Properties

Label 450.6.c.g
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.

\(f(q)\) \(=\) \( q + 4 i q^{2} -16 q^{4} + 79 i q^{7} -64 i q^{8} +O(q^{10})\) \( q + 4 i q^{2} -16 q^{4} + 79 i q^{7} -64 i q^{8} -150 q^{11} + 137 i q^{13} -316 q^{14} + 256 q^{16} -2034 i q^{17} + 1969 q^{19} -600 i q^{22} + 1350 i q^{23} -548 q^{26} -1264 i q^{28} -2946 q^{29} + 713 q^{31} + 1024 i q^{32} + 8136 q^{34} + 3238 i q^{37} + 7876 i q^{38} -6564 q^{41} -19579 i q^{43} + 2400 q^{44} -5400 q^{46} -21150 i q^{47} + 10566 q^{49} -2192 i q^{52} + 25896 i q^{53} + 5056 q^{56} -11784 i q^{58} + 25350 q^{59} + 50615 q^{61} + 2852 i q^{62} -4096 q^{64} + 22519 i q^{67} + 32544 i q^{68} -33900 q^{71} + 82442 i q^{73} -12952 q^{74} -31504 q^{76} -11850 i q^{77} + 81472 q^{79} -26256 i q^{82} -25782 i q^{83} + 78316 q^{86} + 9600 i q^{88} + 103728 q^{89} -10823 q^{91} -21600 i q^{92} + 84600 q^{94} + 57343 i q^{97} + 42264 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + O(q^{10}) \) \( 2q - 32q^{4} - 300q^{11} - 632q^{14} + 512q^{16} + 3938q^{19} - 1096q^{26} - 5892q^{29} + 1426q^{31} + 16272q^{34} - 13128q^{41} + 4800q^{44} - 10800q^{46} + 21132q^{49} + 10112q^{56} + 50700q^{59} + 101230q^{61} - 8192q^{64} - 67800q^{71} - 25904q^{74} - 63008q^{76} + 162944q^{79} + 156632q^{86} + 207456q^{89} - 21646q^{91} + 169200q^{94} + O(q^{100}) \)

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 79.0000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 79.0000i 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.g 2
3.b odd 2 1 150.6.c.c 2
5.b even 2 1 inner 450.6.c.g 2
5.c odd 4 1 450.6.a.e 1
5.c odd 4 1 450.6.a.t 1
15.d odd 2 1 150.6.c.c 2
15.e even 4 1 150.6.a.g 1
15.e even 4 1 150.6.a.i yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.g 1 15.e even 4 1
150.6.a.i yes 1 15.e even 4 1
150.6.c.c 2 3.b odd 2 1
150.6.c.c 2 15.d odd 2 1
450.6.a.e 1 5.c odd 4 1
450.6.a.t 1 5.c odd 4 1
450.6.c.g 2 1.a even 1 1 trivial
450.6.c.g 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 \(S_{6}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{2} + 6241 \)
\( T_{11} + 150 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 6241 + T^{2} \)
$11$ \( ( 150 + T )^{2} \)
$13$ \( 18769 + T^{2} \)
$17$ \( 4137156 + T^{2} \)
$19$ \( ( -1969 + T )^{2} \)
$23$ \( 1822500 + T^{2} \)
$29$ \( ( 2946 + T )^{2} \)
$31$ \( ( -713 + T )^{2} \)
$37$ \( 10484644 + T^{2} \)
$41$ \( ( 6564 + T )^{2} \)
$43$ \( 383337241 + T^{2} \)
$47$ \( 447322500 + T^{2} \)
$53$ \( 670602816 + T^{2} \)
$59$ \( ( -25350 + T )^{2} \)
$61$ \( ( -50615 + T )^{2} \)
$67$ \( 507105361 + T^{2} \)
$71$ \( ( 33900 + T )^{2} \)
$73$ \( 6796683364 + T^{2} \)
$79$ \( ( -81472 + T )^{2} \)
$83$ \( 664711524 + T^{2} \)
$89$ \( ( -103728 + T )^{2} \)
$97$ \( 3288219649 + T^{2} \)
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