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Properties

Label 126.9.s.b
Level $126$
Weight $9$
Character orbit 126.s
Analytic conductor $51.330$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,9,Mod(53,126)]
 
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([3, 4]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.53");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 126.s (of order \(6\), 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: \(51.3297048677\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 10 x^{19} + 3140415 x^{18} - 28263450 x^{17} + 4166681580501 x^{16} - 33332812007100 x^{15} + \cdots + 75\!\cdots\!79 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{18}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 \beta_{2} q^{2} + (128 \beta_{6} + 128) q^{4} + ( - \beta_{4} - 93 \beta_{2}) q^{5} + ( - 3 \beta_{17} - 2 \beta_{14} + \cdots + 313) q^{7}+ \cdots + ( - 1024 \beta_{3} - 1024 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 \beta_{2} q^{2} + (128 \beta_{6} + 128) q^{4} + ( - \beta_{4} - 93 \beta_{2}) q^{5} + ( - 3 \beta_{17} - 2 \beta_{14} + \cdots + 313) q^{7}+ \cdots + (3920 \beta_{15} + \cdots + 9938568 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 1280 q^{4} + 3778 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 1280 q^{4} + 3778 q^{7} + 14816 q^{10} + 17700 q^{13} - 163840 q^{16} - 267794 q^{19} + 453568 q^{22} - 765890 q^{25} - 232448 q^{28} - 2604342 q^{31} + 4087936 q^{34} + 1127530 q^{37} - 1896448 q^{40} - 4460924 q^{43} - 180416 q^{46} - 25278034 q^{49} + 1132800 q^{52} - 18833120 q^{55} + 5146496 q^{58} - 25730232 q^{61} - 41943040 q^{64} + 58134374 q^{67} - 76724864 q^{70} + 5811002 q^{73} - 68555264 q^{76} + 74799798 q^{79} - 9883296 q^{82} + 119739328 q^{85} + 29028352 q^{88} - 479039766 q^{91} + 169364448 q^{94} - 658133608 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 10 x^{19} + 3140415 x^{18} - 28263450 x^{17} + 4166681580501 x^{16} - 33332812007100 x^{15} + \cdots + 75\!\cdots\!79 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 38\!\cdots\!44 \nu^{18} + \cdots + 91\!\cdots\!65 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 21\!\cdots\!49 \nu^{19} + \cdots - 38\!\cdots\!08 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21\!\cdots\!49 \nu^{19} + \cdots + 38\!\cdots\!79 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29\!\cdots\!19 \nu^{19} + \cdots + 62\!\cdots\!48 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 59\!\cdots\!38 \nu^{19} + \cdots + 43\!\cdots\!01 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!40 \nu^{19} + \cdots - 60\!\cdots\!75 ) / 78\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22\!\cdots\!57 \nu^{19} + \cdots + 86\!\cdots\!81 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!83 \nu^{19} + \cdots + 17\!\cdots\!35 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21\!\cdots\!68 \nu^{19} + \cdots + 20\!\cdots\!43 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 84\!\cdots\!61 \nu^{19} + \cdots - 27\!\cdots\!84 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15\!\cdots\!96 \nu^{19} + \cdots - 26\!\cdots\!03 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 28\!\cdots\!62 \nu^{19} + \cdots + 21\!\cdots\!18 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 36\!\cdots\!54 \nu^{19} + \cdots - 29\!\cdots\!24 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 36\!\cdots\!54 \nu^{19} + \cdots - 96\!\cdots\!59 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 30\!