LMFDB - Embedded newform 1045.4.a.e.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.520930 q^{2} +10.1782 q^{3} -7.72863 q^{4} +5.00000 q^{5} +5.30216 q^{6} +12.5313 q^{7} -8.19352 q^{8} +76.5967 q^{9} +O(q^{10})$	\(q+0.520930 q^{2} +10.1782 q^{3} -7.72863 q^{4} +5.00000 q^{5} +5.30216 q^{6} +12.5313 q^{7} -8.19352 q^{8} +76.5967 q^{9} +2.60465 q^{10} +11.0000 q^{11} -78.6639 q^{12} +1.68456 q^{13} +6.52792 q^{14} +50.8912 q^{15} +57.5608 q^{16} -50.1482 q^{17} +39.9016 q^{18} -19.0000 q^{19} -38.6432 q^{20} +127.547 q^{21} +5.73023 q^{22} -37.1628 q^{23} -83.3957 q^{24} +25.0000 q^{25} +0.877539 q^{26} +504.808 q^{27} -96.8497 q^{28} +198.095 q^{29} +26.5108 q^{30} +52.3364 q^{31} +95.5333 q^{32} +111.961 q^{33} -26.1237 q^{34} +62.6564 q^{35} -591.988 q^{36} +56.0823 q^{37} -9.89767 q^{38} +17.1459 q^{39} -40.9676 q^{40} +315.829 q^{41} +66.4428 q^{42} +274.440 q^{43} -85.0150 q^{44} +382.984 q^{45} -19.3592 q^{46} -195.047 q^{47} +585.868 q^{48} -185.967 q^{49} +13.0233 q^{50} -510.421 q^{51} -13.0194 q^{52} -133.916 q^{53} +262.970 q^{54} +55.0000 q^{55} -102.675 q^{56} -193.387 q^{57} +103.194 q^{58} -355.118 q^{59} -393.320 q^{60} -166.004 q^{61} +27.2636 q^{62} +959.856 q^{63} -410.720 q^{64} +8.42281 q^{65} +58.3237 q^{66} +492.547 q^{67} +387.577 q^{68} -378.253 q^{69} +32.6396 q^{70} -73.9504 q^{71} -627.597 q^{72} +7.76522 q^{73} +29.2150 q^{74} +254.456 q^{75} +146.844 q^{76} +137.844 q^{77} +8.93181 q^{78} -601.857 q^{79} +287.804 q^{80} +3069.95 q^{81} +164.525 q^{82} +448.424 q^{83} -985.760 q^{84} -250.741 q^{85} +142.964 q^{86} +2016.26 q^{87} -90.1287 q^{88} -1349.03 q^{89} +199.508 q^{90} +21.1097 q^{91} +287.218 q^{92} +532.693 q^{93} -101.606 q^{94} -95.0000 q^{95} +972.362 q^{96} -623.032 q^{97} -96.8758 q^{98} +842.564 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9}+O(q^{10}) $ 22 * q + 12 * q^2 + 21 * q^3 + 96 * q^4 + 110 * q^5 + 27 * q^6 + 93 * q^7 + 114 * q^8 + 209 * q^9	\( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9} + 60 q^{10} + 242 q^{11} + 164 q^{12} + 207 q^{13} + 129 q^{14} + 105 q^{15} + 604 q^{16} + 209 q^{17} + 869 q^{18} - 418 q^{19} + 480 q^{20} + 45 q^{21} + 132 q^{22} + 191 q^{23} + 458 q^{24} + 550 q^{25} + 363 q^{26} + 45 q^{27} + 577 q^{28} + 389 q^{29} + 135 q^{30} + 198 q^{31} + 1149 q^{32} + 231 q^{33} + 467 q^{34} + 465 q^{35} + 1315 q^{36} + 312 q^{37} - 228 q^{38} + 137 q^{39} + 570 q^{40} + 632 q^{41} - 1794 q^{42} + 1584 q^{43} + 1056 q^{44} + 1045 q^{45} + 681 q^{46} - 54 q^{47} + 2491 q^{48} + 1063 q^{49} + 300 q^{50} + 37 q^{51} + 1246 q^{52} + 343 q^{53} + 1078 q^{54} + 1210 q^{55} - 87 q^{56} - 399 q^{57} - 1424 q^{58} + 2787 q^{59} + 820 q^{60} + 2070 q^{61} - 446 q^{62} + 1696 q^{63} + 1758 q^{64} + 1035 q^{65} + 297 q^{66} + 2423 q^{67} - 524 q^{68} - 997 q^{69} + 645 q^{70} + 2538 q^{71} + 6010 q^{72} + 1397 q^{73} - 1977 q^{74} + 525 q^{75} - 1824 q^{76} + 1023 q^{77} + 202 q^{78} + 878 q^{79} + 3020 q^{80} + 2030 q^{81} - 190 q^{82} + 4932 q^{83} - 4580 q^{84} + 1045 q^{85} - 3394 q^{86} + 6009 q^{87} + 1254 q^{88} + 1812 q^{89} + 4345 q^{90} + 4349 q^{91} - 788 q^{92} - 4848 q^{93} - 2152 q^{94} - 2090 q^{95} + 4032 q^{96} + 988 q^{97} + 1366 q^{98} + 2299 q^{99}+O(q^{100}) \) 22 * q + 12 * q^2 + 21 * q^3 + 96 * q^4 + 110 * q^5 + 27 * q^6 + 93 * q^7 + 114 * q^8 + 209 * q^9 + 60 * q^10 + 242 * q^11 + 164 * q^12 + 207 * q^13 + 129 * q^14 + 105 * q^15 + 604 * q^16 + 209 * q^17 + 869 * q^18 - 418 * q^19 + 480 * q^20 + 45 * q^21 + 132 * q^22 + 191 * q^23 + 458 * q^24 + 550 * q^25 + 363 * q^26 + 45 * q^27 + 577 * q^28 + 389 * q^29 + 135 * q^30 + 198 * q^31 + 1149 * q^32 + 231 * q^33 + 467 * q^34 + 465 * q^35 + 1315 * q^36 + 312 * q^37 - 228 * q^38 + 137 * q^39 + 570 * q^40 + 632 * q^41 - 1794 * q^42 + 1584 * q^43 + 1056 * q^44 + 1045 * q^45 + 681 * q^46 - 54 * q^47 + 2491 * q^48 + 1063 * q^49 + 300 * q^50 + 37 * q^51 + 1246 * q^52 + 343 * q^53 + 1078 * q^54 + 1210 * q^55 - 87 * q^56 - 399 * q^57 - 1424 * q^58 + 2787 * q^59 + 820 * q^60 + 2070 * q^61 - 446 * q^62 + 1696 * q^63 + 1758 * q^64 + 1035 * q^65 + 297 * q^66 + 2423 * q^67 - 524 * q^68 - 997 * q^69 + 645 * q^70 + 2538 * q^71 + 6010 * q^72 + 1397 * q^73 - 1977 * q^74 + 525 * q^75 - 1824 * q^76 + 1023 * q^77 + 202 * q^78 + 878 * q^79 + 3020 * q^80 + 2030 * q^81 - 190 * q^82 + 4932 * q^83 - 4580 * q^84 + 1045 * q^85 - 3394 * q^86 + 6009 * q^87 + 1254 * q^88 + 1812 * q^89 + 4345 * q^90 + 4349 * q^91 - 788 * q^92 - 4848 * q^93 - 2152 * q^94 - 2090 * q^95 + 4032 * q^96 + 988 * q^97 + 1366 * q^98 + 2299 * 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 $

$2$	0.520930	0.184177	0.0920883	−	0.995751i	$-0.470646\pi$
$2$	0.520930	0.184177	0.0920883	+	0.995751i	$0.470646\pi$
$3$	10.1782	1.95880	0.979402	−	0.201918i	$-0.0647174\pi$
$3$	10.1782	1.95880	0.979402	+	0.201918i	$0.0647174\pi$
$4$	−7.72863	−0.966079
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	5.30216	0.360766
$7$	12.5313	0.676626	0.338313	−	0.941034i	$-0.390144\pi$
$7$	12.5313	0.676626	0.338313	+	0.941034i	$0.390144\pi$
$8$	−8.19352	−0.362106
$9$	76.5967	2.83692
$10$	2.60465	0.0823663
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−78.6639	−1.89236
$13$	1.68456	0.0359395	0.0179697	−	0.999839i	$-0.494280\pi$
$13$	1.68456	0.0359395	0.0179697	+	0.999839i	$0.494280\pi$
$14$	6.52792	0.124619
$15$	50.8912	0.876004
$16$	57.5608	0.899388
$17$	−50.1482	−0.715455	−0.357727	−	0.933826i	$-0.616448\pi$
$17$	−50.1482	−0.715455	−0.357727	+	0.933826i	$0.616448\pi$
$18$	39.9016	0.522494
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−38.6432	−0.432044
$21$	127.547	1.32538
$22$	5.73023	0.0555313
$23$	−37.1628	−0.336913	−0.168456	−	0.985709i	$-0.553878\pi$
$23$	−37.1628	−0.336913	−0.168456	+	0.985709i	$0.553878\pi$
$24$	−83.3957	−0.709294
$25$	25.0000	0.200000
$26$	0.877539	0.00661921
$27$	504.808	3.59816
$28$	−96.8497	−0.653674
$29$	198.095	1.26846	0.634231	−	0.773144i	$-0.281317\pi$
$29$	198.095	1.26846	0.634231	+	0.773144i	$0.281317\pi$
$30$	26.5108	0.161339
$31$	52.3364	0.303223	0.151611	−	0.988440i	$-0.451554\pi$
$31$	52.3364	0.303223	0.151611	+	0.988440i	$0.451554\pi$
$32$	95.5333	0.527752
$33$	111.961	0.590602
