LMFDB - Embedded newform 1849.4.a.g.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,4,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.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:	$109.094531601$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.32038 q^{2} +4.35495 q^{3} -6.25659 q^{4} +15.6111 q^{5} +5.75020 q^{6} -4.97789 q^{7} -18.8242 q^{8} -8.03441 q^{9} +O(q^{10})$	\(q+1.32038 q^{2} +4.35495 q^{3} -6.25659 q^{4} +15.6111 q^{5} +5.75020 q^{6} -4.97789 q^{7} -18.8242 q^{8} -8.03441 q^{9} +20.6126 q^{10} +3.73387 q^{11} -27.2471 q^{12} +34.7034 q^{13} -6.57272 q^{14} +67.9853 q^{15} +25.1976 q^{16} -90.1847 q^{17} -10.6085 q^{18} -19.2643 q^{19} -97.6719 q^{20} -21.6785 q^{21} +4.93014 q^{22} +34.2277 q^{23} -81.9782 q^{24} +118.705 q^{25} +45.8217 q^{26} -152.573 q^{27} +31.1446 q^{28} +29.2747 q^{29} +89.7667 q^{30} -224.399 q^{31} +183.864 q^{32} +16.2608 q^{33} -119.078 q^{34} -77.7101 q^{35} +50.2680 q^{36} -214.118 q^{37} -25.4362 q^{38} +151.131 q^{39} -293.865 q^{40} +487.539 q^{41} -28.6238 q^{42} -23.3613 q^{44} -125.426 q^{45} +45.1936 q^{46} -114.819 q^{47} +109.734 q^{48} -318.221 q^{49} +156.736 q^{50} -392.750 q^{51} -217.125 q^{52} -683.378 q^{53} -201.455 q^{54} +58.2897 q^{55} +93.7045 q^{56} -83.8949 q^{57} +38.6538 q^{58} -370.581 q^{59} -425.356 q^{60} +854.289 q^{61} -296.292 q^{62} +39.9944 q^{63} +41.1893 q^{64} +541.756 q^{65} +21.4705 q^{66} -475.209 q^{67} +564.249 q^{68} +149.060 q^{69} -102.607 q^{70} -879.804 q^{71} +151.241 q^{72} +670.576 q^{73} -282.717 q^{74} +516.954 q^{75} +120.529 q^{76} -18.5868 q^{77} +199.551 q^{78} -115.793 q^{79} +393.362 q^{80} -447.519 q^{81} +643.738 q^{82} -939.943 q^{83} +135.633 q^{84} -1407.88 q^{85} +127.490 q^{87} -70.2870 q^{88} +598.117 q^{89} -165.610 q^{90} -172.749 q^{91} -214.149 q^{92} -977.246 q^{93} -151.605 q^{94} -300.736 q^{95} +800.717 q^{96} +664.593 q^{97} -420.173 q^{98} -29.9995 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9}+O(q^{10}) $ 30 * q - 6 * q^2 - 2 * q^3 + 114 * q^4 - 27 * q^5 + 8 * q^6 - 48 * q^7 - 90 * q^8 + 216 * q^9	\( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9} - 27 q^{10} + 80 q^{11} + 36 q^{12} - 13 q^{13} + 36 q^{14} + 16 q^{15} + 318 q^{16} + 66 q^{17} - 80 q^{18} - 254 q^{19} - 312 q^{20} - 548 q^{21} - 305 q^{22} - 105 q^{23} + 123 q^{24} + 523 q^{25} - 549 q^{26} + 10 q^{27} - 578 q^{28} - 793 q^{29} - 1560 q^{30} - 359 q^{31} - 676 q^{32} - 208 q^{33} - 1007 q^{34} - 514 q^{35} + 776 q^{36} - 510 q^{37} - 2066 q^{38} - 898 q^{39} - 1248 q^{40} - 270 q^{41} + 915 q^{42} + 3256 q^{44} - 807 q^{45} - 1960 q^{46} + 1421 q^{47} + 632 q^{48} + 386 q^{49} + 141 q^{50} - 209 q^{51} + 2825 q^{52} - 21 q^{53} + 2368 q^{54} - 2258 q^{55} + 2521 q^{56} - 1723 q^{57} - 347 q^{58} + 1752 q^{59} + 2711 q^{60} - 1759 q^{61} - 395 q^{62} - 2204 q^{63} + 222 q^{64} - 1151 q^{65} + 160 q^{66} - 3001 q^{67} + 1921 q^{68} - 1660 q^{69} - 1597 q^{70} - 727 q^{71} - 9100 q^{72} - 4623 q^{73} - 2649 q^{74} - 1027 q^{75} - 874 q^{76} - 3556 q^{77} - 4979 q^{78} + 546 q^{79} - 5809 q^{80} - 410 q^{81} + 4397 q^{82} - 492 q^{83} - 10611 q^{84} + 1723 q^{85} + 5937 q^{87} - 3974 q^{88} - 5218 q^{89} + 10492 q^{90} - 1104 q^{91} + 1060 q^{92} - 1997 q^{93} + 2134 q^{94} + 6346 q^{95} - 11984 q^{96} + 2590 q^{97} - 6270 q^{98} - 2693 q^{99}+O(q^{100}) \) 30 * q - 6 * q^2 - 2 * q^3 + 114 * q^4 - 27 * q^5 + 8 * q^6 - 48 * q^7 - 90 * q^8 + 216 * q^9 - 27 * q^10 + 80 * q^11 + 36 * q^12 - 13 * q^13 + 36 * q^14 + 16 * q^15 + 318 * q^16 + 66 * q^17 - 80 * q^18 - 254 * q^19 - 312 * q^20 - 548 * q^21 - 305 * q^22 - 105 * q^23 + 123 * q^24 + 523 * q^25 - 549 * q^26 + 10 * q^27 - 578 * q^28 - 793 * q^29 - 1560 * q^30 - 359 * q^31 - 676 * q^32 - 208 * q^33 - 1007 * q^34 - 514 * q^35 + 776 * q^36 - 510 * q^37 - 2066 * q^38 - 898 * q^39 - 1248 * q^40 - 270 * q^41 + 915 * q^42 + 3256 * q^44 - 807 * q^45 - 1960 * q^46 + 1421 * q^47 + 632 * q^48 + 386 * q^49 + 141 * q^50 - 209 * q^51 + 2825 * q^52 - 21 * q^53 + 2368 * q^54 - 2258 * q^55 + 2521 * q^56 - 1723 * q^57 - 347 * q^58 + 1752 * q^59 + 2711 * q^60 - 1759 * q^61 - 395 * q^62 - 2204 * q^63 + 222 * q^64 - 1151 * q^65 + 160 * q^66 - 3001 * q^67 + 1921 * q^68 - 1660 * q^69 - 1597 * q^70 - 727 * q^71 - 9100 * q^72 - 4623 * q^73 - 2649 * q^74 - 1027 * q^75 - 874 * q^76 - 3556 * q^77 - 4979 * q^78 + 546 * q^79 - 5809 * q^80 - 410 * q^81 + 4397 * q^82 - 492 * q^83 - 10611 * q^84 + 1723 * q^85 + 5937 * q^87 - 3974 * q^88 - 5218 * q^89 + 10492 * q^90 - 1104 * q^91 + 1060 * q^92 - 1997 * q^93 + 2134 * q^94 + 6346 * q^95 - 11984 * q^96 + 2590 * q^97 - 6270 * q^98 - 2693 * 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 $

$2$	1.32038	0.466826	0.233413	−	0.972378i	$-0.425011\pi$
$2$	1.32038	0.466826	0.233413	+	0.972378i	$0.425011\pi$
$3$	4.35495	0.838110	0.419055	−	0.907961i	$-0.362361\pi$
$3$	4.35495	0.838110	0.419055	+	0.907961i	$0.362361\pi$
$4$	−6.25659	−0.782074
$5$	15.6111	1.39629	0.698147	−	0.715954i	$-0.254008\pi$
$5$	15.6111	1.39629	0.698147	+	0.715954i	$0.254008\pi$
$6$	5.75020	0.391252
$7$	−4.97789	−0.268781	−0.134390	−	0.990928i	$-0.542908\pi$
$7$	−4.97789	−0.268781	−0.134390	+	0.990928i	$0.542908\pi$
$8$	−18.8242	−0.831918
$9$	−8.03441	−0.297571
$10$	20.6126	0.651826
$11$	3.73387	0.102346	0.0511730	−	0.998690i	$-0.483704\pi$
$11$	3.73387	0.102346	0.0511730	+	0.998690i	$0.483704\pi$
$12$	−27.2471	−0.655464
$13$	34.7034	0.740383	0.370192	−	0.928955i	$-0.379292\pi$
$13$	34.7034	0.740383	0.370192	+	0.928955i	$0.379292\pi$
$14$	−6.57272	−0.125474
$15$	67.9853	1.17025
$16$	25.1976	0.393713
$17$	−90.1847	−1.28665	−0.643324	−	0.765594i	$-0.722445\pi$
$17$	−90.1847	−1.28665	−0.643324	+	0.765594i	$0.722445\pi$
$18$	−10.6085	−0.138914
$19$	−19.2643	−0.232607	−0.116303	−	0.993214i	$-0.537104\pi$
$19$	−19.2643	−0.232607	−0.116303	+	0.993214i	$0.537104\pi$
$20$	−97.6719	−1.09201
$21$	−21.6785	−0.225268
$22$	4.93014	0.0477777
$23$	34.2277	0.310303	0.155151	−	0.987891i	$-0.450413\pi$
$23$	34.2277	0.310303	0.155151	+	0.987891i	$0.450413\pi$
$24$	−81.9782	−0.697239
$25$	118.705	0.949639
$26$	45.8217	0.345630
$27$	−152.573	−1.08751
$28$	31.1446	0.210206
