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Properties

Label 372.2.j.b
Level $372$
Weight $2$
Character orbit 372.j
Analytic conductor $2.970$
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] = [372,2,Mod(97,372)] 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("372.97"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(372, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 372 = 2^{2} \cdot 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 372.j (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] 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: \(2.97043495519\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.1903140625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 6x^{6} + x^{5} + 29x^{4} + 43x^{3} + 194x^{2} + 209x + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + ( - \beta_{6} - \beta_{4} + \beta_{2}) q^{5} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} + \cdots - 2) q^{7} + ( - \beta_{6} + \beta_{3} + \beta_{2} - 1) q^{9} + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1) q^{11}+ \cdots + (\beta_{6} - \beta_{5} - \beta_{4} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 2 q^{5} - 9 q^{7} - 2 q^{9} - 2 q^{11} + 9 q^{13} + 8 q^{15} - 13 q^{17} - 4 q^{19} - 6 q^{21} - 13 q^{23} + 22 q^{25} + 2 q^{27} + 14 q^{29} - 3 q^{31} - 3 q^{33} - 7 q^{35} + 11 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 6x^{6} + x^{5} + 29x^{4} + 43x^{3} + 194x^{2} + 209x + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16228 \nu^{7} + 164686 \nu^{6} - 1074875 \nu^{5} + 2192108 \nu^{4} + 7497629 \nu^{3} + \cdots + 55384240 ) / 205159891 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 294510 \nu^{7} + 348813 \nu^{6} + 3198495 \nu^{5} - 17987557 \nu^{4} + 10405212 \nu^{3} + \cdots - 65903001 ) / 205159891 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28143 \nu^{7} + 261345 \nu^{6} - 931213 \nu^{5} + 997158 \nu^{4} + 325779 \nu^{3} + \cdots + 5595690 ) / 10797889 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 45075 \nu^{7} - 64855 \nu^{6} + 453785 \nu^{5} - 989293 \nu^{4} - 2983130 \nu^{3} + \cdots - 23973516 ) / 10797889 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 967254 \nu^{7} - 4292904 \nu^{6} + 9536834 \nu^{5} - 8103878 \nu^{4} + 28199801 \nu^{3} + \cdots - 119034259 ) / 205159891 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 56305 \nu^{7} + 3158 \nu^{6} - 339265 \nu^{5} + 1359136 \nu^{4} + 462427 \nu^{3} + \cdots + 23665184 ) / 10797889 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{5} - \beta_{3} - 5\beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -7\beta_{7} - \beta_{6} - 7\beta_{5} - 2\beta_{4} - 5\beta_{3} - 5\beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{7} - 15\beta_{6} - 15\beta_{5} - 15\beta_{4} - 22\beta_{3} + 3\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -63\beta_{6} - 30\beta_{5} - 64\beta_{4} + 63\beta_{2} - 87 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 127\beta_{7} - 166\beta_{6} - 127\beta_{4} + 166\beta_{3} + 350\beta_{2} - 54\beta _1 - 350 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 658\beta_{7} + 365\beta_{5} + 689\beta_{3} + 1032\beta_{2} - 365\beta _1 - 689 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/372\mathbb{Z}\right)^\times\).

\(n\) \(125\) \(187\) \(313\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{2} + \beta_{3} - \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
2.68070 + 1.94764i
−1.37168 − 0.996583i
−0.480762 − 1.47963i
0.671745 + 2.06742i
−0.480762 + 1.47963i
0.671745 − 2.06742i
2.68070 − 1.94764i
−1.37168 + 0.996583i
0 −0.309017 + 0.951057i 0 −3.93156 0 0.0626612 − 0.0455260i 0 −0.809017 − 0.587785i 0
97.2 0 −0.309017 + 0.951057i 0 1.07745 0 −3.98971 + 2.89870i 0 −0.809017 − 0.587785i 0
109.1 0 0.809017 + 0.587785i 0 0.0622568 0 −0.862728 + 2.65520i 0 0.309017 + 0.951057i 0
109.2 0 0.809017 + 0.587785i 0 3.79185 0 0.289779 − 0.891847i 0 0.309017 + 0.951057i 0
157.1 0 0.809017 − 0.587785i 0 0.0622568 0 −0.862728 − 2.65520i 0 0.309017 − 0.951057i 0
157.2 0 0.809017 − 0.587785i 0 3.79185 0 0.289779 + 0.891847i 0 0.309017 − 0.951057i 0
349.1 0 −0.309017 − 0.951057i 0 −3.93156 0 0.0626612 + 0.0455260i 0 −0.809017 + 0.587785i 0
349.2 0 −0.309017 − 0.951057i 0 1.07745 0 −3.98971 − 2.89870i 0 −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.2
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 372.2.j.b ✓ 8
3.b odd 2 1 1116.2.m.d 8
31.d even 5 1 inner 372.2.j.b ✓ 8
93.l odd 10 1 1116.2.m.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
372.2.j.b ✓ 8 1.a even 1 1 trivial
372.2.j.b ✓ 8 31.d even 5 1 inner
1116.2.m.d 8 3.b odd 2 1
1116.2.m.d 8 93.l odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - T_{5}^{3} - 15T_{5}^{2} + 17T_{5} - 1 \) acting on \(S_{2}^{\mathrm{new}}(372, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} - 15 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 9 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} + 2 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{8} - 9 T^{7} + \cdots + 961 \) Copy content Toggle raw display
$17$ \( T^{8} + 13 T^{7} + \cdots + 7921 \) Copy content Toggle raw display
$19$ \( T^{8} + 4 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$23$ \( T^{8} + 13 T^{7} + \cdots + 58081 \) Copy content Toggle raw display
$29$ \( T^{8} - 14 T^{7} + \cdots + 28100601 \) Copy content Toggle raw display
$31$ \( T^{8} + 3 T^{7} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( (T^{4} - 56 T^{2} + \cdots + 144)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 17 T^{7} + \cdots + 477481 \) Copy content Toggle raw display
$43$ \( T^{8} + 15 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$47$ \( T^{8} + 22 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$53$ \( (T^{4} - T^{3} + 51 T^{2} + \cdots + 961)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 21 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$61$ \( (T^{4} + 6 T^{3} + \cdots + 4581)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 12 T + 16)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + 9 T^{7} + \cdots + 66080641 \) Copy content Toggle raw display
$73$ \( T^{8} - 19 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$79$ \( T^{8} + 25 T^{7} + \cdots + 241081 \) Copy content Toggle raw display
$83$ \( T^{8} + 27 T^{7} + \cdots + 6241 \) Copy content Toggle raw display
$89$ \( T^{8} - 29 T^{7} + \cdots + 44368921 \) Copy content Toggle raw display
$97$ \( T^{8} + 15 T^{7} + \cdots + 625 \) Copy content Toggle raw display
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