$34$	−26.1237	−0.131770
$35$	62.6564	0.302596
$36$	−591.988	−2.74069
$37$	56.0823	0.249186	0.124593	−	0.992208i	$-0.460238\pi$
$37$	56.0823	0.249186	0.124593	+	0.992208i	$0.460238\pi$
$38$	−9.89767	−0.0422530
$39$	17.1459	0.0703985
$40$	−40.9676	−0.161939
$41$	315.829	1.20303	0.601514	−	0.798862i	$-0.294564\pi$
$41$	315.829	1.20303	0.601514	+	0.798862i	$0.294564\pi$
$42$	66.4428	0.244104
$43$	274.440	0.973294	0.486647	−	0.873599i	$-0.338220\pi$
$43$	274.440	0.973294	0.486647	+	0.873599i	$0.338220\pi$
$44$	−85.0150	−0.291284
$45$	382.984	1.26871
$46$	−19.3592	−0.0620514
$47$	−195.047	−0.605330	−0.302665	−	0.953097i	$-0.597876\pi$
$47$	−195.047	−0.605330	−0.302665	+	0.953097i	$0.597876\pi$
$48$	585.868	1.76172
$49$	−185.967	−0.542178
$50$	13.0233	0.0368353
$51$	−510.421	−1.40144
$52$	−13.0194	−0.0347204
$53$	−133.916	−0.347071	−0.173535	−	0.984828i	$-0.555519\pi$
$53$	−133.916	−0.347071	−0.173535	+	0.984828i	$0.555519\pi$
$54$	262.970	0.662697
$55$	55.0000	0.134840
$56$	−102.675	−0.245010
$57$	−193.387	−0.449381
$58$	103.194	0.233621
$59$	−355.118	−0.783601	−0.391800	−	0.920050i	$-0.628148\pi$
$59$	−355.118	−0.783601	−0.391800	+	0.920050i	$0.628148\pi$
$60$	−393.320	−0.846289
$61$	−166.004	−0.348436	−0.174218	−	0.984707i	$-0.555740\pi$
$61$	−166.004	−0.348436	−0.174218	+	0.984707i	$0.555740\pi$
$62$	27.2636	0.0558465
$63$	959.856	1.91953
$64$	−410.720	−0.802188
$65$	8.42281	0.0160726
$66$	58.3237	0.108775
$67$	492.547	0.898122	0.449061	−	0.893501i	$-0.351759\pi$
$67$	492.547	0.898122	0.449061	+	0.893501i	$0.351759\pi$
$68$	387.577	0.691186
$69$	−378.253	−0.659946
$70$	32.6396	0.0557311
$71$	−73.9504	−0.123610	−0.0618049	−	0.998088i	$-0.519686\pi$
$71$	−73.9504	−0.123610	−0.0618049	+	0.998088i	$0.519686\pi$
$72$	−627.597	−1.02726
$73$	7.76522	0.0124500	0.00622500	−	0.999981i	$-0.498019\pi$
$73$	7.76522	0.0124500	0.00622500	+	0.999981i	$0.498019\pi$
$74$	29.2150	0.0458942
$75$	254.456	0.391761
$76$	146.844	0.221634
$77$	137.844	0.204010
$78$	8.93181	0.0129657
$79$	−601.857	−0.857141	−0.428571	−	0.903508i	$-0.640983\pi$
$79$	−601.857	−0.857141	−0.428571	+	0.903508i	$0.640983\pi$
$80$	287.804	0.402218
$81$	3069.95	4.21118
$82$	164.525	0.221570
$83$	448.424	0.593024	0.296512	−	0.955029i	$-0.404177\pi$
$83$	448.424	0.593024	0.296512	+	0.955029i	$0.404177\pi$
$84$	−985.760	−1.28042
$85$	−250.741	−0.319961
$86$	142.964	0.179258
$87$	2016.26	2.48467
$88$	−90.1287	−0.109179
$89$	−1349.03	−1.60670	−0.803352	−	0.595505i	$-0.796952\pi$
$89$	−1349.03	−1.60670	−0.803352	+	0.595505i	$0.796952\pi$
$90$	199.508	0.233666
$91$	21.1097	0.0243176
$92$	287.218	0.325484
$93$	532.693	0.593954
$94$	−101.606	−0.111488
$95$	−95.0000	−0.102598
$96$	972.362	1.03376
$97$	−623.032	−0.652157	−0.326079	−	0.945343i	$-0.605727\pi$
$97$	−623.032	−0.652157	−0.326079	+	0.945343i	$0.605727\pi$
$98$	−96.8758	−0.0998564
$99$	842.564	0.855363
$100$	−193.216	−0.193216
$101$	425.057	0.418760	0.209380	−	0.977834i	$-0.432855\pi$
$101$	425.057	0.418760	0.209380	+	0.977834i	$0.432855\pi$
$102$	−265.894	−0.258112
$103$	16.9512	0.0162161	0.00810803	−	0.999967i	$-0.497419\pi$
$103$	16.9512	0.0162161	0.00810803	+	0.999967i	$0.497419\pi$
$104$	−13.8025	−0.0130139
$105$	637.733	0.592727
$106$	−69.7607	−0.0639223
$107$	1140.52	1.03045	0.515224	−	0.857055i	$-0.327709\pi$
$107$	1140.52	1.03045	0.515224	+	0.857055i	$0.327709\pi$
$108$	−3901.48	−3.47611
$109$	1094.30	0.961609	0.480804	−	0.876828i	$-0.340345\pi$
$109$	1094.30	0.961609	0.480804	+	0.876828i	$0.340345\pi$
$110$	28.6512	0.0248344
$111$	570.820	0.488106
$112$	721.311	0.608549
$113$	1705.99	1.42023	0.710117	−	0.704084i	$-0.248642\pi$
$113$	1705.99	1.42023	0.710117	+	0.704084i	$0.248642\pi$
$114$	−100.741	−0.0827654
$115$	−185.814	−0.150672
$116$	−1531.01	−1.22543
$117$	129.032	0.101957
$118$	−184.992	−0.144321
$119$	−628.422	−0.484095
$120$	−416.978	−0.317206
$121$	121.000	0.0909091
$122$	−86.4764	−0.0641738
$123$	3214.58	2.35650
$124$	−404.489	−0.292937
$125$	125.000	0.0894427
$126$	500.018	0.353533
$127$	−1087.55	−0.759876	−0.379938	−	0.925012i	$-0.624055\pi$
$127$	−1087.55	−0.759876	−0.379938	+	0.925012i	$0.624055\pi$
$128$	−978.223	−0.675496
$129$	2793.31	1.90649
$130$	4.38769	0.00296020
$131$	1051.30	0.701163	0.350582	−	0.936532i	$-0.385984\pi$
$131$	1051.30	0.701163	0.350582	+	0.936532i	$0.385984\pi$
$132$	−865.303	−0.570568
$133$	−238.094	−0.155229
$134$	256.582	0.165413
$135$	2524.04	1.60915
$136$	410.890	0.259070
$137$	2062.91	1.28647	0.643235	−	0.765669i	$-0.277592\pi$
$137$	2062.91	1.28647	0.643235	+	0.765669i	$0.277592\pi$
$138$	−197.043	−0.121547
$139$	−266.803	−0.162805	−0.0814026	−	0.996681i	$-0.525940\pi$
$139$	−266.803	−0.162805	−0.0814026	+	0.996681i	$0.525940\pi$
$140$	−484.248	−0.292332
$141$	−1985.24	−1.18572
$142$	−38.5230	−0.0227660
$143$	18.5302	0.0108362
$144$	4408.97	2.55149
$145$	990.477	0.567273
$146$	4.04514	0.00229300
$147$	−1892.82	−1.06202
$148$	−433.439	−0.240733
$149$	3536.64	1.94451	0.972257	−	0.233916i	$-0.0751540\pi$
$149$	3536.64	1.94451	0.972257	+	0.233916i	$0.0751540\pi$
$150$	132.554	0.0721532
$151$	2370.73	1.27766	0.638832	−	0.769346i	$-0.279418\pi$
$151$	2370.73	1.27766	0.638832	+	0.769346i	$0.279418\pi$
$152$	155.677	0.0830728
$153$	−3841.19	−2.02969
$154$	71.8071	0.0375739
$155$	261.682	0.135605
$156$	−132.514	−0.0680105
$157$	−1221.03	−0.620691	−0.310346	−	0.950624i	$-0.600445\pi$
$157$	−1221.03	−0.620691	−0.310346	+	0.950624i	$0.600445\pi$
$158$	−313.525	−0.157865
$159$	−1363.03	−0.679844
$160$	477.666	0.236018
$161$	−465.698	−0.227964
$162$	1599.23	0.775601
$163$	−1993.92	−0.958132	−0.479066	−	0.877779i	$-0.659025\pi$
$163$	−1993.92	−0.958132	−0.479066	+	0.877779i	$0.659025\pi$
$164$	−2440.92	−1.16222
$165$	559.804	0.264125
$166$	233.598	0.109221
$167$	−2746.19	−1.27249	−0.636246	−	0.771486i	$-0.719514\pi$
$167$	−2746.19	−1.27249	−0.636246	+	0.771486i	$0.719514\pi$
$168$	−1045.05	−0.479927
$169$	−2194.16	−0.998708
$170$	−130.619	−0.0589293
$171$	−1455.34	−0.650833
$172$	−2121.04	−0.940279
$173$	−1630.25	−0.716450	−0.358225	−	0.933635i	$-0.616618\pi$
$173$	−1630.25	−0.716450	−0.358225	+	0.933635i	$0.616618\pi$
$174$	1050.33	0.457618
$175$	313.282	0.135325
$176$	633.169	0.271176
