LMFDB - Embedded newform 2023.4.a.h.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2023,4,Mod(1,2023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2023, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2023.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2023 = 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2023.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:	$119.360863942$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 53x^{7} + 90x^{6} + 880x^{5} - 1087x^{4} - 4674x^{3} + 2515x^{2} + 1814x + 36 $ x^9 - 2x^8 - 53x^7 + 90x^6 + 880x^5 - 1087x^4 - 4674x^3 + 2515x^2 + 1814x + 36
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 119)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$4.42628$ of defining polynomial
Character	$\chi$	$=$	2023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.42628 q^{2} +5.22352 q^{3} +11.5919 q^{4} -16.9037 q^{5} +23.1207 q^{6} -7.00000 q^{7} +15.8988 q^{8} +0.285172 q^{9} +O(q^{10})$	\(q+4.42628 q^{2} +5.22352 q^{3} +11.5919 q^{4} -16.9037 q^{5} +23.1207 q^{6} -7.00000 q^{7} +15.8988 q^{8} +0.285172 q^{9} -74.8205 q^{10} -22.2449 q^{11} +60.5506 q^{12} +69.0502 q^{13} -30.9839 q^{14} -88.2969 q^{15} -22.3627 q^{16} +1.26225 q^{18} +6.54805 q^{19} -195.946 q^{20} -36.5646 q^{21} -98.4621 q^{22} +202.353 q^{23} +83.0479 q^{24} +160.735 q^{25} +305.635 q^{26} -139.545 q^{27} -81.1434 q^{28} +8.82223 q^{29} -390.826 q^{30} +297.162 q^{31} -226.174 q^{32} -116.197 q^{33} +118.326 q^{35} +3.30569 q^{36} +343.891 q^{37} +28.9835 q^{38} +360.685 q^{39} -268.749 q^{40} +151.543 q^{41} -161.845 q^{42} +150.940 q^{43} -257.861 q^{44} -4.82046 q^{45} +895.670 q^{46} -308.563 q^{47} -116.812 q^{48} +49.0000 q^{49} +711.459 q^{50} +800.424 q^{52} +570.022 q^{53} -617.667 q^{54} +376.021 q^{55} -111.292 q^{56} +34.2039 q^{57} +39.0496 q^{58} +539.811 q^{59} -1023.53 q^{60} -27.4019 q^{61} +1315.32 q^{62} -1.99620 q^{63} -822.208 q^{64} -1167.20 q^{65} -514.319 q^{66} +19.7566 q^{67} +1057.00 q^{69} +523.743 q^{70} +69.4533 q^{71} +4.53390 q^{72} -320.158 q^{73} +1522.16 q^{74} +839.605 q^{75} +75.9044 q^{76} +155.714 q^{77} +1596.49 q^{78} +372.464 q^{79} +378.013 q^{80} -736.618 q^{81} +670.772 q^{82} +312.615 q^{83} -423.854 q^{84} +668.101 q^{86} +46.0831 q^{87} -353.668 q^{88} -443.861 q^{89} -21.3367 q^{90} -483.351 q^{91} +2345.66 q^{92} +1552.23 q^{93} -1365.79 q^{94} -110.686 q^{95} -1181.43 q^{96} -1415.74 q^{97} +216.888 q^{98} -6.34361 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 2 q^{2} - 11 q^{3} + 38 q^{4} + 3 q^{5} - 9 q^{6} - 63 q^{7} + 24 q^{8} + 74 q^{9}+O(q^{10}) $ 9 * q + 2 * q^2 - 11 * q^3 + 38 * q^4 + 3 * q^5 - 9 * q^6 - 63 * q^7 + 24 * q^8 + 74 * q^9	\( 9 q + 2 q^{2} - 11 q^{3} + 38 q^{4} + 3 q^{5} - 9 q^{6} - 63 q^{7} + 24 q^{8} + 74 q^{9} - 134 q^{10} + 8 q^{11} - 56 q^{12} + 164 q^{13} - 14 q^{14} + 34 q^{15} + 178 q^{16} + 98 q^{18} + 244 q^{19} + 41 q^{20} + 77 q^{21} + 80 q^{22} + 14 q^{23} - 298 q^{24} + 684 q^{25} + 326 q^{26} - 218 q^{27} - 266 q^{28} + 234 q^{29} - 335 q^{30} - 555 q^{31} - 181 q^{32} + 458 q^{33} - 21 q^{35} - 1221 q^{36} + 364 q^{37} - 714 q^{38} + 52 q^{39} - 123 q^{40} + 45 q^{41} + 63 q^{42} - 135 q^{43} + 748 q^{44} + 844 q^{45} + 1576 q^{46} - 172 q^{47} + 949 q^{48} + 441 q^{49} - 2901 q^{50} - 1596 q^{52} + 101 q^{53} + 1163 q^{54} + 1260 q^{55} - 168 q^{56} + 602 q^{57} - 1062 q^{58} + 280 q^{59} - 1727 q^{60} - 639 q^{61} + 1708 q^{62} - 518 q^{63} - 2390 q^{64} - 638 q^{65} - 2476 q^{66} + 35 q^{67} + 1288 q^{69} + 938 q^{70} + 1616 q^{71} + 1335 q^{72} - 1049 q^{73} + 370 q^{74} - 1260 q^{75} + 4964 q^{76} - 56 q^{77} + 4714 q^{78} - 2304 q^{79} + 3996 q^{80} - 791 q^{81} + 215 q^{82} + 2508 q^{83} + 392 q^{84} + 623 q^{86} + 166 q^{87} + 416 q^{88} + 2762 q^{89} - 2935 q^{90} - 1148 q^{91} + 2392 q^{92} + 2784 q^{93} - 862 q^{94} + 3462 q^{95} - 2928 q^{96} - 3107 q^{97} + 98 q^{98} + 2396 q^{99}+O(q^{100}) \) 9 * q + 2 * q^2 - 11 * q^3 + 38 * q^4 + 3 * q^5 - 9 * q^6 - 63 * q^7 + 24 * q^8 + 74 * q^9 - 134 * q^10 + 8 * q^11 - 56 * q^12 + 164 * q^13 - 14 * q^14 + 34 * q^15 + 178 * q^16 + 98 * q^18 + 244 * q^19 + 41 * q^20 + 77 * q^21 + 80 * q^22 + 14 * q^23 - 298 * q^24 + 684 * q^25 + 326 * q^26 - 218 * q^27 - 266 * q^28 + 234 * q^29 - 335 * q^30 - 555 * q^31 - 181 * q^32 + 458 * q^33 - 21 * q^35 - 1221 * q^36 + 364 * q^37 - 714 * q^38 + 52 * q^39 - 123 * q^40 + 45 * q^41 + 63 * q^42 - 135 * q^43 + 748 * q^44 + 844 * q^45 + 1576 * q^46 - 172 * q^47 + 949 * q^48 + 441 * q^49 - 2901 * q^50 - 1596 * q^52 + 101 * q^53 + 1163 * q^54 + 1260 * q^55 - 168 * q^56 + 602 * q^57 - 1062 * q^58 + 280 * q^59 - 1727 * q^60 - 639 * q^61 + 1708 * q^62 - 518 * q^63 - 2390 * q^64 - 638 * q^65 - 2476 * q^66 + 35 * q^67 + 1288 * q^69 + 938 * q^70 + 1616 * q^71 + 1335 * q^72 - 1049 * q^73 + 370 * q^74 - 1260 * q^75 + 4964 * q^76 - 56 * q^77 + 4714 * q^78 - 2304 * q^79 + 3996 * q^80 - 791 * q^81 + 215 * q^82 + 2508 * q^83 + 392 * q^84 + 623 * q^86 + 166 * q^87 + 416 * q^88 + 2762 * q^89 - 2935 * q^90 - 1148 * q^91 + 2392 * q^92 + 2784 * q^93 - 862 * q^94 + 3462 * q^95 - 2928 * q^96 - 3107 * q^97 + 98 * q^98 + 2396 * 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 $

$2$	4.42628	1.56492	0.782462	−	0.622698i	$-0.213963\pi$
$2$	4.42628	1.56492	0.782462	+	0.622698i	$0.213963\pi$
$3$	5.22352	1.00527	0.502634	−	0.864500i	$-0.332365\pi$
$3$	5.22352	1.00527	0.502634	+	0.864500i	$0.332365\pi$
$4$	11.5919	1.44899
$5$	−16.9037	−1.51191	−0.755957	−	0.654621i	$-0.772828\pi$
$5$	−16.9037	−1.51191	−0.755957	+	0.654621i	$0.772828\pi$
$6$	23.1207	1.57317
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	15.8988	0.702636
$9$	0.285172	0.0105619
$10$	−74.8205	−2.36603
$11$	−22.2449	−0.609735	−0.304868	−	0.952395i	$-0.598612\pi$
$11$	−22.2449	−0.609735	−0.304868	+	0.952395i	$0.598612\pi$
$12$	60.5506	1.45662
$13$	69.0502	1.47316	0.736580	−	0.676350i	$-0.236439\pi$
$13$	69.0502	1.47316	0.736580	+	0.676350i	$0.236439\pi$
$14$	−30.9839	−0.591486
$15$	−88.2969	−1.51988
$16$	−22.3627	−0.349418
$17$	0	0
$17$	0	0
$18$	1.26225	0.0165286
$19$	6.54805	0.0790645	0.0395322	−	0.999218i	$-0.487413\pi$