$29$	29.2747	0.187454	0.0937272	−	0.995598i	$-0.470122\pi$
$29$	29.2747	0.187454	0.0937272	+	0.995598i	$0.470122\pi$
$30$	89.7667	0.546302
$31$	−224.399	−1.30010	−0.650052	−	0.759890i	$-0.725253\pi$
$31$	−224.399	−1.30010	−0.650052	+	0.759890i	$0.725253\pi$
$32$	183.864	1.01571
$33$	16.2608	0.0857772
$34$	−119.078	−0.600640
$35$	−77.7101	−0.375297
$36$	50.2680	0.232722
$37$	−214.118	−0.951371	−0.475685	−	0.879615i	$-0.657800\pi$
$37$	−214.118	−0.951371	−0.475685	+	0.879615i	$0.657800\pi$
$38$	−25.4362	−0.108587
$39$	151.131	0.620523
$40$	−293.865	−1.16160
$41$	487.539	1.85709	0.928546	−	0.371218i	$-0.121060\pi$
$41$	487.539	1.85709	0.928546	+	0.371218i	$0.121060\pi$
$42$	−28.6238	−0.105161
$43$	0	0
$43$	0	0
$44$	−23.3613	−0.0800421
$45$	−125.426	−0.415497
$46$	45.1936	0.144857
$47$	−114.819	−0.356343	−0.178171	−	0.983999i	$-0.557018\pi$
$47$	−114.819	−0.356343	−0.178171	+	0.983999i	$0.557018\pi$
$48$	109.734	0.329975
$49$	−318.221	−0.927757
$50$	156.736	0.443316
$51$	−392.750	−1.07835
$52$	−217.125	−0.579034
$53$	−683.378	−1.77112	−0.885558	−	0.464528i	$-0.846224\pi$
$53$	−683.378	−1.77112	−0.885558	+	0.464528i	$0.846224\pi$
$54$	−201.455	−0.507677
$55$	58.2897	0.142905
$56$	93.7045	0.223603
$57$	−83.8949	−0.194950
$58$	38.6538	0.0875085
$59$	−370.581	−0.817721	−0.408861	−	0.912597i	$-0.634074\pi$
$59$	−370.581	−0.817721	−0.408861	+	0.912597i	$0.634074\pi$
$60$	−425.356	−0.915221
$61$	854.289	1.79312	0.896562	−	0.442919i	$-0.146057\pi$
$61$	854.289	1.79312	0.896562	+	0.442919i	$0.146057\pi$
$62$	−296.292	−0.606922
$63$	39.9944	0.0799813
$64$	41.1893	0.0804479
$65$	541.756	1.03379
$66$	21.4705	0.0400430
$67$	−475.209	−0.866508	−0.433254	−	0.901272i	$-0.642635\pi$
$67$	−475.209	−0.866508	−0.433254	+	0.901272i	$0.642635\pi$
$68$	564.249	1.00625
$69$	149.060	0.260068
$70$	−102.607	−0.175198
$71$	−879.804	−1.47061	−0.735306	−	0.677735i	$-0.762962\pi$
$71$	−879.804	−1.47061	−0.735306	+	0.677735i	$0.762962\pi$
$72$	151.241	0.247555
$73$	670.576	1.07514	0.537568	−	0.843220i	$-0.319343\pi$
$73$	670.576	1.07514	0.537568	+	0.843220i	$0.319343\pi$
$74$	−282.717	−0.444124
$75$	516.954	0.795903
$76$	120.529	0.181916
$77$	−18.5868	−0.0275086
$78$	199.551	0.289676
$79$	−115.793	−0.164908	−0.0824539	−	0.996595i	$-0.526276\pi$
$79$	−115.793	−0.164908	−0.0824539	+	0.996595i	$0.526276\pi$
$80$	393.362	0.549740
$81$	−447.519	−0.613881
$82$	643.738	0.866938
$83$	−939.943	−1.24304	−0.621519	−	0.783399i	$-0.713484\pi$
$83$	−939.943	−1.24304	−0.621519	+	0.783399i	$0.713484\pi$
$84$	135.633	0.176176
$85$	−1407.88	−1.79654
$86$	0	0
$87$	127.490	0.157107
$88$	−70.2870	−0.0851434
$89$	598.117	0.712362	0.356181	−	0.934417i	$-0.384079\pi$
$89$	598.117	0.712362	0.356181	+	0.934417i	$0.384079\pi$
$90$	−165.610	−0.193965
$91$	−172.749	−0.199001
$92$	−214.149	−0.242680
$93$	−977.246	−1.08963
$94$	−151.605	−0.166350
$95$	−300.736	−0.324788
$96$	800.717	0.851280
$97$	664.593	0.695661	0.347831	−	0.937557i	$-0.386918\pi$
$97$	664.593	0.695661	0.347831	+	0.937557i	$0.386918\pi$
$98$	−420.173	−0.433101
$99$	−29.9995	−0.0304552
$100$	−742.688	−0.742688
$101$	−1130.18	−1.11344	−0.556719	−	0.830701i	$-0.687940\pi$
$101$	−1130.18	−1.11344	−0.556719	+	0.830701i	$0.687940\pi$
$102$	−518.580	−0.503403
$103$	−1560.33	−1.49266	−0.746329	−	0.665577i	$-0.768185\pi$
$103$	−1560.33	−1.49266	−0.746329	+	0.665577i	$0.768185\pi$
$104$	−653.261	−0.615938
$105$	−338.423	−0.314540
$106$	−902.320	−0.826803
$107$	964.626	0.871532	0.435766	−	0.900060i	$-0.356477\pi$
$107$	964.626	0.871532	0.435766	+	0.900060i	$0.356477\pi$
$108$	954.587	0.850511
$109$	1.57769	0.00138637	0.000693187	−	1.00000i	$-0.499779\pi$
$109$	1.57769	0.00138637	0.000693187		1.00000i	$0.499779\pi$
$110$	76.9647	0.0667118
$111$	−932.472	−0.797354
$112$	−125.431	−0.105822
$113$	461.586	0.384269	0.192134	−	0.981369i	$-0.438459\pi$
$113$	461.586	0.384269	0.192134	+	0.981369i	$0.438459\pi$
$114$	−110.773	−0.0910077
$115$	534.330	0.433274
$116$	−183.160	−0.146603
$117$	−278.821	−0.220316
$118$	−489.309	−0.381733
$119$	448.929	0.345826
$120$	−1279.77	−0.973551
$121$	−1317.06	−0.989525
$122$	1127.99	0.837076
$123$	2123.21	1.55645
$124$	1403.97	1.01678
$125$	−98.2728	−0.0703183
$126$	52.8079	0.0373373
$127$	−1877.59	−1.31188	−0.655942	−	0.754811i	$-0.727728\pi$
$127$	−1877.59	−1.31188	−0.655942	+	0.754811i	$0.727728\pi$
$128$	−1416.52	−0.978158
$129$	0	0
$130$	715.325	0.482601
$131$	−812.688	−0.542022	−0.271011	−	0.962576i	$-0.587358\pi$
$131$	−812.688	−0.542022	−0.271011	+	0.962576i	$0.587358\pi$
$132$	−101.737	−0.0670841
$133$	95.8954	0.0625202
$134$	−627.458	−0.404508
$135$	−2381.83	−1.51848
$136$	1697.65	1.07038
$137$	−898.575	−0.560368	−0.280184	−	0.959946i	$-0.590395\pi$
$137$	−898.575	−0.560368	−0.280184	+	0.959946i	$0.590395\pi$
$138$	196.816	0.121406
$139$	−423.030	−0.258136	−0.129068	−	0.991636i	$-0.541199\pi$
$139$	−423.030	−0.258136	−0.129068	+	0.991636i	$0.541199\pi$
$140$	486.200	0.293510
$141$	−500.032	−0.298654
$142$	−1161.68	−0.686520
$143$	129.578	0.0757752
$144$	−202.448	−0.117158
$145$	457.009	0.261742
$146$	885.417	0.501901
$147$	−1385.83	−0.777563
$148$	1339.65	0.744042
$149$	473.514	0.260348	0.130174	−	0.991491i	$-0.458446\pi$
$149$	473.514	0.260348	0.130174	+	0.991491i	$0.458446\pi$
$150$	682.577	0.371548
$151$	1307.13	0.704453	0.352227	−	0.935915i	$-0.385425\pi$
$151$	1307.13	0.704453	0.352227	+	0.935915i	$0.385425\pi$
$152$	362.634	0.193510
$153$	724.581	0.382869
$154$	−24.5417	−0.0128417
$155$	−3503.10	−1.81533
$156$	−945.567	−0.485295
$157$	−2801.81	−1.42426	−0.712129	−	0.702049i	$-0.752269\pi$
$157$	−2801.81	−1.42426	−0.712129	+	0.702049i	$0.752269\pi$
$158$	−152.891	−0.0769832
$159$	−2976.08	−1.48439
$160$	2870.31	1.41824
$161$	−170.382	−0.0834034
$162$	−590.896	−0.286575
$163$	2190.20	1.05245	0.526227	−	0.850344i	$-0.323606\pi$
$163$	2190.20	1.05245	0.526227	+	0.850344i	$0.323606\pi$
$164$	−3050.33	−1.45238
$165$	253.849	0.119770
$166$	−1241.08	−0.580282
$167$	2913.56	1.35005	0.675025	−	0.737795i	$-0.264133\pi$
$167$	2913.56	1.35005	0.675025	+	0.737795i	$0.264133\pi$
$168$	408.078	0.187404
$169$	−992.677	−0.451833
$170$	−1858.94	−0.838671
$171$	154.777	0.0692170
$172$	0	0
