	0.991361i	$-0.541871\pi$
$263$	−294257.	−0.262323	−0.131162	+	0.991361i	$0.541871\pi$
$264$	549221.	0.484995
$265$	550878.	0.481882
$266$	0	0
$267$	1.10197e6	0.945999
$268$	−133574.	−0.113602
$269$	−289738.	−0.244132	−0.122066	−	0.992522i	$-0.538952\pi$
$269$	−289738.	−0.244132	−0.122066	+	0.992522i	$0.538952\pi$
$270$	3.11780e6	2.60279
$271$	−1.63179e6	−1.34971	−0.674855	−	0.737950i	$-0.735794\pi$
$271$	−1.63179e6	−1.34971	−0.674855	+	0.737950i	$0.735794\pi$
$272$	−480722.	−0.393978
$273$	0	0
$274$	−2.54056e6	−2.04434
$275$	3.07116e6	2.44889
$276$	−1.10490e6	−0.873076
$277$	398868.	0.312341	0.156171	−	0.987730i	$-0.450085\pi$
$277$	398868.	0.312341	0.156171	+	0.987730i	$0.450085\pi$
$278$	1.56021e6	1.21079
$279$	−19880.4	−0.0152902
$280$	0	0
$281$	274611.	0.207468	0.103734	−	0.994605i	$-0.466921\pi$
$281$	274611.	0.207468	0.103734	+	0.994605i	$0.466921\pi$
$282$	−742414.	−0.555934
$283$	794683.	0.589831	0.294916	−	0.955523i	$-0.404708\pi$
$283$	794683.	0.589831	0.294916	+	0.955523i	$0.404708\pi$
$284$	1.85415e6	1.36411
$285$	2.23395e6	1.62915
$286$	835619.	0.604078
$287$	0	0
$288$	−1.07654e6	−0.764805
$289$	−687856.	−0.484454
$290$	−4.38268e6	−3.06016
$291$	436746.	0.302340
$292$	−291381.	−0.199988
$293$	1.55323e6	1.05698	0.528490	−	0.848940i	$-0.322759\pi$
$293$	1.55323e6	1.05698	0.528490	+	0.848940i	$0.322759\pi$
$294$	0	0
$295$	194574.	0.130176
$296$	48185.0	0.0319656
$297$	2.23590e6	1.47082
$298$	−4.46274e6	−2.91113
$299$	−424133.	−0.274362
$300$	−2.36502e6	−1.51716
$301$	0	0
$302$	2.88664e6	1.82127
$303$	−904711.	−0.566113
$304$	−1.32372e6	−0.821508
$305$	−1.60337e6	−0.986926
$306$	1.01505e6	0.619702
$307$	−2.07136e6	−1.25433	−0.627163	−	0.778888i	$-0.715784\pi$
$307$	−2.07136e6	−1.25433	−0.627163	+	0.778888i	$0.715784\pi$
$308$	0	0
$309$	996660.	0.593815
$310$	115535.	0.0682824
$311$	2.50827e6	1.47053	0.735263	−	0.677782i	$-0.237059\pi$
$311$	2.50827e6	1.47053	0.735263	+	0.677782i	$0.237059\pi$
$312$	−162352.	−0.0944215
$313$	−832384.	−0.480245	−0.240123	−	0.970743i	$-0.577188\pi$
$313$	−832384.	−0.480245	−0.240123	+	0.970743i	$0.577188\pi$
$314$	−2.29931e6	−1.31606
$315$	0	0
$316$	−54815.6	−0.0308806
$317$	−1.76537e6	−0.986708	−0.493354	−	0.869829i	$-0.664229\pi$
$317$	−1.76537e6	−0.986708	−0.493354	+	0.869829i	$0.664229\pi$
$318$	−531689.	−0.294842
$319$	−3.14299e6	−1.72928
$320$	4.59897e6	2.51065
$321$	−1.09565e6	−0.593487
$322$	0	0
$323$	2.01564e6	1.07500
$324$	−295165.	−0.156208
$325$	−907845.	−0.476764
$326$	2.00143e6	1.04303
$327$	1.46609e6	0.758215
$328$	−1.30077e6	−0.667598
$329$	0	0
$330$	−4.68854e6	−2.37003
$331$	−285068.	−0.143014	−0.0715070	−	0.997440i	$-0.522781\pi$
$331$	−285068.	−0.143014	−0.0715070	+	0.997440i	$0.522781\pi$
$332$	−305938.	−0.152331
$333$	70780.6	0.0349787
$334$	−5.67542e6	−2.78376
$335$	287693.	0.140061
$336$	0	0
$337$	1.96932e6	0.944584	0.472292	−	0.881442i	$-0.343427\pi$
$337$	1.96932e6	0.944584	0.472292	+	0.881442i	$0.343427\pi$
$338$	−247012.	−0.117605
$339$	2.15553e6	1.01872
$340$	−3.37526e6	−1.58347
$341$	82854.5	0.0385860
$342$	2.79504e6	1.29218
$343$	0	0
$344$	−724079.	−0.329906
$345$	2.37975e6	1.07643
$346$	390151.	0.175204
$347$	1.89753e6	0.845988	0.422994	−	0.906133i	$-0.360979\pi$
$347$	1.89753e6	0.845988	0.422994	+	0.906133i	$0.360979\pi$
$348$	2.42033e6	1.07134
$349$	365808.	0.160764	0.0803822	−	0.996764i	$-0.474386\pi$
$349$	365808.	0.160764	0.0803822	+	0.996764i	$0.474386\pi$
$350$	0	0
$351$	−660939.	−0.286348
$352$	4.48666e6	1.93004
$353$	−1.56672e6	−0.669196	−0.334598	−	0.942361i	$-0.608601\pi$
$353$	−1.56672e6	−0.669196	−0.334598	+	0.942361i	$0.608601\pi$
$354$	−187796.	−0.0796487
$355$	−3.99348e6	−1.68182
$356$	−4.58463e6	−1.91725
$357$	0	0
$358$	−4.51322e6	−1.86114
$359$	−4.03307e6	−1.65158	−0.825789	−	0.563979i	$-0.809270\pi$
$359$	−4.03307e6	−1.65158	−0.825789	+	0.563979i	$0.809270\pi$
$360$	−1.18086e6	−0.480222
$361$	3.07418e6	1.24154
$362$	−1.06187e6	−0.425894
$363$	−1.70561e6	−0.679380
$364$	0	0
$365$	627579.	0.246568
$366$	1.54752e6	0.603857
$367$	−683844.	−0.265028	−0.132514	−	0.991181i	$-0.542305\pi$
$367$	−683844.	−0.265028	−0.132514	+	0.991181i	$0.542305\pi$
$368$	−1.41011e6	−0.542793
