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Properties

Label 3380.1.co.b
Level $3380$
Weight $1$
Character orbit 3380.co
Analytic conductor $1.687$
Analytic rank $0$
Dimension $24$
Projective image $D_{78}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3380,1,Mod(179,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(78))
 
chi = DirichletCharacter(H, H._module([39, 39, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3380.179");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3380.co (of order \(78\), degree \(24\), 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: \(1.68683974270\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{39})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{17} + x^{15} - x^{14} + x^{12} - x^{10} + x^{9} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{78}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{78} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{78}^{17} q^{2} + \zeta_{78}^{34} q^{4} - \zeta_{78}^{32} q^{5} + \zeta_{78}^{12} q^{8} - \zeta_{78}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{78}^{17} q^{2} + \zeta_{78}^{34} q^{4} - \zeta_{78}^{32} q^{5} + \zeta_{78}^{12} q^{8} - \zeta_{78}^{4} q^{9} - \zeta_{78}^{10} q^{10} - \zeta_{78}^{14} q^{13} - \zeta_{78}^{29} q^{16} + ( - \zeta_{78}^{30} + \zeta_{78}^{20}) q^{17} + \zeta_{78}^{21} q^{18} + \zeta_{78}^{27} q^{20} - \zeta_{78}^{25} q^{25} + \zeta_{78}^{31} q^{26} + ( - \zeta_{78}^{28} - \zeta_{78}^{6}) q^{29} - \zeta_{78}^{7} q^{32} + ( - \zeta_{78}^{37} - \zeta_{78}^{8}) q^{34} - \zeta_{78}^{38} q^{36} + (\zeta_{78}^{16} - \zeta_{78}^{9}) q^{37} + \zeta_{78}^{5} q^{40} + ( - \zeta_{78}^{24} - \zeta_{78}^{19}) q^{41} + \zeta_{78}^{36} q^{45} - \zeta_{78}^{11} q^{49} - \zeta_{78}^{3} q^{50} + \zeta_{78}^{9} q^{52} + (\zeta_{78}^{37} + \zeta_{78}^{26}) q^{53} + (\zeta_{78}^{23} - \zeta_{78}^{6}) q^{58} + ( - \zeta_{78}^{38} + \zeta_{78}^{3}) q^{61} + \zeta_{78}^{24} q^{64} - \zeta_{78}^{7} q^{65} + (\zeta_{78}^{25} - \zeta_{78}^{15}) q^{68} - \zeta_{78}^{16} q^{72} + \zeta_{78}^{9} q^{73} + ( - \zeta_{78}^{33} + \zeta_{78}^{26}) q^{74} - \zeta_{78}^{22} q^{80} + \zeta_{78}^{8} q^{81} + (\zeta_{78}^{36} - \zeta_{78}^{2}) q^{82} + ( - \zeta_{78}^{23} + \zeta_{78}^{13}) q^{85} + (\zeta_{78}^{29} - \zeta_{78}^{23}) q^{89} + \zeta_{78}^{14} q^{90} + ( - \zeta_{78}^{22} - \zeta_{78}^{10}) q^{97} + \zeta_{78}^{28} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + q^{2} + q^{4} - q^{5} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + q^{2} + q^{4} - q^{5} - 2 q^{8} - q^{9} - q^{10} - q^{13} + q^{16} + 3 q^{17} + 2 q^{18} + 2 q^{20} + q^{25} - q^{26} + q^{29} + q^{32} - q^{36} - q^{37} - q^{40} + 3 q^{41} - 2 q^{45} + q^{49} - 2 q^{50} + 2 q^{52} - 13 q^{53} + q^{58} + q^{61} - 2 q^{64} + q^{65} - 3 q^{68} - q^{72} + 2 q^{73} - 14 q^{74} - q^{80} + q^{81} - 3 q^{82} + 13 q^{85} + q^{90} - 2 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{78}^{34}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1
