Properties

Label 336.4.h.a
Level $336$
Weight $4$
Character orbit 336.h
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 152 x^{10} + 8222 x^{8} + 194132 x^{6} + 1882697 x^{4} + 5152508 x^{2} + 4008004\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 7^{4} \)
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.

\(f(q)\) \(=\) \( q -\beta_{5} q^{3} + \beta_{8} q^{5} -\beta_{1} q^{7} + ( 6 - \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{5} q^{3} + \beta_{8} q^{5} -\beta_{1} q^{7} + ( 6 - \beta_{2} ) q^{9} + ( \beta_{1} + 3 \beta_{5} + \beta_{6} + \beta_{9} ) q^{11} + ( -8 - \beta_{2} - \beta_{3} ) q^{13} + ( -2 \beta_{1} - \beta_{4} + 3 \beta_{7} + \beta_{9} ) q^{15} + ( \beta_{2} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{17} + ( -7 \beta_{5} + \beta_{7} + 4 \beta_{9} ) q^{19} + ( 9 + \beta_{8} - \beta_{11} ) q^{21} + ( 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{23} + ( -89 + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{10} + 2 \beta_{11} ) q^{25} + ( -4 \beta_{1} - 2 \beta_{4} - 10 \beta_{5} - 3 \beta_{7} + 2 \beta_{9} ) q^{27} + ( -2 - 2 \beta_{2} + \beta_{3} + 2 \beta_{8} + \beta_{10} - 3 \beta_{11} ) q^{29} + ( 9 \beta_{1} + 5 \beta_{5} - \beta_{7} - 3 \beta_{9} ) q^{31} + ( -70 + 3 \beta_{2} + 5 \beta_{3} + \beta_{8} + 4 \beta_{10} ) q^{33} + ( \beta_{1} + 2 \beta_{4} + 6 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{35} + ( -60 - 2 \beta_{2} + \beta_{3} + 3 \beta_{10} + 3 \beta_{11} ) q^{37} + ( -7 \beta_{1} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - 10 \beta_{9} ) q^{39} + ( -4 - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{8} + 3 \beta_{10} - 7 \beta_{11} ) q^{41} + ( 11 \beta_{1} + 19 \beta_{5} + 13 \beta_{7} - 3 \beta_{9} ) q^{43} + ( 34 - 5 \beta_{3} + 9 \beta_{8} + 5 \beta_{10} - \beta_{11} ) q^{45} + ( -4 \beta_{1} + 4 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} - 4 \beta_{9} ) q^{47} -49 q^{49} + ( -38 \beta_{1} + 4 \beta_{4} - 10 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 8 \beta_{9} ) q^{51} + ( -6 - 10 \beta_{2} + 3 \beta_{3} + 20 \beta_{8} - \beta_{10} - 5 \beta_{11} ) q^{53} + ( 39 \beta_{1} + 3 \beta_{5} - 31 \beta_{7} - 17 \beta_{9} ) q^{55} + ( -199 - 11 \beta_{2} - \beta_{3} + 10 \beta_{8} + \beta_{10} - 3 \beta_{11} ) q^{57} + ( 9 \beta_{1} - 8 \beta_{4} + 22 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + 9 \beta_{9} ) q^{59} + ( 36 - 11 \beta_{2} - 9 \beta_{3} + 2 \beta_{10} + 2 \beta_{11} ) q^{61} + ( -8 \beta_{1} + \beta_{4} - 11 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + 4 \beta_{9} ) q^{63} + ( 2 + 14 \beta_{2} - \beta_{3} + 