Properties

Label 343.2.g.d
Level $343$
Weight $2$
Character orbit 343.g
Analytic conductor $2.739$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [343,2,Mod(30,343)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(343, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([32])) 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("343.30"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 343 = 7^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 343.g (of order \(21\), degree \(12\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,1,0,-1,14] 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: \(2.73886878933\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 49)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-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 \(\zeta_{21}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{21}^{9} + \zeta_{21}^{8} + \cdots + \zeta_{21}) q^{2} + (\zeta_{21}^{11} - \zeta_{21}^{10} + \cdots - 1) q^{3} + (\zeta_{21}^{11} + \zeta_{21}^{9}) q^{4} + ( - \zeta_{21}^{11} + \zeta_{21}^{10} + \cdots + 2) q^{5}+ \cdots + ( - 11 \zeta_{21}^{11} + 4 \zeta_{21}^{10} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} - q^{4} + 14 q^{5} - 7 q^{6} + 6 q^{8} + 25 q^{9} - 7 q^{10} + 4 q^{11} + 7 q^{12} + 7 q^{13} - 7 q^{15} - 29 q^{16} - 10 q^{18} + 7 q^{19} + 7 q^{20} + 13 q^{22} + q^{23} + 9 q^{25} - 7 q^{26}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/343\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\zeta_{21}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1
−0.733052 + 0.680173i
0.365341 + 0.930874i
0.826239 + 0.563320i
0.955573 + 0.294755i
0.365341 0.930874i
0.826239 0.563320i
−0.988831 0.149042i
0.0747301 0.997204i
0.0747301 + 0.997204i
−0.733052 0.680173i
0.955573 0.294755i
−0.988831 + 0.149042i
−0.603718 + 0.411608i 0.842614 0.781831i −0.535628 + 1.36476i 0.247176 + 0.0762438i −0.186893 + 0.818832i 0 −0.563561 2.46912i −0.125452 + 1.67405i −0.180607 + 0.0557099i
67.1 −1.57906 + 0.487076i 0.170287 + 0.433884i 0.603718 0.411608i 3.84537 + 0.579597i −0.480228 0.602187i 0 1.30778 1.63991i 2.03990 1.89275i −6.35439 + 0.957770i
79.1 1.88980 0.284841i −1.42996 0.974928i 1.57906 0.487076i 0.158960 2.12117i −2.98003 1.43511i 0 −0.598393 + 0.288171i −0.00173159 0.00441201i −0.303796 4.05387i
116.1 0.147791 + 1.97213i −2.53464 0.781831i −1.88980 + 0.284841i 1.18429 1.09886i 1.16728 5.11418i 0 0.0391023 + 0.171318i 3.33440 + 2.27336i 2.34213 + 2.17318i
128.1 −1.57906 0.487076i 0.170287 0.433884i 0.603718 + 0.411608i 3.84537 0.579597i −0.480228 + 0.602187i 0 1.30778 + 1.63991i 2.03990 + 1.89275i −6.35439 0.957770i
165.1 1.88980 + 0.284841i −1.42996 + 0.974928i 1.57906 + 0.487076i 0.158960 + 2.12117i −2.98003 + 1.43511i 0 −0.598393 0.288171i −0.00173159 + 0.00441201i −0.303796 + 4.05387i
177.1 0.109562 + 0.101659i 2.87863 + 0.433884i −0.147791 1.97213i 0.802576 + 2.04493i 0.271281 + 0.340175i 0 0.370666 0.464800i 5.23154 + 1.61372i −0.119953 + 0.305636i
214.1 0.535628 1.36476i 0.0730607 0.974928i −0.109562 0.101659i 0.761623 + 0.519266i −1.29141 0.621909i 0 2.44440 1.17716i 2.02135 + 0.304669i 1.11662 0.761298i
226.1 0.535628 + 1.36476i 0.0730607 + 0.974928i −0.109562 + 0.101659i 0.761623 0.519266i −1.29141 + 0.621909i 0 2.44440 + 1.17716i 2.02135 0.304669i 1.11662 + 0.761298i
263.1 −0.603718 0.411608i 0.842614 + 0.781831i −0.535628 1.36476i 0.247176 0.0762438i −0.186893 0.818832i 0 −0.563561 + 2.46912i −0.125452 1.67405i −0.180607 0.0557099i
