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Properties

Label 115.5.c.c
Level $115$
Weight $5$
Character orbit 115.c
Analytic conductor $11.888$
Analytic rank $0$
Dimension $44$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,5,Mod(114,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.114");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 115.c (of order \(2\), degree \(1\), 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: \(11.8875457546\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 360 q^{4} + 160 q^{6} - 1128 q^{9} + 2136 q^{16} - 3940 q^{24} - 2540 q^{25} + 3144 q^{26} - 348 q^{29} - 4480 q^{31} - 1440 q^{35} - 3728 q^{36} - 1800 q^{39} + 7176 q^{41} + 5372 q^{46} + 9580 q^{49}+ \cdots - 39960 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
114.1 − 7.82075i 2.35396i −45.1642 −21.7947 − 12.2470i 18.4097 −44.2956 228.086i 75.4589 −95.7809 + 170.451i
114.2 − 7.82075i 2.35396i −45.1642 21.7947 + 12.2470i 18.4097 44.2956 228.086i 75.4589 95.7809 − 170.451i
114.3 − 6.76854i − 12.7750i −29.8132 −13.2033 + 21.2290i −86.4679 5.97783 93.4949i −82.1996 143.690 + 89.3674i
114.4 − 6.76854i − 12.7750i −29.8132 13.2033 − 21.2290i −86.4679 −5.97783 93.4949i −82.1996 −143.690 − 89.3674i
114.5 − 6.47969i 15.9529i −25.9863 −3.25644 − 24.7870i 103.370 26.8346 64.7082i −173.496 −160.612 + 21.1007i
114.6 − 6.47969i 15.9529i −25.9863 3.25644 + 24.7870i 103.370 −26.8346 64.7082i −173.496 160.612 − 21.1007i
114.7 − 6.07514i 5.74843i −20.9073 −19.8103 + 15.2497i 34.9225 44.4256 29.8125i 47.9555 92.6440 + 120.350i
114.8 − 6.07514i 5.74843i −20.9073 19.8103 − 15.2497i 34.9225 −44.4256 29.8125i 47.9555 −92.6440 − 120.350i
114.9 − 5.17567i 1.68656i −10.7876 −10.7683 − 22.5620i 8.72909 91.0414 − 26.9778i 78.1555 −116.773 + 55.7333i
114.10 − 5.17567i 1.68656i −10.7876 10.7683 + 22.5620i 8.72909 −91.0414 − 26.9778i 78.1555 116.773 − 55.7333i
114.11 − 4.94563i − 6.82837i −8.45927 −22.1235 − 11.6426i −33.7706 −47.4193 − 37.2937i 34.3734 −57.5800 + 109.415i
114.12 − 4.94563i − 6.82837i −8.45927 22.1235 + 11.6426i −33.7706 47.4193 − 37.2937i 34.3734 57.5800 − 109.415i
114.13 − 3.09098i 9.49430i 6.44587 −21.4801 + 12.7909i 29.3466 12.9929 − 69.3796i −9.14173 39.5363 + 66.3944i
114.14 − 3.09098i 9.49430i 6.44587 21.4801 − 12.7909i 29.3466 −12.9929 − 69.3796i −9.14173 −39.5363 − 66.3944i
114.15 − 2.69449i − 17.0210i 8.73974 −22.5851 − 10.7197i −45.8630 63.1360 − 66.6609i −208.716 −28.8842 + 60.8553i
114.16 − 2.69449i − 17.0210i 8.73974 22.5851 + 10.7197i −45.8630 −63.1360 − 66.6609i −208.716 28.8842 − 60.8553i
114.17 − 2.39227i 13.3085i 10.2771 −19.8434 − 15.2066i 31.8375 −60.8570 − 62.8617i −96.1161 −36.3782 + 47.4707i
114.18 − 2.39227i 13.3085i 10.2771 19.8434 + 15.2066i 31.8375 60.8570 − 62.8617i −96.1161 36.3782 − 47.4707i
114.19 − 2.28507i − 4.59850i 10.7785 −9.51726 + 23.1176i −10.5079 −33.6997 − 61.1906i 59.8538 52.8252 + 21.7476i
114.20 − 2.28507i − 4.59850i 10.7785 9.51726 − 23.1176i −10.5079 33.6997 − 61.1906i 59.8538 −52.8252 − 21.7476i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 114.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.b odd 2 1 inner
115.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.5.c.c ✓ 44
5.b even 2 1 inner 115.5.c.c ✓ 44
23.b odd 2 1 inner 115.5.c.c ✓ 44
115.c odd 2 1 inner 115.5.c.c ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.5.c.c ✓ 44 1.a even 1 1 trivial
115.5.c.c ✓ 44 5.b even 2 1 inner
115.5.c.c ✓ 44 23.b odd 2 1 inner
115.5.c.c ✓ 44 115.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(115, [\chi])\):

\( T_{2}^{22} + 266 T_{2}^{20} + 30135 T_{2}^{18} + 1903779 T_{2}^{16} + 73772780 T_{2}^{14} + \cdots + 6624707045184 \) Copy content Toggle raw display
\( T_{7}^{22} - 28806 T_{7}^{20} + 347917924 T_{7}^{18} - 2323948644785 T_{7}^{16} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display