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Properties

Label 375.2.g.b
Level $375$
Weight $2$
Character orbit 375.g
Analytic conductor $2.994$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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] = [375,2,Mod(76,375)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("375.76"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(375, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.99439007580\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.26265625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{3}) q^{2} + ( - \beta_{6} - \beta_{3} - \beta_1 + 1) q^{3} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + \cdots - 1) q^{4} + (\beta_{5} - \beta_1) q^{6} + ( - \beta_{7} + 2 \beta_{6} + \beta_{3} + \cdots - 1) q^{7}+ \cdots + ( - 3 \beta_{6} + \beta_{5} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 2 q^{3} + q^{4} - q^{6} - 4 q^{7} - 2 q^{9} + 16 q^{11} + 9 q^{12} + 8 q^{13} - 8 q^{14} - 17 q^{16} + q^{17} - 4 q^{18} - 5 q^{19} - 11 q^{21} - 13 q^{22} - 7 q^{23} + 30 q^{24} + 6 q^{26}+ \cdots - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - \nu^{6} + \nu^{3} + 2\nu^{2} + 4\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} + \nu^{3} - 2\nu^{2} + 12\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{7} + 9\nu^{6} + 2\nu^{5} + 4\nu^{4} + \nu^{3} - 4\nu^{2} - 60\nu + 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} - 15\nu^{6} - 4\nu^{5} - 8\nu^{4} + 3\nu^{3} + 2\nu^{2} + 92\nu - 96 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{7} - 13\nu^{6} - 6\nu^{5} - 8\nu^{4} + 3\nu^{3} + 4\nu^{2} + 96\nu - 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\nu^{7} - 21\nu^{6} - 4\nu^{5} - 4\nu^{4} + 5\nu^{3} + 10\nu^{2} + 120\nu - 144 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{7} - 31\nu^{6} - 4\nu^{5} - 8\nu^{4} + 11\nu^{3} + 18\nu^{2} + 180\nu - 224 ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} + \beta_{2} - 4\beta _1 + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} - 4\beta_{4} - 2\beta_{3} - 3\beta_{2} + 2\beta _1 + 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} + 8\beta_{3} + 2\beta_{2} + 7\beta _1 + 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{7} - 11\beta_{6} - 9\beta_{5} + 3\beta_{4} - 11\beta_{3} + 6\beta_{2} + 6\beta _1 + 19 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{7} - 7\beta_{6} + 7\beta_{5} - 9\beta_{4} - 7\beta_{3} + 7\beta_{2} + 12\beta _1 - 12 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -8\beta_{7} + 3\beta_{6} - 3\beta_{5} + 6\beta_{4} - 7\beta_{3} + 7\beta_{2} + 57\beta _1 + 3 ) / 5 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/375\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(-1 + \beta_{1} + \beta_{3} + \beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
−0.0272949 − 1.41395i
1.33631 + 0.462894i
−1.21700 − 0.720348i
1.40799 + 0.132563i
−1.21700 + 0.720348i
1.40799 − 0.132563i
−0.0272949 + 1.41395i
1.33631 − 0.462894i
−1.38048 − 1.00297i −0.309017 − 0.951057i 0.281722 + 0.867051i 0 −0.527295 + 1.62285i 3.94243 −0.573870 + 1.76619i −0.809017 + 0.587785i 0
76.2 2.18949 + 1.59076i −0.309017 − 0.951057i 1.64533 + 5.06380i 0 0.836312 − 2.57390i −0.470294 −2.78023 + 8.55667i −0.809017 + 0.587785i 0
151.1 −0.655837 − 2.01846i 0.809017 − 0.587785i −2.02602 + 1.47199i 0 −1.71700 − 1.24748i −4.35840 0.865884 + 0.629102i 0.309017 − 0.951057i 0
