Properties

Label 450.6.c.p
Level $450$
Weight $6$
Character orbit 450.c
Analytic conductor $72.173$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{4081})\)
Defining polynomial: \(x^{4} + 2041 x^{2} + 1040400\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 \beta_{1} q^{2} -16 q^{4} + ( -50 \beta_{1} - \beta_{2} ) q^{7} + 64 \beta_{1} q^{8} +O(q^{10})\) \( q -4 \beta_{1} q^{2} -16 q^{4} + ( -50 \beta_{1} - \beta_{2} ) q^{7} + 64 \beta_{1} q^{8} + ( -270 - 2 \beta_{3} ) q^{11} + ( -445 \beta_{1} - 2 \beta_{2} ) q^{13} + ( -200 - 4 \beta_{3} ) q^{14} + 256 q^{16} + ( 246 \beta_{1} + 10 \beta_{2} ) q^{17} + ( -296 - 5 \beta_{3} ) q^{19} + ( 1080 \beta_{1} + 8 \beta_{2} ) q^{22} + ( -1830 \beta_{1} - 10 \beta_{2} ) q^{23} + ( -1780 - 8 \beta_{3} ) q^{26} + ( 800 \beta_{1} + 16 \beta_{2} ) q^{28} + ( 2850 - 2 \beta_{3} ) q^{29} + ( -2854 + 5 \beta_{3} ) q^{31} -1024 \beta_{1} q^{32} + ( 984 + 40 \beta_{3} ) q^{34} + ( 5650 \beta_{1} - 24 \beta_{2} ) q^{37} + ( 1184 \beta_{1} + 20 \beta_{2} ) q^{38} + ( -7710 + 30 \beta_{3} ) q^{41} + ( -3160 \beta_{1} - 41 \beta_{2} ) q^{43} + ( 4320 + 32 \beta_{3} ) q^{44} + ( -7320 - 40 \beta_{3} ) q^{46} + ( -3900 \beta_{1} - 100 \beta_{2} ) q^{47} + ( -22422 - 100 \beta_{3} ) q^{49} + ( 7120 \beta_{1} + 32 \beta_{2} ) q^{52} + ( 13914 \beta_{1} + 70 \beta_{2} ) q^{53} + ( 3200 + 64 \beta_{3} ) q^{56} + ( -11400 \beta_{1} + 8 \beta_{2} ) q^{58} + ( 25260 + 52 \beta_{3} ) q^{59} + ( -14563 + 90 \beta_{3} ) q^{61} + ( 11416 \beta_{1} - 20 \beta_{2} ) q^{62} -4096 q^{64} + ( 48700 \beta_{1} - 3 \beta_{2} ) q^{67} + ( -3936 \beta_{1} - 160 \beta_{2} ) q^{68} + ( -3090 - 126 \beta_{3} ) q^{71} + ( -16450 \beta_{1} + 120 \beta_{2} ) q^{73} + ( 22600 - 96 \beta_{3} ) q^{74} + ( 4736 + 80 \beta_{3} ) q^{76} + ( 86958 \beta_{1} + 370 \beta_{2} ) q^{77} + ( -3956 + 360 \beta_{3} ) q^{79} + ( 30840 \beta_{1} - 120 \beta_{2} ) q^{82} + ( 81732 \beta_{1} - 100 \beta_{2} ) q^{83} + ( -12640 - 164 \beta_{3} ) q^{86} + ( -17280 \beta_{1} - 128 \beta_{2} ) q^{88} + ( 82320 - 336 \beta_{3} ) q^{89} + ( -95708 - 545 \beta_{3} ) q^{91} + ( 29280 \beta_{1} + 160 \beta_{2} ) q^{92} + ( -15600 - 400 \beta_{3} ) q^{94} + ( 26215 \beta_{1} + 364 \beta_{2} ) q^{97} + ( 89688 \beta_{1} + 400 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 64q^{4} + O(q^{10}) \) \( 4q - 64q^{4} - 1080q^{11} - 800q^{14} + 1024q^{16} - 1184q^{19} - 7120q^{26} + 11400q^{29} - 11416q^{31} + 3936q^{34} - 30840q^{41} + 17280q^{44} - 29280q^{46} - 89688q^{49} + 12800q^{56} + 101040q^{59} - 58252q^{61} - 16384q^{64} - 12360q^{71} + 90400q^{74} + 18944q^{76} - 15824q^{79} - 50560q^{86} + 329280q^{89} - 382832q^{91} - 62400q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2041 x^{2} + 1040400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 1021 \nu \)\()/1020\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 3061 \nu \)\()/340\)
\(\beta_{3}\)\(=\)\( 6 \nu^{2} + 6123 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 6123\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(-1021 \beta_{2} + 9183 \beta_{1}\)\()/6\)

