Properties

Label 441.2.p.a
Level $441$
Weight $2$
Character orbit 441.p
Analytic conductor $3.521$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( \beta_{1} - 2 \beta_{3} ) q^{5} -2 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( \beta_{1} - 2 \beta_{3} ) q^{5} -2 \beta_{3} q^{8} + ( 4 - 2 \beta_{2} ) q^{10} + ( -\beta_{1} + \beta_{3} ) q^{11} + ( 3 - 6 \beta_{2} ) q^{13} + ( 4 - 4 \beta_{2} ) q^{16} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -1 - \beta_{2} ) q^{19} -2 q^{22} + 4 \beta_{1} q^{23} -\beta_{2} q^{25} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{26} + 2 \beta_{3} q^{29} + ( -2 + \beta_{2} ) q^{31} + ( -4 + 8 \beta_{2} ) q^{34} + ( 1 - \beta_{2} ) q^{37} + ( -\beta_{1} - \beta_{3} ) q^{38} + ( -4 - 4 \beta_{2} ) q^{40} + ( 6 \beta_{1} - 3 \beta_{3} ) q^{41} - q^{43} + 8 \beta_{2} q^{46} + ( -5 \beta_{1} + 10 \beta_{3} ) q^{47} -\beta_{3} q^{50} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{53} + ( -2 + 4 \beta_{2} ) q^{55} + ( -4 + 4 \beta_{2} ) q^{58} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{59} + ( 2 + 2 \beta_{2} ) q^{61} + ( -2 \beta_{1} + \beta_{3} ) q^{62} -8 q^{64} -9 \beta_{1} q^{65} -11 \beta_{2} q^{67} + 5 \beta_{3} q^{71} + ( -2 + \beta_{2} ) q^{73} + ( \beta_{1} - \beta_{3} ) q^{74} + ( -5 + 5 \beta_{2} ) q^{79} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{80} + ( 6 + 6 \beta_{2} ) q^{82} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{83} + 12 q^{85} -\beta_{1} q^{86} + 4 \beta_{2} q^{88} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{89} + ( -20 + 10 \beta_{2} ) q^{94} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{95} + ( 6 - 12 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 12q^{10} + 8q^{16} - 6q^{19} - 8q^{22} - 2q^{25} - 6q^{31} + 2q^{37} - 24q^{40} - 4q^{43} + 16q^{46} - 8q^{58} + 12q^{61} - 32q^{64} - 22q^{67} - 6q^{73} - 10q^{79} + 36q^{82} + 48q^{85} + 8q^{88} - 60q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i 0 0 −1.22474 2.12132i 0 0 2.82843i 0 3.00000 + 1.73205i
80.2 1.22474 0.707107i 0 0 1.22474 + 2.12132i 0 0 2.82843i 0 3.00000 + 1.73205i
215.1 −1.22474 0.707107i 0 0 −1.22474 + 2.12132i 0 0 2.82843i 0 3.00000 1.73205i
215.2 1.22474 + 0.707107i 0 0 1.22474 2.12132i 0 0 2.82843i 0 3.00000 1.73205i
\(n\): e.g. 2-40 or 990-1000
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 441.2.p.a 4
3.b odd 2 1 inner 441.2.p.a 4
7.b odd 2 1 63.2.p.a 4
7.c even 3 1 63.2.p.a 4
7.c even 3 1 441.2.c.a 4
7.d odd 6 1 441.2.c.a 4
7.d odd 6 1 inner 441.2.p.a 4
21.c even 2 1 63.2.p.a 4
21.g even 6 1 441.2.c.a 4
21.g even 6 1 inner 441.2.p.a 4
