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Properties

Label 126.12.g.c
Level $126$
Weight $12$
Character orbit 126.g
Analytic conductor $96.811$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,12,Mod(37,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.37");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 126.g (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(96.8112407505\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 149344 x^{6} + 5578711 x^{5} + 20557200983 x^{4} + 408905884576 x^{3} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 32 \beta_{2} - 32) q^{2} + 1024 \beta_{2} q^{4} + (\beta_{4} - 952 \beta_{2} - 952) q^{5} + ( - \beta_{7} + \beta_{6} + \cdots + 14756) q^{7}+ \cdots + 32768 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 32 \beta_{2} - 32) q^{2} + 1024 \beta_{2} q^{4} + (\beta_{4} - 952 \beta_{2} - 952) q^{5} + ( - \beta_{7} + \beta_{6} + \cdots + 14756) q^{7}+ \cdots + ( - 278368 \beta_{7} + \cdots - 139337888) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{2} - 4096 q^{4} - 3808 q^{5} + 110328 q^{7} + 262144 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 128 q^{2} - 4096 q^{4} - 3808 q^{5} + 110328 q^{7} + 262144 q^{8} - 121856 q^{10} - 920150 q^{11} + 997472 q^{13} - 1396800 q^{14} - 4194304 q^{16} - 1333724 q^{17} - 21551726 q^{19} + 7798784 q^{20} + 58889600 q^{22} - 72510158 q^{23} - 77154744 q^{25} - 15959552 q^{26} - 68278272 q^{28} - 213575104 q^{29} - 194359774 q^{31} - 134217728 q^{32} + 85358336 q^{34} + 981719074 q^{35} + 171517048 q^{37} - 689655232 q^{38} - 124780544 q^{40} + 3312095136 q^{41} + 850279648 q^{43} - 942233600 q^{44} - 2320325056 q^{46} - 2223880974 q^{47} - 4232044312 q^{49} + 4937903616 q^{50} - 510705664 q^{52} + 7185483360 q^{53} + 624915620 q^{55} + 3615227904 q^{56} + 3417201664 q^{58} - 6997401502 q^{59} - 6476463280 q^{61} + 12439025536 q^{62} + 8589934592 q^{64} - 30625528248 q^{65} - 18660972186 q^{67} - 1365733376 q^{68} - 27279047488 q^{70} - 23224449248 q^{71} + 3731641452 q^{73} + 5488545536 q^{74} + 44137934848 q^{76} + 29345595440 q^{77} + 12221157926 q^{79} - 3992977408 q^{80} - 52993522176 q^{82} - 316739523968 q^{83} - 147794862544 q^{85} - 13604474368 q^{86} - 30151475200 q^{88} + 71204406084 q^{89} + 189816116528 q^{91} + 148500803584 q^{92} - 71164191168 q^{94} + 215091896614 q^{95} + 688251797184 q^{97} - 137152279424 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 149344 x^{6} + 5578711 x^{5} + 20557200983 x^{4} + 408905884576 x^{3} + \cdots + 30\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 35\!\cdots\!25 \nu^{7} + \cdots - 55\!\cdots\!48 ) / 52\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35\!\cdots\!25 \nu^{7} + \cdots + 26\!\cdots\!44 ) / 52\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!57 \nu^{7} + \cdots - 65\!\cdots\!44 ) / 52\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 76\!\cdots\!75 \nu^{7} + \cdots + 72\!\cdots\!68 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 20\!\cdots\!37 \nu^{7} + \cdots + 17\!\cdots\!60 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 72\!\cdots\!25 \nu^{7} + \cdots + 34\!\cdots\!39 ) / 19\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!41 \nu^{7} + \cdots + 21\!