Properties

Label 450.2.c.d
Level $450$
Weight $2$
Character orbit 450.c
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} -2 i q^{7} -i q^{8} +O(q^{10})\) \( q + i q^{2} - q^{4} -2 i q^{7} -i q^{8} + 6 q^{11} -4 i q^{13} + 2 q^{14} + q^{16} + 6 i q^{17} + 4 q^{19} + 6 i q^{22} + 4 q^{26} + 2 i q^{28} + 6 q^{29} -4 q^{31} + i q^{32} -6 q^{34} -8 i q^{37} + 4 i q^{38} + 8 i q^{43} -6 q^{44} + 3 q^{49} + 4 i q^{52} -6 i q^{53} -2 q^{56} + 6 i q^{58} -6 q^{59} + 2 q^{61} -4 i q^{62} - q^{64} + 4 i q^{67} -6 i q^{68} -12 q^{71} -10 i q^{73} + 8 q^{74} -4 q^{76} -12 i q^{77} + 4 q^{79} + 12 i q^{83} -8 q^{86} -6 i q^{88} -12 q^{89} -8 q^{91} -2 i q^{97} + 3 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 12q^{11} + 4q^{14} + 2q^{16} + 8q^{19} + 8q^{26} + 12q^{29} - 8q^{31} - 12q^{34} - 12q^{44} + 6q^{49} - 4q^{56} - 12q^{59} + 4q^{61} - 2q^{64} - 24q^{71} + 16q^{74} - 8q^{76} + 8q^{79} - 16q^{86} - 24q^{89} - 16q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 2.00000i 1.00000i 0 0
199.2 1.00000i 0 −1.00000 0 0 2.00000i 1.00000i 0 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.c.d 2
3.b odd 2 1 450.2.c.a 2
4.b odd 2 1 3600.2.f.a 2
5.b even 2 1 inner 450.2.c.d 2
5.c odd 4 1 90.2.a.a 1
5.c odd 4 1 450.2.a.e 1
12.b even 2 1 3600.2.f.u 2
15.d odd 2 1 450.2.c.a 2
15.e even 4 1 90.2.a.b yes 1
15.e even 4 1 450.2.a.a 1
20.d odd 2 1 3600.2.f.a 2
20.e even 4 1 720.2.a.g 1
20.e even 4 1 3600.2.a.ba 1
35.f even 4 1 4410.2.a.k 1
40.i odd 4 1 2880.2.a.k 1
40.k even 4 1 2880.2.a.h 1
45.k odd 12 2 810.2.e.h 2
45.l even 12 2 810.2.e.e 2
60.h even 2 1 3600.2.f.u 2
60.l odd 4 1 720.2.a.b 1
60.l odd 4 1 3600.2.a.bj 1
105.k odd 4 1 4410.2.a.bf 1
120.q odd 4 1 2880.2.a.u 1
120.w even 4 1 2880.2.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.a.a 1 5.c odd 4 1
90.2.a.b yes 1 15.e even 4 1
450.2.a.a 1 15.e even 4 1
450.2.a.e 1 5.c odd 4 1
450.2.c.a 2 3.b odd 2 1
450.2.c.a 2 15.d odd 2 1
450.2.c.d 2 1.a even 1 1 trivial
450.2.c.d 2 5.b even 2 1 inner
720.2.a.b 1 60.l odd 4 1
720.2.a.g 1 20.e even 4 1
810.2.e.e 2 45.l even 12 2
810.2.e.h 2 45.k odd 12 2
2880.2.a.h 1 40.k even 4 1
2880.2.a.k 1 40.i odd 4 1
2880.2.a.u 1 120.q odd 4 1
2880.2.a.bf 1 120.w even 4 1
3600.2.a.ba 1 20.e even 4 1
3600.2.a.bj 1 60.l odd 4 1
3600.2.f.a 2 4.b odd 2 1
3600.2.f.a 2 20.d odd 2 1
3600.2.f.u 2 12.b even 2 1
3600.2.f.u 2 60.h even 2 1
4410.2.a.k 1 35.f even 4 1
4410.2.a.bf 1 105.k odd 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( 6 + T )^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( 12 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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