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Properties

Label 126.9.n.d
Level $126$
Weight $9$
Character orbit 126.n
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(19,126)]
 
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, 5]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.19");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 126.n (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: \( x^{20} - 8 x^{19} - 26382 x^{18} + 177344 x^{17} + 298653216 x^{16} - 1823810808 x^{15} + \cdots + 42\!\cdots\!32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{18}\cdot 7^{12} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (128 \beta_1 - 128) q^{4} + ( - \beta_{12} + \beta_{11} + \cdots - 5 \beta_{2}) q^{5}+ \cdots + (128 \beta_{3} - 128 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (128 \beta_1 - 128) q^{4} + ( - \beta_{12} + \beta_{11} + \cdots - 5 \beta_{2}) q^{5}+ \cdots + ( - 602 \beta_{19} + \cdots - 1250641 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 1280 q^{4} + 4186 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 1280 q^{4} + 4186 q^{7} - 17664 q^{10} - 163840 q^{16} - 250890 q^{19} + 420864 q^{22} + 258962 q^{25} - 1189888 q^{28} + 342762 q^{31} - 4806598 q^{37} + 2260992 q^{40} + 6968252 q^{43} - 4357632 q^{46} - 26046538 q^{49} + 2075904 q^{52} + 2455296 q^{58} - 15410424 q^{61} + 41943040 q^{64} - 70041074 q^{67} - 25804800 q^{70} + 220264098 q^{73} + 12860578 q^{79} + 12085248 q^{82} + 29161632 q^{85} - 26935296 q^{88} - 311022894 q^{91} + 332230656 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{19} - 26382 x^{18} + 177344 x^{17} + 298653216 x^{16} - 1823810808 x^{15} + \cdots + 42\!\cdots\!32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13\!\cdots\!18 \nu^{19} + \cdots + 18\!\cdots\!92 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 68\!\cdots\!09 \nu^{19} + \cdots + 33\!\cdots\!64 ) / 95\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 96\!\cdots\!17 \nu^{19} + \cdots + 23\!\cdots\!32 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 56\!\cdots\!67 \nu^{19} + \cdots + 95\!\cdots\!08 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 63\!\cdots\!87 \nu^{19} + \cdots + 88\!\cdots\!28 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\!\cdots\!90 \nu^{19} + \cdots - 20\!\cdots\!28 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 47\!\cdots\!46 \nu^{19} + \cdots - 48\!\cdots\!92 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 54\!\cdots\!61 \nu^{19} + \cdots - 43\!\cdots\!20 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23\!\cdots\!29 \nu^{19} + \cdots + 71\!\cdots\!68 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 54\!\cdots\!83 \nu^{19} + \cdots - 18\!\cdots\!08 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\!\cdots\!86 \nu^{19} + \cdots - 52\!\cdots\!68 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 22\!\cdots\!10 \nu^{19} + \cdots + 65\!\cdots\!72 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 28\!\cdots\!45 \nu^{19} + \cdots - 18\!\cdots\!56 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 33\!\cdots\!45 \nu^{19} + \cdots + 24\!\cdots\!84 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 89\!\cdots\!93 \nu^{19} + \cdots + 24\!\cdots\!08 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 29\!\cdots\!52 \nu^{19} + \cdots + 17\!\cdots\!56 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 30\!\cdots\!90 \nu^{19} + \cdots + 23\!