$177$	−3614.48	−1.53492
$178$	−702.749	−0.295917
$179$	−865.375	−0.361347	−0.180674	−	0.983543i	$-0.557828\pi$
$179$	−865.375	−0.361347	−0.180674	+	0.983543i	$0.557828\pi$
$180$	−2959.94	−1.22567
$181$	−3250.60	−1.33489	−0.667444	−	0.744660i	$-0.732612\pi$
$181$	−3250.60	−1.33489	−0.667444	+	0.744660i	$0.732612\pi$
$182$	10.9967	0.00447873
$183$	−1689.63	−0.682518
$184$	304.494	0.121998
$185$	280.412	0.111439
$186$	277.496	0.109392
$187$	−551.630	−0.215718
$188$	1507.45	0.584797
$189$	6325.89	2.43461
$190$	−49.4884	−0.0188961
$191$	−3872.07	−1.46687	−0.733437	−	0.679757i	$-0.762085\pi$
$191$	−3872.07	−1.46687	−0.733437	+	0.679757i	$0.762085\pi$
$192$	−4180.41	−1.57133
$193$	−2435.41	−0.908315	−0.454157	−	0.890921i	$-0.650060\pi$
$193$	−2435.41	−0.908315	−0.454157	+	0.890921i	$0.650060\pi$
$194$	−324.556	−0.120112
$195$	85.7295	0.0314831
$196$	1437.27	0.523786
$197$	−2672.19	−0.966425	−0.483213	−	0.875503i	$-0.660530\pi$
$197$	−2672.19	−0.966425	−0.483213	+	0.875503i	$0.660530\pi$
$198$	438.917	0.157538
$199$	1459.07	0.519753	0.259876	−	0.965642i	$-0.416318\pi$
$199$	1459.07	0.519753	0.259876	+	0.965642i	$0.416318\pi$
$200$	−204.838	−0.0724211
$201$	5013.26	1.75925
$202$	221.425	0.0771258
$203$	2482.39	0.858274
$204$	3944.86	1.35390
$205$	1579.14	0.538010
$206$	8.83041	0.00298662
$207$	−2846.55	−0.955793
$208$	96.9647	0.0323235
$209$	−209.000	−0.0691714
$210$	332.214	0.109166
$211$	3361.28	1.09668	0.548342	−	0.836254i	$-0.315259\pi$
$211$	3361.28	1.09668	0.548342	+	0.836254i	$0.315259\pi$
$212$	1034.99	0.335298
$213$	−752.686	−0.242128
$214$	594.130	0.189785
$215$	1372.20	0.435270
$216$	−4136.15	−1.30291
$217$	655.843	0.205168
$218$	570.056	0.177106
$219$	79.0364	0.0243871
$220$	−425.075	−0.130266
$221$	−84.4778	−0.0257131
$222$	297.357	0.0898978
$223$	6370.48	1.91300	0.956500	−	0.291731i	$-0.0942312\pi$
$223$	6370.48	1.91300	0.956500	+	0.291731i	$0.0942312\pi$
$224$	1197.15	0.357090
$225$	1914.92	0.567383
$226$	888.703	0.261574
$227$	−5380.62	−1.57323	−0.786617	−	0.617442i	$-0.788169\pi$
$227$	−5380.62	−1.57323	−0.786617	+	0.617442i	$0.788169\pi$
$228$	1494.61	0.434137
$229$	394.866	0.113945	0.0569727	−	0.998376i	$-0.481855\pi$
$229$	394.866	0.113945	0.0569727	+	0.998376i	$0.481855\pi$
$230$	−96.7962	−0.0277502
$231$	1403.01	0.399616
$232$	−1623.10	−0.459317
$233$	190.781	0.0536414	0.0268207	−	0.999640i	$-0.491462\pi$
$233$	190.781	0.0536414	0.0268207	+	0.999640i	$0.491462\pi$
$234$	67.2166	0.0187782
$235$	−975.235	−0.270712
$236$	2744.58	0.757020
$237$	−6125.85	−1.67897
$238$	−327.364	−0.0891590
$239$	−5068.06	−1.37166	−0.685828	−	0.727764i	$-0.740560\pi$
$239$	−5068.06	−1.37166	−0.685828	+	0.727764i	$0.740560\pi$
$240$	2929.34	0.787867
$241$	372.088	0.0994535	0.0497268	−	0.998763i	$-0.484165\pi$
$241$	372.088	0.0994535	0.0497268	+	0.998763i	$0.484165\pi$
$242$	63.0325	0.0167433
$243$	17616.9	4.65072
$244$	1282.98	0.336617
$245$	−929.835	−0.242469
$246$	1674.57	0.434011
$247$	−32.0067	−0.00824509
$248$	−428.820	−0.109799
$249$	4564.17	1.16162
$250$	65.1163	0.0164733
$251$	−3485.16	−0.876419	−0.438209	−	0.898873i	$-0.644387\pi$
$251$	−3485.16	−0.876419	−0.438209	+	0.898873i	$0.644387\pi$
$252$	−7418.37	−1.85442
$253$	−408.791	−0.101583
$254$	−566.537	−0.139951
$255$	−2552.11	−0.626741
$256$	2776.18	0.677777
$257$	−2839.20	−0.689121	−0.344561	−	0.938764i	$-0.611972\pi$
$257$	−2839.20	−0.689121	−0.344561	+	0.938764i	$0.611972\pi$
$258$	1455.12	0.351131
$259$	702.783	0.168606
$260$	−65.0968	−0.0155274
$261$	15173.5	3.59852
$262$	547.653	0.129138
$263$	7678.72	1.80034	0.900171	−	0.435537i	$-0.143441\pi$
$263$	7678.72	1.80034	0.900171	+	0.435537i	$0.143441\pi$
$264$	−917.352	−0.213860
$265$	−669.579	−0.155215
$266$	−124.031	−0.0285895
$267$	−13730.7	−3.14722
$268$	−3806.71	−0.867657
$269$	−3177.71	−0.720253	−0.360127	−	0.932903i	$-0.617267\pi$
$269$	−3177.71	−0.720253	−0.360127	+	0.932903i	$0.617267\pi$
$270$	1314.85	0.296367
$271$	7686.10	1.72287	0.861434	−	0.507869i	$-0.169567\pi$
$271$	7686.10	1.72287	0.861434	+	0.507869i	$0.169567\pi$
$272$	−2886.57	−0.643471
$273$	214.860	0.0476334
$274$	1074.63	0.236938
$275$	275.000	0.0603023
$276$	2923.38	0.637560
$277$	−1634.99	−0.354645	−0.177323	−	0.984153i	$-0.556744\pi$
$277$	−1634.99	−0.354645	−0.177323	+	0.984153i	$0.556744\pi$
$278$	−138.986	−0.0299849
$279$	4008.80	0.860217
$280$	−513.376	−0.109572
$281$	−1826.30	−0.387715	−0.193858	−	0.981030i	$-0.562100\pi$
$281$	−1826.30	−0.387715	−0.193858	+	0.981030i	$0.562100\pi$
$282$	−1034.17	−0.218383
$283$	950.491	0.199650	0.0998248	−	0.995005i	$-0.468172\pi$
$283$	950.491	0.199650	0.0998248	+	0.995005i	$0.468172\pi$
$284$	571.536	0.119417
$285$	−966.934	−0.200969
$286$	9.65293	0.00199577
$287$	3957.74	0.813999
$288$	7317.54	1.49719
$289$	−2398.16	−0.488124
$290$	515.969	0.104478
$291$	−6341.37	−1.27745
$292$	−60.0146	−0.0120277
$293$	−5272.85	−1.05134	−0.525671	−	0.850688i	$-0.676186\pi$
$293$	−5272.85	−1.05134	−0.525671	+	0.850688i	$0.676186\pi$
$294$	−986.026	−0.195599
$295$	−1775.59	−0.350437
$296$	−459.511	−0.0902316
$297$	5552.89	1.08489
$298$	1842.34	0.358134
$299$	−62.6031	−0.0121085
$300$	−1966.60	−0.378472
$301$	3439.08	0.658556
$302$	1234.98	0.235316
$303$	4326.34	0.820270
$304$	−1093.66	−0.206334
$305$	−830.019	−0.155825
$306$	−2000.99	−0.373821
$307$	3255.99	0.605307	0.302653	−	0.953101i	$-0.402128\pi$
$307$	3255.99	0.605307	0.302653	+	0.953101i	$0.402128\pi$
$308$	−1065.35	−0.197090
$309$	172.534	0.0317641
$310$	136.318	0.0249753
$311$	4926.02	0.898164	0.449082	−	0.893490i	$-0.351751\pi$
$311$	4926.02	0.898164	0.449082	+	0.893490i	$0.351751\pi$
$312$	−140.485	−0.0254917
$313$	2181.32	0.393914	0.196957	−	0.980412i	$-0.436894\pi$
$313$	2181.32	0.393914	0.196957	+	0.980412i	$0.436894\pi$
$314$	−636.069	−0.114317
$315$	4799.28	0.858440
$316$	4651.53	0.828066
$317$	−9023.40	−1.59875	−0.799376	−	0.600831i	$-0.794837\pi$
$317$	−9023.40	−1.59875	−0.799376	+	0.600831i	$0.794837\pi$
$318$	−710.042	−0.125211
$319$	2179.05	0.382456
$320$	−2053.60	−0.358749
$321$	11608.5	2.01845
$322$	−242.596	−0.0419856
$323$	952.816	0.164137
$324$	−23726.5	−4.06833
$325$	42.1140	0.00718790
$326$	−1038.69	−0.176465
$327$	11138.1	1.88360
$328$	−2587.75	−0.435623
