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Properties

Label 441.3.q.b
Level $441$
Weight $3$
Character orbit 441.q
Analytic conductor $12.016$
Analytic rank $0$
Dimension $8$
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] = [441,3,Mod(116,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.116");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.q (of order \(6\), degree \(2\), not 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: \(12.0163796583\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 63)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{3}) q^{4} + 2 \beta_1 q^{5} + (\beta_{7} + 2 \beta_{5} + \cdots - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{3}) q^{4} + 2 \beta_1 q^{5} + (\beta_{7} + 2 \beta_{5} + \cdots - 2 \beta_1) q^{8}+ \cdots + (2 \beta_{3} + 40) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 32 q^{10} + 48 q^{13} + 36 q^{16} - 24 q^{19} + 152 q^{22} - 36 q^{25} + 152 q^{31} + 48 q^{34} + 128 q^{37} + 72 q^{40} + 160 q^{43} - 132 q^{46} + 112 q^{52} + 304 q^{55} - 148 q^{58} + 48 q^{61} - 64 q^{64} + 24 q^{67} - 16 q^{73} - 560 q^{76} + 88 q^{79} + 256 q^{82} + 96 q^{85} - 108 q^{88} - 216 q^{94} + 320 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 258\nu ) / 55 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 148 ) / 55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{6} + 55\nu^{4} - 440\nu^{2} + 576 ) / 495 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -8\nu^{7} + 55\nu^{5} - 440\nu^{3} + 576\nu ) / 495 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{6} + 220\nu^{4} - 1265\nu^{2} + 1656 ) / 495 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -13\nu^{7} + 110\nu^{5} - 715\nu^{3} + 936\nu ) / 165 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 4\beta_{4} + \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 6\beta_{5} + \beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{6} - 23\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{7} - 39\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -55\beta_{3} - 148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -55\beta_{2} - 258\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-\beta_{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−2.23256 + 1.28897i
−1.00781 + 0.581861i
1.00781 − 0.581861i
2.23256 − 1.28897i
−2.23256 − 1.28897i
−1.00781 − 0.581861i
1.00781 + 0.581861i
2.23256 + 1.28897i
−2.23256 + 1.28897i 0 1.32288 − 2.29129i −4.46512 + 2.57794i 0 0 − 3.49117i 0 6.64575 − 11.5108i
116.2 −1.00781 + 0.581861i 0 −1.32288 + 2.29129i −2.01563 + 1.16372i 0 0 − 7.73381i 0 1.35425 − 2.34563i
116.3 1.00781 − 0.581861i 0 −1.32288 + 2.29129i 2.01563 − 1.16372i 0 0 7.73381i 0 1.35425 − 2.34563i
116.4 2.23256 − 1.28897i 0 1.32288 − 2.29129i 4.46512 − 2.57794i 0 0 3.49117i 0 6.64575 − 11.5108i
422.1 −2.23256 − 1.28897i 0 1.32288 + 2.29129i −4.46512 − 2.57794i 0 0 3.49117i 0 6.64575 + 11.5108i
422.2 −1.00781 − 0.581861i 0 −1.32288 − 2.29129i −2.01563 − 1.16372i 0 0 7.73381i 0 1.35425 + 2.34563i
422.3 1.00781 + 0.581861i 0 −1.32288 − 2.29129i 2.01563 + 1.16372i 0 0 − 7.73381i 0 1.35425 + 2.34563i
422.4 2.23256 + 1.28897i 0 1.32288 + 2.29129i 4.46512 + 2.57794i 0 0 − 3.49117i 0 6.64575 + 11.5108i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.4
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 441.3.q.b 8
3.b odd 2 1 inner 441.3.q.b 8
7.b odd 2 1 441.3.q.a 8
7.c even 3 1 63.3.b.a ✓ 4
7.c even 3 1 inner 441.3.q.b 8
7.d odd 6 1 441.3.b.b 4
7.d odd 6 1 441.3.q.a 8
21.c even 2 1 441.3.q.a 8
21.g even 6 1 441.3.b.b 4
21.g even 6 1 441.3.q.a 8
21.h odd 6 1 63.3.b.a ✓ 4
21.h odd 6 1 inner 441.3.q.b 8
28.g odd 6 1 1008.3.d.d 4
35.j even 6 1 1575.3.c.a 4
35.l odd 12 2 1575.3.f.a 8
56.k odd 6 1 4032.3.d.c 4
56.p even 6 1 4032.3.d.b 4
63.g even 3 1 567.3.r.a 8
63.h even 3 1 567.3.r.a 8
63.j odd 6 1 567.3.r.a 8
63.n odd 6 1 567.3.r.a 8
84.n even 6 1 1008.3.d.d 4
105.o odd 6 1 1575.3.c.a 4
105.x even 12 2 1575.3.f.a 8
168.s odd 6 1 4032.3.d.b 4
168.v even 6 1 4032.3.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.b.a ✓ 4 7.c even 3 1
63.3.b.a ✓ 4 21.h odd 6 1
441.3.b.b 4 7.d odd 6 1
441.3.b.b 4 21.g even 6 1
441.3.q.a 8 7.b odd 2 1
441.3.q.a 8 7.d odd 6 1
441.3.q.a 8 21.c even 2 1
441.3.q.a 8 21.g even 6 1
441.3.q.b 8 1.a even 1 1 trivial
441.3.q.b 8 3.b odd 2 1 inner
441.3.q.b 8 7.c even 3 1 inner
441.3.q.b 8 21.h odd 6 1 inner
567.3.r.a 8 63.g even 3 1
567.3.r.a 8 63.h even 3 1
567.3.r.a 8 63.j odd 6 1
567.3.r.a 8 63.n odd 6 1
1008.3.d.d 4 28.g odd 6 1
1008.3.d.d 4 84.n even 6 1
1575.3.c.a 4 35.j even 6 1
1575.3.c.a 4 105.o odd 6 1
1575.3.f.a 8 35.l odd 12 2
1575.3.f.a 8 105.x even 12 2
4032.3.d.b 4 56.p even 6 1
4032.3.d.b 4 168.s odd 6 1
4032.3.d.c 4 56.k odd 6 1
4032.3.d.c 4 168.v even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{8} - 8T_{2}^{6} + 55T_{2}^{4} - 72T_{2}^{2} + 81 \) Copy content Toggle raw display
\( T_{13}^{2} - 12T_{13} - 76 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 8 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 32 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 212 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$13$ \( (T^{2} - 12 T - 76)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 11019960576 \) Copy content Toggle raw display
$19$ \( (T^{4} + 12 T^{3} + \cdots + 440896)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 13680577296 \) Copy content Toggle raw display
$29$ \( (T^{4} + 2228 T^{2} + 1004004)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 76 T^{3} + \cdots + 2005056)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 64 T^{3} + \cdots + 831744)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 4064 T^{2} + 1838736)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 40 T - 608)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 2176782336 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 15936095936016 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 19007367745536 \) Copy content Toggle raw display
$61$ \( (T^{4} - 24 T^{3} + \cdots + 99281296)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 12 T^{3} + \cdots + 29724304)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 5364 T^{2} + 2802276)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} + \cdots + 1838736)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 22 T + 484)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1152 T^{2} + 186624)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} - 80 T + 1572)^{4} \) Copy content Toggle raw display
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