LMFDB - Embedded newform 46.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [46,4,Mod(1,46)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(46, 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("46.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 46 = 2 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	46.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:	$2.71408786026$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	46.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +9.10469 q^{3} +4.00000 q^{4} -1.40312 q^{5} -18.2094 q^{6} -3.40312 q^{7} -8.00000 q^{8} +55.8953 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +9.10469 q^{3} +4.00000 q^{4} -1.40312 q^{5} -18.2094 q^{6} -3.40312 q^{7} -8.00000 q^{8} +55.8953 q^{9} +2.80625 q^{10} +61.6125 q^{11} +36.4187 q^{12} -77.9109 q^{13} +6.80625 q^{14} -12.7750 q^{15} +16.0000 q^{16} -14.8375 q^{17} -111.791 q^{18} -46.6281 q^{19} -5.61250 q^{20} -30.9844 q^{21} -123.225 q^{22} -23.0000 q^{23} -72.8375 q^{24} -123.031 q^{25} +155.822 q^{26} +263.083 q^{27} -13.6125 q^{28} -200.602 q^{29} +25.5500 q^{30} -20.9266 q^{31} -32.0000 q^{32} +560.962 q^{33} +29.6750 q^{34} +4.77501 q^{35} +223.581 q^{36} +233.109 q^{37} +93.2562 q^{38} -709.355 q^{39} +11.2250 q^{40} +309.733 q^{41} +61.9688 q^{42} -399.287 q^{43} +246.450 q^{44} -78.4281 q^{45} +46.0000 q^{46} -190.245 q^{47} +145.675 q^{48} -331.419 q^{49} +246.062 q^{50} -135.091 q^{51} -311.644 q^{52} +516.994 q^{53} -526.166 q^{54} -86.4500 q^{55} +27.2250 q^{56} -424.534 q^{57} +401.203 q^{58} +498.450 q^{59} -51.1000 q^{60} +906.731 q^{61} +41.8531 q^{62} -190.219 q^{63} +64.0000 q^{64} +109.319 q^{65} -1121.92 q^{66} -33.6750 q^{67} -59.3500 q^{68} -209.408 q^{69} -9.55001 q^{70} -61.1578 q^{71} -447.163 q^{72} -79.3453 q^{73} -466.219 q^{74} -1120.16 q^{75} -186.512 q^{76} -209.675 q^{77} +1418.71 q^{78} +533.916 q^{79} -22.4500 q^{80} +886.113 q^{81} -619.466 q^{82} +766.878 q^{83} -123.938 q^{84} +20.8188 q^{85} +798.575 q^{86} -1826.41 q^{87} -492.900 q^{88} +57.9468 q^{89} +156.856 q^{90} +265.141 q^{91} -92.0000 q^{92} -190.530 q^{93} +380.491 q^{94} +65.4250 q^{95} -291.350 q^{96} +1601.06 q^{97} +662.837 q^{98} +3443.85 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - q^{3} + 8 q^{4} + 10 q^{5} + 2 q^{6} + 6 q^{7} - 16 q^{8} + 131 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 - q^3 + 8 * q^4 + 10 * q^5 + 2 * q^6 + 6 * q^7 - 16 * q^8 + 131 * q^9	\( 2 q - 4 q^{2} - q^{3} + 8 q^{4} + 10 q^{5} + 2 q^{6} + 6 q^{7} - 16 q^{8} + 131 q^{9} - 20 q^{10} + 72 q^{11} - 4 q^{12} - 111 q^{13} - 12 q^{14} - 128 q^{15} + 32 q^{16} + 124 q^{17} - 262 q^{18} + 22 q^{19} + 40 q^{20} - 126 q^{21} - 144 q^{22} - 46 q^{23} + 8 q^{24} - 118 q^{25} + 222 q^{26} - 223 q^{27} + 24 q^{28} + 15 q^{29} + 256 q^{30} + 67 q^{31} - 64 q^{32} + 456 q^{33} - 248 q^{34} + 112 q^{35} + 524 q^{36} + 18 q^{37} - 44 q^{38} - 375 q^{39} - 80 q^{40} + 485 q^{41} + 252 q^{42} - 440 q^{43} + 288 q^{44} + 778 q^{45} + 92 q^{46} + 215 q^{47} - 16 q^{48} - 586 q^{49} + 236 q^{50} - 1538 q^{51} - 444 q^{52} + 240 q^{53} + 446 q^{54} + 32 q^{55} - 48 q^{56} - 1118 q^{57} - 30 q^{58} + 792 q^{59} - 512 q^{60} + 456 q^{61} - 134 q^{62} + 516 q^{63} + 128 q^{64} - 268 q^{65} - 912 q^{66} + 240 q^{67} + 496 q^{68} + 23 q^{69} - 224 q^{70} - 705 q^{71} - 1048 q^{72} + 27 q^{73} - 36 q^{74} - 1171 q^{75} + 88 q^{76} - 112 q^{77} + 750 q^{78} + 594 q^{79} + 160 q^{80} + 3770 q^{81} - 970 q^{82} + 394 q^{83} - 504 q^{84} + 1604 q^{85} + 880 q^{86} - 4005 q^{87} - 576 q^{88} - 486 q^{89} - 1556 q^{90} - 46 q^{91} - 184 q^{92} - 1079 q^{93} - 430 q^{94} + 848 q^{95} + 32 q^{96} + 2152 q^{97} + 1172 q^{98} + 4224 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - q^3 + 8 * q^4 + 10 * q^5 + 2 * q^6 + 6 * q^7 - 16 * q^8 + 131 * q^9 - 20 * q^10 + 72 * q^11 - 4 * q^12 - 111 * q^13 - 12 * q^14 - 128 * q^15 + 32 * q^16 + 124 * q^17 - 262 * q^18 + 22 * q^19 + 40 * q^20 - 126 * q^21 - 144 * q^22 - 46 * q^23 + 8 * q^24 - 118 * q^25 + 222 * q^26 - 223 * q^27 + 24 * q^28 + 15 * q^29 + 256 * q^30 + 67 * q^31 - 64 * q^32 + 456 * q^33 - 248 * q^34 + 112 * q^35 + 524 * q^36 + 18 * q^37 - 44 * q^38 - 375 * q^39 - 80 * q^40 + 485 * q^41 + 252 * q^42 - 440 * q^43 + 288 * q^44 + 778 * q^45 + 92 * q^46 + 215 * q^47 - 16 * q^48 - 586 * q^49 + 236 * q^50 - 1538 * q^51 - 444 * q^52 + 240 * q^53 + 446 * q^54 + 32 * q^55 - 48 * q^56 - 1118 * q^57 - 30 * q^58 + 792 * q^59 - 512 * q^60 + 456 * q^61 - 134 * q^62 + 516 * q^63 + 128 * q^64 - 268 * q^65 - 912 * q^66 + 240 * q^67 + 496 * q^68 + 23 * q^69 - 224 * q^70 - 705 * q^71 - 1048 * q^72 + 27 * q^73 - 36 * q^74 - 1171 * q^75 + 88 * q^76 - 112 * q^77 + 750 * q^78 + 594 * q^79 + 160 * q^80 + 3770 * q^81 - 970 * q^82 + 394 * q^83 - 504 * q^84 + 1604 * q^85 + 880 * q^86 - 4005 * q^87 - 576 * q^88 - 486 * q^89 - 1556 * q^90 - 46 * q^91 - 184 * q^92 - 1079 * q^93 - 430 * q^94 + 848 * q^95 + 32 * q^96 + 2152 * q^97 + 1172 * q^98 + 4224 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	9.10469	1.75220	0.876099	−	0.482132i	$-0.160137\pi$
$3$	9.10469	1.75220	0.876099	+	0.482132i	$0.160137\pi$
$4$	4.00000	0.500000
$5$	−1.40312	−0.125499	−0.0627496	−	0.998029i	$-0.519987\pi$
$5$	−1.40312	−0.125499	−0.0627496	+	0.998029i	$0.519987\pi$
$6$	−18.2094	−1.23899
$7$	−3.40312	−0.183751	−0.0918757	−	0.995770i	$-0.529286\pi$
$7$	−3.40312	−0.183751	−0.0918757	+	0.995770i	$0.529286\pi$
$8$	−8.00000	−0.353553
$9$	55.8953	2.07020
$10$	2.80625	0.0887414
$11$	61.6125	1.68881	0.844403	−	0.535708i	$-0.179955\pi$
$11$	61.6125	1.68881	0.844403	+	0.535708i	$0.179955\pi$
$12$	36.4187	0.876099
$13$	−77.9109	−1.66220	−0.831100	−	0.556123i	$-0.812288\pi$
$13$	−77.9109	−1.66220	−0.831100	+	0.556123i	$0.812288\pi$
$14$	6.80625	0.129932
$15$	−12.7750	−0.219899
$16$	16.0000	0.250000
$17$	−14.8375	−0.211684	−0.105842	−	0.994383i	$-0.533754\pi$
$17$	−14.8375	−0.211684	−0.105842	+	0.994383i	$0.533754\pi$
$18$	−111.791	−1.46385
$19$	−46.6281	−0.563012	−0.281506	−	0.959559i	$-0.590834\pi$
$19$	−46.6281	−0.563012	−0.281506	+	0.959559i	$0.590834\pi$
$20$	−5.61250	−0.0627496
$21$	−30.9844	−0.321969
$22$	−123.225	−1.19417
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−72.8375	−0.619495
$25$	−123.031	−0.984250
$26$	155.822	1.17535
$27$	263.083	1.87520
