Properties

Label 450.6.a.z
Level $450$
Weight $6$
Character orbit 450.a
Self dual yes
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.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Defining polynomial: \(x^{2} - 19\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 90)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 q^{2} + 16 q^{4} + \beta q^{7} -64 q^{8} +O(q^{10})\) \( q -4 q^{2} + 16 q^{4} + \beta q^{7} -64 q^{8} + 37 \beta q^{11} + 63 \beta q^{13} -4 \beta q^{14} + 256 q^{16} + 1166 q^{17} -2244 q^{19} -148 \beta q^{22} + 500 q^{23} -252 \beta q^{26} + 16 \beta q^{28} -27 \beta q^{29} + 3856 q^{31} -1024 q^{32} -4664 q^{34} -401 \beta q^{37} + 8976 q^{38} + 550 \beta q^{41} + 304 \beta q^{43} + 592 \beta q^{44} -2000 q^{46} + 19900 q^{47} -16503 q^{49} + 1008 \beta q^{52} + 1146 q^{53} -64 \beta q^{56} + 108 \beta q^{58} -2143 \beta q^{59} -38158 q^{61} -15424 q^{62} + 4096 q^{64} + 2078 \beta q^{67} + 18656 q^{68} -954 \beta q^{71} -3980 \beta q^{73} + 1604 \beta q^{74} -35904 q^{76} + 11248 q^{77} -20664 q^{79} -2200 \beta q^{82} + 96968 q^{83} -1216 \beta q^{86} -2368 \beta q^{88} + 3424 \beta q^{89} + 19152 q^{91} + 8000 q^{92} -79600 q^{94} -334 \beta q^{97} + 66012 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{2} + 32q^{4} - 128q^{8} + O(q^{10}) \) \( 2q - 8q^{2} + 32q^{4} - 128q^{8} + 512q^{16} + 2332q^{17} - 4488q^{19} + 1000q^{23} + 7712q^{31} - 2048q^{32} - 9328q^{34} + 17952q^{38} - 4000q^{46} + 39800q^{47} - 33006q^{49} + 2292q^{53} - 76316q^{61} - 30848q^{62} + 8192q^{64} + 37312q^{68} - 71808q^{76} + 22496q^{77} - 41328q^{79} + 193936q^{83} + 38304q^{91} + 16000q^{92} - 159200q^{94} + 132024q^{98} + O(q^{100}) \)

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
−4.35890
4.35890
−4.00000 0 16.0000 0 0 −17.4356 −64.0000 0 0
1.2 −4.00000 0 16.0000 0 0 17.4356 −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

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner

Twists

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

Hecke kernels

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

\( T_{7}^{2} - 304 \)
\( T_{11}^{2} - 416176 \)
\( T_{17} - 1166 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -304 + T^{2} \)
$11$ \( -416176 + T^{2} \)
$13$ \( -1206576 + T^{2} \)
$17$ \( ( -1166 + T )^{2} \)
$19$ \( ( 2244 + T )^{2} \)
$23$ \( ( -500 + T )^{2} \)
$29$ \( -221616 + T^{2} \)
$31$ \( ( -3856 + T )^{2} \)
$37$ \( -48883504 + T^{2} \)
$41$ \( -91960000 + T^{2} \)
$43$ \( -28094464 + T^{2} \)
$47$ \( ( -19900 + T )^{2} \)
$53$ \( ( -1146 + T )^{2} \)
$59$ \( -1396104496 + T^{2} \)
$61$ \( ( 38158 + T )^{2} \)
$67$ \( -1312697536 + T^{2} \)
$71$ \( -276675264 + T^{2} \)
$73$ \( -4815481600 + T^{2} \)
$79$ \( ( 20664 + T )^{2} \)
$83$ \( ( -96968 + T )^{2} \)
$89$ \( -3564027904 + T^{2} \)
$97$ \( -33913024 + T^{2} \)
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