LMFDB - Embedded newform 1849.4.a.l.1.4

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:	$0$
Dimension:	$60$
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-4.73263 q^{2} -0.112092 q^{3} +14.3978 q^{4} +9.26635 q^{5} +0.530489 q^{6} -26.6989 q^{7} -30.2784 q^{8} -26.9874 q^{9} +O(q^{10})$	\(q-4.73263 q^{2} -0.112092 q^{3} +14.3978 q^{4} +9.26635 q^{5} +0.530489 q^{6} -26.6989 q^{7} -30.2784 q^{8} -26.9874 q^{9} -43.8542 q^{10} -51.7239 q^{11} -1.61387 q^{12} +19.6138 q^{13} +126.356 q^{14} -1.03868 q^{15} +28.1139 q^{16} -26.0364 q^{17} +127.722 q^{18} -28.7769 q^{19} +133.415 q^{20} +2.99272 q^{21} +244.790 q^{22} +122.891 q^{23} +3.39396 q^{24} -39.1348 q^{25} -92.8250 q^{26} +6.05155 q^{27} -384.404 q^{28} -207.874 q^{29} +4.91570 q^{30} -263.665 q^{31} +109.174 q^{32} +5.79782 q^{33} +123.221 q^{34} -247.401 q^{35} -388.559 q^{36} +73.6862 q^{37} +136.191 q^{38} -2.19855 q^{39} -280.570 q^{40} -169.184 q^{41} -14.1635 q^{42} -744.709 q^{44} -250.075 q^{45} -581.597 q^{46} -496.303 q^{47} -3.15134 q^{48} +369.829 q^{49} +185.210 q^{50} +2.91847 q^{51} +282.396 q^{52} -539.979 q^{53} -28.6398 q^{54} -479.291 q^{55} +808.397 q^{56} +3.22566 q^{57} +983.789 q^{58} -773.017 q^{59} -14.9547 q^{60} +157.445 q^{61} +1247.83 q^{62} +720.534 q^{63} -741.591 q^{64} +181.749 q^{65} -27.4390 q^{66} +158.656 q^{67} -374.867 q^{68} -13.7751 q^{69} +1170.86 q^{70} +964.538 q^{71} +817.135 q^{72} -934.552 q^{73} -348.729 q^{74} +4.38669 q^{75} -414.324 q^{76} +1380.97 q^{77} +10.4049 q^{78} -260.007 q^{79} +260.514 q^{80} +727.982 q^{81} +800.686 q^{82} +632.194 q^{83} +43.0886 q^{84} -241.262 q^{85} +23.3009 q^{87} +1566.11 q^{88} -375.457 q^{89} +1183.51 q^{90} -523.667 q^{91} +1769.36 q^{92} +29.5547 q^{93} +2348.82 q^{94} -266.657 q^{95} -12.2375 q^{96} -988.952 q^{97} -1750.26 q^{98} +1395.89 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 60 q + 15 q^{2} + 37 q^{3} + 213 q^{4} + 51 q^{5} + 22 q^{6} + 54 q^{7} + 204 q^{8} + 387 q^{9}+O(q^{10}) $ 60 * q + 15 * q^2 + 37 * q^3 + 213 * q^4 + 51 * q^5 + 22 * q^6 + 54 * q^7 + 204 * q^8 + 387 * q^9	\( 60 q + 15 q^{2} + 37 q^{3} + 213 q^{4} + 51 q^{5} + 22 q^{6} + 54 q^{7} + 204 q^{8} + 387 q^{9} + 27 q^{10} + 43 q^{11} + 432 q^{12} + 13 q^{13} + 96 q^{14} + 65 q^{15} + 717 q^{16} - 138 q^{17} + 625 q^{18} + 610 q^{19} + 345 q^{20} + 611 q^{21} + 118 q^{22} - 243 q^{23} + 258 q^{24} + 899 q^{25} + 1071 q^{26} + 1609 q^{27} + 46 q^{28} + 773 q^{29} + 375 q^{30} - 97 q^{31} + 1967 q^{32} + 500 q^{33} + 217 q^{34} + 247 q^{35} + 175 q^{36} + 228 q^{37} + 1253 q^{38} + 1493 q^{39} + 2220 q^{40} - 951 q^{41} + 2643 q^{42} - 1378 q^{44} + 1086 q^{45} - 565 q^{46} - 2 q^{47} + 2303 q^{48} + 1264 q^{49} + 3273 q^{50} + 3076 q^{51} - 2825 q^{52} - 39 q^{53} + 5201 q^{54} + 1306 q^{55} + 3683 q^{56} + 1342 q^{57} + 2588 q^{58} - 1065 q^{59} + 2803 q^{60} + 2999 q^{61} + 5569 q^{62} + 2377 q^{63} + 2082 q^{64} + 5578 q^{65} + 3338 q^{66} + 961 q^{67} - 3754 q^{68} + 1817 q^{69} + 2738 q^{70} + 8003 q^{71} + 1412 q^{72} - 1011 q^{73} - 1413 q^{74} + 7457 q^{75} + 5516 q^{76} + 4052 q^{77} + 1091 q^{78} - 4422 q^{79} + 1610 q^{80} + 2108 q^{81} + 4676 q^{82} - 297 q^{83} - 54 q^{84} + 4333 q^{85} + 1377 q^{87} + 3652 q^{88} + 2480 q^{89} - 1414 q^{90} + 4551 q^{91} - 3286 q^{92} + 4 q^{93} + 4609 q^{94} - 835 q^{95} + 5864 q^{96} + 3785 q^{97} + 753 q^{98} + 5072 q^{99}+O(q^{100}) \) 60 * q + 15 * q^2 + 37 * q^3 + 213 * q^4 + 51 * q^5 + 22 * q^6 + 54 * q^7 + 204 * q^8 + 387 * q^9 + 27 * q^10 + 43 * q^11 + 432 * q^12 + 13 * q^13 + 96 * q^14 + 65 * q^15 + 717 * q^16 - 138 * q^17 + 625 * q^18 + 610 * q^19 + 345 * q^20 + 611 * q^21 + 118 * q^22 - 243 * q^23 + 258 * q^24 + 899 * q^25 + 1071 * q^26 + 1609 * q^27 + 46 * q^28 + 773 * q^29 + 375 * q^30 - 97 * q^31 + 1967 * q^32 + 500 * q^33 + 217 * q^34 + 247 * q^35 + 175 * q^36 + 228 * q^37 + 1253 * q^38 + 1493 * q^39 + 2220 * q^40 - 951 * q^41 + 2643 * q^42 - 1378 * q^44 + 1086 * q^45 - 565 * q^46 - 2 * q^47 + 2303 * q^48 + 1264 * q^49 + 3273 * q^50 + 3076 * q^51 - 2825 * q^52 - 39 * q^53 + 5201 * q^54 + 1306 * q^55 + 3683 * q^56 + 1342 * q^57 + 2588 * q^58 - 1065 * q^59 + 2803 * q^60 + 2999 * q^61 + 5569 * q^62 + 2377 * q^63 + 2082 * q^64 + 5578 * q^65 + 3338 * q^66 + 961 * q^67 - 3754 * q^68 + 1817 * q^69 + 2738 * q^70 + 8003 * q^71 + 1412 * q^72 - 1011 * q^73 - 1413 * q^74 + 7457 * q^75 + 5516 * q^76 + 4052 * q^77 + 1091 * q^78 - 4422 * q^79 + 1610 * q^80 + 2108 * q^81 + 4676 * q^82 - 297 * q^83 - 54 * q^84 + 4333 * q^85 + 1377 * q^87 + 3652 * q^88 + 2480 * q^89 - 1414 * q^90 + 4551 * q^91 - 3286 * q^92 + 4 * q^93 + 4609 * q^94 - 835 * q^95 + 5864 * q^96 + 3785 * q^97 + 753 * q^98 + 5072 * 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 $

$2$	−4.73263	−1.67324	−0.836619	−	0.547786i	$-0.815471\pi$
$2$	−4.73263	−1.67324	−0.836619	+	0.547786i	$0.815471\pi$
$3$	−0.112092	−0.0215721	−0.0107860	−	0.999942i	$-0.503433\pi$
$3$	−0.112092	−0.0215721	−0.0107860	+	0.999942i	$0.503433\pi$
$4$	14.3978	1.79972
$5$	9.26635	0.828807	0.414404	−	0.910093i	$-0.363990\pi$
$5$	9.26635	0.828807	0.414404	+	0.910093i	$0.363990\pi$
$6$	0.530489	0.0360952
$7$	−26.6989	−1.44160	−0.720801	−	0.693142i	$-0.756226\pi$
$7$	−26.6989	−1.44160	−0.720801	+	0.693142i	$0.756226\pi$
$8$	−30.2784	−1.33813
$9$	−26.9874	−0.999535
$10$	−43.8542	−1.38679
$11$	−51.7239	−1.41776	−0.708879	−	0.705330i	$-0.750799\pi$
$11$	−51.7239	−1.41776	−0.708879	+	0.705330i	$0.750799\pi$
$12$	−1.61387	−0.0388238
$13$	19.6138	0.418454	0.209227	−	0.977867i	$-0.432905\pi$
$13$	19.6138	0.418454	0.209227	+	0.977867i	$0.432905\pi$
$14$	126.356	2.41214
$15$	−1.03868	−0.0178791
$16$	28.1139	0.439280
$17$	−26.0364	−0.371456	−0.185728	−	0.982601i	$-0.559464\pi$
$17$	−26.0364	−0.371456	−0.185728	+	0.982601i	$0.559464\pi$
$18$	127.722	1.67246
$19$	−28.7769	−0.347468	−0.173734	−	0.984793i	$-0.555583\pi$
$19$	−28.7769	−0.347468	−0.173734	+	0.984793i	$0.555583\pi$
$20$	133.415	1.49162
$21$	2.99272	0.0310984
$22$	244.790	2.37224
$23$	122.891	1.11411	0.557055	−	0.830476i	$-0.311931\pi$
$23$	122.891	1.11411	0.557055	+	0.830476i	$0.311931\pi$
$24$	3.39396	0.0288662
$25$	−39.1348	−0.313078
$26$	−92.8250	−0.700172
