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Properties

Label 121.8.a.i
Level $121$
Weight $8$
Character orbit 121.a
Self dual yes
Analytic conductor $37.799$
Analytic rank $0$
Dimension $12$
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] = [121,8,Mod(1,121)] 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("121.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 121.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(37.7985880836\)
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: \( x^{12} - 6 x^{11} - 995 x^{10} + 4070 x^{9} + 370502 x^{8} - 918126 x^{7} - 61207003 x^{6} + \cdots + 7839497781 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 11^{5} \)
Twist minimal: no (minimal twist has level 11)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + ( - \beta_{5} - 1) q^{3} + (\beta_{3} + \beta_{2} - 3 \beta_1 + 46) q^{4} + (\beta_{8} + \beta_{5} - \beta_1 + 12) q^{5} + ( - \beta_{10} - \beta_{7} - 4 \beta_{5} + \cdots + 58) q^{6}+ \cdots + ( - 1001 \beta_{11} - 1225 \beta_{10} + \cdots - 2177789) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{2} - 12 q^{3} + 550 q^{4} + 144 q^{5} + 649 q^{6} + 2244 q^{7} + 3810 q^{8} + 9094 q^{9} + 2120 q^{10} + 5819 q^{12} + 8688 q^{13} + 23988 q^{14} - 29008 q^{15} - 32238 q^{16} + 26214 q^{17}+ \cdots - 25767018 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 995 x^{10} + 4070 x^{9} + 370502 x^{8} - 918126 x^{7} - 61207003 x^{6} + \cdots + 7839497781 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 37\!\cdots\!75 \nu^{11} + \cdots + 24\!\cdots\!47 ) / 92\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 37\!\cdots\!75 \nu^{11} + \cdots - 52\!\cdots\!07 ) / 84\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 65\!\cdots\!77 \nu^{11} + \cdots - 15\!\cdots\!07 ) / 92\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 37\!\cdots\!61 \nu^{11} + \cdots + 32\!\cdots\!10 ) / 92\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 27\!\cdots\!89 \nu^{11} + \cdots + 30\!\cdots\!39 ) / 41\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 65\!\cdots\!26 \nu^{11} + \cdots + 14\!\cdots\!25 ) / 31\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23\!\cdots\!99 \nu^{11} + \cdots + 62\!\cdots\!91 ) / 68\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 86\!\cdots\!80 \nu^{11} + \cdots + 19\!\cdots\!77 ) / 22\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 33\!\cdots\!74 \nu^{11} + \cdots - 83\!\cdots\!31 ) / 68\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 34\!\cdots\!29 \nu^{11} + \cdots - 30\!\cdots\!57 ) / 62\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!86 \nu^{11} + \cdots + 12\!\cdots\!07 ) / 13\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 11\beta _1 + 6 ) / 11 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{4} + 11\beta_{3} + 12\beta_{2} + 23\beta _1 + 1859 ) / 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{11} - 8 \beta_{9} - 11 \beta_{8} - 8 \beta_{7} + 3 \beta_{6} + 8 \beta_{5} - 6 \beta_{4} + \cdots + 5610 ) / 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 26 \beta_{11} + 22 \beta_{10} - 117 \beta_{9} - 119 \beta_{8} - 115 \beta_{7} + 134 \beta_{6} + \cdots + 500425 ) / 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1331 \beta_{11} + 473 \beta_{10} - 3960 \beta_{9} - 4107 \beta_{8} - 4239 \beta_{7} + 2589 \beta_{6} + \cdots + 3054090 ) / 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 15618 \beta_{11} + 20361 \beta_{10} - 63208 \beta_{9} - 68322 \beta_{8} - 63979 \beta_{7} + \cdots + 146888213 ) / 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 517896 \beta_{11} + 375375 \beta_{10} - 1604389 \beta_{9} - 1529749 \beta_{8} - 1723058 \beta_{7} + \cdots + 1353878039 ) / 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7148646 \beta_{11} + 11031020 \beta_{10} - 27193585 \beta_{9} - 29683055 \beta_{8} - 27652049 \beta_{7} + \cdots + 45788839442 ) / 11 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 193441537 \beta_{11} + 199836175 \beta_{10} - 613718014 \beta_{9} - 576949919 \beta_{8} + \cdots + 548296972098 ) / 11 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 269116550 \beta_{11} + 450853723 \beta_{10} - 985695396 \beta_{9} - 1067280418 \beta_{8} + \cdots + 1359599062680 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 71087790213 \beta_{11} + 90317921683 \beta_{10} - 229833613229 \beta_{9} - 217837577914 \beta_{8} + \cdots + 212007363132065 ) / 11 \) 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
19.1359
19.0846
15.8590
11.0625
2.60309
2.26024
−0.100216
−4.18079
−14.3300
−15.3713
−14.9182
−15.1049
−17.7540 −21.8235 187.203 −65.9636 387.453 −464.038 −1051.09 −1710.74 1171.11
1.2 −15.4666 32.1948 111.215 −143.040 −497.943 558.447 259.605 −1150.50 2212.34
1.3 −12.2409 −80.7053 21.8407 102.520 987.908 −743.309 1299.49 4326.34 −1254.94
1.4 −9.68052 76.2585 −34.2874 −11.9711 −738.222 1684.96 1571.03 3628.35 115.886
1.5 −1.22112 −61.1299 −126.509 419.570 74.6473 1011.58 310.787 1549.87 −512.347
1.6 1.35779 28.0947 −126.156 291.418 38.1468 −253.966 −345.091 −1397.69 395.685
1.7 1.48218 6.96323 −125.803 −211.474 10.3208 −1499.53 −376.182 −2138.51 −313.442
1.8 7.79882 −29.3525 −67.1783 −451.713 −228.915 −49.6496 −1522.16 −1325.43 −3522.83
1.9 15.7120 53.8279 118.867 331.685 845.744 −233.117 −143.503 710.446 5211.43
1.10 16.7532 −46.6798 152.671 −389.846 −782.037 1154.33 413.314 −7.99635 −6531.19
1.11 18.5362 87.7129 215.591 −216.330 1625.86 405.350 1623.60 5506.55 −4009.94
1.12 18.7229 −57.3611 222.547 489.146 −1073.97 672.943 1770.20 1103.30 9158.23
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

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

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 121.8.a.i 12
11.b odd 2 1 121.8.a.g 12
11.c even 5 2 11.8.c.a ✓ 24
33.h odd 10 2 99.8.f.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.8.c.a ✓ 24 11.c even 5 2
99.8.f.a 24 33.h odd 10 2
121.8.a.g 12 11.b odd 2 1
121.8.a.i 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 24 T_{2}^{11} - 755 T_{2}^{10} + 19410 T_{2}^{9} + 193952 T_{2}^{8} - 5536584 T_{2}^{7} + \cdots - 56970563584 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(121))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots - 56970563584 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 19\!\cdots\!61 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots - 26\!\cdots\!79 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 32\!\cdots\!05 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots - 17\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 43\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 52\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots - 78\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 36\!\cdots\!79 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 26\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 74\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 74\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 15\!\cdots\!95 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 72\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 72\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 96\!\cdots\!59 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 18\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 11\!\cdots\!99 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 22\!\cdots\!39 \) Copy content Toggle raw display
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