$28$	−13.6125	−0.0918757
$29$	−200.602	−1.28451	−0.642255	−	0.766491i	$-0.722001\pi$
$29$	−200.602	−1.28451	−0.642255	+	0.766491i	$0.722001\pi$
$30$	25.5500	0.155492
$31$	−20.9266	−0.121243	−0.0606213	−	0.998161i	$-0.519308\pi$
$31$	−20.9266	−0.121243	−0.0606213	+	0.998161i	$0.519308\pi$
$32$	−32.0000	−0.176777
$33$	560.962	2.95912
$34$	29.6750	0.149683
$35$	4.77501	0.0230607
$36$	223.581	1.03510
$37$	233.109	1.03576	0.517878	−	0.855455i	$-0.326722\pi$
$37$	233.109	1.03576	0.517878	+	0.855455i	$0.326722\pi$
$38$	93.2562	0.398109
$39$	−709.355	−2.91250
$40$	11.2250	0.0443707
$41$	309.733	1.17981	0.589904	−	0.807473i	$-0.299166\pi$
$41$	309.733	1.17981	0.589904	+	0.807473i	$0.299166\pi$
$42$	61.9688	0.227666
$43$	−399.287	−1.41606	−0.708032	−	0.706180i	$-0.750417\pi$
$43$	−399.287	−1.41606	−0.708032	+	0.706180i	$0.750417\pi$
$44$	246.450	0.844403
$45$	−78.4281	−0.259808
$46$	46.0000	0.147442
$47$	−190.245	−0.590428	−0.295214	−	0.955431i	$-0.595391\pi$
$47$	−190.245	−0.590428	−0.295214	+	0.955431i	$0.595391\pi$
$48$	145.675	0.438049
$49$	−331.419	−0.966235
$50$	246.062	0.695970
$51$	−135.091	−0.370911
$52$	−311.644	−0.831100
$53$	516.994	1.33990	0.669949	−	0.742408i	$-0.266316\pi$
$53$	516.994	1.33990	0.669949	+	0.742408i	$0.266316\pi$
$54$	−526.166	−1.32596
$55$	−86.4500	−0.211944
$56$	27.2250	0.0649659
$57$	−424.534	−0.986508
$58$	401.203	0.908285
$59$	498.450	1.09988	0.549938	−	0.835206i	$-0.314651\pi$
$59$	498.450	1.09988	0.549938	+	0.835206i	$0.314651\pi$
$60$	−51.1000	−0.109950
$61$	906.731	1.90320	0.951599	−	0.307344i	$-0.0994401\pi$
$61$	906.731	1.90320	0.951599	+	0.307344i	$0.0994401\pi$
$62$	41.8531	0.0857315
$63$	−190.219	−0.380402
$64$	64.0000	0.125000
$65$	109.319	0.208605
$66$	−1121.92	−2.09242
$67$	−33.6750	−0.0614038	−0.0307019	−	0.999529i	$-0.509774\pi$
$67$	−33.6750	−0.0614038	−0.0307019	+	0.999529i	$0.509774\pi$
$68$	−59.3500	−0.105842
$69$	−209.408	−0.365358
$70$	−9.55001	−0.0163064
$71$	−61.1578	−0.102227	−0.0511134	−	0.998693i	$-0.516277\pi$
$71$	−61.1578	−0.102227	−0.0511134	+	0.998693i	$0.516277\pi$
$72$	−447.163	−0.731925
$73$	−79.3453	−0.127215	−0.0636073	−	0.997975i	$-0.520261\pi$
$73$	−79.3453	−0.127215	−0.0636073	+	0.997975i	$0.520261\pi$
$74$	−466.219	−0.732389
$75$	−1120.16	−1.72460
$76$	−186.512	−0.281506
$77$	−209.675	−0.310321
$78$	1418.71	2.05945
$79$	533.916	0.760382	0.380191	−	0.924908i	$-0.375858\pi$
$79$	533.916	0.760382	0.380191	+	0.924908i	$0.375858\pi$
$80$	−22.4500	−0.0313748
$81$	886.113	1.21552
$82$	−619.466	−0.834250
$83$	766.878	1.01417	0.507083	−	0.861897i	$-0.330724\pi$
$83$	766.878	1.01417	0.507083	+	0.861897i	$0.330724\pi$
$84$	−123.938	−0.160984
$85$	20.8188	0.0265661
$86$	798.575	1.00131
$87$	−1826.41	−2.25071
$88$	−492.900	−0.597083
$89$	57.9468	0.0690152	0.0345076	−	0.999404i	$-0.489014\pi$
$89$	57.9468	0.0690152	0.0345076	+	0.999404i	$0.489014\pi$
$90$	156.856	0.183712
$91$	265.141	0.305432
$92$	−92.0000	−0.104257
$93$	−190.530	−0.212441
$94$	380.491	0.417496
$95$	65.4250	0.0706576
$96$	−291.350	−0.309748
$97$	1601.06	1.67590	0.837952	−	0.545744i	$-0.183753\pi$
$97$	1601.06	1.67590	0.837952	+	0.545744i	$0.183753\pi$
$98$	662.837	0.683232
$99$	3443.85	3.49616
$100$	−492.125	−0.492125
$101$	367.425	0.361982	0.180991	−	0.983485i	$-0.442070\pi$
$101$	367.425	0.361982	0.180991	+	0.983485i	$0.442070\pi$
$102$	270.181	0.262274
$103$	−1051.19	−1.00560	−0.502800	−	0.864403i	$-0.667697\pi$
$103$	−1051.19	−1.00560	−0.502800	+	0.864403i	$0.667697\pi$
$104$	623.287	0.587677
$105$	43.4749	0.0404068
$106$	−1033.99	−0.947450
$107$	−1104.18	−0.997616	−0.498808	−	0.866712i	$-0.666229\pi$
$107$	−1104.18	−0.997616	−0.498808	+	0.866712i	$0.666229\pi$
$108$	1052.33	0.937598
$109$	−1264.69	−1.11133	−0.555667	−	0.831405i	$-0.687537\pi$
$109$	−1264.69	−1.11133	−0.555667	+	0.831405i	$0.687537\pi$
$110$	172.900	0.149867
$111$	2122.39	1.81485
$112$	−54.4500	−0.0459379
$113$	−2102.16	−1.75004	−0.875021	−	0.484085i	$-0.839153\pi$
$113$	−2102.16	−1.75004	−0.875021	+	0.484085i	$0.839153\pi$
$114$	849.069	0.697566
$115$	32.2719	0.0261684
$116$	−802.406	−0.642255
$117$	−4354.86	−3.44108
$118$	−996.900	−0.777730
$119$	50.4938	0.0388972
$120$	102.200	0.0777462
$121$	2465.10	1.85207
$122$	−1813.46	−1.34576
$123$	2820.02	2.06726
$124$	−83.7062	−0.0606213
$125$	348.019	0.249022
$126$	380.437	0.268985
$127$	349.639	0.244295	0.122147	−	0.992512i	$-0.461022\pi$
$127$	349.639	0.244295	0.122147	+	0.992512i	$0.461022\pi$
$128$	−128.000	−0.0883883
$129$	−3635.39	−2.48122
$130$	−218.637	−0.147506
$131$	−412.864	−0.275360	−0.137680	−	0.990477i	$-0.543964\pi$
$131$	−412.864	−0.275360	−0.137680	+	0.990477i	$0.543964\pi$
$132$	2243.85	1.47956
$133$	158.681	0.103454
$134$	67.3500	0.0434190
$135$	−369.138	−0.235336
$136$	118.700	0.0748414
$137$	2357.69	1.47030	0.735150	−	0.677905i	$-0.237112\pi$
$137$	2357.69	1.47030	0.735150	+	0.677905i	$0.237112\pi$
$138$	418.816	0.258347
$139$	85.9515	0.0524483	0.0262241	−	0.999656i	$-0.491652\pi$
$139$	85.9515	0.0524483	0.0262241	+	0.999656i	$0.491652\pi$
$140$	19.1000	0.0115303
$141$	−1732.12	−1.03455
$142$	122.316	0.0722852
$143$	−4800.29	−2.80713
$144$	894.325	0.517549
$145$	281.469	0.161205
$146$	158.691	0.0899543
$147$	−3017.46	−1.69304
$148$	932.437	0.517878
$149$	−559.531	−0.307642	−0.153821	−	0.988099i	$-0.549158\pi$
$149$	−559.531	−0.307642	−0.153821	+	0.988099i	$0.549158\pi$
$150$	2240.32	1.21948
$151$	210.467	0.113428	0.0567138	−	0.998390i	$-0.481938\pi$
$151$	210.467	0.113428	0.0567138	+	0.998390i	$0.481938\pi$
$152$	373.025	0.199055
$153$	−829.346	−0.438227
$154$	419.350	0.219430
$155$	29.3626	0.0152159
$156$	−2837.42	−1.45625
$157$	−436.734	−0.222008	−0.111004	−	0.993820i	$-0.535407\pi$
$157$	−436.734	−0.222008	−0.111004	+	0.993820i	$0.535407\pi$
$158$	−1067.83	−0.537671
$159$	4707.07	2.34776
$160$	44.9000	0.0221853
$161$	78.2719	0.0383148
$162$	−1772.23	−0.859501
$163$	−1199.47	−0.576377	−0.288189	−	0.957574i	$-0.593053\pi$
$163$	−1199.47	−0.576377	−0.288189	+	0.957574i	$0.593053\pi$
$164$	1238.93	0.589904
$165$	−787.100	−0.371368
$166$	−1533.76	−0.717124
$167$	1388.58	0.643423	0.321711	−	0.946838i	$-0.395742\pi$
$167$	1388.58	0.643423	0.321711	+	0.946838i	$0.395742\pi$
$168$	247.875	0.113833
$169$	3873.11	1.76291
$170$	−41.6377	−0.0187851
$171$	−2606.29	−1.16555
$172$	−1597.15	−0.708032
