↑

Properties

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

Related objects

Downloads

Learn more

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: \( x^{8} - x^{7} + 101803 x^{6} + 6576400 x^{5} + 8539617914 x^{4} + 333205096780 x^{3} + \cdots + 33\!\cdots\!81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 7^{4} \)
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.

\(f(q)\) \(=\) \( q + 32 \beta_{2} q^{2} + (1024 \beta_{2} - 1024) q^{4} + (\beta_{4} - 1876 \beta_{2}) q^{5} + (\beta_{6} + \beta_{5} + 14 \beta_{3} + \cdots - 6789) q^{7}+ \cdots - 32768 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 32 \beta_{2} q^{2} + (1024 \beta_{2} - 1024) q^{4} + (\beta_{4} - 1876 \beta_{2}) q^{5} + (\beta_{6} + \beta_{5} + 14 \beta_{3} + \cdots - 6789) q^{7}+ \cdots + ( - 119392 \beta_{7} + \cdots - 4081690592) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 128 q^{2} - 4096 q^{4} - 7504 q^{5} - 42224 q^{7} - 262144 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 128 q^{2} - 4096 q^{4} - 7504 q^{5} - 42224 q^{7} - 262144 q^{8} + 240128 q^{10} - 213026 q^{11} - 2609712 q^{13} - 1257536 q^{14} - 4194304 q^{16} - 8854244 q^{17} + 7232806 q^{19} + 15368192 q^{20} - 13633664 q^{22} + 10649134 q^{23} - 119407256 q^{25} - 41755392 q^{26} + 2996224 q^{28} + 221414576 q^{29} + 486231270 q^{31} + 134217728 q^{32} - 566671616 q^{34} + 756166390 q^{35} + 463131040 q^{37} - 231449792 q^{38} + 245891072 q^{40} + 1617723408 q^{41} + 2926896352 q^{43} - 218138624 q^{44} - 340772288 q^{46} + 894091254 q^{47} - 11608958872 q^{49} - 7642064384 q^{50} + 1336172544 q^{52} + 1448863512 q^{53} - 25123348964 q^{55} + 1383596032 q^{56} + 3542633216 q^{58} - 14386900738 q^{59} + 10854402216 q^{61} + 31118801280 q^{62} + 8589934592 q^{64} - 5495584080 q^{65} + 19629545546 q^{67} - 9066745856 q^{68} + 13390987456 q^{70} - 1474804928 q^{71} + 21420158732 q^{73} - 14820193280 q^{74} - 14812786688 q^{76} - 28657880944 q^{77} - 60246238086 q^{79} - 7868514304 q^{80} + 25883574528 q^{82} + 139893582304 q^{83} + 109585554592 q^{85} + 46830341632 q^{86} + 6980435968 q^{88} - 126354105612 q^{89} + 44156589120 q^{91} - 21809426432 q^{92} - 28610920128 q^{94} - 368499346550 q^{95} - 538182884176 q^{97} - 210290620288 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 101803 x^{6} + 6576400 x^{5} + 8539617914 x^{4} + 333205096780 x^{3} + \cdots + 33\!\cdots\!81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13\!\cdots\!87 \nu^{7} + \cdots - 11\!\cdots\!07 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13\!\cdots\!87 \nu^{7} + \cdots - 12\!\cdots\!31 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22\!\cdots\!71 \nu^{7} + \cdots - 15\!\cdots\!89 ) / 61\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 42\!\cdots\!41 \nu^{7} + \cdots - 24\!\cdots\!95 ) / 28\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 70\!\cdots\!39 \nu^{7} + \cdots + 47\!\cdots\!29 ) / 13\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!91 \nu^{7} + \cdots + 59\!\cdots\!47 ) / 28\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\!\cdots\!43 \nu^{7} + \cdots + 29\!\cdots\!31 ) / 46\!\cdots\!92 \) 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
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 2\beta_{5} + 9\beta_{4} + 35\beta_{3} - 203588\beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3390 \beta_{7} + 7398 \beta_{6} + 1695 \beta_{5} + 7398 \beta_{4} + 266293 \beta_{3} + \cdots - 59969830 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 204169 \beta_{7} + 1834902 \beta_{6} - 204169 \beta_{5} + 26909097962 \beta_{2} - 11531303 \beta _1 - 26909302131 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 192590691 \beta_{7} - 385181382 \beta_{5} - 933450048 \beta_{4} - 20540418721 \beta_{3} + \cdots - 192590691 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9518249694 \beta_{7} - 40579191729 \beta_{6} - 4759124847 \beta_{5} - 40579191729 \beta_{4} + \cdots + 516527974429175 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 9279926448597 \beta_{7} - 49932096083460 \beta_{6} + 9279926448597 \beta_{5} + \cdots + 59\!\cdots\!99 ) / 12 \) Copy content Toggle raw display

