Properties

Label 450.6.c.i
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 30)
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.

\(f(q)\) \(=\) \( q -4 i q^{2} -16 q^{4} + 32 i q^{7} + 64 i q^{8} +O(q^{10})\) \( q -4 i q^{2} -16 q^{4} + 32 i q^{7} + 64 i q^{8} -12 q^{11} + 154 i q^{13} + 128 q^{14} + 256 q^{16} + 918 i q^{17} + 1060 q^{19} + 48 i q^{22} -4224 i q^{23} + 616 q^{26} -512 i q^{28} -7890 q^{29} + 5192 q^{31} -1024 i q^{32} + 3672 q^{34} + 16382 i q^{37} -4240 i q^{38} -3642 q^{41} -15116 i q^{43} + 192 q^{44} -16896 q^{46} -23592 i q^{47} + 15783 q^{49} -2464 i q^{52} -16074 i q^{53} -2048 q^{56} + 31560 i q^{58} -14340 q^{59} -47938 q^{61} -20768 i q^{62} -4096 q^{64} + 33092 i q^{67} -14688 i q^{68} -51912 q^{71} -12026 i q^{73} + 65528 q^{74} -16960 q^{76} -384 i q^{77} -25160 q^{79} + 14568 i q^{82} + 35796 i q^{83} -60464 q^{86} -768 i q^{88} -75510 q^{89} -4928 q^{91} + 67584 i q^{92} -94368 q^{94} -44158 i q^{97} -63132 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + O(q^{10}) \) \( 2q - 32q^{4} - 24q^{11} + 256q^{14} + 512q^{16} + 2120q^{19} + 1232q^{26} - 15780q^{29} + 10384q^{31} + 7344q^{34} - 7284q^{41} + 384q^{44} - 33792q^{46} + 31566q^{49} - 4096q^{56} - 28680q^{59} - 95876q^{61} - 8192q^{64} - 103824q^{71} + 131056q^{74} - 33920q^{76} - 50320q^{79} - 120928q^{86} - 151020q^{89} - 9856q^{91} - 188736q^{94} + O(q^{100}) \)

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 32.0000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 32.0000i 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.i 2
3.b odd 2 1 150.6.c.f 2
5.b even 2 1 inner 450.6.c.i 2
5.c odd 4 1 90.6.a.a 1
5.c odd 4 1 450.6.a.q 1
15.d odd 2 1 150.6.c.f 2
15.e even 4 1 30.6.a.b 1
15.e even 4 1 150.6.a.b 1
20.e even 4 1 720.6.a.e 1
60.l odd 4 1 240.6.a.f 1
120.q odd 4 1 960.6.a.q 1
120.w even 4 1 960.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.a.b 1 15.e even 4 1
90.6.a.a 1 5.c odd 4 1
150.6.a.b 1 15.e even 4 1
150.6.c.f 2 3.b odd 2 1
150.6.c.f 2 15.d odd 2 1
240.6.a.f 1 60.l odd 4 1
450.6.a.q 1 5.c odd 4 1
450.6.c.i 2 1.a even 1 1 trivial
450.6.c.i 2 5.b even 2 1 inner
720.6.a.e 1 20.e even 4 1
960.6.a.d 1 120.w even 4 1
960.6.a.q 1 120.q odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 1024 \)
\( T_{11} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1024 + T^{2} \)
$11$ \( ( 12 + T )^{2} \)
$13$ \( 23716 + T^{2} \)
$17$ \( 842724 + T^{2} \)
$19$ \( ( -1060 + T )^{2} \)
$23$ \( 17842176 + T^{2} \)
$29$ \( ( 7890 + T )^{2} \)
$31$ \( ( -5192 + T )^{2} \)
$37$ \( 268369924 + T^{2} \)
$41$ \( ( 3642 + T )^{2} \)
$43$ \( 228493456 + T^{2} \)
$47$ \( 556582464 + T^{2} \)
$53$ \( 258373476 + T^{2} \)
$59$ \( ( 14340 + T )^{2} \)
$61$ \( ( 47938 + T )^{2} \)
$67$ \( 1095080464 + T^{2} \)
$71$ \( ( 51912 + T )^{2} \)
$73$ \( 144624676 + T^{2} \)
$79$ \( ( 25160 + T )^{2} \)
$83$ \( 1281353616 + T^{2} \)
$89$ \( ( 75510 + T )^{2} \)
$97$ \( 1949928964 + T^{2} \)
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