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
31.4414i
32.4414i
32.4414i
31.4414i
4.00000i 0 −16.0000 0 0 241.648i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 141.648i 64.0000i 0 0
199.3 4.00000i 0 −16.0000 0 0 141.648i 64.0000i 0 0
199.4 4.00000i 0 −16.0000 0 0 241.648i 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.p 4
3.b odd 2 1 450.6.c.q 4
5.b even 2 1 inner 450.6.c.p 4
5.c odd 4 1 450.6.a.y 2
5.c odd 4 1 450.6.a.bf yes 2
15.d odd 2 1 450.6.c.q 4
15.e even 4 1 450.6.a.ba yes 2
15.e even 4 1 450.6.a.bd yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.6.a.y 2 5.c odd 4 1
450.6.a.ba yes 2 15.e even 4 1
450.6.a.bd yes 2 15.e even 4 1
450.6.a.bf yes 2 5.c odd 4 1
450.6.c.p 4 1.a even 1 1 trivial
450.6.c.p 4 5.b even 2 1 inner
450.6.c.q 4 3.b odd 2 1
450.6.c.q 4 15.d odd 2 1

Hecke kernels

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

\( T_{7}^{4} + 78458 T_{7}^{2} + 1171624441 \)
\( T_{11}^{2} + 540 T_{11} - 74016 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 1171624441 + 78458 T^{2} + T^{4} \)
$11$ \( ( -74016 + 540 T + T^{2} )^{2} \)
$13$ \( 2612129881 + 689882 T^{2} + T^{4} \)
$17$ \( 13049318163456 + 7466832 T^{2} + T^{4} \)
$19$ \( ( -830609 + 592 T + T^{2} )^{2} \)
$23$ \( 104976000000 + 14043600 T^{2} + T^{4} \)
$29$ \( ( 7975584 - 5700 T + T^{2} )^{2} \)
$31$ \( ( 7227091 + 5708 T + T^{2} )^{2} \)
$37$ \( 115919589427216 + 106156808 T^{2} + T^{4} \)
$41$ \( ( 26388000 + 15420 T + T^{2} )^{2} \)
$43$ \( 2678667905710801 + 143454098 T^{2} + T^{4} \)
$47$ \( 123960326400000000 + 765000000 T^{2} + T^{4} \)
$53$ \( 185703196271616 + 747142992 T^{2} + T^{4} \)
$59$ \( ( 538752384 - 50520 T + T^{2} )^{2} \)
$61$ \( ( -85423931 + 29126 T + T^{2} )^{2} \)
$67$ \( 5623345588934394721 + 4744041122 T^{2} + T^{4} \)
$71$ \( ( -573561504 + 6180 T + T^{2} )^{2} \)
$73$ \( 66716358684010000 + 1599000200 T^{2} + T^{4} \)
$79$ \( ( -4744428464 + 7912 T + T^{2} )^{2} \)
$83$ \( 39851820386783870976 + 14094819648 T^{2} + T^{4} \)
$89$ \( ( 2630025216 - 164640 T + T^{2} )^{2} \)
$97$ \( 17465874450640370881 + 11107343618 T^{2} + T^{4} \)
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