Properties

Label 14.3.d.a
Level 14
Weight 3
Character orbit 14.d
Analytic conductor 0.381
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 14 = 2 \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 14.d (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.381472370104\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{3} \) \( + 2 \beta_{2} q^{4} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{5} \) \( + ( -2 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{6} \) \( + ( 3 + 4 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{7} \) \( + 2 \beta_{3} q^{8} \) \( + 6 \beta_{1} q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{3} \) \( + 2 \beta_{2} q^{4} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{5} \) \( + ( -2 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{6} \) \( + ( 3 + 4 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{7} \) \( + 2 \beta_{3} q^{8} \) \( + 6 \beta_{1} q^{9} \) \( + ( 8 - \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{10} \) \( + ( -3 \beta_{1} - 9 \beta_{2} - 3 \beta_{3} ) q^{11} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{12} \) \( + ( 6 - 4 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} ) q^{13} \) \( + ( -10 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{14} \) \( + ( -9 + 3 \beta_{3} ) q^{15} \) \( + ( -4 - 4 \beta_{2} ) q^{16} \) \( + ( -10 - 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{17} \) \( + 12 \beta_{2} q^{18} \) \( + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{19} \) \( + ( -2 + 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{20} \) \( + ( 8 - 8 \beta_{1} - 11 \beta_{2} - 10 \beta_{3} ) q^{21} \) \( + ( 6 - 9 \beta_{3} ) q^{22} \) \( + ( 15 - 9 \beta_{1} + 15 \beta_{2} ) q^{23} \) \( + ( 8 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{24} \) \( + ( 12 \beta_{1} - 2 \beta_{2} + 12 \beta_{3} ) q^{25} \) \( + ( 4 + 6 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} ) q^{26} \) \( + ( -3 + 6 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{27} \) \( + ( -4 - 10 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{28} \) \( + ( 12 - 6 \beta_{3} ) q^{29} \) \( + ( -6 - 9 \beta_{1} - 6 \beta_{2} ) q^{30} \) \( + ( -14 + 15 \beta_{1} - 7 \beta_{2} - 15 \beta_{3} ) q^{31} \) \( + ( -4 \beta_{1} - 4 \beta_{3} ) q^{32} \) \( + ( -15 + 12 \beta_{1} + 15 \beta_{2} + 24 \beta_{3} ) q^{33} \) \( + ( -4 - 10 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} ) q^{34} \) \( + ( 7 - 7 \beta_{1} + 35 \beta_{2} - 14 \beta_{3} ) q^{35} \) \( + 12 \beta_{3} q^{36} \) \( + ( -31 - 24 \beta_{1} - 31 \beta_{2} ) q^{37} \) \( + ( -4 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{38} \) \( + ( -12 \beta_{1} - 6 \beta_{2} - 12 \beta_{3} ) q^{39} \) \( + ( -8 - 2 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{40} \) \( + ( -2 + 20 \beta_{1} - 4 \beta_{2} + 10 \beta_{3} ) q^{41} \) \( + ( 20 + 8 \beta_{1} + 4 \beta_{2} - 11 \beta_{3} ) q^{42} \) \( + ( -2 - 6 \beta_{3} ) q^{43} \) \( + ( 18 + 6 \beta_{1} + 18 \beta_{2} ) q^{44} \) \( + ( 48 - 6 \beta_{1} + 24 \beta_{2} + 6 \beta_{3} ) q^{45} \) \( + ( 15 \beta_{1} - 18 \beta_{2} + 15 \beta_{3} ) q^{46} \) \( + ( 29 + \beta_{1} - 29 \beta_{2} + 2 \beta_{3} ) q^{47} \) \( + ( 4 + 8 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} ) q^{48} \) \( + ( -25 + 4 \beta_{1} - 40 \beta_{2} + 26 \beta_{3} ) q^{49} \) \( + ( -24 - 2 \beta_{3} ) q^{50} \) \( + ( 27 + 21 \beta_{1} + 27 \beta_{2} ) q^{51} \) \( + ( -24 + 4 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{52} \) \( + ( -12 \beta_{1} + 39 \beta_{2} - 12 \beta_{3} ) q^{53} \) \( + ( -6 - 3 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{54} \) \( + ( -3 - 30 \beta_{1} - 6 \beta_{2} - 15 \beta_{3} ) q^{55} \) \( + ( 4 - 4 \beta_{1} - 16 \beta_{2} + 2 \beta_{3} ) q^{56} \) \( + 3 q^{57} \) \( + ( 12 + 12 \beta_{1} + 12 \beta_{2} ) q^{58} \) \( + ( -26 - 25 \beta_{1} - 13 \beta_{2} + 25 \beta_{3} ) q^{59} \) \( + ( -6 \beta_{1} - 