LMFDB - Embedded newform 147.6.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,6,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("147.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	147.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:	$23.5764215125$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 $ x^6 - 2x^5 - 59x^4 + 122x^3 + 941x^2 - 1856*x - 2338
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-5.10089$ of defining polynomial
Character	$\chi$	$=$	147.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.38033 q^{2} -9.00000 q^{3} -20.5734 q^{4} +54.5253 q^{5} -30.4230 q^{6} -177.715 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+3.38033 q^{2} -9.00000 q^{3} -20.5734 q^{4} +54.5253 q^{5} -30.4230 q^{6} -177.715 q^{8} +81.0000 q^{9} +184.313 q^{10} +481.911 q^{11} +185.160 q^{12} -512.622 q^{13} -490.728 q^{15} +57.6111 q^{16} -590.744 q^{17} +273.807 q^{18} -2451.02 q^{19} -1121.77 q^{20} +1629.02 q^{22} +1774.90 q^{23} +1599.44 q^{24} -151.993 q^{25} -1732.83 q^{26} -729.000 q^{27} -4246.44 q^{29} -1658.82 q^{30} -9767.37 q^{31} +5881.64 q^{32} -4337.20 q^{33} -1996.91 q^{34} -1666.44 q^{36} -9969.65 q^{37} -8285.24 q^{38} +4613.60 q^{39} -9689.98 q^{40} +3377.54 q^{41} -18223.4 q^{43} -9914.53 q^{44} +4416.55 q^{45} +5999.76 q^{46} -1320.64 q^{47} -518.500 q^{48} -513.788 q^{50} +5316.69 q^{51} +10546.4 q^{52} +34837.0 q^{53} -2464.26 q^{54} +26276.3 q^{55} +22059.1 q^{57} -14354.4 q^{58} -11592.5 q^{59} +10095.9 q^{60} -31406.1 q^{61} -33016.9 q^{62} +18038.3 q^{64} -27950.9 q^{65} -14661.2 q^{66} +27555.9 q^{67} +12153.6 q^{68} -15974.1 q^{69} -22868.5 q^{71} -14394.9 q^{72} -15936.4 q^{73} -33700.7 q^{74} +1367.94 q^{75} +50425.6 q^{76} +15595.5 q^{78} +87165.3 q^{79} +3141.26 q^{80} +6561.00 q^{81} +11417.2 q^{82} -90307.5 q^{83} -32210.5 q^{85} -61601.2 q^{86} +38218.0 q^{87} -85643.0 q^{88} +126570. q^{89} +14929.4 q^{90} -36515.7 q^{92} +87906.4 q^{93} -4464.20 q^{94} -133642. q^{95} -52934.7 q^{96} +16573.1 q^{97} +39034.8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} - 54 q^{3} + 150 q^{4} - 100 q^{5} - 18 q^{6} - 114 q^{8} + 486 q^{9}+O(q^{10}) $ 6 * q + 2 * q^2 - 54 * q^3 + 150 * q^4 - 100 * q^5 - 18 * q^6 - 114 * q^8 + 486 * q^9	\( 6 q + 2 q^{2} - 54 q^{3} + 150 q^{4} - 100 q^{5} - 18 q^{6} - 114 q^{8} + 486 q^{9} - 864 q^{10} + 604 q^{11} - 1350 q^{12} - 1352 q^{13} + 900 q^{15} + 4578 q^{16} - 3028 q^{17} + 162 q^{18} - 1728 q^{19} - 452 q^{20} - 4116 q^{22} - 4484 q^{23} + 1026 q^{24} + 4806 q^{25} - 14172 q^{26} - 4374 q^{27} - 5320 q^{29} + 7776 q^{30} - 3976 q^{31} - 37326 q^{32} - 5436 q^{33} + 16336 q^{34} + 12150 q^{36} + 22680 q^{37} - 52744 q^{38} + 12168 q^{39} - 100600 q^{40} - 28756 q^{41} - 6768 q^{43} - 64940 q^{44} - 8100 q^{45} + 540 q^{46} - 51552 q^{47} - 41202 q^{48} - 40622 q^{50} + 27252 q^{51} - 119296 q^{52} + 80884 q^{53} - 1458 q^{54} - 11656 q^{55} + 15552 q^{57} - 70464 q^{58} - 8872 q^{59} + 4068 q^{60} - 50896 q^{61} - 11824 q^{62} + 199590 q^{64} + 3492 q^{65} + 37044 q^{66} + 6480 q^{67} - 37348 q^{68} + 40356 q^{69} - 110852 q^{71} - 9234 q^{72} - 64232 q^{73} - 27464 q^{74} - 43254 q^{75} + 194864 q^{76} + 127548 q^{78} + 111696 q^{79} + 308940 q^{80} + 39366 q^{81} + 189640 q^{82} - 101128 q^{83} - 23292 q^{85} + 3824 q^{86} + 47880 q^{87} - 97788 q^{88} + 35012 q^{89} - 69984 q^{90} - 449260 q^{92} + 35784 q^{93} + 121016 q^{94} - 119080 q^{95} + 335934 q^{96} - 70952 q^{97} + 48924 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 - 54 * q^3 + 150 * q^4 - 100 * q^5 - 18 * q^6 - 114 * q^8 + 486 * q^9 - 864 * q^10 + 604 * q^11 - 1350 * q^12 - 1352 * q^13 + 900 * q^15 + 4578 * q^16 - 3028 * q^17 + 162 * q^18 - 1728 * q^19 - 452 * q^20 - 4116 * q^22 - 4484 * q^23 + 1026 * q^24 + 4806 * q^25 - 14172 * q^26 - 4374 * q^27 - 5320 * q^29 + 7776 * q^30 - 3976 * q^31 - 37326 * q^32 - 5436 * q^33 + 16336 * q^34 + 12150 * q^36 + 22680 * q^37 - 52744 * q^38 + 12168 * q^39 - 100600 * q^40 - 28756 * q^41 - 6768 * q^43 - 64940 * q^44 - 8100 * q^45 + 540 * q^46 - 51552 * q^47 - 41202 * q^48 - 40622 * q^50 + 27252 * q^51 - 119296 * q^52 + 80884 * q^53 - 1458 * q^54 - 11656 * q^55 + 15552 * q^57 - 70464 * q^58 - 8872 * q^59 + 4068 * q^60 - 50896 * q^61 - 11824 * q^62 + 199590 * q^64 + 3492 * q^65 + 37044 * q^66 + 6480 * q^67 - 37348 * q^68 + 40356 * q^69 - 110852 * q^71 - 9234 * q^72 - 64232 * q^73 - 27464 * q^74 - 43254 * q^75 + 194864 * q^76 + 127548 * q^78 + 111696 * q^79 + 308940 * q^80 + 39366 * q^81 + 189640 * q^82 - 101128 * q^83 - 23292 * q^85 + 3824 * q^86 + 47880 * q^87 - 97788 * q^88 + 35012 * q^89 - 69984 * q^90 - 449260 * q^92 + 35784 * q^93 + 121016 * q^94 - 119080 * q^95 + 335934 * q^96 - 70952 * q^97 + 48924 * 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 $

$2$	3.38033	0.597564	0.298782	−	0.954321i	$-0.403420\pi$
$2$	3.38033	0.597564	0.298782	+	0.954321i	$0.403420\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−20.5734	−0.642918
$5$	54.5253	0.975378	0.487689	−	0.873017i	$-0.337840\pi$
$5$	54.5253	0.975378	0.487689	+	0.873017i	$0.337840\pi$
$6$	−30.4230	−0.345004
$7$	0	0
$7$	0	0
$8$	−177.715	−0.981748
$9$	81.0000	0.333333
$10$	184.313	0.582850
$11$	481.911	1.20084	0.600420	−	0.799685i	$-0.295000\pi$
$11$	481.911	1.20084	0.600420	+	0.799685i	$0.295000\pi$
$12$	185.160	0.371189
$13$	−512.622	−0.841277	−0.420638	−	0.907228i	$-0.638194\pi$
$13$	−512.622	−0.841277	−0.420638	+	0.907228i	$0.638194\pi$
$14$	0	0
$15$	−490.728	−0.563135
$16$	57.6111	0.0562608
$17$	−590.744	−0.495766	−0.247883	−	0.968790i	$-0.579735\pi$
$17$	−590.744	−0.495766	−0.247883	+	0.968790i	$0.579735\pi$
$18$	273.807	0.199188
$19$	−2451.02	−1.55762	−0.778811	−	0.627259i	$-0.784177\pi$
$19$	−2451.02	−1.55762	−0.778811	+	0.627259i	$0.784177\pi$
$20$	−1121.77	−0.627088
$21$	0	0
$22$	1629.02	0.717579
$23$	1774.90	0.699609	0.349804	−	0.936823i	$-0.386248\pi$
$23$	1774.90	0.699609	0.349804	+	0.936823i	$0.386248\pi$
$24$	1599.44	0.566812
$25$	−151.993	−0.0486379
$26$	−1732.83	−0.502716
$27$	−729.000	−0.192450
$28$	0	0
$29$	−4246.44	−0.937627	−0.468814	−	0.883297i	$-0.655318\pi$
