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 $4$

Related objects

Downloads

Learn more

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: \(x^{12} - 14 x^{10} - 32 x^{9} + 70 x^{8} + 224 x^{7} - 50 x^{6} + 2016 x^{5} + 5670 x^{4} - 23328 x^{3} - 91854 x^{2} + 531441\)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{3} + \beta_{4} ) q^{3} -\beta_{11} q^{5} + ( 2 - 2 \beta_{1} - 2 \beta_{6} - \beta_{9} ) q^{7} + ( 1 + 12 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{3} + \beta_{4} ) q^{3} -\beta_{11} q^{5} + ( 2 - 2 \beta_{1} - 2 \beta_{6} - \beta_{9} ) q^{7} + ( 1 + 12 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{9} + ( 2 - 2 \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{11} + ( 6 + 32 \beta_{1} + 8 \beta_{2} - 8 \beta_{6} - 2 \beta_{7} + 6 \beta_{9} ) q^{13} + ( -11 - 2 \beta_{3} + \beta_{4} + 8 \beta_{7} - 8 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{15} + ( 3 - 4 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} - 5 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{17} + ( -27 - 11 \beta_{1} + 4 \beta_{2} - 5 \beta_{6} - 9 \beta_{7} + 9 \beta_{9} ) q^{19} + ( -34 - 3 \beta_{1} - 8 \beta_{2} - 7 \beta_{3} + 8 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{21} + ( -3 + \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + 16 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + \beta_{7} + 5 \beta_{9} ) q^{23} + ( -80 - 74 \beta_{1} + 6 \beta_{2} + 10 \beta_{6} + 10 \beta_{7} + 6 \beta_{9} ) q^{25} + ( -38 - 43 \beta_{1} + 7 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 7 \beta_{6} + 6 \beta_{7} + 2 \beta_{8} + 13 \beta_{9} + 3 \beta_{10} ) q^{27} + ( 2 - 4 \beta_{1} + 4 \beta_{2} + 12 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} - 2 \beta_{8} + \beta_{9} + 5 \beta_{10} + 10 \beta_{11} ) q^{29} + ( -73 + 75 \beta_{1} - 2 \beta_{2} + 11 \beta_{6} - 11 \beta_{7} + 2 \beta_{9} ) q^{31} + ( 6 \beta_{1} - 18 \beta_{2} - 15 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 18 \beta_{6} - 6 \beta_{7} - 6 \beta_{8} + 3 \beta_{9} ) q^{33} + ( 9 - 17 \beta_{1} + 10 \beta_{2} + 42 \beta_{3} - 38 \beta_{4} - 6 \beta_{5} + 15 \beta_{6} - 18 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} + 9 \beta_{10} - 4 \beta_{11} ) q^{35} + ( 1 - 48 \beta_{1} + 7 \beta_{2} + \beta_{6} - 8 \beta_{7} - 8 \beta_{9} ) q^{37} + ( 82 + 72 \beta_{1} - 6 \beta_{2} - 8 \beta_{3} - 12 \beta_{4} + 14 \beta_{6} + 10 \beta_{7} - 4 \beta_{8} - 6 \beta_{9} + 10 \beta_{10} - 10 \beta_{11} ) q^{39} + ( -8 + 8 \beta_{1} + 4 \beta_{2} + 28 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} - 4 \beta_{8} + 10 \beta_{9} + 4 \beta_{10} ) q^{41} + ( 16 \beta_{1} - 16 \beta_{2} - 16 \beta_{6} - 12 \beta_{7} + 28 \beta_{9} ) q^{43} + ( -1 + 15 \beta_{1} - 13 \beta_{2} + 9 \beta_{3} - 8 \beta_{4} - 2 \beta_{5} + 19 \beta_{6} - 20 \beta_{7} - \beta_{8} + 15 \beta_{9} + 21 \beta_{10} + 21 \beta_{11} ) q^{45} + ( -15 - 15 \beta_{1} + 30 \beta_{2} + 90 \beta_{3} + 15 \beta_{6} - 15 \beta_{7} + 15 \beta_{9} - 14 \beta_{11} ) q^{47} + ( -119 + 63 \beta_{1} + 21 \beta_{2} - 35 \beta_{6} + 35 \beta_{7} + 7 \beta_{9} ) q^{49} + ( -61 - 8 \beta_{1} + 19 \beta_{2} - 23 \beta_{3} + 52 \beta_{4} - 6 \beta_{5} - 61 \beta_{6} + 42 \beta_{7} + 48 \beta_{9} + 10 \beta_{10} + 5 \beta_{11} ) q^{51} + ( -24 + 24 \beta_{2} + 48 \beta_{3} + 48 \beta_{4} + 24 \beta_{9} + 13 \beta_{10} - 13 \beta_{11} ) q^{53} + ( 119 + 263 \beta_{1} + 35 \beta_{2} - 35 \beta_{6} - 40 \beta_{7} - 5 \beta_{9} ) q^{55} + ( 14 + 2 \beta_{1} - 2 \beta_{2} + 30 \beta_{3} - 15 \beta_{4} + \beta_{5} - 2 \beta_{6} + 16 \beta_{7} + \beta_{8} - 14 \beta_{9} - 14 \beta_{10} - 28 \beta_{11} ) q^{57} + ( -2 + 6 \beta_{1} - 2 \beta_{2} - 24 \beta_{3} + 26 \beta_{4} - 4 \beta_{5} - 7 \beta_{6} + 5 \beta_{7} - 2 \beta_{8} + 6 \beta_{9} + 31 \beta_{10} + 31 \beta_{11} ) q^{59} + ( 245 + 138 \beta_{1} - 25 \beta_{2} - 31 \beta_{6} - 6 \beta_{7} + 6 \beta_{9} ) q^{61} + ( -77 + 27 \beta_{1} - 11 \beta_{2} + 49 \beta_{3} - 52 \beta_{4} + 8 \beta_{5} + 52 \beta_{6} + 10 \beta_{7} + 12 \beta_{8} - 52 \beta_{9} - 12 \beta_{10} + 3 \beta_{11} ) q^{63} + ( 54 - 64 \beta_{1} + 10 \beta_{2} + 108 \beta_{3} - 196 \beta_{4} - 20 \beta_{5} + 54 \beta_{6} - 64 \beta_{7} - 44 \beta_{9} + 28 \beta_{10} + 14 \beta_{11} ) q^{65} + ( -259 - 330 \beta_{1} - 71 \beta_{2} + 26 \beta_{6} + 26 \beta_{7} - 71 \beta_{9} ) q^{67} + ( 98 + 332 \beta_{1} + 32 \beta_{2} + 45 \beta_{4} - 9 \beta_{5} - 32 \beta_{6} + 20 \beta_{7} + 9 \beta_{8} + 52 \beta_{9} + 6 \beta_{10} ) q^{69} + ( 8 + 26 \beta_{1} - 26 \beta_{2} - 120 \beta_{3} + 60 \beta_{4} - 8 \beta_{5} - 26 \beta_{6} + 22 \beta_{7} - 8 \beta_{8} + 4 \beta_{9} ) q^{71} + ( -147 + 130 \beta_{1} + 17 \beta_{2} + 45 \beta_{6} - 45 \beta_{7} - 17 \beta_{9} ) q^{73} + ( 382 + 156 \beta_{1} + 58 \beta_{2} + 112 \beta_{3} - 16 \beta_{4} - 16 \beta_{5} + 38 \beta_{6} - 52 \beta_{7} - 32 \beta_{8} + 36 \beta_{9} + 30 \beta_{11} ) q^{75} + ( -30 + 24 \beta_{1} + 4 \beta_{2} - 84 \beta_{3} + 150 \beta_{4} - 22 \beta_{5} - 36 \beta_{6} + 25 \beta_{7} + 2 \beta_{8} + 39 \beta_{9} - 16 \beta_{10} - 17 \beta_{11} ) q^{77} + ( -44 - 30 \beta_{1} + \beta_{2} - 44 \beta_{6} + 43 \beta_{7} + 43 \beta_{9} ) q^{79} + ( 141 + 168 \beta_{1} + 42 \beta_{2} + 63 \beta_{3} + 48 \beta_{4} + 42 \beta_{6} + 27 \beta_{7} - 15 \beta_{8} + 42 \beta_{9} ) q^{81} + ( 12 + 12 \beta_{1} - 6 \beta_{2} - 18 \beta_{4} + 18 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} - 18 \beta_{8} - 3 \beta_{9} - 7 \beta_{10} ) q^{83} + ( -512 - 32 \beta_{1} + 32 \beta_{2} + 32 \beta_{6} - 105 \beta_{7} + 73 \beta_{9} ) q^{85} + ( 184 - 125 \beta_{1} - 59 \beta_{2} - 12 \beta_{3} + 12 \beta_{4} + 5 \beta_{6} - 5 \beta_{7} + 59 \beta_{9} - 12 \beta_{10} - 12 \beta_{11} ) q^{87} + ( 49 + 34 \beta_{1} - 71 \beta_{2} - 222 