Properties

Label 336.4.bc.c
Level $336$
Weight $4$
Character orbit 336.bc
Analytic conductor $19.825$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 14 x^{10} - 32 x^{9} + 70 x^{8} + 224 x^{7} - 50 x^{6} + 2016 x^{5} + 5670 x^{4} - 23328 x^{3} - 91854 x^{2} + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{7} \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{3}) q^{3} - \beta_{11} q^{5} + ( - \beta_{9} - 2 \beta_{6} - 2 \beta_1 + 2) q^{7} + ( - \beta_{9} - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 12 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_{3}) q^{3} - \beta_{11} q^{5} + ( - \beta_{9} - 2 \beta_{6} - 2 \beta_1 + 2) q^{7} + ( - \beta_{9} - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 12 \beta_1 + 1) q^{9} + (\beta_{11} - \beta_{10} - 2 \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{4} + 2) q^{11} + (6 \beta_{9} - 2 \beta_{7} - 8 \beta_{6} + 8 \beta_{2} + 32 \beta_1 + 6) q^{13} + (2 \beta_{11} + \beta_{10} - 8 \beta_{9} + 8 \beta_{7} + \beta_{4} - 2 \beta_{3} - 11) q^{15} + ( - 2 \beta_{11} - 2 \beta_{10} + \beta_{9} - 2 \beta_{8} - 5 \beta_{7} + 3 \beta_{6} + \cdots + 3) q^{17}+ \cdots + ( - 36 \beta_{11} - 18 \beta_{10} - 18 \beta_{9} + 12 \beta_{8} + 123 \beta_{7} + \cdots - 780) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 42 q^{7} - 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 42 q^{7} - 84 q^{9} - 132 q^{15} - 204 q^{19} - 378 q^{21} - 444 q^{25} - 1458 q^{31} - 108 q^{33} + 240 q^{37} + 432 q^{39} - 342 q^{45} - 1218 q^{49} + 300 q^{51} + 180 q^{57} + 2148 q^{61} - 1596 q^{63} - 1980 q^{67} - 3084 q^{73} + 3384 q^{75} + 438 q^{79} + 1008 q^{81} - 6144 q^{85} + 2898 q^{87} - 3780 q^{91} + 882 q^{93} - 9216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 14 x^{10} - 32 x^{9} + 70 x^{8} + 224 x^{7} - 50 x^{6} + 2016 x^{5} + 5670 x^{4} - 23328 x^{3} - 91854 x^{2} + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 36 \nu^{11} - 109 \nu^{10} + 180 \nu^{9} + 1949 \nu^{8} + 2588 \nu^{7} - 1681 \nu^{6} - 7388 \nu^{5} - 64615 \nu^{4} - 433908 \nu^{3} - 556065 \nu^{2} + 2452356 \nu + 6436341 ) / 2571912 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 871 \nu^{11} + 2952 \nu^{10} + 7489 \nu^{9} - 31292 \nu^{8} - 131909 \nu^{7} - 50236 \nu^{6} - 103283 \nu^{5} + 1112616 \nu^{4} + 5111019 \nu^{3} + \cdots - 192499740 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 109 \nu^{11} + 324 \nu^{10} - 797 \nu^{9} - 5108 \nu^{8} - 6383 \nu^{7} + 9188 \nu^{6} - 7961 \nu^{5} + 229788 \nu^{4} + 1395873 \nu^{3} + 854388 \nu^{2} + \cdots - 19131876 ) / 2571912 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 109 \nu^{11} - 324 \nu^{10} + 797 \nu^{9} + 5108 \nu^{8} + 6383 \nu^{7} - 9188 \nu^{6} + 7961 \nu^{5} - 229788 \nu^{4} - 1395873 \nu^{3} - 854388 \nu^{2} + \cdots + 19131876 ) / 2571912 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1522 \nu^{11} - 11709 \nu^{10} + 3326 \nu^{9} + 159413 \nu^{8} + 296822 \nu^{7} - 135341 \nu^{6} - 987142 \nu^{5} - 3274839 \nu^{4} - 41038974 \nu^{3} + \cdots + 653377185 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1643 \nu^{11} - 4941 \nu^{10} + 403 \nu^{9} + 56869 \nu^{8} + 156097 \nu^{7} + 149963 \nu^{6} + 486583 \nu^{5} - 669015 \nu^{4} - 18160767 \nu^{3} + \cdots + 250426809 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 58 \nu^{11} - 36 \nu^{10} + 569 \nu^{9} + 1631 \nu^{8} + 494 \nu^{7} + 2470 \nu^{6} + 1154 \nu^{5} - 102492 \nu^{4} - 375435 \nu^{3} + 400221 \nu^{2} + 1430298 \nu + 2834352 ) / 826686 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1931 \nu^{11} + 774 \nu^{10} - 425 \nu^{9} + 5758 \nu^{8} - 83879 \nu^{7} + 542606 \nu^{6} + 1335967 \nu^{5} - 3732822 \nu^{4} - 10260027 \nu^{3} + \cdots - 65544390 