$29$	−4246.44	−0.937627	−0.468814	+	0.883297i	$0.655318\pi$
$30$	−1658.82	−0.336509
$31$	−9767.37	−1.82547	−0.912733	−	0.408558i	$-0.866032\pi$
$31$	−9767.37	−1.82547	−0.912733	+	0.408558i	$0.866032\pi$
$32$	5881.64	1.01537
$33$	−4337.20	−0.693306
$34$	−1996.91	−0.296252
$35$	0	0
$36$	−1666.44	−0.214306
$37$	−9969.65	−1.19722	−0.598612	−	0.801039i	$-0.704281\pi$
$37$	−9969.65	−1.19722	−0.598612	+	0.801039i	$0.704281\pi$
$38$	−8285.24	−0.930778
$39$	4613.60	0.485711
$40$	−9689.98	−0.957575
$41$	3377.54	0.313791	0.156896	−	0.987615i	$-0.449851\pi$
$41$	3377.54	0.313791	0.156896	+	0.987615i	$0.449851\pi$
$42$	0	0
$43$	−18223.4	−1.50300	−0.751500	−	0.659733i	$-0.770669\pi$
$43$	−18223.4	−1.50300	−0.751500	+	0.659733i	$0.770669\pi$
$44$	−9914.53	−0.772042
$45$	4416.55	0.325126
$46$	5999.76	0.418061
$47$	−1320.64	−0.0872046	−0.0436023	−	0.999049i	$-0.513883\pi$
$47$	−1320.64	−0.0872046	−0.0436023	+	0.999049i	$0.513883\pi$
$48$	−518.500	−0.0324822
$49$	0	0
$50$	−513.788	−0.0290642
$51$	5316.69	0.286231
$52$	10546.4	0.540872
$53$	34837.0	1.70353	0.851767	−	0.523921i	$-0.175531\pi$
$53$	34837.0	1.70353	0.851767	+	0.523921i	$0.175531\pi$
$54$	−2464.26	−0.115001
$55$	26276.3	1.17127
$56$	0	0
$57$	22059.1	0.899293
$58$	−14354.4	−0.560292
$59$	−11592.5	−0.433558	−0.216779	−	0.976221i	$-0.569555\pi$
$59$	−11592.5	−0.433558	−0.216779	+	0.976221i	$0.569555\pi$
$60$	10095.9	0.362049
$61$	−31406.1	−1.08066	−0.540331	−	0.841452i	$-0.681701\pi$
$61$	−31406.1	−1.08066	−0.540331	+	0.841452i	$0.681701\pi$
$62$	−33016.9	−1.09083
$63$	0	0
$64$	18038.3	0.550486
$65$	−27950.9	−0.820563
$66$	−14661.2	−0.414294
$67$	27555.9	0.749942	0.374971	−	0.927036i	$-0.377653\pi$
$67$	27555.9	0.749942	0.374971	+	0.927036i	$0.377653\pi$
$68$	12153.6	0.318737
$69$	−15974.1	−0.403919
$70$	0	0
$71$	−22868.5	−0.538384	−0.269192	−	0.963086i	$-0.586757\pi$
$71$	−22868.5	−0.538384	−0.269192	+	0.963086i	$0.586757\pi$
$72$	−14394.9	−0.327249
$73$	−15936.4	−0.350012	−0.175006	−	0.984567i	$-0.555995\pi$
$73$	−15936.4	−0.350012	−0.175006	+	0.984567i	$0.555995\pi$
$74$	−33700.7	−0.715418
$75$	1367.94	0.0280811
$76$	50425.6	1.00142
$77$	0	0
$78$	15595.5	0.290243
$79$	87165.3	1.57136	0.785680	−	0.618633i	$-0.212313\pi$
$79$	87165.3	1.57136	0.785680	+	0.618633i	$0.212313\pi$
$80$	3141.26	0.0548755
$81$	6561.00	0.111111
$82$	11417.2	0.187510
$83$	−90307.5	−1.43889	−0.719447	−	0.694548i	$-0.755605\pi$
$83$	−90307.5	−1.43889	−0.719447	+	0.694548i	$0.755605\pi$
$84$	0	0
$85$	−32210.5	−0.483559
$86$	−61601.2	−0.898138
$87$	38218.0	0.541339
$88$	−85643.0	−1.17892
$89$	126570.	1.69377	0.846884	−	0.531777i	$-0.178476\pi$
$89$	126570.	1.69377	0.846884	+	0.531777i	$0.178476\pi$
$90$	14929.4	0.194283
$91$	0	0
$92$	−36515.7	−0.449791
$93$	87906.4	1.05393
$94$	−4464.20	−0.0521103
$95$	−133642.	−1.51927
$96$	−52934.7	−0.586223
$97$	16573.1	0.178844	0.0894219	−	0.995994i	$-0.471498\pi$
$97$	16573.1	0.178844	0.0894219	+	0.995994i	$0.471498\pi$
$98$	0	0
$99$	39034.8	0.400280
$100$	3127.02	0.0312702
$101$	76610.1	0.747279	0.373639	−	0.927574i	$-0.378110\pi$
$101$	76610.1	0.747279	0.373639	+	0.927574i	$0.378110\pi$
$102$	17972.2	0.171041
$103$	−80435.8	−0.747062	−0.373531	−	0.927618i	$-0.621853\pi$
$103$	−80435.8	−0.747062	−0.373531	+	0.927618i	$0.621853\pi$
$104$	91100.8	0.825922
$105$	0	0
$106$	117760.	1.01797
$107$	148042.	1.25005	0.625023	−	0.780606i	$-0.285090\pi$
$107$	148042.	1.25005	0.625023	+	0.780606i	$0.285090\pi$
$108$	14998.0	0.123730
$109$	−124982.	−1.00759	−0.503793	−	0.863825i	$-0.668062\pi$
$109$	−124982.	−1.00759	−0.503793	+	0.863825i	$0.668062\pi$
$110$	88822.7	0.699910
$111$	89726.9	0.691218
$112$	0	0
$113$	−194474.	−1.43273	−0.716366	−	0.697724i	$-0.754196\pi$
$113$	−194474.	−1.43273	−0.716366	+	0.697724i	$0.754196\pi$
$114$	74567.2	0.537385
$115$	96777.1	0.682383
$116$	87363.6	0.602817
$117$	−41522.4	−0.280426
$118$	−39186.5	−0.259078
$119$	0	0
$120$	87209.8	0.552856
$121$	71187.4	0.442018
$122$	−106163.	−0.645765
$123$	−30397.9	−0.181168
$124$	200948.	1.17362
$125$	−178679.	−1.02282
$126$	0	0
$127$	239766.	1.31910	0.659551	−	0.751660i	$-0.270746\pi$
$127$	239766.	1.31910	0.659551	+	0.751660i	$0.270746\pi$
$128$	−127237.	−0.686417
$129$	164011.	0.867758
$130$	−94483.2	−0.490339
$131$	349576.	1.77977	0.889883	−	0.456188i	$-0.150786\pi$
$131$	349576.	1.77977	0.889883	+	0.456188i	$0.150786\pi$
$132$	89230.8	0.445738
$133$	0	0
$134$	93148.1	0.448138
$135$	−39748.9	−0.187712
$136$	104984.	0.486717
$137$	−280966.	−1.27894	−0.639472	−	0.768814i	$-0.720847\pi$
$137$	−280966.	−1.27894	−0.639472	+	0.768814i	$0.720847\pi$
$138$	−53997.8	−0.241367
$139$	−50042.9	−0.219688	−0.109844	−	0.993949i	$-0.535035\pi$
$139$	−50042.9	−0.219688	−0.109844	+	0.993949i	$0.535035\pi$
$140$	0	0
$141$	11885.8	0.0503476
$142$	−77303.2	−0.321719
$143$	−247038.	−1.01024
$144$	4666.50	0.0187536
$145$	−231538.	−0.914541
$146$	−53870.3	−0.209155
$147$	0	0
$148$	205109.	0.769717
$149$	−328813.	−1.21334	−0.606671	−	0.794953i	$-0.707495\pi$
$149$	−328813.	−1.21334	−0.606671	+	0.794953i	$0.707495\pi$
$150$	4624.09	0.0167802
$151$	511304.	1.82489	0.912445	−	0.409200i	$-0.134192\pi$
$151$	511304.	1.82489	0.912445	+	0.409200i	$0.134192\pi$
$152$	435583.	1.52919
$153$	−47850.2	−0.165255
$154$	0	0
$155$	−532569.	−1.78052
$156$	−94917.3	−0.312272
$157$	75442.1	0.244267	0.122134	−	0.992514i	$-0.461026\pi$
$157$	75442.1	0.244267	0.122134	+	0.992514i	$0.461026\pi$
$158$	294647.	0.938988
$159$	−313533.	−0.983536
$160$	320698.	0.990367
$161$	0	0
$162$	22178.3	0.0663960
$163$	340753.	1.00455	0.502274	−	0.864709i	$-0.332497\pi$
$163$	340753.	1.00455	0.502274	+	0.864709i	$0.332497\pi$
$164$	−69487.4	−0.201742
$165$	−236487.	−0.676235
$166$	−305269.	−0.859831
$167$	132392.	0.367342	0.183671	−	0.982988i	$-0.441202\pi$
$167$	132392.	0.367342	0.183671	+	0.982988i	$0.441202\pi$
$168$	0	0
$169$	−108512.	−0.292253
$170$	−108882.	−0.288957
$171$	−198532.	−0.519207
$172$	374917.	0.966306
$173$	−449322.	−1.14141	−0.570706	−	0.821154i	$-0.693330\pi$
$173$	−449322.	−1.14141	−0.570706	+	0.821154i	$0.693330\pi$
$174$	129189.	0.323485
$175$	0	0
$176$	27763.4	0.0675602
$177$	104332.	0.250315
$178$	427847.	1.01213
