LMFDB - Embedded newform 57.6.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [57,6,Mod(1,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, 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("57.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 57 = 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	57.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:	$9.14187772934$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 90x^{2} + 118x + 1412 $ x^4 - x^3 - 90x^2 + 118x + 1412
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.82184$ of defining polynomial
Character	$\chi$	$=$	57.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.82184 q^{2} -9.00000 q^{3} +1.89382 q^{4} +23.6174 q^{5} -52.3966 q^{6} -250.612 q^{7} -175.273 q^{8} +81.0000 q^{9} +137.497 q^{10} +414.956 q^{11} -17.0444 q^{12} -920.892 q^{13} -1459.02 q^{14} -212.557 q^{15} -1081.02 q^{16} -238.401 q^{17} +471.569 q^{18} -361.000 q^{19} +44.7271 q^{20} +2255.50 q^{21} +2415.81 q^{22} +1138.97 q^{23} +1577.46 q^{24} -2567.22 q^{25} -5361.29 q^{26} -729.000 q^{27} -474.613 q^{28} +7031.41 q^{29} -1237.47 q^{30} +495.673 q^{31} -684.752 q^{32} -3734.61 q^{33} -1387.93 q^{34} -5918.80 q^{35} +153.399 q^{36} +2981.99 q^{37} -2101.68 q^{38} +8288.03 q^{39} -4139.51 q^{40} +9339.27 q^{41} +13131.2 q^{42} -11537.4 q^{43} +785.851 q^{44} +1913.01 q^{45} +6630.91 q^{46} -19955.6 q^{47} +9729.14 q^{48} +45999.2 q^{49} -14945.9 q^{50} +2145.61 q^{51} -1744.00 q^{52} -20089.6 q^{53} -4244.12 q^{54} +9800.20 q^{55} +43925.5 q^{56} +3249.00 q^{57} +40935.7 q^{58} -48844.4 q^{59} -402.544 q^{60} +24559.4 q^{61} +2885.73 q^{62} -20299.5 q^{63} +30606.0 q^{64} -21749.1 q^{65} -21742.3 q^{66} -59482.1 q^{67} -451.487 q^{68} -10250.7 q^{69} -34458.3 q^{70} -4635.76 q^{71} -14197.1 q^{72} -26670.8 q^{73} +17360.6 q^{74} +23105.0 q^{75} -683.668 q^{76} -103993. q^{77} +48251.6 q^{78} -68240.6 q^{79} -25530.8 q^{80} +6561.00 q^{81} +54371.8 q^{82} +2260.79 q^{83} +4271.51 q^{84} -5630.41 q^{85} -67169.0 q^{86} -63282.7 q^{87} -72730.8 q^{88} +749.444 q^{89} +11137.3 q^{90} +230786. q^{91} +2157.00 q^{92} -4461.06 q^{93} -116179. q^{94} -8525.89 q^{95} +6162.77 q^{96} +88106.3 q^{97} +267800. q^{98} +33611.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} - 36 q^{3} + 53 q^{4} - 8 q^{5} - 9 q^{6} - 142 q^{7} - 147 q^{8} + 324 q^{9} - 496 q^{10} - 714 q^{11} - 477 q^{12} - 74 q^{13} - 2790 q^{14} + 72 q^{15} - 2839 q^{16} - 3690 q^{17} + 81 q^{18}+ \cdots - 57834 q^{99}+O(q^{100}) $ 4 * q + q^2 - 36 * q^3 + 53 * q^4 - 8 * q^5 - 9 * q^6 - 142 * q^7 - 147 * q^8 + 324 * q^9 - 496 * q^10 - 714 * q^11 - 477 * q^12 - 74 * q^13 - 2790 * q^14 + 72 * q^15 - 2839 * q^16 - 3690 * q^17 + 81 * q^18 - 1444 * q^19 - 5158 * q^20 + 1278 * q^21 - 1582 * q^22 + 862 * q^23 + 1323 * q^24 + 3282 * q^25 + 870 * q^26 - 2916 * q^27 + 5488 * q^28 + 992 * q^29 + 4464 * q^30 - 7382 * q^31 - 6103 * q^32 + 6426 * q^33 + 10288 * q^34 - 10242 * q^35 + 4293 * q^36 + 19114 * q^37 - 361 * q^38 + 666 * q^39 + 19470 * q^40 - 5792 * q^41 + 25110 * q^42 - 18634 * q^43 - 5316 * q^44 - 648 * q^45 + 62256 * q^46 - 35424 * q^47 + 25551 * q^48 + 15910 * q^49 + 14309 * q^50 + 33210 * q^51 + 48170 * q^52 - 20256 * q^53 - 729 * q^54 + 23538 * q^55 + 27516 * q^56 + 12996 * q^57 + 37566 * q^58 - 76944 * q^59 + 46422 * q^60 + 11562 * q^61 - 104740 * q^62 - 11502 * q^63 - 15455 * q^64 - 157464 * q^65 + 14238 * q^66 - 110044 * q^67 - 70926 * q^68 - 7758 * q^69 + 24138 * q^70 - 71184 * q^71 - 11907 * q^72 - 138830 * q^73 + 91586 * q^74 - 29538 * q^75 - 19133 * q^76 - 76122 * q^77 - 7830 * q^78 - 59470 * q^79 + 81842 * q^80 + 26244 * q^81 + 17030 * q^82 - 191600 * q^83 - 49392 * q^84 + 23682 * q^85 + 56246 * q^86 - 8928 * q^87 - 134640 * q^88 + 158920 * q^89 - 40176 * q^90 + 267196 * q^91 + 34460 * q^92 + 66438 * q^93 + 35946 * q^94 + 2888 * q^95 + 54927 * q^96 + 47460 * q^97 + 178319 * q^98 - 57834 * q^99

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 $

$2$	5.82184	1.02917	0.514583	−	0.857441i	$-0.327947\pi$
$2$	5.82184	1.02917	0.514583	+	0.857441i	$0.327947\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	1.89382	0.0591818
$5$	23.6174	0.422482	0.211241	−	0.977434i	$-0.432250\pi$
$5$	23.6174	0.422482	0.211241	+	0.977434i	$0.432250\pi$
$6$	−52.3966	−0.594189
$7$	−250.612	−1.93311	−0.966554	−	0.256464i	$-0.917443\pi$
$7$	−250.612	−1.93311	−0.966554	+	0.256464i	$0.917443\pi$
$8$	−175.273	−0.968258
$9$	81.0000	0.333333
$10$	137.497	0.434803
$11$	414.956	1.03400	0.517000	−	0.855985i	$-0.327049\pi$
$11$	414.956	1.03400	0.517000	+	0.855985i	$0.327049\pi$
$12$	−17.0444	−0.0341686
$13$	−920.892	−1.51130	−0.755650	−	0.654976i	$-0.772679\pi$
$13$	−920.892	−1.51130	−0.755650	+	0.654976i	$0.772679\pi$
$14$	−1459.02	−1.98949
$15$	−212.557	−0.243920
$16$	−1081.02	−1.05568
$17$	−238.401	−0.200071	−0.100036	−	0.994984i	$-0.531896\pi$
$17$	−238.401	−0.200071	−0.100036	+	0.994984i	$0.531896\pi$
$18$	471.569	0.343055
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	44.7271	0.0250032
$21$	2255.50	1.11608
$22$	2415.81	1.06416
$23$	1138.97	0.448945	0.224472	−	0.974480i	$-0.427934\pi$
$23$	1138.97	0.448945	0.224472	+	0.974480i	$0.427934\pi$
$24$	1577.46	0.559024
$25$	−2567.22	−0.821509
$26$	−5361.29	−1.55538
$27$	−729.000	−0.192450
$28$	−474.613	−0.114405
$29$	7031.41	1.55256	0.776278	−	0.630391i	$-0.217105\pi$
$29$	7031.41	1.55256	0.776278	+	0.630391i	$0.217105\pi$
$30$	−1237.47	−0.251034
$31$	495.673	0.0926384	0.0463192	−	0.998927i	$-0.485251\pi$
$31$	495.673	0.0926384	0.0463192	+	0.998927i	$0.485251\pi$
$32$	−684.752	−0.118211
$33$	−3734.61	−0.596980
$34$	−1387.93	−0.205907
$35$	−5918.80	−0.816702
$36$	153.399	0.0197273
$37$	2981.99	0.358098	0.179049	−	0.983840i	$-0.442698\pi$
$37$	2981.99	0.358098	0.179049	+	0.983840i	$0.442698\pi$
$38$	−2101.68	−0.236107
$39$	8288.03	0.872549
$40$	−4139.51	−0.409071
$41$	9339.27	0.867668	0.433834	−	0.900993i	$-0.357160\pi$
$41$	9339.27	0.867668	0.433834	+	0.900993i	$0.357160\pi$
$42$	13131.2	1.14863
$43$	−11537.4	−0.951563	−0.475781	−	0.879564i	$-0.657835\pi$
$43$	−11537.4	−0.951563	−0.475781	+	0.879564i	$0.657835\pi$
$44$	785.851	0.0611940