$369$	−1.91074e6	−0.730527
$370$	−411341.	−0.156206
$371$	0	0
$372$	−63804.1	−0.0239051
$373$	−3.09158e6	−1.15056	−0.575279	−	0.817957i	$-0.695107\pi$
$373$	−3.09158e6	−1.15056	−0.575279	+	0.817957i	$0.695107\pi$
$374$	−4.23036e6	−1.56386
$375$	2.13056e6	0.782374
$376$	779288.	0.284268
$377$	929078.	0.336666
$378$	0	0
$379$	−5.24600e6	−1.87599	−0.937995	−	0.346649i	$-0.887319\pi$
$379$	−5.24600e6	−1.87599	−0.937995	+	0.346649i	$0.887319\pi$
$380$	−9.29414e6	−3.30179
$381$	560312.	0.197751
$382$	−3.02880e6	−1.06197
$383$	−1.57009e6	−0.546926	−0.273463	−	0.961883i	$-0.588169\pi$
$383$	−1.57009e6	−0.546926	−0.273463	+	0.961883i	$0.588169\pi$
$384$	−1.85543e6	−0.642120
$385$	0	0
$386$	−7.57889e6	−2.58903
$387$	−1.06362e6	−0.361003
$388$	−1.81704e6	−0.612751
$389$	1.54482e6	0.517611	0.258805	−	0.965929i	$-0.416671\pi$
$389$	1.54482e6	0.517611	0.258805	+	0.965929i	$0.416671\pi$
$390$	1.38595e6	0.461409
$391$	2.14719e6	0.710280
$392$	0	0
$393$	2.65663e6	0.867659
$394$	−2.65150e6	−0.860500
$395$	118062.	0.0380731
$396$	−3.35649e6	−1.07559
$397$	−5.06711e6	−1.61356	−0.806778	−	0.590855i	$-0.798790\pi$
$397$	−5.06711e6	−1.61356	−0.806778	+	0.590855i	$0.798790\pi$
$398$	−2.83543e6	−0.897244
$399$	0	0
$400$	−3.01831e6	−0.943221
$401$	−138120.	−0.0428939	−0.0214469	−	0.999770i	$-0.506827\pi$
$401$	−138120.	−0.0428939	−0.0214469	+	0.999770i	$0.506827\pi$
$402$	−277672.	−0.0856973
$403$	−24492.1	−0.00751213
$404$	3.76396e6	1.14734
$405$	635729.	0.192590
$406$	0	0
$407$	−294989.	−0.0882713
$408$	821914.	0.244442
$409$	4.13317e6	1.22173	0.610865	−	0.791735i	$-0.290822\pi$
$409$	4.13317e6	1.22173	0.610865	+	0.791735i	$0.290822\pi$
$410$	1.11043e7	3.26235
$411$	−3.02184e6	−0.882403
$412$	−4.14651e6	−1.20348
$413$	0	0
$414$	2.97746e6	0.853778
$415$	658932.	0.187811
$416$	−1.32627e6	−0.375750
$417$	1.85577e6	0.522618
$418$	−1.16487e7	−3.26091
$419$	2.96703e6	0.825633	0.412817	−	0.910814i	$-0.364545\pi$
$419$	2.96703e6	0.825633	0.412817	+	0.910814i	$0.364545\pi$
$420$	0	0
$421$	−2.98125e6	−0.819773	−0.409887	−	0.912137i	$-0.634432\pi$
$421$	−2.98125e6	−0.819773	−0.409887	+	0.912137i	$0.634432\pi$
$422$	4.40630e6	1.20446
$423$	1.14472e6	0.311064
$424$	558096.	0.150763
$425$	4.59601e6	1.23426
$426$	3.85437e6	1.02903
$427$	0	0
$428$	4.55836e6	1.20282
$429$	993918.	0.260740
$430$	6.18125e6	1.61215
$431$	−3.34790e6	−0.868119	−0.434059	−	0.900884i	$-0.642919\pi$
$431$	−3.34790e6	−0.868119	−0.434059	+	0.900884i	$0.642919\pi$
$432$	−2.19742e6	−0.566505
$433$	3.76782e6	0.965764	0.482882	−	0.875686i	$-0.339590\pi$
$433$	3.76782e6	0.965764	0.482882	+	0.875686i	$0.339590\pi$
$434$	0	0
$435$	−5.21293e6	−1.32087
$436$	−6.09954e6	−1.53667
$437$	5.91252e6	1.48105
$438$	−605719.	−0.150864
$439$	−5.63075e6	−1.39446	−0.697228	−	0.716849i	$-0.745584\pi$
$439$	−5.63075e6	−1.39446	−0.697228	+	0.716849i	$0.745584\pi$
$440$	4.92141e6	1.21188
$441$	0	0
$442$	1.25051e6	0.304461
$443$	3.69803e6	0.895284	0.447642	−	0.894213i	$-0.352264\pi$
$443$	3.69803e6	0.895284	0.447642	+	0.894213i	$0.352264\pi$
$444$	227163.	0.0546865
$445$	9.87441e6	2.36380
$446$	−130260.	−0.0310079
$447$	−5.30816e6	−1.25654
$448$	0	0
$449$	−505514.	−0.118336	−0.0591681	−	0.998248i	$-0.518845\pi$
$449$	−505514.	−0.118336	−0.0591681	+	0.998248i	$0.518845\pi$
$450$	6.37317e6	1.48362
$451$	7.96330e6	1.84354
$452$	−8.96787e6	−2.06464
$453$	3.43348e6	0.786121
$454$	2.36723e6	0.539015
$455$	0	0
$456$	2.26323e6	0.509701
$457$	783627.	0.175517	0.0877584	−	0.996142i	$-0.472030\pi$
$457$	783627.	0.175517	0.0877584	+	0.996142i	$0.472030\pi$
$458$	6.42638e6	1.43154
$459$	3.34603e6	0.741308
$460$	−9.90073e6	−2.18159
$461$	−4.64856e6	−1.01875	−0.509373	−	0.860546i	$-0.670123\pi$
$461$	−4.64856e6	−1.01875	−0.509373	+	0.860546i	$0.670123\pi$
$462$	0	0
$463$	2.64872e6	0.574226	0.287113	−	0.957897i	$-0.407304\pi$
$463$	2.64872e6	0.574226	0.287113	+	0.957897i	$0.407304\pi$
$464$	3.08890e6	0.666053
$465$	137422.	0.0294729
$466$	−9.69698e6	−2.06858
$467$	2.42172e6	0.513845	0.256923	−	0.966432i	$-0.417291\pi$
$467$	2.42172e6	0.513845	0.256923	+	0.966432i	$0.417291\pi$
$468$	992190.	0.209402
$469$	0	0
$470$	−6.65255e6	−1.38913
$471$	−2.73490e6	−0.568052
$472$	197124.	0.0407271