−0.996757 + 0.0804666i
0.692724 + 0.721202i
−0.845190 + 0.534466i
−0.919979 + 0.391967i
−0.0402659 − 0.999189i
−0.0402659 + 0.999189i
0.948536 + 0.316668i
0.428693 + 0.903450i
−0.632445 + 0.774605i
0.799443 + 0.600742i
0.987050 + 0.160411i
0.987050 − 0.160411i
0.948536 − 0.316668i
−0.200026 + 0.979791i
0.692724 − 0.721202i
−0.845190 − 0.534466i
0.278217 − 0.960518i
0.799443 − 0.600742i
−0.200026 − 0.979791i
0.278217 + 0.960518i
−0.200026 + 0.979791i 0 −0.919979 − 0.391967i 0.845190 + 0.534466i 0 0 0.568065 − 0.822984i −0.948536 + 0.316668i −0.692724 + 0.721202i
199.1 0.428693 + 0.903450i 0 −0.632445 + 0.774605i −0.799443 − 0.600742i 0 0 −0.970942 − 0.239316i 0.996757 + 0.0804666i 0.200026 − 0.979791i
439.1 0.987050 − 0.160411i 0 0.948536 − 0.316668i −0.692724 − 0.721202i 0 0 0.885456 − 0.464723i 0.632445 + 0.774605i −0.799443 − 0.600742i
459.1 −0.845190 + 0.534466i 0 0.428693 − 0.903450i −0.948536 + 0.316668i 0 0 0.120537 + 0.992709i 0.0402659 + 0.999189i 0.632445 − 0.774605i
719.1 −0.632445 − 0.774605i 0 −0.200026 + 0.979791i −0.278217 + 0.960518i 0 0 0.885456 − 0.464723i −0.987050 + 0.160411i 0.919979 − 0.391967i
959.1 −0.632445 + 0.774605i 0 −0.200026 − 0.979791i −0.278217 − 0.960518i 0 0 0.885456 + 0.464723i −0.987050 − 0.160411i 0.919979 + 0.391967i
979.1 0.692724 − 0.721202i 0 −0.0402659 − 0.999189i 0.632445 + 0.774605i 0 0 −0.748511 − 0.663123i −0.278217 − 0.960518i 0.996757 + 0.0804666i
1219.1 0.948536 + 0.316668i 0 0.799443 + 0.600742i 0.0402659 + 0.999189i 0 0 0.568065 + 0.822984i 0.200026 + 0.979791i −0.278217 + 0.960518i
1239.1 0.799443 + 0.600742i 0 0.278217 + 0.960518i 0.996757 − 0.0804666i 0 0 −0.354605 + 0.935016i 0.919979 − 0.391967i 0.845190 + 0.534466i
1479.1 −0.0402659 − 0.999189i 0 −0.996757 + 0.0804666i 0.200026 − 0.979791i 0 0 0.120537 + 0.992709i 0.845190 − 0.534466i −0.987050 − 0.160411i
1739.1 −0.919979 + 0.391967i 0 0.692724 − 0.721202i −0.428693 + 0.903450i 0 0 −0.354605 + 0.935016i −0.799443 − 0.600742i 0.0402659 − 0.999189i
1759.1 −0.919979 − 0.391967i 0 0.692724 + 0.721202i −0.428693 − 0.903450i 0 0 −0.354605 − 0.935016i −0.799443 + 0.600742i 0.0402659 + 0.999189i
1999.1 0.692724 + 0.721202i 0 −0.0402659 + 0.999189i 0.632445 − 0.774605i 0 0 −0.748511 + 0.663123i −0.278217 + 0.960518i 0.996757 − 0.0804666i
2019.1 0.278217 − 0.960518i 0 −0.845190 − 0.534466i −0.987050 − 0.160411i 0 0 −0.748511 + 0.663123i −0.692724 − 0.721202i −0.428693 + 0.903450i
2259.1 0.428693 − 0.903450i 0 −0.632445 − 0.774605i −0.799443 + 0.600742i 0 0 −0.970942 + 0.239316i 0.996757 − 0.0804666i 0.200026 + 0.979791i
2279.1 0.987050 + 0.160411i 0 0.948536 + 0.316668i −0.692724 + 0.721202i 0 0 0.885456 + 0.464723i 0.632445 − 0.774605i −0.799443 + 0.600742i
2519.1 −0.996757 − 0.0804666i 0 0.987050 + 0.160411i 0.919979 − 0.391967i 0 0 −0.970942 − 0.239316i −0.428693 − 0.903450i −0.948536 + 0.316668i