$179$	−93666.8	−0.218501	−0.109250	−	0.994014i	$-0.534845\pi$
$179$	−93666.8	−0.218501	−0.109250	+	0.994014i	$0.534845\pi$
$180$	−90863.3	−0.209029
$181$	−399795.	−0.907071	−0.453535	−	0.891238i	$-0.649837\pi$
$181$	−399795.	−0.907071	−0.453535	+	0.891238i	$0.649837\pi$
$182$	0	0
$183$	282655.	0.623921
$184$	−315428.	−0.686839
$185$	−543598.	−1.16775
$186$	297153.	0.629792
$187$	−284686.	−0.595336
$188$	27170.0	0.0560654
$189$	0	0
$190$	−451755.	−0.907861
$191$	27822.7	0.0551844	0.0275922	−	0.999619i	$-0.491216\pi$
$191$	27822.7	0.0551844	0.0275922	+	0.999619i	$0.491216\pi$
$192$	−162345.	−0.317823
$193$	−66464.3	−0.128438	−0.0642192	−	0.997936i	$-0.520456\pi$
$193$	−66464.3	−0.128438	−0.0642192	+	0.997936i	$0.520456\pi$
$194$	56022.5	0.106871
$195$	251558.	0.473752
$196$	0	0
$197$	−492506.	−0.904162	−0.452081	−	0.891977i	$-0.649318\pi$
$197$	−492506.	−0.904162	−0.452081	+	0.891977i	$0.649318\pi$
$198$	131951.	0.239193
$199$	−349206.	−0.625099	−0.312549	−	0.949902i	$-0.601183\pi$
$199$	−349206.	−0.625099	−0.312549	+	0.949902i	$0.601183\pi$
$200$	27011.6	0.0477501
$201$	−248003.	−0.432979
$202$	258967.	0.446547
$203$	0	0
$204$	−109382.	−0.184023
$205$	184161.	0.306065
$206$	−271900.	−0.446417
$207$	143767.	0.233203
$208$	−29532.7	−0.0473309
$209$	−1.18117e6	−1.87046
$210$	0	0
$211$	−218250.	−0.337480	−0.168740	−	0.985661i	$-0.553970\pi$
$211$	−218250.	−0.337480	−0.168740	+	0.985661i	$0.553970\pi$
$212$	−716714.	−1.09523
$213$	205817.	0.310836
$214$	500431.	0.746982
$215$	−993638.	−1.46599
$216$	129554.	0.188937
$217$	0	0
$218$	−422481.	−0.602096
$219$	143428.	0.202080
$220$	−540593.	−0.753032
$221$	302828.	0.417076
$222$	303306.	0.413047
$223$	−1.24807e6	−1.68065	−0.840324	−	0.542084i	$-0.817635\pi$
$223$	−1.24807e6	−1.68065	−0.840324	+	0.542084i	$0.817635\pi$
$224$	0	0
$225$	−12311.5	−0.0162126
$226$	−657386.	−0.856149
$227$	520973.	0.671044	0.335522	−	0.942032i	$-0.391087\pi$
$227$	520973.	0.671044	0.335522	+	0.942032i	$0.391087\pi$
$228$	−453831.	−0.578172
$229$	525651.	0.662383	0.331191	−	0.943564i	$-0.392549\pi$
$229$	525651.	0.662383	0.331191	+	0.943564i	$0.392549\pi$
$230$	327139.	0.407767
$231$	0	0
$232$	754658.	0.920514
$233$	210205.	0.253661	0.126831	−	0.991924i	$-0.459520\pi$
$233$	210205.	0.253661	0.126831	+	0.991924i	$0.459520\pi$
$234$	−140359.	−0.167572
$235$	−72008.2	−0.0850575
$236$	238497.	0.278742
$237$	−784488.	−0.907225
$238$	0	0
$239$	805791.	0.912489	0.456244	−	0.889855i	$-0.349194\pi$
$239$	805791.	0.912489	0.456244	+	0.889855i	$0.349194\pi$
$240$	−28271.3	−0.0316824
$241$	−1.13795e6	−1.26206	−0.631030	−	0.775758i	$-0.717368\pi$
$241$	−1.13795e6	−1.26206	−0.631030	+	0.775758i	$0.717368\pi$
$242$	240637.	0.264134
$243$	−59049.0	−0.0641500
$244$	646130.	0.694777
$245$	0	0
$246$	−102755.	−0.108259
$247$	1.25644e6	1.31039
$248$	1.73581e6	1.79215
$249$	812768.	0.830746
$250$	−603994.	−0.611199
$251$	1.16566e6	1.16785	0.583927	−	0.811806i	$-0.301516\pi$
$251$	1.16566e6	1.16785	0.583927	+	0.811806i	$0.301516\pi$
$252$	0	0
$253$	855346.	0.840118
$254$	810489.	0.788248
$255$	289894.	0.279183
$256$	−1.00733e6	−0.960664
$257$	1.02881e6	0.971635	0.485817	−	0.874060i	$-0.338522\pi$
$257$	1.02881e6	0.971635	0.485817	+	0.874060i	$0.338522\pi$
$258$	554411.	0.518540
$259$	0	0
$260$	575043.	0.527554
$261$	−343962.	−0.312542
$262$	1.18168e6	1.06352
$263$	−657345.	−0.586009	−0.293004	−	0.956111i	$-0.594655\pi$
$263$	−657345.	−0.586009	−0.293004	+	0.956111i	$0.594655\pi$
$264$	770787.	0.680651
$265$	1.89950e6	1.66159
$266$	0	0
$267$	−1.13913e6	−0.977898
$268$	−566918.	−0.482151
$269$	−5475.86	−0.00461393	−0.00230697	−	0.999997i	$-0.500734\pi$
$269$	−5475.86	−0.00461393	−0.00230697	+	0.999997i	$0.500734\pi$
$270$	−134365.	−0.112170
$271$	769147.	0.636189	0.318094	−	0.948059i	$-0.396957\pi$
$271$	769147.	0.636189	0.318094	+	0.948059i	$0.396957\pi$
$272$	−34033.4	−0.0278922
$273$	0	0
$274$	−949757.	−0.764251
$275$	−73247.3	−0.0584063
$276$	328642.	0.259687
$277$	−1.17384e6	−0.919196	−0.459598	−	0.888127i	$-0.652007\pi$
$277$	−1.17384e6	−0.919196	−0.459598	+	0.888127i	$0.652007\pi$
$278$	−169162.	−0.131277
$279$	−791157.	−0.608488
$280$	0	0
$281$	837649.	0.632843	0.316422	−	0.948619i	$-0.397519\pi$
$281$	837649.	0.632843	0.316422	+	0.948619i	$0.397519\pi$
$282$	40177.8	0.0300859
$283$	597221.	0.443271	0.221635	−	0.975130i	$-0.428861\pi$
$283$	597221.	0.443271	0.221635	+	0.975130i	$0.428861\pi$
$284$	470483.	0.346137
$285$	1.20278e6	0.877151
$286$	−835071.	−0.603682
$287$	0	0
$288$	476412.	0.338456
$289$	−1.07088e6	−0.754216
$290$	−782677.	−0.546496
$291$	−149158.	−0.103256
$292$	327866.	0.225029
$293$	851321.	0.579328	0.289664	−	0.957128i	$-0.406457\pi$
$293$	851321.	0.579328	0.289664	+	0.957128i	$0.406457\pi$
$294$	0	0
$295$	−632084.	−0.422883
$296$	1.77176e6	1.17537
$297$	−351313.	−0.231102
$298$	−1.11150e6	−0.725049
$299$	−909855.	−0.588565
$300$	−28143.1	−0.0180538
$301$	0	0
$302$	1.72838e6	1.09049
$303$	−689491.	−0.431441
$304$	−141206.	−0.0876331
$305$	−1.71243e6	−1.05405
$306$	−161750.	−0.0987506
$307$	600906.	0.363882	0.181941	−	0.983309i	$-0.441762\pi$
$307$	600906.	0.363882	0.181941	+	0.983309i	$0.441762\pi$
$308$	0	0
$309$	723923.	0.431316
$310$	−1.80026e6	−1.06397
$311$	465866.	0.273124	0.136562	−	0.990632i	$-0.456395\pi$
$311$	465866.	0.273124	0.136562	+	0.990632i	$0.456395\pi$
$312$	−819907.	−0.476846
$313$	2.97366e6	1.71565	0.857827	−	0.513938i	$-0.171814\pi$
$313$	2.97366e6	1.71565	0.857827	+	0.513938i	$0.171814\pi$
$314$	255019.	0.145965
$315$	0	0
$316$	−1.79328e6	−1.01026
$317$	2.55521e6	1.42816	0.714082	−	0.700062i	$-0.246844\pi$
$317$	2.55521e6	1.42816	0.714082	+	0.700062i	$0.246844\pi$
$318$	−1.05984e6	−0.587725
$319$	−2.04641e6	−1.12594
$320$	983544.	0.536932
$321$	−1.33238e6	−0.721714
$322$	0	0
$323$	1.44792e6	0.772216
$324$	−134982.	−0.0714353
$325$	77915.2	0.0409179
$326$	1.15186e6	0.600281
$327$	1.12484e6	0.581730
$328$	−600241.	−0.308064
$329$	0	0
$330$	−799405.	−0.404093
$331$	2.56138e6	1.28500	0.642502	−	0.766284i	$-0.277896\pi$
$331$	2.56138e6	1.28500	0.642502	+	0.766284i	$0.277896\pi$
$332$	1.85793e6	0.925090
$333$	−807542.	−0.399075
