LMFDB - Embedded newform 355.4.a.d.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [355,4,Mod(1,355)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(355, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("355.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 355 = 5 \cdot 71 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	355.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:	$20.9456780520$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 9 x^{19} - 89 x^{18} + 952 x^{17} + 2911 x^{16} - 41549 x^{15} - 37799 x^{14} + \cdots - 106929408 $ x^20 - 9x^19 - 89x^18 + 952x^17 + 2911x^16 - 41549x^15 - 37799x^14 + 974485x^13 - 43478x^12 - 13402754x^11 + 6726996x^10 + 110353987x^9 - 76601582x^8 - 528296868x^7 + 380540584x^6 + 1335398700x^5 - 833784408x^4 - 1402468992x^3 + 573826752x^2 + 308840768*x - 106929408
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.316870$ of defining polynomial
Character	$\chi$	$=$	355.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.316870 q^{2} -6.27919 q^{3} -7.89959 q^{4} +5.00000 q^{5} -1.98969 q^{6} -28.3648 q^{7} -5.03811 q^{8} +12.4282 q^{9} +O(q^{10})$	\(q+0.316870 q^{2} -6.27919 q^{3} -7.89959 q^{4} +5.00000 q^{5} -1.98969 q^{6} -28.3648 q^{7} -5.03811 q^{8} +12.4282 q^{9} +1.58435 q^{10} -50.5142 q^{11} +49.6030 q^{12} -32.5335 q^{13} -8.98796 q^{14} -31.3959 q^{15} +61.6003 q^{16} +7.29433 q^{17} +3.93812 q^{18} -146.330 q^{19} -39.4980 q^{20} +178.108 q^{21} -16.0064 q^{22} -69.5381 q^{23} +31.6352 q^{24} +25.0000 q^{25} -10.3089 q^{26} +91.4991 q^{27} +224.070 q^{28} +37.4635 q^{29} -9.94844 q^{30} -90.3143 q^{31} +59.8242 q^{32} +317.188 q^{33} +2.31135 q^{34} -141.824 q^{35} -98.1776 q^{36} -235.949 q^{37} -46.3675 q^{38} +204.284 q^{39} -25.1905 q^{40} +1.88573 q^{41} +56.4371 q^{42} +440.555 q^{43} +399.041 q^{44} +62.1409 q^{45} -22.0346 q^{46} +125.651 q^{47} -386.800 q^{48} +461.562 q^{49} +7.92176 q^{50} -45.8024 q^{51} +257.002 q^{52} +253.785 q^{53} +28.9934 q^{54} -252.571 q^{55} +142.905 q^{56} +918.830 q^{57} +11.8711 q^{58} -840.342 q^{59} +248.015 q^{60} +696.852 q^{61} -28.6179 q^{62} -352.523 q^{63} -473.846 q^{64} -162.668 q^{65} +100.507 q^{66} +472.644 q^{67} -57.6222 q^{68} +436.643 q^{69} -44.9398 q^{70} +71.0000 q^{71} -62.6145 q^{72} +228.829 q^{73} -74.7651 q^{74} -156.980 q^{75} +1155.94 q^{76} +1432.83 q^{77} +64.7316 q^{78} -550.939 q^{79} +308.002 q^{80} -910.101 q^{81} +0.597533 q^{82} -1195.87 q^{83} -1406.98 q^{84} +36.4716 q^{85} +139.599 q^{86} -235.240 q^{87} +254.496 q^{88} -1029.01 q^{89} +19.6906 q^{90} +922.808 q^{91} +549.323 q^{92} +567.101 q^{93} +39.8152 q^{94} -731.648 q^{95} -375.647 q^{96} +367.762 q^{97} +146.255 q^{98} -627.800 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 9 q^{2} + 14 q^{3} + 99 q^{4} + 100 q^{5} + 9 q^{6} + 40 q^{7} + 132 q^{8} + 236 q^{9}+O(q^{10}) $ 20 * q + 9 * q^2 + 14 * q^3 + 99 * q^4 + 100 * q^5 + 9 * q^6 + 40 * q^7 + 132 * q^8 + 236 * q^9	\( 20 q + 9 q^{2} + 14 q^{3} + 99 q^{4} + 100 q^{5} + 9 q^{6} + 40 q^{7} + 132 q^{8} + 236 q^{9} + 45 q^{10} + 62 q^{11} + 160 q^{12} + 232 q^{13} + 56 q^{14} + 70 q^{15} + 387 q^{16} + 314 q^{17} + 224 q^{18} + 54 q^{19} + 495 q^{20} + 58 q^{21} - 123 q^{22} + 344 q^{23} + 419 q^{24} + 500 q^{25} - 387 q^{26} + 326 q^{27} + 786 q^{28} + 748 q^{29} + 45 q^{30} - 18 q^{31} + 345 q^{32} + 746 q^{33} - 142 q^{34} + 200 q^{35} + 90 q^{36} + 858 q^{37} + 723 q^{38} + 210 q^{39} + 660 q^{40} + 386 q^{41} - 237 q^{42} + 1210 q^{43} + 1105 q^{44} + 1180 q^{45} - 57 q^{46} + 320 q^{47} + 2280 q^{48} + 1576 q^{49} + 225 q^{50} - 350 q^{51} + 780 q^{52} + 2066 q^{53} - 536 q^{54} + 310 q^{55} + 938 q^{56} + 710 q^{57} + 1852 q^{58} + 1198 q^{59} + 800 q^{60} + 284 q^{61} + 2444 q^{62} + 732 q^{63} + 1118 q^{64} + 1160 q^{65} - 219 q^{66} + 976 q^{67} + 2904 q^{68} + 1026 q^{69} + 280 q^{70} + 1420 q^{71} + 3374 q^{72} + 4310 q^{73} + 955 q^{74} + 350 q^{75} + 740 q^{76} + 5196 q^{77} + 61 q^{78} - 340 q^{79} + 1935 q^{80} + 2556 q^{81} + 1191 q^{82} - 354 q^{83} - 1955 q^{84} + 1570 q^{85} + 1248 q^{86} + 3392 q^{87} + 135 q^{88} - 1446 q^{89} + 1120 q^{90} - 240 q^{91} + 4114 q^{92} - 1066 q^{93} - 3985 q^{94} + 270 q^{95} - 2145 q^{96} + 3282 q^{97} - 4708 q^{98} - 6506 q^{99}+O(q^{100}) \) 20 * q + 9 * q^2 + 14 * q^3 + 99 * q^4 + 100 * q^5 + 9 * q^6 + 40 * q^7 + 132 * q^8 + 236 * q^9 + 45 * q^10 + 62 * q^11 + 160 * q^12 + 232 * q^13 + 56 * q^14 + 70 * q^15 + 387 * q^16 + 314 * q^17 + 224 * q^18 + 54 * q^19 + 495 * q^20 + 58 * q^21 - 123 * q^22 + 344 * q^23 + 419 * q^24 + 500 * q^25 - 387 * q^26 + 326 * q^27 + 786 * q^28 + 748 * q^29 + 45 * q^30 - 18 * q^31 + 345 * q^32 + 746 * q^33 - 142 * q^34 + 200 * q^35 + 90 * q^36 + 858 * q^37 + 723 * q^38 + 210 * q^39 + 660 * q^40 + 386 * q^41 - 237 * q^42 + 1210 * q^43 + 1105 * q^44 + 1180 * q^45 - 57 * q^46 + 320 * q^47 + 2280 * q^48 + 1576 * q^49 + 225 * q^50 - 350 * q^51 + 780 * q^52 + 2066 * q^53 - 536 * q^54 + 310 * q^55 + 938 * q^56 + 710 * q^57 + 1852 * q^58 + 1198 * q^59 + 800 * q^60 + 284 * q^61 + 2444 * q^62 + 732 * q^63 + 1118 * q^64 + 1160 * q^65 - 219 * q^66 + 976 * q^67 + 2904 * q^68 + 1026 * q^69 + 280 * q^70 + 1420 * q^71 + 3374 * q^72 + 4310 * q^73 + 955 * q^74 + 350 * q^75 + 740 * q^76 + 5196 * q^77 + 61 * q^78 - 340 * q^79 + 1935 * q^80 + 2556 * q^81 + 1191 * q^82 - 354 * q^83 - 1955 * q^84 + 1570 * q^85 + 1248 * q^86 + 3392 * q^87 + 135 * q^88 - 1446 * q^89 + 1120 * q^90 - 240 * q^91 + 4114 * q^92 - 1066 * q^93 - 3985 * q^94 + 270 * q^95 - 2145 * q^96 + 3282 * q^97 - 4708 * q^98 - 6506 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.316870	0.112031	0.0560153	−	0.998430i	$-0.482160\pi$
$2$	0.316870	0.112031	0.0560153	+	0.998430i	$0.482160\pi$
$3$	−6.27919	−1.20843	−0.604215	−	0.796821i	$-0.706513\pi$
$3$	−6.27919	−1.20843	−0.604215	+	0.796821i	$0.706513\pi$
$4$	−7.89959	−0.987449
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−1.98969	−0.135381
$7$	−28.3648	−1.53156	−0.765778	−	0.643105i	$-0.777646\pi$
$7$	−28.3648	−1.53156	−0.765778	+	0.643105i	$0.777646\pi$
$8$	−5.03811	−0.222655
$9$	12.4282	0.460303
$10$	1.58435	0.0501016
$11$	−50.5142	−1.38460	−0.692300	−	0.721610i	$-0.743402\pi$
$11$	−50.5142	−1.38460	−0.692300	+	0.721610i	$0.743402\pi$
$12$	49.6030	1.19326
$13$	−32.5335	−0.694091	−0.347045	−	0.937848i	$-0.612815\pi$
