Properties

Label 450.6.a.bd
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: \(\Q(\sqrt{4081}) \)
Defining polynomial: \(x^{2} - x - 1020\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{4081}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} + ( -50 - \beta ) q^{7} + 64 q^{8} +O(q^{10})\) \( q + 4 q^{2} + 16 q^{4} + ( -50 - \beta ) q^{7} + 64 q^{8} + ( 270 + 2 \beta ) q^{11} + ( 445 + 2 \beta ) q^{13} + ( -200 - 4 \beta ) q^{14} + 256 q^{16} + ( -246 - 10 \beta ) q^{17} + ( 296 + 5 \beta ) q^{19} + ( 1080 + 8 \beta ) q^{22} + ( -1830 - 10 \beta ) q^{23} + ( 1780 + 8 \beta ) q^{26} + ( -800 - 16 \beta ) q^{28} + ( 2850 - 2 \beta ) q^{29} + ( -2854 + 5 \beta ) q^{31} + 1024 q^{32} + ( -984 - 40 \beta ) q^{34} + ( 5650 - 24 \beta ) q^{37} + ( 1184 + 20 \beta ) q^{38} + ( 7710 - 30 \beta ) q^{41} + ( 3160 + 41 \beta ) q^{43} + ( 4320 + 32 \beta ) q^{44} + ( -7320 - 40 \beta ) q^{46} + ( 3900 + 100 \beta ) q^{47} + ( 22422 + 100 \beta ) q^{49} + ( 7120 + 32 \beta ) q^{52} + ( 13914 + 70 \beta ) q^{53} + ( -3200 - 64 \beta ) q^{56} + ( 11400 - 8 \beta ) q^{58} + ( 25260 + 52 \beta ) q^{59} + ( -14563 + 90 \beta ) q^{61} + ( -11416 + 20 \beta ) q^{62} + 4096 q^{64} + ( 48700 - 3 \beta ) q^{67} + ( -3936 - 160 \beta ) q^{68} + ( 3090 + 126 \beta ) q^{71} + ( 16450 - 120 \beta ) q^{73} + ( 22600 - 96 \beta ) q^{74} + ( 4736 + 80 \beta ) q^{76} + ( -86958 - 370 \beta ) q^{77} + ( 3956 - 360 \beta ) q^{79} + ( 30840 - 120 \beta ) q^{82} + ( 81732 - 100 \beta ) q^{83} + ( 12640 + 164 \beta ) q^{86} + ( 17280 + 128 \beta ) q^{88} + ( 82320 - 336 \beta ) q^{89} + ( -95708 - 545 \beta ) q^{91} + ( -29280 - 160 \beta ) q^{92} + ( 15600 + 400 \beta ) q^{94} + ( 26215 + 364 \beta ) q^{97} + ( 89688 + 400 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{2} + 32q^{4} - 100q^{7} + 128q^{8} + O(q^{10}) \) \( 2q + 8q^{2} + 32q^{4} - 100q^{7} + 128q^{8} + 540q^{11} + 890q^{13} - 400q^{14} + 512q^{16} - 492q^{17} + 592q^{19} + 2160q^{22} - 3660q^{23} + 3560q^{26} - 1600q^{28} + 5700q^{29} - 5708q^{31} + 2048q^{32} - 1968q^{34} + 11300q^{37} + 2368q^{38} + 15420q^{41} + 6320q^{43} + 8640q^{44} - 14640q^{46} + 7800q^{47} + 44844q^{49} + 14240q^{52} + 27828q^{53} - 6400q^{56} + 22800q^{58} + 50520q^{59} - 29126q^{61} - 22832q^{62} + 8192q^{64} + 97400q^{67} - 7872q^{68} + 6180q^{71} + 32900q^{73} + 45200q^{74} + 9472q^{76} - 173916q^{77} + 7912q^{79} + 61680q^{82} + 163464q^{83} + 25280q^{86} + 34560q^{88} + 164640q^{89} - 191416q^{91} - 58560q^{92} + 31200q^{94} + 52430q^{97} + 179376q^{98} + O(q^{100}) \)

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
32.4414
−31.4414
4.00000 0 16.0000 0 0 −241.648 64.0000 0 0
1.2 4.00000 0 16.0000 0 0 141.648 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.bd yes 2
3.b odd 2 1 450.6.a.y 2
5.b even 2 1 450.6.a.ba yes 2
5.c odd 4 2 450.6.c.q 4
15.d odd 2 1 450.6.a.bf yes 2
15.e even 4 2 450.6.c.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.6.a.y 2 3.b odd 2 1
450.6.a.ba yes 2 5.b even 2 1
450.6.a.bd yes 2 1.a even 1 1 trivial
450.6.a.bf yes 2 15.d odd 2 1
450.6.c.p 4 15.e even 4 2
450.6.c.q 4 5.c odd 4 2

Hecke kernels

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

\( T_{7}^{2} + 100 T_{7} - 34229 \)
\( T_{11}^{2} - 540 T_{11} - 74016 \)
\( T_{17}^{2} + 492 T_{17} - 3612384 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -4 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -34229 + 100 T + T^{2} \)
$11$ \( -74016 - 540 T + T^{2} \)
$13$ \( 51109 - 890 T + T^{2} \)
$17$ \( -3612384 + 492 T + T^{2} \)
$19$ \( -830609 - 592 T + T^{2} \)
$23$ \( -324000 + 3660 T + T^{2} \)
$29$ \( 7975584 - 5700 T + T^{2} \)
$31$ \( 7227091 + 5708 T + T^{2} \)
$37$ \( 10766596 - 11300 T + T^{2} \)
$41$ \( 26388000 - 15420 T + T^{2} \)
$43$ \( -51755849 - 6320 T + T^{2} \)
$47$ \( -352080000 - 7800 T + T^{2} \)
$53$ \( 13627296 - 27828 T + T^{2} \)
$59$ \( 538752384 - 50520 T + T^{2} \)
$61$ \( -85423931 + 29126 T + T^{2} \)
$67$ \( 2371359439 - 97400 T + T^{2} \)
$71$ \( -573561504 - 6180 T + T^{2} \)
$73$ \( -258295100 - 32900 T + T^{2} \)
$79$ \( -4744428464 - 7912 T + T^{2} \)
$83$ \( 6312829824 - 163464 T + T^{2} \)
$89$ \( 2630025216 - 164640 T + T^{2} \)
$97$ \( -4179219359 - 52430 T + T^{2} \)
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