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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [679,2,Mod(1,679)] 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("679.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(679, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 679 = 7 \cdot 97 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 679.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,2,4,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.42184229724\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 20 x^{12} + 41 x^{11} + 147 x^{10} - 310 x^{9} - 484 x^{8} + 1064 x^{7} + 689 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{11} + 1) q^{5} + (\beta_{13} - 2 \beta_{12} + \cdots + \beta_1) q^{6} + q^{7} + (\beta_{12} + \beta_{10} + \cdots + \beta_1) q^{8}+ \cdots + (4 \beta_{13} - 4 \beta_{12} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{2} + 4 q^{3} + 16 q^{4} + 19 q^{5} - q^{6} + 14 q^{7} - 3 q^{8} + 24 q^{9} - 8 q^{10} + 5 q^{11} + 15 q^{12} + 9 q^{13} + 2 q^{14} - 2 q^{15} + 12 q^{16} + 29 q^{17} - 32 q^{18} + 4 q^{19}+ \cdots - 31 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 20 x^{12} + 41 x^{11} + 147 x^{10} - 310 x^{9} - 484 x^{8} + 1064 x^{7} + 689 x^{6} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23429 \nu^{13} + 90142 \nu^{12} - 464774 \nu^{11} - 1722699 \nu^{10} + 3617445 \nu^{9} + \cdots + 2142107 ) / 668281 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25475 \nu^{13} + 76608 \nu^{12} + 641391 \nu^{11} - 1381241 \nu^{10} - 5976099 \nu^{9} + \cdots - 243322 ) / 668281 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 49023 \nu^{13} + 150333 \nu^{12} + 1108925 \nu^{11} - 2812696 \nu^{10} - 9338775 \nu^{9} + \cdots + 984430 ) / 668281 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 64989 \nu^{13} - 189479 \nu^{12} - 1591545 \nu^{11} + 3381907 \nu^{10} + 14518100 \nu^{9} + \cdots + 1701056 ) / 668281 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 113596 \nu^{13} - 87067 \nu^{12} - 2359489 \nu^{11} + 1727579 \nu^{10} + 18530798 \nu^{9} + \cdots - 1059584 ) / 668281 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 137000 \nu^{13} + 3806 \nu^{12} - 2683288 \nu^{11} + 173382 \nu^{10} + 19612879 \nu^{9} + \cdots - 1359991 ) / 668281 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 211274 \nu^{13} - 189854 \nu^{12} - 4507030 \nu^{11} + 3658404 \nu^{10} + 36429395 \nu^{9} + \cdots - 435802 ) / 668281 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 228246 \nu^{13} + 98028 \nu^{12} + 4647955 \nu^{11} - 2178524 \nu^{10} - 35664289 \nu^{9} + \cdots - 588341 ) / 668281 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 282254 \nu^{13} - 126810 \nu^{12} - 5799493 \nu^{11} + 2667472 \nu^{10} + 44933241 \nu^{9} + \cdots + 399947 ) / 668281 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 300698 \nu^{13} - 158219 \nu^{12} - 6221654 \nu^{11} + 3268521 \nu^{10} + 48620509 \nu^{9} + \cdots + 1746018 ) / 668281 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 429330 \nu^{13} - 390895 \nu^{12} - 9165741 \nu^{11} + 7556587 \nu^{10} + 74113899 \nu^{9} + \cdots + 589835 ) / 668281 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{10} + \beta_{5} - \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{11} - 2\beta_{9} - 2\beta_{8} + \beta_{7} + 7\beta_{2} - \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + 9 \beta_{12} - 2 \beta_{11} + 9 \beta_{10} - \beta_{9} + 2 \beta_{8} + 8 \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{12} + 10 \beta_{11} + \beta_{10} - 19 \beta_{9} - 20 \beta_{8} + 10 \beta_{7} - 2 \beta_{6} + \cdots + 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14 \beta_{13} + 64 \beta_{12} - 23 \beta_{11} + 67 \beta_{10} - 13 \beta_{9} + 25 \beta_{8} - \beta_{7} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3 \beta_{13} + 82 \beta_{12} + 78 \beta_{11} + 12 \beta_{10} - 141 \beta_{9} - 160 \beta_{8} + \cdots + 418 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 140 \beta_{13} + 421 \beta_{12} - 202 \beta_{11} + 470 \beta_{10} - 120 \beta_{9} + 232 \beta_{8} + \cdots - 156 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 57 \beta_{13} + 629 \beta_{12} + 566 \beta_{11} + 98 \beta_{10} - 957 \beta_{9} - 1200 \beta_{8} + \cdots + 2463 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1216 \beta_{13} + 2659 \beta_{12} - 1611 \beta_{11} + 3206 \beta_{10} - 964 \beta_{9} + 1940 \beta_{8} + \cdots - 1382 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 715 \beta_{13} + 4653 \beta_{12} + 4020 \beta_{11} + 661 \beta_{10} - 6209 \beta_{9} - 8781 \beta_{8} + \cdots + 14948 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 9804 \beta_{13} + 16322 \beta_{12} - 12259 \beta_{11} + 21526 \beta_{10} - 7209 \beta_{9} + 15444 \beta_{8} + \cdots - 11223 \) 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