$13$	−32.5335	−0.694091	−0.347045	+	0.937848i	$0.612815\pi$
$14$	−8.98796	−0.171581
$15$	−31.3959	−0.540426
$16$	61.6003	0.962505
$17$	7.29433	0.104067	0.0520334	−	0.998645i	$-0.483430\pi$
$17$	7.29433	0.104067	0.0520334	+	0.998645i	$0.483430\pi$
$18$	3.93812	0.0515680
$19$	−146.330	−1.76686	−0.883429	−	0.468565i	$-0.844771\pi$
$19$	−146.330	−1.76686	−0.883429	+	0.468565i	$0.844771\pi$
$20$	−39.4980	−0.441601
$21$	178.108	1.85078
$22$	−16.0064	−0.155117
$23$	−69.5381	−0.630422	−0.315211	−	0.949022i	$-0.602075\pi$
$23$	−69.5381	−0.630422	−0.315211	+	0.949022i	$0.602075\pi$
$24$	31.6352	0.269063
$25$	25.0000	0.200000
$26$	−10.3089	−0.0777594
$27$	91.4991	0.652186
$28$	224.070	1.51233
$29$	37.4635	0.239889	0.119945	−	0.992781i	$-0.461728\pi$
$29$	37.4635	0.239889	0.119945	+	0.992781i	$0.461728\pi$
$30$	−9.94844	−0.0605443
$31$	−90.3143	−0.523256	−0.261628	−	0.965169i	$-0.584259\pi$
$31$	−90.3143	−0.523256	−0.261628	+	0.965169i	$0.584259\pi$
$32$	59.8242	0.330485
$33$	317.188	1.67319
$34$	2.31135	0.0116586
$35$	−141.824	−0.684933
$36$	−98.1776	−0.454526
$37$	−235.949	−1.04837	−0.524185	−	0.851604i	$-0.675630\pi$
$37$	−235.949	−1.04837	−0.524185	+	0.851604i	$0.675630\pi$
$38$	−46.3675	−0.197942
$39$	204.284	0.838760
$40$	−25.1905	−0.0995744
$41$	1.88573	0.00718298	0.00359149	−	0.999994i	$-0.498857\pi$
$41$	1.88573	0.00718298	0.00359149	+	0.999994i	$0.498857\pi$
$42$	56.4371	0.207344
$43$	440.555	1.56242	0.781210	−	0.624268i	$-0.214603\pi$
$43$	440.555	1.56242	0.781210	+	0.624268i	$0.214603\pi$
$44$	399.041	1.36722
$45$	62.1409	0.205854
$46$	−22.0346	−0.0706265
$47$	125.651	0.389960	0.194980	−	0.980807i	$-0.437536\pi$
$47$	125.651	0.389960	0.194980	+	0.980807i	$0.437536\pi$
$48$	−386.800	−1.16312
$49$	461.562	1.34566
$50$	7.92176	0.0224061
$51$	−45.8024	−0.125757
$52$	257.002	0.685379
$53$	253.785	0.657738	0.328869	−	0.944376i	$-0.393333\pi$
$53$	253.785	0.657738	0.328869	+	0.944376i	$0.393333\pi$
$54$	28.9934	0.0730647
$55$	−252.571	−0.619212
$56$	142.905	0.341009
$57$	918.830	2.13512
$58$	11.8711	0.0268749
$59$	−840.342	−1.85429	−0.927146	−	0.374701i	$-0.877745\pi$
$59$	−840.342	−1.85429	−0.927146	+	0.374701i	$0.877745\pi$
$60$	248.015	0.533644
$61$	696.852	1.46267	0.731334	−	0.682020i	$-0.238898\pi$
$61$	696.852	1.46267	0.731334	+	0.682020i	$0.238898\pi$
$62$	−28.6179	−0.0586206
$63$	−352.523	−0.704980
$64$	−473.846	−0.925481
$65$	−162.668	−0.310407
$66$	100.507	0.187449
$67$	472.644	0.861831	0.430916	−	0.902392i	$-0.358191\pi$
$67$	472.644	0.861831	0.430916	+	0.902392i	$0.358191\pi$
$68$	−57.6222	−0.102761
$69$	436.643	0.761821
$70$	−44.9398	−0.0767334
$71$	71.0000	0.118678
$71$	71.0000	0.118678
$72$	−62.6145	−0.102489
$73$	228.829	0.366882	0.183441	−	0.983031i	$-0.441276\pi$
$73$	228.829	0.366882	0.183441	+	0.983031i	$0.441276\pi$
$74$	−74.7651	−0.117450
$75$	−156.980	−0.241686
$76$	1155.94	1.74468
$77$	1432.83	2.12059
$78$	64.7316	0.0939668
$79$	−550.939	−0.784626	−0.392313	−	0.919832i	$-0.628325\pi$
$79$	−550.939	−0.784626	−0.392313	+	0.919832i	$0.628325\pi$
$80$	308.002	0.430445
$81$	−910.101	−1.24842
$82$	0.597533	0.000804713 0
$83$	−1195.87	−1.58149	−0.790743	−	0.612148i	$-0.790306\pi$
$83$	−1195.87	−1.58149	−0.790743	+	0.612148i	$0.790306\pi$
$84$	−1406.98	−1.82755
$85$	36.4716	0.0465400
$86$	139.599	0.175039
$87$	−235.240	−0.289889
$88$	254.496	0.308288
$89$	−1029.01	−1.22555	−0.612777	−	0.790256i	$-0.709948\pi$
$89$	−1029.01	−1.22555	−0.612777	+	0.790256i	$0.709948\pi$
$90$	19.6906	0.0230619
$91$	922.808	1.06304
$92$	549.323	0.622509
$93$	567.101	0.632318
$94$	39.8152	0.0436875
$95$	−731.648	−0.790163
$96$	−375.647	−0.399368
$97$	367.762	0.384954	0.192477	−	0.981301i	$-0.438348\pi$
$97$	367.762	0.384954	0.192477	+	0.981301i	$0.438348\pi$
$98$	146.255	0.150755
$99$	−627.800	−0.637336
$100$	−197.490	−0.197490
$101$	−638.154	−0.628700	−0.314350	−	0.949307i	$-0.601787\pi$
$101$	−638.154	−0.628700	−0.314350	+	0.949307i	$0.601787\pi$
$102$	−14.5134	−0.0140887
$103$	−935.971	−0.895378	−0.447689	−	0.894189i	$-0.647753\pi$
$103$	−935.971	−0.895378	−0.447689	+	0.894189i	$0.647753\pi$
$104$	163.907	0.154543
$105$	890.540	0.827693
$106$	80.4171	0.0736868
$107$	−1103.23	−0.996759	−0.498379	−	0.866959i	$-0.666071\pi$
$107$	−1103.23	−0.996759	−0.498379	+	0.866959i	$0.666071\pi$
$108$	−722.806	−0.644000
$109$	−1274.15	−1.11965	−0.559824	−	0.828611i	$-0.689131\pi$
$109$	−1274.15	−1.11965	−0.559824	+	0.828611i	$0.689131\pi$
$110$	−80.0322	−0.0693706
$111$	1481.57	1.26688
$112$	−1747.28	−1.47413
$113$	−1616.79	−1.34597	−0.672987	−	0.739654i	$-0.734989\pi$
$113$	−1616.79	−1.34597	−0.672987	+	0.739654i	$0.734989\pi$
$114$	291.150	0.239199
$115$	−347.691	−0.281933
$116$	−295.946	−0.236879
$117$	−404.333	−0.319492
$118$	−266.279	−0.207737
$119$	−206.902	−0.159384
$120$	158.176	0.120329
$121$	1220.68	0.917117
$122$	220.812	0.163863
$123$	−11.8409	−0.00868013
$124$	713.447	0.516689
$125$	125.000	0.0894427
$126$	−111.704	−0.0789793
$127$	−244.433	−0.170787	−0.0853935	−	0.996347i	$-0.527215\pi$
$127$	−244.433	−0.170787	−0.0853935	+	0.996347i	$0.527215\pi$
$128$	−628.741	−0.434167
$129$	−2766.33	−1.88808
$130$	−51.5445	−0.0347750
$131$	2792.96	1.86276	0.931382	−	0.364043i	$-0.118604\pi$
$131$	2792.96	1.86276	0.931382	+	0.364043i	$0.118604\pi$
$132$	−2505.66	−1.65219
$133$	4150.61	2.70604
$134$	149.767	0.0965514
$135$	457.496	0.291666
$136$	−36.7496	−0.0231710
$137$	299.268	0.186629	0.0933145	−	0.995637i	$-0.470254\pi$
$137$	299.268	0.186629	0.0933145	+	0.995637i	$0.470254\pi$
$138$	138.359	0.0853472
$139$	1359.94	0.829844	0.414922	−	0.909857i	$-0.363809\pi$
$139$	1359.94	0.829844	0.414922	+	0.909857i	$0.363809\pi$
$140$	1120.35	0.676336
$141$	−788.988	−0.471240
$142$	22.4978	0.0132956
$143$	1643.40	0.961038
$144$	765.580	0.443044
$145$	187.317	0.107282
$146$	72.5089	0.0411019
$147$	−2898.24	−1.62614
$148$	1863.90	1.03521
$149$	−1663.60	−0.914678	−0.457339	−	0.889292i	$-0.651197\pi$
$149$	−1663.60	−0.914678	−0.457339	+	0.889292i	$0.651197\pi$
$150$	−49.7422	−0.0270762
$151$	−2499.63	−1.34713	−0.673567	−	0.739127i	$-0.735239\pi$
$151$	−2499.63	−1.34713	−0.673567	+	0.739127i	$0.735239\pi$
$152$	737.224	0.393400
$153$	90.6552	0.0479022
$154$	454.020	0.237571
$155$	−451.572	−0.234007
$156$	−1613.76	−0.828233
$157$	3241.09	1.64756	0.823781	−	0.566908i	$-0.191861\pi$
$157$	3241.09	1.64756	0.823781	+	0.566908i	$0.191861\pi$
