Properties

Label 441.3.j.b
Level $441$
Weight $3$
Character orbit 441.j
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.j (of order \(6\), degree \(2\), not 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + q^{4} + ( 4 - 2 \zeta_{6} ) q^{5} + ( -3 - 3 \zeta_{6} ) q^{6} + ( 5 - 10 \zeta_{6} ) q^{8} -9 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + q^{4} + ( 4 - 2 \zeta_{6} ) q^{5} + ( -3 - 3 \zeta_{6} ) q^{6} + ( 5 - 10 \zeta_{6} ) q^{8} -9 \zeta_{6} q^{9} -6 \zeta_{6} q^{10} + ( 1 + \zeta_{6} ) q^{11} + ( 3 - 3 \zeta_{6} ) q^{12} + ( -4 + 4 \zeta_{6} ) q^{13} + ( 6 - 12 \zeta_{6} ) q^{15} -11 q^{16} + ( -18 + 9 \zeta_{6} ) q^{17} + ( -18 + 9 \zeta_{6} ) q^{18} + ( 11 - 11 \zeta_{6} ) q^{19} + ( 4 - 2 \zeta_{6} ) q^{20} + ( 3 - 3 \zeta_{6} ) q^{22} + ( 32 - 16 \zeta_{6} ) q^{23} + ( -15 - 15 \zeta_{6} ) q^{24} + ( -13 + 13 \zeta_{6} ) q^{25} + ( 4 + 4 \zeta_{6} ) q^{26} -27 q^{27} + ( 52 - 26 \zeta_{6} ) q^{29} -18 q^{30} -32 q^{31} + ( 9 - 18 \zeta_{6} ) q^{32} + ( 6 - 3 \zeta_{6} ) q^{33} + 27 \zeta_{6} q^{34} -9 \zeta_{6} q^{36} + ( 34 - 34 \zeta_{6} ) q^{37} + ( -11 - 11 \zeta_{6} ) q^{38} + 12 \zeta_{6} q^{39} -30 \zeta_{6} q^{40} + ( 7 + 7 \zeta_{6} ) q^{41} + 61 \zeta_{6} q^{43} + ( 1 + \zeta_{6} ) q^{44} + ( -18 - 18 \zeta_{6} ) q^{45} -48 \zeta_{6} q^{46} + ( -28 + 56 \zeta_{6} ) q^{47} + ( -33 + 33 \zeta_{6} ) q^{48} + ( 13 + 13 \zeta_{6} ) q^{50} + ( -27 + 54 \zeta_{6} ) q^{51} + ( -4 + 4 \zeta_{6} ) q^{52} + ( -27 + 54 \zeta_{6} ) q^{54} + 6 q^{55} -33 \zeta_{6} q^{57} -78 \zeta_{6} q^{58} + ( -29 + 58 \zeta_{6} ) q^{59} + ( 6 - 12 \zeta_{6} ) q^{60} -56 q^{61} + ( -32 + 64 \zeta_{6} ) q^{62} -71 q^{64} + ( -8 + 16 \zeta_{6} ) q^{65} -9 \zeta_{6} q^{66} -31 q^{67} + ( -18 + 9 \zeta_{6} ) q^{68} + ( 48 - 96 \zeta_{6} ) q^{69} + ( 18 - 36 \zeta_{6} ) q^{71} + ( -90 + 45 \zeta_{6} ) q^{72} + 65 \zeta_{6} q^{73} + ( -34 - 34 \zeta_{6} ) q^{74} + 39 \zeta_{6} q^{75} + ( 11 - 11 \zeta_{6} ) q^{76} + ( 24 - 12 \zeta_{6} ) q^{78} + 38 q^{79} + ( -44 + 22 \zeta_{6} ) q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} + ( 21 - 21 \zeta_{6} ) q^{82} + ( 56 - 28 \zeta_{6} ) q^{83} + ( -54 + 54 \zeta_{6} ) q^{85} + ( 122 - 61 \zeta_{6} ) q^{86} + ( 78 - 156 \zeta_{6} ) q^{87} + ( 15 - 15 \zeta_{6} ) q^{88} + ( 