$173$	2726.37	1.19816	0.599082	−	0.800688i	$-0.295532\pi$
$173$	2726.37	1.19816	0.599082	+	0.800688i	$0.295532\pi$
$174$	3652.83	1.59150
$175$	418.691	0.180857
$176$	985.800	0.422202
$177$	4538.23	1.92720
$178$	−115.894	−0.0488011
$179$	1197.97	0.500226	0.250113	−	0.968217i	$-0.419532\pi$
$179$	1197.97	0.500226	0.250113	+	0.968217i	$0.419532\pi$
$180$	−313.712	−0.129904
$181$	486.384	0.199738	0.0998692	−	0.995001i	$-0.468158\pi$
$181$	486.384	0.199738	0.0998692	+	0.995001i	$0.468158\pi$
$182$	−530.281	−0.215973
$183$	8255.50	3.33478
$184$	184.000	0.0737210
$185$	−327.081	−0.129986
$186$	381.059	0.150218
$187$	−914.175	−0.357492
$188$	−760.981	−0.295214
$189$	−895.303	−0.344570
$190$	−130.850	−0.0499624
$191$	−2208.62	−0.836704	−0.418352	−	0.908285i	$-0.637392\pi$
$191$	−2208.62	−0.836704	−0.418352	+	0.908285i	$0.637392\pi$
$192$	582.700	0.219025
$193$	961.280	0.358520	0.179260	−	0.983802i	$-0.442630\pi$
$193$	961.280	0.358520	0.179260	+	0.983802i	$0.442630\pi$
$194$	−3202.11	−1.18504
$195$	995.313	0.365517
$196$	−1325.67	−0.483118
$197$	−5228.47	−1.89093	−0.945465	−	0.325725i	$-0.894392\pi$
$197$	−5228.47	−1.89093	−0.945465	+	0.325725i	$0.894392\pi$
$198$	−6887.70	−2.47216
$199$	3656.52	1.30253	0.651266	−	0.758850i	$-0.274238\pi$
$199$	3656.52	1.30253	0.651266	+	0.758850i	$0.274238\pi$
$200$	984.250	0.347985
$201$	−306.600	−0.107592
$202$	−734.850	−0.255960
$203$	682.672	0.236030
$204$	−540.363	−0.185456
$205$	−434.594	−0.148065
$206$	2102.38	0.711067
$207$	−1285.59	−0.431666
$208$	−1246.57	−0.415550
$209$	−2872.87	−0.950818
$210$	−86.9499	−0.0285720
$211$	−2849.63	−0.929747	−0.464874	−	0.885377i	$-0.653900\pi$
$211$	−2849.63	−0.929747	−0.464874	+	0.885377i	$0.653900\pi$
$212$	2067.97	0.669949
$213$	−556.823	−0.179122
$214$	2208.36	0.705421
$215$	560.250	0.177715
$216$	−2104.66	−0.662982
$217$	71.2157	0.0222785
$218$	2529.38	0.785832
$219$	−722.414	−0.222905
$220$	−345.800	−0.105972
$221$	1156.00	0.351860
$222$	−4244.78	−1.28329
$223$	−3924.56	−1.17851	−0.589256	−	0.807946i	$-0.700579\pi$
$223$	−3924.56	−1.17851	−0.589256	+	0.807946i	$0.700579\pi$
$224$	108.900	0.0324830
$225$	−6876.87	−2.03759
$226$	4204.32	1.23747
$227$	−2692.10	−0.787142	−0.393571	−	0.919294i	$-0.628760\pi$
$227$	−2692.10	−0.787142	−0.393571	+	0.919294i	$0.628760\pi$
$228$	−1698.14	−0.493254
$229$	−2226.63	−0.642533	−0.321267	−	0.946989i	$-0.604109\pi$
$229$	−2226.63	−0.642533	−0.321267	+	0.946989i	$0.604109\pi$
$230$	−64.5437	−0.0185039
$231$	−1909.02	−0.543743
$232$	1604.81	0.454143
$233$	−138.836	−0.0390363	−0.0195181	−	0.999810i	$-0.506213\pi$
$233$	−138.836	−0.0390363	−0.0195181	+	0.999810i	$0.506213\pi$
$234$	8709.71	2.43321
$235$	266.938	0.0740983
$236$	1993.80	0.549938
$237$	4861.13	1.33234
$238$	−100.988	−0.0275044
$239$	2165.37	0.586052	0.293026	−	0.956105i	$-0.405338\pi$
$239$	2165.37	0.586052	0.293026	+	0.956105i	$0.405338\pi$
$240$	−204.400	−0.0549749
$241$	4147.85	1.10866	0.554329	−	0.832297i	$-0.312975\pi$
$241$	4147.85	1.10866	0.554329	+	0.832297i	$0.312975\pi$
$242$	−4930.20	−1.30961
$243$	964.543	0.254631
$244$	3626.92	0.951599
$245$	465.022	0.121262
$246$	−5640.04	−1.46177
$247$	3632.84	0.935838
$248$	167.412	0.0428657
$249$	6982.18	1.77702
$250$	−696.037	−0.176085
$251$	1007.93	0.253466	0.126733	−	0.991937i	$-0.459551\pi$
$251$	1007.93	0.253466	0.126733	+	0.991937i	$0.459551\pi$
$252$	−760.875	−0.190201
$253$	−1417.09	−0.352140
$254$	−699.278	−0.172743
$255$	189.549	0.0465491
$256$	256.000	0.0625000
$257$	−3690.68	−0.895791	−0.447895	−	0.894086i	$-0.647826\pi$
$257$	−3690.68	−0.895791	−0.447895	+	0.894086i	$0.647826\pi$
$258$	7270.77	1.75449
$259$	−793.300	−0.190321
$260$	437.275	0.104302
$261$	−11212.7	−2.65919
$262$	825.728	0.194709
$263$	2881.53	0.675601	0.337800	−	0.941218i	$-0.390317\pi$
$263$	2881.53	0.675601	0.337800	+	0.941218i	$0.390317\pi$
$264$	−4487.70	−1.04621
$265$	−725.406	−0.168156
$266$	−317.363	−0.0731532
$267$	527.588	0.120928
$268$	−134.700	−0.0307019
$269$	2417.30	0.547902	0.273951	−	0.961744i	$-0.411669\pi$
$269$	2417.30	0.547902	0.273951	+	0.961744i	$0.411669\pi$
$270$	738.276	0.166407
$271$	2756.05	0.617779	0.308889	−	0.951098i	$-0.400043\pi$
$271$	2756.05	0.617779	0.308889	+	0.951098i	$0.400043\pi$
$272$	−237.400	−0.0529209
$273$	2414.02	0.535177
$274$	−4715.38	−1.03966
$275$	−7580.26	−1.66221
$276$	−837.631	−0.182679
$277$	−3117.08	−0.676126	−0.338063	−	0.941123i	$-0.609772\pi$
$277$	−3117.08	−0.676126	−0.338063	+	0.941123i	$0.609772\pi$
$278$	−171.903	−0.0370865
$279$	−1169.70	−0.250996
$280$	−38.2000	−0.00815318
$281$	2917.44	0.619360	0.309680	−	0.950841i	$-0.399778\pi$
$281$	2917.44	0.619360	0.309680	+	0.950841i	$0.399778\pi$
$282$	3464.25	0.731535
$283$	−6795.66	−1.42742	−0.713710	−	0.700441i	$-0.752987\pi$
$283$	−6795.66	−1.42742	−0.713710	+	0.700441i	$0.752987\pi$
$284$	−244.631	−0.0511134
$285$	595.674	0.123806
$286$	9600.57	1.98494
$287$	−1054.06	−0.216791
$288$	−1788.65	−0.365963
$289$	−4692.85	−0.955190
$290$	−562.938	−0.113989
$291$	14577.1	2.93651
$292$	−317.381	−0.0636073
$293$	−805.147	−0.160536	−0.0802682	−	0.996773i	$-0.525578\pi$
$293$	−805.147	−0.160536	−0.0802682	+	0.996773i	$0.525578\pi$
$294$	6034.93	1.19716
$295$	−699.387	−0.138034
$296$	−1864.87	−0.366195
$297$	16209.2	3.16684
$298$	1119.06	0.217535
$299$	1791.95	0.346593
$300$	−4480.64	−0.862300
$301$	1358.82	0.260204
$302$	−420.934	−0.0802055
$303$	3345.29	0.634264
$304$	−746.050	−0.140753
$305$	−1272.26	−0.238850
$306$	1658.69	0.309873
$307$	−4728.10	−0.878980	−0.439490	−	0.898247i	$-0.644841\pi$
$307$	−4728.10	−0.878980	−0.439490	+	0.898247i	$0.644841\pi$
$308$	−838.700	−0.155160
$309$	−9570.76	−1.76201
$310$	−58.7251	−0.0107592
$311$	5833.88	1.06369	0.531847	−	0.846840i	$-0.321498\pi$
$311$	5833.88	1.06369	0.531847	+	0.846840i	$0.321498\pi$
$312$	5674.84	1.02973
$313$	8024.33	1.44908	0.724540	−	0.689233i	$-0.242052\pi$
$313$	8024.33	1.44908	0.724540	+	0.689233i	$0.242052\pi$
$314$	873.469	0.156983
$315$	266.900	0.0477401
$316$	2135.66	0.380191
$317$	−4527.26	−0.802134	−0.401067	−	0.916049i	$-0.631360\pi$
$317$	−4527.26	−0.802134	−0.401067	+	0.916049i	$0.631360\pi$
$318$	−9414.13	−1.66012
$319$	−12359.6	−2.16929
$320$	−89.8000	−0.0156874
$321$	−10053.2	−1.74802
$322$	−156.544	−0.0270927
$323$	691.844	0.119180
$324$	3544.45	0.607759
$325$	9585.48	1.63602