$19$	6.54805	0.0790645	0.0395322	+	0.999218i	$0.487413\pi$
$20$	−195.946	−2.19075
$21$	−36.5646	−0.379955
$22$	−98.4621	−0.954190
$23$	202.353	1.83450	0.917251	−	0.398311i	$-0.130403\pi$
$23$	202.353	1.83450	0.917251	+	0.398311i	$0.130403\pi$
$24$	83.0479	0.706337
$25$	160.735	1.28588
$26$	305.635	2.30538
$27$	−139.545	−0.994650
$28$	−81.1434	−0.547667
$29$	8.82223	0.0564913	0.0282456	−	0.999601i	$-0.491008\pi$
$29$	8.82223	0.0564913	0.0282456	+	0.999601i	$0.491008\pi$
$30$	−390.826	−2.37849
$31$	297.162	1.72168	0.860838	−	0.508879i	$-0.169940\pi$
$31$	297.162	1.72168	0.860838	+	0.508879i	$0.169940\pi$
$32$	−226.174	−1.24945
$33$	−116.197	−0.612947
$34$	0	0
$35$	118.326	0.571450
$36$	3.30569	0.0153041
$37$	343.891	1.52798	0.763992	−	0.645226i	$-0.223237\pi$
$37$	343.891	1.52798	0.763992	+	0.645226i	$0.223237\pi$
$38$	28.9835	0.123730
$39$	360.685	1.48092
$40$	−268.749	−1.06232
$41$	151.543	0.577245	0.288623	−	0.957443i	$-0.406803\pi$
$41$	151.543	0.577245	0.288623	+	0.957443i	$0.406803\pi$
$42$	−161.845	−0.594601
$43$	150.940	0.535305	0.267652	−	0.963516i	$-0.413752\pi$
$43$	150.940	0.535305	0.267652	+	0.963516i	$0.413752\pi$
$44$	−257.861	−0.883500
$45$	−4.82046	−0.0159687
$46$	895.670	2.87086
$47$	−308.563	−0.957629	−0.478815	−	0.877916i	$-0.658933\pi$
$47$	−308.563	−0.957629	−0.478815	+	0.877916i	$0.658933\pi$
$48$	−116.812	−0.351258
$49$	49.0000	0.142857
$50$	711.459	2.01231
$51$	0	0
$52$	800.424	2.13459
$53$	570.022	1.47733	0.738665	−	0.674073i	$-0.235457\pi$
$53$	570.022	1.47733	0.738665	+	0.674073i	$0.235457\pi$
$54$	−617.667	−1.55655
$55$	376.021	0.921867
$56$	−111.292	−0.265571
$57$	34.2039	0.0794809
$58$	39.0496	0.0884046
$59$	539.811	1.19114	0.595571	−	0.803302i	$-0.296926\pi$
$59$	539.811	1.19114	0.595571	+	0.803302i	$0.296926\pi$
$60$	−1023.53	−2.20229
$61$	−27.4019	−0.0575157	−0.0287579	−	0.999586i	$-0.509155\pi$
$61$	−27.4019	−0.0575157	−0.0287579	+	0.999586i	$0.509155\pi$
$62$	1315.32	2.69429
$63$	−1.99620	−0.00399203
$64$	−822.208	−1.60588
$65$	−1167.20	−2.22729
$66$	−514.319	−0.959216
$67$	19.7566	0.0360246	0.0180123	−	0.999838i	$-0.494266\pi$
$67$	19.7566	0.0360246	0.0180123	+	0.999838i	$0.494266\pi$
$68$	0	0
$69$	1057.00	1.84416
$70$	523.743	0.894276
$71$	69.4533	0.116093	0.0580464	−	0.998314i	$-0.481513\pi$
$71$	69.4533	0.116093	0.0580464	+	0.998314i	$0.481513\pi$
$72$	4.53390	0.00742118
$73$	−320.158	−0.513310	−0.256655	−	0.966503i	$-0.582620\pi$
$73$	−320.158	−0.513310	−0.256655	+	0.966503i	$0.582620\pi$
$74$	1522.16	2.39118
$75$	839.605	1.29266
$76$	75.9044	0.114564
$77$	155.714	0.230458
$78$	1596.49	2.31753
$79$	372.464	0.530449	0.265225	−	0.964187i	$-0.414554\pi$
$79$	372.464	0.530449	0.265225	+	0.964187i	$0.414554\pi$
$80$	378.013	0.528290
$81$	−736.618	−1.01045
$82$	670.772	0.903346
$83$	312.615	0.413421	0.206711	−	0.978402i	$-0.433724\pi$
$83$	312.615	0.413421	0.206711	+	0.978402i	$0.433724\pi$
$84$	−423.854	−0.550551
$85$	0	0
$86$	668.101	0.837712
$87$	46.0831	0.0567888
$88$	−353.668	−0.428422
$89$	−443.861	−0.528642	−0.264321	−	0.964435i	$-0.585148\pi$
$89$	−443.861	−0.528642	−0.264321	+	0.964435i	$0.585148\pi$
$90$	−21.3367	−0.0249898
$91$	−483.351	−0.556802
$92$	2345.66	2.65817
$93$	1552.23	1.73074
$94$	−1365.79	−1.49862
$95$	−110.686	−0.119539
$96$	−1181.43	−1.25603
$97$	−1415.74	−1.48193	−0.740964	−	0.671545i	$-0.765631\pi$
$97$	−1415.74	−1.48193	−0.740964	+	0.671545i	$0.765631\pi$
$98$	216.888	0.223561
$99$	−6.34361	−0.00643997
$100$	1863.23	1.86323
$101$	1119.77	1.10318	0.551592	−	0.834114i	$-0.314020\pi$
$101$	1119.77	1.10318	0.551592	+	0.834114i	$0.314020\pi$
$102$	0	0
$103$	−930.310	−0.889963	−0.444981	−	0.895540i	$-0.646790\pi$
$103$	−930.310	−0.889963	−0.444981	+	0.895540i	$0.646790\pi$
$104$	1097.82	1.03510
$105$	618.078	0.574460
$106$	2523.07	2.31191
$107$	939.841	0.849139	0.424569	−	0.905395i	$-0.360426\pi$
$107$	939.841	0.849139	0.424569	+	0.905395i	$0.360426\pi$
$108$	−1617.60	−1.44124
$109$	−250.597	−0.220210	−0.110105	−	0.993920i	$-0.535119\pi$
$109$	−250.597	−0.220210	−0.110105	+	0.993920i	$0.535119\pi$
$110$	1664.37	1.44265
$111$	1796.32	1.53603
$112$	156.539	0.132068
$113$	−758.427	−0.631387	−0.315694	−	0.948861i	$-0.602237\pi$
$113$	−758.427	−0.631387	−0.315694	+	0.948861i	$0.602237\pi$
$114$	151.396	0.124382
$115$	−3420.52	−2.77361
$116$	102.267	0.0818553
$117$	19.6912	0.0155594
$118$	2389.35	1.86405
$119$	0	0
$120$	−1403.82	−1.06792
$121$	−836.165	−0.628223
$122$	−121.289	−0.0900078
$123$	791.589	0.580286
$124$	3444.68	2.49469
$125$	−604.060	−0.432230
$126$	−8.83574	−0.00624722
$127$	−794.380	−0.555038	−0.277519	−	0.960720i	$-0.589512\pi$
$127$	−794.380	−0.555038	−0.277519	+	0.960720i	$0.589512\pi$
$128$	−1829.93	−1.26363
$129$	788.437	0.538124
$130$	−5166.37	−3.48554
$131$	−1633.80	−1.08966	−0.544831	−	0.838546i	$-0.683406\pi$
$131$	−1633.80	−1.08966	−0.544831	+	0.838546i	$0.683406\pi$
$132$	−1346.94	−0.888154
$133$	−45.8363	−0.0298836
$134$	87.4480	0.0563758
$135$	2358.84	1.50382
$136$	0	0
$137$	−810.805	−0.505633	−0.252817	−	0.967514i	$-0.581357\pi$
$137$	−810.805	−0.505633	−0.252817	+	0.967514i	$0.581357\pi$
$138$	4678.55	2.88598
$139$	3035.90	1.85253	0.926266	−	0.376870i	$-0.123000\pi$
$139$	3035.90	1.85253	0.926266	+	0.376870i	$0.123000\pi$
$140$	1371.63	0.828025
$141$	−1611.79	−0.962673
$142$	307.419	0.181676
$143$	−1536.01	−0.898238
$144$	−6.37722	−0.00369052
$145$	−149.128	−0.0854099
$146$	−1417.11	−0.803291
$147$	255.953	0.143610
$148$	3986.36	2.21403
$149$	2210.48	1.21537	0.607684	−	0.794179i	$-0.292099\pi$
$149$	2210.48	1.21537	0.607684	+	0.794179i	$0.292099\pi$
$150$	3716.32	2.02291
$151$	13.3482	0.00719378	0.00359689	−	0.999994i	$-0.498855\pi$
$151$	13.3482	0.00719378	0.00359689	+	0.999994i	$0.498855\pi$
$152$	104.106	0.0555535
$153$	0	0
$154$	689.234	0.360650
$155$	−5023.15	−2.60303
$156$	4181.03	2.14584
$157$	1189.54	0.604684	0.302342	−	0.953200i	$-0.402232\pi$
$157$	1189.54	0.604684	0.302342	+	0.953200i	$0.402232\pi$
$158$	1648.63	0.830113
$159$	2977.52	1.48511
$160$	3823.18	1.88906
$161$	−1416.47	−0.693376
$162$	−3260.48	−1.58128
$163$	−3487.28	−1.67574	−0.837868	−	0.545873i	$-0.816198\pi$
$163$	−3487.28	−1.67574	−0.837868	+	0.545873i	$0.816198\pi$
