LMFDB - Embedded newform 1849.4.a.i.1.29

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:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.29
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)

$f(q)$	$=$	$q+1.05440 q^{2} -4.53621 q^{3} -6.88824 q^{4} -14.4725 q^{5} -4.78298 q^{6} -15.4664 q^{7} -15.6982 q^{8} -6.42280 q^{9} +O(q^{10})$	\(q+1.05440 q^{2} -4.53621 q^{3} -6.88824 q^{4} -14.4725 q^{5} -4.78298 q^{6} -15.4664 q^{7} -15.6982 q^{8} -6.42280 q^{9} -15.2598 q^{10} -62.4686 q^{11} +31.2465 q^{12} -21.6145 q^{13} -16.3078 q^{14} +65.6505 q^{15} +38.5538 q^{16} +25.1556 q^{17} -6.77220 q^{18} +45.2306 q^{19} +99.6903 q^{20} +70.1588 q^{21} -65.8669 q^{22} +195.169 q^{23} +71.2102 q^{24} +84.4542 q^{25} -22.7903 q^{26} +151.613 q^{27} +106.536 q^{28} -39.6769 q^{29} +69.2218 q^{30} -254.845 q^{31} +166.236 q^{32} +283.371 q^{33} +26.5241 q^{34} +223.838 q^{35} +44.2418 q^{36} -110.141 q^{37} +47.6911 q^{38} +98.0478 q^{39} +227.192 q^{40} -62.1285 q^{41} +73.9754 q^{42} +430.299 q^{44} +92.9542 q^{45} +205.786 q^{46} +586.112 q^{47} -174.888 q^{48} -103.791 q^{49} +89.0485 q^{50} -114.111 q^{51} +148.886 q^{52} -515.058 q^{53} +159.861 q^{54} +904.079 q^{55} +242.794 q^{56} -205.175 q^{57} -41.8353 q^{58} +250.178 q^{59} -452.216 q^{60} +812.162 q^{61} -268.709 q^{62} +99.3374 q^{63} -133.151 q^{64} +312.816 q^{65} +298.786 q^{66} -558.014 q^{67} -173.278 q^{68} -885.326 q^{69} +236.014 q^{70} +227.436 q^{71} +100.826 q^{72} -568.115 q^{73} -116.133 q^{74} -383.102 q^{75} -311.559 q^{76} +966.164 q^{77} +103.382 q^{78} -1336.40 q^{79} -557.971 q^{80} -514.332 q^{81} -65.5083 q^{82} -16.2755 q^{83} -483.270 q^{84} -364.066 q^{85} +179.983 q^{87} +980.643 q^{88} +858.571 q^{89} +98.0109 q^{90} +334.298 q^{91} -1344.37 q^{92} +1156.03 q^{93} +617.997 q^{94} -654.601 q^{95} -754.083 q^{96} -312.788 q^{97} -109.437 q^{98} +401.223 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) $ 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} - 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} - 386 q^{18} + 12 q^{19} + 108 q^{20} - 408 q^{21} + 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} + 1493 q^{26} + 10 q^{27} - 242 q^{28} + 208 q^{29} - 48 q^{30} - 932 q^{31} - 1124 q^{32} + 254 q^{33} + 765 q^{34} - 1452 q^{35} + 747 q^{36} - 90 q^{37} - 1213 q^{38} - 1610 q^{39} - 1693 q^{40} - 1354 q^{41} - 16 q^{42} - 2704 q^{44} + 4508 q^{45} + 233 q^{46} - 3484 q^{47} - 376 q^{48} + 1324 q^{49} - 408 q^{50} + 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} - 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} + 1172 q^{61} - 1546 q^{62} - 3686 q^{63} + 606 q^{64} + 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} + 136 q^{69} - 1310 q^{70} + 162 q^{71} - 5814 q^{72} + 746 q^{73} - 4332 q^{74} + 236 q^{75} + 1338 q^{76} + 2024 q^{77} - 2782 q^{78} - 2656 q^{79} - 5713 q^{80} - 86 q^{81} - 4168 q^{82} - 3514 q^{83} - 4269 q^{84} - 7558 q^{85} - 10278 q^{87} + 11692 q^{88} + 2640 q^{89} - 8286 q^{90} - 5946 q^{91} - 4271 q^{92} - 2 q^{93} + 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} - 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 - 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 - 386 * q^18 + 12 * q^19 + 108 * q^20 - 408 * q^21 + 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 + 1493 * q^26 + 10 * q^27 - 242 * q^28 + 208 * q^29 - 48 * q^30 - 932 * q^31 - 1124 * q^32 + 254 * q^33 + 765 * q^34 - 1452 * q^35 + 747 * q^36 - 90 * q^37 - 1213 * q^38 - 1610 * q^39 - 1693 * q^40 - 1354 * q^41 - 16 * q^42 - 2704 * q^44 + 4508 * q^45 + 233 * q^46 - 3484 * q^47 - 376 * q^48 + 1324 * q^49 - 408 * q^50 + 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 - 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 + 1172 * q^61 - 1546 * q^62 - 3686 * q^63 + 606 * q^64 + 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 + 136 * q^69 - 1310 * q^70 + 162 * q^71 - 5814 * q^72 + 746 * q^73 - 4332 * q^74 + 236 * q^75 + 1338 * q^76 + 2024 * q^77 - 2782 * q^78 - 2656 * q^79 - 5713 * q^80 - 86 * q^81 - 4168 * q^82 - 3514 * q^83 - 4269 * q^84 - 7558 * q^85 - 10278 * q^87 + 11692 * q^88 + 2640 * q^89 - 8286 * q^90 - 5946 * q^91 - 4271 * q^92 - 2 * q^93 + 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 - 2826 * q^98 - 8174 * 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 $

$2$	1.05440	0.372787	0.186393	−	0.982475i	$-0.440320\pi$
$2$	1.05440	0.372787	0.186393	+	0.982475i	$0.440320\pi$
$3$	−4.53621	−0.872994	−0.436497	−	0.899706i	$-0.643781\pi$
$3$	−4.53621	−0.872994	−0.436497	+	0.899706i	$0.643781\pi$
$4$	−6.88824	−0.861030
$5$	−14.4725	−1.29446	−0.647231	−	0.762294i	$-0.724073\pi$
$5$	−14.4725	−1.29446	−0.647231	+	0.762294i	$0.724073\pi$
$6$	−4.78298	−0.325441
$7$	−15.4664	−0.835106	−0.417553	−	0.908653i	$-0.637112\pi$
$7$	−15.4664	−0.835106	−0.417553	+	0.908653i	$0.637112\pi$
$8$	−15.6982	−0.693767
$9$	−6.42280	−0.237881
$10$	−15.2598	−0.482559
$11$	−62.4686	−1.71227	−0.856136	−	0.516750i	$-0.827142\pi$
$11$	−62.4686	−1.71227	−0.856136	+	0.516750i	$0.827142\pi$
$12$	31.2465	0.751674
$13$	−21.6145	−0.461137	−0.230568	−	0.973056i	$-0.574059\pi$
$13$	−21.6145	−0.461137	−0.230568	+	0.973056i	$0.574059\pi$
$14$	−16.3078	−0.311316
$15$	65.6505	1.13006
$16$	38.5538	0.602403
$17$	25.1556	0.358891	0.179445	−	0.983768i	$-0.442570\pi$
$17$	25.1556	0.358891	0.179445	+	0.983768i	$0.442570\pi$
$18$	−6.77220	−0.0886790
$19$	45.2306	0.546137	0.273069	−	0.961995i	$-0.411961\pi$
$19$	45.2306	0.546137	0.273069	+	0.961995i	$0.411961\pi$
$20$	99.6903	1.11457
$21$	70.1588	0.729043
$22$	−65.8669	−0.638313
$23$	195.169	1.76937	0.884684	−	0.466191i	$-0.154374\pi$
$23$	195.169	1.76937	0.884684	+	0.466191i	$0.154374\pi$
$24$	71.2102	0.605655
$25$	84.4542	0.675634
$26$	−22.7903	−0.171906
$27$	151.613	1.08066
$28$	106.536	0.719051
$29$	−39.6769	−0.254063	−0.127031	−	0.991899i	$-0.540545\pi$
$29$	−39.6769	−0.254063	−0.127031	+	0.991899i	$0.540545\pi$
$30$	69.2218	0.421271
$31$	−254.845	−1.47650	−0.738250	−	0.674527i	$-0.764348\pi$
$31$	−254.845	−1.47650	−0.738250	+	0.674527i	$0.764348\pi$
