LMFDB - Newform orbit 74.8.f.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,8,Mod(7,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([16]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("74.7");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$74 = 2 \cdot 37$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	74.f (of order $9$ , degree $6$ , minimal)

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:	no
Analytic conductor:	$23.1164918858$
Analytic rank:	$0$
Dimension:	$66$
Relative dimension:	$11$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$ -expansion

The algebraic $q$ -expansion of this newform has not been computed, but we have computed the trace expansion.

\operatorname{Tr}(f)(q) =

66 q - 39 q^{3} + 459 q^{5} - 918 q^{7} + 16896 q^{8} + 3237 q^{9} + 4704 q^{10} + 4539 q^{11} - 2496 q^{12} - 10542 q^{13} + 6744 q^{14} - 39894 q^{15} - 78822 q^{17} + 51792 q^{18} + 79461 q^{19} + 29376 q^{20}+ \cdots + 66155463 q^{99}+O(q^{100})

66 * q - 39 * q^3 + 459 * q^5 - 918 * q^7 + 16896 * q^8 + 3237 * q^9 + 4704 * q^10 + 4539 * q^11 - 2496 * q^12 - 10542 * q^13 + 6744 * q^14 - 39894 * q^15 - 78822 * q^17 + 51792 * q^18 + 79461 * q^19 + 29376 * q^20 - 5835 * q^21 + 18168 * q^22 - 199479 * q^23 + 19968 * q^24 + 356301 * q^25 + 186744 * q^26 - 91689 * q^27 - 91584 * q^28 - 38526 * q^29 + 319152 * q^30 + 1917246 * q^31 - 902847 * q^33 - 205776 * q^34 - 1574793 * q^35 + 3079296 * q^36 - 396513 * q^37 - 692880 * q^38 + 2514306 * q^39 + 470016 * q^40 + 2989350 * q^41 + 46680 * q^42 + 6006348 * q^43 - 145344 * q^44 + 1083726 * q^45 - 588744 * q^46 - 894135 * q^47 + 7973742 * q^49 + 2012400 * q^50 - 524880 * q^51 - 2262720 * q^52 + 1457844 * q^53 + 1452312 * q^54 + 2415690 * q^55 - 1202688 * q^56 + 2734338 * q^57 - 3423360 * q^58 + 3099522 * q^59 - 3102912 * q^60 - 10968396 * q^61 - 3067320 * q^62 - 10727970 * q^63 - 8650752 * q^64 - 13497300 * q^65 - 4598712 * q^66 - 1978119 * q^67 + 5026176 * q^68 - 7251108 * q^69 + 435744 * q^70 - 19555002 * q^71 + 3314688 * q^72 - 31332840 * q^73 + 7607472 * q^74 + 44758074 * q^75 + 5085504 * q^76 + 1982007 * q^77 - 22512264 * q^78 + 10522902 * q^79 + 4816896 * q^80 - 7548876 * q^81 - 7150320 * q^82 + 19771926 * q^83 + 3556224 * q^84 - 6684072 * q^85 + 22982496 * q^86 - 67769667 * q^87 - 2323968 * q^88 - 42018963 * q^89 - 25158672 * q^90 + 4847565 * q^91 - 1096704 * q^92 - 14558475 * q^93 + 3175680 * q^94 + 4508415 * q^95 - 2555904 * q^96 - 39286662 * q^97 + 71630904 * q^98 + 66155463 * q^99