$173$	3189.01	1.40148	0.700740	−	0.713416i	$-0.252853\pi$
$173$	3189.01	1.40148	0.700740	+	0.713416i	$0.252853\pi$
$174$	168.335	0.0733418
$175$	−590.900	−0.255245
$176$	94.0848	0.0402949
$177$	−1613.86	−0.685341
$178$	789.743	0.332549
$179$	−2902.73	−1.21207	−0.606034	−	0.795439i	$-0.707240\pi$
$179$	−2902.73	−1.21207	−0.606034	+	0.795439i	$0.707240\pi$
$180$	784.737	0.324949
$181$	4262.88	1.75059	0.875296	−	0.483587i	$-0.160666\pi$
$181$	4262.88	1.75059	0.875296	+	0.483587i	$0.160666\pi$
$182$	−228.095	−0.0928986
$183$	3720.39	1.50284
$184$	−644.307	−0.258146
$185$	−3342.60	−1.32839
$186$	−1290.34	−0.508668
$187$	−336.738	−0.131683
$188$	718.377	0.278686
$189$	759.492	0.292301
$190$	−397.086	−0.151619
$191$	−1817.08	−0.688374	−0.344187	−	0.938901i	$-0.611846\pi$
$191$	−1817.08	−0.688374	−0.344187	+	0.938901i	$0.611846\pi$
$192$	179.377	0.0674242
$193$	−891.204	−0.332385	−0.166192	−	0.986093i	$-0.553147\pi$
$193$	−891.204	−0.332385	−0.166192	+	0.986093i	$0.553147\pi$
$194$	877.516	0.324753
$195$	2359.32	0.866433
$196$	1990.98	0.725574
$197$	−1697.67	−0.613980	−0.306990	−	0.951713i	$-0.599322\pi$
$197$	−1697.67	−0.613980	−0.306990	+	0.951713i	$0.599322\pi$
$198$	−39.6108	−0.0142173
$199$	2475.28	0.881750	0.440875	−	0.897569i	$-0.354668\pi$
$199$	2475.28	0.881750	0.440875	+	0.897569i	$0.354668\pi$
$200$	−2234.52	−0.790022
$201$	−2069.51	−0.726229
$202$	−1492.27	−0.519781
$203$	−145.726	−0.0503841
$204$	2457.27	0.843351
$205$	7610.99	2.59305
$206$	−2060.23	−0.696811
$207$	−274.999	−0.0923371
$208$	874.443	0.291499
$209$	−71.9304	−0.0238064
$210$	−446.848	−0.146836
$211$	4547.34	1.48366	0.741829	−	0.670589i	$-0.233959\pi$
$211$	4547.34	1.48366	0.741829	+	0.670589i	$0.233959\pi$
$212$	4275.62	1.38514
$213$	−3831.50	−1.23254
$214$	1273.68	0.406854
$215$	0	0
$216$	2872.06	0.904717
$217$	1117.03	0.349443
$218$	2.08315	0.000647195 0
$219$	2920.32	0.901083
$220$	−364.695	−0.111762
$221$	−3129.71	−0.952612
$222$	−1231.22	−0.372225
$223$	−3359.59	−1.00885	−0.504427	−	0.863454i	$-0.668296\pi$
$223$	−3359.59	−1.00885	−0.504427	+	0.863454i	$0.668296\pi$
$224$	−915.253	−0.273004
$225$	−953.724	−0.282585
$226$	609.470	0.179386
$227$	−3617.71	−1.05778	−0.528889	−	0.848691i	$-0.677391\pi$
$227$	−3617.71	−1.05778	−0.528889	+	0.848691i	$0.677391\pi$
$228$	524.896	0.152465
$229$	2149.30	0.620218	0.310109	−	0.950701i	$-0.399634\pi$
$229$	2149.30	0.620218	0.310109	+	0.950701i	$0.399634\pi$
$230$	705.520	0.202264
$231$	−80.9446	−0.0230553
$232$	−551.071	−0.155947
$233$	1356.11	0.381296	0.190648	−	0.981658i	$-0.438941\pi$
$233$	1356.11	0.381296	0.190648	+	0.981658i	$0.438941\pi$
$234$	−368.151	−0.102849
$235$	−1792.45	−0.497559
$236$	2318.57	0.639518
$237$	−504.272	−0.138211
$238$	592.759	0.161440
$239$	−4542.30	−1.22936	−0.614679	−	0.788777i	$-0.710715\pi$
$239$	−4542.30	−1.22936	−0.614679	+	0.788777i	$0.710715\pi$
$240$	1713.07	0.460743
$241$	−344.024	−0.0919524	−0.0459762	−	0.998943i	$-0.514640\pi$
$241$	−344.024	−0.0919524	−0.0459762	+	0.998943i	$0.514640\pi$
$242$	−1739.02	−0.461936
$243$	2170.55	0.573008
$244$	−5344.94	−1.40236
$245$	−4967.76	−1.29542
$246$	2803.45	0.726590
$247$	−668.535	−0.172218
$248$	4224.12	1.08158
$249$	−4093.40	−1.04180
$250$	−129.758	−0.0328264
$251$	1574.04	0.395826	0.197913	−	0.980220i	$-0.436584\pi$
$251$	1574.04	0.395826	0.197913	+	0.980220i	$0.436584\pi$
$252$	−250.229	−0.0625513
$253$	127.802	0.0317582
$254$	−2479.14	−0.612421
$255$	−6131.24	−1.50570
$256$	−2199.87	−0.537077
$257$	−2254.21	−0.547136	−0.273568	−	0.961853i	$-0.588204\pi$
$257$	−2254.21	−0.547136	−0.273568	+	0.961853i	$0.588204\pi$
$258$	0	0
$259$	1065.85	0.255710
$260$	−3389.55	−0.808503
$261$	−235.205	−0.0557809
$262$	−1073.06	−0.253030
$263$	3718.26	0.871777	0.435889	−	0.900001i	$-0.356434\pi$
$263$	3718.26	0.871777	0.435889	+	0.900001i	$0.356434\pi$
$264$	−306.096	−0.0713596
$265$	−10668.3	−2.47300
$266$	126.619	0.0291860
$267$	2604.77	0.597038
$268$	2973.19	0.677673
$269$	−5900.56	−1.33741	−0.668706	−	0.743527i	$-0.733151\pi$
$269$	−5900.56	−1.33741	−0.668706	+	0.743527i	$0.733151\pi$
$270$	−3144.92	−0.708866
$271$	−6867.08	−1.53928	−0.769641	−	0.638476i	$-0.779565\pi$
$271$	−6867.08	−1.53928	−0.769641	+	0.638476i	$0.779565\pi$
$272$	−2272.44	−0.506570
$273$	−752.315	−0.166785
$274$	−1186.46	−0.261594
$275$	443.229	0.0971917
$276$	−932.606	−0.203392
$277$	1041.33	0.225876	0.112938	−	0.993602i	$-0.463974\pi$
$277$	1041.33	0.225876	0.112938	+	0.993602i	$0.463974\pi$
$278$	−558.562	−0.120505
$279$	1802.91	0.386873
$280$	1462.83	0.312216
$281$	−2022.29	−0.429323	−0.214662	−	0.976689i	$-0.568865\pi$
$281$	−2022.29	−0.429323	−0.214662	+	0.976689i	$0.568865\pi$
$282$	−660.233	−0.139420
$283$	8496.85	1.78475	0.892377	−	0.451292i	$-0.149037\pi$
$283$	8496.85	1.78475	0.892377	+	0.451292i	$0.149037\pi$
$284$	5504.57	1.15013
$285$	−1309.69	−0.272208
$286$	171.093	0.0353738
$287$	−2426.91	−0.499150
$288$	−1477.24	−0.302247
$289$	3220.28	0.655461
$290$	603.426	0.122188
$291$	2894.27	0.583041
$292$	−4195.52	−0.840836
$293$	1114.98	0.222313	0.111157	−	0.993803i	$-0.464544\pi$
$293$	1114.98	0.222313	0.111157	+	0.993803i	$0.464544\pi$
$294$	−1829.83	−0.362986
$295$	−5785.16	−1.14178
$296$	4030.58	0.791462
$297$	−569.689	−0.111302
$298$	625.220	0.121537
$299$	1187.82	0.229743
$300$	−3234.37	−0.622455
$301$	0	0
$302$	1725.91	0.328857
$303$	−4921.88	−0.933184
$304$	−485.414	−0.0915803
$305$	13336.4	2.50373
$306$	956.724	0.178733
$307$	−4992.32	−0.928100	−0.464050	−	0.885809i	$-0.653604\pi$
$307$	−4992.32	−0.928100	−0.464050	+	0.885809i	$0.653604\pi$
$308$	116.290	0.0215138
$309$	−6795.15	−1.25101
$310$	−4625.44	−0.847442
$311$	−7610.06	−1.38755	−0.693773	−	0.720194i	$-0.744053\pi$
$311$	−7610.06	−1.38755	−0.693773	+	0.720194i	$0.744053\pi$
$312$	−2844.92	−0.516224
$313$	−4478.36	−0.808728	−0.404364	−	0.914598i	$-0.632507\pi$
$313$	−4478.36	−0.808728	−0.404364	+	0.914598i	$0.632507\pi$
$314$	−3699.46	−0.664880
$315$	624.355	0.111677
$316$	724.469	0.128970
$317$	−5059.65	−0.896461	−0.448230	−	0.893918i	$-0.647946\pi$
$317$	−5059.65	−0.896461	−0.448230	+	0.893918i	$0.647946\pi$
$318$	−3929.56	−0.692952
$319$	109.308	0.0191852
$320$	643.009	0.112329
