Properties

Label 450.2.j.c
Level $450$
Weight $2$
Character orbit 450.j
Analytic conductor $3.593$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + ( -1 + 2 \zeta_{12}^{2} ) q^{6} -\zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + ( -1 + 2 \zeta_{12}^{2} ) q^{6} -\zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -6 + 6 \zeta_{12}^{2} ) q^{11} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{12} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{13} -\zeta_{12}^{2} q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{18} + 4 q^{19} + ( 1 - 2 \zeta_{12}^{2} ) q^{21} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{22} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{23} + ( -2 + \zeta_{12}^{2} ) q^{24} + 2 q^{26} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} -\zeta_{12}^{3} q^{28} + ( 3 - 3 \zeta_{12}^{2} ) q^{29} + 4 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{33} -3 q^{36} -8 \zeta_{12}^{3} q^{37} + 4 \zeta_{12} q^{38} + ( 2 + 2 \zeta_{12}^{2} ) q^{39} + 3 \zeta_{12}^{2} q^{41} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{42} -8 \zeta_{12} q^{43} -6 q^{44} + 9 q^{46} -3 \zeta_{12} q^{47} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{48} -6 \zeta_{12}^{2} q^{49} + 2 \zeta_{12} q^{52} + 6 \zeta_{12}^{3} q^{53} + ( -3 - 3 \zeta_{12}^{2} ) q^{54} + ( 1 - \zeta_{12}^{2} ) q^{56} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{57} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{58} + 6 \zeta_{12}^{2} q^{59} + ( 13 - 13 \zeta_{12}^{2} ) q^{61} + 4 \zeta_{12}^{3} q^{62} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{63} - q^{64} + ( -6 - 6 \zeta_{12}^{2} ) q^{66} + ( 13 \zeta_{12} - 13 \zeta_{12}^{3} ) q^{67} + ( 9 + 9 \zeta_{12}^{2} ) q^{69} -6 q^{71} -3 \zeta_{12} q^{72} -4 \zeta_{12}^{3} q^{73} + ( 8 - 8 \zeta_{12}^{2} ) q^{74} + 4 \zeta_{12}^{2} q^{76} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{78} + ( -10 + 10 \zeta_{12}^{2} ) q^{79} -9 \zeta_{12}^{2} q^{81} + 3 \zeta_{12}^{3} q^{82} + 9 \zeta_{12} q^{83} + ( 2 - \zeta_{12}^{2} ) q^{84} -8 \zeta_{12}^{2} q^{86} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{87} -6 \zeta_{12} q^{88} -9 q^{89} -2 q^{91} + 9 \zeta_{12} q^{92} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{93} -3 \zeta_{12}^{2} q^{94} + ( -1 - \zeta_{12}^{2} ) q^{96} + 2 \zeta_{12} q^{97} -6 \zeta_{12}^{3} q^{98} -18 \zeta_{12}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 6q^{9} - 12q^{11} - 2q^{14} - 2q^{16} + 16q^{19} - 6q^{24} + 8q^{26} + 6q^{29} + 8q^{31} - 12q^{36} + 12q^{39} + 6q^{41} - 24q^{44} + 36q^{46} - 12q^{49} - 18q^{54} + 2q^{56} + 12q^{59} + 26q^{61} - 4q^{64} - 36q^{66} + 54q^{69} - 24q^{71} + 16q^{74} + 8q^{76} - 20q^{79} - 18q^{81} + 6q^{84} - 16q^{86} - 36q^{89} - 8q^{91} - 6q^{94} - 6q^{96} - 36q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i −0.866025 + 1.50000i 0.500000 0.866025i 0 1.73205i 0.866025 0.500000i 1.00000i −1.50000 2.59808i 0