$27$	6.05155	0.0431341
$28$	−384.404	−2.59449
$29$	−207.874	−1.33107	−0.665537	−	0.746365i	$-0.731797\pi$
$29$	−207.874	−1.33107	−0.665537	+	0.746365i	$0.731797\pi$
$30$	4.91570	0.0299160
$31$	−263.665	−1.52760	−0.763802	−	0.645451i	$-0.776669\pi$
$31$	−263.665	−1.52760	−0.763802	+	0.645451i	$0.776669\pi$
$32$	109.174	0.603106
$33$	5.79782	0.0305840
$34$	123.221	0.621534
$35$	−247.401	−1.19481
$36$	−388.559	−1.79889
$37$	73.6862	0.327404	0.163702	−	0.986510i	$-0.447657\pi$
$37$	73.6862	0.327404	0.163702	+	0.986510i	$0.447657\pi$
$38$	136.191	0.581396
$39$	−2.19855	−0.00902692
$40$	−280.570	−1.10905
$41$	−169.184	−0.644442	−0.322221	−	0.946665i	$-0.604429\pi$
$41$	−169.184	−0.644442	−0.322221	+	0.946665i	$0.604429\pi$
$42$	−14.1635	−0.0520350
$43$	0	0
$43$	0	0
$44$	−744.709	−2.55157
$45$	−250.075	−0.828422
$46$	−581.597	−1.86417
$47$	−496.303	−1.54028	−0.770141	−	0.637874i	$-0.779814\pi$
$47$	−496.303	−1.54028	−0.770141	+	0.637874i	$0.779814\pi$
$48$	−3.15134	−0.00947619
$49$	369.829	1.07822
$50$	185.210	0.523854
$51$	2.91847	0.00801309
$52$	282.396	0.753101
$53$	−539.979	−1.39947	−0.699734	−	0.714403i	$-0.746698\pi$
$53$	−539.979	−1.39947	−0.699734	+	0.714403i	$0.746698\pi$
$54$	−28.6398	−0.0721737
$55$	−479.291	−1.17505
$56$	808.397	1.92905
$57$	3.22566	0.00749560
$58$	983.789	2.22720
$59$	−773.017	−1.70573	−0.852867	−	0.522129i	$-0.825138\pi$
$59$	−773.017	−1.70573	−0.852867	+	0.522129i	$0.825138\pi$
$60$	−14.9547	−0.0321774
$61$	157.445	0.330472	0.165236	−	0.986254i	$-0.447161\pi$
$61$	157.445	0.330472	0.165236	+	0.986254i	$0.447161\pi$
$62$	1247.83	2.55604
$63$	720.534	1.44093
$64$	−741.591	−1.44842
$65$	181.749	0.346817
$66$	−27.4390	−0.0511743
$67$	158.656	0.289297	0.144648	−	0.989483i	$-0.453795\pi$
$67$	158.656	0.289297	0.144648	+	0.989483i	$0.453795\pi$
$68$	−374.867	−0.668518
$69$	−13.7751	−0.0240337
$70$	1170.86	1.99920
$71$	964.538	1.61225	0.806124	−	0.591746i	$-0.201561\pi$
$71$	964.538	1.61225	0.806124	+	0.591746i	$0.201561\pi$
$72$	817.135	1.33750
$73$	−934.552	−1.49837	−0.749185	−	0.662361i	$-0.769555\pi$
$73$	−934.552	−1.49837	−0.749185	+	0.662361i	$0.769555\pi$
$74$	−348.729	−0.547824
$75$	4.38669	0.00675375
$76$	−414.324	−0.625346
$77$	1380.97	2.04384
$78$	10.4049	0.0151042
$79$	−260.007	−0.370292	−0.185146	−	0.982711i	$-0.559276\pi$
$79$	−260.007	−0.370292	−0.185146	+	0.982711i	$0.559276\pi$
$80$	260.514	0.364079
$81$	727.982	0.998604
$82$	800.686	1.07830
$83$	632.194	0.836052	0.418026	−	0.908435i	$-0.362722\pi$
$83$	632.194	0.836052	0.418026	+	0.908435i	$0.362722\pi$
$84$	43.0886	0.0559685
$85$	−241.262	−0.307866
$86$	0	0
$87$	23.3009	0.0287140
$88$	1566.11	1.89714
$89$	−375.457	−0.447173	−0.223587	−	0.974684i	$-0.571777\pi$
$89$	−375.457	−0.447173	−0.223587	+	0.974684i	$0.571777\pi$
$90$	1183.51	1.38615
$91$	−523.667	−0.603244
$92$	1769.36	2.00509
$93$	29.5547	0.0329536
$94$	2348.82	2.57726
$95$	−266.657	−0.287984
$96$	−12.2375	−0.0130103
$97$	−988.952	−1.03518	−0.517592	−	0.855627i	$-0.673172\pi$
$97$	−988.952	−1.03518	−0.517592	+	0.855627i	$0.673172\pi$
$98$	−1750.26	−1.80412
$99$	1395.89	1.41710
$100$	−563.454	−0.563454
$101$	−1662.06	−1.63743	−0.818716	−	0.574198i	$-0.805314\pi$
$101$	−1662.06	−1.63743	−0.818716	+	0.574198i	$0.805314\pi$
$102$	−13.8120	−0.0134078
$103$	1299.24	1.24289	0.621447	−	0.783456i	$-0.286545\pi$
$103$	1299.24	1.24289	0.621447	+	0.783456i	$0.286545\pi$
$104$	−593.874	−0.559944
$105$	27.7316	0.0257746
$106$	2555.52	2.34164
$107$	−441.630	−0.399009	−0.199505	−	0.979897i	$-0.563933\pi$
$107$	−441.630	−0.399009	−0.199505	+	0.979897i	$0.563933\pi$
$108$	87.1290	0.0776295
$109$	640.229	0.562595	0.281297	−	0.959621i	$-0.409235\pi$
$109$	640.229	0.562595	0.281297	+	0.959621i	$0.409235\pi$
$110$	2268.31	1.96613
$111$	−8.25962	−0.00706278
$112$	−750.610	−0.633268
$113$	−10.8037	−0.00899400	−0.00449700	−	0.999990i	$-0.501431\pi$
$113$	−10.8037	−0.00899400	−0.00449700	+	0.999990i	$0.501431\pi$
$114$	−15.2659	−0.0125419
$115$	1138.75	0.923382
$116$	−2992.92	−2.39556
$117$	−529.327	−0.418259
$118$	3658.40	2.85410
$119$	695.142	0.535492
$120$	31.4496	0.0239245
$121$	1344.36	1.01004
$122$	−745.129	−0.552957
$123$	18.9642	0.0139020
$124$	−3796.20	−2.74926
$125$	−1520.93	−1.08829
$126$	−3410.02	−2.41102
$127$	−270.687	−0.189130	−0.0945652	−	0.995519i	$-0.530146\pi$
$127$	−270.687	−0.189130	−0.0945652	+	0.995519i	$0.530146\pi$
$128$	2636.29	1.82044
$129$	0	0
$130$	−860.149	−0.580308
$131$	1395.97	0.931040	0.465520	−	0.885037i	$-0.345867\pi$
$131$	1395.97	0.931040	0.465520	+	0.885037i	$0.345867\pi$
$132$	83.4758	0.0550427
$133$	768.312	0.500910
$134$	−750.859	−0.484062
$135$	56.0758	0.0357499
$136$	788.339	0.497055
$137$	−689.995	−0.430294	−0.215147	−	0.976582i	$-0.569023\pi$
$137$	−689.995	−0.430294	−0.215147	+	0.976582i	$0.569023\pi$
$138$	65.1923	0.0402140
$139$	−2294.45	−1.40009	−0.700046	−	0.714098i	$-0.746837\pi$
$139$	−2294.45	−1.40009	−0.700046	+	0.714098i	$0.746837\pi$
$140$	−3562.03	−2.15033
$141$	55.6315	0.0332271
$142$	−4564.80	−2.69767
$143$	−1014.50	−0.593266
$144$	−758.723	−0.439076
$145$	−1926.23	−1.10320
$146$	4422.89	2.50713
$147$	−41.4548	−0.0232594
$148$	1060.92	0.589236
$149$	738.215	0.405885	0.202943	−	0.979191i	$-0.434949\pi$
$149$	738.215	0.405885	0.202943	+	0.979191i	$0.434949\pi$
$150$	−20.7606	−0.0113006
$151$	−2340.34	−1.26129	−0.630644	−	0.776072i	$-0.717209\pi$
$151$	−2340.34	−1.26129	−0.630644	+	0.776072i	$0.717209\pi$
$152$	871.319	0.464956
$153$	702.656	0.371283
$154$	−6535.61	−3.41983
$155$	−2443.22	−1.26609
$156$	−31.6543	−0.0162460
$157$	−3433.57	−1.74540	−0.872702	−	0.488253i	$-0.837634\pi$
$157$	−3433.57	−1.74540	−0.872702	+	0.488253i	$0.837634\pi$
$158$	1230.52	0.619587
$159$	60.5273	0.0301895
$160$	1011.64	0.499859
$161$	−3281.05	−1.60610
$162$	−3445.27	−1.67090
$163$	1381.29	0.663749	0.331874	−	0.943324i	$-0.392319\pi$
$163$	1381.29	0.663749	0.331874	+	0.943324i	$0.392319\pi$
$164$	−2435.88	−1.15982
$165$	53.7247	0.0253482
$166$	−2991.94	−1.39891
$167$	1563.46	0.724455	0.362227	−	0.932090i	$-0.382016\pi$
$167$	1563.46	0.724455	0.362227	+	0.932090i	$0.382016\pi$
$168$	−90.6148	−0.0416136
$169$	−1812.30	−0.824897
$170$	1141.81	0.515132
$171$	776.616	0.347306
$172$	0	0