$321$	4200.90	0.730440
$322$	−224.969	−0.0389349
$323$	1737.34	0.299283
$324$	2799.94	0.480100
$325$	4119.46	0.703097
$326$	2891.91	0.491312
$327$	6.87074	0.00116194
$328$	−9177.50	−1.54495
$329$	571.557	0.0957780
$330$	335.177	0.0559118
$331$	10063.6	1.67113	0.835566	−	0.549390i	$-0.185140\pi$
$331$	10063.6	1.67113	0.835566	+	0.549390i	$0.185140\pi$
$332$	5880.84	0.972147
$333$	1720.31	0.283100
$334$	3847.02	0.630238
$335$	−7418.51	−1.20990
$336$	−546.246	−0.0886909
$337$	−961.257	−0.155380	−0.0776899	−	0.996978i	$-0.524754\pi$
$337$	−961.257	−0.155380	−0.0776899	+	0.996978i	$0.524754\pi$
$338$	−1310.71	−0.210927
$339$	2010.18	0.322060
$340$	8808.52	1.40503
$341$	−837.877	−0.133060
$342$	204.365	0.0323123
$343$	3291.48	0.518144
$344$	0	0
$345$	2326.98	0.363132
$346$	4210.72	0.654247
$347$	7415.19	1.14717	0.573585	−	0.819146i	$-0.305552\pi$
$347$	7415.19	1.14717	0.573585	+	0.819146i	$0.305552\pi$
$348$	−797.652	−0.122870
$349$	−7957.18	−1.22045	−0.610226	−	0.792227i	$-0.708922\pi$
$349$	−7957.18	−1.22045	−0.610226	+	0.792227i	$0.708922\pi$
$350$	−780.214	−0.119155
$351$	−5294.80	−0.805172
$352$	686.524	0.103954
$353$	4679.48	0.705563	0.352781	−	0.935706i	$-0.385236\pi$
$353$	4679.48	0.705563	0.352781	+	0.935706i	$0.385236\pi$
$354$	−2130.91	−0.319935
$355$	−13734.7	−2.05341
$356$	−3742.17	−0.557120
$357$	1955.06	0.289840
$358$	−3832.71	−0.565824
$359$	4626.84	0.680209	0.340105	−	0.940388i	$-0.389538\pi$
$359$	4626.84	0.680209	0.340105	+	0.940388i	$0.389538\pi$
$360$	2361.03	0.345659
$361$	−6487.89	−0.945894
$362$	5628.63	0.817222
$363$	−5735.72	−0.829332
$364$	1080.82	0.155633
$365$	10468.4	1.50121
$366$	4912.33	0.701562
$367$	−7285.27	−1.03621	−0.518103	−	0.855318i	$-0.673362\pi$
$367$	−7285.27	−1.03621	−0.518103	+	0.855318i	$0.673362\pi$
$368$	862.457	0.122170
$369$	−3917.09	−0.552616
$370$	−4413.51	−0.620129
$371$	3401.78	0.476042
$372$	6114.23	0.852172
$373$	−9225.62	−1.28066	−0.640328	−	0.768102i	$-0.721201\pi$
$373$	−9225.62	−1.28066	−0.640328	+	0.768102i	$0.721201\pi$
$374$	−444.623	−0.0614731
$375$	−427.973	−0.0589345
$376$	2161.37	0.296448
$377$	1015.93	0.138788
$378$	1002.82	0.136454
$379$	−7678.83	−1.04073	−0.520363	−	0.853945i	$-0.674203\pi$
$379$	−7678.83	−1.04073	−0.520363	+	0.853945i	$0.674203\pi$
$380$	1881.58	0.254008
$381$	−8176.81	−1.09950
$382$	−2399.24	−0.321351
$383$	12068.6	1.61013	0.805063	−	0.593189i	$-0.202131\pi$
$383$	12068.6	1.61013	0.805063	+	0.593189i	$0.202131\pi$
$384$	−6168.89	−0.819805
$385$	−290.160	−0.0384101
$386$	−1176.73	−0.155166
$387$	0	0
$388$	−4158.08	−0.544059
$389$	779.907	0.101653	0.0508263	−	0.998708i	$-0.483815\pi$
$389$	779.907	0.101653	0.0508263	+	0.998708i	$0.483815\pi$
$390$	3115.20	0.404473
$391$	−3086.81	−0.399250
$392$	5990.23	0.771818
$393$	−3539.22	−0.454274
$394$	−2241.58	−0.286622
$395$	−1807.65	−0.230260
$396$	187.695	0.0238182
$397$	6572.49	0.830891	0.415446	−	0.909618i	$-0.363626\pi$
$397$	6572.49	0.830891	0.415446	+	0.909618i	$0.363626\pi$
$398$	3268.32	0.411624
$399$	417.620	0.0523988
$400$	2991.08	0.373886
$401$	8199.25	1.02108	0.510538	−	0.859855i	$-0.329446\pi$
$401$	8199.25	1.02108	0.510538	+	0.859855i	$0.329446\pi$
$402$	−2732.55	−0.339023
$403$	−7787.40	−0.962575
$404$	7071.08	0.870790
$405$	−6986.24	−0.857159
$406$	−192.414	−0.0235206
$407$	−799.488	−0.0973690
$408$	7393.18	0.897101
$409$	−2566.69	−0.310305	−0.155152	−	0.987891i	$-0.549587\pi$
$409$	−2566.69	−0.310305	−0.155152	+	0.987891i	$0.549587\pi$
$410$	10049.4	1.21050
$411$	−3913.25	−0.469650
$412$	9762.34	1.16737
$413$	1844.71	0.219788
$414$	−363.104	−0.0431053
$415$	−14673.5	−1.73565
$416$	6380.69	0.752017
$417$	−1842.28	−0.216347
$418$	−94.9756	−0.0111134
$419$	3189.67	0.371899	0.185949	−	0.982559i	$-0.440464\pi$
$419$	3189.67	0.371899	0.185949	+	0.982559i	$0.440464\pi$
$420$	2117.38	0.245994
$421$	8593.75	0.994855	0.497427	−	0.867506i	$-0.334278\pi$
$421$	8593.75	0.994855	0.497427	+	0.867506i	$0.334278\pi$
$422$	6004.23	0.692610
$423$	922.505	0.106037
$424$	12864.0	1.47342
$425$	−10705.4	−1.22185
$426$	−5059.05	−0.575379
$427$	−4252.56	−0.481957
$428$	−6035.27	−0.681603
$429$	564.306	0.0635080
$430$	0	0
$431$	14083.4	1.57396	0.786978	−	0.616980i	$-0.211644\pi$
$431$	14083.4	1.57396	0.786978	+	0.616980i	$0.211644\pi$
$432$	−3844.48	−0.428166
$433$	4030.11	0.447285	0.223643	−	0.974671i	$-0.428205\pi$
$433$	4030.11	0.447285	0.223643	+	0.974671i	$0.428205\pi$
$434$	1474.91	0.163129
$435$	1990.25	0.219368
$436$	−9.87093	−0.00108425
$437$	−659.372	−0.0721785
$438$	3855.95	0.420649
$439$	−1907.40	−0.207370	−0.103685	−	0.994610i	$-0.533063\pi$
$439$	−1907.40	−0.207370	−0.103685	+	0.994610i	$0.533063\pi$
$440$	−1097.25	−0.118885
$441$	2556.72	0.276073
$442$	−4132.42	−0.444704
$443$	−383.394	−0.0411187	−0.0205594	−	0.999789i	$-0.506545\pi$
$443$	−383.394	−0.0411187	−0.0205594	+	0.999789i	$0.506545\pi$
$444$	5834.09	0.623590
$445$	9337.23	0.994668
$446$	−4435.94	−0.470959
$447$	2062.13	0.218200
$448$	−205.036	−0.0216228
$449$	−6486.48	−0.681773	−0.340886	−	0.940105i	$-0.610727\pi$
$449$	−6486.48	−0.681773	−0.340886	+	0.940105i	$0.610727\pi$
$450$	−1259.28	−0.131918
$451$	1820.41	0.190066
$452$	−2887.95	−0.300526
$453$	5692.47	0.590409
$454$	−4776.76	−0.493798
$455$	−2696.80	−0.277864
$456$	1579.25	0.162182
$457$	−6968.83	−0.713322	−0.356661	−	0.934234i	$-0.616085\pi$
$457$	−6968.83	−0.713322	−0.356661	+	0.934234i	$0.616085\pi$
$458$	2837.90	0.289534
$459$	13759.8	1.39924
$460$	−3343.08	−0.338852
$461$	−9235.71	−0.933080	−0.466540	−	0.884500i	$-0.654500\pi$
$461$	−9235.71	−0.933080	−0.466540	+	0.884500i	$0.654500\pi$
$462$	−106.878	−0.0107628
$463$	−6346.35	−0.637019	−0.318510	−	0.947920i	$-0.603182\pi$
$463$	−6346.35	−0.637019	−0.318510	+	0.947920i	$0.603182\pi$
$464$	737.653	0.0738032
$465$	−15255.8	−1.52145
$466$	1790.59	0.177999
$467$	12576.7	1.24621	0.623103	−	0.782140i	$-0.285872\pi$
$467$	12576.7	1.24621	0.623103	+	0.782140i	$0.285872\pi$
$468$	1744.47	0.172304
$469$	2365.54	0.232901
$470$	−2366.72	−0.232274
$471$	−12201.7	−1.19369
$472$	6975.87	0.680277
$473$	0	0
$474$	−665.832	−0.0645204
$475$	−2286.76	−0.220893
$476$	−2808.77	−0.270461
$477$	5490.54	0.527033
$478$	−5997.57	−0.573896
