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Properties

Label 1638.2.x.d
Level $1638$
Weight $2$
Character orbit 1638.x
Analytic conductor $13.079$
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] = [1638,2,Mod(307,1638)] 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("1638.307"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,4,0,0,0,0,4,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.7442857984.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{6} q^{4} + ( - \beta_{7} + \beta_{3} + \beta_1 + 1) q^{5} + \beta_{5} q^{7} - \beta_{2} q^{8} + (\beta_{7} + \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{10} + (\beta_{6} + \beta_{3} + 1) q^{11}+ \cdots + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5} + 4 q^{10} + 8 q^{11} + 16 q^{13} - 8 q^{16} + 12 q^{17} + 4 q^{19} + 4 q^{20} + 4 q^{22} + 4 q^{28} + 12 q^{29} + 20 q^{31} - 24 q^{34} - 32 q^{35} - 8 q^{37} - 12 q^{38} - 16 q^{41} + 8 q^{44}+ \cdots + 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{7} + 3\nu^{6} + 52\nu^{5} + 60\nu^{4} + 374\nu^{3} + 219\nu^{2} + 612\nu - 162 ) / 432 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 8\nu^{5} + 24\nu^{4} + 155\nu^{3} - 249\nu^{2} + 774\nu - 810 ) / 432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 3\nu^{6} + 52\nu^{5} - 60\nu^{4} + 374\nu^{3} - 219\nu^{2} + 612\nu + 162 ) / 432 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 3\nu^{6} - 8\nu^{5} - 24\nu^{4} + 155\nu^{3} + 249\nu^{2} + 1206\nu + 810 ) / 432 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} - 112\nu^{5} - 665\nu^{3} - 954\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 20\nu^{4} + 97\nu^{2} + 114 ) / 24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 3\beta_{4} - 3\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -6\beta_{6} + 3\beta_{5} - 6\beta_{4} + 3\beta_{3} - 6\beta_{2} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -13\beta_{7} + 6\beta_{5} - 45\beta_{4} - 6\beta_{3} + 45\beta_{2} - 6\beta _1 + 73 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 114\beta_{6} - 45\beta_{5} + 120\beta_{4} - 45\beta_{3} + 120\beta_{2} + 118\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 187\beta_{7} - 120\beta_{5} + 609\beta_{4} + 120\beta_{3} - 609\beta_{2} + 120\beta _1 - 895 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -1842\beta_{6} + 609\beta_{5} - 1890\beta_{4} + 609\beta_{3} - 1890\beta_{2} - 1504\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-\beta_{6}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
0.916813i
− 1.91681i
2.73923i
− 3.73923i
− 0.916813i
1.91681i
− 2.73923i
3.73923i
−0.707107 − 0.707107i 0 1.00000i −2.27220 + 2.27220i 0 1.35539 + 2.27220i 0.707107 − 0.707107i 0 3.21338
307.2 −0.707107 − 0.707107i 0 1.00000i 2.56510 − 2.56510i 0 −0.648285 − 2.56510i 0.707107 − 0.707107i 0 −3.62760
307.3 0.707107 + 0.707107i 0 1.00000i −0.0951965 + 0.0951965i 0 −2.64404 + 0.0951965i −0.707107 + 0.707107i 0 −0.134628
307.4 0.707107 + 0.707107i 0 1.00000i 1.80230 − 1.80230i 0 1.93693 − 1.80230i −0.707107 + 0.707107i 0 2.54884
811.1 −0.707107 + 0.707107i 0 − 1.00000i −2.27220 − 2.27220i 0 1.35539 − 2.27220i 0.707107 + 0.707107i 0 3.21338
811.2 −0.707107 + 0.707107i 0 − 1.00000i 2.56510 + 2.56510i 0 −0.648285 + 2.56510i 0.707107 + 0.707107i 0 −3.62760
811.3 0.707107 − 0.707107i 0 − 1.00000i −0.0951965 − 0.0951965i 0 −2.64404 − 0.0951965i −0.707107 − 0.707107i 0 −0.134628
811.4 0.707107 − 0.707107i 0 − 1.00000i 1.80230 + 1.80230i 0 1.93693 + 1.80230i −0.707107 − 0.707107i 0 2.54884
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.x.d 8
3.b odd 2 1 546.2.o.a ✓ 8
7.b odd 2 1 1638.2.x.b 8
13.d odd 4 1 1638.2.x.b 8
21.c even 2 1 546.2.o.d yes 8
39.f even 4 1 546.2.o.d yes 8
91.i even 4 1 inner 1638.2.x.d 8
273.o odd 4 1 546.2.o.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.o.a ✓ 8 3.b odd 2 1
546.2.o.a ✓ 8 273.o odd 4 1
546.2.o.d yes 8 21.c even 2 1
546.2.o.d yes 8 39.f even 4 1
1638.2.x.b 8 7.b odd 2 1
1638.2.x.b 8 13.d odd 4 1
1638.2.x.d 8 1.a even 1 1 trivial
1638.2.x.d 8 91.i even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 4T_{5}^{7} + 8T_{5}^{6} + 4T_{5}^{5} + 113T_{5}^{4} - 424T_{5}^{3} + 800T_{5}^{2} + 160T_{5} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T + 13)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 6 T^{3} - 35 T^{2} + \cdots - 92)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{7} + \cdots + 777924 \) Copy content Toggle raw display
$23$ \( T^{8} + 102 T^{6} + \cdots + 135424 \) Copy content Toggle raw display
$29$ \( (T^{4} - 6 T^{3} + \cdots + 356)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 20 T^{7} + \cdots + 5184 \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{7} + \cdots + 777924 \) Copy content Toggle raw display
$41$ \( T^{8} + 16 T^{7} + \cdots + 3936256 \) Copy content Toggle raw display
$43$ \( T^{8} + 254 T^{6} + \cdots + 14961424 \) Copy content Toggle raw display
$47$ \( T^{8} - 16 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( (T^{4} - 12 T^{3} + \cdots + 128)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 28 T^{7} + \cdots + 73984 \) Copy content Toggle raw display
$61$ \( T^{8} + 106 T^{6} + \cdots + 8464 \) Copy content Toggle raw display
$67$ \( T^{8} - 8 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( T^{8} + 52 T^{7} + \cdots + 1327104 \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{7} + \cdots + 777924 \) Copy content Toggle raw display
$79$ \( (T^{4} + 24 T^{3} + \cdots - 20744)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 32 T^{7} + \cdots + 541696 \) Copy content Toggle raw display
$89$ \( T^{8} + 4 T^{7} + \cdots + 45050944 \) Copy content Toggle raw display
$97$ \( T^{8} + 36 T^{7} + \cdots + 485409024 \) Copy content Toggle raw display
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