$164$	1756.68	0.836423
$165$	1964.15	0.926723
$166$	1383.72	0.646973
$167$	3091.88	1.43268	0.716338	−	0.697754i	$-0.245817\pi$
$167$	3091.88	1.43268	0.716338	+	0.697754i	$0.245817\pi$
$168$	−581.335	−0.266970
$169$	2570.93	1.17020
$170$	0	0
$171$	1.86732	0.000835072 0
$172$	1749.68	0.775651
$173$	3139.53	1.37973	0.689867	−	0.723936i	$-0.257669\pi$
$173$	3139.53	1.37973	0.689867	+	0.723936i	$0.257669\pi$
$174$	203.977	0.0888702
$175$	−1125.15	−0.486018
$176$	497.457	0.213052
$177$	2819.71	1.19742
$178$	−1964.65	−0.827285
$179$	3822.18	1.59600	0.797998	−	0.602660i	$-0.205893\pi$
$179$	3822.18	1.59600	0.797998	+	0.602660i	$0.205893\pi$
$180$	−55.8784	−0.0231385
$181$	−2158.69	−0.886487	−0.443244	−	0.896401i	$-0.646172\pi$
$181$	−2158.69	−0.886487	−0.443244	+	0.896401i	$0.646172\pi$
$182$	−2139.45	−0.871354
$183$	−143.135	−0.0578187
$184$	3217.18	1.28899
$185$	−5813.04	−2.31018
$186$	6870.62	2.70848
$187$	0	0
$188$	−3576.84	−1.38760
$189$	976.818	0.375942
$190$	−489.928	−0.187069
$191$	−1055.33	−0.399797	−0.199899	−	0.979817i	$-0.564061\pi$
$191$	−1055.33	−0.399797	−0.199899	+	0.979817i	$0.564061\pi$
$192$	−4294.82	−1.61433
$193$	−102.145	−0.0380960	−0.0190480	−	0.999819i	$-0.506064\pi$
$193$	−102.145	−0.0380960	−0.0190480	+	0.999819i	$0.506064\pi$
$194$	−6266.47	−2.31911
$195$	−6096.92	−2.23902
$196$	568.004	0.206999
$197$	4360.12	1.57688	0.788440	−	0.615111i	$-0.210889\pi$
$197$	4360.12	1.57688	0.788440	+	0.615111i	$0.210889\pi$
$198$	−28.0786	−0.0100781
$199$	−3775.79	−1.34502	−0.672510	−	0.740088i	$-0.734784\pi$
$199$	−3775.79	−1.34502	−0.672510	+	0.740088i	$0.734784\pi$
$200$	2555.50	0.903507
$201$	103.199	0.0362144
$202$	4956.43	1.72640
$203$	−61.7556	−0.0213517
$204$	0	0
$205$	−2561.64	−0.872745
$206$	−4117.81	−1.39272
$207$	57.7053	0.0193758
$208$	−1544.15	−0.514748
$209$	−145.661	−0.0482084
$210$	2735.78	0.898986
$211$	−5842.83	−1.90634	−0.953168	−	0.302443i	$-0.902198\pi$
$211$	−5842.83	−1.90634	−0.953168	+	0.302443i	$0.902198\pi$
$212$	6607.65	2.14064
$213$	362.791	0.116704
$214$	4159.99	1.32884
$215$	−2551.44	−0.809334
$216$	−2218.61	−0.698876
$217$	−2080.14	−0.650732
$218$	−1109.21	−0.344612
$219$	−1672.35	−0.516013
$220$	4358.81	1.33578
$221$	0	0
$222$	7951.03	2.40377
$223$	6090.44	1.82891	0.914454	−	0.404691i	$-0.132621\pi$
$223$	6090.44	1.82891	0.914454	+	0.404691i	$0.132621\pi$
$224$	1583.22	0.472247
$225$	45.8372	0.0135814
$226$	−3357.01	−0.988074
$227$	5109.99	1.49410	0.747052	−	0.664765i	$-0.231468\pi$
$227$	5109.99	1.49410	0.747052	+	0.664765i	$0.231468\pi$
$228$	396.488	0.115167
$229$	2493.77	0.719619	0.359809	−	0.933026i	$-0.382842\pi$
$229$	2493.77	0.719619	0.359809	+	0.933026i	$0.382842\pi$
$230$	−15140.2	−4.34049
$231$	813.377	0.231672
$232$	140.263	0.0396928
$233$	−2113.26	−0.594182	−0.297091	−	0.954849i	$-0.596017\pi$
$233$	−2113.26	−0.594182	−0.297091	+	0.954849i	$0.596017\pi$
$234$	87.1585	0.0243493
$235$	5215.86	1.44785
$236$	6257.45	1.72595
$237$	1945.57	0.533243
$238$	0	0
$239$	−4230.50	−1.14497	−0.572486	−	0.819915i	$-0.694021\pi$
$239$	−4230.50	−1.14497	−0.572486	+	0.819915i	$0.694021\pi$
$240$	1974.56	0.531072
$241$	−3922.05	−1.04830	−0.524152	−	0.851625i	$-0.675618\pi$
$241$	−3922.05	−1.04830	−0.524152	+	0.851625i	$0.675618\pi$
$242$	−3701.10	−0.983122
$243$	−80.0136	−0.0211229
$244$	−317.641	−0.0833397
$245$	−828.282	−0.215988
$246$	3503.79	0.908104
$247$	452.144	0.116475
$248$	4724.54	1.20971
$249$	1632.95	0.415599
$250$	−2673.74	−0.676408
$251$	0.674614	0.000169646 0	8.48232e−5	−	1.00000i	$-0.499973\pi$
$251$	0.674614	0.000169646 0	8.48232e−5		1.00000i	$0.499973\pi$
$252$	−23.1398	−0.00578441
$253$	−4501.32	−1.11856
$254$	−3516.15	−0.868593
$255$	0	0
$256$	−1522.09	−0.371604
$257$	5160.55	1.25255	0.626277	−	0.779601i	$-0.284578\pi$
$257$	5160.55	1.25255	0.626277	+	0.779601i	$0.284578\pi$
$258$	3489.84	0.842124
$259$	−2407.24	−0.577524
$260$	−13530.1	−3.22732
$261$	2.51585	0.000596656 0
$262$	−7231.64	−1.70524
$263$	−3100.41	−0.726918	−0.363459	−	0.931610i	$-0.618404\pi$
$263$	−3100.41	−0.726918	−0.363459	+	0.931610i	$0.618404\pi$
$264$	−1847.39	−0.430678
$265$	−9635.48	−2.23360
$266$	−202.884	−0.0467655
$267$	−2318.52	−0.531427
$268$	229.017	0.0521993
$269$	2982.89	0.676097	0.338048	−	0.941129i	$-0.390233\pi$
$269$	2982.89	0.676097	0.338048	+	0.941129i	$0.390233\pi$
$270$	10440.9	2.35337
$271$	−630.039	−0.141226	−0.0706128	−	0.997504i	$-0.522495\pi$
$271$	−630.039	−0.141226	−0.0706128	+	0.997504i	$0.522495\pi$
$272$	0	0
$273$	−2524.80	−0.559735
$274$	−3588.85	−0.791278
$275$	−3575.54	−0.784048
$276$	12252.6	2.67217
$277$	6451.11	1.39931	0.699656	−	0.714480i	$-0.253336\pi$
$277$	6451.11	1.39931	0.699656	+	0.714480i	$0.253336\pi$
$278$	13437.7	2.89907
$279$	84.7423	0.0181842
$280$	1881.24	0.401521
$281$	−6936.02	−1.47248	−0.736242	−	0.676718i	$-0.763402\pi$
$281$	−6936.02	−1.47248	−0.736242	+	0.676718i	$0.763402\pi$
$282$	−7134.21	−1.50651
$283$	3194.87	0.671078	0.335539	−	0.942026i	$-0.391082\pi$
$283$	3194.87	0.671078	0.335539	+	0.942026i	$0.391082\pi$
$284$	805.097	0.168217
$285$	−578.172	−0.120168
$286$	−6798.83	−1.40567
$287$	−1060.80	−0.218178
$288$	−64.4985	−0.0131966
$289$	0	0
$290$	−660.083	−0.133660
$291$	−7395.17	−1.48973
$292$	−3711.24	−0.743781
$293$	9221.06	1.83857	0.919283	−	0.393597i	$-0.128770\pi$
$293$	9221.06	1.83857	0.919283	+	0.393597i	$0.128770\pi$
$294$	1132.92	0.224738
$295$	−9124.81	−1.80090
$296$	5467.47	1.07362
$297$	3104.17	0.606473
$298$	9784.21	1.90196
$299$	13972.5	2.70251
$300$	9732.63	1.87305
$301$	−1056.58	−0.202326
$302$	59.0828	0.0112577
$303$	5849.16	1.10899
$304$	−146.432	−0.0276265
$305$	463.195	0.0869588
$306$	0	0
$307$	166.875	0.0310230	0.0155115	−	0.999880i	$-0.495062\pi$
$307$	166.875	0.0310230	0.0155115	+	0.999880i	$0.495062\pi$
$308$	1805.03	0.333932
$309$	−4859.49	−0.894650
$310$	−22233.8	−4.07354
$311$	604.942	0.110299	0.0551497	−	0.998478i	$-0.482436\pi$
$311$	604.942	0.110299	0.0551497	+	0.998478i	$0.482436\pi$
$312$	5734.47	1.04055
$313$	6192.74	1.11832	0.559160	−	0.829060i	$-0.311124\pi$
$313$	6192.74	1.11832	0.559160	+	0.829060i	$0.311124\pi$
$314$	5265.22	0.946285
$315$	33.7432	0.00603560
$316$	4317.58	0.768616