$479$	10394.1	0.991478	0.495739	−	0.868471i	$-0.334897\pi$
$479$	10394.1	0.991478	0.495739	+	0.868471i	$0.334897\pi$
$480$	12500.0	1.18864
$481$	−7430.60	−0.704379
$482$	−454.243	−0.0429257
$483$	−742.003	−0.0699013
$484$	8240.29	0.773882
$485$	10375.0	0.971348
$486$	2865.96	0.267495
$487$	10010.4	0.931449	0.465724	−	0.884930i	$-0.345794\pi$
$487$	10010.4	0.931449	0.465724	+	0.884930i	$0.345794\pi$
$488$	−16081.3	−1.49173
$489$	9538.23	0.882073
$490$	−6559.34	−0.604736
$491$	−7102.70	−0.652832	−0.326416	−	0.945226i	$-0.605841\pi$
$491$	−7102.70	−0.652832	−0.326416	+	0.945226i	$0.605841\pi$
$492$	−13284.0	−1.21726
$493$	−2640.13	−0.241188
$494$	−882.722	−0.0803958
$495$	−468.324	−0.0425244
$496$	−5654.32	−0.511868
$497$	4379.57	0.395272
$498$	−5404.86	−0.486340
$499$	−20381.3	−1.82844	−0.914222	−	0.405213i	$-0.867197\pi$
$499$	−20381.3	−1.82844	−0.914222	+	0.405213i	$0.867197\pi$
$500$	614.852	0.0549941
$501$	12688.4	1.13149
$502$	2078.33	0.184782
$503$	3122.92	0.276827	0.138414	−	0.990375i	$-0.455800\pi$
$503$	3122.92	0.276827	0.138414	+	0.990375i	$0.455800\pi$
$504$	−752.861	−0.0665379
$505$	−17643.3	−1.55469
$506$	168.747	0.0148256
$507$	−4323.06	−0.378686
$508$	11747.3	1.02599
$509$	16509.6	1.43768	0.718838	−	0.695178i	$-0.244674\pi$
$509$	16509.6	1.43768	0.718838	+	0.695178i	$0.244674\pi$
$510$	−8095.58	−0.702899
$511$	−3338.05	−0.288976
$512$	8427.53	0.727437
$513$	2939.21	0.252962
$514$	−2976.42	−0.255417
$515$	−24358.4	−2.08419
$516$	0	0
$517$	−428.721	−0.0364702
$518$	1407.33	0.119372
$519$	13888.0	1.17460
$520$	−10198.1	−0.860031
$521$	4809.95	0.404468	0.202234	−	0.979337i	$-0.435180\pi$
$521$	4809.95	0.404468	0.202234	+	0.979337i	$0.435180\pi$
$522$	−310.561	−0.0260400
$523$	16802.3	1.40481	0.702405	−	0.711778i	$-0.252110\pi$
$523$	16802.3	1.40481	0.702405	+	0.711778i	$0.252110\pi$
$524$	5084.66	0.423901
$525$	−2573.34	−0.213923
$526$	4909.52	0.406968
$527$	20237.3	1.67278
$528$	409.735	0.0337716
$529$	−10995.5	−0.903712
$530$	−14086.2	−1.15446
$531$	2977.40	0.243330
$532$	−599.978	−0.0488954
$533$	16919.2	1.37496
$534$	3439.29	0.278713
$535$	15058.8	1.21692
$536$	8945.41	0.720863
$537$	−12641.2	−1.01585
$538$	−7791.00	−0.624338
$539$	−1188.20	−0.0949522
$540$	14902.1	1.18756
$541$	7216.68	0.573511	0.286755	−	0.958004i	$-0.407423\pi$
$541$	7216.68	0.573511	0.286755	+	0.958004i	$0.407423\pi$
$542$	−9067.18	−0.718577
$543$	18564.6	1.46719
$544$	−16581.7	−1.30686
$545$	24.6293	0.00193579
$546$	−993.344	−0.0778593
$547$	−2417.37	−0.188957	−0.0944784	−	0.995527i	$-0.530118\pi$
$547$	−2417.37	−0.188957	−0.0944784	+	0.995527i	$0.530118\pi$
$548$	5622.01	0.438249
$549$	−6863.71	−0.533581
$550$	585.232	0.0453716
$551$	−563.956	−0.0436031
$552$	−2805.93	−0.216355
$553$	576.404	0.0443240
$554$	1374.96	0.105445
$555$	−14556.9	−1.11334
$556$	2646.73	0.201882
$557$	2933.69	0.223168	0.111584	−	0.993755i	$-0.464408\pi$
$557$	2933.69	0.223168	0.111584	+	0.993755i	$0.464408\pi$
$558$	2380.54	0.180602
$559$	0	0
$560$	−1958.11	−0.147759
$561$	−1466.48	−0.110365
$562$	−2670.20	−0.200419
$563$	1621.04	0.121348	0.0606740	−	0.998158i	$-0.480675\pi$
$563$	1621.04	0.121348	0.0606740	+	0.998158i	$0.480675\pi$
$564$	3128.49	0.233570
$565$	7205.84	0.536552
$566$	11219.1	0.833169
$567$	2227.70	0.164999
$568$	16561.6	1.22343
$569$	12415.4	0.914730	0.457365	−	0.889279i	$-0.348793\pi$
$569$	12415.4	0.914730	0.457365	+	0.889279i	$0.348793\pi$
$570$	−1729.29	−0.127074
$571$	24621.9	1.80454	0.902271	−	0.431168i	$-0.141899\pi$
$571$	24621.9	1.80454	0.902271	+	0.431168i	$0.141899\pi$
$572$	−810.716	−0.0592618
$573$	−7913.31	−0.576934
$574$	−3204.45	−0.233016
$575$	4062.99	0.294676
$576$	−330.932	−0.0239390
$577$	−9543.80	−0.688585	−0.344293	−	0.938862i	$-0.611881\pi$
$577$	−9543.80	−0.688585	−0.344293	+	0.938862i	$0.611881\pi$
$578$	4252.00	0.305986
$579$	−3881.15	−0.278575
$580$	−2859.32	−0.204701
$581$	4678.93	0.334105
$582$	3821.54	0.272179
$583$	−2551.65	−0.181267
$584$	−12623.0	−0.894425
$585$	−4352.69	−0.307627
$586$	1472.20	0.103782
$587$	−10249.6	−0.720692	−0.360346	−	0.932819i	$-0.617341\pi$
$587$	−10249.6	−0.720692	−0.360346	+	0.932819i	$0.617341\pi$
$588$	8670.60	0.608111
$589$	4322.88	0.302413
$590$	−7638.62	−0.533012
$591$	−7393.27	−0.514583
$592$	−5395.26	−0.374567
$593$	−4080.19	−0.282552	−0.141276	−	0.989970i	$-0.545121\pi$
$593$	−4080.19	−0.282552	−0.141276	+	0.989970i	$0.545121\pi$
$594$	−752.207	−0.0519586
$595$	7008.26	0.482875
$596$	−2962.58	−0.203611
$597$	10779.7	0.739004
$598$	1568.37	0.107250
$599$	18792.0	1.28184	0.640918	−	0.767609i	$-0.278554\pi$
$599$	18792.0	1.28184	0.640918	+	0.767609i	$0.278554\pi$
$600$	−9731.22	−0.662126
$601$	11550.6	0.783961	0.391981	−	0.919974i	$-0.371790\pi$
$601$	11550.6	0.783961	0.391981	+	0.919974i	$0.371790\pi$
$602$	0	0
$603$	3818.03	0.257848
$604$	−8178.15	−0.550934
$605$	−20560.7	−1.38167
$606$	−6498.77	−0.435634
$607$	−15936.5	−1.06564	−0.532819	−	0.846229i	$-0.678867\pi$
$607$	−15936.5	−1.06564	−0.532819	+	0.846229i	$0.678867\pi$
$608$	−3542.00	−0.236262
$609$	−634.630	−0.0422274
$610$	17609.1	1.16881
$611$	−3984.61	−0.263830
$612$	−4533.41	−0.299432
$613$	−5661.66	−0.373038	−0.186519	−	0.982451i	$-0.559721\pi$
$613$	−5661.66	−0.373038	−0.186519	+	0.982451i	$0.559721\pi$
$614$	−6591.77	−0.433261
$615$	33145.5	2.17326
$616$	349.881	0.0228849
$617$	730.404	0.0476579	0.0238290	−	0.999716i	$-0.492414\pi$
$617$	730.404	0.0476579	0.0238290	+	0.999716i	$0.492414\pi$
$618$	−8972.20	−0.584005
$619$	−19348.7	−1.25636	−0.628181	−	0.778067i	$-0.716200\pi$
$619$	−19348.7	−1.25636	−0.628181	+	0.778067i	$0.716200\pi$
$620$	21917.5	1.41972
$621$	−5222.22	−0.337457
$622$	−10048.2	−0.647742
$623$	−2977.36	−0.191469
$624$	3808.15	0.244308
$625$	−16372.3	−1.04782
$626$	−5913.15	−0.377535
$627$	−313.253	−0.0199524
$628$	17529.8	1.11387
$629$	19310.1	1.22408
$630$	824.387	0.0521339
$631$	1817.20	0.114646	0.0573230	−	0.998356i	$-0.481744\pi$
$631$	1817.20	0.114646	0.0573230	+	0.998356i	$0.481744\pi$
$632$	2179.70	0.137190
$633$	19803.4	1.24347
$634$	−6680.67	−0.418491
$635$	−29311.2	−1.83178
$636$	18620.1	1.16090
$637$	−11043.3	−0.686896
$638$	144.328	0.00895614
$639$	7068.71	0.437612
