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Properties

Label 343.4.a.c
Level $343$
Weight $4$
Character orbit 343.a
Self dual yes
Analytic conductor $20.238$
Analytic rank $1$
Dimension $12$
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] = [343,4,Mod(1,343)] 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("343.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(343, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 343 = 7^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 343.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-22,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(20.2376551320\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 59x^{10} + 1038x^{8} - 7419x^{6} + 20458x^{4} - 11940x^{2} + 1912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 7^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{3} - 1) q^{2} + \beta_{4} q^{3} + ( - 3 \beta_{6} + 3 \beta_{3} + \cdots + 1) q^{4} + ( - \beta_{11} - \beta_{8} - \beta_{4}) q^{5} + ( - \beta_{9} + \beta_{8} + \cdots - 2 \beta_{4}) q^{6}+ \cdots + (113 \beta_{6} - 61 \beta_{5} + \cdots - 164) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 22 q^{2} + 46 q^{4} - 138 q^{8} + 106 q^{9} - 134 q^{11} - 332 q^{15} + 150 q^{16} - 284 q^{18} + 296 q^{22} - 64 q^{23} + 414 q^{25} - 1642 q^{29} + 332 q^{30} - 426 q^{32} + 592 q^{36} - 1170 q^{37}+ \cdots - 3098 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 59x^{10} + 1038x^{8} - 7419x^{6} + 20458x^{4} - 11940x^{2} + 1912 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -1611\nu^{10} + 109207\nu^{8} - 2360688\nu^{6} + 19350073\nu^{4} - 53684536\nu^{2} + 15401868 ) / 3350704 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2037\nu^{10} + 156581\nu^{8} - 3960324\nu^{6} + 37487711\nu^{4} - 120957996\nu^{2} + 41010844 ) / 3350704 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -1571\nu^{10} + 88185\nu^{8} - 1405678\nu^{6} + 9010125\nu^{4} - 22250702\nu^{2} + 8712256 ) / 1675352 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4841\nu^{11} - 295925\nu^{9} + 5537880\nu^{7} - 41904739\nu^{5} + 121615232\nu^{3} - 82510516\nu ) / 3350704 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9733\nu^{10} + 539373\nu^{8} - 8321916\nu^{6} + 49930479\nu^{4} - 106662180\nu^{2} + 20408156 ) / 3350704 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6635\nu^{10} - 368177\nu^{8} + 5693478\nu^{6} - 34177437\nu^{4} + 73875422\nu^{2} - 22799656 ) / 1675352 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34239 \nu^{11} + 1993199 \nu^{9} - 34125436 \nu^{7} + 234643981 \nu^{5} - 598584932 \nu^{3} + 194484340 \nu ) / 3350704 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -9442\nu^{11} + 555469\nu^{9} - 9683665\nu^{7} + 67303464\nu^{5} - 169777993\nu^{3} + 49193350\nu ) / 837676 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9839\nu^{11} - 568156\nu^{9} + 9551303\nu^{7} - 63671239\nu^{5} + 155667983\nu^{3} - 55984110\nu ) / 837676 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 42955 \nu^{11} + 2523131 \nu^{9} - 43962156 \nu^{7} + 308576337 \nu^{5} - 805132996 \nu^{3} + 288162964 \nu ) / 3350704 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22931\nu^{11} - 1381489\nu^{9} + 25169934\nu^{7} - 183776117\nu^{5} + 489391238\nu^{3} - 155588232\nu ) / 1675352 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{9} + 2\beta_{8} - 2\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{6} + 3\beta_{5} + 3\beta_{3} - \beta_{2} + 2\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 17\beta_{11} - 19\beta_{10} + 25\beta_{9} + 40\beta_{8} + 25\beta_{7} - 44\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 106\beta_{6} + 82\beta_{5} + 177\beta_{3} - 51\beta_{2} + 97\beta _1 + 201 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 444\beta_{11} - 823\beta_{10} + 754\beta_{9} + 1109\beta_{8} + 1068\beta_{7} - 1433\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3669\beta_{6} + 2618\beta_{5} + 