$32$	166.236	0.918335
$33$	283.371	1.49480
$34$	26.5241	0.133790
$35$	223.838	1.08101
$36$	44.2418	0.204823
$37$	−110.141	−0.489380	−0.244690	−	0.969601i	$-0.578686\pi$
$37$	−110.141	−0.489380	−0.244690	+	0.969601i	$0.578686\pi$
$38$	47.6911	0.203593
$39$	98.0478	0.402570
$40$	227.192	0.898056
$41$	−62.1285	−0.236655	−0.118327	−	0.992975i	$-0.537753\pi$
$41$	−62.1285	−0.236655	−0.118327	+	0.992975i	$0.537753\pi$
$42$	73.9754	0.271777
$43$	0	0
$43$	0	0
$44$	430.299	1.47432
$45$	92.9542	0.307929
$46$	205.786	0.659597
$47$	586.112	1.81900	0.909502	−	0.415698i	$-0.136463\pi$
$47$	586.112	1.81900	0.909502	+	0.415698i	$0.136463\pi$
$48$	−174.888	−0.525894
$49$	−103.791	−0.302598
$50$	89.0485	0.251867
$51$	−114.111	−0.313309
$52$	148.886	0.397053
$53$	−515.058	−1.33488	−0.667441	−	0.744663i	$-0.732610\pi$
$53$	−515.058	−1.33488	−0.667441	+	0.744663i	$0.732610\pi$
$54$	159.861	0.402857
$55$	904.079	2.21647
$56$	242.794	0.579369
$57$	−205.175	−0.476775
$58$	−41.8353	−0.0947112
$59$	250.178	0.552041	0.276020	−	0.961152i	$-0.410984\pi$
$59$	250.178	0.552041	0.276020	+	0.961152i	$0.410984\pi$
$60$	−452.216	−0.973014
$61$	812.162	1.70470	0.852350	−	0.522971i	$-0.175177\pi$
$61$	812.162	1.70470	0.852350	+	0.522971i	$0.175177\pi$
$62$	−268.709	−0.550420
$63$	99.3374	0.198656
$64$	−133.151	−0.260060
$65$	312.816	0.596924
$66$	298.786	0.557243
$67$	−558.014	−1.01750	−0.508748	−	0.860915i	$-0.669892\pi$
$67$	−558.014	−1.01750	−0.508748	+	0.860915i	$0.669892\pi$
$68$	−173.278	−0.309016
$69$	−885.326	−1.54465
$70$	236.014	0.402988
$71$	227.436	0.380165	0.190083	−	0.981768i	$-0.439124\pi$
$71$	227.436	0.380165	0.190083	+	0.981768i	$0.439124\pi$
$72$	100.826	0.165034
$73$	−568.115	−0.910860	−0.455430	−	0.890272i	$-0.650515\pi$
$73$	−568.115	−0.910860	−0.455430	+	0.890272i	$0.650515\pi$
$74$	−116.133	−0.182434
$75$	−383.102	−0.589824
$76$	−311.559	−0.470241
$77$	966.164	1.42993
$78$	103.382	0.150073
$79$	−1336.40	−1.90325	−0.951624	−	0.307264i	$-0.900586\pi$
$79$	−1336.40	−1.90325	−0.951624	+	0.307264i	$0.900586\pi$
$80$	−557.971	−0.779788
$81$	−514.332	−0.705531
$82$	−65.5083	−0.0882217
$83$	−16.2755	−0.0215237	−0.0107619	−	0.999942i	$-0.503426\pi$
$83$	−16.2755	−0.0215237	−0.0107619	+	0.999942i	$0.503426\pi$
$84$	−483.270	−0.627728
$85$	−364.066	−0.464570
$86$	0	0
$87$	179.983	0.221795
$88$	980.643	1.18792
$89$	858.571	1.02257	0.511283	−	0.859412i	$-0.329170\pi$
$89$	858.571	1.02257	0.511283	+	0.859412i	$0.329170\pi$
$90$	98.0109	0.114792
$91$	334.298	0.385098
$92$	−1344.37	−1.52348
$93$	1156.03	1.28898
$94$	617.997	0.678101
$95$	−654.601	−0.706954
$96$	−754.083	−0.801701
$97$	−312.788	−0.327410	−0.163705	−	0.986509i	$-0.552345\pi$
$97$	−312.788	−0.327410	−0.163705	+	0.986509i	$0.552345\pi$
$98$	−109.437	−0.112804
$99$	401.223	0.407318
$100$	−581.741	−0.581741
$101$	−524.823	−0.517048	−0.258524	−	0.966005i	$-0.583236\pi$
$101$	−524.823	−0.517048	−0.258524	+	0.966005i	$0.583236\pi$
$102$	−120.319	−0.116798
$103$	517.358	0.494920	0.247460	−	0.968898i	$-0.420404\pi$
$103$	517.358	0.494920	0.247460	+	0.968898i	$0.420404\pi$
$104$	339.308	0.319922
$105$	−1015.37	−0.943718
$106$	−543.078	−0.497626
$107$	1251.80	1.13099	0.565494	−	0.824752i	$-0.308685\pi$
$107$	1251.80	1.13099	0.565494	+	0.824752i	$0.308685\pi$
$108$	−1044.35	−0.930483
$109$	−775.264	−0.681256	−0.340628	−	0.940198i	$-0.610640\pi$
$109$	−775.264	−0.681256	−0.340628	+	0.940198i	$0.610640\pi$
$110$	953.261	0.826272
$111$	499.622	0.427226
$112$	−596.287	−0.503070
$113$	−64.5912	−0.0537719	−0.0268859	−	0.999639i	$-0.508559\pi$
$113$	−64.5912	−0.0537719	−0.0268859	+	0.999639i	$0.508559\pi$
$114$	−216.337	−0.177735
$115$	−2824.58	−2.29038
$116$	273.304	0.218756
$117$	138.825	0.109696
$118$	263.788	0.205793
$119$	−389.067	−0.299712
$120$	−1030.59	−0.783997
$121$	2571.33	1.93188
$122$	856.344	0.635490
$123$	281.828	0.206598
$124$	1755.43	1.27131
$125$	586.800	0.419880
$126$	104.741	0.0740564
$127$	−231.994	−0.162095	−0.0810477	−	0.996710i	$-0.525827\pi$
$127$	−231.994	−0.162095	−0.0810477	+	0.996710i	$0.525827\pi$
$128$	−1470.29	−1.01528
$129$	0	0
$130$	329.833	0.222525
$131$	378.667	0.252552	0.126276	−	0.991995i	$-0.459698\pi$
$131$	378.667	0.252552	0.126276	+	0.991995i	$0.459698\pi$
$132$	−1951.93	−1.28707
$133$	−699.553	−0.456082
$134$	−588.370	−0.379309
$135$	−2194.22	−1.39888
$136$	−394.897	−0.248987
$137$	−189.827	−0.118380	−0.0591899	−	0.998247i	$-0.518852\pi$
$137$	−189.827	−0.118380	−0.0591899	+	0.998247i	$0.518852\pi$
$138$	−933.488	−0.575824
$139$	1933.32	1.17973	0.589865	−	0.807502i	$-0.299181\pi$
$139$	1933.32	1.17973	0.589865	+	0.807502i	$0.299181\pi$
$140$	−1541.85	−0.930785
$141$	−2658.73	−1.58798
$142$	239.809	0.141721
$143$	1350.23	0.789592
$144$	−247.623	−0.143300
$145$	574.225	0.328875
$146$	−599.020	−0.339557
$147$	470.818	0.264166
$148$	758.677	0.421371
$149$	3149.21	1.73150	0.865749	−	0.500479i	$-0.166843\pi$
$149$	3149.21	1.73150	0.865749	+	0.500479i	$0.166843\pi$
$150$	−403.943	−0.219879
$151$	2085.86	1.12414	0.562068	−	0.827091i	$-0.310006\pi$
$151$	2085.86	1.12414	0.562068	+	0.827091i	$0.310006\pi$
$152$	−710.037	−0.378892
$153$	−161.570	−0.0853734
$154$	1018.72	0.533059
$155$	3688.25	1.91128
$156$	−675.377	−0.346624
$157$	−3511.99	−1.78527	−0.892634	−	0.450782i	$-0.851145\pi$
$157$	−3511.99	−1.78527	−0.892634	+	0.450782i	$0.851145\pi$
$158$	−1409.10	−0.709506
$159$	2336.41	1.16534
$160$	−2405.86	−1.18875
$161$	−3018.55	−1.47761
$162$	−542.312	−0.263013
$163$	2706.27	1.30044	0.650219	−	0.759747i	$-0.274677\pi$
$163$	2706.27	1.30044	0.650219	+	0.759747i	$0.274677\pi$
$164$	427.956	0.203767
$165$	−4101.09	−1.93497
$166$	−17.1609	−0.00802376
$167$	1012.98	0.469380	0.234690	−	0.972070i	$-0.424593\pi$
$167$	1012.98	0.469380	0.234690	+	0.972070i	$0.424593\pi$
$168$	−1101.36	−0.505786
$169$	−1729.81	−0.787353
$170$	−383.871	−0.173186
$171$	−290.507	−0.129916
$172$	0	0
$173$	456.153	0.200466	0.100233	−	0.994964i	$-0.468041\pi$
$173$	456.153	0.200466	0.100233	+	0.994964i	$0.468041\pi$
$174$	189.774	0.0826823
