LMFDB - Embedded newform 2015.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2015,4,Mod(1,2015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2015, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2015.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2015 = 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2015.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:	$118.888848662$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.37200 q^{2} +7.54523 q^{3} +20.8584 q^{4} -5.00000 q^{5} -40.5329 q^{6} -10.2525 q^{7} -69.0752 q^{8} +29.9304 q^{9} +O(q^{10})$	\(q-5.37200 q^{2} +7.54523 q^{3} +20.8584 q^{4} -5.00000 q^{5} -40.5329 q^{6} -10.2525 q^{7} -69.0752 q^{8} +29.9304 q^{9} +26.8600 q^{10} -20.0035 q^{11} +157.381 q^{12} +13.0000 q^{13} +55.0766 q^{14} -37.7261 q^{15} +204.205 q^{16} +57.8944 q^{17} -160.786 q^{18} -1.10627 q^{19} -104.292 q^{20} -77.3577 q^{21} +107.459 q^{22} +138.257 q^{23} -521.188 q^{24} +25.0000 q^{25} -69.8360 q^{26} +22.1107 q^{27} -213.851 q^{28} -146.040 q^{29} +202.665 q^{30} -31.0000 q^{31} -544.387 q^{32} -150.931 q^{33} -311.009 q^{34} +51.2627 q^{35} +624.300 q^{36} -358.983 q^{37} +5.94289 q^{38} +98.0879 q^{39} +345.376 q^{40} +108.680 q^{41} +415.565 q^{42} +148.439 q^{43} -417.241 q^{44} -149.652 q^{45} -742.716 q^{46} -257.742 q^{47} +1540.77 q^{48} -237.886 q^{49} -134.300 q^{50} +436.826 q^{51} +271.159 q^{52} +507.082 q^{53} -118.779 q^{54} +100.018 q^{55} +708.196 q^{56} -8.34707 q^{57} +784.528 q^{58} +406.129 q^{59} -786.906 q^{60} -108.561 q^{61} +166.532 q^{62} -306.863 q^{63} +1290.81 q^{64} -65.0000 q^{65} +810.802 q^{66} -476.610 q^{67} +1207.58 q^{68} +1043.18 q^{69} -275.383 q^{70} +468.068 q^{71} -2067.45 q^{72} +119.783 q^{73} +1928.45 q^{74} +188.631 q^{75} -23.0750 q^{76} +205.087 q^{77} -526.928 q^{78} -432.194 q^{79} -1021.02 q^{80} -641.291 q^{81} -583.827 q^{82} +557.434 q^{83} -1613.56 q^{84} -289.472 q^{85} -797.415 q^{86} -1101.91 q^{87} +1381.75 q^{88} +506.942 q^{89} +803.931 q^{90} -133.283 q^{91} +2883.82 q^{92} -233.902 q^{93} +1384.59 q^{94} +5.53136 q^{95} -4107.52 q^{96} +169.351 q^{97} +1277.92 q^{98} -598.715 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 9 q^{2} + q^{3} + 149 q^{4} - 200 q^{5} + 11 q^{6} - 64 q^{7} - 87 q^{8} + 247 q^{9}+O(q^{10}) $ 40 * q - 9 * q^2 + q^3 + 149 * q^4 - 200 * q^5 + 11 * q^6 - 64 * q^7 - 87 * q^8 + 247 * q^9	\( 40 q - 9 q^{2} + q^{3} + 149 q^{4} - 200 q^{5} + 11 q^{6} - 64 q^{7} - 87 q^{8} + 247 q^{9} + 45 q^{10} - q^{11} - 226 q^{12} + 520 q^{13} + 138 q^{14} - 5 q^{15} + 413 q^{16} + 6 q^{17} - 306 q^{18} - 265 q^{19} - 745 q^{20} - 344 q^{21} - 313 q^{22} + 24 q^{23} - 321 q^{24} + 1000 q^{25} - 117 q^{26} + 28 q^{27} - 675 q^{28} - 9 q^{29} - 55 q^{30} - 1240 q^{31} - 160 q^{32} - 986 q^{33} - 298 q^{34} + 320 q^{35} + 238 q^{36} - 1403 q^{37} - 314 q^{38} + 13 q^{39} + 435 q^{40} - 84 q^{41} + 744 q^{42} - 421 q^{43} + 812 q^{44} - 1235 q^{45} - 527 q^{46} - 460 q^{47} - 1573 q^{48} + 748 q^{49} - 225 q^{50} - 160 q^{51} + 1937 q^{52} - 649 q^{53} + 1184 q^{54} + 5 q^{55} + 1518 q^{56} - 468 q^{57} - 1111 q^{58} + 49 q^{59} + 1130 q^{60} - 161 q^{61} + 279 q^{62} - 244 q^{63} + 421 q^{64} - 2600 q^{65} + 2027 q^{66} - 1981 q^{67} + 2041 q^{68} + 1664 q^{69} - 690 q^{70} - 1510 q^{71} - 876 q^{72} - 3562 q^{73} - 1005 q^{74} + 25 q^{75} - 2776 q^{76} - 1096 q^{77} + 143 q^{78} - 1094 q^{79} - 2065 q^{80} + 712 q^{81} + 228 q^{82} - 817 q^{83} - 1837 q^{84} - 30 q^{85} + 1125 q^{86} - 1288 q^{87} - 4011 q^{88} - 2326 q^{89} + 1530 q^{90} - 832 q^{91} + 4425 q^{92} - 31 q^{93} - 2590 q^{94} + 1325 q^{95} - 3706 q^{96} - 7166 q^{97} - 7363 q^{98} - 2161 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + q^3 + 149 * q^4 - 200 * q^5 + 11 * q^6 - 64 * q^7 - 87 * q^8 + 247 * q^9 + 45 * q^10 - q^11 - 226 * q^12 + 520 * q^13 + 138 * q^14 - 5 * q^15 + 413 * q^16 + 6 * q^17 - 306 * q^18 - 265 * q^19 - 745 * q^20 - 344 * q^21 - 313 * q^22 + 24 * q^23 - 321 * q^24 + 1000 * q^25 - 117 * q^26 + 28 * q^27 - 675 * q^28 - 9 * q^29 - 55 * q^30 - 1240 * q^31 - 160 * q^32 - 986 * q^33 - 298 * q^34 + 320 * q^35 + 238 * q^36 - 1403 * q^37 - 314 * q^38 + 13 * q^39 + 435 * q^40 - 84 * q^41 + 744 * q^42 - 421 * q^43 + 812 * q^44 - 1235 * q^45 - 527 * q^46 - 460 * q^47 - 1573 * q^48 + 748 * q^49 - 225 * q^50 - 160 * q^51 + 1937 * q^52 - 649 * q^53 + 1184 * q^54 + 5 * q^55 + 1518 * q^56 - 468 * q^57 - 1111 * q^58 + 49 * q^59 + 1130 * q^60 - 161 * q^61 + 279 * q^62 - 244 * q^63 + 421 * q^64 - 2600 * q^65 + 2027 * q^66 - 1981 * q^67 + 2041 * q^68 + 1664 * q^69 - 690 * q^70 - 1510 * q^71 - 876 * q^72 - 3562 * q^73 - 1005 * q^74 + 25 * q^75 - 2776 * q^76 - 1096 * q^77 + 143 * q^78 - 1094 * q^79 - 2065 * q^80 + 712 * q^81 + 228 * q^82 - 817 * q^83 - 1837 * q^84 - 30 * q^85 + 1125 * q^86 - 1288 * q^87 - 4011 * q^88 - 2326 * q^89 + 1530 * q^90 - 832 * q^91 + 4425 * q^92 - 31 * q^93 - 2590 * q^94 + 1325 * q^95 - 3706 * q^96 - 7166 * q^97 - 7363 * q^98 - 2161 * 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 $

$2$	−5.37200	−1.89929	−0.949644	−	0.313330i	$-0.898555\pi$
$2$	−5.37200	−1.89929	−0.949644	+	0.313330i	$0.898555\pi$
$3$	7.54523	1.45208	0.726040	−	0.687653i	$-0.241359\pi$
$3$	7.54523	1.45208	0.726040	+	0.687653i	$0.241359\pi$
$4$	20.8584	2.60730
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−40.5329	−2.75792
$7$	−10.2525	−0.553585	−0.276792	−	0.960930i	$-0.589271\pi$
$7$	−10.2525	−0.553585	−0.276792	+	0.960930i	$0.589271\pi$
$8$	−69.0752	−3.05272
$9$	29.9304	1.10853
$10$	26.8600	0.849388
$11$	−20.0035	−0.548300	−0.274150	−	0.961687i	$-0.588396\pi$
$11$	−20.0035	−0.548300	−0.274150	+	0.961687i	$0.588396\pi$
$12$	157.381	3.78600
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	55.0766	1.05142
$15$	−37.7261	−0.649390
$16$	204.205	3.19070
$17$	57.8944	0.825968	0.412984	−	0.910738i	$-0.364487\pi$
$17$	57.8944	0.825968	0.412984	+	0.910738i	$0.364487\pi$
$18$	−160.786	−2.10543
$19$	−1.10627	−0.0133577	−0.00667884	−	0.999978i	$-0.502126\pi$
$19$	−1.10627	−0.0133577	−0.00667884	+	0.999978i	$0.502126\pi$
$20$	−104.292	−1.16602
$21$	−77.3577	−0.803849
$22$	107.459	1.04138
$23$	138.257	1.25342	0.626708	−	0.779254i	$-0.284402\pi$