$329$	−2444.19	−0.409582
$330$	291.619	0.0486457
$331$	−11264.1	−1.87049	−0.935246	−	0.353998i	$-0.884822\pi$
$331$	−11264.1	−1.87049	−0.935246	+	0.353998i	$0.884822\pi$
$332$	−3465.71	−0.572908
$333$	4295.72	0.706919
$334$	−1430.57	−0.234363
$335$	2462.73	0.401652
$336$	7341.68	1.19203
$337$	−2336.00	−0.377596	−0.188798	−	0.982016i	$-0.560459\pi$
$337$	−2336.00	−0.377596	−0.188798	+	0.982016i	$0.560459\pi$
$338$	−1143.01	−0.183939
$339$	17364.0	2.78196
$340$	1937.89	0.309108
$341$	575.701	0.0914251
$342$	−758.129	−0.119868
$343$	−6628.63	−1.04348
$344$	−2248.63	−0.352435
$345$	−1891.26	−0.295137
$346$	−849.247	−0.131953
$347$	−9151.40	−1.41577	−0.707886	−	0.706327i	$-0.750351\pi$
$347$	−9151.40	−1.41577	−0.707886	+	0.706327i	$0.750351\pi$
$348$	−15583.0	−2.40039
$349$	−11428.2	−1.75283	−0.876416	−	0.481555i	$-0.840072\pi$
$349$	−11428.2	−1.75283	−0.876416	+	0.481555i	$0.840072\pi$
$350$	163.198	0.0249237
$351$	850.380	0.129316
$352$	1050.87	0.159123
$353$	−1337.41	−0.201652	−0.100826	−	0.994904i	$-0.532149\pi$
$353$	−1337.41	−0.201652	−0.100826	+	0.994904i	$0.532149\pi$
$354$	−1882.89	−0.282696
$355$	−369.752	−0.0552800
$356$	10426.1	1.55220
$357$	−6396.23	−0.948248
$358$	−450.800	−0.0665517
$359$	−5359.05	−0.787854	−0.393927	−	0.919142i	$-0.628884\pi$
$359$	−5359.05	−0.787854	−0.393927	+	0.919142i	$0.628884\pi$
$360$	−3137.98	−0.459406
$361$	361.000	0.0526316
$362$	−1693.33	−0.245855
$363$	1231.57	0.178073
$364$	−163.149	−0.0234927
$365$	38.8261	0.00556781
$366$	−880.178	−0.125704
$367$	−1821.17	−0.259031	−0.129516	−	0.991577i	$-0.541342\pi$
$367$	−1821.17	−0.259031	−0.129516	+	0.991577i	$0.541342\pi$
$368$	−2139.12	−0.303015
$369$	24191.4	3.41289
$370$	146.075	0.0205245
$371$	−1678.14	−0.234837
$372$	−4116.99	−0.573806
$373$	−392.096	−0.0544288	−0.0272144	−	0.999630i	$-0.508664\pi$
$373$	−392.096	−0.0544288	−0.0272144	+	0.999630i	$0.508664\pi$
$374$	−287.361	−0.0397302
$375$	1272.28	0.175201
$376$	1598.12	0.219194
$377$	333.704	0.0455879
$378$	3295.35	0.448398
$379$	−5348.35	−0.724871	−0.362436	−	0.932009i	$-0.618055\pi$
$379$	−5348.35	−0.724871	−0.362436	+	0.932009i	$0.618055\pi$
$380$	734.220	0.0991176
$381$	−11069.3	−1.48845
$382$	−2017.08	−0.270164
$383$	12644.6	1.68696	0.843482	−	0.537157i	$-0.180502\pi$
$383$	12644.6	1.68696	0.843482	+	0.537157i	$0.180502\pi$
$384$	−9956.60	−1.32317
$385$	689.221	0.0912362
$386$	−1268.68	−0.167290
$387$	21021.2	2.76115
$388$	4815.18	0.630036
$389$	13575.0	1.76936	0.884678	−	0.466203i	$-0.154378\pi$
$389$	13575.0	1.76936	0.884678	+	0.466203i	$0.154378\pi$
$390$	44.6591	0.00579846
$391$	1863.65	0.241046
$392$	1523.72	0.196326
$393$	10700.4	1.37344
$394$	−1392.03	−0.177993
$395$	−3009.28	−0.383325
$396$	−6511.87	−0.826348
$397$	−7212.69	−0.911825	−0.455912	−	0.890025i	$-0.650687\pi$
$397$	−7212.69	−0.911825	−0.455912	+	0.890025i	$0.650687\pi$
$398$	760.074	0.0957263
$399$	−2423.38	−0.304063
$400$	1439.02	0.179878
$401$	−13580.4	−1.69120	−0.845600	−	0.533817i	$-0.820757\pi$
$401$	−13580.4	−1.69120	−0.845600	+	0.533817i	$0.820757\pi$
$402$	2611.56	0.324012
$403$	88.1640	0.0108977
$404$	−3285.11	−0.404556
$405$	15349.7	1.88330
$406$	1293.15	0.158074
$407$	616.905	0.0751323
$408$	4182.14	0.507468
$409$	7428.61	0.898095	0.449048	−	0.893508i	$-0.351763\pi$
$409$	7428.61	0.898095	0.449048	+	0.893508i	$0.351763\pi$
$410$	822.623	0.0990889
$411$	20996.8	2.51994
$412$	−131.010	−0.0156660
$413$	−4450.09	−0.530204
$414$	−1482.86	−0.176035
$415$	2242.12	0.265208
$416$	160.932	0.0189671
$417$	−2715.58	−0.318903
$418$	−108.874	−0.0127398
$419$	−11401.9	−1.32941	−0.664703	−	0.747108i	$-0.731442\pi$
$419$	−11401.9	−1.32941	−0.664703	+	0.747108i	$0.731442\pi$
$420$	−4928.80	−0.572621
$421$	−9777.48	−1.13189	−0.565944	−	0.824443i	$-0.691488\pi$
$421$	−9777.48	−1.13189	−0.565944	+	0.824443i	$0.691488\pi$
$422$	1750.99	0.201983
$423$	−14940.0	−1.71727
$424$	1097.24	0.125676
$425$	−1253.71	−0.143091
$426$	−392.097	−0.0445942
$427$	−2080.24	−0.235761
$428$	−8814.64	−0.995495
$429$	188.605	0.0212259
$430$	714.819	0.0801666
$431$	−3930.18	−0.439235	−0.219617	−	0.975586i	$-0.570481\pi$
$431$	−3930.18	−0.439235	−0.219617	+	0.975586i	$0.570481\pi$
$432$	29057.2	3.23614
$433$	6730.54	0.746996	0.373498	−	0.927631i	$-0.378158\pi$
$433$	6730.54	0.746996	0.373498	+	0.927631i	$0.378158\pi$
$434$	341.648	0.0377872
$435$	10081.3	1.11118
$436$	−8457.48	−0.928990
$437$	706.094	0.0772930
$438$	41.1724	0.00449154
$439$	−17094.6	−1.85850	−0.929250	−	0.369451i	$-0.879546\pi$
$439$	−17094.6	−1.85850	−0.929250	+	0.369451i	$0.879546\pi$
$440$	−450.643	−0.0488263
$441$	−14244.5	−1.53811
$442$	−44.0070	−0.00473575
$443$	−2919.19	−0.313081	−0.156540	−	0.987672i	$-0.550034\pi$
$443$	−2919.19	−0.313081	−0.156540	+	0.987672i	$0.550034\pi$
$444$	−4411.65	−0.471549
$445$	−6745.14	−0.718540
$446$	3318.58	0.352330
$447$	35996.8	3.80892
$448$	−5146.85	−0.542781
$449$	10364.6	1.08939	0.544696	−	0.838634i	$-0.316645\pi$
$449$	10364.6	1.08939	0.544696	+	0.838634i	$0.316645\pi$
$450$	997.539	0.104499
$451$	3474.11	0.362726
$452$	−13185.0	−1.37206
$453$	24129.9	2.50269
$454$	−2802.92	−0.289753
$455$	105.549	0.0108752
$456$	1584.52	0.162723
$457$	−1116.20	−0.114253	−0.0571267	−	0.998367i	$-0.518194\pi$
$457$	−1116.20	−0.114253	−0.0571267	+	0.998367i	$0.518194\pi$
$458$	205.698	0.0209861
$459$	−25315.2	−2.57432
$460$	1436.09	0.145561
$461$	2419.01	0.244391	0.122196	−	0.992506i	$-0.461006\pi$
$461$	2419.01	0.244391	0.122196	+	0.992506i	$0.461006\pi$
$462$	730.871	0.0736000
$463$	−6105.69	−0.612863	−0.306432	−	0.951893i	$-0.599135\pi$
$463$	−6105.69	−0.612863	−0.306432	+	0.951893i	$0.599135\pi$
$464$	11402.5	1.14084
$465$	2663.47	0.265624
$466$	99.3834	0.00987950
$467$	−7889.10	−0.781722	−0.390861	−	0.920450i	$-0.627823\pi$
$467$	−7889.10	−0.781722	−0.390861	+	0.920450i	$0.627823\pi$
$468$	−997.241	−0.0984989
$469$	6172.24	0.607692
$470$	−508.029	−0.0498588
$471$	−12427.9	−1.21581
$472$	2909.67	0.283746
$473$	3018.84	0.293459
$474$	−3191.14	−0.309227
$475$	−475.000	−0.0458831
$476$	4856.84	0.467674
$477$	−10257.5	−0.984610
$478$	−2640.11	−0.252627
$479$	−1079.01	−0.102925	−0.0514626	−	0.998675i	$-0.516388\pi$