$45$	1913.01	0.140827
$46$	6630.91	0.462039
$47$	−19955.6	−1.31771	−0.658857	−	0.752268i	$-0.728960\pi$
$47$	−19955.6	−1.31771	−0.658857	+	0.752268i	$0.728960\pi$
$48$	9729.14	0.609497
$49$	45999.2	2.73690
$50$	−14945.9	−0.845469
$51$	2145.61	0.115511
$52$	−1744.00	−0.0894414
$53$	−20089.6	−0.982385	−0.491193	−	0.871051i	$-0.663439\pi$
$53$	−20089.6	−0.982385	−0.491193	+	0.871051i	$0.663439\pi$
$54$	−4244.12	−0.198063
$55$	9800.20	0.436846
$56$	43925.5	1.87175
$57$	3249.00	0.132453
$58$	40935.7	1.59784
$59$	−48844.4	−1.82677	−0.913387	−	0.407093i	$-0.866543\pi$
$59$	−48844.4	−1.82677	−0.913387	+	0.407093i	$0.866543\pi$
$60$	−402.544	−0.0144356
$61$	24559.4	0.845069	0.422535	−	0.906347i	$-0.361140\pi$
$61$	24559.4	0.845069	0.422535	+	0.906347i	$0.361140\pi$
$62$	2885.73	0.0953402
$63$	−20299.5	−0.644369
$64$	30606.0	0.934021
$65$	−21749.1	−0.638496
$66$	−21742.3	−0.614391
$67$	−59482.1	−1.61882	−0.809411	−	0.587243i	$-0.800213\pi$
$67$	−59482.1	−1.61882	−0.809411	+	0.587243i	$0.800213\pi$
$68$	−451.487	−0.0118406
$69$	−10250.7	−0.259199
$70$	−34458.3	−0.840522
$71$	−4635.76	−0.109138	−0.0545689	−	0.998510i	$-0.517378\pi$
$71$	−4635.76	−0.109138	−0.0545689	+	0.998510i	$0.517378\pi$
$72$	−14197.1	−0.322753
$73$	−26670.8	−0.585772	−0.292886	−	0.956147i	$-0.594616\pi$
$73$	−26670.8	−0.585772	−0.292886	+	0.956147i	$0.594616\pi$
$74$	17360.6	0.368542
$75$	23105.0	0.474299
$76$	−683.668	−0.0135772
$77$	−103993.	−1.99883
$78$	48251.6	0.897997
$79$	−68240.6	−1.23020	−0.615100	−	0.788449i	$-0.710884\pi$
$79$	−68240.6	−1.23020	−0.615100	+	0.788449i	$0.710884\pi$
$80$	−25530.8	−0.446005
$81$	6561.00	0.111111
$82$	54371.8	0.892974
$83$	2260.79	0.0360217	0.0180109	−	0.999838i	$-0.494267\pi$
$83$	2260.79	0.0360217	0.0180109	+	0.999838i	$0.494267\pi$
$84$	4271.51	0.0660516
$85$	−5630.41	−0.0845265
$86$	−67169.0	−0.979315
$87$	−63282.7	−0.896368
$88$	−72730.8	−1.00118
$89$	749.444	0.0100291	0.00501457	−	0.999987i	$-0.498404\pi$
$89$	749.444	0.0100291	0.00501457	+	0.999987i	$0.498404\pi$
$90$	11137.3	0.144934
$91$	230786.	2.92150
$92$	2157.00	0.0265694
$93$	−4461.06	−0.0534848
$94$	−116179.	−1.35615
$95$	−8525.89	−0.0969239
$96$	6162.77	0.0682492
$97$	88106.3	0.950775	0.475387	−	0.879777i	$-0.342308\pi$
$97$	88106.3	0.950775	0.475387	+	0.879777i	$0.342308\pi$
$98$	267800.	2.81673
$99$	33611.5	0.344667
$100$	−4861.84	−0.0486184
$101$	−125404.	−1.22323	−0.611613	−	0.791157i	$-0.709479\pi$
$101$	−125404.	−1.22323	−0.611613	+	0.791157i	$0.709479\pi$
$102$	12491.4	0.118880
$103$	−110218.	−1.02367	−0.511835	−	0.859084i	$-0.671034\pi$
$103$	−110218.	−1.02367	−0.511835	+	0.859084i	$0.671034\pi$
$104$	161408.	1.46333
$105$	53269.2	0.471523
$106$	−116958.	−1.01104
$107$	160783.	1.35763	0.678813	−	0.734311i	$-0.262495\pi$
$107$	160783.	1.35763	0.678813	+	0.734311i	$0.262495\pi$
$108$	−1380.59	−0.0113895
$109$	−111776.	−0.901122	−0.450561	−	0.892746i	$-0.648776\pi$
$109$	−111776.	−0.901122	−0.450561	+	0.892746i	$0.648776\pi$
$110$	57055.2	0.449587
$111$	−26837.9	−0.206748
$112$	270915.	2.04074
$113$	116128.	0.855544	0.427772	−	0.903887i	$-0.359299\pi$
$113$	116128.	0.855544	0.427772	+	0.903887i	$0.359299\pi$
$114$	18915.2	0.136316
$115$	26899.6	0.189671
$116$	13316.2	0.0918830
$117$	−74592.3	−0.503766
$118$	−284364.	−1.88005
$119$	59746.0	0.386760
$120$	37255.6	0.236177
$121$	11137.7	0.0691561
$122$	142981.	0.869716
$123$	−84053.5	−0.500948
$124$	938.714	0.00548250
$125$	−134436.	−0.769554
$126$	−118181.	−0.663163
$127$	−6062.47	−0.0333534	−0.0166767	−	0.999861i	$-0.505309\pi$
$127$	−6062.47	−0.0333534	−0.0166767	+	0.999861i	$0.505309\pi$
$128$	200095.	1.07947
$129$	103837.	0.549385
$130$	−126620.	−0.657118
$131$	115145.	0.586228	0.293114	−	0.956078i	$-0.405308\pi$
$131$	115145.	0.586228	0.293114	+	0.956078i	$0.405308\pi$
$132$	−7072.66	−0.0353304
$133$	90470.8	0.443485
$134$	−346295.	−1.66604
$135$	−17217.1	−0.0813066
$136$	41785.3	0.193721
$137$	326643.	1.48687	0.743434	−	0.668810i	$-0.233196\pi$
$137$	326643.	1.48687	0.743434	+	0.668810i	$0.233196\pi$
$138$	−59678.2	−0.266758
$139$	−228617.	−1.00362	−0.501812	−	0.864977i	$-0.667333\pi$
$139$	−228617.	−1.00362	−0.501812	+	0.864977i	$0.667333\pi$
$140$	−11209.1	−0.0483339
$141$	179601.	0.760782
$142$	−26988.6	−0.112321
$143$	−382130.	−1.56268
$144$	−87562.3	−0.351893
$145$	166064.	0.655926
$146$	−155273.	−0.602857
$147$	−413992.	−1.58015
$148$	5647.34	0.0211929
$149$	225286.	0.831321	0.415661	−	0.909520i	$-0.363550\pi$
$149$	225286.	0.831321	0.415661	+	0.909520i	$0.363550\pi$
$150$	134513.	0.488132
$151$	102524.	0.365918	0.182959	−	0.983121i	$-0.441432\pi$
$151$	102524.	0.365918	0.182959	+	0.983121i	$0.441432\pi$
$152$	63273.7	0.222134
$153$	−19310.5	−0.0666905
$154$	−605430.	−2.05713
$155$	11706.5	0.0391380
$156$	15696.0	0.0516390
$157$	441599.	1.42981	0.714906	−	0.699220i	$-0.246469\pi$
$157$	441599.	1.42981	0.714906	+	0.699220i	$0.246469\pi$
$158$	−397286.	−1.26608
$159$	180806.	0.567180
$160$	−16172.1	−0.0499420
$161$	−285439.	−0.867859
$162$	38197.1	0.114352
$163$	−451041.	−1.32968	−0.664839	−	0.746986i	$-0.731500\pi$
$163$	−451041.	−1.32968	−0.664839	+	0.746986i	$0.731500\pi$
$164$	17686.9	0.0513501
$165$	−88201.8	−0.252213
$166$	13161.9	0.0370723
$167$	447874.	1.24270	0.621348	−	0.783535i	$-0.286585\pi$
$167$	447874.	1.24270	0.621348	+	0.783535i	$0.286585\pi$
$168$	−395330.	−1.08065
$169$	476750.	1.28403
$170$	−32779.4	−0.0869918
$171$	−29241.0	−0.0764719
$172$	−21849.8	−0.0563152
$173$	−381949.	−0.970266	−0.485133	−	0.874440i	$-0.661229\pi$
$173$	−381949.	−0.970266	−0.485133	+	0.874440i	$0.661229\pi$
$174$	−368421.	−0.922511
$175$	643374.	1.58807
$176$	−448574.	−1.09157
$177$	439600.	1.05469
$178$	4363.14	0.0103217
$179$	193497.	0.451379	0.225690	−	0.974199i	$-0.427536\pi$
$179$	193497.	0.451379	0.225690	+	0.974199i	$0.427536\pi$
$180$	3622.90	0.00833441
$181$	52655.8	0.119468	0.0597338	−	0.998214i	$-0.480975\pi$
$181$	52655.8	0.119468	0.0597338	+	0.998214i	$0.480975\pi$
$182$	1.34360e6	3.00671
$183$	−221034.	−0.487901
$184$	−199631.	−0.434694
$185$	70426.9	0.151290
$186$	−25971.5	−0.0550447
$187$	−98925.8	−0.206874
$188$	−37792.4	−0.0779847
$189$	182696.	0.372027
