Properties

Label 450.2.e.e
Level $450$
Weight $2$
Character orbit 450.e
Analytic conductor $3.593$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i −1.50000 + 0.866025i −0.500000 0.866025i 0 1.73205i −0.500000 + 0.866025i −1.00000 1.50000 2.59808i 0
301.1 0.500000 + 0.866025i −1.50000 0.866025i −0.500000 + 0.866025i 0 1.73205i −0.500000 0.866025i −1.00000 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
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.e 2
3.b odd 2 1 1350.2.e.b 2
5.b even 2 1 90.2.e.a 2
5.c odd 4 2 450.2.j.c 4
9.c even 3 1 inner 450.2.e.e 2
9.c even 3 1 4050.2.a.n 1
9.d odd 6 1 1350.2.e.b 2
9.d odd 6 1 4050.2.a.ba 1
15.d odd 2 1 270.2.e.b 2
15.e even 4 2 1350.2.j.e 4
20.d odd 2 1 720.2.q.b 2
45.h odd 6 1 270.2.e.b 2
45.h odd 6 1 810.2.a.b 1
45.j even 6 1 90.2.e.a 2
45.j even 6 1 810.2.a.g 1
45.k odd 12 2 450.2.j.c 4
45.k odd 12 2 4050.2.c.t 2
45.l even 12 2 1350.2.j.e 4
45.l even 12 2 4050.2.c.a 2
60.h even 2 1 2160.2.q.b 2
180.n even 6 1 2160.2.q.b 2
180.n even 6 1 6480.2.a.v 1
180.p odd 6 1 720.2.q.b 2
180.p odd 6 1 6480.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.a 2 5.b even 2 1
90.2.e.a 2 45.j even 6 1
270.2.e.b 2 15.d odd 2 1
270.2.e.b 2 45.h odd 6 1
450.2.e.e 2 1.a even 1 1 trivial
450.2.e.e 2 9.c even 3 1 inner
450.2.j.c 4 5.c odd 4 2
450.2.j.c 4 45.k odd 12 2
720.2.q.b 2 20.d odd 2 1
720.2.q.b 2 180.p odd 6 1
810.2.a.b 1 45.h odd 6 1
810.2.a.g 1 45.j even 6 1
1350.2.e.b 2 3.b odd 2 1
1350.2.e.b 2 9.d odd 6 1
1350.2.j.e 4 15.e even 4 2
1350.2.j.e 4 45.l even 12 2
2160.2.q.b 2 60.h even 2 1
2160.2.q.b 2 180.n even 6 1
4050.2.a.n 1 9.c even 3 1
4050.2.a.ba 1 9.d odd 6 1
4050.2.c.a 2 45.l even 12 2
4050.2.c.t 2 45.k odd 12 2
6480.2.a.g 1 180.p odd 6 1
6480.2.a.v 1 180.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 \(S_{2}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{2} + T_{7} + 1 \)
\( T_{11}^{2} + 6 T_{11} + 36 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 3 + 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T + T^{2} \)
$11$ \( 36 + 6 T + T^{2} \)
$13$ \( 4 - 2 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( 81 - 9 T + T^{2} \)
$29$ \( 9 + 3 T + T^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( ( 8 + T )^{2} \)
$41$ \( 9 - 3 T + T^{2} \)
$43$ \( 64 - 8 T + T^{2} \)
$47$ \( 9 + 3 T + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 36 + 6 T + T^{2} \)
$61$ \( 169 - 13 T + T^{2} \)
$67$ \( 169 + 13 T + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( ( -4 + T )^{2} \)
$79$ \( 100 - 10 T + T^{2} \)
$83$ \( 81 + 9 T + T^{2} \)
$89$ \( ( -9 + T )^{2} \)
$97$ \( 4 - 2 T + T^{2} \)
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