$158$	−174.576	−0.0879021
$159$	−1593.57	−0.794830
$160$	299.121	0.147797
$161$	1972.44	0.965526
$162$	−288.384	−0.139862
$163$	3064.29	1.47247	0.736237	−	0.676724i	$-0.236601\pi$
$163$	3064.29	1.47247	0.736237	+	0.676724i	$0.236601\pi$
$164$	−14.8965	−0.00709283
$165$	1585.94	0.748274
$166$	−378.934	−0.177175
$167$	2590.44	1.20032	0.600161	−	0.799879i	$-0.295103\pi$
$167$	2590.44	1.20032	0.600161	+	0.799879i	$0.295103\pi$
$168$	−897.327	−0.412085
$169$	−1138.57	−0.518238
$170$	11.5568	0.00521391
$171$	−1818.61	−0.813290
$172$	−3480.21	−1.54281
$173$	−3587.04	−1.57640	−0.788201	−	0.615418i	$-0.788987\pi$
$173$	−3587.04	−1.57640	−0.788201	+	0.615418i	$0.788987\pi$
$174$	−74.5406	−0.0324765
$175$	−709.120	−0.306311
$176$	−3111.69	−1.33268
$177$	5276.66	2.24078
$178$	−326.061	−0.137300
$179$	−1340.29	−0.559654	−0.279827	−	0.960050i	$-0.590277\pi$
$179$	−1340.29	−0.559654	−0.279827	+	0.960050i	$0.590277\pi$
$180$	−490.888	−0.203270
$181$	993.285	0.407902	0.203951	−	0.978981i	$-0.434622\pi$
$181$	993.285	0.407902	0.203951	+	0.978981i	$0.434622\pi$
$182$	292.410	0.119093
$183$	−4375.66	−1.76753
$184$	350.341	0.140367
$185$	−1179.74	−0.468846
$186$	179.697	0.0708390
$187$	−368.467	−0.144091
$188$	−992.595	−0.385066
$189$	−2595.36	−0.998859
$190$	−231.837	−0.0885224
$191$	1605.36	0.608168	0.304084	−	0.952645i	$-0.401650\pi$
$191$	1605.36	0.608168	0.304084	+	0.952645i	$0.401650\pi$
$192$	2975.37	1.11838
$193$	1762.47	0.657335	0.328668	−	0.944446i	$-0.393400\pi$
$193$	1762.47	0.657335	0.328668	+	0.944446i	$0.393400\pi$
$194$	116.533	0.0431266
$195$	1021.42	0.375105
$196$	−3646.16	−1.32877
$197$	3466.40	1.25366	0.626829	−	0.779157i	$-0.284352\pi$
$197$	3466.40	1.25366	0.626829	+	0.779157i	$0.284352\pi$
$198$	−198.931	−0.0714011
$199$	−3884.74	−1.38383	−0.691915	−	0.721979i	$-0.743233\pi$
$199$	−3884.74	−1.38383	−0.691915	+	0.721979i	$0.743233\pi$
$200$	−125.953	−0.0445310
$201$	−2967.82	−1.04146
$202$	−202.212	−0.0704336
$203$	−1062.64	−0.367404
$204$	361.821	0.124179
$205$	9.42867	0.00321233
$206$	−296.581	−0.100310
$207$	−864.233	−0.290185
$208$	−2004.08	−0.668066
$209$	7391.72	2.44639
$210$	282.186	0.0927269
$211$	−5597.56	−1.82631	−0.913156	−	0.407610i	$-0.866362\pi$
$211$	−5597.56	−1.82631	−0.913156	+	0.407610i	$0.866362\pi$
$212$	−2004.80	−0.649483
$213$	−445.822	−0.143414
$214$	−349.580	−0.111667
$215$	2202.78	0.698736
$216$	−460.983	−0.145212
$217$	2561.75	0.801396
$218$	−403.741	−0.125435
$219$	−1436.86	−0.443351
$220$	1995.21	0.611440
$221$	−237.310	−0.0722317
$222$	469.464	0.141930
$223$	−1113.74	−0.334446	−0.167223	−	0.985919i	$-0.553480\pi$
$223$	−1113.74	−0.334446	−0.167223	+	0.985919i	$0.553480\pi$
$224$	−1696.90	−0.506156
$225$	310.705	0.0920606
$226$	−512.314	−0.150790
$227$	−1105.87	−0.323345	−0.161673	−	0.986844i	$-0.551689\pi$
$227$	−1105.87	−0.323345	−0.161673	+	0.986844i	$0.551689\pi$
$228$	−7258.39	−2.10833
$229$	−259.975	−0.0750202	−0.0375101	−	0.999296i	$-0.511943\pi$
$229$	−259.975	−0.0750202	−0.0375101	+	0.999296i	$0.511943\pi$
$230$	−110.173	−0.0315851
$231$	−8996.98	−2.56259
$232$	−188.745	−0.0534126
$233$	3497.60	0.983412	0.491706	−	0.870761i	$-0.336373\pi$
$233$	3497.60	0.983412	0.491706	+	0.870761i	$0.336373\pi$
$234$	−128.121	−0.0357929
$235$	628.257	0.174396
$236$	6638.36	1.83102
$237$	3459.45	0.948166
$238$	−65.5611	−0.0178559
$239$	2415.05	0.653625	0.326813	−	0.945089i	$-0.394025\pi$
$239$	2415.05	0.653625	0.326813	+	0.945089i	$0.394025\pi$
$240$	−1934.00	−0.520163
$241$	−5878.33	−1.57119	−0.785595	−	0.618741i	$-0.787643\pi$
$241$	−5878.33	−1.57119	−0.785595	+	0.618741i	$0.787643\pi$
$242$	386.798	0.102745
$243$	3244.22	0.856447
$244$	−5504.85	−1.44431
$245$	2307.81	0.601799
$246$	−3.75202	−0.000972440 0
$247$	4760.62	1.22636
$248$	455.013	0.116506
$249$	7509.07	1.91112
$250$	39.6088	0.0100203
$251$	1912.94	0.481051	0.240526	−	0.970643i	$-0.422680\pi$
$251$	1912.94	0.481051	0.240526	+	0.970643i	$0.422680\pi$
$252$	2784.79	0.696132
$253$	3512.66	0.872882
$254$	−77.4537	−0.0191334
$255$	−229.012	−0.0562404
$256$	3591.54	0.876841
$257$	4256.73	1.03318	0.516590	−	0.856233i	$-0.327201\pi$
$257$	4256.73	1.03318	0.516590	+	0.856233i	$0.327201\pi$
$258$	−876.568	−0.211522
$259$	6692.64	1.60564
$260$	1285.01	0.306511
$261$	465.603	0.110422
$262$	885.006	0.208686
$263$	4087.04	0.958241	0.479121	−	0.877749i	$-0.340956\pi$
$263$	4087.04	0.958241	0.479121	+	0.877749i	$0.340956\pi$
$264$	−1598.03	−0.372545
$265$	1268.93	0.294149
$266$	1315.20	0.303159
$267$	6461.32	1.48100
$268$	−3733.70	−0.851014
$269$	−3177.67	−0.720246	−0.360123	−	0.932905i	$-0.617265\pi$
$269$	−3177.67	−0.720246	−0.360123	+	0.932905i	$0.617265\pi$
$270$	144.967	0.0326755
$271$	3929.49	0.880810	0.440405	−	0.897799i	$-0.354835\pi$
$271$	3929.49	0.880810	0.440405	+	0.897799i	$0.354835\pi$
$272$	449.333	0.100165
$273$	−5794.48	−1.28461
$274$	94.8290	0.0209081
$275$	−1262.85	−0.276920
$276$	−3449.30	−0.752259
$277$	−7419.08	−1.60928	−0.804638	−	0.593766i	$-0.797640\pi$
$277$	−7419.08	−1.60928	−0.804638	+	0.593766i	$0.797640\pi$
$278$	430.923	0.0929679
$279$	−1122.44	−0.240856
$280$	714.525	0.152504
$281$	4965.27	1.05410	0.527052	−	0.849833i	$-0.323297\pi$
$281$	4965.27	1.05410	0.527052	+	0.849833i	$0.323297\pi$
$282$	−250.007	−0.0527932
$283$	−3609.31	−0.758132	−0.379066	−	0.925370i	$-0.623755\pi$
$283$	−3609.31	−0.758132	−0.379066	+	0.925370i	$0.623755\pi$
$284$	−560.871	−0.117189
$285$	4594.15	0.954856
$286$	520.746	0.107666
$287$	−53.4885	−0.0110011
$288$	743.506	0.152123
$289$	−4859.79	−0.989170
$290$	59.3553	0.0120188
$291$	−2309.24	−0.465190
$292$	−1807.65	−0.362277
$293$	4766.33	0.950349	0.475175	−	0.879892i	$-0.342385\pi$
$293$	4766.33	0.950349	0.475175	+	0.879892i	$0.342385\pi$
$294$	−918.365	−0.182177
$295$	−4201.71	−0.829264
$296$	1188.73	0.233425
$297$	−4622.00	−0.903016
$298$	−527.144	−0.102472
$299$	2262.32	0.437570
$300$	1240.08	0.238653
$301$	−12496.3	−2.39293
$302$	−792.059	−0.150920
$303$	4007.09	0.759740
$304$	−9013.95	−1.70061
$305$	3484.26	0.654125
$306$	28.7259	0.00536651
$307$	1554.85	0.289055	0.144528	−	0.989501i	$-0.453834\pi$
$307$	1554.85	0.289055	0.144528	+	0.989501i	$0.453834\pi$
$308$	−11318.7	−2.09398
$309$	5877.14	1.08200
$310$	−143.090	−0.0262160
$311$	−1499.51	−0.273406	−0.136703	−	0.990612i	$-0.543651\pi$
$311$	−1499.51	−0.273406	−0.136703	+	0.990612i	$0.543651\pi$
