Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} ) q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} -\beta_{2} q^{4} + \beta_{1} q^{6} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{7} + q^{8} + ( -3 + \beta_{1} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} ) q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} -\beta_{2} q^{4} + \beta_{1} q^{6} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{7} + q^{8} + ( -3 + \beta_{1} - \beta_{3} ) q^{9} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{11} -\beta_{3} q^{12} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{13} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{14} + ( -1 + \beta_{2} ) q^{16} + ( 4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{17} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{18} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( 3 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{21} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{22} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{23} + ( -\beta_{1} + \beta_{3} ) q^{24} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{26} + ( 3 + 2 \beta_{1} - 2 \beta_{3} ) q^{27} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{28} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{29} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{31} -\beta_{2} q^{32} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{33} + ( -5 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{34} + ( 3 \beta_{2} + \beta_{3} ) q^{36} + 4 q^{37} + ( -1 + \beta_{1} - 2 \beta_{3} ) q^{38} + ( 12 - 2 \beta_{1} - 6 \beta_{2} ) q^{39} + 3 \beta_{2} q^{41} + ( -6 + \beta_{1} + 3 \beta_{2} ) q^{42} + ( -9 + \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{44} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{46} + ( -4 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{47} + \beta_{1} q^{48} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{49} + ( 3 - 5 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{51} + ( -2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{52} + ( 2 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{53} + ( -3 - 2 \beta_{1} + 3 \beta_{2} ) q^{54} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{56} + ( 3 - \beta_{1} - 6 \beta_{2} ) q^{57} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{58} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{59} + ( -2 + 3 \beta_{1} - \beta_{2} - 6 \beta_{3} ) q^{61} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{62} + ( -3 - 2 \beta_{1} + 5 \beta_{3} ) q^{63} + q^{64} + ( -6 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{66} -7 \beta_{2} q^{67} + ( 1 - 2 \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{68} + ( -6 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{69} -6 q^{71} + ( -3 + \beta_{1} - \beta_{3} ) q^{72} + ( 4 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{73} + ( -4 + 4 \beta_{2} ) q^{74} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{76} + ( -2 + 4 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{77} + ( -6 + 12 \beta_{2} + 2 \beta_{3} ) q^{78} + ( -2 + 2 \beta_{2} ) q^{79} + ( 6 - 5 \beta_{1} + 5 \beta_{3} ) q^{81} -3 q^{82} + ( 4 + \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{83} + ( 3 - 6 \beta_{2} - \beta_{3} ) q^{84} + ( 1 - 2 \beta_{1} - 9 \beta_{2} + \beta_{3} ) q^{86} + ( -3 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{87} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{88} + ( 9 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{89} + ( -18 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{91} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{92} + ( 12 - 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{93} + ( -1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{94} -\beta_{3} q^{96} + ( -5 - \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{97} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{98} + ( -6 - \beta_{1} + 3 \beta_{2} + 5 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{3} - 2q^{4} + q^{6} + q^{7} + 4q^{8} - 10q^{9} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{3} - 2q^{4} + q^{6} + q^{7} + 4q^{8} - 10q^{9} + 3q^{11} + q^{12} - 