$23$	138.257	1.25342	0.626708	+	0.779254i	$0.284402\pi$
$24$	−521.188	−4.43279
$25$	25.0000	0.200000
$26$	−69.8360	−0.526768
$27$	22.1107	0.157601
$28$	−213.851	−1.44336
$29$	−146.040	−0.935138	−0.467569	−	0.883957i	$-0.654870\pi$
$29$	−146.040	−0.935138	−0.467569	+	0.883957i	$0.654870\pi$
$30$	202.665	1.23338
$31$	−31.0000	−0.179605
$31$	−31.0000	−0.179605
$32$	−544.387	−3.00734
$33$	−150.931	−0.796174
$34$	−311.009	−1.56875
$35$	51.2627	0.247571
$36$	624.300	2.89028
$37$	−358.983	−1.59504	−0.797519	−	0.603294i	$-0.793855\pi$
$37$	−358.983	−1.59504	−0.797519	+	0.603294i	$0.793855\pi$
$38$	5.94289	0.0253701
$39$	98.0879	0.402734
$40$	345.376	1.36522
$41$	108.680	0.413973	0.206987	−	0.978344i	$-0.433634\pi$
$41$	108.680	0.413973	0.206987	+	0.978344i	$0.433634\pi$
$42$	415.565	1.52674
$43$	148.439	0.526436	0.263218	−	0.964736i	$-0.415216\pi$
$43$	148.439	0.526436	0.263218	+	0.964736i	$0.415216\pi$
$44$	−417.241	−1.42958
$45$	−149.652	−0.495752
$46$	−742.716	−2.38060
$47$	−257.742	−0.799906	−0.399953	−	0.916536i	$-0.630974\pi$
$47$	−257.742	−0.799906	−0.399953	+	0.916536i	$0.630974\pi$
$48$	1540.77	4.63315
$49$	−237.886	−0.693544
$50$	−134.300	−0.379858
$51$	436.826	1.19937
$52$	271.159	0.723134
$53$	507.082	1.31421	0.657104	−	0.753800i	$-0.271781\pi$
$53$	507.082	1.31421	0.657104	+	0.753800i	$0.271781\pi$
$54$	−118.779	−0.299329
$55$	100.018	0.245207
$56$	708.196	1.68994
$57$	−8.34707	−0.0193964
$58$	784.528	1.77610
$59$	406.129	0.896161	0.448080	−	0.893993i	$-0.352108\pi$
$59$	406.129	0.896161	0.448080	+	0.893993i	$0.352108\pi$
$60$	−786.906	−1.69315
$61$	−108.561	−0.227867	−0.113933	−	0.993488i	$-0.536345\pi$
$61$	−108.561	−0.227867	−0.113933	+	0.993488i	$0.536345\pi$
$62$	166.532	0.341122
$63$	−306.863	−0.613668
$64$	1290.81	2.52111
$65$	−65.0000	−0.124035
$66$	810.802	1.51216
$67$	−476.610	−0.869062	−0.434531	−	0.900657i	$-0.643086\pi$
$67$	−476.610	−0.869062	−0.434531	+	0.900657i	$0.643086\pi$
$68$	1207.58	2.15354
$69$	1043.18	1.82006
$70$	−275.383	−0.470208
$71$	468.068	0.782387	0.391194	−	0.920308i	$-0.372062\pi$
$71$	468.068	0.782387	0.391194	+	0.920308i	$0.372062\pi$
$72$	−2067.45	−3.38405
$73$	119.783	0.192048	0.0960239	−	0.995379i	$-0.469387\pi$
$73$	119.783	0.192048	0.0960239	+	0.995379i	$0.469387\pi$
$74$	1928.45	3.02944
$75$	188.631	0.290416
$76$	−23.0750	−0.0348275
$77$	205.087	0.303530
$78$	−526.928	−0.764909
$79$	−432.194	−0.615514	−0.307757	−	0.951465i	$-0.599578\pi$
$79$	−432.194	−0.615514	−0.307757	+	0.951465i	$0.599578\pi$
$80$	−1021.02	−1.42692
$81$	−641.291	−0.879686
$82$	−583.827	−0.786255
$83$	557.434	0.737185	0.368592	−	0.929591i	$-0.379840\pi$
$83$	557.434	0.737185	0.368592	+	0.929591i	$0.379840\pi$
$84$	−1613.56	−2.09587
$85$	−289.472	−0.369384
$86$	−797.415	−0.999854
$87$	−1101.91	−1.35789
$88$	1381.75	1.67381
$89$	506.942	0.603772	0.301886	−	0.953344i	$-0.402384\pi$
$89$	506.942	0.603772	0.301886	+	0.953344i	$0.402384\pi$
$90$	803.931	0.941575
$91$	−133.283	−0.153537
$92$	2883.82	3.26803
$93$	−233.902	−0.260801
$94$	1384.59	1.51925
$95$	5.53136	0.00597374
$96$	−4107.52	−4.36690
$97$	169.351	0.177267	0.0886337	−	0.996064i	$-0.471750\pi$
$97$	169.351	0.177267	0.0886337	+	0.996064i	$0.471750\pi$
$98$	1277.92	1.31724
$99$	−598.715	−0.607809
$100$	521.459	0.521459
$101$	1080.80	1.06479	0.532396	−	0.846495i	$-0.321292\pi$
$101$	1080.80	1.06479	0.532396	+	0.846495i	$0.321292\pi$
$102$	−2346.63	−2.27795
$103$	−518.827	−0.496326	−0.248163	−	0.968718i	$-0.579827\pi$
$103$	−518.827	−0.496326	−0.248163	+	0.968718i	$0.579827\pi$
$104$	−897.977	−0.846672
$105$	386.788	0.359492
$106$	−2724.04	−2.49606
$107$	−1609.04	−1.45375	−0.726877	−	0.686767i	$-0.759029\pi$
$107$	−1609.04	−1.45375	−0.726877	+	0.686767i	$0.759029\pi$
$108$	461.194	0.410911
$109$	681.355	0.598734	0.299367	−	0.954138i	$-0.403225\pi$
$109$	681.355	0.598734	0.299367	+	0.954138i	$0.403225\pi$
$110$	−537.295	−0.465719
$111$	−2708.61	−2.31612
$112$	−2093.62	−1.76632
$113$	−1009.26	−0.840206	−0.420103	−	0.907476i	$-0.638006\pi$
$113$	−1009.26	−0.840206	−0.420103	+	0.907476i	$0.638006\pi$
$114$	44.8404	0.0368394
$115$	−691.285	−0.560545
$116$	−3046.16	−2.43818
$117$	389.096	0.307452
$118$	−2181.72	−1.70207
$119$	−593.564	−0.457243
$120$	2605.94	1.98240
$121$	−930.858	−0.699368
$122$	583.192	0.432785
$123$	820.012	0.601122
$124$	−646.610	−0.468284
$125$	−125.000	−0.0894427
$126$	1648.47	1.16553
$127$	−1354.65	−0.946501	−0.473251	−	0.880928i	$-0.656920\pi$
$127$	−1354.65	−0.946501	−0.473251	+	0.880928i	$0.656920\pi$
$128$	−2579.12	−1.78097
$129$	1120.01	0.764427
$130$	349.180	0.235578
$131$	1101.12	0.734390	0.367195	−	0.930144i	$-0.380318\pi$
$131$	1101.12	0.734390	0.367195	+	0.930144i	$0.380318\pi$
$132$	−3148.18	−2.07586
$133$	11.3421	0.00739461
$134$	2560.35	1.65060
$135$	−110.554	−0.0704811
$136$	−3999.07	−2.52145
$137$	−2621.20	−1.63463	−0.817314	−	0.576193i	$-0.804538\pi$
$137$	−2621.20	−1.63463	−0.817314	+	0.576193i	$0.804538\pi$
$138$	−5603.96	−3.45682
$139$	−255.799	−0.156090	−0.0780451	−	0.996950i	$-0.524868\pi$
$139$	−255.799	−0.156090	−0.0780451	+	0.996950i	$0.524868\pi$
$140$	1069.26	0.645490
$141$	−1944.72	−1.16153
$142$	−2514.46	−1.48598
$143$	−260.046	−0.152071
$144$	6111.94	3.53700
$145$	730.201	0.418206
$146$	−643.472	−0.364754
$147$	−1794.90	−1.00708
$148$	−7487.80	−4.15874
$149$	683.078	0.375570	0.187785	−	0.982210i	$-0.439869\pi$
$149$	683.078	0.375570	0.187785	+	0.982210i	$0.439869\pi$
$150$	−1013.32	−0.551584
$151$	2116.98	1.14091	0.570454	−	0.821330i	$-0.306767\pi$
$151$	2116.98	1.14091	0.570454	+	0.821330i	$0.306767\pi$
$152$	76.4159	0.0407773
$153$	1732.80	0.915614
$154$	−1101.73	−0.576492
$155$	155.000	0.0803219
$156$	2045.95	1.05005
$157$	−909.247	−0.462203	−0.231101	−	0.972930i	$-0.574233\pi$
$157$	−909.247	−0.462203	−0.231101	+	0.972930i	$0.574233\pi$
$158$	2321.74	1.16904
$159$	3826.05	1.90833
$160$	2721.93	1.34492
$161$	−1417.48	−0.693872
$162$	3445.01	1.67078
$163$	−4035.05	−1.93895	−0.969477	−	0.245181i	$-0.921152\pi$
$163$	−4035.05	−1.93895	−0.969477	+	0.245181i	$0.921152\pi$
$164$	2266.88	1.07935
$165$	754.656	0.356060
$166$	−2994.53	−1.40013
$167$	−1930.33	−0.894453	−0.447227	−	0.894421i	$-0.647588\pi$
$167$	−1930.33	−0.894453	−0.447227	+	0.894421i	$0.647588\pi$