151.2 0.346820 + 1.06740i 0.809017 − 0.587785i 0.598970 − 0.435177i 0 0.907987 + 0.659691i −1.11373 2.48822 + 1.80780i 0.309017 − 0.951057i 0
226.1 −0.655837 + 2.01846i 0.809017 + 0.587785i −2.02602 − 1.47199i 0 −1.71700 + 1.24748i −4.35840 0.865884 − 0.629102i 0.309017 + 0.951057i 0
226.2 0.346820 − 1.06740i 0.809017 + 0.587785i 0.598970 + 0.435177i 0 0.907987 − 0.659691i −1.11373 2.48822 − 1.80780i 0.309017 + 0.951057i 0
301.1 −1.38048 + 1.00297i −0.309017 + 0.951057i 0.281722 − 0.867051i 0 −0.527295 − 1.62285i 3.94243 −0.573870 − 1.76619i −0.809017 − 0.587785i 0
301.2 2.18949 − 1.59076i −0.309017 + 0.951057i 1.64533 − 5.06380i 0 0.836312 + 2.57390i −0.470294 −2.78023 − 8.55667i −0.809017 − 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.2.g.b 8
5.b even 2 1 75.2.g.b ✓ 8
5.c odd 4 2 375.2.i.b 16
15.d odd 2 1 225.2.h.c 8
25.d even 5 1 inner 375.2.g.b 8
25.d even 5 1 1875.2.a.e 4
25.e even 10 1 75.2.g.b ✓ 8
25.e even 10 1 1875.2.a.h 4
25.f odd 20 2 375.2.i.b 16
25.f odd 20 2 1875.2.b.c 8
75.h odd 10 1 225.2.h.c 8
75.h odd 10 1 5625.2.a.i 4
75.j odd 10 1 5625.2.a.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.b ✓ 8 5.b even 2 1
75.2.g.b ✓ 8 25.e even 10 1
225.2.h.c 8 15.d odd 2 1
225.2.h.c 8 75.h odd 10 1
375.2.g.b 8 1.a even 1 1 trivial
375.2.g.b 8 25.d even 5 1 inner
375.2.i.b 16 5.c odd 4 2
375.2.i.b 16 25.f odd 20 2
1875.2.a.e 4 25.d even 5 1
1875.2.a.h 4 25.e even 10 1
1875.2.b.c 8 25.f odd 20 2
5625.2.a.i 4 75.h odd 10 1
5625.2.a.n 4 75.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} + 2T_{2}^{6} - 3T_{2}^{5} + 25T_{2}^{4} + 43T_{2}^{3} + 82T_{2}^{2} + 11T_{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} - 16 T^{2} + \cdots - 9)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 16 T^{7} + \cdots + 11881 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{8} - T^{7} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{8} + 5 T^{7} + \cdots + 75625 \) Copy content Toggle raw display
$23$ \( T^{8} + 7 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$29$ \( T^{8} - 5 T^{7} + \cdots + 164025 \) Copy content Toggle raw display
$31$ \( T^{8} + 19 T^{7} + \cdots + 505521 \) Copy content Toggle raw display
$37$ \( T^{8} - T^{7} + 77 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{4} + 7 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 16 T^{3} + \cdots - 99)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - T^{7} + \cdots + 96721 \) Copy content Toggle raw display
$53$ \( T^{8} - 3 T^{7} + \cdots + 68121 \) Copy content Toggle raw display
$59$ \( T^{8} - 30 T^{7} + \cdots + 13286025 \) Copy content Toggle raw display
$61$ \( T^{8} + 14 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$67$ \( T^{8} + 4 T^{7} + \cdots + 29241 \) Copy content Toggle raw display
$71$ \( T^{8} - 21 T^{7} + \cdots + 829921 \) Copy content Toggle raw display
$73$ \( T^{8} + 2 T^{7} + \cdots + 23707161 \) Copy content Toggle raw display
$79$ \( T^{8} + 30 T^{7} + \cdots + 96924025 \) Copy content Toggle raw display
$83$ \( T^{8} + 2 T^{7} + \cdots + 958441 \) Copy content Toggle raw display
$89$ \( T^{8} + 255 T^{6} + \cdots + 10923025 \) Copy content Toggle raw display
$97$ \( T^{8} - 6 T^{7} + \cdots + 4414201 \) Copy content Toggle raw display
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