$334$	447528.	0.219510
$335$	1.50249e6	0.731477
$336$	0	0
$337$	−1.21525e6	−0.582894	−0.291447	−	0.956587i	$-0.594137\pi$
$337$	−1.21525e6	−0.582894	−0.291447	+	0.956587i	$0.594137\pi$
$338$	−366805.	−0.174640
$339$	1.75026e6	0.827188
$340$	662678.	0.310889
$341$	−4.70701e6	−2.19209
$342$	−671105.	−0.310259
$343$	0	0
$344$	3.23858e6	1.47557
$345$	−870994.	−0.393974
$346$	−1.51886e6	−0.682066
$347$	1.25318e6	0.558714	0.279357	−	0.960187i	$-0.409879\pi$
$347$	1.25318e6	0.558714	0.279357	+	0.960187i	$0.409879\pi$
$348$	−786272.	−0.348037
$349$	−450403.	−0.197942	−0.0989709	−	0.995090i	$-0.531555\pi$
$349$	−450403.	−0.197942	−0.0989709	+	0.995090i	$0.531555\pi$
$350$	0	0
$351$	373702.	0.161904
$352$	2.83443e6	1.21929
$353$	−545281.	−0.232907	−0.116454	−	0.993196i	$-0.537153\pi$
$353$	−545281.	−0.232907	−0.116454	+	0.993196i	$0.537153\pi$
$354$	352678.	0.149579
$355$	−1.24691e6	−0.525128
$356$	−2.60396e6	−1.08895
$357$	0	0
$358$	−316625.	−0.130568
$359$	−1.01159e6	−0.414254	−0.207127	−	0.978314i	$-0.566411\pi$
$359$	−1.01159e6	−0.414254	−0.207127	+	0.978314i	$0.566411\pi$
$360$	−784888.	−0.319192
$361$	3.53138e6	1.42619
$362$	−1.35144e6	−0.542033
$363$	−640686.	−0.255199
$364$	0	0
$365$	−868937.	−0.341394
$366$	955468.	0.372832
$367$	1.09038e6	0.422584	0.211292	−	0.977423i	$-0.432233\pi$
$367$	1.09038e6	0.422584	0.211292	+	0.977423i	$0.432233\pi$
$368$	102254.	0.0393605
$369$	273581.	0.104597
$370$	−1.83754e6	−0.697803
$371$	0	0
$372$	−1.80853e6	−0.677592
$373$	−3.13003e6	−1.16487	−0.582433	−	0.812879i	$-0.697899\pi$
$373$	−3.13003e6	−1.16487	−0.582433	+	0.812879i	$0.697899\pi$
$374$	−962333.	−0.355751
$375$	1.60811e6	0.590524
$376$	234698.	0.0856130
$377$	2.17682e6	0.788804
$378$	0	0
$379$	−1.27933e6	−0.457492	−0.228746	−	0.973486i	$-0.573463\pi$
$379$	−1.27933e6	−0.457492	−0.228746	+	0.973486i	$0.573463\pi$
$380$	2.74947e6	0.976766
$381$	−2.15790e6	−0.761584
$382$	94050.0	0.0329762
$383$	−1.30666e6	−0.455161	−0.227580	−	0.973759i	$-0.573081\pi$
$383$	−1.30666e6	−0.455161	−0.227580	+	0.973759i	$0.573081\pi$
$384$	1.14513e6	0.396303
$385$	0	0
$386$	−224671.	−0.0767501
$387$	−1.47610e6	−0.501000
$388$	−340964.	−0.114982
$389$	−36488.2	−0.0122258	−0.00611292	−	0.999981i	$-0.501946\pi$
$389$	−36488.2	−0.0122258	−0.00611292	+	0.999981i	$0.501946\pi$
$390$	850348.	0.283097
$391$	−1.04851e6	−0.346842
$392$	0	0
$393$	−3.14618e6	−1.02755
$394$	−1.66483e6	−0.540294
$395$	4.75271e6	1.53267
$396$	−803077.	−0.257347
$397$	−640204.	−0.203865	−0.101932	−	0.994791i	$-0.532503\pi$
$397$	−640204.	−0.203865	−0.101932	+	0.994791i	$0.532503\pi$
$398$	−1.18043e6	−0.373536
$399$	0	0
$400$	−8756.50	−0.00273641
$401$	−2.69149e6	−0.835858	−0.417929	−	0.908480i	$-0.637244\pi$
$401$	−2.69149e6	−0.835858	−0.417929	+	0.908480i	$0.637244\pi$
$402$	−838333.	−0.258733
$403$	5.00697e6	1.53572
$404$	−1.57613e6	−0.480439
$405$	357740.	0.108375
$406$	0	0
$407$	−4.80449e6	−1.43768
$408$	−944858.	−0.281006
$409$	−2.92572e6	−0.864817	−0.432409	−	0.901678i	$-0.642336\pi$
$409$	−2.92572e6	−0.864817	−0.432409	+	0.901678i	$0.642336\pi$
$410$	622526.	0.182893
$411$	2.52869e6	0.738399
$412$	1.65484e6	0.480299
$413$	0	0
$414$	485981.	0.139354
$415$	−4.92404e6	−1.40347
$416$	−3.01506e6	−0.854205
$417$	450386.	0.126837
$418$	−3.99275e6	−1.11772
$419$	5.52810e6	1.53830	0.769150	−	0.639068i	$-0.220680\pi$
$419$	5.52810e6	1.53830	0.769150	+	0.639068i	$0.220680\pi$
$420$	0	0
$421$	5.77477e6	1.58792	0.793961	−	0.607968i	$-0.208015\pi$
$421$	5.77477e6	1.58792	0.793961	+	0.607968i	$0.208015\pi$
$422$	−737756.	−0.201666
$423$	−106972.	−0.0290682
$424$	−6.19106e6	−1.67244
$425$	89789.1	0.0241130
$426$	695729.	0.185744
$427$	0	0
$428$	−3.04573e6	−0.803677
$429$	2.22334e6	0.583262
$430$	−3.35882e6	−0.876024
$431$	2.20964e6	0.572965	0.286483	−	0.958085i	$-0.407514\pi$
$431$	2.20964e6	0.572965	0.286483	+	0.958085i	$0.407514\pi$
$432$	−41998.5	−0.0108274
$433$	−4.97215e6	−1.27446	−0.637228	−	0.770676i	$-0.719919\pi$
$433$	−4.97215e6	−1.27446	−0.637228	+	0.770676i	$0.719919\pi$
$434$	0	0
$435$	2.08385e6	0.528010
$436$	2.57130e6	0.647794
$437$	−4.35032e6	−1.08973
$438$	484833.	0.120755
$439$	2.86405e6	0.709282	0.354641	−	0.935002i	$-0.384603\pi$
$439$	2.86405e6	0.709282	0.354641	+	0.935002i	$0.384603\pi$
$440$	−4.66971e6	−1.14990
$441$	0	0
$442$	1.02366e6	0.249230
$443$	−5.66234e6	−1.37084	−0.685419	−	0.728149i	$-0.740381\pi$
$443$	−5.66234e6	−1.37084	−0.685419	+	0.728149i	$0.740381\pi$
$444$	−1.84598e6	−0.444396
$445$	6.90124e6	1.65206
$446$	−4.21889e6	−1.00429
$447$	2.95932e6	0.700523
$448$	0	0
$449$	7.54480e6	1.76617	0.883084	−	0.469215i	$-0.155463\pi$
$449$	7.54480e6	1.76617	0.883084	+	0.469215i	$0.155463\pi$
$450$	−41616.8	−0.00968808
$451$	1.62767e6	0.376813
$452$	4.00098e6	0.921129
$453$	−4.60173e6	−1.05360
$454$	1.76106e6	0.400991
$455$	0	0
$456$	−3.92025e6	−0.882879
$457$	−589014.	−0.131928	−0.0659638	−	0.997822i	$-0.521012\pi$
$457$	−589014.	−0.131928	−0.0659638	+	0.997822i	$0.521012\pi$
$458$	1.77687e6	0.395816
$459$	430652.	0.0954102
$460$	−1.99103e6	−0.438716
$461$	5.94951e6	1.30385	0.651927	−	0.758282i	$-0.273961\pi$
$461$	5.94951e6	1.30385	0.651927	+	0.758282i	$0.273961\pi$
$462$	0	0
$463$	−2.16083e6	−0.468456	−0.234228	−	0.972182i	$-0.575256\pi$
$463$	−2.16083e6	−0.468456	−0.234228	+	0.972182i	$0.575256\pi$
$464$	−244642.	−0.0527517
$465$	4.79312e6	1.02798
$466$	710563.	0.151579
$467$	−505766.	−0.107314	−0.0536571	−	0.998559i	$-0.517088\pi$
$467$	−505766.	−0.107314	−0.0536571	+	0.998559i	$0.517088\pi$
$468$	854255.	0.180291
$469$	0	0
$470$	−243412.	−0.0508272
$471$	−678979.	−0.141028
$472$	2.06016e6	0.425644
$473$	−8.78208e6	−1.80486
$474$	−2.65183e6	−0.542125
$475$	372538.	0.0757594
$476$	0	0
$477$	2.82179e6	0.567845
$478$	2.72384e6	0.545270
$479$	547067.	0.108944	0.0544718	−	0.998515i	$-0.482652\pi$
$479$	547067.	0.108944	0.0544718	+	0.998515i	$0.482652\pi$
$480$	−2.88628e6	−0.571789
$481$	5.11066e6	1.00720
$482$	−3.84665e6	−0.754162
$483$	0	0
$484$	−1.46456e6	−0.284181
$485$	903652.	0.174440
$486$	−199605.	−0.0383337
$487$	2.37889e6	0.454519	0.227259	−	0.973834i	$-0.427024\pi$
$487$	2.37889e6	0.454519	0.227259	+	0.973834i	$0.427024\pi$