$479$	−1079.01	−0.102925	−0.0514626	+	0.998675i	$0.516388\pi$
$480$	4861.81	0.462313
$481$	94.4741	0.00895561
$482$	193.832	0.0183170
$483$	−4739.99	−0.446536
$484$	−935.164	−0.0878254
$485$	−3115.16	−0.291654
$486$	9177.17	0.856553
$487$	−11956.9	−1.11256	−0.556281	−	0.830994i	$-0.687772\pi$
$487$	−11956.9	−1.11256	−0.556281	+	0.830994i	$0.687772\pi$
$488$	1360.15	0.126171
$489$	−20294.6	−1.87679
$490$	−484.379	−0.0446572
$491$	−6547.84	−0.601833	−0.300916	−	0.953651i	$-0.597292\pi$
$491$	−6547.84	−0.601833	−0.300916	+	0.953651i	$0.597292\pi$
$492$	−24844.3	−2.27656
$493$	−9934.13	−0.907527
$494$	−16.6732	−0.00151855
$495$	4212.82	0.382530
$496$	3012.53	0.272715
$497$	−926.694	−0.0836376
$498$	2377.62	0.213943
$499$	8040.02	0.721284	0.360642	−	0.932704i	$-0.382558\pi$
$499$	8040.02	0.721284	0.360642	+	0.932704i	$0.382558\pi$
$500$	−966.079	−0.0864087
$501$	−27951.4	−2.49257
$502$	−1815.52	−0.161416
$503$	−13733.0	−1.21735	−0.608674	−	0.793420i	$-0.708298\pi$
$503$	−13733.0	−1.21735	−0.608674	+	0.793420i	$0.708298\pi$
$504$	−7864.59	−0.695073
$505$	2125.29	0.187275
$506$	−212.952	−0.0187092
$507$	−22332.7	−1.95627
$508$	8405.26	0.734101
$509$	7404.84	0.644821	0.322411	−	0.946600i	$-0.395507\pi$
$509$	7404.84	0.644821	0.322411	+	0.946600i	$0.395507\pi$
$510$	−1329.47	−0.115431
$511$	97.3082	0.00842400
$512$	9271.98	0.800327
$513$	−9591.35	−0.825475
$514$	−1479.02	−0.126920
$515$	84.7562	0.00725204
$516$	−21588.5	−1.84182
$517$	−2145.52	−0.182514
$518$	366.101	0.0310532
$519$	−16593.1	−1.40338
$520$	−69.0124	−0.00581999
$521$	−12856.8	−1.08113	−0.540565	−	0.841302i	$-0.681789\pi$
$521$	−12856.8	−1.08113	−0.540565	+	0.841302i	$0.681789\pi$
$522$	7904.32	0.662763
$523$	1731.10	0.144734	0.0723669	−	0.997378i	$-0.476945\pi$
$523$	1731.10	0.144734	0.0723669	+	0.997378i	$0.476945\pi$
$524$	−8125.10	−0.677379
$525$	3188.66	0.265076
$526$	4000.07	0.331581
$527$	−2624.58	−0.216942
$528$	6444.55	0.531180
$529$	−10785.9	−0.886490
$530$	−348.804	−0.0285869
$531$	−27200.9	−2.22301
$532$	1840.14	0.149963
$533$	532.033	0.0432362
$534$	−7152.75	−0.579644
$535$	5702.59	0.460831
$536$	−4035.69	−0.325215
$537$	−8808.00	−0.707809
$538$	−1655.36	−0.132654
$539$	−2045.64	−0.163473
$540$	−19507.4	−1.55456
$541$	1712.00	0.136053	0.0680266	−	0.997684i	$-0.478330\pi$
$541$	1712.00	0.136053	0.0680266	+	0.997684i	$0.478330\pi$
$542$	4003.92	0.317312
$543$	−33085.4	−2.61479
$544$	−4790.83	−0.377583
$545$	5471.52	0.430045
$546$	111.927	0.00877296
$547$	−21135.2	−1.65206	−0.826028	−	0.563630i	$-0.809404\pi$
$547$	−21135.2	−1.65206	−0.826028	+	0.563630i	$0.809404\pi$
$548$	−15943.5	−1.24283
$549$	−12715.4	−0.988484
$550$	143.256	0.0111063
$551$	−3763.81	−0.291005
$552$	3099.22	0.238970
$553$	−7542.04	−0.579964
$554$	−851.714	−0.0653174
$555$	2854.10	0.218288
$556$	2062.02	0.157283
$557$	5268.54	0.400782	0.200391	−	0.979716i	$-0.435779\pi$
$557$	5268.54	0.400782	0.200391	+	0.979716i	$0.435779\pi$
$558$	2088.30	0.158432
$559$	462.311	0.0349797
$560$	3606.55	0.272151
$561$	−5614.63	−0.422549
$562$	−951.375	−0.0714081
$563$	18539.4	1.38782	0.693911	−	0.720061i	$-0.255886\pi$
$563$	18539.4	1.38782	0.693911	+	0.720061i	$0.255886\pi$
$564$	15343.2	1.14550
$565$	8529.97	0.635148
$566$	495.139	0.0367708
$567$	38470.4	2.84939
$568$	605.914	0.0447598
$569$	4459.84	0.328587	0.164294	−	0.986411i	$-0.447466\pi$
$569$	4459.84	0.328587	0.164294	+	0.986411i	$0.447466\pi$
$570$	−503.705	−0.0370138
$571$	2341.02	0.171574	0.0857868	−	0.996314i	$-0.472660\pi$
$571$	2341.02	0.171574	0.0857868	+	0.996314i	$0.472660\pi$
$572$	−143.213	−0.0104686
$573$	−39410.9	−2.87332
$574$	2061.70	0.149920
$575$	−929.071	−0.0673825
$576$	−31459.8	−2.27574
$577$	23376.3	1.68660	0.843301	−	0.537442i	$-0.180609\pi$
$577$	23376.3	1.68660	0.843301	+	0.537442i	$0.180609\pi$
$578$	−1249.27	−0.0899011
$579$	−24788.2	−1.77921
$580$	−7655.03	−0.548031
$581$	5619.33	0.401255
$582$	−3303.41	−0.235276
$583$	−1473.07	−0.104646
$584$	−63.6245	−0.00450822
$585$	645.160	0.0455967
$586$	−2746.79	−0.193633
$587$	−3290.06	−0.231338	−0.115669	−	0.993288i	$-0.536901\pi$
$587$	−3290.06	−0.231338	−0.115669	+	0.993288i	$0.536901\pi$
$588$	14628.9	1.02600
$589$	−994.392	−0.0695640
$590$	−924.958	−0.0645423
$591$	−27198.2	−1.89304
$592$	3228.14	0.224115
$593$	21509.5	1.48953	0.744765	−	0.667327i	$-0.232562\pi$
$593$	21509.5	1.48953	0.744765	+	0.667327i	$0.232562\pi$
$594$	2892.67	0.199811
$595$	−3142.11	−0.216494
$596$	−27333.4	−1.87855
$597$	14850.8	1.01809
$598$	−32.6118	−0.00223010
$599$	−21935.0	−1.49623	−0.748114	−	0.663570i	$-0.769040\pi$
$599$	−21935.0	−1.49623	−0.748114	+	0.663570i	$0.769040\pi$
$600$	−2084.89	−0.141859
$601$	21103.5	1.43233	0.716163	−	0.697933i	$-0.245897\pi$
$601$	21103.5	1.43233	0.716163	+	0.697933i	$0.245897\pi$
$602$	1791.52	0.121291
$603$	37727.5	2.54790
$604$	−18322.5	−1.23432
$605$	605.000	0.0406558
$606$	2253.72	0.151074
$607$	7644.76	0.511188	0.255594	−	0.966784i	$-0.417729\pi$
$607$	7644.76	0.511188	0.255594	+	0.966784i	$0.417729\pi$
$608$	−1815.13	−0.121075
$609$	25266.4	1.68119
$610$	−432.382	−0.0286994
$611$	−328.569	−0.0217553
$612$	29687.2	1.96084
$613$	−1507.52	−0.0993278	−0.0496639	−	0.998766i	$-0.515815\pi$
$613$	−1507.52	−0.0993278	−0.0496639	+	0.998766i	$0.515815\pi$
$614$	1696.14	0.111483
$615$	16072.9	1.05386
$616$	−1129.43	−0.0738733
$617$	12144.2	0.792392	0.396196	−	0.918166i	$-0.370330\pi$
$617$	12144.2	0.792392	0.396196	+	0.918166i	$0.370330\pi$
$618$	89.8781	0.00585020
$619$	25828.5	1.67712	0.838560	−	0.544810i	$-0.183398\pi$
$619$	25828.5	1.67712	0.838560	+	0.544810i	$0.183398\pi$
$620$	−2022.45	−0.131005
$621$	−18760.1	−1.21227
$622$	2566.11	0.165421
$623$	−16905.0	−1.08714
$624$	986.931	0.0633155
$625$	625.000	0.0400000
$626$	1136.31	0.0725498
$627$	−2127.25	−0.135493
$628$	9436.86	0.599637
$629$	−2812.43	−0.178281
$630$	2500.09	0.158105
$631$	2641.05	0.166622	0.0833112	−	0.996524i	$-0.473450\pi$
$631$	2641.05	0.166622	0.0833112	+	0.996524i	$0.473450\pi$
$632$	4931.32	0.310376
$633$	34212.0	2.14819
$634$	−4700.56	−0.294453
$635$	−5437.74	−0.339827
$636$	10534.3	0.656783
$637$	−313.273	−0.0194856
$638$	1135.13	0.0704394
$639$	−5664.36	−0.350671
$640$	−4891.11	−0.302091