$473$	4.43281e6	0.911017
$474$	−113950.	−0.0232953
$475$	1.26556e7	2.57364
$476$	0	0
$477$	819807.	0.164974
$478$	−2.42020e6	−0.484486
$479$	−5.58315e6	−1.11184	−0.555918	−	0.831237i	$-0.687633\pi$
$479$	−5.58315e6	−1.11184	−0.555918	+	0.831237i	$0.687633\pi$
$480$	7.44154e6	1.47421
$481$	87199.7	0.0171851
$482$	7.06492e6	1.38513
$483$	0	0
$484$	7.09602e6	1.37690
$485$	3.91355e6	0.755468
$486$	7.60554e6	1.46063
$487$	7.25183e6	1.38556	0.692780	−	0.721149i	$-0.256386\pi$
$487$	7.25183e6	1.38556	0.692780	+	0.721149i	$0.256386\pi$
$488$	−1.62438e6	−0.308773
$489$	2.38058e6	0.450205
$490$	0	0
$491$	4.17410e6	0.781373	0.390687	−	0.920524i	$-0.372238\pi$
$491$	4.17410e6	0.781373	0.390687	+	0.920524i	$0.372238\pi$
$492$	−6.13233e6	−1.14212
$493$	−4.70350e6	−0.871574
$494$	3.44341e6	0.634850
$495$	7.22923e6	1.32611
$496$	−81428.7	−0.0148619
$497$	0	0
$498$	−635979.	−0.114913
$499$	−7.83551e6	−1.40869	−0.704345	−	0.709857i	$-0.748759\pi$
$499$	−7.83551e6	−1.40869	−0.704345	+	0.709857i	$0.748759\pi$
$500$	−8.86396e6	−1.58563
$501$	−6.75057e6	−1.20156
$502$	−6.63084e6	−1.17438
$503$	3.72420e6	0.656315	0.328158	−	0.944623i	$-0.393572\pi$
$503$	3.72420e6	0.656315	0.328158	+	0.944623i	$0.393572\pi$
$504$	0	0
$505$	−8.10685e6	−1.41457
$506$	−1.24090e7	−2.15457
$507$	−293806.	−0.0507622
$508$	−2.33112e6	−0.400780
$509$	6.76675e6	1.15767	0.578836	−	0.815444i	$-0.303507\pi$
$509$	6.76675e6	1.15767	0.578836	+	0.815444i	$0.303507\pi$
$510$	−7.01644e6	−1.19451
$511$	0	0
$512$	−6.08853e6	−1.02645
$513$	9.21366e6	1.54575
$514$	−1.57762e7	−2.63387
$515$	8.93078e6	1.48379
$516$	−3.41359e6	−0.564401
$517$	−4.77080e6	−0.784992
$518$	0	0
$519$	464062.	0.0756236
$520$	−1.45479e6	−0.235934
$521$	−9.08380e6	−1.46613	−0.733066	−	0.680157i	$-0.761912\pi$
$521$	−9.08380e6	−1.46613	−0.733066	+	0.680157i	$0.761912\pi$
$522$	−6.52223e6	−1.04766
$523$	1.09284e7	1.74703	0.873515	−	0.486796i	$-0.161835\pi$
$523$	1.09284e7	1.74703	0.873515	+	0.486796i	$0.161835\pi$
$524$	−1.10526e7	−1.75848
$525$	0	0
$526$	2.54490e6	0.401057
$527$	123992.	0.0194477
$528$	3.30447e6	0.515843
$529$	−137934.	−0.0214305
$530$	−4.76431e6	−0.736733
$531$	289562.	0.0445661
$532$	0	0
$533$	−2.35398e6	−0.358910
$534$	−9.53045e6	−1.44631
$535$	−9.81783e6	−1.48297
$536$	291463.	0.0438200
$537$	−5.36821e6	−0.803329
$538$	2.50582e6	0.373245
$539$	0	0
$540$	−1.54286e7	−2.27689
$541$	1.17853e6	0.173120	0.0865598	−	0.996247i	$-0.472413\pi$
$541$	1.17853e6	0.173120	0.0865598	+	0.996247i	$0.472413\pi$
$542$	1.41126e7	2.06353
$543$	−1.26304e6	−0.183830
$544$	6.71432e6	0.972758
$545$	1.31372e7	1.89458
$546$	0	0
$547$	9.57702e6	1.36855	0.684277	−	0.729222i	$-0.260118\pi$
$547$	9.57702e6	1.36855	0.684277	+	0.729222i	$0.260118\pi$
$548$	1.25721e7	1.78836
$549$	−2.38611e6	−0.337878
$550$	−2.65611e7	−3.74403
$551$	−1.29516e7	−1.81737
$552$	2.41094e6	0.336774
$553$	0	0
$554$	−3.44964e6	−0.477528
$555$	−489266.	−0.0674237
$556$	−7.72075e6	−1.05919
$557$	2.46873e6	0.337159	0.168580	−	0.985688i	$-0.446082\pi$
$557$	2.46873e6	0.337159	0.168580	+	0.985688i	$0.446082\pi$
$558$	171937.	0.0233767
$559$	−1.31036e6	−0.177362
$560$	0	0
$561$	−5.03176e6	−0.675014
$562$	−2.37499e6	−0.317191
$563$	−7.81681e6	−1.03934	−0.519671	−	0.854367i	$-0.673945\pi$
$563$	−7.81681e6	−1.03934	−0.519671	+	0.854367i	$0.673945\pi$
$564$	3.67387e6	0.486325
$565$	1.93151e7	2.54551
$566$	−6.87288e6	−0.901774
$567$	0	0
$568$	−4.04581e6	−0.526180
$569$	−9.64303e6	−1.24863	−0.624314	−	0.781174i	$-0.714621\pi$
$569$	−9.64303e6	−1.24863	−0.624314	+	0.781174i	$0.714621\pi$
$570$	−1.93205e7	−2.49076
$571$	1.12378e7	1.44242	0.721211	−	0.692715i	$-0.243586\pi$
$571$	1.12378e7	1.44242	0.721211	+	0.692715i	$0.243586\pi$
$572$	−4.13510e6	−0.528440
$573$	−3.60258e6	−0.458381
$574$	0	0
$575$	1.34816e7	1.70048
$576$	6.84412e6	0.859530
$577$	5.44674e6	0.681078	0.340539	−	0.940230i	$-0.389390\pi$
$577$	5.44674e6	0.681078	0.340539	+	0.940230i	$0.389390\pi$
$578$	5.94897e6	0.740666
$579$	−9.01463e6	−1.11751
$580$	2.16879e7	2.67699
$581$	0	0
$582$	−3.77723e6	−0.462238
$583$	−3.41667e6	−0.416324
$584$	635803.	0.0771420
$585$	−2.13699e6	−0.258174
$586$	−1.34332e7	−1.61598
$587$	−9.96240e6	−1.19335	−0.596676	−	0.802482i	$-0.703512\pi$
