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Properties

Label 441.2.p.c
Level $441$
Weight $2$
Character orbit 441.p
Analytic conductor $3.521$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,2,Mod(80,441)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.80");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.p (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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{3} + \beta_1) q^{2} + (2 \beta_{8} - 2 \beta_{4} - \beta_{2} + 1) q^{4} + ( - \beta_{13} + \beta_{10}) q^{5} + ( - \beta_{6} + \beta_{5} - 3 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{3} + \beta_1) q^{2} + (2 \beta_{8} - 2 \beta_{4} - \beta_{2} + 1) q^{4} + ( - \beta_{13} + \beta_{10}) q^{5} + ( - \beta_{6} + \beta_{5} - 3 \beta_{3}) q^{8} + ( - \beta_{9} - \beta_{7}) q^{10} + ( - 2 \beta_{6} + 2 \beta_1) q^{11} + ( - \beta_{12} + \beta_{9}) q^{13} - 3 \beta_{2} q^{16} + (\beta_{15} + \beta_{14} + \beta_{10}) q^{17} + 2 \beta_{12} q^{19} + (2 \beta_{15} - \beta_{13}) q^{20} - 2 q^{22} + ( - 2 \beta_{3} + 2 \beta_1) q^{23} + ( - \beta_{8} + \beta_{4} + 5 \beta_{2} - 5) q^{25} + (\beta_{14} + \beta_{13} - \beta_{10}) q^{26} + ( - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{3}) q^{29} + 2 \beta_{9} q^{31} + ( - \beta_{6} + 3 \beta_1) q^{32} + ( - \beta_{12} + \beta_{9}) q^{34} + ( - \beta_{4} + 4 \beta_{2}) q^{37} + ( - 2 \beta_{15} - 2 \beta_{14} + 2 \beta_{10}) q^{38} + ( - 3 \beta_{12} + \beta_{11} - \beta_{7}) q^{40} + ( - \beta_{15} + \beta_{13}) q^{41} + ( - 6 \beta_{8} - 4) q^{43} + (2 \beta_{5} + 6 \beta_{3} - 6 \beta_1) q^{44} + (2 \beta_{8} - 2 \beta_{4} - 2 \beta_{2} + 2) q^{46} + ( - 2 \beta_{13} + 2 \beta_{10}) q^{47} + (6 \beta_{6} - 6 \beta_{5} + 7 \beta_{3}) q^{50} + (\beta_{9} + 2 \beta_{7}) q^{52} + 5 \beta_{6} q^{53} + (2 \beta_{12} + 2 \beta_{11} - 2 \beta_{9}) q^{55} + ( - 4 \beta_{4} - 6 \beta_{2}) q^{58} - 2 \beta_{10} q^{59} + ( - \beta_{12} + \beta_{11} - \beta_{7}) q^{61} + ( - 2 \beta_{15} + 2 \beta_{13}) q^{62} + (2 \beta_{8} + 7) q^{64} + (\beta_{5} - 10 \beta_{3} + 10 \beta_1) q^{65} + ( - 6 \beta_{8} + 6 \beta_{4}) q^{67} + ( - \beta_{14} + 3 \beta_{13} - 3 \beta_{10}) q^{68} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3}) q^{71} + ( - \beta_{9} + \beta_{7}) q^{73} + (3 \beta_{6} + 2 \beta_1) q^{74} + (2 \beta_{12} - 4 \beta_{11} - 2 \beta_{9}) q^{76} + (4 \beta_{4} + 4 \beta_{2}) q^{79} - 3 \beta_{10} q^{80} + (3 \beta_{12} - 2 \beta_{11} + 2 \beta_{7}) q^{82} - 4 \beta_{13} q^{83} + (9 \beta_{8} - 8) q^{85} + ( - 10 \beta_{5} + 16 \beta_{3} - 16 \beta_1) q^{86} + ( - 4 \beta_{8} + 4 \beta_{4} + \cdots + 2) q^{88}+ \cdots + (\beta_{12} + \beta_{11} - \beta_{9}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 24 q^{16} - 32 q^{22} - 40 q^{25} + 32 q^{37} - 64 q^{43} + 16 q^{46} - 48 q^{58} + 112 q^{64} + 32 q^{79} - 128 q^{85} + 16 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{48}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{48}^{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{48}^{12} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{48}^{14} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{48}^{14} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{48}^{14} + \zeta_{48}^{10} + \zeta_{48}^{6} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{48}^{13} + \zeta_{48}^{11} + 3\zeta_{48}^{7} - \zeta_{48}^{3} + 3\zeta_{48} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\zeta_{48}^{10} + \zeta_{48}^{6} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 2\zeta_{48}^{13} - 2\zeta_{48}^{11} + \zeta_{48}^{7} + 2\zeta_{48}^{3} + \zeta_{48} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\zeta_{48}^{13} - \zeta_{48}^{11} - 