$173$	2369.63	1.04139	0.520693	−	0.853744i	$-0.325674\pi$
$173$	2369.63	1.04139	0.520693	+	0.853744i	$0.325674\pi$
$174$	−110.275	−0.0480454
$175$	1044.85	0.451335
$176$	−1454.16	−0.622793
$177$	86.6490	0.0367962
$178$	1776.90	0.748227
$179$	−2438.65	−1.01829	−0.509144	−	0.860681i	$-0.670038\pi$
$179$	−2438.65	−1.01829	−0.509144	+	0.860681i	$0.670038\pi$
$180$	−3600.53	−1.49093
$181$	3136.26	1.28793	0.643967	−	0.765053i	$-0.277287\pi$
$181$	3136.26	1.28793	0.643967	+	0.765053i	$0.277287\pi$
$182$	2478.32	1.00937
$183$	−17.6483	−0.00712896
$184$	−3720.93	−1.49082
$185$	682.802	0.271355
$186$	−139.872	−0.0551392
$187$	1346.70	0.526635
$188$	−7145.67	−2.77208
$189$	−161.570	−0.0621823
$190$	1261.99	0.481865
$191$	2667.97	1.01072	0.505360	−	0.862909i	$-0.331360\pi$
$191$	2667.97	1.01072	0.505360	+	0.862909i	$0.331360\pi$
$192$	83.1264	0.0312455
$193$	1624.05	0.605710	0.302855	−	0.953037i	$-0.402060\pi$
$193$	1624.05	0.605710	0.302855	+	0.953037i	$0.402060\pi$
$194$	4680.34	1.73211
$195$	−20.3725	−0.00748158
$196$	5324.72	1.94050
$197$	−4235.27	−1.53173	−0.765864	−	0.643002i	$-0.777689\pi$
$197$	−4235.27	−1.53173	−0.765864	+	0.643002i	$0.777689\pi$
$198$	−6606.25	−2.37114
$199$	−1016.79	−0.362203	−0.181101	−	0.983464i	$-0.557966\pi$
$199$	−1016.79	−0.362203	−0.181101	+	0.983464i	$0.557966\pi$
$200$	1184.94	0.418938
$201$	−17.7840	−0.00624074
$202$	7865.89	2.73981
$203$	5549.99	1.91888
$204$	42.0195	0.0144213
$205$	−1567.72	−0.534118
$206$	−6148.83	−2.07966
$207$	−3316.51	−1.11359
$208$	551.422	0.183818
$209$	1488.45	0.492625
$210$	−131.244	−0.0431270
$211$	1199.85	0.391476	0.195738	−	0.980656i	$-0.437290\pi$
$211$	1199.85	0.391476	0.195738	+	0.980656i	$0.437290\pi$
$212$	−7774.50	−2.51866
$213$	−108.117	−0.0347796
$214$	2090.07	0.667637
$215$	0	0
$216$	−183.231	−0.0577189
$217$	7039.56	2.20220
$218$	−3029.97	−0.941354
$219$	104.756	0.0323230
$220$	−6900.73	−2.11476
$221$	−510.673	−0.155437
$222$	39.0897	0.0118177
$223$	−871.285	−0.261639	−0.130820	−	0.991406i	$-0.541761\pi$
$223$	−871.285	−0.261639	−0.130820	+	0.991406i	$0.541761\pi$
$224$	−2914.82	−0.869440
$225$	1056.15	0.312933
$226$	51.1297	0.0150491
$227$	−509.533	−0.148982	−0.0744910	−	0.997222i	$-0.523733\pi$
$227$	−509.533	−0.148982	−0.0744910	+	0.997222i	$0.523733\pi$
$228$	46.4424	0.0134900
$229$	29.2888	0.00845178	0.00422589	−	0.999991i	$-0.498655\pi$
$229$	29.2888	0.00845178	0.00422589	+	0.999991i	$0.498655\pi$
$230$	−5389.28	−1.54504
$231$	−154.795	−0.0440900
$232$	6294.07	1.78115
$233$	−362.476	−0.101917	−0.0509583	−	0.998701i	$-0.516228\pi$
$233$	−362.476	−0.101917	−0.0509583	+	0.998701i	$0.516228\pi$
$234$	2505.11	0.699846
$235$	−4598.92	−1.27660
$236$	−11129.7	−3.06985
$237$	29.1447	0.00798798
$238$	−3289.85	−0.896006
$239$	−4490.45	−1.21533	−0.607664	−	0.794195i	$-0.707893\pi$
$239$	−4490.45	−1.21533	−0.607664	+	0.794195i	$0.707893\pi$
$240$	−29.2015	−0.00785394
$241$	2537.70	0.678290	0.339145	−	0.940734i	$-0.389862\pi$
$241$	2537.70	0.678290	0.339145	+	0.940734i	$0.389862\pi$
$242$	−6362.35	−1.69003
$243$	−244.993	−0.0646761
$244$	2266.86	0.594757
$245$	3426.96	0.893636
$246$	−89.7503	−0.0232613
$247$	−564.426	−0.145399
$248$	7983.35	2.04413
$249$	−70.8638	−0.0180354
$250$	7198.00	1.82097
$251$	602.417	0.151491	0.0757455	−	0.997127i	$-0.475866\pi$
$251$	602.417	0.151491	0.0757455	+	0.997127i	$0.475866\pi$
$252$	10374.1	2.59328
$253$	−6356.39	−1.57954
$254$	1281.06	0.316460
$255$	27.0435	0.00664130
$256$	−6543.83	−1.59762
$257$	458.487	0.111283	0.0556413	−	0.998451i	$-0.482280\pi$
$257$	458.487	0.111283	0.0556413	+	0.998451i	$0.482280\pi$
$258$	0	0
$259$	−1967.34	−0.471986
$260$	2616.78	0.624175
$261$	5609.97	1.33045
$262$	−6606.60	−1.55785
$263$	2731.45	0.640413	0.320207	−	0.947348i	$-0.396248\pi$
$263$	2731.45	0.640413	0.320207	+	0.947348i	$0.396248\pi$
$264$	−175.549	−0.0409253
$265$	−5003.63	−1.15989
$266$	−3636.13	−0.838142
$267$	42.0857	0.00964646
$268$	2284.29	0.520654
$269$	59.7747	0.0135484	0.00677421	−	0.999977i	$-0.497844\pi$
$269$	59.7747	0.0135484	0.00677421	+	0.999977i	$0.497844\pi$
$270$	−265.386	−0.0598181
$271$	313.484	0.0702685	0.0351343	−	0.999383i	$-0.488814\pi$
$271$	313.484	0.0702685	0.0351343	+	0.999383i	$0.488814\pi$
$272$	−731.986	−0.163173
$273$	58.6988	0.0130132
$274$	3265.49	0.719983
$275$	2024.20	0.443869
$276$	−198.331	−0.0432540
$277$	4038.08	0.875902	0.437951	−	0.898999i	$-0.355704\pi$
$277$	4038.08	0.875902	0.437951	+	0.898999i	$0.355704\pi$
$278$	10858.8	2.34269
$279$	7115.65	1.52689
$280$	7490.89	1.59881
$281$	−3089.92	−0.655975	−0.327988	−	0.944682i	$-0.606370\pi$
$281$	−3089.92	−0.655975	−0.327988	+	0.944682i	$0.606370\pi$
$282$	−263.284	−0.0555968
$283$	1265.62	0.265843	0.132921	−	0.991127i	$-0.457564\pi$
$283$	1265.62	0.265843	0.132921	+	0.991127i	$0.457564\pi$
$284$	13887.2	2.90160
$285$	29.8901	0.00621241
$286$	4801.27	0.992674
$287$	4517.02	0.929029
$288$	−2946.32	−0.602826
$289$	−4235.11	−0.862020
$290$	9116.13	1.84592
$291$	110.854	0.0223311
$292$	−13455.5	−2.69665
$293$	−1806.92	−0.360277	−0.180138	−	0.983641i	$-0.557655\pi$
$293$	−1806.92	−0.360277	−0.180138	+	0.983641i	$0.557655\pi$
$294$	196.190	0.0389186
$295$	−7163.05	−1.41372
$296$	−2231.10	−0.438108
$297$	−313.010	−0.0611537
$298$	−3493.70	−0.679143
$299$	2410.36	0.466203
$300$	63.1586	0.0121549
$301$	0	0
$302$	11076.0	2.11043
$303$	186.303	0.0353229
$304$	−809.033	−0.152636
$305$	1458.94	0.273897
$306$	−3325.41	−0.621245
$307$	−1852.10	−0.344316	−0.172158	−	0.985069i	$-0.555074\pi$
$307$	−1852.10	−0.344316	−0.172158	+	0.985069i	$0.555074\pi$
$308$	19882.9	3.67835
$309$	−145.634	−0.0268118
$310$	11562.8	2.11847
$311$	6373.79	1.16214	0.581069	−	0.813855i	$-0.302635\pi$
$311$	6373.79	1.16214	0.581069	+	0.813855i	$0.302635\pi$
$312$	66.5685	0.0120792
$313$	3481.92	0.628785	0.314393	−	0.949293i	$-0.398199\pi$
$313$	3481.92	0.628785	0.314393	+	0.949293i	$0.398199\pi$
$314$	16249.8	2.92048
$315$	6676.72	1.19426
$316$	−3743.53	−0.666424
$317$	3510.02	0.621900	0.310950	−	0.950426i	$-0.399353\pi$
$317$	3510.02	0.621900	0.310950	+	0.950426i	$0.399353\pi$
$318$	−286.453	−0.0505141
$319$	10752.0	1.88714
$320$	−6871.84	−1.20046
$321$	49.5031	0.00860746
$322$	15528.0	2.68739
$323$	749.248	0.129069
