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Properties

Label 245.4.f.b
Level $245$
Weight $4$
Character orbit 245.f
Analytic conductor $14.455$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,4,Mod(48,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.48");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.f (of order \(4\), degree \(2\), 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: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 32 q^{11} - 1152 q^{16} - 64 q^{18} + 576 q^{22} + 768 q^{23} + 288 q^{25} - 2512 q^{30} + 1840 q^{32} - 4640 q^{36} + 864 q^{37} + 608 q^{43} - 3552 q^{46} - 2480 q^{50} + 1056 q^{51} - 1384 q^{53} + 7872 q^{57} + 5296 q^{58} + 9104 q^{60} + 736 q^{65} + 1856 q^{67} - 6816 q^{71} - 8528 q^{72} - 5408 q^{78} - 12616 q^{81} + 5600 q^{85} + 8672 q^{86} - 10080 q^{88} + 10624 q^{92} + 416 q^{93} + 5888 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
48.1 −3.87967 + 3.87967i −4.10057 + 4.10057i − 22.1036i −10.5738 − 3.63242i − 31.8177i 0 54.7173 + 54.7173i − 6.62940i 55.1154 − 26.9303i
48.2 −3.87967 + 3.87967i 4.10057 − 4.10057i − 22.1036i 10.5738 + 3.63242i 31.8177i 0 54.7173 + 54.7173i − 6.62940i −55.1154 + 26.9303i
48.3 −3.43814 + 3.43814i −1.63468 + 1.63468i − 15.6416i 10.9126 − 2.43206i − 11.2405i 0 26.2728 + 26.2728i 21.6556i −29.1573 + 45.8808i
48.4 −3.43814 + 3.43814i 1.63468 − 1.63468i − 15.6416i −10.9126 + 2.43206i 11.2405i 0 26.2728 + 26.2728i 21.6556i 29.1573 − 45.8808i
48.5 −3.07628 + 3.07628i −7.21462 + 7.21462i − 10.9270i 3.77798 − 10.5227i − 44.3884i 0 9.00424 + 9.00424i − 77.1016i 20.7486 + 43.9928i
48.6 −3.07628 + 3.07628i 7.21462 − 7.21462i − 10.9270i −3.77798 + 10.5227i 44.3884i 0 9.00424 + 9.00424i − 77.1016i −20.7486 − 43.9928i
48.7 −2.80000 + 2.80000i −3.95087 + 3.95087i − 7.68004i 10.9464 + 2.27495i − 22.1249i 0 −0.895902 − 0.895902i − 4.21879i −37.0199 + 24.2802i
48.8 −2.80000 + 2.80000i 3.95087 − 3.95087i − 7.68004i −10.9464 − 2.27495i 22.1249i 0 −0.895902 − 0.895902i − 4.21879i 37.0199 − 24.2802i
48.9 −2.26795 + 2.26795i −0.903970 + 0.903970i − 2.28720i −7.11962 + 8.62038i − 4.10032i 0 −12.9563 − 12.9563i 25.3657i −3.40366 − 35.6976i
48.10 −2.26795 + 2.26795i 0.903970 − 0.903970i − 2.28720i 7.11962 − 8.62038i 4.10032i 0 −12.9563 − 12.9563i 25.3657i 3.40366 + 35.6976i
48.11 −1.25791 + 1.25791i −6.21110 + 6.21110i 4.83534i 4.89978 + 10.0495i − 15.6260i 0 −16.1457 − 16.1457i − 50.1554i −18.8048 − 6.47785i
48.12 −1.25791 + 1.25791i 6.21110 − 6.21110i 4.83534i −4.89978 − 10.0495i 15.6260i 0 −16.1457 − 16.1457i − 50.1554i 18.8048 + 6.47785i
48.13 −1.17475 + 1.17475i −1.49701 + 1.49701i 5.23990i 0.670040 + 11.1602i − 3.51724i 0 −15.5536 − 15.5536i 22.5179i −13.8977 − 12.3234i
48.14 −1.17475 + 1.17475i 1.49701 − 1.49701i 5.23990i −0.670040 − 11.1602i 3.51724i 0 −15.5536 − 15.5536i 22.5179i 13.8977 + 12.3234i
48.15 −0.753059 + 0.753059i −5.33387 + 5.33387i 6.86581i −0.872788 − 11.1462i − 8.03343i 0 −11.1948 − 11.1948i − 29.9003i 9.05102 + 7.73650i
48.16 −0.753059 + 0.753059i 5.33387 − 5.33387i 6.86581i 0.872788 + 11.1462i 8.03343i 0 −11.1948 − 11.1948i − 29.9003i −9.05102 − 7.73650i
48.17 −0.547180 + 0.547180i −0.435810 + 0.435810i 7.40119i 10.9041 − 2.47015i − 0.476933i 0 −8.42723 − 8.42723i 26.6201i −4.61487 + 7.31810i
48.18 −0.547180 + 0.547180i 0.435810 − 0.435810i 7.40119i −10.9041 + 2.47015i 0.476933i 0 −8.42723 − 8.42723i 26.6201i 4.61487 − 7.31810i
48.19 −0.412101 + 0.412101i −6.71998 + 6.71998i 7.66034i −11.1765 − 0.294346i − 5.53862i 0 −6.45365 − 6.45365i − 63.3162i 4.72714 − 4.48454i
48.20 −0.412101 + 0.412101i 6.71998 − 6.71998i 7.66034i 11.1765 + 0.294346i 5.53862i 0 −6.45365 − 6.45365i − 63.3162i −4.72714 + 4.48454i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 48.36
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.f.b ✓ 72
5.c odd 4 1 inner 245.4.f.b ✓ 72
7.b odd 2 1 inner 245.4.f.b ✓ 72
7.c even 3 2 245.4.l.d 144
7.d odd 6 2 245.4.l.d 144
35.f even 4 1 inner 245.4.f.b ✓ 72
35.k even 12 2 245.4.l.d 144
35.l odd 12 2 245.4.l.d 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.f.b ✓ 72 1.a even 1 1 trivial
245.4.f.b ✓ 72 5.c odd 4 1 inner
245.4.f.b ✓ 72 7.b odd 2 1 inner
245.4.f.b ✓ 72 35.f even 4 1 inner
245.4.l.d 144 7.c even 3 2
245.4.l.d 144 7.d odd 6 2
245.4.l.d 144 35.k even 12 2
245.4.l.d 144 35.l odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 2160 T_{2}^{32} - 184 T_{2}^{31} + 6344 T_{2}^{29} + 1715834 T_{2}^{28} + \cdots + 535191579787264 \) acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\). Copy content Toggle raw display