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Properties

Label 2116.4.a.h
Level $2116$
Weight $4$
Character orbit 2116.a
Self dual yes
Analytic conductor $124.848$
Analytic rank $1$
Dimension $20$
Inner twists $2$

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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] = [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 = [20,0,0,0,0] 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: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 268 x^{18} + 30325 x^{16} - 1885780 x^{14} + 70316500 x^{12} - 1612065448 x^{10} + \cdots + 540816101604 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{2}\cdot 23^{2} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + \beta_{11} q^{5} + (\beta_{15} + \beta_{12}) q^{7} + (\beta_{6} + \beta_{4} + 3 \beta_{2} + 7) q^{9} + ( - \beta_{19} - \beta_{15} + \cdots + 2 \beta_{9}) q^{11} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2} - 4) q^{13}+ \cdots + ( - 15 \beta_{19} + \cdots + 135 \beta_{9}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 128 q^{9} - 88 q^{13} + 300 q^{25} - 552 q^{27} - 1200 q^{29} + 136 q^{31} - 1216 q^{35} - 648 q^{39} - 1136 q^{41} + 1080 q^{47} + 132 q^{49} + 2488 q^{55} - 1564 q^{59} + 88 q^{71} - 1060 q^{73}+ \cdots - 4264 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 268 x^{18} + 30325 x^{16} - 1885780 x^{14} + 70316500 x^{12} - 1612065448 x^{10} + \cdots + 540816101604 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 14\!\cdots\!10 \nu^{18} + \cdots - 12\!\cdots\!00 ) / 94\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 128821430608881 \nu^{18} + \cdots - 83\!\cdots\!50 ) / 55\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 24\!\cdots\!99 \nu^{18} + \cdots - 75\!\cdots\!78 ) / 57\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27\!\cdots\!25 \nu^{18} + \cdots - 27\!\cdots\!06 ) / 28\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 66\!\cdots\!01 \nu^{18} + \cdots - 54\!\cdots\!34 ) / 57\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 40\!\cdots\!41 \nu^{18} + \cdots + 23\!\cdots\!18 ) / 17\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 74\!\cdots\!59 \nu^{18} + \cdots + 21\!\cdots\!10 ) / 28\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!01 \nu^{18} + \cdots + 61\!\cdots\!86 ) / 28\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 27\!\cdots\!51 \nu^{19} + \cdots + 13\!\cdots\!10 \nu ) / 84\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 87\!\cdots\!31 \nu^{18} + \cdots - 56\!\cdots\!78 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 64\!\cdots\!11 \nu^{19} + \cdots + 11\!\cdots\!46 \nu ) / 42\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 22\!\cdots\!37 \nu^{19} + \cdots + 11\!\cdots\!78 \nu ) / 84\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!93 \nu^{19} + \cdots + 36\!\cdots\!62 \nu ) / 42\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 77\!\cdots\!19 \nu^{19} + \cdots - 26\!\cdots\!02 \nu ) / 21\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 34\!\cdots\!09 \nu^{19} + \cdots - 20\!\cdots\!74 \nu ) / 84\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 20\!\cdots\!07 \nu^{19} + \cdots + 19\!\cdots\!54 \nu ) / 42\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 23\!\cdots\!71 \nu^{19} + \cdots - 24\!\cdots\!56 \nu ) / 15\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 66\!\cdots\!81 \nu^{19} + \cdots + 74\!\cdots\!14 \nu ) / 42\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 53\!\cdots\!23 \nu^{19} + \cdots - 51\!\cdots\!50 \nu ) / 25\!\cdots\!48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 23\beta_{16} - 23\beta_{15} + 3\beta_{14} - 18\beta_{12} - 23\beta_{11} + 23\beta_{9} ) / 92 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 71 \beta_{10} + 8 \beta_{8} + 18 \beta_{7} - 46 \beta_{6} + 156 \beta_{5} + 23 \beta_{4} + \cdots + 4919 ) / 184 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 115 \beta_{19} - 92 \beta_{17} + 667 \beta_{16} - 851 \beta_{15} - 30 \beta_{14} + \cdots + 1173 \beta_{9} ) / 92 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2107 \beta_{10} - 59 \beta_{8} + 333 \beta_{7} - 1380 \beta_{6} + 4450 \beta_{5} - 46 \beta_{4} + \cdots + 102237 ) / 92 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 17894 \beta_{19} - 759 \beta_{18} - 15019 \beta_{17} + 59501 \beta_{16} - 75003 \beta_{15} + \cdots + 114724 \beta_{9} ) / 184 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 115103 \beta_{10} - 14584 \beta_{8} + 17234 \beta_{7} - 71530 \beta_{6} + 225936 \beta_{5} + \cdots + 4796415 ) / 92 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 557658 \beta_{19} - 55775 \beta_{18} - 484656 \beta_{17} + 1540057 \beta_{16} + \cdots + 2904532 \beta_{9} ) / 92 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 12408343 \beta_{10} - 2515862 \beta_{8} + 1990962 \beta_{7} - 7511432 \beta_{6} + 21345876 \beta_{5} + \cdots + 473795509 ) / 184 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 32275187 \beta_{19} - 5066946 \beta_{18} - 29047942 \beta_{17} + 83234447 \beta_{16} + \cdots + 150502731 \beta_{9} ) / 92 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 334356698 \beta_{10} - 88180552 \beta_{8} + 59661141 \beta_{7} - 204726312 \beta_{6} + \cdots + 12067501503 ) / 92 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3618949720 \beta_{19} - 765073035 \beta_{18} - 3378955829 \beta_{17} + 9116948449 \beta_{16} + \cdots + 15781417340 \beta_{9} ) / 184 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 9040984673 \beta_{10} - 2841373252 \beta_{8} + 1810854266 \beta_{7} - 5768586346 \beta_{6} + \cdots + 314250459495 ) / 46 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 99981385296 \beta_{19} - 26288451358 \beta_{18} - 96945647910 \beta_{17} + \cdots + 416584376393 \beta_{9} ) / 92 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1965089438309 \beta_{10} - 702336885928 \beta_{8} + 440712323934 \beta_{7} - 1330356450538 \beta_{6} + \cdots + 66599045686397 ) / 184 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 5491493453137 \beta_{19} - 1709642554770 \beta_{18} - 5528816628134 \beta_{17} + \cdots + 22112651686719 \beta_{9} ) / 92 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 53671568646091 \beta_{10} - 21203483897909 \beta_{8} + 13371637612227 \beta_{7} + \cdots + 17\!\cdots\!77 ) / 92 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 602494458417098 \beta_{19} - 214667484560781 \beta_{18} - 629024711395465 \beta_{17} + \cdots + 23\!\cdots\!52 \beta_{9} ) / 184 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 29\!\cdots\!75 \beta_{10} + \cdots + 97\!\cdots\!95 ) / 92 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 16\!\cdots\!90 \beta_{19} + \cdots + 63\!\cdots\!32 \beta_{9} ) / 92 \) 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
−4.35977
4.35977
−2.72837
2.72837
−0.661407
0.661407
7.60721
−7.60721
5.90493
−5.90493
6.03655
−6.03655
6.89323
−6.89323
−2.93981
2.93981
−2.40601
2.40601
−7.07001
7.07001
0 −9.85887 0 −2.08503 0 2.78903 0 70.1973 0
1.2 0 −9.85887 0 2.08503 0 −2.78903 0 70.1973 0
1.3 0 −6.53889 0 −3.36225 0 6.13068 0 15.7571 0
1.4 0 −6.53889 0 3.36225 0 −6.13068 0 15.7571 0
1.5 0 −4.71337 0 −21.0356 0 30.8813 0 −4.78414 0
1.6 0 −4.71337 0 21.0356 0 −30.8813 0 −4.78414 0
1.7 0 −2.76847 0 −18.0392 0 −9.35214 0 −19.3356 0
1.8 0 −2.76847 0 18.0392 0 9.35214 0 −19.3356 0
1.9 0 −1.96921 0 −9.86992 0 −15.2432 0 −23.1222 0
1.10 0 −1.96921 0 9.86992 0 15.2432 0 −23.1222 0
1.11 0 1.87752 0 −18.1474 0 −9.78721 0 −23.4749 0
1.12 0 1.87752 0 18.1474 0 9.78721 0 −23.4749 0
1.13 0 3.41931 0 −4.72164 0 −19.8610 0 −15.3084 0
1.14 0 3.41931 0 4.72164 0 19.8610 0 −15.3084 0
1.15 0 5.16226 0 −8.87296 0 32.9157 0 −0.351111 0
1.16 0 5.16226 0 8.87296 0 −32.9157 0 −0.351111 0
1.17 0 7.68749 0 −7.34353 0 17.7485 0 32.0975 0
1.18 0 7.68749 0 7.34353 0 −17.7485 0 32.0975 0
1.19 0 7.70224 0 −5.89450 0 16.9842 0 32.3244 0
1.20 0 7.70224 0 5.89450 0 −16.9842 0 32.3244 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.20
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.h ✓ 20
23.b odd 2 1 inner 2116.4.a.h ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2116.4.a.h ✓ 20 1.a even 1 1 trivial
2116.4.a.h ✓ 20 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2116))\):

\( T_{3}^{10} - 167 T_{3}^{8} + 92 T_{3}^{7} + 9016 T_{3}^{6} - 5464 T_{3}^{5} - 189560 T_{3}^{4} + \cdots - 3250544 \) Copy content Toggle raw display
\( T_{5}^{20} - 1400 T_{5}^{18} + 764700 T_{5}^{16} - 208727480 T_{5}^{14} + 30826147294 T_{5}^{12} + \cdots + 74\!\cdots\!36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} - 167 T^{8} + \cdots - 3250544)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 74\!\cdots\!36 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 51\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots - 19488725599536)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 55\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 29\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 14\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 68\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 58\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 30\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 81\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 21\!\cdots\!39)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 66\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 77\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 89\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 39\!\cdots\!44 \) Copy content Toggle raw display
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