Properties

Label 441.3.m.e
Level $441$
Weight $3$
Character orbit 441.m
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.m (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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{4} + ( -6 + 3 \zeta_{6} ) q^{5} + 7 q^{8} +O(q^{10})\) \( q + \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{4} + ( -6 + 3 \zeta_{6} ) q^{5} + 7 q^{8} + ( -3 - 3 \zeta_{6} ) q^{10} + ( -11 + 11 \zeta_{6} ) q^{11} + ( -4 + 8 \zeta_{6} ) q^{13} -5 \zeta_{6} q^{16} + ( 14 + 14 \zeta_{6} ) q^{17} + ( 4 - 2 \zeta_{6} ) q^{19} + ( -9 + 18 \zeta_{6} ) q^{20} -11 q^{22} + 28 \zeta_{6} q^{23} + ( 2 - 2 \zeta_{6} ) q^{25} + ( -8 + 4 \zeta_{6} ) q^{26} -25 q^{29} + ( 19 + 19 \zeta_{6} ) q^{31} + ( 33 - 33 \zeta_{6} ) q^{32} + ( -14 + 28 \zeta_{6} ) q^{34} + 58 \zeta_{6} q^{37} + ( 2 + 2 \zeta_{6} ) q^{38} + ( -42 + 21 \zeta_{6} ) q^{40} + ( 2 - 4 \zeta_{6} ) q^{41} + 26 q^{43} + 33 \zeta_{6} q^{44} + ( -28 + 28 \zeta_{6} ) q^{46} + ( -88 + 44 \zeta_{6} ) q^{47} + 2 q^{50} + ( 12 + 12 \zeta_{6} ) q^{52} + ( 31 - 31 \zeta_{6} ) q^{53} + ( 33 - 66 \zeta_{6} ) q^{55} -25 \zeta_{6} q^{58} + ( -5 - 5 \zeta_{6} ) q^{59} + ( -16 + 8 \zeta_{6} ) q^{61} + ( -19 + 38 \zeta_{6} ) q^{62} + 13 q^{64} -36 \zeta_{6} q^{65} + ( 52 - 52 \zeta_{6} ) q^{67} + ( 84 - 42 \zeta_{6} ) q^{68} -64 q^{71} + ( -4 - 4 \zeta_{6} ) q^{73} + ( -58 + 58 \zeta_{6} ) q^{74} + ( 6 - 12 \zeta_{6} ) q^{76} -17 \zeta_{6} q^{79} + ( 15 + 15 \zeta_{6} ) q^{80} + ( 4 - 2 \zeta_{6} ) q^{82} + ( 31 - 62 \zeta_{6} ) q^{83} -126 q^{85} + 26 \zeta_{6} q^{86} + ( -77 + 77 \zeta_{6} ) q^{88} + ( -92 + 46 \zeta_{6} ) q^{89} + 84 q^{92} + ( -44 - 44 \zeta_{6} ) q^{94} + ( -18 + 18 \zeta_{6} ) q^{95} + ( -53 + 106 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 3q^{4} - 9q^{5} + 14q^{8} + O(q^{10}) \) \( 2q + q^{2} + 3q^{4} - 9q^{5} + 14q^{8} - 9q^{10} - 11q^{11} - 5q^{16} + 42q^{17} + 6q^{19} - 22q^{22} + 28q^{23} + 2q^{25} - 12q^{26} - 50q^{29} + 57q^{31} + 33q^{32} + 58q^{37} + 6q^{38} - 63q^{40} + 52q^{43} + 33q^{44} - 28q^{46} - 132q^{47} + 4q^{50} + 36q^{52} + 31q^{53} - 25q^{58} - 15q^{59} - 24q^{61} + 26q^{64} - 36q^{65} + 52q^{67} + 126q^{68} - 128q^{71} - 12q^{73} - 58q^{74} - 17q^{79} + 45q^{80} + 6q^{82} - 252q^{85} + 26q^{86} - 77q^{88} - 138q^{89} + 168q^{92} - 132q^{94} - 18q^{95} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 0 1.50000 + 2.59808i −4.50000 2.59808i 0 0 7.00000 0 −4.50000 + 2.59808i
325.1 0.500000 + 0.866025i 0 1.50000 2.59808i −4.50000 + 2.59808i 0 0 7.00000 0 −4.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{2}^{2} - T_{2} + 1 \)
\( T_{5}^{2} + 9 T_{5} + 27 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 27 + 9 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 121 + 11 T + T^{2} \)
$13$ \( 48 + T^{2} \)
$17$ \( 588 - 42 T + T^{2} \)
$19$ \( 12 - 6 T + T^{2} \)
$23$ \( 784 - 28 T + T^{2} \)
$29$ \( ( 25 + T )^{2} \)
$31$ \( 1083 - 57 T + T^{2} \)
$37$ \( 3364 - 58 T + T^{2} \)
$41$ \( 12 + T^{2} \)
$43$ \( ( -26 + T )^{2} \)
$47$ \( 5808 + 132 T + T^{2} \)
$53$ \( 961 - 31 T + T^{2} \)
$59$ \( 75 + 15 T + T^{2} \)
$61$ \( 192 + 24 T + T^{2} \)
$67$ \( 2704 - 52 T + T^{2} \)
$71$ \( ( 64 + T )^{2} \)
$73$ \( 48 + 12 T + T^{2} \)
$79$ \( 289 + 17 T + T^{2} \)
$83$ \( 2883 + T^{2} \)
$89$ \( 6348 + 138 T + T^{2} \)
$97$ \( 8427 + T^{2} \)
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