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Properties

Label 1911.4.a.bd
Level $1911$
Weight $4$
Character orbit 1911.a
Self dual yes
Analytic conductor $112.753$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1911,4,Mod(1,1911)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1911, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("1911.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1911.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,-6,-42,50,-4] 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: yes
Analytic conductor: \(112.752650021\)
Analytic rank: \(1\)
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: \( x^{14} - 6 x^{13} - 63 x^{12} + 408 x^{11} + 1393 x^{10} - 10374 x^{9} - 12229 x^{8} + 122556 x^{7} + \cdots - 43904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} - \beta_{6} q^{5} + 3 \beta_1 q^{6} + ( - \beta_{11} - \beta_{9} + \beta_{5} + \cdots - 1) q^{8} + 9 q^{9} + ( - \beta_{9} - \beta_{8} - \beta_{7} + \cdots + 3) q^{10}+ \cdots + (9 \beta_{12} - 9 \beta_{7} + \cdots - 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 6 q^{2} - 42 q^{3} + 50 q^{4} - 4 q^{5} + 18 q^{6} - 30 q^{8} + 126 q^{9} + 32 q^{10} - 68 q^{11} - 150 q^{12} + 182 q^{13} + 12 q^{15} - 50 q^{16} - 54 q^{18} + 24 q^{19} + 96 q^{20} - 300 q^{22}+ \cdots - 612 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 63 x^{12} + 408 x^{11} + 1393 x^{10} - 10374 x^{9} - 12229 x^{8} + 122556 x^{7} + \cdots - 43904 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1441879 \nu^{13} - 5183881 \nu^{12} - 57257672 \nu^{11} + 294023024 \nu^{10} + \cdots - 75859424384 ) / 15119367168 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 208118 \nu^{13} - 1851177 \nu^{12} - 14002120 \nu^{11} + 126899616 \nu^{10} + \cdots + 16208102016 ) / 1889920896 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12859563 \nu^{13} + 44786941 \nu^{12} + 916906920 \nu^{11} - 2938218608 \nu^{10} + \cdots - 237258238336 ) / 15119367168 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8211189 \nu^{13} + 30478559 \nu^{12} + 582158952 \nu^{11} - 1992880720 \nu^{10} + \cdots - 258237275264 ) / 7559683584 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21058747 \nu^{13} - 77354973 \nu^{12} - 1519349384 \nu^{11} + 5132765040 \nu^{10} + \cdots + 447877051776 ) / 15119367168 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14444211 \nu^{13} + 62891401 \nu^{12} + 1039908264 \nu^{11} - 4194643280 \nu^{10} + \cdots - 384946232704 ) / 7559683584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24577 \nu^{13} - 98839 \nu^{12} - 1751192 \nu^{11} + 6537680 \nu^{10} + 47690081 \nu^{9} + \cdots + 254131840 ) / 12212736 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37828163 \nu^{13} + 136952037 \nu^{12} + 2696682856 \nu^{11} - 9015542064 \nu^{10} + \cdots - 550670171520 ) / 15119367168 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 43285889 \nu^{13} + 167149623 \nu^{12} + 3084882616 \nu^{11} - 11031866448 \nu^{10} + \cdots - 566992823424 ) / 15119367168 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 57659429 \nu^{13} + 212433323 \nu^{12} + 4132599160 \nu^{11} - 13999270096 \nu^{10} + \cdots - 1898192027264 ) / 15119367168 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 36961217 \nu^{13} - 146367419 \nu^{12} - 2637740872 \nu^{11} + 9668605744 \nu^{10} + \cdots + 697785266816 ) / 7559683584 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{5} + 18\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} + 2\beta_{11} - \beta_{10} - 2\beta_{8} - \beta_{7} - 2\beta_{6} - 4\beta_{5} + \beta_{4} + 25\beta_{2} + 219 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{12} + 35 \beta_{11} - 8 \beta_{10} + 27 \beta_{9} - 2 \beta_{8} - 6 \beta_{7} - 4 \beta_{6} + \cdots + 45 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 31 \beta_{13} + 16 \beta_{12} + 94 \beta_{11} - 63 \beta_{10} - 4 \beta_{9} - 82 \beta_{8} + \cdots + 4565 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 16 \beta_{13} + 102 \beta_{12} + 1027 \beta_{11} - 