$312$	−1029.21	−0.186754
$313$	8397.03	1.51638	0.758192	−	0.652031i	$-0.226083\pi$
$313$	8397.03	1.51638	0.758192	+	0.652031i	$0.226083\pi$
$314$	1027.01	0.184577
$315$	−1762.62	−0.315277
$316$	4352.19	0.774778
$317$	3014.91	0.534178	0.267089	−	0.963672i	$-0.413938\pi$
$317$	3014.91	0.534178	0.267089	+	0.963672i	$0.413938\pi$
$318$	−504.954	−0.0890453
$319$	−1892.44	−0.332151
$320$	−2369.23	−0.413887
$321$	6927.38	1.20451
$322$	625.006	0.108168
$323$	−1067.38	−0.183871
$324$	7189.43	1.23276
$325$	−813.338	−0.138818
$326$	970.981	0.164962
$327$	8000.64	1.35302
$328$	−9.50053	−0.00159933
$329$	−3564.08	−0.597246
$330$	502.537	0.0838296
$331$	7905.51	1.31277	0.656384	−	0.754427i	$-0.272085\pi$
$331$	7905.51	1.31277	0.656384	+	0.754427i	$0.272085\pi$
$332$	9446.86	1.56164
$333$	−2932.41	−0.482568
$334$	820.832	0.134473
$335$	2363.22	0.385423
$336$	10971.5	1.78138
$337$	3090.81	0.499606	0.249803	−	0.968297i	$-0.419634\pi$
$337$	3090.81	0.499606	0.249803	+	0.968297i	$0.419634\pi$
$338$	−360.779	−0.0580585
$339$	10152.1	1.62652
$340$	−288.111	−0.0459559
$341$	4562.15	0.724500
$342$	−576.264	−0.0911133
$343$	−3363.00	−0.529402
$344$	−2219.57	−0.347881
$345$	2183.21	0.340697
$346$	−1136.63	−0.176605
$347$	−9727.12	−1.50484	−0.752419	−	0.658685i	$-0.771113\pi$
$347$	−9727.12	−1.50484	−0.752419	+	0.658685i	$0.771113\pi$
$348$	1858.30	0.286251
$349$	−5173.45	−0.793491	−0.396745	−	0.917929i	$-0.629860\pi$
$349$	−5173.45	−0.793491	−0.396745	+	0.917929i	$0.629860\pi$
$350$	−224.699	−0.0343162
$351$	−2976.79	−0.452676
$352$	−3021.97	−0.457589
$353$	−6557.60	−0.988742	−0.494371	−	0.869251i	$-0.664602\pi$
$353$	−6557.60	−0.988742	−0.494371	+	0.869251i	$0.664602\pi$
$354$	1672.02	0.251036
$355$	355.000	0.0530745
$356$	8128.72	1.21017
$357$	1299.18	0.192604
$358$	−424.698	−0.0626983
$359$	11071.7	1.62769	0.813846	−	0.581081i	$-0.197370\pi$
$359$	11071.7	1.62769	0.813846	+	0.581081i	$0.197370\pi$
$360$	−313.073	−0.0458344
$361$	14553.3	2.12179
$362$	314.742	0.0456975
$363$	−7664.89	−1.10827
$364$	−7289.80	−1.04970
$365$	1144.14	0.164074
$366$	−1386.52	−0.198018
$367$	−1055.13	−0.150074	−0.0750371	−	0.997181i	$-0.523908\pi$
$367$	−1055.13	−0.150074	−0.0750371	+	0.997181i	$0.523908\pi$
$368$	−4283.57	−0.606784
$369$	23.4363	0.00330635
$370$	−373.826	−0.0525250
$371$	−7198.58	−1.00736
$372$	−4479.86	−0.624382
$373$	7549.83	1.04803	0.524015	−	0.851709i	$-0.324434\pi$
$373$	7549.83	1.04803	0.524015	+	0.851709i	$0.324434\pi$
$374$	−116.756	−0.0161426
$375$	−784.898	−0.108085
$376$	−633.045	−0.0868266
$377$	−1218.82	−0.166505
$378$	−822.391	−0.111903
$379$	8861.23	1.20098	0.600489	−	0.799633i	$-0.294973\pi$
$379$	8861.23	1.20098	0.600489	+	0.799633i	$0.294973\pi$
$380$	5779.72	0.780246
$381$	1534.84	0.206384
$382$	508.692	0.0681334
$383$	−4439.20	−0.592252	−0.296126	−	0.955149i	$-0.595695\pi$
$383$	−4439.20	−0.592252	−0.296126	+	0.955149i	$0.595695\pi$
$384$	3947.98	0.524661
$385$	7164.13	0.948358
$386$	558.476	0.0736416
$387$	5475.31	0.719187
$388$	−2905.17	−0.380123
$389$	4697.46	0.612264	0.306132	−	0.951989i	$-0.400965\pi$
$389$	4697.46	0.612264	0.306132	+	0.951989i	$0.400965\pi$
$390$	323.658	0.0420232
$391$	−507.234	−0.0656059
$392$	−2325.40	−0.299619
$393$	−17537.5	−2.25102
$394$	1098.40	0.140448
$395$	−2754.69	−0.350895
$396$	4959.36	0.629337
$397$	4653.38	0.588279	0.294139	−	0.955763i	$-0.404967\pi$
$397$	4653.38	0.588279	0.294139	+	0.955763i	$0.404967\pi$
$398$	−1230.96	−0.155031
$399$	−26062.5	−3.27006
$400$	1540.01	0.192501
$401$	−12107.8	−1.50782	−0.753908	−	0.656980i	$-0.771833\pi$
$401$	−12107.8	−1.50782	−0.753908	+	0.656980i	$0.771833\pi$
$402$	−940.414	−0.116676
$403$	2938.24	0.363187
$404$	5041.16	0.620809
$405$	−4550.51	−0.558312
$406$	−336.720	−0.0411605
$407$	11918.8	1.45157
$408$	230.758	0.0280005
$409$	−4572.66	−0.552820	−0.276410	−	0.961040i	$-0.589145\pi$
$409$	−4572.66	−0.552820	−0.276410	+	0.961040i	$0.589145\pi$
$410$	2.98767	0.000359879 0
$411$	−1879.16	−0.225528
$412$	7393.79	0.884140
$413$	23836.1	2.83995
$414$	−273.850	−0.0325096
$415$	−5979.33	−0.707262
$416$	−1946.29	−0.229387
$417$	−8539.30	−1.00281
$418$	2342.21	0.274071
$419$	8568.59	0.999053	0.499527	−	0.866299i	$-0.333507\pi$
$419$	8568.59	0.999053	0.499527	+	0.866299i	$0.333507\pi$
$420$	−7034.90	−0.817305
$421$	−4973.90	−0.575803	−0.287901	−	0.957660i	$-0.592958\pi$
$421$	−4973.90	−0.575803	−0.287901	+	0.957660i	$0.592958\pi$
$422$	−1773.70	−0.204603
$423$	1561.62	0.179500
$424$	−1278.60	−0.146449
$425$	182.358	0.0208133
$426$	−141.268	−0.0160668
$427$	−19766.1	−2.24016
$428$	8715.06	0.984248
$429$	−10319.2	−1.16135
$430$	697.995	0.0782797
$431$	−7519.93	−0.840423	−0.420212	−	0.907426i	$-0.638044\pi$
$431$	−7519.93	−0.840423	−0.420212	+	0.907426i	$0.638044\pi$
$432$	5636.38	0.627732
$433$	−10509.3	−1.16639	−0.583195	−	0.812332i	$-0.698198\pi$
$433$	−10509.3	−1.16639	−0.583195	+	0.812332i	$0.698198\pi$
$434$	811.742	0.0897808
$435$	−1176.20	−0.129643
$436$	10065.3	1.10560
$437$	10175.5	1.11387
$438$	−455.297	−0.0496688
$439$	−682.186	−0.0741662	−0.0370831	−	0.999312i	$-0.511807\pi$
$439$	−682.186	−0.0741662	−0.0370831	+	0.999312i	$0.511807\pi$
$440$	1272.48	0.137871
$441$	5736.38	0.619413
$442$	−75.1965	−0.00809216
$443$	−2580.52	−0.276759	−0.138380	−	0.990379i	$-0.544189\pi$
$443$	−2580.52	−0.276759	−0.138380	+	0.990379i	$0.544189\pi$
$444$	−11703.8	−1.25098
$445$	−5145.03	−0.548085
$446$	−352.911	−0.0374682
$447$	10446.0	1.10532
$448$	13440.6	1.41743
$449$	−13785.0	−1.44890	−0.724448	−	0.689329i	$-0.757905\pi$
$449$	−13785.0	−1.44890	−0.724448	+	0.689329i	$0.757905\pi$
$450$	98.4531	0.0103136
$451$	−95.2563	−0.00994556
$452$	12772.0	1.32908
$453$	15695.7	1.62792
$454$	−350.418	−0.0362245
$455$	4614.04	0.475405
$456$	−4629.17	−0.475396
$457$	−3310.31	−0.338840	−0.169420	−	0.985544i	$-0.554189\pi$
$457$	−3310.31	−0.338840	−0.169420	+	0.985544i	$0.554189\pi$
$458$	−82.3783	−0.00840455
$459$	667.425	0.0678708
$460$	2746.61	0.278395
$461$	−10738.2	−1.08487	−0.542437	−	0.840096i	$-0.682498\pi$
$461$	−10738.2	−1.08487	−0.542437	+	0.840096i	$0.682498\pi$
$462$	−2850.87	−0.287088
$463$	−11191.2	−1.12333	−0.561664	−	0.827365i	$-0.689839\pi$
$463$	−11191.2	−1.12333	−0.561664	+	0.827365i	$0.689839\pi$
$464$	2307.76	0.230895
$465$	2835.50	0.282781
$466$	1108.28	0.110172
$467$	−976.399	−0.0967503	−0.0483751	−	0.998829i	$-0.515404\pi$
