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Properties

Label 3.21.b.a
Level $3$
Weight $21$
Character orbit 3.b
Analytic conductor $7.605$
Analytic rank $0$
Dimension $6$
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] = [3,21,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 21, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 21);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 21 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), 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: \(7.60541295308\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 116898x^{4} + 3059043456x^{2} + 18153107947520 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{14}\cdot 3^{19}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 11 \beta_1 - 687) q^{3} + (\beta_{3} + 3 \beta_{2} + \cdots - 354200) q^{4}+ \cdots + ( - 204 \beta_{5} + 117 \beta_{4} + \cdots - 181247247) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 11 \beta_1 - 687) q^{3} + (\beta_{3} + 3 \beta_{2} + \cdots - 354200) q^{4}+ \cdots + ( - 1784433170439 \beta_{5} + \cdots - 29\!\cdots\!80) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4122 q^{3} - 2125200 q^{4} - 96401232 q^{6} + 559607916 q^{7} - 1087483482 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4122 q^{3} - 2125200 q^{4} - 96401232 q^{6} + 559607916 q^{7} - 1087483482 q^{9} - 10880304480 q^{10} + 63957886128 q^{12} + 17847879276 q^{13} - 487764624960 q^{15} - 317472843648 q^{16} + 9564155313120 q^{18} - 14958020195124 q^{19} + 25978999888044 q^{21} - 71051286019680 q^{22} + 294048705803904 q^{24} - 397204775017050 q^{25} + 10\!\cdots\!98 q^{27}+ \cdots - 17\!\cdots\!80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 116898x^{4} + 3059043456x^{2} + 18153107947520 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -17\nu^{5} - 5728\nu^{4} - 2170562\nu^{3} - 526987456\nu^{2} - 55742610560\nu - 6124748591104 ) / 64336896 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\nu^{5} + 5728\nu^{4} + 2170562\nu^{3} + 1299030208\nu^{2} + 55871284352\nu + 36208166465536 ) / 21445632 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 373\nu^{5} + 2864\nu^{4} + 36271114\nu^{3} + 263493728\nu^{2} + 493897505920\nu + 3062374295552 ) / 1787136 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1343 \nu^{5} + 475424 \nu^{4} - 171474398 \nu^{3} + 45284044352 \nu^{2} + \cdots + 568520968810496 ) / 21445632 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3\beta_{2} - \beta _1 - 1402776 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -27\beta_{5} - 17\beta_{4} + 54\beta_{3} - 6867\beta_{2} - 1158985\beta_1 ) / 108 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 416\beta_{5} - 46833\beta_{3} - 239091\beta_{2} + 91761\beta _1 + 45282332952 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1303179\beta_{5} + 1085281\beta_{4} - 2606358\beta_{3} + 336207555\beta_{2} + 44432621465\beta_1 ) / 54 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
− 287.188i
− 161.041i
− 92.1239i
92.1239i
161.041i
287.188i
− 1723.13i −25942.2 − 53045.1i −1.92059e6 − 9.20415e6i −9.14036e7 + 4.47017e7i 4.08861e8 1.50260e9i −2.14079e9 + 2.75221e9i −1.58599e10
2.2 − 966.247i 56662.8 + 16616.6i 114943. 1.69009e7i 1.60557e7 − 5.47503e7i 2.16509e8 − 1.12425e9i 2.93456e9 + 1.88308e9i 1.63305e10
2.3 − 552.743i −32781.6 + 49113.6i 743051. − 1.06933e7i 2.71472e7 + 1.81198e7i −3.45566e8 − 9.90310e8i −1.33751e9 − 3.22005e9i −5.91067e9
2.4 552.743i −32781.6 − 49113.6i 743051. 1.06933e7i 2.71472e7 − 1.81198e7i −3.45566e8 9.90310e8i −1.33751e9 + 3.22005e9i −5.91067e9
2.5 966.247i 56662.8 − 16616.6i 114943. − 1.69009e7i 1.60557e7 + 5.47503e7i 2.16509e8 1.12425e9i 2.93456e9 − 1.88308e9i 1.63305e10
2.6 1723.13i −25942.2 + 53045.1i −1.92059e6 9.20415e6i −9.14036e7 − 4.47017e7i 4.08861e8 − 1.50260e9i −2.14079e9 − 2.75221e9i −1.58599e10
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.21.b.a ✓ 6
3.b odd 2 1 inner 3.21.b.a ✓ 6
4.b odd 2 1 48.21.e.c 6
12.b even 2 1 48.21.e.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.21.b.a ✓ 6 1.a even 1 1 trivial
3.21.b.a ✓ 6 3.b odd 2 1 inner
48.21.e.c 6 4.b odd 2 1
48.21.e.c 6 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{21}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + \cdots + 84\!\cdots\!20 \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{3} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{3} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 21\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots - 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 24\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots - 58\!\cdots\!88)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{3} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 15\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 96\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 70\!\cdots\!48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 24\!\cdots\!12)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 26\!\cdots\!80 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
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