$175$	−1306.20	−0.564226
$176$	−2408.40	−1.03148
$177$	−1134.86	−0.481928
$178$	905.278	0.381199
$179$	2095.61	0.875045	0.437523	−	0.899207i	$-0.355856\pi$
$179$	2095.61	0.875045	0.437523	+	0.899207i	$0.355856\pi$
$180$	−640.291	−0.265136
$181$	3117.69	1.28031	0.640155	−	0.768246i	$-0.278870\pi$
$181$	3117.69	1.28031	0.640155	+	0.768246i	$0.278870\pi$
$182$	352.484	0.143559
$183$	−3684.14	−1.48819
$184$	−3063.79	−1.22753
$185$	1594.02	0.633484
$186$	1218.92	0.480513
$187$	−1571.44	−0.614519
$188$	−4037.28	−1.56622
$189$	−2344.90	−0.902468
$190$	−690.211	−0.263543
$191$	−3068.03	−1.16228	−0.581138	−	0.813805i	$-0.697392\pi$
$191$	−3068.03	−1.16228	−0.581138	+	0.813805i	$0.697392\pi$
$192$	603.999	0.227031
$193$	−2193.07	−0.817933	−0.408966	−	0.912549i	$-0.634111\pi$
$193$	−2193.07	−0.817933	−0.408966	+	0.912549i	$0.634111\pi$
$194$	−329.804	−0.122054
$195$	−1419.00	−0.521111
$196$	714.938	0.260546
$197$	302.542	0.109417	0.0547087	−	0.998502i	$-0.482577\pi$
$197$	302.542	0.109417	0.0547087	+	0.998502i	$0.482577\pi$
$198$	423.050	0.151843
$199$	1003.68	0.357533	0.178766	−	0.983892i	$-0.442789\pi$
$199$	1003.68	0.357533	0.178766	+	0.983892i	$0.442789\pi$
$200$	−1325.78	−0.468733
$201$	2531.27	0.888268
$202$	−553.373	−0.192749
$203$	613.658	0.212169
$204$	786.026	0.269769
$205$	899.157	0.306341
$206$	545.502	0.184500
$207$	−1253.53	−0.420900
$208$	−833.320	−0.277790
$209$	−2825.49	−0.935136
$210$	−1070.61	−0.351806
$211$	4888.64	1.59501	0.797506	−	0.603311i	$-0.206152\pi$
$211$	4888.64	1.59501	0.797506	+	0.603311i	$0.206152\pi$
$212$	3547.85	1.14937
$213$	−1031.70	−0.331882
$214$	1319.90	0.421618
$215$	0	0
$216$	−2380.04	−0.749729
$217$	3941.53	1.23303
$218$	−817.439	−0.253963
$219$	2577.09	0.795176
$220$	−6227.52	−1.90845
$221$	−543.726	−0.165498
$222$	526.802	0.159264
$223$	3321.24	0.997340	0.498670	−	0.866792i	$-0.333822\pi$
$223$	3321.24	0.997340	0.498670	+	0.866792i	$0.333822\pi$
$224$	−2571.08	−0.766907
$225$	−542.432	−0.160721
$226$	−68.1049	−0.0200455
$227$	−3632.36	−1.06206	−0.531032	−	0.847352i	$-0.678195\pi$
$227$	−3632.36	−1.06206	−0.531032	+	0.847352i	$0.678195\pi$
$228$	1413.30	0.410517
$229$	3958.77	1.14237	0.571185	−	0.820821i	$-0.306484\pi$
$229$	3958.77	1.14237	0.571185	+	0.820821i	$0.306484\pi$
$230$	−2978.24	−0.853824
$231$	−4382.72	−1.24832
$232$	622.855	0.176260
$233$	−4538.90	−1.27620	−0.638098	−	0.769955i	$-0.720278\pi$
$233$	−4538.90	−1.27620	−0.638098	+	0.769955i	$0.720278\pi$
$234$	146.378	0.0408932
$235$	−8482.53	−2.35463
$236$	−1723.29	−0.475324
$237$	6062.19	1.66152
$238$	−410.232	−0.111729
$239$	−6078.04	−1.64500	−0.822502	−	0.568762i	$-0.807422\pi$
$239$	−6078.04	−1.64500	−0.822502	+	0.568762i	$0.807422\pi$
$240$	2531.07	0.680750
$241$	4552.97	1.21694	0.608471	−	0.793576i	$-0.291783\pi$
$241$	4552.97	1.21694	0.608471	+	0.793576i	$0.291783\pi$
$242$	2711.21	0.720179
$243$	−1760.43	−0.464739
$244$	−5594.37	−1.46780
$245$	1502.12	0.391702
$246$	297.159	0.0770171
$247$	−977.635	−0.251844
$248$	4000.60	1.02435
$249$	73.8292	0.0187901
$250$	618.722	0.156526
$251$	−5848.23	−1.47067	−0.735333	−	0.677706i	$-0.762974\pi$
$251$	−5848.23	−1.47067	−0.735333	+	0.677706i	$0.762974\pi$
$252$	−684.260	−0.171049
$253$	−12191.9	−3.02964
$254$	−244.614	−0.0604270
$255$	1651.48	0.405567
$256$	−485.064	−0.118424
$257$	1935.75	0.469840	0.234920	−	0.972015i	$-0.424517\pi$
$257$	1935.75	0.469840	0.234920	+	0.972015i	$0.424517\pi$
$258$	0	0
$259$	1703.48	0.408684
$260$	−2154.75	−0.513970
$261$	254.837	0.0604368
$262$	399.267	0.0941480
$263$	−6747.04	−1.58190	−0.790951	−	0.611879i	$-0.790414\pi$
$263$	−6747.04	−1.58190	−0.790951	+	0.611879i	$0.790414\pi$
$264$	−4448.40	−1.03705
$265$	7454.20	1.72795
$266$	−737.609	−0.170021
$267$	−3894.66	−0.892694
$268$	3843.74	0.876095
$269$	−198.559	−0.0450051	−0.0225025	−	0.999747i	$-0.507163\pi$
$269$	−198.559	−0.0450051	−0.0225025	+	0.999747i	$0.507163\pi$
$270$	−2313.59	−0.521483
$271$	2023.47	0.453570	0.226785	−	0.973945i	$-0.427179\pi$
$271$	2023.47	0.453570	0.226785	+	0.973945i	$0.427179\pi$
$272$	969.845	0.216197
$273$	−1516.44	−0.336188
$274$	−200.154	−0.0441304
$275$	−5275.74	−1.15687
$276$	6098.34	1.32999
$277$	8288.84	1.79794	0.898968	−	0.438014i	$-0.144318\pi$
$277$	8288.84	1.79794	0.898968	+	0.438014i	$0.144318\pi$
$278$	2038.50	0.439787
$279$	1636.82	0.351232
$280$	−3513.84	−0.749972
$281$	5413.58	1.14928	0.574639	−	0.818407i	$-0.305142\pi$
$281$	5413.58	1.14928	0.574639	+	0.818407i	$0.305142\pi$
$282$	−2803.36	−0.591978
$283$	−7149.31	−1.50170	−0.750852	−	0.660471i	$-0.770357\pi$
$283$	−7149.31	−1.50170	−0.750852	+	0.660471i	$0.770357\pi$
$284$	−1566.64	−0.327334
$285$	2969.41	0.617167
$286$	1423.68	0.294349
$287$	960.903	0.197632
$288$	−1067.70	−0.218455
$289$	−4280.19	−0.871198
$290$	605.463	0.122600
$291$	1418.87	0.285827
$292$	3913.31	0.784278
$293$	4475.99	0.892458	0.446229	−	0.894919i	$-0.352767\pi$
$293$	4475.99	0.892458	0.446229	+	0.894919i	$0.352767\pi$
$294$	496.431	0.0984776
$295$	−3620.71	−0.714596
$296$	1729.01	0.339516
$297$	−9471.05	−1.85039
$298$	3320.52	0.645479
$299$	−4218.47	−0.815920
$300$	2638.90	0.507856
$301$	0	0
$302$	2199.33	0.419063
$303$	2380.71	0.451380
$304$	1743.81	0.328995
$305$	−11754.0	−2.20667
$306$	−170.359	−0.0318261
$307$	−3974.32	−0.738848	−0.369424	−	0.929261i	$-0.620445\pi$
$307$	−3974.32	−0.738848	−0.369424	+	0.929261i	$0.620445\pi$
$308$	−6655.17	−1.23121
$309$	−2346.84	−0.432062
$310$	3888.90	0.712498
$311$	−5015.11	−0.914408	−0.457204	−	0.889362i	$-0.651149\pi$
$311$	−5015.11	−0.914408	−0.457204	+	0.889362i	$0.651149\pi$
$312$	−1539.17	−0.279290
$313$	466.135	0.0841774	0.0420887	−	0.999114i	$-0.486599\pi$
$313$	466.135	0.0841774	0.0420887	+	0.999114i	$0.486599\pi$
$314$	−3703.04	−0.665524
$315$	−1437.66	−0.257153
$316$	9205.44	1.63875
$317$	−1289.75	−0.228515	−0.114258	−	0.993451i	$-0.536449\pi$
$317$	−1289.75	−0.228515	−0.114258	+	0.993451i	$0.536449\pi$
$318$	2463.51	0.434425
$319$	2478.56	0.435025
$320$	1927.03	0.336638
$321$	−5678.42	−0.987347
$322$	−3182.76	−0.550833
$323$	1137.80	0.196003
$324$	3542.84	0.607483