$641$	15882.9	0.978687	0.489344	−	0.872091i	$-0.337236\pi$
$641$	15882.9	0.978687	0.489344	+	0.872091i	$0.337236\pi$
$642$	6047.20	0.371751
$643$	−20328.1	−1.24675	−0.623377	−	0.781922i	$-0.714240\pi$
$643$	−20328.1	−1.24675	−0.623377	+	0.781922i	$0.714240\pi$
$644$	3599.21	0.220231
$645$	13966.6	0.852610
$646$	496.351	0.0302301
$647$	−11913.4	−0.723899	−0.361950	−	0.932198i	$-0.617889\pi$
$647$	−11913.4	−0.723899	−0.361950	+	0.932198i	$0.617889\pi$
$648$	−25153.7	−1.52489
$649$	−3906.30	−0.236264
$650$	21.9385	0.00132384
$651$	6675.33	0.401885
$652$	15410.2	0.925631
$653$	19071.3	1.14291	0.571453	−	0.820635i	$-0.306380\pi$
$653$	19071.3	1.14291	0.571453	+	0.820635i	$0.306380\pi$
$654$	5802.17	0.346916
$655$	5256.49	0.313570
$656$	18179.3	1.08199
$657$	594.791	0.0353196
$658$	−1273.25	−0.0754354
$659$	−6056.00	−0.357979	−0.178989	−	0.983851i	$-0.557283\pi$
$659$	−6056.00	−0.357979	−0.178989	+	0.983851i	$0.557283\pi$
$660$	−4326.52	−0.255166
$661$	−430.094	−0.0253082	−0.0126541	−	0.999920i	$-0.504028\pi$
$661$	−430.094	−0.0253082	−0.0126541	+	0.999920i	$0.504028\pi$
$662$	−5867.83	−0.344501
$663$	−859.836	−0.0503669
$664$	−3674.17	−0.214737
$665$	−1190.47	−0.0694203
$666$	2237.77	0.130198
$667$	−7361.79	−0.427361
$668$	21224.3	1.22933
$669$	64840.4	3.74720
$670$	1282.91	0.0739749
$671$	−1826.04	−0.105057
$672$	12184.9	0.699471
$673$	−7283.74	−0.417188	−0.208594	−	0.978002i	$-0.566889\pi$
$673$	−7283.74	−0.417188	−0.208594	+	0.978002i	$0.566889\pi$
$674$	−1216.89	−0.0695443
$675$	12620.2	0.719632
$676$	16957.9	0.964831
$677$	−11296.1	−0.641276	−0.320638	−	0.947202i	$-0.603897\pi$
$677$	−11296.1	−0.641276	−0.320638	+	0.947202i	$0.603897\pi$
$678$	9045.44	0.512372
$679$	−7807.39	−0.441267
$680$	2054.45	0.115860
$681$	−54765.2	−3.08166
$682$	299.900	0.0168384
$683$	−12641.9	−0.708239	−0.354119	−	0.935200i	$-0.615219\pi$
$683$	−12641.9	−0.708239	−0.354119	+	0.935200i	$0.615219\pi$
$684$	11247.8	0.628756
$685$	10314.6	0.575327
$686$	−3453.06	−0.192184
$687$	4019.05	0.223197
$688$	15797.0	0.875369
$689$	−225.589	−0.0124735
$690$	−985.216	−0.0543573
$691$	18050.4	0.993732	0.496866	−	0.867827i	$-0.334484\pi$
$691$	18050.4	0.993732	0.496866	+	0.867827i	$0.334484\pi$
$692$	12599.6	0.692147
$693$	10558.4	0.578760
$694$	−4767.24	−0.260752
$695$	−1334.01	−0.0728087
$696$	−16520.3	−0.899713
$697$	−15838.2	−0.860712
$698$	−5953.30	−0.322830
$699$	1941.81	0.105073
$700$	−2421.24	−0.130735
$701$	5455.07	0.293916	0.146958	−	0.989143i	$-0.453052\pi$
$701$	5455.07	0.293916	0.146958	+	0.989143i	$0.453052\pi$
$702$	442.989	0.0238170
$703$	−1065.56	−0.0571671
$704$	−4517.92	−0.241869
$705$	−9926.18	−0.530272
$706$	−696.696	−0.0371395
$707$	5326.51	0.283344
$708$	27935.0	1.48285
$709$	−29298.0	−1.55192	−0.775959	−	0.630783i	$-0.782734\pi$
$709$	−29298.0	−1.55192	−0.775959	+	0.630783i	$0.782734\pi$
$710$	−192.615	−0.0101813
$711$	−46100.3	−2.43164
$712$	11053.3	0.581797
$713$	−1944.97	−0.102160
$714$	−3331.99	−0.174645
$715$	92.6509	0.00484608
$716$	6688.17	0.349090
$717$	−51584.0	−2.68681
$718$	−2791.69	−0.145104
$719$	14796.3	0.767465	0.383732	−	0.923444i	$-0.374639\pi$
$719$	14796.3	0.767465	0.383732	+	0.923444i	$0.374639\pi$
$720$	22044.9	1.14106
$721$	212.421	0.0109722
$722$	188.056	0.00969351
$723$	3787.21	0.194810
$724$	25122.7	1.28961
$725$	4952.39	0.253692
$726$	641.561	0.0327969
$727$	−5015.08	−0.255845	−0.127922	−	0.991784i	$-0.540831\pi$
$727$	−5015.08	−0.255845	−0.127922	+	0.991784i	$0.540831\pi$
$728$	−172.963	−0.00880554
$729$	96420.5	4.89867
$730$	20.2257	0.00102546
$731$	−13762.7	−0.696348
$732$	13058.5	0.659367
$733$	−29603.6	−1.49173	−0.745863	−	0.666099i	$-0.767963\pi$
$733$	−29603.6	−1.49173	−0.745863	+	0.666099i	$0.767963\pi$
$734$	−948.704	−0.0477075
$735$	−9464.09	−0.474950
$736$	−3550.29	−0.177806
$737$	5418.01	0.270794
$738$	12602.0	0.628574
$739$	26583.1	1.32324	0.661621	−	0.749838i	$-0.269869\pi$
$739$	26583.1	1.32324	0.661621	+	0.749838i	$0.269869\pi$
$740$	−2167.20	−0.107659
$741$	−325.772	−0.0161505
$742$	−874.192	−0.0432515
$743$	3324.48	0.164150	0.0820750	−	0.996626i	$-0.473845\pi$
$743$	3324.48	0.164150	0.0820750	+	0.996626i	$0.473845\pi$
$744$	−4364.63	−0.215074
$745$	17683.2	0.869613
$746$	−204.255	−0.0100245
$747$	34347.8	1.68236
$748$	4263.35	0.208400
$749$	14292.2	0.697228
$750$	662.769	0.0322679
$751$	−11185.1	−0.543475	−0.271737	−	0.962371i	$-0.587598\pi$
$751$	−11185.1	−0.543475	−0.271737	+	0.962371i	$0.587598\pi$
$752$	−11227.1	−0.544426
$753$	−35472.8	−1.71673
$754$	173.836	0.00839622
$755$	11853.6	0.571389
$756$	−48890.5	−2.35202
$757$	−14346.3	−0.688806	−0.344403	−	0.938822i	$-0.611919\pi$
$757$	−14346.3	−0.688806	−0.344403	+	0.938822i	$0.611919\pi$
$758$	−2786.11	−0.133504
$759$	−4160.78	−0.198981
$760$	778.384	0.0371513
$761$	17688.8	0.842599	0.421300	−	0.906922i	$-0.361574\pi$
$761$	17688.8	0.842599	0.421300	+	0.906922i	$0.361574\pi$
$762$	−5766.35	−0.274138
$763$	13713.0	0.650649
$764$	29925.8	1.41712
$765$	−19206.0	−0.907703
$766$	6586.94	0.310699
$767$	−598.218	−0.0281622
$768$	28256.6	1.32763
$769$	27253.4	1.27800	0.639001	−	0.769206i	$-0.279348\pi$
$769$	27253.4	1.27800	0.639001	+	0.769206i	$0.279348\pi$
$770$	359.036	0.0168036
$771$	−28898.0	−1.34985
$772$	18822.4	0.877504
$773$	14672.5	0.682709	0.341354	−	0.939935i	$-0.389114\pi$
$773$	14672.5	0.682709	0.341354	+	0.939935i	$0.389114\pi$
$774$	10950.6	0.508540
$775$	1308.41	0.0606445
$776$	5104.82	0.236150
$777$	7153.10	0.330265
$778$	7071.62	0.325874
$779$	−6000.74	−0.275993
$780$	−662.571	−0.0304152
$781$	−813.455	−0.0372698
$782$	970.832	0.0443950
$783$	100000.	4.56413
$784$	−10704.4	−0.487628
$785$	−6105.13	−0.277582
$786$	5574.15	0.252956
$787$	21544.6	0.975835	0.487917	−	0.872890i	$-0.337757\pi$
$787$	21544.6	0.975835	0.487917	+	0.872890i	$0.337757\pi$
$788$	20652.4	0.933643
$789$	78155.9	3.52652
$790$	−1567.63	−0.0705995
$791$	21378.3	0.960967
$792$	−6903.56	−0.309732
$793$	−279.644	−0.0125226
$794$	−3757.31	−0.167937
$795$	−6815.14	−0.304035
$796$	−11276.6	−0.502122
$797$	34577.0	1.53674	0.768369	−	0.640007i	$-0.221068\pi$
$797$	34577.0	1.53674	0.768369	+	0.640007i	$0.221068\pi$
$798$	−1262.41	−0.0560012
$799$	9781.26	0.433086
$800$	2388.33	0.105550
