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Properties

Label 115.7.c.c
Level $115$
Weight $7$
Character orbit 115.c
Analytic conductor $26.456$
Analytic rank $0$
Dimension $68$
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] = [115,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.114");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(26.4562196163\)
Analytic rank: \(0\)
Dimension: \(68\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 2440 q^{4} + 352 q^{6} - 16908 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 2440 q^{4} + 352 q^{6} - 16908 q^{9} + 66968 q^{16} - 30916 q^{24} + 32588 q^{25} - 22072 q^{26} + 103360 q^{29} - 17256 q^{31} - 358168 q^{35} + 451984 q^{36} + 192432 q^{39} - 183552 q^{41} - 397956 q^{46} + 806756 q^{49} - 749960 q^{50} - 1638436 q^{54} - 1752 q^{55} - 505552 q^{59} - 4095100 q^{64} + 1354876 q^{69} + 1196604 q^{70} + 493688 q^{71} + 3178568 q^{75} + 2473820 q^{81} + 3306336 q^{85} - 3770196 q^{94} + 896144 q^{95} + 16928136 q^{96}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
114.1 − 4.94808i 27.5624i 39.5165 31.3018 − 121.017i 136.381 −630.980 − 512.208i −30.6878 −598.804 − 154.884i
114.2 4.94808i − 27.5624i 39.5165 31.3018 + 121.017i 136.381 −630.980 512.208i −30.6878 −598.804 + 154.884i
114.3 − 15.5178i 42.9237i −176.801 113.739 − 51.8501i 666.080 −624.197 1750.41i −1113.45 −804.597 − 1764.97i
114.4 15.5178i − 42.9237i −176.801 113.739 + 51.8501i 666.080 −624.197 − 1750.41i −1113.45 −804.597 + 1764.97i
114.5 − 11.1333i − 17.3572i −59.9514 −121.447 + 29.5914i −193.244 504.412 − 45.0744i 427.728 329.451 + 1352.11i
114.6 11.1333i 17.3572i −59.9514 −121.447 − 29.5914i −193.244 504.412 45.0744i 427.728 329.451 − 1352.11i
114.7 − 10.4494i − 30.8468i −45.1897 41.2139 + 118.010i −322.330 505.735 − 196.556i −222.524 1233.13 − 430.660i
114.8 10.4494i 30.8468i −45.1897 41.2139 − 118.010i −322.330 505.735 196.556i −222.524 1233.13 + 430.660i
114.9 − 14.4471i − 3.70784i −144.720 95.5216 − 80.6265i −53.5677 454.114 1166.17i 715.252 −1164.82 − 1380.01i
114.10 14.4471i 3.70784i −144.720 95.5216 + 80.6265i −53.5677 454.114 − 1166.17i 715.252 −1164.82 + 1380.01i
114.11 − 8.96761i − 45.1542i −16.4180 −49.9957 + 114.566i −404.925 −395.172 − 426.697i −1309.90 1027.39 + 448.342i
114.12 8.96761i 45.1542i −16.4180 −49.9957 − 114.566i −404.925 −395.172 426.697i −1309.90 1027.39 − 448.342i
114.13 − 4.35786i 24.6249i 45.0090 108.536 + 62.0072i 107.312 −356.743 − 475.046i 122.613 270.219 − 472.986i
114.14 4.35786i − 24.6249i 45.0090 108.536 − 62.0072i 107.312 −356.743 475.046i 122.613 270.219 + 472.986i
114.15 − 11.1327i 32.3873i −59.9364 122.088 + 26.8239i 360.557 301.337 − 45.2393i −319.938 298.622 − 1359.17i
114.16 11.1327i − 32.3873i −59.9364 122.088 − 26.8239i 360.557 301.337 45.2393i −319.938 298.622 + 1359.17i
114.17 − 0.0499926i 38.9668i 63.9975 95.0255 + 81.2106i 1.94805 −273.692 − 6.39893i −789.409 4.05993 − 4.75057i
114.18 0.0499926i − 38.9668i 63.9975 95.0255 − 81.2106i 1.94805 −273.692 6.39893i −789.409 4.05993 + 4.75057i
114.19 − 7.58308i 53.1699i 6.49696 −65.6908 + 106.347i 403.191 −232.063 − 534.584i −2098.04 806.439 + 498.138i
114.20 7.58308i − 53.1699i 6.49696 −65.6908 − 106.347i 403.191 −232.063 534.584i −2098.04 806.439 − 498.138i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 114.68
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 115.7.c.c ✓ 68
5.b even 2 1 inner 115.7.c.c ✓ 68
23.b odd 2 1 inner 115.7.c.c ✓ 68
115.c odd 2 1 inner 115.7.c.c ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.7.c.c ✓ 68 1.a even 1 1 trivial
115.7.c.c ✓ 68 5.b even 2 1 inner
115.7.c.c ✓ 68 23.b odd 2 1 inner
115.7.c.c ✓ 68 115.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 \(S_{7}^{\mathrm{new}}(115, [\chi])\):

\( T_{2}^{34} + 1698 T_{2}^{32} + 1305039 T_{2}^{30} + 601112923 T_{2}^{28} + 185197947420 T_{2}^{26} + 40339513938099 T_{2}^{24} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{34} - 2201722 T_{7}^{32} + 2149633757328 T_{7}^{30} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display