$168$	5343.50	2.45393
$169$	169.000	0.0769231
$170$	1555.04	0.701567
$171$	−33.1112	−0.0148075
$172$	3096.20	1.37258
$173$	1794.61	0.788682	0.394341	−	0.918964i	$-0.370973\pi$
$173$	1794.61	0.788682	0.394341	+	0.918964i	$0.370973\pi$
$174$	5919.44	2.57903
$175$	−256.313	−0.110717
$176$	−4084.82	−1.74946
$177$	3064.33	1.30130
$178$	−2723.29	−1.14674
$179$	−1348.40	−0.563038	−0.281519	−	0.959556i	$-0.590838\pi$
$179$	−1348.40	−0.563038	−0.281519	+	0.959556i	$0.590838\pi$
$180$	−3121.50	−1.29257
$181$	−3078.20	−1.26409	−0.632047	−	0.774930i	$-0.717785\pi$
$181$	−3078.20	−1.26409	−0.632047	+	0.774930i	$0.717785\pi$
$182$	715.996	0.291611
$183$	−819.121	−0.330881
$184$	−9550.13	−3.82633
$185$	1794.91	0.713322
$186$	1256.52	0.495337
$187$	−1158.09	−0.452878
$188$	−5376.08	−2.08559
$189$	−226.691	−0.0872452
$190$	−29.7144	−0.0113459
$191$	2368.83	0.897394	0.448697	−	0.893684i	$-0.351888\pi$
$191$	2368.83	0.897394	0.448697	+	0.893684i	$0.351888\pi$
$192$	9739.42	3.66084
$193$	4695.65	1.75130	0.875649	−	0.482949i	$-0.160434\pi$
$193$	4695.65	1.75130	0.875649	+	0.482949i	$0.160434\pi$
$194$	−909.751	−0.336682
$195$	−490.440	−0.180108
$196$	−4961.91	−1.80827
$197$	−1257.69	−0.454857	−0.227428	−	0.973795i	$-0.573032\pi$
$197$	−1257.69	−0.454857	−0.227428	+	0.973795i	$0.573032\pi$
$198$	3216.29	1.15440
$199$	1825.56	0.650304	0.325152	−	0.945662i	$-0.394585\pi$
$199$	1825.56	0.650304	0.325152	+	0.945662i	$0.394585\pi$
$200$	−1726.88	−0.610544
$201$	−3596.13	−1.26195
$202$	−5806.08	−2.02235
$203$	1497.28	0.517678
$204$	9111.49	3.12712
$205$	−543.398	−0.185134
$206$	2787.14	0.942666
$207$	4138.09	1.38946
$208$	2654.66	0.884941
$209$	22.1293	0.00732401
$210$	−2077.83	−0.682779
$211$	−4113.94	−1.34225	−0.671126	−	0.741343i	$-0.734189\pi$
$211$	−4113.94	−1.34225	−0.671126	+	0.741343i	$0.734189\pi$
$212$	10576.9	3.42653
$213$	3531.68	1.13609
$214$	8643.76	2.76110
$215$	−742.195	−0.235429
$216$	−1527.30	−0.481110
$217$	317.829	0.0994268
$218$	−3660.24	−1.13717
$219$	903.787	0.278869
$220$	2086.21	0.639327
$221$	752.627	0.229082
$222$	14550.6	4.39898
$223$	120.684	0.0362404	0.0181202	−	0.999836i	$-0.494232\pi$
$223$	120.684	0.0362404	0.0181202	+	0.999836i	$0.494232\pi$
$224$	5581.34	1.66482
$225$	748.261	0.221707
$226$	5421.75	1.59579
$227$	−4571.93	−1.33678	−0.668391	−	0.743810i	$-0.733017\pi$
$227$	−4571.93	−1.33678	−0.668391	+	0.743810i	$0.733017\pi$
$228$	−174.106	−0.0505722
$229$	241.657	0.0697343	0.0348672	−	0.999392i	$-0.488899\pi$
$229$	241.657	0.0697343	0.0348672	+	0.999392i	$0.488899\pi$
$230$	3713.58	1.06464
$231$	1547.43	0.440750
$232$	10087.8	2.85471
$233$	1219.91	0.343000	0.171500	−	0.985184i	$-0.445139\pi$
$233$	1219.91	0.343000	0.171500	+	0.985184i	$0.445139\pi$
$234$	−2090.22	−0.583940
$235$	1288.71	0.357729
$236$	8471.19	2.33656
$237$	−3261.00	−0.893775
$238$	3188.63	0.868437
$239$	491.918	0.133136	0.0665680	−	0.997782i	$-0.478795\pi$
$239$	491.918	0.133136	0.0665680	+	0.997782i	$0.478795\pi$
$240$	−7703.86	−2.07201
$241$	4811.59	1.28607	0.643033	−	0.765839i	$-0.277676\pi$
$241$	4811.59	1.28607	0.643033	+	0.765839i	$0.277676\pi$
$242$	5000.57	1.32830
$243$	−5435.68	−1.43497
$244$	−2264.42	−0.594116
$245$	1189.43	0.310162
$246$	−4405.10	−1.14170
$247$	−14.3815	−0.00370476
$248$	2141.33	0.548285
$249$	4205.96	1.07045
$250$	671.500	0.169878
$251$	−1881.67	−0.473187	−0.236594	−	0.971609i	$-0.576031\pi$
$251$	−1881.67	−0.473187	−0.236594	+	0.971609i	$0.576031\pi$
$252$	−6400.66	−1.60001
$253$	−2765.63	−0.687248
$254$	7277.18	1.79768
$255$	−2184.13	−0.536375
$256$	3528.56	0.861465
$257$	−645.895	−0.156770	−0.0783849	−	0.996923i	$-0.524976\pi$
$257$	−645.895	−0.156770	−0.0783849	+	0.996923i	$0.524976\pi$
$258$	−6016.67	−1.45187
$259$	3680.48	0.882989
$260$	−1355.79	−0.323395
$261$	−4371.05	−1.03663
$262$	−5915.20	−1.39482
$263$	3136.97	0.735490	0.367745	−	0.929927i	$-0.380130\pi$
$263$	3136.97	0.735490	0.367745	+	0.929927i	$0.380130\pi$
$264$	10425.6	2.43050
$265$	−2535.41	−0.587732
$266$	−60.9297	−0.0140445
$267$	3824.99	0.876725
$268$	−9941.31	−2.26590
$269$	−3891.46	−0.882032	−0.441016	−	0.897499i	$-0.645382\pi$
$269$	−3891.46	−0.882032	−0.441016	+	0.897499i	$0.645382\pi$
$270$	593.894	0.133864
$271$	1956.83	0.438630	0.219315	−	0.975654i	$-0.429618\pi$
$271$	1956.83	0.438630	0.219315	+	0.975654i	$0.429618\pi$
$272$	11822.3	2.63542
$273$	−1005.65	−0.222948
$274$	14081.1	3.10463
$275$	−500.089	−0.109660
$276$	21759.0	4.74544
$277$	−5757.62	−1.24889	−0.624443	−	0.781070i	$-0.714674\pi$
$277$	−5757.62	−1.24889	−0.624443	+	0.781070i	$0.714674\pi$
$278$	1374.15	0.296460
$279$	−927.843	−0.199099
$280$	−3540.98	−0.755764
$281$	−6537.33	−1.38784	−0.693922	−	0.720050i	$-0.744119\pi$
$281$	−6537.33	−1.38784	−0.693922	+	0.720050i	$0.744119\pi$
$282$	10447.1	2.20607
$283$	307.040	0.0644935	0.0322467	−	0.999480i	$-0.489734\pi$
$283$	307.040	0.0644935	0.0322467	+	0.999480i	$0.489734\pi$
$284$	9763.14	2.03992
$285$	41.7353	0.00867434
$286$	1396.97	0.288827
$287$	−1114.24	−0.229169
$288$	−16293.7	−3.33374
$289$	−1561.24	−0.317777
$290$	−3922.64	−0.794294
$291$	1277.79	0.257406
$292$	2498.47	0.500726
$293$	−2698.11	−0.537970	−0.268985	−	0.963144i	$-0.586688\pi$
$293$	−2698.11	−0.537970	−0.268985	+	0.963144i	$0.586688\pi$
$294$	9642.20	1.91274
$295$	−2030.64	−0.400775
$296$	24796.8	4.86920
$297$	−442.293	−0.0864123
$298$	−3669.49	−0.713316
$299$	1797.34	0.347635
$300$	3934.53	0.757200
$301$	−1521.88	−0.291427
$302$	−11372.4	−2.16691
$303$	8154.91	1.54616
$304$	−225.906	−0.0426204
$305$	542.807	0.101905
$306$	−9308.62	−1.73902
$307$	−3093.68	−0.575133	−0.287567	−	0.957761i	$-0.592846\pi$
$307$	−3093.68	−0.575133	−0.287567	+	0.957761i	$0.592846\pi$
$308$	4277.78	0.791394
$309$	−3914.67	−0.720704
$310$	−832.660	−0.152555
$311$	−5156.55	−0.940196	−0.470098	−	0.882614i	$-0.655781\pi$
$311$	−5156.55	−0.940196	−0.470098	+	0.882614i	$0.655781\pi$
$312$	−6775.44	−1.22944
$313$	−9745.11	−1.75983	−0.879914	−	0.475132i	$-0.842400\pi$
$313$	−9745.11	−1.75983	−0.879914	+	0.475132i	$0.842400\pi$
$314$	4884.48	0.877856
$315$	1534.31	0.274441
$316$	−9014.86	−1.60483
$317$	−133.181	−0.0235968	−0.0117984	−	0.999930i	$-0.503756\pi$
$317$	−133.181	−0.0235968	−0.0117984	+	0.999930i	$0.503756\pi$