6754\beta_{3} - 1957\beta_{2} + 3707\beta _1 + 5776 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 14159\beta_{11} - 30202\beta_{10} + 25104\beta_{9} + 36254\beta_{8} + 38769\beta_{7} - 49169\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 127472\beta_{6} + 88943\beta_{5} + 240177\beta_{3} - 69988\beta_{2} + 132713\beta _1 + 190630 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 481414 \beta_{11} - 1066717 \beta_{10} + 863637 \beta_{9} + 1242828 \beta_{8} + 1363518 \beta_{7} - 1707109 \beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4437942\beta_{6} + 3077945\beta_{5} + 8410200\beta_{3} - 2455917\beta_{2} + 4660680\beta _1 + 6549173 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 16664398 \beta_{11} - 37304294 \beta_{10} + 29991147 \beta_{9} + 43128763 \beta_{8} + \cdots - 59434868 \beta_{4} ) / 7 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.536631
−0.536631
−3.04868
3.04868
2.37737
−2.37737
5.90311
−5.90311
0.653547
−0.653547
2.91408
−2.91408
−5.32460 −8.14122 20.3514 13.9048 43.3488 0 −65.7661 39.2795 −74.0376
1.2 −5.32460 8.14122 20.3514 −13.9048 −43.3488 0 −65.7661 39.2795 74.0376
1.3 −4.20622 −1.51747 9.69231 17.0765 6.38284 0 −7.11821 −24.6973 −71.8275
1.4 −4.20622 1.51747 9.69231 −17.0765 −6.38284 0 −7.11821 −24.6973 71.8275
1.5 −3.65095 −5.65703 5.32941 −14.0452 20.6535 0 9.75020 5.00198 51.2783
1.6 −3.65095 5.65703 5.32941 14.0452 −20.6535 0 9.75020 5.00198 −51.2783
1.7 −1.27927 −6.12018 −6.36346 −3.28676 7.82939 0 18.3748 10.4566 4.20466
1.8 −1.27927 6.12018 −6.36346 3.28676 −7.82939 0 18.3748 10.4566 −4.20466
1.9 0.316139 −7.37064 −7.90006 8.04565 −2.33014 0 −5.02662 27.3264 2.54354
1.10 0.316139 7.37064 −7.90006 −8.04565 2.33014 0 −5.02662 27.3264 −2.54354
1.11 3.14490 −4.75739 1.89043 14.1155 −14.9615 0 −19.2140 −4.36723 44.3918
1.12 3.14490 4.75739 1.89043 −14.1155 14.9615 0 −19.2140 −4.36723 −44.3918
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 343.4.a.c ✓ 12
7.b odd 2 1 inner 343.4.a.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
343.4.a.c ✓ 12 1.a even 1 1 trivial
343.4.a.c ✓ 12 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(343))\):

\( T_{2}^{6} + 11T_{2}^{5} + 25T_{2}^{4} - 87T_{2}^{3} - 358T_{2}^{2} - 208T_{2} + 104 \) Copy content Toggle raw display
\( T_{3}^{12} - 215T_{3}^{10} + 17968T_{3}^{8} - 733139T_{3}^{6} + 14844158T_{3}^{4} - 128194192T_{3}^{2} + 224944888 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 11 T^{5} + \cdots + 104)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 224944888 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 1549645333432 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + 67 T^{5} + \cdots + 151514152)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 13\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 52\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 20\!\cdots\!72 \) Copy content Toggle raw display
$23$ \( (T^{6} + 32 T^{5} + \cdots - 166987717)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 821 T^{5} + \cdots - 709504835416)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 17\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 5791328343352)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 89\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 34956080379688)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 53\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 227365385023512)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 31\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 35\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 26\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 96156629133923)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 10\!\cdots\!81)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 38\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 25\!\cdots\!88 \) Copy content Toggle raw display
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