$640$	−22113.4	−1.36580
$641$	−18827.8	−1.16014	−0.580072	−	0.814565i	$-0.696976\pi$
$641$	−18827.8	−1.16014	−0.580072	+	0.814565i	$0.696976\pi$
$642$	5546.79	0.340988
$643$	29918.2	1.83493	0.917463	−	0.397822i	$-0.130234\pi$
$643$	29918.2	1.83493	0.917463	+	0.397822i	$0.130234\pi$
$644$	1066.01	0.0652276
$645$	0	0
$646$	2293.96	0.139713
$647$	−2632.68	−0.159971	−0.0799856	−	0.996796i	$-0.525487\pi$
$647$	−2632.68	−0.159971	−0.0799856	+	0.996796i	$0.525487\pi$
$648$	8424.17	0.510698
$649$	−1383.70	−0.0836904
$650$	5439.26	0.328224
$651$	4864.62	0.292872
$652$	−13703.2	−0.823096
$653$	10367.7	0.621315	0.310658	−	0.950522i	$-0.399451\pi$
$653$	10367.7	0.621315	0.310658	+	0.950522i	$0.399451\pi$
$654$	9.07201	0.000542421 0
$655$	−12686.9	−0.756823
$656$	12284.8	0.731161
$657$	−5387.68	−0.319929
$658$	754.674	0.0447116
$659$	−14731.7	−0.870815	−0.435407	−	0.900234i	$-0.643396\pi$
$659$	−14731.7	−0.870815	−0.435407	+	0.900234i	$0.643396\pi$
$660$	−1588.23	−0.0936692
$661$	17743.4	1.04408	0.522042	−	0.852920i	$-0.325171\pi$
$661$	17743.4	1.04408	0.522042	+	0.852920i	$0.325171\pi$
$662$	13287.8	0.780127
$663$	−13629.7	−0.798394
$664$	17693.6	1.03411
$665$	1497.03	0.0872966
$666$	2271.47	0.132158
$667$	1002.01	0.0581676
$668$	−18229.0	−1.05584
$669$	−14630.8	−0.845532
$670$	−9795.28	−0.564813
$671$	3189.81	0.183519
$672$	−3985.88	−0.228808
$673$	16842.3	0.964668	0.482334	−	0.875987i	$-0.339789\pi$
$673$	16842.3	0.964668	0.482334	+	0.875987i	$0.339789\pi$
$674$	−1269.23	−0.0725353
$675$	−18111.2	−1.03274
$676$	6210.77	0.353367
$677$	7750.69	0.440005	0.220002	−	0.975499i	$-0.429394\pi$
$677$	7750.69	0.440005	0.220002	+	0.975499i	$0.429394\pi$
$678$	2654.21	0.150346
$679$	−3308.27	−0.186980
$680$	26502.1	1.49457
$681$	−15754.9	−0.886535
$682$	−1106.32	−0.0621160
$683$	−3639.03	−0.203871	−0.101935	−	0.994791i	$-0.532503\pi$
$683$	−3639.03	−0.203871	−0.101935	+	0.994791i	$0.532503\pi$
$684$	−968.377	−0.0541328
$685$	−14027.7	−0.782439
$686$	4346.02	0.241883
$687$	9360.11	0.519811
$688$	0	0
$689$	−23715.5	−1.31131
$690$	3072.50	0.169519
$691$	−21688.0	−1.19399	−0.596997	−	0.802243i	$-0.703640\pi$
$691$	−21688.0	−1.19399	−0.596997	+	0.802243i	$0.703640\pi$
$692$	−19952.3	−1.09606
$693$	149.334	0.00818576
$694$	9790.89	0.535529
$695$	−6603.95	−0.360435
$696$	−2399.89	−0.130700
$697$	−43968.5	−2.38942
$698$	−10506.5	−0.569739
$699$	5905.80	0.319568
$700$	3697.02	0.199620
$701$	−5751.17	−0.309870	−0.154935	−	0.987925i	$-0.549517\pi$
$701$	−5751.17	−0.309870	−0.154935	+	0.987925i	$0.549517\pi$
$702$	−6991.16	−0.375875
$703$	4124.82	0.221295
$704$	153.796	0.00823352
$705$	−7806.02	−0.417010
$706$	6178.70	0.329375
$707$	5625.91	0.299271
$708$	10097.3	0.535987
$709$	−23130.5	−1.22522	−0.612612	−	0.790384i	$-0.709881\pi$
$709$	−23130.5	−1.22522	−0.612612	+	0.790384i	$0.709881\pi$
$710$	−18135.0	−0.958584
$711$	930.328	0.0490718
$712$	−11259.0	−0.592627
$713$	−7680.65	−0.403426
$714$	2581.43	0.135305
$715$	2022.85	0.105805
$716$	18161.2	0.947926
$717$	−19781.5	−1.03034
$718$	6109.19	0.317539
$719$	−32891.3	−1.70604	−0.853018	−	0.521882i	$-0.825230\pi$
$719$	−32891.3	−1.70604	−0.853018	+	0.521882i	$0.825230\pi$
$720$	−3160.43	−0.163586
$721$	7767.14	0.401198
$722$	−8566.49	−0.441568
$723$	−1498.21	−0.0770662
$724$	−26671.1	−1.36909
$725$	3475.05	0.178014
$726$	−7573.35	−0.387153
$727$	−7446.63	−0.379890	−0.189945	−	0.981795i	$-0.560831\pi$
$727$	−7446.63	−0.379890	−0.189945	+	0.981795i	$0.560831\pi$
$728$	3251.86	0.165552
$729$	21535.7	1.09412
$730$	13822.3	0.700802
$731$	0	0
$732$	−23276.9	−1.17533
$733$	26197.6	1.32009	0.660047	−	0.751224i	$-0.270536\pi$
$733$	26197.6	1.32009	0.660047	+	0.751224i	$0.270536\pi$
$734$	−9619.34	−0.483728
$735$	−21634.3	−1.08571
$736$	6293.23	0.315179
$737$	−1774.37	−0.0886836
$738$	−5172.05	−0.257976
$739$	−9936.92	−0.494635	−0.247318	−	0.968934i	$-0.579549\pi$
$739$	−9936.92	−0.494635	−0.247318	+	0.968934i	$0.579549\pi$
$740$	20913.3	1.03890
$741$	−2911.44	−0.144338
$742$	4491.65	0.222229
$743$	6039.26	0.298195	0.149098	−	0.988822i	$-0.452363\pi$
$743$	6039.26	0.298195	0.149098	+	0.988822i	$0.452363\pi$
$744$	18395.8	0.906483
$745$	7392.05	0.363522
$746$	−12181.3	−0.597843
$747$	7551.89	0.369892
$748$	2106.83	0.102986
$749$	−4801.80	−0.234251
$750$	−565.088	−0.0275121
$751$	−26396.0	−1.28256	−0.641280	−	0.767307i	$-0.721596\pi$
$751$	−26396.0	−1.28256	−0.641280	+	0.767307i	$0.721596\pi$
$752$	−2893.17	−0.140297
$753$	6854.86	0.331746
$754$	1341.42	0.0647898
$755$	20405.6	0.983624
$756$	−4751.83	−0.228601
$757$	37477.2	1.79938	0.899691	−	0.436526i	$-0.143791\pi$
$757$	37477.2	1.79938	0.899691	+	0.436526i	$0.143791\pi$
$758$	−10139.0	−0.485838
$759$	556.571	0.0266169
$760$	5661.09	0.270197
$761$	20551.1	0.978944	0.489472	−	0.872019i	$-0.337190\pi$
$761$	20551.1	0.978944	0.489472	+	0.872019i	$0.337190\pi$
$762$	−10796.5	−0.513276
$763$	−7.85354	−0.000372631 0
$764$	11368.7	0.538360
$765$	11311.5	0.534598
$766$	15935.2	0.751648
$767$	−12860.4	−0.605427
$768$	−9580.32	−0.450130
$769$	36513.5	1.71224	0.856119	−	0.516778i	$-0.172869\pi$
$769$	36513.5	1.71224	0.856119	+	0.516778i	$0.172869\pi$
$770$	−383.122	−0.0179308
$771$	−9816.98	−0.458560
$772$	5575.90	0.259949
$773$	21597.5	1.00493	0.502464	−	0.864598i	$-0.332427\pi$
$773$	21597.5	1.00493	0.502464	+	0.864598i	$0.332427\pi$
$774$	0	0
$775$	−26637.3	−1.23463
$776$	−12510.4	−0.578733
$777$	4641.74	0.214313
$778$	1029.78	0.0474540
$779$	−9392.08	−0.431972
$780$	−14761.3	−0.677614
$781$	−3285.08	−0.150511
$782$	−4075.78	−0.186380
$783$	−4466.53	−0.203858
$784$	−8018.41	−0.365270
$785$	−43739.1	−1.98868
$786$	−4673.12	−0.212067
$787$	24574.8	1.11308	0.556542	−	0.830820i	$-0.312128\pi$
$787$	24574.8	1.11308	0.556542	+	0.830820i	$0.312128\pi$
$788$	10621.6	0.480178
$789$	16192.8	0.730646
$790$	−2386.79	−0.107491
$791$	−2297.72	−0.103284
$792$	564.715	0.0253362
$793$	29646.7	1.32760
$794$	8678.20	0.387881
$795$	−46459.7	−2.07265
$796$	−15486.8	−0.689594
$797$	21154.6	0.940194	0.470097	−	0.882615i	$-0.344219\pi$
$797$	21154.6	0.940194	0.470097	+	0.882615i	$0.344219\pi$
$798$	551.418	0.0244611
$799$	10354.9	0.458487
$800$	21825.5	0.964561
$801$	−4805.52	−0.211978