$325$	−1825.43	−0.311559
$326$	2853.49	0.484786
$327$	3516.76	0.594732
$328$	975.303	0.164183
$329$	−9065.03	−1.51906
$330$	−4324.19	−0.721330
$331$	6084.71	1.01041	0.505205	−	0.862999i	$-0.331417\pi$
$331$	6084.71	1.01041	0.505205	+	0.862999i	$0.331417\pi$
$332$	112.110	0.0185326
$333$	707.413	0.116414
$334$	1068.08	0.174978
$335$	8075.88	1.31711
$336$	2704.89	0.439177
$337$	−2709.51	−0.437972	−0.218986	−	0.975728i	$-0.570275\pi$
$337$	−2709.51	−0.437972	−0.218986	+	0.975728i	$0.570275\pi$
$338$	−1823.92	−0.293515
$339$	292.999	0.0469425
$340$	2507.77	0.400009
$341$	15919.8	2.52817
$342$	−306.310	−0.0484309
$343$	6910.24	1.08781
$344$	0	0
$345$	12812.9	1.99949
$346$	480.967	0.0747311
$347$	817.917	0.126536	0.0632681	−	0.997997i	$-0.479848\pi$
$347$	817.917	0.126536	0.0632681	+	0.997997i	$0.479848\pi$
$348$	−1239.76	−0.190972
$349$	2334.83	0.358111	0.179055	−	0.983839i	$-0.442696\pi$
$349$	2334.83	0.358111	0.179055	+	0.983839i	$0.442696\pi$
$350$	−1377.26	−0.210336
$351$	−3277.03	−0.498333
$352$	−10384.6	−1.57244
$353$	−6169.22	−0.930182	−0.465091	−	0.885263i	$-0.653978\pi$
$353$	−6169.22	−0.930182	−0.465091	+	0.885263i	$0.653978\pi$
$354$	−1196.60	−0.179656
$355$	−3291.58	−0.492110
$356$	−5914.05	−0.880460
$357$	1764.89	0.261647
$358$	2209.61	0.326205
$359$	183.060	0.0269123	0.0134562	−	0.999909i	$-0.495717\pi$
$359$	183.060	0.0269123	0.0134562	+	0.999909i	$0.495717\pi$
$360$	−1459.21	−0.213631
$361$	−4813.19	−0.701734
$362$	3287.29	0.477283
$363$	−11664.1	−1.68652
$364$	−2302.72	−0.331581
$365$	8222.06	1.17907
$366$	−3884.56	−0.554779
$367$	4878.74	0.693919	0.346960	−	0.937880i	$-0.387214\pi$
$367$	4878.74	0.693919	0.346960	+	0.937880i	$0.387214\pi$
$368$	7524.49	1.06587
$369$	399.039	0.0562958
$370$	1680.73	0.236154
$371$	7966.09	1.11477
$372$	−7963.02	−1.10985
$373$	−2998.62	−0.416254	−0.208127	−	0.978102i	$-0.566737\pi$
$373$	−2998.62	−0.416254	−0.208127	+	0.978102i	$0.566737\pi$
$374$	−1656.93	−0.229084
$375$	−2661.85	−0.366553
$376$	−9200.88	−1.26197
$377$	857.596	0.117158
$378$	−2472.46	−0.336428
$379$	10795.8	1.46317	0.731585	−	0.681750i	$-0.238781\pi$
$379$	10795.8	1.46317	0.731585	+	0.681750i	$0.238781\pi$
$380$	4509.05	0.608709
$381$	1052.37	0.141508
$382$	−3234.93	−0.433281
$383$	−9850.38	−1.31418	−0.657090	−	0.753812i	$-0.728213\pi$
$383$	−9850.38	−1.31418	−0.657090	+	0.753812i	$0.728213\pi$
$384$	6669.52	0.886335
$385$	−13982.8	−1.85099
$386$	−2312.38	−0.304914
$387$	0	0
$388$	2154.56	0.281910
$389$	547.776	0.0713967	0.0356984	−	0.999363i	$-0.488634\pi$
$389$	547.776	0.0713967	0.0356984	+	0.999363i	$0.488634\pi$
$390$	−1496.19	−0.194263
$391$	4909.59	0.635010
$392$	1629.33	0.209933
$393$	−1717.71	−0.220476
$394$	319.000	0.0407894
$395$	19341.1	2.46368
$396$	−2763.72	−0.350713
$397$	3433.56	0.434069	0.217035	−	0.976164i	$-0.430362\pi$
$397$	3433.56	0.434069	0.217035	+	0.976164i	$0.430362\pi$
$398$	1058.28	0.133283
$399$	3173.32	0.398157
$400$	3256.03	0.407004
$401$	2541.81	0.316539	0.158269	−	0.987396i	$-0.449409\pi$
$401$	2541.81	0.316539	0.158269	+	0.987396i	$0.449409\pi$
$402$	2668.97	0.331135
$403$	5508.34	0.680869
$404$	3615.11	0.445194
$405$	7443.69	0.913284
$406$	647.041	0.0790939
$407$	6880.35	0.837952
$408$	1791.34	0.217364
$409$	2769.69	0.334847	0.167423	−	0.985885i	$-0.446455\pi$
$409$	2769.69	0.334847	0.167423	+	0.985885i	$0.446455\pi$
$410$	948.071	0.114200
$411$	861.096	0.103345
$412$	−3563.68	−0.426141
$413$	−3869.35	−0.461013
$414$	−1321.72	−0.156906
$415$	235.548	0.0278617
$416$	−3593.11	−0.423478
$417$	−8769.96	−1.02990
$418$	−2979.20	−0.348606
$419$	3091.89	0.360498	0.180249	−	0.983621i	$-0.442310\pi$
$419$	3091.89	0.360498	0.180249	+	0.983621i	$0.442310\pi$
$420$	6994.15	0.812570
$421$	−7883.68	−0.912653	−0.456327	−	0.889812i	$-0.650835\pi$
$421$	−7883.68	−0.912653	−0.456327	+	0.889812i	$0.650835\pi$
$422$	5154.58	0.594599
$423$	−3764.48	−0.432707
$424$	8085.47	0.926097
$425$	2124.50	0.242479
$426$	−1087.82	−0.123721
$427$	−12561.2	−1.42361
$428$	−8622.68	−0.973815
$429$	−6124.91	−0.689309
$430$	0	0
$431$	−12359.0	−1.38124	−0.690618	−	0.723219i	$-0.742661\pi$
$431$	−12359.0	−1.38124	−0.690618	+	0.723219i	$0.742661\pi$
$432$	5845.25	0.650994
$433$	14075.5	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$433$	14075.5	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$434$	4155.95	0.459659
$435$	−2604.81	−0.287106
$436$	5340.21	0.586582
$437$	8827.59	0.966318
$438$	2717.28	0.296431
$439$	3314.21	0.360316	0.180158	−	0.983638i	$-0.442339\pi$
$439$	3314.21	0.360316	0.180158	+	0.983638i	$0.442339\pi$
$440$	−14192.4	−1.53772
$441$	666.629	0.0719824
$442$	−573.305	−0.0616953
$443$	−6141.24	−0.658643	−0.329322	−	0.944218i	$-0.606820\pi$
$443$	−6141.24	−0.658643	−0.329322	+	0.944218i	$0.606820\pi$
$444$	−3441.52	−0.367854
$445$	−12425.7	−1.32367
$446$	3501.92	0.371795
$447$	−14285.5	−1.51159
$448$	2059.36	0.217177
$449$	4718.97	0.495995	0.247998	−	0.968761i	$-0.420227\pi$
$449$	4718.97	0.495995	0.247998	+	0.968761i	$0.420227\pi$
$450$	−571.941	−0.0599145
$451$	3881.08	0.405217
$452$	444.919	0.0462992
$453$	−9461.88	−0.981364
$454$	−3829.96	−0.395923
$455$	−4838.13	−0.498495
$456$	3220.88	0.330771
$457$	4042.78	0.413815	0.206907	−	0.978361i	$-0.433660\pi$
$457$	4042.78	0.413815	0.206907	+	0.978361i	$0.433660\pi$
$458$	4174.13	0.425861
$459$	3813.92	0.387840
$460$	19456.4	1.97209
$461$	17986.7	1.81719	0.908595	−	0.417678i	$-0.137156\pi$
$461$	17986.7	1.81719	0.908595	+	0.417678i	$0.137156\pi$
$462$	−4621.14	−0.465357
$463$	−5977.08	−0.599953	−0.299977	−	0.953947i	$-0.596979\pi$
$463$	−5977.08	−0.599953	−0.299977	+	0.953947i	$0.596979\pi$
$464$	−1529.69	−0.153048
$465$	−16730.7	−1.66853
$466$	−4785.82	−0.475749
$467$	−7804.47	−0.773336	−0.386668	−	0.922219i	$-0.626374\pi$
$467$	−7804.47	−0.773336	−0.386668	+	0.922219i	$0.626374\pi$
$468$	−956.263	−0.0944514
$469$	8630.46	0.849718
$470$	−8943.98	−0.877776
$471$	15931.1	1.55853
$472$	−3927.34	−0.382988
$473$	0	0
$474$	6391.97	0.619394
$475$	3819.91	0.368989
$476$	2679.99	0.258061
$477$	3308.12	0.317544