$317$	−1803.66	−0.319571	−0.159785	−	0.987152i	$-0.551080\pi$
$317$	−1803.66	−0.319571	−0.159785	+	0.987152i	$0.551080\pi$
$318$	13179.3	2.32409
$319$	−196.250	−0.0344447
$320$	13898.4	2.42794
$321$	4909.28	0.853611
$322$	−6269.69	−1.08508
$323$	0	0
$324$	−8538.82	−1.46413
$325$	11098.8	1.89431
$326$	−15435.7	−2.62240
$327$	−1309.00	−0.221370
$328$	2409.36	0.405593
$329$	2159.94	0.361950
$330$	8693.89	1.45025
$331$	5111.31	0.848771	0.424385	−	0.905482i	$-0.360490\pi$
$331$	5111.31	0.848771	0.424385	+	0.905482i	$0.360490\pi$
$332$	3623.81	0.599043
$333$	98.0681	0.0161384
$334$	13685.5	2.24203
$335$	−333.959	−0.0544661
$336$	817.686	0.132763
$337$	−3865.00	−0.624748	−0.312374	−	0.949959i	$-0.601124\pi$
$337$	−3865.00	−0.624748	−0.312374	+	0.949959i	$0.601124\pi$
$338$	11379.6	1.83128
$339$	−3961.66	−0.634713
$340$	0	0
$341$	−6610.35	−1.04977
$342$	8.26526	0.00130683
$343$	−343.000	−0.0539949
$344$	2399.77	0.376124
$345$	−17867.1	−2.78822
$346$	13896.4	2.15918
$347$	−27.8871	−0.00431428	−0.00215714	−	0.999998i	$-0.500687\pi$
$347$	−27.8871	−0.00431428	−0.00215714	+	0.999998i	$0.500687\pi$
$348$	534.192	0.0822864
$349$	4895.68	0.750888	0.375444	−	0.926845i	$-0.377490\pi$
$349$	4895.68	0.750888	0.375444	+	0.926845i	$0.377490\pi$
$350$	−4980.21	−0.760582
$351$	−9635.64	−1.46528
$352$	5031.22	0.761833
$353$	3996.67	0.602609	0.301305	−	0.953528i	$-0.402578\pi$
$353$	3996.67	0.602609	0.301305	+	0.953528i	$0.402578\pi$
$354$	12480.8	1.87387
$355$	−1174.02	−0.175522
$356$	−5145.20	−0.765997
$357$	0	0
$358$	16918.0	2.49761
$359$	482.472	0.0709300	0.0354650	−	0.999371i	$-0.488709\pi$
$359$	482.472	0.0709300	0.0354650	+	0.999371i	$0.488709\pi$
$360$	−76.6397	−0.0112202
$361$	−6816.12	−0.993749
$362$	−9554.96	−1.38729
$363$	−4367.72	−0.631532
$364$	−5602.97	−0.806801
$365$	5411.85	0.776080
$366$	−633.553	−0.0904819
$367$	−5495.14	−0.781592	−0.390796	−	0.920477i	$-0.627800\pi$
$367$	−5495.14	−0.781592	−0.390796	+	0.920477i	$0.627800\pi$
$368$	−4525.17	−0.641007
$369$	43.2158	0.00609681
$370$	−25730.1	−3.61526
$371$	−3990.15	−0.558378
$372$	17993.4	2.50783
$373$	−3038.90	−0.421845	−0.210922	−	0.977503i	$-0.567647\pi$
$373$	−3038.90	−0.421845	−0.210922	+	0.977503i	$0.567647\pi$
$374$	0	0
$375$	−3155.32	−0.434507
$376$	−4905.80	−0.672865
$377$	609.177	0.0832207
$378$	4323.67	0.588321
$379$	−13910.5	−1.88531	−0.942654	−	0.333770i	$-0.891679\pi$
$379$	−13910.5	−1.88531	−0.942654	+	0.333770i	$0.891679\pi$
$380$	−1283.07	−0.173210
$381$	−4149.46	−0.557962
$382$	−4671.20	−0.625653
$383$	4187.26	0.558640	0.279320	−	0.960198i	$-0.409891\pi$
$383$	4187.26	0.558640	0.279320	+	0.960198i	$0.409891\pi$
$384$	−9558.65	−1.27028
$385$	−2632.15	−0.348433
$386$	−452.120	−0.0596174
$387$	43.0437	0.00565384
$388$	−16411.2	−2.14730
$389$	−7416.22	−0.966626	−0.483313	−	0.875448i	$-0.660567\pi$
$389$	−7416.22	−0.966626	−0.483313	+	0.875448i	$0.660567\pi$
$390$	−26986.6	−3.50390
$391$	0	0
$392$	779.043	0.100377
$393$	−8534.18	−1.09540
$394$	19299.1	2.46770
$395$	−6296.03	−0.801993
$396$	−73.5347	−0.00933145
$397$	−11937.9	−1.50918	−0.754590	−	0.656197i	$-0.772164\pi$
$397$	−11937.9	−1.50918	−0.754590	+	0.656197i	$0.772164\pi$
$398$	−16712.7	−2.10485
$399$	−239.427	−0.0300410
$400$	−3594.48	−0.449310
$401$	15219.6	1.89533	0.947667	−	0.319260i	$-0.103434\pi$
$401$	15219.6	1.89533	0.947667	+	0.319260i	$0.103434\pi$
$402$	456.787	0.0566728
$403$	20519.1	2.53630
$404$	12980.3	1.59850
$405$	12451.6	1.52771
$406$	−273.347	−0.0334138
$407$	−7649.83	−0.931666
$408$	0	0
$409$	7534.85	0.910940	0.455470	−	0.890251i	$-0.349471\pi$
$409$	7534.85	0.910940	0.455470	+	0.890251i	$0.349471\pi$
$410$	−11338.5	−1.36578
$411$	−4235.26	−0.508296
$412$	−10784.1	−1.28955
$413$	−3778.68	−0.450210
$414$	255.420	0.0303217
$415$	−5284.35	−0.625057
$416$	−15617.4	−1.84064
$417$	15858.1	1.86229
$418$	−644.734	−0.0754425
$419$	−3895.36	−0.454179	−0.227089	−	0.973874i	$-0.572921\pi$
$419$	−3895.36	−0.454179	−0.227089	+	0.973874i	$0.572921\pi$
$420$	7164.71	0.832386
$421$	−8636.29	−0.999779	−0.499889	−	0.866089i	$-0.666626\pi$
$421$	−8636.29	−0.999779	−0.499889	+	0.866089i	$0.666626\pi$
$422$	−25862.0	−2.98327
$423$	−87.9935	−0.0101144
$424$	9062.68	1.03803
$425$	0	0
$426$	1605.81	0.182633
$427$	191.814	0.0217389
$428$	10894.6	1.23039
$429$	−8023.40	−0.902969
$430$	−11293.4	−1.26655
$431$	−9927.45	−1.10949	−0.554743	−	0.832022i	$-0.687183\pi$
$431$	−9927.45	−1.10949	−0.554743	+	0.832022i	$0.687183\pi$
$432$	3120.62	0.347548
$433$	−6576.00	−0.729844	−0.364922	−	0.931038i	$-0.618904\pi$
$433$	−6576.00	−0.729844	−0.364922	+	0.931038i	$0.618904\pi$
$434$	−9207.26	−1.01835
$435$	−778.975	−0.0858598
$436$	−2904.90	−0.319082
$437$	1325.02	0.145044
$438$	−7402.28	−0.807522
$439$	−9719.62	−1.05670	−0.528351	−	0.849026i	$-0.677189\pi$
$439$	−9719.62	−1.05670	−0.528351	+	0.849026i	$0.677189\pi$
$440$	5978.30	0.647737
$441$	13.9734	0.00150884
$442$	0	0
$443$	−11347.5	−1.21702	−0.608508	−	0.793548i	$-0.708232\pi$
$443$	−11347.5	−1.21702	−0.608508	+	0.793548i	$0.708232\pi$
$444$	20822.8	2.22570
$445$	7502.90	0.799261
$446$	26958.0	2.86210
$447$	11546.5	1.22177
$448$	5755.46	0.606964
$449$	−16453.2	−1.72934	−0.864672	−	0.502338i	$-0.832473\pi$
$449$	−16453.2	−1.72934	−0.864672	+	0.502338i	$0.832473\pi$
$450$	202.888	0.0212538
$451$	−3371.06	−0.351967
$452$	−8791.62	−0.914874
$453$	69.7246	0.00723167
$454$	22618.2	2.33816
$455$	8170.43	0.841837
$456$	543.802	0.0558461
$457$	−2402.41	−0.245908	−0.122954	−	0.992412i	$-0.539237\pi$
$457$	−2402.41	−0.245908	−0.122954	+	0.992412i	$0.539237\pi$
$458$	11038.1	1.12615
$459$	0	0
$460$	−39650.4	−4.01893
$461$	12582.2	1.27117	0.635586	−	0.772030i	$-0.280759\pi$
$461$	12582.2	1.27117	0.635586	+	0.772030i	$0.280759\pi$
$462$	3600.23	0.362550
$463$	−4320.33	−0.433656	−0.216828	−	0.976210i	$-0.569571\pi$
$463$	−4320.33	−0.433656	−0.216828	+	0.976210i	$0.569571\pi$
$464$	−197.289	−0.0197391
$465$	−26238.5	−2.61674
$466$	−9353.88	−0.929851
$467$	−453.197	−0.0449068	−0.0224534	−	0.999748i	$-0.507148\pi$
$467$	−453.197	−0.0449068	−0.0224534	+	0.999748i	$0.507148\pi$
$468$	228.258	0.0225454
$469$	−138.296	−0.0136160
$470$	23086.9	2.26578
$471$	6213.57	0.607869