$324$	10481.3	1.79721
$325$	−767.583	−0.131009
$326$	−6537.14	−1.11061
$327$	−71.7645	−0.0121363
$328$	5122.61	0.862345
$329$	13250.7	2.22048
$330$	−254.259	−0.0424136
$331$	−8790.82	−1.45978	−0.729890	−	0.683565i	$-0.760429\pi$
$331$	−8790.82	−1.45978	−0.729890	+	0.683565i	$0.760429\pi$
$332$	9102.20	1.50466
$333$	−1988.60	−0.327251
$334$	−7399.27	−1.21218
$335$	1470.16	0.239771
$336$	84.1373	0.0136609
$337$	6257.33	1.01145	0.505725	−	0.862695i	$-0.331225\pi$
$337$	6257.33	1.01145	0.505725	+	0.862695i	$0.331225\pi$
$338$	8576.94	1.38025
$339$	1.21100	0.000194019 0
$340$	−3473.64	−0.554073
$341$	13637.8	2.16577
$342$	−3675.44	−0.581125
$343$	−716.305	−0.112760
$344$	0	0
$345$	−127.645	−0.0199193
$346$	−11214.6	−1.74249
$347$	−3588.48	−0.555158	−0.277579	−	0.960703i	$-0.589532\pi$
$347$	−3588.48	−0.555158	−0.277579	+	0.960703i	$0.589532\pi$
$348$	335.482	0.0516773
$349$	2838.19	0.435315	0.217657	−	0.976025i	$-0.430158\pi$
$349$	2838.19	0.435315	0.217657	+	0.976025i	$0.430158\pi$
$350$	−4944.91	−0.755190
$351$	118.694	0.0180496
$352$	−5646.90	−0.855059
$353$	5179.05	0.780887	0.390443	−	0.920627i	$-0.372322\pi$
$353$	5179.05	0.780887	0.390443	+	0.920627i	$0.372322\pi$
$354$	−410.077	−0.0615688
$355$	8937.75	1.33624
$356$	−5405.75	−0.804788
$357$	−77.9198	−0.0115517
$358$	11541.2	1.70384
$359$	13098.4	1.92565	0.962827	−	0.270120i	$-0.0870632\pi$
$359$	13098.4	1.92565	0.962827	+	0.270120i	$0.0870632\pi$
$360$	7571.86	1.10853
$361$	−6030.89	−0.879266
$362$	−14842.7	−2.15502
$363$	−150.692	−0.0217886
$364$	−7539.64	−1.08567
$365$	−8659.88	−1.24186
$366$	83.5229	0.0119284
$367$	−2611.45	−0.371435	−0.185717	−	0.982603i	$-0.559461\pi$
$367$	−2611.45	−0.371435	−0.185717	+	0.982603i	$0.559461\pi$
$368$	3454.95	0.489407
$369$	4565.84	0.644142
$370$	−3231.45	−0.454041
$371$	14416.8	2.01748
$372$	425.523	0.0593074
$373$	−5408.49	−0.750780	−0.375390	−	0.926867i	$-0.622491\pi$
$373$	−5408.49	−0.750780	−0.375390	+	0.926867i	$0.622491\pi$
$374$	−6373.45	−0.881185
$375$	170.484	0.0234767
$376$	15027.2	2.06109
$377$	−4077.20	−0.556993
$378$	764.649	0.104046
$379$	2453.85	0.332574	0.166287	−	0.986077i	$-0.446822\pi$
$379$	2453.85	0.332574	0.166287	+	0.986077i	$0.446822\pi$
$380$	−3839.27	−0.518291
$381$	30.3418	0.00407994
$382$	−12626.5	−1.69117
$383$	1835.07	0.244824	0.122412	−	0.992479i	$-0.460937\pi$
$383$	1835.07	0.244824	0.122412	+	0.992479i	$0.460937\pi$
$384$	−295.506	−0.0392708
$385$	12796.5	1.69395
$386$	−7686.05	−1.01350
$387$	0	0
$388$	−14238.7	−1.86305
$389$	−11261.1	−1.46776	−0.733882	−	0.679277i	$-0.762293\pi$
$389$	−11261.1	−1.46776	−0.733882	+	0.679277i	$0.762293\pi$
$390$	96.4157	0.0125185
$391$	−3199.64	−0.413843
$392$	−11197.8	−1.44279
$393$	−156.477	−0.0200845
$394$	20044.0	2.56295
$395$	−2409.32	−0.306901
$396$	20097.8	2.55038
$397$	−14228.7	−1.79879	−0.899396	−	0.437135i	$-0.855993\pi$
$397$	−14228.7	−1.79879	−0.899396	+	0.437135i	$0.855993\pi$
$398$	4812.09	0.606051
$399$	−86.1215	−0.0108057
$400$	−1100.23	−0.137529
$401$	8238.25	1.02593	0.512966	−	0.858409i	$-0.328547\pi$
$401$	8238.25	1.02593	0.512966	+	0.858409i	$0.328547\pi$
$402$	84.1652	0.0104422
$403$	−5171.49	−0.639231
$404$	−23929.9	−2.94693
$405$	6745.74	0.827650
$406$	−26266.0	−3.21074
$407$	−3811.33	−0.464179
$408$	−88.3664	−0.0107225
$409$	10717.4	1.29570	0.647851	−	0.761767i	$-0.275668\pi$
$409$	10717.4	1.29570	0.647851	+	0.761767i	$0.275668\pi$
$410$	7419.43	0.893706
$411$	77.3428	0.00928233
$412$	18706.2	2.23687
$413$	20638.7	2.45899
$414$	15695.8	1.86330
$415$	5858.13	0.692926
$416$	2141.32	0.252372
$417$	257.189	0.0302029
$418$	−7044.31	−0.824278
$419$	5689.14	0.663324	0.331662	−	0.943398i	$-0.392391\pi$
$419$	5689.14	0.663324	0.331662	+	0.943398i	$0.392391\pi$
$420$	399.274	0.0463871
$421$	−9088.34	−1.05211	−0.526055	−	0.850451i	$-0.676329\pi$
$421$	−9088.34	−1.05211	−0.526055	+	0.850451i	$0.676329\pi$
$422$	−5678.47	−0.655032
$423$	13393.9	1.53957
$424$	16349.7	1.87267
$425$	1018.93	0.116295
$426$	511.677	0.0581945
$427$	−4203.60	−0.476409
$428$	−6358.49	−0.718106
$429$	113.718	0.0127980
$430$	0	0
$431$	6799.33	0.759889	0.379944	−	0.925009i	$-0.375943\pi$
$431$	6799.33	0.759889	0.379944	+	0.925009i	$0.375943\pi$
$432$	170.133	0.0189480
$433$	−16399.2	−1.82008	−0.910041	−	0.414518i	$-0.863950\pi$
$433$	−16399.2	−1.82008	−0.910041	+	0.414518i	$0.863950\pi$
$434$	−33315.7	−3.68480
$435$	215.915	0.0237984
$436$	9217.88	1.01251
$437$	−3536.43	−0.387117
$438$	−495.770	−0.0540840
$439$	3647.81	0.396584	0.198292	−	0.980143i	$-0.436461\pi$
$439$	3647.81	0.396584	0.198292	+	0.980143i	$0.436461\pi$
$440$	14512.2	1.57236
$441$	−9980.74	−1.07772
$442$	2416.83	0.260083
$443$	11100.7	1.19055	0.595273	−	0.803523i	$-0.297044\pi$
$443$	11100.7	1.19055	0.595273	+	0.803523i	$0.297044\pi$
$444$	−118.920	−0.0127111
$445$	−3479.12	−0.370620
$446$	4123.47	0.437785
$447$	−82.7479	−0.00875580
$448$	19799.6	2.08805
$449$	−1760.76	−0.185067	−0.0925336	−	0.995710i	$-0.529497\pi$
$449$	−1760.76	−0.185067	−0.0925336	+	0.995710i	$0.529497\pi$
$450$	−4998.36	−0.523611
$451$	8750.85	0.913662
$452$	−155.549	−0.0161867
$453$	262.334	0.0272086
$454$	2411.43	0.249282
$455$	−4852.48	−0.499973
$456$	−97.6677	−0.0100301
$457$	−18784.9	−1.92280	−0.961401	−	0.275152i	$-0.911272\pi$
$457$	−18784.9	−1.92280	−0.961401	+	0.275152i	$0.911272\pi$
$458$	−138.613	−0.0141418
$459$	−157.561	−0.0160224
$460$	16395.5	1.66183
$461$	−846.007	−0.0854717	−0.0427358	−	0.999086i	$-0.513607\pi$
$461$	−846.007	−0.0854717	−0.0427358	+	0.999086i	$0.513607\pi$
$462$	732.589	0.0737730
$463$	−11515.9	−1.15592	−0.577958	−	0.816067i	$-0.696150\pi$
$463$	−11515.9	−1.15592	−0.577958	+	0.816067i	$0.696150\pi$
$464$	−5844.14	−0.584715
$465$	273.865	0.0273122
$466$	1715.46	0.170531
$467$	−8315.37	−0.823961	−0.411980	−	0.911193i	$-0.635163\pi$
$467$	−8315.37	−0.823961	−0.411980	+	0.911193i	$0.635163\pi$
$468$	−7621.13	−0.752750
$469$	−4235.93	−0.417051
$470$	21765.0	2.13605
$471$	384.875	0.0376520
$472$	23405.7	2.28249
$473$	0	0
$474$	−137.931	−0.0133658
$475$	1126.18	0.108785
$476$	10008.5	0.963738
$477$	14572.7	1.39882
$478$	21251.6	2.03353
$479$	3210.25	0.306221	0.153111	−	0.988209i	$-0.451071\pi$
$479$	3210.25	0.306221	0.153111	+	0.988209i	$0.451071\pi$
