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Properties

Label 714.2.i.o
Level $714$
Weight $2$
Character orbit 714.i
Analytic conductor $5.701$
Analytic rank $0$
Dimension $6$
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(205,714)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(714, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) 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("714.205"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,3,3,-3,3,6,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.11337408.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 18x^{4} + 81x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{2} + 1) q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{5} + q^{6} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{7} - q^{8}+ \cdots + ( - 2 \beta_{4} - 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 3 q^{3} - 3 q^{4} + 3 q^{5} + 6 q^{6} - 6 q^{7} - 6 q^{8} - 3 q^{9} - 3 q^{10} + 3 q^{12} - 18 q^{13} - 3 q^{14} + 6 q^{15} - 3 q^{16} + 3 q^{17} + 3 q^{18} + 6 q^{19} - 6 q^{20} - 3 q^{21}+ \cdots + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 18x^{4} + 81x^{2} + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 9\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{2} + \nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 9\nu^{2} - 2\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + \nu^{4} + 15\nu^{3} + 11\nu^{2} + 48\nu + 12 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{3} - \beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 9\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} - 18\beta_{3} + 11\beta _1 + 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{5} - 4\beta_{4} - 4\beta_{3} - 60\beta_{2} + 87\beta _1 + 30 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(239\) \(409\) \(547\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
205.1
3.17656i
− 2.78499i
− 0.391571i
− 3.17656i
2.78499i
0.391571i
0.500000 − 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i −1.89812 + 3.28764i 1.00000 −0.647140 − 2.56539i −1.00000 −0.500000 + 0.866025i 1.89812 + 3.28764i
205.2 0.500000 − 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i 1.26690 − 2.19433i 1.00000 −2.64497 − 0.0641892i −1.00000 −0.500000 + 0.866025i −1.26690 − 2.19433i
205.3 0.500000 − 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i 2.13122 − 3.69139i 1.00000 0.292113 + 2.62958i −1.00000 −0.500000 + 0.866025i −2.13122 − 3.69139i
613.1 0.500000 + 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i −1.89812 − 3.28764i 1.00000 −0.647140 + 2.56539i −1.00000 −0.500000 − 0.866025i 1.89812 − 3.28764i
613.2 0.500000 + 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i 1.26690 + 2.19433i 1.00000 −2.64497 + 0.0641892i −1.00000 −0.500000 − 0.866025i −1.26690 + 2.19433i
613.3 0.500000 + 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i 2.13122 + 3.69139i 1.00000 0.292113 − 2.62958i −1.00000 −0.500000 − 0.866025i −2.13122 + 3.69139i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 205.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.i.o ✓ 6
7.c even 3 1 inner 714.2.i.o ✓ 6
7.c even 3 1 4998.2.a.cg 3
7.d odd 6 1 4998.2.a.ch 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.i.o ✓ 6 1.a even 1 1 trivial
714.2.i.o ✓ 6 7.c even 3 1 inner
4998.2.a.cg 3 7.c even 3 1
4998.2.a.ch 3 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\):

\( T_{5}^{6} - 3T_{5}^{5} + 24T_{5}^{4} - 37T_{5}^{3} + 348T_{5}^{2} - 615T_{5} + 1681 \) Copy content Toggle raw display
\( T_{11}^{6} + 36T_{11}^{4} - 96T_{11}^{3} + 1296T_{11}^{2} - 1728T_{11} + 2304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + \cdots + 1681 \) Copy content Toggle raw display
$7$ \( T^{6} + 6 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} + 36 T^{4} + \cdots + 2304 \) Copy content Toggle raw display
$13$ \( (T^{3} + 9 T^{2} - 96)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$19$ \( T^{6} - 6 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{6} + 9 T^{4} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( (T^{3} + 3 T^{2} - 24 T - 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 9 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$37$ \( T^{6} + 6 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( (T^{3} + 3 T^{2} - 24 T + 16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 6 T^{2} - 15 T + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 3 T^{5} + \cdots + 204304 \) Copy content Toggle raw display
$53$ \( T^{6} + 12 T^{5} + \cdots + 16384 \) Copy content Toggle raw display
$59$ \( (T^{2} - 3 T + 9)^{3} \) Copy content Toggle raw display
$61$ \( T^{6} + 6 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( T^{6} + 99 T^{4} + \cdots + 125316 \) Copy content Toggle raw display
$71$ \( (T^{3} - 153 T + 186)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 12 T^{5} + \cdots + 2458624 \) Copy content Toggle raw display
$79$ \( T^{6} - 24 T^{5} + \cdots + 473344 \) Copy content Toggle raw display
$83$ \( (T^{3} - 27 T^{2} + \cdots - 528)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 6 T^{5} + \cdots + 274576 \) Copy content Toggle raw display
$97$ \( (T^{3} - 144 T - 384)^{2} \) Copy content Toggle raw display
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