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} - 31 \beta_{6} + 28 \beta_{7} - 12 \beta_{8} - 34 \beta_{9} - 18 \beta_{11} ) q^{89} + ( -588 - 546 \beta_{1} + 154 \beta_{2} + 42 \beta_{6} - 70 \beta_{7} - 42 \beta_{9} ) q^{91} + ( -6 - 165 \beta_{1} + 24 \beta_{2} + 95 \beta_{3} - 181 \beta_{4} - 9 \beta_{5} - 6 \beta_{6} - 18 \beta_{7} - 9 \beta_{9} - 30 \beta_{10} - 15 \beta_{11} ) q^{93} + ( 41 - 69 \beta_{2} - 138 \beta_{3} - 110 \beta_{4} - 14 \beta_{6} + 14 \beta_{7} + 28 \beta_{8} - 69 \beta_{9} ) q^{95} + ( -387 - 596 \beta_{1} + 108 \beta_{2} - 108 \beta_{6} - 73 \beta_{7} + 35 \beta_{9} ) q^{97} + ( -780 + 105 \beta_{1} - 105 \beta_{2} - 78 \beta_{3} + 39 \beta_{4} + 12 \beta_{5} - 105 \beta_{6} + 123 \beta_{7} + 12 \beta_{8} - 18 \beta_{9} - 18 \beta_{10} - 36 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 42q^{7} - 84q^{9} + O(q^{10}) \) \( 12q + 42q^{7} - 84q^{9} - 132q^{15} - 204q^{19} - 378q^{21} - 444q^{25} - 1458q^{31} - 108q^{33} + 240q^{37} + 432q^{39} - 342q^{45} - 1218q^{49} + 300q^{51} + 180q^{57} + 2148q^{61} - 1596q^{63} - 1980q^{67} - 3084q^{73} + 3384q^{75} + 438q^{79} + 1008q^{81} - 6144q^{85} + 2898q^{87} - 3780q^{91} + 882q^{93} - 9216q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -36 \nu^{11} - 109 \nu^{10} + 180 \nu^{9} + 1949 \nu^{8} + 2588 \nu^{7} - 1681 \nu^{6} - 7388 \nu^{5} - 64615 \nu^{4} - 433908 \nu^{3} - 556065 \nu^{2} + 2452356 \nu + 6436341 \)\()/2571912\)
\(\beta_{2}\)\(=\)\((\)\( 871 \nu^{11} + 2952 \nu^{10} + 7489 \nu^{9} - 31292 \nu^{8} - 131909 \nu^{7} - 50236 \nu^{6} - 103283 \nu^{5} + 1112616 \nu^{4} + 5111019 \nu^{3} + 24048252 \nu^{2} - 26447391 \nu - 192499740 \)\()/23147208\)
\(\beta_{3}\)\(=\)\((\)\( 109 \nu^{11} + 324 \nu^{10} - 797 \nu^{9} - 5108 \nu^{8} - 6383 \nu^{7} + 9188 \nu^{6} - 7961 \nu^{5} + 229788 \nu^{4} + 1395873 \nu^{3} + 854388 \nu^{2} - 3864429 \nu - 19131876 \)\()/2571912\)
\(\beta_{4}\)\(=\)\((\)\( -109 \nu^{11} - 324 \nu^{10} + 797 \nu^{9} + 5108 \nu^{8} + 6383 \nu^{7} - 9188 \nu^{6} + 7961 \nu^{5} - 229788 \nu^{4} - 1395873 \nu^{3} - 854388 \nu^{2} + 11580165 \nu + 19131876 \)\()/2571912\)
\(\beta_{5}\)\(=\)\((\)\(-1522 \nu^{11} - 11709 \nu^{10} + 3326 \nu^{9} + 159413 \nu^{8} + 296822 \nu^{7} - 135341 \nu^{6} - 987142 \nu^{5} - 3274839 \nu^{4} - 41038974 \nu^{3} - 17899137 \nu^{2} + 216631098 \nu + 653377185\)\()/23147208\)
\(\beta_{6}\)\(=\)\((\)\( -1643 \nu^{11} - 4941 \nu^{10} + 403 \nu^{9} + 56869 \nu^{8} + 156097 \nu^{7} + 149963 \nu^{6} + 486583 \nu^{5} - 669015 \nu^{4} - 18160767 \nu^{3} - 31563513 \nu^{2} + 19125315 \nu + 250426809 \)\()/23147208\)
\(\beta_{7}\)\(=\)\((\)\( -58 \nu^{11} - 36 \nu^{10} + 569 \nu^{9} + 1631 \nu^{8} + 494 \nu^{7} + 2470 \nu^{6} + 1154 \nu^{5} - 102492 \nu^{4} - 375435 \nu^{3} + 400221 \nu^{2} + 1430298 \nu + 2834352 \)\()/826686\)