\cdots\!76 ) / 77\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{7} - 5\beta_{6} - 26\beta_{5} + \beta_{4} + 16\beta_{3} + 298676\beta_{2} + \beta _1 - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6585 \beta_{7} - 1317 \beta_{6} + 11220 \beta_{5} - 9903 \beta_{4} + 501493 \beta_{3} - 1317 \beta_{2} + \cdots - 51299770 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 299125 \beta_{7} + 1196500 \beta_{6} + 299125 \beta_{5} + 7463410 \beta_{4} + 299125 \beta_{3} + \cdots - 75283730943 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 427740420 \beta_{7} + 534675525 \beta_{6} - 516082710 \beta_{5} - 106935105 \beta_{4} + \cdots + 106935105 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 212242146285 \beta_{7} + 42448429257 \beta_{6} + 1016868977160 \beta_{5} - 1059317406417 \beta_{4} + \cdots + 10\!\cdots\!54 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 32207374766007 \beta_{7} - 128829499064028 \beta_{6} - 32207374766007 \beta_{5} + \cdots + 24\!\cdots\!93 ) / 24 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−178.705 + 309.527i
190.275 − 329.566i
51.3434 − 88.9294i
−62.4133 + 108.103i
−178.705 − 309.527i
190.275 + 329.566i
51.3434 + 88.9294i
−62.4133 − 108.103i
−16.0000 + 27.7128i 0 −512.000 − 886.810i −4992.28 + 8646.88i 0 24330.9 − 37220.1i 32768.0 0 −159753. − 276700.i
37.2 −16.0000 + 27.7128i 0 −512.000 − 886.810i −2646.67 + 4584.17i 0 22119.1 + 38575.5i 32768.0 0 −84693.5 − 146693.i
37.3 −16.0000 + 27.7128i 0 −512.000 − 886.810i −274.515 + 475.474i 0 −25454.3 − 36461.0i 32768.0 0 −8784.49 − 15215.2i
37.4 −16.0000 + 27.7128i 0 −512.000 − 886.810i 6009.47 − 10408.7i 0 34168.3 + 28457.9i 32768.0 0 192303. + 333079.i
109.1 −16.0000 − 27.7128i 0 −512.000 + 886.810i −4992.28 − 8646.88i 0 24330.9 + 37220.1i 32768.0 0 −159753. + 276700.i
109.2 −16.0000 − 27.7128i 0 −512.000 + 886.810i −2646.67 − 4584.17i 0 22119.1 − 38575.5i 32768.0 0 −84693.5 + 146693.i
109.3 −16.0000 − 27.7128i 0 −512.000 + 886.810i −274.515 − 475.474i 0 −25454.3 + 36461.0i 32768.0 0 −8784.49 + 15215.2i
109.4 −16.0000 − 27.7128i 0 −512.000 + 886.810i 6009.47 + 10408.7i 0 34168.3 − 28457.9i 32768.0 0 192303. − 333079.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.12.g.c 8
3.b odd 2 1 14.12.c.b ✓ 8
7.c even 3 1 inner 126.12.g.c 8
12.b even 2 1 112.12.i.b 8
21.c even 2 1 98.12.c.n 8
21.g even 6 1 98.12.a.h 4
21.g even 6 1 98.12.c.n 8
21.h odd 6 1 14.12.c.b ✓ 8
21.h odd 6 1 98.12.a.i 4
84.n even 6 1 112.12.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.12.c.b ✓ 8 3.b odd 2 1
14.12.c.b ✓ 8 21.h odd 6 1
98.12.a.h 4 21.g even 6 1
98.12.a.i 4 21.h odd 6 1
98.12.c.n 8 21.c even 2 1
98.12.c.n 8 21.g even 6 1
112.12.i.b 8 12.b even 2 1
112.12.i.b 8 84.n even 6 1
126.12.g.c 8 1.a even 1 1 trivial
126.12.g.c 8 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 3808 T_{5}^{7} + 143484054 T_{5}^{6} + 922871052880 T_{5}^{5} + \cdots + 12\!\cdots\!25 \) acting on \(S_{12}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32 T + 1024)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 87\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 24\!\cdots\!40)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 87\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 40\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 20\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 45\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 25\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 13\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 42\!\cdots\!40)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 51\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 23\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 87\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
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