Properties

Label 450.6.a.bc
Level $450$
Weight $6$
Character orbit 450.a
Self dual yes
Analytic conductor $72.173$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1249}) \)
Defining polynomial: \(x^{2} - x - 312\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 5\sqrt{1249}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} + ( -57 - \beta ) q^{7} + 64 q^{8} +O(q^{10})\) \( q + 4 q^{2} + 16 q^{4} + ( -57 - \beta ) q^{7} + 64 q^{8} + ( -87 + \beta ) q^{11} + ( -321 + 3 \beta ) q^{13} + ( -228 - 4 \beta ) q^{14} + 256 q^{16} + ( 757 - 3 \beta ) q^{17} + ( -60 - 12 \beta ) q^{19} + ( -348 + 4 \beta ) q^{22} + ( 2176 + 12 \beta ) q^{23} + ( -1284 + 12 \beta ) q^{26} + ( -912 - 16 \beta ) q^{28} + ( 1215 - 27 \beta ) q^{29} + ( 5842 + 6 \beta ) q^{31} + 1024 q^{32} + ( 3028 - 12 \beta ) q^{34} + ( 8793 - 19 \beta ) q^{37} + ( -240 - 48 \beta ) q^{38} + ( -12492 + 34 \beta ) q^{41} + ( 12084 + 56 \beta ) q^{43} + ( -1392 + 16 \beta ) q^{44} + ( 8704 + 48 \beta ) q^{46} + ( -6658 - 6 \beta ) q^{47} + ( 17667 + 114 \beta ) q^{49} + ( -5136 + 48 \beta ) q^{52} + ( -6849 + 111 \beta ) q^{53} + ( -3648 - 64 \beta ) q^{56} + ( 4860 - 108 \beta ) q^{58} + ( 11865 + 185 \beta ) q^{59} + ( 28562 - 96 \beta ) q^{61} + ( 23368 + 24 \beta ) q^{62} + 4096 q^{64} + ( 19158 - 86 \beta ) q^{67} + ( 12112 - 48 \beta ) q^{68} + ( 5538 - 294 \beta ) q^{71} + ( 44274 + 98 \beta ) q^{73} + ( 35172 - 76 \beta ) q^{74} + ( -960 - 192 \beta ) q^{76} + ( -26266 + 30 \beta ) q^{77} + ( 7110 - 198 \beta ) q^{79} + ( -49968 + 136 \beta ) q^{82} + ( 25396 - 432 \beta ) q^{83} + ( 48336 + 224 \beta ) q^{86} + ( -5568 + 64 \beta ) q^{88} + ( 30210 - 200 \beta ) q^{89} + ( -75378 + 150 \beta ) q^{91} + ( 34816 + 192 \beta ) q^{92} + ( -26632 - 24 \beta ) q^{94} + ( 85608 + 16 \beta ) q^{97} + ( 70668 + 456 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{2} + 32q^{4} - 114q^{7} + 128q^{8} + O(q^{10}) \) \( 2q + 8q^{2} + 32q^{4} - 114q^{7} + 128q^{8} - 174q^{11} - 642q^{13} - 456q^{14} + 512q^{16} + 1514q^{17} - 120q^{19} - 696q^{22} + 4352q^{23} - 2568q^{26} - 1824q^{28} + 2430q^{29} + 11684q^{31} + 2048q^{32} + 6056q^{34} + 17586q^{37} - 480q^{38} - 24984q^{41} + 24168q^{43} - 2784q^{44} + 17408q^{46} - 13316q^{47} + 35334q^{49} - 10272q^{52} - 13698q^{53} - 7296q^{56} + 9720q^{58} + 23730q^{59} + 57124q^{61} + 46736q^{62} + 8192q^{64} + 38316q^{67} + 24224q^{68} + 11076q^{71} + 88548q^{73} + 70344q^{74} - 1920q^{76} - 52532q^{77} + 14220q^{79} - 99936q^{82} + 50792q^{83} + 96672q^{86} - 11136q^{88} + 60420q^{89} - 150756q^{91} + 69632q^{92} - 53264q^{94} + 171216q^{97} + 141336q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
18.1706
−17.1706
4.00000 0 16.0000 0 0 −233.706 64.0000 0 0
1.2 4.00000 0 16.0000 0 0 119.706 64.0000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.6.a.bc 2
3.b odd 2 1 150.6.a.n 2
5.b even 2 1 450.6.a.bb 2
5.c odd 4 2 90.6.c.c 4
15.d odd 2 1 150.6.a.o 2
15.e even 4 2 30.6.c.b 4
20.e even 4 2 720.6.f.i 4
60.l odd 4 2 240.6.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.b 4 15.e even 4 2
90.6.c.c 4 5.c odd 4 2
150.6.a.n 2 3.b odd 2 1
150.6.a.o 2 15.d odd 2 1
240.6.f.b 4 60.l odd 4 2
450.6.a.bb 2 5.b even 2 1
450.6.a.bc 2 1.a even 1 1 trivial
720.6.f.i 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7}^{2} + 114 T_{7} - 27976 \)
\( T_{11}^{2} + 174 T_{11} - 23656 \)
\( T_{17}^{2} - 1514 T_{17} + 292024 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -4 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -27976 + 114 T + T^{2} \)
$11$ \( -23656 + 174 T + T^{2} \)
$13$ \( -177984 + 642 T + T^{2} \)
$17$ \( 292024 - 1514 T + T^{2} \)
$19$ \( -4492800 + 120 T + T^{2} \)
$23$ \( 238576 - 4352 T + T^{2} \)
$29$ \( -21286800 - 2430 T + T^{2} \)
$31$ \( 33004864 - 11684 T + T^{2} \)
$37$ \( 66044624 - 17586 T + T^{2} \)
$41$ \( 119953964 + 24984 T + T^{2} \)
$43$ \( 48101456 - 24168 T + T^{2} \)
$47$ \( 43204864 + 13316 T + T^{2} \)
$53$ \( -337814424 + 13698 T + T^{2} \)
$59$ \( -927897400 - 23730 T + T^{2} \)
$61$ \( 528018244 - 57124 T + T^{2} \)
$67$ \( 136088864 - 38316 T + T^{2} \)
$71$ \( -2668294656 - 11076 T + T^{2} \)
$73$ \( 1660302176 - 88548 T + T^{2} \)
$79$ \( -1173592800 - 14220 T + T^{2} \)
$83$ \( -5182377584 - 50792 T + T^{2} \)
$89$ \( -336355900 - 60420 T + T^{2} \)
$97$ \( 7320736064 - 171216 T + T^{2} \)
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