\cdots\!84 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 99\!\cdots\!72 \nu^{19} + \cdots - 11\!\cdots\!40 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 24\!\cdots\!35 \nu^{19} + \cdots + 59\!\cdots\!60 ) / 63\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} - \beta_{13} + 4\beta_{8} - 4\beta_{7} - 14\beta_{5} + 14\beta_{4} - 126\beta_{2} + 398 ) / 1008 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{19} + \beta_{15} - 5 \beta_{14} - 5 \beta_{13} + 28 \beta_{10} + 28 \beta_{9} + \cdots + 332715 ) / 126 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16 \beta_{19} + 82 \beta_{17} - 82 \beta_{16} - 16 \beta_{15} - 1675 \beta_{14} - 1669 \beta_{13} + \cdots + 1706020 ) / 336 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 7522 \beta_{19} - 1140 \beta_{17} + 1140 \beta_{16} + 7510 \beta_{15} - 18450 \beta_{14} + \cdots + 1248623035 ) / 126 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 235760 \beta_{19} - 2700 \beta_{18} + 1087830 \beta_{17} - 1082910 \beta_{16} + 236720 \beta_{15} + \cdots + 82160942238 ) / 1008 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 46482399 \beta_{19} - 30900 \beta_{18} - 11269770 \beta_{17} + 11235570 \beta_{16} + \cdots + 5125264972995 ) / 126 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4849687696 \beta_{19} - 30273810 \beta_{18} + 4528730178 \beta_{17} - 4483130862 \beta_{16} + \cdots + 645119109051556 ) / 1008 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 268640003668 \beta_{19} - 574364280 \beta_{18} - 74014415160 \beta_{17} + 73384137960 \beta_{16} + \cdots + 22\!\cdots\!31 ) / 126 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 45463978287120 \beta_{19} - 275514462720 \beta_{18} + 18457946828070 \beta_{17} + \cdots + 40\!\cdots\!54 ) / 1008 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 15\!\cdots\!37 \beta_{19} - 6599826691920 \beta_{18} - 421785264094830 \beta_{17} + \cdots + 98\!\cdots\!27 ) / 126 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 11\!\cdots\!00 \beta_{19} - 801253145693550 \beta_{18} + \cdots + 77\!\cdots\!36 ) / 336 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 82\!\cdots\!06 \beta_{19} + \cdots + 44\!\cdots\!67 ) / 126 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 22\!\cdots\!84 \beta_{19} + \cdots + 12\!\cdots\!50 ) / 1008 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 44\!\cdots\!59 \beta_{19} + \cdots + 20\!\cdots\!59 ) / 126 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 14\!\cdots\!24 \beta_{19} + \cdots + 65\!\cdots\!56 ) / 1008 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 23\!\cdots\!72 \beta_{19} + \cdots + 95\!\cdots\!23 ) / 126 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 83\!\cdots\!80 \beta_{19} + \cdots + 33\!\cdots\!18 ) / 1008 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 12\!\cdots\!97 \beta_{19} + \cdots + 44\!\cdots\!43 ) / 126 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 16\!\cdots\!76 \beta_{19} + \cdots + 56\!\cdots\!64 ) / 336 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
42.0992 − 1.22474i
−49.5297 − 1.22474i
5.49836 − 1.22474i
−63.1006 − 1.22474i
70.5683 − 1.22474i
69.1541 + 1.22474i
−64.5148 + 1.22474i
4.08415 + 1.22474i
−50.9439 + 1.22474i
40.6850 + 1.22474i
42.0992 + 1.22474i
−49.5297 + 1.22474i
5.49836 + 1.22474i
−63.1006 + 1.22474i
70.5683 + 1.22474i
69.1541 − 1.22474i
−64.5148 − 1.22474i
4.08415 − 1.22474i
−50.9439 − 1.22474i
40.6850 − 1.22474i