$318$	−20553.5	−3.62448
$319$	2921.32	0.512736
$320$	−6454.03	−1.12747
$321$	−12140.6	−2.11097
$322$	7614.73	1.31786
$323$	−64.0469	−0.0110330
$324$	−13376.3	−2.29360
$325$	325.000	0.0554700
$326$	21676.3	3.68263
$327$	5140.98	0.869409
$328$	−7507.06	−1.26374
$329$	2642.51	0.442816
$330$	−4054.01	−0.676261
$331$	481.808	0.0800077	0.0400039	−	0.999200i	$-0.487263\pi$
$331$	481.808	0.0800077	0.0400039	+	0.999200i	$0.487263\pi$
$332$	11627.2	1.92206
$333$	−10744.5	−1.76815
$334$	10369.7	1.69882
$335$	2383.05	0.388656
$336$	−15796.8	−2.56484
$337$	−1908.50	−0.308495	−0.154247	−	0.988032i	$-0.549295\pi$
$337$	−1908.50	−0.308495	−0.154247	+	0.988032i	$0.549295\pi$
$338$	−907.868	−0.146099
$339$	−7615.10	−1.22005
$340$	−6037.92	−0.963094
$341$	620.110	0.0984775
$342$	177.873	0.0281236
$343$	5955.55	0.937520
$344$	−10253.5	−1.60706
$345$	−5215.90	−0.813956
$346$	−9640.66	−1.49793
$347$	2296.33	0.355255	0.177628	−	0.984098i	$-0.443158\pi$
$347$	2296.33	0.355255	0.177628	+	0.984098i	$0.443158\pi$
$348$	−22984.0	−3.54043
$349$	−5032.24	−0.771833	−0.385916	−	0.922534i	$-0.626115\pi$
$349$	−5032.24	−0.771833	−0.385916	+	0.922534i	$0.626115\pi$
$350$	1376.92	0.210283
$351$	287.440	0.0437105
$352$	10889.7	1.64892
$353$	−6898.34	−1.04012	−0.520059	−	0.854130i	$-0.674090\pi$
$353$	−6898.34	−1.04012	−0.520059	+	0.854130i	$0.674090\pi$
$354$	−16461.6	−2.47154
$355$	−2340.34	−0.349894
$356$	10574.0	1.57421
$357$	−4478.58	−0.663954
$358$	7243.58	1.06937
$359$	−10425.7	−1.53272	−0.766361	−	0.642410i	$-0.777935\pi$
$359$	−10425.7	−1.53272	−0.766361	+	0.642410i	$0.777935\pi$
$360$	10337.2	1.51339
$361$	−6857.78	−0.999822
$362$	16536.1	2.40088
$363$	−7023.54	−1.01554
$364$	−2780.07	−0.400316
$365$	−598.913	−0.0858864
$366$	4400.32	0.628438
$367$	−1222.21	−0.173839	−0.0869196	−	0.996215i	$-0.527702\pi$
$367$	−1222.21	−0.173839	−0.0869196	+	0.996215i	$0.527702\pi$
$368$	28232.7	3.99928
$369$	3252.83	0.458904
$370$	−9642.27	−1.35481
$371$	−5198.87	−0.727526
$372$	−4878.82	−0.679986
$373$	2132.36	0.296004	0.148002	−	0.988987i	$-0.452716\pi$
$373$	2132.36	0.296004	0.148002	+	0.988987i	$0.452716\pi$
$374$	6221.28	0.860146
$375$	−943.153	−0.129878
$376$	17803.6	2.44189
$377$	−1898.52	−0.259361
$378$	1217.78	0.165704
$379$	−4459.10	−0.604350	−0.302175	−	0.953253i	$-0.597713\pi$
$379$	−4459.10	−0.604350	−0.302175	+	0.953253i	$0.597713\pi$
$380$	115.375	0.0155753
$381$	−10221.1	−1.37440
$382$	−12725.3	−1.70441
$383$	−5331.73	−0.711328	−0.355664	−	0.934614i	$-0.615745\pi$
$383$	−5331.73	−0.711328	−0.355664	+	0.934614i	$0.615745\pi$
$384$	−19460.0	−2.58611
$385$	−1025.43	−0.135743
$386$	−25225.0	−3.32622
$387$	4442.85	0.583572
$388$	3532.38	0.462189
$389$	6632.16	0.864431	0.432215	−	0.901770i	$-0.357732\pi$
$389$	6632.16	0.864431	0.432215	+	0.901770i	$0.357732\pi$
$390$	2634.64	0.342078
$391$	8004.31	1.03528
$392$	16432.0	2.11720
$393$	8308.17	1.06639
$394$	6756.31	0.863904
$395$	2160.97	0.275266
$396$	−12488.2	−1.58474
$397$	−7564.38	−0.956285	−0.478142	−	0.878282i	$-0.658690\pi$
$397$	−7564.38	−0.956285	−0.478142	+	0.878282i	$0.658690\pi$
$398$	−9806.90	−1.23511
$399$	85.5786	0.0107376
$400$	5105.12	0.638140
$401$	−1420.86	−0.176944	−0.0884719	−	0.996079i	$-0.528198\pi$
$401$	−1420.86	−0.176944	−0.0884719	+	0.996079i	$0.528198\pi$
$402$	19318.4	2.39680
$403$	−403.000	−0.0498135
$404$	22543.8	2.77623
$405$	3206.46	0.393408
$406$	−8043.40	−0.983220
$407$	7180.93	0.874558
$408$	−30173.9	−3.66135
$409$	−316.840	−0.0383049	−0.0191525	−	0.999817i	$-0.506097\pi$
$409$	−316.840	−0.0383049	−0.0191525	+	0.999817i	$0.506097\pi$
$410$	2919.13	0.351624
$411$	−19777.5	−2.37361
$412$	−10821.9	−1.29407
$413$	−4163.85	−0.496101
$414$	−22229.8	−2.63898
$415$	−2787.17	−0.329679
$416$	−7077.03	−0.834086
$417$	−1930.06	−0.226655
$418$	−118.879	−0.0139104
$419$	−4288.33	−0.499997	−0.249999	−	0.968246i	$-0.580430\pi$
$419$	−4288.33	−0.499997	−0.249999	+	0.968246i	$0.580430\pi$
$420$	8067.78	0.937303
$421$	14046.6	1.62610	0.813051	−	0.582192i	$-0.197805\pi$
$421$	14046.6	1.62610	0.813051	+	0.582192i	$0.197805\pi$
$422$	22100.1	2.54932
$423$	−7714.33	−0.886723
$424$	−35026.8	−4.01191
$425$	1447.36	0.165194
$426$	−18972.2	−2.15776
$427$	1113.03	0.126144
$428$	−33562.0	−3.79037
$429$	−1962.11	−0.220819
$430$	3987.07	0.447148
$431$	6906.48	0.771864	0.385932	−	0.922527i	$-0.373880\pi$
$431$	6906.48	0.771864	0.385932	+	0.922527i	$0.373880\pi$
$432$	4515.12	0.502856
$433$	−9058.31	−1.00535	−0.502673	−	0.864477i	$-0.667650\pi$
$433$	−9058.31	−1.00535	−0.502673	+	0.864477i	$0.667650\pi$
$434$	−1707.37	−0.188840
$435$	5509.53	0.607269
$436$	14212.0	1.56108
$437$	−152.950	−0.0167427
$438$	−4855.14	−0.529652
$439$	−14691.6	−1.59725	−0.798624	−	0.601830i	$-0.794438\pi$
$439$	−14691.6	−1.59725	−0.798624	+	0.601830i	$0.794438\pi$
$440$	−6908.74	−0.748548
$441$	−7120.02	−0.768817
$442$	−4043.11	−0.435093
$443$	6882.41	0.738134	0.369067	−	0.929403i	$-0.379677\pi$
$443$	6882.41	0.738134	0.369067	+	0.929403i	$0.379677\pi$
$444$	−56497.1	−6.03882
$445$	−2534.71	−0.270015
$446$	−648.315	−0.0688310
$447$	5153.98	0.545357
$448$	−13234.0	−1.39565
$449$	7172.33	0.753860	0.376930	−	0.926242i	$-0.376980\pi$
$449$	7172.33	0.753860	0.376930	+	0.926242i	$0.376980\pi$
$450$	−4019.66	−0.421085
$451$	−2173.98	−0.226981
$452$	−21051.5	−2.19067
$453$	15973.1	1.65669
$454$	24560.4	2.53894
$455$	666.415	0.0686637
$456$	576.575	0.0592119
$457$	−17703.4	−1.81210	−0.906052	−	0.423167i	$-0.860918\pi$
$457$	−17703.4	−1.81210	−0.906052	+	0.423167i	$0.860918\pi$
$458$	−1298.18	−0.132446
$459$	1280.09	0.130173
$460$	−14419.1	−1.46151
$461$	−7103.00	−0.717613	−0.358807	−	0.933412i	$-0.616816\pi$
$461$	−7103.00	−0.717613	−0.358807	+	0.933412i	$0.616816\pi$
$462$	−8312.78	−0.837112
$463$	1780.65	0.178734	0.0893672	−	0.995999i	$-0.471516\pi$
$463$	1780.65	0.178734	0.0893672	+	0.995999i	$0.471516\pi$
$464$	−29822.1	−2.98374
$465$	1169.51	0.116634
$466$	−6553.36	−0.651456
$467$	15985.6	1.58399	0.791997	−	0.610525i	$-0.209041\pi$
$467$	15985.6	1.58399	0.791997	+	0.610525i	$0.209041\pi$
$468$	8115.90	0.801619
$469$	4886.46	0.481100
$470$	−6922.95	−0.679430
$471$	−6860.48	−0.671155
$472$	−28053.4	−2.73573
$473$	−2969.31	−0.288645
$474$	17518.1	1.69754
$475$	−27.6568	−0.00267154
