Properties

Label 441.3.d.d
Level $441$
Weight $3$
Character orbit 441.d
Analytic conductor $12.016$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.0163796583\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( 2 - 4 \zeta_{6} ) q^{5} -8 q^{8} +O(q^{10})\) \( q + 2 q^{2} + ( 2 - 4 \zeta_{6} ) q^{5} -8 q^{8} + ( 4 - 8 \zeta_{6} ) q^{10} -10 q^{11} + ( 7 - 14 \zeta_{6} ) q^{13} -16 q^{16} + ( -4 + 8 \zeta_{6} ) q^{17} + ( 19 - 38 \zeta_{6} ) q^{19} -20 q^{22} -40 q^{23} + 13 q^{25} + ( 14 - 28 \zeta_{6} ) q^{26} -16 q^{29} + ( -3 + 6 \zeta_{6} ) q^{31} + ( -8 + 16 \zeta_{6} ) q^{34} + 5 q^{37} + ( 38 - 76 \zeta_{6} ) q^{38} + ( -16 + 32 \zeta_{6} ) q^{40} + ( -14 + 28 \zeta_{6} ) q^{41} -19 q^{43} -80 q^{46} + ( 30 - 60 \zeta_{6} ) q^{47} + 26 q^{50} + 32 q^{53} + ( -20 + 40 \zeta_{6} ) q^{55} -32 q^{58} + ( 24 - 48 \zeta_{6} ) q^{59} + ( 12 - 24 \zeta_{6} ) q^{61} + ( -6 + 12 \zeta_{6} ) q^{62} + 64 q^{64} -42 q^{65} + 59 q^{67} + 26 q^{71} + ( 11 - 22 \zeta_{6} ) q^{73} + 10 q^{74} + 47 q^{79} + ( -32 + 64 \zeta_{6} ) q^{80} + ( -28 + 56 \zeta_{6} ) q^{82} + ( 14 - 28 \zeta_{6} ) q^{83} + 24 q^{85} -38 q^{86} + 80 q^{88} + ( -68 + 136 \zeta_{6} ) q^{89} + ( 60 - 120 \zeta_{6} ) q^{94} -114 q^{95} + ( -28 + 56 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} - 16q^{8} + O(q^{10}) \) \( 2q + 4q^{2} - 16q^{8} - 20q^{11} - 32q^{16} - 40q^{22} - 80q^{23} + 26q^{25} - 32q^{29} + 10q^{37} - 38q^{43} - 160q^{46} + 52q^{50} + 64q^{53} - 64q^{58} + 128q^{64} - 84q^{65} + 118q^{67} + 52q^{71} + 20q^{74} + 94q^{79} + 48q^{85} - 76q^{86} + 160q^{88} - 228q^{95} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
0.500000 + 0.866025i
0.500000 0.866025i
2.00000 0 0 3.46410i 0 0 −8.00000 0 6.92820i
244.2 2.00000 0 0 3.46410i 0 0 −8.00000 0 6.92820i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.3.d.d 2
3.b odd 2 1 147.3.d.a 2
7.b odd 2 1 inner 441.3.d.d 2
7.c even 3 1 63.3.m.a 2
7.c even 3 1 441.3.m.b 2
7.d odd 6 1 63.3.m.a 2
7.d odd 6 1 441.3.m.b 2
12.b even 2 1 2352.3.f.b 2
21.c even 2 1 147.3.d.a 2
21.g even 6 1 21.3.f.c 2
21.g even 6 1 147.3.f.e 2
21.h odd 6 1 21.3.f.c 2
21.h odd 6 1 147.3.f.e 2
28.f even 6 1 1008.3.cg.f 2
28.g odd 6 1 1008.3.cg.f 2
84.h odd 2 1 2352.3.f.b 2
84.j odd 6 1 336.3.bh.c 2
84.n even 6 1 336.3.bh.c 2
105.o odd 6 1 525.3.o.b 2
105.p even 6 1 525.3.o.b 2
105.w odd 12 2 525.3.s.d 4
105.x even 12 2 525.3.s.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.c 2 21.g even 6 1
21.3.f.c 2 21.h odd 6 1
63.3.m.a 2 7.c even 3 1
63.3.m.a 2 7.d odd 6 1
147.3.d.a 2 3.b odd 2 1
147.3.d.a 2 21.c even 2 1
147.3.f.e 2 21.g even 6 1
147.3.f.e 2 21.h odd 6 1
336.3.bh.c 2 84.j odd 6 1
336.3.bh.c 2 84.n even 6 1
441.3.d.d 2 1.a even 1 1 trivial
441.3.d.d 2 7.b odd 2 1 inner
441.3.m.b 2 7.c even 3 1
441.3.m.b 2 7.d odd 6 1
525.3.o.b 2 105.o odd 6 1
525.3.o.b 2 105.p even 6 1
525.3.s.d 4 105.w odd 12 2
525.3.s.d 4 105.x even 12 2
1008.3.cg.f 2 28.f even 6 1
1008.3.cg.f 2 28.g odd 6 1
2352.3.f.b 2 12.b even 2 1
2352.3.f.b 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 12 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 10 + T )^{2} \)
$13$ \( 147 + T^{2} \)
$17$ \( 48 + T^{2} \)
$19$ \( 1083 + T^{2} \)
$23$ \( ( 40 + T )^{2} \)
$29$ \( ( 16 + T )^{2} \)
$31$ \( 27 + T^{2} \)
$37$ \( ( -5 + T )^{2} \)
$41$ \( 588 + T^{2} \)
$43$ \( ( 19 + T )^{2} \)
$47$ \( 2700 + T^{2} \)
$53$ \( ( -32 + T )^{2} \)
$59$ \( 1728 + T^{2} \)
$61$ \( 432 + T^{2} \)
$67$ \( ( -59 + T )^{2} \)
$71$ \( ( -26 + T )^{2} \)
$73$ \( 363 + T^{2} \)
$79$ \( ( -47 + T )^{2} \)
$83$ \( 588 + T^{2} \)
$89$ \( 13872 + T^{2} \)
$97$ \( 2352 + T^{2} \)
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