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Properties

Label 121.8.a.h
Level $121$
Weight $8$
Character orbit 121.a
Self dual yes
Analytic conductor $37.799$
Analytic rank $0$
Dimension $12$
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] = [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 = [12,0] 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: \( x^{12} - 1178x^{10} + 491649x^{8} - 90964856x^{6} + 8088778192x^{4} - 339064394496x^{2} + 5382548481024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 11^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + 7) q^{3} + (\beta_{2} + 68) q^{4} + ( - \beta_{6} - \beta_{3} + 67) q^{5} + ( - \beta_{10} - \beta_{9} + \cdots + 8 \beta_1) q^{6} + ( - 2 \beta_{10} - \beta_{9} + \cdots + 26 \beta_1) q^{7}+ \cdots + (8302 \beta_{11} - 9324 \beta_{10} + \cdots + 484667 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 80 q^{3} + 820 q^{4} + 804 q^{5} + 6540 q^{9} + 10780 q^{12} + 64548 q^{14} + 32960 q^{15} + 100676 q^{16} + 4176 q^{20} + 240720 q^{23} - 5200 q^{25} + 208500 q^{26} - 248560 q^{27} + 929008 q^{31}+ \cdots - 7806660 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 1178x^{10} + 491649x^{8} - 90964856x^{6} + 8088778192x^{4} - 339064394496x^{2} + 5382548481024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 196 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 36973 \nu^{10} - 41703998 \nu^{8} + 16100229429 \nu^{6} - 2566210169252 \nu^{4} + \cdots - 40\!\cdots\!96 ) / 595157207040 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 139775 \nu^{10} - 163886794 \nu^{8} + 67239732375 \nu^{6} - 11762464048012 \nu^{4} + \cdots - 22\!\cdots\!08 ) / 595157207040 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 190577 \nu^{10} - 213789142 \nu^{8} + 81671279193 \nu^{6} - 12746766753076 \nu^{4} + \cdots - 18\!\cdots\!88 ) / 595157207040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6107 \nu^{10} - 6876754 \nu^{8} + 2645160147 \nu^{6} - 417970725148 \nu^{4} + \cdots - 629802299281920 ) / 6763150080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12187 \nu^{11} + 13697186 \nu^{9} - 5251688883 \nu^{7} + 825392913068 \nu^{5} + \cdots + 12\!\cdots\!16 \nu ) / 1723876577280 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23480267 \nu^{11} + 26259377938 \nu^{9} - 9980725385667 \nu^{7} + \cdots + 21\!\cdots\!68 \nu ) / 290568974192640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 40564223 \nu^{11} + 45691959466 \nu^{9} - 17581151442903 \nu^{7} + \cdots + 41\!\cdots\!60 \nu ) / 290568974192640 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 416611295 \nu^{11} - 469803215434 \nu^{9} + 181173898884087 \nu^{7} + \cdots - 43\!\cdots\!72 \nu ) / 26\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 571580627 \nu^{11} + 644333046946 \nu^{9} - 248306657786955 \nu^{7} + \cdots + 60\!\cdots\!40 \nu ) / 26\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 196 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{11} - 3\beta_{10} + 2\beta_{9} - \beta_{8} + 28\beta_{7} + 341\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 26\beta_{6} - 44\beta_{5} - 4\beta_{4} - 136\beta_{3} + 505\beta_{2} + 67270 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2440\beta_{11} - 1611\beta_{10} + 1242\beta_{9} - 885\beta_{8} + 24724\beta_{7} + 143037\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20714\beta_{6} - 40196\beta_{5} - 4156\beta_{4} - 78184\beta_{3} + 232281\beta_{2} + 28293318 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -1294584\beta_{11} - 738987\beta_{10} + 723010\beta_{9} - 574301\beta_{8} + 15658884\beta_{7} + 64154821\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12596802 \beta_{6} - 26580588 \beta_{5} - 3025668 \beta_{4} - 34651272 \beta_{3} + 107210681 \beta_{2} + 12699135374 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 663907064 \beta_{11} - 332728395 \beta_{10} + 403780906 \beta_{9} - 332860181 \beta_{8} + \cdots + 29629119373 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6993165754 \beta_{6} - 15532015636 \beta_{5} - 1880689676 \beta_{4} - 14462598824 \beta_{3} + \cdots + 5865388531574 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 335765128760 \beta_{11} - 151271803179 \beta_{10} + 217471812498 \beta_{9} + \cdots + 13926019877397 \beta_1 \) 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
−22.2131
−18.9481
−11.9558
−8.86084
−7.29884
−7.12875
7.12875
7.29884
8.86084
11.9558
18.9481
22.2131
−22.2131 12.9476 365.423 −177.844 −287.607 −878.904 −5273.90 −2019.36 3950.48
1.2 −18.9481 35.9043 231.031 301.406 −680.318 −1563.34 −1952.24 −897.883 −5711.07
1.3 −11.9558 −81.0918 14.9400 208.288 969.513 1059.07 1351.72 4388.87 −2490.24
1.4 −8.86084 −32.6618 −49.4856 22.9312 289.411 885.467 1572.67 −1120.21 −203.189
1.5 −7.29884 22.5310 −74.7270 −362.817 −164.450 479.616 1479.67 −1679.36 2648.14
1.6 −7.12875 82.3707 −77.1810 410.036 −587.200 −1001.17 1462.68 4597.93 −2923.04
1.7 7.12875 82.3707 −77.1810 410.036 587.200 1001.17 −1462.68 4597.93 2923.04
1.8 7.29884 22.5310 −74.7270 −362.817 164.450 −479.616 −1479.67 −1679.36 −2648.14
1.9 8.86084 −32.6618 −49.4856 22.9312 −289.411 −885.467 −1572.67 −1120.21 203.189
1.10 11.9558 −81.0918 14.9400 208.288 −969.513 −1059.07 −1351.72 4388.87 2490.24
1.11 18.9481 35.9043 231.031 301.406 680.318 1563.34 1952.24 −897.883 5711.07
1.12 22.2131 12.9476 365.423 −177.844 287.607 878.904 5273.90 −2019.36 −3950.48
\(n\): 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

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.8.a.h ✓ 12
11.b odd 2 1 inner 121.8.a.h ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.8.a.h ✓ 12 1.a even 1 1 trivial
121.8.a.h ✓ 12 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 1178 T_{2}^{10} + 491649 T_{2}^{8} - 90964856 T_{2}^{6} + 8088778192 T_{2}^{4} + \cdots + 5382548481024 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(121))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 5382548481024 \) Copy content Toggle raw display
$3$ \( (T^{6} - 40 T^{5} + \cdots + 2285099856)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots + 38088404947125)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 43\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 12\!\cdots\!89 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 27\!\cdots\!04)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 37\!\cdots\!29 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 14\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 13\!\cdots\!99)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 23\!\cdots\!49 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 31\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 24\!\cdots\!56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 74\!\cdots\!21)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 19\!\cdots\!96)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 27\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 31\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 71\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 21\!\cdots\!39)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 68\!\cdots\!51)^{2} \) Copy content Toggle raw display
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