−2.65920
−2.48752
−1.89050
−1.55535
−0.952506
−0.374274
0.00870050
0.431416
1.21055
1.23525
1.99300
2.25872
2.29940
2.48231
−2.65920 3.25573 5.07134 2.44880 −8.65765 1.00000 −8.16731 7.59981 −6.51185
1.2 −2.48752 −3.28960 4.18776 2.12084 8.18296 1.00000 −5.44210 7.82149 −5.27564
1.3 −1.89050 0.613394 1.57397 0.111652 −1.15962 1.00000 0.805399 −2.62375 −0.211078
1.4 −1.55535 2.61909 0.419128 0.934549 −4.07362 1.00000 2.45882 3.85965 −1.45356
1.5 −0.952506 1.72777 −1.09273 4.19646 −1.64572 1.00000 2.94585 −0.0147939 −3.99716
1.6 −0.374274 −2.88523 −1.85992 4.06241 1.07986 1.00000 1.44467 5.32454 −1.52045
1.7 0.00870050 −0.317372 −1.99992 1.80743 −0.00276129 1.00000 −0.0348014 −2.89928 0.0157256
1.8 0.431416 −0.224854 −1.81388 −3.59694 −0.0970054 1.00000 −1.64537 −2.94944 −1.55178
1.9 1.21055 −2.59004 −0.534559 −0.0635542 −3.13539 1.00000 −3.06822 3.70832 −0.0769358
1.10 1.23525 1.79684 −0.474151 3.74098 2.21955 1.00000 −3.05620 0.228617 4.62105
1.11 1.99300 3.09503 1.97204 −0.216572 6.16839 1.00000 −0.0557173 6.57922 −0.431627
1.12 2.25872 0.230018 3.10180 2.07162 0.519545 1.00000 2.48865 −2.94709 4.67921
1.13 2.29940 1.76115 3.28724 −1.54585 4.04960 1.00000 2.95987 0.101663 −3.55452
1.14 2.48231 −1.79194 4.16188 2.92816 −4.44815 1.00000 5.36647 0.211039 7.26861
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( -1 \)
\(97\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 679.2.a.e ✓ 14
3.b odd 2 1 6111.2.a.r 14
7.b odd 2 1 4753.2.a.k 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
679.2.a.e ✓ 14 1.a even 1 1 trivial
4753.2.a.k 14 7.b odd 2 1
6111.2.a.r 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 2 T_{2}^{13} - 20 T_{2}^{12} + 41 T_{2}^{11} + 147 T_{2}^{10} - 310 T_{2}^{9} - 484 T_{2}^{8} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(679))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 2 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{14} - 4 T^{13} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{14} - 19 T^{13} + \cdots + 29 \) Copy content Toggle raw display
$7$ \( (T - 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} - 5 T^{13} + \cdots - 476464 \) Copy content Toggle raw display
$13$ \( T^{14} - 9 T^{13} + \cdots + 197008 \) Copy content Toggle raw display
$17$ \( T^{14} - 29 T^{13} + \cdots + 1154513 \) Copy content Toggle raw display
$19$ \( T^{14} - 4 T^{13} + \cdots - 43357309 \) Copy content Toggle raw display
$23$ \( T^{14} - 17 T^{13} + \cdots - 100672 \) Copy content Toggle raw display
$29$ \( T^{14} - 5 T^{13} + \cdots + 23592128 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 591039424 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 280212928 \) Copy content Toggle raw display
$41$ \( T^{14} - 30 T^{13} + \cdots - 2935583 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 13343994719 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 173760448 \) Copy content Toggle raw display
$53$ \( T^{14} - 16 T^{13} + \cdots + 84516253 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 429552433 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 17209581376 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 504567488 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 99982357568 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 884400923072 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 229699871753 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 128585431541 \) Copy content Toggle raw display
$89$ \( T^{14} - 71 T^{13} + \cdots + 95936 \) Copy content Toggle raw display
$97$ \( (T + 1)^{14} \) Copy content Toggle raw display
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