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 74 \nu^{11} + 207 \nu^{10} - 793 \nu^{9} - 3808 \nu^{8} - 4846 \nu^{7} + 2878 \nu^{6} + 13022 \nu^{5} + 236277 \nu^{4} + 974187 \nu^{3} + 454896 \nu^{2} - 6364170 \nu - 13581270 ) / 826686 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 125 \nu^{11} - 477 \nu^{10} + 454 \nu^{9} + 9949 \nu^{8} + 18097 \nu^{7} - 9712 \nu^{6} - 17087 \nu^{5} - 300483 \nu^{4} - 2110050 \nu^{3} - 3421197 \nu^{2} + \cdots + 40743810 ) / 826686 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3554 \nu^{11} - 16407 \nu^{10} + 12982 \nu^{9} + 145867 \nu^{8} + 174346 \nu^{7} + 403685 \nu^{6} + 1680538 \nu^{5} - 4010661 \nu^{4} - 40201758 \nu^{3} + \cdots + 264480471 ) / 23147208 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + 2\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 12\beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} + 4 \beta_{7} - 5 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} + 5 \beta _1 + 26 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14\beta_{9} - 5\beta_{8} + 9\beta_{7} + 14\beta_{6} + 16\beta_{4} + 21\beta_{3} + 14\beta_{2} + 56\beta _1 + 47 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14 \beta_{11} + 28 \beta_{10} + 33 \beta_{9} + 21 \beta_{7} - 19 \beta_{6} - 12 \beta_{5} + 78 \beta_{4} - 33 \beta_{3} - 2 \beta_{2} - 539 \beta _1 - 19 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 32 \beta_{11} + 16 \beta_{10} - 110 \beta_{9} + 24 \beta_{8} + 50 \beta_{7} + 60 \beta_{6} + 24 \beta_{5} - 200 \beta_{4} + 400 \beta_{3} + 60 \beta_{2} - 60 \beta _1 + 721 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 70 \beta_{11} + 70 \beta_{10} - 266 \beta_{9} - 164 \beta_{8} + 214 \beta_{7} + 378 \beta_{6} + 157 \beta_{4} + 321 \beta_{3} - 266 \beta_{2} - 3920 \beta _1 - 3490 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 448 \beta_{11} + 896 \beta_{10} + 1521 \beta_{9} + 1566 \beta_{7} - 1703 \beta_{6} + 45 \beta_{5} - 755 \beta_{4} + 355 \beta_{3} + 137 \beta_{2} - 562 \beta _1 - 1703 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 622 \beta_{11} + 311 \beta_{10} - 3721 \beta_{9} - 1966 \beta_{8} + 194 \beta_{7} + 3527 \beta_{6} - 1966 \beta_{5} - 2078 \beta_{4} + 4156 \beta_{3} + 3527 \beta_{2} - 3527 \beta _1 + 12070 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5376 \beta_{11} + 5376 \beta_{10} + 6104 \beta_{9} + 3415 \beta_{8} + 8175 \beta_{7} + 4760 \beta_{6} + 13472 \beta_{4} + 10057 \beta_{3} + 6104 \beta_{2} - 39466 \beta _1 - 48985 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 10136 \beta_{11} + 20272 \beta_{10} - 28127 \beta_{9} - 25519 \beta_{7} - 10121 \beta_{6} + 2608 \beta_{5} - 39374 \beta_{4} + 18383 \beta_{3} + 35640 \beta_{2} - 286703 \beta _1 - 10121 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−1.67777 2.48698i
−2.14770 2.09461i
2.99268 + 0.209499i
−2.92030 + 0.686929i
2.88784 + 0.812653i
0.865250 + 2.87252i
−1.67777 + 2.48698i
−2.14770 + 2.09461i
2.99268 0.209499i
−2.92030 0.686929i
2.88784 0.812653i
0.865250 2.87252i
0 −4.30758 + 2.90598i 0 −7.41560 12.8442i 0 16.6528 + 8.10467i 0 10.1105 25.0355i 0
17.2 0 −3.62798 + 3.71992i 0 9.68891 + 16.7817i 0 2.66851 18.3270i 0 −0.675594 26.9915i 0
17.3 0 0.362863 5.18347i 0 7.41560 + 12.8442i 0 16.6528 + 8.10467i 0 −26.7367 3.76177i 0
17.4 0 1.18980 + 5.05810i 0 0.619556 + 1.07310i 0 −8.82127 + 16.2845i 0 −24.1688 + 12.0362i 0
17.5 0 1.40756 5.00188i 0 −9.68891 16.7817i 0 2.66851 18.3270i 0 −23.0376 14.0809i 0
17.6 0 4.97534 1.49866i 0 −0.619556 1.07310i 0 −8.82127 + 16.2845i 0 22.5081 14.9127i 0
257.1 0 −4.30758 2.90598i 0 −7.41560 + 12.8442i 0 16.6528 8.10467i 0 10.1105 + 25.0355i 0
257.2 0 −3.62798 3.71992i 0 9.68891 16.7817i 0 2.66851 + 18.3270i 0 −0.675594 + 26.9915i 0