\cdots\!11 \nu^{19} + \cdots + 15\!\cdots\!62 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 80\!\cdots\!22 \nu^{19} + \cdots - 16\!\cdots\!07 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 15\!\cdots\!54 \nu^{19} + \cdots + 27\!\cdots\!26 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 19\!\cdots\!56 \nu^{19} + \cdots + 82\!\cdots\!19 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 26\!\cdots\!53 \nu^{19} + \cdots + 11\!\cdots\!88 ) / 50\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{6} + \beta_{5} - 93\beta_{3} - 93\beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - 9 \beta_{18} + 28 \beta_{17} - 19 \beta_{16} - 22 \beta_{14} - 8 \beta_{13} + \beta_{12} + \cdots - 313980 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 24 \beta_{19} - 57 \beta_{18} - 66 \beta_{17} - 30 \beta_{16} - 2217 \beta_{15} - 150 \beta_{14} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 48 \beta_{19} - 357999 \beta_{18} - 3055676 \beta_{17} + 3413369 \beta_{16} - 8868 \beta_{15} + \cdots + 152842079429 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 20791420 \beta_{19} + 17066665 \beta_{18} + 35777110 \beta_{17} + 18856460 \beta_{16} + \cdots + 6864636 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 62374380 \beta_{19} + 4773113528292 \beta_{18} - 6726235124852 \beta_{17} + 1953336698030 \beta_{16} + \cdots - 81\!\cdots\!71 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15680842838136 \beta_{19} + 13673331829640 \beta_{18} - 43533288460928 \beta_{17} + \cdots - 34703575575774 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 62723662433096 \beta_{19} + \cdots + 44\!\cdots\!98 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 11\!\cdots\!64 \beta_{19} + \cdots + 46\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 57\!\cdots\!40 \beta_{19} + \cdots - 25\!\cdots\!17 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 82\!\cdots\!48 \beta_{19} + \cdots - 46\!\cdots\!06 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 49\!\cdots\!64 \beta_{19} + \cdots + 14\!\cdots\!13 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 58\!\cdots\!40 \beta_{19} + \cdots + 41\!\cdots\!28 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 40\!\cdots\!24 \beta_{19} + \cdots - 84\!\cdots\!00 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 40\!\cdots\!04 \beta_{19} + \cdots - 34\!\cdots\!56 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 32\!\cdots\!84 \beta_{19} + \cdots + 49\!\cdots\!41 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 28\!\cdots\!16 \beta_{19} + \cdots + 27\!\cdots\!96 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 25\!\cdots\!00 \beta_{19} + \cdots - 28\!\cdots\!63 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 19\!\cdots\!44 \beta_{19} + \cdots - 20\!\cdots\!74 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.500000 + 780.566i
0.500000 + 725.430i
0.500000 + 50.5839i
0.500000 − 325.242i
0.500000 − 572.227i
0.500000 + 573.959i
0.500000 + 326.974i
0.500000 − 48.8519i
0.500000 − 723.698i
0.500000 − 778.834i
0.500000 − 780.566i
0.500000 − 725.430i
0.500000 − 50.5839i
0.500000 + 325.242i
0.500000 + 572.227i
0.500000 − 573.959i
0.500000 − 326.974i
0.500000 + 48.8519i
0.500000 + 723.698i
0.500000 + 778.834i
−9.79796 + 5.65685i 0 64.0000 − 110.851i −675.240 + 389.850i 0 −2257.29 + 818.185i 1448.15i 0 4410.65 − 7639.47i
53.2 −9.79796 + 5.65685i 0 64.0000 − 110.851i −627.491 + 362.282i 0 539.994 − 2339.49i 1448.15i 0 4098.76 − 7099.25i