$802$	10826.2	0.476664
$803$	2503.85	0.110036
$804$	12948.1	0.567965
$805$	−2659.84	−0.116456
$806$	−10282.3	−0.449355
$807$	−25696.7	−1.12090
$808$	21274.7	0.926289
$809$	28704.3	1.24745	0.623726	−	0.781643i	$-0.285618\pi$
$809$	28704.3	1.24745	0.623726	+	0.781643i	$0.285618\pi$
$810$	−9224.51	−0.400144
$811$	−17593.2	−0.761753	−0.380876	−	0.924626i	$-0.624378\pi$
$811$	−17593.2	−0.761753	−0.380876	+	0.924626i	$0.624378\pi$
$812$	911.749	0.0394041
$813$	−29905.8	−1.29009
$814$	−1055.63	−0.0454543
$815$	34191.4	1.46954
$816$	−9896.37	−0.424562
$817$	0	0
$818$	−3389.01	−0.144858
$819$	1387.94	0.0592168
$820$	−47618.9	−2.02795
$821$	30584.0	1.30011	0.650055	−	0.759887i	$-0.274746\pi$
$821$	30584.0	1.30011	0.650055	+	0.759887i	$0.274746\pi$
$822$	−5166.98	−0.219245
$823$	15398.4	0.652191	0.326096	−	0.945337i	$-0.394267\pi$
$823$	15398.4	0.652191	0.326096	+	0.945337i	$0.394267\pi$
$824$	29371.9	1.24177
$825$	1930.24	0.0814574
$826$	2435.72	0.102603
$827$	20703.4	0.870529	0.435264	−	0.900303i	$-0.356655\pi$
$827$	20703.4	0.870529	0.435264	+	0.900303i	$0.356655\pi$
$828$	1720.56	0.0722144
$829$	−6121.19	−0.256451	−0.128225	−	0.991745i	$-0.540928\pi$
$829$	−6121.19	−0.256451	−0.128225	+	0.991745i	$0.540928\pi$
$830$	−19374.6	−0.810245
$831$	4534.95	0.189309
$832$	1429.41	0.0595623
$833$	28698.6	1.19370
$834$	−2432.51	−0.100996
$835$	45483.8	1.88507
$836$	450.039	0.0186183
$837$	34237.2	1.41387
$838$	4211.58	0.173612
$839$	−5023.30	−0.206703	−0.103351	−	0.994645i	$-0.532957\pi$
$839$	−5023.30	−0.206703	−0.103351	+	0.994645i	$0.532957\pi$
$840$	6370.53	0.261672
$841$	−23532.0	−0.964861
$842$	11347.0	0.464424
$843$	−8806.98	−0.359820
$844$	−28450.8	−1.16033
$845$	−15496.7	−0.630892
$846$	1218.06	0.0495009
$847$	6556.17	0.265965
$848$	−17219.5	−0.697312
$849$	37003.3	1.49582
$850$	−14135.2	−0.570391
$851$	−7328.75	−0.295213
$852$	23972.1	0.963934
$853$	−15719.7	−0.630986	−0.315493	−	0.948928i	$-0.602170\pi$
$853$	−15719.7	−0.630986	−0.315493	+	0.948928i	$0.602170\pi$
$854$	−5615.00	−0.224990
$855$	2416.23	0.0966473
$856$	−18158.3	−0.725043
$857$	49442.8	1.97075	0.985375	−	0.170398i	$-0.0545055\pi$
$857$	49442.8	1.97075	0.985375	+	0.170398i	$0.0545055\pi$
$858$	745.099	0.0296472
$859$	−1836.36	−0.0729402	−0.0364701	−	0.999335i	$-0.511611\pi$
$859$	−1836.36	−0.0729402	−0.0364701	+	0.999335i	$0.511611\pi$
$860$	0	0
$861$	−10569.1	−0.418343
$862$	18595.5	0.734764
$863$	29206.5	1.15203	0.576014	−	0.817440i	$-0.304607\pi$
$863$	29206.5	1.15203	0.576014	+	0.817440i	$0.304607\pi$
$864$	−28052.7	−1.10460
$865$	49783.8	1.95688
$866$	5321.28	0.208804
$867$	14024.2	0.549349
$868$	−6988.82	−0.273290
$869$	−432.356	−0.0168776
$870$	2627.89	0.102407
$871$	−16491.4	−0.641548
$872$	−29.6986	−0.00115335
$873$	−5339.61	−0.207009
$874$	−870.623	−0.0336948
$875$	489.191	0.0189002
$876$	−18271.3	−0.704714
$877$	45448.1	1.74991	0.874957	−	0.484201i	$-0.160890\pi$
$877$	45448.1	1.74991	0.874957	+	0.484201i	$0.160890\pi$
$878$	−2518.50	−0.0968055
$879$	4855.68	0.186323
$880$	1468.76	0.0562636
$881$	−4104.22	−0.156952	−0.0784760	−	0.996916i	$-0.525005\pi$
$881$	−4104.22	−0.156952	−0.0784760	+	0.996916i	$0.525005\pi$
$882$	3375.84	0.128878
$883$	−17943.1	−0.683843	−0.341922	−	0.939728i	$-0.611078\pi$
$883$	−17943.1	−0.683843	−0.341922	+	0.939728i	$0.611078\pi$
$884$	19581.3	0.745013
$885$	−25194.1	−0.956938
$886$	−506.227	−0.0191953
$887$	5704.94	0.215956	0.107978	−	0.994153i	$-0.465562\pi$
$887$	5704.94	0.215956	0.107978	+	0.994153i	$0.465562\pi$
$888$	17553.0	0.663333
$889$	9346.44	0.352609
$890$	12328.7	0.464336
$891$	−1670.98	−0.0628282
$892$	21019.6	0.788999
$893$	2211.91	0.0828877
$894$	2722.80	0.101861
$895$	−45314.6	−1.69240
$896$	7051.30	0.262910
$897$	5172.88	0.192550
$898$	−8564.63	−0.318269
$899$	−6569.21	−0.243710
$900$	5967.06	0.221002
$901$	61630.3	2.27880
$902$	2403.64	0.0887276
$903$	0	0
$904$	−8688.96	−0.319680
$905$	66548.0	2.44434
$906$	7516.24	0.275618
$907$	−7250.35	−0.265429	−0.132714	−	0.991154i	$-0.542369\pi$
$907$	−7250.35	−0.265429	−0.132714	+	0.991154i	$0.542369\pi$
$908$	22634.5	0.827261
$909$	9080.34	0.331327
$910$	−3560.81	−0.129714
$911$	24710.5	0.898679	0.449340	−	0.893361i	$-0.351659\pi$
$911$	24710.5	0.898679	0.449340	+	0.893361i	$0.351659\pi$
$912$	−2113.95	−0.0767544
$913$	−3509.63	−0.127220
$914$	−9201.53	−0.332997
$915$	58079.2	2.09840
$916$	−13447.3	−0.485057
$917$	4045.47	0.145685
$918$	18168.1	0.653201
$919$	18905.6	0.678604	0.339302	−	0.940678i	$-0.389809\pi$
$919$	18905.6	0.678604	0.339302	+	0.940678i	$0.389809\pi$
$920$	−10058.3	−0.360449
$921$	−21741.3	−0.777850
$922$	−12194.7	−0.435586
$923$	−30532.2	−1.08882
$924$	506.437	0.0180309
$925$	−25416.8	−0.903459
$926$	−8379.61	−0.297377
$927$	12536.3	0.444171
$928$	5382.56	0.190400
$929$	40793.3	1.44067	0.720336	−	0.693625i	$-0.243987\pi$
$929$	40793.3	1.44067	0.720336	+	0.693625i	$0.243987\pi$
$930$	−20143.5	−0.710250
$931$	6130.29	0.215803
$932$	−8484.64	−0.298201
$933$	−33141.4	−1.16292
$934$	16606.0	0.581761
$935$	−5256.84	−0.183869
$936$	5248.57	0.183285
$937$	30424.2	1.06074	0.530370	−	0.847766i	$-0.322053\pi$
$937$	30424.2	1.06074	0.530370	+	0.847766i	$0.322053\pi$
$938$	3123.41	0.108724
$939$	−19503.0	−0.677803
$940$	11214.6	0.389128
$941$	13717.9	0.475231	0.237615	−	0.971359i	$-0.423634\pi$
$941$	13717.9	0.475231	0.237615	+	0.971359i	$0.423634\pi$
$942$	−16110.9	−0.557243
$943$	16687.3	0.576261
$944$	−9337.77	−0.321948
$945$	11856.5	0.408138
$946$	0	0
$947$	−6731.42	−0.230984	−0.115492	−	0.993308i	$-0.536844\pi$
$947$	−6731.42	−0.230984	−0.115492	+	0.993308i	$0.536844\pi$
$948$	3155.02	0.108091
$949$	23271.2	0.796013
$950$	−3019.40	−0.103118
$951$	−22034.5	−0.751333
$952$	−8450.72	−0.287699
$953$	22468.9	0.763735	0.381868	−	0.924217i	$-0.375281\pi$
$953$	22468.9	0.763735	0.381868	+	0.924217i	$0.375281\pi$
$954$	7249.62	0.246032
$955$	−28366.6	−0.961174
$956$	28419.3	0.961449
$957$	476.031	0.0160793
$958$	13724.2	0.462848
$959$	4473.00	0.150616
$960$	2800.27	0.0941441
$961$	20563.9	0.690271
$962$	−9811.24	−0.328822
$963$	−7750.21	−0.259343
$964$	2152.42	0.0719135
$965$	−13912.6	−0.464107
$966$	−979.728	−0.0326317