$467$	−976.399	−0.0967503	−0.0483751	+	0.998829i	$0.515404\pi$
$468$	3194.06	0.315482
$469$	−13406.5	−1.31994
$470$	199.076	0.0195376
$471$	−20351.4	−1.99096
$472$	4233.73	0.412867
$473$	−22254.3	−2.16333
$474$	1096.20	0.106224
$475$	−3658.24	−0.353372
$476$	1634.44	0.157384
$477$	3154.09	0.302759
$478$	765.257	0.0732260
$479$	−7790.22	−0.743098	−0.371549	−	0.928413i	$-0.621173\pi$
$479$	−7790.22	−0.743098	−0.371549	+	0.928413i	$0.621173\pi$
$480$	−1878.24	−0.178603
$481$	7676.24	0.727664
$482$	−1862.67	−0.176021
$483$	−12385.3	−1.16677
$484$	−9642.89	−0.905606
$485$	1838.81	0.172157
$486$	1028.00	0.0959483
$487$	3226.21	0.300192	0.150096	−	0.988671i	$-0.452042\pi$
$487$	3226.21	0.300192	0.150096	+	0.988671i	$0.452042\pi$
$488$	−3510.82	−0.325670
$489$	−19241.2	−1.77938
$490$	731.277	0.0674199
$491$	−458.852	−0.0421746	−0.0210873	−	0.999778i	$-0.506713\pi$
$491$	−458.852	−0.0421746	−0.0210873	+	0.999778i	$0.506713\pi$
$492$	93.5381	0.00857119
$493$	273.271	0.0249645
$494$	1508.50	0.137390
$495$	−3139.00	−0.285025
$496$	−5563.39	−0.503636
$497$	−2013.90	−0.181762
$498$	2379.40	0.214103
$499$	−738.125	−0.0662185	−0.0331093	−	0.999452i	$-0.510541\pi$
$499$	−738.125	−0.0662185	−0.0331093	+	0.999452i	$0.510541\pi$
$500$	−987.449	−0.0883201
$501$	−16265.8	−1.45051
$502$	606.155	0.0538925
$503$	17123.8	1.51791	0.758957	−	0.651140i	$-0.225709\pi$
$503$	17123.8	1.51791	0.758957	+	0.651140i	$0.225709\pi$
$504$	1776.05	0.156967
$505$	−3190.77	−0.281163
$506$	1113.06	0.0977894
$507$	7149.29	0.626254
$508$	1930.92	0.168644
$509$	19592.6	1.70614	0.853072	−	0.521794i	$-0.174737\pi$
$509$	19592.6	1.70614	0.853072	+	0.521794i	$0.174737\pi$
$510$	−72.5671	−0.00630064
$511$	−6490.68	−0.561900
$512$	6167.98	0.532400
$513$	−13389.0	−1.15232
$514$	1348.83	0.115748
$515$	−4679.85	−0.400425
$516$	21852.9	1.86438
$517$	−6347.17	−0.539939
$518$	2120.70	0.179881
$519$	22523.7	1.90497
$520$	819.537	0.0691136
$521$	3086.25	0.259522	0.129761	−	0.991545i	$-0.458579\pi$
$521$	3086.25	0.259522	0.129761	+	0.991545i	$0.458579\pi$
$522$	147.536	0.0123706
$523$	1733.48	0.144933	0.0724663	−	0.997371i	$-0.476913\pi$
$523$	1733.48	0.144933	0.0724663	+	0.997371i	$0.476913\pi$
$524$	−22063.3	−1.83938
$525$	4452.70	0.370156
$526$	1295.06	0.107352
$527$	−658.782	−0.0544535
$528$	19538.9	1.61046
$529$	−7331.45	−0.602568
$530$	402.085	0.0329537
$531$	−10443.9	−0.853536
$532$	−32788.1	−2.67208
$533$	−61.3496	−0.00498564
$534$	2047.40	0.165917
$535$	−5516.15	−0.445764
$536$	−2381.23	−0.191891
$537$	8415.93	0.676302
$538$	−1006.91	−0.0806895
$539$	−23315.4	−1.86321
$540$	−3614.03	−0.288006
$541$	1662.07	0.132085	0.0660424	−	0.997817i	$-0.478963\pi$
$541$	1662.07	0.132085	0.0660424	+	0.997817i	$0.478963\pi$
$542$	1245.14	0.0986776
$543$	−6237.02	−0.492921
$544$	436.377	0.0343925
$545$	−6370.76	−0.500722
$546$	−1836.10	−0.143915
$547$	−9495.08	−0.742194	−0.371097	−	0.928594i	$-0.621018\pi$
$547$	−9495.08	−0.742194	−0.371097	+	0.928594i	$0.621018\pi$
$548$	−2364.09	−0.184287
$549$	8660.60	0.673271
$550$	−400.161	−0.0310235
$551$	−5482.01	−0.423850
$552$	−2199.85	−0.169623
$553$	15627.3	1.20170
$554$	−2350.89	−0.180288
$555$	7407.83	0.566567
$556$	−10742.9	−0.819429
$557$	−19462.4	−1.48052	−0.740261	−	0.672320i	$-0.765298\pi$
$557$	−19462.4	−1.48052	−0.740261	+	0.672320i	$0.765298\pi$
$558$	−355.669	−0.0269833
$559$	−14332.8	−1.08446
$560$	−8736.41	−0.659251
$561$	2313.67	0.174124
$562$	1573.35	0.118092
$563$	−2388.08	−0.178767	−0.0893834	−	0.995997i	$-0.528490\pi$
$563$	−2388.08	−0.178767	−0.0893834	+	0.995997i	$0.528490\pi$
$564$	6232.69	0.465325
$565$	−8083.97	−0.601938
$566$	−1143.68	−0.0849340
$567$	25814.8	1.91203
$568$	−357.706	−0.0264243
$569$	17365.7	1.27945	0.639727	−	0.768602i	$-0.279047\pi$
$569$	17365.7	1.27945	0.639727	+	0.768602i	$0.279047\pi$
$570$	1455.75	0.106973
$571$	18405.5	1.34894	0.674471	−	0.738302i	$-0.264372\pi$
$571$	18405.5	1.34894	0.674471	+	0.738302i	$0.264372\pi$
$572$	−12982.2	−0.948976
$573$	−10080.4	−0.734929
$574$	−16.9489	−0.00123246
$575$	−1738.45	−0.126084
$576$	−5889.05	−0.426002
$577$	12392.7	0.894131	0.447065	−	0.894501i	$-0.352469\pi$
$577$	12392.7	0.894131	0.447065	+	0.894501i	$0.352469\pi$
$578$	−1539.92	−0.110817
$579$	−11066.9	−0.794344
$580$	−1479.73	−0.105935
$581$	33920.5	2.42213
$582$	−731.731	−0.0521155
$583$	−12819.8	−0.910704
$584$	−1152.86	−0.0816880
$585$	−2021.66	−0.142881
$586$	1510.31	0.106468
$587$	7306.10	0.513722	0.256861	−	0.966448i	$-0.417312\pi$
$587$	7306.10	0.513722	0.256861	+	0.966448i	$0.417312\pi$
$588$	22894.9	1.60573
$589$	13215.7	0.924519
$590$	−1331.40	−0.0929029
$591$	−21766.2	−1.51496
$592$	−14534.5	−1.00906
$593$	18344.3	1.27034	0.635170	−	0.772372i	$-0.280930\pi$
$593$	18344.3	1.27034	0.635170	+	0.772372i	$0.280930\pi$
$594$	−1464.58	−0.101165
$595$	−1034.51	−0.0712787
$596$	13141.7	0.903198
$597$	24393.0	1.67226
$598$	716.862	0.0490212
$599$	9264.73	0.631964	0.315982	−	0.948765i	$-0.397666\pi$
$599$	9264.73	0.631964	0.315982	+	0.948765i	$0.397666\pi$
$600$	790.881	0.0538126
$601$	7922.12	0.537687	0.268844	−	0.963184i	$-0.413359\pi$
$601$	7922.12	0.537687	0.268844	+	0.963184i	$0.413359\pi$
$602$	−3959.70	−0.268082
$603$	5874.11	0.396704
$604$	19746.1	1.33023
$605$	6103.41	0.410147
$606$	1269.73	0.0851141
$607$	−9526.36	−0.637006	−0.318503	−	0.947922i	$-0.603180\pi$
$607$	−9526.36	−0.637006	−0.318503	+	0.947922i	$0.603180\pi$
$608$	−8754.04	−0.583920
$609$	6672.54	0.443982
$610$	1104.06	0.0732820
$611$	−4087.88	−0.270668
$612$	−716.140	−0.0473010
$613$	15408.5	1.01524	0.507622	−	0.861580i	$-0.330525\pi$
$613$	15408.5	1.01524	0.507622	+	0.861580i	$0.330525\pi$
$614$	492.686	0.0323830
$615$	−59.2044	−0.00388187
$616$	−7218.73	−0.472160
$617$	−13181.4	−0.860073	−0.430036	−	0.902812i	$-0.641499\pi$
$617$	−13181.4	−0.860073	−0.430036	+	0.902812i	$0.641499\pi$
$618$	1862.29	0.121217
$619$	9468.93	0.614844	0.307422	−	0.951573i	$-0.400534\pi$
$619$	9468.93	0.614844	0.307422	+	0.951573i	$0.400534\pi$
$620$	3567.23	0.231070
$621$	−6362.68	−0.411152
$622$	−475.150	−0.0306299
$623$	29187.5	1.87701
$624$	12584.0	0.807311
$625$	625.000	0.0400000
$626$	2660.77	0.169881
$627$	−46414.0	−2.95629
$628$	−25603.3	−1.62688
$629$	−1721.09	−0.109100
$630$	−558.520	−0.0353206
$631$	9724.07	0.613485	0.306743	−	0.951792i	$-0.400761\pi$
$631$	9724.07	0.613485	0.306743	+	0.951792i	$0.400761\pi$