$480$	−113.397	−0.0107830
$481$	1445.27	0.137003
$482$	−12010.0	−1.13494
$483$	367.779	0.0346470
$484$	19355.8	1.81779
$485$	−9163.98	−0.857968
$486$	1159.46	0.108218
$487$	−7485.57	−0.696517	−0.348258	−	0.937399i	$-0.613227\pi$
$487$	−7485.57	−0.696517	−0.348258	+	0.937399i	$0.613227\pi$
$488$	−4767.18	−0.442213
$489$	−154.831	−0.0143184
$490$	−16218.6	−1.49526
$491$	9840.84	0.904503	0.452252	−	0.891890i	$-0.350621\pi$
$491$	9840.84	0.904503	0.452252	+	0.891890i	$0.350621\pi$
$492$	273.042	0.0250197
$493$	5412.28	0.494436
$494$	2671.22	0.243287
$495$	12934.8	1.17450
$496$	−7412.67	−0.671046
$497$	−25752.1	−2.32422
$498$	335.372	0.0301775
$499$	18529.8	1.66234	0.831170	−	0.556018i	$-0.187671\pi$
$499$	18529.8	1.66234	0.831170	+	0.556018i	$0.187671\pi$
$500$	−21898.0	−1.95862
$501$	−175.251	−0.0156280
$502$	−2851.02	−0.253480
$503$	9865.58	0.874522	0.437261	−	0.899335i	$-0.355949\pi$
$503$	9865.58	0.874522	0.437261	+	0.899335i	$0.355949\pi$
$504$	−21816.6	−1.92815
$505$	−15401.2	−1.35712
$506$	30082.5	2.64294
$507$	203.144	0.0177947
$508$	−3897.29	−0.340382
$509$	4643.48	0.404359	0.202179	−	0.979349i	$-0.435198\pi$
$509$	4643.48	0.404359	0.202179	+	0.979349i	$0.435198\pi$
$510$	−127.987	−0.0111125
$511$	24951.5	2.16005
$512$	9879.26	0.852746
$513$	−174.145	−0.0149877
$514$	−2169.85	−0.186202
$515$	12039.2	1.03012
$516$	0	0
$517$	25670.7	2.18375
$518$	9310.68	0.789745
$519$	−265.616	−0.0224649
$520$	−5503.05	−0.464086
$521$	15883.1	1.33561	0.667805	−	0.744336i	$-0.267234\pi$
$521$	15883.1	1.33561	0.667805	+	0.744336i	$0.267234\pi$
$522$	−26549.9	−2.22617
$523$	−15729.1	−1.31507	−0.657537	−	0.753422i	$-0.728402\pi$
$523$	−15729.1	−1.31507	−0.657537	+	0.753422i	$0.728402\pi$
$524$	20098.8	1.67561
$525$	−117.120	−0.00973623
$526$	−12927.0	−1.07156
$527$	6864.90	0.567438
$528$	163.000	0.0134349
$529$	2935.18	0.241241
$530$	23680.3	1.94077
$531$	20861.8	1.70494
$532$	11062.0	0.901500
$533$	−3318.35	−0.269669
$534$	−199.176	−0.0161408
$535$	−4092.30	−0.330702
$536$	−4803.83	−0.387116
$537$	273.353	0.0219666
$538$	−282.891	−0.0226697
$539$	−19129.0	−1.52865
$540$	807.367	0.0643399
$541$	−12963.9	−1.03024	−0.515120	−	0.857118i	$-0.672253\pi$
$541$	−12963.9	−1.03024	−0.515120	+	0.857118i	$0.672253\pi$
$542$	−1483.60	−0.117576
$543$	−351.549	−0.0277834
$544$	−2842.50	−0.224028
$545$	5932.58	0.466283
$546$	−277.800	−0.0217742
$547$	12649.2	0.988740	0.494370	−	0.869251i	$-0.335399\pi$
$547$	12649.2	0.988740	0.494370	+	0.869251i	$0.335399\pi$
$548$	−9934.39	−0.774409
$549$	−4249.04	−0.330318
$550$	−9579.80	−0.742698
$551$	5981.97	0.462505
$552$	417.086	0.0321601
$553$	6941.90	0.533815
$554$	−19110.8	−1.46559
$555$	−76.5365	−0.00585368
$556$	−33035.0	−2.51978
$557$	−9926.99	−0.755153	−0.377576	−	0.925978i	$-0.623242\pi$
$557$	−9926.99	−0.755153	−0.377576	+	0.925978i	$0.623242\pi$
$558$	−33675.8	−2.55485
$559$	0	0
$560$	−6955.41	−0.524857
$561$	−150.954	−0.0113606
$562$	14623.4	1.09760
$563$	−8142.55	−0.609534	−0.304767	−	0.952427i	$-0.598579\pi$
$563$	−8142.55	−0.609534	−0.304767	+	0.952427i	$0.598579\pi$
$564$	800.971	0.0597996
$565$	−100.110	−0.00745429
$566$	−5989.73	−0.444818
$567$	−19436.3	−1.43959
$568$	−29204.6	−2.15739
$569$	3555.31	0.261944	0.130972	−	0.991386i	$-0.458190\pi$
$569$	3555.31	0.261944	0.130972	+	0.991386i	$0.458190\pi$
$570$	−141.459	−0.0103948
$571$	8157.20	0.597843	0.298921	−	0.954278i	$-0.403373\pi$
$571$	8157.20	0.597843	0.298921	+	0.954278i	$0.403373\pi$
$572$	−14606.6	−1.06771
$573$	−299.058	−0.0218033
$574$	−21377.4	−1.55449
$575$	−4809.31	−0.348804
$576$	20013.6	1.44775
$577$	−4139.33	−0.298652	−0.149326	−	0.988788i	$-0.547710\pi$
$577$	−4139.33	−0.298652	−0.149326	+	0.988788i	$0.547710\pi$
$578$	20043.2	1.44236
$579$	−182.043	−0.0130664
$580$	−27733.4	−1.98546
$581$	−16878.9	−1.20526
$582$	−524.629	−0.0373652
$583$	27929.8	1.98411
$584$	28296.7	2.00501
$585$	−4904.93	−0.346656
$586$	8551.46	0.602829
$587$	12187.7	0.856965	0.428483	−	0.903550i	$-0.359048\pi$
$587$	12187.7	0.856965	0.428483	+	0.903550i	$0.359048\pi$
$588$	−596.858	−0.0418605
$589$	7587.49	0.530793
$590$	33900.1	2.36550
$591$	474.740	0.0330426
$592$	2071.61	0.143822
$593$	2112.35	0.146279	0.0731397	−	0.997322i	$-0.476698\pi$
$593$	2112.35	0.146279	0.0731397	+	0.997322i	$0.476698\pi$
$594$	1481.36	0.102325
$595$	6441.43	0.443820
$596$	10628.7	0.730481
$597$	113.974	0.00781347
$598$	−11407.3	−0.780069
$599$	11709.3	0.798714	0.399357	−	0.916795i	$-0.369233\pi$
$599$	11709.3	0.798714	0.399357	+	0.916795i	$0.369233\pi$
$600$	−132.822	−0.00903738
$601$	12356.6	0.838665	0.419332	−	0.907833i	$-0.362264\pi$
$601$	12356.6	0.838665	0.419332	+	0.907833i	$0.362264\pi$
$602$	0	0
$603$	−4281.71	−0.289162
$604$	−33695.8	−2.26997
$605$	12457.3	0.837125
$606$	−881.703	−0.0591035
$607$	−18660.7	−1.24780	−0.623899	−	0.781505i	$-0.714452\pi$
$607$	−18660.7	−1.24780	−0.623899	+	0.781505i	$0.714452\pi$
$608$	−3141.69	−0.209560
$609$	−622.108	−0.0413943
$610$	−6904.62	−0.458295
$611$	−9734.40	−0.644537
$612$	10116.7	0.668207
$613$	11643.8	0.767194	0.383597	−	0.923501i	$-0.374685\pi$
$613$	11643.8	0.767194	0.383597	+	0.923501i	$0.374685\pi$
$614$	8765.32	0.576123
$615$	175.729	0.0115220
$616$	−41813.4	−2.73492
$617$	−16473.2	−1.07486	−0.537429	−	0.843309i	$-0.680605\pi$
$617$	−16473.2	−1.07486	−0.537429	+	0.843309i	$0.680605\pi$
$618$	689.234	0.0448625
$619$	139.905	0.00908440	0.00454220	−	0.999990i	$-0.498554\pi$
$619$	139.905	0.00908440	0.00454220	+	0.999990i	$0.498554\pi$
$620$	−35176.9	−2.27861
$621$	743.681	0.0480562
$622$	−30164.8	−1.94453
$623$	10024.3	0.644646
$624$	−61.8099	−0.00396535
$625$	−9201.62	−0.588904
$626$	−16478.6	−1.05211
$627$	−166.844	−0.0106269
$628$	−49435.8	−3.14124
$629$	−1918.52	−0.121616
$630$	−31598.4	−1.99827
$631$	−2625.62	−0.165649	−0.0828243	−	0.996564i	$-0.526394\pi$
$631$	−2625.62	−0.165649	−0.0828243	+	0.996564i	$0.526394\pi$
$632$	7872.59	0.495498
$633$	−134.494	−0.00844495
$634$	−16611.6	−1.04059
$635$	−2508.28	−0.156753
$636$	871.459	0.0543327
$637$	7253.76	0.451184
$638$	−50885.3	−3.15763
$639$	−26030.4	−1.61150
$640$	24428.7	1.50880
$641$	−2373.14	−0.146230	−0.0731149	−	0.997324i	$-0.523294\pi$