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

Character values

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

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

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
−115.837 − 200.636i
152.619 + 264.344i
64.1153 + 111.051i
−100.397 − 173.893i
−115.837 + 200.636i
152.619 − 264.344i
64.1153 − 111.051i
−100.397 + 173.893i
16.0000 − 27.7128i 0 −512.000 − 886.810i −6622.39 + 11470.3i 0 4344.64 − 44254.4i −32768.0 0 211917. + 367050.i
37.2 16.0000 − 27.7128i 0 −512.000 − 886.810i −3667.11 + 6351.63i 0 −28136.3 + 34433.6i −32768.0 0 117348. + 203252.i
37.3 16.0000 − 27.7128i 0 −512.000 − 886.810i 3211.13 − 5561.84i 0 −9575.58 − 43423.9i −32768.0 0 −102756. − 177979.i
37.4 16.0000 − 27.7128i 0 −512.000 − 886.810i 3326.37 − 5761.45i 0 12255.3 + 42745.0i −32768.0 0 −106444. − 184366.i
109.1 16.0000 + 27.7128i 0 −512.000 + 886.810i −6622.39 − 11470.3i 0 4344.64 + 44254.4i −32768.0 0 211917. − 367050.i
109.2 16.0000 + 27.7128i 0 −512.000 + 886.810i −3667.11 − 6351.63i 0 −28136.3 − 34433.6i −32768.0 0 117348. − 203252.i
109.3 16.0000 + 27.7128i 0 −512.000 + 886.810i 3211.13 + 5561.84i 0 −9575.58 + 43423.9i −32768.0 0 −102756. + 177979.i
109.4 16.0000 + 27.7128i 0 −512.000 + 886.810i 3326.37 + 5761.45i 0 12255.3 − 42745.0i −32768.0 0 −106444. + 184366.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.e 8
3.b odd 2 1 14.12.c.a ✓ 8
7.c even 3 1 inner 126.12.g.e 8
12.b even 2 1 112.12.i.a 8
21.c even 2 1 98.12.c.l 8
21.g even 6 1 98.12.a.l 4
21.g even 6 1 98.12.c.l 8
21.h odd 6 1 14.12.c.a ✓ 8
21.h odd 6 1 98.12.a.j 4
84.n even 6 1 112.12.i.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.12.c.a ✓ 8 3.b odd 2 1
14.12.c.a ✓ 8 21.h odd 6 1
98.12.a.j 4 21.h odd 6 1
98.12.a.l 4 21.g even 6 1
98.12.c.l 8 21.c even 2 1
98.12.c.l 8 21.g even 6 1
112.12.i.a 8 12.b even 2 1
112.12.i.a 8 84.n even 6 1
126.12.g.e 8 1.a even 1 1 trivial
126.12.g.e 8 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 7504 T_{5}^{7} + 185514886 T_{5}^{6} - 187838634480 T_{5}^{5} + \cdots + 17\!\cdots\!25 \) 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
$5$ \( T^{8} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 84\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 48\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 25\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 37\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 52\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 67\!\cdots\!89 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 17\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 24\!\cdots\!81 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 31\!\cdots\!01 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 84\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 34\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 29\!\cdots\!96)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 73\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 47\!\cdots\!84)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 59\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less