Properties

Label 450.2.f.b
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: no (minimal twist has level 90)
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.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( -2 + 2 \zeta_{8}^{2} ) q^{7} + \zeta_{8}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( -2 + 2 \zeta_{8}^{2} ) q^{7} + \zeta_{8}^{3} q^{8} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{11} + ( 3 + 3 \zeta_{8}^{2} ) q^{13} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{14} - q^{16} -4 \zeta_{8} q^{17} + 8 \zeta_{8}^{2} q^{19} + ( -2 + 2 \zeta_{8}^{2} ) q^{22} -4 \zeta_{8}^{3} q^{23} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{26} + ( -2 - 2 \zeta_{8}^{2} ) q^{28} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{29} + 4 q^{31} -\zeta_{8} q^{32} -4 \zeta_{8}^{2} q^{34} + ( 3 - 3 \zeta_{8}^{2} ) q^{37} + 8 \zeta_{8}^{3} q^{38} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{41} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{44} + 4 q^{46} -\zeta_{8}^{2} q^{49} + ( -3 + 3 \zeta_{8}^{2} ) q^{52} + 4 \zeta_{8}^{3} q^{53} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{56} + ( -1 - \zeta_{8}^{2} ) q^{58} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{59} + 4 \zeta_{8} q^{62} -\zeta_{8}^{2} q^{64} + ( 4 - 4 \zeta_{8}^{2} ) q^{67} -4 \zeta_{8}^{3} q^{68} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} + ( -1 - \zeta_{8}^{2} ) q^{73} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{74} -8 q^{76} -8 \zeta_{8} q^{77} -12 \zeta_{8}^{2} q^{79} + ( 7 - 7 \zeta_{8}^{2} ) q^{82} -12 \zeta_{8}^{3} q^{83} + ( -2 - 2 \zeta_{8}^{2} ) q^{88} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{89} -12 q^{91} + 4 \zeta_{8} q^{92} + ( 3 - 3 \zeta_{8}^{2} ) q^{97} -\zeta_{8}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{7} + O(q^{10}) \) \( 4q - 8q^{7} + 12q^{13} - 4q^{16} - 8q^{22} - 8q^{28} + 16q^{31} + 12q^{37} + 16q^{46} - 12q^{52} - 4q^{58} + 16q^{67} - 4q^{73} - 32q^{76} + 28q^{82} - 8q^{88} - 48q^{91} + 12q^{97} + O(q^{100}) \)

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 −2.00000 2.00000i 0.707107 + 0.707107i 0 0
107.2 0.707107 0.707107i 0 1.00000i 0 0 −2.00000 2.00000i −0.707107 0.707107i 0 0
143.1 −0.707107 0.707107i 0 1.00000i 0 0 −2.00000 + 2.00000i 0.707107 0.707107i 0 0
143.2 0.707107 + 0.707107i 0 1.00000i 0 0 −2.00000 + 2.00000i −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.b 4
3.b odd 2 1 inner 450.2.f.b 4
4.b odd 2 1 3600.2.w.g 4
5.b even 2 1 90.2.f.a 4
5.c odd 4 1 90.2.f.a 4
5.c odd 4 1 inner 450.2.f.b 4
12.b even 2 1 3600.2.w.g 4
15.d odd 2 1 90.2.f.a 4
15.e even 4 1 90.2.f.a 4
15.e even 4 1 inner 450.2.f.b 4
20.d odd 2 1 720.2.w.a 4
20.e even 4 1 720.2.w.a 4
20.e even 4 1 3600.2.w.g 4
40.e odd 2 1 2880.2.w.c 4
40.f even 2 1 2880.2.w.l 4
40.i odd 4 1 2880.2.w.l 4
40.k even 4 1 2880.2.w.c 4
45.h odd 6 2 810.2.m.c 8
45.j even 6 2 810.2.m.c 8
45.k odd 12 2 810.2.m.c 8
45.l even 12 2 810.2.m.c 8
60.h even 2 1 720.2.w.a 4
60.l odd 4 1 720.2.w.a 4
60.l odd 4 1 3600.2.w.g 4
120.i odd 2 1 2880.2.w.l 4
120.m even 2 1 2880.2.w.c 4
120.q odd 4 1 2880.2.w.c 4
120.w even 4 1 2880.2.w.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.f.a 4 5.b even 2 1
90.2.f.a 4 5.c odd 4 1
90.2.f.a 4 15.d odd 2 1
90.2.f.a 4 15.e even 4 1
450.2.f.b 4 1.a even 1 1 trivial
450.2.f.b 4 3.b odd 2 1 inner
450.2.f.b 4 5.c odd 4 1 inner
450.2.f.b 4 15.e even 4 1 inner
720.2.w.a 4 20.d odd 2 1
720.2.w.a 4 20.e even 4 1
720.2.w.a 4 60.h even 2 1
720.2.w.a 4 60.l odd 4 1
810.2.m.c 8 45.h odd 6 2
810.2.m.c 8 45.j even 6 2
810.2.m.c 8 45.k odd 12 2
810.2.m.c 8 45.l even 12 2
2880.2.w.c 4 40.e odd 2 1
2880.2.w.c 4 40.k even 4 1
2880.2.w.c 4 120.m even 2 1
2880.2.w.c 4 120.q odd 4 1
2880.2.w.l 4 40.f even 2 1
2880.2.w.l 4 40.i odd 4 1
2880.2.w.l 4 120.i odd 2 1
2880.2.w.l 4 120.w even 4 1
3600.2.w.g 4 4.b odd 2 1
3600.2.w.g 4 12.b even 2 1
3600.2.w.g 4 20.e even 4 1
3600.2.w.g 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 4 T_{7} + 8 \) acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 8 + 4 T + T^{2} )^{2} \)
$11$ \( ( 8 + T^{2} )^{2} \)
$13$ \( ( 18 - 6 T + T^{2} )^{2} \)
$17$ \( 256 + T^{4} \)
$19$ \( ( 64 + T^{2} )^{2} \)
$23$ \( 256 + T^{4} \)
$29$ \( ( -2 + T^{2} )^{2} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( ( 18 - 6 T + T^{2} )^{2} \)
$41$ \( ( 98 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( 256 + T^{4} \)
$59$ \( ( -8 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( 32 - 8 T + T^{2} )^{2} \)
$71$ \( ( 32 + T^{2} )^{2} \)
$73$ \( ( 2 + 2 T + T^{2} )^{2} \)
$79$ \( ( 144 + T^{2} )^{2} \)
$83$ \( 20736 + T^{4} \)
$89$ \( ( -98 + T^{2} )^{2} \)
$97$ \( ( 18 - 6 T + T^{2} )^{2} \)
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