\(\beta_{8}\)\(=\)\((\)\( -1931 \nu^{11} + 774 \nu^{10} - 425 \nu^{9} + 5758 \nu^{8} - 83879 \nu^{7} + 542606 \nu^{6} + 1335967 \nu^{5} - 3732822 \nu^{4} - 10260027 \nu^{3} - 13569606 \nu^{2} - 12734901 \nu - 65544390 \)\()/23147208\)
\(\beta_{9}\)\(=\)\((\)\( 74 \nu^{11} + 207 \nu^{10} - 793 \nu^{9} - 3808 \nu^{8} - 4846 \nu^{7} + 2878 \nu^{6} + 13022 \nu^{5} + 236277 \nu^{4} + 974187 \nu^{3} + 454896 \nu^{2} - 6364170 \nu - 13581270 \)\()/826686\)
\(\beta_{10}\)\(=\)\((\)\( -125 \nu^{11} - 477 \nu^{10} + 454 \nu^{9} + 9949 \nu^{8} + 18097 \nu^{7} - 9712 \nu^{6} - 17087 \nu^{5} - 300483 \nu^{4} - 2110050 \nu^{3} - 3421197 \nu^{2} + 10517283 \nu + 40743810 \)\()/826686\)
\(\beta_{11}\)\(=\)\((\)\(-3554 \nu^{11} - 16407 \nu^{10} + 12982 \nu^{9} + 145867 \nu^{8} + 174346 \nu^{7} + 403685 \nu^{6} + 1680538 \nu^{5} - 4010661 \nu^{4} - 40201758 \nu^{3} - 41698071 \nu^{2} + 165953934 \nu + 264480471\)\()/23147208\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{9} + 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 12 \beta_{1} - 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} + 4 \beta_{7} - 5 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} + 5 \beta_{1} + 26\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(14 \beta_{9} - 5 \beta_{8} + 9 \beta_{7} + 14 \beta_{6} + 16 \beta_{4} + 21 \beta_{3} + 14 \beta_{2} + 56 \beta_{1} + 47\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(14 \beta_{11} + 28 \beta_{10} + 33 \beta_{9} + 21 \beta_{7} - 19 \beta_{6} - 12 \beta_{5} + 78 \beta_{4} - 33 \beta_{3} - 2 \beta_{2} - 539 \beta_{1} - 19\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(32 \beta_{11} + 16 \beta_{10} - 110 \beta_{9} + 24 \beta_{8} + 50 \beta_{7} + 60 \beta_{6} + 24 \beta_{5} - 200 \beta_{4} + 400 \beta_{3} + 60 \beta_{2} - 60 \beta_{1} + 721\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-70 \beta_{11} + 70 \beta_{10} - 266 \beta_{9} - 164 \beta_{8} + 214 \beta_{7} + 378 \beta_{6} + 157 \beta_{4} + 321 \beta_{3} - 266 \beta_{2} - 3920 \beta_{1} - 3490\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(448 \beta_{11} + 896 \beta_{10} + 1521 \beta_{9} + 1566 \beta_{7} - 1703 \beta_{6} + 45 \beta_{5} - 755 \beta_{4} + 355 \beta_{3} + 137 \beta_{2} - 562 \beta_{1} - 1703\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(622 \beta_{11} + 311 \beta_{10} - 3721 \beta_{9} - 1966 \beta_{8} + 194 \beta_{7} + 3527 \beta_{6} - 1966 \beta_{5} - 2078 \beta_{4} + 4156 \beta_{3} + 3527 \beta_{2} - 3527 \beta_{1} + 12070\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-5376 \beta_{11} + 5376 \beta_{10} + 6104 \beta_{9} + 3415 \beta_{8} + 8175 \beta_{7} + 4760 \beta_{6} + 13472 \beta_{4} + 10057 \beta_{3} + 6104 \beta_{2} - 39466 \beta_{1} - 48985\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(10136 \beta_{11} + 20272 \beta_{10} - 28127 \beta_{9} - 25519 \beta_{7} - 10121 \beta_{6} + 2608 \beta_{5} - 39374 \beta_{4} + 18383 \beta_{3} + 35640 \beta_{2} - 286703 \beta_{1} - 10121\)\()/3\)

