Properties

Label 336.4.q.i
Level $336$
Weight $4$
Character orbit 336.q
Analytic conductor $19.825$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{193})\)
Defining polynomial: \(x^{4} - x^{3} + 49 x^{2} + 48 x + 2304\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - 3 \beta_{2} ) q^{3} + ( \beta_{1} - 6 \beta_{2} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{7} -9 \beta_{2} q^{9} +O(q^{10})\) \( q + ( 3 - 3 \beta_{2} ) q^{3} + ( \beta_{1} - 6 \beta_{2} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{7} -9 \beta_{2} q^{9} + ( 1 - 7 \beta_{1} + 6 \beta_{2} - 7 \beta_{3} ) q^{11} + 5 \beta_{3} q^{13} + ( -15 - 3 \beta_{3} ) q^{15} + ( -52 + 4 \beta_{1} + 48 \beta_{2} + 4 \beta_{3} ) q^{17} + ( -3 \beta_{1} + 35 \beta_{2} ) q^{19} + ( -6 - 9 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{21} + ( -20 \beta_{1} + 48 \beta_{2} ) q^{23} + ( 52 - 11 \beta_{1} - 41 \beta_{2} - 11 \beta_{3} ) q^{25} -27 q^{27} + ( 143 - 11 \beta_{3} ) q^{29} + ( 171 + 20 \beta_{1} - 191 \beta_{2} + 20 \beta_{3} ) q^{31} + ( -21 \beta_{1} + 18 \beta_{2} ) q^{33} + ( 81 - 8 \beta_{1} + 66 \beta_{2} + 9 \beta_{3} ) q^{35} + ( -45 \beta_{1} + 25 \beta_{2} ) q^{37} + ( 15 \beta_{1} - 15 \beta_{2} + 15 \beta_{3} ) q^{39} + ( -72 - 18 \beta_{3} ) q^{41} + ( -362 + 3 \beta_{3} ) q^{43} + ( -45 - 9 \beta_{1} + 54 \beta_{2} - 9 \beta_{3} ) q^{45} + ( -36 \beta_{1} - 90 \beta_{2} ) q^{47} + ( 232 - 2 \beta_{1} - 379 \beta_{2} + 11 \beta_{3} ) q^{49} + ( 12 \beta_{1} + 144 \beta_{2} ) q^{51} + ( -243 - 9 \beta_{1} + 252 \beta_{2} - 9 \beta_{3} ) q^{53} + ( 331 + 41 \beta_{3} ) q^{55} + ( 96 + 9 \beta_{3} ) q^{57} + ( 113 - 53 \beta_{1} - 60 \beta_{2} - 53 \beta_{3} ) q^{59} + ( -40 \beta_{1} + 286 \beta_{2} ) q^{61} + ( -27 - 9 \beta_{1} + 27 \beta_{2} + 18 \beta_{3} ) q^{63} + ( 30 \beta_{1} - 270 \beta_{2} ) q^{65} + ( 94 - 77 \beta_{1} - 17 \beta_{2} - 77 \beta_{3} ) q^{67} + ( 84 + 60 \beta_{3} ) q^{69} + ( -778 - 44 \beta_{3} ) q^{71} + ( -634 + 53 \beta_{1} + 581 \beta_{2} + 53 \beta_{3} ) q^{73} + ( -33 \beta_{1} - 123 \beta_{2} ) q^{75} + ( 334 + 11 \beta_{1} - 1020 \beta_{2} + 20 \beta_{3} ) q^{77} + ( -62 \beta_{1} + 761 \beta_{2} ) q^{79} + ( -81 + 81 \beta_{2} ) q^{81} + ( 755 - 101 \beta_{3} ) q^{83} + ( 68 + 28 \beta_{3} ) q^{85} + ( 429 - 33 \beta_{1} - 396 \beta_{2} - 33 \beta_{3} ) q^{87} + ( -42 \beta_{1} + 1008 \beta_{2} ) q^{89} + ( -720 + 5 \beta_{1} + 475 \beta_{2} - 10 \beta_{3} ) q^{91} + ( 60 \beta_{1} - 573 \beta_{2} ) q^{93} + ( 304 + 50 \beta_{1} - 354 \beta_{2} + 50 \beta_{3} ) q^{95} + ( 295 - 29 \beta_{3} ) q^{97} + ( -9 + 63 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{3} - 11q^{5} - 6q^{7} - 18q^{9} + O(q^{10}) \) \( 4q + 6q^{3} - 11q^{5} - 6q^{7} - 18q^{9} - 5q^{11} + 10q^{13} - 66q^{15} - 100q^{17} + 67q^{19} - 27q^{21} + 76q^{23} + 93q^{25} - 108q^{27} + 550q^{29} + 362q^{31} + 15q^{33} + 466q^{35} + 5q^{37} + 15q^{39} - 324q^{41} - 1442q^{43} - 99q^{45} - 216q^{47} + 190q^{49} + 300q^{51} - 495q^{53} + 1406q^{55} + 402q^{57} + 173q^{59} + 532q^{61} - 27q^{63} - 510q^{65} + 111q^{67} + 456q^{69} - 3200q^{71} - 1215q^{73} - 279q^{75} - 653q^{77} + 1460q^{79} - 162q^{81} + 2818q^{83} + 328q^{85} + 825q^{87} + 1974q^{89} - 1945q^{91} - 1086q^{93} + 658q^{95} + 1122q^{97} + 90q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 49 x^{2} + 48 x + 2304\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 49 \nu^{2} - 49 \nu + 2304 \)\()/2352\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 97 \)\()/49\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 48 \beta_{2} + \beta_{1} - 49\)