$476$	−12380.8	−1.19217
$477$	15177.2	1.45685
$478$	−2642.58	−0.252864
$479$	4985.67	0.475576	0.237788	−	0.971317i	$-0.423578\pi$
$479$	4985.67	0.475576	0.237788	+	0.971317i	$0.423578\pi$
$480$	20537.6	1.95294
$481$	−4666.77	−0.442384
$482$	−25847.9	−2.44261
$483$	−10695.2	−1.00756
$484$	−19416.2	−1.82346
$485$	−846.753	−0.0792764
$486$	29200.4	2.72543
$487$	3537.24	0.329133	0.164567	−	0.986366i	$-0.447377\pi$
$487$	3537.24	0.329133	0.164567	+	0.986366i	$0.447377\pi$
$488$	7498.90	0.695614
$489$	−30445.4	−2.81552
$490$	−6389.60	−0.589088
$491$	−1338.68	−0.123042	−0.0615211	−	0.998106i	$-0.519595\pi$
$491$	−1338.68	−0.123042	−0.0615211	+	0.998106i	$0.519595\pi$
$492$	17104.1	1.56730
$493$	−8454.92	−0.772394
$494$	77.2576	0.00703640
$495$	2993.57	0.271820
$496$	−6330.35	−0.573067
$497$	−4798.89	−0.433118
$498$	−22594.4	−2.03309
$499$	−12171.7	−1.09195	−0.545974	−	0.837802i	$-0.683840\pi$
$499$	−12171.7	−1.09195	−0.545974	+	0.837802i	$0.683840\pi$
$500$	−2607.30	−0.233204
$501$	−14564.8	−1.29882
$502$	10108.3	0.898719
$503$	2606.03	0.231008	0.115504	−	0.993307i	$-0.463152\pi$
$503$	2606.03	0.231008	0.115504	+	0.993307i	$0.463152\pi$
$504$	21196.6	1.87336
$505$	−5404.02	−0.476190
$506$	14857.0	1.30528
$507$	1275.14	0.111698
$508$	−28255.8	−2.46781
$509$	−10653.1	−0.927684	−0.463842	−	0.885918i	$-0.653530\pi$
$509$	−10653.1	−0.927684	−0.463842	+	0.885918i	$0.653530\pi$
$510$	11733.2	1.01873
$511$	−1228.07	−0.106315
$512$	1677.51	0.144797
$513$	−24.4605	−0.00210518
$514$	3469.75	0.297751
$515$	2594.13	0.221964
$516$	23361.5	1.99309
$517$	5155.76	0.438588
$518$	−19771.5	−1.67705
$519$	13540.8	1.14523
$520$	4489.89	0.378643
$521$	14412.4	1.21194	0.605969	−	0.795488i	$-0.292786\pi$
$521$	14412.4	1.21194	0.605969	+	0.795488i	$0.292786\pi$
$522$	23481.3	1.96886
$523$	15632.9	1.30703	0.653517	−	0.756912i	$-0.273293\pi$
$523$	15632.9	1.30703	0.653517	+	0.756912i	$0.273293\pi$
$524$	22967.5	1.91477
$525$	−1933.94	−0.160770
$526$	−16851.8	−1.39691
$527$	−1794.73	−0.148348
$528$	−30820.9	−2.54035
$529$	6947.99	0.571052
$530$	13620.2	1.11627
$531$	12155.6	0.993425
$532$	236.577	0.0192800
$533$	1412.83	0.114816
$534$	−20547.8	−1.66515
$535$	8045.20	0.650139
$536$	32921.9	2.65300
$537$	−10174.0	−0.817576
$538$	20904.9	1.67523
$539$	4758.55	0.380270
$540$	−2305.97	−0.183765
$541$	20178.0	1.60355	0.801776	−	0.597625i	$-0.203889\pi$
$541$	20178.0	1.60355	0.801776	+	0.597625i	$0.203889\pi$
$542$	−10512.1	−0.833084
$543$	−23225.7	−1.83557
$544$	−31516.9	−2.48397
$545$	−3406.78	−0.267762
$546$	5402.35	0.423442
$547$	2725.69	0.213056	0.106528	−	0.994310i	$-0.466027\pi$
$547$	2725.69	0.213056	0.106528	+	0.994310i	$0.466027\pi$
$548$	−54673.9	−4.26196
$549$	−3249.29	−0.252598
$550$	2686.48	0.208276
$551$	161.560	0.0124913
$552$	−72057.9	−5.55613
$553$	4431.08	0.340739
$554$	30929.9	2.37200
$555$	13543.0	1.03580
$556$	−5335.54	−0.406974
$557$	17961.8	1.36637	0.683184	−	0.730247i	$-0.260595\pi$
$557$	17961.8	1.36637	0.683184	+	0.730247i	$0.260595\pi$
$558$	4984.37	0.378146
$559$	1929.71	0.146007
$560$	10468.1	0.789924
$561$	−8738.08	−0.657615
$562$	35118.5	2.63592
$563$	−13380.4	−1.00163	−0.500815	−	0.865554i	$-0.666966\pi$
$563$	−13380.4	−1.00163	−0.500815	+	0.865554i	$0.666966\pi$
$564$	−40563.8	−3.02844
$565$	5046.30	0.375751
$566$	−1649.42	−0.122492
$567$	6574.86	0.486981
$568$	−32331.9	−2.38841
$569$	13260.6	0.977000	0.488500	−	0.872564i	$-0.337544\pi$
$569$	13260.6	0.977000	0.488500	+	0.872564i	$0.337544\pi$
$570$	−224.202	−0.0164751
$571$	13986.1	1.02505	0.512523	−	0.858673i	$-0.328711\pi$
$571$	13986.1	1.02505	0.512523	+	0.858673i	$0.328711\pi$
$572$	−5424.14	−0.396494
$573$	17873.3	1.30309
$574$	5985.70	0.435259
$575$	3456.42	0.250683
$576$	38634.4	2.79473
$577$	−15445.0	−1.11435	−0.557177	−	0.830394i	$-0.688116\pi$
$577$	−15445.0	−1.11435	−0.557177	+	0.830394i	$0.688116\pi$
$578$	8386.96	0.603550
$579$	35429.7	2.54302
$580$	15230.8	1.09039
$581$	−5715.11	−0.408094
$582$	−6864.28	−0.488889
$583$	−10143.4	−0.720580
$584$	−8274.00	−0.586268
$585$	−1945.48	−0.137497
$586$	14494.2	1.02176
$587$	−21464.0	−1.50922	−0.754611	−	0.656173i	$-0.772174\pi$
$587$	−21464.0	−1.50922	−0.754611	+	0.656173i	$0.772174\pi$
$588$	−37438.7	−2.62576
$589$	34.2944	0.00239911
$590$	10908.6	0.761188
$591$	−9489.56	−0.660488
$592$	−73306.0	−5.08929
$593$	−1103.75	−0.0764341	−0.0382170	−	0.999269i	$-0.512168\pi$
$593$	−1103.75	−0.0764341	−0.0382170	+	0.999269i	$0.512168\pi$
$594$	2376.00	0.164122
$595$	2967.82	0.204485
$596$	14247.9	0.979222
$597$	13774.3	0.944293
$598$	−9655.31	−0.660259
$599$	16213.5	1.10595	0.552977	−	0.833196i	$-0.313492\pi$
$599$	16213.5	1.10595	0.552977	+	0.833196i	$0.313492\pi$
$600$	−13029.7	−0.886558
$601$	16232.2	1.10170	0.550851	−	0.834603i	$-0.314303\pi$
$601$	16232.2	1.10170	0.550851	+	0.834603i	$0.314303\pi$
$602$	8175.52	0.553504
$603$	−14265.1	−0.963385
$604$	44156.7	2.97468
$605$	4654.29	0.312767
$606$	−43808.2	−2.93661
$607$	−27159.0	−1.81606	−0.908032	−	0.418901i	$-0.862416\pi$
$607$	−27159.0	−1.81606	−0.908032	+	0.418901i	$0.862416\pi$
$608$	602.239	0.0401711
$609$	11297.3	0.751710
$610$	−2915.96	−0.193547
$611$	−3350.65	−0.221854
$612$	36143.5	2.38728
$613$	−19453.1	−1.28173	−0.640866	−	0.767653i	$-0.721425\pi$
$613$	−19453.1	−1.28173	−0.640866	+	0.767653i	$0.721425\pi$
$614$	16619.3	1.09234
$615$	−4100.06	−0.268830
$616$	−14166.4	−0.926593
$617$	−29253.2	−1.90874	−0.954369	−	0.298631i	$-0.903470\pi$
$617$	−29253.2	−1.90874	−0.954369	+	0.298631i	$0.903470\pi$
$618$	21029.6	1.36883
$619$	7869.89	0.511014	0.255507	−	0.966807i	$-0.417758\pi$
$619$	7869.89	0.511014	0.255507	+	0.966807i	$0.417758\pi$
$620$	3233.05	0.209423
$621$	3056.96	0.197539
$622$	27701.0	1.78570
$623$	−5197.44	−0.334239
$624$	20030.0	1.28500
$625$	625.000	0.0400000
$626$	52350.7	3.34242
$627$	166.971	0.0106350
$628$	−18965.4	−1.20510
$629$	−20783.1	−1.31745
$630$	−8242.33	−0.521242
$631$	−17330.2	−1.09335	−0.546676	−	0.837344i	$-0.684107\pi$
$631$	−17330.2	−1.09335	−0.546676	+	0.837344i	$0.684107\pi$
$632$	29853.9	1.87899
$633$	−31040.6	−1.94906
$634$	715.449	0.0448172
$635$	6773.25	0.423288
$636$	79805.1	4.97559
$637$	−3092.51	−0.192354
$638$	−15693.3	−0.973833
$639$	14009.5	0.867303
