Properties

Label 450.6.f.b
Level $450$
Weight $6$
Character orbit 450.f
Analytic conductor $72.173$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more about

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: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + 4 \zeta_{8}^{3} q^{2} -16 \zeta_{8}^{2} q^{4} + ( 22 + 22 \zeta_{8}^{2} ) q^{7} + 64 \zeta_{8} q^{8} +O(q^{10})\) \( q + 4 \zeta_{8}^{3} q^{2} -16 \zeta_{8}^{2} q^{4} + ( 22 + 22 \zeta_{8}^{2} ) q^{7} + 64 \zeta_{8} q^{8} + ( 122 \zeta_{8} + 122 \zeta_{8}^{3} ) q^{11} + ( 837 - 837 \zeta_{8}^{2} ) q^{13} + ( -88 \zeta_{8} + 88 \zeta_{8}^{3} ) q^{14} -256 q^{16} -286 \zeta_{8}^{3} q^{17} + 1012 \zeta_{8}^{2} q^{19} + ( -488 - 488 \zeta_{8}^{2} ) q^{22} -3736 \zeta_{8} q^{23} + ( 3348 \zeta_{8} + 3348 \zeta_{8}^{3} ) q^{26} + ( 352 - 352 \zeta_{8}^{2} ) q^{28} + ( 1951 \zeta_{8} - 1951 \zeta_{8}^{3} ) q^{29} -4136 q^{31} -1024 \zeta_{8}^{3} q^{32} + 1144 \zeta_{8}^{2} q^{34} + ( 2667 + 2667 \zeta_{8}^{2} ) q^{37} -4048 \zeta_{8} q^{38} + ( 7343 \zeta_{8} + 7343 \zeta_{8}^{3} ) q^{41} + ( 10560 - 10560 \zeta_{8}^{2} ) q^{43} + ( 1952 \zeta_{8} - 1952 \zeta_{8}^{3} ) q^{44} + 14944 q^{46} + 22140 \zeta_{8}^{3} q^{47} -15839 \zeta_{8}^{2} q^{49} + ( -13392 - 13392 \zeta_{8}^{2} ) q^{52} + 19066 \zeta_{8} q^{53} + ( 1408 \zeta_{8} + 1408 \zeta_{8}^{3} ) q^{56} + ( -7804 + 7804 \zeta_{8}^{2} ) q^{58} + ( -17402 \zeta_{8} + 17402 \zeta_{8}^{3} ) q^{59} -11040 q^{61} -16544 \zeta_{8}^{3} q^{62} + 4096 \zeta_{8}^{2} q^{64} + ( -27944 - 27944 \zeta_{8}^{2} ) q^{67} -4576 \zeta_{8} q^{68} + ( -29996 \zeta_{8} - 29996 \zeta_{8}^{3} ) q^{71} + ( -2839 + 2839 \zeta_{8}^{2} ) q^{73} + ( -10668 \zeta_{8} + 10668 \zeta_{8}^{3} ) q^{74} + 16192 q^{76} + 5368 \zeta_{8}^{3} q^{77} -98688 \zeta_{8}^{2} q^{79} + ( -29372 - 29372 \zeta_{8}^{2} ) q^{82} -14748 \zeta_{8} q^{83} + ( 42240 \zeta_{8} + 42240 \zeta_{8}^{3} ) q^{86} + ( -7808 + 7808 \zeta_{8}^{2} ) q^{88} + ( 81263 \zeta_{8} - 81263 \zeta_{8}^{3} ) q^{89} + 36828 q^{91} + 59776 \zeta_{8}^{3} q^{92} -88560 \zeta_{8}^{2} q^{94} + ( -23043 - 23043 \zeta_{8}^{2} ) q^{97} + 63356 \zeta_{8} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 88q^{7} + O(q^{10}) \) \( 4q + 88q^{7} + 3348q^{13} - 1024q^{16} - 1952q^{22} + 1408q^{28} - 16544q^{31} + 10668q^{37} + 42240q^{43} + 59776q^{46} - 53568q^{52} - 31216q^{58} - 44160q^{61} - 111776q^{67} - 11356q^{73} + 64768q^{76} - 117488q^{82} - 31232q^{88} + 147312q^{91} - 92172q^{97} + O(q^{100}) \)

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 22.0000 + 22.0000i 45.2548 + 45.2548i 0 0
107.2 2.82843 2.82843i 0 16.0000i 0 0 22.0000 + 22.0000i −45.2548 45.2548i 0 0
143.1 −2.82843 2.82843i 0 16.0000i 0 0 22.0000 22.0000i 45.2548 45.2548i 0 0
143.2 2.82843 + 2.82843i 0 16.0000i 0 0 22.0000 22.0000i −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.b 4
3.b odd 2 1 inner 450.6.f.b 4
5.b even 2 1 90.6.f.b 4
5.c odd 4 1 90.6.f.b 4
5.c odd 4 1 inner 450.6.f.b 4
15.d odd 2 1 90.6.f.b 4
15.e even 4 1 90.6.f.b 4
15.e even 4 1 inner 450.6.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.6.f.b 4 5.b even 2 1
90.6.f.b 4 5.c odd 4 1
90.6.f.b 4 15.d odd 2 1
90.6.f.b 4 15.e even 4 1
450.6.f.b 4 1.a even 1 1 trivial
450.6.f.b 4 3.b odd 2 1 inner
450.6.f.b 4 5.c odd 4 1 inner
450.6.f.b 4 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 44 T_{7} + 968 \) acting on \(S_{6}^{\mathrm{new}}(450, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 968 - 44 T + T^{2} )^{2} \)
$11$ \( ( 29768 + T^{2} )^{2} \)
$13$ \( ( 1401138 - 1674 T + T^{2} )^{2} \)
$17$ \( 6690585616 + T^{4} \)
$19$ \( ( 1024144 + T^{2} )^{2} \)
$23$ \( 194817277628416 + T^{4} \)
$29$ \( ( -7612802 + T^{2} )^{2} \)
$31$ \( ( 4136 + T )^{4} \)
$37$ \( ( 14225778 - 5334 T + T^{2} )^{2} \)
$41$ \( ( 107839298 + T^{2} )^{2} \)
$43$ \( ( 223027200 - 21120 T + T^{2} )^{2} \)
$47$ \( 240276040256160000 + T^{4} \)
$53$ \( 132141232964670736 + T^{4} \)
$59$ \( ( -605659208 + T^{2} )^{2} \)
$61$ \( ( 11040 + T )^{4} \)
$67$ \( ( 1561734272 + 55888 T + T^{2} )^{2} \)
$71$ \( ( 1799520032 + T^{2} )^{2} \)
$73$ \( ( 16119842 + 5678 T + T^{2} )^{2} \)
$79$ \( ( 9739321344 + T^{2} )^{2} \)
$83$ \( 47307774252278016 + T^{4} \)
$89$ \( ( -13207350338 + T^{2} )^{2} \)
$97$ \( ( 1061959698 + 46086 T + T^{2} )^{2} \)
show more
show less