Properties

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

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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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + ( -2 + \zeta_{12}^{2} ) q^{6} + 4 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} -3 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{2} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + ( -2 + \zeta_{12}^{2} ) q^{6} + 4 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} -3 \zeta_{12}^{2} q^{9} + ( -3 + 3 \zeta_{12}^{2} ) q^{11} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{12} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{13} + 4 \zeta_{12}^{2} q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 3 \zeta_{12}^{3} q^{17} -3 \zeta_{12}^{3} q^{18} -5 q^{19} + ( -8 + 4 \zeta_{12}^{2} ) q^{21} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{22} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{23} + ( -1 - \zeta_{12}^{2} ) q^{24} + 4 q^{26} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + 4 \zeta_{12}^{3} q^{28} + ( 6 - 6 \zeta_{12}^{2} ) q^{29} -2 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{33} + ( -3 + 3 \zeta_{12}^{2} ) q^{34} + ( 3 - 3 \zeta_{12}^{2} ) q^{36} -4 \zeta_{12}^{3} q^{37} -5 \zeta_{12} q^{38} + ( -4 + 8 \zeta_{12}^{2} ) q^{39} + 3 \zeta_{12}^{2} q^{41} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{42} + 11 \zeta_{12} q^{43} -3 q^{44} -6 q^{46} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{48} + 9 \zeta_{12}^{2} q^{49} + ( -3 - 3 \zeta_{12}^{2} ) q^{51} + 4 \zeta_{12} q^{52} -6 \zeta_{12}^{3} q^{53} + ( 3 + 3 \zeta_{12}^{2} ) q^{54} + ( -4 + 4 \zeta_{12}^{2} ) q^{56} + ( 5 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{57} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{58} -3 \zeta_{12}^{2} q^{59} + ( 10 - 10 \zeta_{12}^{2} ) q^{61} -2 \zeta_{12}^{3} q^{62} -12 \zeta_{12}^{3} q^{63} - q^{64} + ( 3 - 6 \zeta_{12}^{2} ) q^{66} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{67} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{68} + ( 6 - 12 \zeta_{12}^{2} ) q^{69} + 6 q^{71} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{72} + 7 \zeta_{12}^{3} q^{73} + ( 4 - 4 \zeta_{12}^{2} ) q^{74} -5 \zeta_{12}^{2} q^{76} + ( -12 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{77} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{78} + ( 14 - 14 \zeta_{12}^{2} ) q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + 3 \zeta_{12}^{3} q^{82} + 12 \zeta_{12} q^{83} + ( -4 - 4 \zeta_{12}^{2} ) q^{84} + 11 \zeta_{12}^{2} q^{86} + ( 6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{87} -3 \zeta_{12} q^{88} -6 q^{89} + 16 q^{91} -6 \zeta_{12} q^{92} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{93} + ( 1 - 2 \zeta_{12}^{2} ) q^{96} -11 \zeta_{12} q^{97} + 9 \zeta_{12}^{3} q^{98} + 9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 6q^{6} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 6q^{6} - 6q^{9} - 6q^{11} + 8q^{14} - 2q^{16} - 20q^{19} - 24q^{21} - 6q^{24} + 16q^{26} + 12q^{29} - 4q^{31} - 6q^{34} + 6q^{36} + 6q^{41} - 12q^{44} - 24q^{46} + 18q^{49} - 18q^{51} + 18q^{54} - 8q^{56} - 6q^{59} + 20q^{61} - 4q^{64} + 24q^{71} + 8q^{74} - 10q^{76} + 28q^{79} - 18q^{81} - 24q^{84} + 22q^{86} - 24q^{89} + 64q^{91} + 36q^{99} + O(q^{100}) \)