$488$	5.58135e6	1.06094
$489$	−3.06677e6	−0.579976
$490$	0	0
$491$	−7.19719e6	−1.34728	−0.673642	−	0.739058i	$-0.735271\pi$
$491$	−7.19719e6	−1.34728	−0.673642	+	0.739058i	$0.735271\pi$
$492$	625386.	0.116476
$493$	2.50856e6	0.464844
$494$	4.24720e6	0.783042
$495$	2.12838e6	0.390424
$496$	−562709.	−0.102702
$497$	0	0
$498$	2.74742e6	0.496423
$499$	2.22236e6	0.399543	0.199771	−	0.979843i	$-0.435980\pi$
$499$	2.22236e6	0.399543	0.199771	+	0.979843i	$0.435980\pi$
$500$	3.67603e6	0.657588
$501$	−1.19153e6	−0.212085
$502$	3.94032e6	0.697867
$503$	305857.	0.0539013	0.0269506	−	0.999637i	$-0.491420\pi$
$503$	305857.	0.0539013	0.0269506	+	0.999637i	$0.491420\pi$
$504$	0	0
$505$	4.17719e6	0.728879
$506$	2.89135e6	0.502024
$507$	976604.	0.168732
$508$	−4.93280e6	−0.848074
$509$	9.06420e6	1.55073	0.775363	−	0.631516i	$-0.217567\pi$
$509$	9.06420e6	1.55073	0.775363	+	0.631516i	$0.217567\pi$
$510$	979938.	0.166830
$511$	0	0
$512$	666475.	0.112359
$513$	1.78679e6	0.299764
$514$	3.47772e6	0.580614
$515$	−4.38579e6	−0.728668
$516$	−3.37426e6	−0.557897
$517$	−636431.	−0.104719
$518$	0	0
$519$	4.04390e6	0.658995
$520$	4.96730e6	0.805586
$521$	−8.44649e6	−1.36327	−0.681635	−	0.731692i	$-0.738731\pi$
$521$	−8.44649e6	−1.36327	−0.681635	+	0.731692i	$0.738731\pi$
$522$	−1.16270e6	−0.186764
$523$	−3.91585e6	−0.625997	−0.312998	−	0.949754i	$-0.601333\pi$
$523$	−3.91585e6	−0.625997	−0.312998	+	0.949754i	$0.601333\pi$
$524$	−7.19195e6	−1.14424
$525$	0	0
$526$	−2.22204e6	−0.350178
$527$	5.77001e6	0.905003
$528$	−249871.	−0.0390059
$529$	−3.28606e6	−0.510548
$530$	6.42092e6	0.992905
$531$	−938992.	−0.144519
$532$	0	0
$533$	−1.73140e6	−0.263985
$534$	−3.85062e6	−0.584356
$535$	8.07204e6	1.21927
$536$	−4.89711e6	−0.736254
$537$	843001.	0.126152
$538$	−18510.2	−0.00275712
$539$	0	0
$540$	817769.	0.120683
$541$	−2.94802e6	−0.433049	−0.216524	−	0.976277i	$-0.569472\pi$
$541$	−2.94802e6	−0.433049	−0.216524	+	0.976277i	$0.569472\pi$
$542$	2.59997e6	0.380163
$543$	3.59816e6	0.523698
$544$	−3.47454e6	−0.503385
$545$	−6.81469e6	−0.982776
$546$	0	0
$547$	−5.50670e6	−0.786906	−0.393453	−	0.919345i	$-0.628720\pi$
$547$	−5.50670e6	−0.786906	−0.393453	+	0.919345i	$0.628720\pi$
$548$	5.78041e6	0.822256
$549$	−2.54390e6	−0.360221
$550$	−247600.	−0.0349015
$551$	1.04081e7	1.46047
$552$	2.83885e6	0.396547
$553$	0	0
$554$	−3.96796e6	−0.549278
$555$	4.89238e6	0.674199
$556$	1.02955e6	0.141241
$557$	−4.97235e6	−0.679085	−0.339542	−	0.940591i	$-0.610272\pi$
$557$	−4.97235e6	−0.679085	−0.339542	+	0.940591i	$0.610272\pi$
$558$	−2.67437e6	−0.363611
$559$	9.34174e6	1.26444
$560$	0	0
$561$	2.56217e6	0.343717
$562$	2.83153e6	0.378164
$563$	−1.31397e7	−1.74709	−0.873544	−	0.486745i	$-0.838184\pi$
$563$	−1.31397e7	−1.74709	−0.873544	+	0.486745i	$0.838184\pi$
$564$	−244530.	−0.0323694
$565$	−1.06037e7	−1.39746
$566$	2.01880e6	0.264882
$567$	0	0
$568$	4.06409e6	0.528558
$569$	−1.38589e7	−1.79452	−0.897261	−	0.441500i	$-0.854447\pi$
$569$	−1.38589e7	−1.79452	−0.897261	+	0.441500i	$0.854447\pi$
$570$	4.06580e6	0.524154
$571$	−1.06977e7	−1.37309	−0.686547	−	0.727085i	$-0.740874\pi$
$571$	−1.06977e7	−1.37309	−0.686547	+	0.727085i	$0.740874\pi$
$572$	5.08241e6	0.649501
$573$	−250404.	−0.0318607
$574$	0	0
$575$	−269774.	−0.0340275
$576$	1.46110e6	0.183495
$577$	5.61898e6	0.702616	0.351308	−	0.936260i	$-0.385737\pi$
$577$	5.61898e6	0.702616	0.351308	+	0.936260i	$0.385737\pi$
$578$	−3.61993e6	−0.450692
$579$	598178.	0.0741540
$580$	4.76353e6	0.587975
$581$	0	0
$582$	−504203.	−0.0617018
$583$	1.67883e7	2.04567
$584$	2.83214e6	0.343624
$585$	−2.26402e6	−0.273521
$586$	2.87775e6	0.346185
$587$	6.86832e6	0.822726	0.411363	−	0.911471i	$-0.365053\pi$
$587$	6.86832e6	0.822726	0.411363	+	0.911471i	$0.365053\pi$
$588$	0	0
$589$	2.39400e7	2.84338
$590$	−2.13665e6	−0.252699
$591$	4.43256e6	0.522018
$592$	−574362.	−0.0673568
$593$	−4.07737e6	−0.476149	−0.238075	−	0.971247i	$-0.576516\pi$
$593$	−4.07737e6	−0.476149	−0.238075	+	0.971247i	$0.576516\pi$
$594$	−1.18755e6	−0.138098
$595$	0	0
$596$	6.76479e6	0.780079
$597$	3.14285e6	0.360901
$598$	−3.07561e6	−0.351705
$599$	582993.	0.0663890	0.0331945	−	0.999449i	$-0.489432\pi$
$599$	582993.	0.0663890	0.0331945	+	0.999449i	$0.489432\pi$
$600$	−243104.	−0.0275686
$601$	6.87250e6	0.776119	0.388060	−	0.921634i	$-0.373145\pi$
$601$	6.87250e6	0.776119	0.388060	+	0.921634i	$0.373145\pi$
$602$	0	0
$603$	2.23203e6	0.249981
$604$	−1.05192e7	−1.17325
$605$	3.88151e6	0.431134
$606$	−2.33071e6	−0.257814
$607$	2.40563e6	0.265006	0.132503	−	0.991183i	$-0.457699\pi$
$607$	2.40563e6	0.265006	0.132503	+	0.991183i	$0.457699\pi$
$608$	−1.44160e7	−1.58156
$609$	0	0
$610$	−5.78858e6	−0.629865
$611$	676989.	0.0733632
$612$	984440.	0.106246
$613$	−1.16245e7	−1.24946	−0.624731	−	0.780840i	$-0.714791\pi$
$613$	−1.16245e7	−1.24946	−0.624731	+	0.780840i	$0.714791\pi$
$614$	2.03126e6	0.217443
$615$	−1.65745e6	−0.176707
$616$	0	0
$617$	−1.37370e7	−1.45271	−0.726353	−	0.687322i	$-0.758786\pi$
$617$	−1.37370e7	−1.45271	−0.726353	+	0.687322i	$0.758786\pi$
$618$	2.44710e6	0.257739
$619$	−1.64008e7	−1.72044	−0.860220	−	0.509923i	$-0.829674\pi$
$619$	−1.64008e7	−1.72044	−0.860220	+	0.509923i	$0.829674\pi$
$620$	1.09567e7	1.14473
$621$	−1.29390e6	−0.134640
$622$	1.57478e6	0.163209
$623$	0	0
$624$	265794.	0.0273265
$625$	−9.26754e6	−0.948996
$626$	1.00519e7	1.02521
$627$	1.06305e7	1.07991
$628$	−1.55210e6	−0.157044
$629$	5.88951e6	0.593543
$630$	0	0
$631$	−7.28252e6	−0.728129	−0.364065	−	0.931374i	$-0.618611\pi$
$631$	−7.28252e6	−0.728129	−0.364065	+	0.931374i	$0.618611\pi$
$632$	−1.54906e7	−1.54268
$633$	1.96425e6	0.194844
$634$	8.63745e6	0.853419
$635$	1.30733e7	1.28662
$636$	6.45042e6	0.632332
$637$	0	0
$638$	−6.91754e6	−0.672821
$639$	−1.85235e6	−0.179461
$640$	−6.93763e6	−0.669516
$641$	−107514.	−0.0103352	−0.00516762	−	0.999987i	$-0.501645\pi$
$641$	−107514.	−0.0103352	−0.00516762	+	0.999987i	$0.501645\pi$
$642$	−4.50388e6	−0.431270
$643$	−9.75913e6	−0.930858	−0.465429	−	0.885085i	$-0.654100\pi$
$643$	−9.75913e6	−0.930858	−0.465429	+	0.885085i	$0.654100\pi$
$644$	0	0
$645$	8.94274e6	0.846392
$646$	4.89445e6	0.461448