72 + 72 \zeta_{6} ) q^{89} + ( -54 + 54 \zeta_{6} ) q^{90} + ( 32 - 16 \zeta_{6} ) q^{92} + ( -96 + 96 \zeta_{6} ) q^{93} + 84 q^{94} + ( 22 - 44 \zeta_{6} ) q^{95} + ( -27 - 27 \zeta_{6} ) q^{96} -115 \zeta_{6} q^{97} + ( 9 - 18 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} + 2q^{4} + 6q^{5} - 9q^{6} - 9q^{9} + O(q^{10}) \) \( 2q + 3q^{3} + 2q^{4} + 6q^{5} - 9q^{6} - 9q^{9} - 6q^{10} + 3q^{11} + 3q^{12} - 4q^{13} - 22q^{16} - 27q^{17} - 27q^{18} + 11q^{19} + 6q^{20} + 3q^{22} + 48q^{23} - 45q^{24} - 13q^{25} + 12q^{26} - 54q^{27} + 78q^{29} - 36q^{30} - 64q^{31} + 9q^{33} + 27q^{34} - 9q^{36} + 34q^{37} - 33q^{38} + 12q^{39} - 30q^{40} + 21q^{41} + 61q^{43} + 3q^{44} - 54q^{45} - 48q^{46} - 33q^{48} + 39q^{50} - 4q^{52} + 12q^{55} - 33q^{57} - 78q^{58} - 112q^{61} - 142q^{64} - 9q^{66} - 62q^{67} - 27q^{68} - 135q^{72} + 65q^{73} - 102q^{74} + 39q^{75} + 11q^{76} + 36q^{78} + 76q^{79} - 66q^{80} - 81q^{81} + 21q^{82} + 84q^{83} - 54q^{85} + 183q^{86} + 15q^{88} + 216q^{89} - 54q^{90} + 48q^{92} - 96q^{93} + 168q^{94} - 81q^{96} - 115q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(1 - \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1
0.500000 0.866025i
0.500000 + 0.866025i
1.73205i 1.50000 + 2.59808i 1.00000 3.00000 + 1.73205i −4.50000 + 2.59808i 0 8.66025i −4.50000 + 7.79423i −3.00000 + 5.19615i
275.1 1.73205i 1.50000 2.59808i 1.00000 3.00000 1.73205i −4.50000 2.59808i 0 8.66025i −4.50000 7.79423i −3.00000 5.19615i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.j 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.j.b 2
7.b odd 2 1 441.3.j.a 2
7.c even 3 1 441.3.n.a 2
7.c even 3 1 441.3.r.a 2
7.d odd 6 1 9.3.d.a 2
7.d odd 6 1 441.3.n.b 2
9.d odd 6 1 441.3.n.a 2
21.g even 6 1 27.3.d.a 2
28.f even 6 1 144.3.q.a 2
35.i odd 6 1 225.3.j.a 2
35.k even 12 2 225.3.i.a 4
56.j odd 6 1 576.3.q.b 2
56.m even 6 1 576.3.q.a 2
63.i even 6 1 81.3.b.a 2
63.i even 6 1 441.3.j.a 2
63.j odd 6 1 inner 441.3.j.b 2
63.k odd 6 1 27.3.d.a 2
63.n odd 6 1 441.3.r.a 2
63.o even 6 1 441.3.n.b 2
63.s even 6 1 9.3.d.a 2
63.t odd 6 1 81.3.b.a 2
84.j odd 6 1 432.3.q.a 2
105.p even 6 1 675.3.j.a 2
105.w odd 12 2 675.3.i.a 4
168.ba even 6 1 1728.3.q.a 2
168.be odd 6 1 1728.3.q.b 2
252.n even 6 1 432.3.q.a 2
