Properties

Label 336.2.s.b
Level 336
Weight 2
Character orbit 336.s
Analytic conductor 2.683
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: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{8}^{2} ) q^{2} + ( -\zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + 2 \zeta_{8}^{2} q^{4} + ( 1 - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{5} + ( 1 - 2 \zeta_{8} - \zeta_{8}^{2} ) q^{6} - q^{7} + ( -2 + 2 \zeta_{8}^{2} ) q^{8} + ( 1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{8}^{2} ) q^{2} + ( -\zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + 2 \zeta_{8}^{2} q^{4} + ( 1 - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{5} + ( 1 - 2 \zeta_{8} - \zeta_{8}^{2} ) q^{6} - q^{7} + ( -2 + 2 \zeta_{8}^{2} ) q^{8} + ( 1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{10} + ( 1 + 4 \zeta_{8} + \zeta_{8}^{2} ) q^{11} + ( 2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{12} + ( -3 + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{13} + ( -1 - \zeta_{8}^{2} ) q^{14} + ( 1 + 2 \zeta_{8} - 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{15} -4 q^{16} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{17} + ( 1 + \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{18} + ( -1 + 2 \zeta_{8} - \zeta_{8}^{2} ) q^{19} + ( 2 - 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{20} + ( \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{21} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{22} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{23} + ( 2 + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{24} + ( 4 \zeta_{8} - \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{25} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{26} + ( \zeta_{8} - 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} -2 \zeta_{8}^{2} q^{28} + ( 3 + 3 \zeta_{8}^{2} ) q^{29} + ( 4 - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{30} + ( -6 \zeta_{8} + 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{31} + ( -4 - 4 \zeta_{8}^{2} ) q^{32} + ( -3 - 2 \zeta_{8} - 5 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{33} + ( -4 + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{34} + ( -1 + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{35} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{36} + ( -1 + 4 \zeta_{8} - \zeta_{8}^{2} ) q^{37} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{38} + ( 5 + 2 \zeta_{8} + \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{39} + ( -4 \zeta_{8} + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{40} + ( 6 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{41} + ( -1 + 2 \zeta_{8} + \zeta_{8}^{2} ) q^{42} + ( -1 + \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{43} + ( -2 + 2 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{44} + ( -3 + 4 \zeta_{8} - 5 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{45} + ( 2 - 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{46} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{47} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{48} + q^{49} + ( 1 - \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{50} + ( 4 - 6 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{51} + ( -6 - 4 \zeta_{8} - 6 \zeta_{8}^{2} ) q^{52} + ( 3 - 3 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{53} + ( 5 + 2 \zeta_{8} - 5 \zeta_{8}^{2} ) q^{54} + ( -6 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{55} + ( 2 - 2 \zeta_{8}^{2} ) q^{56} + ( -3 + 2 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{57} + 6 \zeta_{8}^{2} q^{58} + ( 3 + 10 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{59} + ( 6 - 4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{60} + ( 1 - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{61} + ( -2 + 2 \zeta_{8}^{2} - 12 \zeta_{8}^{3} ) q^{62} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{63} -8 \zeta_{8}^{2} q^{64} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{65} + ( 2 + 2 \zeta_{8} - 8 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{66} + ( -7 - 7 \zeta_{8}^{2} ) q^{67} + ( -8 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{68} + ( -2 - 2 \zeta_{8} + 8 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{69} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{70} -6 \zeta_{8}^{2} q^{71} + ( -2 - 8 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{72} + ( 2 \zeta_{8} - 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{73} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{74} + ( -1 + 5 \zeta_{8} - 8 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{75} + ( 2 - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{76} + ( -1 - 4 \zeta_{8} - \zeta_{8}^{2} ) q^{77} + ( 4 + 8 \zeta_{8} + 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{78} -2 \zeta_{8}^{2} q^{79} + ( -4 + 4 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{80} + ( -7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} + ( 6 - 8 \zeta_{8} + 6 \zeta_{8}^{2} ) q^{82} + ( 1 - \zeta_{8}^{2} + 14 \zeta_{8}^{3} ) q^{83} + ( -2 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{84} + ( 