$641$	−2373.14	−0.146230	−0.0731149	+	0.997324i	$0.523294\pi$
$642$	−234.280	−0.0144023
$643$	16611.8	1.01883	0.509414	−	0.860521i	$-0.329862\pi$
$643$	16611.8	1.01883	0.509414	+	0.860521i	$0.329862\pi$
$644$	−47239.8	−2.89054
$645$	0	0
$646$	−3545.91	−0.215963
$647$	26021.2	1.58114	0.790570	−	0.612372i	$-0.209784\pi$
$647$	26021.2	1.58114	0.790570	+	0.612372i	$0.209784\pi$
$648$	−22042.1	−1.33626
$649$	39983.4	2.41832
$650$	3632.69	0.219209
$651$	−789.078	−0.0475060
$652$	19887.5	1.19456
$653$	3783.97	0.226766	0.113383	−	0.993551i	$-0.463831\pi$
$653$	3783.97	0.226766	0.113383	+	0.993551i	$0.463831\pi$
$654$	339.635	0.0203070
$655$	12935.5	0.771653
$656$	−4756.43	−0.283091
$657$	25221.2	1.49767
$658$	−62710.8	−3.71538
$659$	−13511.3	−0.798670	−0.399335	−	0.916805i	$-0.630759\pi$
$659$	−13511.3	−0.798670	−0.399335	+	0.916805i	$0.630759\pi$
$660$	773.516	0.0456198
$661$	−30399.5	−1.78881	−0.894404	−	0.447260i	$-0.852400\pi$
$661$	−30399.5	−1.78881	−0.894404	+	0.447260i	$0.852400\pi$
$662$	41603.7	2.44256
$663$	57.2423	0.00335310
$664$	−19141.8	−1.11874
$665$	7119.44	0.415158
$666$	9411.31	0.547569
$667$	−25545.8	−1.48296
$668$	22510.3	1.30382
$669$	97.6640	0.00564411
$670$	−6957.72	−0.401194
$671$	−8143.66	−0.468529
$672$	326.728	0.0187556
$673$	14949.3	0.856248	0.428124	−	0.903720i	$-0.359175\pi$
$673$	14949.3	0.856248	0.428124	+	0.903720i	$0.359175\pi$
$674$	−29613.6	−1.69240
$675$	−236.826	−0.0135044
$676$	−26093.1	−1.48459
$677$	−6020.73	−0.341795	−0.170898	−	0.985289i	$-0.554667\pi$
$677$	−6020.73	−0.341795	−0.170898	+	0.985289i	$0.554667\pi$
$678$	−5.73122	−0.000324640 0
$679$	26403.9	1.49232
$680$	7305.03	0.411963
$681$	57.1146	0.00321385
$682$	−64542.6	−3.62385
$683$	12622.0	0.707126	0.353563	−	0.935411i	$-0.384970\pi$
$683$	12622.0	0.707126	0.353563	+	0.935411i	$0.384970\pi$
$684$	11181.6	0.625055
$685$	−6393.73	−0.356630
$686$	3390.00	0.188675
$687$	−3.28303	−0.000182323 0
$688$	0	0
$689$	−10591.1	−0.585613
$690$	604.095	0.0333297
$691$	22807.2	1.25561	0.627805	−	0.778371i	$-0.283954\pi$
$691$	22807.2	1.25561	0.627805	+	0.778371i	$0.283954\pi$
$692$	34117.4	1.87421
$693$	−37268.8	−2.04289
$694$	16983.0	0.928911
$695$	−21261.2	−1.16041
$696$	−705.514	−0.0384230
$697$	4404.94	0.239382
$698$	−13432.1	−0.728385
$699$	40.6306	0.00219856
$700$	15043.6	0.812277
$701$	23450.4	1.26349	0.631747	−	0.775175i	$-0.282338\pi$
$701$	23450.4	1.26349	0.631747	+	0.775175i	$0.282338\pi$
$702$	−561.735	−0.0302013
$703$	−2120.46	−0.113762
$704$	38358.0	2.05351
$705$	515.501	0.0275389
$706$	−24510.5	−1.30661
$707$	44375.0	2.36053
$708$	1247.55	0.0662230
$709$	19708.7	1.04397	0.521987	−	0.852954i	$-0.325191\pi$
$709$	19708.7	1.04397	0.521987	+	0.852954i	$0.325191\pi$
$710$	−42299.1	−2.23585
$711$	7016.93	0.370120
$712$	11368.2	0.598374
$713$	−32402.1	−1.70192
$714$	368.766	0.0193287
$715$	−9400.74	−0.491703
$716$	−35111.2	−1.83264
$717$	503.343	0.0262171
$718$	−61990.1	−3.22208
$719$	22063.2	1.14439	0.572197	−	0.820116i	$-0.306091\pi$
$719$	22063.2	1.14439	0.572197	+	0.820116i	$0.306091\pi$
$720$	−7030.59	−0.363909
$721$	−34688.3	−1.79176
$722$	28542.0	1.47122
$723$	−284.456	−0.0146321
$724$	45155.1	2.31792
$725$	8135.09	0.416730
$726$	713.168	0.0364575
$727$	25888.0	1.32068	0.660339	−	0.750967i	$-0.270412\pi$
$727$	25888.0	1.32068	0.660339	+	0.750967i	$0.270412\pi$
$728$	15855.8	0.807217
$729$	−19628.1	−0.997209
$730$	40984.0	2.07793
$731$	0	0
$732$	−254.097	−0.0128302
$733$	7039.38	0.354714	0.177357	−	0.984147i	$-0.443245\pi$
$733$	7039.38	0.354714	0.177357	+	0.984147i	$0.443245\pi$
$734$	12359.0	0.621498
$735$	−384.135	−0.0192776
$736$	13416.5	0.671927
$737$	−8206.29	−0.410153
$738$	−21608.5	−1.07780
$739$	−2258.79	−0.112437	−0.0562185	−	0.998418i	$-0.517904\pi$
$739$	−2258.79	−0.112437	−0.0562185	+	0.998418i	$0.517904\pi$
$740$	9830.83	0.488363
$741$	63.2676	0.00313656
$742$	−68229.5	−3.37572
$743$	−6115.36	−0.301952	−0.150976	−	0.988537i	$-0.548242\pi$
$743$	−6115.36	−0.301952	−0.150976	+	0.988537i	$0.548242\pi$
$744$	−894.869	−0.0440961
$745$	6840.56	0.336401
$746$	25596.4	1.25623
$747$	−17061.3	−0.835663
$748$	19389.5	0.947797
$749$	11791.0	0.575213
$750$	−806.837	−0.0392820
$751$	−2383.93	−0.115833	−0.0579166	−	0.998321i	$-0.518446\pi$
$751$	−2383.93	−0.115833	−0.0579166	+	0.998321i	$0.518446\pi$
$752$	−13953.0	−0.676616
$753$	−67.5261	−0.00326798
$754$	19295.9	0.931981
$755$	−21686.4	−1.04536
$756$	−2326.24	−0.111911
$757$	−7043.84	−0.338194	−0.169097	−	0.985599i	$-0.554085\pi$
$757$	−7043.84	−0.338194	−0.169097	+	0.985599i	$0.554085\pi$
$758$	−11613.2	−0.556476
$759$	712.500	0.0340739
$760$	8073.94	0.385359
$761$	29501.1	1.40528	0.702638	−	0.711548i	$-0.252006\pi$
$761$	29501.1	1.40528	0.702638	+	0.711548i	$0.252006\pi$
$762$	−143.596	−0.00682671
$763$	−17093.4	−0.811038
$764$	38412.9	1.81902
$765$	6511.05	0.307722
$766$	−8684.70	−0.409649
$767$	−15161.8	−0.713770
$768$	733.511	0.0344639
$769$	−35816.4	−1.67955	−0.839775	−	0.542935i	$-0.817313\pi$
$769$	−35816.4	−1.67955	−0.839775	+	0.542935i	$0.817313\pi$
$770$	−60561.2	−2.83438
$771$	−51.3926	−0.00240060
$772$	23382.8	1.09011
$773$	5118.55	0.238165	0.119082	−	0.992884i	$-0.462005\pi$
$773$	5118.55	0.238165	0.119082	+	0.992884i	$0.462005\pi$
$774$	0	0
$775$	10318.5	0.478259
$776$	29943.8	1.38521
$777$	220.522	0.0101817
$778$	53294.6	2.45592
$779$	4868.60	0.223923
$780$	−293.319	−0.0134648
$781$	−49889.6	−2.28578
$782$	15142.7	0.692458
$783$	−1257.96	−0.0574147
$784$	10397.4	0.473640
$785$	−31816.6	−1.44660
$786$	740.546	0.0336061
$787$	−17449.9	−0.790371	−0.395186	−	0.918601i	$-0.629320\pi$
$787$	−17449.9	−0.790371	−0.395186	+	0.918601i	$0.629320\pi$
$788$	−60978.5	−2.75669
$789$	−306.174	−0.0138151
$790$	11402.4	0.513518
$791$	288.445	0.0129658
$792$	−42265.4	−1.89626
$793$	3088.10	0.138287
$794$	67339.4	3.00980
$795$	560.867	0.0250212
$796$	−14639.5	−0.651865
$797$	−34176.4	−1.51893	−0.759466	−	0.650546i	$-0.774540\pi$
$797$	−34176.4	−1.51893	−0.759466	+	0.650546i	$0.774540\pi$
$798$	407.581	0.0180805
$799$	12921.9	0.572147
$800$	−4272.50	−0.188820
$801$	10132.6	0.446965
$802$	−38988.6	−1.71663
$803$	48338.6	2.12433
$804$	−256.051	−0.0112316
$805$	−30403.3	−1.33115
$806$	24474.7	1.06959
$807$	−6.70026	−0.000292268 0
