Properties

Label 450.2.j.b
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: yes
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + ( -1 - \zeta_{12}^{2} ) q^{6} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + 3 q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{2} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + ( -1 - \zeta_{12}^{2} ) q^{6} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + 3 q^{9} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{12} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{13} + 2 \zeta_{12}^{2} q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 6 \zeta_{12}^{3} q^{17} + 3 \zeta_{12} q^{18} + 7 q^{19} + ( -2 - 2 \zeta_{12}^{2} ) q^{21} + ( 1 - 2 \zeta_{12}^{2} ) q^{24} -4 q^{26} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + 2 \zeta_{12}^{3} q^{28} + ( -6 + 6 \zeta_{12}^{2} ) q^{29} + 10 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + ( -6 + 6 \zeta_{12}^{2} ) q^{34} + 3 \zeta_{12}^{2} q^{36} -2 \zeta_{12}^{3} q^{37} + 7 \zeta_{12} q^{38} + ( 8 - 4 \zeta_{12}^{2} ) q^{39} -9 \zeta_{12}^{2} q^{41} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{42} + \zeta_{12} q^{43} + 6 \zeta_{12} q^{47} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{48} -3 \zeta_{12}^{2} q^{49} + ( 6 - 12 \zeta_{12}^{2} ) q^{51} -4 \zeta_{12} q^{52} -12 \zeta_{12}^{3} q^{53} + ( -3 - 3 \zeta_{12}^{2} ) q^{54} + ( -2 + 2 \zeta_{12}^{2} ) q^{56} + ( -14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{57} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{58} -9 \zeta_{12}^{2} q^{59} + ( 4 - 4 \zeta_{12}^{2} ) q^{61} + 10 \zeta_{12}^{3} q^{62} + 6 \zeta_{12} q^{63} - q^{64} + ( 13 \zeta_{12} - 13 \zeta_{12}^{3} ) q^{67} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{68} + 6 q^{71} + 3 \zeta_{12}^{3} q^{72} -\zeta_{12}^{3} q^{73} + ( 2 - 2 \zeta_{12}^{2} ) q^{74} + 7 \zeta_{12}^{2} q^{76} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{78} + ( 2 - 2 \zeta_{12}^{2} ) q^{79} + 9 q^{81} -9 \zeta_{12}^{3} q^{82} -9 \zeta_{12} q^{83} + ( 2 - 4 \zeta_{12}^{2} ) q^{84} + \zeta_{12}^{2} q^{86} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{87} -15 q^{89} -8 q^{91} + ( -10 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{93} + 6 \zeta_{12}^{2} q^{94} + ( 2 - \zeta_{12}^{2} ) q^{96} + 17 \zeta_{12} q^{97} -3 \zeta_{12}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 6q^{6} + 12q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 6q^{6} + 12q^{9} + 4q^{14} - 2q^{16} + 28q^{19} - 12q^{21} - 16q^{26} - 12q^{29} + 20q^{31} - 12q^{34} + 6q^{36} + 24q^{39} - 18q^{41} - 6q^{49} - 18q^{54} - 4q^{56} - 18q^{59} + 8q^{61} - 4q^{64} + 24q^{71} + 4q^{74} + 14q^{76} + 4q^{79} + 36q^{81} + 2q^{86} - 60q^{89} - 32q^{91} + 12q^{94} + 6q^{96} + O(q^{100}) \)