$967$	38313.9	1.27414	0.637069	−	0.770807i	$-0.280147\pi$
$967$	38313.9	1.27414	0.637069	+	0.770807i	$0.280147\pi$
$968$	24792.5	0.823204
$969$	7566.04	0.250832
$970$	13699.0	0.453450
$971$	−54694.4	−1.80765	−0.903825	−	0.427902i	$-0.859253\pi$
$971$	−54694.4	−1.80765	−0.903825	+	0.427902i	$0.859253\pi$
$972$	−13580.2	−0.448134
$973$	2105.80	0.0693821
$974$	13217.6	0.434824
$975$	17940.0	0.589273
$976$	21526.1	0.705976
$977$	−14933.3	−0.489005	−0.244502	−	0.969649i	$-0.578625\pi$
$977$	−14933.3	−0.489005	−0.244502	+	0.969649i	$0.578625\pi$
$978$	12594.1	0.411774
$979$	2233.29	0.0729074
$980$	31081.2	1.01312
$981$	−12.6758	−0.000412545 0
$982$	−9378.28	−0.304759
$983$	17906.2	0.580997	0.290499	−	0.956875i	$-0.406179\pi$
$983$	17906.2	0.580997	0.290499	+	0.956875i	$0.406179\pi$
$984$	−39967.6	−1.29484
$985$	−26502.4	−0.857297
$986$	−3485.98	−0.112593
$987$	2489.10	0.0802726
$988$	4182.75	0.134687
$989$	0	0
$990$	−618.366	−0.0198515
$991$	−5800.06	−0.185918	−0.0929591	−	0.995670i	$-0.529633\pi$
$991$	−5800.06	−0.185918	−0.0929591	+	0.995670i	$0.529633\pi$
$992$	−41258.8	−1.32053
$993$	43826.4	1.40059
$994$	5782.70	0.184523
$995$	38641.8	1.23118
$996$	25610.7	0.814767
$997$	29540.8	0.938383	0.469191	−	0.883097i	$-0.344545\pi$
$997$	29540.8	0.938383	0.469191	+	0.883097i	$0.344545\pi$
$998$	−26911.2	−0.853565
$999$	32668.6	1.03462

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.g.1.18		30
43.32	odd	14		43.4.e.a.35.6	yes	60
43.39	odd	14		43.4.e.a.16.6	✓	60
43.42	odd	2		1849.4.a.h.1.13		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.16.6	✓	60	43.39	odd	14
43.4.e.a.35.6	yes	60	43.32	odd	14
1849.4.a.g.1.18		30	1.1	even	1	trivial
1849.4.a.h.1.13		30	43.42	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+1.32038 q^{2} +4.35495 q^{3} -6.25659 q^{4} +15.6111 q^{5} +5.75020 q^{6} -4.97789 q^{7} -18.8242 q^{8} -8.03441 q^{9} +O(q^{10})\)	\(q+1.32038 q^{2} +4.35495 q^{3} -6.25659 q^{4} +15.6111 q^{5} +5.75020 q^{6} -4.97789 q^{7} -18.8242 q^{8} -8.03441 q^{9} +20.6126 q^{10} +3.73387 q^{11} -27.2471 q^{12} +34.7034 q^{13} -6.57272 q^{14} +67.9853 q^{15} +25.1976 q^{16} -90.1847 q^{17} -10.6085 q^{18} -19.2643 q^{19} -97.6719 q^{20} -21.6785 q^{21} +4.93014 q^{22} +34.2277 q^{23} -81.9782 q^{24} +118.705 q^{25} +45.8217 q^{26} -152.573 q^{27} +31.1446 q^{28} +29.2747 q^{29} +89.7667 q^{30} -224.399 q^{31} +183.864 q^{32} +16.2608 q^{33} -119.078 q^{34} -77.7101 q^{35} +50.2680 q^{36} -214.118 q^{37} -25.4362 q^{38} +151.131 q^{39} -293.865 q^{40} +487.539 q^{41} -28.6238 q^{42} -23.3613 q^{44} -125.426 q^{45} +45.1936 q^{46} -114.819 q^{47} +109.734 q^{48} -318.221 q^{49} +156.736 q^{50} -392.750 q^{51} -217.125 q^{52} -683.378 q^{53} -201.455 q^{54} +58.2897 q^{55} +93.7045 q^{56} -83.8949 q^{57} +38.6538 q^{58} -370.581 q^{59} -425.356 q^{60} +854.289 q^{61} -296.292 q^{62} +39.9944 q^{63} +41.1893 q^{64} +541.756 q^{65} +21.4705 q^{66} -475.209 q^{67} +564.249 q^{68} +149.060 q^{69} -102.607 q^{70} -879.804 q^{71} +151.241 q^{72} +670.576 q^{73} -282.717 q^{74} +516.954 q^{75} +120.529 q^{76} -18.5868 q^{77} +199.551 q^{78} -115.793 q^{79} +393.362 q^{80} -447.519 q^{81} +643.738 q^{82} -939.943 q^{83} +135.633 q^{84} -1407.88 q^{85} +127.490 q^{87} -70.2870 q^{88} +598.117 q^{89} -165.610 q^{90} -172.749 q^{91} -214.149 q^{92} -977.246 q^{93} -151.605 q^{94} -300.736 q^{95} +800.717 q^{96} +664.593 q^{97} -420.173 q^{98} -29.9995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9}+O(q^{10}) \) 30 * q - 6 * q^2 - 2 * q^3 + 114 * q^4 - 27 * q^5 + 8 * q^6 - 48 * q^7 - 90 * q^8 + 216 * q^9	\( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9} - 27 q^{10} + 80 q^{11} + 36 q^{12} - 13 q^{13} + 36 q^{14} + 16 q^{15} + 318 q^{16} + 66 q^{17} - 80 q^{18} - 254 q^{19} - 312 q^{20} - 548 q^{21} - 305 q^{22} - 105 q^{23} + 123 q^{24} + 523 q^{25} - 549 q^{26} + 10 q^{27} - 578 q^{28} - 793 q^{29} - 1560 q^{30} - 359 q^{31} - 676 q^{32} - 208 q^{33} - 1007 q^{34} - 514 q^{35} + 776 q^{36} - 510 q^{37} - 2066 q^{38} - 898 q^{39} - 1248 q^{40} - 270 q^{41} + 915 q^{42} + 3256 q^{44} - 807 q^{45} - 1960 q^{46} + 1421 q^{47} + 632 q^{48} + 386 q^{49} + 141 q^{50} - 209 q^{51} + 2825 q^{52} - 21 q^{53} + 2368 q^{54} - 2258 q^{55} + 2521 q^{56} - 1723 q^{57} - 347 q^{58} + 1752 q^{59} + 2711 q^{60} - 1759 q^{61} - 395 q^{62} - 2204 q^{63} + 222 q^{64} - 1151 q^{65} + 160 q^{66} - 3001 q^{67} + 1921 q^{68} - 1660 q^{69} - 1597 q^{70} - 727 q^{71} - 9100 q^{72} - 4623 q^{73} - 2649 q^{74} - 1027 q^{75} - 874 q^{76} - 3556 q^{77} - 4979 q^{78} + 546 q^{79} - 5809 q^{80} - 410 q^{81} + 4397 q^{82} - 492 q^{83} - 10611 q^{84} + 1723 q^{85} + 5937 q^{87} - 3974 q^{88} - 5218 q^{89} + 10492 q^{90} - 1104 q^{91} + 1060 q^{92} - 1997 q^{93} + 2134 q^{94} + 6346 q^{95} - 11984 q^{96} + 2590 q^{97} - 6270 q^{98} - 2693 q^{99}+O(q^{100}) \) 30 * q - 6 * q^2 - 2 * q^3 + 114 * q^4 - 27 * q^5 + 8 * q^6 - 48 * q^7 - 90 * q^8 + 216 * q^9 - 27 * q^10 + 80 * q^11 + 36 * q^12 - 13 * q^13 + 36 * q^14 + 16 * q^15 + 318 * q^16 + 66 * q^17 - 80 * q^18 - 254 * q^19 - 312 * q^20 - 548 * q^21 - 305 * q^22 - 105 * q^23 + 123 * q^24 + 523 * q^25 - 549 * q^26 + 10 * q^27 - 578 * q^28 - 793 * q^29 - 1560 * q^30 - 359 * q^31 - 676 * q^32 - 208 * q^33 - 1007 * q^34 - 514 * q^35 + 776 * q^36 - 510 * q^37 - 2066 * q^38 - 898 * q^39 - 1248 * q^40 - 270 * q^41 + 915 * q^42 + 3256 * q^44 - 807 * q^45 - 1960 * q^46 + 1421 * q^47 + 632 * q^48 + 386 * q^49 + 141 * q^50 - 209 * q^51 + 2825 * q^52 - 21 * q^53 + 2368 * q^54 - 2258 * q^55 + 2521 * q^56 - 1723 * q^57 - 347 * q^58 + 1752 * q^59 + 2711 * q^60 - 1759 * q^61 - 395 * q^62 - 2204 * q^63 + 222 * q^64 - 1151 * q^65 + 160 * q^66 - 3001 * q^67 + 1921 * q^68 - 1660 * q^69 - 1597 * q^70 - 727 * q^71 - 9100 * q^72 - 4623 * q^73 - 2649 * q^74 - 1027 * q^75 - 874 * q^76 - 3556 * q^77 - 4979 * q^78 + 546 * q^79 - 5809 * q^80 - 410 * q^81 + 4397 * q^82 - 492 * q^83 - 10611 * q^84 + 1723 * q^85 + 5937 * q^87 - 3974 * q^88 - 5218 * q^89 + 10492 * q^90 - 1104 * q^91 + 1060 * q^92 - 1997 * q^93 + 2134 * q^94 + 6346 * q^95 - 11984 * q^96 + 2590 * q^97 - 6270 * q^98 - 2693 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more