\(\nu^{3}\)\(=\)\(49 \beta_{3} - 97\)

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
−3.22311 5.58259i
3.72311 + 6.44862i
−3.22311 + 5.58259i
3.72311 6.44862i
0 1.50000 2.59808i 0 −6.22311 10.7787i 0 −15.3924 + 10.2992i 0 −4.50000 7.79423i 0
193.2 0 1.50000 2.59808i 0 0.723111 + 1.25246i 0 12.3924 13.7633i 0 −4.50000 7.79423i 0
289.1 0 1.50000 + 2.59808i 0 −6.22311 + 10.7787i 0 −15.3924 10.2992i 0 −4.50000 + 7.79423i 0
289.2 0 1.50000 + 2.59808i 0 0.723111 1.25246i 0 12.3924 + 13.7633i 0 −4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.q.i 4
4.b odd 2 1 84.4.i.a 4
7.c even 3 1 inner 336.4.q.i 4
7.c even 3 1 2352.4.a.bt 2
7.d odd 6 1 2352.4.a.bx 2
12.b even 2 1 252.4.k.f 4
28.d even 2 1 588.4.i.j 4
28.f even 6 1 588.4.a.f 2
28.f even 6 1 588.4.i.j 4
28.g odd 6 1 84.4.i.a 4
28.g odd 6 1 588.4.a.i 2
84.h odd 2 1 1764.4.k.q 4
84.j odd 6 1 1764.4.a.y 2
84.j odd 6 1 1764.4.k.q 4
84.n even 6 1 252.4.k.f 4
84.n even 6 1 1764.4.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.a 4 4.b odd 2 1
84.4.i.a 4 28.g odd 6 1
252.4.k.f 4 12.b even 2 1
252.4.k.f 4 84.n even 6 1
336.4.q.i 4 1.a even 1 1 trivial
336.4.q.i 4 7.c even 3 1 inner
588.4.a.f 2 28.f even 6 1
588.4.a.i 2 28.g odd 6 1
588.4.i.j 4 28.d even 2 1
588.4.i.j 4 28.f even 6 1
1764.4.a.o 2 84.n even 6 1
1764.4.a.y 2 84.j odd 6 1
1764.4.k.q 4 84.h odd 2 1
1764.4.k.q 4 84.j odd 6 1
2352.4.a.bt 2 7.c even 3 1
2352.4.a.bx 2 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 11 T_{5}^{3} + 139 T_{5}^{2} - 198 T_{5} + 324 \) acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 9 - 3 T + T^{2} )^{2} \)
$5$ \( 324 - 198 T + 139 T^{2} + 11 T^{3} + T^{4} \)
$7$ \( 117649 + 2058 T - 77 T^{2} + 6 T^{3} + T^{4} \)
$11$ \( 5560164 - 11790 T + 2383 T^{2} + 5 T^{3} + T^{4} \)
$13$ \( ( -1200 - 5 T + T^{2} )^{2} \)
$17$ \( 2985984 + 172800 T + 8272 T^{2} + 100 T^{3} + T^{4} \)
$19$ \( 473344 - 46096 T + 3801 T^{2} - 67 T^{3} + T^{4} \)
$23$ \( 318836736 + 1357056 T + 23632 T^{2} - 76 T^{3} + T^{4} \)
$29$ \( ( 13068 - 275 T + T^{2} )^{2} \)
$31$ \( 181198521 - 4872882 T + 117583 T^{2} - 362 T^{3} + T^{4} \)
$37$ \( 9545290000 + 488500 T + 97725 T^{2} - 5 T^{3} + T^{4} \)
$41$ \( ( -9072 + 162 T + T^{2} )^{2} \)
$43$ \( ( 129526 + 721 T + T^{2} )^{2} \)
$47$ \( 2587553424 - 10987488 T + 97524 T^{2} + 216 T^{3} + T^{4} \)
$53$ \( 3288793104 + 28387260 T + 187677 T^{2} + 495 T^{3} + T^{4} \)
$59$ \( 16397314704 + 22152996 T + 157981 T^{2} - 173 T^{3} + T^{4} \)
$61$ \( 41525136 + 3428208 T + 289468 T^{2} - 532 T^{3} + T^{4} \)
$67$ \( 80085604036 + 31412334 T + 295315 T^{2} - 111 T^{3} + T^{4} \)
$71$ \( ( 546588 + 1600 T + T^{2} )^{2} \)
$73$ \( 54532524484 + 283729230 T + 1242703 T^{2} + 1215 T^{3} + T^{4} \)
$79$ \( 120705520329 - 507243420 T + 1784173 T^{2} - 1460 T^{3} + T^{4} \)
$83$ \( ( 4122 - 1409 T + T^{2} )^{2} \)
$89$ \( 790420571136 - 1754996544 T + 3007620 T^{2} - 1974 T^{3} + T^{4} \)
$97$ \( ( 38102 - 561 T + T^{2} )^{2} \)
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