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$	$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
7.1	7.51754	−	2.73616i	−83.1185	−	30.2527i	49.0268	−	41.1384i	77.4784	+	439.402i	−707.623	−71.3442	−	404.613i	256.000	−	443.405i	4318.12	+	3623.33i	1784.72	+	3091.23i
7.2	7.51754	−	2.73616i	−62.7580	−	22.8420i	49.0268	−	41.1384i	1.66323	+	9.43265i	−534.285	141.286	+	801.274i	256.000	−	443.405i	1741.46	+	1461.26i	38.3126	+	66.3594i
7.3	7.51754	−	2.73616i	−56.1799	−	20.4478i	49.0268	−	41.1384i	−76.5259	−	434.000i	−478.283	−144.408	−	818.976i	256.000	−	443.405i	1062.73	+	891.735i	−1762.78	−	3053.23i
7.4	7.51754	−	2.73616i	−36.4042	−	13.2501i	49.0268	−	41.1384i	−1.10281	−	6.25436i	−309.925	125.532	+	711.929i	256.000	−	443.405i	−525.635	−	441.061i	−25.4034	−	43.9999i
7.5	7.51754	−	2.73616i	−19.5402	−	7.11207i	49.0268	−	41.1384i	69.5532	+	394.456i	−166.354	−65.8495	−	373.451i	256.000	−	443.405i	−1344.10	−	1127.83i	1602.16	+	2775.03i
7.6	7.51754	−	2.73616i	6.89307	+	2.50887i	49.0268	−	41.1384i	−27.8528	−	157.961i	58.6836	117.132	+	664.288i	256.000	−	443.405i	−1634.12	−	1371.19i	−641.591	−	1111.27i
7.7	7.51754	−	2.73616i	25.6856	+	9.34879i	49.0268	−	41.1384i	46.1726	+	261.858i	218.672	−206.971	−	1173.79i	256.000	−	443.405i	−1102.99	−	925.518i	1063.59	+	1842.19i
7.8	7.51754	−	2.73616i	28.9563	+	10.5392i	49.0268	−	41.1384i	−48.7600	−	276.532i	246.517	−219.425	−	1244.42i	256.000	−	443.405i	−947.948	−	795.422i	−1123.19	−	1945.43i
7.9	7.51754	−	2.73616i	46.7880	+	17.0295i	49.0268	−	41.1384i	58.4927	+	331.728i	398.326	201.162	+	1140.85i	256.000	−	443.405i	223.779	+	187.773i	1347.38	+	2333.74i
7.10	7.51754	−	2.73616i	65.8840	+	23.9798i	49.0268	−	41.1384i	42.5352	+	241.229i	560.898	44.8916	+	254.593i	256.000	−	443.405i	2090.33	+	1754.00i	979.802	+	1697.07i
7.11	7.51754	−	2.73616i	79.5512	+	28.9543i	49.0268	−	41.1384i	−57.6869	−	327.158i	677.253	−6.84167	−	38.8010i	256.000	−	443.405i	3814.71	+	3200.92i	−1328.82	−	2301.59i
9.1	−1.38919	−	7.87846i	−14.0898	+	79.9071i	−60.1403	+	21.8893i	−46.9471	−	39.3933i	649.118	−514.898	−	432.050i	256.000	+	443.405i	−4131.51	−	1503.75i	−245.140	+	424.595i
9.2	−1.38919	−	7.87846i	−11.0757	+	62.8133i	−60.1403	+	21.8893i	48.5860	+	40.7685i	510.258	745.438	+	625.497i	256.000	+	443.405i	−1767.73	−	643.401i	253.698	−	439.418i
9.3	−1.38919	−	7.87846i	−8.32711	+	47.2254i	−60.1403	+	21.8893i	−300.074	−	251.792i	383.631	−303.488	−	254.657i	256.000	+	443.405i	−105.789	−	38.5041i	−1566.88	+	2713.91i
9.4	−1.38919	−	7.87846i	−6.48463	+	36.7762i	−60.1403	+	21.8893i	381.143	+	319.817i	298.748	975.665	+	818.680i	256.000	+	443.405i	744.672	+	271.038i	1990.19	−	3447.11i
9.5	−1.38919	−	7.87846i	−2.39984	+	13.6102i	−60.1403	+	21.8893i	−34.7022	−	29.1186i	110.561	−140.013	−	117.485i	256.000	+	443.405i	1875.63	+	682.674i	−181.202	+	313.852i
9.6	−1.38919	−	7.87846i	−0.307798	+	1.74561i	−60.1403	+	21.8893i	323.926	+	271.806i	14.1803	−1225.86	−	1028.62i	256.000	+	443.405i	2052.16	+	746.923i	1691.42	−	2929.63i
9.7	−1.38919	−	7.87846i	2.48533	−	14.0950i	−60.1403	+	21.8893i	−356.271	−	298.947i	−114.500	1364.62	+	1145.05i	256.000	+	443.405i	1862.62	+	677.936i	−1860.32	+	3222.16i
9.8	−1.38919	−	7.87846i	6.70779	−	38.0417i	−60.1403	+	21.8893i	124.780	+	104.703i	−309.029	25.1771	+	21.1261i	256.000	+	443.405i	652.928	+	237.646i	651.556	−	1128.53i
9.9	−1.38919	−	7.87846i	9.76159	−	55.3608i	−60.1403	+	21.8893i	−282.513	−	237.057i	−449.718	−1337.20	−	1122.04i	256.000	+	443.405i	−914.416	−	332.820i	−1475.18	+	2555.09i

See all 66 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.f	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	74.8.f.b	✓	66
37.f	even	9	1	inner	74.8.f.b	✓	66

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
74.8.f.b	✓	66	1.a	even	1	1	trivial
74.8.f.b	✓	66	37.f	even	9	1	inner

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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 7.11
Significant digits:
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