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Properties

Label 1911.4.a.w
Level $1911$
Weight $4$
Character orbit 1911.a
Self dual yes
Analytic conductor $112.753$
Analytic rank $1$
Dimension $11$
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 = [11,-1,33,31,-17] 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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - x^{10} - 59 x^{9} + 36 x^{8} + 1220 x^{7} - 339 x^{6} - 10807 x^{5} + 58 x^{4} + 40509 x^{3} + \cdots - 27208 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 273)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 3) q^{4} + (\beta_{6} + \beta_1 - 2) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{9} + \beta_{6} - \beta_{4} + \cdots - 5) q^{8} + 9 q^{9} + ( - \beta_{10} - \beta_{5} + \beta_{3} + \cdots - 7) q^{10}+ \cdots + (18 \beta_{10} + 9 \beta_{9} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - q^{2} + 33 q^{3} + 31 q^{4} - 17 q^{5} - 3 q^{6} - 54 q^{8} + 99 q^{9} - 75 q^{10} + 7 q^{11} + 93 q^{12} + 143 q^{13} - 51 q^{15} + 23 q^{16} + 20 q^{17} - 9 q^{18} - 242 q^{19} - 254 q^{20} - 290 q^{22}+ \cdots + 63 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - x^{10} - 59 x^{9} + 36 x^{8} + 1220 x^{7} - 339 x^{6} - 10807 x^{5} + 58 x^{4} + 40509 x^{3} + \cdots - 27208 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 406747 \nu^{10} + 1891310 \nu^{9} + 42954303 \nu^{8} - 175153971 \nu^{7} + \cdots + 135192928552 ) / 2660299344 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 220847 \nu^{10} - 1362646 \nu^{9} - 10452323 \nu^{8} + 69768919 \nu^{7} + 147631585 \nu^{6} + \cdots + 4366351208 ) / 295588816 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1418323 \nu^{10} - 9422489 \nu^{9} - 76662543 \nu^{8} + 481991916 \nu^{7} + 1522054520 \nu^{6} + \cdots - 73910801872 ) / 665074836 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5679841 \nu^{10} + 945218 \nu^{9} + 326446365 \nu^{8} + 40652439 \nu^{7} + \cdots + 64540024264 ) / 2660299344 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3085409 \nu^{10} + 12354292 \nu^{9} + 170438973 \nu^{8} - 624596415 \nu^{7} + \cdots + 121026513776 ) / 1330149672 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6269743 \nu^{10} - 4955378 \nu^{9} - 320012571 \nu^{8} + 55857243 \nu^{7} + 5181778613 \nu^{6} + \cdots + 39953543528 ) / 2660299344 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 958433 \nu^{10} + 1651129 \nu^{9} + 52564659 \nu^{8} - 73408479 \nu^{7} - 955841770 \nu^{6} + \cdots + 1492670834 ) / 332537418 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5407331 \nu^{10} + 15458506 \nu^{9} + 297365595 \nu^{8} - 745882755 \nu^{7} + \cdots + 143470109744 ) / 1330149672 \) 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
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{6} + \beta_{4} + 18\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{10} - \beta_{8} - 3\beta_{7} - 3\beta_{6} + 2\beta_{4} + 2\beta_{3} + 27\beta_{2} + 2\beta _1 + 203 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{10} + 29 \beta_{9} - 5 \beta_{8} - 7 \beta_{7} - 30 \beta_{6} - 4 \beta_{5} + 28 \beta_{4} + \cdots + 201 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 116 \beta_{10} + 17 \beta_{9} - 36 \beta_{8} - 138 \beta_{7} - 121 \beta_{6} - 10 \beta_{5} + \cdots + 4518 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 15 \beta_{10} + 755 \beta_{9} - 223 \beta_{8} - 397 \beta_{7} - 794 \beta_{6} - 220 \beta_{5} + \cdots + 6949 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3476 \beta_{10} + 934 \beta_{9} - 1050 \beta_{8} - 4818 \beta_{7} - 3728 \beta_{6} - 672 \beta_{5} + \cdots + 109711 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 630 \beta_{10} + 19728 \beta_{9} - 7430 \beta_{8} - 15848 \beta_{7} - 21178 \beta_{6} - 8438 \beta_{5} + \cdots + 224640 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 96444 \beta_{10} + 36579 \beta_{9} - 30022 \beta_{8} - 152064 \beta_{7} - 107535 \beta_{6} + \cdots + 2798120 \) 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
5.37078
4.32055
2.93112
2.46025
1.67966
−0.614465
−1.37626
−1.96658
−3.16084
−3.80688
−4.83732
−5.37078 3.00000 20.8452 −8.52330 −16.1123 0 −68.9889 9.00000 45.7767
1.2 −4.32055 3.00000 10.6672 3.35894 −12.9617 0 −11.5237 9.00000 −14.5125
1.3 −2.93112 3.00000 0.591475 −0.344609 −8.79337 0 21.7153 9.00000 1.01009
1.4 −2.46025 3.00000 −1.94719 21.0905 −7.38074 0 24.4725 9.00000 −51.8878
1.5 −1.67966 3.00000 −5.17875 −10.6494 −5.03897 0 22.1358 9.00000 17.8873
1.6 0.614465 3.00000 −7.62243 9.49056 1.84339 0 −9.59944 9.00000 5.83161
1.7 1.37626 3.00000 −6.10590 −18.6871 4.12879 0 −19.4134 9.00000 −25.7184
1.8 1.96658 3.00000 −4.13255 −10.3068 5.89975 0 −23.8597 9.00000 −20.2692
1.9 3.16084 3.00000 1.99089 10.8277 9.48251 0 −18.9938 9.00000 34.2246
1.10 3.80688 3.00000 6.49236 3.12156 11.4206 0 −5.73941 9.00000 11.8834
1.11 4.83732 3.00000 15.3997 −16.3780 14.5120 0 35.7947 9.00000 −79.2258
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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.w 11
7.b odd 2 1 1911.4.a.v 11
7.d odd 6 2 273.4.i.d ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.4.i.d ✓ 22 7.d odd 6 2
1911.4.a.v 11 7.b odd 2 1
1911.4.a.w 11 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{11} + T_{2}^{10} - 59 T_{2}^{9} - 36 T_{2}^{8} + 1220 T_{2}^{7} + 339 T_{2}^{6} - 10807 T_{2}^{5} + \cdots + 27208 \) Copy content Toggle raw display
\( T_{5}^{11} + 17 T_{5}^{10} - 647 T_{5}^{9} - 11881 T_{5}^{8} + 104376 T_{5}^{7} + 2187198 T_{5}^{6} + \cdots - 2242210608 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} + T^{10} + \cdots + 27208 \) Copy content Toggle raw display
$3$ \( (T - 3)^{11} \) Copy content Toggle raw display
$5$ \( T^{11} + \cdots - 2242210608 \) Copy content Toggle raw display
$7$ \( T^{11} \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots - 48364591092795 \) Copy content Toggle raw display
$13$ \( (T - 13)^{11} \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots + 15\!\cdots\!14 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots - 68\!\cdots\!92 \) Copy content Toggle raw display
$23$ \( T^{11} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots - 23\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots + 57\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{11} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots - 80\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots + 34\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots - 61\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 22\!\cdots\!47 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots + 79\!\cdots\!07 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots - 39\!\cdots\!02 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 12\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots + 17\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots + 39\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots - 52\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots - 13\!\cdots\!96 \) Copy content Toggle raw display
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