Properties

Label 42.9.g.a
Level $42$
Weight $9$
Character orbit 42.g
Analytic conductor $17.110$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 42.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.1099016226\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2x^{7} + 243x^{6} - 2x^{5} + 49033x^{4} - 23088x^{3} + 2105352x^{2} + 2056320x + 73410624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2}) q^{2} + (27 \beta_1 + 54) q^{3} + 128 \beta_1 q^{4} + ( - \beta_{7} - 5 \beta_{6} - 15 \beta_{3} - 16 \beta_{2} + 186 \beta_1 - 185) q^{5} + ( - 54 \beta_{3} + 27 \beta_{2}) q^{6} + ( - 14 \beta_{7} - 7 \beta_{6} + 7 \beta_{5} - 21 \beta_{4} - 7 \beta_{3} + \cdots - 434) q^{7}+ \cdots + (2187 \beta_1 + 2187) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2}) q^{2} + (27 \beta_1 + 54) q^{3} + 128 \beta_1 q^{4} + ( - \beta_{7} - 5 \beta_{6} - 15 \beta_{3} - 16 \beta_{2} + 186 \beta_1 - 185) q^{5} + ( - 54 \beta_{3} + 27 \beta_{2}) q^{6} + ( - 14 \beta_{7} - 7 \beta_{6} + 7 \beta_{5} - 21 \beta_{4} - 7 \beta_{3} + \cdots - 434) q^{7}+ \cdots + (231822 \beta_{7} + 367416 \beta_{6} - 115911 \beta_{5} + \cdots - 4122495) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 324 q^{3} - 512 q^{4} - 2226 q^{5} - 140 q^{7} + 8748 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 324 q^{3} - 512 q^{4} - 2226 q^{5} - 140 q^{7} + 8748 q^{9} - 21888 q^{10} - 7434 q^{11} - 41472 q^{12} + 32256 q^{14} - 120204 q^{15} - 65536 q^{16} + 7932 q^{17} - 773082 q^{19} + 130410 q^{21} + 557568 q^{22} + 391200 q^{23} - 138886 q^{25} - 805632 q^{26} + 654080 q^{28} - 2318196 q^{29} - 590976 q^{30} - 3212544 q^{31} - 602154 q^{33} + 3340932 q^{35} - 2239488 q^{36} + 544802 q^{37} - 1570560 q^{38} + 1341522 q^{39} + 2801664 q^{40} - 72576 q^{42} + 552284 q^{43} - 951552 q^{44} - 4868262 q^{45} + 7878912 q^{46} - 15735252 q^{47} - 588196 q^{49} + 25400832 q^{50} + 214164 q^{51} + 6359808 q^{52} + 10935222 q^{53} - 8601600 q^{56} - 41746428 q^{57} - 8158848 q^{58} + 63840318 q^{59} + 7693056 q^{60} + 43108176 q^{61} + 10869390 q^{63} + 16777216 q^{64} - 9118644 q^{65} + 22581504 q^{66} + 48827290 q^{67} - 1015296 q^{68} + 11053056 q^{70} - 164526744 q^{71} + 150530022 q^{73} - 31064064 q^{74} - 11249766 q^{75} - 137090394 q^{77} - 43504128 q^{78} - 18689696 q^{79} + 36470784 q^{80} - 19131876 q^{81} - 33914880 q^{82} + 18144000 q^{84} - 190617576 q^{85} - 37077504 q^{86} - 93886938 q^{87} - 35684352 q^{88} - 191605284 q^{89} + 110138574 q^{91} - 100147200 q^{92} - 86738688 q^{93} + 302275584 q^{94} + 260649552 q^{95} - 282428160 q^{98} - 32516316 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 243x^{6} - 2x^{5} + 49033x^{4} - 23088x^{3} + 2105352x^{2} + 2056320x + 73410624 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6934582 \nu^{7} - 213073991 \nu^{6} + 1776935352 \nu^{5} - 40879197461 \nu^{4} + 277529721052 \nu^{3} + \cdots - 413175598845912 ) / 362820220623000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 239 \nu^{7} + 16582 \nu^{6} - 49029 \nu^{5} - 74978 \nu^{4} + 91321 \nu^{3} - 6267024 \nu^{2} - 77523264 \nu - 63227678976 ) / 5451279750 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 87595352 \nu^{7} - 3004572376 \nu^{6} + 25056699072 \nu^{5} - 833613645796 \nu^{4} + 3913467474272 \nu^{3} + \cdots - 58\!\cdots\!32 ) / 408172748200875 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5452777121 \nu^{7} + 23659464973 \nu^{6} - 2162183841681 \nu^{5} + 4145347854883 \nu^{4} - 387341147347031 \nu^{3} + \cdots + 10\!\cdots\!36 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2668426 \nu^{7} + 5906063 \nu^{6} - 783660036 \nu^{5} + 409326773 \nu^{4} - 158695101136 \nu^{3} + 89328125259 \nu^{2} + \cdots - 967483677384 ) / 370687023000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2720810176 \nu^{7} + 1044395813 \nu^{6} - 445348695786 \nu^{5} - 1121515139977 \nu^{4} - 80588018609386 \nu^{3} + \cdots - 58\!\cdots\!84 ) / 362820220623000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9283131394 \nu^{7} - 5722683347 \nu^{6} + 1651704201984 \nu^{5} + 3375232761163 \nu^{4} + 336750861568384 \nu^{3} + \cdots + 31\!