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 \(S_{4}^{\mathrm{new}}(336, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 387420489 + 22320522 T^{2} + 459270 T^{4} - 163296 T^{5} + 15174 T^{6} - 6048 T^{7} + 630 T^{8} + 42 T^{10} + T^{12} \)
$5$ \( 16083058761 + 10590781509 T^{2} + 6898376178 T^{4} + 49602429 T^{6} + 272898 T^{8} + 597 T^{10} + T^{12} \)
$7$ \( ( 40353607 - 2470629 T + 180075 T^{2} - 11270 T^{3} + 525 T^{4} - 21 T^{5} + T^{6} )^{2} \)
$11$ \( 11724549836769 - 25352019656271 T^{2} + 54798295654818 T^{4} - 44172622863 T^{6} + 28201122 T^{8} - 5967 T^{10} + T^{12} \)
$13$ \( ( 49422707712 + 42650496 T^{2} + 11724 T^{4} + T^{6} )^{2} \)
$17$ \( 12212399445462404496 + 345318682065762636 T^{2} + 9694705791341673 T^{4} + 1959611934870 T^{6} + 297275355 T^{8} + 19902 T^{10} + T^{12} \)
$19$ \( ( 33716904588 + 1897756614 T + 24791661 T^{2} - 608634 T^{3} - 2499 T^{4} + 102 T^{5} + T^{6} )^{2} \)
$23$ \( 450410306529317904 - 25898608390188996 T^{2} + 1468139137683705 T^{4} - 1207984179330 T^{6} + 943480467 T^{8} - 31338 T^{10} + T^{12} \)
$29$ \( ( 42952073472 + 165491424 T^{2} + 55377 T^{4} + T^{6} )^{2} \)
$31$ \( ( 2248345829547 + 119324724345 T + 2742043428 T^{2} + 33493905 T^{3} + 223092 T^{4} + 729 T^{5} + T^{6} )^{2} \)
$37$ \( ( 236777613604 - 2973600378 T + 95736081 T^{2} - 239876 T^{3} + 20511 T^{4} - 120 T^{5} + T^{6} )^{2} \)
$41$ \( ( -18213269969664 + 3004260624 T^{2} - 129432 T^{4} + T^{6} )^{2} \)
$43$ \( ( -12209024 - 100848 T + T^{3} )^{4} \)
$47$ \( \)\(18\!\cdots\!36\)\( + \)\(78\!\cdots\!44\)\( T^{2} + \)\(26\!\cdots\!13\)\( T^{4} + 27989833646485314 T^{6} + 225284539203 T^{8} + 532002 T^{10} + T^{12} \)
$53$ \( \)\(17\!\cdots\!89\)\( - \)\(57\!\cdots\!99\)\( T^{2} + \)\(18\!\cdots\!14\)\( T^{4} - 26016338421043479 T^{6} + 321973295922 T^{8} - 604215 T^{10} + T^{12} \)
$59$ \( \)\(36\!\cdots\!01\)\( + \)\(15\!\cdots\!57\)\( T^{2} + \)\(57\!\cdots\!18\)\( T^{4} + 41488136310689469 T^{6} + 219666412818 T^{8} + 549669 T^{10} + T^{12} \)
$61$ \( ( 687780167594688 - 905255055432 T - 15864629823 T^{2} + 21403746 T^{3} + 364563 T^{4} - 1074 T^{5} + T^{6} )^{2} \)
$67$ \( ( 11409580521347044 + 2924718984078 T + 106497200781 T^{2} + 186524086 T^{3} + 1007481 T^{4} + 990 T^{5} + T^{6} )^{2} \)
$71$ \( ( 15147749091670272 + 202724447760 T^{2} + 821736 T^{4} + T^{6} )^{2} \)
$73$ \( ( 1316942318068272 - 6387323622876 T - 21981359295 T^{2} + 156696498 T^{3} + 894207 T^{4} + 1542 T^{5} + T^{6} )^{2} \)
$79$ \( ( 2399094376263601 - 9894316204755 T + 51532760694 T^{2} - 53722007 T^{3} + 249966 T^{4} - 219 T^{5} + T^{6} )^{2} \)
$83$ \( ( -18264889109263104 + 282198968928 T^{2} - 1217091 T^{4} + T^{6} )^{2} \)
$89$ \( \)\(14\!\cdots\!76\)\( + \)\(51\!\cdots\!04\)\( T^{2} + \)\(14\!\cdots\!33\)\( T^{4} + 14622864529157921430 T^{6} + 10780696444635 T^{8} + 3892158 T^{10} + T^{12} \)
$97$ \( ( 597566408536829952 + 2519767249344 T^{2} + 3137499 T^{4} + T^{6} )^{2} \)
show more
show less