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Properties

Label 3060.2.w.a
Level $3060$
Weight $2$
Character orbit 3060.w
Analytic conductor $24.434$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3060,2,Mod(953,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3060.953"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3060.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.4342230185\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{8}^{3} + 2 \zeta_{8}) q^{5} + (2 \zeta_{8}^{2} - 2) q^{7} + (4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{11} + ( - \zeta_{8}^{2} - 1) q^{13} - \zeta_{8} q^{17} + 8 \zeta_{8}^{2} q^{19} - 4 \zeta_{8}^{3} q^{23} + \cdots + (11 \zeta_{8}^{2} - 11) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7} - 4 q^{13} - 16 q^{25} + 12 q^{37} - 8 q^{43} - 48 q^{55} - 16 q^{61} + 24 q^{67} + 36 q^{73} + 4 q^{85} + 16 q^{91} - 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times\).

\(n\) \(1261\) \(1361\) \(1531\) \(1837\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-\zeta_{8}^{2}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
953.1
−0.707107 − 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 − 0.707107i
0 0 0 −0.707107 − 2.12132i 0 −2.00000 + 2.00000i 0 0 0
953.2 0 0 0 0.707107 + 2.12132i 0 −2.00000 + 2.00000i 0 0 0
2177.1 0 0 0 −0.707107 + 2.12132i 0 −2.00000 − 2.00000i 0 0 0
2177.2 0 0 0 0.707107 − 2.12132i 0 −2.00000 − 2.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3060.2.w.a ✓ 4
3.b odd 2 1 inner 3060.2.w.a ✓ 4
5.c odd 4 1 inner 3060.2.w.a ✓ 4
15.e even 4 1 inner 3060.2.w.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3060.2.w.a ✓ 4 1.a even 1 1 trivial
3060.2.w.a ✓ 4 3.b odd 2 1 inner
3060.2.w.a ✓ 4 5.c odd 4 1 inner
3060.2.w.a ✓ 4 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 4T_{7} + 8 \) acting on \(S_{2}^{\mathrm{new}}(3060, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 8T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 20736 \) Copy content Toggle raw display
$53$ \( T^{4} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} - 200)^{2} \) Copy content Toggle raw display
$61$ \( (T + 4)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 12 T + 72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 18 T + 162)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 20736 \) Copy content Toggle raw display
$89$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 22 T + 242)^{2} \) Copy content Toggle raw display
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