Properties

Label 42.9.c.a
Level $42$
Weight $9$
Character orbit 42.c
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.1099016226\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 2 x^{11} + 7731 x^{10} + 218714 x^{9} + 46944238 x^{8} + 954612102 x^{7} + 95335059229 x^{6} + 576178351086 x^{5} + 122405480888457 x^{4} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{18}\cdot 7^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{2} q^{3} + 128 q^{4} + ( - \beta_{3} + 2 \beta_{2}) q^{5} - \beta_{4} q^{6} + (\beta_{6} - 9 \beta_{2} - 8 \beta_1 + 535) q^{7} + 128 \beta_1 q^{8} - 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{2} q^{3} + 128 q^{4} + ( - \beta_{3} + 2 \beta_{2}) q^{5} - \beta_{4} q^{6} + (\beta_{6} - 9 \beta_{2} - 8 \beta_1 + 535) q^{7} + 128 \beta_1 q^{8} - 2187 q^{9} + (\beta_{9} + \beta_{7} + 4 \beta_{4} + 5 \beta_{3} + 28 \beta_{2}) q^{10} + ( - \beta_{10} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - 331 \beta_1 + 362) q^{11} - 128 \beta_{2} q^{12} + (\beta_{8} + 3 \beta_{7} + \beta_{6} + 8 \beta_{4} - 13 \beta_{3} - 213 \beta_{2}) q^{13} + (2 \beta_{9} - \beta_{8} - 3 \beta_{7} + \beta_{6} + \beta_{5} - 10 \beta_{4} + \beta_{3} + \cdots - 1024) q^{14}+ \cdots + (2187 \beta_{10} - 2187 \beta_{6} - 2187 \beta_{5} + 2187 \beta_{4} + \cdots - 791694) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 1536 q^{4} + 6420 q^{7} - 26244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 1536 q^{4} + 6420 q^{7} - 26244 q^{9} + 4344 q^{11} - 12288 q^{14} + 59616 q^{15} + 196608 q^{16} - 224856 q^{21} - 508416 q^{22} + 499800 q^{23} - 3001476 q^{25} + 821760 q^{28} - 1278408 q^{29} + 705024 q^{30} + 2028912 q^{35} - 3359232 q^{36} + 7068648 q^{37} - 5473008 q^{39} + 1513728 q^{42} - 11388024 q^{43} + 556032 q^{44} + 8171520 q^{46} - 12346788 q^{49} + 30019584 q^{50} + 16727472 q^{51} + 19714968 q^{53} - 1572864 q^{56} - 10386144 q^{57} - 17696256 q^{58} + 7630848 q^{60} - 14040540 q^{63} + 25165824 q^{64} - 93770592 q^{65} - 9394008 q^{67} + 11218944 q^{70} + 5393208 q^{71} + 58512384 q^{74} + 24982968 q^{77} + 32638464 q^{78} + 134560968 q^{79} + 57395628 q^{81} - 28781568 q^{84} - 102074640 q^{85} - 282934272 q^{86} - 65077248 q^{88} - 96105408 q^{91} + 63974400 q^{92} + 202339296 q^{93} - 378351840 q^{95} + 387747840 q^{98} - 9500328 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 7731 x^{10} + 218714 x^{9} + 46944238 x^{8} + 954612102 x^{7} + 95335059229 x^{6} + 576178351086 x^{5} + 122405480888457 x^{4} + \cdots + 37\!\cdots\!84 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 63\!\cdots\!07 \nu^{11} + \cdots - 16\!\cdots\!08 ) / 21\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22\!\cdots\!69 \nu^{11} + \cdots + 27\!\cdots\!64 ) / 92\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 69\!\cdots\!27 \nu^{11} + \cdots + 59\!\cdots\!32 ) / 25\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12\!\cdots\!53 \nu^{11} + \cdots - 38\!\cdots\!52 ) / 28\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!59 \nu^{11} + \cdots - 47\!\cdots\!20 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\!\cdots\!06 \nu^{11} + \cdots + 17\!\cdots\!64 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17\!\cdots\!61 \nu^{11} + \cdots - 15\!\cdots\!48 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13\!\cdots\!06 \nu^{11} + \cdots - 17\!\cdots\!68 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 27\!\cdots\!53 \nu^{11} + \cdots - 53\!\cdots\!96 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 42\!\cdots\!84 \nu^{11} + \cdots + 23\!\cdots\!68 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 56\!\cdots\!46 \nu^{11} + \cdots + 64\!\cdots\!04 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2 \beta_{11} + 24 \beta_{10} + 137 \beta_{9} - 159 \beta_{8} + 132 \beta_{7} + 113 \beta_{6} + 11 \beta_{5} + 131 \beta_{4} + 508 \beta_{3} + 236 \beta_{2} - 757 \beta _1 + 6048 ) / 36288 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 182 \beta_{11} + 56 \beta_{10} + 7467 \beta_{9} - 3215 \beta_{8} - 1510 \beta_{7} + 13101 \beta_{6} + 1799 \beta_{5} + 50105 \beta_{4} - 48478 \beta_{3} + 1711980 \beta_{2} - 1772155 \beta _1 - 46744992 ) / 36288 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1018 \beta_{11} - 55376 \beta_{10} + 106626 \beta_{9} + 107135 \beta_{8} - 107135 \beta_{7} + 538905 \beta_{6} + 29812 \beta_{5} - 350708 \beta_{4} - 213761 \beta_{3} + \cdots - 531105120 ) / 9072 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 883638 \beta_{11} - 3028648 \beta_{10} - 50161667 \beta_{9} + 48791677 \beta_{8} - 12055936 \beta_{7} - 7089335 \beta_{6} + 12627951 \beta_{5} - 465121683 \beta_{4} + \cdots - 212023592928 ) / 36288 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 35913818 \beta_{11} + 606616184 \beta_{10} - 7070671369 \beta_{9} + 2703179005 \beta_{8} + 457009346 \beta_{7} - 16616533471 \beta_{6} + \cdots + 11505977991648 ) / 36288 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 361359674 \beta_{11} + 2408334552 \beta_{10} - 5896286276 \beta_{9} - 6076966113 \beta_{8} + 6076966113 \beta_{7} - 39588217505 \beta_{6} + \cdots + 92345788388736 ) / 1296 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 317909957062 \beta_{11} + 3825593556088 \beta_{10} + 32880588483733 \beta_{9} - 35898291037075 \beta_{8} + 12957397876660 \beta_{7} + \cdots + 94\!