$587$	−9.96240e6	−1.19335	−0.596676	+	0.802482i	$0.703512\pi$
$588$	0	0
$589$	341426.	0.0405516
$590$	−1.68279e6	−0.199021
$591$	−3.15380e6	−0.371420
$592$	289912.	0.0339987
$593$	3.17929e6	0.371273	0.185636	−	0.982619i	$-0.440565\pi$
$593$	3.17929e6	0.371273	0.185636	+	0.982619i	$0.440565\pi$
$594$	−1.93373e7	−2.24869
$595$	0	0
$596$	2.20841e7	2.54662
$597$	−3.37257e6	−0.387280
$598$	3.66815e6	0.419463
$599$	−1.66017e7	−1.89054	−0.945271	−	0.326285i	$-0.894203\pi$
$599$	−1.66017e7	−1.89054	−0.945271	+	0.326285i	$0.894203\pi$
$600$	5.16054e6	0.585217
$601$	−1.05771e6	−0.119449	−0.0597244	−	0.998215i	$-0.519022\pi$
$601$	−1.05771e6	−0.119449	−0.0597244	+	0.998215i	$0.519022\pi$
$602$	0	0
$603$	428141.	0.0479505
$604$	−1.42847e7	−1.59323
$605$	−1.52835e7	−1.69759
$606$	7.82446e6	0.865512
$607$	−2.42026e6	−0.266619	−0.133309	−	0.991074i	$-0.542560\pi$
$607$	−2.42026e6	−0.266619	−0.133309	+	0.991074i	$0.542560\pi$
$608$	1.84886e7	2.02836
$609$	0	0
$610$	1.38669e7	1.50888
$611$	1.41027e6	0.152826
$612$	−5.02301e6	−0.542108
$613$	110410.	0.0118675	0.00593373	−	0.999982i	$-0.498111\pi$
$613$	110410.	0.0118675	0.00593373	+	0.999982i	$0.498111\pi$
$614$	1.79144e7	1.91770
$615$	1.32079e7	1.40814
$616$	0	0
$617$	−1.06544e7	−1.12672	−0.563361	−	0.826211i	$-0.690492\pi$
$617$	−1.06544e7	−1.12672	−0.563361	+	0.826211i	$0.690492\pi$
$618$	−8.61969e6	−0.907864
$619$	1.30456e7	1.36848	0.684240	−	0.729256i	$-0.260134\pi$
$619$	1.30456e7	1.36848	0.684240	+	0.729256i	$0.260134\pi$
$620$	−571730.	−0.0597326
$621$	9.81499e6	1.02132
$622$	−2.16929e7	−2.24824
$623$	0	0
$624$	−976815.	−0.100427
$625$	2.30421e6	0.235951
$626$	7.19894e6	0.734231
$627$	−1.38555e7	−1.40751
$628$	1.13783e7	1.15127
$629$	−441453.	−0.0444895
$630$	0	0
$631$	−1.53156e6	−0.153130	−0.0765651	−	0.997065i	$-0.524395\pi$
$631$	−1.53156e6	−0.153130	−0.0765651	+	0.997065i	$0.524395\pi$
$632$	119609.	0.0119117
$633$	5.24103e6	0.519885
$634$	1.52680e7	1.50855
$635$	5.02079e6	0.494126
$636$	2.63109e6	0.257924
$637$	0	0
$638$	2.71824e7	2.64384
$639$	−5.94302e6	−0.575778
$640$	−1.66259e7	−1.60449
$641$	8.09995e6	0.778641	0.389321	−	0.921102i	$-0.372710\pi$
$641$	8.09995e6	0.778641	0.389321	+	0.921102i	$0.372710\pi$
$642$	9.47585e6	0.907362
$643$	5.29175e6	0.504745	0.252373	−	0.967630i	$-0.418789\pi$
$643$	5.29175e6	0.504745	0.252373	+	0.967630i	$0.418789\pi$
$644$	0	0
$645$	7.35223e6	0.695856
$646$	−1.74324e7	−1.64353
$647$	−1.80626e7	−1.69637	−0.848185	−	0.529700i	$-0.822305\pi$
$647$	−1.80626e7	−1.69637	−0.848185	+	0.529700i	$0.822305\pi$
$648$	644060.	0.0602544
$649$	−1.20679e6	−0.112466
$650$	7.85156e6	0.728908
$651$	0	0
$652$	−9.90416e6	−0.912428
$653$	−7.10212e6	−0.651786	−0.325893	−	0.945407i	$-0.605665\pi$
$653$	−7.10212e6	−0.651786	−0.325893	+	0.945407i	$0.605665\pi$
$654$	−1.26796e7	−1.15921
$655$	2.38052e7	2.16805
$656$	−7.82627e6	−0.710060
$657$	933954.	0.0844135
$658$	0	0
$659$	1.41430e7	1.26861	0.634303	−	0.773084i	$-0.281287\pi$
$659$	1.41430e7	1.26861	0.634303	+	0.773084i	$0.281287\pi$
$660$	2.32015e7	2.07327
$661$	675807.	0.0601615	0.0300808	−	0.999547i	$-0.490424\pi$
$661$	675807.	0.0601615	0.0300808	+	0.999547i	$0.490424\pi$
$662$	2.46543e6	0.218649
$663$	1.48741e6	0.131415
$664$	667566.	0.0587590
$665$	0	0
$666$	−612152.	−0.0534778
$667$	−1.37969e7	−1.20079
$668$	2.80851e7	2.43520
$669$	−154936.	−0.0133840
$670$	−2.48814e6	−0.214135
$671$	9.94447e6	0.852659
$672$	0	0
$673$	2.66714e6	0.226991	0.113496	−	0.993539i	$-0.463795\pi$
$673$	2.66714e6	0.226991	0.113496	+	0.993539i	$0.463795\pi$
$674$	−1.70318e7	−1.44414
$675$	2.10087e7	1.77476
$676$	1.22235e6	0.102880
$677$	1.20319e7	1.00893	0.504465	−	0.863432i	$-0.331690\pi$
$677$	1.20319e7	1.00893	0.504465	+	0.863432i	$0.331690\pi$
$678$	−1.86423e7	−1.55749
$679$	0	0
$680$	7.36493e6	0.610796
$681$	2.81568e6	0.232656
$682$	−716573.	−0.0589929
$683$	−1.83758e7	−1.50728	−0.753640	−	0.657287i	$-0.771704\pi$
$683$	−1.83758e7	−1.50728	−0.753640	+	0.657287i	$0.771704\pi$
$684$	−1.38314e7	−1.13038
$685$	−2.70778e7	−2.20489
$686$	0	0
$687$	7.64379e6	0.617898
$688$	−4.35653e6	−0.350889
$689$	1.00998e6	0.0810521
$690$	−2.05815e7	−1.64571