257.3 0 0.362863 + 5.18347i 0 7.41560 12.8442i 0 16.6528 8.10467i 0 −26.7367 + 3.76177i 0
257.4 0 1.18980 5.05810i 0 0.619556 1.07310i 0 −8.82127 16.2845i 0 −24.1688 12.0362i 0
257.5 0 1.40756 + 5.00188i 0 −9.68891 + 16.7817i 0 2.66851 + 18.3270i 0 −23.0376 + 14.0809i 0
257.6 0 4.97534 + 1.49866i 0 −0.619556 + 1.07310i 0 −8.82127 16.2845i 0 22.5081 + 14.9127i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.bc.c 12
3.b odd 2 1 inner 336.4.bc.c 12
4.b odd 2 1 84.4.k.c 12
7.d odd 6 1 inner 336.4.bc.c 12
12.b even 2 1 84.4.k.c 12
21.g even 6 1 inner 336.4.bc.c 12
28.d even 2 1 588.4.k.c 12
28.f even 6 1 84.4.k.c 12
28.f even 6 1 588.4.f.c 12
28.g odd 6 1 588.4.f.c 12
28.g odd 6 1 588.4.k.c 12
84.h odd 2 1 588.4.k.c 12
84.j odd 6 1 84.4.k.c 12
84.j odd 6 1 588.4.f.c 12
84.n even 6 1 588.4.f.c 12
84.n even 6 1 588.4.k.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.k.c 12 4.b odd 2 1
84.4.k.c 12 12.b even 2 1
84.4.k.c 12 28.f even 6 1
84.4.k.c 12 84.j odd 6 1
336.4.bc.c 12 1.a even 1 1 trivial
336.4.bc.c 12 3.b odd 2 1 inner
336.4.bc.c 12 7.d odd 6 1 inner
336.4.bc.c 12 21.g even 6 1 inner
588.4.f.c 12 28.f even 6 1
588.4.f.c 12 28.g odd 6 1
588.4.f.c 12 84.j odd 6 1
588.4.f.c 12 84.n even 6 1
588.4.k.c 12 28.d even 2 1
588.4.k.c 12 28.g odd 6 1
588.4.k.c 12 84.h odd 2 1
588.4.k.c 12 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{12} + 597T_{5}^{10} + 272898T_{5}^{8} + 49602429T_{5}^{6} + 6898376178T_{5}^{4} + 10590781509T_{5}^{2} + 16083058761 \) Copy content Toggle raw display
\( T_{13}^{6} + 11724T_{13}^{4} + 42650496T_{13}^{2} + 49422707712 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 42 T^{10} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( T^{12} + 597 T^{10} + \cdots + 16083058761 \) Copy content Toggle raw display
$7$ \( (T^{6} - 21 T^{5} + 525 T^{4} + \cdots + 40353607)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 5967 T^{10} + \cdots + 11724549836769 \) Copy content Toggle raw display
$13$ \( (T^{6} + 11724 T^{4} + \cdots + 49422707712)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 19902 T^{10} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{6} + 102 T^{5} + \cdots + 33716904588)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 31338 T^{10} + \cdots + 45\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{6} + 55377 T^{4} + \cdots + 42952073472)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 729 T^{5} + \cdots + 2248345829547)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 120 T^{5} + \cdots + 236777613604)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 129432 T^{4} + \cdots - 18213269969664)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 100848 T - 12209024)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + 532002 T^{10} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{12} - 604215 T^{10} + \cdots + 17\!\cdots\!89 \) Copy content Toggle raw display
$59$ \( T^{12} + 549669 T^{10} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( (T^{6} - 1074 T^{5} + \cdots + 687780167594688)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 990 T^{5} + \cdots + 11\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 821736 T^{4} + \cdots + 15\!\cdots\!72)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 1542 T^{5} + \cdots + 13\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 219 T^{5} + \cdots + 23\!\cdots\!01)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 1217091 T^{4} + \cdots - 18\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 3892158 T^{10} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{6} + 3137499 T^{4} + \cdots + 59\!\cdots\!52)^{2} \) Copy content Toggle raw display
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