$640$	12895.6	0.796473
$641$	4157.52	0.256181	0.128091	−	0.991762i	$-0.459115\pi$
$641$	4157.52	0.256181	0.128091	+	0.991762i	$0.459115\pi$
$642$	65219.1	4.00934
$643$	7960.61	0.488236	0.244118	−	0.969746i	$-0.421502\pi$
$643$	7960.61	0.488236	0.244118	+	0.969746i	$0.421502\pi$
$644$	−29566.4	−1.80913
$645$	−5600.03	−0.341862
$646$	344.060	0.0209549
$647$	−11168.9	−0.678661	−0.339331	−	0.940667i	$-0.610201\pi$
$647$	−11168.9	−0.678661	−0.339331	+	0.940667i	$0.610201\pi$
$648$	44297.3	2.68544
$649$	−8124.02	−0.491364
$650$	−1745.90	−0.105354
$651$	2398.09	0.144376
$652$	−84164.6	−5.05543
$653$	−11581.3	−0.694042	−0.347021	−	0.937857i	$-0.612807\pi$
$653$	−11581.3	−0.694042	−0.347021	+	0.937857i	$0.612807\pi$
$654$	−27617.3	−1.65126
$655$	−5505.58	−0.328429
$656$	22192.9	1.32086
$657$	3585.14	0.212892
$658$	−14195.6	−0.841035
$659$	−23976.5	−1.41729	−0.708643	−	0.705567i	$-0.750692\pi$
$659$	−23976.5	−1.41729	−0.708643	+	0.705567i	$0.750692\pi$
$660$	15740.9	0.928354
$661$	−10724.3	−0.631051	−0.315526	−	0.948917i	$-0.602181\pi$
$661$	−10724.3	−0.631051	−0.315526	+	0.948917i	$0.602181\pi$
$662$	−2588.27	−0.151958
$663$	5678.74	0.332646
$664$	−38504.8	−2.25042
$665$	−56.7104	−0.00330697
$666$	57719.5	3.35823
$667$	−20191.1	−1.17212
$668$	−40263.6	−2.33211
$669$	910.589	0.0526239
$670$	−12801.7	−0.738171
$671$	2171.61	0.124939
$672$	42112.5	2.41745
$673$	−5220.28	−0.299000	−0.149500	−	0.988762i	$-0.547766\pi$
$673$	−5220.28	−0.299000	−0.149500	+	0.988762i	$0.547766\pi$
$674$	10252.5	0.585920
$675$	552.768	0.0315201
$676$	3525.07	0.200561
$677$	−1282.28	−0.0727946	−0.0363973	−	0.999337i	$-0.511588\pi$
$677$	−1282.28	−0.0727946	−0.0363973	+	0.999337i	$0.511588\pi$
$678$	40908.3	2.31722
$679$	−1736.27	−0.0981326
$680$	19995.3	1.12763
$681$	−34496.2	−1.94111
$682$	−3331.23	−0.187037
$683$	−503.295	−0.0281963	−0.0140981	−	0.999901i	$-0.504488\pi$
$683$	−503.295	−0.0281963	−0.0140981	+	0.999901i	$0.504488\pi$
$684$	−690.645	−0.0386074
$685$	13106.0	0.731028
$686$	−31993.2	−1.78062
$687$	1823.36	0.101260
$688$	30312.0	1.67970
$689$	6592.06	0.364496
$690$	28019.8	1.54594
$691$	24408.5	1.34377	0.671883	−	0.740658i	$-0.265486\pi$
$691$	24408.5	1.34377	0.671883	+	0.740658i	$0.265486\pi$
$692$	37432.7	2.05633
$693$	6138.34	0.336474
$694$	−12335.9	−0.674732
$695$	1278.99	0.0698057
$696$	76114.4	4.14527
$697$	6291.94	0.341929
$698$	27033.2	1.46593
$699$	9204.50	0.498063
$700$	−5346.28	−0.288672
$701$	21390.3	1.15250	0.576249	−	0.817274i	$-0.304516\pi$
$701$	21390.3	1.15250	0.576249	+	0.817274i	$0.304516\pi$
$702$	−1544.13	−0.0830189
$703$	397.132	0.0213060
$704$	−25820.7	−1.38232
$705$	9723.62	0.519450
$706$	37057.9	1.97548
$707$	−11081.0	−0.589453
$708$	63917.0	3.39287
$709$	939.506	0.0497657	0.0248829	−	0.999690i	$-0.492079\pi$
$709$	939.506	0.0497657	0.0248829	+	0.999690i	$0.492079\pi$
$710$	12572.3	0.664550
$711$	−12935.7	−0.682318
$712$	−35017.1	−1.84315
$713$	−4285.97	−0.225120
$714$	24058.9	1.26104
$715$	1300.23	0.0680082
$716$	−28125.3	−1.46801
$717$	3711.63	0.193324
$718$	56006.9	2.91108
$719$	25533.2	1.32438	0.662189	−	0.749337i	$-0.269628\pi$
$719$	25533.2	1.32438	0.662189	+	0.749337i	$0.269628\pi$
$720$	−30559.7	−1.58179
$721$	5319.29	0.274758
$722$	36840.0	1.89895
$723$	36304.5	1.86747
$724$	−64206.3	−3.29587
$725$	−3651.01	−0.187028
$726$	37730.4	1.92880
$727$	−5443.44	−0.277697	−0.138849	−	0.990314i	$-0.544340\pi$
$727$	−5443.44	−0.277697	−0.138849	+	0.990314i	$0.544340\pi$
$728$	9206.54	0.468705
$729$	−23698.5	−1.20401
$730$	3217.36	0.163123
$731$	8593.79	0.434819
$732$	−17085.5	−0.862704
$733$	33829.7	1.70468	0.852340	−	0.522988i	$-0.175183\pi$
$733$	33829.7	1.70468	0.852340	+	0.522988i	$0.175183\pi$
$734$	6565.72	0.330171
$735$	8974.50	0.450380
$736$	−75265.3	−3.76945
$737$	9533.88	0.476506
$738$	−17474.2	−0.871590
$739$	1935.05	0.0963219	0.0481610	−	0.998840i	$-0.484664\pi$
$739$	1935.05	0.0963219	0.0481610	+	0.998840i	$0.484664\pi$
$740$	37439.0	1.85984
$741$	−108.512	−0.00537960
$742$	27928.3	1.38178
$743$	−23913.4	−1.18075	−0.590375	−	0.807129i	$-0.701020\pi$
$743$	−23913.4	−1.18075	−0.590375	+	0.807129i	$0.701020\pi$
$744$	16156.8	0.796153
$745$	−3415.39	−0.167960
$746$	−11455.1	−0.562197
$747$	16684.2	0.817194
$748$	−24155.9	−1.18079
$749$	16496.7	0.804776
$750$	5066.62	0.246676
$751$	−16732.1	−0.813001	−0.406500	−	0.913651i	$-0.633251\pi$
$751$	−16732.1	−0.813001	−0.406500	+	0.913651i	$0.633251\pi$
$752$	−52632.2	−2.55226
$753$	−14197.6	−0.687106
$754$	10198.9	0.492601
$755$	−10584.9	−0.510229
$756$	−4728.41	−0.227474
$757$	30909.1	1.48403	0.742016	−	0.670382i	$-0.233870\pi$
$757$	30909.1	1.48403	0.742016	+	0.670382i	$0.233870\pi$
$758$	23954.3	1.14783
$759$	−20867.3	−0.997938
$760$	−382.079	−0.0182362
$761$	22707.3	1.08165	0.540827	−	0.841134i	$-0.318111\pi$
$761$	22707.3	1.08165	0.540827	+	0.841134i	$0.318111\pi$
$762$	54907.9	2.61037
$763$	−6985.62	−0.331450
$764$	49409.9	2.33977
$765$	−8664.02	−0.409475
$766$	28642.0	1.35102
$767$	5279.67	0.248550
$768$	26623.8	1.25091
$769$	−41886.5	−1.96420	−0.982098	−	0.188369i	$-0.939680\pi$
$769$	−41886.5	−1.96420	−0.982098	+	0.188369i	$0.939680\pi$
$770$	5508.64	0.257815
$771$	−4873.42	−0.227642
$772$	97943.7	4.56615
$773$	−14669.1	−0.682551	−0.341276	−	0.939963i	$-0.610859\pi$
$773$	−14669.1	−0.682551	−0.341276	+	0.939963i	$0.610859\pi$
$774$	−23867.0	−1.10837
$775$	−775.000	−0.0359211
$776$	−11697.9	−0.541148
$777$	27770.1	1.28217
$778$	−35627.9	−1.64180
$779$	−120.229	−0.00552972
$780$	−10229.8	−0.469596
$781$	−9363.02	−0.428982
$782$	−42999.1	−1.96630
$783$	−3229.06	−0.147378
$784$	−48577.4	−2.21289
$785$	4546.24	0.206703
$786$	−44631.5	−2.02539
$787$	−235.281	−0.0106567	−0.00532837	−	0.999986i	$-0.501696\pi$
$787$	−235.281	−0.0106567	−0.00532837	+	0.999986i	$0.501696\pi$
$788$	−26233.4	−1.18595
$789$	23669.2	1.06799
$790$	−11608.7	−0.522810
$791$	10347.5	0.465125
$792$	41356.3	1.85547
$793$	−1411.30	−0.0631989
$794$	40635.8	1.81626
$795$	−19130.2	−0.853433
$796$	38078.2	1.69554
$797$	−29924.0	−1.32994	−0.664970	−	0.746870i	$-0.731556\pi$
$797$	−29924.0	−1.32994	−0.664970	+	0.746870i	$0.731556\pi$
$798$	−459.728	−0.0203937
$799$	−14921.8	−0.660697
$800$	−13609.7	−0.601468
$801$	15173.0	0.669302
$802$	7632.87	0.336067