275.1 0.147791 1.97213i −2.53464 + 0.781831i −1.88980 0.284841i 1.18429 + 1.09886i 1.16728 + 5.11418i 0 0.0391023 0.171318i 3.33440 2.27336i 2.34213 2.17318i
312.1 0.109562 0.101659i 2.87863 0.433884i −0.147791 + 1.97213i 0.802576 2.04493i 0.271281 0.340175i 0 0.370666 + 0.464800i 5.23154 1.61372i −0.119953 0.305636i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 30.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 343.2.g.d 12
7.b odd 2 1 343.2.g.a 12
7.c even 3 1 49.2.e.b 12
7.c even 3 1 343.2.g.b 12
7.d odd 6 1 343.2.e.b 12
7.d odd 6 1 343.2.g.c 12
21.h odd 6 1 441.2.u.b 12
28.g odd 6 1 784.2.u.b 12
49.e even 7 1 343.2.g.b 12
49.f odd 14 1 343.2.g.c 12
49.g even 21 1 49.2.e.b 12
49.g even 21 1 inner 343.2.g.d 12
49.g even 21 1 2401.2.a.c 6
49.h odd 42 1 343.2.e.b 12
49.h odd 42 1 343.2.g.a 12
49.h odd 42 1 2401.2.a.d 6
147.n odd 42 1 441.2.u.b 12
196.o odd 42 1 784.2.u.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.e.b 12 7.c even 3 1
49.2.e.b 12 49.g even 21 1
343.2.e.b 12 7.d odd 6 1
343.2.e.b 12 49.h odd 42 1
343.2.g.a 12 7.b odd 2 1
343.2.g.a 12 49.h odd 42 1
343.2.g.b 12 7.c even 3 1
343.2.g.b 12 49.e even 7 1
343.2.g.c 12 7.d odd 6 1
343.2.g.c 12 49.f odd 14 1
343.2.g.d 12 1.a even 1 1 trivial
343.2.g.d 12 49.g even 21 1 inner
441.2.u.b 12 21.h odd 6 1
441.2.u.b 12 147.n odd 42 1
784.2.u.b 12 28.g odd 6 1
784.2.u.b 12 196.o odd 42 1
2401.2.a.c 6 49.g even 21 1
2401.2.a.d 6 49.h odd 42 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(343, [\chi])\):

\( T_{2}^{12} - T_{2}^{11} + T_{2}^{9} - 15T_{2}^{8} + 22T_{2}^{6} - 21T_{2}^{5} + 48T_{2}^{4} + 71T_{2}^{3} + 28T_{2}^{2} - 8T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{12} - 14 T_{3}^{10} - 7 T_{3}^{9} + 63 T_{3}^{8} + 35 T_{3}^{7} - 35 T_{3}^{6} + 49 T_{3}^{5} + \cdots + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - 14 T^{10} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{12} - 14 T^{11} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 1849 \) Copy content Toggle raw display
$13$ \( T^{12} - 7 T^{11} + \cdots + 49 \) Copy content Toggle raw display
$17$ \( T^{12} - 7 T^{10} + \cdots + 3087049 \) Copy content Toggle raw display
$19$ \( T^{12} - 7 T^{11} + \cdots + 4439449 \) Copy content Toggle raw display
$23$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{12} + 11 T^{11} + \cdots + 1681 \) Copy content Toggle raw display
$31$ \( T^{12} - 7 T^{11} + \cdots + 66961489 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 295118041 \) Copy content Toggle raw display
$41$ \( T^{12} - 21 T^{11} + \cdots + 10413529 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 200307409 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 3021810841 \) Copy content Toggle raw display
$53$ \( T^{12} + 24 T^{11} + \cdots + 2985984 \) Copy content Toggle raw display
$59$ \( T^{12} + 7 T^{11} + \cdots + 54125449 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 261242569 \) Copy content Toggle raw display
$67$ \( T^{12} + 24 T^{11} + \cdots + 85849 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 312925003609 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 460917961 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 467813641 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 118178641 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 89755965649 \) Copy content Toggle raw display
$97$ \( (T^{6} + 14 T^{5} + \cdots + 18571)^{2} \) Copy content Toggle raw display
show more
show less