$190$	−49636.4	−0.0997507
$191$	−303888.	−0.602739	−0.301370	−	0.953507i	$-0.597444\pi$
$191$	−303888.	−0.602739	−0.301370	+	0.953507i	$0.597444\pi$
$192$	−275454.	−0.539257
$193$	134586.	0.260080	0.130040	−	0.991509i	$-0.458489\pi$
$193$	134586.	0.260080	0.130040	+	0.991509i	$0.458489\pi$
$194$	512941.	0.978505
$195$	195742.	0.368636
$196$	87114.0	0.161975
$197$	−651096.	−1.19531	−0.597654	−	0.801754i	$-0.703900\pi$
$197$	−651096.	−1.19531	−0.597654	+	0.801754i	$0.703900\pi$
$198$	195680.	0.354719
$199$	−291770.	−0.522286	−0.261143	−	0.965300i	$-0.584099\pi$
$199$	−291770.	−0.522286	−0.261143	+	0.965300i	$0.584099\pi$
$200$	449965.	0.795433
$201$	535339.	0.934627
$202$	−730080.	−1.25890
$203$	−1.76215e6	−3.00126
$204$	4063.39	0.00683617
$205$	220570.	0.366574
$206$	−641673.	−1.05353
$207$	92256.7	0.149648
$208$	995499.	1.59545
$209$	−149799.	−0.237216
$210$	310125.	0.485275
$211$	1.05072e6	1.62472	0.812361	−	0.583154i	$-0.198182\pi$
$211$	1.05072e6	1.62472	0.812361	+	0.583154i	$0.198182\pi$
$212$	−38046.1	−0.0581393
$213$	41721.8	0.0630107
$214$	936052.	1.39722
$215$	−272484.	−0.402018
$216$	127774.	0.186341
$217$	−124221.	−0.179080
$218$	−650744.	−0.927404
$219$	240037.	0.338196
$220$	18559.8	0.0258533
$221$	219541.	0.302368
$222$	−156246.	−0.212778
$223$	−81993.8	−0.110413	−0.0552063	−	0.998475i	$-0.517582\pi$
$223$	−81993.8	−0.110413	−0.0552063	+	0.998475i	$0.517582\pi$
$224$	171607.	0.228515
$225$	−207945.	−0.273836
$226$	676081.	0.880497
$227$	911528.	1.17410	0.587051	−	0.809550i	$-0.300289\pi$
$227$	911528.	1.17410	0.587051	+	0.809550i	$0.300289\pi$
$228$	6153.01	0.00783882
$229$	−660815.	−0.832705	−0.416353	−	0.909203i	$-0.636692\pi$
$229$	−660815.	−0.832705	−0.416353	+	0.909203i	$0.636692\pi$
$230$	156605.	0.195203
$231$	935935.	1.15403
$232$	−1.23242e6	−1.50327
$233$	866250.	1.04533	0.522665	−	0.852538i	$-0.324938\pi$
$233$	866250.	1.04533	0.522665	+	0.852538i	$0.324938\pi$
$234$	−434264.	−0.518459
$235$	−471301.	−0.556710
$236$	−92502.4	−0.108112
$237$	614166.	0.710256
$238$	347831.	0.398040
$239$	−385761.	−0.436841	−0.218421	−	0.975855i	$-0.570090\pi$
$239$	−385761.	−0.436841	−0.218421	+	0.975855i	$0.570090\pi$
$240$	229777.	0.257501
$241$	−1.27179e6	−1.41050	−0.705252	−	0.708957i	$-0.749166\pi$
$241$	−1.27179e6	−1.41050	−0.705252	+	0.708957i	$0.749166\pi$
$242$	64841.7	0.0711731
$243$	−59049.0	−0.0641500
$244$	46510.9	0.0500127
$245$	1.08638e6	1.15629
$246$	−489346.	−0.515559
$247$	332442.	0.346716
$248$	−86878.2	−0.0896978
$249$	−20347.1	−0.0207972
$250$	−782662.	−0.791999
$251$	−1.13475e6	−1.13689	−0.568444	−	0.822722i	$-0.692454\pi$
$251$	−1.13475e6	−1.13689	−0.568444	+	0.822722i	$0.692454\pi$
$252$	−38443.6	−0.0381349
$253$	472623.	0.464209
$254$	−35294.7	−0.0343262
$255$	50673.7	0.0488014
$256$	185531.	0.176936
$257$	−129194.	−0.122014	−0.0610071	−	0.998137i	$-0.519431\pi$
$257$	−129194.	−0.122014	−0.0610071	+	0.998137i	$0.519431\pi$
$258$	604521.	0.565408
$259$	−747320.	−0.692241
$260$	−41188.9	−0.0377873
$261$	569544.	0.517519
$262$	670355.	0.603326
$263$	913758.	0.814595	0.407298	−	0.913295i	$-0.366471\pi$
$263$	913758.	0.814595	0.407298	+	0.913295i	$0.366471\pi$
$264$	654577.	0.578031
$265$	−474465.	−0.415040
$266$	526706.	0.456420
$267$	−6744.99	−0.00579033
$268$	−112648.	−0.0958048
$269$	−460736.	−0.388214	−0.194107	−	0.980980i	$-0.562181\pi$
$269$	−460736.	−0.388214	−0.194107	+	0.980980i	$0.562181\pi$
$270$	−100235.	−0.0836780
$271$	−726289.	−0.600739	−0.300370	−	0.953823i	$-0.597110\pi$
$271$	−726289.	−0.600739	−0.300370	+	0.953823i	$0.597110\pi$
$272$	257715.	0.211211
$273$	−2.07708e6	−1.68673
$274$	1.90166e6	1.53023
$275$	−1.06528e6	−0.849441
$276$	−19413.0	−0.0153398
$277$	−588940.	−0.461181	−0.230590	−	0.973051i	$-0.574066\pi$
$277$	−588940.	−0.461181	−0.230590	+	0.973051i	$0.574066\pi$
$278$	−1.33097e6	−1.03290
$279$	40149.5	0.0308795
$280$	1.03741e6	0.790778
$281$	111634.	0.0843396	0.0421698	−	0.999110i	$-0.486573\pi$
$281$	111634.	0.0843396	0.0421698	+	0.999110i	$0.486573\pi$
$282$	1.04561e6	0.782971
$283$	−686287.	−0.509377	−0.254689	−	0.967023i	$-0.581973\pi$
$283$	−686287.	−0.509377	−0.254689	+	0.967023i	$0.581973\pi$
$284$	−8779.28	−0.00645897
$285$	76733.0	0.0559590
$286$	−2.22470e6	−1.60826
$287$	−2.34053e6	−1.67730
$288$	−55464.9	−0.0394037
$289$	−1.36302e6	−0.959971
$290$	966797.	0.675056
$291$	−792957.	−0.548930
$292$	−50509.6	−0.0346670
$293$	−1.87734e6	−1.27754	−0.638769	−	0.769399i	$-0.720556\pi$
$293$	−1.87734e6	−1.27754	−0.638769	+	0.769399i	$0.720556\pi$
$294$	−2.41020e6	−1.62624
$295$	−1.15358e6	−0.771778
$296$	−522663.	−0.346731
$297$	−302503.	−0.198993
$298$	1.31158e6	0.855567
$299$	−1.04887e6	−0.678490
$300$	43756.6	0.0280698
$301$	2.89141e6	1.83947
$302$	596879.	0.376590
$303$	1.12863e6	0.706230
$304$	390247.	0.242189
$305$	580029.	0.357026
$306$	−112422.	−0.0686356
$307$	396709.	0.240229	0.120115	−	0.992760i	$-0.461674\pi$
$307$	396709.	0.240229	0.120115	+	0.992760i	$0.461674\pi$
$308$	−196943.	−0.118295
$309$	991964.	0.591016
$310$	68153.5	0.0402795
$311$	−1.18936e6	−0.697290	−0.348645	−	0.937255i	$-0.613358\pi$
$311$	−1.18936e6	−0.697290	−0.348645	+	0.937255i	$0.613358\pi$
$312$	−1.45267e6	−0.844852
$313$	−778041.	−0.448892	−0.224446	−	0.974487i	$-0.572057\pi$
$313$	−778041.	−0.448892	−0.224446	+	0.974487i	$0.572057\pi$
$314$	2.57092e6	1.47151
$315$	−479423.	−0.272234
$316$	−129235.	−0.0728054
$317$	3.52584e6	1.97067	0.985335	−	0.170630i	$-0.0545802\pi$
$317$	3.52584e6	1.97067	0.985335	+	0.170630i	$0.0545802\pi$
$318$	1.05263e6	0.583722
$319$	2.91773e6	1.60534
$320$	722835.	0.394606
$321$	−1.44705e6	−0.783826
$322$	−1.66178e6	−0.893171
$323$	86062.7	0.0458995
$324$	12425.3	0.00657576
$325$	2.36413e6	1.24155
$326$	−2.62589e6	−1.36846
$327$	1.00599e6	0.520263
$328$	−1.63693e6	−0.840126
$329$	5.00112e6	2.54728
$330$	−513497.	−0.259569
$331$	2.21135e6	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$331$	2.21135e6	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$332$	4281.52	0.00213183
$333$	241541.	0.119366
$334$	2.60745e6	1.27894
$335$	−1.40481e6	−0.683922
$336$	−2.43824e6	−1.17822
$337$	3.52355e6	1.69008	0.845038	−	0.534706i	$-0.179578\pi$
