Properties

Label 450.2.f.a
Level $450$
Weight $2$
Character orbit 450.f
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.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( -3 + 3 \zeta_{8}^{2} ) q^{7} + \zeta_{8}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( -3 + 3 \zeta_{8}^{2} ) q^{7} + \zeta_{8}^{3} q^{8} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{11} + ( -3 - 3 \zeta_{8}^{2} ) q^{13} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{14} - q^{16} + 6 \zeta_{8} q^{17} -2 \zeta_{8}^{2} q^{19} + ( -3 + 3 \zeta_{8}^{2} ) q^{22} + 6 \zeta_{8}^{3} q^{23} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{26} + ( -3 - 3 \zeta_{8}^{2} ) q^{28} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{29} + 4 q^{31} -\zeta_{8} q^{32} + 6 \zeta_{8}^{2} q^{34} + ( -3 + 3 \zeta_{8}^{2} ) q^{37} -2 \zeta_{8}^{3} q^{38} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{41} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{44} -6 q^{46} -11 \zeta_{8}^{2} q^{49} + ( 3 - 3 \zeta_{8}^{2} ) q^{52} -6 \zeta_{8}^{3} q^{53} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{56} + ( 6 + 6 \zeta_{8}^{2} ) q^{58} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{59} -10 q^{61} + 4 \zeta_{8} q^{62} -\zeta_{8}^{2} q^{64} + ( 6 - 6 \zeta_{8}^{2} ) q^{67} + 6 \zeta_{8}^{3} q^{68} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{71} + ( 6 + 6 \zeta_{8}^{2} ) q^{73} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{74} + 2 q^{76} -18 \zeta_{8} q^{77} + 8 \zeta_{8}^{2} q^{79} + ( 3 - 3 \zeta_{8}^{2} ) q^{82} -12 \zeta_{8}^{3} q^{83} + ( -3 - 3 \zeta_{8}^{2} ) q^{88} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{89} + 18 q^{91} -6 \zeta_{8} q^{92} + ( 12 - 12 \zeta_{8}^{2} ) q^{97} -11 \zeta_{8}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{7} + O(q^{10}) \) \( 4q - 12q^{7} - 12q^{13} - 4q^{16} - 12q^{22} - 12q^{28} + 16q^{31} - 12q^{37} - 24q^{46} + 12q^{52} + 24q^{58} - 40q^{61} + 24q^{67} + 24q^{73} + 8q^{76} + 12q^{82} - 12q^{88} + 72q^{91} + 48q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i 0 1.00000i 0 0 −3.00000 3.00000i 0.707107 + 0.707107i 0 0
107.2 0.707107 0.707107i 0 1.00000i 0 0 −3.00000 3.00000i −0.707107 0.707107i 0 0
143.1 −0.707107 0.707107i 0 1.00000i 0 0 −3.00000 + 3.00000i 0.707107 0.707107i 0 0
143.2 0.707107 + 0.707107i 0 1.00000i 0 0 −3.00000 + 3.00000i −0.707107 + 0.707107i 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
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.f.a 4
3.b odd 2 1 inner 450.2.f.a 4
4.b odd 2 1 3600.2.w.h 4
5.b even 2 1 450.2.f.c yes 4
5.c odd 4 1 inner 450.2.f.a 4
5.c odd 4 1 450.2.f.c yes 4
12.b even 2 1 3600.2.w.h 4
15.d odd 2 1 450.2.f.c yes 4
15.e even 4 1 inner 450.2.f.a 4
15.e even 4 1 450.2.f.c yes 4
20.d odd 2 1 3600.2.w.a 4
20.e even 4 1 3600.2.w.a 4
20.e even 4 1 3600.2.w.h 4
60.h even 2 1 3600.2.w.a 4
60.l odd 4 1 3600.2.w.a 4
60.l odd 4 1 3600.2.w.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.f.a 4 1.a even 1 1 trivial
450.2.f.a 4 3.b odd 2 1 inner
450.2.f.a 4 5.c odd 4 1 inner
450.2.f.a 4 15.e even 4 1 inner
450.2.f.c yes 4 5.b even 2 1
450.2.f.c yes 4 5.c odd 4 1
450.2.f.c yes 4 15.d odd 2 1
450.2.f.c yes 4 15.e even 4 1
3600.2.w.a 4 20.d odd 2 1
3600.2.w.a 4 20.e even 4 1
3600.2.w.a 4 60.h even 2 1
3600.2.w.a 4 60.l odd 4 1
3600.2.w.h 4 4.b odd 2 1
3600.2.w.h 4 12.b even 2 1
3600.2.w.h 4 20.e even 4 1
3600.2.w.h 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 6 T_{7} + 18 \) acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 18 + 6 T + T^{2} )^{2} \)
$11$ \( ( 18 + T^{2} )^{2} \)
$13$ \( ( 18 + 6 T + T^{2} )^{2} \)
$17$ \( 1296 + T^{4} \)
$19$ \( ( 4 + T^{2} )^{2} \)
$23$ \( 1296 + T^{4} \)
$29$ \( ( -72 + T^{2} )^{2} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( ( 18 + 6 T + T^{2} )^{2} \)
$41$ \( ( 18 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( 1296 + T^{4} \)
$59$ \( ( -18 + T^{2} )^{2} \)
$61$ \( ( 10 + T )^{4} \)
$67$ \( ( 72 - 12 T + T^{2} )^{2} \)
$71$ \( ( 72 + T^{2} )^{2} \)
$73$ \( ( 72 - 12 T + T^{2} )^{2} \)
$79$ \( ( 64 + T^{2} )^{2} \)
$83$ \( 20736 + T^{4} \)
$89$ \( ( -18 + T^{2} )^{2} \)
$97$ \( ( 288 - 24 T + T^{2} )^{2} \)
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