$803$	−2396.08	−0.105300
$804$	−75009.4	−3.29027
$805$	7087.42	0.310309
$806$	2164.92	0.0946103
$807$	−29362.0	−1.28078
$808$	−74656.7	−3.25051
$809$	−27189.8	−1.18164	−0.590818	−	0.806805i	$-0.701195\pi$
$809$	−27189.8	−1.18164	−0.590818	+	0.806805i	$0.701195\pi$
$810$	−17225.1	−0.747194
$811$	−5851.99	−0.253380	−0.126690	−	0.991942i	$-0.540435\pi$
$811$	−5851.99	−0.253380	−0.126690	+	0.991942i	$0.540435\pi$
$812$	31230.9	1.34974
$813$	14764.7	0.636925
$814$	−38575.9	−1.66104
$815$	20175.3	0.867127
$816$	89202.1	3.82683
$817$	−164.214	−0.00703197
$818$	1702.06	0.0727521
$819$	−3989.22	−0.170201
$820$	−11334.4	−0.482700
$821$	−2646.10	−0.112484	−0.0562421	−	0.998417i	$-0.517912\pi$
$821$	−2646.10	−0.112484	−0.0562421	+	0.998417i	$0.517912\pi$
$822$	106245.	4.50817
$823$	−8521.32	−0.360917	−0.180458	−	0.983583i	$-0.557758\pi$
$823$	−8521.32	−0.360917	−0.180458	+	0.983583i	$0.557758\pi$
$824$	35838.1	1.51514
$825$	−3773.28	−0.159235
$826$	22368.2	0.942239
$827$	−12587.4	−0.529271	−0.264636	−	0.964349i	$-0.585252\pi$
$827$	−12587.4	−0.529271	−0.264636	+	0.964349i	$0.585252\pi$
$828$	86313.8	3.62272
$829$	26356.0	1.10420	0.552101	−	0.833777i	$-0.313826\pi$
$829$	26356.0	1.10420	0.552101	+	0.833777i	$0.313826\pi$
$830$	14972.7	0.626155
$831$	−43442.5	−1.81348
$832$	16780.5	0.699229
$833$	−13772.2	−0.572845
$834$	10368.3	0.430484
$835$	9651.67	0.400012
$836$	461.582	0.0190959
$837$	−685.433	−0.0283059
$838$	23036.9	0.949639
$839$	39500.8	1.62541	0.812704	−	0.582676i	$-0.197994\pi$
$839$	39500.8	1.62541	0.812704	+	0.582676i	$0.197994\pi$
$840$	−26717.5	−1.09743
$841$	−3061.24	−0.125517
$842$	−75458.3	−3.08844
$843$	−49325.6	−2.01526
$844$	−85810.1	−3.49965
$845$	−845.000	−0.0344010
$846$	41441.4	1.68414
$847$	9543.66	0.387159
$848$	103549.	4.19324
$849$	2316.69	0.0936496
$850$	−7775.22	−0.313750
$851$	−49631.9	−1.99925
$852$	73665.1	2.96212
$853$	−12241.8	−0.491385	−0.245692	−	0.969348i	$-0.579015\pi$
$853$	−12241.8	−0.491385	−0.245692	+	0.969348i	$0.579015\pi$
$854$	−5979.20	−0.239583
$855$	165.556	0.00662210
$856$	111145.	4.43791
$857$	−2523.43	−0.100582	−0.0502910	−	0.998735i	$-0.516015\pi$
$857$	−2523.43	−0.100582	−0.0502910	+	0.998735i	$0.516015\pi$
$858$	10540.4	0.419399
$859$	32904.1	1.30695	0.653477	−	0.756946i	$-0.273310\pi$
$859$	32904.1	1.30695	0.653477	+	0.756946i	$0.273310\pi$
$860$	−15481.0	−0.613834
$861$	−8407.20	−0.332772
$862$	−37101.6	−1.46599
$863$	−1690.00	−0.0666607	−0.0333303	−	0.999444i	$-0.510611\pi$
$863$	−1690.00	−0.0666607	−0.0333303	+	0.999444i	$0.510611\pi$
$864$	−12036.8	−0.473958
$865$	−8973.07	−0.352709
$866$	48661.2	1.90944
$867$	−11779.9	−0.461437
$868$	6629.39	0.259235
$869$	8645.40	0.337486
$870$	−29597.2	−1.15338
$871$	−6195.93	−0.241034
$872$	−47064.7	−1.82777
$873$	5068.74	0.196507
$874$	821.646	0.0317993
$875$	1281.57	0.0495141
$876$	18851.5	0.727093
$877$	50982.5	1.96300	0.981502	−	0.191450i	$-0.0613189\pi$
$877$	50982.5	1.96300	0.981502	+	0.191450i	$0.0613189\pi$
$878$	78923.3	3.03364
$879$	−20357.8	−0.781175
$880$	20424.1	0.782382
$881$	−37519.1	−1.43479	−0.717395	−	0.696667i	$-0.754666\pi$
$881$	−37519.1	−1.43479	−0.717395	+	0.696667i	$0.754666\pi$
$882$	38248.7	1.46021
$883$	−32223.3	−1.22809	−0.614044	−	0.789272i	$-0.710458\pi$
$883$	−32223.3	−1.22809	−0.614044	+	0.789272i	$0.710458\pi$
$884$	15698.6	0.597286
$885$	−15321.7	−0.581957
$886$	−36972.3	−1.40193
$887$	−19027.4	−0.720269	−0.360134	−	0.932900i	$-0.617269\pi$
$887$	−19027.4	−0.720269	−0.360134	+	0.932900i	$0.617269\pi$
$888$	187097.	7.07047
$889$	13888.6	0.523969
$890$	13616.5	0.512837
$891$	12828.1	0.482331
$892$	2517.28	0.0944895
$893$	285.133	0.0106849
$894$	−27687.2	−1.03579
$895$	6741.98	0.251798
$896$	26442.5	0.985916
$897$	13561.3	0.504794
$898$	−38529.7	−1.43180
$899$	4527.25	0.167956
$900$	15607.5	0.578056
$901$	29357.2	1.08549
$902$	11678.6	0.431103
$903$	−11482.9	−0.423175
$904$	69714.9	2.56491
$905$	15391.0	0.565320
$906$	−85807.2	−3.14653
$907$	−26743.4	−0.979052	−0.489526	−	0.871989i	$-0.662830\pi$
$907$	−26743.4	−0.979052	−0.489526	+	0.871989i	$0.662830\pi$
$908$	−95363.0	−3.48539
$909$	32348.9	1.18036
$910$	−3579.98	−0.130412
$911$	1964.74	0.0714543	0.0357271	−	0.999362i	$-0.488625\pi$
$911$	1964.74	0.0714543	0.0357271	+	0.999362i	$0.488625\pi$
$912$	−1704.51	−0.0618882
$913$	−11150.7	−0.404198
$914$	95102.8	3.44171
$915$	4095.60	0.147974
$916$	5040.58	0.181818
$917$	−11289.2	−0.406547
$918$	−6876.63	−0.247236
$919$	−25013.2	−0.897833	−0.448917	−	0.893574i	$-0.648190\pi$
$919$	−25013.2	−0.897833	−0.448917	+	0.893574i	$0.648190\pi$
$920$	47750.6	1.71119
$921$	−23342.5	−0.835139
$922$	38157.3	1.36295
$923$	6084.89	0.216995
$924$	32276.8	1.14917
$925$	−8974.57	−0.319008
$926$	−9565.67	−0.339468
$927$	−15528.7	−0.550194
$928$	79502.4	2.81228
$929$	35069.5	1.23853	0.619264	−	0.785183i	$-0.287431\pi$
$929$	35069.5	1.23853	0.619264	+	0.785183i	$0.287431\pi$
$930$	−6282.61	−0.221521
$931$	263.166	0.00926414
$932$	25445.4	0.894303
$933$	−38907.3	−1.36524
$934$	−85874.6	−3.00846
$935$	5790.47	0.202533
$936$	−26876.8	−0.938565
$937$	−23176.0	−0.808033	−0.404017	−	0.914752i	$-0.632386\pi$
$937$	−23176.0	−0.808033	−0.404017	+	0.914752i	$0.632386\pi$
$938$	−26250.0	−0.913747
$939$	−73529.1	−2.55541
$940$	26880.4	0.932705
$941$	−16514.8	−0.572123	−0.286062	−	0.958211i	$-0.592346\pi$
$941$	−16514.8	−0.572123	−0.286062	+	0.958211i	$0.592346\pi$
$942$	36854.5	1.27472
$943$	15025.7	0.518881
$944$	82933.5	2.85938
$945$	1133.46	0.0390173
$946$	15951.1	0.548219
$947$	19593.7	0.672345	0.336173	−	0.941800i	$-0.390867\pi$
$947$	19593.7	0.672345	0.336173	+	0.941800i	$0.390867\pi$
$948$	−68019.1	−2.33034
$949$	1557.17	0.0532645
$950$	148.572	0.00507402
$951$	−1004.88	−0.0342645
$952$	41000.6	1.39584
$953$	29884.9	1.01581	0.507905	−	0.861413i	$-0.330420\pi$
$953$	29884.9	1.01581	0.507905	+	0.861413i	$0.330420\pi$
$954$	−81531.8	−2.76697
$955$	−11844.1	−0.401327
$956$	10260.6	0.347125
$957$	22042.0	0.744533
$958$	−26783.0	−0.903257
$959$	26873.9	0.904905
$960$	−48697.1	−1.63718
$961$	961.000	0.0322581
$962$	25069.9	0.840215
$963$	−48159.2	−1.61154
$964$	100362.	3.35315
$965$	−23478.3	−0.783204
$966$	57454.8	1.91364
$967$	−22287.8	−0.741186	−0.370593	−	0.928795i	$-0.620846\pi$
