Properties

Label 441.2.c.a
Level $441$
Weight $2$
Character orbit 441.c
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + \beta_{3} q^{5} -2 \beta_{1} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + \beta_{3} q^{5} -2 \beta_{1} q^{8} -2 \beta_{2} q^{10} -\beta_{1} q^{11} + 3 \beta_{2} q^{13} -4 q^{16} + 2 \beta_{3} q^{17} -\beta_{2} q^{19} -2 q^{22} -4 \beta_{1} q^{23} + q^{25} + 3 \beta_{3} q^{26} + 2 \beta_{1} q^{29} + \beta_{2} q^{31} -4 \beta_{2} q^{34} - q^{37} -\beta_{3} q^{38} -4 \beta_{2} q^{40} -3 \beta_{3} q^{41} - q^{43} -8 q^{46} -5 \beta_{3} q^{47} -\beta_{1} q^{50} + 2 \beta_{1} q^{53} -2 \beta_{2} q^{55} + 4 q^{58} + 2 \beta_{3} q^{59} + 2 \beta_{2} q^{61} + \beta_{3} q^{62} -8 q^{64} + 9 \beta_{1} q^{65} + 11 q^{67} + 5 \beta_{1} q^{71} + \beta_{2} q^{73} + \beta_{1} q^{74} + 5 q^{79} -4 \beta_{3} q^{80} + 6 \beta_{2} q^{82} + 3 \beta_{3} q^{83} + 12 q^{85} + \beta_{1} q^{86} -4 q^{88} -2 \beta_{3} q^{89} + 10 \beta_{2} q^{94} -3 \beta_{1} q^{95} + 6 \beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 16q^{16} - 8q^{22} + 4q^{25} - 4q^{37} - 4q^{43} - 32q^{46} + 16q^{58} - 32q^{64} + 44q^{67} + 20q^{79} + 48q^{85} - 16q^{88} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{2}\)\(=\)\( \nu^{2} - 1 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{1}\)

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
440.1
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
1.41421i 0 0 −2.44949 0 0 2.82843i 0 3.46410i
440.2 1.41421i 0 0 2.44949 0 0 2.82843i 0 3.46410i
440.3 1.41421i 0 0 −2.44949 0 0 2.82843i 0 3.46410i
440.4 1.41421i 0 0 2.44949 0 0 2.82843i 0 3.46410i
\(n\): 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.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.c.a 4
3.b odd 2 1 inner 441.2.c.a 4
4.b odd 2 1 7056.2.k.b 4
7.b odd 2 1 inner 441.2.c.a 4
7.c even 3 1 63.2.p.a 4
7.c even 3 1 441.2.p.a 4
7.d odd 6 1 63.2.p.a 4
7.d odd 6 1 441.2.p.a 4
12.b even 2 1 7056.2.k.b 4
21.c even 2 1 inner 441.2.c.a 4
21.g even 6 1 63.2.p.a 4
21.g even 6 1 441.2.p.a 4
21.h odd 6 1 63.2.p.a 4
21.h odd 6 1 441.2.p.a 4
28.d even 2 1 7056.2.k.b 4
28.f even 6 1 1008.2.bt.b 4
28.g odd 6 1 1008.2.bt.b 4
35.i odd 6 1 1575.2.bk.c 4
35.j even 6 1 1575.2.bk.c 4
35.k even 12 2 1575.2.bc.a 8
35.l odd 12 2 1575.2.bc.a 8
63.g even 3 1 567.2.s.d 4
63.h even 3 1 567.2.i.d 4
63.i even 6 1 567.2.i.d 4
63.j odd 6 1 567.2.i.d 4
63.k odd 6 1 567.2.s.d 4
63.n odd 6 1 567.2.s.d 4
63.s even 6 1 567.2.s.d 4
63.t odd 6 1 567.2.i.d 4
84.h odd 2 1 7056.2.k.b 4
84.j odd 6 1 1008.2.bt.b 4
84.n even 6 1 1008.2.bt.b 4
105.o odd 6 1 1575.2.bk.c 4
105.p even 6 1 1575.2.bk.c 4
105.w odd 12 2 1575.2.bc.a 8
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.c even 3 1
63.2.p.a 4 7.d odd 6 1
63.2.p.a 4 21.g even 6 1
63.2.p.a 4 21.h odd 6 1
441.2.c.a 4 1.a even 1 1 trivial
441.2.c.a 4 3.b odd 2 1 inner
441.2.c.a 4 7.b odd 2 1 inner
441.2.c.a 4 21.c even 2 1 inner
441.2.p.a 4 7.c even 3 1
441.2.p.a 4 7.d odd 6 1
441.2.p.a 4 21.g even 6 1
441.2.p.a 4 21.h odd 6 1
567.2.i.d 4 63.h even 3 1
567.2.i.d 4 63.i even 6 1
567.2.i.d 4 63.j odd 6 1
567.2.i.d 4 63.t odd 6 1
567.2.s.d 4 63.g even 3 1
567.2.s.d 4 63.k odd 6 1
567.2.s.d 4 63.n odd 6 1
567.2.s.d 4 63.s even 6 1
1008.2.bt.b 4 28.f even 6 1
1008.2.bt.b 4 28.g odd 6 1
1008.2.bt.b 4 84.j odd 6 1
1008.2.bt.b 4 84.n even 6 1
1575.2.bc.a 8 35.k even 12 2
1575.2.bc.a 8 35.l odd 12 2
1575.2.bc.a 8 105.w odd 12 2
1575.2.bc.a 8 105.x even 12 2
1575.2.bk.c 4 35.i odd 6 1
1575.2.bk.c 4 35.j even 6 1
1575.2.bk.c 4 105.o odd 6 1
1575.2.bk.c 4 105.p even 6 1
7056.2.k.b 4 4.b odd 2 1
7056.2.k.b 4 12.b even 2 1
7056.2.k.b 4 28.d even 2 1
7056.2.k.b 4 84.h odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

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