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Properties

Label 209.4.a.a
Level $209$
Weight $4$
Character orbit 209.a
Self dual yes
Analytic conductor $12.331$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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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] = [209,4,Mod(1,209)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(209, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("209.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 209.a (trivial)

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: yes
Analytic conductor: \(12.3313991912\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 43x^{6} - 12x^{5} + 545x^{4} + 264x^{3} - 2017x^{2} - 660x + 2178 \) 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 3) q^{4} + ( - \beta_{5} + \beta_1 - 2) q^{5} + (\beta_{6} - \beta_{3} - \beta_{2} + \beta_1) q^{6} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 7) q^{7}+ \cdots + ( - 11 \beta_{7} - 11 \beta_{6} + \cdots + 22) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} + 22 q^{4} - 12 q^{5} + q^{6} - 59 q^{7} - 36 q^{8} - 19 q^{9} - 120 q^{10} - 88 q^{11} - 55 q^{12} - 61 q^{13} - 93 q^{14} - 62 q^{15} - 34 q^{16} + 107 q^{17} - 183 q^{18} + 152 q^{19} - 82 q^{20}+ \cdots + 209 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 43x^{6} - 12x^{5} + 545x^{4} + 264x^{3} - 2017x^{2} - 660x + 2178 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{7} - 165\nu^{6} - 334\nu^{5} + 6318\nu^{4} + 6389\nu^{3} - 64911\nu^{2} - 45634\nu + 129690 ) / 6864 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17\nu^{7} - 33\nu^{6} - 566\nu^{5} + 390\nu^{4} + 4975\nu^{3} + 6237\nu^{2} - 12278\nu - 22110 ) / 3432 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 43\nu^{5} - 12\nu^{4} + 512\nu^{3} + 264\nu^{2} - 1258\nu - 462 ) / 66 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43\nu^{7} - 33\nu^{6} - 1684\nu^{5} + 936\nu^{4} + 18287\nu^{3} - 6633\nu^{2} - 46702\nu + 17358 ) / 1716 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} - 12\nu^{6} - 194\nu^{5} + 390\nu^{4} + 2023\nu^{3} - 3114\nu^{2} - 4070\nu + 5688 ) / 156 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{4} + 2\beta_{3} + \beta_{2} + 18\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} - 2\beta_{5} - 4\beta_{4} + 23\beta_{2} + 2\beta _1 + 193 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -25\beta_{7} + 33\beta_{6} - 14\beta_{5} + 29\beta_{4} + 42\beta_{3} + 29\beta_{2} + 362\beta _1 + 192 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{7} + 88\beta_{6} - 72\beta_{5} - 144\beta_{4} - 16\beta_{3} + 499\beta_{2} + 88\beta _1 + 3907 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -563\beta_{7} + 931\beta_{6} - 560\beta_{5} + 687\beta_{4} + 782\beta_{3} + 747\beta_{2} + 7632\beta _1 + 5058 \) 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
4.92700
4.06018
1.73547
1.14081
−1.60842
−2.09608
−3.50438
−4.65458
−4.92700 −2.96147 16.2753 9.76654 14.5912 −16.5465 −40.7725 −18.2297 −48.1197
1.2 −4.06018 5.61545 8.48503 −6.03806 −22.7997 4.30385 −1.96930 4.53328 24.5156
1.3 −1.73547 −9.28551 −4.98814 −1.59392 16.1147 8.61614 22.5405 59.2207 2.76620
1.4 −1.14081 1.79882 −6.69855 12.6906 −2.05212 −10.6895 16.7683 −23.7642 −14.4776
1.5 1.60842 7.48922 −5.41298 −10.7091 12.0458 −24.0540 −21.5737 29.0884 −17.2248
1.6 2.09608 0.225665 −3.60644 −3.33943 0.473012 21.2626 −24.3281 −26.9491 −6.99972
1.7 3.50438 −0.605984 4.28065 0.861140 −2.12360 −30.2736 −13.0340 −26.6328 3.01776
1.8 4.65458 −3.27619 13.6651 −13.6377 −15.2493 −11.6190 26.3687 −16.2666 −63.4778
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( +1 \)
\(19\) \( -1 \)

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 209.4.a.a ✓ 8
3.b odd 2 1 1881.4.a.a 8
11.b odd 2 1 2299.4.a.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.4.a.a ✓ 8 1.a even 1 1 trivial
1881.4.a.a 8 3.b odd 2 1
2299.4.a.i 8 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 43T_{2}^{6} + 12T_{2}^{5} + 545T_{2}^{4} - 264T_{2}^{3} - 2017T_{2}^{2} + 660T_{2} + 2178 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(209))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 43 T^{6} + \cdots + 2178 \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + \cdots + 932 \) Copy content Toggle raw display
$5$ \( T^{8} + 12 T^{7} + \cdots - 500988 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots - 1179970928 \) Copy content Toggle raw display
$11$ \( (T + 11)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots - 928182024856 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots - 5700281587968 \) Copy content Toggle raw display
$19$ \( (T - 19)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 43\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 80\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 50\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 60\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 64\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 19\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 14\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 15\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots - 29\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 69\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 31\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 20\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
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