21.h odd 6 1 63.2.p.a 4
21.h odd 6 1 441.2.c.a 4
28.d even 2 1 1008.2.bt.b 4
28.f even 6 1 7056.2.k.b 4
28.g odd 6 1 1008.2.bt.b 4
28.g odd 6 1 7056.2.k.b 4
35.c odd 2 1 1575.2.bk.c 4
35.f even 4 2 1575.2.bc.a 8
35.j even 6 1 1575.2.bk.c 4
35.l odd 12 2 1575.2.bc.a 8
63.g even 3 1 567.2.i.d 4
63.h even 3 1 567.2.s.d 4
63.j odd 6 1 567.2.s.d 4
63.l odd 6 1 567.2.i.d 4
63.l odd 6 1 567.2.s.d 4
63.n odd 6 1 567.2.i.d 4
63.o even 6 1 567.2.i.d 4
63.o even 6 1 567.2.s.d 4
84.h odd 2 1 1008.2.bt.b 4
84.j odd 6 1 7056.2.k.b 4
84.n even 6 1 1008.2.bt.b 4
84.n even 6 1 7056.2.k.b 4
105.g even 2 1 1575.2.bk.c 4
105.k odd 4 2 1575.2.bc.a 8
105.o odd 6 1 1575.2.bk.c 4
105.x even 12 2 1575.2.bc.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.p.a 4 7.b odd 2 1
63.2.p.a 4 7.c even 3 1
63.2.p.a 4 21.c even 2 1
63.2.p.a 4 21.h odd 6 1
441.2.c.a 4 7.c even 3 1
441.2.c.a 4 7.d odd 6 1
441.2.c.a 4 21.g even 6 1
441.2.c.a 4 21.h odd 6 1
441.2.p.a 4 1.a even 1 1 trivial
441.2.p.a 4 3.b odd 2 1 inner
441.2.p.a 4 7.d odd 6 1 inner
441.2.p.a 4 21.g even 6 1 inner
567.2.i.d 4 63.g even 3 1
567.2.i.d 4 63.l odd 6 1
567.2.i.d 4 63.n odd 6 1
567.2.i.d 4 63.o even 6 1
567.2.s.d 4 63.h even 3 1
567.2.s.d 4 63.j odd 6 1
567.2.s.d 4 63.l odd 6 1
567.2.s.d 4 63.o even 6 1
1008.2.bt.b 4 28.d even 2 1
1008.2.bt.b 4 28.g odd 6 1
1008.2.bt.b 4 84.h odd 2 1
1008.2.bt.b 4 84.n even 6 1
1575.2.bc.a 8 35.f even 4 2
1575.2.bc.a 8 35.l odd 12 2
1575.2.bc.a 8 105.k odd 4 2
1575.2.bc.a 8 105.x even 12 2
1575.2.bk.c 4 35.c odd 2 1
1575.2.bk.c 4 35.j even 6 1
1575.2.bk.c 4 105.g even 2 1
1575.2.bk.c 4 105.o odd 6 1
7056.2.k.b 4 28.f even 6 1
7056.2.k.b 4 28.g odd 6 1
7056.2.k.b 4 84.j odd 6 1
7056.2.k.b 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2 T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 36 + 6 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 4 - 2 T^{2} + T^{4} \)
$13$ \( ( 27 + T^{2} )^{2} \)
$17$ \( 576 + 24 T^{2} + T^{4} \)
$19$ \( ( 3 + 3 T + T^{2} )^{2} \)
$23$ \( 1024 - 32 T^{2} + T^{4} \)
$29$ \( ( 8 + T^{2} )^{2} \)
$31$ \( ( 3 + 3 T + T^{2} )^{2} \)
$37$ \( ( 1 - T + T^{2} )^{2} \)
$41$ \( ( -54 + T^{2} )^{2} \)
$43$ \( ( 1 + T )^{4} \)
$47$ \( 22500 + 150 T^{2} + T^{4} \)
$53$ \( 64 - 8 T^{2} + T^{4} \)
$59$ \( 576 + 24 T^{2} + T^{4} \)
$61$ \( ( 12 - 6 T + T^{2} )^{2} \)
$67$ \( ( 121 + 11 T + T^{2} )^{2} \)
$71$ \( ( 50 + T^{2} )^{2} \)
$73$ \( ( 3 + 3 T + T^{2} )^{2} \)
$79$ \( ( 25 + 5 T + T^{2} )^{2} \)
$83$ \( ( -54 + T^{2} )^{2} \)
$89$ \( 576 + 24 T^{2} + T^{4} \)
$97$ \( ( 108 + T^{2} )^{2} \)
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