Properties

Label 162.7.d.c
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: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 6 \beta_1 q^{5} + 205 \beta_{2} q^{7} + 32 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 6 \beta_1 q^{5} + 205 \beta_{2} q^{7} + 32 \beta_{3} q^{8} - 192 q^{10} + ( - 186 \beta_{3} + 186 \beta_1) q^{11} + ( - 2041 \beta_{2} + 2041) q^{13} - 205 \beta_1 q^{14} - 1024 \beta_{2} q^{16} + 1458 \beta_{3} q^{17} - 1501 q^{19} + ( - 192 \beta_{3} + 192 \beta_1) q^{20} + (5952 \beta_{2} - 5952) q^{22} - 1254 \beta_1 q^{23} - 14473 \beta_{2} q^{25} + 2041 \beta_{3} q^{26} + 6560 q^{28} + ( - 5028 \beta_{3} + 5028 \beta_1) q^{29} + ( - 34990 \beta_{2} + 34990) q^{31} + 1024 \beta_1 q^{32} - 46656 \beta_{2} q^{34} + 1230 \beta_{3} q^{35} - 57625 q^{37} + ( - 1501 \beta_{3} + 1501 \beta_1) q^{38} + (6144 \beta_{2} - 6144) q^{40} + 23748 \beta_1 q^{41} + 62566 \beta_{2} q^{43} - 5952 \beta_{3} q^{44} + 40128 q^{46} + (9174 \beta_{3} - 9174 \beta_1) q^{47} + ( - 75624 \beta_{2} + 75624) q^{49} + 14473 \beta_1 q^{50} - 65312 \beta_{2} q^{52} + 13644 \beta_{3} q^{53} + 35712 q^{55} + (6560 \beta_{3} - 6560 \beta_1) q^{56} + (160896 \beta_{2} - 160896) q^{58} + 65634 \beta_1 q^{59} + 61297 \beta_{2} q^{61} + 34990 \beta_{3} q^{62} - 32768 q^{64} + ( - 12246 \beta_{3} + 12246 \beta_1) q^{65} + (67691 \beta_{2} - 67691) q^{67} + 46656 \beta_1 q^{68} - 39360 \beta_{2} q^{70} + 90072 \beta_{3} q^{71} + 423983 q^{73} + ( - 57625 \beta_{3} + 57625 \beta_1) q^{74} + (48032 \beta_{2} - 48032) q^{76} + 38130 \beta_1 q^{77} + 707533 \beta_{2} q^{79} - 6144 \beta_{3} q^{80} - 759936 q^{82} + ( - 152196 \beta_{3} + 152196 \beta_1) q^{83} + (279936 \beta_{2} - 279936) q^{85} - 62566 \beta_1 q^{86} + 190464 \beta_{2} q^{88} + 118026 \beta_{3} q^{89} + 418405 q^{91} + (40128 \beta_{3} - 40128 \beta_1) q^{92} + ( - 293568 \beta_{2} + 293568) q^{94} - 9006 \beta_1 q^{95} - 526151 \beta_{2} q^{97} + 75624 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 64 q^{4} + 410 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 64 q^{4} + 410 q^{7} - 768 q^{10} + 4082 q^{13} - 2048 q^{16} - 6004 q^{19} - 11904 q^{22} - 28946 q^{25} + 26240 q^{28} + 69980 q^{31} - 93312 q^{34} - 230500 q^{37} - 12288 q^{40} + 125132 q^{43} + 160512 q^{46} + 151248 q^{49} - 130624 q^{52} + 142848 q^{55} - 321792 q^{58} + 122594 q^{61} - 131072 q^{64} - 135382 q^{67} - 78720 q^{70} + 1695932 q^{73} - 96064 q^{76} + 1415066 q^{79} - 3039744 q^{82} - 559872 q^{85} + 380928 q^{88} + 1673620 q^{91} + 587136 q^{94} - 1052302 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{3} \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) 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 29.3939 + 16.9706i 0 102.500 + 177.535i 181.019i 0 −192.000
53.2 4.89898 2.82843i 0 16.0000 27.7128i −29.3939 16.9706i 0 102.500 + 177.535i 181.019i 0 −192.000
107.1 −4.89898 2.82843i 0 16.0000 + 27.7128i 29.3939 16.9706i 0 102.500 177.535i 181.019i 0 −192.000
107.2 4.89898 + 2.82843i 0 16.0000 + 27.7128i −29.3939 + 16.9706i 0 102.500 177.535i 181.019i 0 −192.000
\(n\): 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.c 4
3.b odd 2 1 inner 162.7.d.c 4
9.c even 3 1 54.7.b.a 2
9.c even 3 1 inner 162.7.d.c 4
9.d odd 6 1 54.7.b.a 2
9.d odd 6 1 inner 162.7.d.c 4
36.f odd 6 1 432.7.e.f 2
36.h even 6 1 432.7.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.7.b.a 2 9.c even 3 1
54.7.b.a 2 9.d odd 6 1
162.7.d.c 4 1.a even 1 1 trivial
162.7.d.c 4 3.b odd 2 1 inner
162.7.d.c 4 9.c even 3 1 inner
162.7.d.c 4 9.d odd 6 1 inner
432.7.e.f 2 36.f odd 6 1
432.7.e.f 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 1152T_{5}^{2} + 1327104 \) 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
$5$ \( T^{4} - 1152 T^{2} + \cdots + 1327104 \) Copy content Toggle raw display
$7$ \( (T^{2} - 205 T + 42025)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 1107072 T^{2} + \cdots + 1225608413184 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2041 T + 4165681)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 68024448)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1501)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 50320512 T^{2} + \cdots + 25\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{4} - 808985088 T^{2} + \cdots + 65\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( (T^{2} - 34990 T + 1224300100)^{2} \) Copy content Toggle raw display
$37$ \( (T + 57625)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 18046960128 T^{2} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{2} - 62566 T + 3914504356)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 2693192832 T^{2} + \cdots + 72\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{2} + 5957079552)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 137850302592 T^{2} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{2} - 61297 T + 3757322209)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 67691 T + 4582071481)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 259614885888)^{2} \) Copy content Toggle raw display
$73$ \( (T - 423983)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 707533 T + 500602946089)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 741235917312 T^{2} + \cdots + 54\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{2} + 445764373632)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 526151 T + 276834874801)^{2} \) Copy content Toggle raw display
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