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Properties

Label 714.2.m.c
Level $714$
Weight $2$
Character orbit 714.m
Analytic conductor $5.701$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [714,2,Mod(421,714)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(714, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) 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("714.421"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-8,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.836829184.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_1 q^{3} - q^{4} + (\beta_{6} + \beta_{4} + \beta_1 + 1) q^{5} + \beta_{2} q^{6} - \beta_{2} q^{7} - \beta_{5} q^{8} + \beta_{5} q^{9} + (\beta_{7} - \beta_{3} - \beta_{2} - 1) q^{10}+ \cdots + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 4 q^{5} - 4 q^{10} - 8 q^{11} - 8 q^{13} + 8 q^{16} + 4 q^{17} - 8 q^{18} - 4 q^{20} + 8 q^{21} - 8 q^{22} - 12 q^{23} + 8 q^{30} - 32 q^{31} - 8 q^{33} + 8 q^{34} - 8 q^{35} - 20 q^{37}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + \nu^{4} + 11\nu^{3} + 7\nu^{2} + 26\nu + 6 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 11\nu^{3} + 7\nu^{2} - 26\nu + 6 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + \nu^{5} + 11\nu^{4} + 7\nu^{3} + 34\nu^{2} + 6\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - \nu^{5} + 11\nu^{4} - 7\nu^{3} + 34\nu^{2} - 6\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 12\nu^{5} + 41\nu^{3} + 38\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + \nu^{6} + 14\nu^{5} + 10\nu^{4} + 59\nu^{3} + 19\nu^{2} + 78\nu - 14 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - \nu^{6} + 14\nu^{5} - 10\nu^{4} + 59\nu^{3} - 19\nu^{2} + 78\nu + 14 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{7} - 5\beta_{6} + 10\beta_{5} - 3\beta_{4} + 3\beta_{3} - 7\beta_{2} + 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{7} + 7\beta_{6} - 7\beta_{4} - 7\beta_{3} + 15\beta_{2} + 15\beta _1 + 44 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 29\beta_{7} + 29\beta_{6} - 58\beta_{5} + 7\beta_{4} - 7\beta_{3} + 43\beta_{2} - 43\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 43\beta_{7} - 43\beta_{6} + 51\beta_{4} + 51\beta_{3} - 131\beta_{2} - 131\beta _1 - 260 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -181\beta_{7} - 181\beta_{6} + 378\beta_{5} + \beta_{4} - \beta_{3} - 267\beta_{2} + 267\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/714\mathbb{Z}\right)^\times\).

\(n\) \(239\) \(409\) \(547\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
421.1
− 2.63640i
0.222191i
− 1.65222i
2.06644i
2.63640i
− 0.222191i
1.65222i
− 2.06644i
1.00000i −0.707107 − 0.707107i −1.00000 −0.222191 − 0.222191i 0.707107 − 0.707107i −0.707107 + 0.707107i − 1.00000i 1.00000i 0.222191 − 0.222191i
421.2 1.00000i −0.707107 − 0.707107i −1.00000 2.63640 + 2.63640i 0.707107 − 0.707107i −0.707107 + 0.707107i − 1.00000i 1.00000i −2.63640 + 2.63640i
421.3 1.00000i 0.707107 + 0.707107i −1.00000 −2.06644 − 2.06644i −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i 1.00000i 2.06644 − 2.06644i
421.4 1.00000i 0.707107 + 0.707107i −1.00000 1.65222 + 1.65222i −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i 1.00000i −1.65222 + 1.65222i
463.1 − 1.00000i −0.707107 + 0.707107i −1.00000 −0.222191 + 0.222191i 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i − 1.00000i 0.222191 + 0.222191i
463.2 − 1.00000i −0.707107 + 0.707107i −1.00000 2.63640 − 2.63640i 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i − 1.00000i −2.63640 − 2.63640i
463.3 − 1.00000i 0.707107 − 0.707107i −1.00000 −2.06644 + 2.06644i −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i − 1.00000i 2.06644 + 2.06644i
463.4 − 1.00000i 0.707107 − 0.707107i −1.00000 1.65222 − 1.65222i −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i − 1.00000i −1.65222 − 1.65222i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 421.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.m.c ✓ 8
3.b odd 2 1 2142.2.p.d 8
17.c even 4 1 inner 714.2.m.c ✓ 8
51.f odd 4 1 2142.2.p.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.m.c ✓ 8 1.a even 1 1 trivial
714.2.m.c ✓ 8 17.c even 4 1 inner
2142.2.p.d 8 3.b odd 2 1
2142.2.p.d 8 51.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 4T_{5}^{7} + 8T_{5}^{6} + 8T_{5}^{5} + 84T_{5}^{4} - 288T_{5}^{3} + 512T_{5}^{2} + 256T_{5} + 64 \) acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 8 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} - 14 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( T^{8} + 56 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( T^{8} - 16 T^{5} + \cdots + 8464 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 32)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + 20 T^{7} + \cdots + 1721344 \) Copy content Toggle raw display
$41$ \( T^{8} - 20 T^{7} + \cdots + 16384 \) Copy content Toggle raw display
$43$ \( T^{8} + 232 T^{6} + \cdots + 1183744 \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 272 T^{6} + \cdots + 4734976 \) Copy content Toggle raw display
$59$ \( T^{8} + 228 T^{6} + \cdots + 295936 \) Copy content Toggle raw display
$61$ \( T^{8} + 44 T^{7} + \cdots + 49336576 \) Copy content Toggle raw display
$67$ \( (T^{4} + 8 T^{3} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 12 T^{7} + \cdots + 43454464 \) Copy content Toggle raw display
$73$ \( T^{8} - 28 T^{7} + \cdots + 72318016 \) Copy content Toggle raw display
$79$ \( T^{8} - 24 T^{7} + \cdots + 262144 \) Copy content Toggle raw display
$83$ \( T^{8} + 100 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( (T^{4} - 50 T^{2} + \cdots - 128)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 4 T^{7} + \cdots + 712676416 \) Copy content Toggle raw display
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