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 −1.50000 0.866025i −3.46410 + 2.00000i 1.00000i −1.50000 + 2.59808i 0
49.2 0.866025 0.500000i −0.866025 1.50000i 0.500000 0.866025i 0 −1.50000 0.866025i 3.46410 2.00000i 1.00000i −1.50000 + 2.59808i 0
349.1 −0.866025 0.500000i 0.866025 1.50000i 0.500000 + 0.866025i 0 −1.50000 + 0.866025i −3.46410 2.00000i 1.00000i −1.50000 2.59808i 0
349.2 0.866025 + 0.500000i −0.866025 + 1.50000i 0.500000 + 0.866025i 0 −1.50000 + 0.866025i 3.46410 + 2.00000i 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.a 4
3.b odd 2 1 1350.2.j.c 4
5.b even 2 1 inner 450.2.j.a 4
5.c odd 4 1 90.2.e.b 2
5.c odd 4 1 450.2.e.d 2
9.c even 3 1 inner 450.2.j.a 4
9.c even 3 1 4050.2.c.p 2
9.d odd 6 1 1350.2.j.c 4
9.d odd 6 1 4050.2.c.d 2
15.d odd 2 1 1350.2.j.c 4
15.e even 4 1 270.2.e.a 2
15.e even 4 1 1350.2.e.g 2
20.e even 4 1 720.2.q.c 2
45.h odd 6 1 1350.2.j.c 4
45.h odd 6 1 4050.2.c.d 2
45.j even 6 1 inner 450.2.j.a 4
45.j even 6 1 4050.2.c.p 2
45.k odd 12 1 90.2.e.b 2
45.k odd 12 1 450.2.e.d 2
45.k odd 12 1 810.2.a.a 1
45.k odd 12 1 4050.2.a.bi 1
45.l even 12 1 270.2.e.a 2
45.l even 12 1 810.2.a.e 1
45.l even 12 1 1350.2.e.g 2
45.l even 12 1 4050.2.a.q 1
60.l odd 4 1 2160.2.q.d 2
180.v odd 12 1 2160.2.q.d 2
180.v odd 12 1 6480.2.a.l 1
180.x even 12 1 720.2.q.c 2
180.x even 12 1 6480.2.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.b 2 5.c odd 4 1
90.2.e.b 2 45.k odd 12 1
270.2.e.a 2 15.e even 4 1
270.2.e.a 2 45.l even 12 1
450.2.e.d 2 5.c odd 4 1
450.2.e.d 2 45.k odd 12 1
450.2.j.a 4 1.a even 1 1 trivial
450.2.j.a 4 5.b even 2 1 inner
450.2.j.a 4 9.c even 3 1 inner
450.2.j.a 4 45.j even 6 1 inner
720.2.q.c 2 20.e even 4 1
720.2.q.c 2 180.x even 12 1
810.2.a.a 1 45.k odd 12 1
810.2.a.e 1 45.l even 12 1
1350.2.e.g 2 15.e even 4 1
1350.2.e.g 2 45.l even 12 1
1350.2.j.c 4 3.b odd 2 1
1350.2.j.c 4 9.d odd 6 1
1350.2.j.c 4 15.d odd 2 1
1350.2.j.c 4 45.h odd 6 1
2160.2.q.d 2 60.l odd 4 1
2160.2.q.d 2 180.v odd 12 1
4050.2.a.q 1 45.l even 12 1
4050.2.a.bi 1 45.k odd 12 1
4050.2.c.d 2 9.d odd 6 1
4050.2.c.d 2 45.h odd 6 1
4050.2.c.p 2 9.c even 3 1
4050.2.c.p 2 45.j even 6 1
6480.2.a.l 1 180.v odd 12 1
6480.2.a.z 1 180.x even 12 1

Hecke kernels

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

\( T_{7}^{4} - 16 T_{7}^{2} + 256 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{19} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 256 - 16 T^{2} + T^{4} \)
$11$ \( ( 9 + 3 T + T^{2} )^{2} \)
$13$ \( 256 - 16 T^{2} + T^{4} \)
$17$ \( ( 9 + T^{2} )^{2} \)
$19$ \( ( 5 + T )^{4} \)
$23$ \( 1296 - 36 T^{2} + T^{4} \)
$29$ \( ( 36 - 6 T + T^{2} )^{2} \)
$31$ \( ( 4 + 2 T + T^{2} )^{2} \)
$37$ \( ( 16 + T^{2} )^{2} \)
$41$ \( ( 9 - 3 T + T^{2} )^{2} \)
$43$ \( 14641 - 121 T^{2} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( 9 + 3 T + T^{2} )^{2} \)
$61$ \( ( 100 - 10 T + T^{2} )^{2} \)
$67$ \( 625 - 25 T^{2} + T^{4} \)
$71$ \( ( -6 + T )^{4} \)
$73$ \( ( 49 + T^{2} )^{2} \)
$79$ \( ( 196 - 14 T + T^{2} )^{2} \)
$83$ \( 20736 - 144 T^{2} + T^{4} \)
$89$ \( ( 6 + T )^{4} \)
$97$ \( 14641 - 121 T^{2} + T^{4} \)
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