Properties

Label 450.6.c.f
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: \( 2 \)
Twist minimal: no (minimal twist has level 10)
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.

\(f(q)\) \(=\) \( q + 4 i q^{2} -16 q^{4} + 118 i q^{7} -64 i q^{8} +O(q^{10})\) \( q + 4 i q^{2} -16 q^{4} + 118 i q^{7} -64 i q^{8} -192 q^{11} + 1106 i q^{13} -472 q^{14} + 256 q^{16} + 762 i q^{17} + 2740 q^{19} -768 i q^{22} -1566 i q^{23} -4424 q^{26} -1888 i q^{28} + 5910 q^{29} -6868 q^{31} + 1024 i q^{32} -3048 q^{34} + 5518 i q^{37} + 10960 i q^{38} + 378 q^{41} -2434 i q^{43} + 3072 q^{44} + 6264 q^{46} + 13122 i q^{47} + 2883 q^{49} -17696 i q^{52} + 9174 i q^{53} + 7552 q^{56} + 23640 i q^{58} -34980 q^{59} -9838 q^{61} -27472 i q^{62} -4096 q^{64} -33722 i q^{67} -12192 i q^{68} -70212 q^{71} + 21986 i q^{73} -22072 q^{74} -43840 q^{76} -22656 i q^{77} -4520 q^{79} + 1512 i q^{82} + 109074 i q^{83} + 9736 q^{86} + 12288 i q^{88} + 38490 q^{89} -130508 q^{91} + 25056 i q^{92} -52488 q^{94} + 1918 i q^{97} + 11532 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + O(q^{10}) \) \( 2q - 32q^{4} - 384q^{11} - 944q^{14} + 512q^{16} + 5480q^{19} - 8848q^{26} + 11820q^{29} - 13736q^{31} - 6096q^{34} + 756q^{41} + 6144q^{44} + 12528q^{46} + 5766q^{49} + 15104q^{56} - 69960q^{59} - 19676q^{61} - 8192q^{64} - 140424q^{71} - 44144q^{74} - 87680q^{76} - 9040q^{79} + 19472q^{86} + 76980q^{89} - 261016q^{91} - 104976q^{94} + O(q^{100}) \)

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 0 118.000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 118.000i 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.f 2
3.b odd 2 1 50.6.b.b 2
5.b even 2 1 inner 450.6.c.f 2
5.c odd 4 1 90.6.a.b 1
5.c odd 4 1 450.6.a.u 1
12.b even 2 1 400.6.c.i 2
15.d odd 2 1 50.6.b.b 2
15.e even 4 1 10.6.a.c 1
15.e even 4 1 50.6.a.b 1
20.e even 4 1 720.6.a.v 1
60.h even 2 1 400.6.c.i 2
60.l odd 4 1 80.6.a.c 1
60.l odd 4 1 400.6.a.i 1
105.k odd 4 1 490.6.a.k 1
120.q odd 4 1 320.6.a.k 1
120.w even 4 1 320.6.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.c 1 15.e even 4 1
50.6.a.b 1 15.e even 4 1
50.6.b.b 2 3.b odd 2 1
50.6.b.b 2 15.d odd 2 1
80.6.a.c 1 60.l odd 4 1
90.6.a.b 1 5.c odd 4 1
320.6.a.f 1 120.w even 4 1
320.6.a.k 1 120.q odd 4 1
400.6.a.i 1 60.l odd 4 1
400.6.c.i 2 12.b even 2 1
400.6.c.i 2 60.h even 2 1
450.6.a.u 1 5.c odd 4 1
450.6.c.f 2 1.a even 1 1 trivial
450.6.c.f 2 5.b even 2 1 inner
490.6.a.k 1 105.k odd 4 1
720.6.a.v 1 20.e even 4 1

Hecke kernels

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

\( T_{7}^{2} + 13924 \)
\( T_{11} + 192 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 13924 + T^{2} \)
$11$ \( ( 192 + T )^{2} \)
$13$ \( 1223236 + T^{2} \)
$17$ \( 580644 + T^{2} \)
$19$ \( ( -2740 + T )^{2} \)
$23$ \( 2452356 + T^{2} \)
$29$ \( ( -5910 + T )^{2} \)
$31$ \( ( 6868 + T )^{2} \)
$37$ \( 30448324 + T^{2} \)
$41$ \( ( -378 + T )^{2} \)
$43$ \( 5924356 + T^{2} \)
$47$ \( 172186884 + T^{2} \)
$53$ \( 84162276 + T^{2} \)
$59$ \( ( 34980 + T )^{2} \)
$61$ \( ( 9838 + T )^{2} \)
$67$ \( 1137173284 + T^{2} \)
$71$ \( ( 70212 + T )^{2} \)
$73$ \( 483384196 + T^{2} \)
$79$ \( ( 4520 + T )^{2} \)
$83$ \( 11897137476 + T^{2} \)
$89$ \( ( -38490 + T )^{2} \)
$97$ \( 3678724 + T^{2} \)
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