$647$	6.75588e6	0.634485	0.317243	−	0.948344i	$-0.397243\pi$
$647$	6.75588e6	0.634485	0.317243	+	0.948344i	$0.397243\pi$
$648$	−1.16599e6	−0.109083
$649$	−5.58655e6	−0.520633
$650$	263379.	0.0244511
$651$	0	0
$652$	−7.01043e6	−0.645841
$653$	2.20346e6	0.202219	0.101110	−	0.994875i	$-0.467761\pi$
$653$	2.20346e6	0.202219	0.101110	+	0.994875i	$0.467761\pi$
$654$	3.80233e6	0.347620
$655$	1.90607e7	1.73595
$656$	194584.	0.0176542
$657$	−1.29085e6	−0.116671
$658$	0	0
$659$	−1.39630e7	−1.25247	−0.626233	−	0.779636i	$-0.715404\pi$
$659$	−1.39630e7	−1.25247	−0.626233	+	0.779636i	$0.715404\pi$
$660$	4.86534e6	0.434763
$661$	51052.8	0.00454482	0.00227241	−	0.999997i	$-0.499277\pi$
$661$	51052.8	0.00454482	0.00227241	+	0.999997i	$0.499277\pi$
$662$	8.65832e6	0.767872
$663$	−2.72545e6	−0.240799
$664$	1.60490e7	1.41263
$665$	0	0
$666$	−2.72976e6	−0.238473
$667$	−7.53703e6	−0.655972
$668$	−2.72375e6	−0.236170
$669$	1.12326e7	0.970323
$670$	5.07893e6	0.437104
$671$	−1.51350e7	−1.29770
$672$	0	0
$673$	6.71688e6	0.571650	0.285825	−	0.958282i	$-0.407732\pi$
$673$	6.71688e6	0.571650	0.285825	+	0.958282i	$0.407732\pi$
$674$	−4.10793e6	−0.348316
$675$	110803.	0.00936036
$676$	2.23245e6	0.187895
$677$	−1.45655e7	−1.22139	−0.610693	−	0.791867i	$-0.709109\pi$
$677$	−1.45655e7	−1.22139	−0.610693	+	0.791867i	$0.709109\pi$
$678$	5.91647e6	0.494298
$679$	0	0
$680$	5.72429e6	0.474733
$681$	−4.68876e6	−0.387427
$682$	−1.59112e7	−1.30991
$683$	−1.62565e6	−0.133344	−0.0666722	−	0.997775i	$-0.521238\pi$
$683$	−1.62565e6	−0.133344	−0.0666722	+	0.997775i	$0.521238\pi$
$684$	4.08448e6	0.333808
$685$	−1.53197e7	−1.24745
$686$	0	0
$687$	−4.73086e6	−0.382427
$688$	−1.04987e6	−0.0845600
$689$	−1.78582e7	−1.43314
$690$	−2.94425e6	−0.235424
$691$	2.81556e6	0.224321	0.112160	−	0.993690i	$-0.464223\pi$
$691$	2.81556e6	0.224321	0.112160	+	0.993690i	$0.464223\pi$
$692$	9.24407e6	0.733834
$693$	0	0
$694$	4.23616e6	0.333867
$695$	−2.72860e6	−0.214278
$696$	−6.79192e6	−0.531459
$697$	−1.99526e6	−0.155567
$698$	−1.52251e6	−0.118283
$699$	−1.89185e6	−0.146451
$700$	0	0
$701$	1.40996e7	1.08371	0.541855	−	0.840472i	$-0.317722\pi$
$701$	1.40996e7	1.08371	0.541855	+	0.840472i	$0.317722\pi$
$702$	1.26323e6	0.0967478
$703$	2.44358e7	1.86482
$704$	8.69287e6	0.661046
$705$	648074.	0.0491079
$706$	−1.84323e6	−0.139177
$707$	0	0
$708$	−2.14647e6	−0.160932
$709$	−8.44943e6	−0.631265	−0.315633	−	0.948881i	$-0.602217\pi$
$709$	−8.44943e6	−0.631265	−0.315633	+	0.948881i	$0.602217\pi$
$710$	−4.21498e6	−0.313797
$711$	7.06039e6	0.523787
$712$	−2.24933e7	−1.66285
$713$	−1.73361e7	−1.27711
$714$	0	0
$715$	−1.34698e7	−0.985365
$716$	1.92704e6	0.140478
$717$	−7.25212e6	−0.526826
$718$	−3.41950e6	−0.247543
$719$	−1.82657e7	−1.31769	−0.658846	−	0.752278i	$-0.728955\pi$
$719$	−1.82657e7	−1.31769	−0.658846	+	0.752278i	$0.728955\pi$
$720$	254442.	0.0182918
$721$	0	0
$722$	1.19372e7	0.852237
$723$	1.02415e7	0.728651
$724$	8.22513e6	0.583172
$725$	645431.	0.0456042
$726$	−2.16573e6	−0.152498
$727$	−1.37476e6	−0.0964700	−0.0482350	−	0.998836i	$-0.515360\pi$
$727$	−1.37476e6	−0.0964700	−0.0482350	+	0.998836i	$0.515360\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	−2.93729e6	−0.204005
$731$	1.07654e7	0.745136
$732$	−5.81517e6	−0.401130
$733$	−1.64625e7	−1.13171	−0.565857	−	0.824503i	$-0.691455\pi$
$733$	−1.64625e7	−1.13171	−0.565857	+	0.824503i	$0.691455\pi$
$734$	3.68585e6	0.252521
$735$	0	0
$736$	1.04393e7	0.710360
$737$	1.32795e7	0.900561
$738$	924793.	0.0625034
$739$	2.72997e6	0.183885	0.0919425	−	0.995764i	$-0.470692\pi$
$739$	2.72997e6	0.183885	0.0919425	+	0.995764i	$0.470692\pi$
$740$	1.11836e7	0.750765
$741$	−1.13080e7	−0.756555
$742$	0	0
$743$	−2.34396e7	−1.55768	−0.778839	−	0.627224i	$-0.784191\pi$
$743$	−2.34396e7	−1.55768	−0.778839	+	0.627224i	$0.784191\pi$
$744$	−1.56223e7	−1.03470
$745$	−1.79286e7	−1.18347
$746$	−1.05805e7	−0.696082
$747$	−7.31491e6	−0.479631
$748$	5.85695e6	0.382752
$749$	0	0
$750$	5.43595e6	0.352876
$751$	1.79407e7	1.16075	0.580376	−	0.814349i	$-0.302906\pi$
$751$	1.79407e7	1.16075	0.580376	+	0.814349i	$0.302906\pi$
$752$	−76083.4	−0.00490620
$753$	−1.04910e7	−0.674260
$754$	7.35837e6	0.471361
$755$	2.78790e7	1.77996
$756$	0	0
$757$	−2.30823e7	−1.46399	−0.731997	−	0.681308i	$-0.761412\pi$
$757$	−2.30823e7	−1.46399	−0.731997	+	0.681308i	$0.761412\pi$
$758$	−4.32455e6	−0.273381
$759$	−7.69811e6	−0.485043
$760$	2.37503e7	1.49154
$761$	−1.91587e7	−1.19924	−0.599618	−	0.800286i	$-0.704681\pi$
$761$	−1.91587e7	−1.19924	−0.599618	+	0.800286i	$0.704681\pi$
$762$	−7.29440e6	−0.455095
$763$	0	0
$764$	−572407.	−0.0354790
$765$	−2.60905e6	−0.161186
$766$	−4.41693e6	−0.271988
$767$	5.94257e6	0.364742
$768$	9.06596e6	0.554639
$769$	−1.73017e7	−1.05505	−0.527526	−	0.849539i	$-0.676880\pi$
$769$	−1.73017e7	−1.05505	−0.527526	+	0.849539i	$0.676880\pi$
$770$	0	0
$771$	−9.25931e6	−0.560974
$772$	1.36739e6	0.0825753
$773$	−3.12287e7	−1.87977	−0.939885	−	0.341491i	$-0.889068\pi$
$773$	−3.12287e7	−1.87977	−0.939885	+	0.341491i	$0.889068\pi$
$774$	−4.98970e6	−0.299379
$775$	1.48458e6	0.0887868
$776$	−2.94529e6	−0.175580
$777$	0	0
$778$	−123342.	−0.00730572
$779$	−8.27840e6	−0.488768
$780$	−5.17539e6	−0.304584
$781$	−1.10206e7	−0.646514
$782$	−3.54432e6	−0.207260
$783$	3.09566e6	0.180446
$784$	0	0
$785$	4.11350e6	0.238253
$786$	−1.06351e7	−0.614026
$787$	2.48691e7	1.43128	0.715638	−	0.698471i	$-0.246136\pi$
$787$	2.48691e7	1.43128	0.715638	+	0.698471i	$0.246136\pi$
$788$	1.01325e7	0.581302
$789$	5.91611e6	0.338332
$790$	1.60657e7	0.915868
$791$	0	0
$792$	−6.93708e6	−0.392974
$793$	1.60995e7	0.909136
$794$	−2.16410e6	−0.121822
$795$	−1.70955e7	−0.959319
$796$	7.18433e6	0.401887
$797$	−584574.	−0.0325982	−0.0162991	−	0.999867i	$-0.505188\pi$
$797$	−584574.	−0.0325982	−0.0162991	+	0.999867i	$0.505188\pi$
$798$	0	0
$799$	780159.	0.0432331
$800$	−893970.	−0.0493853
$801$	1.02521e7	0.564590
$802$	−9.09814e6	−0.499478
$803$	−7.67993e6	−0.420309
$804$	5.10226e6	0.278370
$805$	0	0
$806$	1.69252e7	0.917691
$807$	49282.7	0.00266386
$808$	−1.36148e7	−0.733639
$809$	−1.10552e7	−0.593878	−0.296939	−	0.954897i	$-0.595966\pi$