$691$	−1.33412e7	−1.06292	−0.531459	−	0.847084i	$-0.678356\pi$
$691$	−1.33412e7	−1.06292	−0.531459	+	0.847084i	$0.678356\pi$
$692$	−1.93068e6	−0.153266
$693$	0	0
$694$	−1.64109e7	−1.29340
$695$	1.66290e7	1.30588
$696$	−5.28124e6	−0.413250
$697$	1.19171e7	0.929159
$698$	−3.16372e6	−0.245787
$699$	−1.15340e7	−0.892865
$700$	0	0
$701$	2.51952e6	0.193652	0.0968262	−	0.995301i	$-0.469131\pi$
$701$	2.51952e6	0.193652	0.0968262	+	0.995301i	$0.469131\pi$
$702$	5.71618e6	0.437787
$703$	−1.21559e6	−0.0927679
$704$	−2.85239e7	−2.16909
$705$	−7.91281e6	−0.599595
$706$	1.35499e7	1.02311
$707$	0	0
$708$	929319.	0.0696757
$709$	2.51441e7	1.87854	0.939272	−	0.343174i	$-0.111502\pi$
$709$	2.51441e7	1.87854	0.939272	+	0.343174i	$0.111502\pi$
$710$	3.45379e7	2.57128
$711$	175698.	0.0130345
$712$	1.00038e7	0.739546
$713$	363709.	0.0267936
$714$	0	0
$715$	8.90621e6	0.651520
$716$	2.23339e7	1.62810
$717$	−2.87868e6	−0.209120
$718$	3.48803e7	2.52504
$719$	−415865.	−0.0300006	−0.0150003	−	0.999887i	$-0.504775\pi$
$719$	−415865.	−0.0300006	−0.0150003	+	0.999887i	$0.504775\pi$
$720$	−7.10483e6	−0.510766
$721$	0	0
$722$	−2.65873e7	−1.89815
$723$	8.40330e6	0.597866
$724$	5.25474e6	0.372567
$725$	−2.95318e7	−2.08663
$726$	1.47511e7	1.03868
$727$	1.53987e7	1.08056	0.540278	−	0.841486i	$-0.318319\pi$
$727$	1.53987e7	1.08056	0.540278	+	0.841486i	$0.318319\pi$
$728$	0	0
$729$	1.07222e7	0.747251
$730$	−5.42767e6	−0.376970
$731$	6.63374e6	0.459161
$732$	−7.65798e6	−0.528246
$733$	1.82125e7	1.25202	0.626008	−	0.779817i	$-0.284688\pi$
$733$	1.82125e7	1.25202	0.626008	+	0.779817i	$0.284688\pi$
$734$	5.91428e6	0.405193
$735$	0	0
$736$	1.96953e7	1.34019
$737$	−1.78434e6	−0.121007
$738$	1.65252e7	1.11688
$739$	−1.88107e7	−1.26705	−0.633524	−	0.773723i	$-0.718392\pi$
$739$	−1.88107e7	−1.26705	−0.633524	+	0.773723i	$0.718392\pi$
$740$	2.03554e6	0.136647
$741$	4.09573e6	0.274022
$742$	0	0
$743$	2.61292e7	1.73642	0.868209	−	0.496199i	$-0.165271\pi$
$743$	2.61292e7	1.73642	0.868209	+	0.496199i	$0.165271\pi$
$744$	139223.	0.00922098
$745$	−4.75649e7	−3.13976
$746$	2.67378e7	1.75905
$747$	980611.	0.0642977
$748$	2.09341e7	1.36805
$749$	0	0
$750$	−1.84263e7	−1.19615
$751$	−1.90282e7	−1.23111	−0.615557	−	0.788092i	$-0.711069\pi$
$751$	−1.90282e7	−1.23111	−0.615557	+	0.788092i	$0.711069\pi$
$752$	4.68871e6	0.302349
$753$	−7.88699e6	−0.506902
$754$	−8.03520e6	−0.514717
$755$	3.07664e7	1.96431
$756$	0	0
$757$	−1.91639e7	−1.21547	−0.607734	−	0.794140i	$-0.707921\pi$
$757$	−1.91639e7	−1.21547	−0.607734	+	0.794140i	$0.707921\pi$
$758$	4.53705e7	2.86814
$759$	−1.47598e7	−0.929983
$760$	2.02801e7	1.27361
$761$	468091.	0.0293001	0.0146500	−	0.999893i	$-0.495337\pi$
$761$	468091.	0.0293001	0.0146500	+	0.999893i	$0.495337\pi$
$762$	−4.84590e6	−0.302334
$763$	0	0
$764$	1.49882e7	0.928999
$765$	1.08186e7	0.668371
$766$	1.35791e7	0.836177
$767$	356732.	0.0218954
$768$	−376805.	−0.0230523
$769$	1.92729e7	1.17525	0.587627	−	0.809132i	$-0.300062\pi$
$769$	1.92729e7	1.17525	0.587627	+	0.809132i	$0.300062\pi$
$770$	0	0
$771$	−1.87648e7	−1.13686
$772$	3.75045e7	2.26485
$773$	−2.42083e7	−1.45719	−0.728593	−	0.684947i	$-0.759825\pi$
$773$	−2.42083e7	−1.45719	−0.728593	+	0.684947i	$0.759825\pi$
$774$	9.19884e6	0.551926
$775$	778509.	0.0465596
$776$	3.96483e6	0.236358
$777$	0	0
$778$	−1.33605e7	−0.791358
$779$	3.28151e7	1.93745
$780$	−6.85844e6	−0.403635
$781$	2.47684e7	1.45302
$782$	−1.85702e7	−1.08592
$783$	−2.15001e7	−1.25324
$784$	0	0
$785$	−2.45066e7	−1.41941
$786$	−2.29760e7	−1.32653
$787$	1.48087e7	0.852276	0.426138	−	0.904658i	$-0.359874\pi$
$787$	1.48087e7	0.852276	0.426138	+	0.904658i	$0.359874\pi$
$788$	1.31211e7	0.752755
$789$	3.02701e6	0.173110
$790$	−1.02107e6	−0.0582087
$791$	0	0
$792$	7.32396e6	0.414890
$793$	−2.93962e6	−0.166000
$794$	4.38233e7	2.46691
$795$	−5.66686e6	−0.317998
$796$	1.40312e7	0.784898
$797$	2.90994e7	1.62270	0.811351	−	0.584560i	$-0.198733\pi$
$797$	2.90994e7	1.62270	0.811351	+	0.584560i	$0.198733\pi$
$798$	0	0
$799$	−7.13954e6	−0.395643
$800$	4.21571e7	2.32887
$801$	1.46949e7	0.809257
$802$	1.19454e6	0.0655790
$803$	−3.89239e6	−0.213023
$804$	1.37407e6	0.0749669
$805$	0	0
$806$	211822.	0.0114850
$807$	2.98052e6	0.161105
