Properties

Label 162.7.d.g
Level $162$
Weight $7$
Character orbit 162.d
Analytic conductor $37.269$
Analytic rank $0$
Dimension $16$
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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + 485774x^{12} + 87183614355x^{8} + 6839940225440174x^{4} + 198392288899684017121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{36} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 \beta_{8} q^{2} + ( - 32 \beta_{2} + 32) q^{4} + (\beta_{13} + 12 \beta_{9} - 12 \beta_{8}) q^{5} + ( - \beta_{6} - \beta_{3} - 120 \beta_{2}) q^{7} + 128 \beta_{9} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta_{8} q^{2} + ( - 32 \beta_{2} + 32) q^{4} + (\beta_{13} + 12 \beta_{9} - 12 \beta_{8}) q^{5} + ( - \beta_{6} - \beta_{3} - 120 \beta_{2}) q^{7} + 128 \beta_{9} q^{8} + (4 \beta_{5} + 4 \beta_{4} - 96) q^{10} + (\beta_{15} - 5 \beta_{14} + \beta_{11} - 11 \beta_{10} - 421 \beta_{8}) q^{11} + ( - \beta_{7} + 5 \beta_{5} - 11 \beta_{3} + 568 \beta_{2} - 11 \beta_1 - 568) q^{13} + (4 \beta_{15} + 8 \beta_{14} + 8 \beta_{12} + 488 \beta_{9} - 488 \beta_{8}) q^{14} - 1024 \beta_{2} q^{16} + (24 \beta_{13} + 7 \beta_{12} - \beta_{11} - 24 \beta_{10} - 818 \beta_{9}) q^{17} + (9 \beta_{7} + 9 \beta_{6} - 19 \beta_{5} - 19 \beta_{4} - 9 \beta_{2} + \cdots - 2956) q^{19}+ \cdots + (4888 \beta_{13} + 3144 \beta_{12} + 776 \beta_{11} - 4888 \beta_{10} - 647860 \beta_{9}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 256 q^{4} - 964 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 256 q^{4} - 964 q^{7} - 1536 q^{10} - 4540 q^{13} - 8192 q^{16} - 47368 q^{19} - 27072 q^{22} + 32392 q^{25} - 61696 q^{28} - 77056 q^{31} + 52608 q^{34} - 22696 q^{37} - 24576 q^{40} + 226604 q^{43} - 325440 q^{46} - 1298088 q^{49} + 145280 q^{52} - 2921832 q^{55} - 867456 q^{58} + 327476 q^{61} - 524288 q^{64} - 1713292 q^{67} + 176352 q^{70} - 4378432 q^{73} - 757888 q^{76} + 1326884 q^{79} - 2317632 q^{82} - 3483180 q^{85} + 866304 q^{88} + 2260648 q^{91} + 26400 q^{94} + 2200064 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 485774x^{12} + 87183614355x^{8} + 6839940225440174x^{4} + 198392288899684017121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -346\nu^{12} - 125696265\nu^{8} - 14422204736895\nu^{4} - 515812550322387299 ) / 142209872480800 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 110729 \nu^{14} - 39704089485 \nu^{10} + \cdots + 13\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4593352202400 \nu^{14} + \cdots + 12\!\cdots\!09 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 62\!\cdots\!13 \nu^{14} + \cdots - 21\!\cdots\!33 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 63\!\cdots\!13 \nu^{14} + \cdots + 23\!\cdots\!59 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17\!\cdots\!25 \nu^{14} + \cdots + 10\!\cdots\!71 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 35\!\cdots\!65 \nu^{14} + \cdots - 21\!\cdots\!52 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 65\!\cdots\!76 \nu^{15} + \cdots + 25\!\cdots\!43 \nu ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 27\!\cdots\!47 \nu^{15} + \cdots + 17\!\cdots\!96 \nu ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 55\!\cdots\!91 \nu^{15} + \cdots - 10\!\cdots\!83 \nu ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 62\!\cdots\!45 \nu^{15} + \cdots + 37\!\cdots\!03 \nu ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!76 \nu^{15} + \cdots - 10\!\cdots\!43 \nu ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 60\!\cdots\!15 \nu^{15} + \cdots - 21\!\cdots\!41 \nu ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 20\!\cdots\!96 \nu^{15} + \cdots + 21\!\cdots\!71 \nu ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 29\!\cdots\!00 \nu^{15} + \cdots - 21\!