\cdots\!04 ) / 36288 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 11051294283638 \beta_{11} - 92753784863432 \beta_{10} + \cdots - 29\!\cdots\!64 ) / 12096 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 12\!\cdots\!16 \beta_{11} + \cdots - 35\!\cdots\!00 ) / 9072 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 23\!\cdots\!98 \beta_{11} + \cdots - 63\!\cdots\!32 ) / 36288 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 18\!\cdots\!06 \beta_{11} + \cdots + 52\!\cdots\!84 ) / 36288 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
16.2382 + 28.1253i
11.5485 + 20.0026i
−26.5796 46.0372i
−26.5796 + 46.0372i
11.5485 20.0026i
16.2382 28.1253i
−18.3977 31.8658i
−24.4982 42.4322i
42.6889 + 73.9393i
42.6889 73.9393i
−24.4982 + 42.4322i
−18.3977 + 31.8658i
−11.3137 46.7654i 128.000 1031.06i 529.090i −283.934 2384.15i −1448.15 −2187.00 11665.2i
13.2 −11.3137 46.7654i 128.000 200.812i 529.090i 630.018 + 2316.87i −1448.15 −2187.00 2271.93i
13.3 −11.3137 46.7654i 128.000 1217.44i 529.090i 1530.45 1850.01i −1448.15 −2187.00 13773.8i
13.4 −11.3137 46.7654i 128.000 1217.44i 529.090i 1530.45 + 1850.01i −1448.15 −2187.00 13773.8i
13.5 −11.3137 46.7654i 128.000 200.812i 529.090i 630.018 2316.87i −1448.15 −2187.00 2271.93i
13.6 −11.3137 46.7654i 128.000 1031.06i 529.090i −283.934 + 2384.15i −1448.15 −2187.00 11665.2i
13.7 11.3137 46.7654i 128.000 640.234i 529.090i 2336.76 + 551.670i 1448.15 −2187.00 7243.42i
13.8 11.3137 46.7654i 128.000 561.514i 529.090i 1145.19 2110.29i 1448.15 −2187.00 6352.81i
13.9 11.3137 46.7654i 128.000 730.548i 529.090i −2148.48 + 1071.83i 1448.15 −2187.00 8265.21i
13.10 11.3137 46.7654i 128.000 730.548i 529.090i −2148.48 1071.83i 1448.15 −2187.00 8265.21i
13.11 11.3137 46.7654i 128.000 561.514i 529.090i 1145.19 + 2110.29i 1448.15 −2187.00 6352.81i
13.12 11.3137 46.7654i 128.000 640.234i 529.090i 2336.76 551.670i 1448.15 −2187.00 7243.42i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.9.c.a 12
3.b odd 2 1 126.9.c.c 12
4.b odd 2 1 336.9.f.c 12
7.b odd 2 1 inner 42.9.c.a 12
21.c even 2 1 126.9.c.c 12
28.d even 2 1 336.9.f.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.9.c.a 12 1.a even 1 1 trivial
42.9.c.a 12 7.b odd 2 1 inner
126.9.c.c 12 3.b odd 2 1
126.9.c.c 12 21.c even 2 1
336.9.f.c 12 4.b odd 2 1
336.9.f.c 12 28.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(42, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 128)^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2187)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 3844488 T^{10} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} - 6420 T^{11} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{6} - 2172 T^{5} + \cdots + 58\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 3291860544 T^{10} + \cdots + 95\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{12} + 49633345128 T^{10} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + 51783512208 T^{10} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{6} - 249900 T^{5} + \cdots - 92\!\cdots\!72)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 639204 T^{5} + \cdots - 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 4654633419072 T^{10} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( (T^{6} - 3534324 T^{5} + \cdots + 45\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 17887757094696 T^{10} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( (T^{6} + 5694012 T^{5} + \cdots + 10\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 72986678806080 T^{10} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{6} - 9857484 T^{5} + \cdots - 40\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( (T^{6} + 4697004 T^{5} + \cdots - 32\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 2696604 T^{5} + \cdots - 11\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 67\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{6} - 67280484 T^{5} + \cdots + 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
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