252.r odd 6 1 1296.3.e.a 2
252.bj even 6 1 1296.3.e.a 2
252.bn odd 6 1 144.3.q.a 2
315.u even 6 1 225.3.j.a 2
315.bn odd 6 1 675.3.j.a 2
315.bw odd 12 2 225.3.i.a 4
315.cg even 12 2 675.3.i.a 4
504.u odd 6 1 576.3.q.a 2
504.y even 6 1 576.3.q.b 2
504.cw odd 6 1 1728.3.q.a 2
504.cz even 6 1 1728.3.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 7.d odd 6 1
9.3.d.a 2 63.s even 6 1
27.3.d.a 2 21.g even 6 1
27.3.d.a 2 63.k odd 6 1
81.3.b.a 2 63.i even 6 1
81.3.b.a 2 63.t odd 6 1
144.3.q.a 2 28.f even 6 1
144.3.q.a 2 252.bn odd 6 1
225.3.i.a 4 35.k even 12 2
225.3.i.a 4 315.bw odd 12 2
225.3.j.a 2 35.i odd 6 1
225.3.j.a 2 315.u even 6 1
432.3.q.a 2 84.j odd 6 1
432.3.q.a 2 252.n even 6 1
441.3.j.a 2 7.b odd 2 1
441.3.j.a 2 63.i even 6 1
441.3.j.b 2 1.a even 1 1 trivial
441.3.j.b 2 63.j odd 6 1 inner
441.3.n.a 2 7.c even 3 1
441.3.n.a 2 9.d odd 6 1
441.3.n.b 2 7.d odd 6 1
441.3.n.b 2 63.o even 6 1
441.3.r.a 2 7.c even 3 1
441.3.r.a 2 63.n odd 6 1
576.3.q.a 2 56.m even 6 1
576.3.q.a 2 504.u odd 6 1
576.3.q.b 2 56.j odd 6 1
576.3.q.b 2 504.y even 6 1
675.3.i.a 4 105.w odd 12 2
675.3.i.a 4 315.cg even 12 2
675.3.j.a 2 105.p even 6 1
675.3.j.a 2 315.bn odd 6 1
1296.3.e.a 2 252.r odd 6 1
1296.3.e.a 2 252.bj even 6 1
1728.3.q.a 2 168.ba even 6 1
1728.3.q.a 2 504.cw odd 6 1
1728.3.q.b 2 168.be odd 6 1
1728.3.q.b 2 504.cz even 6 1

Hecke kernels

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

\( T_{2}^{2} + 3 \)
\( T_{5}^{2} - 6 T_{5} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + T^{2} \)
$3$ \( 9 - 3 T + T^{2} \)
$5$ \( 12 - 6 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 3 - 3 T + T^{2} \)
$13$ \( 16 + 4 T + T^{2} \)
$17$ \( 243 + 27 T + T^{2} \)
$19$ \( 121 - 11 T + T^{2} \)
$23$ \( 768 - 48 T + T^{2} \)
$29$ \( 2028 - 78 T + T^{2} \)
$31$ \( ( 32 + T )^{2} \)
$37$ \( 1156 - 34 T + T^{2} \)
$41$ \( 147 - 21 T + T^{2} \)
$43$ \( 3721 - 61 T + T^{2} \)
$47$ \( 2352 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( 2523 + T^{2} \)
$61$ \( ( 56 + T )^{2} \)
$67$ \( ( 31 + T )^{2} \)
$71$ \( 972 + T^{2} \)
$73$ \( 4225 - 65 T + T^{2} \)
$79$ \( ( -38 + T )^{2} \)
$83$ \( 2352 - 84 T + T^{2} \)
$89$ \( 15552 - 216 T + T^{2} \)
$97$ \( 13225 + 115 T + T^{2} \)
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