22 \beta_{8} + 11 \beta_{10} - 9 \beta_{11} ) q^{65} + ( 57 \beta_{1} - 41 \beta_{5} - 11 \beta_{7} + 15 \beta_{9} ) q^{67} + ( -38 + \beta_{2} + 10 \beta_{3} + 19 \beta_{8} - \beta_{10} + \beta_{11} ) q^{69} + ( -11 \beta_{1} + 8 \beta_{4} - 27 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} - 11 \beta_{9} ) q^{71} + ( 138 - 10 \beta_{2} + 2 \beta_{3} + 12 \beta_{10} + 12 \beta_{11} ) q^{73} + ( -36 \beta_{1} + 6 \beta_{4} + 81 \beta_{5} + 18 \beta_{6} + 18 \beta_{9} ) q^{75} + ( -4 - 9 \beta_{2} + 2 \beta_{3} - 9 \beta_{8} - 3 \beta_{10} - \beta_{11} ) q^{77} + ( 12 \beta_{1} + 32 \beta_{5} + 32 \beta_{7} ) q^{79} + ( -79 - 10 \beta_{2} - \beta_{3} - 36 \beta_{8} + \beta_{10} - 11 \beta_{11} ) q^{81} + ( -11 \beta_{1} - 20 \beta_{4} - 70 \beta_{5} - 4 \beta_{6} + 10 \beta_{7} - 11 \beta_{9} ) q^{83} + ( -122 + 26 \beta_{2} + 23 \beta_{3} - 3 \beta_{10} - 3 \beta_{11} ) q^{85} + ( -61 \beta_{1} + 2 \beta_{4} - 23 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} + 17 \beta_{9} ) q^{87} + ( 3 \beta_{2} + 29 \beta_{8} + 3 \beta_{10} - 3 \beta_{11} ) q^{89} + ( 11 \beta_{1} + 14 \beta_{7} + 7 \beta_{9} ) q^{91} + ( 64 + 8 \beta_{2} + \beta_{3} - 18 \beta_{8} - \beta_{10} + 11 \beta_{11} ) q^{93} + ( 23 \beta_{1} - 14 \beta_{4} + 61 \beta_{5} + 10 \beta_{6} + 7 \beta_{7} + 23 \beta_{9} ) q^{95} + ( -522 + 4 \beta_{2} + 4 \beta_{3} ) q^{97} + ( -111 \beta_{1} + 14 \beta_{4} + 39 \beta_{5} + 15 \beta_{6} + 15 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 76 q^{9} + O(q^{10}) \) \( 12 q + 76 q^{9} - 96 q^{13} + 112 q^{21} - 1068 q^{25} - 832 q^{33} - 720 q^{37} + 392 q^{45} - 588 q^{49} - 2336 q^{57} + 432 q^{61} - 424 q^{69} + 1656 q^{73} - 868 q^{81} - 1464 q^{85} + 696 q^{93} - 6264 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 152 x^{10} + 8222 x^{8} + 194132 x^{6} + 1882697 x^{4} + 5152508 x^{2} + 4008004\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -25835 \nu^{11} - 663660 \nu^{9} + 233331932 \nu^{7} + 14784806206 \nu^{5} + 272808035391 \nu^{3} + 1258402345082 \nu \)\()/ 152656104744 \)
\(\beta_{2}\)\(=\)\((\)\(11075 \nu^{11} - 81642 \nu^{10} + 1831163 \nu^{9} - 15354372 \nu^{8} + 110542037 \nu^{7} - 1002264802 \nu^{6} + 3016463563 \nu^{5} - 26413350216 \nu^{4} + 36258659442 \nu^{3} - 245395469352 \nu^{2} + 131131719184 \nu - 333587622676\)\()/ 4566635612 \)
\(\beta_{3}\)\(=\)\((\)\(-33225 \nu^{11} + 427768 \nu^{10} - 5493489 \nu^{9} + 47224804 \nu^{8} - 331626111 \nu^{7} + 1461277048 \nu^{6} - 9049390689 \nu^{5} + 13371095848 \nu^{4} - 108775978326 \nu^{3} + 61211427904 \nu^{2} - 393395157552 \nu + 420228241972\)\()/ 13699906836 \)