$337$	3.52355e6	1.69008	0.845038	+	0.534706i	$0.179578\pi$
$338$	2.77556e6	1.32147
$339$	−1.04516e6	−0.493949
$340$	−10663.0	−0.00500243
$341$	205683.	0.0957881
$342$	−170236.	−0.0787023
$343$	−7.31589e6	−3.35762
$344$	2.02220e6	0.921358
$345$	−242096.	−0.109507
$346$	−2.22365e6	−0.998564
$347$	3.53103e6	1.57426	0.787131	−	0.616785i	$-0.211565\pi$
$347$	3.53103e6	1.57426	0.787131	+	0.616785i	$0.211565\pi$
$348$	−119846.	−0.0530487
$349$	−1.25322e6	−0.550760	−0.275380	−	0.961335i	$-0.588804\pi$
$349$	−1.25322e6	−0.550760	−0.275380	+	0.961335i	$0.588804\pi$
$350$	3.74562e6	1.63438
$351$	671330.	0.290850
$352$	−284142.	−0.122230
$353$	964634.	0.412027	0.206014	−	0.978549i	$-0.433951\pi$
$353$	964634.	0.412027	0.206014	+	0.978549i	$0.433951\pi$
$354$	2.55928e6	1.08545
$355$	−109485.	−0.0461087
$356$	1419.31	0.000593543 0
$357$	−537714.	−0.223296
$358$	1.12651e6	0.464544
$359$	−1.26957e6	−0.519900	−0.259950	−	0.965622i	$-0.583706\pi$
$359$	−1.26957e6	−0.519900	−0.259950	+	0.965622i	$0.583706\pi$
$360$	−335300.	−0.136357
$361$	130321.	0.0526316
$362$	306554.	0.122952
$363$	−100239.	−0.0399273
$364$	437067.	0.172900
$365$	−629896.	−0.247478
$366$	−1.28683e6	−0.502131
$367$	−1.26919e6	−0.491881	−0.245941	−	0.969285i	$-0.579097\pi$
$367$	−1.26919e6	−0.491881	−0.245941	+	0.969285i	$0.579097\pi$
$368$	−1.23125e6	−0.473942
$369$	756481.	0.289223
$370$	410014.	0.155702
$371$	5.03469e6	1.89906
$372$	−8448.42	−0.00316533
$373$	3.47150e6	1.29195	0.645975	−	0.763359i	$-0.276451\pi$
$373$	3.47150e6	1.29195	0.645975	+	0.763359i	$0.276451\pi$
$374$	−575930.	−0.212908
$375$	1.20992e6	0.444302
$376$	3.49769e6	1.27589
$377$	−6.47517e6	−2.34638
$378$	1.06363e6	0.382877
$379$	−2.84273e6	−1.01657	−0.508286	−	0.861188i	$-0.669721\pi$
$379$	−2.84273e6	−1.01657	−0.508286	+	0.861188i	$0.669721\pi$
$380$	−16146.5	−0.00573613
$381$	54562.2	0.0192566
$382$	−1.76918e6	−0.620319
$383$	−1.59297e6	−0.554894	−0.277447	−	0.960741i	$-0.589488\pi$
$383$	−1.59297e6	−0.554894	−0.277447	+	0.960741i	$0.589488\pi$
$384$	−1.80086e6	−0.623234
$385$	−2.45604e6	−0.844470
$386$	783538.	0.267665
$387$	−934531.	−0.317188
$388$	166857.	0.0562686
$389$	−5.24604e6	−1.75775	−0.878876	−	0.477050i	$-0.841706\pi$
$389$	−5.24604e6	−1.75775	−0.878876	+	0.477050i	$0.841706\pi$
$390$	1.13958e6	0.379387
$391$	−271532.	−0.0898211
$392$	−8.06243e6	−2.65003
$393$	−1.03630e6	−0.338459
$394$	−3.79058e6	−1.23017
$395$	−1.61167e6	−0.519736
$396$	63654.0	0.0203980
$397$	−361755.	−0.115196	−0.0575982	−	0.998340i	$-0.518344\pi$
$397$	−361755.	−0.115196	−0.0575982	+	0.998340i	$0.518344\pi$
$398$	−1.69864e6	−0.537518
$399$	−814237.	−0.256046
$400$	2.77520e6	0.867250
$401$	−3.07851e6	−0.956049	−0.478025	−	0.878346i	$-0.658647\pi$
$401$	−3.07851e6	−0.956049	−0.478025	+	0.878346i	$0.658647\pi$
$402$	3.11666e6	0.961886
$403$	−456461.	−0.140004
$404$	−237492.	−0.0723927
$405$	154954.	0.0469424
$406$	−1.02590e7	−3.08879
$407$	1.23739e6	0.370273
$408$	−376068.	−0.111845
$409$	4.13583e6	1.22252	0.611258	−	0.791432i	$-0.290664\pi$
$409$	4.13583e6	1.22252	0.611258	+	0.791432i	$0.290664\pi$
$410$	1.28412e6	0.377265
$411$	−2.93979e6	−0.858443
$412$	−208733.	−0.0605827
$413$	1.22410e7	3.53135
$414$	537103.	0.154013
$415$	53394.0	0.0152185
$416$	630583.	0.178652
$417$	2.05755e6	0.579443
$418$	−872107.	−0.244134
$419$	3.19560e6	0.889238	0.444619	−	0.895720i	$-0.353339\pi$
$419$	3.19560e6	0.889238	0.444619	+	0.895720i	$0.353339\pi$
$420$	100882.	0.0279056
$421$	996574.	0.274034	0.137017	−	0.990569i	$-0.456249\pi$
$421$	996574.	0.274034	0.137017	+	0.990569i	$0.456249\pi$
$422$	6.11710e6	1.67211
$423$	−1.61641e6	−0.439238
$424$	3.52117e6	0.951202
$425$	612026.	0.164361
$426$	242898.	0.0648484
$427$	−6.15486e6	−1.63361
$428$	304493.	0.0803467
$429$	3.43917e6	0.902216
$430$	−1.58636e6	−0.413743
$431$	−2.51955e6	−0.653325	−0.326662	−	0.945141i	$-0.605924\pi$
$431$	−2.51955e6	−0.653325	−0.326662	+	0.945141i	$0.605924\pi$
$432$	788060.	0.203166
$433$	−3.82101e6	−0.979395	−0.489698	−	0.871892i	$-0.662893\pi$
$433$	−3.82101e6	−0.979395	−0.489698	+	0.871892i	$0.662893\pi$
$434$	−723197.	−0.184303
$435$	−1.49457e6	−0.378699
$436$	−211684.	−0.0533300
$437$	−411169.	−0.102995
$438$	1.39746e6	0.348059
$439$	619538.	0.153429	0.0767144	−	0.997053i	$-0.475557\pi$
$439$	619538.	0.153429	0.0767144	+	0.997053i	$0.475557\pi$
$440$	−1.71771e6	−0.422979
$441$	3.72593e6	0.912302
$442$	1.27813e6	0.311187
$443$	−1.08715e6	−0.263198	−0.131599	−	0.991303i	$-0.542011\pi$
$443$	−1.08715e6	−0.263198	−0.131599	+	0.991303i	$0.542011\pi$
$444$	−50826.0	−0.0122357
$445$	17699.9	0.00423713
$446$	−477355.	−0.113633
$447$	−2.02758e6	−0.479963
$448$	−7.67021e6	−1.80556
$449$	−4.56352e6	−1.06828	−0.534139	−	0.845397i	$-0.679364\pi$
$449$	−4.56352e6	−1.06828	−0.534139	+	0.845397i	$0.679364\pi$
$450$	−1.21062e6	−0.281823
$451$	3.87539e6	0.897169
$452$	219926.	0.0506327
$453$	−922717.	−0.211263
$454$	5.30677e6	1.20834
$455$	5.45058e6	1.23428
$456$	−569463.	−0.128249
$457$	605485.	0.135617	0.0678083	−	0.997698i	$-0.478399\pi$
$457$	605485.	0.135617	0.0678083	+	0.997698i	$0.478399\pi$
$458$	−3.84716e6	−0.856992
$459$	173794.	0.0385038
$460$	50942.9	0.0112251
$461$	−7.26112e6	−1.59130	−0.795649	−	0.605759i	$-0.792870\pi$
$461$	−7.26112e6	−1.59130	−0.795649	+	0.605759i	$0.792870\pi$
$462$	5.44887e6	1.18768
$463$	1.84692e6	0.400400	0.200200	−	0.979755i	$-0.435841\pi$
$463$	1.84692e6	0.400400	0.200200	+	0.979755i	$0.435841\pi$
$464$	−7.60106e6	−1.63900
$465$	−105359.	−0.0225963
$466$	5.04317e6	1.07582
$467$	6.63166e6	1.40712	0.703559	−	0.710637i	$-0.251593\pi$
$467$	6.63166e6	1.40712	0.703559	+	0.710637i	$0.251593\pi$
$468$	−141264.	−0.0298138
$469$	1.49069e7	3.12936
$470$	−2.74384e6	−0.572947
$471$	−3.97439e6	−0.825503
$472$	8.56112e6	1.76879
$473$	−4.78752e6	−0.983916
$474$	3.57557e6	0.730971
$475$	926765.	0.188467
$476$	113148.	0.0228891
$477$	−1.62726e6	−0.327462
$478$	−2.24584e6	−0.449582
$479$	−2.39693e6	−0.477328	−0.238664	−	0.971102i	$-0.576709\pi$
$479$	−2.39693e6	−0.477328	−0.238664	+	0.971102i	$0.576709\pi$
$480$	145549.	0.0288340
$481$	−2.74609e6	−0.541193
$482$	−7.40418e6	−1.45164
$483$	2.56895e6	0.501059
