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Properties

Label 42.9.g.b
Level $42$
Weight $9$
Character orbit 42.g
Analytic conductor $17.110$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 2 x^{11} + 6683 x^{10} + 105006 x^{9} + 34690411 x^{8} + 548172728 x^{7} + 69245553226 x^{6} + 2257669296800 x^{5} + 109555419264148 x^{4} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{6}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2}) q^{2} + (27 \beta_1 - 27) q^{3} + ( - 128 \beta_1 - 128) q^{4} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 63 \beta_1 - 126) q^{5} + ( - 54 \beta_{3} + 27 \beta_{2}) q^{6} + (\beta_{10} - \beta_{6} - \beta_{5} + 45 \beta_{3} - 64 \beta_{2} + 100 \beta_1 - 318) q^{7} + 128 \beta_{2} q^{8} - 2187 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2}) q^{2} + (27 \beta_1 - 27) q^{3} + ( - 128 \beta_1 - 128) q^{4} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 63 \beta_1 - 126) q^{5} + ( - 54 \beta_{3} + 27 \beta_{2}) q^{6} + (\beta_{10} - \beta_{6} - \beta_{5} + 45 \beta_{3} - 64 \beta_{2} + 100 \beta_1 - 318) q^{7} + 128 \beta_{2} q^{8} - 2187 \beta_1 q^{9} + ( - 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - 4 \beta_{7} + 6 \beta_{4} + \cdots + 236) q^{10}+ \cdots + (8748 \beta_{10} + 4374 \beta_{9} + 6561 \beta_{8} + \cdots + 3525444) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 486 q^{3} - 768 q^{4} - 1122 q^{5} - 4434 q^{7} + 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 486 q^{3} - 768 q^{4} - 1122 q^{5} - 4434 q^{7} + 13122 q^{9} + 4224 q^{10} + 9606 q^{11} + 62208 q^{12} - 19968 q^{14} + 60588 q^{15} - 98304 q^{16} - 227388 q^{17} + 315408 q^{19} + 155358 q^{21} + 557568 q^{22} - 663480 q^{23} + 601008 q^{25} - 585984 q^{26} + 398592 q^{28} + 5944884 q^{29} - 114048 q^{30} - 3214278 q^{31} - 778086 q^{33} - 1019784 q^{35} - 3359232 q^{36} + 1607760 q^{37} - 986880 q^{38} + 903636 q^{39} - 540672 q^{40} + 2788992 q^{42} + 12627648 q^{43} + 1229568 q^{44} - 2453814 q^{45} - 4608768 q^{46} - 7830240 q^{47} + 16477242 q^{49} - 16565760 q^{50} + 6139476 q^{51} - 4283904 q^{52} - 33200358 q^{53} - 8110080 q^{56} - 17032032 q^{57} + 5576064 q^{58} - 29657382 q^{59} - 3877632 q^{60} + 58344096 q^{61} - 2886840 q^{63} + 25165824 q^{64} + 30909216 q^{65} - 22581504 q^{66} + 7977444 q^{67} + 29105664 q^{68} - 59774976 q^{70} + 117213264 q^{71} - 170286636 q^{73} - 80311296 q^{74} - 48681648 q^{75} - 3340686 q^{77} + 31643136 q^{78} + 35420490 q^{79} + 18382848 q^{80} - 28697814 q^{81} + 154232832 q^{82} - 26085888 q^{84} + 416634600 q^{85} - 71958528 q^{86} - 240767802 q^{87} - 35684352 q^{88} + 78064092 q^{89} + 335513292 q^{91} + 169850880 q^{92} + 86785506 q^{93} - 18456576 q^{94} - 250727796 q^{95} - 150533376 q^{98} + 42016644 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 6683 x^{10} + 105006 x^{9} + 34690411 x^{8} + 548172728 x^{7} + 69245553226 x^{6} + 2257669296800 x^{5} + 109555419264148 x^{4} + \cdots + 30\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 31\!\cdots\!86 \nu^{11} + \cdots + 48\!\cdots\!92 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17\!\cdots\!43 \nu^{11} + \cdots + 43\!\cdots\!24 ) / 38\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 61\!\cdots\!37 \nu^{11} + \cdots + 68\!\cdots\!04 ) / 49\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 30\!\cdots\!64 \nu^{11} + \cdots - 86\!\cdots\!68 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22\!\cdots\!12 \nu^{11} + \cdots + 22\!\cdots\!16 ) / 43\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!62 \nu^{11} + \cdots + 29\!\cdots\!16 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29\!\cdots\!52 \nu^{11} + \cdots - 24\!\cdots\!64 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31\!\cdots\!04 \nu^{11} + \cdots - 75\!