$808$	50324.3	2.19109
$809$	−16262.0	−0.706725	−0.353363	−	0.935486i	$-0.614962\pi$
$809$	−16262.0	−0.706725	−0.353363	+	0.935486i	$0.614962\pi$
$810$	−31925.1	−1.38486
$811$	25310.1	1.09588	0.547939	−	0.836518i	$-0.315412\pi$
$811$	25310.1	1.09588	0.547939	+	0.836518i	$0.315412\pi$
$812$	79907.5	3.45345
$813$	−35.1390	−0.00151584
$814$	18037.6	0.776682
$815$	12799.5	0.550120
$816$	82.0497	0.00351999
$817$	0	0
$818$	−50721.6	−2.16802
$819$	14132.4	0.602963
$820$	−22571.7	−0.961265
$821$	11014.3	0.468211	0.234106	−	0.972211i	$-0.424784\pi$
$821$	11014.3	0.468211	0.234106	+	0.972211i	$0.424784\pi$
$822$	−366.035	−0.0155315
$823$	−12830.1	−0.543414	−0.271707	−	0.962380i	$-0.587588\pi$
$823$	−12830.1	−0.543414	−0.271707	+	0.962380i	$0.587588\pi$
$824$	−39338.9	−1.66315
$825$	−226.897	−0.00957518
$826$	−97675.2	−4.11447
$827$	−14250.4	−0.599194	−0.299597	−	0.954066i	$-0.596852\pi$
$827$	−14250.4	−0.599194	−0.299597	+	0.954066i	$0.596852\pi$
$828$	−47750.4	−2.00416
$829$	−29686.8	−1.24375	−0.621873	−	0.783118i	$-0.713628\pi$
$829$	−29686.8	−1.24375	−0.621873	+	0.783118i	$0.713628\pi$
$830$	−27724.4	−1.15943
$831$	−452.636	−0.0188950
$832$	−14545.4	−0.606097
$833$	−9629.02	−0.400511
$834$	−1217.18	−0.0505366
$835$	14487.5	0.600434
$836$	21430.5	0.886588
$837$	−1595.58	−0.0658919
$838$	−26924.6	−1.10990
$839$	−10879.0	−0.447657	−0.223828	−	0.974629i	$-0.571856\pi$
$839$	−10879.0	−0.447657	−0.223828	+	0.974629i	$0.571856\pi$
$840$	−839.668	−0.0344896
$841$	18822.4	0.771758
$842$	43011.7	1.76043
$843$	346.355	0.0141508
$844$	17275.3	0.704548
$845$	−16793.4	−0.683680
$846$	−63388.6	−2.57606
$847$	−35892.8	−1.45607
$848$	−15180.9	−0.614759
$849$	−141.866	−0.00573478
$850$	−4822.21	−0.194589
$851$	9055.36	0.364764
$852$	−1556.64	−0.0625936
$853$	22404.3	0.899307	0.449654	−	0.893203i	$-0.351547\pi$
$853$	22404.3	0.899307	0.449654	+	0.893203i	$0.351547\pi$
$854$	19894.1	0.797145
$855$	7196.39	0.287850
$856$	13371.8	0.533925
$857$	23406.0	0.932945	0.466472	−	0.884536i	$-0.345525\pi$
$857$	23406.0	0.932945	0.466472	+	0.884536i	$0.345525\pi$
$858$	−538.183	−0.0214141
$859$	−9181.11	−0.364674	−0.182337	−	0.983236i	$-0.558366\pi$
$859$	−9181.11	−0.364674	−0.182337	+	0.983236i	$0.558366\pi$
$860$	0	0
$861$	−506.321	−0.0200411
$862$	−32178.7	−1.27147
$863$	−19475.2	−0.768184	−0.384092	−	0.923295i	$-0.625485\pi$
$863$	−19475.2	−0.768184	−0.384092	+	0.923295i	$0.625485\pi$
$864$	660.672	0.0260145
$865$	21957.8	0.863108
$866$	77611.4	3.04543
$867$	474.721	0.0185956
$868$	101354.	3.96335
$869$	13448.6	0.524985
$870$	−1021.84	−0.0398204
$871$	3111.85	0.121057
$872$	−19385.1	−0.752823
$873$	26689.3	1.03470
$874$	16736.6	0.647739
$875$	40607.1	1.56888
$876$	1508.25	0.0581724
$877$	−27910.4	−1.07465	−0.537325	−	0.843375i	$-0.680565\pi$
$877$	−27910.4	−1.07465	−0.537325	+	0.843375i	$0.680565\pi$
$878$	−17263.7	−0.663579
$879$	202.540	0.00777192
$880$	−13474.8	−0.516175
$881$	−40334.8	−1.54247	−0.771234	−	0.636552i	$-0.780360\pi$
$881$	−40334.8	−1.54247	−0.771234	+	0.636552i	$0.780360\pi$
$882$	47235.1	1.80328
$883$	15179.9	0.578534	0.289267	−	0.957248i	$-0.406588\pi$
$883$	15179.9	0.578534	0.289267	+	0.957248i	$0.406588\pi$
$884$	−7352.57	−0.279744
$885$	802.919	0.0304970
$886$	−52535.7	−1.99207
$887$	−18153.2	−0.687175	−0.343588	−	0.939121i	$-0.611642\pi$
$887$	−18153.2	−0.687175	−0.343588	+	0.939121i	$0.611642\pi$
$888$	250.088	0.00945090
$889$	7227.03	0.272651
$890$	16465.4	0.620136
$891$	−37654.1	−1.41578
$892$	−12544.6	−0.470878
$893$	14282.1	0.535198
$894$	391.615	0.0146505
$895$	−22597.4	−0.843965
$896$	−70385.8	−2.62436
$897$	−270.182	−0.0100570
$898$	8333.01	0.309662
$899$	54809.1	2.03335
$900$	15206.2	0.563192
$901$	14059.1	0.519841
$902$	−41414.6	−1.52877
$903$	0	0
$904$	327.117	0.0120351
$905$	29061.6	1.06745
$906$	−1241.53	−0.0455265
$907$	−45094.3	−1.65086	−0.825430	−	0.564504i	$-0.809068\pi$
$907$	−45094.3	−1.65086	−0.825430	+	0.564504i	$0.809068\pi$
$908$	−7336.15	−0.268126
$909$	44854.6	1.63667
$910$	22965.0	0.836573
$911$	34601.4	1.25839	0.629197	−	0.777246i	$-0.283384\pi$
$911$	34601.4	1.25839	0.629197	+	0.777246i	$0.283384\pi$
$912$	90.6861	0.00329267
$913$	−32699.5	−1.18532
$914$	88902.0	3.21730
$915$	−163.535	−0.00590854
$916$	421.694	0.0152109
$917$	−37270.7	−1.34219
$918$	745.676	0.0268094
$919$	−23235.9	−0.834039	−0.417020	−	0.908898i	$-0.636925\pi$
$919$	−23235.9	−0.834039	−0.417020	+	0.908898i	$0.636925\pi$
$920$	−34479.5	−1.23560
$921$	207.606	0.00742762
$922$	4003.84	0.143014
$923$	18918.3	0.674651
$924$	−2228.71	−0.0793497
$925$	−2883.69	−0.102503
$926$	54500.4	1.93412
$927$	−35063.2	−1.24232
$928$	−22694.4	−0.802779
$929$	−5363.32	−0.189413	−0.0947067	−	0.995505i	$-0.530191\pi$
$929$	−5363.32	−0.189413	−0.0947067	+	0.995505i	$0.530191\pi$
$930$	−1296.10	−0.0456998
$931$	−10642.6	−0.374646
$932$	−5218.85	−0.183422
$933$	−714.451	−0.0250697
$934$	39353.6	1.37868
$935$	12479.0	0.436479
$936$	16027.1	0.559683
$937$	36653.5	1.27793	0.638964	−	0.769236i	$-0.279363\pi$
$937$	36653.5	1.27793	0.638964	+	0.769236i	$0.279363\pi$
$938$	20047.1	0.697825
$939$	−390.295	−0.0135642
$940$	−66214.2	−2.29752
$941$	32423.9	1.12326	0.561631	−	0.827388i	$-0.310174\pi$
$941$	32423.9	1.12326	0.561631	+	0.827388i	$0.310174\pi$
$942$	−1821.47	−0.0630008
$943$	−20791.2	−0.717979
$944$	−21732.6	−0.749295
$945$	−1497.16	−0.0515371
$946$	0	0
$947$	7763.95	0.266414	0.133207	−	0.991088i	$-0.457472\pi$
$947$	7763.95	0.266414	0.133207	+	0.991088i	$0.457472\pi$
$948$	419.619	0.0143762
$949$	−18330.1	−0.626998
$950$	−5329.79	−0.182022
$951$	−393.444	−0.0134157
$952$	−21047.8	−0.716557
$953$	25844.0	0.878455	0.439228	−	0.898376i	$-0.355252\pi$
$953$	25844.0	0.878455	0.439228	+	0.898376i	$0.355252\pi$
$954$	−68967.0	−2.34055
$955$	24722.3	0.837692
$956$	−64652.6	−2.18725
$957$	−1205.21	−0.0407096
$958$	−15192.9	−0.512381
$959$	18422.1	0.620312
$960$	770.278	0.0258965
$961$	39728.4	1.33357
$962$	−6839.92	−0.229239
$963$	11918.5	0.398823
$964$	36537.3	1.22073
$965$	15049.1	0.502017
$966$	−1740.56	−0.0579727
$967$	−51441.7	−1.71071	−0.855353	−	0.518045i	$-0.826660\pi$
$967$	−51441.7	−1.71071	−0.855353	+	0.518045i	$0.826660\pi$
$968$	−40704.9	−1.35156