49.2 0.866025 0.500000i 0.866025 1.50000i 0.500000 0.866025i 0 1.73205i −0.866025 + 0.500000i 1.00000i −1.50000 2.59808i 0
349.1 −0.866025 0.500000i −0.866025 1.50000i 0.500000 + 0.866025i 0 1.73205i 0.866025 + 0.500000i 1.00000i −1.50000 + 2.59808i 0
349.2 0.866025 + 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i 0 1.73205i −0.866025 0.500000i 1.00000i −1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.j.c 4
3.b odd 2 1 1350.2.j.e 4
5.b even 2 1 inner 450.2.j.c 4
5.c odd 4 1 90.2.e.a 2
5.c odd 4 1 450.2.e.e 2
9.c even 3 1 inner 450.2.j.c 4
9.c even 3 1 4050.2.c.t 2
9.d odd 6 1 1350.2.j.e 4
9.d odd 6 1 4050.2.c.a 2
15.d odd 2 1 1350.2.j.e 4
15.e even 4 1 270.2.e.b 2
15.e even 4 1 1350.2.e.b 2
20.e even 4 1 720.2.q.b 2
45.h odd 6 1 1350.2.j.e 4
45.h odd 6 1 4050.2.c.a 2
45.j even 6 1 inner 450.2.j.c 4
45.j even 6 1 4050.2.c.t 2
45.k odd 12 1 90.2.e.a 2
45.k odd 12 1 450.2.e.e 2
45.k odd 12 1 810.2.a.g 1
45.k odd 12 1 4050.2.a.n 1
45.l even 12 1 270.2.e.b 2
45.l even 12 1 810.2.a.b 1
45.l even 12 1 1350.2.e.b 2
45.l even 12 1 4050.2.a.ba 1
60.l odd 4 1 2160.2.q.b 2
180.v odd 12 1 2160.2.q.b 2
180.v odd 12 1 6480.2.a.v 1
180.x even 12 1 720.2.q.b 2
180.x even 12 1 6480.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.a 2 5.c odd 4 1
90.2.e.a 2 45.k odd 12 1
270.2.e.b 2 15.e even 4 1
270.2.e.b 2 45.l even 12 1
450.2.e.e 2 5.c odd 4 1
450.2.e.e 2 45.k odd 12 1
450.2.j.c 4 1.a even 1 1 trivial
450.2.j.c 4 5.b even 2 1 inner
450.2.j.c 4 9.c even 3 1 inner
450.2.j.c 4 45.j even 6 1 inner
720.2.q.b 2 20.e even 4 1
720.2.q.b 2 180.x even 12 1
810.2.a.b 1 45.l even 12 1
810.2.a.g 1 45.k odd 12 1
1350.2.e.b 2 15.e even 4 1
1350.2.e.b 2 45.l even 12 1
1350.2.j.e 4 3.b odd 2 1
1350.2.j.e 4 9.d odd 6 1
1350.2.j.e 4 15.d odd 2 1
1350.2.j.e 4 45.h odd 6 1
2160.2.q.b 2 60.l odd 4 1
2160.2.q.b 2 180.v odd 12 1
4050.2.a.n 1 45.k odd 12 1
4050.2.a.ba 1 45.l even 12 1
4050.2.c.a 2 9.d odd 6 1
4050.2.c.a 2 45.h odd 6 1
4050.2.c.t 2 9.c even 3 1
4050.2.c.t 2 45.j even 6 1
6480.2.a.g 1 180.x even 12 1
6480.2.a.v 1 180.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{4} - T_{7}^{2} + 1 \)
\( T_{11}^{2} + 6 T_{11} + 36 \)
\( T_{19} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 1 - T^{2} + T^{4} \)
$11$ \( ( 36 + 6 T + T^{2} )^{2} \)
$13$ \( 16 - 4 T^{2} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( ( -4 + T )^{4} \)
$23$ \( 6561 - 81 T^{2} + T^{4} \)
$29$ \( ( 9 - 3 T + T^{2} )^{2} \)
$31$ \( ( 16 - 4 T + T^{2} )^{2} \)
$37$ \( ( 64 + T^{2} )^{2} \)
$41$ \( ( 9 - 3 T + T^{2} )^{2} \)
$43$ \( 4096 - 64 T^{2} + T^{4} \)
$47$ \( 81 - 9 T^{2} + T^{4} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( 36 - 6 T + T^{2} )^{2} \)
$61$ \( ( 169 - 13 T + T^{2} )^{2} \)
$67$ \( 28561 - 169 T^{2} + T^{4} \)
$71$ \( ( 6 + T )^{4} \)
$73$ \( ( 16 + T^{2} )^{2} \)
$79$ \( ( 100 + 10 T + T^{2} )^{2} \)
$83$ \( 6561 - 81 T^{2} + T^{4} \)
$89$ \( ( 9 + T )^{4} \)
$97$ \( 16 - 4 T^{2} + T^{4} \)
show more
show less