$478$	−6408.69	−0.613236
$479$	−271.734	−0.0259203	−0.0129602	−	0.999916i	$-0.504125\pi$
$479$	−271.734	−0.0259203	−0.0129602	+	0.999916i	$0.504125\pi$
$480$	10913.5	1.03777
$481$	2380.64	0.225671
$482$	4800.66	0.453659
$483$	13692.8	1.28994
$484$	−17711.9	−1.66341
$485$	4526.84	0.423821
$486$	−1856.20	−0.173248
$487$	−259.217	−0.0241196	−0.0120598	−	0.999927i	$-0.503839\pi$
$487$	−259.217	−0.0241196	−0.0120598	+	0.999927i	$0.503839\pi$
$488$	−12749.5	−1.18267
$489$	−12276.2	−1.13528
$490$	1583.84	0.146021
$491$	16271.5	1.49557	0.747784	−	0.663942i	$-0.231118\pi$
$491$	16271.5	1.49557	0.747784	+	0.663942i	$0.231118\pi$
$492$	−1941.30	−0.177887
$493$	−998.098	−0.0911807
$494$	−1030.82	−0.0938841
$495$	−5806.72	−0.527258
$496$	−9825.24	−0.889448
$497$	−3517.62	−0.317478
$498$	77.8455	0.00700470
$499$	−8634.67	−0.774631	−0.387316	−	0.921947i	$-0.626598\pi$
$499$	−8634.67	−0.774631	−0.387316	+	0.921947i	$0.626598\pi$
$500$	−4042.02	−0.361529
$501$	−4595.07	−0.409766
$502$	−6166.38	−0.548245
$503$	−10084.7	−0.893943	−0.446971	−	0.894548i	$-0.647497\pi$
$503$	−10084.7	−0.893943	−0.446971	+	0.894548i	$0.647497\pi$
$504$	−1559.42	−0.137821
$505$	7595.52	0.669299
$506$	−12855.2	−1.12941
$507$	7846.80	0.687354
$508$	1598.03	0.139569
$509$	−2063.68	−0.179707	−0.0898536	−	0.995955i	$-0.528640\pi$
$509$	−2063.68	−0.179707	−0.0898536	+	0.995955i	$0.528640\pi$
$510$	1741.32	0.151190
$511$	8786.68	0.760665
$512$	11250.8	0.971135
$513$	6857.54	0.590190
$514$	2041.06	0.175150
$515$	−7487.48	−0.640656
$516$	0	0
$517$	−36613.6	−3.11463
$518$	1796.15	0.152352
$519$	−2069.20	−0.175006
$520$	−4910.64	−0.414127
$521$	−12140.1	−1.02086	−0.510430	−	0.859919i	$-0.670514\pi$
$521$	−12140.1	−1.02086	−0.510430	+	0.859919i	$0.670514\pi$
$522$	268.700	0.0225300
$523$	5578.83	0.466434	0.233217	−	0.972425i	$-0.425075\pi$
$523$	5578.83	0.466434	0.233217	+	0.972425i	$0.425075\pi$
$524$	−2608.35	−0.217455
$525$	5925.20	0.492566
$526$	−7114.08	−0.589712
$527$	−6410.79	−0.529902
$528$	10925.0	0.900474
$529$	25923.8	2.13066
$530$	7859.71	0.644158
$531$	−1606.84	−0.131320
$532$	4818.69	0.392701
$533$	1342.88	0.109130
$534$	−4106.53	−0.332784
$535$	−18116.7	−1.46402
$536$	8759.80	0.705906
$537$	−9506.12	−0.763909
$538$	−209.361	−0.0167773
$539$	6483.69	0.518130
$540$	15114.3	1.20448
$541$	5469.14	0.434633	0.217317	−	0.976101i	$-0.430270\pi$
$541$	5469.14	0.434633	0.217317	+	0.976101i	$0.430270\pi$
$542$	2133.55	0.169085
$543$	−14142.5	−1.11770
$544$	4181.78	0.329582
$545$	11220.0	0.881860
$546$	−1598.94	−0.125327
$547$	−11040.8	−0.863019	−0.431510	−	0.902108i	$-0.642019\pi$
$547$	−11040.8	−0.863019	−0.431510	+	0.902108i	$0.642019\pi$
$548$	1307.58	0.101929
$549$	−5216.35	−0.405517
$550$	−5562.74	−0.431266
$551$	−1794.61	−0.138753
$552$	13898.0	1.07163
$553$	20669.3	1.58941
$554$	8739.76	0.670247
$555$	−7230.80	−0.553028
$556$	−13317.2	−1.01578
$557$	18238.0	1.38738	0.693688	−	0.720275i	$-0.255984\pi$
$557$	18238.0	1.38738	0.693688	+	0.720275i	$0.255984\pi$
$558$	1725.86	0.130935
$559$	0	0
$560$	8629.79	0.651206
$561$	7128.38	0.536471
$562$	5708.08	0.428436
$563$	20112.6	1.50559	0.752793	−	0.658258i	$-0.228706\pi$
$563$	20112.6	1.50559	0.752793	+	0.658258i	$0.228706\pi$
$564$	18314.0	1.36730
$565$	934.798	0.0696057
$566$	−7538.23	−0.559815
$567$	7954.86	0.589193
$568$	−3570.33	−0.263746
$569$	11537.1	0.850019	0.425010	−	0.905189i	$-0.360271\pi$
$569$	11537.1	0.850019	0.425010	+	0.905189i	$0.360271\pi$
$570$	3130.94	0.230072
$571$	−2585.01	−0.189456	−0.0947281	−	0.995503i	$-0.530198\pi$
$571$	−2585.01	−0.189456	−0.0947281	+	0.995503i	$0.530198\pi$
$572$	−9300.69	−0.679862
$573$	13917.2	1.01466
$574$	1013.18	0.0736745
$575$	16482.8	1.19544
$576$	855.199	0.0618634
$577$	−9197.18	−0.663576	−0.331788	−	0.943354i	$-0.607652\pi$
$577$	−9197.18	−0.663576	−0.331788	+	0.943354i	$0.607652\pi$
$578$	−4513.04	−0.324771
$579$	9948.25	0.714050
$580$	−3955.40	−0.283171
$581$	251.723	0.0179746
$582$	1496.06	0.106553
$583$	32175.0	2.28568
$584$	8918.36	0.631925
$585$	−2009.16	−0.141997
$586$	4719.49	0.332697
$587$	11366.3	0.799210	0.399605	−	0.916687i	$-0.369147\pi$
$587$	11366.3	0.799210	0.399605	+	0.916687i	$0.369147\pi$
$588$	−3243.11	−0.227455
$589$	−11526.8	−0.806372
$590$	−3817.68	−0.266392
$591$	−1372.39	−0.0955208
$592$	−4246.35	−0.294804
$593$	8866.85	0.614027	0.307013	−	0.951705i	$-0.400670\pi$
$593$	8866.85	0.614027	0.307013	+	0.951705i	$0.400670\pi$
$594$	−9986.27	−0.689801
$595$	5630.78	0.387966
$596$	−21692.5	−1.49087
$597$	−4552.90	−0.312124
$598$	−4447.95	−0.304164
$599$	−15941.8	−1.08742	−0.543709	−	0.839274i	$-0.682980\pi$
$599$	−15941.8	−1.08742	−0.543709	+	0.839274i	$0.682980\pi$
$600$	6014.00	0.409201
$601$	−590.214	−0.0400588	−0.0200294	−	0.999799i	$-0.506376\pi$
$601$	−590.214	−0.0400588	−0.0200294	+	0.999799i	$0.506376\pi$
$602$	0	0
$603$	3584.01	0.242044
$604$	−14367.9	−0.967915
$605$	−37213.7	−2.50074
$606$	2510.22	0.168268
$607$	24494.4	1.63788	0.818942	−	0.573876i	$-0.194561\pi$
$607$	24494.4	1.63788	0.818942	+	0.573876i	$0.194561\pi$
$608$	7518.97	0.501537
$609$	−2783.68	−0.185223
$610$	−12393.5	−0.822618
$611$	−12668.5	−0.838810
$612$	1112.93	0.0735091
$613$	18011.0	1.18672	0.593358	−	0.804939i	$-0.297802\pi$
$613$	18011.0	1.18672	0.593358	+	0.804939i	$0.297802\pi$
$614$	−4190.52	−0.275433
$615$	−4078.76	−0.267434
$616$	−15167.0	−0.992038
$617$	−3697.23	−0.241240	−0.120620	−	0.992699i	$-0.538488\pi$
$617$	−3697.23	−0.241240	−0.120620	+	0.992699i	$0.538488\pi$
$618$	−2474.51	−0.161067
$619$	−776.148	−0.0503974	−0.0251987	−	0.999682i	$-0.508022\pi$
$619$	−776.148	−0.0503974	−0.0251987	+	0.999682i	$0.508022\pi$
$620$	−25405.6	−1.64567
$621$	29590.1	1.91209
$622$	−5287.93	−0.340879
$623$	−13279.0	−0.853951
$624$	3780.11	0.242509
$625$	−19049.3	−1.21915
$626$	491.493	0.0313802
$627$	12817.0	0.816368
$628$	24191.4	1.53717
$629$	−2770.67	−0.175634
$630$	−1515.87	−0.0958632
$631$	−2827.64	−0.178394	−0.0891968	−	0.996014i	$-0.528430\pi$
$631$	−2827.64	−0.178394	−0.0891968	+	0.996014i	$0.528430\pi$
$632$	20979.0	1.32041
$633$	−22175.9	−1.39244