392 \beta_{10} + 635 \beta_{9} - 134 \beta_{8} + \cdots + 1977 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 807 \beta_{13} + 760 \beta_{12} + 3398 \beta_{11} - 2511 \beta_{10} - 108 \beta_{9} - 2626 \beta_{8} + \cdots + 101541 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 912 \beta_{13} + 3726 \beta_{12} + 28987 \beta_{11} - 13904 \beta_{10} + 14723 \beta_{9} + \cdots + 77225 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 20487 \beta_{13} + 26176 \beta_{12} + 110606 \beta_{11} - 84823 \beta_{10} - 588 \beta_{9} + \cdots + 2361149 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 36400 \beta_{13} + 120294 \beta_{12} + 809771 \beta_{11} - 439944 \beta_{10} + 345219 \beta_{9} + \cdots + 2763889 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 523663 \beta_{13} + 804584 \beta_{12} + 3411558 \beta_{11} - 2648743 \beta_{10} + 79060 \beta_{9} + \cdots + 56909421 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1260880 \beta_{13} + 3666382 \beta_{12} + 22579531 \beta_{11} - 13238848 \beta_{10} + 8245235 \beta_{9} + \cdots + 93082233 \) 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
5.31718
4.29566
4.21197
3.96203
2.30041
2.24751
2.06054
−0.0267877
−1.11764
−2.00578
−2.27160
−3.63893
−4.61525
−4.71932
−5.31718 −3.00000 20.2724 −6.77406 15.9515 0 −65.2547 9.00000 36.0189
1.2 −4.29566 −3.00000 10.4527 14.9360 12.8870 0 −10.5360 9.00000 −64.1599
1.3 −4.21197 −3.00000 9.74066 8.49708 12.6359 0 −7.33162 9.00000 −35.7894
1.4 −3.96203 −3.00000 7.69771 −16.3547 11.8861 0 1.19769 9.00000 64.7978
1.5 −2.30041 −3.00000 −2.70812 6.49112 6.90123 0 24.6331 9.00000 −14.9322
1.6 −2.24751 −3.00000 −2.94870 −6.25819 6.74253 0 24.6073 9.00000 14.0653
1.7 −2.06054 −3.00000 −3.75416 −12.1609 6.18163 0 24.2200 9.00000 25.0581
1.8 0.0267877 −3.00000 −7.99928 15.2159 −0.0803631 0 −0.428584 9.00000 0.407598
1.9 1.11764 −3.00000 −6.75088 1.89518 −3.35291 0 −16.4862 9.00000 2.11813
1.10 2.00578 −3.00000 −3.97684 −18.3233 −6.01734 0 −24.0229 9.00000 −36.7526
1.11 2.27160 −3.00000 −2.83982 4.25461 −6.81481 0 −24.6238 9.00000 9.66478
1.12 3.63893 −3.00000 5.24182 −9.58367 −10.9168 0 −10.0368 9.00000 −34.8743
1.13 4.61525 −3.00000 13.3005 4.52975 −13.8457 0 24.4632 9.00000 20.9059
1.14 4.71932 −3.00000 14.2720 9.63526 −14.1580 0 29.5994 9.00000 45.4719
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( +1 \)
\(13\) \( -1 \)

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 1911.4.a.bd ✓ 14
7.b odd 2 1 1911.4.a.be yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.4.a.bd ✓ 14 1.a even 1 1 trivial
1911.4.a.be yes 14 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1911))\):

\( T_{2}^{14} + 6 T_{2}^{13} - 63 T_{2}^{12} - 408 T_{2}^{11} + 1393 T_{2}^{10} + 10374 T_{2}^{9} + \cdots - 43904 \) Copy content Toggle raw display
\( T_{5}^{14} + 4 T_{5}^{13} - 808 T_{5}^{12} - 1604 T_{5}^{11} + 248520 T_{5}^{10} + 33428 T_{5}^{9} + \cdots + 6531484301176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 6 T^{13} + \cdots - 43904 \) Copy content Toggle raw display
$3$ \( (T + 3)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 6531484301176 \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 14\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( (T - 13)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 60\!\cdots\!88 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 14\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 48\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 27\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 24\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 28\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 52\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 44\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 63\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 18\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 64\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 15\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
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