$472$	8582.37	0.836939
$473$	−3357.64	−0.326394
$474$	8611.65	0.834486
$475$	1052.50	0.101668
$476$	0	0
$477$	162.554	0.0156034
$478$	−18725.4	−1.79180
$479$	7262.14	0.692726	0.346363	−	0.938101i	$-0.387417\pi$
$479$	7262.14	0.692726	0.346363	+	0.938101i	$0.387417\pi$
$480$	19970.5	1.89901
$481$	23745.8	2.25096
$482$	−17360.1	−1.64052
$483$	−7398.97	−0.697028
$484$	−9692.75	−0.910289
$485$	23931.3	2.24055
$486$	−354.162	−0.0330558
$487$	10655.0	0.991425	0.495712	−	0.868487i	$-0.334907\pi$
$487$	10655.0	0.991425	0.495712	+	0.868487i	$0.334907\pi$
$488$	−435.659	−0.0404126
$489$	−18215.9	−1.68456
$490$	−3666.20	−0.338004
$491$	21324.2	1.95998	0.979990	−	0.199048i	$-0.0637851\pi$
$491$	21324.2	1.95998	0.979990	+	0.199048i	$0.0637851\pi$
$492$	9176.03	0.840828
$493$	0	0
$494$	2001.31	0.182274
$495$	107.231	0.00973668
$496$	−6645.37	−0.601584
$497$	−486.173	−0.0438789
$498$	7227.89	0.650381
$499$	6043.52	0.542175	0.271087	−	0.962555i	$-0.412617\pi$
$499$	6043.52	0.542175	0.271087	+	0.962555i	$0.412617\pi$
$500$	−7002.22	−0.626297
$501$	16150.5	1.44022
$502$	2.98603	0.000265484 0
$503$	1741.30	0.154355	0.0771775	−	0.997017i	$-0.475409\pi$
$503$	1741.30	0.154355	0.0771775	+	0.997017i	$0.475409\pi$
$504$	−31.7373	−0.00280494
$505$	−18928.3	−1.66792
$506$	−19924.1	−1.75046
$507$	13429.3	1.17636
$508$	−9208.39	−0.804245
$509$	−7405.78	−0.644902	−0.322451	−	0.946586i	$-0.604507\pi$
$509$	−7405.78	−0.644902	−0.322451	+	0.946586i	$0.604507\pi$
$510$	0	0
$511$	2241.10	0.194013
$512$	7902.21	0.682093
$513$	−913.750	−0.0786414
$514$	22842.0	1.96015
$515$	15725.7	1.34555
$516$	9139.50	0.779737
$517$	6863.96	0.583900
$518$	−10655.1	−0.903781
$519$	16399.4	1.38700
$520$	−18557.2	−1.56497
$521$	19487.0	1.63866	0.819331	−	0.573321i	$-0.194345\pi$
$521$	19487.0	1.63866	0.819331	+	0.573321i	$0.194345\pi$
$522$	11.1358	0.000933722 0
$523$	4971.34	0.415643	0.207822	−	0.978167i	$-0.433363\pi$
$523$	4971.34	0.415643	0.207822	+	0.978167i	$0.433363\pi$
$524$	−18938.9	−1.57891
$525$	−5877.23	−0.488578
$526$	−13723.3	−1.13757
$527$	0	0
$528$	2598.48	0.214175
$529$	28779.8	2.36539
$530$	−42649.3	−3.49541
$531$	153.939	0.0125807
$532$	−531.331	−0.0433010
$533$	10464.1	0.850375
$534$	−10262.4	−0.831643
$535$	−15886.8	−1.28382
$536$	314.106	0.0253122
$537$	19965.2	1.60440
$538$	13203.1	1.05804
$539$	−1090.00	−0.0871050
$540$	27343.4	2.17903
$541$	3403.45	0.270472	0.135236	−	0.990813i	$-0.456821\pi$
$541$	3403.45	0.270472	0.135236	+	0.990813i	$0.456821\pi$
$542$	−2788.73	−0.221008
$543$	−11276.0	−0.891157
$544$	0	0
$545$	4236.02	0.332938
$546$	−11175.4	−0.875943
$547$	9897.33	0.773637	0.386818	−	0.922156i	$-0.373574\pi$
$547$	9897.33	0.773637	0.386818	+	0.922156i	$0.373574\pi$
$548$	−9398.79	−0.732658
$549$	−7.81426	−0.000607476 0
$550$	−15826.3	−1.22698
$551$	57.7684	0.00446645
$552$	16805.0	1.29578
$553$	−2607.25	−0.200491
$554$	28554.4	2.18982
$555$	−30364.5	−2.32235
$556$	35192.0	2.68430
$557$	−13722.2	−1.04386	−0.521930	−	0.852988i	$-0.674788\pi$
$557$	−13722.2	−1.04386	−0.521930	+	0.852988i	$0.674788\pi$
$558$	375.093	0.0284569
$559$	10422.4	0.788589
$560$	−2646.09	−0.199675
$561$	0	0
$562$	−30700.7	−2.30433
$563$	−6833.01	−0.511504	−0.255752	−	0.966742i	$-0.582323\pi$
$563$	−6833.01	−0.511504	−0.255752	+	0.966742i	$0.582323\pi$
$564$	−18683.7	−1.39490
$565$	12820.2	0.954603
$566$	14141.4	1.05019
$567$	5156.33	0.381914
$568$	1104.23	0.0815709
$569$	8285.15	0.610425	0.305212	−	0.952284i	$-0.401273\pi$
$569$	8285.15	0.610425	0.305212	+	0.952284i	$0.401273\pi$
$570$	−2559.15	−0.188054
$571$	−12246.2	−0.897526	−0.448763	−	0.893651i	$-0.648135\pi$
$571$	−12246.2	−0.897526	−0.448763	+	0.893651i	$0.648135\pi$
$572$	−17805.4	−1.30154
$573$	−5512.56	−0.401903
$574$	−4695.40	−0.341433
$575$	32525.3	2.35895
$576$	−234.470	−0.0169611
$577$	552.177	0.0398396	0.0199198	−	0.999802i	$-0.493659\pi$
$577$	552.177	0.0398396	0.0199198	+	0.999802i	$0.493659\pi$
$578$	0	0
$579$	−533.554	−0.0382966
$580$	−1728.68	−0.123758
$581$	−2188.30	−0.156258
$582$	−32733.1	−2.33132
$583$	−12680.1	−0.900780
$584$	−5090.13	−0.360670
$585$	−332.854	−0.0235244
$586$	40814.9	2.87722
$587$	−16616.0	−1.16834	−0.584170	−	0.811631i	$-0.698580\pi$
$587$	−16616.0	−1.16834	−0.584170	+	0.811631i	$0.698580\pi$
$588$	2966.98	0.208089
$589$	1945.83	0.136123
$590$	−40388.9	−2.81828
$591$	22775.2	1.58519
$592$	−7690.35	−0.533905
$593$	8807.53	0.609919	0.304959	−	0.952365i	$-0.401357\pi$
$593$	8807.53	0.609919	0.304959	+	0.952365i	$0.401357\pi$
$594$	13739.9	0.949085
$595$	0	0
$596$	25623.7	1.76106
$597$	−19722.9	−1.35210
$598$	61846.2	4.22923
$599$	1706.32	0.116391	0.0581957	−	0.998305i	$-0.481465\pi$
$599$	1706.32	0.116391	0.0581957	+	0.998305i	$0.481465\pi$
$600$	13348.7	0.908266
$601$	−7869.44	−0.534111	−0.267056	−	0.963681i	$-0.586051\pi$
$601$	−7869.44	−0.534111	−0.267056	+	0.963681i	$0.586051\pi$
$602$	−4676.71	−0.316625
$603$	5.63401	0.000380489 0
$604$	154.731	0.0104237
$605$	14134.3	0.949819
$606$	25890.0	1.73549
$607$	−9589.90	−0.641256	−0.320628	−	0.947205i	$-0.603894\pi$
$607$	−9589.90	−0.641256	−0.320628	+	0.947205i	$0.603894\pi$
$608$	−1481.00	−0.0987870
$609$	−322.582	−0.0214642
$610$	2050.23	0.136084
$611$	−21306.4	−1.41074
$612$	0	0
$613$	−19106.4	−1.25889	−0.629446	−	0.777045i	$-0.716718\pi$
$613$	−19106.4	−1.25889	−0.629446	+	0.777045i	$0.716718\pi$
$614$	738.636	0.0485487
$615$	−13380.8	−0.877342
$616$	2475.68	0.161928
$617$	20503.2	1.33781	0.668905	−	0.743348i	$-0.266763\pi$
$617$	20503.2	1.33781	0.668905	+	0.743348i	$0.266763\pi$
$618$	−21509.5	−1.40006
$619$	−2652.77	−0.172252	−0.0861258	−	0.996284i	$-0.527449\pi$
$619$	−2652.77	−0.172252	−0.0861258	+	0.996284i	$0.527449\pi$
$620$	−58227.9	−3.77176
$621$	−28237.4	−1.82469
$622$	2677.64	0.172610
$623$	3107.03	0.199808
$624$	−8065.91	−0.517460
$625$	−9881.06	−0.632388
$626$	27410.8	1.75009
$627$	−760.861	−0.0484623
$628$	13789.0	0.876181
$629$	0	0
$630$	149.357	0.00944526
$631$	15223.5	0.960440	0.480220	−	0.877148i	$-0.340557\pi$
$631$	15223.5	0.960440	0.480220	+	0.877148i	$0.340557\pi$
$632$	5921.75	0.372713
$633$	−30520.1	−1.91638
$634$	−7983.51	−0.500104
$635$	13428.0	0.839170
$636$	34515.2	2.15191