2\zeta_{48}^{7} + \zeta_{48}^{3} + 2\zeta_{48} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( 3\zeta_{48}^{15} - \zeta_{48}^{13} + 3\zeta_{48}^{9} + \zeta_{48}^{5} - \zeta_{48}^{3} \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( -\zeta_{48}^{15} - 2\zeta_{48}^{11} - \zeta_{48}^{9} + \zeta_{48}^{7} + 2\zeta_{48}^{5} + \zeta_{48} \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( -2\zeta_{48}^{15} - \zeta_{48}^{13} + 2\zeta_{48}^{9} + \zeta_{48}^{5} + \zeta_{48}^{3} \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( \zeta_{48}^{15} + 3\zeta_{48}^{11} - \zeta_{48}^{9} - \zeta_{48}^{7} + 3\zeta_{48}^{5} + \zeta_{48} \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( -\zeta_{48}^{15} + 3\zeta_{48}^{13} + \zeta_{48}^{9} - 3\zeta_{48}^{5} - 3\zeta_{48}^{3} \) Copy content Toggle raw display

Display \(\zeta_{48}^j\) in terms of \(\beta_i\)

\(\zeta_{48}\)\(=\) \( ( \beta_{15} + \beta_{14} + 3\beta_{10} + \beta_{9} + 2\beta_{7} ) / 14 \) Copy content Toggle raw display
\(\zeta_{48}^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{3}\)\(=\) \( ( -2\beta_{15} + \beta_{13} - 3\beta_{12} - \beta_{11} + 3\beta_{9} ) / 14 \) Copy content Toggle raw display
\(\zeta_{48}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{48}^{5}\)\(=\) \( ( 2\beta_{14} + \beta_{13} + 3\beta_{12} + \beta_{11} - \beta_{10} - \beta_{7} ) / 14 \) Copy content Toggle raw display
\(\zeta_{48}^{6}\)\(=\) \( ( \beta_{8} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{7}\)\(=\) \( ( -\beta_{15} - \beta_{14} - 3\beta_{10} + \beta_{9} + 2\beta_{7} ) / 14 \) Copy content Toggle raw display
\(\zeta_{48}^{8}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{48}^{9}\)\(=\) \( ( \beta_{15} + 3\beta_{13} - \beta_{12} + 2\beta_{11} + \beta_{9} ) / 14 \) Copy content Toggle raw display
\(\zeta_{48}^{10}\)\(=\) \( ( -\beta_{8} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{11}\)\(=\) \( ( 2\beta_{14} + \beta_{13} - 3\beta_{12} - \beta_{11} - \beta_{10} + \beta_{7} ) / 14 \) Copy content Toggle raw display
\(\zeta_{48}^{12}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{48}^{13}\)\(=\) \( ( 2\beta_{15} + 2\beta_{14} - \beta_{10} + 3\beta_{9} - \beta_{7} ) / 14 \) Copy content Toggle raw display
\(\zeta_{48}^{14}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{15}\)\(=\) \( ( -\beta_{15} - 3\beta_{13} - \beta_{12} + 2\beta_{11} + \beta_{9} ) / 14 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{48}^j\)

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
−0.608761 − 0.793353i
0.608761 + 0.793353i
0.130526 − 0.991445i
−0.130526 + 0.991445i
0.793353 − 0.608761i
−0.793353 + 0.608761i
0.991445 + 0.130526i
−0.991445 − 0.130526i
−0.608761 + 0.793353i
0.608761 − 0.793353i
0.130526 + 0.991445i
−0.130526 − 0.991445i
0.793353 + 0.608761i
−0.793353 − 0.608761i
0.991445 − 0.130526i
−0.991445 + 0.130526i
−2.09077 + 1.20711i 0 1.91421 − 3.31552i −1.68925 − 2.92586i 0 0 4.41421i 0 7.06365 + 4.07820i
80.2 −2.09077 + 1.20711i 0 1.91421 − 3.31552i 1.68925 + 2.92586i 0 0 4.41421i 0 −7.06365 − 4.07820i
80.3 −0.358719 + 0.207107i 0 −0.914214 + 1.58346i −1.46508 − 2.53759i 0 0 − 1.58579i 0 1.05110 + 0.606854i
80.4 −0.358719 + 0.207107i 0 −0.914214 + 1.58346i 1.46508 + 2.53759i 0 0 − 1.58579i 0 −1.05110 − 0.606854i
80.5 0.358719 − 0.207107i 0 −0.914214 + 1.58346i −1.46508 − 2.53759i 0 0 1.58579i 0 −1.05110 − 0.606854i
80.6 0.358719 − 0.207107i 0 −0.914214 + 1.58346i 1.46508 + 2.53759i 0 0 1.58579i 0 1.05110 + 0.606854i
80.7 2.09077 − 1.20711i 0 1.91421 − 3.31552i −1.68925 − 2.92586i 0 0 − 4.41421i 0 −7.06365 − 4.07820i
80.8 2.09077 − 1.20711i 0 1.91421 − 3.31552i 1.68925 + 2.92586i 0 0 − 4.41421i 0 7.06365 + 4.07820i
