Properties

Label 450.6.f.a
Level $450$
Weight $6$
Character orbit 450.f
Analytic conductor $72.173$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q -4 \zeta_{8} q^{2} + 16 \zeta_{8}^{2} q^{4} + ( -33 + 33 \zeta_{8}^{2} ) q^{7} -64 \zeta_{8}^{3} q^{8} +O(q^{10})\) \( q -4 \zeta_{8} q^{2} + 16 \zeta_{8}^{2} q^{4} + ( -33 + 33 \zeta_{8}^{2} ) q^{7} -64 \zeta_{8}^{3} q^{8} + ( 183 \zeta_{8} + 183 \zeta_{8}^{3} ) q^{11} + ( -213 - 213 \zeta_{8}^{2} ) q^{13} + ( 132 \zeta_{8} - 132 \zeta_{8}^{3} ) q^{14} -256 q^{16} -654 \zeta_{8} q^{17} -722 \zeta_{8}^{2} q^{19} + ( 732 - 732 \zeta_{8}^{2} ) q^{22} + 1146 \zeta_{8}^{3} q^{23} + ( 852 \zeta_{8} + 852 \zeta_{8}^{3} ) q^{26} + ( -528 - 528 \zeta_{8}^{2} ) q^{28} + ( -1794 \zeta_{8} + 1794 \zeta_{8}^{3} ) q^{29} + 2764 q^{31} + 1024 \zeta_{8} q^{32} + 2616 \zeta_{8}^{2} q^{34} + ( 4467 - 4467 \zeta_{8}^{2} ) q^{37} + 2888 \zeta_{8}^{3} q^{38} + ( 5397 \zeta_{8} + 5397 \zeta_{8}^{3} ) q^{41} + ( -3240 - 3240 \zeta_{8}^{2} ) q^{43} + ( -2928 \zeta_{8} + 2928 \zeta_{8}^{3} ) q^{44} + 4584 q^{46} -14040 \zeta_{8} q^{47} + 14629 \zeta_{8}^{2} q^{49} + ( 3408 - 3408 \zeta_{8}^{2} ) q^{52} -11226 \zeta_{8}^{3} q^{53} + ( 2112 \zeta_{8} + 2112 \zeta_{8}^{3} ) q^{56} + ( 7176 + 7176 \zeta_{8}^{2} ) q^{58} + ( -8097 \zeta_{8} + 8097 \zeta_{8}^{3} ) q^{59} + 29750 q^{61} -11056 \zeta_{8} q^{62} -4096 \zeta_{8}^{2} q^{64} + ( 19866 - 19866 \zeta_{8}^{2} ) q^{67} -10464 \zeta_{8}^{3} q^{68} + ( 50406 \zeta_{8} + 50406 \zeta_{8}^{3} ) q^{71} + ( 45066 + 45066 \zeta_{8}^{2} ) q^{73} + ( -17868 \zeta_{8} + 17868 \zeta_{8}^{3} ) q^{74} + 11552 q^{76} -12078 \zeta_{8} q^{77} -2992 \zeta_{8}^{2} q^{79} + ( 21588 - 21588 \zeta_{8}^{2} ) q^{82} -40452 \zeta_{8}^{3} q^{83} + ( 12960 \zeta_{8} + 12960 \zeta_{8}^{3} ) q^{86} + ( 11712 + 11712 \zeta_{8}^{2} ) q^{88} + ( -8277 \zeta_{8} + 8277 \zeta_{8}^{3} ) q^{89} + 14058 q^{91} -18336 \zeta_{8} q^{92} + 56160 \zeta_{8}^{2} q^{94} + ( 111372 - 111372 \zeta_{8}^{2} ) q^{97} -58516 \zeta_{8}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 132q^{7} + O(q^{10}) \) \( 4q - 132q^{7} - 852q^{13} - 1024q^{16} + 2928q^{22} - 2112q^{28} + 11056q^{31} + 17868q^{37} - 12960q^{43} + 18336q^{46} + 13632q^{52} + 28704q^{58} + 119000q^{61} + 79464q^{67} + 180264q^{73} + 46208q^{76} + 86352q^{82} + 46848q^{88} + 56232q^{91} + 445488q^{97} + O(q^{100}) \)

Character values

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

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

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
−2.82843 + 2.82843i 0 16.0000i 0 0 −33.0000 33.0000i 45.2548 + 45.2548i 0 0
107.2 2.82843 2.82843i 0 16.0000i 0 0 −33.0000 33.0000i −45.2548 45.2548i 0 0
143.1 −2.82843 2.82843i 0 16.0000i 0 0 −33.0000 + 33.0000i 45.2548 45.2548i 0 0
143.2 2.82843 + 2.82843i 0 16.0000i 0 0 −33.0000 + 33.0000i −45.2548 + 45.2548i 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 450.6.f.a 4
3.b odd 2 1 inner 450.6.f.a 4
5.b even 2 1 450.6.f.c yes 4
5.c odd 4 1 inner 450.6.f.a 4
5.c odd 4 1 450.6.f.c yes 4
15.d odd 2 1 450.6.f.c yes 4
15.e even 4 1 inner 450.6.f.a 4
15.e even 4 1 450.6.f.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.6.f.a 4 1.a even 1 1 trivial
450.6.f.a 4 3.b odd 2 1 inner
450.6.f.a 4 5.c odd 4 1 inner
450.6.f.a 4 15.e even 4 1 inner
450.6.f.c yes 4 5.b even 2 1
450.6.f.c yes 4 5.c odd 4 1
450.6.f.c yes 4 15.d odd 2 1
450.6.f.c yes 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 66 T_{7} + 2178 \) acting on \(S_{6}^{\mathrm{new}}(450, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 2178 + 66 T + T^{2} )^{2} \)
$11$ \( ( 66978 + T^{2} )^{2} \)
$13$ \( ( 90738 + 426 T + T^{2} )^{2} \)
$17$ \( 182940976656 + T^{4} \)
$19$ \( ( 521284 + T^{2} )^{2} \)
$23$ \( 1724798915856 + T^{4} \)
$29$ \( ( -6436872 + T^{2} )^{2} \)
$31$ \( ( -2764 + T )^{4} \)
$37$ \( ( 39908178 - 8934 T + T^{2} )^{2} \)
$41$ \( ( 58255218 + T^{2} )^{2} \)
$43$ \( ( 20995200 + 6480 T + T^{2} )^{2} \)
$47$ \( 38856925186560000 + T^{4} \)
$53$ \( 15881815684501776 + T^{4} \)
$59$ \( ( -131122818 + T^{2} )^{2} \)
$61$ \( ( -29750 + T )^{4} \)
$67$ \( ( 789315912 - 39732 T + T^{2} )^{2} \)
$71$ \( ( 5081529672 + T^{2} )^{2} \)
$73$ \( ( 4061888712 - 90132 T + T^{2} )^{2} \)
$79$ \( ( 8952064 + T^{2} )^{2} \)
$83$ \( 2677688135405404416 + T^{4} \)
$89$ \( ( -137017458 + T^{2} )^{2} \)
$97$ \( ( 24807444768 - 222744 T + T^{2} )^{2} \)
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