$969$	−83.9846	−0.00278429
$970$	43369.7	1.43558
$971$	357.934	0.0118297	0.00591486	−	0.999983i	$-0.498117\pi$
$971$	357.934	0.0118297	0.00591486	+	0.999983i	$0.498117\pi$
$972$	−3527.35	−0.116399
$973$	61259.2	2.01838
$974$	35426.4	1.16544
$975$	86.0398	0.00282613
$976$	4426.40	0.145170
$977$	−27063.7	−0.886226	−0.443113	−	0.896466i	$-0.646126\pi$
$977$	−27063.7	−0.886226	−0.443113	+	0.896466i	$0.646126\pi$
$978$	732.760	0.0239582
$979$	19420.1	0.633983
$980$	49340.7	1.60830
$981$	−17278.1	−0.562333
$982$	−46573.1	−1.51345
$983$	−29234.2	−0.948550	−0.474275	−	0.880377i	$-0.657290\pi$
$983$	−29234.2	−0.948550	−0.474275	+	0.880377i	$0.657290\pi$
$984$	−574.203	−0.0186026
$985$	−39245.5	−1.26951
$986$	−25614.3	−0.827308
$987$	−1485.30	−0.0479003
$988$	−8126.48	−0.261678
$989$	0	0
$990$	−61215.8	−1.96522
$991$	−7027.91	−0.225277	−0.112638	−	0.993636i	$-0.535930\pi$
$991$	−7027.91	−0.225277	−0.112638	+	0.993636i	$0.535930\pi$
$992$	−28785.4	−0.921307
$993$	985.379	0.0314905
$994$	121875.	3.88897
$995$	−9421.93	−0.300196
$996$	−1020.28	−0.0324587
$997$	26506.6	0.841997	0.420999	−	0.907061i	$-0.361680\pi$
$997$	26506.6	0.841997	0.420999	+	0.907061i	$0.361680\pi$
$998$	−87694.7	−2.78149
$999$	445.916	0.0141223

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.l.1.4		60
43.15	even	21		43.4.g.a.10.10	✓	120
43.23	even	21		43.4.g.a.13.10	yes	120
43.42	odd	2		1849.4.a.k.1.57		60

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.10.10	✓	120	43.15	even	21
43.4.g.a.13.10	yes	120	43.23	even	21
1849.4.a.k.1.57		60	43.42	odd	2
1849.4.a.l.1.4		60	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q-4.73263 q^{2} -0.112092 q^{3} +14.3978 q^{4} +9.26635 q^{5} +0.530489 q^{6} -26.6989 q^{7} -30.2784 q^{8} -26.9874 q^{9} +O(q^{10})\)	\(q-4.73263 q^{2} -0.112092 q^{3} +14.3978 q^{4} +9.26635 q^{5} +0.530489 q^{6} -26.6989 q^{7} -30.2784 q^{8} -26.9874 q^{9} -43.8542 q^{10} -51.7239 q^{11} -1.61387 q^{12} +19.6138 q^{13} +126.356 q^{14} -1.03868 q^{15} +28.1139 q^{16} -26.0364 q^{17} +127.722 q^{18} -28.7769 q^{19} +133.415 q^{20} +2.99272 q^{21} +244.790 q^{22} +122.891 q^{23} +3.39396 q^{24} -39.1348 q^{25} -92.8250 q^{26} +6.05155 q^{27} -384.404 q^{28} -207.874 q^{29} +4.91570 q^{30} -263.665 q^{31} +109.174 q^{32} +5.79782 q^{33} +123.221 q^{34} -247.401 q^{35} -388.559 q^{36} +73.6862 q^{37} +136.191 q^{38} -2.19855 q^{39} -280.570 q^{40} -169.184 q^{41} -14.1635 q^{42} -744.709 q^{44} -250.075 q^{45} -581.597 q^{46} -496.303 q^{47} -3.15134 q^{48} +369.829 q^{49} +185.210 q^{50} +2.91847 q^{51} +282.396 q^{52} -539.979 q^{53} -28.6398 q^{54} -479.291 q^{55} +808.397 q^{56} +3.22566 q^{57} +983.789 q^{58} -773.017 q^{59} -14.9547 q^{60} +157.445 q^{61} +1247.83 q^{62} +720.534 q^{63} -741.591 q^{64} +181.749 q^{65} -27.4390 q^{66} +158.656 q^{67} -374.867 q^{68} -13.7751 q^{69} +1170.86 q^{70} +964.538 q^{71} +817.135 q^{72} -934.552 q^{73} -348.729 q^{74} +4.38669 q^{75} -414.324 q^{76} +1380.97 q^{77} +10.4049 q^{78} -260.007 q^{79} +260.514 q^{80} +727.982 q^{81} +800.686 q^{82} +632.194 q^{83} +43.0886 q^{84} -241.262 q^{85} +23.3009 q^{87} +1566.11 q^{88} -375.457 q^{89} +1183.51 q^{90} -523.667 q^{91} +1769.36 q^{92} +29.5547 q^{93} +2348.82 q^{94} -266.657 q^{95} -12.2375 q^{96} -988.952 q^{97} -1750.26 q^{98} +1395.89 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 15 q^{2} + 37 q^{3} + 213 q^{4} + 51 q^{5} + 22 q^{6} + 54 q^{7} + 204 q^{8} + 387 q^{9}+O(q^{10}) \) 60 * q + 15 * q^2 + 37 * q^3 + 213 * q^4 + 51 * q^5 + 22 * q^6 + 54 * q^7 + 204 * q^8 + 387 * q^9	\( 60 q + 15 q^{2} + 37 q^{3} + 213 q^{4} + 51 q^{5} + 22 q^{6} + 54 q^{7} + 204 q^{8} + 387 q^{9} + 27 q^{10} + 43 q^{11} + 432 q^{12} + 13 q^{13} + 96 q^{14} + 65 q^{15} + 717 q^{16} - 138 q^{17} + 625 q^{18} + 610 q^{19} + 345 q^{20} + 611 q^{21} + 118 q^{22} - 243 q^{23} + 258 q^{24} + 899 q^{25} + 1071 q^{26} + 1609 q^{27} + 46 q^{28} + 773 q^{29} + 375 q^{30} - 97 q^{31} + 1967 q^{32} + 500 q^{33} + 217 q^{34} + 247 q^{35} + 175 q^{36} + 228 q^{37} + 1253 q^{38} + 1493 q^{39} + 2220 q^{40} - 951 q^{41} + 2643 q^{42} - 1378 q^{44} + 1086 q^{45} - 565 q^{46} - 2 q^{47} + 2303 q^{48} + 1264 q^{49} + 3273 q^{50} + 3076 q^{51} - 2825 q^{52} - 39 q^{53} + 5201 q^{54} + 1306 q^{55} + 3683 q^{56} + 1342 q^{57} + 2588 q^{58} - 1065 q^{59} + 2803 q^{60} + 2999 q^{61} + 5569 q^{62} + 2377 q^{63} + 2082 q^{64} + 5578 q^{65} + 3338 q^{66} + 961 q^{67} - 3754 q^{68} + 1817 q^{69} + 2738 q^{70} + 8003 q^{71} + 1412 q^{72} - 1011 q^{73} - 1413 q^{74} + 7457 q^{75} + 5516 q^{76} + 4052 q^{77} + 1091 q^{78} - 4422 q^{79} + 1610 q^{80} + 2108 q^{81} + 4676 q^{82} - 297 q^{83} - 54 q^{84} + 4333 q^{85} + 1377 q^{87} + 3652 q^{88} + 2480 q^{89} - 1414 q^{90} + 4551 q^{91} - 3286 q^{92} + 4 q^{93} + 4609 q^{94} - 835 q^{95} + 5864 q^{96} + 3785 q^{97} + 753 q^{98} + 5072 q^{99}+O(q^{100}) \) 60 * q + 15 * q^2 + 37 * q^3 + 213 * q^4 + 51 * q^5 + 22 * q^6 + 54 * q^7 + 204 * q^8 + 387 * q^9 + 27 * q^10 + 43 * q^11 + 432 * q^12 + 13 * q^13 + 96 * q^14 + 65 * q^15 + 717 * q^16 - 138 * q^17 + 625 * q^18 + 610 * q^19 + 345 * q^20 + 611 * q^21 + 118 * q^22 - 243 * q^23 + 258 * q^24 + 899 * q^25 + 1071 * q^26 + 1609 * q^27 + 46 * q^28 + 773 * q^29 + 375 * q^30 - 97 * q^31 + 1967 * q^32 + 500 * q^33 + 217 * q^34 + 247 * q^35 + 175 * q^36 + 228 * q^37 + 1253 * q^38 + 1493 * q^39 + 2220 * q^40 - 951 * q^41 + 2643 * q^42 - 1378 * q^44 + 1086 * q^45 - 565 * q^46 - 2 * q^47 + 2303 * q^48 + 1264 * q^49 + 3273 * q^50 + 3076 * q^51 - 2825 * q^52 - 39 * q^53 + 5201 * q^54 + 1306 * q^55 + 3683 * q^56 + 1342 * q^57 + 2588 * q^58 - 1065 * q^59 + 2803 * q^60 + 2999 * q^61 + 5569 * q^62 + 2377 * q^63 + 2082 * q^64 + 5578 * q^65 + 3338 * q^66 + 961 * q^67 - 3754 * q^68 + 1817 * q^69 + 2738 * q^70 + 8003 * q^71 + 1412 * q^72 - 1011 * q^73 - 1413 * q^74 + 7457 * q^75 + 5516 * q^76 + 4052 * q^77 + 1091 * q^78 - 4422 * q^79 + 1610 * q^80 + 2108 * q^81 + 4676 * q^82 - 297 * q^83 - 54 * q^84 + 4333 * q^85 + 1377 * q^87 + 3652 * q^88 + 2480 * q^89 - 1414 * q^90 + 4551 * q^91 - 3286 * q^92 + 4 * q^93 + 4609 * q^94 - 835 * q^95 + 5864 * q^96 + 3785 * q^97 + 753 * q^98 + 5072 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more