$801$	−103331.	−4.55808
$802$	−7074.42	−0.311479
$803$	85.4175	0.00375382
$804$	−38745.7	−1.69957
$805$	−2328.49	−0.101948
$806$	45.9273	0.00200710
$807$	−32343.5	−1.41084
$808$	−3482.72	−0.151636
$809$	−41395.3	−1.79899	−0.899495	−	0.436932i	$-0.856065\pi$
$809$	−41395.3	−1.79899	−0.899495	+	0.436932i	$0.856065\pi$
$810$	7996.15	0.346859
$811$	35100.9	1.51980	0.759900	−	0.650040i	$-0.225248\pi$
$811$	35100.9	1.51980	0.759900	+	0.650040i	$0.225248\pi$
$812$	−19185.5	−0.829161
$813$	78231.1	3.37476
$814$	321.365	0.0138376
$815$	−9969.58	−0.428490
$816$	−29380.2	−1.26043
$817$	−5214.35	−0.223289
$818$	3869.79	0.165408
$819$	1616.94	0.0689870
$820$	−12204.6	−0.519760
$821$	3592.25	0.152704	0.0763521	−	0.997081i	$-0.475673\pi$
$821$	3592.25	0.152704	0.0763521	+	0.997081i	$0.475673\pi$
$822$	10937.9	0.464115
$823$	8017.53	0.339579	0.169790	−	0.985480i	$-0.445691\pi$
$823$	8017.53	0.339579	0.169790	+	0.985480i	$0.445691\pi$
$824$	−138.890	−0.00587193
$825$	2799.02	0.118120
$826$	−2318.18	−0.0976512
$827$	35157.5	1.47829	0.739145	−	0.673547i	$-0.235230\pi$
$827$	35157.5	1.47829	0.739145	+	0.673547i	$0.235230\pi$
$828$	22000.0	0.923371
$829$	18396.3	0.770724	0.385362	−	0.922765i	$-0.374077\pi$
$829$	18396.3	0.770724	0.385362	+	0.922765i	$0.374077\pi$
$830$	1167.99	0.0488451
$831$	−16641.3	−0.694681
$832$	−691.884	−0.0288302
$833$	9325.91	0.387904
$834$	−1414.63	−0.0587346
$835$	−13730.9	−0.569076
$836$	1615.28	0.0668251
$837$	26419.9	1.09104
$838$	−5939.61	−0.244845
$839$	−24368.5	−1.00274	−0.501368	−	0.865234i	$-0.667170\pi$
$839$	−24368.5	−1.00274	−0.501368	+	0.865234i	$0.667170\pi$
$840$	−5225.27	−0.214630
$841$	14852.8	0.608996
$842$	−5093.38	−0.208467
$843$	−18588.5	−0.759459
$844$	−25978.1	−1.05948
$845$	−10970.8	−0.446636
$846$	−7782.67	−0.316281
$847$	1516.29	0.0615114
$848$	−7708.30	−0.312151
$849$	9674.34	0.391075
$850$	−653.093	−0.0263540
$851$	−2084.18	−0.0839538
$852$	5817.23	0.233914
$853$	−29967.2	−1.20288	−0.601440	−	0.798918i	$-0.705406\pi$
$853$	−29967.2	−1.20288	−0.601440	+	0.798918i	$0.705406\pi$
$854$	−1083.66	−0.0434216
$855$	−7276.69	−0.291062
$856$	−9344.85	−0.373131
$857$	7525.89	0.299976	0.149988	−	0.988688i	$-0.452076\pi$
$857$	7525.89	0.299976	0.149988	+	0.988688i	$0.452076\pi$
$858$	98.2499	0.00390932
$859$	−32403.5	−1.28707	−0.643535	−	0.765417i	$-0.722533\pi$
$859$	−32403.5	−1.28707	−0.643535	+	0.765417i	$0.722533\pi$
$860$	−10605.2	−0.420506
$861$	40282.8	1.59447
$862$	−2047.35	−0.0808967
$863$	−19810.6	−0.781414	−0.390707	−	0.920515i	$-0.627769\pi$
$863$	−19810.6	−0.781414	−0.390707	+	0.920515i	$0.627769\pi$
$864$	48226.0	1.89894
$865$	−8151.26	−0.320406
$866$	3506.14	0.137579
$867$	−24409.0	−0.956141
$868$	−5068.77	−0.198209
$869$	−6620.42	−0.258438
$870$	5251.66	0.204653
$871$	829.725	0.0322780
$872$	−8966.20	−0.348204
$873$	−47722.2	−1.85012
$874$	367.826	0.0142356
$875$	1566.41	0.0605192
$876$	−610.843	−0.0235599
$877$	−16480.3	−0.634548	−0.317274	−	0.948334i	$-0.602768\pi$
$877$	−16480.3	−0.634548	−0.317274	+	0.948334i	$0.602768\pi$
$878$	−8905.10	−0.342292
$879$	−53668.4	−2.05938
$880$	3165.84	0.121273
$881$	219.786	0.00840496	0.00420248	−	0.999991i	$-0.498662\pi$
$881$	219.786	0.00840496	0.00420248	+	0.999991i	$0.498662\pi$
$882$	−7420.37	−0.283284
$883$	−720.008	−0.0274408	−0.0137204	−	0.999906i	$-0.504367\pi$
$883$	−720.008	−0.0274408	−0.0137204	+	0.999906i	$0.504367\pi$
$884$	652.898	0.0248409
$885$	−18072.4	−0.686437
$886$	−1520.69	−0.0576621
$887$	47938.4	1.81467	0.907336	−	0.420406i	$-0.138112\pi$
$887$	47938.4	1.81467	0.907336	+	0.420406i	$0.138112\pi$
$888$	−4677.02	−0.176746
$889$	−13628.4	−0.514152
$890$	−3513.74	−0.132338
$891$	33769.4	1.26972
$892$	−49235.1	−1.84811
$893$	3705.89	0.138872
$894$	18751.8	0.701514
$895$	−4326.88	−0.161599
$896$	−12258.4	−0.457058
$897$	−637.190	−0.0237181
$898$	5399.24	0.200640
$899$	10367.6	0.384626
$900$	−14799.7	−0.548137
$901$	6715.64	0.248313
$902$	1809.77	0.0668057
$903$	35003.8	1.28998
$904$	−13978.1	−0.514275
$905$	−16253.0	−0.596980
$906$	12570.0	0.460938
$907$	−32243.3	−1.18040	−0.590198	−	0.807258i	$-0.700950\pi$
$907$	−32243.3	−1.18040	−0.590198	+	0.807258i	$0.700950\pi$
$908$	41584.8	1.51987
$909$	32558.0	1.18799
$910$	54.9835	0.00200295
$911$	21746.2	0.790872	0.395436	−	0.918494i	$-0.370594\pi$
$911$	21746.2	0.790872	0.395436	+	0.918494i	$0.370594\pi$
$912$	−11131.5	−0.404167
$913$	4932.67	0.178803
$914$	−581.464	−0.0210428
$915$	−8448.14	−0.305232
$916$	−3051.78	−0.110080
$917$	13174.1	0.474425
$918$	−13187.5	−0.474130
$919$	22827.6	0.819383	0.409692	−	0.912224i	$-0.365636\pi$
$919$	22827.6	0.819383	0.409692	+	0.912224i	$0.365636\pi$
$920$	1522.47	0.0545591
$921$	33140.3	1.18568
$922$	1260.13	0.0450111
$923$	−124.574	−0.00444248
$924$	−10843.4	−0.386061
$925$	1402.06	0.0498372
$926$	−3180.64	−0.112875
$927$	1298.41	0.0460036
$928$	18924.7	0.669433
$929$	31035.8	1.09607	0.548036	−	0.836455i	$-0.315376\pi$
$929$	31035.8	1.09607	0.548036	+	0.836455i	$0.315376\pi$
$930$	1387.48	0.0489218
$931$	3533.37	0.124384
$932$	−1474.47	−0.0518219
$933$	50138.3	1.75933
$934$	−4109.67	−0.143975
$935$	−2758.15	−0.0964719
$936$	−1057.23	−0.0369193
$937$	−19647.7	−0.685018	−0.342509	−	0.939515i	$-0.611277\pi$
$937$	−19647.7	−0.685018	−0.342509	+	0.939515i	$0.611277\pi$
$938$	3215.31	0.111923
$939$	22202.0	0.771602
$940$	7537.23	0.261529
$941$	28336.2	0.981652	0.490826	−	0.871258i	$-0.336695\pi$
$941$	28336.2	0.981652	0.490826	+	0.871258i	$0.336695\pi$
$942$	−6474.07	−0.223924
$943$	−11737.1	−0.405315
$944$	−20440.9	−0.704761
$945$	31629.5	1.08879
$946$	1572.60	0.0540483
$947$	27436.9	0.941479	0.470740	−	0.882272i	$-0.343987\pi$
$947$	27436.9	0.941479	0.470740	+	0.882272i	$0.343987\pi$
$948$	47344.4	1.62202
$949$	13.0810	0.000447447 0
$950$	−247.442	−0.00845060
$951$	−91842.4	−3.13164
$952$	5148.98	0.175294
$953$	9541.01	0.324306	0.162153	−	0.986766i	$-0.448156\pi$
$953$	9541.01	0.324306	0.162153	+	0.986766i	$0.448156\pi$
$954$	−5343.45	−0.181342
$955$	−19360.3	−0.656006
$956$	39169.2	1.32513
$957$	22178.9	0.749156
$958$	−562.088	−0.0189564
$959$	25850.9	0.870459
$960$	−20902.1	−0.702720
$961$	−27051.9	−0.908056
$962$	49.2144	0.00164941
$963$	87360.0	2.92330