$967$	−22287.8	−0.741186	−0.370593	+	0.928795i	$0.620846\pi$
$968$	64299.2	2.13497
$969$	−483.249	−0.0160208
$970$	4548.76	0.150569
$971$	26242.1	0.867301	0.433650	−	0.901081i	$-0.357225\pi$
$971$	26242.1	0.867301	0.433650	+	0.901081i	$0.357225\pi$
$972$	−113379.	−3.74140
$973$	2622.58	0.0864092
$974$	−19002.1	−0.625119
$975$	2452.20	0.0805469
$976$	−22168.8	−0.727055
$977$	−18442.2	−0.603907	−0.301953	−	0.953323i	$-0.597639\pi$
$977$	−18442.2	−0.603907	−0.301953	+	0.953323i	$0.597639\pi$
$978$	163552.	5.34748
$979$	−10140.6	−0.331048
$980$	24809.5	0.808685
$981$	20393.3	0.663717
$982$	7191.39	0.233693
$983$	−8692.82	−0.282053	−0.141026	−	0.990006i	$-0.545040\pi$
$983$	−8692.82	−0.282053	−0.141026	+	0.990006i	$0.545040\pi$
$984$	−56642.5	−1.83506
$985$	6288.45	0.203418
$986$	45419.8	1.46700
$987$	19938.3	0.643003
$988$	−299.975	−0.00965940
$989$	20522.7	0.659844
$990$	−16081.5	−0.516265
$991$	−22709.4	−0.727940	−0.363970	−	0.931411i	$-0.618579\pi$
$991$	−22709.4	−0.727940	−0.363970	+	0.931411i	$0.618579\pi$
$992$	16876.0	0.540134
$993$	3635.35	0.116178
$994$	25779.6	0.822615
$995$	−9127.80	−0.290825
$996$	87729.6	2.79098
$997$	−20520.8	−0.651857	−0.325929	−	0.945394i	$-0.605677\pi$
$997$	−20520.8	−0.651857	−0.325929	+	0.945394i	$0.605677\pi$
$998$	65386.6	2.07392
$999$	−7937.37	−0.251379

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2015.4.a.c.1.1	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2015.4.a.c.1.1	✓	40	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2015 = 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2015.a (trivial)

\(f(q)\)	\(=\)	\(q-5.37200 q^{2} +7.54523 q^{3} +20.8584 q^{4} -5.00000 q^{5} -40.5329 q^{6} -10.2525 q^{7} -69.0752 q^{8} +29.9304 q^{9} +O(q^{10})\)	\(q-5.37200 q^{2} +7.54523 q^{3} +20.8584 q^{4} -5.00000 q^{5} -40.5329 q^{6} -10.2525 q^{7} -69.0752 q^{8} +29.9304 q^{9} +26.8600 q^{10} -20.0035 q^{11} +157.381 q^{12} +13.0000 q^{13} +55.0766 q^{14} -37.7261 q^{15} +204.205 q^{16} +57.8944 q^{17} -160.786 q^{18} -1.10627 q^{19} -104.292 q^{20} -77.3577 q^{21} +107.459 q^{22} +138.257 q^{23} -521.188 q^{24} +25.0000 q^{25} -69.8360 q^{26} +22.1107 q^{27} -213.851 q^{28} -146.040 q^{29} +202.665 q^{30} -31.0000 q^{31} -544.387 q^{32} -150.931 q^{33} -311.009 q^{34} +51.2627 q^{35} +624.300 q^{36} -358.983 q^{37} +5.94289 q^{38} +98.0879 q^{39} +345.376 q^{40} +108.680 q^{41} +415.565 q^{42} +148.439 q^{43} -417.241 q^{44} -149.652 q^{45} -742.716 q^{46} -257.742 q^{47} +1540.77 q^{48} -237.886 q^{49} -134.300 q^{50} +436.826 q^{51} +271.159 q^{52} +507.082 q^{53} -118.779 q^{54} +100.018 q^{55} +708.196 q^{56} -8.34707 q^{57} +784.528 q^{58} +406.129 q^{59} -786.906 q^{60} -108.561 q^{61} +166.532 q^{62} -306.863 q^{63} +1290.81 q^{64} -65.0000 q^{65} +810.802 q^{66} -476.610 q^{67} +1207.58 q^{68} +1043.18 q^{69} -275.383 q^{70} +468.068 q^{71} -2067.45 q^{72} +119.783 q^{73} +1928.45 q^{74} +188.631 q^{75} -23.0750 q^{76} +205.087 q^{77} -526.928 q^{78} -432.194 q^{79} -1021.02 q^{80} -641.291 q^{81} -583.827 q^{82} +557.434 q^{83} -1613.56 q^{84} -289.472 q^{85} -797.415 q^{86} -1101.91 q^{87} +1381.75 q^{88} +506.942 q^{89} +803.931 q^{90} -133.283 q^{91} +2883.82 q^{92} -233.902 q^{93} +1384.59 q^{94} +5.53136 q^{95} -4107.52 q^{96} +169.351 q^{97} +1277.92 q^{98} -598.715 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 9 q^{2} + q^{3} + 149 q^{4} - 200 q^{5} + 11 q^{6} - 64 q^{7} - 87 q^{8} + 247 q^{9}+O(q^{10}) \) 40 * q - 9 * q^2 + q^3 + 149 * q^4 - 200 * q^5 + 11 * q^6 - 64 * q^7 - 87 * q^8 + 247 * q^9	\( 40 q - 9 q^{2} + q^{3} + 149 q^{4} - 200 q^{5} + 11 q^{6} - 64 q^{7} - 87 q^{8} + 247 q^{9} + 45 q^{10} - q^{11} - 226 q^{12} + 520 q^{13} + 138 q^{14} - 5 q^{15} + 413 q^{16} + 6 q^{17} - 306 q^{18} - 265 q^{19} - 745 q^{20} - 344 q^{21} - 313 q^{22} + 24 q^{23} - 321 q^{24} + 1000 q^{25} - 117 q^{26} + 28 q^{27} - 675 q^{28} - 9 q^{29} - 55 q^{30} - 1240 q^{31} - 160 q^{32} - 986 q^{33} - 298 q^{34} + 320 q^{35} + 238 q^{36} - 1403 q^{37} - 314 q^{38} + 13 q^{39} + 435 q^{40} - 84 q^{41} + 744 q^{42} - 421 q^{43} + 812 q^{44} - 1235 q^{45} - 527 q^{46} - 460 q^{47} - 1573 q^{48} + 748 q^{49} - 225 q^{50} - 160 q^{51} + 1937 q^{52} - 649 q^{53} + 1184 q^{54} + 5 q^{55} + 1518 q^{56} - 468 q^{57} - 1111 q^{58} + 49 q^{59} + 1130 q^{60} - 161 q^{61} + 279 q^{62} - 244 q^{63} + 421 q^{64} - 2600 q^{65} + 2027 q^{66} - 1981 q^{67} + 2041 q^{68} + 1664 q^{69} - 690 q^{70} - 1510 q^{71} - 876 q^{72} - 3562 q^{73} - 1005 q^{74} + 25 q^{75} - 2776 q^{76} - 1096 q^{77} + 143 q^{78} - 1094 q^{79} - 2065 q^{80} + 712 q^{81} + 228 q^{82} - 817 q^{83} - 1837 q^{84} - 30 q^{85} + 1125 q^{86} - 1288 q^{87} - 4011 q^{88} - 2326 q^{89} + 1530 q^{90} - 832 q^{91} + 4425 q^{92} - 31 q^{93} - 2590 q^{94} + 1325 q^{95} - 3706 q^{96} - 7166 q^{97} - 7363 q^{98} - 2161 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + q^3 + 149 * q^4 - 200 * q^5 + 11 * q^6 - 64 * q^7 - 87 * q^8 + 247 * q^9 + 45 * q^10 - q^11 - 226 * q^12 + 520 * q^13 + 138 * q^14 - 5 * q^15 + 413 * q^16 + 6 * q^17 - 306 * q^18 - 265 * q^19 - 745 * q^20 - 344 * q^21 - 313 * q^22 + 24 * q^23 - 321 * q^24 + 1000 * q^25 - 117 * q^26 + 28 * q^27 - 675 * q^28 - 9 * q^29 - 55 * q^30 - 1240 * q^31 - 160 * q^32 - 986 * q^33 - 298 * q^34 + 320 * q^35 + 238 * q^36 - 1403 * q^37 - 314 * q^38 + 13 * q^39 + 435 * q^40 - 84 * q^41 + 744 * q^42 - 421 * q^43 + 812 * q^44 - 1235 * q^45 - 527 * q^46 - 460 * q^47 - 1573 * q^48 + 748 * q^49 - 225 * q^50 - 160 * q^51 + 1937 * q^52 - 649 * q^53 + 1184 * q^54 + 5 * q^55 + 1518 * q^56 - 468 * q^57 - 1111 * q^58 + 49 * q^59 + 1130 * q^60 - 161 * q^61 + 279 * q^62 - 244 * q^63 + 421 * q^64 - 2600 * q^65 + 2027 * q^66 - 1981 * q^67 + 2041 * q^68 + 1664 * q^69 - 690 * q^70 - 1510 * q^71 - 876 * q^72 - 3562 * q^73 - 1005 * q^74 + 25 * q^75 - 2776 * q^76 - 1096 * q^77 + 143 * q^78 - 1094 * q^79 - 2065 * q^80 + 712 * q^81 + 228 * q^82 - 817 * q^83 - 1837 * q^84 - 30 * q^85 + 1125 * q^86 - 1288 * q^87 - 4011 * q^88 - 2326 * q^89 + 1530 * q^90 - 832 * q^91 + 4425 * q^92 - 31 * q^93 - 2590 * q^94 + 1325 * q^95 - 3706 * q^96 - 7166 * q^97 - 7363 * q^98 - 2161 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more