\cdots\!36 \nu ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 4\beta_{14} + 6\beta_{13} + 2\beta_{12} - 3\beta_{11} + 12\beta_{10} - 38\beta_{9} - 2\beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} - 154\beta_{3} - 2594\beta_{2} - 77\beta _1 + 1297 ) / 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 72 \beta_{15} - 1154 \beta_{14} - 2436 \beta_{13} - 625 \beta_{12} - 483 \beta_{11} - 750 \beta_{10} + 19567 \beta_{9} + 3457 \beta_{8} ) / 81 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 96\beta_{7} - 96\beta_{6} + 2127\beta_{5} - 2127\beta_{4} + 1078\beta_{3} + 11611\beta _1 - 6557901 ) / 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 99504 \beta_{15} - 159152 \beta_{14} - 328566 \beta_{13} + 119672 \beta_{12} + 413043 \beta_{11} - 1950684 \beta_{10} + 27933562 \beta_{9} - 6706136 \beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 365024 \beta_{7} - 365024 \beta_{6} - 406936 \beta_{5} - 406936 \beta_{4} + 9562710 \beta_{3} + 466711184 \beta_{2} + 4723171 \beta _1 - 233355592 ) / 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 51666624 \beta_{15} + 402203792 \beta_{14} + 772529316 \beta_{13} + 200636200 \beta_{12} + 102715737 \beta_{11} + 50431602 \beta_{10} - 9802759642 \beta_{9} - 5095485928 \beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 69354528 \beta_{7} + 69354528 \beta_{6} - 553988649 \beta_{5} + 553988649 \beta_{4} - 269272570 \beta_{3} - 2800707517 \beta _1 + 831688852677 ) / 54 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 11945304648 \beta_{15} - 11077715194 \beta_{14} - 1834011510 \beta_{13} - 37653432149 \beta_{12} - 28946255865 \beta_{11} + 151620937404 \beta_{10} + \cdots + 1483960759397 \beta_{8} ) / 81 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 155759031134 \beta_{7} + 155759031134 \beta_{6} + 259402944703 \beta_{5} + 259402944703 \beta_{4} - 2472136817594 \beta_{3} - 187985751037982 \beta_{2} + \cdots + 93992875518991 ) / 54 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 10374491746224 \beta_{15} - 67426468882732 \beta_{14} - 118152875712756 \beta_{13} - 26352036126998 \beta_{12} - 11382858372297 \beta_{11} + \cdots + 16\!\cdots\!30 \beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 3532308996000 \beta_{7} - 3532308996000 \beta_{6} + 18766028206320 \beta_{5} - 18766028206320 \beta_{4} + 8814749330240 \beta_{3} + \cdots - 18\!\cdots\!81 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 43\!\cdots\!60 \beta_{15} + \cdots - 79\!\cdots\!02 \beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 29\!\cdots\!38 \beta_{7} + \cdots - 16\!\cdots\!53 ) / 54 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 88\!\cdots\!08 \beta_{15} + \cdots - 18\!\cdots\!53 \beta_{8} ) / 81 \) 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
−14.2877 + 13.5806i
−11.9992 12.7063i
13.5806 14.2877i
12.7063 + 11.9992i
−12.7063 11.9992i
−13.5806 + 14.2877i
11.9992 + 12.7063i
14.2877 13.5806i
−14.2877 13.5806i
−11.9992 + 12.7063i
13.5806 + 14.2877i
12.7063 11.9992i
−12.7063 + 11.9992i
−13.5806 14.2877i
11.9992 12.7063i
14.2877 + 13.5806i
−4.89898 + 2.82843i 0 16.0000 27.7128i −132.984 76.7783i 0 −336.136 582.205i 181.019i 0 868.648
53.2 −4.89898 + 2.82843i 0 16.0000 27.7128i −74.2039 42.8417i 0 282.029 + 488.488i 181.019i 0 484.698
53.3 −4.89898 + 2.82843i 0 16.0000 27.7128i 117.830 + 68.0293i 0 93.5265 + 161.993i 181.019i 0 −769.663
53.4 −4.89898 + 2.82843i 0 16.0000 27.7128i 148.146 + 85.5319i 0 −280.419 485.701i 181.019i 0 −967.683
53.5 4.89898 2.82843i 0 16.0000 27.7128i −148.146 85.5319i 0 −280.419 485.701i 181.019i 0 −967.683
53.6 4.89898 2.82843i 0 16.0000 27.7128i −117.830 68.0293i 0 93.5265 + 161.993i 181.019i 0 −769.663
53.7 4.89898 2.82843i 0 16.0000 27.7128i 74.2039 + 42.8417i 0 282.029 + 488.488i 181.019i 0 484.698
