Properties

Label 162.7.d.b
Level $162$
Weight $7$
Character orbit 162.d
Analytic conductor $37.269$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.2687615464\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{3} ) q^{2} + ( 32 - 32 \beta_{2} ) q^{4} + 30 \beta_{1} q^{5} -2 \beta_{2} q^{7} -32 \beta_{3} q^{8} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{3} ) q^{2} + ( 32 - 32 \beta_{2} ) q^{4} + 30 \beta_{1} q^{5} -2 \beta_{2} q^{7} -32 \beta_{3} q^{8} + 960 q^{10} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{11} + ( 2950 - 2950 \beta_{2} ) q^{13} -2 \beta_{1} q^{14} -1024 \beta_{2} q^{16} + 792 \beta_{3} q^{17} + 5258 q^{19} + ( 960 \beta_{1} - 960 \beta_{3} ) q^{20} + ( -192 + 192 \beta_{2} ) q^{22} + 1812 \beta_{1} q^{23} + 13175 \beta_{2} q^{25} -2950 \beta_{3} q^{26} -64 q^{28} + ( 390 \beta_{1} - 390 \beta_{3} ) q^{29} + ( -22898 + 22898 \beta_{2} ) q^{31} -1024 \beta_{1} q^{32} + 25344 \beta_{2} q^{34} -60 \beta_{3} q^{35} + 34058 q^{37} + ( 5258 \beta_{1} - 5258 \beta_{3} ) q^{38} + ( 30720 - 30720 \beta_{2} ) q^{40} + 2964 \beta_{1} q^{41} + 6406 \beta_{2} q^{43} + 192 \beta_{3} q^{44} + 57984 q^{46} + ( 31800 \beta_{1} - 31800 \beta_{3} ) q^{47} + ( 117645 - 117645 \beta_{2} ) q^{49} + 13175 \beta_{1} q^{50} -94400 \beta_{2} q^{52} -34038 \beta_{3} q^{53} -5760 q^{55} + ( -64 \beta_{1} + 64 \beta_{3} ) q^{56} + ( 12480 - 12480 \beta_{2} ) q^{58} + 57774 \beta_{1} q^{59} + 62566 \beta_{2} q^{61} + 22898 \beta_{3} q^{62} -32768 q^{64} + ( 88500 \beta_{1} - 88500 \beta_{3} ) q^{65} + ( -438698 + 438698 \beta_{2} ) q^{67} + 25344 \beta_{1} q^{68} -1920 \beta_{2} q^{70} -12060 \beta_{3} q^{71} -730510 q^{73} + ( 34058 \beta_{1} - 34058 \beta_{3} ) q^{74} + ( 168256 - 168256 \beta_{2} ) q^{76} + 12 \beta_{1} q^{77} -340562 \beta_{2} q^{79} -30720 \beta_{3} q^{80} + 94848 q^{82} + ( 87726 \beta_{1} - 87726 \beta_{3} ) q^{83} + ( -760320 + 760320 \beta_{2} ) q^{85} + 6406 \beta_{1} q^{86} + 6144 \beta_{2} q^{88} -68364 \beta_{3} q^{89} -5900 q^{91} + ( 57984 \beta_{1} - 57984 \beta_{3} ) q^{92} + ( 1017600 - 1017600 \beta_{2} ) q^{94} + 157740 \beta_{1} q^{95} + 281086 \beta_{2} q^{97} -117645 \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 64q^{4} - 4q^{7} + O(q^{10}) \) \( 4q + 64q^{4} - 4q^{7} + 3840q^{10} + 5900q^{13} - 2048q^{16} + 21032q^{19} - 384q^{22} + 26350q^{25} - 256q^{28} - 45796q^{31} + 50688q^{34} + 136232q^{37} + 61440q^{40} + 12812q^{43} + 231936q^{46} + 235290q^{49} - 188800q^{52} - 23040q^{55} + 24960q^{58} + 125132q^{61} - 131072q^{64} - 877396q^{67} - 3840q^{70} - 2922040q^{73} + 336512q^{76} - 681124q^{79} + 379392q^{82} - 1520640q^{85} + 12288q^{88} - 23600q^{91} + 2035200q^{94} + 562172q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( 2 \nu^{3} \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/4\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)\(/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
