Properties

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

Related objects

Downloads

Learn more

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.

\(f(q)\) \(=\) \( q - q^{2} -3 q^{4} + ( -3 + 6 \zeta_{6} ) q^{5} + 7 q^{8} +O(q^{10})\) \( q - q^{2} -3 q^{4} + ( -3 + 6 \zeta_{6} ) q^{5} + 7 q^{8} + ( 3 - 6 \zeta_{6} ) q^{10} + 11 q^{11} + ( 4 - 8 \zeta_{6} ) q^{13} + 5 q^{16} + ( -14 + 28 \zeta_{6} ) q^{17} + ( 2 - 4 \zeta_{6} ) q^{19} + ( 9 - 18 \zeta_{6} ) q^{20} -11 q^{22} -28 q^{23} -2 q^{25} + ( -4 + 8 \zeta_{6} ) q^{26} -25 q^{29} + ( -19 + 38 \zeta_{6} ) q^{31} -33 q^{32} + ( 14 - 28 \zeta_{6} ) q^{34} -58 q^{37} + ( -2 + 4 \zeta_{6} ) q^{38} + ( -21 + 42 \zeta_{6} ) q^{40} + ( -2 + 4 \zeta_{6} ) q^{41} + 26 q^{43} -33 q^{44} + 28 q^{46} + ( -44 + 88 \zeta_{6} ) q^{47} + 2 q^{50} + ( -12 + 24 \zeta_{6} ) q^{52} -31 q^{53} + ( -33 + 66 \zeta_{6} ) q^{55} + 25 q^{58} + ( 5 - 10 \zeta_{6} ) q^{59} + ( -8 + 16 \zeta_{6} ) q^{61} + ( 19 - 38 \zeta_{6} ) q^{62} + 13 q^{64} + 36 q^{65} -52 q^{67} + ( 42 - 84 \zeta_{6} ) q^{68} -64 q^{71} + ( 4 - 8 \zeta_{6} ) q^{73} + 58 q^{74} + ( -6 + 12 \zeta_{6} ) q^{76} + 17 q^{79} + ( -15 + 30 \zeta_{6} ) q^{80} + ( 2 - 4 \zeta_{6} ) q^{82} + ( -31 + 62 \zeta_{6} ) q^{83} -126 q^{85} -26 q^{86} + 77 q^{88} + ( -46 + 92 \zeta_{6} ) q^{89} + 84 q^{92} + ( 44 - 88 \zeta_{6} ) q^{94} + 18 q^{95} + ( 53 - 106 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 6q^{4} + 14q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 6q^{4} + 14q^{8} + 22q^{11} + 10q^{16} - 22q^{22} - 56q^{23} - 4q^{25} - 50q^{29} - 66q^{32} - 116q^{37} + 52q^{43} - 66q^{44} + 56q^{46} + 4q^{50} - 62q^{53} + 50q^{58} + 26q^{64} + 72q^{65} - 104q^{67} - 128q^{71} + 116q^{74} + 34q^{79} - 252q^{85} - 52q^{86} + 154q^{88} + 168q^{92} + 36q^{95} + O(q^{100}) \)

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
−1.00000 0 −3.00000 5.19615i 0 0 7.00000 0 5.19615i
244.2 −1.00000 0 −3.00000 5.19615i 0 0 7.00000 0 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
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.b 2
3.b odd 2 1 147.3.d.b 2
7.b odd 2 1 inner 441.3.d.b 2
7.c even 3 1 63.3.m.c 2
7.c even 3 1 441.3.m.e 2
7.d odd 6 1 63.3.m.c 2
7.d odd 6 1 441.3.m.e 2
12.b even 2 1 2352.3.f.d 2
21.c even 2 1 147.3.d.b 2
21.g even 6 1 21.3.f.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.f.c 2
28.f even 6 1 1008.3.cg.g 2
28.g odd 6 1 1008.3.cg.g 2
84.h odd 2 1 2352.3.f.d 2
84.j odd 6 1 336.3.bh.a 2
84.n even 6 1 336.3.bh.a 2
105.o odd 6 1 525.3.o.g 2
105.p even 6 1 525.3.o.g 2
105.w odd 12 2 525.3.s.c 4
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.g even 6 1
21.3.f.b 2 21.h odd 6 1
63.3.m.c 2 7.c even 3 1
63.3.m.c 2 7.d odd 6 1
147.3.d.b 2 3.b odd 2 1
147.3.d.b 2 21.c even 2 1
147.3.f.c 2 21.g even 6 1
147.3.f.c 2 21.h odd 6 1
336.3.bh.a 2 84.j odd 6 1
336.3.bh.a 2 84.n even 6 1
441.3.d.b 2 1.a even 1 1 trivial
441.3.d.b 2 7.b odd 2 1 inner
441.3.m.e 2 7.c even 3 1
441.3.m.e 2 7.d odd 6 1
525.3.o.g 2 105.o odd 6 1
525.3.o.g 2 105.p even 6 1
525.3.s.c 4 105.w odd 12 2
525.3.s.c 4 105.x even 12 2
1008.3.cg.g 2 28.f even 6 1
1008.3.cg.g 2 28.g odd 6 1
2352.3.f.d 2 12.b even 2 1
2352.3.f.d 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 27 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -11 + T )^{2} \)
$13$ \( 48 + T^{2} \)
$17$ \( 588 + T^{2} \)
$19$ \( 12 + T^{2} \)
$23$ \( ( 28 + T )^{2} \)
$29$ \( ( 25 + T )^{2} \)
$31$ \( 1083 + T^{2} \)
$37$ \( ( 58 + T )^{2} \)
$41$ \( 12 + T^{2} \)
$43$ \( ( -26 + T )^{2} \)
$47$ \( 5808 + T^{2} \)
$53$ \( ( 31 + T )^{2} \)
$59$ \( 75 + T^{2} \)
$61$ \( 192 + T^{2} \)
$67$ \( ( 52 + T )^{2} \)
$71$ \( ( 64 + T )^{2} \)
$73$ \( 48 + T^{2} \)
$79$ \( ( -17 + T )^{2} \)
$83$ \( 2883 + T^{2} \)
$89$ \( 6348 + T^{2} \)
$97$ \( 8427 + T^{2} \)
show more
show less