$637$	3383.46	0.210451
$638$	−868.655	−0.0539034
$639$	19.8061	0.00122616
$640$	30932.5	1.91049
$641$	−8978.54	−0.553247	−0.276623	−	0.960978i	$-0.589215\pi$
$641$	−8978.54	−0.553247	−0.276623	+	0.960978i	$0.589215\pi$
$642$	21729.8	1.33584
$643$	13807.1	0.846810	0.423405	−	0.905940i	$-0.360835\pi$
$643$	13807.1	0.846810	0.423405	+	0.905940i	$0.360835\pi$
$644$	−16419.6	−1.00470
$645$	−13327.5	−0.813597
$646$	0	0
$647$	5033.05	0.305826	0.152913	−	0.988240i	$-0.451135\pi$
$647$	5033.05	0.305826	0.152913	+	0.988240i	$0.451135\pi$
$648$	−11711.4	−0.709979
$649$	−12008.0	−0.726282
$650$	49126.4	2.96446
$651$	−10865.6	−0.654160
$652$	−40424.3	−2.42812
$653$	−10208.1	−0.611754	−0.305877	−	0.952071i	$-0.598950\pi$
$653$	−10208.1	−0.611754	−0.305877	+	0.952071i	$0.598950\pi$
$654$	−5794.00	−0.346427
$655$	27617.2	1.64747
$656$	−3388.92	−0.201700
$657$	−91.2999	−0.00542153
$658$	9560.50	0.566424
$659$	15428.3	0.911990	0.455995	−	0.889982i	$-0.349284\pi$
$659$	15428.3	0.911990	0.455995	+	0.889982i	$0.349284\pi$
$660$	22768.3	1.34281
$661$	22651.9	1.33291	0.666457	−	0.745544i	$-0.267810\pi$
$661$	22651.9	1.33291	0.666457	+	0.745544i	$0.267810\pi$
$662$	22624.1	1.32826
$663$	0	0
$664$	4970.21	0.290484
$665$	774.804	0.0451814
$666$	434.076	0.0252554
$667$	1785.20	0.103633
$668$	35840.8	2.07593
$669$	31813.6	1.83854
$670$	−1478.20	−0.0852354
$671$	609.553	0.0350694
$672$	8269.98	0.474734
$673$	17291.4	0.990393	0.495196	−	0.868781i	$-0.335096\pi$
$673$	17291.4	0.990393	0.495196	+	0.868781i	$0.335096\pi$
$674$	−17107.6	−0.977684
$675$	−22429.9	−1.27900
$676$	29802.0	1.69561
$677$	8539.36	0.484777	0.242389	−	0.970179i	$-0.422069\pi$
$677$	8539.36	0.484777	0.242389	+	0.970179i	$0.422069\pi$
$678$	−17535.4	−0.993278
$679$	9910.21	0.560116
$680$	0	0
$681$	26692.1	1.50197
$682$	−29259.2	−1.64281
$683$	−1930.75	−0.108167	−0.0540835	−	0.998536i	$-0.517224\pi$
$683$	−1930.75	−0.108167	−0.0540835	+	0.998536i	$0.517224\pi$
$684$	21.6458	0.00121001
$685$	13705.6	0.764474
$686$	−1518.21	−0.0844980
$687$	13026.2	0.723409
$688$	−3375.43	−0.187045
$689$	39360.1	2.17634
$690$	−79084.9	−4.36335
$691$	−32951.2	−1.81407	−0.907036	−	0.421054i	$-0.861660\pi$
$691$	−32951.2	−1.81407	−0.907036	+	0.421054i	$0.861660\pi$
$692$	36393.2	1.99922
$693$	44.4053	0.00243408
$694$	−123.436	−0.00675153
$695$	−51318.0	−2.80087
$696$	732.668	0.0399019
$697$	0	0
$698$	21669.6	1.17508
$699$	−11038.7	−0.597312
$700$	−13042.6	−0.704235
$701$	−21147.5	−1.13941	−0.569707	−	0.821848i	$-0.692943\pi$
$701$	−21147.5	−1.13941	−0.569707	+	0.821848i	$0.692943\pi$
$702$	−42650.0	−2.29305
$703$	2251.82	0.120809
$704$	18289.9	0.979159
$705$	27245.2	1.45548
$706$	17690.3	0.943038
$707$	−7838.41	−0.416965
$708$	32685.9	1.73504
$709$	14069.8	0.745279	0.372639	−	0.927976i	$-0.378453\pi$
$709$	14069.8	0.745279	0.372639	+	0.927976i	$0.378453\pi$
$710$	−5196.53	−0.274679
$711$	106.216	0.00560256
$712$	−7056.87	−0.371443
$713$	60131.7	3.15842
$714$	0	0
$715$	25964.3	1.35806
$716$	44306.4	2.31258
$717$	−22098.1	−1.15100
$718$	2135.55	0.111000
$719$	10892.1	0.564959	0.282479	−	0.959273i	$-0.408843\pi$
$719$	10892.1	0.564959	0.282479	+	0.959273i	$0.408843\pi$
$720$	107.799	0.00557975
$721$	6512.17	0.336374
$722$	−30170.0	−1.55514
$723$	−20486.9	−1.05383
$724$	−25023.4	−1.28451
$725$	1418.04	0.0726412
$726$	−19332.7	−0.988300
$727$	−4947.43	−0.252394	−0.126197	−	0.992005i	$-0.540277\pi$
$727$	−4947.43	−0.252394	−0.126197	+	0.992005i	$0.540277\pi$
$728$	−7684.72	−0.391229
$729$	19470.7	0.989216
$730$	23954.3	1.21451
$731$	0	0
$732$	−1659.21	−0.0837787
$733$	−3023.86	−0.152372	−0.0761861	−	0.997094i	$-0.524274\pi$
$733$	−3023.86	−0.152372	−0.0761861	+	0.997094i	$0.524274\pi$
$734$	−24323.0	−1.22313
$735$	−4326.55	−0.217125
$736$	−45767.1	−2.29211
$737$	−439.483	−0.0219655
$738$	191.285	0.00954106
$739$	22602.2	1.12508	0.562541	−	0.826769i	$-0.309824\pi$
$739$	22602.2	1.12508	0.562541	+	0.826769i	$0.309824\pi$
$740$	−67384.3	−3.34743
$741$	2361.78	0.117088
$742$	−17661.5	−0.873820
$743$	4371.41	0.215843	0.107922	−	0.994159i	$-0.465580\pi$
$743$	4371.41	0.215843	0.107922	+	0.994159i	$0.465580\pi$
$744$	24678.7	1.21608
$745$	−37365.4	−1.83753
$746$	−13451.0	−0.660155
$747$	89.1489	0.00436652
$748$	0	0
$749$	−6578.88	−0.320944
$750$	−13966.3	−0.679971
$751$	9415.34	0.457484	0.228742	−	0.973487i	$-0.426539\pi$
$751$	9415.34	0.457484	0.228742	+	0.973487i	$0.426539\pi$
$752$	6900.32	0.334613
$753$	3.52386	0.000170540 0
$754$	2696.38	0.130234
$755$	−225.634	−0.0108764
$756$	11323.2	0.544737
$757$	−20071.8	−0.963704	−0.481852	−	0.876253i	$-0.660036\pi$
$757$	−20071.8	−0.963704	−0.481852	+	0.876253i	$0.660036\pi$
$758$	−61571.5	−2.95037
$759$	−23512.8	−1.12445
$760$	−1759.78	−0.0839921
$761$	420.505	0.0200306	0.0100153	−	0.999950i	$-0.496812\pi$
$761$	420.505	0.0200306	0.0100153	+	0.999950i	$0.496812\pi$
$762$	−18366.7	−0.873168
$763$	1754.18	0.0832315
$764$	−12233.4	−0.579303
$765$	0	0
$766$	18534.0	0.874229
$767$	37274.1	1.75474
$768$	−7950.68	−0.373562
$769$	−21370.8	−1.00215	−0.501074	−	0.865405i	$-0.667061\pi$
$769$	−21370.8	−1.00215	−0.501074	+	0.865405i	$0.667061\pi$
$770$	−11650.6	−0.545272
$771$	26956.2	1.25915
$772$	−1184.05	−0.0552007
$773$	−34813.4	−1.61986	−0.809929	−	0.586527i	$-0.800495\pi$
$773$	−34813.4	−1.61986	−0.809929	+	0.586527i	$0.800495\pi$
$774$	190.523	0.00884784
$775$	47764.5	2.21387
$776$	−22508.7	−1.04126
$777$	−12574.3	−0.580566
$778$	−32826.3	−1.51270
$779$	992.312	0.0456396
$780$	−70675.0	−3.24432
$781$	−1544.98	−0.0707859
$782$	0	0
$783$	−1231.10	−0.0561890
$784$	−1095.77	−0.0499168
$785$	−20107.6	−0.914230
$786$	−37774.6	−1.71422
$787$	23937.8	1.08423	0.542115	−	0.840304i	$-0.317624\pi$
$787$	23937.8	1.08423	0.542115	+	0.840304i	$0.317624\pi$
$788$	50542.1	2.28488
$789$	−16195.1	−0.730747
$790$	−27867.9	−1.25506
$791$	5308.99	0.238642
$792$	−100.856	−0.00452495
$793$	−1892.11	−0.0847299
$794$	−52840.3	−2.36175
$795$	−50331.1	−2.24536
$796$	−43768.7	−1.94892
$797$	−14091.2	−0.626267	−0.313134	−	0.949709i	$-0.601379\pi$
$797$	−14091.2	−0.626267	−0.313134	+	0.949709i	$0.601379\pi$
$798$	−1059.77	−0.0470119
$799$	0	0
$800$	−36354.2	−1.60664
$801$	−126.577	−0.00558347