−5.65685 + 9.79796i 0 −64.0000 − 110.851i −565.865 − 326.703i 0 2120.69 − 1125.82i 1448.15 0 6402.04 − 3696.22i
19.2 −5.65685 + 9.79796i 0 −64.0000 − 110.851i −268.774 − 155.177i 0 −1576.44 + 1810.98i 1448.15 0 3040.83 − 1755.63i
19.3 −5.65685 + 9.79796i 0 −64.0000 − 110.851i −254.021 − 146.659i 0 −1180.12 − 2090.96i 1448.15 0 2873.92 − 1659.26i
19.4 −5.65685 + 9.79796i 0 −64.0000 − 110.851i 664.367 + 383.572i 0 1667.57 + 1727.43i 1448.15 0 −7516.45 + 4339.62i
19.5 −5.65685 + 9.79796i 0 −64.0000 − 110.851i 814.617 + 470.320i 0 14.7985 − 2400.95i 1448.15 0 −9216.34 + 5321.06i
19.6 5.65685 − 9.79796i 0 −64.0000 − 110.851i −814.617 − 470.320i 0 14.7985 − 2400.95i −1448.15 0 −9216.34 + 5321.06i
19.7 5.65685 − 9.79796i 0 −64.0000 − 110.851i −664.367 − 383.572i 0 1667.57 + 1727.43i −1448.15 0 −7516.45 + 4339.62i
19.8 5.65685 − 9.79796i 0 −64.0000 − 110.851i 254.021 + 146.659i 0 −1180.12 − 2090.96i −1448.15 0 2873.92 − 1659.26i
19.9 5.65685 − 9.79796i 0 −64.0000 − 110.851i 268.774 + 155.177i 0 −1576.44 + 1810.98i −1448.15 0 3040.83 − 1755.63i
19.10 5.65685 − 9.79796i 0 −64.0000 − 110.851i 565.865 + 326.703i 0 2120.69 − 1125.82i −1448.15 0 6402.04 − 3696.22i
73.1 −5.65685 − 9.79796i 0 −64.0000 + 110.851i −565.865 + 326.703i 0 2120.69 + 1125.82i 1448.15 0 6402.04 + 3696.22i
73.2 −5.65685 − 9.79796i 0 −64.0000 + 110.851i −268.774 + 155.177i 0 −1576.44 − 1810.98i 1448.15 0 3040.83 + 1755.63i
73.3 −5.65685 − 9.79796i 0 −64.0000 + 110.851i −254.021 + 146.659i 0 −1180.12 + 2090.96i 1448.15 0 2873.92 + 1659.26i
73.4 −5.65685 − 9.79796i 0 −64.0000 + 110.851i 664.367 − 383.572i 0 1667.57 − 1727.43i 1448.15 0 −7516.45 − 4339.62i
73.5 −5.65685 − 9.79796i 0 −64.0000 + 110.851i 814.617 − 470.320i 0 14.7985 + 2400.95i 1448.15 0 −9216.34 − 5321.06i
73.6 5.65685 + 9.79796i 0 −64.0000 + 110.851i −814.617 + 470.320i 0 14.7985 + 2400.95i −1448.15 0 −9216.34 − 5321.06i
73.7 5.65685 + 9.79796i 0 −64.0000 + 110.851i −664.367 + 383.572i 0 1667.57 − 1727.43i −1448.15 0 −7516.45 − 4339.62i
73.8 5.65685 + 9.79796i 0 −64.0000 + 110.851i 254.021 − 146.659i 0 −1180.12 + 2090.96i −1448.15 0 2873.92 + 1659.26i
73.9 5.65685 + 9.79796i 0 −64.0000 + 110.851i 268.774 − 155.177i 0 −1576.44 − 1810.98i −1448.15 0 3040.83 + 1755.63i
73.10 5.65685 + 9.79796i 0 −64.0000 + 110.851i 565.865 − 326.703i 0 2120.69 + 1125.82i −1448.15 0 6402.04 + 3696.22i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.10
Significant digits:
Format:

Inner twists

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

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} - 2082606 T_{5}^{18} + 2832711417027 T_{5}^{16} + \cdots + 33\!\cdots\!00 \) 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
$5$ \( T^{20} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 63\!\cdots\!01)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 68\!\cdots\!52)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 18\!\cdots\!48)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 38\!\cdots\!27)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 20\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 81\!\cdots\!20)^{4} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 76\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 16\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 38\!\cdots\!29)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 18\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 54\!\cdots\!48)^{2} \) Copy content Toggle raw display
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