Properties

Label 450.2.e.c
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: yes
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} + ( -1 + 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( -1 - \zeta_{6} ) q^{6} + ( -2 + 2 \zeta_{6} ) q^{7} + q^{8} -3 q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} + ( -1 + 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( -1 - \zeta_{6} ) q^{6} + ( -2 + 2 \zeta_{6} ) q^{7} + q^{8} -3 q^{9} + ( 2 - \zeta_{6} ) q^{12} + 4 \zeta_{6} q^{13} -2 \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} -6 q^{17} + ( 3 - 3 \zeta_{6} ) q^{18} -7 q^{19} + ( -2 - 2 \zeta_{6} ) q^{21} + ( -1 + 2 \zeta_{6} ) q^{24} -4 q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + 2 q^{28} + ( 6 - 6 \zeta_{6} ) q^{29} + 10 \zeta_{6} q^{31} -\zeta_{6} q^{32} + ( 6 - 6 \zeta_{6} ) q^{34} + 3 \zeta_{6} q^{36} + 2 q^{37} + ( 7 - 7 \zeta_{6} ) q^{38} + ( -8 + 4 \zeta_{6} ) q^{39} -9 \zeta_{6} q^{41} + ( 4 - 2 \zeta_{6} ) q^{42} + ( 1 - \zeta_{6} ) q^{43} + ( -6 + 6 \zeta_{6} ) q^{47} + ( -1 - \zeta_{6} ) q^{48} + 3 \zeta_{6} q^{49} + ( 6 - 12 \zeta_{6} ) q^{51} + ( 4 - 4 \zeta_{6} ) q^{52} -12 q^{53} + ( 3 + 3 \zeta_{6} ) q^{54} + ( -2 + 2 \zeta_{6} ) q^{56} + ( 7 - 14 \zeta_{6} ) q^{57} + 6 \zeta_{6} q^{58} + 9 \zeta_{6} q^{59} + ( 4 - 4 \zeta_{6} ) q^{61} -10 q^{62} + ( 6 - 6 \zeta_{6} ) q^{63} + q^{64} + 13 \zeta_{6} q^{67} + 6 \zeta_{6} q^{68} + 6 q^{71} -3 q^{72} - q^{73} + ( -2 + 2 \zeta_{6} ) q^{74} + 7 \zeta_{6} q^{76} + ( 4 - 8 \zeta_{6} ) q^{78} + ( -2 + 2 \zeta_{6} ) q^{79} + 9 q^{81} + 9 q^{82} + ( -9 + 9 \zeta_{6} ) q^{83} + ( -2 + 4 \zeta_{6} ) q^{84} + \zeta_{6} q^{86} + ( 6 + 6 \zeta_{6} ) q^{87} + 15 q^{89} -8 q^{91} + ( -20 + 10 \zeta_{6} ) q^{93} -6 \zeta_{6} q^{94} + ( 2 - \zeta_{6} ) q^{96} + ( -17 + 17 \zeta_{6} ) q^{97} -3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - 3q^{6} - 2q^{7} + 2q^{8} - 6q^{9} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - 3q^{6} - 2q^{7} + 2q^{8} - 6q^{9} + 3q^{12} + 4q^{13} - 2q^{14} - q^{16} - 12q^{17} + 3q^{18} - 14q^{19} - 6q^{21} - 8q^{26} + 4q^{28} + 6q^{29} + 10q^{31} - q^{32} + 6q^{34} + 3q^{36} + 4q^{37} + 7q^{38} - 12q^{39} - 9q^{41} + 6q^{42} + q^{43} - 6q^{47} - 3q^{48} + 3q^{49} + 4q^{52} - 24q^{53} + 9q^{54} - 2q^{56} + 6q^{58} + 9q^{59} + 4q^{61} - 20q^{62} + 6q^{63} + 2q^{64} + 13q^{67} + 6q^{68} + 12q^{71} - 6q^{72} - 2q^{73} - 2q^{74} + 7q^{76} - 2q^{79} + 18q^{81} + 18q^{82} - 9q^{83} + q^{86} + 18q^{87} + 30q^{89} - 16q^{91} - 30q^{93} - 6q^{94} + 3q^{96} - 17q^{97} - 6q^{98} + O(q^{100}) \)

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 1.73205i −0.500000 0.866025i 0 −1.50000 0.866025i −1.00000 + 1.73205i 1.00000 −3.00000 0
301.1 −0.500000 0.866025i 1.73205i −0.500000 + 0.866025i 0 −1.50000 + 0.866025i −1.00000 1.73205i 1.00000 −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
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.c 2
3.b odd 2 1 1350.2.e.h 2
5.b even 2 1 450.2.e.f yes 2
5.c odd 4 2 450.2.j.b 4
9.c even 3 1 inner 450.2.e.c 2
9.c even 3 1 4050.2.a.bg 1
9.d odd 6 1 1350.2.e.h 2
9.d odd 6 1 4050.2.a.o 1
15.d odd 2 1 1350.2.e.d 2
15.e even 4 2 1350.2.j.b 4
45.h odd 6 1 1350.2.e.d 2
45.h odd 6 1 4050.2.a.u 1
45.j even 6 1 450.2.e.f yes 2
45.j even 6 1 4050.2.a.d 1
45.k odd 12 2 450.2.j.b 4
45.k odd 12 2 4050.2.c.h 2
45.l even 12 2 1350.2.j.b 4
45.l even 12 2 4050.2.c.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.c 2 1.a even 1 1 trivial
450.2.e.c 2 9.c even 3 1 inner
450.2.e.f yes 2 5.b even 2 1
450.2.e.f yes 2 45.j even 6 1
450.2.j.b 4 5.c odd 4 2
450.2.j.b 4 45.k odd 12 2
1350.2.e.d 2 15.d odd 2 1
1350.2.e.d 2 45.h odd 6 1
1350.2.e.h 2 3.b odd 2 1
1350.2.e.h 2 9.d odd 6 1
1350.2.j.b 4 15.e even 4 2
1350.2.j.b 4 45.l even 12 2
4050.2.a.d 1 45.j even 6 1
4050.2.a.o 1 9.d odd 6 1
4050.2.a.u 1 45.h odd 6 1
4050.2.a.bg 1 9.c even 3 1
4050.2.c.h 2 45.k odd 12 2
4050.2.c.m 2 45.l even 12 2

Hecke kernels

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

\( T_{7}^{2} + 2 T_{7} + 4 \)
\( T_{11} \)
\( T_{17} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 3 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + 2 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 16 - 4 T + T^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( ( 7 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( 36 - 6 T + T^{2} \)
$31$ \( 100 - 10 T + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( 81 + 9 T + T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( 36 + 6 T + T^{2} \)
$53$ \( ( 12 + T )^{2} \)
$59$ \( 81 - 9 T + T^{2} \)
$61$ \( 16 - 4 T + T^{2} \)
$67$ \( 169 - 13 T + T^{2} \)
$71$ \( ( -6 + T )^{2} \)
$73$ \( ( 1 + T )^{2} \)
$79$ \( 4 + 2 T + T^{2} \)
$83$ \( 81 + 9 T + T^{2} \)
$89$ \( ( -15 + T )^{2} \)
$97$ \( 289 + 17 T + T^{2} \)
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