2q^{13} + q^{14} - 2q^{16} + 18q^{17} + 5q^{18} + 2q^{19} + 16q^{21} + 3q^{22} - 3q^{23} - 2q^{24} + 4q^{26} + 16q^{27} - 2q^{28} + 3q^{29} + 2q^{31} - 2q^{32} + 15q^{33} - 9q^{34} + 5q^{36} + 16q^{37} - q^{38} + 34q^{39} + 6q^{41} - 17q^{42} - 17q^{43} - 6q^{44} + 6q^{46} - 9q^{47} + q^{48} - 3q^{49} - 9q^{51} - 2q^{52} - 8q^{54} + q^{56} - q^{57} + 3q^{58} - 3q^{59} - q^{61} - 4q^{62} - 19q^{63} + 4q^{64} - 18q^{66} - 14q^{67} - 9q^{68} - 15q^{69} - 24q^{71} - 10q^{72} + 22q^{73} - 8q^{74} - q^{76} - 18q^{77} - 2q^{78} - 4q^{79} + 14q^{81} - 12q^{82} + 9q^{83} + q^{84} - 17q^{86} - 18q^{87} + 3q^{88} + 30q^{89} - 68q^{91} - 3q^{92} + 32q^{93} - 9q^{94} + q^{96} - 11q^{97} + 6q^{98} - 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 2 \beta_{1} + 3\)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
1.68614 + 0.396143i
−1.18614 1.26217i
−1.18614 + 1.26217i
1.68614 0.396143i
−0.500000 + 0.866025i −0.500000 1.65831i −0.500000 0.866025i 0 1.68614 + 0.396143i −1.18614 + 2.05446i 1.00000 −2.50000 + 1.65831i 0
151.2 −0.500000 + 0.866025i −0.500000 + 1.65831i −0.500000 0.866025i 0 −1.18614 1.26217i 1.68614 2.92048i 1.00000 −2.50000 1.65831i 0
301.1 −0.500000 0.866025i −0.500000 1.65831i −0.500000 + 0.866025i 0 −1.18614 + 1.26217i 1.68614 + 2.92048i 1.00000 −2.50000 + 1.65831i 0
301.2 −0.500000 0.866025i −0.500000 + 1.65831i −0.500000 + 0.866025i 0 1.68614 0.396143i −1.18614 2.05446i 1.00000 −2.50000 1.65831i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.e.j 4
3.b odd 2 1 1350.2.e.l 4
5.b even 2 1 90.2.e.c 4
5.c odd 4 2 450.2.j.g 8
9.c even 3 1 inner 450.2.e.j 4
9.c even 3 1 4050.2.a.bw 2
9.d odd 6 1 1350.2.e.l 4
9.d odd 6 1 4050.2.a.bo 2
15.d odd 2 1 270.2.e.c 4
15.e even 4 2 1350.2.j.f 8
20.d odd 2 1 720.2.q.f 4
45.h odd 6 1 270.2.e.c 4
45.h odd 6 1 810.2.a.k 2
45.j even 6 1 90.2.e.c 4
45.j even 6 1 810.2.a.i 2
45.k odd 12 2 450.2.j.g 8
45.k odd 12 2 4050.2.c.v 4
45.l even 12 2 1350.2.j.f 8
45.l even 12 2 4050.2.c.ba 4
60.h even 2 1 2160.2.q.f 4
180.n even 6 1 2160.2.q.f 4
180.n even 6 1 6480.2.a.bn 2
180.p odd 6 1 720.2.q.f 4
180.p odd 6 1 6480.2.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.c 4 5.b even 2 1
90.2.e.c 4 45.j even 6 1
270.2.e.c 4 15.d odd 2 1
270.2.e.c 4 45.h odd 6 1
450.2.e.j 4 1.a even 1 1 trivial
450.2.e.j 4 9.c even 3 1 inner
450.2.j.g 8 5.c odd 4 2
450.2.j.g 8 45.k odd 12 2
720.2.q.f 4 20.d odd 2 1
720.2.q.f 4 180.p odd 6 1
810.2.a.i 2 45.j even 6 1
810.2.a.k 2 45.h odd 6 1
1350.2.e.l 4 3.b odd 2 1
1350.2.e.l 4 9.d odd 6 1
1350.2.j.f 8 15.e even 4 2
1350.2.j.f 8 45.l even 12 2
2160.2.q.f 4 60.h even 2 1
2160.2.q.f 4 180.n even 6 1
4050.2.a.bo 2 9.d odd 6 1
4050.2.a.bw 2 9.c even 3 1
4050.2.c.v 4 45.k odd 12 2
4050.2.c.ba 4 45.l even 12 2
6480.2.a.be 2 180.p odd 6 1
6480.2.a.bn 2 180.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 \(S_{2}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{4} - T_{7}^{3} + 9 T_{7}^{2} + 8 T_{7} + 64 \)
\( T_{11}^{4} - 3 T_{11}^{3} + 15 T_{11}^{2} + 18 T_{11} + 36 \)
\( T_{17}^{2} - 9 T_{17} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 3 + T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 64 + 8 T + 9 T^{2} - T^{3} + T^{4} \)
$11$ \( 36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4} \)
$13$ \( 1024 - 64 T + 36 T^{2} + 2 T^{3} + T^{4} \)
$17$ \( ( 12 - 9 T + T^{2} )^{2} \)
$19$ \( ( -8 - T + T^{2} )^{2} \)
$23$ \( 36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4} \)
$29$ \( 36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4} \)
$31$ \( 1024 + 64 T + 36 T^{2} - 2 T^{3} + T^{4} \)
$37$ \( ( -4 + T )^{4} \)
$41$ \( ( 9 - 3 T + T^{2} )^{2} \)
$43$ \( 4096 + 1088 T + 225 T^{2} + 17 T^{3} + T^{4} \)
$47$ \( 144 + 108 T + 69 T^{2} + 9 T^{3} + T^{4} \)
$53$ \( ( -132 + T^{2} )^{2} \)
$59$ \( 36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4} \)
$61$ \( 5476 - 74 T + 75 T^{2} + T^{3} + T^{4} \)
$67$ \( ( 49 + 7 T + T^{2} )^{2} \)
$71$ \( ( 6 + T )^{4} \)
$73$ \( ( -44 - 11 T + T^{2} )^{2} \)
$79$ \( ( 4 + 2 T + T^{2} )^{2} \)
$83$ \( 144 - 108 T + 69 T^{2} - 9 T^{3} + T^{4} \)
$89$ \( ( -18 - 15 T + T^{2} )^{2} \)
$97$ \( 484 + 242 T + 99 T^{2} + 11 T^{3} + T^{4} \)
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