53.8 4.89898 2.82843i 0 16.0000 27.7128i 132.984 + 76.7783i 0 −336.136 582.205i 181.019i 0 868.648
107.1 −4.89898 2.82843i 0 16.0000 + 27.7128i −132.984 + 76.7783i 0 −336.136 + 582.205i 181.019i 0 868.648
107.2 −4.89898 2.82843i 0 16.0000 + 27.7128i −74.2039 + 42.8417i 0 282.029 488.488i 181.019i 0 484.698
107.3 −4.89898 2.82843i 0 16.0000 + 27.7128i 117.830 68.0293i 0 93.5265 161.993i 181.019i 0 −769.663
107.4 −4.89898 2.82843i 0 16.0000 + 27.7128i 148.146 85.5319i 0 −280.419 + 485.701i 181.019i 0 −967.683
107.5 4.89898 + 2.82843i 0 16.0000 + 27.7128i −148.146 + 85.5319i 0 −280.419 + 485.701i 181.019i 0 −967.683
107.6 4.89898 + 2.82843i 0 16.0000 + 27.7128i −117.830 + 68.0293i 0 93.5265 161.993i 181.019i 0 −769.663
107.7 4.89898 + 2.82843i 0 16.0000 + 27.7128i 74.2039 42.8417i 0 282.029 488.488i 181.019i 0 484.698
107.8 4.89898 + 2.82843i 0 16.0000 + 27.7128i 132.984 76.7783i 0 −336.136 + 582.205i 181.019i 0 868.648
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.8
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.g 16
3.b odd 2 1 inner 162.7.d.g 16
9.c even 3 1 162.7.b.b 8
9.c even 3 1 inner 162.7.d.g 16
9.d odd 6 1 162.7.b.b 8
9.d odd 6 1 inner 162.7.d.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.7.b.b 8 9.c even 3 1
162.7.b.b 8 9.d odd 6 1
162.7.d.g 16 1.a even 1 1 trivial
162.7.d.g 16 3.b odd 2 1 inner
162.7.d.g 16 9.c even 3 1 inner
162.7.d.g 16 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 78696 T_{5}^{14} + 4000981230 T_{5}^{12} - 122466234531456 T_{5}^{10} + \cdots + 87\!\cdots\!25 \) acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 32 T^{2} + 1024)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 78696 T^{14} + \cdots + 87\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + 482 T^{7} + \cdots + 15\!\cdots\!16)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 12104748 T^{14} + \cdots + 51\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T^{8} + 2270 T^{7} + \cdots + 10\!\cdots\!21)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 51916752 T^{6} + \cdots + 89\!\cdots\!25)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 11842 T^{3} + \cdots - 41\!\cdots\!44)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} - 582093972 T^{14} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} - 1686224736 T^{14} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( (T^{8} + 38528 T^{7} + \cdots + 62\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 5674 T^{3} + \cdots + 68\!\cdots\!73)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} - 13639103856 T^{14} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{8} - 113302 T^{7} + \cdots + 91\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 40347462384 T^{14} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{8} + 79583256720 T^{6} + \cdots + 57\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} - 268690196304 T^{14} + \cdots + 64\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{8} - 163738 T^{7} + \cdots + 69\!\cdots\!49)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 856646 T^{7} + \cdots + 86\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 812781263700 T^{6} + \cdots + 41\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1094608 T^{3} + \cdots - 40\!\cdots\!71)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 663442 T^{7} + \cdots + 66\!\cdots\!16)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 705097881360 T^{14} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{8} + 2921642775864 T^{6} + \cdots + 10\!\cdots\!61)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 1100032 T^{7} + \cdots + 52\!\cdots\!24)^{2} \) Copy content Toggle raw display
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