$632$	2775.69	0.174701
$633$	35148.1	2.20697
$634$	955.336	0.0598442
$635$	−1222.17	−0.0763783
$636$	12588.5	0.784855
$637$	−15016.3	−0.934012
$638$	−599.657	−0.0372110
$639$	882.401	0.0546279
$640$	−3143.71	−0.194165
$641$	8046.70	0.495827	0.247914	−	0.968782i	$-0.420255\pi$
$641$	8046.70	0.495827	0.247914	+	0.968782i	$0.420255\pi$
$642$	2195.08	0.134942
$643$	−20586.6	−1.26260	−0.631302	−	0.775537i	$-0.717479\pi$
$643$	−20586.6	−1.26260	−0.631302	+	0.775537i	$0.717479\pi$
$644$	−15581.4	−0.953408
$645$	−13831.7	−0.844373
$646$	−338.219	−0.0205992
$647$	−25462.4	−1.54719	−0.773594	−	0.633682i	$-0.781543\pi$
$647$	−25462.4	−1.54719	−0.773594	+	0.633682i	$0.781543\pi$
$648$	4585.19	0.277968
$649$	42449.2	2.56745
$650$	−257.723	−0.0155519
$651$	−16085.7	−0.968431
$652$	−24206.6	−1.45399
$653$	7573.48	0.453864	0.226932	−	0.973911i	$-0.427130\pi$
$653$	7573.48	0.453864	0.226932	+	0.973911i	$0.427130\pi$
$654$	2535.16	0.151579
$655$	13964.8	0.833053
$656$	116.162	0.00691366
$657$	2843.92	0.168877
$658$	−1129.35	−0.0669098
$659$	19.5228	0.00115402	0.000577010	−	1.00000i	$-0.499816\pi$
$659$	19.5228	0.00115402	0.000577010		1.00000i	$0.499816\pi$
$660$	−12528.3	−0.738883
$661$	−8720.18	−0.513125	−0.256562	−	0.966528i	$-0.582590\pi$
$661$	−8720.18	−0.513125	−0.256562	+	0.966528i	$0.582590\pi$
$662$	2505.02	0.147070
$663$	1490.12	0.0872870
$664$	6024.90	0.352126
$665$	20753.0	1.21018
$666$	−929.195	−0.0540624
$667$	−2605.14	−0.151231
$668$	−20463.4	−1.18526
$669$	6993.37	0.404155
$670$	748.835	0.0431791
$671$	−35200.9	−2.02521
$672$	10655.2	0.611654
$673$	−20070.3	−1.14956	−0.574778	−	0.818309i	$-0.694912\pi$
$673$	−20070.3	−1.14956	−0.574778	+	0.818309i	$0.694912\pi$
$674$	979.387	0.0559712
$675$	2287.48	0.130437
$676$	8994.23	0.511734
$677$	−7754.25	−0.440207	−0.220103	−	0.975477i	$-0.570639\pi$
$677$	−7754.25	−0.440207	−0.220103	+	0.975477i	$0.570639\pi$
$678$	3216.91	0.182219
$679$	−10431.5	−0.589579
$680$	−183.748	−0.0103624
$681$	6943.98	0.390740
$682$	1445.61	0.0811661
$683$	28456.1	1.59420	0.797102	−	0.603845i	$-0.206365\pi$
$683$	28456.1	1.59420	0.797102	+	0.603845i	$0.206365\pi$
$684$	14366.3	0.803083
$685$	1496.34	0.0834630
$686$	−1065.64	−0.0593093
$687$	1632.43	0.0906567
$688$	27138.4	1.50384
$689$	−8256.54	−0.456530
$690$	691.796	0.0381684
$691$	−16980.0	−0.934806	−0.467403	−	0.884044i	$-0.654810\pi$
$691$	−16980.0	−0.934806	−0.467403	+	0.884044i	$0.654810\pi$
$692$	28336.1	1.55662
$693$	17807.4	0.976115
$694$	−3082.23	−0.168588
$695$	6799.68	0.371118
$696$	1185.16	0.0645453
$697$	13.7552	0.000747509 0
$698$	−1639.31	−0.0888952
$699$	−21962.1	−1.18838
$700$	5601.76	0.302467
$701$	20805.7	1.12100	0.560499	−	0.828155i	$-0.310609\pi$
$701$	20805.7	1.12100	0.560499	+	0.828155i	$0.310609\pi$
$702$	−943.256	−0.0507136
$703$	34526.3	1.85232
$704$	23935.9	1.28142
$705$	−3944.94	−0.210745
$706$	−2077.91	−0.110769
$707$	18101.1	0.962889
$708$	−41683.5	−2.21266
$709$	−19891.2	−1.05364	−0.526820	−	0.849977i	$-0.676616\pi$
$709$	−19891.2	−1.05364	−0.526820	+	0.849977i	$0.676616\pi$
$710$	112.489	0.00594596
$711$	−6847.17	−0.361166
$712$	5184.24	0.272876
$713$	6280.29	0.329872
$714$	411.671	0.0215776
$715$	8217.02	0.429789
$716$	10587.7	0.552630
$717$	−15164.5	−0.789860
$718$	3508.29	0.182351
$719$	8228.12	0.426783	0.213392	−	0.976967i	$-0.431549\pi$
$719$	8228.12	0.426783	0.213392	+	0.976967i	$0.431549\pi$
$720$	3827.90	0.198135
$721$	26548.6	1.37132
$722$	4611.52	0.237705
$723$	36911.1	1.89867
$724$	−7846.55	−0.402783
$725$	936.587	0.0479779
$726$	−2428.78	−0.124160
$727$	−24617.3	−1.25585	−0.627927	−	0.778272i	$-0.716097\pi$
$727$	−24617.3	−1.25585	−0.627927	+	0.778272i	$0.716097\pi$
$728$	−4649.20	−0.236691
$729$	4201.68	0.213467
$730$	362.545	0.0183813
$731$	3213.56	0.162596
$732$	34566.0	1.74535
$733$	−27927.4	−1.40726	−0.703630	−	0.710567i	$-0.748439\pi$
$733$	−27927.4	−1.40726	−0.703630	+	0.710567i	$0.748439\pi$
$734$	−334.338	−0.0168129
$735$	−14491.2	−0.727232
$736$	−4160.06	−0.208345
$737$	−23875.2	−1.19329
$738$	7.42625	0.000370412 0
$739$	11145.2	0.554780	0.277390	−	0.960757i	$-0.410531\pi$
$739$	11145.2	0.554780	0.277390	+	0.960757i	$0.410531\pi$
$740$	9319.49	0.462961
$741$	−29892.8	−1.48197
$742$	−2281.01	−0.112855
$743$	−2334.72	−0.115279	−0.0576396	−	0.998337i	$-0.518357\pi$
$743$	−2334.72	−0.115279	−0.0576396	+	0.998337i	$0.518357\pi$
$744$	−2857.11	−0.140789
$745$	−8317.98	−0.409057
$746$	2392.32	0.117411
$747$	−14862.4	−0.727963
$748$	2910.74	0.142282
$749$	31292.9	1.52659
$750$	−248.711	−0.0121089
$751$	−1451.93	−0.0705484	−0.0352742	−	0.999378i	$-0.511230\pi$
$751$	−1451.93	−0.0705484	−0.0352742	+	0.999378i	$0.511230\pi$
$752$	7740.16	0.375339
$753$	−12011.7	−0.581317
$754$	−386.207	−0.0186536
$755$	−12498.2	−0.602456
$756$	20502.3	0.986323
$757$	25545.8	1.22652	0.613261	−	0.789880i	$-0.289857\pi$
$757$	25545.8	1.22652	0.613261	+	0.789880i	$0.289857\pi$
$758$	2807.86	0.134546
$759$	−22056.7	−1.05482
$760$	3686.12	0.175934
$761$	−18110.7	−0.862696	−0.431348	−	0.902186i	$-0.641962\pi$
$761$	−18110.7	−0.862696	−0.431348	+	0.902186i	$0.641962\pi$
$762$	486.346	0.0231213
$763$	36141.1	1.71480
$764$	−12681.7	−0.600535
$765$	453.276	0.0214225
$766$	−1406.65	−0.0663504
$767$	27339.3	1.28705
$768$	−22551.9	−1.05960
$769$	−5860.26	−0.274807	−0.137403	−	0.990515i	$-0.543876\pi$
$769$	−5860.26	−0.274807	−0.137403	+	0.990515i	$0.543876\pi$
$770$	2270.10	0.106245
$771$	−26728.8	−1.24853
$772$	−13922.8	−0.649085
$773$	−25911.3	−1.20565	−0.602824	−	0.797875i	$-0.705958\pi$
$773$	−25911.3	−1.20565	−0.602824	+	0.797875i	$0.705958\pi$
$774$	1734.96	0.0805709
$775$	−2257.86	−0.104651
$776$	−1852.82	−0.0857120
$777$	−42024.3	−1.94030
$778$	1488.49	0.0685923
$779$	−275.939	−0.0126913
$780$	−8068.81	−0.370397
$781$	−3586.51	−0.164322
$782$	−160.727	−0.00734987
$783$	3427.87	0.156452
$784$	28432.4	1.29521
$785$	16205.5	0.736812
$786$	−5557.12	−0.252183
$787$	−39421.5	−1.78555	−0.892773	−	0.450507i	$-0.851243\pi$
$787$	−39421.5	−1.78555	−0.892773	+	0.450507i	$0.851243\pi$
$788$	−27383.1	−1.23792
$789$	−25663.3	−1.15797
$790$	−872.881	−0.0393110
$791$	45860.0	2.06144
$792$	3162.92	0.141906
$793$	−22671.1	−1.01522
$794$	1474.52	0.0659052
$795$	−7967.83	−0.355459
$796$	30687.9	1.36646
$797$	−19868.2	−0.883022	−0.441511	−	0.897256i	$-0.645557\pi$
$797$	−19868.2	−0.883022	−0.441511	+	0.897256i	$0.645557\pi$