$634$	−1359.91	−0.0851875
$635$	3357.54	0.209826
$636$	−16093.8	−1.00340
$637$	2243.39	0.139539
$638$	2613.40	0.162171
$639$	−1460.78	−0.0904343
$640$	21278.8	1.31424
$641$	12066.0	0.743491	0.371746	−	0.928335i	$-0.378759\pi$
$641$	12066.0	0.743491	0.371746	+	0.928335i	$0.378759\pi$
$642$	−5987.32	−0.368070
$643$	−6253.55	−0.383540	−0.191770	−	0.981440i	$-0.561423\pi$
$643$	−6253.55	−0.383540	−0.191770	+	0.981440i	$0.561423\pi$
$644$	20792.5	1.27227
$645$	0	0
$646$	1199.70	0.0730675
$647$	−8121.27	−0.493478	−0.246739	−	0.969082i	$-0.579359\pi$
$647$	−8121.27	−0.493478	−0.246739	+	0.969082i	$0.579359\pi$
$648$	8074.07	0.489474
$649$	−15628.3	−0.945245
$650$	−1924.74	−0.116145
$651$	−17879.6	−1.07643
$652$	−18641.4	−1.11972
$653$	−10001.3	−0.599356	−0.299678	−	0.954040i	$-0.596879\pi$
$653$	−10001.3	−0.599356	−0.299678	+	0.954040i	$0.596879\pi$
$654$	3708.07	0.221708
$655$	−5480.27	−0.326919
$656$	−2395.29	−0.142561
$657$	3648.89	0.216677
$658$	−9558.17	−0.566286
$659$	8841.69	0.522646	0.261323	−	0.965251i	$-0.415841\pi$
$659$	8841.69	0.522646	0.261323	+	0.965251i	$0.415841\pi$
$660$	28249.3	1.66607
$661$	1408.36	0.0828728	0.0414364	−	0.999141i	$-0.486807\pi$
$661$	1408.36	0.0828728	0.0414364	+	0.999141i	$0.486807\pi$
$662$	6415.72	0.376668
$663$	2466.46	0.144478
$664$	255.496	0.0149325
$665$	10124.3	0.590382
$666$	745.896	0.0433977
$667$	−7743.69	−0.449530
$668$	−6977.62	−0.404150
$669$	−15065.9	−0.870672
$670$	8515.21	0.491002
$671$	−50734.7	−2.91891
$672$	11662.9	0.669505
$673$	20128.9	1.15291	0.576457	−	0.817127i	$-0.304435\pi$
$673$	20128.9	1.15291	0.576457	+	0.817127i	$0.304435\pi$
$674$	−2856.91	−0.163270
$675$	12804.3	0.730132
$676$	11915.4	0.677935
$677$	2419.42	0.137350	0.0686750	−	0.997639i	$-0.478123\pi$
$677$	2419.42	0.137350	0.0686750	+	0.997639i	$0.478123\pi$
$678$	308.938	0.0174996
$679$	4837.70	0.273422
$680$	5715.17	0.322304
$681$	16477.2	0.927175
$682$	16785.9	0.942469
$683$	12041.6	0.674611	0.337306	−	0.941395i	$-0.390484\pi$
$683$	12041.6	0.674611	0.337306	+	0.941395i	$0.390484\pi$
$684$	2001.08	0.111861
$685$	2747.28	0.153238
$686$	7286.16	0.405520
$687$	−17957.8	−0.997283
$688$	0	0
$689$	11132.7	0.615563
$690$	13509.9	0.745383
$691$	1693.27	0.0932198	0.0466099	−	0.998913i	$-0.485158\pi$
$691$	1693.27	0.0932198	0.0466099	+	0.998913i	$0.485158\pi$
$692$	−3142.09	−0.172607
$693$	−6205.47	−0.340154
$694$	862.411	0.0471710
$695$	−27980.1	−1.52712
$696$	−2825.40	−0.153874
$697$	−1562.88	−0.0849331
$698$	2461.85	0.133499
$699$	20589.4	1.11411
$700$	8997.43	0.485815
$701$	−26393.7	−1.42208	−0.711039	−	0.703152i	$-0.751775\pi$
$701$	−26393.7	−1.42208	−0.711039	+	0.703152i	$0.751775\pi$
$702$	−3455.30	−0.185772
$703$	−4981.74	−0.267269
$704$	8317.73	0.445293
$705$	38478.5	2.05558
$706$	−6504.82	−0.346760
$707$	8117.11	0.431790
$708$	7817.19	0.414955
$709$	21874.3	1.15868	0.579341	−	0.815085i	$-0.303310\pi$
$709$	21874.3	1.15868	0.579341	+	0.815085i	$0.303310\pi$
$710$	−3470.64	−0.183452
$711$	8583.42	0.452747
$712$	−13478.0	−0.709423
$713$	−49737.8	−2.61247
$714$	1860.90	0.0975383
$715$	−19541.2	−1.02210
$716$	−14435.1	−0.753440
$717$	27571.3	1.43608
$718$	193.018	0.0100326
$719$	17951.1	0.931103	0.465552	−	0.885021i	$-0.345856\pi$
$719$	17951.1	0.931103	0.465552	+	0.885021i	$0.345856\pi$
$720$	3583.73	0.185497
$721$	−8001.65	−0.413311
$722$	−5075.03	−0.261597
$723$	−20653.2	−1.06238
$724$	−21475.4	−1.10239
$725$	−3350.88	−0.171653
$726$	−12298.6	−0.628712
$727$	21470.8	1.09534	0.547668	−	0.836696i	$-0.315516\pi$
$727$	21470.8	1.09534	0.547668	+	0.836696i	$0.315516\pi$
$728$	−5247.86	−0.267168
$729$	21872.6	1.11125
$730$	8669.34	0.439543
$731$	0	0
$732$	25377.2	1.28138
$733$	−32219.7	−1.62355	−0.811775	−	0.583970i	$-0.801499\pi$
$733$	−32219.7	−1.62355	−0.811775	+	0.583970i	$0.801499\pi$
$734$	5144.15	0.258684
$735$	−6813.93	−0.341953
$736$	32444.1	1.62487
$737$	34858.4	1.74223
$738$	420.747	0.0209863
$739$	−14452.3	−0.719402	−0.359701	−	0.933068i	$-0.617121\pi$
$739$	−14452.3	−0.719402	−0.359701	+	0.933068i	$0.617121\pi$
$740$	−10980.0	−0.545449
$741$	4434.76	0.219858
$742$	8399.45	0.415571
$743$	24465.7	1.20802	0.604011	−	0.796976i	$-0.293568\pi$
$743$	24465.7	1.20802	0.604011	+	0.796976i	$0.293568\pi$
$744$	−18147.6	−0.894250
$745$	−45577.0	−2.24136
$746$	−3161.75	−0.155174
$747$	104.534	0.00512010
$748$	10824.4	0.529119
$749$	−19360.8	−0.944496
$750$	−2806.65	−0.136646
$751$	−38635.2	−1.87725	−0.938626	−	0.344935i	$-0.887901\pi$
$751$	−38635.2	−1.87725	−0.938626	+	0.344935i	$0.887901\pi$
$752$	22596.8	1.09577
$753$	26528.8	1.28388
$754$	904.249	0.0436748
$755$	−30187.6	−1.45515
$756$	16152.2	0.777052
$757$	6833.97	0.328118	0.164059	−	0.986451i	$-0.447541\pi$
$757$	6833.97	0.328118	0.164059	+	0.986451i	$0.447541\pi$
$758$	11383.1	0.545450
$759$	55305.1	2.64486
$760$	10276.0	0.490462
$761$	−30885.3	−1.47121	−0.735606	−	0.677410i	$-0.763103\pi$
$761$	−30885.3	−1.47121	−0.735606	+	0.677410i	$0.763103\pi$
$762$	1109.62	0.0527524
$763$	11990.5	0.568921
$764$	21133.3	1.00075
$765$	2338.32	0.110513
$766$	−10386.2	−0.489909
$767$	−5407.47	−0.254566
$768$	2200.35	0.103383
$769$	−31913.9	−1.49654	−0.748272	−	0.663392i	$-0.769116\pi$
$769$	−31913.9	−1.49654	−0.748272	+	0.663392i	$0.769116\pi$
$770$	−14743.5	−0.690025
$771$	−8780.97	−0.410167
$772$	15106.4	0.704265
$773$	−11641.0	−0.541651	−0.270825	−	0.962629i	$-0.587297\pi$
$773$	−11641.0	−0.541651	−0.270825	+	0.962629i	$0.587297\pi$
$774$	0	0
$775$	−21522.7	−0.997574
$776$	4910.20	0.227147
$777$	−7727.35	−0.356779
$778$	577.575	0.0266157
$779$	−2810.11	−0.129246
$780$	9774.41	0.448692
$781$	−14207.6	−0.650947
$782$	5176.67	0.236723
$783$	−6015.53	−0.274556
$784$	−4001.54	−0.182286
$785$	50827.4	2.31096
$786$	−1811.16	−0.0821907
$787$	29770.6	1.34842	0.674211	−	0.738539i	$-0.264484\pi$
$787$	29770.6	1.34842	0.674211	+	0.738539i	$0.264484\pi$
$788$	−2083.98	−0.0942117
$789$	30606.0	1.38099
$790$	20393.2	0.918429
$791$	998.991	0.0449052
$792$	−6298.47	−0.282584
$793$	−17554.5	−0.786100
$794$	3620.35	0.161815
$795$	−33813.8	−1.50849