53.3 −9.79796 + 5.65685i 0 64.0000 − 110.851i −43.0570 + 24.8590i 0 547.483 + 2337.75i 1448.15i 0 281.247 − 487.134i
53.4 −9.79796 + 5.65685i 0 64.0000 − 110.851i 282.418 − 163.054i 0 −233.264 − 2389.64i 1448.15i 0 −1844.75 + 3195.20i
53.5 −9.79796 + 5.65685i 0 64.0000 − 110.851i 496.313 − 286.547i 0 2347.58 + 503.657i 1448.15i 0 −3241.90 + 5615.14i
53.6 9.79796 − 5.65685i 0 64.0000 − 110.851i −496.313 + 286.547i 0 2347.58 + 503.657i − 1448.15i 0 −3241.90 + 5615.14i
53.7 9.79796 − 5.65685i 0 64.0000 − 110.851i −282.418 + 163.054i 0 −233.264 − 2389.64i − 1448.15i 0 −1844.75 + 3195.20i
53.8 9.79796 − 5.65685i 0 64.0000 − 110.851i 43.0570 − 24.8590i 0 547.483 + 2337.75i − 1448.15i 0 281.247 − 487.134i
53.9 9.79796 − 5.65685i 0 64.0000 − 110.851i 627.491 − 362.282i 0 539.994 − 2339.49i − 1448.15i 0 4098.76 − 7099.25i
53.10 9.79796 − 5.65685i 0 64.0000 − 110.851i 675.240 − 389.850i 0 −2257.29 + 818.185i − 1448.15i 0 4410.65 − 7639.47i
107.1 −9.79796 − 5.65685i 0 64.0000 + 110.851i −675.240 − 389.850i 0 −2257.29 − 818.185i − 1448.15i 0 4410.65 + 7639.47i
107.2 −9.79796 − 5.65685i 0 64.0000 + 110.851i −627.491 − 362.282i 0 539.994 + 2339.49i − 1448.15i 0 4098.76 + 7099.25i
107.3 −9.79796 − 5.65685i 0 64.0000 + 110.851i −43.0570 − 24.8590i 0 547.483 − 2337.75i − 1448.15i 0 281.247 + 487.134i
107.4 −9.79796 − 5.65685i 0 64.0000 + 110.851i 282.418 + 163.054i 0 −233.264 + 2389.64i − 1448.15i 0 −1844.75 − 3195.20i
107.5 −9.79796 − 5.65685i 0 64.0000 + 110.851i 496.313 + 286.547i 0 2347.58 − 503.657i − 1448.15i 0 −3241.90 − 5615.14i
107.6 9.79796 + 5.65685i 0 64.0000 + 110.851i −496.313 − 286.547i 0 2347.58 − 503.657i 1448.15i 0 −3241.90 − 5615.14i
107.7 9.79796 + 5.65685i 0 64.0000 + 110.851i −282.418 − 163.054i 0 −233.264 + 2389.64i 1448.15i 0 −1844.75 − 3195.20i
107.8 9.79796 + 5.65685i 0 64.0000 + 110.851i 43.0570 + 24.8590i 0 547.483 − 2337.75i 1448.15i 0 281.247 + 487.134i
107.9 9.79796 + 5.65685i 0 64.0000 + 110.851i 627.491 + 362.282i 0 539.994 + 2339.49i 1448.15i 0 4098.76 + 7099.25i
107.10 9.79796 + 5.65685i 0 64.0000 + 110.851i 675.240 + 389.850i 0 −2257.29 − 818.185i 1448.15i 0 4410.65 + 7639.47i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.9.s.b ✓ 20
3.b odd 2 1 inner 126.9.s.b ✓ 20
7.c even 3 1 inner 126.9.s.b ✓ 20
21.h odd 6 1 inner 126.9.s.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.9.s.b ✓ 20 1.a even 1 1 trivial
126.9.s.b ✓ 20 3.b odd 2 1 inner
126.9.s.b ✓ 20 7.c even 3 1 inner
126.9.s.b ✓ 20 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} - 1570180 T_{5}^{18} + 1614925361487 T_{5}^{16} + \cdots + 75\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 128 T^{2} + 16384)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 63\!\cdots\!01)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots - 38\!\cdots\!90)^{4} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 32\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 11\!\cdots\!01)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 45\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 10\!\cdots\!10)^{4} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 47\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 46\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 13\!\cdots\!49)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 75\!\cdots\!72)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 14\!\cdots\!12)^{4} \) Copy content Toggle raw display
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