$798$	−8258.42	−0.366347
$799$	916.542	0.0405819
$800$	1495.60	0.0660970
$801$	−12788.7	−0.564127
$802$	−3836.60	−0.168921
$803$	−11559.1	−0.507984
$804$	23444.6	1.02839
$805$	9862.18	0.431796
$806$	931.042	0.0406880
$807$	19953.2	0.870367
$808$	3215.09	0.139983
$809$	40900.3	1.77747	0.888737	−	0.458417i	$-0.151583\pi$
$809$	40900.3	1.77747	0.888737	+	0.458417i	$0.151583\pi$
$810$	−1441.92	−0.0625480
$811$	26916.9	1.16545	0.582725	−	0.812669i	$-0.301986\pi$
$811$	26916.9	1.16545	0.582725	+	0.812669i	$0.301986\pi$
$812$	8394.46	0.362793
$813$	−24674.0	−1.06440
$814$	3776.70	0.162621
$815$	15321.4	0.658511
$816$	−2821.44	−0.121042
$817$	−64466.3	−2.76057
$818$	−1448.94	−0.0619327
$819$	11468.8	0.489320
$820$	−74.4827	−0.00317201
$821$	26979.7	1.14689	0.573447	−	0.819243i	$-0.305606\pi$
$821$	26979.7	1.14689	0.573447	+	0.819243i	$0.305606\pi$
$822$	−595.449	−0.0252660
$823$	10179.8	0.431160	0.215580	−	0.976486i	$-0.430836\pi$
$823$	10179.8	0.431160	0.215580	+	0.976486i	$0.430836\pi$
$824$	4715.52	0.199360
$825$	7929.70	0.334638
$826$	7552.96	0.318161
$827$	25419.2	1.06882	0.534409	−	0.845226i	$-0.320534\pi$
$827$	25419.2	1.06882	0.534409	+	0.845226i	$0.320534\pi$
$828$	6827.09	0.286543
$829$	26132.7	1.09484	0.547421	−	0.836857i	$-0.315610\pi$
$829$	26132.7	1.09484	0.547421	+	0.836857i	$0.315610\pi$
$830$	−1894.67	−0.0792350
$831$	46585.8	1.94470
$832$	15415.9	0.642367
$833$	3366.79	0.140039
$834$	−2705.85	−0.112345
$835$	12952.2	0.536801
$836$	−58391.5	−2.41569
$837$	−8263.68	−0.341260
$838$	2715.13	0.111924
$839$	−45199.3	−1.85990	−0.929948	−	0.367690i	$-0.880149\pi$
$839$	−45199.3	−1.85990	−0.929948	+	0.367690i	$0.880149\pi$
$840$	−4486.64	−0.184290
$841$	−22985.5	−0.942453
$842$	−1576.08	−0.0645075
$843$	−31177.9	−1.27381
$844$	44218.4	1.80339
$845$	−5692.85	−0.231763
$846$	494.830	0.0201095
$847$	−34624.4	−1.40462
$848$	15633.3	0.633076
$849$	22663.6	0.916150
$850$	57.7839	0.00233173
$851$	16407.4	0.660916
$852$	3521.81	0.141614
$853$	2581.44	0.103619	0.0518095	−	0.998657i	$-0.483501\pi$
$853$	2581.44	0.103619	0.0518095	+	0.998657i	$0.483501\pi$
$854$	−6263.28	−0.250966
$855$	−9093.05	−0.363714
$856$	5558.19	0.221933
$857$	28817.5	1.14864	0.574322	−	0.818630i	$-0.305266\pi$
$857$	28817.5	1.14864	0.574322	+	0.818630i	$0.305266\pi$
$858$	−3269.86	−0.130106
$859$	37518.9	1.49025	0.745127	−	0.666923i	$-0.232389\pi$
$859$	37518.9	1.49025	0.745127	+	0.666923i	$0.232389\pi$
$860$	−17401.0	−0.689966
$861$	335.864	0.0132941
$862$	−2382.84	−0.0941531
$863$	−21412.4	−0.844598	−0.422299	−	0.906457i	$-0.638777\pi$
$863$	−21412.4	−0.844598	−0.422299	+	0.906457i	$0.638777\pi$
$864$	5473.86	0.215538
$865$	−17935.2	−0.704988
$866$	−3330.10	−0.130671
$867$	30515.5	1.19534
$868$	−20236.8	−0.791337
$869$	27830.2	1.08639
$870$	−372.703	−0.0145239
$871$	−15376.8	−0.598189
$872$	6419.32	0.249295
$873$	4570.61	0.177196
$874$	3224.31	0.124787
$875$	−3545.60	−0.136987
$876$	11350.6	0.437786
$877$	28165.4	1.08447	0.542234	−	0.840228i	$-0.317579\pi$
$877$	28165.4	1.08447	0.542234	+	0.840228i	$0.317579\pi$
$878$	−216.164	−0.00830888
$879$	−29928.7	−1.14843
$880$	−15558.4	−0.595995
$881$	33221.2	1.27043	0.635215	−	0.772335i	$-0.280911\pi$
$881$	33221.2	1.27043	0.635215	+	0.772335i	$0.280911\pi$
$882$	1817.69	0.0693932
$883$	41589.4	1.58505	0.792523	−	0.609842i	$-0.208767\pi$
$883$	41589.4	1.58505	0.792523	+	0.609842i	$0.208767\pi$
$884$	1874.65	0.0713252
$885$	26383.3	1.00211
$886$	−817.691	−0.0310055
$887$	−33075.4	−1.25204	−0.626022	−	0.779805i	$-0.715318\pi$
$887$	−33075.4	−1.25204	−0.626022	+	0.779805i	$0.715318\pi$
$888$	−7464.29	−0.282078
$889$	6933.31	0.261570
$890$	−1630.31	−0.0614022
$891$	45973.0	1.72857
$892$	8798.08	0.330248
$893$	−18386.5	−0.689004
$894$	3310.04	0.123830
$895$	−6701.45	−0.250285
$896$	17834.1	0.664951
$897$	−14205.5	−0.528773
$898$	−4368.06	−0.162321
$899$	−3383.49	−0.125524
$900$	−2454.44	−0.0909052
$901$	1851.19	0.0684486
$902$	−30.1839	−0.00111421
$903$	78466.4	2.89169
$904$	8145.58	0.299688
$905$	4966.43	0.182419
$906$	4973.48	0.182376
$907$	−30202.5	−1.10569	−0.552844	−	0.833285i	$-0.686457\pi$
$907$	−30202.5	−1.10569	−0.552844	+	0.833285i	$0.686457\pi$
$908$	8735.94	0.319287
$909$	−7931.10	−0.289393
$910$	1462.05	0.0532599
$911$	24941.4	0.907074	0.453537	−	0.891237i	$-0.350162\pi$
$911$	24941.4	0.907074	0.453537	+	0.891237i	$0.350162\pi$
$912$	56600.2	2.05507
$913$	60408.2	2.18973
$914$	−1048.94	−0.0379604
$915$	−21878.3	−0.790464
$916$	2053.70	0.0740786
$917$	−79221.8	−2.85293
$918$	211.487	0.00760361
$919$	22225.7	0.797777	0.398889	−	0.916999i	$-0.369396\pi$
$919$	22225.7	0.797777	0.398889	+	0.916999i	$0.369396\pi$
$920$	1751.70	0.0627738
$921$	−9763.20	−0.349303
$922$	−3402.61	−0.121539
$923$	−2309.88	−0.0823734
$924$	71072.5	2.53042
$925$	−5898.72	−0.209674
$926$	−3546.17	−0.125847
$927$	−11632.4	−0.412145
$928$	2241.22	0.0792798
$929$	5322.68	0.187978	0.0939889	−	0.995573i	$-0.470038\pi$
$929$	5322.68	0.187978	0.0939889	+	0.995573i	$0.470038\pi$
$930$	898.486	0.0316801
$931$	−67540.2	−2.37760
$932$	−27629.6	−0.971070
$933$	9415.70	0.330392
$934$	−309.392	−0.0108390
$935$	−1842.33	−0.0644393
$936$	2037.07	0.0711365
$937$	41806.5	1.45759	0.728793	−	0.684734i	$-0.240082\pi$
$937$	41806.5	1.45759	0.728793	+	0.684734i	$0.240082\pi$
$938$	−4248.11	−0.147874
$939$	−52726.5	−1.83244
$940$	−4962.97	−0.172207
$941$	23455.4	0.812566	0.406283	−	0.913747i	$-0.366825\pi$
$941$	23455.4	0.812566	0.406283	+	0.913747i	$0.366825\pi$
$942$	−6448.76	−0.223049
$943$	−131.130	−0.00452831
$944$	−51765.3	−1.78476
$945$	−12976.8	−0.446703
$946$	−7051.73	−0.242359
$947$	−22992.0	−0.788954	−0.394477	−	0.918906i	$-0.629074\pi$
$947$	−22992.0	−0.788954	−0.394477	+	0.918906i	$0.629074\pi$
$948$	−27328.2	−0.936266
$949$	−7444.60	−0.254649
$950$	−1159.19	−0.0395884
$951$	−18931.2	−0.645516
$952$	1042.40	0.0354876
$953$	−4988.54	−0.169564	−0.0847821	−	0.996400i	$-0.527019\pi$
$953$	−4988.54	−0.169564	−0.0847821	+	0.996400i	$0.527019\pi$
$954$	999.438	0.0339182
$955$	8026.82	0.271981
$956$	−19077.9	−0.645422
$957$	11883.0	0.401381
$958$	−2468.49	−0.0832497
$959$	−8488.67	−0.285833
$960$	14876.8	0.500154
$961$	−21634.3	−0.726203
$962$	2432.37	0.0815207
$963$	−13711.1	−0.458811
$964$	46436.4	1.55147
$965$	8812.37	0.293969
$966$	−3924.53	−0.130714
$967$	13401.3	0.445662	0.222831	−	0.974857i	$-0.428470\pi$
