Properties

Label 450.6.c.m
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 18)
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.

\(f(q)\) \(=\) \( q + 4 i q^{2} -16 q^{4} + 148 i q^{7} -64 i q^{8} +O(q^{10})\) \( q + 4 i q^{2} -16 q^{4} + 148 i q^{7} -64 i q^{8} + 384 q^{11} -334 i q^{13} -592 q^{14} + 256 q^{16} -576 i q^{17} + 664 q^{19} + 1536 i q^{22} -3840 i q^{23} + 1336 q^{26} -2368 i q^{28} -96 q^{29} -4564 q^{31} + 1024 i q^{32} + 2304 q^{34} -5798 i q^{37} + 2656 i q^{38} -6720 q^{41} -14872 i q^{43} -6144 q^{44} + 15360 q^{46} + 19200 i q^{47} -5097 q^{49} + 5344 i q^{52} + 7776 i q^{53} + 9472 q^{56} -384 i q^{58} + 13056 q^{59} + 42782 q^{61} -18256 i q^{62} -4096 q^{64} -36656 i q^{67} + 9216 i q^{68} + 64512 q^{71} -16810 i q^{73} + 23192 q^{74} -10624 q^{76} + 56832 i q^{77} -28076 q^{79} -26880 i q^{82} -66432 i q^{83} + 59488 q^{86} -24576 i q^{88} + 81792 q^{89} + 49432 q^{91} + 61440 i q^{92} -76800 q^{94} + 29938 i q^{97} -20388 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + O(q^{10}) \) \( 2q - 32q^{4} + 768q^{11} - 1184q^{14} + 512q^{16} + 1328q^{19} + 2672q^{26} - 192q^{29} - 9128q^{31} + 4608q^{34} - 13440q^{41} - 12288q^{44} + 30720q^{46} - 10194q^{49} + 18944q^{56} + 26112q^{59} + 85564q^{61} - 8192q^{64} + 129024q^{71} + 46384q^{74} - 21248q^{76} - 56152q^{79} + 118976q^{86} + 163584q^{89} + 98864q^{91} - 153600q^{94} + O(q^{100}) \)

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 0 148.000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 148.000i 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.m 2
3.b odd 2 1 450.6.c.c 2
5.b even 2 1 inner 450.6.c.m 2
5.c odd 4 1 18.6.a.a 1
5.c odd 4 1 450.6.a.v 1
15.d odd 2 1 450.6.c.c 2
15.e even 4 1 18.6.a.c yes 1
15.e even 4 1 450.6.a.k 1
20.e even 4 1 144.6.a.a 1
35.f even 4 1 882.6.a.k 1
40.i odd 4 1 576.6.a.bh 1
40.k even 4 1 576.6.a.bi 1
45.k odd 12 2 162.6.c.l 2
45.l even 12 2 162.6.c.a 2
60.l odd 4 1 144.6.a.l 1
105.k odd 4 1 882.6.a.l 1
120.q odd 4 1 576.6.a.b 1
120.w even 4 1 576.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.a.a 1 5.c odd 4 1
18.6.a.c yes 1 15.e even 4 1
144.6.a.a 1 20.e even 4 1
144.6.a.l 1 60.l odd 4 1
162.6.c.a 2 45.l even 12 2
162.6.c.l 2 45.k odd 12 2
450.6.a.k 1 15.e even 4 1
450.6.a.v 1 5.c odd 4 1
450.6.c.c 2 3.b odd 2 1
450.6.c.c 2 15.d odd 2 1
450.6.c.m 2 1.a even 1 1 trivial
450.6.c.m 2 5.b even 2 1 inner
576.6.a.a 1 120.w even 4 1
576.6.a.b 1 120.q odd 4 1
576.6.a.bh 1 40.i odd 4 1
576.6.a.bi 1 40.k even 4 1
882.6.a.k 1 35.f even 4 1
882.6.a.l 1 105.k odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 21904 \)
\( T_{11} - 384 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 21904 + T^{2} \)
$11$ \( ( -384 + T )^{2} \)
$13$ \( 111556 + T^{2} \)
$17$ \( 331776 + T^{2} \)
$19$ \( ( -664 + T )^{2} \)
$23$ \( 14745600 + T^{2} \)
$29$ \( ( 96 + T )^{2} \)
$31$ \( ( 4564 + T )^{2} \)
$37$ \( 33616804 + T^{2} \)
$41$ \( ( 6720 + T )^{2} \)
$43$ \( 221176384 + T^{2} \)
$47$ \( 368640000 + T^{2} \)
$53$ \( 60466176 + T^{2} \)
$59$ \( ( -13056 + T )^{2} \)
$61$ \( ( -42782 + T )^{2} \)
$67$ \( 1343662336 + T^{2} \)
$71$ \( ( -64512 + T )^{2} \)
$73$ \( 282576100 + T^{2} \)
$79$ \( ( 28076 + T )^{2} \)
$83$ \( 4413210624 + T^{2} \)
$89$ \( ( -81792 + T )^{2} \)
$97$ \( 896283844 + T^{2} \)
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