215.1 −2.09077 − 1.20711i 0 1.91421 + 3.31552i −1.68925 + 2.92586i 0 0 − 4.41421i 0 7.06365 − 4.07820i
215.2 −2.09077 − 1.20711i 0 1.91421 + 3.31552i 1.68925 − 2.92586i 0 0 − 4.41421i 0 −7.06365 + 4.07820i
215.3 −0.358719 − 0.207107i 0 −0.914214 − 1.58346i −1.46508 + 2.53759i 0 0 1.58579i 0 1.05110 − 0.606854i
215.4 −0.358719 − 0.207107i 0 −0.914214 − 1.58346i 1.46508 − 2.53759i 0 0 1.58579i 0 −1.05110 + 0.606854i
215.5 0.358719 + 0.207107i 0 −0.914214 − 1.58346i −1.46508 + 2.53759i 0 0 − 1.58579i 0 −1.05110 + 0.606854i
215.6 0.358719 + 0.207107i 0 −0.914214 − 1.58346i 1.46508 − 2.53759i 0 0 − 1.58579i 0 1.05110 − 0.606854i
215.7 2.09077 + 1.20711i 0 1.91421 + 3.31552i −1.68925 + 2.92586i 0 0 4.41421i 0 −7.06365 + 4.07820i
215.8 2.09077 + 1.20711i 0 1.91421 + 3.31552i 1.68925 − 2.92586i 0 0 4.41421i 0 7.06365 − 4.07820i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 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 441.2.p.c 16
3.b odd 2 1 inner 441.2.p.c 16
7.b odd 2 1 inner 441.2.p.c 16
7.c even 3 1 441.2.c.b ✓ 8
7.c even 3 1 inner 441.2.p.c 16
7.d odd 6 1 441.2.c.b ✓ 8
7.d odd 6 1 inner 441.2.p.c 16
21.c even 2 1 inner 441.2.p.c 16
21.g even 6 1 441.2.c.b ✓ 8
21.g even 6 1 inner 441.2.p.c 16
21.h odd 6 1 441.2.c.b ✓ 8
21.h odd 6 1 inner 441.2.p.c 16
28.f even 6 1 7056.2.k.g 8
28.g odd 6 1 7056.2.k.g 8
84.j odd 6 1 7056.2.k.g 8
84.n even 6 1 7056.2.k.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.c.b ✓ 8 7.c even 3 1
441.2.c.b ✓ 8 7.d odd 6 1
441.2.c.b ✓ 8 21.g even 6 1
441.2.c.b ✓ 8 21.h odd 6 1
441.2.p.c 16 1.a even 1 1 trivial
441.2.p.c 16 3.b odd 2 1 inner
441.2.p.c 16 7.b odd 2 1 inner
441.2.p.c 16 7.c even 3 1 inner
441.2.p.c 16 7.d odd 6 1 inner
441.2.p.c 16 21.c even 2 1 inner
441.2.p.c 16 21.g even 6 1 inner
441.2.p.c 16 21.h odd 6 1 inner
7056.2.k.g 8 28.f even 6 1
7056.2.k.g 8 28.g odd 6 1
7056.2.k.g 8 84.j odd 6 1
7056.2.k.g 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 6T_{2}^{6} + 35T_{2}^{4} - 6T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 6 T^{6} + 35 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 20 T^{6} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 24 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 20 T^{2} + 98)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 52 T^{6} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 80 T^{6} + \cdots + 2458624)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{2} + 16)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 24 T^{2} + 16)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 80 T^{6} + \cdots + 2458624)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} + \cdots + 196)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 68 T^{2} + 98)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 56)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + 80 T^{6} + \cdots + 2458624)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 50 T^{2} + 2500)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 80 T^{6} + \cdots + 2458624)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 68 T^{6} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 72 T^{2} + 5184)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 24 T^{2} + 16)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 52 T^{6} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots + 256)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 320 T^{2} + 25088)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 164 T^{6} + \cdots + 23059204)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 52 T^{2} + 98)^{4} \) Copy content Toggle raw display
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