$802$	67366.0	2.96606
$803$	7121.87	0.312983
$804$	1196.27	0.0524743
$805$	23943.6	1.04832
$806$	90823.3	3.96913
$807$	15581.2	0.679658
$808$	17803.1	0.775137
$809$	35107.7	1.52574	0.762868	−	0.646554i	$-0.223790\pi$
$809$	35107.7	1.52574	0.762868	+	0.646554i	$0.223790\pi$
$810$	55114.1	2.39076
$811$	7397.43	0.320295	0.160147	−	0.987093i	$-0.448803\pi$
$811$	7397.43	0.320295	0.160147	+	0.987093i	$0.448803\pi$
$812$	−715.866	−0.0309384
$813$	−3291.02	−0.141969
$814$	−33860.3	−1.45799
$815$	58948.0	2.53357
$816$	0	0
$817$	988.361	0.0423236
$818$	33351.3	1.42555
$819$	−137.838	−0.00588090
$820$	−29694.3	−1.26460
$821$	19148.5	0.813993	0.406997	−	0.913430i	$-0.366576\pi$
$821$	19148.5	0.813993	0.406997	+	0.913430i	$0.366576\pi$
$822$	−18746.4	−0.795446
$823$	−6580.95	−0.278733	−0.139367	−	0.990241i	$-0.544507\pi$
$823$	−6580.95	−0.278733	−0.139367	+	0.990241i	$0.544507\pi$
$824$	−14790.8	−0.625320
$825$	−18676.9	−0.788178
$826$	−16725.5	−0.704544
$827$	5487.32	0.230729	0.115364	−	0.993323i	$-0.463196\pi$
$827$	5487.32	0.230729	0.115364	+	0.993323i	$0.463196\pi$
$828$	668.916	0.0280754
$829$	−21781.2	−0.912537	−0.456269	−	0.889842i	$-0.650814\pi$
$829$	−21781.2	−0.912537	−0.456269	+	0.889842i	$0.650814\pi$
$830$	−23390.0	−0.978167
$831$	33697.5	1.40668
$832$	−56773.6	−2.36571
$833$	0	0
$834$	70192.4	2.91434
$835$	−52264.2	−2.16608
$836$	−1688.49	−0.0698535
$837$	−41467.7	−1.71246
$838$	−17242.0	−0.710756
$839$	−9781.70	−0.402505	−0.201252	−	0.979539i	$-0.564501\pi$
$839$	−9781.70	−0.402505	−0.201252	+	0.979539i	$0.564501\pi$
$840$	9826.72	0.403636
$841$	−24311.2	−0.996809
$842$	−38226.6	−1.56458
$843$	−36230.4	−1.48024
$844$	−67729.6	−2.76226
$845$	−43458.3	−1.76924
$846$	−389.484	−0.0158283
$847$	5853.15	0.237446
$848$	−12747.2	−0.516205
$849$	16688.5	0.674613
$850$	0	0
$851$	69587.5	2.80309
$852$	4205.44	0.169103
$853$	−42400.6	−1.70196	−0.850978	−	0.525202i	$-0.823990\pi$
$853$	−42400.6	−1.70196	−0.850978	+	0.525202i	$0.823990\pi$
$854$	849.020	0.0340198
$855$	−31.5646	−0.00126256
$856$	14942.4	0.596635
$857$	−33492.7	−1.33499	−0.667497	−	0.744612i	$-0.732634\pi$
$857$	−33492.7	−1.33499	−0.667497	+	0.744612i	$0.732634\pi$
$858$	−35513.8	−1.41308
$859$	17039.5	0.676810	0.338405	−	0.941001i	$-0.390113\pi$
$859$	17039.5	0.676810	0.338405	+	0.941001i	$0.390113\pi$
$860$	−29576.1	−1.17272
$861$	−5541.12	−0.219327
$862$	−43941.7	−1.73626
$863$	13005.4	0.512988	0.256494	−	0.966546i	$-0.417433\pi$
$863$	13005.4	0.512988	0.256494	+	0.966546i	$0.417433\pi$
$864$	31561.6	1.24276
$865$	−53069.7	−2.08604
$866$	−29107.2	−1.14215
$867$	0	0
$868$	−24112.8	−0.942905
$869$	−8285.43	−0.323434
$870$	−3447.96	−0.134364
$871$	1364.20	0.0530700
$872$	−3984.21	−0.154727
$873$	−403.730	−0.0156520
$874$	5864.89	0.226983
$875$	4228.42	0.163368
$876$	−19385.8	−0.747698
$877$	24871.7	0.957647	0.478824	−	0.877911i	$-0.341063\pi$
$877$	24871.7	0.957647	0.478824	+	0.877911i	$0.341063\pi$
$878$	−43021.7	−1.65366
$879$	48166.4	1.84825
$880$	−8408.86	−0.322117
$881$	5392.80	0.206229	0.103115	−	0.994669i	$-0.467119\pi$
$881$	5392.80	0.206229	0.103115	+	0.994669i	$0.467119\pi$
$882$	61.8502	0.00236123
$883$	−24850.0	−0.947077	−0.473539	−	0.880773i	$-0.657024\pi$
$883$	−24850.0	−0.947077	−0.473539	+	0.880773i	$0.657024\pi$
$884$	0	0
$885$	−47663.6	−1.81039
$886$	−50227.4	−1.90454
$887$	23725.5	0.898111	0.449056	−	0.893504i	$-0.351761\pi$
$887$	23725.5	0.898111	0.449056	+	0.893504i	$0.351761\pi$
$888$	28559.5	1.07927
$889$	5560.66	0.209785
$890$	33209.9	1.25078
$891$	16386.0	0.616107
$892$	70599.9	2.65007
$893$	−2020.49	−0.0757145
$894$	51108.0	1.91198
$895$	−64609.0	−2.41301
$896$	12809.5	0.477606
$897$	72985.7	2.71675
$898$	−72826.5	−2.70629
$899$	2621.64	0.0972597
$900$	531.341	0.0196793
$901$	0	0
$902$	−14921.2	−0.550802
$903$	−5519.06	−0.203392
$904$	−12058.1	−0.443635
$905$	36489.9	1.34029
$906$	308.620	0.0113170
$907$	−47638.2	−1.74399	−0.871996	−	0.489513i	$-0.837175\pi$
$907$	−47638.2	−1.74399	−0.871996	+	0.489513i	$0.837175\pi$
$908$	59234.6	2.16494
$909$	319.328	0.0116517
$910$	36164.6	1.31741
$911$	20622.7	0.750013	0.375007	−	0.927022i	$-0.377640\pi$
$911$	20622.7	0.750013	0.375007	+	0.927022i	$0.377640\pi$
$912$	−764.892	−0.0277720
$913$	−6954.09	−0.252077
$914$	−10633.7	−0.384827
$915$	2419.51	0.0874168
$916$	28907.5	1.04272
$917$	11436.6	0.411853
$918$	0	0
$919$	−4491.12	−0.161206	−0.0806030	−	0.996746i	$-0.525685\pi$
$919$	−4491.12	−0.161206	−0.0806030	+	0.996746i	$0.525685\pi$
$920$	−54382.2	−1.94884
$921$	871.676	0.0311864
$922$	55692.1	1.98929
$923$	4795.76	0.171023
$924$	9428.60	0.335691
$925$	55275.5	1.96481
$926$	−19123.0	−0.678639
$927$	−265.298	−0.00939971
$928$	−1995.36	−0.0705829
$929$	19354.9	0.683544	0.341772	−	0.939783i	$-0.388973\pi$
$929$	19354.9	0.683544	0.341772	+	0.939783i	$0.388973\pi$
$930$	−116139.	−4.09500
$931$	320.854	0.0112949
$932$	−24496.8	−0.860964
$933$	3159.93	0.110880
$934$	−2005.98	−0.0702757
$935$	0	0
$936$	313.066	0.0109326
$937$	−18723.3	−0.652788	−0.326394	−	0.945234i	$-0.605834\pi$
$937$	−18723.3	−0.652788	−0.326394	+	0.945234i	$0.605834\pi$
$938$	−612.136	−0.0213081
$939$	32347.9	1.12421
$940$	60461.9	2.09792
$941$	28104.8	0.973634	0.486817	−	0.873504i	$-0.338158\pi$
$941$	28104.8	0.973634	0.486817	+	0.873504i	$0.338158\pi$
$942$	27503.0	0.951269
$943$	30665.2	1.05896
$944$	−12071.7	−0.416206
$945$	−16511.9	−0.568392
$946$	−14861.8	−0.510782
$947$	47298.0	1.62300	0.811498	−	0.584355i	$-0.198652\pi$
$947$	47298.0	1.62300	0.811498	+	0.584355i	$0.198652\pi$
$948$	22552.9	0.772664
$949$	−22107.0	−0.756188
$950$	4658.67	0.159102
$951$	−9421.48	−0.321254
$952$	0	0
$953$	−31810.9	−1.08128	−0.540639	−	0.841255i	$-0.681817\pi$
$953$	−31810.9	−1.08128	−0.540639	+	0.841255i	$0.681817\pi$
$954$	719.509	0.0244182
$955$	17839.1	0.604459
$956$	−49039.6	−1.65905
$957$	−1025.11	−0.0346261
$958$	32144.2	1.08406
$959$	5675.64	0.191111
$960$	72598.4	2.44073
$961$	58514.5	1.96417
$962$	105105.	3.52259
$963$	268.016	0.00896853
$964$	−45464.1	−1.51898
$965$	1726.62	0.0575978
$966$	−32749.9	−1.09080
$967$	5938.61	0.197490	0.0987450	−	0.995113i	$-0.468517\pi$
$967$	5938.61	0.197490	0.0987450	+	0.995113i	$0.468517\pi$