−4.89898 + 2.82843i 0 16.0000 27.7128i −146.969 84.8528i 0 −1.00000 1.73205i 181.019i 0 960.000
53.2 4.89898 2.82843i 0 16.0000 27.7128i 146.969 + 84.8528i 0 −1.00000 1.73205i 181.019i 0 960.000
107.1 −4.89898 2.82843i 0 16.0000 + 27.7128i −146.969 + 84.8528i 0 −1.00000 + 1.73205i 181.019i 0 960.000
107.2 4.89898 + 2.82843i 0 16.0000 + 27.7128i 146.969 84.8528i 0 −1.00000 + 1.73205i 181.019i 0 960.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.7.d.b 4
3.b odd 2 1 inner 162.7.d.b 4
9.c even 3 1 6.7.b.a 2
9.c even 3 1 inner 162.7.d.b 4
9.d odd 6 1 6.7.b.a 2
9.d odd 6 1 inner 162.7.d.b 4
36.f odd 6 1 48.7.e.b 2
36.h even 6 1 48.7.e.b 2
45.h odd 6 1 150.7.d.a 2
45.j even 6 1 150.7.d.a 2
45.k odd 12 2 150.7.b.a 4
45.l even 12 2 150.7.b.a 4
63.l odd 6 1 294.7.b.a 2
63.o even 6 1 294.7.b.a 2
72.j odd 6 1 192.7.e.c 2
72.l even 6 1 192.7.e.f 2
72.n even 6 1 192.7.e.c 2
72.p odd 6 1 192.7.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.7.b.a 2 9.c even 3 1
6.7.b.a 2 9.d odd 6 1
48.7.e.b 2 36.f odd 6 1
48.7.e.b 2 36.h even 6 1
150.7.b.a 4 45.k odd 12 2
150.7.b.a 4 45.l even 12 2
150.7.d.a 2 45.h odd 6 1
150.7.d.a 2 45.j even 6 1
162.7.d.b 4 1.a even 1 1 trivial
162.7.d.b 4 3.b odd 2 1 inner
162.7.d.b 4 9.c even 3 1 inner
162.7.d.b 4 9.d odd 6 1 inner
192.7.e.c 2 72.j odd 6 1
192.7.e.c 2 72.n even 6 1
192.7.e.f 2 72.l even 6 1
192.7.e.f 2 72.p odd 6 1
294.7.b.a 2 63.l odd 6 1
294.7.b.a 2 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 28800 T_{5}^{2} + 829440000 \) acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1024 - 32 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 829440000 - 28800 T^{2} + T^{4} \)
$7$ \( ( 4 + 2 T + T^{2} )^{2} \)
$11$ \( 1327104 - 1152 T^{2} + T^{4} \)
$13$ \( ( 8702500 - 2950 T + T^{2} )^{2} \)
$17$ \( ( 20072448 + T^{2} )^{2} \)
$19$ \( ( -5258 + T )^{4} \)
$23$ \( 11039076170072064 - 105067008 T^{2} + T^{4} \)
$29$ \( 23689635840000 - 4867200 T^{2} + T^{4} \)
$31$ \( ( 524318404 + 22898 T + T^{2} )^{2} \)
$37$ \( ( -34058 + T )^{4} \)
$41$ \( 79033780026998784 - 281129472 T^{2} + T^{4} \)
$43$ \( ( 41036836 - 6406 T + T^{2} )^{2} \)
$47$ \( \)\(10\!\cdots\!00\)\( - 32359680000 T^{2} + T^{4} \)
$53$ \( ( 37074734208 + T^{2} )^{2} \)
$59$ \( \)\(11\!\cdots\!24\)\( - 106810722432 T^{2} + T^{4} \)
$61$ \( ( 3914504356 - 62566 T + T^{2} )^{2} \)
$67$ \( ( 192455935204 + 438698 T + T^{2} )^{2} \)
$71$ \( ( 4654195200 + T^{2} )^{2} \)
$73$ \( ( 730510 + T )^{4} \)
$79$ \( ( 115982475844 + 340562 T + T^{2} )^{2} \)
$83$ \( \)\(60\!\cdots\!24\)\( - 246267234432 T^{2} + T^{4} \)
$89$ \( ( 149556367872 + T^{2} )^{2} \)
$97$ \( ( 79009339396 - 281086 T + T^{2} )^{2} \)
show more
show less