$808$	−8.21308e6	−0.442566
$809$	3.37792e7	1.81459	0.907295	−	0.420496i	$-0.138144\pi$
$809$	3.37792e7	1.81459	0.907295	+	0.420496i	$0.138144\pi$
$810$	−5.49815e6	−0.294445
$811$	2.42986e7	1.29727	0.648633	−	0.761101i	$-0.275341\pi$
$811$	2.42986e7	1.29727	0.648633	+	0.761101i	$0.275341\pi$
$812$	0	0
$813$	1.67861e7	0.890686
$814$	2.55123e6	0.134955
$815$	2.13317e7	1.12494
$816$	4.94517e6	0.259990
$817$	1.82667e7	0.957425
$818$	−3.57460e7	−1.86786
$819$	0	0
$820$	−5.49500e7	−2.85386
$821$	−7.42442e6	−0.384419	−0.192209	−	0.981354i	$-0.561565\pi$
$821$	−7.42442e6	−0.384419	−0.192209	+	0.981354i	$0.561565\pi$
$822$	2.61346e7	1.34908
$823$	−2.98540e7	−1.53640	−0.768198	−	0.640212i	$-0.778847\pi$
$823$	−2.98540e7	−1.53640	−0.768198	+	0.640212i	$0.778847\pi$
$824$	9.04781e6	0.464222
$825$	−3.15929e7	−1.61605
$826$	0	0
$827$	2.47154e7	1.25662	0.628311	−	0.777962i	$-0.283747\pi$
$827$	2.47154e7	1.25662	0.628311	+	0.777962i	$0.283747\pi$
$828$	−1.47341e7	−0.746875
$829$	1.14800e7	0.580169	0.290084	−	0.957001i	$-0.406317\pi$
$829$	1.14800e7	0.580169	0.290084	+	0.957001i	$0.406317\pi$
$830$	−5.69882e6	−0.287137
$831$	−4.10314e6	−0.206117
$832$	8.43176e6	0.422289
$833$	0	0
$834$	−1.60498e7	−0.799013
$835$	−6.04899e7	−3.00239
$836$	5.76443e7	2.85260
$837$	566779.	0.0279640
$838$	−2.56606e7	−1.26228
$839$	−2.41188e7	−1.18291	−0.591454	−	0.806339i	$-0.701446\pi$
$839$	−2.41188e7	−1.18291	−0.591454	+	0.806339i	$0.701446\pi$
$840$	0	0
$841$	9.71140e6	0.473469
$842$	2.57836e7	1.25332
$843$	−2.82491e6	−0.136910
$844$	−2.18048e7	−1.05365
$845$	−2.63271e6	−0.126841
$846$	−9.90022e6	−0.475575
$847$	0	0
$848$	3.35787e6	0.160352
$849$	−8.17487e6	−0.389235
$850$	−3.97489e7	−1.88703
$851$	−1.29492e6	−0.0612943
$852$	−1.90735e7	−0.900186
$853$	2.02681e6	0.0953764	0.0476882	−	0.998862i	$-0.484815\pi$
$853$	2.02681e6	0.0953764	0.0476882	+	0.998862i	$0.484815\pi$
$854$	0	0
$855$	2.97901e7	1.39366
$856$	−9.94649e6	−0.463965
$857$	−1.52859e7	−0.710950	−0.355475	−	0.934686i	$-0.615681\pi$
$857$	−1.52859e7	−0.710950	−0.355475	+	0.934686i	$0.615681\pi$
$858$	−8.59598e6	−0.398637
$859$	−1.78567e7	−0.825693	−0.412846	−	0.910801i	$-0.635465\pi$
$859$	−1.78567e7	−0.825693	−0.412846	+	0.910801i	$0.635465\pi$
$860$	−3.05882e7	−1.41029
$861$	0	0
$862$	2.89546e7	1.32724
$863$	−1.12315e7	−0.513347	−0.256673	−	0.966498i	$-0.582626\pi$
$863$	−1.12315e7	−0.513347	−0.256673	+	0.966498i	$0.582626\pi$
$864$	3.06917e7	1.39874
$865$	4.15832e6	0.188963
$866$	−3.25863e7	−1.47652
$867$	7.07595e6	0.319696
$868$	0	0
$869$	−732249.	−0.0328934
$870$	4.50844e7	2.01943
$871$	527457.	0.0235582
$872$	1.33094e7	0.592744
$873$	5.82408e6	0.258638
$874$	−5.11349e7	−2.26433
$875$	0	0
$876$	2.99743e6	0.131974
$877$	1.75233e6	0.0769336	0.0384668	−	0.999260i	$-0.487753\pi$
$877$	1.75233e6	0.0769336	0.0384668	+	0.999260i	$0.487753\pi$
$878$	4.86980e7	2.13194
$879$	−1.59780e7	−0.697510
$880$	2.96104e7	1.28896
$881$	1.04009e7	0.451474	0.225737	−	0.974188i	$-0.427521\pi$
$881$	1.04009e7	0.451474	0.225737	+	0.974188i	$0.427521\pi$
$882$	0	0
$883$	4.25645e7	1.83715	0.918577	−	0.395241i	$-0.129339\pi$
$883$	4.25645e7	1.83715	0.918577	+	0.395241i	$0.129339\pi$
$884$	−6.18821e6	−0.266339
$885$	−2.00157e6	−0.0859040
$886$	−3.19827e7	−1.36877
$887$	−201500.	−0.00859935	−0.00429968	−	0.999991i	$-0.501369\pi$
$887$	−201500.	−0.00859935	−0.00429968	+	0.999991i	$0.501369\pi$
$888$	−495677.	−0.0210944
$889$	0	0
$890$	−8.53996e7	−3.61394
$891$	−3.94294e6	−0.166389
$892$	644596.	0.0271253
$893$	−1.96595e7	−0.824980
$894$	4.59081e7	1.92108
$895$	−4.81029e7	−2.00731
$896$	0	0
$897$	4.36304e6	0.181054
$898$	4.37198e6	0.180920
$899$	−796718.	−0.0328780
$900$	−3.15379e7	−1.29786
$901$	−5.11307e6	−0.209831
$902$	−6.88712e7	−2.81852
$903$	0	0
$904$	1.95682e7	0.796397
$905$	−1.13177e7	−0.459342
$906$	−2.96947e7	−1.20187
$907$	1.47930e7	0.597088	0.298544	−	0.954396i	$-0.403499\pi$
$907$	1.47930e7	0.597088	0.298544	+	0.954396i	$0.403499\pi$
$908$	−1.17143e7	−0.471523
$909$	−1.20645e7	−0.484283
$910$	0	0
$911$	−3.50070e7	−1.39752	−0.698762	−	0.715354i	$-0.746265\pi$
$911$	−3.50070e7	−1.39752	−0.698762	+	0.715354i	$0.746265\pi$
$912$	1.36170e7	0.542121