$968$	−13294.0	−0.441412
$969$	0	0
$970$	105927.	3.50629
$971$	16153.1	0.533860	0.266930	−	0.963716i	$-0.413991\pi$
$971$	16153.1	0.533860	0.266930	+	0.963716i	$0.413991\pi$
$972$	−927.511	−0.0306069
$973$	−21251.3	−0.700191
$974$	47161.9	1.55151
$975$	57974.9	1.90429
$976$	612.783	0.0200970
$977$	−49414.9	−1.61814	−0.809070	−	0.587713i	$-0.800029\pi$
$977$	−49414.9	−1.61814	−0.809070	+	0.587713i	$0.800029\pi$
$978$	−80628.5	−2.63621
$979$	9873.64	0.322332
$980$	−9601.38	−0.312964
$981$	−71.4632	−0.00232584
$982$	94387.0	3.06722
$983$	43923.2	1.42516	0.712581	−	0.701590i	$-0.247526\pi$
$983$	43923.2	1.42516	0.712581	+	0.701590i	$0.247526\pi$
$984$	12585.3	0.407730
$985$	−73702.1	−2.38411
$986$	0	0
$987$	11282.5	0.363856
$988$	5241.22	0.168771
$989$	30543.1	0.982017
$990$	474.632	0.0152372
$991$	−24956.2	−0.799961	−0.399980	−	0.916524i	$-0.630983\pi$
$991$	−24956.2	−0.799961	−0.399980	+	0.916524i	$0.630983\pi$
$992$	−67210.5	−2.15115
$993$	26699.0	0.853241
$994$	−2151.94	−0.0686672
$995$	63824.9	2.03355
$996$	18929.0	0.602198
$997$	−51762.8	−1.64428	−0.822138	−	0.569288i	$-0.807219\pi$
$997$	−51762.8	−1.64428	−0.822138	+	0.569288i	$0.807219\pi$
$998$	26750.3	0.848463
$999$	−47988.5	−1.51981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2023.4.a.h.1.8		9
17.16	even	2		119.4.a.e.1.8	✓	9
51.50	odd	2		1071.4.a.r.1.2		9
68.67	odd	2		1904.4.a.s.1.8		9
119.118	odd	2		833.4.a.g.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.4.a.e.1.8	✓	9	17.16	even	2
833.4.a.g.1.8		9	119.118	odd	2
1071.4.a.r.1.2		9	51.50	odd	2
1904.4.a.s.1.8		9	68.67	odd	2
2023.4.a.h.1.8		9	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:	\( N \)	\(=\)	\( 2023 = 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2023.a (trivial)

\(f(q)\)	\(=\)	\(q+4.42628 q^{2} +5.22352 q^{3} +11.5919 q^{4} -16.9037 q^{5} +23.1207 q^{6} -7.00000 q^{7} +15.8988 q^{8} +0.285172 q^{9} +O(q^{10})\)	\(q+4.42628 q^{2} +5.22352 q^{3} +11.5919 q^{4} -16.9037 q^{5} +23.1207 q^{6} -7.00000 q^{7} +15.8988 q^{8} +0.285172 q^{9} -74.8205 q^{10} -22.2449 q^{11} +60.5506 q^{12} +69.0502 q^{13} -30.9839 q^{14} -88.2969 q^{15} -22.3627 q^{16} +1.26225 q^{18} +6.54805 q^{19} -195.946 q^{20} -36.5646 q^{21} -98.4621 q^{22} +202.353 q^{23} +83.0479 q^{24} +160.735 q^{25} +305.635 q^{26} -139.545 q^{27} -81.1434 q^{28} +8.82223 q^{29} -390.826 q^{30} +297.162 q^{31} -226.174 q^{32} -116.197 q^{33} +118.326 q^{35} +3.30569 q^{36} +343.891 q^{37} +28.9835 q^{38} +360.685 q^{39} -268.749 q^{40} +151.543 q^{41} -161.845 q^{42} +150.940 q^{43} -257.861 q^{44} -4.82046 q^{45} +895.670 q^{46} -308.563 q^{47} -116.812 q^{48} +49.0000 q^{49} +711.459 q^{50} +800.424 q^{52} +570.022 q^{53} -617.667 q^{54} +376.021 q^{55} -111.292 q^{56} +34.2039 q^{57} +39.0496 q^{58} +539.811 q^{59} -1023.53 q^{60} -27.4019 q^{61} +1315.32 q^{62} -1.99620 q^{63} -822.208 q^{64} -1167.20 q^{65} -514.319 q^{66} +19.7566 q^{67} +1057.00 q^{69} +523.743 q^{70} +69.4533 q^{71} +4.53390 q^{72} -320.158 q^{73} +1522.16 q^{74} +839.605 q^{75} +75.9044 q^{76} +155.714 q^{77} +1596.49 q^{78} +372.464 q^{79} +378.013 q^{80} -736.618 q^{81} +670.772 q^{82} +312.615 q^{83} -423.854 q^{84} +668.101 q^{86} +46.0831 q^{87} -353.668 q^{88} -443.861 q^{89} -21.3367 q^{90} -483.351 q^{91} +2345.66 q^{92} +1552.23 q^{93} -1365.79 q^{94} -110.686 q^{95} -1181.43 q^{96} -1415.74 q^{97} +216.888 q^{98} -6.34361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 2 q^{2} - 11 q^{3} + 38 q^{4} + 3 q^{5} - 9 q^{6} - 63 q^{7} + 24 q^{8} + 74 q^{9}+O(q^{10}) \) 9 * q + 2 * q^2 - 11 * q^3 + 38 * q^4 + 3 * q^5 - 9 * q^6 - 63 * q^7 + 24 * q^8 + 74 * q^9	\( 9 q + 2 q^{2} - 11 q^{3} + 38 q^{4} + 3 q^{5} - 9 q^{6} - 63 q^{7} + 24 q^{8} + 74 q^{9} - 134 q^{10} + 8 q^{11} - 56 q^{12} + 164 q^{13} - 14 q^{14} + 34 q^{15} + 178 q^{16} + 98 q^{18} + 244 q^{19} + 41 q^{20} + 77 q^{21} + 80 q^{22} + 14 q^{23} - 298 q^{24} + 684 q^{25} + 326 q^{26} - 218 q^{27} - 266 q^{28} + 234 q^{29} - 335 q^{30} - 555 q^{31} - 181 q^{32} + 458 q^{33} - 21 q^{35} - 1221 q^{36} + 364 q^{37} - 714 q^{38} + 52 q^{39} - 123 q^{40} + 45 q^{41} + 63 q^{42} - 135 q^{43} + 748 q^{44} + 844 q^{45} + 1576 q^{46} - 172 q^{47} + 949 q^{48} + 441 q^{49} - 2901 q^{50} - 1596 q^{52} + 101 q^{53} + 1163 q^{54} + 1260 q^{55} - 168 q^{56} + 602 q^{57} - 1062 q^{58} + 280 q^{59} - 1727 q^{60} - 639 q^{61} + 1708 q^{62} - 518 q^{63} - 2390 q^{64} - 638 q^{65} - 2476 q^{66} + 35 q^{67} + 1288 q^{69} + 938 q^{70} + 1616 q^{71} + 1335 q^{72} - 1049 q^{73} + 370 q^{74} - 1260 q^{75} + 4964 q^{76} - 56 q^{77} + 4714 q^{78} - 2304 q^{79} + 3996 q^{80} - 791 q^{81} + 215 q^{82} + 2508 q^{83} + 392 q^{84} + 623 q^{86} + 166 q^{87} + 416 q^{88} + 2762 q^{89} - 2935 q^{90} - 1148 q^{91} + 2392 q^{92} + 2784 q^{93} - 862 q^{94} + 3462 q^{95} - 2928 q^{96} - 3107 q^{97} + 98 q^{98} + 2396 q^{99}+O(q^{100}) \) 9 * q + 2 * q^2 - 11 * q^3 + 38 * q^4 + 3 * q^5 - 9 * q^6 - 63 * q^7 + 24 * q^8 + 74 * q^9 - 134 * q^10 + 8 * q^11 - 56 * q^12 + 164 * q^13 - 14 * q^14 + 34 * q^15 + 178 * q^16 + 98 * q^18 + 244 * q^19 + 41 * q^20 + 77 * q^21 + 80 * q^22 + 14 * q^23 - 298 * q^24 + 684 * q^25 + 326 * q^26 - 218 * q^27 - 266 * q^28 + 234 * q^29 - 335 * q^30 - 555 * q^31 - 181 * q^32 + 458 * q^33 - 21 * q^35 - 1221 * q^36 + 364 * q^37 - 714 * q^38 + 52 * q^39 - 123 * q^40 + 45 * q^41 + 63 * q^42 - 135 * q^43 + 748 * q^44 + 844 * q^45 + 1576 * q^46 - 172 * q^47 + 949 * q^48 + 441 * q^49 - 2901 * q^50 - 1596 * q^52 + 101 * q^53 + 1163 * q^54 + 1260 * q^55 - 168 * q^56 + 602 * q^57 - 1062 * q^58 + 280 * q^59 - 1727 * q^60 - 639 * q^61 + 1708 * q^62 - 518 * q^63 - 2390 * q^64 - 638 * q^65 - 2476 * q^66 + 35 * q^67 + 1288 * q^69 + 938 * q^70 + 1616 * q^71 + 1335 * q^72 - 1049 * q^73 + 370 * q^74 - 1260 * q^75 + 4964 * q^76 - 56 * q^77 + 4714 * q^78 - 2304 * q^79 + 3996 * q^80 - 791 * q^81 + 215 * q^82 + 2508 * q^83 + 392 * q^84 + 623 * q^86 + 166 * q^87 + 416 * q^88 + 2762 * q^89 - 2935 * q^90 - 1148 * q^91 + 2392 * q^92 + 2784 * q^93 - 862 * q^94 + 3462 * q^95 - 2928 * q^96 - 3107 * q^97 + 98 * q^98 + 2396 * q^99

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