$796$	−6913.59	−0.307846
$797$	12956.7	0.575845	0.287923	−	0.957654i	$-0.407035\pi$
$797$	12956.7	0.575845	0.287923	+	0.957654i	$0.407035\pi$
$798$	3345.95	0.148428
$799$	14744.0	0.652824
$800$	14039.4	0.620458
$801$	−5514.43	−0.243249
$802$	2680.09	0.118001
$803$	35489.4	1.55964
$804$	−17436.0	−0.764826
$805$	43686.1	1.91271
$806$	5808.00	0.253819
$807$	900.706	0.0392892
$808$	8238.75	0.358711
$809$	−20078.1	−0.872571	−0.436285	−	0.899808i	$-0.643706\pi$
$809$	−20078.1	−0.872571	−0.436285	+	0.899808i	$0.643706\pi$
$810$	7848.63	0.340460
$811$	−11737.7	−0.508221	−0.254111	−	0.967175i	$-0.581783\pi$
$811$	−11737.7	−0.508221	−0.254111	+	0.967175i	$0.581783\pi$
$812$	−4227.03	−0.182684
$813$	−9178.91	−0.395963
$814$	7254.64	0.312377
$815$	−39166.6	−1.68337
$816$	−4399.42	−0.188738
$817$	0	0
$818$	2920.36	0.124826
$819$	−2147.13	−0.0916077
$820$	−6193.61	−0.263769
$821$	480.443	0.0204234	0.0102117	−	0.999948i	$-0.496749\pi$
$821$	480.443	0.0204234	0.0102117	+	0.999948i	$0.496749\pi$
$822$	907.940	0.0385256
$823$	6403.13	0.271202	0.135601	−	0.990764i	$-0.456704\pi$
$823$	6403.13	0.271202	0.135601	+	0.990764i	$0.456704\pi$
$824$	−8121.56	−0.343359
$825$	23931.9	1.00994
$826$	−4079.84	−0.171859
$827$	−13707.5	−0.576370	−0.288185	−	0.957575i	$-0.593052\pi$
$827$	−13707.5	−0.576370	−0.288185	+	0.957575i	$0.593052\pi$
$828$	8634.61	0.362407
$829$	−32923.0	−1.37933	−0.689665	−	0.724128i	$-0.742242\pi$
$829$	−32923.0	−1.37933	−0.689665	+	0.724128i	$0.742242\pi$
$830$	248.362	0.0103865
$831$	−37599.9	−1.56959
$832$	2877.98	0.119923
$833$	−2610.93	−0.108600
$834$	−9247.05	−0.383932
$835$	−14660.3	−0.607594
$836$	19462.7	0.805180
$837$	−38637.8	−1.59560
$838$	3260.09	0.134389
$839$	−4172.23	−0.171682	−0.0858410	−	0.996309i	$-0.527358\pi$
$839$	−4172.23	−0.171682	−0.0858410	+	0.996309i	$0.527358\pi$
$840$	15939.5	0.654721
$841$	−22814.7	−0.935452
$842$	−8312.55	−0.340225
$843$	−24557.1	−1.00331
$844$	−33674.1	−1.37335
$845$	25034.8	1.01920
$846$	−3969.27	−0.161308
$847$	−39769.2	−1.61332
$848$	−19857.5	−0.804136
$849$	32430.7	1.31098
$850$	2240.07	0.0903928
$851$	−21496.0	−0.865893
$852$	7106.60	0.285761
$853$	39769.7	1.59635	0.798177	−	0.602424i	$-0.205798\pi$
$853$	39769.7	1.59635	0.798177	+	0.602424i	$0.205798\pi$
$854$	−13244.5	−0.530701
$855$	4204.37	0.168171
$856$	−19650.9	−0.784643
$857$	20869.8	0.831855	0.415927	−	0.909398i	$-0.363457\pi$
$857$	20869.8	0.831855	0.415927	+	0.909398i	$0.363457\pi$
$858$	−6458.11	−0.256965
$859$	−10297.0	−0.408999	−0.204499	−	0.978867i	$-0.565557\pi$
$859$	−10297.0	−0.408999	−0.204499	+	0.978867i	$0.565557\pi$
$860$	0	0
$861$	−4358.86	−0.172531
$862$	−13031.3	−0.514907
$863$	24683.3	0.973615	0.486808	−	0.873509i	$-0.338161\pi$
$863$	24683.3	0.973615	0.486808	+	0.873509i	$0.338161\pi$
$864$	25203.6	0.992411
$865$	−6601.68	−0.259496
$866$	14841.3	0.582363
$867$	19415.9	0.760550
$868$	−27150.2	−1.06168
$869$	83483.0	3.25888
$870$	−2746.51	−0.107029
$871$	12061.2	0.469205
$872$	12170.2	0.472633
$873$	2008.97	0.0778849
$874$	9307.81	0.360230
$875$	−9075.68	−0.350644
$876$	−17751.6	−0.684670
$877$	−12580.6	−0.484398	−0.242199	−	0.970227i	$-0.577869\pi$
$877$	−12580.6	−0.484398	−0.242199	+	0.970227i	$0.577869\pi$
$878$	3494.50	0.134321
$879$	−20304.0	−0.779111
$880$	34855.7	1.33521
$881$	−8858.61	−0.338768	−0.169384	−	0.985550i	$-0.554178\pi$
$881$	−8858.61	−0.338768	−0.169384	+	0.985550i	$0.554178\pi$
$882$	702.894	0.0268341
$883$	6626.72	0.252556	0.126278	−	0.991995i	$-0.459697\pi$
$883$	6626.72	0.252556	0.126278	+	0.991995i	$0.459697\pi$
$884$	3745.32	0.142498
$885$	16424.3	0.623838
$886$	−6475.32	−0.245534
$887$	−11414.3	−0.432079	−0.216039	−	0.976385i	$-0.569314\pi$
$887$	−11414.3	−0.432079	−0.216039	+	0.976385i	$0.569314\pi$
$888$	−7843.15	−0.296395
$889$	3588.10	0.135367
$890$	−13101.7	−0.493448
$891$	32129.6	1.20806
$892$	−22877.5	−0.858740
$893$	26510.2	0.993426
$894$	−15062.6	−0.563500
$895$	−30328.8	−1.13271
$896$	22740.0	0.847868
$897$	19135.9	0.712294
$898$	4975.68	0.184900
$899$	10111.5	0.375124
$900$	3736.40	0.138385
$901$	−12956.6	−0.479076
$902$	4092.21	0.151060
$903$	0	0
$904$	1013.96	0.0373052
$905$	−45120.9	−1.65731
$906$	−9976.61	−0.365840
$907$	11407.7	0.417626	0.208813	−	0.977956i	$-0.433040\pi$
$907$	11407.7	0.417626	0.208813	+	0.977956i	$0.433040\pi$
$908$	25020.6	0.914469
$909$	3370.83	0.122996
$910$	−5101.33	−0.185832
$911$	−49077.0	−1.78484	−0.892422	−	0.451201i	$-0.850996\pi$
$911$	−49077.0	−1.78484	−0.892422	+	0.451201i	$0.850996\pi$
$912$	−7910.29	−0.287210
$913$	1016.71	0.0368545
$914$	4262.71	0.154265
$915$	53318.8	1.92641
$916$	−27269.0	−0.983615
$917$	−5856.61	−0.210908
$918$	4021.40	0.144582
$919$	−2643.59	−0.0948901	−0.0474451	−	0.998874i	$-0.515108\pi$
$919$	−2643.59	−0.0948901	−0.0474451	+	0.998874i	$0.515108\pi$
$920$	44340.8	1.58899
$921$	18028.3	0.645010
$922$	18965.2	0.677424
$923$	−4915.92	−0.175308
$924$	30189.2	1.07484
$925$	−9301.86	−0.330642
$926$	−6302.23	−0.223655
$927$	−3322.88	−0.117732
$928$	−6595.75	−0.233315
$929$	5048.62	0.178299	0.0891495	−	0.996018i	$-0.471585\pi$
$929$	5048.62	0.178299	0.0891495	+	0.996018i	$0.471585\pi$
$930$	−17640.8	−0.622007
$931$	−4694.53	−0.165260
$932$	31265.1	1.09884
$933$	22749.6	0.798272
$934$	−8229.03	−0.288289
$935$	22742.7	0.795471
$936$	−2179.30	−0.0761034
$937$	−25866.9	−0.901850	−0.450925	−	0.892562i	$-0.648906\pi$
$937$	−25866.9	−0.901850	−0.450925	+	0.892562i	$0.648906\pi$
$938$	9099.96	0.316763
$939$	−2114.49	−0.0734864
$940$	58429.7	2.02741
$941$	19865.7	0.688207	0.344103	−	0.938932i	$-0.388183\pi$
$941$	19865.7	0.688207	0.344103	+	0.938932i	$0.388183\pi$
$942$	16797.8	0.580999
$943$	−12125.5	−0.418729
$944$	9645.31	0.332551
$945$	33936.7	1.16821
$946$	0	0
$947$	−15772.5	−0.541223	−0.270611	−	0.962689i	$-0.587226\pi$
$947$	−15772.5	−0.541223	−0.270611	+	0.962689i	$0.587226\pi$
$948$	−41757.8	−1.43062
$949$	12279.5	0.420031
$950$	4027.72	0.137554
$951$	5850.56	0.199492
$952$	6107.63	0.207930
$953$	−5260.78	−0.178818	−0.0894089	−	0.995995i	$-0.528498\pi$
$953$	−5260.78	−0.178818	−0.0894089	+	0.995995i	$0.528498\pi$