$326$	2398.93	0.407560
$327$	−11514.6	−1.94728
$328$	−2477.86	−0.417125
$329$	647.428	0.108492
$330$	1574.20	0.262597
$331$	−935.443	−0.155337	−0.0776686	−	0.996979i	$-0.524748\pi$
$331$	−935.443	−0.155337	−0.0776686	+	0.996979i	$0.524748\pi$
$332$	3067.51	0.507083
$333$	13029.7	2.14422
$334$	−2777.16	−0.454969
$335$	47.2502	0.00770613
$336$	−495.750	−0.0804922
$337$	−1650.15	−0.266735	−0.133367	−	0.991067i	$-0.542579\pi$
$337$	−1650.15	−0.266735	−0.133367	+	0.991067i	$0.542579\pi$
$338$	−7746.23	−1.24657
$339$	−19139.5	−3.06642
$340$	83.2754	0.0132831
$341$	−1289.34	−0.204755
$342$	5212.59	0.824165
$343$	2295.13	0.361299
$344$	3194.30	0.500654
$345$	293.825	0.0458522
$346$	−5452.75	−0.847230
$347$	8343.39	1.29077	0.645384	−	0.763858i	$-0.276697\pi$
$347$	8343.39	1.29077	0.645384	+	0.763858i	$0.276697\pi$
$348$	−7305.66	−1.12536
$349$	6870.72	1.05381	0.526907	−	0.849923i	$-0.323351\pi$
$349$	6870.72	1.05381	0.526907	+	0.849923i	$0.323351\pi$
$350$	−837.381	−0.127885
$351$	−20497.0	−3.11695
$352$	−1971.60	−0.298542
$353$	3698.10	0.557592	0.278796	−	0.960350i	$-0.410065\pi$
$353$	3698.10	0.557592	0.278796	+	0.960350i	$0.410065\pi$
$354$	−9076.46	−1.36274
$355$	85.8121	0.0128294
$356$	231.787	0.0345076
$357$	459.730	0.0681555
$358$	−2395.94	−0.353713
$359$	−1539.79	−0.226371	−0.113185	−	0.993574i	$-0.536105\pi$
$359$	−1539.79	−0.226371	−0.113185	+	0.993574i	$0.536105\pi$
$360$	627.425	0.0918560
$361$	−4684.82	−0.683018
$362$	−972.768	−0.141236
$363$	22444.0	3.24519
$364$	1060.56	0.152716
$365$	111.331	0.0159653
$366$	−16511.0	−2.35804
$367$	−3516.58	−0.500174	−0.250087	−	0.968223i	$-0.580459\pi$
$367$	−3516.58	−0.500174	−0.250087	+	0.968223i	$0.580459\pi$
$368$	−368.000	−0.0521286
$369$	17312.6	2.44244
$370$	654.163	0.0919143
$371$	−1759.39	−0.246208
$372$	−762.119	−0.106220
$373$	190.852	0.0264932	0.0132466	−	0.999912i	$-0.495783\pi$
$373$	190.852	0.0264932	0.0132466	+	0.999912i	$0.495783\pi$
$374$	1828.35	0.252785
$375$	3168.60	0.436336
$376$	1521.96	0.208748
$377$	15629.1	2.13511
$378$	1790.61	0.243648
$379$	−9899.32	−1.34167	−0.670836	−	0.741606i	$-0.734065\pi$
$379$	−9899.32	−1.34167	−0.670836	+	0.741606i	$0.734065\pi$
$380$	261.700	0.0353288
$381$	3183.35	0.428053
$382$	4417.25	0.591639
$383$	12121.1	1.61713	0.808565	−	0.588407i	$-0.200245\pi$
$383$	12121.1	1.61713	0.808565	+	0.588407i	$0.200245\pi$
$384$	−1165.40	−0.154874
$385$	294.200	0.0389450
$386$	−1922.56	−0.253512
$387$	−22318.3	−2.93153
$388$	6404.22	0.837952
$389$	−2479.68	−0.323200	−0.161600	−	0.986856i	$-0.551665\pi$
$389$	−2479.68	−0.323200	−0.161600	+	0.986856i	$0.551665\pi$
$390$	−1990.63	−0.258460
$391$	341.262	0.0441391
$392$	2651.35	0.341616
$393$	−3759.00	−0.482484
$394$	10456.9	1.33709
$395$	−749.150	−0.0954274
$396$	13775.4	1.74808
$397$	4988.43	0.630635	0.315318	−	0.948986i	$-0.397889\pi$
$397$	4988.43	0.630635	0.315318	+	0.948986i	$0.397889\pi$
$398$	−7313.04	−0.921029
$399$	1444.74	0.181272
$400$	−1968.50	−0.246062
$401$	6200.67	0.772186	0.386093	−	0.922460i	$-0.373824\pi$
$401$	6200.67	0.772186	0.386093	+	0.922460i	$0.373824\pi$
$402$	613.200	0.0760787
$403$	1630.41	0.201529
$404$	1469.70	0.180991
$405$	−1243.33	−0.152547
$406$	−1365.34	−0.166899
$407$	14362.4	1.74919
$408$	1080.73	0.131137
$409$	−12275.7	−1.48410	−0.742049	−	0.670346i	$-0.766146\pi$
$409$	−12275.7	−1.48410	−0.742049	+	0.670346i	$0.766146\pi$
$410$	869.187	0.104698
$411$	21466.0	2.57626
$412$	−4204.76	−0.502800
$413$	−1696.29	−0.202104
$414$	2571.18	0.305234
$415$	−1076.03	−0.127277
$416$	2493.15	0.293838
$417$	782.561	0.0918998
$418$	5745.75	0.672330
$419$	7354.79	0.857530	0.428765	−	0.903416i	$-0.358949\pi$
$419$	7354.79	0.857530	0.428765	+	0.903416i	$0.358949\pi$
$420$	173.900	0.0202034
$421$	−2676.45	−0.309839	−0.154919	−	0.987927i	$-0.549512\pi$
$421$	−2676.45	−0.309839	−0.154919	+	0.987927i	$0.549512\pi$
$422$	5699.26	0.657431
$423$	−10633.8	−1.22230
$424$	−4135.95	−0.473725
$425$	1825.47	0.208350
$426$	1113.65	0.126658
$427$	−3085.72	−0.349715
$428$	−4416.71	−0.498808
$429$	−43705.1	−4.91865
$430$	−1120.50	−0.125663
$431$	−880.806	−0.0984383	−0.0492192	−	0.998788i	$-0.515673\pi$
$431$	−880.806	−0.0984383	−0.0492192	+	0.998788i	$0.515673\pi$
$432$	4209.32	0.468799
$433$	728.134	0.0808127	0.0404063	−	0.999183i	$-0.487135\pi$
$433$	728.134	0.0808127	0.0404063	+	0.999183i	$0.487135\pi$
$434$	−142.431	−0.0157533
$435$	2562.69	0.282463
$436$	−5058.76	−0.555667
$437$	1072.45	0.117396
$438$	1444.83	0.157618
$439$	−8143.35	−0.885332	−0.442666	−	0.896687i	$-0.645967\pi$
$439$	−8143.35	−0.885332	−0.442666	+	0.896687i	$0.645967\pi$
$440$	691.600	0.0749335
$441$	−18524.8	−2.00030
$442$	−2312.01	−0.248803
$443$	15146.3	1.62443	0.812213	−	0.583360i	$-0.198262\pi$
$443$	15146.3	1.62443	0.812213	+	0.583360i	$0.198262\pi$
$444$	8489.55	0.907424
$445$	−81.3066	−0.00866136
$446$	7849.12	0.833334
$447$	−5094.36	−0.539049
$448$	−217.800	−0.0229689
$449$	−9991.34	−1.05016	−0.525079	−	0.851054i	$-0.675964\pi$
$449$	−9991.34	−1.05016	−0.525079	+	0.851054i	$0.675964\pi$
$450$	13753.7	1.44079
$451$	19083.4	1.99247
$452$	−8408.65	−0.875021
$453$	1916.24	0.198748
$454$	5384.21	0.556593
$455$	−372.025	−0.0383315
$456$	3396.28	0.348783
$457$	9744.14	0.997400	0.498700	−	0.866775i	$-0.333811\pi$
$457$	9744.14	0.997400	0.498700	+	0.866775i	$0.333811\pi$
$458$	4453.27	0.454340
$459$	−3903.49	−0.396948
$460$	129.087	0.0130842
$461$	13262.0	1.33985	0.669925	−	0.742429i	$-0.266326\pi$
$461$	13262.0	1.33985	0.669925	+	0.742429i	$0.266326\pi$
$462$	3818.05	0.384484
$463$	−3249.19	−0.326139	−0.163070	−	0.986615i	$-0.552140\pi$
$463$	−3249.19	−0.326139	−0.163070	+	0.986615i	$0.552140\pi$
$464$	−3209.62	−0.321127
$465$	267.337	0.0266612
$466$	277.672	0.0276028
$467$	−15841.9	−1.56975	−0.784877	−	0.619652i	$-0.787274\pi$
$467$	−15841.9	−1.56975	−0.784877	+	0.619652i	$0.787274\pi$
$468$	−17419.4	−1.72054
$469$	114.600	0.0112830
$470$	−533.876	−0.0523954
$471$	−3976.33	−0.389001
$472$	−3987.60	−0.388865
$473$	−24601.1	−2.39146
$474$	−9722.27	−0.942107
$475$	5736.72	0.554144
$476$	201.975	0.0194486
$477$	28897.5	2.77385
$478$	−4330.75	−0.414401
$479$	10766.3	1.02698	0.513491	−	0.858095i	$-0.328352\pi$
$479$	10766.3	1.02698	0.513491	+	0.858095i	$0.328352\pi$
$480$	408.800	0.0388731
$481$	−18161.8	−1.72163
$482$	−8295.71	−0.783940