\(\beta_{4}\)\(=\)\((\)\(-1960682 \nu^{11} + 82836182 \nu^{10} - 356890329 \nu^{9} + 11268239697 \nu^{8} - 22932451609 \nu^{7} + 519686539372 \nu^{6} - 608627339477 \nu^{5} + 9934149962165 \nu^{4} - 5757508799817 \nu^{3} + 70963309344642 \nu^{2} - 5037235696882 \nu + 79758915468232\)\()/ 534296366604 \)
\(\beta_{5}\)\(=\)\((\)\(465665 \nu^{11} - 247214 \nu^{10} + 70293516 \nu^{9} - 45174954 \nu^{8} + 3735683236 \nu^{7} - 2969047312 \nu^{6} + 84349141874 \nu^{5} - 82883803046 \nu^{4} + 721869749607 \nu^{3} - 865719051714 \nu^{2} + 1015402744858 \nu - 1423599464980\)\()/ 82199441016 \)
\(\beta_{6}\)\(=\)\((\)\(465665 \nu^{11} + 5913864 \nu^{10} + 70293516 \nu^{9} + 828220206 \nu^{8} + 3735683236 \nu^{7} + 39696452796 \nu^{6} + 84349141874 \nu^{5} + 790602608598 \nu^{4} + 721869749607 \nu^{3} + 6061899930900 \nu^{2} + 1015402744858 \nu + 11961336451680\)\()/ 82199441016 \)
\(\beta_{7}\)\(=\)\((\)\(10318207 \nu^{11} + 3213782 \nu^{10} + 1313885808 \nu^{9} + 587274402 \nu^{8} + 53961839768 \nu^{7} + 38597615056 \nu^{6} + 855107175178 \nu^{5} + 1077489439598 \nu^{4} + 5122885035405 \nu^{3} + 11254347672282 \nu^{2} + 19451764261934 \nu + 18506793044740\)\()/ 1068592733208 \)
\(\beta_{8}\)\(=\)\((\)\( 253923 \nu^{11} + 36440350 \nu^{9} + 1823770458 \nu^{7} + 39066975904 \nu^{5} + 324933491115 \nu^{3} + 412302833662 \nu \)\()/ 16190798988 \)
\(\beta_{9}\)\(=\)\((\)\(-2184885 \nu^{11} - 584324 \nu^{10} - 331874292 \nu^{9} - 106777164 \nu^{8} - 17808077436 \nu^{7} - 7017748192 \nu^{6} - 408149910990 \nu^{5} - 195907170836 \nu^{4} - 3586080293391 \nu^{3} - 2046245031324 \nu^{2} - 5600887195290 \nu - 3364871462680\)\()/ 97144793928 \)
\(\beta_{10}\)\(=\)\((\)\(-9393673 \nu^{11} + 10740587 \nu^{10} - 1394772549 \nu^{9} + 1804012496 \nu^{8} - 72680418595 \nu^{7} + 107219119007 \nu^{6} - 1608638914983 \nu^{5} + 2662213769072 \nu^{4} - 13426814594356 \nu^{3} + 24323932543196 \nu^{2} - 17536296677124 \nu + 35078177994860\)\()/ 178098788868 \)
\(\beta_{11}\)\(=\)\((\)\(9825598 \nu^{11} + 7556549 \nu^{10} + 1466187906 \nu^{9} + 1205191988 \nu^{8} + 76991558038 \nu^{7} + 68130791729 \nu^{6} + 1726280993940 \nu^{5} + 1632093110648 \nu^{4} + 14840902312594 \nu^{3} + 14753509238468 \nu^{2} + 22650433725300 \nu + 22068260710496\)\()/ 178098788868 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(4 \beta_{11} - 2 \beta_{10} + 7 \beta_{9} - 6 \beta_{8} - 14 \beta_{5} - \beta_{3} + \beta_{2} + \beta_{1} + 2\)\()/42\)
\(\nu^{2}\)\(=\)\((\)\(-4 \beta_{11} - 4 \beta_{10} - 3 \beta_{9} + 3 \beta_{7} + 18 \beta_{6} - 39 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_{1} - 1064\)\()/42\)