2539.1 −0.0402659 + 0.999189i 0 −0.996757 − 0.0804666i 0.200026 + 0.979791i 0 0 0.120537 − 0.992709i 0.845190 + 0.534466i −0.987050 + 0.160411i
2779.1 0.278217 + 0.960518i 0 −0.845190 + 0.534466i −0.987050 + 0.160411i 0 0 −0.748511 − 0.663123i −0.692724 + 0.721202i −0.428693 − 0.903450i
2799.1 −0.996757 + 0.0804666i 0 0.987050 − 0.160411i 0.919979 + 0.391967i 0 0 −0.970942 + 0.239316i −0.428693 + 0.903450i −0.948536 − 0.316668i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
845.bh even 78 1 inner
3380.co odd 78 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.1.co.b yes 24
4.b odd 2 1 CM 3380.1.co.b yes 24
5.b even 2 1 3380.1.co.a ✓ 24
20.d odd 2 1 3380.1.co.a ✓ 24
169.k even 78 1 3380.1.co.a ✓ 24
676.u odd 78 1 3380.1.co.a ✓ 24
845.bh even 78 1 inner 3380.1.co.b yes 24
3380.co odd 78 1 inner 3380.1.co.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3380.1.co.a ✓ 24 5.b even 2 1
3380.1.co.a ✓ 24 20.d odd 2 1
3380.1.co.a ✓ 24 169.k even 78 1
3380.1.co.a ✓ 24 676.u odd 78 1
3380.1.co.b yes 24 1.a even 1 1 trivial
3380.1.co.b yes 24 4.b odd 2 1 CM
3380.1.co.b yes 24 845.bh even 78 1 inner
3380.1.co.b yes 24 3380.co odd 78 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{24} - 3 T_{17}^{23} + 6 T_{17}^{22} - 9 T_{17}^{21} + 9 T_{17}^{20} - 26 T_{17}^{19} + 38 T_{17}^{18} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(3380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} - T^{23} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{24} \) Copy content Toggle raw display
$5$ \( T^{24} + T^{23} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{24} \) Copy content Toggle raw display
$11$ \( T^{24} \) Copy content Toggle raw display
$13$ \( T^{24} + T^{23} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{24} - 3 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{24} \) Copy content Toggle raw display
$23$ \( T^{24} \) Copy content Toggle raw display
$29$ \( T^{24} - T^{23} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{24} \) Copy content Toggle raw display
$37$ \( T^{24} + T^{23} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{24} - 3 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{24} \) Copy content Toggle raw display
$47$ \( T^{24} \) Copy content Toggle raw display
$53$ \( T^{24} + 13 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{24} \) Copy content Toggle raw display
$61$ \( T^{24} - T^{23} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{24} \) Copy content Toggle raw display
$71$ \( T^{24} \) Copy content Toggle raw display
$73$ \( (T^{12} - T^{11} + T^{10} + \cdots + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{24} \) Copy content Toggle raw display
$83$ \( T^{24} \) Copy content Toggle raw display
$89$ \( T^{24} - 13 T^{22} + \cdots + 169 \) Copy content Toggle raw display
$97$ \( T^{24} + 2 T^{23} + \cdots + 1 \) Copy content Toggle raw display
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