$483$	712.641	0.0671351
$484$	9860.40	0.926033
$485$	−2246.48	−0.210325
$486$	−1929.09	−0.180052
$487$	−18053.6	−1.67985	−0.839925	−	0.542703i	$-0.817401\pi$
$487$	−18053.6	−1.67985	−0.839925	+	0.542703i	$0.817401\pi$
$488$	−7253.85	−0.672882
$489$	−10920.8	−1.00993
$490$	−930.043	−0.0857451
$491$	−2531.32	−0.232662	−0.116331	−	0.993210i	$-0.537113\pi$
$491$	−2531.32	−0.232662	−0.116331	+	0.993210i	$0.537113\pi$
$492$	11280.1	1.03363
$493$	2976.42	0.271909
$494$	−7265.68	−0.661738
$495$	−4832.15	−0.438766
$496$	−334.825	−0.0303106
$497$	208.128	0.0187843
$498$	−13964.4	−1.25654
$499$	7554.37	0.677716	0.338858	−	0.940838i	$-0.389959\pi$
$499$	7554.37	0.677716	0.338858	+	0.940838i	$0.389959\pi$
$500$	1392.07	0.124511
$501$	12642.6	1.12740
$502$	−2015.86	−0.179227
$503$	−12465.0	−1.10495	−0.552474	−	0.833530i	$-0.686316\pi$
$503$	−12465.0	−1.10495	−0.552474	+	0.833530i	$0.686316\pi$
$504$	1521.75	0.134492
$505$	−515.543	−0.0454284
$506$	2834.17	0.249001
$507$	35263.5	3.08897
$508$	1398.56	0.122147
$509$	2161.55	0.188230	0.0941149	−	0.995561i	$-0.469998\pi$
$509$	2161.55	0.188230	0.0941149	+	0.995561i	$0.469998\pi$
$510$	−379.098	−0.0329152
$511$	270.022	0.0233759
$512$	−512.000	−0.0441942
$513$	−12267.1	−1.05576
$514$	7381.36	0.633420
$515$	1474.95	0.126202
$516$	−14541.5	−1.24061
$517$	−11721.5	−0.997119
$518$	1586.60	0.134578
$519$	24822.8	2.09942
$520$	−874.550	−0.0737530
$521$	−2880.90	−0.242255	−0.121127	−	0.992637i	$-0.538651\pi$
$521$	−2880.90	−0.242255	−0.121127	+	0.992637i	$0.538651\pi$
$522$	22425.4	1.88033
$523$	−16965.4	−1.41844	−0.709222	−	0.704985i	$-0.750954\pi$
$523$	−16965.4	−1.41844	−0.709222	+	0.704985i	$0.750954\pi$
$524$	−1651.46	−0.137680
$525$	3812.05	0.316898
$526$	−5763.07	−0.477722
$527$	310.498	0.0256651
$528$	8975.40	0.739781
$529$	529.000	0.0434783
$530$	1450.81	0.118904
$531$	27861.0	2.27696
$532$	634.725	0.0517271
$533$	−24131.6	−1.96108
$534$	−1055.18	−0.0855092
$535$	1549.30	0.125200
$536$	269.400	0.0217095
$537$	10907.1	0.876495
$538$	−4834.61	−0.387425
$539$	−20419.5	−1.63178
$540$	−1476.55	−0.117668
$541$	−2210.68	−0.175683	−0.0878417	−	0.996134i	$-0.527997\pi$
$541$	−2210.68	−0.175683	−0.0878417	+	0.996134i	$0.527997\pi$
$542$	−5512.10	−0.436836
$543$	4428.38	0.349981
$544$	474.800	0.0374207
$545$	1774.52	0.139472
$546$	−4828.04	−0.378427
$547$	12826.0	1.00256	0.501280	−	0.865285i	$-0.332863\pi$
$547$	12826.0	1.00256	0.501280	+	0.865285i	$0.332863\pi$
$548$	9430.76	0.735150
$549$	50682.0	3.93999
$550$	15160.5	1.17536
$551$	9353.67	0.723194
$552$	1675.26	0.129174
$553$	−1816.98	−0.139721
$554$	6234.15	0.478093
$555$	−2977.97	−0.227762
$556$	343.806	0.0262241
$557$	−13291.7	−1.01111	−0.505554	−	0.862795i	$-0.668712\pi$
$557$	−13291.7	−1.01111	−0.505554	+	0.862795i	$0.668712\pi$
$558$	2339.39	0.177481
$559$	31108.9	2.35378
$560$	76.4001	0.00576517
$561$	−8323.28	−0.626397
$562$	−5834.89	−0.437953
$563$	−6781.89	−0.507678	−0.253839	−	0.967246i	$-0.581693\pi$
$563$	−6781.89	−0.507678	−0.253839	+	0.967246i	$0.581693\pi$
$564$	−6928.49	−0.517273
$565$	2949.59	0.219629
$566$	13591.3	1.00934
$567$	−3015.55	−0.223353
$568$	489.263	0.0361426
$569$	−9189.69	−0.677068	−0.338534	−	0.940954i	$-0.609931\pi$
$569$	−9189.69	−0.677068	−0.338534	+	0.940954i	$0.609931\pi$
$570$	−1191.35	−0.0875441
$571$	−14790.2	−1.08398	−0.541988	−	0.840386i	$-0.682328\pi$
$571$	−14790.2	−1.08398	−0.541988	+	0.840386i	$0.682328\pi$
$572$	−19201.1	−1.40357
$573$	−20108.8	−1.46607
$574$	2108.12	0.153295
$575$	2829.72	0.205230
$576$	3577.30	0.258775
$577$	1346.56	0.0971540	0.0485770	−	0.998819i	$-0.484531\pi$
$577$	1346.56	0.0971540	0.0485770	+	0.998819i	$0.484531\pi$
$578$	9385.70	0.675421
$579$	8752.15	0.628198
$580$	1125.88	0.0806025
$581$	−2609.78	−0.186355
$582$	−29154.2	−2.07643
$583$	31853.3	2.26283
$584$	634.762	0.0449771
$585$	6110.40	0.431853
$586$	1610.29	0.113516
$587$	−5862.71	−0.412232	−0.206116	−	0.978528i	$-0.566082\pi$
$587$	−5862.71	−0.412232	−0.206116	+	0.978528i	$0.566082\pi$
$588$	−12069.9	−0.846518
$589$	975.766	0.0682610
$590$	1398.77	0.0976045
$591$	−47603.6	−3.31328
$592$	3729.75	0.258939
$593$	−2663.02	−0.184413	−0.0922066	−	0.995740i	$-0.529392\pi$
$593$	−2663.02	−0.184413	−0.0922066	+	0.995740i	$0.529392\pi$
$594$	−32418.4	−2.23930
$595$	−70.8491	−0.00488156
$596$	−2238.13	−0.153821
$597$	33291.5	2.28229
$598$	−3583.90	−0.245078
$599$	−16378.3	−1.11720	−0.558598	−	0.829439i	$-0.688660\pi$
$599$	−16378.3	−1.11720	−0.558598	+	0.829439i	$0.688660\pi$
$600$	8961.29	0.609738
$601$	11584.6	0.786268	0.393134	−	0.919481i	$-0.371391\pi$
$601$	11584.6	0.786268	0.393134	+	0.919481i	$0.371391\pi$
$602$	−2717.65	−0.183992
$603$	−1882.27	−0.127118
$604$	841.869	0.0567138
$605$	−3458.84	−0.232433
$606$	−6690.58	−0.448492
$607$	−20218.9	−1.35200	−0.675998	−	0.736904i	$-0.736287\pi$
$607$	−20218.9	−1.35200	−0.675998	+	0.736904i	$0.736287\pi$
$608$	1492.10	0.0995274
$609$	6215.51	0.413572
$610$	2544.51	0.168892
$611$	14822.2	0.981410
$612$	−3317.38	−0.219113
$613$	8024.50	0.528722	0.264361	−	0.964424i	$-0.414839\pi$
$613$	8024.50	0.528722	0.264361	+	0.964424i	$0.414839\pi$
$614$	9456.20	0.621533
$615$	−3956.84	−0.259439
$616$	1677.40	0.109715
$617$	8933.48	0.582899	0.291449	−	0.956586i	$-0.405863\pi$
$617$	8933.48	0.582899	0.291449	+	0.956586i	$0.405863\pi$
$618$	19141.5	1.24593
$619$	−10355.7	−0.672421	−0.336211	−	0.941787i	$-0.609145\pi$
$619$	−10355.7	−0.672421	−0.336211	+	0.941787i	$0.609145\pi$
$620$	117.450	0.00760793
$621$	−6050.90	−0.391005
$622$	−11667.8	−0.752146
$623$	−197.200	−0.0126816
$624$	−11349.7	−0.728126
$625$	14890.6	0.952998
$626$	−16048.7	−1.02465
$627$	−26156.6	−1.66602
$628$	−1746.94	−0.111004
$629$	−3458.76	−0.219252
$630$	−533.801	−0.0337574
$631$	19288.8	1.21692	0.608459	−	0.793586i	$-0.291788\pi$
$631$	19288.8	1.21692	0.608459	+	0.793586i	$0.291788\pi$
$632$	−4271.32	−0.268836
$633$	−25945.0	−1.62910
$634$	9054.52	0.567194
$635$	−490.587	−0.0306588
$636$	18828.3	1.17388
$637$	25821.1	1.60608
$638$	24719.1	1.53392
$639$	−3418.44	−0.211630
$640$	179.600	0.0110927
$641$	−12912.1	−0.795628	−0.397814	−	0.917466i	$-0.630231\pi$
$641$	−12912.1	−0.795628	−0.397814	+	0.917466i	$0.630231\pi$
$642$	20106.4	1.23604
$643$	−30943.4	−1.89781	−0.948903	−	0.315568i	$-0.897805\pi$
$643$	−30943.4	−1.89781	−0.948903	+	0.315568i	$0.897805\pi$
