Properties

Label 162.7.d.d
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
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.

\(f(q)\) \(=\) \( q + (4 \beta_{3} - 4 \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 123 \beta_1 q^{5} + 484 \beta_{2} q^{7} + 128 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (4 \beta_{3} - 4 \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 123 \beta_1 q^{5} + 484 \beta_{2} q^{7} + 128 \beta_{3} q^{8} - 984 q^{10} + (948 \beta_{3} - 948 \beta_1) q^{11} + (3368 \beta_{2} - 3368) q^{13} - 1936 \beta_1 q^{14} - 1024 \beta_{2} q^{16} - 9 \beta_{3} q^{17} + 5744 q^{19} + ( - 3936 \beta_{3} + 3936 \beta_1) q^{20} + ( - 7584 \beta_{2} + 7584) q^{22} - 2388 \beta_1 q^{23} + 14633 \beta_{2} q^{25} - 13472 \beta_{3} q^{26} + 15488 q^{28} + (20757 \beta_{3} - 20757 \beta_1) q^{29} + ( - 39796 \beta_{2} + 39796) q^{31} + 4096 \beta_1 q^{32} + 72 \beta_{2} q^{34} + 59532 \beta_{3} q^{35} + 52526 q^{37} + (22976 \beta_{3} - 22976 \beta_1) q^{38} + (31488 \beta_{2} - 31488) q^{40} - 26193 \beta_1 q^{41} - 3800 \beta_{2} q^{43} + 30336 \beta_{3} q^{44} + 19104 q^{46} + (54300 \beta_{3} - 54300 \beta_1) q^{47} + (116607 \beta_{2} - 116607) q^{49} - 58532 \beta_1 q^{50} + 107776 \beta_{2} q^{52} - 168813 \beta_{3} q^{53} - 233208 q^{55} + (61952 \beta_{3} - 61952 \beta_1) q^{56} + ( - 166056 \beta_{2} + 166056) q^{58} - 176664 \beta_1 q^{59} - 13250 \beta_{2} q^{61} + 159184 \beta_{3} q^{62} - 32768 q^{64} + (414264 \beta_{3} - 414264 \beta_1) q^{65} + (168968 \beta_{2} - 168968) q^{67} - 288 \beta_1 q^{68} - 476256 \beta_{2} q^{70} + 375804 \beta_{3} q^{71} + 236144 q^{73} + (210104 \beta_{3} - 210104 \beta_1) q^{74} + ( - 183808 \beta_{2} + 183808) q^{76} - 458832 \beta_1 q^{77} + 35116 \beta_{2} q^{79} - 125952 \beta_{3} q^{80} + 209544 q^{82} + ( - 7764 \beta_{3} + 7764 \beta_1) q^{83} + ( - 2214 \beta_{2} + 2214) q^{85} + 15200 \beta_1 q^{86} - 242688 \beta_{2} q^{88} + 91449 \beta_{3} q^{89} - 1630112 q^{91} + (76416 \beta_{3} - 76416 \beta_1) q^{92} + ( - 434400 \beta_{2} + 434400) q^{94} + 706512 \beta_1 q^{95} + 321424 \beta_{2} q^{97} - 466428 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 64 q^{4} + 968 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 64 q^{4} + 968 q^{7} - 3936 q^{10} - 6736 q^{13} - 2048 q^{16} + 22976 q^{19} + 15168 q^{22} + 29266 q^{25} + 61952 q^{28} + 79592 q^{31} + 144 q^{34} + 210104 q^{37} - 62976 q^{40} - 7600 q^{43} + 76416 q^{46} - 233214 q^{49} + 215552 q^{52} - 932832 q^{55} + 332112 q^{58} - 26500 q^{61} - 131072 q^{64} - 337936 q^{67} - 952512 q^{70} + 944576 q^{73} + 367616 q^{76} + 70232 q^{79} + 838176 q^{82} + 4428 q^{85} - 485376 q^{88} - 6520448 q^{91} + 868800 q^{94} + 642848 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) 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 150.644 + 86.9741i 0 242.000 + 419.156i 181.019i 0 −984.000
53.2 4.89898 2.82843i 0 16.0000 27.7128i −150.644 86.9741i 0 242.000 + 419.156i 181.019i 0 −984.000
107.1 −4.89898 2.82843i 0 16.0000 + 27.7128i 150.644 86.9741i 0 242.000 419.156i 181.019i 0 −984.000
107.2 4.89898 + 2.82843i 0 16.0000 + 27.7128i −150.644 + 86.9741i 0 242.000 419.156i 181.019i 0 −984.000
\(n\): 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.d 4
3.b odd 2 1 inner 162.7.d.d 4
9.c even 3 1 18.7.b.a 2
9.c even 3 1 inner 162.7.d.d 4
9.d odd 6 1 18.7.b.a 2
9.d odd 6 1 inner 162.7.d.d 4
36.f odd 6 1 144.7.e.d 2
36.h even 6 1 144.7.e.d 2
45.h odd 6 1 450.7.d.a 2
45.j even 6 1 450.7.d.a 2
45.k odd 12 2 450.7.b.a 4
45.l even 12 2 450.7.b.a 4
72.j odd 6 1 576.7.e.b 2
72.l even 6 1 576.7.e.k 2
72.n even 6 1 576.7.e.b 2
72.p odd 6 1 576.7.e.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.b.a 2 9.c even 3 1
18.7.b.a 2 9.d odd 6 1
144.7.e.d 2 36.f odd 6 1
144.7.e.d 2 36.h even 6 1
162.7.d.d 4 1.a even 1 1 trivial
162.7.d.d 4 3.b odd 2 1 inner
162.7.d.d 4 9.c even 3 1 inner
162.7.d.d 4 9.d odd 6 1 inner
450.7.b.a 4 45.k odd 12 2
450.7.b.a 4 45.l even 12 2
450.7.d.a 2 45.h odd 6 1
450.7.d.a 2 45.j even 6 1
576.7.e.b 2 72.j odd 6 1
576.7.e.b 2 72.n even 6 1
576.7.e.k 2 72.l even 6 1
576.7.e.k 2 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 30258T_{5}^{2} + 915546564 \) 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
$5$ \( T^{4} - 30258 T^{2} + \cdots + 915546564 \) Copy content Toggle raw display
$7$ \( (T^{2} - 484 T + 234256)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 1797408 T^{2} + \cdots + 3230675518464 \) Copy content Toggle raw display
$13$ \( (T^{2} + 3368 T + 11343424)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$19$ \( (T - 5744)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 130076032287744 \) Copy content Toggle raw display
$29$ \( T^{4} - 861706098 T^{2} + \cdots + 74\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( (T^{2} - 39796 T + 1583721616)^{2} \) Copy content Toggle raw display
$37$ \( (T - 52526)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 1372146498 T^{2} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{2} + 3800 T + 14440000)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 5896980000 T^{2} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{2} + 56995657938)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 62420337792 T^{2} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{2} + 13250 T + 175562500)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 168968 T + 28550185024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 282457292832)^{2} \) Copy content Toggle raw display
$73$ \( (T - 236144)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 35116 T + 1233133456)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 120559392 T^{2} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{2} + 16725839202)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 321424 T + 103313387776)^{2} \) Copy content Toggle raw display
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