$964$	−2875.73	−0.0960800
$965$	−12177.1	−0.406211
$966$	−2469.20	−0.0822416
$967$	10357.4	0.344438	0.172219	−	0.985059i	$-0.444906\pi$
$967$	10357.4	0.344438	0.172219	+	0.985059i	$0.444906\pi$
$968$	−991.416	−0.0329187
$969$	9698.00	0.321512
$970$	−1622.78	−0.0537158
$971$	−12983.7	−0.429110	−0.214555	−	0.976712i	$-0.568830\pi$
$971$	−12983.7	−0.429110	−0.214555	+	0.976712i	$0.568830\pi$
$972$	−136154.	−4.49296
$973$	−3343.38	−0.110158
$974$	−6228.70	−0.204908
$975$	428.647	0.0140797
$976$	−9555.31	−0.313379
$977$	−8486.82	−0.277909	−0.138955	−	0.990299i	$-0.544374\pi$
$977$	−8486.82	−0.277909	−0.138955	+	0.990299i	$0.544374\pi$
$978$	−10572.1	−0.345661
$979$	−14839.3	−0.484439
$980$	7186.35	0.234244
$981$	83820.2	2.72800
$982$	−3410.97	−0.110843
$983$	−9199.52	−0.298494	−0.149247	−	0.988800i	$-0.547685\pi$
$983$	−9199.52	−0.298494	−0.149247	+	0.988800i	$0.547685\pi$
$984$	−26338.7	−0.853301
$985$	−13361.0	−0.432199
$986$	−5174.99	−0.167145
$987$	−24877.6	−0.802291
$988$	247.368	0.00796540
$989$	−10199.0	−0.327915
$990$	2194.59	0.0704530
$991$	−55582.3	−1.78166	−0.890832	−	0.454333i	$-0.849878\pi$
$991$	−55582.3	−1.78166	−0.890832	+	0.454333i	$0.849878\pi$
$992$	4999.87	0.160026
$993$	−114649.	−3.66393
$994$	−482.743	−0.0154041
$995$	7295.36	0.232441
$996$	−35274.8	−1.12221
$997$	369.574	0.0117397	0.00586987	−	0.999983i	$-0.498132\pi$
$997$	369.574	0.0117397	0.00586987	+	0.999983i	$0.498132\pi$
$998$	4188.29	0.132844
$999$	28310.8	0.896611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.e.1.11	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.e.1.11	✓	22	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+0.520930 q^{2} +10.1782 q^{3} -7.72863 q^{4} +5.00000 q^{5} +5.30216 q^{6} +12.5313 q^{7} -8.19352 q^{8} +76.5967 q^{9} +O(q^{10})\)	\(q+0.520930 q^{2} +10.1782 q^{3} -7.72863 q^{4} +5.00000 q^{5} +5.30216 q^{6} +12.5313 q^{7} -8.19352 q^{8} +76.5967 q^{9} +2.60465 q^{10} +11.0000 q^{11} -78.6639 q^{12} +1.68456 q^{13} +6.52792 q^{14} +50.8912 q^{15} +57.5608 q^{16} -50.1482 q^{17} +39.9016 q^{18} -19.0000 q^{19} -38.6432 q^{20} +127.547 q^{21} +5.73023 q^{22} -37.1628 q^{23} -83.3957 q^{24} +25.0000 q^{25} +0.877539 q^{26} +504.808 q^{27} -96.8497 q^{28} +198.095 q^{29} +26.5108 q^{30} +52.3364 q^{31} +95.5333 q^{32} +111.961 q^{33} -26.1237 q^{34} +62.6564 q^{35} -591.988 q^{36} +56.0823 q^{37} -9.89767 q^{38} +17.1459 q^{39} -40.9676 q^{40} +315.829 q^{41} +66.4428 q^{42} +274.440 q^{43} -85.0150 q^{44} +382.984 q^{45} -19.3592 q^{46} -195.047 q^{47} +585.868 q^{48} -185.967 q^{49} +13.0233 q^{50} -510.421 q^{51} -13.0194 q^{52} -133.916 q^{53} +262.970 q^{54} +55.0000 q^{55} -102.675 q^{56} -193.387 q^{57} +103.194 q^{58} -355.118 q^{59} -393.320 q^{60} -166.004 q^{61} +27.2636 q^{62} +959.856 q^{63} -410.720 q^{64} +8.42281 q^{65} +58.3237 q^{66} +492.547 q^{67} +387.577 q^{68} -378.253 q^{69} +32.6396 q^{70} -73.9504 q^{71} -627.597 q^{72} +7.76522 q^{73} +29.2150 q^{74} +254.456 q^{75} +146.844 q^{76} +137.844 q^{77} +8.93181 q^{78} -601.857 q^{79} +287.804 q^{80} +3069.95 q^{81} +164.525 q^{82} +448.424 q^{83} -985.760 q^{84} -250.741 q^{85} +142.964 q^{86} +2016.26 q^{87} -90.1287 q^{88} -1349.03 q^{89} +199.508 q^{90} +21.1097 q^{91} +287.218 q^{92} +532.693 q^{93} -101.606 q^{94} -95.0000 q^{95} +972.362 q^{96} -623.032 q^{97} -96.8758 q^{98} +842.564 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9}+O(q^{10}) \) 22 * q + 12 * q^2 + 21 * q^3 + 96 * q^4 + 110 * q^5 + 27 * q^6 + 93 * q^7 + 114 * q^8 + 209 * q^9	\( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9} + 60 q^{10} + 242 q^{11} + 164 q^{12} + 207 q^{13} + 129 q^{14} + 105 q^{15} + 604 q^{16} + 209 q^{17} + 869 q^{18} - 418 q^{19} + 480 q^{20} + 45 q^{21} + 132 q^{22} + 191 q^{23} + 458 q^{24} + 550 q^{25} + 363 q^{26} + 45 q^{27} + 577 q^{28} + 389 q^{29} + 135 q^{30} + 198 q^{31} + 1149 q^{32} + 231 q^{33} + 467 q^{34} + 465 q^{35} + 1315 q^{36} + 312 q^{37} - 228 q^{38} + 137 q^{39} + 570 q^{40} + 632 q^{41} - 1794 q^{42} + 1584 q^{43} + 1056 q^{44} + 1045 q^{45} + 681 q^{46} - 54 q^{47} + 2491 q^{48} + 1063 q^{49} + 300 q^{50} + 37 q^{51} + 1246 q^{52} + 343 q^{53} + 1078 q^{54} + 1210 q^{55} - 87 q^{56} - 399 q^{57} - 1424 q^{58} + 2787 q^{59} + 820 q^{60} + 2070 q^{61} - 446 q^{62} + 1696 q^{63} + 1758 q^{64} + 1035 q^{65} + 297 q^{66} + 2423 q^{67} - 524 q^{68} - 997 q^{69} + 645 q^{70} + 2538 q^{71} + 6010 q^{72} + 1397 q^{73} - 1977 q^{74} + 525 q^{75} - 1824 q^{76} + 1023 q^{77} + 202 q^{78} + 878 q^{79} + 3020 q^{80} + 2030 q^{81} - 190 q^{82} + 4932 q^{83} - 4580 q^{84} + 1045 q^{85} - 3394 q^{86} + 6009 q^{87} + 1254 q^{88} + 1812 q^{89} + 4345 q^{90} + 4349 q^{91} - 788 q^{92} - 4848 q^{93} - 2152 q^{94} - 2090 q^{95} + 4032 q^{96} + 988 q^{97} + 1366 q^{98} + 2299 q^{99}+O(q^{100}) \) 22 * q + 12 * q^2 + 21 * q^3 + 96 * q^4 + 110 * q^5 + 27 * q^6 + 93 * q^7 + 114 * q^8 + 209 * q^9 + 60 * q^10 + 242 * q^11 + 164 * q^12 + 207 * q^13 + 129 * q^14 + 105 * q^15 + 604 * q^16 + 209 * q^17 + 869 * q^18 - 418 * q^19 + 480 * q^20 + 45 * q^21 + 132 * q^22 + 191 * q^23 + 458 * q^24 + 550 * q^25 + 363 * q^26 + 45 * q^27 + 577 * q^28 + 389 * q^29 + 135 * q^30 + 198 * q^31 + 1149 * q^32 + 231 * q^33 + 467 * q^34 + 465 * q^35 + 1315 * q^36 + 312 * q^37 - 228 * q^38 + 137 * q^39 + 570 * q^40 + 632 * q^41 - 1794 * q^42 + 1584 * q^43 + 1056 * q^44 + 1045 * q^45 + 681 * q^46 - 54 * q^47 + 2491 * q^48 + 1063 * q^49 + 300 * q^50 + 37 * q^51 + 1246 * q^52 + 343 * q^53 + 1078 * q^54 + 1210 * q^55 - 87 * q^56 - 399 * q^57 - 1424 * q^58 + 2787 * q^59 + 820 * q^60 + 2070 * q^61 - 446 * q^62 + 1696 * q^63 + 1758 * q^64 + 1035 * q^65 + 297 * q^66 + 2423 * q^67 - 524 * q^68 - 997 * q^69 + 645 * q^70 + 2538 * q^71 + 6010 * q^72 + 1397 * q^73 - 1977 * q^74 + 525 * q^75 - 1824 * q^76 + 1023 * q^77 + 202 * q^78 + 878 * q^79 + 3020 * q^80 + 2030 * q^81 - 190 * q^82 + 4932 * q^83 - 4580 * q^84 + 1045 * q^85 - 3394 * q^86 + 6009 * q^87 + 1254 * q^88 + 1812 * q^89 + 4345 * q^90 + 4349 * q^91 - 788 * q^92 - 4848 * q^93 - 2152 * q^94 - 2090 * q^95 + 4032 * q^96 + 988 * q^97 + 1366 * q^98 + 2299 * q^99

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