\(\nu^{3}\)\(=\)\((\)\(-194 \beta_{11} + 76 \beta_{10} - 343 \beta_{9} + 144 \beta_{8} + 84 \beta_{7} + 770 \beta_{5} + 59 \beta_{3} - 101 \beta_{2} + 275 \beta_{1} - 118\)\()/42\)
\(\nu^{4}\)\(=\)\((\)\(153 \beta_{11} + 153 \beta_{10} + 172 \beta_{9} - 32 \beta_{7} - 360 \beta_{6} + 1144 \beta_{5} + 64 \beta_{4} + 157 \beta_{3} + 4 \beta_{2} + 172 \beta_{1} + 15540\)\()/14\)
\(\nu^{5}\)\(=\)\((\)\(11134 \beta_{11} - 3152 \beta_{10} + 18949 \beta_{9} - 5340 \beta_{8} - 5964 \beta_{7} - 43862 \beta_{5} - 3991 \beta_{3} + 8821 \beta_{2} - 26129 \beta_{1} + 7982\)\()/42\)
\(\nu^{6}\)\(=\)\((\)\(-35698 \beta_{11} - 35698 \beta_{10} - 48717 \beta_{9} + 2853 \beta_{7} + 63990 \beta_{6} - 267417 \beta_{5} - 5706 \beta_{4} - 38912 \beta_{3} - 3214 \beta_{2} - 48717 \beta_{1} - 2414804\)\()/42\)
\(\nu^{7}\)\(=\)\((\)\(-676904 \beta_{11} + 149410 \beta_{10} - 1114855 \beta_{9} + 266244 \beta_{8} + 383670 \beta_{7} + 2613380 \beta_{5} + 263747 \beta_{3} - 641831 \beta_{2} + 1999115 \beta_{1} - 527494\)\()/42\)
\(\nu^{8}\)\(=\)\((\)\(831543 \beta_{11} + 831543 \beta_{10} + 1242088 \beta_{9} - 24928 \beta_{7} - 1299696 \beta_{6} + 6342832 \beta_{5} + 49856 \beta_{4} + 906923 \beta_{3} + 75380 \beta_{2} + 1242088 \beta_{1} + 45860500\)\()/14\)
\(\nu^{9}\)\(=\)\((\)\(42318904 \beta_{11} - 7917818 \beta_{10} + 68114557 \beta_{9} - 15612804 \beta_{8} - 24483522 \beta_{7} - 160712636 \beta_{5} - 17200543 \beta_{3} + 43683811 \beta_{2} - 140030249 \beta_{1} + 34401086\)\()/42\)
\(\nu^{10}\)\(=\)\((\)\(-167007082 \beta_{11} - 167007082 \beta_{10} - 260652747 \beta_{9} + 1081491 \beta_{7} + 243041850 \beta_{6} - 1288897311 \beta_{5} - 2162982 \beta_{4} - 180757880 \beta_{3} - 13750798 \beta_{2} - 260652747 \beta_{1} - 8285451524\)\()/42\)
\(\nu^{11}\)\(=\)\((\)\(-2682632054 \beta_{11} + 454103344 \beta_{10} - 4255592215 \beta_{9} + 978049092 \beta_{8} + 1568046732 \beta_{7} + 10079231162 \beta_{5} + 1114264355 \beta_{3} - 2888689721 \beta_{2} + 9403781483 \beta_{1} - 2228528710\)\()/42\)

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
4.19902i
4.19902i
8.01905i
8.01905i
1.52642i
1.52642i
5.90052i
5.90052i
1.17553i
1.17553i
5.61561i
5.61561i
0 −4.90731 1.70829i 0 20.5781i 0 7.00000i 0 21.1635 + 16.7663i 0
239.2 0 −4.90731 + 1.70829i 0 20.5781i 0 7.00000i 0 21.1635 16.7663i 0
239.3 0 −4.59729 2.42176i 0 2.41112i 0 7.00000i 0 15.2701 + 22.2671i 0
239.4 0 −4.59729 + 2.42176i 0 2.41112i 0 7.00000i 0 15.2701 22.2671i 0
239.5 0 −2.18705 4.71347i 0 14.5852i 0 7.00000i 0 −17.4336 + 20.6172i 0
239.6 0 −2.18705 + 4.71347i 0 14.5852i 0 7.00000i 0 −17.4336 20.6172i 0