$484$	21092.7	0.00409278
$485$	2.08084e6	0.401685
$486$	−343774.	−0.0660210
$487$	−2.17359e6	−0.415294	−0.207647	−	0.978204i	$-0.566580\pi$
$487$	−2.17359e6	−0.415294	−0.207647	+	0.978204i	$0.566580\pi$
$488$	−4.30460e6	−0.818245
$489$	4.05937e6	0.767690
$490$	6.32474e6	1.19002
$491$	9.09470e6	1.70249	0.851245	−	0.524768i	$-0.175848\pi$
$491$	9.09470e6	1.70249	0.851245	+	0.524768i	$0.175848\pi$
$492$	−159182.	−0.0296470
$493$	−1.67629e6	−0.310622
$494$	1.93542e6	0.356828
$495$	793816.	0.145615
$496$	−535830.	−0.0977964
$497$	1.16177e6	0.210975
$498$	−118458.	−0.0214037
$499$	8.69614e6	1.56342	0.781709	−	0.623644i	$-0.214348\pi$
$499$	8.69614e6	1.56342	0.781709	+	0.623644i	$0.214348\pi$
$500$	−254596.	−0.0455436
$501$	−4.03087e6	−0.717470
$502$	−6.60636e6	−1.17005
$503$	−2.72206e6	−0.479708	−0.239854	−	0.970809i	$-0.577100\pi$
$503$	−2.72206e6	−0.479708	−0.239854	+	0.970809i	$0.577100\pi$
$504$	3.55797e6	0.623915
$505$	−2.96171e6	−0.516790
$506$	2.75154e6	0.477748
$507$	−4.29075e6	−0.741332
$508$	−11481.2	−0.00197391
$509$	7.17949e6	1.22828	0.614142	−	0.789195i	$-0.289502\pi$
$509$	7.17949e6	1.22828	0.614142	+	0.789195i	$0.289502\pi$
$510$	295014.	0.0502247
$511$	6.68401e6	1.13236
$512$	−5.32292e6	−0.897377
$513$	263169.	0.0441511
$514$	−752149.	−0.125573
$515$	−2.60307e6	−0.432482
$516$	196648.	0.0325136
$517$	−8.28072e6	−1.36252
$518$	−4.35078e6	−0.712431
$519$	3.43755e6	0.560183
$520$	3.81204e6	0.618229
$521$	−2.71363e6	−0.437982	−0.218991	−	0.975727i	$-0.570277\pi$
$521$	−2.71363e6	−0.437982	−0.218991	+	0.975727i	$0.570277\pi$
$522$	3.31579e6	0.532612
$523$	−8.12813e6	−1.29938	−0.649690	−	0.760199i	$-0.725101\pi$
$523$	−8.12813e6	−1.29938	−0.649690	+	0.760199i	$0.725101\pi$
$524$	218063.	0.0346940
$525$	−5.79037e6	−0.916870
$526$	5.31975e6	0.838353
$527$	−118169.	−0.0185343
$528$	4.03717e6	0.630220
$529$	−5.13909e6	−0.798448
$530$	−2.76226e6	−0.427144
$531$	−3.95640e6	−0.608925
$532$	171335.	0.0262463
$533$	−8.60047e6	−1.31131
$534$	−39268.3	−0.00595921
$535$	3.79728e6	0.573572
$536$	1.04256e7	1.56744
$537$	−1.74147e6	−0.260604
$538$	−2.68233e6	−0.399537
$539$	1.90876e7	2.82996
$540$	−32606.1	−0.00481187
$541$	6.55456e6	0.962831	0.481416	−	0.876492i	$-0.340123\pi$
$541$	6.55456e6	0.962831	0.481416	+	0.876492i	$0.340123\pi$
$542$	−4.22834e6	−0.618260
$543$	−473903.	−0.0689747
$544$	163245.	0.0236507
$545$	−2.63987e6	−0.380707
$546$	−1.20924e7	−1.73593
$547$	4.13403e6	0.590752	0.295376	−	0.955381i	$-0.404555\pi$
$547$	4.13403e6	0.590752	0.295376	+	0.955381i	$0.404555\pi$
$548$	618603.	0.0879955
$549$	1.98931e6	0.281690
$550$	−6.20190e6	−0.874215
$551$	−2.53834e6	−0.356181
$552$	1.79668e6	0.250971
$553$	1.71019e7	2.37811
$554$	−3.42871e6	−0.474632
$555$	−633842.	−0.0873471
$556$	−432959.	−0.0593963
$557$	−1.13647e7	−1.55211	−0.776053	−	0.630668i	$-0.782781\pi$
$557$	−1.13647e7	−1.55211	−0.776053	+	0.630668i	$0.782781\pi$
$558$	233744.	0.0317801
$559$	1.06247e7	1.43810
$560$	6.39832e6	0.862176
$561$	890333.	0.119439
$562$	649917.	0.0867994
$563$	1.00795e7	1.34019	0.670095	−	0.742276i	$-0.266254\pi$
$563$	1.00795e7	1.34019	0.670095	+	0.742276i	$0.266254\pi$
$564$	340131.	0.0450245
$565$	2.74266e6	0.361452
$566$	−3.99545e6	−0.524233
$567$	−1.64426e6	−0.214790
$568$	812525.	0.105673
$569$	1.25833e7	1.62934	0.814671	−	0.579924i	$-0.196918\pi$
$569$	1.25833e7	1.62934	0.814671	+	0.579924i	$0.196918\pi$
$570$	446727.	0.0575911
$571$	−1.32744e7	−1.70382	−0.851909	−	0.523690i	$-0.824555\pi$
$571$	−1.32744e7	−1.70382	−0.851909	+	0.523690i	$0.824555\pi$
$572$	−723684.	−0.0924824
$573$	2.73499e6	0.347992
$574$	−1.36262e7	−1.72621
$575$	−2.92399e6	−0.368813
$576$	2.47908e6	0.311340
$577$	−7.77405e6	−0.972093	−0.486046	−	0.873933i	$-0.661561\pi$
$577$	−7.77405e6	−0.972093	−0.486046	+	0.873933i	$0.661561\pi$
$578$	−7.93530e6	−0.987970
$579$	−1.21127e6	−0.150157
$580$	314494.	0.0388189
$581$	−566580.	−0.0696339
$582$	−4.61647e6	−0.564940
$583$	−8.33631e6	−1.01579
$584$	4.67468e6	0.567178
$585$	−1.76168e6	−0.212832
$586$	−1.09296e7	−1.31480
$587$	−9.72112e6	−1.16445	−0.582225	−	0.813027i	$-0.697818\pi$
$587$	−9.72112e6	−1.16445	−0.582225	+	0.813027i	$0.697818\pi$
$588$	−784026.	−0.0935163
$589$	−178938.	−0.0212527
$590$	−6.71595e6	−0.794287
$591$	5.85986e6	0.690111
$592$	−3.22357e6	−0.378036
$593$	−9.51207e6	−1.11081	−0.555403	−	0.831581i	$-0.687436\pi$
$593$	−9.51207e6	−1.11081	−0.555403	+	0.831581i	$0.687436\pi$
$594$	−1.76112e6	−0.204797
$595$	1.41105e6	0.163399
$596$	426651.	0.0491991
$597$	2.62593e6	0.301542
$598$	−6.10635e6	−0.698279
$599$	2.79130e6	0.317863	0.158931	−	0.987290i	$-0.449195\pi$
$599$	2.79130e6	0.317863	0.158931	+	0.987290i	$0.449195\pi$
$600$	−4.04968e6	−0.459243
$601$	−1.71913e6	−0.194143	−0.0970716	−	0.995277i	$-0.530948\pi$
$601$	−1.71913e6	−0.194143	−0.0970716	+	0.995277i	$0.530948\pi$
$602$	1.68333e7	1.89312
$603$	−4.81805e6	−0.539607
$604$	194162.	0.0216557
$605$	263043.	0.0292172
$606$	6.57072e6	0.726827
$607$	−1.11303e7	−1.22612	−0.613061	−	0.790035i	$-0.710062\pi$
$607$	−1.11303e7	−1.22612	−0.613061	+	0.790035i	$0.710062\pi$
$608$	247195.	0.0271195
$609$	1.58594e7	1.73278
$610$	3.37684e6	0.367439
$611$	1.83770e7	1.99146
$612$	−36570.5	−0.00394686
$613$	998180.	0.107290	0.0536448	−	0.998560i	$-0.482916\pi$
$613$	998180.	0.107290	0.0536448	+	0.998560i	$0.482916\pi$
$614$	2.30958e6	0.247236
$615$	−1.98513e6	−0.211641
$616$	1.82272e7	1.93539
$617$	−5.14322e6	−0.543904	−0.271952	−	0.962311i	$-0.587669\pi$
$617$	−5.14322e6	−0.543904	−0.271952	+	0.962311i	$0.587669\pi$
$618$	5.77505e6	0.608254
$619$	−1.51122e7	−1.58526	−0.792632	−	0.609701i	$-0.791290\pi$
$619$	−1.51122e7	−1.58526	−0.792632	+	0.609701i	$0.791290\pi$
$620$	22170.0	0.00231626
$621$	−830310.	−0.0863995
$622$	−6.92428e6	−0.717626
$623$	−187819.	−0.0193874
$624$	−8.95949e6	−0.921132
$625$	4.84753e6	0.496387
$626$	−4.52963e6	−0.461984
$627$	1.34819e6	0.136957
$628$	836308.	0.0846189
$629$	−710908.	−0.0716451
$630$	−2.79112e6	−0.280174
$631$	−3.02455e6	−0.302404	−0.151202	−	0.988503i	$-0.548314\pi$
$631$	−3.02455e6	−0.302404	−0.151202	+	0.988503i	$0.548314\pi$
$632$	1.19608e7	1.19115
$633$	−9.45644e6	−0.938034
