↑

Properties

Label 3.23.b.a
Level $3$
Weight $23$
Character orbit 3.b
Analytic conductor $9.201$
Analytic rank $0$
Dimension $6$
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] = [3,23,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, 23, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 23);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 23 \)
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: \(9.20122304526\)
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: \( x^{6} + 126474x^{4} + 3861674040x^{2} + 9831214131200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{21}\cdot 3^{22} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 8 \beta_1 + 14445) q^{3} + (\beta_{4} - 6 \beta_{2} - 1876448) q^{4} + (\beta_{3} - 6 \beta_{2} + 5130 \beta_1) q^{5} + ( - \beta_{5} - 26 \beta_{4} + \cdots + 48929184) q^{6}+ \cdots + ( - 1188 \beta_{5} - 4644 \beta_{4} + \cdots + 9556619193) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 8 \beta_1 + 14445) q^{3} + (\beta_{4} - 6 \beta_{2} - 1876448) q^{4} + (\beta_{3} - 6 \beta_{2} + 5130 \beta_1) q^{5} + ( - \beta_{5} - 26 \beta_{4} + \cdots + 48929184) q^{6}+ \cdots + (233297081440461 \beta_{5} + \cdots - 10\!\cdots\!60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 86670 q^{3} - 11258688 q^{4} + 293575104 q^{6} - 3447063060 q^{7} + 57339715158 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 86670 q^{3} - 11258688 q^{4} + 293575104 q^{6} - 3447063060 q^{7} + 57339715158 q^{9} - 186869600640 q^{10} - 1325972980800 q^{12} + 2025132496860 q^{13} + 2628031314240 q^{15} - 9703467227136 q^{16} + 42127150135680 q^{18} + 100485688668636 q^{19} - 789079193287812 q^{21} + 10\!\cdots\!00 q^{22}+ \cdots - 60\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 126474x^{4} + 3861674040x^{2} + 9831214131200 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 12\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 65\nu^{5} + 12256\nu^{4} + 8098250\nu^{3} + 909321664\nu^{2} + 227790065720\nu + 4539979294720 ) / 61954080 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1369\nu^{5} + 6128\nu^{4} + 101935546\nu^{3} + 454660832\nu^{2} + 309897772600\nu + 2269989647360 ) / 5162840 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 65\nu^{5} + 12256\nu^{4} + 8098250\nu^{3} + 2396219584\nu^{2} + 227790065720\nu + 67224621806080 ) / 10325680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1625 \nu^{5} + 189968 \nu^{4} - 202456250 \nu^{3} + 16324832672 \nu^{2} + \cdots + 164396642835200 ) / 15488520 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - 6\beta_{2} - 6070752 ) / 144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -99\beta_{5} + 99\beta_{4} - 65\beta_{3} + 5934\beta_{2} - 4708940\beta_1 ) / 864 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6740\beta_{5} - 118031\beta_{4} + 1382186\beta_{2} + 40440\beta _1 + 595607370912 ) / 216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3625423\beta_{5} - 3625423\beta_{4} + 4049125\beta_{3} - 227318438\beta_{2} + 167163893020\beta_1 ) / 432 \) 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
− 281.771i
− 210.435i
− 52.8797i
52.8797i
210.435i
281.771i
− 3381.25i 169408. + 51788.6i −7.23853e6 − 6.36810e7i 1.75110e8 − 5.72810e8i −2.60308e9 1.02933e10i 2.60170e10 + 1.75468e10i −2.15321e11
2.2 − 2525.22i −174303. + 31617.4i −2.18245e6 4.89124e7i 7.98410e7 + 4.40153e8i −1.80726e7 − 5.08037e9i 2.93817e10 − 1.10220e10i 1.23515e11
2.3 − 634.557i 48229.8 − 170455.i 3.79164e6 − 2.56634e6i −1.08163e8 − 3.06045e7i 8.97619e8 − 5.06753e9i −2.67288e10 − 1.64420e10i −1.62849e9
2.4 634.557i 48229.8 + 170455.i 3.79164e6 2.56634e6i −1.08163e8 + 3.06045e7i 8.97619e8 5.06753e9i −2.67288e10 + 1.64420e10i −1.62849e9
2.5 2525.22i −174303. − 31617.4i −2.18245e6 − 4.89124e7i 7.98410e7 − 4.40153e8i −1.80726e7 5.08037e9i 2.93817e10 + 1.10220e10i 1.23515e11
2.6 3381.25i 169408. − 51788.6i −7.23853e6 6.36810e7i 1.75110e8 + 5.72810e8i −2.60308e9 − 1.02933e10i 2.60170e10 − 1.75468e10i −2.15321e11
\(n\): 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 3.23.b.a ✓ 6
3.b odd 2 1 inner 3.23.b.a ✓ 6
4.b odd 2 1 48.23.e.b 6
12.b even 2 1 48.23.e.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.23.b.a ✓ 6 1.a even 1 1 trivial
3.23.b.a ✓ 6 3.b odd 2 1 inner
48.23.e.b 6 4.b odd 2 1
48.23.e.b 6 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{23}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots + 30\!\cdots\!29 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{3} + \cdots - 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{3} + \cdots - 23\!\cdots\!80)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots + 73\!\cdots\!92)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots - 66\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots + 17\!\cdots\!20)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{3} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 56\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + \cdots - 38\!\cdots\!60)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots + 76\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 16\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots - 29\!\cdots\!40)^{2} \) Copy content Toggle raw display
show more
show less