Properties

Label 450.6.f.d
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: \(\Z[a_1, \ldots, a_{29}]\)
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.

\(f(q)\) \(=\) \( q -4 \zeta_{8} q^{2} + 16 \zeta_{8}^{2} q^{4} + ( 52 - 52 \zeta_{8}^{2} ) q^{7} -64 \zeta_{8}^{3} q^{8} +O(q^{10})\) \( q -4 \zeta_{8} q^{2} + 16 \zeta_{8}^{2} q^{4} + ( 52 - 52 \zeta_{8}^{2} ) q^{7} -64 \zeta_{8}^{3} q^{8} + ( 88 \zeta_{8} + 88 \zeta_{8}^{3} ) q^{11} + ( -183 - 183 \zeta_{8}^{2} ) q^{13} + ( -208 \zeta_{8} + 208 \zeta_{8}^{3} ) q^{14} -256 q^{16} + 496 \zeta_{8} q^{17} + 2108 \zeta_{8}^{2} q^{19} + ( 352 - 352 \zeta_{8}^{2} ) q^{22} -2564 \zeta_{8}^{3} q^{23} + ( 732 \zeta_{8} + 732 \zeta_{8}^{3} ) q^{26} + ( 832 + 832 \zeta_{8}^{2} ) q^{28} + ( -4949 \zeta_{8} + 4949 \zeta_{8}^{3} ) q^{29} + 6784 q^{31} + 1024 \zeta_{8} q^{32} -1984 \zeta_{8}^{2} q^{34} + ( -3723 + 3723 \zeta_{8}^{2} ) q^{37} -8432 \zeta_{8}^{3} q^{38} + ( -8633 \zeta_{8} - 8633 \zeta_{8}^{3} ) q^{41} + ( -3180 - 3180 \zeta_{8}^{2} ) q^{43} + ( -1408 \zeta_{8} + 1408 \zeta_{8}^{3} ) q^{44} -10256 q^{46} -10980 \zeta_{8} q^{47} + 11399 \zeta_{8}^{2} q^{49} + ( 2928 - 2928 \zeta_{8}^{2} ) q^{52} + 32324 \zeta_{8}^{3} q^{53} + ( -3328 \zeta_{8} - 3328 \zeta_{8}^{3} ) q^{56} + ( 19796 + 19796 \zeta_{8}^{2} ) q^{58} + ( 25348 \zeta_{8} - 25348 \zeta_{8}^{3} ) q^{59} + 13740 q^{61} -27136 \zeta_{8} q^{62} -4096 \zeta_{8}^{2} q^{64} + ( -34064 + 34064 \zeta_{8}^{2} ) q^{67} + 7936 \zeta_{8}^{3} q^{68} + ( -33964 \zeta_{8} - 33964 \zeta_{8}^{3} ) q^{71} + ( 29891 + 29891 \zeta_{8}^{2} ) q^{73} + ( 14892 \zeta_{8} - 14892 \zeta_{8}^{3} ) q^{74} -33728 q^{76} + 9152 \zeta_{8} q^{77} + 59328 \zeta_{8}^{2} q^{79} + ( -34532 + 34532 \zeta_{8}^{2} ) q^{82} + 53928 \zeta_{8}^{3} q^{83} + ( 12720 \zeta_{8} + 12720 \zeta_{8}^{3} ) q^{86} + ( 5632 + 5632 \zeta_{8}^{2} ) q^{88} + ( 12053 \zeta_{8} - 12053 \zeta_{8}^{3} ) q^{89} -19032 q^{91} + 41024 \zeta_{8} q^{92} + 43920 \zeta_{8}^{2} q^{94} + ( -96093 + 96093 \zeta_{8}^{2} ) q^{97} -45596 \zeta_{8}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 208q^{7} + O(q^{10}) \) \( 4q + 208q^{7} - 732q^{13} - 1024q^{16} + 1408q^{22} + 3328q^{28} + 27136q^{31} - 14892q^{37} - 12720q^{43} - 41024q^{46} + 11712q^{52} + 79184q^{58} + 54960q^{61} - 136256q^{67} + 119564q^{73} - 134912q^{76} - 138128q^{82} + 22528q^{88} - 76128q^{91} - 384372q^{97} + O(q^{100}) \)

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 52.0000 + 52.0000i 45.2548 + 45.2548i 0 0
107.2 2.82843 2.82843i 0 16.0000i 0 0 52.0000 + 52.0000i −45.2548 45.2548i 0 0
143.1 −2.82843 2.82843i 0 16.0000i 0 0 52.0000 52.0000i 45.2548 45.2548i 0 0
143.2 2.82843 + 2.82843i 0 16.0000i 0 0 52.0000 52.0000i −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.d 4
3.b odd 2 1 inner 450.6.f.d 4
5.b even 2 1 90.6.f.a 4
5.c odd 4 1 90.6.f.a 4
5.c odd 4 1 inner 450.6.f.d 4
15.d odd 2 1 90.6.f.a 4
15.e even 4 1 90.6.f.a 4
15.e even 4 1 inner 450.6.f.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.6.f.a 4 5.b even 2 1
90.6.f.a 4 5.c odd 4 1
90.6.f.a 4 15.d odd 2 1
90.6.f.a 4 15.e even 4 1
450.6.f.d 4 1.a even 1 1 trivial
450.6.f.d 4 3.b odd 2 1 inner
450.6.f.d 4 5.c odd 4 1 inner
450.6.f.d 4 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 104 T_{7} + 5408 \) acting on \(S_{6}^{\mathrm{new}}(450, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 5408 - 104 T + T^{2} )^{2} \)
$11$ \( ( 15488 + T^{2} )^{2} \)
$13$ \( ( 66978 + 366 T + T^{2} )^{2} \)
$17$ \( 60523872256 + T^{4} \)
$19$ \( ( 4443664 + T^{2} )^{2} \)
$23$ \( 43218738217216 + T^{4} \)
$29$ \( ( -48985202 + T^{2} )^{2} \)
$31$ \( ( -6784 + T )^{4} \)
$37$ \( ( 27721458 + 7446 T + T^{2} )^{2} \)
$41$ \( ( 149057378 + T^{2} )^{2} \)
$43$ \( ( 20224800 + 6360 T + T^{2} )^{2} \)
$47$ \( 14534810048160000 + T^{4} \)
$53$ \( 1091692665128632576 + T^{4} \)
$59$ \( ( -1285042208 + T^{2} )^{2} \)
$61$ \( ( -13740 + T )^{4} \)
$67$ \( ( 2320712192 + 68128 T + T^{2} )^{2} \)
$71$ \( ( 2307106592 + T^{2} )^{2} \)
$73$ \( ( 1786943762 - 59782 T + T^{2} )^{2} \)
$79$ \( ( 3519811584 + T^{2} )^{2} \)
$83$ \( 8457796986669305856 + T^{4} \)
$89$ \( ( -290549618 + T^{2} )^{2} \)
$97$ \( ( 18467729298 + 192186 T + T^{2} )^{2} \)
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