$634$	2.05269e7	2.02815
$635$	−143180.	−0.0140912
$636$	342414.	0.0335668
$637$	−4.23603e7	−4.13628
$638$	1.69865e7	1.65216
$639$	−375496.	−0.0363792
$640$	4.72574e6	0.456057
$641$	−6.09294e6	−0.585709	−0.292855	−	0.956157i	$-0.594605\pi$
$641$	−6.09294e6	−0.585709	−0.292855	+	0.956157i	$0.594605\pi$
$642$	−8.42446e6	−0.806686
$643$	7.43059e6	0.708755	0.354377	−	0.935103i	$-0.384693\pi$
$643$	7.43059e6	0.708755	0.354377	+	0.935103i	$0.384693\pi$
$644$	−540570.	−0.0513615
$645$	2.45236e6	0.232105
$646$	501043.	0.0472382
$647$	−1.75431e6	−0.164758	−0.0823789	−	0.996601i	$-0.526252\pi$
$647$	−1.75431e6	−0.164758	−0.0823789	+	0.996601i	$0.526252\pi$
$648$	−1.14997e6	−0.107584
$649$	−2.02683e7	−1.88888
$650$	1.37636e7	1.27776
$651$	1.11799e6	0.103392
$652$	−854189.	−0.0786928
$653$	−6.02593e6	−0.553021	−0.276510	−	0.961011i	$-0.589178\pi$
$653$	−6.02593e6	−0.553021	−0.276510	+	0.961011i	$0.589178\pi$
$654$	5.85669e6	0.535437
$655$	2.71943e6	0.247670
$656$	−1.00959e7	−0.915979
$657$	−2.16033e6	−0.195257
$658$	2.91157e7	2.62158
$659$	−1.15725e7	−1.03804	−0.519018	−	0.854763i	$-0.673702\pi$
$659$	−1.15725e7	−1.03804	−0.519018	+	0.854763i	$0.673702\pi$
$660$	−167038.	−0.0149264
$661$	−8.34628e6	−0.743001	−0.371500	−	0.928433i	$-0.621157\pi$
$661$	−8.34628e6	−0.743001	−0.371500	+	0.928433i	$0.621157\pi$
$662$	1.28741e7	1.14176
$663$	−1.97587e6	−0.174572
$664$	−396256.	−0.0348783
$665$	2.13669e6	0.187364
$666$	1.40621e6	0.122847
$667$	8.00857e6	0.697012
$668$	848192.	0.0735449
$669$	737944.	0.0637468
$670$	−8.17860e6	−0.703869
$671$	1.01911e7	0.873802
$672$	−1.54446e6	−0.131933
$673$	−5.33266e6	−0.453843	−0.226922	−	0.973913i	$-0.572866\pi$
$673$	−5.33266e6	−0.453843	−0.226922	+	0.973913i	$0.572866\pi$
$674$	2.05136e7	1.73937
$675$	1.87150e6	0.158100
$676$	902877.	0.0759909
$677$	2.14247e7	1.79656	0.898282	−	0.439418i	$-0.144815\pi$
$677$	2.14247e7	1.79656	0.898282	+	0.439418i	$0.144815\pi$
$678$	−6.08473e6	−0.508355
$679$	−2.20805e7	−1.83795
$680$	986861.	0.0818434
$681$	−8.20376e6	−0.677868
$682$	1.19745e6	0.0985818
$683$	2.01614e7	1.65375	0.826874	−	0.562388i	$-0.190117\pi$
$683$	2.01614e7	1.65375	0.826874	+	0.562388i	$0.190117\pi$
$684$	−55377.1	−0.00452575
$685$	7.71448e6	0.628174
$686$	−4.25920e7	−3.45555
$687$	5.94734e6	0.480763
$688$	1.24721e7	1.00454
$689$	1.85004e7	1.48468
$690$	−1.40945e6	−0.112700
$691$	1.54453e7	1.23056	0.615279	−	0.788310i	$-0.289043\pi$
$691$	1.54453e7	1.23056	0.615279	+	0.788310i	$0.289043\pi$
$692$	−723343.	−0.0574221
$693$	−8.42342e6	−0.666278
$694$	2.05571e7	1.62018
$695$	−5.39934e6	−0.424013
$696$	1.10918e7	0.867916
$697$	−2.22649e6	−0.173596
$698$	−7.29603e6	−0.566823
$699$	−7.79625e6	−0.603521
$700$	1.21843e6	0.0939846
$701$	−3.36415e6	−0.258571	−0.129285	−	0.991607i	$-0.541268\pi$
$701$	−3.36415e6	−0.258571	−0.129285	+	0.991607i	$0.541268\pi$
$702$	3.90838e6	0.299332
$703$	−1.07650e6	−0.0821532
$704$	1.27001e7	0.965777
$705$	4.24171e6	0.321417
$706$	5.61595e6	0.424044
$707$	3.14276e7	2.36463
$708$	832521.	0.0624183
$709$	9.46957e6	0.707481	0.353740	−	0.935344i	$-0.384910\pi$
$709$	9.46957e6	0.707481	0.353740	+	0.935344i	$0.384910\pi$
$710$	−637402.	−0.0474535
$711$	−5.52749e6	−0.410066
$712$	−131358.	−0.00971080
$713$	564557.	0.0415895
$714$	−3.13048e6	−0.229808
$715$	−9.02493e6	−0.660205
$716$	366448.	0.0267135
$717$	3.47185e6	0.252210
$718$	−7.39122e6	−0.535063
$719$	−1.17104e7	−0.844790	−0.422395	−	0.906412i	$-0.638811\pi$
$719$	−1.17104e7	−0.844790	−0.422395	+	0.906412i	$0.638811\pi$
$720$	−2.06800e6	−0.148668
$721$	2.76219e7	1.97886
$722$	758708.	0.0541666
$723$	1.14461e7	0.814355
$724$	99720.6	0.00707031
$725$	−1.80511e7	−1.27544
$726$	−583575.	−0.0410918
$727$	1.55978e7	1.09453	0.547265	−	0.836959i	$-0.315669\pi$
$727$	1.55978e7	1.09453	0.547265	+	0.836959i	$0.315669\pi$
$728$	−4.04507e7	−2.82877
$729$	531441.	0.0370370
$730$	−3.66715e6	−0.254696
$731$	2.75053e6	0.190381
$732$	−418598.	−0.0288749
$733$	172411.	0.0118523	0.00592617	−	0.999982i	$-0.498114\pi$
$733$	172411.	0.0118523	0.00592617	+	0.999982i	$0.498114\pi$
$734$	−7.38900e6	−0.506227
$735$	−9.77744e6	−0.667585
$736$	−779913.	−0.0530703
$737$	−2.46825e7	−1.67386
$738$	4.40411e6	0.297658
$739$	−1.29022e6	−0.0869064	−0.0434532	−	0.999055i	$-0.513836\pi$
$739$	−1.29022e6	−0.0869064	−0.0434532	+	0.999055i	$0.513836\pi$
$740$	133376.	0.00895359
$741$	−2.99198e6	−0.200176
$742$	2.93111e7	1.95444
$743$	−3.69707e6	−0.245689	−0.122844	−	0.992426i	$-0.539202\pi$
$743$	−3.69707e6	−0.245689	−0.122844	+	0.992426i	$0.539202\pi$
$744$	781904.	0.0517871
$745$	5.32068e6	0.351218
$746$	2.02105e7	1.32963
$747$	183124.	0.0120072
$748$	−187348.	−0.0122432
$749$	−4.02940e7	−2.62444
$750$	7.04396e6	0.457261
$751$	−188742.	−0.0122115	−0.00610575	−	0.999981i	$-0.501944\pi$
$751$	−188742.	−0.0122115	−0.00610575	+	0.999981i	$0.501944\pi$
$752$	2.15724e7	1.39108
$753$	1.02128e7	0.656382
$754$	−3.76974e7	−2.41481
$755$	2.42136e6	0.154594
$756$	345993.	0.0220172
$757$	−2.13322e7	−1.35299	−0.676497	−	0.736446i	$-0.736503\pi$
$757$	−2.13322e7	−1.35299	−0.676497	+	0.736446i	$0.736503\pi$
$758$	−1.65499e7	−1.04622
$759$	−4.25361e6	−0.268011
$760$	1.49436e6	0.0938473
$761$	−2.83469e7	−1.77437	−0.887185	−	0.461414i	$-0.847342\pi$
$761$	−2.83469e7	−1.77437	−0.887185	+	0.461414i	$0.847342\pi$
$762$	317652.	0.0198182
$763$	2.80124e7	1.74197
$764$	−575508.	−0.0356712
$765$	−456063.	−0.0281755
$766$	−9.27400e6	−0.571078
$767$	4.49804e7	2.76080
$768$	−1.66978e6	−0.102154
$769$	2.37153e7	1.44615	0.723073	−	0.690772i	$-0.242729\pi$
$769$	2.37153e7	1.44615	0.723073	+	0.690772i	$0.242729\pi$
$770$	−1.42987e7	−0.869100
$771$	1.16275e6	0.0704450
$772$	254881.	0.0153920
$773$	92234.1	0.00555192	0.00277596	−	0.999996i	$-0.499116\pi$
$773$	92234.1	0.00555192	0.00277596	+	0.999996i	$0.499116\pi$
$774$	−5.44069e6	−0.326438
$775$	−1.27250e6	−0.0761033
$776$	−1.54427e7	−0.920595
$777$	6.72588e6	0.399666
$778$	−3.05416e7	−1.80902
$779$	−3.37148e6	−0.199057
$780$	370700.	0.0218165
$781$	−1.92364e6	−0.112848
$782$	−1.58081e6	−0.0924408
$783$	−5.12589e6	−0.298789
$784$	−4.97258e7	−2.88929
$785$	1.04294e7	0.604070
$786$	−6.03320e6	−0.348330
