Properties

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

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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: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 54)
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{3} + 2 \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 15 \beta_1 q^{5} - 389 \beta_{2} q^{7} - 64 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{3} + 2 \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 15 \beta_1 q^{5} - 389 \beta_{2} q^{7} - 64 \beta_{3} q^{8} + 240 q^{10} + ( - 735 \beta_{3} + 735 \beta_1) q^{11} + (1415 \beta_{2} - 1415) q^{13} - 778 \beta_1 q^{14} - 1024 \beta_{2} q^{16} + 837 \beta_{3} q^{17} - 3067 q^{19} + ( - 480 \beta_{3} + 480 \beta_1) q^{20} + ( - 11760 \beta_{2} + 11760) q^{22} - 7401 \beta_1 q^{23} - 13825 \beta_{2} q^{25} + 2830 \beta_{3} q^{26} - 12448 q^{28} + ( - 4578 \beta_{3} + 4578 \beta_1) q^{29} + ( - 11338 \beta_{2} + 11338) q^{31} - 2048 \beta_1 q^{32} + 13392 \beta_{2} q^{34} - 5835 \beta_{3} q^{35} + 47135 q^{37} + (6134 \beta_{3} - 6134 \beta_1) q^{38} + ( - 7680 \beta_{2} + 7680) q^{40} - 2190 \beta_1 q^{41} - 145118 \beta_{2} q^{43} - 23520 \beta_{3} q^{44} - 118416 q^{46} + (63705 \beta_{3} - 63705 \beta_1) q^{47} + (33672 \beta_{2} - 33672) q^{49} - 27650 \beta_1 q^{50} + 45280 \beta_{2} q^{52} - 93978 \beta_{3} q^{53} + 88200 q^{55} + (24896 \beta_{3} - 24896 \beta_1) q^{56} + ( - 73248 \beta_{2} + 73248) q^{58} - 127893 \beta_1 q^{59} + 350305 \beta_{2} q^{61} - 22676 \beta_{3} q^{62} - 32768 q^{64} + (21225 \beta_{3} - 21225 \beta_1) q^{65} + (120341 \beta_{2} - 120341) q^{67} + 26784 \beta_1 q^{68} - 93360 \beta_{2} q^{70} - 118476 \beta_{3} q^{71} + 175151 q^{73} + ( - 94270 \beta_{3} + 94270 \beta_1) q^{74} + (98144 \beta_{2} - 98144) q^{76} - 285915 \beta_1 q^{77} + 252259 \beta_{2} q^{79} - 15360 \beta_{3} q^{80} - 35040 q^{82} + ( - 119886 \beta_{3} + 119886 \beta_1) q^{83} + (100440 \beta_{2} - 100440) q^{85} - 290236 \beta_1 q^{86} - 376320 \beta_{2} q^{88} + 279297 \beta_{3} q^{89} + 550435 q^{91} + (236832 \beta_{3} - 236832 \beta_1) q^{92} + (1019280 \beta_{2} - 1019280) q^{94} - 46005 \beta_1 q^{95} + 1297105 \beta_{2} q^{97} + 67344 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 64 q^{4} - 778 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 64 q^{4} - 778 q^{7} + 960 q^{10} - 2830 q^{13} - 2048 q^{16} - 12268 q^{19} + 23520 q^{22} - 27650 q^{25} - 49792 q^{28} + 22676 q^{31} + 26784 q^{34} + 188540 q^{37} + 15360 q^{40} - 290236 q^{43} - 473664 q^{46} - 67344 q^{49} + 90560 q^{52} + 352800 q^{55} + 146496 q^{58} + 700610 q^{61} - 131072 q^{64} - 240682 q^{67} - 186720 q^{70} + 700604 q^{73} - 196288 q^{76} + 504518 q^{79} - 140160 q^{82} - 200880 q^{85} - 752640 q^{88} + 2201740 q^{91} - 2038560 q^{94} + 2594210 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display

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 −36.7423 21.2132i 0 −194.500 336.884i 181.019i 0 240.000
53.2 4.89898 2.82843i 0 16.0000 27.7128i 36.7423 + 21.2132i 0 −194.500 336.884i 181.019i 0 240.000
107.1 −4.89898 2.82843i 0 16.0000 + 27.7128i −36.7423 + 21.2132i 0 −194.500 + 336.884i 181.019i 0 240.000
107.2 4.89898 + 2.82843i 0 16.0000 + 27.7128i 36.7423 21.2132i 0 −194.500 + 336.884i 181.019i 0 240.000
\(n\): 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.a 4
3.b odd 2 1 inner 162.7.d.a 4
9.c even 3 1 54.7.b.b 2
9.c even 3 1 inner 162.7.d.a 4
9.d odd 6 1 54.7.b.b 2
9.d odd 6 1 inner 162.7.d.a 4
36.f odd 6 1 432.7.e.c 2
36.h even 6 1 432.7.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.7.b.b 2 9.c even 3 1
54.7.b.b 2 9.d odd 6 1
162.7.d.a 4 1.a even 1 1 trivial
162.7.d.a 4 3.b odd 2 1 inner
162.7.d.a 4 9.c even 3 1 inner
162.7.d.a 4 9.d odd 6 1 inner
432.7.e.c 2 36.f odd 6 1
432.7.e.c 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 1800T_{5}^{2} + 3240000 \) acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 32T^{2} + 1024 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 1800 T^{2} + \cdots + 3240000 \) Copy content Toggle raw display
$7$ \( (T^{2} + 389 T + 151321)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 4321800 T^{2} + \cdots + 18677955240000 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1415 T + 2002225)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 5604552)^{2} \) Copy content Toggle raw display
$19$ \( (T + 3067)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 438198408 T^{2} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{4} - 167664672 T^{2} + \cdots + 28\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{2} - 11338 T + 128550244)^{2} \) Copy content Toggle raw display
$37$ \( (T - 47135)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 38368800 T^{2} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + 145118 T + 21059233924)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 32466616200 T^{2} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{2} + 70654915872)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 130852955592 T^{2} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{2} - 350305 T + 122713593025)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 120341 T + 14481956281)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 112292500608)^{2} \) Copy content Toggle raw display
$73$ \( (T - 175151)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 252259 T + 63634603081)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 114981223968 T^{2} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{2} + 624054513672)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 1297105 T + 1682481381025)^{2} \) Copy content Toggle raw display
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