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 1.73205 0.500000 0.866025i 0 −1.50000 + 0.866025i −1.73205 + 1.00000i 1.00000i 3.00000 0
49.2 0.866025 0.500000i −1.73205 0.500000 0.866025i 0 −1.50000 + 0.866025i 1.73205 1.00000i 1.00000i 3.00000 0
349.1 −0.866025 0.500000i 1.73205 0.500000 + 0.866025i 0 −1.50000 0.866025i −1.73205 1.00000i 1.00000i 3.00000 0
349.2 0.866025 + 0.500000i −1.73205 0.500000 + 0.866025i 0 −1.50000 0.866025i 1.73205 + 1.00000i 1.00000i 3.00000 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.b 4
3.b odd 2 1 1350.2.j.b 4
5.b even 2 1 inner 450.2.j.b 4
5.c odd 4 1 450.2.e.c 2
5.c odd 4 1 450.2.e.f yes 2
9.c even 3 1 inner 450.2.j.b 4
9.c even 3 1 4050.2.c.h 2
9.d odd 6 1 1350.2.j.b 4
9.d odd 6 1 4050.2.c.m 2
15.d odd 2 1 1350.2.j.b 4
15.e even 4 1 1350.2.e.d 2
15.e even 4 1 1350.2.e.h 2
45.h odd 6 1 1350.2.j.b 4
45.h odd 6 1 4050.2.c.m 2
45.j even 6 1 inner 450.2.j.b 4
45.j even 6 1 4050.2.c.h 2
45.k odd 12 1 450.2.e.c 2
45.k odd 12 1 450.2.e.f yes 2
45.k odd 12 1 4050.2.a.d 1
45.k odd 12 1 4050.2.a.bg 1
45.l even 12 1 1350.2.e.d 2
45.l even 12 1 1350.2.e.h 2
45.l even 12 1 4050.2.a.o 1
45.l even 12 1 4050.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.c 2 5.c odd 4 1
450.2.e.c 2 45.k odd 12 1
450.2.e.f yes 2 5.c odd 4 1
450.2.e.f yes 2 45.k odd 12 1
450.2.j.b 4 1.a even 1 1 trivial
450.2.j.b 4 5.b even 2 1 inner
450.2.j.b 4 9.c even 3 1 inner
450.2.j.b 4 45.j even 6 1 inner
1350.2.e.d 2 15.e even 4 1
1350.2.e.d 2 45.l even 12 1
1350.2.e.h 2 15.e even 4 1
1350.2.e.h 2 45.l even 12 1
1350.2.j.b 4 3.b odd 2 1
1350.2.j.b 4 9.d odd 6 1
1350.2.j.b 4 15.d odd 2 1
1350.2.j.b 4 45.h odd 6 1
4050.2.a.d 1 45.k odd 12 1
4050.2.a.o 1 45.l even 12 1
4050.2.a.u 1 45.l even 12 1
4050.2.a.bg 1 45.k odd 12 1
4050.2.c.h 2 9.c even 3 1
4050.2.c.h 2 45.j even 6 1
4050.2.c.m 2 9.d odd 6 1
4050.2.c.m 2 45.h odd 6 1

Hecke kernels

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

\( T_{7}^{4} - 4 T_{7}^{2} + 16 \)
\( T_{11} \)
\( T_{19} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 16 - 4 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 256 - 16 T^{2} + T^{4} \)
$17$ \( ( 36 + T^{2} )^{2} \)
$19$ \( ( -7 + T )^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 36 + 6 T + T^{2} )^{2} \)
$31$ \( ( 100 - 10 T + T^{2} )^{2} \)
$37$ \( ( 4 + T^{2} )^{2} \)
$41$ \( ( 81 + 9 T + T^{2} )^{2} \)
$43$ \( 1 - T^{2} + T^{4} \)
$47$ \( 1296 - 36 T^{2} + T^{4} \)
$53$ \( ( 144 + T^{2} )^{2} \)
$59$ \( ( 81 + 9 T + T^{2} )^{2} \)
$61$ \( ( 16 - 4 T + T^{2} )^{2} \)
$67$ \( 28561 - 169 T^{2} + T^{4} \)
$71$ \( ( -6 + T )^{4} \)
$73$ \( ( 1 + T^{2} )^{2} \)
$79$ \( ( 4 - 2 T + T^{2} )^{2} \)
$83$ \( 6561 - 81 T^{2} + T^{4} \)
$89$ \( ( 15 + T )^{4} \)
$97$ \( 83521 - 289 T^{2} + T^{4} \)
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