$787$	350387.	0.0201656	0.0100828	−	0.999949i	$-0.496790\pi$
$787$	350387.	0.0201656	0.0100828	+	0.999949i	$0.496790\pi$
$788$	−1.23306e6	−0.0707404
$789$	−8.22382e6	−0.470307
$790$	−9.38288e6	−0.534895
$791$	−2.91031e7	−1.65386
$792$	−5.89119e6	−0.333726
$793$	−2.26165e7	−1.27715
$794$	−2.10608e6	−0.118556
$795$	4.27018e6	0.239623
$796$	−552559.	−0.0309098
$797$	9.25461e6	0.516074	0.258037	−	0.966135i	$-0.416924\pi$
$797$	9.25461e6	0.516074	0.258037	+	0.966135i	$0.416924\pi$
$798$	−4.74036e6	−0.263514
$799$	4.75744e6	0.263637
$800$	1.75791e6	0.0971115
$801$	60704.9	0.00334305
$802$	−1.79226e7	−0.983933
$803$	−1.10672e7	−0.605688
$804$	1.01383e6	0.0553129
$805$	−6.74134e6	−0.366654
$806$	−2.65744e6	−0.144088
$807$	4.14663e6	0.224136
$808$	2.19799e7	1.18440
$809$	−2.94050e6	−0.157961	−0.0789805	−	0.996876i	$-0.525166\pi$
$809$	−2.94050e6	−0.157961	−0.0789805	+	0.996876i	$0.525166\pi$
$810$	902117.	0.0483115
$811$	2.12110e7	1.13242	0.566211	−	0.824260i	$-0.308409\pi$
$811$	2.12110e7	1.13242	0.566211	+	0.824260i	$0.308409\pi$
$812$	−3.33719e6	−0.177620
$813$	6.53660e6	0.346837
$814$	7.20391e6	0.381072
$815$	−1.06524e7	−0.561765
$816$	−2.31943e6	−0.121943
$817$	4.16501e6	0.218303
$818$	2.40781e7	1.25817
$819$	1.86937e7	0.973835
$820$	417719.	0.0216945
$821$	−28537.6	−0.00147761	−0.000738804	−	1.00000i	$-0.500235\pi$
$821$	−28537.6	−0.00147761	−0.000738804		1.00000i	$0.500235\pi$
$822$	−1.71150e7	−0.883480
$823$	−1.60864e7	−0.827863	−0.413931	−	0.910308i	$-0.635845\pi$
$823$	−1.60864e7	−0.827863	−0.413931	+	0.910308i	$0.635845\pi$
$824$	1.93183e7	0.991177
$825$	9.58754e6	0.490425
$826$	7.12650e7	3.63434
$827$	−1.43894e7	−0.731607	−0.365803	−	0.930692i	$-0.619206\pi$
$827$	−1.43894e7	−0.731607	−0.365803	+	0.930692i	$0.619206\pi$
$828$	174717.	0.00885646
$829$	2.29442e7	1.15954	0.579771	−	0.814779i	$-0.303142\pi$
$829$	2.29442e7	1.15954	0.579771	+	0.814779i	$0.303142\pi$
$830$	310851.	0.0156624
$831$	5.30046e6	0.266263
$832$	−2.81848e7	−1.41158
$833$	−1.09662e7	−0.547577
$834$	1.19787e7	0.596343
$835$	1.05776e7	0.525016
$836$	−283692.	−0.0140389
$837$	−361345.	−0.0178283
$838$	1.86043e7	0.915173
$839$	−3.34833e7	−1.64219	−0.821095	−	0.570792i	$-0.806636\pi$
$839$	−3.34833e7	−1.64219	−0.821095	+	0.570792i	$0.806636\pi$
$840$	−9.33668e6	−0.456556
$841$	2.89295e7	1.41043
$842$	5.80189e6	0.282026
$843$	−1.00471e6	−0.0486935
$844$	1.98986e6	0.0961540
$845$	1.12596e7	0.542477
$846$	−9.41046e6	−0.452049
$847$	−2.79123e6	−0.133686
$848$	2.17172e7	1.03708
$849$	6.17658e6	0.294089
$850$	3.56312e6	0.169154
$851$	3.39640e6	0.160766
$852$	79013.5	0.00372909
$853$	−9.55702e6	−0.449728	−0.224864	−	0.974390i	$-0.572194\pi$
$853$	−9.55702e6	−0.449728	−0.224864	+	0.974390i	$0.572194\pi$
$854$	−3.58326e7	−1.68126
$855$	−690597.	−0.0323080
$856$	−2.81809e7	−1.31453
$857$	1.69429e7	0.788019	0.394009	−	0.919106i	$-0.371088\pi$
$857$	1.69429e7	0.788019	0.394009	+	0.919106i	$0.371088\pi$
$858$	2.00223e7	0.928529
$859$	−6.40368e6	−0.296106	−0.148053	−	0.988979i	$-0.547301\pi$
$859$	−6.40368e6	−0.296106	−0.148053	+	0.988979i	$0.547301\pi$
$860$	−516035.	−0.0237921
$861$	2.10648e7	0.968387
$862$	−1.46684e7	−0.672379
$863$	3.12340e6	0.142758	0.0713791	−	0.997449i	$-0.477260\pi$
$863$	3.12340e6	0.142758	0.0713791	+	0.997449i	$0.477260\pi$
$864$	499184.	0.0227497
$865$	−9.02067e6	−0.409919
$866$	−2.22453e7	−1.00796
$867$	1.22672e7	0.554240
$868$	−235253.	−0.0105983
$869$	−2.83169e7	−1.27203
$870$	−8.70117e6	−0.389744
$871$	5.47766e7	2.44652
$872$	1.95914e7	0.872518
$873$	7.13661e6	0.316925
$874$	−2.39376e6	−0.105999
$875$	3.36911e7	1.48763
$876$	454586.	0.0200150
$877$	3.31907e7	1.45719	0.728597	−	0.684943i	$-0.240173\pi$
$877$	3.31907e7	1.45719	0.728597	+	0.684943i	$0.240173\pi$
$878$	3.60685e6	0.157904
$879$	1.68961e7	0.737587
$880$	−1.05942e7	−0.461169
$881$	3.73467e7	1.62111	0.810555	−	0.585662i	$-0.199165\pi$
$881$	3.73467e7	1.62111	0.810555	+	0.585662i	$0.199165\pi$
$882$	2.16918e7	0.938909
$883$	−3.95144e7	−1.70551	−0.852753	−	0.522314i	$-0.825069\pi$
$883$	−3.95144e7	−1.70551	−0.852753	+	0.522314i	$0.825069\pi$
$884$	415771.	0.0178947
$885$	1.03822e7	0.445586
$886$	−6.32924e6	−0.270874
$887$	3.39325e7	1.44813	0.724064	−	0.689732i	$-0.242272\pi$
$887$	3.39325e7	1.44813	0.724064	+	0.689732i	$0.242272\pi$
$888$	4.70397e6	0.200185
$889$	1.51932e6	0.0644757
$890$	103046.	0.00436071
$891$	2.72253e6	0.114889
$892$	−155281.	−0.00653442
$893$	7.20399e6	0.302304
$894$	−1.18042e7	−0.493962
$895$	4.56990e6	0.190699
$896$	−5.01462e7	−2.08674
$897$	9.43983e6	0.391726
$898$	−2.65681e7	−1.09943
$899$	3.48528e6	0.143826
$900$	−393809.	−0.0162061
$901$	4.78938e6	0.196547
$902$	2.25619e7	0.923335
$903$	−2.60227e7	−1.06202
$904$	−2.03542e7	−0.828388
$905$	1.24360e6	0.0504729
$906$	−5.37191e6	−0.217424
$907$	−1.00269e7	−0.404716	−0.202358	−	0.979312i	$-0.564860\pi$
$907$	−1.00269e7	−0.404716	−0.202358	+	0.979312i	$0.564860\pi$
$908$	1.72627e6	0.0694854
$909$	−1.01577e7	−0.407742
$910$	3.17324e7	1.27028
$911$	−6.57126e6	−0.262333	−0.131166	−	0.991360i	$-0.541872\pi$
$911$	−6.57126e6	−0.262333	−0.131166	+	0.991360i	$0.541872\pi$
$912$	−3.51222e6	−0.139828
$913$	938128.	0.0372465
$914$	3.52504e6	0.139572
$915$	−5.22026e6	−0.206129
$916$	−1.25146e6	−0.0492810
$917$	−2.88566e7	−1.13324
$918$	1.01180e6	0.0396268
$919$	−7.60638e6	−0.297091	−0.148545	−	0.988906i	$-0.547459\pi$
$919$	−7.60638e6	−0.297091	−0.148545	+	0.988906i	$0.547459\pi$
$920$	−4.71478e6	−0.183650
$921$	−3.57038e6	−0.138696
$922$	−4.22731e7	−1.63771
$923$	4.26903e6	0.164940
$924$	1.77249e6	0.0682974
$925$	−7.65541e6	−0.294181
$926$	1.07524e7	0.412078
$927$	−8.92767e6	−0.341223
$928$	−4.81477e6	−0.183529
$929$	−3.40362e7	−1.29390	−0.646951	−	0.762532i	$-0.723956\pi$
$929$	−3.40362e7	−1.29390	−0.646951	+	0.762532i	$0.723956\pi$
$930$	−613381.	−0.0232554
$931$	−1.66057e7	−0.627889
$932$	1.64052e6	0.0618645
$933$	1.07043e7	0.402580
$934$	3.86085e7	1.44816
$935$	−2.33637e6	−0.0874004
$936$	1.30740e7	0.487776
$937$	1.77553e6	0.0660663	0.0330332	−	0.999454i	$-0.489483\pi$
$937$	1.77553e6	0.0660663	0.0330332	+	0.999454i	$0.489483\pi$
$938$	8.67855e7	3.22063
$939$	7.00237e6	0.259168
$940$	−892558.	−0.0329471