239.7 0 2.18705 4.71347i 0 14.5852i 0 7.00000i 0 −17.4336 20.6172i 0
239.8 0 2.18705 + 4.71347i 0 14.5852i 0 7.00000i 0 −17.4336 + 20.6172i 0
239.9 0 4.59729 2.42176i 0 2.41112i 0 7.00000i 0 15.2701 22.2671i 0
239.10 0 4.59729 + 2.42176i 0 2.41112i 0 7.00000i 0 15.2701 + 22.2671i 0
239.11 0 4.90731 1.70829i 0 20.5781i 0 7.00000i 0 21.1635 16.7663i 0
239.12 0 4.90731 + 1.70829i 0 20.5781i 0 7.00000i 0 21.1635 + 16.7663i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.h.a 12
3.b odd 2 1 inner 336.4.h.a 12
4.b odd 2 1 inner 336.4.h.a 12
12.b even 2 1 inner 336.4.h.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.4.h.a 12 1.a even 1 1 trivial
336.4.h.a 12 3.b odd 2 1 inner
336.4.h.a 12 4.b odd 2 1 inner
336.4.h.a 12 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 642 T_{5}^{4} + 93780 T_{5}^{2} + 523688 \) acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 387420489 - 20194758 T^{2} + 684531 T^{4} - 10332 T^{6} + 939 T^{8} - 38 T^{10} + T^{12} \)
$5$ \( ( 523688 + 93780 T^{2} + 642 T^{4} + T^{6} )^{2} \)
$7$ \( ( 49 + T^{2} )^{6} \)
$11$ \( ( -5604950432 + 11128224 T^{2} - 6474 T^{4} + T^{6} )^{2} \)
$13$ \( ( -78284 - 3174 T + 24 T^{2} + T^{3} )^{4} \)
$17$ \( ( 24639286688 + 56812320 T^{2} + 14634 T^{4} + T^{6} )^{2} \)
$19$ \( ( 22072450624 + 51011940 T^{2} + 14568 T^{4} + T^{6} )^{2} \)
$23$ \( ( -344672294528 + 226562496 T^{2} - 28386 T^{4} + T^{6} )^{2} \)
$29$ \( ( 301614442112 + 210928656 T^{2} + 38064 T^{4} + T^{6} )^{2} \)
$31$ \( ( 15017031936 + 98119872 T^{2} + 20100 T^{4} + T^{6} )^{2} \)
$37$ \( ( -2562416 - 43308 T + 180 T^{2} + T^{3} )^{4} \)
$41$ \( ( 185103027240608 + 12026771328 T^{2} + 220170 T^{4} + T^{6} )^{2} \)
$43$ \( ( 585026061785344 + 29340122256 T^{2} + 319896 T^{4} + T^{6} )^{2} \)
$47$ \( ( -513493070118912 + 26731118592 T^{2} - 313224 T^{4} + T^{6} )^{2} \)
$53$ \( ( 1844458262816768 + 54397638864 T^{2} + 427608 T^{4} + T^{6} )^{2} \)
$59$ \( ( -99385006592 + 633318324 T^{2} - 472464 T^{4} + T^{6} )^{2} \)
$61$ \( ( 9896868 - 386430 T - 108 T^{2} + T^{3} )^{4} \)
$67$ \( ( 3411526238025984 + 258013233936 T^{2} + 1062120 T^{4} + T^{6} )^{2} \)
$71$ \( ( -206806281826208 + 25193854080 T^{2} - 629562 T^{4} + T^{6} )^{2} \)
$73$ \( ( 277691624 - 886284 T - 414 T^{2} + T^{3} )^{4} \)
$79$ \( ( 98061409365037056 + 785982640896 T^{2} + 1733808 T^{4} + T^{6} )^{2} \)
$83$ \( ( -124813819450782368 + 2162701351092 T^{2} - 2901912 T^{4} + T^{6} )^{2} \)
$89$ \( ( 141902464928 + 8756533440 T^{2} + 622794 T^{4} + T^{6} )^{2} \)
$97$ \( ( 117443368 + 763596 T + 1566 T^{2} + T^{3} )^{4} \)
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