$954$	3488.08	0.118376
$955$	44402.1	1.50452
$956$	41867.0	1.41640
$957$	−11243.3	−0.379774
$958$	−286.516	−0.00966276
$959$	2935.94	0.0988597
$960$	−8741.39	−0.293883
$961$	35155.0	1.18006
$962$	2510.14	0.0841271
$963$	−8040.04	−0.269041
$964$	−31362.0	−1.04782
$965$	31739.3	1.05878
$966$	14437.7	0.480874
$967$	−22150.5	−0.736619	−0.368310	−	0.929703i	$-0.620063\pi$
$967$	−22150.5	−0.736619	−0.368310	+	0.929703i	$0.620063\pi$
$968$	−40365.2	−1.34027
$969$	−5161.32	−0.171110
$970$	4773.10	0.157995
$971$	−32180.2	−1.06355	−0.531777	−	0.846884i	$-0.678476\pi$
$971$	−32180.2	−1.06355	−0.531777	+	0.846884i	$0.678476\pi$
$972$	12126.3	0.400154
$973$	−29901.5	−0.985199
$974$	−273.318	−0.00899147
$975$	8280.55	0.271990
$976$	31311.9	1.02692
$977$	−21321.9	−0.698207	−0.349103	−	0.937084i	$-0.613514\pi$
$977$	−21321.9	−0.698207	−0.349103	+	0.937084i	$0.613514\pi$
$978$	−12944.0	−0.423216
$979$	−53633.8	−1.75091
$980$	−10347.0	−0.337267
$981$	4979.37	0.162058
$982$	17156.7	0.557528
$983$	14512.5	0.470881	0.235440	−	0.971889i	$-0.424347\pi$
$983$	14512.5	0.470881	0.235440	+	0.971889i	$0.424347\pi$
$984$	−4424.18	−0.143331
$985$	−4378.55	−0.141637
$986$	−1052.39	−0.0339910
$987$	41120.9	1.32613
$988$	6734.19	0.216845
$989$	0	0
$990$	−6122.61	−0.196555
$991$	30878.2	0.989785	0.494893	−	0.868954i	$-0.335207\pi$
$991$	30878.2	0.989785	0.494893	+	0.868954i	$0.335207\pi$
$992$	−42364.5	−1.35592
$993$	−27601.5	−0.882082
$994$	−3708.98	−0.118352
$995$	−14525.8	−0.462813
$996$	−508.553	−0.0161788
$997$	25585.2	0.812731	0.406366	−	0.913711i	$-0.366796\pi$
$997$	25585.2	0.812731	0.406366	+	0.913711i	$0.366796\pi$
$998$	−9104.40	−0.288772
$999$	−16698.8	−0.528855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.i.1.29	✓	50
43.42	odd	2		1849.4.a.j.1.22	yes	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.29	✓	50	1.1	even	1	trivial
1849.4.a.j.1.22	yes	50	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)

\(f(q)\)	\(=\)	\(q+1.05440 q^{2} -4.53621 q^{3} -6.88824 q^{4} -14.4725 q^{5} -4.78298 q^{6} -15.4664 q^{7} -15.6982 q^{8} -6.42280 q^{9} +O(q^{10})\)	\(q+1.05440 q^{2} -4.53621 q^{3} -6.88824 q^{4} -14.4725 q^{5} -4.78298 q^{6} -15.4664 q^{7} -15.6982 q^{8} -6.42280 q^{9} -15.2598 q^{10} -62.4686 q^{11} +31.2465 q^{12} -21.6145 q^{13} -16.3078 q^{14} +65.6505 q^{15} +38.5538 q^{16} +25.1556 q^{17} -6.77220 q^{18} +45.2306 q^{19} +99.6903 q^{20} +70.1588 q^{21} -65.8669 q^{22} +195.169 q^{23} +71.2102 q^{24} +84.4542 q^{25} -22.7903 q^{26} +151.613 q^{27} +106.536 q^{28} -39.6769 q^{29} +69.2218 q^{30} -254.845 q^{31} +166.236 q^{32} +283.371 q^{33} +26.5241 q^{34} +223.838 q^{35} +44.2418 q^{36} -110.141 q^{37} +47.6911 q^{38} +98.0478 q^{39} +227.192 q^{40} -62.1285 q^{41} +73.9754 q^{42} +430.299 q^{44} +92.9542 q^{45} +205.786 q^{46} +586.112 q^{47} -174.888 q^{48} -103.791 q^{49} +89.0485 q^{50} -114.111 q^{51} +148.886 q^{52} -515.058 q^{53} +159.861 q^{54} +904.079 q^{55} +242.794 q^{56} -205.175 q^{57} -41.8353 q^{58} +250.178 q^{59} -452.216 q^{60} +812.162 q^{61} -268.709 q^{62} +99.3374 q^{63} -133.151 q^{64} +312.816 q^{65} +298.786 q^{66} -558.014 q^{67} -173.278 q^{68} -885.326 q^{69} +236.014 q^{70} +227.436 q^{71} +100.826 q^{72} -568.115 q^{73} -116.133 q^{74} -383.102 q^{75} -311.559 q^{76} +966.164 q^{77} +103.382 q^{78} -1336.40 q^{79} -557.971 q^{80} -514.332 q^{81} -65.5083 q^{82} -16.2755 q^{83} -483.270 q^{84} -364.066 q^{85} +179.983 q^{87} +980.643 q^{88} +858.571 q^{89} +98.0109 q^{90} +334.298 q^{91} -1344.37 q^{92} +1156.03 q^{93} +617.997 q^{94} -654.601 q^{95} -754.083 q^{96} -312.788 q^{97} -109.437 q^{98} +401.223 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} - 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} - 386 q^{18} + 12 q^{19} + 108 q^{20} - 408 q^{21} + 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} + 1493 q^{26} + 10 q^{27} - 242 q^{28} + 208 q^{29} - 48 q^{30} - 932 q^{31} - 1124 q^{32} + 254 q^{33} + 765 q^{34} - 1452 q^{35} + 747 q^{36} - 90 q^{37} - 1213 q^{38} - 1610 q^{39} - 1693 q^{40} - 1354 q^{41} - 16 q^{42} - 2704 q^{44} + 4508 q^{45} + 233 q^{46} - 3484 q^{47} - 376 q^{48} + 1324 q^{49} - 408 q^{50} + 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} - 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} + 1172 q^{61} - 1546 q^{62} - 3686 q^{63} + 606 q^{64} + 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} + 136 q^{69} - 1310 q^{70} + 162 q^{71} - 5814 q^{72} + 746 q^{73} - 4332 q^{74} + 236 q^{75} + 1338 q^{76} + 2024 q^{77} - 2782 q^{78} - 2656 q^{79} - 5713 q^{80} - 86 q^{81} - 4168 q^{82} - 3514 q^{83} - 4269 q^{84} - 7558 q^{85} - 10278 q^{87} + 11692 q^{88} + 2640 q^{89} - 8286 q^{90} - 5946 q^{91} - 4271 q^{92} - 2 q^{93} + 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} - 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 - 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 - 386 * q^18 + 12 * q^19 + 108 * q^20 - 408 * q^21 + 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 + 1493 * q^26 + 10 * q^27 - 242 * q^28 + 208 * q^29 - 48 * q^30 - 932 * q^31 - 1124 * q^32 + 254 * q^33 + 765 * q^34 - 1452 * q^35 + 747 * q^36 - 90 * q^37 - 1213 * q^38 - 1610 * q^39 - 1693 * q^40 - 1354 * q^41 - 16 * q^42 - 2704 * q^44 + 4508 * q^45 + 233 * q^46 - 3484 * q^47 - 376 * q^48 + 1324 * q^49 - 408 * q^50 + 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 - 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 + 1172 * q^61 - 1546 * q^62 - 3686 * q^63 + 606 * q^64 + 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 + 136 * q^69 - 1310 * q^70 + 162 * q^71 - 5814 * q^72 + 746 * q^73 - 4332 * q^74 + 236 * q^75 + 1338 * q^76 + 2024 * q^77 - 2782 * q^78 - 2656 * q^79 - 5713 * q^80 - 86 * q^81 - 4168 * q^82 - 3514 * q^83 - 4269 * q^84 - 7558 * q^85 - 10278 * q^87 + 11692 * q^88 + 2640 * q^89 - 8286 * q^90 - 5946 * q^91 - 4271 * q^92 - 2 * q^93 + 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 - 2826 * q^98 - 8174 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more