$941$	1.16181e7	0.427721	0.213860	−	0.976864i	$-0.431396\pi$
$941$	1.16181e7	0.427721	0.213860	+	0.976864i	$0.431396\pi$
$942$	−2.31383e7	−0.849579
$943$	1.06372e7	0.389535
$944$	5.28016e7	1.92849
$945$	4.31481e6	0.157174
$946$	−2.78722e7	−1.01261
$947$	3.38853e7	1.22782	0.613912	−	0.789374i	$-0.289595\pi$
$947$	3.38853e7	1.22782	0.613912	+	0.789374i	$0.289595\pi$
$948$	1.16312e6	0.0420342
$949$	2.45609e7	0.885277
$950$	5.39548e6	0.193964
$951$	−3.17325e7	−1.13777
$952$	−1.04719e7	−0.374483
$953$	−2.44407e7	−0.871729	−0.435864	−	0.900012i	$-0.643557\pi$
$953$	−2.44407e7	−0.871729	−0.435864	+	0.900012i	$0.643557\pi$
$954$	−9.47364e6	−0.337012
$955$	−7.17705e6	−0.254646
$956$	−730561.	−0.0258530
$957$	−2.62595e7	−0.926845
$958$	−1.39546e7	−0.491250
$959$	−8.18606e7	−2.87427
$960$	−6.50551e6	−0.227826
$961$	−2.83835e7	−0.991418
$962$	−1.59873e7	−0.556977
$963$	1.30234e7	0.452542
$964$	−2.40855e6	−0.0834762
$965$	3.17858e6	0.109879
$966$	1.49560e7	0.515672
$967$	−3.67340e7	−1.26329	−0.631644	−	0.775258i	$-0.717620\pi$
$967$	−3.67340e7	−1.26329	−0.631644	+	0.775258i	$0.717620\pi$
$968$	−1.95214e6	−0.0669609
$969$	−774564.	−0.0265001
$970$	1.21143e7	0.413400
$971$	−3.43205e7	−1.16817	−0.584085	−	0.811693i	$-0.698546\pi$
$971$	−3.43205e7	−1.16817	−0.584085	+	0.811693i	$0.698546\pi$
$972$	−111828.	−0.00379651
$973$	5.72940e7	1.94011
$974$	−1.26543e7	−0.427406
$975$	−2.12772e7	−0.716807
$976$	−2.65490e7	−0.892122
$977$	−8.40409e6	−0.281679	−0.140839	−	0.990032i	$-0.544980\pi$
$977$	−8.40409e6	−0.281679	−0.140839	+	0.990032i	$0.544980\pi$
$978$	2.36330e7	0.790081
$979$	310986.	0.0103701
$980$	2.05741e6	0.0684314
$981$	−9.05388e6	−0.300374
$982$	5.29479e7	1.75215
$983$	1.87755e7	0.619738	0.309869	−	0.950779i	$-0.399715\pi$
$983$	1.87755e7	0.619738	0.309869	+	0.950779i	$0.399715\pi$
$984$	1.47323e7	0.485047
$985$	−1.53772e7	−0.504995
$986$	−9.75910e6	−0.319682
$987$	−4.50100e7	−1.47067
$988$	629585.	0.0205193
$989$	−1.31408e7	−0.427199
$990$	4.62147e6	0.149862
$991$	−1.98114e6	−0.0640814	−0.0320407	−	0.999487i	$-0.510201\pi$
$991$	−1.98114e6	−0.0640814	−0.0320407	+	0.999487i	$0.510201\pi$
$992$	−339413.	−0.0109509
$993$	−1.99022e7	−0.640512
$994$	6.76366e6	0.217128
$995$	−6.89086e6	−0.220656
$996$	−38533.7	−0.00123081
$997$	1.55752e6	0.0496244	0.0248122	−	0.999692i	$-0.492101\pi$
$997$	1.55752e6	0.0496244	0.0248122	+	0.999692i	$0.492101\pi$
$998$	5.06275e7	1.60902
$999$	−2.17387e6	−0.0689159

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.6.a.f.1.3	✓	4
3.2	odd	2		171.6.a.j.1.2		4
4.3	odd	2		912.6.a.r.1.3		4
19.18	odd	2		1083.6.a.g.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.6.a.f.1.3	✓	4	1.1	even	1	trivial
171.6.a.j.1.2		4	3.2	odd	2
912.6.a.r.1.3		4	4.3	odd	2
1083.6.a.g.1.2		4	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	57.a (trivial)

\(f(q)\)	\(=\)	\(q+5.82184 q^{2} -9.00000 q^{3} +1.89382 q^{4} +23.6174 q^{5} -52.3966 q^{6} -250.612 q^{7} -175.273 q^{8} +81.0000 q^{9} +137.497 q^{10} +414.956 q^{11} -17.0444 q^{12} -920.892 q^{13} -1459.02 q^{14} -212.557 q^{15} -1081.02 q^{16} -238.401 q^{17} +471.569 q^{18} -361.000 q^{19} +44.7271 q^{20} +2255.50 q^{21} +2415.81 q^{22} +1138.97 q^{23} +1577.46 q^{24} -2567.22 q^{25} -5361.29 q^{26} -729.000 q^{27} -474.613 q^{28} +7031.41 q^{29} -1237.47 q^{30} +495.673 q^{31} -684.752 q^{32} -3734.61 q^{33} -1387.93 q^{34} -5918.80 q^{35} +153.399 q^{36} +2981.99 q^{37} -2101.68 q^{38} +8288.03 q^{39} -4139.51 q^{40} +9339.27 q^{41} +13131.2 q^{42} -11537.4 q^{43} +785.851 q^{44} +1913.01 q^{45} +6630.91 q^{46} -19955.6 q^{47} +9729.14 q^{48} +45999.2 q^{49} -14945.9 q^{50} +2145.61 q^{51} -1744.00 q^{52} -20089.6 q^{53} -4244.12 q^{54} +9800.20 q^{55} +43925.5 q^{56} +3249.00 q^{57} +40935.7 q^{58} -48844.4 q^{59} -402.544 q^{60} +24559.4 q^{61} +2885.73 q^{62} -20299.5 q^{63} +30606.0 q^{64} -21749.1 q^{65} -21742.3 q^{66} -59482.1 q^{67} -451.487 q^{68} -10250.7 q^{69} -34458.3 q^{70} -4635.76 q^{71} -14197.1 q^{72} -26670.8 q^{73} +17360.6 q^{74} +23105.0 q^{75} -683.668 q^{76} -103993. q^{77} +48251.6 q^{78} -68240.6 q^{79} -25530.8 q^{80} +6561.00 q^{81} +54371.8 q^{82} +2260.79 q^{83} +4271.51 q^{84} -5630.41 q^{85} -67169.0 q^{86} -63282.7 q^{87} -72730.8 q^{88} +749.444 q^{89} +11137.3 q^{90} +230786. q^{91} +2157.00 q^{92} -4461.06 q^{93} -116179. q^{94} -8525.89 q^{95} +6162.77 q^{96} +88106.3 q^{97} +267800. q^{98} +33611.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} - 36 q^{3} + 53 q^{4} - 8 q^{5} - 9 q^{6} - 142 q^{7} - 147 q^{8} + 324 q^{9} - 496 q^{10} - 714 q^{11} - 477 q^{12} - 74 q^{13} - 2790 q^{14} + 72 q^{15} - 2839 q^{16} - 3690 q^{17} + 81 q^{18}+ \cdots - 57834 q^{99}+O(q^{100}) \) 4 * q + q^2 - 36 * q^3 + 53 * q^4 - 8 * q^5 - 9 * q^6 - 142 * q^7 - 147 * q^8 + 324 * q^9 - 496 * q^10 - 714 * q^11 - 477 * q^12 - 74 * q^13 - 2790 * q^14 + 72 * q^15 - 2839 * q^16 - 3690 * q^17 + 81 * q^18 - 1444 * q^19 - 5158 * q^20 + 1278 * q^21 - 1582 * q^22 + 862 * q^23 + 1323 * q^24 + 3282 * q^25 + 870 * q^26 - 2916 * q^27 + 5488 * q^28 + 992 * q^29 + 4464 * q^30 - 7382 * q^31 - 6103 * q^32 + 6426 * q^33 + 10288 * q^34 - 10242 * q^35 + 4293 * q^36 + 19114 * q^37 - 361 * q^38 + 666 * q^39 + 19470 * q^40 - 5792 * q^41 + 25110 * q^42 - 18634 * q^43 - 5316 * q^44 - 648 * q^45 + 62256 * q^46 - 35424 * q^47 + 25551 * q^48 + 15910 * q^49 + 14309 * q^50 + 33210 * q^51 + 48170 * q^52 - 20256 * q^53 - 729 * q^54 + 23538 * q^55 + 27516 * q^56 + 12996 * q^57 + 37566 * q^58 - 76944 * q^59 + 46422 * q^60 + 11562 * q^61 - 104740 * q^62 - 11502 * q^63 - 15455 * q^64 - 157464 * q^65 + 14238 * q^66 - 110044 * q^67 - 70926 * q^68 - 7758 * q^69 + 24138 * q^70 - 71184 * q^71 - 11907 * q^72 - 138830 * q^73 + 91586 * q^74 - 29538 * q^75 - 19133 * q^76 - 76122 * q^77 - 7830 * q^78 - 59470 * q^79 + 81842 * q^80 + 26244 * q^81 + 17030 * q^82 - 191600 * q^83 - 49392 * q^84 + 23682 * q^85 + 56246 * q^86 - 8928 * q^87 - 134640 * q^88 + 158920 * q^89 - 40176 * q^90 + 267196 * q^91 + 34460 * q^92 + 66438 * q^93 + 35946 * q^94 + 2888 * q^95 + 54927 * q^96 + 47460 * q^97 + 178319 * q^98 - 57834 * q^99

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