$809$	−1.10552e7	−0.593878	−0.296939	+	0.954897i	$0.595966\pi$
$810$	1.20928e6	0.0647612
$811$	−1.62780e7	−0.869060	−0.434530	−	0.900657i	$-0.643086\pi$
$811$	−1.62780e7	−0.869060	−0.434530	+	0.900657i	$0.643086\pi$
$812$	0	0
$813$	−6.92232e6	−0.367304
$814$	−1.62408e7	−0.859103
$815$	1.85796e7	0.979813
$816$	306300.	0.0161036
$817$	4.46659e7	2.34111
$818$	−9.88989e6	−0.516783
$819$	0	0
$820$	−3.78882e6	−0.196775
$821$	3.28574e7	1.70128	0.850638	−	0.525752i	$-0.176216\pi$
$821$	3.28574e7	1.70128	0.850638	+	0.525752i	$0.176216\pi$
$822$	8.54781e6	0.441240
$823$	1.06046e7	0.545752	0.272876	−	0.962049i	$-0.412025\pi$
$823$	1.06046e7	0.545752	0.272876	+	0.962049i	$0.412025\pi$
$824$	1.42947e7	0.733426
$825$	659226.	0.0337209
$826$	0	0
$827$	1.57582e7	0.801203	0.400602	−	0.916252i	$-0.368801\pi$
$827$	1.57582e7	0.801203	0.400602	+	0.916252i	$0.368801\pi$
$828$	−2.95777e6	−0.149930
$829$	2.70135e7	1.36520	0.682598	−	0.730794i	$-0.260850\pi$
$829$	2.70135e7	1.36520	0.682598	+	0.730794i	$0.260850\pi$
$830$	−1.66449e7	−0.838660
$831$	1.05645e7	0.530698
$832$	−9.24684e6	−0.463111
$833$	0	0
$834$	1.52245e6	0.0757930
$835$	7.21871e6	0.358297
$836$	2.43007e7	1.20255
$837$	7.12042e6	0.351311
$838$	1.86868e7	0.919232
$839$	3.50490e7	1.71898	0.859489	−	0.511154i	$-0.170782\pi$
$839$	3.50490e7	1.71898	0.859489	+	0.511154i	$0.170782\pi$
$840$	0	0
$841$	−2.47888e6	−0.120855
$842$	1.95206e7	0.948885
$843$	−7.53884e6	−0.365372
$844$	4.49013e6	0.216972
$845$	−5.91663e6	−0.285057
$846$	−361600.	−0.0173701
$847$	0	0
$848$	2.00699e6	0.0958422
$849$	−5.37499e6	−0.255922
$850$	303517.	0.0144091
$851$	−1.76952e7	−0.837589
$852$	−4.23434e6	−0.199842
$853$	538313.	0.0253316	0.0126658	−	0.999920i	$-0.495968\pi$
$853$	538313.	0.0253316	0.0126658	+	0.999920i	$0.495968\pi$
$854$	0	0
$855$	−1.08250e7	−0.506423
$856$	−2.63094e7	−1.22723
$857$	3.68275e6	0.171285	0.0856426	−	0.996326i	$-0.472706\pi$
$857$	3.68275e6	0.171285	0.0856426	+	0.996326i	$0.472706\pi$
$858$	7.51564e6	0.348536
$859$	1.88883e7	0.873394	0.436697	−	0.899609i	$-0.356148\pi$
$859$	1.88883e7	0.873394	0.436697	+	0.899609i	$0.356148\pi$
$860$	2.04425e7	0.942513
$861$	0	0
$862$	7.46931e6	0.342383
$863$	2.78544e7	1.27311	0.636557	−	0.771230i	$-0.280358\pi$
$863$	2.78544e7	1.27311	0.636557	+	0.771230i	$0.280358\pi$
$864$	−4.28771e6	−0.195408
$865$	−2.44994e7	−1.11331
$866$	−1.68075e7	−0.761568
$867$	9.63791e6	0.435447
$868$	0	0
$869$	4.20059e7	1.88695
$870$	7.04409e6	0.315520
$871$	−1.41258e7	−0.630909
$872$	2.22113e7	0.989195
$873$	1.34242e6	0.0596146
$874$	−1.47055e7	−0.651180
$875$	0	0
$876$	−2.95079e6	−0.129921
$877$	−1.75052e7	−0.768544	−0.384272	−	0.923220i	$-0.625548\pi$
$877$	−1.75052e7	−0.768544	−0.384272	+	0.923220i	$0.625548\pi$
$878$	9.68143e6	0.423841
$879$	−7.66189e6	−0.334475
$880$	1.51381e6	0.0658968
$881$	5.66292e6	0.245811	0.122905	−	0.992418i	$-0.460779\pi$
$881$	5.66292e6	0.245811	0.122905	+	0.992418i	$0.460779\pi$
$882$	0	0
$883$	1.43933e7	0.621239	0.310620	−	0.950534i	$-0.399463\pi$
$883$	1.43933e7	0.621239	0.310620	+	0.950534i	$0.399463\pi$
$884$	−6.23020e6	−0.268146
$885$	5.68876e6	0.244151
$886$	−1.91406e7	−0.819163
$887$	−5.50193e6	−0.234804	−0.117402	−	0.993084i	$-0.537457\pi$
$887$	−5.50193e6	−0.234804	−0.117402	+	0.993084i	$0.537457\pi$
$888$	−1.59458e7	−0.678602
$889$	0	0
$890$	2.33285e7	0.987214
$891$	3.16182e6	0.133427
$892$	2.56770e7	1.08052
$893$	3.23691e6	0.135832
$894$	1.00035e7	0.418607
$895$	−5.10721e6	−0.213121
$896$	0	0
$897$	8.18869e6	0.339808
$898$	2.55039e7	1.05540
$899$	4.14766e7	1.71161
$900$	253288.	0.0104234
$901$	−2.05797e7	−0.844554
$902$	5.50208e6	0.225170
$903$	0	0
$904$	3.45610e7	1.40658
$905$	−2.17990e7	−0.884737
$906$	−1.55554e7	−0.629593
$907$	6.01555e6	0.242805	0.121402	−	0.992603i	$-0.461261\pi$
$907$	6.01555e6	0.242805	0.121402	+	0.992603i	$0.461261\pi$
$908$	−1.07182e7	−0.431426
$909$	6.20542e6	0.249093
$910$	0	0
$911$	−1.47513e7	−0.588892	−0.294446	−	0.955668i	$-0.595135\pi$
$911$	−1.47513e7	−0.588892	−0.294446	+	0.955668i	$0.595135\pi$
$912$	1.27085e6	0.0505950
$913$	−4.35202e7	−1.72788
$914$	−1.99106e6	−0.0788351
$915$	1.54119e7	0.608559
$916$	−1.08144e7	−0.425858
$917$	0	0
$918$	1.45575e6	0.0570137
$919$	−3.54001e7	−1.38266	−0.691331	−	0.722538i	$-0.742976\pi$
$919$	−3.54001e7	−1.38266	−0.691331	+	0.722538i	$0.742976\pi$
$920$	−1.71988e7	−0.669928
$921$	−5.40816e6	−0.210088
$922$	2.01113e7	0.779135
$923$	1.17229e7	0.452930
$924$	0	0
$925$	1.51532e6	0.0582305
$926$	−7.30434e6	−0.279932
$927$	−6.51530e6	−0.249021
$928$	−2.49760e7	−0.952036
$929$	3.83266e7	1.45700	0.728502	−	0.685044i	$-0.240217\pi$
$929$	3.83266e7	1.45700	0.728502	+	0.685044i	$0.240217\pi$
$930$	1.62023e7	0.614285
$931$	0	0
$932$	−4.32463e6	−0.163083
$933$	−4.19280e6	−0.157688
$934$	−1.70966e6	−0.0641271
$935$	−1.55226e7	−0.580677
$936$	7.37917e6	0.275307
$937$	−1.48693e7	−0.553274	−0.276637	−	0.960974i	$-0.589220\pi$
$937$	−1.48693e7	−0.553274	−0.276637	+	0.960974i	$0.589220\pi$
$938$	0	0
$939$	−2.67629e7	−0.990534
$940$	1.48145e6	0.0546849
$941$	3.50894e6	0.129182	0.0645909	−	0.997912i	$-0.479426\pi$
$941$	3.50894e6	0.129182	0.0645909	+	0.997912i	$0.479426\pi$
$942$	−2.29517e6	−0.0842730
$943$	5.99481e6	0.219531
$944$	−667856.	−0.0243923
$945$	0	0
$946$	−2.96863e7	−1.07852
$947$	3.89774e7	1.41234	0.706168	−	0.708044i	$-0.250422\pi$
$947$	3.89774e7	1.41234	0.706168	+	0.708044i	$0.250422\pi$
$948$	1.61395e7	0.583271
$949$	8.16935e6	0.294457
$950$	1.25930e6	0.0452711
$951$	−2.29969e7	−0.824551
$952$	0	0
$953$	7.11438e6	0.253749	0.126875	−	0.991919i	$-0.459505\pi$
$953$	7.11438e6	0.253749	0.126875	+	0.991919i	$0.459505\pi$
$954$	9.53860e6	0.339323
$955$	1.51704e6	0.0538256
$956$	−1.65778e7	−0.586655
$957$	1.84177e7	0.650062
$958$	1.84927e6	0.0651008
$959$	0	0
$960$	−8.85190e6	−0.309998
$961$	6.67724e7	2.33232
$962$	1.72757e7	0.601865
$963$	1.19914e7	0.416682
$964$	2.34114e7	0.811401
$965$	−3.62398e6	−0.125276
$966$	0	0
$967$	−1.99357e7	−0.685592	−0.342796	−	0.939410i	$-0.611374\pi$
$967$	−1.99357e7	−0.685592	−0.342796	+	0.939410i	$0.611374\pi$
$968$	−1.26511e7	−0.433950
$969$	−1.30313e7	−0.445839
$970$	3.05464e6	0.104239