\cdots\!46 ) / 816345496401750 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -4\beta_{7} - 4\beta_{6} - 4\beta_{5} + 4\beta_{4} + 3\beta_{3} - 4\beta_{2} - 40\beta _1 + 4 ) / 84 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} - 4\beta_{5} + 4\beta_{4} + 285\beta_{3} - 283\beta_{2} - 5062\beta _1 - 5060 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1184\beta_{7} + 1940\beta_{6} - 592\beta_{5} + 970\beta_{4} + 378\beta_{3} + 173\beta_{2} - 592\beta _1 - 15458 ) / 84 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 295 \beta_{7} + 484 \beta_{6} + 295 \beta_{5} - 484 \beta_{4} - 34572 \beta_{3} + 295 \beta_{2} + 429775 \beta _1 - 295 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 108616 \beta_{7} - 200470 \beta_{6} + 217232 \beta_{5} - 400940 \beta_{4} - 537897 \beta_{3} + 245573 \beta_{2} + 4328270 \beta _1 + 4219654 ) / 84 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 322900 \beta_{7} - 596572 \beta_{6} + 161450 \beta_{5} - 298286 \beta_{4} - 136836 \beta_{3} + 13903163 \beta_{2} + 161450 \beta _1 + 164675554 ) / 42 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 21249568 \beta_{7} - 40329874 \beta_{6} - 21249568 \beta_{5} + 40329874 \beta_{4} + 118965801 \beta_{3} - 21249568 \beta_{2} - 1126597834 \beta _1 + 21249568 ) / 84 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
3.56219 + 6.16990i
−3.06219 5.30387i
−6.76102 11.7104i
7.26102 + 12.5765i
3.56219 6.16990i
−3.06219 + 5.30387i
−6.76102 + 11.7104i
7.26102 12.5765i
−5.65685 + 9.79796i 40.5000 23.3827i −64.0000 110.851i −271.195 156.575i 529.090i 746.403 + 2282.04i 1448.15 1093.50 1894.00i 3068.22 1771.44i
19.2 −5.65685 + 9.79796i 40.5000 23.3827i −64.0000 110.851i 198.356 + 114.521i 529.090i −2266.33 792.819i 1448.15 1093.50 1894.00i −2244.14 + 1295.66i
19.3 5.65685 9.79796i 40.5000 23.3827i −64.0000 110.851i −974.971 562.900i 529.090i −797.874 + 2264.55i −1448.15 1093.50 1894.00i −11030.5 + 6368.49i
19.4 5.65685 9.79796i 40.5000 23.3827i −64.0000 110.851i −65.1898 37.6374i 529.090i 2247.80 843.923i −1448.15 1093.50 1894.00i −737.539 + 425.818i
31.1 −5.65685 9.79796i 40.5000 + 23.3827i −64.0000 + 110.851i −271.195 + 156.575i 529.090i 746.403 2282.04i 1448.15 1093.50 + 1894.00i 3068.22 + 1771.44i
31.2 −5.65685 9.79796i 40.5000 + 23.3827i −64.0000 + 110.851i 198.356 114.521i 529.090i −2266.33 + 792.819i 1448.15 1093.50 + 1894.00i −2244.14 1295.66i
31.3 5.65685 + 9.79796i 40.5000 + 23.3827i −64.0000 + 110.851i −974.971 + 562.900i 529.090i −797.874 2264.55i −1448.15 1093.50 + 1894.00i −11030.5 6368.49i
31.4 5.65685 + 9.79796i 40.5000 + 23.3827i −64.0000 + 110.851i −65.1898 + 37.6374i 529.090i 2247.80 + 843.923i −1448.15 1093.50 + 1894.00i −737.539 425.818i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.9.g.a 8
3.b odd 2 1 126.9.n.a 8
7.c even 3 1 294.9.c.a 8
7.d odd 6 1 inner 42.9.g.a 8
7.d odd 6 1 294.9.c.a 8
21.g even 6 1 126.9.n.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.9.g.a 8 1.a even 1 1 trivial
42.9.g.a 8 7.d odd 6 1 inner
126.9.n.a 8 3.b odd 2 1
126.9.n.a 8 21.g even 6 1
294.9.c.a 8 7.c even 3 1
294.9.c.a 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2226 T_{5}^{7} + 1765731 T_{5}^{6} + 253850814 T_{5}^{5} - 82470162699 T_{5}^{4} - 15607863346140 T_{5}^{3} + \cdots + 36\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(42, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 128 T^{2} + 16384)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 81 T + 2187)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 2226 T^{7} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + 140 T^{7} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + 7434 T^{7} + \cdots + 51\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{8} + 3917293830 T^{6} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{8} - 7932 T^{7} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{8} + 773082 T^{7} + \cdots + 59\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{8} - 391200 T^{7} + \cdots + 96\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{4} + 1159098 T^{3} + \cdots - 32\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 3212544 T^{7} + \cdots + 25\!\cdots\!89 \) Copy content Toggle raw display
$37$ \( T^{8} - 544802 T^{7} + \cdots + 30\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{8} + 9884157950064 T^{6} + \cdots + 46\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{4} - 276142 T^{3} + \cdots + 26\!\cdots\!72)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 15735252 T^{7} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} - 10935222 T^{7} + \cdots + 65\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{8} - 63840318 T^{7} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{8} - 43108176 T^{7} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{8} - 48827290 T^{7} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} + 82263372 T^{3} + \cdots + 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 150530022 T^{7} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{8} + 18689696 T^{7} + \cdots + 58\!\cdots\!29 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{8} + 191605284 T^{7} + \cdots + 46\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 91\!\cdots\!24 \) Copy content Toggle raw display
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