$644$	313.087	0.0191574
$645$	5100.90	0.311392
$646$	−1383.69	−0.0842732
$647$	15938.8	0.968499	0.484250	−	0.874930i	$-0.339093\pi$
$647$	15938.8	0.968499	0.484250	+	0.874930i	$0.339093\pi$
$648$	−7088.90	−0.429751
$649$	30710.7	1.85748
$650$	−19171.0	−1.15684
$651$	648.396	0.0390363
$652$	−4797.87	−0.288189
$653$	3395.93	0.203512	0.101756	−	0.994809i	$-0.467554\pi$
$653$	3395.93	0.203512	0.101756	+	0.994809i	$0.467554\pi$
$654$	23029.2	1.37693
$655$	579.300	0.0345574
$656$	4955.72	0.294952
$657$	−4435.03	−0.263359
$658$	−1294.86	−0.0767155
$659$	−15030.2	−0.888460	−0.444230	−	0.895913i	$-0.646523\pi$
$659$	−15030.2	−0.888460	−0.444230	+	0.895913i	$0.646523\pi$
$660$	−3148.40	−0.185684
$661$	−15082.3	−0.887496	−0.443748	−	0.896152i	$-0.646351\pi$
$661$	−15082.3	−0.887496	−0.443748	+	0.896152i	$0.646351\pi$
$662$	1870.89	0.109840
$663$	10525.0	0.616529
$664$	−6135.02	−0.358562
$665$	−222.650	−0.0129834
$666$	−26059.4	−1.51619
$667$	4613.84	0.267839
$668$	5554.32	0.321711
$669$	−35731.9	−2.06499
$670$	−94.5004	−0.00544906
$671$	55866.0	3.21413
$672$	991.500	0.0569166
$673$	32456.3	1.85899	0.929493	−	0.368839i	$-0.120245\pi$
$673$	32456.3	1.85899	0.929493	+	0.368839i	$0.120245\pi$
$674$	3300.31	0.188610
$675$	−32367.4	−1.84566
$676$	15492.5	0.881455
$677$	−34072.6	−1.93429	−0.967145	−	0.254224i	$-0.918180\pi$
$677$	−34072.6	−1.93429	−0.967145	+	0.254224i	$0.918180\pi$
$678$	38279.1	2.16829
$679$	−5448.59	−0.307950
$680$	−166.551	−0.00939254
$681$	−24510.7	−1.37923
$682$	2578.67	0.144784
$683$	−20945.9	−1.17346	−0.586729	−	0.809784i	$-0.699584\pi$
$683$	−20945.9	−1.17346	−0.586729	+	0.809784i	$0.699584\pi$
$684$	−10425.2	−0.582773
$685$	−3308.13	−0.184522
$686$	−4590.26	−0.255477
$687$	−20272.8	−1.12585
$688$	−6388.60	−0.354016
$689$	−40279.5	−2.22718
$690$	−587.650	−0.0324224
$691$	−34095.6	−1.87707	−0.938536	−	0.345182i	$-0.887817\pi$
$691$	−34095.6	−1.87707	−0.938536	+	0.345182i	$0.887817\pi$
$692$	10905.5	0.599082
$693$	−11719.8	−0.642425
$694$	−16686.8	−0.912711
$695$	−120.601	−0.00658222
$696$	14611.3	0.795748
$697$	−4595.66	−0.249746
$698$	−13741.4	−0.745160
$699$	−1264.06	−0.0683993
$700$	1674.76	0.0904287
$701$	26916.2	1.45023	0.725115	−	0.688628i	$-0.241786\pi$
$701$	26916.2	1.45023	0.725115	+	0.688628i	$0.241786\pi$
$702$	40994.0	2.20402
$703$	−10869.5	−0.583142
$704$	3943.20	0.211101
$705$	2430.38	0.129835
$706$	−7396.20	−0.394277
$707$	−1250.39	−0.0665147
$708$	18152.9	0.963600
$709$	−15391.4	−0.815282	−0.407641	−	0.913142i	$-0.633649\pi$
$709$	−15391.4	−0.815282	−0.407641	+	0.913142i	$0.633649\pi$
$710$	−171.624	−0.00907174
$711$	29843.4	1.57414
$712$	−463.575	−0.0244006
$713$	481.311	0.0252808
$714$	−919.461	−0.0481932
$715$	6735.40	0.352293
$716$	4791.88	0.250113
$717$	19715.0	1.02688
$718$	3079.58	0.160068
$719$	16237.9	0.842240	0.421120	−	0.907005i	$-0.361637\pi$
$719$	16237.9	0.842240	0.421120	+	0.907005i	$0.361637\pi$
$720$	−1254.85	−0.0649520
$721$	3577.33	0.184781
$722$	9369.64	0.482966
$723$	37764.9	1.94259
$724$	1945.54	0.0998692
$725$	24680.3	1.26428
$726$	−44887.9	−2.29469
$727$	13661.2	0.696925	0.348462	−	0.937323i	$-0.386704\pi$
$727$	13661.2	0.696925	0.348462	+	0.937323i	$0.386704\pi$
$728$	−2121.12	−0.107986
$729$	−15143.2	−0.769353
$730$	−222.663	−0.0112892
$731$	5924.42	0.299758
$732$	33022.0	1.66739
$733$	12222.7	0.615900	0.307950	−	0.951403i	$-0.400357\pi$
$733$	12222.7	0.615900	0.307950	+	0.951403i	$0.400357\pi$
$734$	7033.16	0.353676
$735$	4233.88	0.212475
$736$	736.000	0.0368605
$737$	−2074.80	−0.103699
$738$	−34625.2	−1.72706
$739$	10840.1	0.539594	0.269797	−	0.962917i	$-0.413043\pi$
$739$	10840.1	0.539594	0.269797	+	0.962917i	$0.413043\pi$
$740$	−1308.33	−0.0649932
$741$	33075.9	1.63977
$742$	3518.79	0.174095
$743$	11839.8	0.584603	0.292302	−	0.956326i	$-0.405579\pi$
$743$	11839.8	0.584603	0.292302	+	0.956326i	$0.405579\pi$
$744$	1524.24	0.0751092
$745$	785.092	0.0386088
$746$	−381.705	−0.0187335
$747$	42864.9	2.09952
$748$	−3656.70	−0.178746
$749$	3757.66	0.183313
$750$	−6337.20	−0.308536
$751$	24428.6	1.18697	0.593484	−	0.804846i	$-0.297752\pi$
$751$	24428.6	1.18697	0.593484	+	0.804846i	$0.297752\pi$
$752$	−3043.92	−0.147607
$753$	9176.87	0.444122
$754$	−31258.1	−1.50975
$755$	−295.312	−0.0142351
$756$	−3581.21	−0.172285
$757$	22284.6	1.06994	0.534972	−	0.844870i	$-0.320322\pi$
$757$	22284.6	1.06994	0.534972	+	0.844870i	$0.320322\pi$
$758$	19798.6	0.948706
$759$	−12902.1	−0.617020
$760$	−523.400	−0.0249812
$761$	−5150.71	−0.245352	−0.122676	−	0.992447i	$-0.539148\pi$
$761$	−5150.71	−0.245352	−0.122676	+	0.992447i	$0.539148\pi$
$762$	−6366.71	−0.302679
$763$	4303.90	0.204209
$764$	−8834.50	−0.418352
$765$	1163.68	0.0549971
$766$	−24242.3	−1.14348
$767$	−38834.7	−1.82821
$768$	2330.80	0.109512
$769$	20286.3	0.951289	0.475645	−	0.879638i	$-0.342215\pi$
$769$	20286.3	0.951289	0.475645	+	0.879638i	$0.342215\pi$
$770$	−588.400	−0.0275383
$771$	−33602.5	−1.56960
$772$	3845.12	0.179260
$773$	−5981.92	−0.278337	−0.139169	−	0.990269i	$-0.544443\pi$
$773$	−5981.92	−0.278337	−0.139169	+	0.990269i	$0.544443\pi$
$774$	44636.6	2.07291
$775$	2574.62	0.119333
$776$	−12808.4	−0.592521
$777$	−7222.75	−0.333481
$778$	4959.37	0.228537
$779$	−14442.3	−0.664246
$780$	3981.25	0.182759
$781$	−3768.09	−0.172641
$782$	−682.525	−0.0312110
$783$	−52774.8	−2.40871
$784$	−5302.70	−0.241559
$785$	612.793	0.0278618
$786$	7518.00	0.341168
$787$	−1118.63	−0.0506667	−0.0253333	−	0.999679i	$-0.508065\pi$
$787$	−1118.63	−0.0506667	−0.0253333	+	0.999679i	$0.508065\pi$
$788$	−20913.9	−0.945465
$789$	26235.5	1.18379
$790$	1498.30	0.0674774
$791$	7153.92	0.321573
$792$	−27550.8	−1.23608
$793$	−70644.3	−3.16350
$794$	−9976.86	−0.445926
$795$	−6604.60	−0.294643
$796$	14626.1	0.651266
$797$	30098.0	1.33768	0.668838	−	0.743409i	$-0.266792\pi$
$797$	30098.0	1.33768	0.668838	+	0.743409i	$0.266792\pi$
$798$	−2889.49	−0.128179
$799$	2822.76	0.124984
$800$	3937.00	0.173992
$801$	3238.96	0.142875
$802$	−12401.3	−0.546018
$803$	−4888.66	−0.214841
$804$	−1226.40	−0.0537958
$805$	−109.825	−0.00480848
$806$	−3260.82	−0.142503
$807$	22008.8	0.960033
$808$	−2939.40	−0.127980
$809$	730.186	0.0317330	0.0158665	−	0.999874i	$-0.494949\pi$
$809$	730.186	0.0317330	0.0158665	+	0.999874i	$0.494949\pi$
$810$	2486.65	0.107867
$811$	−34542.4	−1.49562	−0.747810	−	0.663912i	$-0.768895\pi$
