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Properties

Label 700.2.bc.a
Level $700$
Weight $2$
Character orbit 700.bc
Analytic conductor $5.590$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [700,2,Mod(157,700)] 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("700.157"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.bc (of order \(12\), 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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{24}^{7} q^{3} + (2 \zeta_{24}^{5} - 3 \zeta_{24}) q^{7} + 2 \zeta_{24}^{2} q^{9} + 2 \zeta_{24}^{3} q^{13} - 6 \zeta_{24} q^{17} + (4 \zeta_{24}^{6} - 2 \zeta_{24}^{2}) q^{19} + ( - 3 \zeta_{24}^{4} + 1) q^{21} + \cdots + ( - 8 \zeta_{24}^{5} + 8 \zeta_{24}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{21} - 24 q^{31} + 24 q^{51} + 36 q^{61} - 48 q^{71} + 4 q^{81} - 40 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(1 - \zeta_{24}^{4}\) \(1\) \(\zeta_{24}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
0 −0.965926 + 0.258819i 0 0 0 1.15539 − 2.38014i 0 −1.73205 + 1.00000i 0
157.2 0 0.965926 − 0.258819i 0 0 0 −1.15539 + 2.38014i 0 −1.73205 + 1.00000i 0
257.1 0 −0.258819 + 0.965926i 0 0 0 −2.38014 + 1.15539i 0 1.73205 + 1.00000i 0
257.2 0 0.258819 − 0.965926i 0 0 0 2.38014 − 1.15539i 0 1.73205 + 1.00000i 0
493.1 0 −0.258819 − 0.965926i 0 0 0 −2.38014 − 1.15539i 0 1.73205 − 1.00000i 0
493.2 0 0.258819 + 0.965926i 0 0 0 2.38014 + 1.15539i 0 1.73205 − 1.00000i 0
593.1 0 −0.965926 − 0.258819i 0 0 0 1.15539 + 2.38014i 0 −1.73205 − 1.00000i 0
593.2 0 0.965926 + 0.258819i 0 0 0 −1.15539 − 2.38014i 0 −1.73205 − 1.00000i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 157.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.d odd 6 1 inner
35.i odd 6 1 inner
35.k even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.bc.a ✓ 8
5.b even 2 1 inner 700.2.bc.a ✓ 8
5.c odd 4 2 inner 700.2.bc.a ✓ 8
7.d odd 6 1 inner 700.2.bc.a ✓ 8
35.i odd 6 1 inner 700.2.bc.a ✓ 8
35.k even 12 2 inner 700.2.bc.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.bc.a ✓ 8 1.a even 1 1 trivial
700.2.bc.a ✓ 8 5.b even 2 1 inner
700.2.bc.a ✓ 8 5.c odd 4 2 inner
700.2.bc.a ✓ 8 7.d odd 6 1 inner
700.2.bc.a ✓ 8 35.i odd 6 1 inner
700.2.bc.a ✓ 8 35.k even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - T_{3}^{4} + 1 \) acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 23T^{4} + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 1296 T^{4} + 1679616 \) Copy content Toggle raw display
$19$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 729 T^{4} + 531441 \) Copy content Toggle raw display
$29$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T + 12)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} - 2304 T^{4} + 5308416 \) Copy content Toggle raw display
$41$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 5625)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} - 11664 T^{4} + 136048896 \) Copy content Toggle raw display
$59$ \( (T^{4} + 108 T^{2} + 11664)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 9 T + 27)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$71$ \( (T + 6)^{8} \) Copy content Toggle raw display
$73$ \( T^{8} - 10000 T^{4} + 100000000 \) Copy content Toggle raw display
$79$ \( (T^{4} - 196 T^{2} + 38416)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 81)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 243 T^{2} + 59049)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4096)^{2} \) Copy content Toggle raw display
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