\cdots\!52 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19\!\cdots\!07 \nu^{11} + \cdots - 53\!\cdots\!44 ) / 98\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 39\!\cdots\!86 \nu^{11} + \cdots - 22\!\cdots\!52 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!94 \nu^{11} + \cdots + 72\!\cdots\!28 ) / 19\!\cdots\!40 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 5 \beta_{11} + 5 \beta_{10} + 10 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 15 \beta_{6} - 6 \beta_{5} - 20 \beta_{4} - 8 \beta_{2} - 225 \beta _1 + 2 ) / 672 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 74 \beta_{11} - 31 \beta_{10} + 31 \beta_{9} - 37 \beta_{8} - 77 \beta_{7} - 3 \beta_{6} + 400 \beta_{5} - 237 \beta_{4} - 15592 \beta_{3} - 40 \beta_{2} - 499011 \beta _1 - 499319 ) / 224 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 23491 \beta_{11} - 46882 \beta_{10} - 23441 \beta_{9} - 23681 \beta_{8} - 46817 \beta_{7} + 190 \beta_{6} - 46114 \beta_{5} + 115569 \beta_{4} + 23516 \beta_{3} + 634974 \beta_{2} + \cdots - 19793895 ) / 672 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 543303 \beta_{11} - 169655 \beta_{10} - 339310 \beta_{9} - 380554 \beta_{8} + 380554 \beta_{7} + 322045 \beta_{6} - 768094 \beta_{5} - 442596 \beta_{4} + 103810108 \beta_{3} + \cdots + 380554 ) / 224 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 22413082 \beta_{11} + 109373171 \beta_{10} - 109373171 \beta_{9} + 92331913 \beta_{8} + 176143197 \beta_{7} + 153730115 \beta_{6} + 433650592 \beta_{5} + \cdots + 96506987983 ) / 672 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 670559331 \beta_{11} + 1842895410 \beta_{10} + 921447705 \beta_{9} + 2638131273 \beta_{8} - 751897467 \beta_{7} - 1967571942 \beta_{6} - 3154371342 \beta_{5} + \cdots + 7896264848363 ) / 224 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 553888077125 \beta_{11} + 507375414005 \beta_{10} + 1014750828010 \beta_{9} + 168033488614 \beta_{8} - 168033488614 \beta_{7} - 710948764275 \beta_{6} + \cdots - 168033488614 ) / 672 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9802335487634 \beta_{11} - 4894242561043 \beta_{10} + 4894242561043 \beta_{9} - 2938413657937 \beta_{8} - 4898912864321 \beta_{7} + \cdots - 34\!\cdots\!83 ) / 224 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 16\!\cdots\!57 \beta_{11} + \cdots - 22\!\cdots\!77 ) / 672 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 61\!\cdots\!93 \beta_{11} + \cdots + 47\!\cdots\!38 ) / 224 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 54\!\cdots\!78 \beta_{11} + \cdots + 11\!\cdots\!49 ) / 672 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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
−32.5038 + 56.2982i
34.6228 − 59.9685i
−1.61905 + 2.80427i
27.0215 − 46.8026i
−13.8894 + 24.0571i
−12.6321 + 21.8795i
−32.5038 − 56.2982i
34.6228 + 59.9685i
−1.61905 − 2.80427i
27.0215 + 46.8026i
−13.8894 − 24.0571i
−12.6321 − 21.8795i
−5.65685 + 9.79796i −40.5000 + 23.3827i −64.0000 − 110.851i −1036.40 − 598.368i − 529.090i −2203.25 + 954.186i 1448.15 1093.50 − 1894.00i 11725.6 − 6769.76i
19.2 −5.65685 + 9.79796i −40.5000 + 23.3827i −64.0000 − 110.851i 124.890 + 72.1052i − 529.090i 1980.48 + 1357.39i 1448.15 1093.50 − 1894.00i −1412.97 + 815.777i
19.3 −5.65685 + 9.79796i −40.5000 + 23.3827i −64.0000 − 110.851i 537.676 + 310.427i − 529.090i −2285.79 − 734.809i 1448.15 1093.50 − 1894.00i −6083.11 + 3512.08i
19.4 5.65685 − 9.79796i −40.5000 + 23.3827i −64.0000 − 110.851i −666.879 − 385.023i 529.090i 1263.93 − 2041.39i −1448.15 1093.50 − 1894.00i −7544.87 + 4356.03i
19.5 5.65685 − 9.79796i −40.5000 + 23.3827i −64.0000 − 110.851i −127.532 − 73.6307i 529.090i 1148.13 + 2108.70i −1448.15 1093.50 − 1894.00i −1442.86 + 833.036i