8 - 12 \zeta_{8} + 8 \zeta_{8}^{2} ) q^{85} + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{86} + ( 3 - 6 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{87} + ( -4 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{88} + ( 10 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{89} + ( 2 + 2 \zeta_{8} - 8 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{90} + ( 3 - 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{91} + ( 4 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{92} + ( 2 - 8 \zeta_{8} + 12 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{93} + 8 \zeta_{8} q^{94} + ( -6 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{95} + ( -4 + 8 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{96} -2 q^{97} + ( 1 + \zeta_{8}^{2} ) q^{98} + ( -7 + 4 \zeta_{8} + 9 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{5} + 4q^{6} - 4q^{7} - 8q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{5} + 4q^{6} - 4q^{7} - 8q^{8} + 4q^{9} + 8q^{10} + 4q^{11} + 8q^{12} - 12q^{13} - 4q^{14} + 4q^{15} - 16q^{16} + 4q^{18} - 4q^{19} + 8q^{20} + 8q^{24} - 24q^{26} + 12q^{29} + 16q^{30} - 16q^{32} - 12q^{33} - 16q^{34} - 4q^{35} - 4q^{37} + 20q^{39} + 24q^{41} - 4q^{42} - 4q^{43} - 8q^{44} - 12q^{45} + 8q^{46} + 4q^{49} + 4q^{50} + 16q^{51} - 24q^{52} + 12q^{53} + 20q^{54} - 24q^{55} + 8q^{56} - 12q^{57} + 12q^{59} + 24q^{60} + 4q^{61} - 8q^{62} - 4q^{63} + 8q^{66} - 28q^{67} - 32q^{68} - 8q^{69} - 8q^{70} - 8q^{72} - 4q^{75} + 8q^{76} - 4q^{77} + 16q^{78} - 16q^{80} - 28q^{81} + 24q^{82} + 4q^{83} - 8q^{84} + 32q^{85} - 8q^{86} + 12q^{87} - 16q^{88} + 40q^{89} + 8q^{90} + 12q^{91} + 16q^{92} + 8q^{93} - 24q^{95} - 16q^{96} - 8q^{97} + 4q^{98} - 28q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(\zeta_{8}\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
1.00000 + 1.00000i −1.41421 1.00000i 2.00000i −0.414214 + 0.414214i −0.414214 2.41421i −1.00000 −2.00000 + 2.00000i 1.00000 + 2.82843i −0.828427
155.2 1.00000 + 1.00000i 1.41421 1.00000i 2.00000i 2.41421 2.41421i 2.41421 + 0.414214i −1.00000 −2.00000 + 2.00000i 1.00000 2.82843i 4.82843
323.1 1.00000 1.00000i −1.41421 + 1.00000i 2.00000i −0.414214 0.414214i −0.414214 + 2.41421i −1.00000 −2.00000 2.00000i 1.00000 2.82843i −0.828427
323.2 1.00000 1.00000i 1.41421 + 1.00000i 2.00000i 2.41421 + 2.41421i 2.41421 0.414214i −1.00000 −2.00000 2.00000i 1.00000 + 2.82843i 4.82843
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.s.b yes 4
3.b odd 2 1 336.2.s.a 4
4.b odd 2 1 1344.2.s.a 4
12.b even 2 1 1344.2.s.b 4
16.e even 4 1 1344.2.s.b 4
16.f odd 4 1 336.2.s.a 4
48.i odd 4 1 1344.2.s.a 4
48.k even 4 1 inner 336.2.s.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.s.a 4 3.b odd 2 1
336.2.s.a 4 16.f odd 4 1
336.2.s.b yes 4 1.a even 1 1 trivial
336.2.s.b yes 4 48.k even 4 1 inner
1344.2.s.a 4 4.b odd 2 1
1344.2.s.a 4 48.i odd 4 1
1344.2.s.b 4 12.b even 2 1
1344.2.s.b 4 16.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 4 T_{5}^{3} + 8 T_{5}^{2} + 8 T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 2 T^{2} )^{2} \)
$3$ \( 1 - 2 T^{2} + 9 T^{4} \)
$5$ \( 1 - 4 T + 8 T^{2} - 12 T^{3} + 14 T^{4} - 60 T^{5} + 200 T^{6} - 500 T^{7} + 625 T^{8} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( 1 - 4 T + 8 T^{2} + 12 T^{3} - 178 T^{4} + 132 T^{5} + 968 T^{6} - 5324 T^{7} + 14641 T^{8} \)
$13$ \( 1 + 12 T + 72 T^{2} + 324 T^{3} + 1262 T^{4} + 4212 T^{5} + 12168 T^{6} + 26364 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 20 T^{2} + 166 T^{4} - 5780 T^{6} + 83521 T^{8} \)
$19$ \( 1 + 4 T + 8 T^{2} + 68 T^{3} + 574 T^{4} + 1292 T^{5} + 2888 T^{6} + 27436 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 10 T + 29 T^{2} )^{2}( 1 + 4 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 28 T^{2} + 966 T^{4} + 26908 T^{6} + 923521 T^{8} \)
$37$ \( 1 + 4 T + 8 T^{2} + 92 T^{3} + 862 T^{4} + 3404 T^{5} + 10952 T^{6} + 202612 T^{7} + 1874161 T^{8} \)
$41$ \( ( 1 - 12 T + 86 T^{2} - 492 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 + 4 T + 8 T^{2} + 116 T^{3} + 1486 T^{4} + 4988 T^{5} + 14792 T^{6} + 318028 T^{7} + 3418801 T^{8} \)
$47$ \( ( 1 + 62 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( 1 - 12 T + 72 T^{2} - 660 T^{3} + 6046 T^{4} - 34980 T^{5} + 202248 T^{6} - 1786524 T^{7} + 7890481 T^{8} \)
$59$ \( ( 1 - 6 T + 59 T^{2} )^{2}( 1 - 82 T^{2} + 3481 T^{4} ) \)
$61$ \( 1 - 4 T + 8 T^{2} - 236 T^{3} + 6958 T^{4} - 14396 T^{5} + 29768 T^{6} - 907924 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 + 14 T + 98 T^{2} + 938 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 106 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( 1 - 148 T^{2} + 14086 T^{4} - 788692 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 - 154 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( 1 - 4 T + 8 T^{2} + 444 T^{3} - 12994 T^{4} + 36852 T^{5} + 55112 T^{6} - 2287148 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 - 20 T + 246 T^{2} - 1780 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{4} \)
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