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Properties

Label 1925.1.cb.b
Level $1925$
Weight $1$
Character orbit 1925.cb
Analytic conductor $0.961$
Analytic rank $0$
Dimension $16$
Projective image $D_{15}$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1925,1,Mod(174,1925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1925, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 5, 6])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1925.174"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1925.cb (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.960700149319\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{60})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} - \cdots)\)

$q$-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)
 

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

\(f(q)\) \(=\) \( q + (\zeta_{60}^{23} - \zeta_{60}) q^{2} + ( - \zeta_{60}^{24} + \cdots + \zeta_{60}^{2}) q^{4} - \zeta_{60}^{9} q^{7} + (\zeta_{60}^{25} + \cdots - \zeta_{60}^{3}) q^{8} - \zeta_{60}^{12} q^{9} + \zeta_{60}^{28} q^{11} + \cdots + \zeta_{60}^{10} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{9} + 2 q^{11} + 6 q^{14} - 6 q^{16} - 4 q^{29} - 20 q^{44} + 16 q^{46} + 4 q^{49} - 4 q^{56} + 14 q^{64} - 6 q^{71} + 4 q^{74} + 6 q^{79} - 4 q^{81} - 14 q^{86} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times\).

\(n\) \(276\) \(1002\) \(1751\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{60}^{24}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
174.1
0.994522 − 0.104528i
0.406737 − 0.913545i
−0.406737 + 0.913545i
−0.994522 + 0.104528i
0.994522 + 0.104528i
0.406737 + 0.913545i
−0.406737 − 0.913545i
−0.994522 − 0.104528i
0.743145 − 0.669131i
−0.207912 − 0.978148i
0.207912 + 0.978148i
−0.743145 + 0.669131i
0.743145 + 0.669131i
−0.207912 + 0.978148i
0.207912 − 0.978148i
−0.743145 − 0.669131i
−1.73767 − 0.564602i 0 1.89169 + 1.37440i 0 0 −0.587785 + 0.809017i −1.43721 − 1.97815i −0.309017 + 0.951057i 0
174.2 −0.198825 − 0.0646021i 0 −0.773659 − 0.562096i 0 0 0.587785 − 0.809017i 0.240391 + 0.330869i −0.309017 + 0.951057i 0
174.3 0.198825 + 0.0646021i 0 −0.773659 − 0.562096i 0 0 −0.587785 + 0.809017i −0.240391 − 0.330869i −0.309017 + 0.951057i 0
174.4 1.73767 + 0.564602i 0 1.89169 + 1.37440i 0 0 0.587785 − 0.809017i 1.43721 + 1.97815i −0.309017 + 0.951057i 0
874.1 −1.73767 + 0.564602i 0 1.89169 − 1.37440i 0 0 −0.587785 − 0.809017i −1.43721 + 1.97815i −0.309017 − 0.951057i 0
874.2 −0.198825 + 0.0646021i 0 −0.773659 + 0.562096i 0 0 0.587785 + 0.809017i 0.240391 − 0.330869i −0.309017 − 0.951057i 0
874.3 0.198825 − 0.0646021i 0 −0.773659 + 0.562096i 0 0 −0.587785 − 0.809017i −0.240391 + 0.330869i −0.309017 − 0.951057i 0
874.4 1.73767 − 0.564602i 0 1.89169 − 1.37440i 0 0 0.587785 + 0.809017i 1.43721 − 1.97815i −0.309017 − 0.951057i 0
1049.1 −1.14988 + 1.58268i 0 −0.873619 − 2.68872i 0 0 −0.951057 + 0.309017i 3.39939 + 1.10453i 0.809017 + 0.587785i 0
1049.2 −0.786610 + 1.08268i 0 −0.244415 − 0.752232i 0 0 0.951057 − 0.309017i −0.266080 − 0.0864545i 0.809017 + 0.587785i 0
1049.3 0.786610 − 1.08268i 0 −0.244415 − 0.752232i 0 0 −0.951057 + 0.309017i 0.266080 + 0.0864545i 0.809017 + 0.587785i 0
1049.4 1.14988 − 1.58268i 0 −0.873619 − 2.68872i 0 0 0.951057 − 0.309017i −3.39939 − 1.10453i 0.809017 + 0.587785i 0
1224.1 −1.14988 − 1.58268i 0 −0.873619 + 2.68872i 0 0 −0.951057 − 0.309017i 3.39939 − 1.10453i 0.809017 − 0.587785i 0
1224.2 −0.786610 − 1.08268i 0 −0.244415 + 0.752232i 0 0 0.951057 + 0.309017i −0.266080 + 0.0864545i 0.809017 − 0.587785i 0
1224.3 0.786610 + 1.08268i 0 −0.244415 + 0.752232i 0 0 −0.951057 − 0.309017i 0.266080 − 0.0864545i 0.809017 − 0.587785i 0
1224.4 1.14988 + 1.58268i 0 −0.873619 + 2.68872i 0 0 0.951057 + 0.309017i −3.39939 + 1.10453i 0.809017 − 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 174.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
5.b even 2 1 inner
11.c even 5 1 inner
35.c odd 2 1 inner
55.j even 10 1 inner
77.j odd 10 1 inner
385.y odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1925.1.cb.b 16
5.b even 2 1 inner 1925.1.cb.b 16
5.c odd 4 1 1925.1.bn.b ✓ 8
5.c odd 4 1 1925.1.bn.d yes 8
7.b odd 2 1 CM 1925.1.cb.b 16
11.c even 5 1 inner 1925.1.cb.b 16
35.c odd 2 1 inner 1925.1.cb.b 16
35.f even 4 1 1925.1.bn.b ✓ 8
35.f even 4 1 1925.1.bn.d yes 8
55.j even 10 1 inner 1925.1.cb.b 16
55.k odd 20 1 1925.1.bn.b ✓ 8
55.k odd 20 1 1925.1.bn.d yes 8
77.j odd 10 1 inner 1925.1.cb.b 16
385.y odd 10 1 inner 1925.1.cb.b 16
385.bk even 20 1 1925.1.bn.b ✓ 8
385.bk even 20 1 1925.1.bn.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1925.1.bn.b ✓ 8 5.c odd 4 1
1925.1.bn.b ✓ 8 35.f even 4 1
1925.1.bn.b ✓ 8 55.k odd 20 1
1925.1.bn.b ✓ 8 385.bk even 20 1
1925.1.bn.d yes 8 5.c odd 4 1
1925.1.bn.d yes 8 35.f even 4 1
1925.1.bn.d yes 8 55.k odd 20 1
1925.1.bn.d yes 8 385.bk even 20 1
1925.1.cb.b 16 1.a even 1 1 trivial
1925.1.cb.b 16 5.b even 2 1 inner
1925.1.cb.b 16 7.b odd 2 1 CM
1925.1.cb.b 16 11.c even 5 1 inner
1925.1.cb.b 16 35.c odd 2 1 inner
1925.1.cb.b 16 55.j even 10 1 inner
1925.1.cb.b 16 77.j odd 10 1 inner
1925.1.cb.b 16 385.y odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 2T_{2}^{14} + 13T_{2}^{12} - 49T_{2}^{10} + 150T_{2}^{8} + T_{2}^{6} + 523T_{2}^{4} - 37T_{2}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1925, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( (T^{8} + 9 T^{6} + 26 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 2 T^{7} + 3 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} - 7 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{8} + 9 T^{6} + 26 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{8} + T^{6} + 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( (T^{8} + 9 T^{6} + 26 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 3 T^{7} + 8 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( (T^{8} - 3 T^{7} + 8 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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