↑

Properties

Label 2116.4.a.g
Level $2116$
Weight $4$
Character orbit 2116.a
Self dual yes
Analytic conductor $124.848$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2116,4,Mod(1,2116)] 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("2116.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2116, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2116 = 2^{2} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2116.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-2,0,0] 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: \(124.848041572\)
Analytic rank: \(0\)
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: \( x^{12} - 2 x^{11} - 91 x^{10} + 122 x^{9} + 2363 x^{8} - 2074 x^{7} - 23617 x^{6} + 12114 x^{5} + \cdots - 4554 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + (\beta_{8} + \beta_1) q^{5} + (\beta_{7} - \beta_1) q^{7} + (\beta_{10} - \beta_{9} + 9) q^{9} + (\beta_{11} - \beta_{8} + \beta_{4} - \beta_1) q^{11} + (\beta_{10} - 2 \beta_{6} + \beta_{3} + 7) q^{13}+ \cdots + ( - 9 \beta_{11} + 7 \beta_{8} + \cdots - 25 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 102 q^{9} + 86 q^{13} + 26 q^{25} + 94 q^{27} - 46 q^{29} + 292 q^{31} + 124 q^{35} + 620 q^{39} + 666 q^{41} + 224 q^{47} + 1532 q^{49} - 1228 q^{55} + 730 q^{59} + 1424 q^{71} - 112 q^{73}+ \cdots + 3240 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 91 x^{10} + 122 x^{9} + 2363 x^{8} - 2074 x^{7} - 23617 x^{6} + 12114 x^{5} + \cdots - 4554 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 62613704029 \nu^{11} + 196889362019 \nu^{10} + 5331816250018 \nu^{9} + \cdots + 899155827903546 ) / 114938065137162 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6319786649326 \nu^{11} + 82637527027135 \nu^{10} + \cdots - 64\!\cdots\!32 ) / 69\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8595448175360 \nu^{11} - 33495022894813 \nu^{10} - 685891930550582 \nu^{9} + \cdots - 48\!\cdots\!73 ) / 69\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18050413258583 \nu^{11} + 66187120234507 \nu^{10} + \cdots - 23\!\cdots\!10 ) / 13\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20963799939023 \nu^{11} - 48252679082140 \nu^{10} + \cdots - 78\!\cdots\!20 ) / 13\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8567536578199 \nu^{11} + 12891554737513 \nu^{10} + 814038886355121 \nu^{9} + \cdots + 16\!\cdots\!42 ) / 46\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 51030168465448 \nu^{11} - 87680552899219 \nu^{10} + \cdots - 11\!\cdots\!74 ) / 13\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26646411492390 \nu^{11} - 68197395581591 \nu^{10} + \cdots - 32\!\cdots\!22 ) / 69\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 36862100949694 \nu^{11} + 109446653930981 \nu^{10} + \cdots + 18\!\cdots\!25 ) / 69\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 47396489450567 \nu^{11} - 101806962244177 \nu^{10} + \cdots - 63\!\cdots\!63 ) / 69\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 176739163936469 \nu^{11} + 492455576018442 \nu^{10} + \cdots + 25\!\cdots\!24 ) / 13\!\cdots\!02 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{3} + 12\beta _1 + 2 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 4 \beta_{11} - 5 \beta_{10} - \beta_{9} - 4 \beta_{8} - 4 \beta_{7} - 6 \beta_{6} - 2 \beta_{4} + \cdots + 188 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 15 \beta_{11} - 9 \beta_{10} - 47 \beta_{9} - 24 \beta_{8} + 12 \beta_{7} + 46 \beta_{6} + \cdots + 186 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 276 \beta_{11} - 269 \beta_{10} - 81 \beta_{9} - 408 \beta_{8} - 264 \beta_{7} - 278 \beta_{6} + \cdots + 7706 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1345 \beta_{11} - 743 \beta_{10} - 2526 \beta_{9} - 1930 \beta_{8} + 650 \beta_{7} + 2906 \beta_{6} + \cdots + 15190 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 8859 \beta_{11} - 7590 \beta_{10} - 3122 \beta_{9} - 13674 \beta_{8} - 7698 \beta_{7} - 6480 \beta_{6} + \cdots + 205687 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 96950 \beta_{11} - 54242 \beta_{10} - 144759 \beta_{9} - 138026 \beta_{8} + 25522 \beta_{7} + \cdots + 1177020 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1111344 \beta_{11} - 894787 \beta_{10} - 467725 \beta_{9} - 1718400 \beta_{8} - 871008 \beta_{7} + \cdots + 23823616 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 6584247 \beta_{11} - 3825365 \beta_{10} - 8560567 \beta_{9} - 9453132 \beta_{8} + 706752 \beta_{7} + \cdots + 86650358 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 69382850 \beta_{11} - 53934735 \beta_{10} - 33769587 \beta_{9} - 106805420 \beta_{8} + \cdots + 1421146442 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 437131255 \beta_{11} - 263651749 \beta_{10} - 515007742 \beta_{9} - 633317542 \beta_{8} + \cdots + 6143674198 ) / 12 \) 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
−2.05914
1.40496
−0.200914
−3.66502
7.99234
4.52824
−0.687950
2.77615
−7.12297
−3.65887
3.07863
−0.385467
0 −8.61943 0 −4.07706 0 −27.4630 0 47.2947 0
1.2 0 −8.61943 0 4.07706 0 27.4630 0 47.2947 0
1.3 0 −5.89893 0 −4.27775 0 −23.5832 0 7.79739 0
1.4 0 −5.89893 0 4.27775 0 23.5832 0 7.79739 0
1.5 0 −0.560111 0 −11.9051 0 0.612597 0 −26.6863 0
1.6 0 −0.560111 0 11.9051 0 −0.612597 0 −26.6863 0
1.7 0 0.365286 0 −21.6125 0 22.1192 0 −26.8666 0
1.8 0 0.365286 0 21.6125 0 −22.1192 0 −26.8666 0
1.9 0 4.68462 0 −7.28677 0 −27.3351 0 −5.05434 0
1.10 0 4.68462 0 7.28677 0 27.3351 0 −5.05434 0
1.11 0 9.02857 0 −8.13328 0 −16.6364 0 54.5151 0
1.12 0 9.02857 0 8.13328 0 16.6364 0 54.5151 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(23\) \( -1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2116.4.a.g ✓ 12
23.b odd 2 1 inner 2116.4.a.g ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2116.4.a.g ✓ 12 1.a even 1 1 trivial
2116.4.a.g ✓ 12 23.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2116))\):

\( T_{3}^{6} + T_{3}^{5} - 106T_{3}^{4} - 104T_{3}^{3} + 2156T_{3}^{2} + 436T_{3} - 440 \) Copy content Toggle raw display
\( T_{5}^{12} - 763T_{5}^{10} + 168046T_{5}^{8} - 15224262T_{5}^{6} + 626181309T_{5}^{4} - 11172001167T_{5}^{2} + 70729402500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{5} - 106 T^{4} + \cdots - 440)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 70729402500 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 15927761721600 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 281719305974016 \) Copy content Toggle raw display
$13$ \( (T^{6} - 43 T^{5} + \cdots - 374631540)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( (T^{6} + 23 T^{5} + \cdots + 571605824592)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 105998158512896)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 56356036761222)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 794010814974000)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 18\!\cdots\!76)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 90\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 39\!\cdots\!25)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
show more
show less