$913$	−4.08684e6	−0.162260
$914$	−6.77726e6	−0.268342
$915$	1.64938e7	0.651281
$916$	−3.18012e7	−1.25229
$917$	0	0
$918$	−2.89384e7	−1.13336
$919$	4.24526e7	1.65812	0.829060	−	0.559160i	$-0.188876\pi$
$919$	4.24526e7	1.65812	0.829060	+	0.559160i	$0.188876\pi$
$920$	2.16037e7	0.841509
$921$	2.13080e7	0.827741
$922$	4.02034e7	1.55753
$923$	−7.32164e6	−0.282881
$924$	0	0
$925$	−2.77175e6	−0.106512
$926$	−2.29076e7	−0.877916
$927$	1.32906e7	0.507980
$928$	−4.31431e7	−1.64453
$929$	3.95487e7	1.50347	0.751733	−	0.659468i	$-0.229218\pi$
$929$	3.95487e7	1.50347	0.751733	+	0.659468i	$0.229218\pi$
$930$	−1.18850e6	−0.0450602
$931$	0	0
$932$	4.79859e7	1.80957
$933$	−2.58024e7	−0.970413
$934$	−2.09445e7	−0.785601
$935$	−4.50881e7	−1.68668
$936$	−2.16499e6	−0.0807730
$937$	4.62850e7	1.72223	0.861116	−	0.508409i	$-0.169766\pi$
$937$	4.62850e7	1.72223	0.861116	+	0.508409i	$0.169766\pi$
$938$	0	0
$939$	8.56270e6	0.316918
$940$	3.29205e7	1.21520
$941$	−5.74465e6	−0.211490	−0.105745	−	0.994393i	$-0.533723\pi$
$941$	−5.74465e6	−0.211490	−0.105745	+	0.994393i	$0.533723\pi$
$942$	2.36530e7	0.868477
$943$	3.49568e7	1.28013
$944$	1.18602e6	0.0433175
$945$	0	0
$946$	−3.83375e7	−1.39282
$947$	5.15966e7	1.86959	0.934795	−	0.355187i	$-0.115583\pi$
$947$	5.15966e7	1.86959	0.934795	+	0.355187i	$0.115583\pi$
$948$	563886.	0.0203784
$949$	1.15060e6	0.0414725
$950$	−1.09453e8	−3.93476
$951$	1.81603e7	0.651137
$952$	0	0
$953$	−2.18333e7	−0.778729	−0.389365	−	0.921084i	$-0.627305\pi$
$953$	−2.18333e7	−0.778729	−0.389365	+	0.921084i	$0.627305\pi$
$954$	−7.09017e6	−0.252223
$955$	−3.22816e7	−1.14537
$956$	1.19765e7	0.423822
$957$	3.23318e7	1.14117
$958$	4.82863e7	1.69985
$959$	0	0
$960$	−4.73095e7	−1.65680
$961$	−2.86081e7	−0.999266
$962$	−754154.	−0.0262738
$963$	−1.46107e7	−0.507699
$964$	−3.49611e7	−1.21169
$965$	−8.07774e7	−2.79236
$966$	0	0
$967$	−7.30509e6	−0.251223	−0.125611	−	0.992080i	$-0.540089\pi$
$967$	−7.30509e6	−0.251223	−0.125611	+	0.992080i	$0.540089\pi$
$968$	−1.54837e7	−0.531114
$969$	−2.07348e7	−0.709400
$970$	−3.38466e7	−1.15501
$971$	1.62933e6	0.0554576	0.0277288	−	0.999615i	$-0.491173\pi$
$971$	1.62933e6	0.0554576	0.0277288	+	0.999615i	$0.491173\pi$
$972$	−3.76364e7	−1.27774
$973$	0	0
$974$	−6.27180e7	−2.11834
$975$	9.33896e6	0.314621
$976$	−9.77335e6	−0.328412
$977$	−2.55515e7	−0.856407	−0.428204	−	0.903682i	$-0.640853\pi$
$977$	−2.55515e7	−0.856407	−0.428204	+	0.903682i	$0.640853\pi$
$978$	−2.05886e7	−0.688304
$979$	−6.12433e7	−2.04222
$980$	0	0
$981$	1.95506e7	0.648617
$982$	−3.61000e7	−1.19462
$983$	8.48095e6	0.279937	0.139969	−	0.990156i	$-0.455300\pi$
$983$	8.48095e6	0.279937	0.139969	+	0.990156i	$0.455300\pi$
$984$	1.33809e7	0.440554
$985$	−2.82603e7	−0.928080
$986$	4.06786e7	1.33252
$987$	0	0
$988$	−1.70399e7	−0.555359
$989$	1.94589e7	0.632597
$990$	−6.25225e7	−2.02744
$991$	−2.74483e7	−0.887833	−0.443916	−	0.896068i	$-0.646411\pi$
$991$	−2.74483e7	−0.887833	−0.443916	+	0.896068i	$0.646411\pi$
$992$	1.13733e6	0.0366949
$993$	2.93248e6	0.0943762
$994$	0	0
$995$	−3.02206e7	−0.967710
$996$	3.14717e6	0.100525
$997$	2.26765e7	0.722501	0.361251	−	0.932469i	$-0.382350\pi$
$997$	2.26765e7	0.722501	0.361251	+	0.932469i	$0.382350\pi$
$998$	6.77660e7	2.15370
$999$	−2.01792e6	−0.0639719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.6.a.b.1.1		3
7.6	odd	2		13.6.a.b.1.1	✓	3
21.20	even	2		117.6.a.d.1.3		3
28.27	even	2		208.6.a.j.1.2		3
35.13	even	4		325.6.b.c.274.5		6
35.27	even	4		325.6.b.c.274.2		6
35.34	odd	2		325.6.a.c.1.3		3
56.13	odd	2		832.6.a.s.1.2		3
56.27	even	2		832.6.a.t.1.2		3
91.34	even	4		169.6.b.b.168.2		6
91.83	even	4		169.6.b.b.168.5		6
91.90	odd	2		169.6.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.b.1.1	✓	3	7.6	odd	2
117.6.a.d.1.3		3	21.20	even	2
169.6.a.b.1.3		3	91.90	odd	2
169.6.b.b.168.2		6	91.34	even	4
169.6.b.b.168.5		6	91.83	even	4
208.6.a.j.1.2		3	28.27	even	2
325.6.a.c.1.3		3	35.34	odd	2
325.6.b.c.274.2		6	35.27	even	4
325.6.b.c.274.5		6	35.13	even	4
637.6.a.b.1.1		3	1.1	even	1	trivial
832.6.a.s.1.2		3	56.13	odd	2
832.6.a.t.1.2		3	56.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)

