Properties

Label 450.6.c.d
Level $450$
Weight $6$
Character orbit 450.c
Analytic conductor $72.173$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 i q^{2} -16 q^{4} + 98 i q^{7} + 64 i q^{8} +O(q^{10})\) \( q -4 i q^{2} -16 q^{4} + 98 i q^{7} + 64 i q^{8} -354 q^{11} -404 i q^{13} + 392 q^{14} + 256 q^{16} -654 i q^{17} -1796 q^{19} + 1416 i q^{22} -1080 i q^{23} -1616 q^{26} -1568 i q^{28} -5754 q^{29} + 10196 q^{31} -1024 i q^{32} -2616 q^{34} + 5552 i q^{37} + 7184 i q^{38} + 12960 q^{41} + 8968 i q^{43} + 5664 q^{44} -4320 q^{46} + 5400 i q^{47} + 7203 q^{49} + 6464 i q^{52} + 8214 i q^{53} -6272 q^{56} + 23016 i q^{58} + 3954 q^{59} + 962 q^{61} -40784 i q^{62} -4096 q^{64} -17956 i q^{67} + 10464 i q^{68} + 56148 q^{71} + 85690 i q^{73} + 22208 q^{74} + 28736 q^{76} -34692 i q^{77} + 26044 q^{79} -51840 i q^{82} -93468 i q^{83} + 35872 q^{86} -22656 i q^{88} + 73428 q^{89} + 39592 q^{91} + 17280 i q^{92} + 21600 q^{94} + 128978 i q^{97} -28812 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + O(q^{10}) \) \( 2q - 32q^{4} - 708q^{11} + 784q^{14} + 512q^{16} - 3592q^{19} - 3232q^{26} - 11508q^{29} + 20392q^{31} - 5232q^{34} + 25920q^{41} + 11328q^{44} - 8640q^{46} + 14406q^{49} - 12544q^{56} + 7908q^{59} + 1924q^{61} - 8192q^{64} + 112296q^{71} + 44416q^{74} + 57472q^{76} + 52088q^{79} + 71744q^{86} + 146856q^{89} + 79184q^{91} + 43200q^{94} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
4.00000i 0 −16.0000 0 0 98.0000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 98.0000i 64.0000i 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.6.c.d 2
3.b odd 2 1 450.6.c.l 2
5.b even 2 1 inner 450.6.c.d 2
5.c odd 4 1 90.6.a.c 1
5.c odd 4 1 450.6.a.o 1
15.d odd 2 1 450.6.c.l 2
15.e even 4 1 90.6.a.e yes 1
15.e even 4 1 450.6.a.d 1
20.e even 4 1 720.6.a.o 1
60.l odd 4 1 720.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.6.a.c 1 5.c odd 4 1
90.6.a.e yes 1 15.e even 4 1
450.6.a.d 1 15.e even 4 1
450.6.a.o 1 5.c odd 4 1
450.6.c.d 2 1.a even 1 1 trivial
450.6.c.d 2 5.b even 2 1 inner
450.6.c.l 2 3.b odd 2 1
450.6.c.l 2 15.d odd 2 1
720.6.a.c 1 60.l odd 4 1
720.6.a.o 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{2} + 9604 \)
\( T_{11} + 354 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 9604 + T^{2} \)
$11$ \( ( 354 + T )^{2} \)
$13$ \( 163216 + T^{2} \)
$17$ \( 427716 + T^{2} \)
$19$ \( ( 1796 + T )^{2} \)
$23$ \( 1166400 + T^{2} \)
$29$ \( ( 5754 + T )^{2} \)
$31$ \( ( -10196 + T )^{2} \)
$37$ \( 30824704 + T^{2} \)
$41$ \( ( -12960 + T )^{2} \)
$43$ \( 80425024 + T^{2} \)
$47$ \( 29160000 + T^{2} \)
$53$ \( 67469796 + T^{2} \)
$59$ \( ( -3954 + T )^{2} \)
$61$ \( ( -962 + T )^{2} \)
$67$ \( 322417936 + T^{2} \)
$71$ \( ( -56148 + T )^{2} \)
$73$ \( 7342776100 + T^{2} \)
$79$ \( ( -26044 + T )^{2} \)
$83$ \( 8736267024 + T^{2} \)
$89$ \( ( -73428 + T )^{2} \)
$97$ \( 16635324484 + T^{2} \)
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