Properties

Label 450.6.c.k
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: \( 1 \)
Twist minimal: no (minimal twist has level 150)
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.

\(f(q)\) \(=\) \( q -4 i q^{2} -16 q^{4} + i q^{7} + 64 i q^{8} +O(q^{10})\) \( q -4 i q^{2} -16 q^{4} + i q^{7} + 64 i q^{8} + 210 q^{11} -667 i q^{13} + 4 q^{14} + 256 q^{16} + 114 i q^{17} -581 q^{19} -840 i q^{22} + 4350 i q^{23} -2668 q^{26} -16 i q^{28} -126 q^{29} + 7583 q^{31} -1024 i q^{32} + 456 q^{34} + 3742 i q^{37} + 2324 i q^{38} + 2856 q^{41} -18241 i q^{43} -3360 q^{44} + 17400 q^{46} -23370 i q^{47} + 16806 q^{49} + 10672 i q^{52} + 21684 i q^{53} -64 q^{56} + 504 i q^{58} -32310 q^{59} -7165 q^{61} -30332 i q^{62} -4096 q^{64} -59579 i q^{67} -1824 i q^{68} + 43080 q^{71} -28942 i q^{73} + 14968 q^{74} + 9296 q^{76} + 210 i q^{77} -27608 q^{79} -11424 i q^{82} + 1782 i q^{83} -72964 q^{86} + 13440 i q^{88} + 50208 q^{89} + 667 q^{91} -69600 i q^{92} -93480 q^{94} -142793 i q^{97} -67224 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + O(q^{10}) \) \( 2q - 32q^{4} + 420q^{11} + 8q^{14} + 512q^{16} - 1162q^{19} - 5336q^{26} - 252q^{29} + 15166q^{31} + 912q^{34} + 5712q^{41} - 6720q^{44} + 34800q^{46} + 33612q^{49} - 128q^{56} - 64620q^{59} - 14330q^{61} - 8192q^{64} + 86160q^{71} + 29936q^{74} + 18592q^{76} - 55216q^{79} - 145928q^{86} + 100416q^{89} + 1334q^{91} - 186960q^{94} + O(q^{100}) \)

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 1.00000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 1.00000i 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.k 2
3.b odd 2 1 150.6.c.a 2
5.b even 2 1 inner 450.6.c.k 2
5.c odd 4 1 450.6.a.g 1
5.c odd 4 1 450.6.a.r 1
15.d odd 2 1 150.6.c.a 2
15.e even 4 1 150.6.a.e 1
15.e even 4 1 150.6.a.k yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.e 1 15.e even 4 1
150.6.a.k yes 1 15.e even 4 1
150.6.c.a 2 3.b odd 2 1
150.6.c.a 2 15.d odd 2 1
450.6.a.g 1 5.c odd 4 1
450.6.a.r 1 5.c odd 4 1
450.6.c.k 2 1.a even 1 1 trivial
450.6.c.k 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 \(S_{6}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{2} + 1 \)
\( T_{11} - 210 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( -210 + T )^{2} \)
$13$ \( 444889 + T^{2} \)
$17$ \( 12996 + T^{2} \)
$19$ \( ( 581 + T )^{2} \)
$23$ \( 18922500 + T^{2} \)
$29$ \( ( 126 + T )^{2} \)
$31$ \( ( -7583 + T )^{2} \)
$37$ \( 14002564 + T^{2} \)
$41$ \( ( -2856 + T )^{2} \)
$43$ \( 332734081 + T^{2} \)
$47$ \( 546156900 + T^{2} \)
$53$ \( 470195856 + T^{2} \)
$59$ \( ( 32310 + T )^{2} \)
$61$ \( ( 7165 + T )^{2} \)
$67$ \( 3549657241 + T^{2} \)
$71$ \( ( -43080 + T )^{2} \)
$73$ \( 837639364 + T^{2} \)
$79$ \( ( 27608 + T )^{2} \)
$83$ \( 3175524 + T^{2} \)
$89$ \( ( -50208 + T )^{2} \)
$97$ \( 20389840849 + T^{2} \)
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