\(f(q)\)	\(=\)	\(q-8.64858 q^{2} -10.2870 q^{3} +42.7979 q^{4} -92.1784 q^{5} +88.9676 q^{6} -93.3863 q^{8} -137.178 q^{9} +O(q^{10})\)	\(q-8.64858 q^{2} -10.2870 q^{3} +42.7979 q^{4} -92.1784 q^{5} +88.9676 q^{6} -93.3863 q^{8} -137.178 q^{9} +797.212 q^{10} +571.711 q^{11} -440.260 q^{12} -169.000 q^{13} +948.236 q^{15} -561.873 q^{16} +855.571 q^{17} +1186.40 q^{18} +2355.90 q^{19} -3945.04 q^{20} -4944.49 q^{22} +2509.66 q^{23} +960.662 q^{24} +5371.86 q^{25} +1461.61 q^{26} +3910.88 q^{27} -5497.50 q^{29} -8200.89 q^{30} +144.924 q^{31} +7847.77 q^{32} -5881.17 q^{33} -7399.47 q^{34} -5870.95 q^{36} -515.975 q^{37} -20375.2 q^{38} +1738.50 q^{39} +8608.21 q^{40} +13928.9 q^{41} +7753.58 q^{43} +24468.0 q^{44} +12644.9 q^{45} -21705.0 q^{46} -8344.77 q^{47} +5779.97 q^{48} -46459.0 q^{50} -8801.22 q^{51} -7232.84 q^{52} -5976.21 q^{53} -33823.6 q^{54} -52699.5 q^{55} -24235.1 q^{57} +47545.6 q^{58} -2110.84 q^{59} +40582.5 q^{60} +17394.2 q^{61} -1253.38 q^{62} -49892.1 q^{64} +15578.2 q^{65} +50863.8 q^{66} -3121.05 q^{67} +36616.6 q^{68} -25816.8 q^{69} +43323.3 q^{71} +12810.6 q^{72} -6808.31 q^{73} +4462.45 q^{74} -55260.1 q^{75} +100828. q^{76} -15035.5 q^{78} -1280.80 q^{79} +51792.6 q^{80} -6896.72 q^{81} -120465. q^{82} -7148.44 q^{83} -78865.2 q^{85} -67057.5 q^{86} +56552.6 q^{87} -53390.0 q^{88} -107123. q^{89} -109360. q^{90} +107408. q^{92} -1490.82 q^{93} +72170.4 q^{94} -217163. q^{95} -80729.7 q^{96} -42456.2 q^{97} -78426.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 7 q^{2} - 8 q^{3} + 121 q^{4} - 56 q^{5} + 199 q^{6} + 327 q^{8} - 191 q^{9}+O(q^{10}) \) 3 * q + 7 * q^2 - 8 * q^3 + 121 * q^4 - 56 * q^5 + 199 * q^6 + 327 * q^8 - 191 * q^9	\( 3 q + 7 q^{2} - 8 q^{3} + 121 q^{4} - 56 q^{5} + 199 q^{6} + 327 q^{8} - 191 q^{9} + 1291 q^{10} + 556 q^{11} + 1091 q^{12} - 507 q^{13} + 1972 q^{15} + 2785 q^{16} - 908 q^{17} + 976 q^{18} - 148 q^{19} + 767 q^{20} - 3410 q^{22} + 3624 q^{23} + 13377 q^{24} + 2023 q^{25} - 1183 q^{26} + 4276 q^{27} - 8758 q^{29} + 1717 q^{30} + 2608 q^{31} + 34871 q^{32} + 1828 q^{33} - 15989 q^{34} - 4678 q^{36} - 20632 q^{37} - 33786 q^{38} + 1352 q^{39} + 43509 q^{40} + 10998 q^{41} + 2032 q^{43} + 49790 q^{44} + 13918 q^{45} - 3408 q^{46} - 34260 q^{47} + 76223 q^{48} - 65038 q^{50} + 13468 q^{51} - 20449 q^{52} - 12570 q^{53} - 55595 q^{54} - 35312 q^{55} + 1716 q^{57} - 19126 q^{58} - 63948 q^{59} + 110465 q^{60} + 12754 q^{61} + 40340 q^{62} + 117393 q^{64} + 9464 q^{65} + 112354 q^{66} + 56132 q^{67} + 49175 q^{68} + 20112 q^{69} + 77580 q^{71} + 28794 q^{72} + 43026 q^{73} - 131519 q^{74} - 23548 q^{75} + 99638 q^{76} - 33631 q^{78} - 61872 q^{79} + 264683 q^{80} - 108221 q^{81} - 142260 q^{82} - 98092 q^{83} - 55226 q^{85} - 69897 q^{86} - 139072 q^{87} + 158010 q^{88} - 33694 q^{89} - 101258 q^{90} + 300864 q^{92} + 105392 q^{93} - 72843 q^{94} - 196560 q^{95} + 243641 q^{96} - 76334 q^{97} - 60332 q^{99}+O(q^{100}) \) 3 * q + 7 * q^2 - 8 * q^3 + 121 * q^4 - 56 * q^5 + 199 * q^6 + 327 * q^8 - 191 * q^9 + 1291 * q^10 + 556 * q^11 + 1091 * q^12 - 507 * q^13 + 1972 * q^15 + 2785 * q^16 - 908 * q^17 + 976 * q^18 - 148 * q^19 + 767 * q^20 - 3410 * q^22 + 3624 * q^23 + 13377 * q^24 + 2023 * q^25 - 1183 * q^26 + 4276 * q^27 - 8758 * q^29 + 1717 * q^30 + 2608 * q^31 + 34871 * q^32 + 1828 * q^33 - 15989 * q^34 - 4678 * q^36 - 20632 * q^37 - 33786 * q^38 + 1352 * q^39 + 43509 * q^40 + 10998 * q^41 + 2032 * q^43 + 49790 * q^44 + 13918 * q^45 - 3408 * q^46 - 34260 * q^47 + 76223 * q^48 - 65038 * q^50 + 13468 * q^51 - 20449 * q^52 - 12570 * q^53 - 55595 * q^54 - 35312 * q^55 + 1716 * q^57 - 19126 * q^58 - 63948 * q^59 + 110465 * q^60 + 12754 * q^61 + 40340 * q^62 + 117393 * q^64 + 9464 * q^65 + 112354 * q^66 + 56132 * q^67 + 49175 * q^68 + 20112 * q^69 + 77580 * q^71 + 28794 * q^72 + 43026 * q^73 - 131519 * q^74 - 23548 * q^75 + 99638 * q^76 - 33631 * q^78 - 61872 * q^79 + 264683 * q^80 - 108221 * q^81 - 142260 * q^82 - 98092 * q^83 - 55226 * q^85 - 69897 * q^86 - 139072 * q^87 + 158010 * q^88 - 33694 * q^89 - 101258 * q^90 + 300864 * q^92 + 105392 * q^93 - 72843 * q^94 - 196560 * q^95 + 243641 * q^96 - 76334 * q^97 - 60332 * q^99

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