Properties

Label 450.6.c.b
Level $450$
Weight $6$
Character orbit 450.c
Analytic conductor $72.173$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

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.

\(f(q)\) \(=\) \( q + 4 i q^{2} -16 q^{4} + 164 i q^{7} -64 i q^{8} +O(q^{10})\) \( q + 4 i q^{2} -16 q^{4} + 164 i q^{7} -64 i q^{8} -720 q^{11} -698 i q^{13} -656 q^{14} + 256 q^{16} + 2226 i q^{17} -356 q^{19} -2880 i q^{22} -1800 i q^{23} + 2792 q^{26} -2624 i q^{28} + 714 q^{29} + 848 q^{31} + 1024 i q^{32} -8904 q^{34} -11302 i q^{37} -1424 i q^{38} -9354 q^{41} + 5956 i q^{43} + 11520 q^{44} + 7200 q^{46} + 11160 i q^{47} -10089 q^{49} + 11168 i q^{52} + 14106 i q^{53} + 10496 q^{56} + 2856 i q^{58} + 7920 q^{59} -13450 q^{61} + 3392 i q^{62} -4096 q^{64} -65476 i q^{67} -35616 i q^{68} -34560 q^{71} -86258 i q^{73} + 45208 q^{74} + 5696 q^{76} -118080 i q^{77} + 108832 q^{79} -37416 i q^{82} + 10668 i q^{83} -23824 q^{86} + 46080 i q^{88} + 10818 q^{89} + 114472 q^{91} + 28800 i q^{92} -44640 q^{94} + 4418 i q^{97} -40356 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + O(q^{10}) \) \( 2q - 32q^{4} - 1440q^{11} - 1312q^{14} + 512q^{16} - 712q^{19} + 5584q^{26} + 1428q^{29} + 1696q^{31} - 17808q^{34} - 18708q^{41} + 23040q^{44} + 14400q^{46} - 20178q^{49} + 20992q^{56} + 15840q^{59} - 26900q^{61} - 8192q^{64} - 69120q^{71} + 90416q^{74} + 11392q^{76} + 217664q^{79} - 47648q^{86} + 21636q^{89} + 228944q^{91} - 89280q^{94} + O(q^{100}) \)

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 0 164.000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 164.000i 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.b 2
3.b odd 2 1 150.6.c.d 2
5.b even 2 1 inner 450.6.c.b 2
5.c odd 4 1 90.6.a.g 1
5.c odd 4 1 450.6.a.b 1
15.d odd 2 1 150.6.c.d 2
15.e even 4 1 30.6.a.a 1
15.e even 4 1 150.6.a.h 1
20.e even 4 1 720.6.a.m 1
60.l odd 4 1 240.6.a.a 1
120.q odd 4 1 960.6.a.u 1
120.w even 4 1 960.6.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.a.a 1 15.e even 4 1
90.6.a.g 1 5.c odd 4 1
150.6.a.h 1 15.e even 4 1
150.6.c.d 2 3.b odd 2 1
150.6.c.d 2 15.d odd 2 1
240.6.a.a 1 60.l odd 4 1
450.6.a.b 1 5.c odd 4 1
450.6.c.b 2 1.a even 1 1 trivial
450.6.c.b 2 5.b even 2 1 inner
720.6.a.m 1 20.e even 4 1
960.6.a.n 1 120.w even 4 1
960.6.a.u 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 \(S_{6}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{2} + 26896 \)
\( T_{11} + 720 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 26896 + T^{2} \)
$11$ \( ( 720 + T )^{2} \)
$13$ \( 487204 + T^{2} \)
$17$ \( 4955076 + T^{2} \)
$19$ \( ( 356 + T )^{2} \)
$23$ \( 3240000 + T^{2} \)
$29$ \( ( -714 + T )^{2} \)
$31$ \( ( -848 + T )^{2} \)
$37$ \( 127735204 + T^{2} \)
$41$ \( ( 9354 + T )^{2} \)
$43$ \( 35473936 + T^{2} \)
$47$ \( 124545600 + T^{2} \)
$53$ \( 198979236 + T^{2} \)
$59$ \( ( -7920 + T )^{2} \)
$61$ \( ( 13450 + T )^{2} \)
$67$ \( 4287106576 + T^{2} \)
$71$ \( ( 34560 + T )^{2} \)
$73$ \( 7440442564 + T^{2} \)
$79$ \( ( -108832 + T )^{2} \)
$83$ \( 113806224 + T^{2} \)
$89$ \( ( -10818 + T )^{2} \)
$97$ \( 19518724 + T^{2} \)
show more
show less