19.6 5.65685 − 9.79796i −40.5000 + 23.3827i −64.0000 − 110.851i 607.249 + 350.595i 529.090i −2120.49 − 1126.19i −1448.15 1093.50 − 1894.00i 6870.24 − 3966.53i
31.1 −5.65685 − 9.79796i −40.5000 − 23.3827i −64.0000 + 110.851i −1036.40 + 598.368i 529.090i −2203.25 − 954.186i 1448.15 1093.50 + 1894.00i 11725.6 + 6769.76i
31.2 −5.65685 − 9.79796i −40.5000 − 23.3827i −64.0000 + 110.851i 124.890 − 72.1052i 529.090i 1980.48 − 1357.39i 1448.15 1093.50 + 1894.00i −1412.97 − 815.777i
31.3 −5.65685 − 9.79796i −40.5000 − 23.3827i −64.0000 + 110.851i 537.676 − 310.427i 529.090i −2285.79 + 734.809i 1448.15 1093.50 + 1894.00i −6083.11 − 3512.08i
31.4 5.65685 + 9.79796i −40.5000 − 23.3827i −64.0000 + 110.851i −666.879 + 385.023i − 529.090i 1263.93 + 2041.39i −1448.15 1093.50 + 1894.00i −7544.87 − 4356.03i
31.5 5.65685 + 9.79796i −40.5000 − 23.3827i −64.0000 + 110.851i −127.532 + 73.6307i − 529.090i 1148.13 − 2108.70i −1448.15 1093.50 + 1894.00i −1442.86 − 833.036i
31.6 5.65685 + 9.79796i −40.5000 − 23.3827i −64.0000 + 110.851i 607.249 − 350.595i − 529.090i −2120.49 + 1126.19i −1448.15 1093.50 + 1894.00i 6870.24 + 3966.53i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.6
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.b ✓ 12
3.b odd 2 1 126.9.n.c 12
7.c even 3 1 294.9.c.b 12
7.d odd 6 1 inner 42.9.g.b ✓ 12
7.d odd 6 1 294.9.c.b 12
21.g even 6 1 126.9.n.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.9.g.b ✓ 12 1.a even 1 1 trivial
42.9.g.b ✓ 12 7.d odd 6 1 inner
126.9.n.c 12 3.b odd 2 1
126.9.n.c 12 21.g even 6 1
294.9.c.b 12 7.c even 3 1
294.9.c.b 12 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 1122 T_{5}^{11} - 842937 T_{5}^{10} - 1416597930 T_{5}^{9} + 921696727761 T_{5}^{8} + 814659406013160 T_{5}^{7} + \cdots + 72\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(42, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 128 T^{2} + 16384)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} + 81 T + 2187)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 1122 T^{11} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + 4434 T^{11} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{12} - 9606 T^{11} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{12} + 6646642092 T^{10} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{12} + 227388 T^{11} + \cdots + 47\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{12} - 315408 T^{11} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{12} + 663480 T^{11} + \cdots + 46\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{6} - 2972442 T^{5} + \cdots - 30\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 3214278 T^{11} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( T^{12} - 1607760 T^{11} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{12} + 61465893903048 T^{10} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{6} - 6313824 T^{5} + \cdots + 34\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 7830240 T^{11} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{12} + 33200358 T^{11} + \cdots + 94\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{12} + 29657382 T^{11} + \cdots + 79\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{12} - 58344096 T^{11} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{12} - 7977444 T^{11} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{6} - 58606632 T^{5} + \cdots - 45\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 170286636 T^{11} + \cdots + 82\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{12} - 35420490 T^{11} + \cdots + 23\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 40\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{12} - 78064092 T^{11} + \cdots + 72\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 73\!\cdots\!36 \) Copy content Toggle raw display
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