$811$	−34542.4	−1.49562	−0.747810	+	0.663912i	$0.768895\pi$
$812$	2730.69	0.118015
$813$	25093.0	1.08247
$814$	−28724.9	−1.23686
$815$	1683.00	0.0723349
$816$	−2161.45	−0.0927279
$817$	18618.0	0.797261
$818$	24551.5	1.04942
$819$	14820.1	0.632304
$820$	−1738.37	−0.0740325
$821$	25416.3	1.08043	0.540217	−	0.841526i	$-0.318342\pi$
$821$	25416.3	1.08043	0.540217	+	0.841526i	$0.318342\pi$
$822$	−42932.1	−1.82169
$823$	−36849.3	−1.56073	−0.780367	−	0.625322i	$-0.784968\pi$
$823$	−36849.3	−1.56073	−0.780367	+	0.625322i	$0.784968\pi$
$824$	8409.52	0.355534
$825$	−69015.9	−2.91252
$826$	3392.57	0.142909
$827$	8488.52	0.356922	0.178461	−	0.983947i	$-0.442888\pi$
$827$	8488.52	0.356922	0.178461	+	0.983947i	$0.442888\pi$
$828$	−5142.37	−0.215833
$829$	−17502.1	−0.733260	−0.366630	−	0.930367i	$-0.619488\pi$
$829$	−17502.1	−0.733260	−0.366630	+	0.930367i	$0.619488\pi$
$830$	2152.05	0.0899985
$831$	−28380.0	−1.18471
$832$	−4986.30	−0.207775
$833$	4917.42	0.204536
$834$	−1565.12	−0.0649829
$835$	−1948.35	−0.0807491
$836$	−11491.5	−0.475409
$837$	−5505.42	−0.227354
$838$	−14709.6	−0.606366
$839$	−6056.79	−0.249229	−0.124615	−	0.992205i	$-0.539769\pi$
$839$	−6056.79	−0.249229	−0.124615	+	0.992205i	$0.539769\pi$
$840$	−347.799	−0.0142860
$841$	15852.0	0.649964
$842$	5352.90	0.219089
$843$	26562.4	1.08524
$844$	−11398.5	−0.464874
$845$	−5434.46	−0.221244
$846$	21267.6	0.864298
$847$	−8389.04	−0.340320
$848$	8271.90	0.334974
$849$	−61872.3	−2.50112
$850$	−3650.95	−0.147325
$851$	−5361.52	−0.215970
$852$	−2227.29	−0.0895608
$853$	30466.7	1.22293	0.611465	−	0.791271i	$-0.290580\pi$
$853$	30466.7	1.22293	0.611465	+	0.791271i	$0.290580\pi$
$854$	6171.44	0.247286
$855$	3656.95	0.146275
$856$	8833.43	0.352711
$857$	42451.4	1.69208	0.846039	−	0.533121i	$-0.178981\pi$
$857$	42451.4	1.69208	0.846039	+	0.533121i	$0.178981\pi$
$858$	87410.2	3.47801
$859$	23261.8	0.923960	0.461980	−	0.886890i	$-0.347139\pi$
$859$	23261.8	0.923960	0.461980	+	0.886890i	$0.347139\pi$
$860$	2241.00	0.0888575
$861$	−9596.88	−0.379861
$862$	1761.61	0.0696064
$863$	37167.9	1.46606	0.733031	−	0.680196i	$-0.238105\pi$
$863$	37167.9	1.46606	0.733031	+	0.680196i	$0.238105\pi$
$864$	−8418.65	−0.331491
$865$	−3825.44	−0.150369
$866$	−1456.27	−0.0571432
$867$	−42726.9	−1.67368
$868$	284.863	0.0111393
$869$	32895.9	1.28414
$870$	−5125.37	−0.199731
$871$	2623.65	0.102065
$872$	10117.5	0.392916
$873$	89491.5	3.46945
$874$	−2144.89	−0.0830116
$875$	−1184.35	−0.0457581
$876$	−2889.66	−0.111453
$877$	−47613.3	−1.83328	−0.916641	−	0.399712i	$-0.869110\pi$
$877$	−47613.3	−1.83328	−0.916641	+	0.399712i	$0.869110\pi$
$878$	16286.7	0.626024
$879$	−7330.61	−0.281292
$880$	−1383.20	−0.0529860
$881$	−8298.27	−0.317339	−0.158670	−	0.987332i	$-0.550720\pi$
$881$	−8298.27	−0.317339	−0.158670	+	0.987332i	$0.550720\pi$
$882$	37049.5	1.41442
$883$	−24325.8	−0.927099	−0.463549	−	0.886071i	$-0.653424\pi$
$883$	−24325.8	−0.927099	−0.463549	+	0.886071i	$0.653424\pi$
$884$	4624.01	0.175930
$885$	−6367.70	−0.241862
$886$	−30292.5	−1.14864
$887$	−21964.2	−0.831439	−0.415720	−	0.909493i	$-0.636470\pi$
$887$	−21964.2	−0.831439	−0.415720	+	0.909493i	$0.636470\pi$
$888$	−16979.1	−0.641646
$889$	−1189.87	−0.0448895
$890$	162.613	0.00612450
$891$	54595.6	2.05277
$892$	−15698.2	−0.589256
$893$	8870.78	0.332418
$894$	10188.7	0.381165
$895$	−1680.90	−0.0627780
$896$	435.600	0.0162415
$897$	16315.2	0.607299
$898$	19982.7	0.742573
$899$	4197.90	0.155737
$900$	−27507.5	−1.01880
$901$	−7670.89	−0.283634
$902$	−38166.8	−1.40889
$903$	12371.7	0.455929
$904$	16817.3	0.618733
$905$	−682.458	−0.0250670
$906$	−3832.48	−0.140536
$907$	−4228.93	−0.154817	−0.0774086	−	0.996999i	$-0.524665\pi$
$907$	−4228.93	−0.154817	−0.0774086	+	0.996999i	$0.524665\pi$
$908$	−10768.4	−0.393571
$909$	20537.3	0.749373
$910$	744.050	0.0271044
$911$	−40118.0	−1.45902	−0.729510	−	0.683970i	$-0.760252\pi$
$911$	−40118.0	−1.45902	−0.729510	+	0.683970i	$0.760252\pi$
$912$	−6792.55	−0.246627
$913$	47249.3	1.71273
$914$	−19488.3	−0.705268
$915$	−11583.5	−0.418512
$916$	−8906.54	−0.321267
$917$	1405.03	0.0505977
$918$	7806.98	0.280685
$919$	38199.4	1.37114	0.685572	−	0.728005i	$-0.259552\pi$
$919$	38199.4	1.37114	0.685572	+	0.728005i	$0.259552\pi$
$920$	−258.175	−0.00925193
$921$	−43047.9	−1.54015
$922$	−26523.9	−0.947417
$923$	4764.87	0.169921
$924$	−7636.10	−0.271871
$925$	−28679.7	−1.01944
$926$	6498.38	0.230615
$927$	−58756.6	−2.08179
$928$	6419.25	0.227071
$929$	13530.5	0.477848	0.238924	−	0.971038i	$-0.423205\pi$
$929$	13530.5	0.477848	0.238924	+	0.971038i	$0.423205\pi$
$930$	−534.674	−0.0188523
$931$	15453.4	0.544002
$932$	−555.344	−0.0195181
$933$	53115.6	1.86380
$934$	31683.8	1.10998
$935$	1282.70	0.0448650
$936$	34838.8	1.21661
$937$	34004.9	1.18558	0.592792	−	0.805356i	$-0.298026\pi$
$937$	34004.9	1.18558	0.592792	+	0.805356i	$0.298026\pi$
$938$	−229.200	−0.00797831
$939$	73059.0	2.53907
$940$	1067.75	0.0370491
$941$	40271.5	1.39513	0.697563	−	0.716523i	$-0.254268\pi$
$941$	40271.5	1.39513	0.697563	+	0.716523i	$0.254268\pi$
$942$	7952.66	0.275065
$943$	−7123.85	−0.246007
$944$	7975.20	0.274969
$945$	1256.22	0.0432433
$946$	49202.2	1.69102
$947$	−30897.3	−1.06022	−0.530109	−	0.847930i	$-0.677849\pi$
$947$	−30897.3	−1.06022	−0.530109	+	0.847930i	$0.677849\pi$
$948$	19444.5	0.666170
$949$	6181.87	0.211456
$950$	−11473.4	−0.391839
$951$	−41219.3	−1.40550
$952$	−403.951	−0.0137522
$953$	−34614.8	−1.17658	−0.588291	−	0.808649i	$-0.700199\pi$
$953$	−34614.8	−1.17658	−0.588291	+	0.808649i	$0.700199\pi$
$954$	−57795.1	−1.96141
$955$	3098.98	0.105006
$956$	8661.49	0.293026
$957$	−112530.	−3.80102
$958$	−21532.6	−0.726185
$959$	−8023.51	−0.270170
$960$	−817.600	−0.0274874
$961$	−29353.1	−0.985300
$962$	36323.5	1.21738
$963$	−61718.4	−2.06526
$964$	16591.4	0.554329
$965$	−1348.79	−0.0449940
$966$	−1425.28	−0.0474717
$967$	13078.3	0.434923	0.217461	−	0.976069i	$-0.430222\pi$
$967$	13078.3	0.434923	0.217461	+	0.976069i	$0.430222\pi$
$968$	−19720.8	−0.654804
$969$	6299.03	0.208827
$970$	4492.96	0.148722
$971$	11408.6	0.377055	0.188527	−	0.982068i	$-0.439629\pi$
$971$	11408.6	0.377055	0.188527	+	0.982068i	$0.439629\pi$
$972$	3858.17	0.127316
$973$	−292.504	−0.00963745
$974$	36107.2	1.18783
$975$	87272.8	2.86663
$976$	14507.7	0.475799