$971$	−4.95888e7	−1.68786	−0.843928	−	0.536456i	$-0.819763\pi$
$971$	−4.95888e7	−1.68786	−0.843928	+	0.536456i	$0.819763\pi$
$972$	1.21484e6	0.0412432
$973$	0	0
$974$	8.04143e6	0.271604
$975$	−701236.	−0.0236240
$976$	−1.80934e6	−0.0607989
$977$	−355348.	−0.0119102	−0.00595508	−	0.999982i	$-0.501896\pi$
$977$	−355348.	−0.0119102	−0.00595508	+	0.999982i	$0.501896\pi$
$978$	−1.03667e7	−0.346572
$979$	6.09953e7	2.03395
$980$	0	0
$981$	−1.01236e7	−0.335862
$982$	−2.43289e7	−0.805088
$983$	991754.	0.0327356	0.0163678	−	0.999866i	$-0.494790\pi$
$983$	991754.	0.0327356	0.0163678	+	0.999866i	$0.494790\pi$
$984$	5.40217e6	0.177861
$985$	−2.68541e7	−0.881900
$986$	8.47976e6	0.277774
$987$	0	0
$988$	−2.58493e7	−0.842474
$989$	−3.23448e7	−1.05151
$990$	7.19464e6	0.233303
$991$	1.06674e7	0.345045	0.172523	−	0.985006i	$-0.444808\pi$
$991$	1.06674e7	0.345045	0.172523	+	0.985006i	$0.444808\pi$
$992$	−5.74481e7	−1.85352
$993$	−2.30524e7	−0.741898
$994$	0	0
$995$	−1.90405e7	−0.609707
$996$	−1.67214e7	−0.534101
$997$	−5.58283e7	−1.77876	−0.889379	−	0.457171i	$-0.848863\pi$
$997$	−5.58283e7	−1.77876	−0.889379	+	0.457171i	$0.848863\pi$
$998$	7.51232e6	0.238752
$999$	7.26788e6	0.230406

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.6.a.n.1.4	✓	6
3.2	odd	2		441.6.a.bb.1.3		6
7.2	even	3		147.6.e.q.67.3		12
7.3	odd	6		147.6.e.p.79.3		12
7.4	even	3		147.6.e.q.79.3		12
7.5	odd	6		147.6.e.p.67.3		12
7.6	odd	2		147.6.a.o.1.4	yes	6
21.20	even	2		441.6.a.ba.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.6.a.n.1.4	✓	6	1.1	even	1	trivial
147.6.a.o.1.4	yes	6	7.6	odd	2
147.6.e.p.67.3		12	7.5	odd	6
147.6.e.p.79.3		12	7.3	odd	6
147.6.e.q.67.3		12	7.2	even	3
147.6.e.q.79.3		12	7.4	even	3
441.6.a.ba.1.3		6	21.20	even	2
441.6.a.bb.1.3		6	3.2	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:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)

\(f(q)\)	\(=\)	\(q+3.38033 q^{2} -9.00000 q^{3} -20.5734 q^{4} +54.5253 q^{5} -30.4230 q^{6} -177.715 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+3.38033 q^{2} -9.00000 q^{3} -20.5734 q^{4} +54.5253 q^{5} -30.4230 q^{6} -177.715 q^{8} +81.0000 q^{9} +184.313 q^{10} +481.911 q^{11} +185.160 q^{12} -512.622 q^{13} -490.728 q^{15} +57.6111 q^{16} -590.744 q^{17} +273.807 q^{18} -2451.02 q^{19} -1121.77 q^{20} +1629.02 q^{22} +1774.90 q^{23} +1599.44 q^{24} -151.993 q^{25} -1732.83 q^{26} -729.000 q^{27} -4246.44 q^{29} -1658.82 q^{30} -9767.37 q^{31} +5881.64 q^{32} -4337.20 q^{33} -1996.91 q^{34} -1666.44 q^{36} -9969.65 q^{37} -8285.24 q^{38} +4613.60 q^{39} -9689.98 q^{40} +3377.54 q^{41} -18223.4 q^{43} -9914.53 q^{44} +4416.55 q^{45} +5999.76 q^{46} -1320.64 q^{47} -518.500 q^{48} -513.788 q^{50} +5316.69 q^{51} +10546.4 q^{52} +34837.0 q^{53} -2464.26 q^{54} +26276.3 q^{55} +22059.1 q^{57} -14354.4 q^{58} -11592.5 q^{59} +10095.9 q^{60} -31406.1 q^{61} -33016.9 q^{62} +18038.3 q^{64} -27950.9 q^{65} -14661.2 q^{66} +27555.9 q^{67} +12153.6 q^{68} -15974.1 q^{69} -22868.5 q^{71} -14394.9 q^{72} -15936.4 q^{73} -33700.7 q^{74} +1367.94 q^{75} +50425.6 q^{76} +15595.5 q^{78} +87165.3 q^{79} +3141.26 q^{80} +6561.00 q^{81} +11417.2 q^{82} -90307.5 q^{83} -32210.5 q^{85} -61601.2 q^{86} +38218.0 q^{87} -85643.0 q^{88} +126570. q^{89} +14929.4 q^{90} -36515.7 q^{92} +87906.4 q^{93} -4464.20 q^{94} -133642. q^{95} -52934.7 q^{96} +16573.1 q^{97} +39034.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} - 54 q^{3} + 150 q^{4} - 100 q^{5} - 18 q^{6} - 114 q^{8} + 486 q^{9}+O(q^{10}) \) 6 * q + 2 * q^2 - 54 * q^3 + 150 * q^4 - 100 * q^5 - 18 * q^6 - 114 * q^8 + 486 * q^9	\( 6 q + 2 q^{2} - 54 q^{3} + 150 q^{4} - 100 q^{5} - 18 q^{6} - 114 q^{8} + 486 q^{9} - 864 q^{10} + 604 q^{11} - 1350 q^{12} - 1352 q^{13} + 900 q^{15} + 4578 q^{16} - 3028 q^{17} + 162 q^{18} - 1728 q^{19} - 452 q^{20} - 4116 q^{22} - 4484 q^{23} + 1026 q^{24} + 4806 q^{25} - 14172 q^{26} - 4374 q^{27} - 5320 q^{29} + 7776 q^{30} - 3976 q^{31} - 37326 q^{32} - 5436 q^{33} + 16336 q^{34} + 12150 q^{36} + 22680 q^{37} - 52744 q^{38} + 12168 q^{39} - 100600 q^{40} - 28756 q^{41} - 6768 q^{43} - 64940 q^{44} - 8100 q^{45} + 540 q^{46} - 51552 q^{47} - 41202 q^{48} - 40622 q^{50} + 27252 q^{51} - 119296 q^{52} + 80884 q^{53} - 1458 q^{54} - 11656 q^{55} + 15552 q^{57} - 70464 q^{58} - 8872 q^{59} + 4068 q^{60} - 50896 q^{61} - 11824 q^{62} + 199590 q^{64} + 3492 q^{65} + 37044 q^{66} + 6480 q^{67} - 37348 q^{68} + 40356 q^{69} - 110852 q^{71} - 9234 q^{72} - 64232 q^{73} - 27464 q^{74} - 43254 q^{75} + 194864 q^{76} + 127548 q^{78} + 111696 q^{79} + 308940 q^{80} + 39366 q^{81} + 189640 q^{82} - 101128 q^{83} - 23292 q^{85} + 3824 q^{86} + 47880 q^{87} - 97788 q^{88} + 35012 q^{89} - 69984 q^{90} - 449260 q^{92} + 35784 q^{93} + 121016 q^{94} - 119080 q^{95} + 335934 q^{96} - 70952 q^{97} + 48924 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 - 54 * q^3 + 150 * q^4 - 100 * q^5 - 18 * q^6 - 114 * q^8 + 486 * q^9 - 864 * q^10 + 604 * q^11 - 1350 * q^12 - 1352 * q^13 + 900 * q^15 + 4578 * q^16 - 3028 * q^17 + 162 * q^18 - 1728 * q^19 - 452 * q^20 - 4116 * q^22 - 4484 * q^23 + 1026 * q^24 + 4806 * q^25 - 14172 * q^26 - 4374 * q^27 - 5320 * q^29 + 7776 * q^30 - 3976 * q^31 - 37326 * q^32 - 5436 * q^33 + 16336 * q^34 + 12150 * q^36 + 22680 * q^37 - 52744 * q^38 + 12168 * q^39 - 100600 * q^40 - 28756 * q^41 - 6768 * q^43 - 64940 * q^44 - 8100 * q^45 + 540 * q^46 - 51552 * q^47 - 41202 * q^48 - 40622 * q^50 + 27252 * q^51 - 119296 * q^52 + 80884 * q^53 - 1458 * q^54 - 11656 * q^55 + 15552 * q^57 - 70464 * q^58 - 8872 * q^59 + 4068 * q^60 - 50896 * q^61 - 11824 * q^62 + 199590 * q^64 + 3492 * q^65 + 37044 * q^66 + 6480 * q^67 - 37348 * q^68 + 40356 * q^69 - 110852 * q^71 - 9234 * q^72 - 64232 * q^73 - 27464 * q^74 - 43254 * q^75 + 194864 * q^76 + 127548 * q^78 + 111696 * q^79 + 308940 * q^80 + 39366 * q^81 + 189640 * q^82 - 101128 * q^83 - 23292 * q^85 + 3824 * q^86 + 47880 * q^87 - 97788 * q^88 + 35012 * q^89 - 69984 * q^90 - 449260 * q^92 + 35784 * q^93 + 121016 * q^94 - 119080 * q^95 + 335934 * q^96 - 70952 * q^97 + 48924 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more