$967$	13401.3	0.445662	0.222831	+	0.974857i	$0.428470\pi$
$968$	−6149.93	−0.204201
$969$	6702.25	0.222195
$970$	582.664	0.0192868
$971$	4483.10	0.148166	0.0740832	−	0.997252i	$-0.476397\pi$
$971$	4483.10	0.148166	0.0740832	+	0.997252i	$0.476397\pi$
$972$	−25628.0	−0.845698
$973$	−38574.3	−1.27095
$974$	1022.29	0.0336307
$975$	5107.10	0.167752
$976$	42926.3	1.40783
$977$	44077.8	1.44337	0.721686	−	0.692221i	$-0.243368\pi$
$977$	44077.8	1.44337	0.721686	+	0.692221i	$0.243368\pi$
$978$	−6096.97	−0.199345
$979$	51979.4	1.69690
$980$	−18230.8	−0.594246
$981$	−15835.4	−0.515378
$982$	−145.397	−0.00472484
$983$	−15648.1	−0.507727	−0.253864	−	0.967240i	$-0.581701\pi$
$983$	−15648.1	−0.507727	−0.253864	+	0.967240i	$0.581701\pi$
$984$	59.6556	0.00193267
$985$	17332.0	0.560653
$986$	86.5914	0.00279679
$987$	22379.5	0.721730
$988$	−37606.9	−1.21097
$989$	−30635.4	−0.984984
$990$	−994.655	−0.0319315
$991$	−48645.9	−1.55932	−0.779661	−	0.626202i	$-0.784609\pi$
$991$	−48645.9	−1.55932	−0.779661	+	0.626202i	$0.784609\pi$
$992$	−5402.98	−0.172928
$993$	−49640.2	−1.58639
$994$	−638.145	−0.0203629
$995$	−19423.7	−0.618867
$996$	−59318.6	−1.88713
$997$	−17365.8	−0.551634	−0.275817	−	0.961210i	$-0.588948\pi$
$997$	−17365.8	−0.551634	−0.275817	+	0.961210i	$0.588948\pi$
$998$	−233.890	−0.00741849
$999$	−21589.1	−0.683733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	355.4.a.d.1.10	✓	20
5.4	even	2		1775.4.a.g.1.11		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
355.4.a.d.1.10	✓	20	1.1	even	1	trivial
1775.4.a.g.1.11		20	5.4	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:	\( N \)	\(=\)	\( 355 = 5 \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	355.a (trivial)

\(f(q)\)	\(=\)	\(q+0.316870 q^{2} -6.27919 q^{3} -7.89959 q^{4} +5.00000 q^{5} -1.98969 q^{6} -28.3648 q^{7} -5.03811 q^{8} +12.4282 q^{9} +O(q^{10})\)	\(q+0.316870 q^{2} -6.27919 q^{3} -7.89959 q^{4} +5.00000 q^{5} -1.98969 q^{6} -28.3648 q^{7} -5.03811 q^{8} +12.4282 q^{9} +1.58435 q^{10} -50.5142 q^{11} +49.6030 q^{12} -32.5335 q^{13} -8.98796 q^{14} -31.3959 q^{15} +61.6003 q^{16} +7.29433 q^{17} +3.93812 q^{18} -146.330 q^{19} -39.4980 q^{20} +178.108 q^{21} -16.0064 q^{22} -69.5381 q^{23} +31.6352 q^{24} +25.0000 q^{25} -10.3089 q^{26} +91.4991 q^{27} +224.070 q^{28} +37.4635 q^{29} -9.94844 q^{30} -90.3143 q^{31} +59.8242 q^{32} +317.188 q^{33} +2.31135 q^{34} -141.824 q^{35} -98.1776 q^{36} -235.949 q^{37} -46.3675 q^{38} +204.284 q^{39} -25.1905 q^{40} +1.88573 q^{41} +56.4371 q^{42} +440.555 q^{43} +399.041 q^{44} +62.1409 q^{45} -22.0346 q^{46} +125.651 q^{47} -386.800 q^{48} +461.562 q^{49} +7.92176 q^{50} -45.8024 q^{51} +257.002 q^{52} +253.785 q^{53} +28.9934 q^{54} -252.571 q^{55} +142.905 q^{56} +918.830 q^{57} +11.8711 q^{58} -840.342 q^{59} +248.015 q^{60} +696.852 q^{61} -28.6179 q^{62} -352.523 q^{63} -473.846 q^{64} -162.668 q^{65} +100.507 q^{66} +472.644 q^{67} -57.6222 q^{68} +436.643 q^{69} -44.9398 q^{70} +71.0000 q^{71} -62.6145 q^{72} +228.829 q^{73} -74.7651 q^{74} -156.980 q^{75} +1155.94 q^{76} +1432.83 q^{77} +64.7316 q^{78} -550.939 q^{79} +308.002 q^{80} -910.101 q^{81} +0.597533 q^{82} -1195.87 q^{83} -1406.98 q^{84} +36.4716 q^{85} +139.599 q^{86} -235.240 q^{87} +254.496 q^{88} -1029.01 q^{89} +19.6906 q^{90} +922.808 q^{91} +549.323 q^{92} +567.101 q^{93} +39.8152 q^{94} -731.648 q^{95} -375.647 q^{96} +367.762 q^{97} +146.255 q^{98} -627.800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 9 q^{2} + 14 q^{3} + 99 q^{4} + 100 q^{5} + 9 q^{6} + 40 q^{7} + 132 q^{8} + 236 q^{9}+O(q^{10}) \) 20 * q + 9 * q^2 + 14 * q^3 + 99 * q^4 + 100 * q^5 + 9 * q^6 + 40 * q^7 + 132 * q^8 + 236 * q^9	\( 20 q + 9 q^{2} + 14 q^{3} + 99 q^{4} + 100 q^{5} + 9 q^{6} + 40 q^{7} + 132 q^{8} + 236 q^{9} + 45 q^{10} + 62 q^{11} + 160 q^{12} + 232 q^{13} + 56 q^{14} + 70 q^{15} + 387 q^{16} + 314 q^{17} + 224 q^{18} + 54 q^{19} + 495 q^{20} + 58 q^{21} - 123 q^{22} + 344 q^{23} + 419 q^{24} + 500 q^{25} - 387 q^{26} + 326 q^{27} + 786 q^{28} + 748 q^{29} + 45 q^{30} - 18 q^{31} + 345 q^{32} + 746 q^{33} - 142 q^{34} + 200 q^{35} + 90 q^{36} + 858 q^{37} + 723 q^{38} + 210 q^{39} + 660 q^{40} + 386 q^{41} - 237 q^{42} + 1210 q^{43} + 1105 q^{44} + 1180 q^{45} - 57 q^{46} + 320 q^{47} + 2280 q^{48} + 1576 q^{49} + 225 q^{50} - 350 q^{51} + 780 q^{52} + 2066 q^{53} - 536 q^{54} + 310 q^{55} + 938 q^{56} + 710 q^{57} + 1852 q^{58} + 1198 q^{59} + 800 q^{60} + 284 q^{61} + 2444 q^{62} + 732 q^{63} + 1118 q^{64} + 1160 q^{65} - 219 q^{66} + 976 q^{67} + 2904 q^{68} + 1026 q^{69} + 280 q^{70} + 1420 q^{71} + 3374 q^{72} + 4310 q^{73} + 955 q^{74} + 350 q^{75} + 740 q^{76} + 5196 q^{77} + 61 q^{78} - 340 q^{79} + 1935 q^{80} + 2556 q^{81} + 1191 q^{82} - 354 q^{83} - 1955 q^{84} + 1570 q^{85} + 1248 q^{86} + 3392 q^{87} + 135 q^{88} - 1446 q^{89} + 1120 q^{90} - 240 q^{91} + 4114 q^{92} - 1066 q^{93} - 3985 q^{94} + 270 q^{95} - 2145 q^{96} + 3282 q^{97} - 4708 q^{98} - 6506 q^{99}+O(q^{100}) \) 20 * q + 9 * q^2 + 14 * q^3 + 99 * q^4 + 100 * q^5 + 9 * q^6 + 40 * q^7 + 132 * q^8 + 236 * q^9 + 45 * q^10 + 62 * q^11 + 160 * q^12 + 232 * q^13 + 56 * q^14 + 70 * q^15 + 387 * q^16 + 314 * q^17 + 224 * q^18 + 54 * q^19 + 495 * q^20 + 58 * q^21 - 123 * q^22 + 344 * q^23 + 419 * q^24 + 500 * q^25 - 387 * q^26 + 326 * q^27 + 786 * q^28 + 748 * q^29 + 45 * q^30 - 18 * q^31 + 345 * q^32 + 746 * q^33 - 142 * q^34 + 200 * q^35 + 90 * q^36 + 858 * q^37 + 723 * q^38 + 210 * q^39 + 660 * q^40 + 386 * q^41 - 237 * q^42 + 1210 * q^43 + 1105 * q^44 + 1180 * q^45 - 57 * q^46 + 320 * q^47 + 2280 * q^48 + 1576 * q^49 + 225 * q^50 - 350 * q^51 + 780 * q^52 + 2066 * q^53 - 536 * q^54 + 310 * q^55 + 938 * q^56 + 710 * q^57 + 1852 * q^58 + 1198 * q^59 + 800 * q^60 + 284 * q^61 + 2444 * q^62 + 732 * q^63 + 1118 * q^64 + 1160 * q^65 - 219 * q^66 + 976 * q^67 + 2904 * q^68 + 1026 * q^69 + 280 * q^70 + 1420 * q^71 + 3374 * q^72 + 4310 * q^73 + 955 * q^74 + 350 * q^75 + 740 * q^76 + 5196 * q^77 + 61 * q^78 - 340 * q^79 + 1935 * q^80 + 2556 * q^81 + 1191 * q^82 - 354 * q^83 - 1955 * q^84 + 1570 * q^85 + 1248 * q^86 + 3392 * q^87 + 135 * q^88 - 1446 * q^89 + 1120 * q^90 - 240 * q^91 + 4114 * q^92 - 1066 * q^93 - 3985 * q^94 + 270 * q^95 - 2145 * q^96 + 3282 * q^97 - 4708 * q^98 - 6506 * q^99

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