Properties

Label 450.6.c.a
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
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.

\(f(q)\) \(=\) \( q -4 i q^{2} -16 q^{4} + 142 i q^{7} + 64 i q^{8} +O(q^{10})\) \( q -4 i q^{2} -16 q^{4} + 142 i q^{7} + 64 i q^{8} -777 q^{11} + 884 i q^{13} + 568 q^{14} + 256 q^{16} -27 i q^{17} -1145 q^{19} + 3108 i q^{22} -1854 i q^{23} + 3536 q^{26} -2272 i q^{28} -4920 q^{29} + 1802 q^{31} -1024 i q^{32} -108 q^{34} -13178 i q^{37} + 4580 i q^{38} + 15123 q^{41} + 7844 i q^{43} + 12432 q^{44} -7416 q^{46} -6732 i q^{47} -3357 q^{49} -14144 i q^{52} -3414 i q^{53} -9088 q^{56} + 19680 i q^{58} + 33960 q^{59} + 47402 q^{61} -7208 i q^{62} -4096 q^{64} + 13177 i q^{67} + 432 i q^{68} + 7548 q^{71} -59821 i q^{73} -52712 q^{74} + 18320 q^{76} -110334 i q^{77} -75830 q^{79} -60492 i q^{82} -46299 i q^{83} + 31376 q^{86} -49728 i q^{88} -30585 q^{89} -125528 q^{91} + 29664 i q^{92} -26928 q^{94} -104018 i q^{97} + 13428 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + O(q^{10}) \) \( 2q - 32q^{4} - 1554q^{11} + 1136q^{14} + 512q^{16} - 2290q^{19} + 7072q^{26} - 9840q^{29} + 3604q^{31} - 216q^{34} + 30246q^{41} + 24864q^{44} - 14832q^{46} - 6714q^{49} - 18176q^{56} + 67920q^{59} + 94804q^{61} - 8192q^{64} + 15096q^{71} - 105424q^{74} + 36640q^{76} - 151660q^{79} + 62752q^{86} - 61170q^{89} - 251056q^{91} - 53856q^{94} + O(q^{100}) \)

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 0 142.000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 142.000i 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.a 2
3.b odd 2 1 50.6.b.c 2
5.b even 2 1 inner 450.6.c.a 2
5.c odd 4 1 450.6.a.j 1
5.c odd 4 1 450.6.a.n 1
12.b even 2 1 400.6.c.g 2
15.d odd 2 1 50.6.b.c 2
15.e even 4 1 50.6.a.a 1
15.e even 4 1 50.6.a.f yes 1
60.h even 2 1 400.6.c.g 2
60.l odd 4 1 400.6.a.e 1
60.l odd 4 1 400.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.6.a.a 1 15.e even 4 1
50.6.a.f yes 1 15.e even 4 1
50.6.b.c 2 3.b odd 2 1
50.6.b.c 2 15.d odd 2 1
400.6.a.e 1 60.l odd 4 1
400.6.a.j 1 60.l odd 4 1
400.6.c.g 2 12.b even 2 1
400.6.c.g 2 60.h even 2 1
450.6.a.j 1 5.c odd 4 1
450.6.a.n 1 5.c odd 4 1
450.6.c.a 2 1.a even 1 1 trivial
450.6.c.a 2 5.b even 2 1 inner

Hecke kernels

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

\( T_{7}^{2} + 20164 \)
\( T_{11} + 777 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 20164 + T^{2} \)
$11$ \( ( 777 + T )^{2} \)
$13$ \( 781456 + T^{2} \)
$17$ \( 729 + T^{2} \)
$19$ \( ( 1145 + T )^{2} \)
$23$ \( 3437316 + T^{2} \)
$29$ \( ( 4920 + T )^{2} \)
$31$ \( ( -1802 + T )^{2} \)
$37$ \( 173659684 + T^{2} \)
$41$ \( ( -15123 + T )^{2} \)
$43$ \( 61528336 + T^{2} \)
$47$ \( 45319824 + T^{2} \)
$53$ \( 11655396 + T^{2} \)
$59$ \( ( -33960 + T )^{2} \)
$61$ \( ( -47402 + T )^{2} \)
$67$ \( 173633329 + T^{2} \)
$71$ \( ( -7548 + T )^{2} \)
$73$ \( 3578552041 + T^{2} \)
$79$ \( ( 75830 + T )^{2} \)
$83$ \( 2143597401 + T^{2} \)
$89$ \( ( 30585 + T )^{2} \)
$97$ \( 10819744324 + T^{2} \)
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