$977$	33058.5	1.08253	0.541267	−	0.840851i	$-0.317945\pi$
$977$	33058.5	1.08253	0.541267	+	0.840851i	$0.317945\pi$
$978$	21841.5	0.714126
$979$	3570.25	0.116553
$980$	1860.09	0.0606309
$981$	−70690.3	−2.30068
$982$	5062.65	0.164517
$983$	26380.8	0.855967	0.427984	−	0.903787i	$-0.359224\pi$
$983$	26380.8	0.855967	0.427984	+	0.903787i	$0.359224\pi$
$984$	−22560.2	−0.730886
$985$	7336.19	0.237310
$986$	−5952.85	−0.192269
$987$	5894.63	0.190099
$988$	14531.4	0.467919
$989$	9183.61	0.295270
$990$	9664.30	0.310254
$991$	−46295.9	−1.48399	−0.741997	−	0.670403i	$-0.766121\pi$
$991$	−46295.9	−1.48399	−0.741997	+	0.670403i	$0.766121\pi$
$992$	669.650	0.0214329
$993$	−8516.91	−0.272181
$994$	−416.256	−0.0132825
$995$	−5130.55	−0.163467
$996$	27928.7	0.888510
$997$	−35436.0	−1.12565	−0.562824	−	0.826577i	$-0.690285\pi$
$997$	−35436.0	−1.12565	−0.562824	+	0.826577i	$0.690285\pi$
$998$	−15108.7	−0.479217
$999$	61327.1	1.94224

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	46.4.a.c.1.2	✓	2
3.2	odd	2		414.4.a.j.1.2		2
4.3	odd	2		368.4.a.g.1.1		2
5.2	odd	4		1150.4.b.i.599.1		4
5.3	odd	4		1150.4.b.i.599.4		4
5.4	even	2		1150.4.a.k.1.1		2
7.6	odd	2		2254.4.a.d.1.1		2
8.3	odd	2		1472.4.a.l.1.2		2
8.5	even	2		1472.4.a.m.1.1		2
23.22	odd	2		1058.4.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.4.a.c.1.2	✓	2	1.1	even	1	trivial
368.4.a.g.1.1		2	4.3	odd	2
414.4.a.j.1.2		2	3.2	odd	2
1058.4.a.f.1.2		2	23.22	odd	2
1150.4.a.k.1.1		2	5.4	even	2
1150.4.b.i.599.1		4	5.2	odd	4
1150.4.b.i.599.4		4	5.3	odd	4
1472.4.a.l.1.2		2	8.3	odd	2
1472.4.a.m.1.1		2	8.5	even	2
2254.4.a.d.1.1		2	7.6	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:	\( N \)	\(=\)	\( 46 = 2 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	46.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +9.10469 q^{3} +4.00000 q^{4} -1.40312 q^{5} -18.2094 q^{6} -3.40312 q^{7} -8.00000 q^{8} +55.8953 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +9.10469 q^{3} +4.00000 q^{4} -1.40312 q^{5} -18.2094 q^{6} -3.40312 q^{7} -8.00000 q^{8} +55.8953 q^{9} +2.80625 q^{10} +61.6125 q^{11} +36.4187 q^{12} -77.9109 q^{13} +6.80625 q^{14} -12.7750 q^{15} +16.0000 q^{16} -14.8375 q^{17} -111.791 q^{18} -46.6281 q^{19} -5.61250 q^{20} -30.9844 q^{21} -123.225 q^{22} -23.0000 q^{23} -72.8375 q^{24} -123.031 q^{25} +155.822 q^{26} +263.083 q^{27} -13.6125 q^{28} -200.602 q^{29} +25.5500 q^{30} -20.9266 q^{31} -32.0000 q^{32} +560.962 q^{33} +29.6750 q^{34} +4.77501 q^{35} +223.581 q^{36} +233.109 q^{37} +93.2562 q^{38} -709.355 q^{39} +11.2250 q^{40} +309.733 q^{41} +61.9688 q^{42} -399.287 q^{43} +246.450 q^{44} -78.4281 q^{45} +46.0000 q^{46} -190.245 q^{47} +145.675 q^{48} -331.419 q^{49} +246.062 q^{50} -135.091 q^{51} -311.644 q^{52} +516.994 q^{53} -526.166 q^{54} -86.4500 q^{55} +27.2250 q^{56} -424.534 q^{57} +401.203 q^{58} +498.450 q^{59} -51.1000 q^{60} +906.731 q^{61} +41.8531 q^{62} -190.219 q^{63} +64.0000 q^{64} +109.319 q^{65} -1121.92 q^{66} -33.6750 q^{67} -59.3500 q^{68} -209.408 q^{69} -9.55001 q^{70} -61.1578 q^{71} -447.163 q^{72} -79.3453 q^{73} -466.219 q^{74} -1120.16 q^{75} -186.512 q^{76} -209.675 q^{77} +1418.71 q^{78} +533.916 q^{79} -22.4500 q^{80} +886.113 q^{81} -619.466 q^{82} +766.878 q^{83} -123.938 q^{84} +20.8188 q^{85} +798.575 q^{86} -1826.41 q^{87} -492.900 q^{88} +57.9468 q^{89} +156.856 q^{90} +265.141 q^{91} -92.0000 q^{92} -190.530 q^{93} +380.491 q^{94} +65.4250 q^{95} -291.350 q^{96} +1601.06 q^{97} +662.837 q^{98} +3443.85 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - q^{3} + 8 q^{4} + 10 q^{5} + 2 q^{6} + 6 q^{7} - 16 q^{8} + 131 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 - q^3 + 8 * q^4 + 10 * q^5 + 2 * q^6 + 6 * q^7 - 16 * q^8 + 131 * q^9	\( 2 q - 4 q^{2} - q^{3} + 8 q^{4} + 10 q^{5} + 2 q^{6} + 6 q^{7} - 16 q^{8} + 131 q^{9} - 20 q^{10} + 72 q^{11} - 4 q^{12} - 111 q^{13} - 12 q^{14} - 128 q^{15} + 32 q^{16} + 124 q^{17} - 262 q^{18} + 22 q^{19} + 40 q^{20} - 126 q^{21} - 144 q^{22} - 46 q^{23} + 8 q^{24} - 118 q^{25} + 222 q^{26} - 223 q^{27} + 24 q^{28} + 15 q^{29} + 256 q^{30} + 67 q^{31} - 64 q^{32} + 456 q^{33} - 248 q^{34} + 112 q^{35} + 524 q^{36} + 18 q^{37} - 44 q^{38} - 375 q^{39} - 80 q^{40} + 485 q^{41} + 252 q^{42} - 440 q^{43} + 288 q^{44} + 778 q^{45} + 92 q^{46} + 215 q^{47} - 16 q^{48} - 586 q^{49} + 236 q^{50} - 1538 q^{51} - 444 q^{52} + 240 q^{53} + 446 q^{54} + 32 q^{55} - 48 q^{56} - 1118 q^{57} - 30 q^{58} + 792 q^{59} - 512 q^{60} + 456 q^{61} - 134 q^{62} + 516 q^{63} + 128 q^{64} - 268 q^{65} - 912 q^{66} + 240 q^{67} + 496 q^{68} + 23 q^{69} - 224 q^{70} - 705 q^{71} - 1048 q^{72} + 27 q^{73} - 36 q^{74} - 1171 q^{75} + 88 q^{76} - 112 q^{77} + 750 q^{78} + 594 q^{79} + 160 q^{80} + 3770 q^{81} - 970 q^{82} + 394 q^{83} - 504 q^{84} + 1604 q^{85} + 880 q^{86} - 4005 q^{87} - 576 q^{88} - 486 q^{89} - 1556 q^{90} - 46 q^{91} - 184 q^{92} - 1079 q^{93} - 430 q^{94} + 848 q^{95} + 32 q^{96} + 2152 q^{97} + 1172 q^{98} + 4224 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - q^3 + 8 * q^4 + 10 * q^5 + 2 * q^6 + 6 * q^7 - 16 * q^8 + 131 * q^9 - 20 * q^10 + 72 * q^11 - 4 * q^12 - 111 * q^13 - 12 * q^14 - 128 * q^15 + 32 * q^16 + 124 * q^17 - 262 * q^18 + 22 * q^19 + 40 * q^20 - 126 * q^21 - 144 * q^22 - 46 * q^23 + 8 * q^24 - 118 * q^25 + 222 * q^26 - 223 * q^27 + 24 * q^28 + 15 * q^29 + 256 * q^30 + 67 * q^31 - 64 * q^32 + 456 * q^33 - 248 * q^34 + 112 * q^35 + 524 * q^36 + 18 * q^37 - 44 * q^38 - 375 * q^39 - 80 * q^40 + 485 * q^41 + 252 * q^42 - 440 * q^43 + 288 * q^44 + 778 * q^45 + 92 * q^46 + 215 * q^47 - 16 * q^48 - 586 * q^49 + 236 * q^50 - 1538 * q^51 - 444 * q^52 + 240 * q^53 + 446 * q^54 + 32 * q^55 - 48 * q^56 - 1118 * q^57 - 30 * q^58 + 792 * q^59 - 512 * q^60 + 456 * q^61 - 134 * q^62 + 516 * q^63 + 128 * q^64 - 268 * q^65 - 912 * q^66 + 240 * q^67 + 496 * q^68 + 23 * q^69 - 224 * q^70 - 705 * q^71 - 1048 * q^72 + 27 * q^73 - 36 * q^74 - 1171 * q^75 + 88 * q^76 - 112 * q^77 + 750 * q^78 + 594 * q^79 + 160 * q^80 + 3770 * q^81 - 970 * q^82 + 394 * q^83 - 504 * q^84 + 1604 * q^85 + 880 * q^86 - 4005 * q^87 - 576 * q^88 - 486 * q^89 - 1556 * q^90 - 46 * q^91 - 184 * q^92 - 1079 * q^93 - 430 * q^94 + 848 * q^95 + 32 * q^96 + 2152 * q^97 + 1172 * q^98 + 4224 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more