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Properties

Label 325.1.u.a
Level $325$
Weight $1$
Character orbit 325.u
Analytic conductor $0.162$
Analytic rank $0$
Dimension $8$
Projective image $A_{4}$
CM/RM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,1,Mod(68,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.68");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 325.u (of order \(12\), degree \(4\), 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: \(0.162196129106\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.4225.1
Artin image: $C_8.A_4$
Artin field: Galois closure of 32.0.24788700386255228556692600250244140625.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{5} q^{2} - \zeta_{24}^{11} q^{3} - \zeta_{24}^{4} q^{6} - \zeta_{24} q^{7} - \zeta_{24}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{5} q^{2} - \zeta_{24}^{11} q^{3} - \zeta_{24}^{4} q^{6} - \zeta_{24} q^{7} - \zeta_{24}^{3} q^{8} + \zeta_{24}^{8} q^{11} + \zeta_{24}^{3} q^{13} + \zeta_{24}^{6} q^{14} + \zeta_{24}^{8} q^{16} + \zeta_{24} q^{17} - \zeta_{24}^{10} q^{19} - q^{21} + \zeta_{24} q^{22} + \zeta_{24}^{11} q^{23} - \zeta_{24}^{2} q^{24} - \zeta_{24}^{8} q^{26} + \zeta_{24}^{9} q^{27} - \zeta_{24}^{2} q^{29} - \zeta_{24} q^{32} + \zeta_{24}^{7} q^{33} - \zeta_{24}^{6} q^{34} + \zeta_{24}^{5} q^{37} - \zeta_{24}^{3} q^{38} + \zeta_{24}^{2} q^{39} - \zeta_{24}^{8} q^{41} + \zeta_{24}^{5} q^{42} - \zeta_{24}^{7} q^{43} + \zeta_{24}^{4} q^{46} + \zeta_{24}^{7} q^{48} + q^{51} + \zeta_{24}^{2} q^{54} + \zeta_{24}^{4} q^{56} - \zeta_{24}^{9} q^{57} + \zeta_{24}^{7} q^{58} + \zeta_{24}^{10} q^{59} - \zeta_{24}^{4} q^{61} + \zeta_{24}^{6} q^{64} + q^{66} + \zeta_{24}^{5} q^{67} + \zeta_{24}^{10} q^{69} - \zeta_{24}^{4} q^{71} - \zeta_{24}^{10} q^{74} - \zeta_{24}^{9} q^{77} - \zeta_{24}^{7} q^{78} + \zeta_{24}^{8} q^{81} - \zeta_{24} q^{82} - q^{86} - \zeta_{24} q^{87} - \zeta_{24}^{11} q^{88} + \zeta_{24}^{2} q^{89} - \zeta_{24}^{4} q^{91} - \zeta_{24} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{6} - 4 q^{11} - 4 q^{16} - 8 q^{21} + 4 q^{26} + 4 q^{41} + 4 q^{46} + 8 q^{51} + 4 q^{56} - 4 q^{61} + 8 q^{66} - 4 q^{71} - 4 q^{81} - 8 q^{86} - 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-\zeta_{24}^{6}\) \(-\zeta_{24}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.965926 − 0.258819i 0.258819 − 0.965926i 0 0 −0.500000 + 0.866025i −0.258819 − 0.965926i 0.707107 + 0.707107i 0 0
68.2 0.965926 + 0.258819i −0.258819 + 0.965926i 0 0 −0.500000 + 0.866025i 0.258819 + 0.965926i −0.707107 − 0.707107i 0 0
107.1 −0.258819 + 0.965926i 0.965926 + 0.258819i 0 0 −0.500000 + 0.866025i −0.965926 + 0.258819i −0.707107 + 0.707107i 0 0
107.2 0.258819 − 0.965926i −0.965926 − 0.258819i 0 0 −0.500000 + 0.866025i 0.965926 − 0.258819i 0.707107 − 0.707107i 0 0
243.1 −0.258819 − 0.965926i 0.965926 − 0.258819i 0 0 −0.500000 − 0.866025i −0.965926 − 0.258819i −0.707107 − 0.707107i 0 0
243.2 0.258819 + 0.965926i −0.965926 + 0.258819i 0 0 −0.500000 − 0.866025i 0.965926 + 0.258819i 0.707107 + 0.707107i 0 0
282.1 −0.965926 + 0.258819i 0.258819 + 0.965926i 0 0 −0.500000 − 0.866025i −0.258819 + 0.965926i 0.707107 − 0.707107i 0 0
282.2 0.965926 − 0.258819i −0.258819 − 0.965926i 0 0 −0.500000 − 0.866025i 0.258819 − 0.965926i −0.707107 + 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
13.c even 3 1 inner
65.n even 6 1 inner
65.q odd 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.1.u.a ✓ 8
3.b odd 2 1 2925.1.cy.b 8
5.b even 2 1 inner 325.1.u.a ✓ 8
5.c odd 4 2 inner 325.1.u.a ✓ 8
13.c even 3 1 inner 325.1.u.a ✓ 8
15.d odd 2 1 2925.1.cy.b 8
15.e even 4 2 2925.1.cy.b 8
39.i odd 6 1 2925.1.cy.b 8
65.n even 6 1 inner 325.1.u.a ✓ 8
65.q odd 12 2 inner 325.1.u.a ✓ 8
195.x odd 6 1 2925.1.cy.b 8
195.bl even 12 2 2925.1.cy.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.1.u.a ✓ 8 1.a even 1 1 trivial
325.1.u.a ✓ 8 5.b even 2 1 inner
325.1.u.a ✓ 8 5.c odd 4 2 inner
325.1.u.a ✓ 8 13.c even 3 1 inner
325.1.u.a ✓ 8 65.n even 6 1 inner
325.1.u.a ✓ 8 65.q odd 12 2 inner
2925.1.cy.b 8 3.b odd 2 1
2925.1.cy.b 8 15.d odd 2 1
2925.1.cy.b 8 15.e even 4 2
2925.1.cy.b 8 39.i odd 6 1
2925.1.cy.b 8 195.x odd 6 1
2925.1.cy.b 8 195.bl even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(325, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$29$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$41$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$71$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
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