18 \beta_{2} - 6 \beta_{3} ) q^{60} \) \( + ( -7 - 32 \beta_{1} + 7 \beta_{2} - 64 \beta_{3} ) q^{61} \) \( + ( 30 - 14 \beta_{1} + 60 \beta_{2} - 7 \beta_{3} ) q^{62} \) \( + ( -60 + 18 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{63} \) \( + 8 q^{64} \) \( + ( -42 + 42 \beta_{1} - 42 \beta_{2} ) q^{65} \) \( + ( -48 - 15 \beta_{1} - 24 \beta_{2} + 15 \beta_{3} ) q^{66} \) \( + ( 45 \beta_{1} + 29 \beta_{2} + 45 \beta_{3} ) q^{67} \) \( + ( 10 - 4 \beta_{1} - 10 \beta_{2} - 8 \beta_{3} ) q^{68} \) \( + ( 3 - 12 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{69} \) \( + ( 28 + 7 \beta_{1} + 14 \beta_{2} + 35 \beta_{3} ) q^{70} \) \( + ( -6 + 30 \beta_{3} ) q^{71} \) \( + ( -24 - 24 \beta_{2} ) q^{72} \) \( + ( 106 + 16 \beta_{1} + 53 \beta_{2} - 16 \beta_{3} ) q^{73} \) \( + ( -31 \beta_{1} - 48 \beta_{2} - 31 \beta_{3} ) q^{74} \) \( + ( 22 - 10 \beta_{1} - 22 \beta_{2} - 20 \beta_{3} ) q^{75} \) \( + ( 2 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{76} \) \( + ( 42 + 42 \beta_{1} + 21 \beta_{2} ) q^{77} \) \( + ( 24 - 6 \beta_{3} ) q^{78} \) \( + ( 55 + 15 \beta_{1} + 55 \beta_{2} ) q^{79} \) \( + ( 8 - 8 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} ) q^{80} \) \( + ( -54 \beta_{1} - 9 \beta_{2} - 54 \beta_{3} ) q^{81} \) \( + ( -20 - 2 \beta_{1} + 20 \beta_{2} - 4 \beta_{3} ) q^{82} \) \( + ( -68 + 8 \beta_{1} - 136 \beta_{2} + 4 \beta_{3} ) q^{83} \) \( + ( 22 + 20 \beta_{1} + 38 \beta_{2} + 4 \beta_{3} ) q^{84} \) \( + ( -9 + 24 \beta_{3} ) q^{85} \) \( + ( 12 - 2 \beta_{1} + 12 \beta_{2} ) q^{86} \) \( + ( -48 - 18 \beta_{1} - 24 \beta_{2} + 18 \beta_{3} ) q^{87} \) \( + ( 18 \beta_{1} + 12 \beta_{2} + 18 \beta_{3} ) q^{88} \) \( + ( -63 + 24 \beta_{1} + 63 \beta_{2} + 48 \beta_{3} ) q^{89} \) \( + ( -12 + 48 \beta_{1} - 24 \beta_{2} + 24 \beta_{3} ) q^{90} \) \( + ( 30 - 44 \beta_{1} + 48 \beta_{2} + 8 \beta_{3} ) q^{91} \) \( + ( -30 - 18 \beta_{3} ) q^{92} \) \( + ( -69 - 24 \beta_{1} - 69 \beta_{2} ) q^{93} \) \( + ( -4 + 29 \beta_{1} - 2 \beta_{2} - 29 \beta_{3} ) q^{94} \) \( + ( -9 \beta_{1} + 15 \beta_{2} - 9 \beta_{3} ) q^{95} \) \( + ( -8 + 4 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{96} \) \( + ( 22 - 52 \beta_{1} + 44 \beta_{2} - 26 \beta_{3} ) q^{97} \) \( + ( -52 - 25 \beta_{1} - 44 \beta_{2} - 40 \beta_{3} ) q^{98} \) \( + ( 36 - 54 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 24q^{10} \) \(\mathstrut +\mathstrut 18q^{11} \) \(\mathstrut +\mathstrut 12q^{12} \) \(\mathstrut -\mathstrut 36q^{14} \) \(\mathstrut -\mathstrut 36q^{15} \) \(\mathstrut -\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 30q^{17} \) \(\mathstrut -\mathstrut 24q^{18} \) \(\mathstrut +\mathstrut 6q^{19} \) \(\mathstrut +\mathstrut 54q^{21} \) \(\mathstrut +\mathstrut 24q^{22} \) \(\mathstrut +\mathstrut 30q^{23} \) \(\mathstrut +\mathstrut 24q^{24} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut +\mathstrut 24q^{26} \) \(\mathstrut -\mathstrut 20q^{28} \) \(\mathstrut +\mathstrut 48q^{29} \) \(\mathstrut -\mathstrut 12q^{30} \) \(\mathstrut -\mathstrut 42q^{31} \) \(\mathstrut -\mathstrut 90q^{33} \) \(\mathstrut -\mathstrut 42q^{35} \) \(\mathstrut -\mathstrut 62q^{37} \) \(\mathstrut -\mathstrut 12q^{38} \) \(\mathstrut +\mathstrut 12q^{39} \) \(\mathstrut -\mathstrut 48q^{40} \) \(\mathstrut +\mathstrut 72q^{42} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 36q^{44} \) \(\mathstrut +\mathstrut 144q^{45} \) \(\mathstrut +\mathstrut 36q^{46} \) \(\mathstrut +\mathstrut 174q^{47} \) \(\mathstrut -\mathstrut 20q^{49} \) \(\mathstrut -\mathstrut 96q^{50} \) \(\mathstrut +\mathstrut 54q^{51} \) \(\mathstrut -\mathstrut 72q^{52} \) \(\mathstrut -\mathstrut 78q^{53} \) \(\mathstrut -\mathstrut 36q^{54} \) \(\mathstrut +\mathstrut 48q^{56} \) \(\mathstrut +\mathstrut 12q^{57} \) \(\mathstrut +\mathstrut 24q^{58} \) \(\mathstrut -\mathstrut 78q^{59} \) \(\mathstrut +\mathstrut 36q^{60} \) \(\mathstrut -\mathstrut 42q^{61} \) \(\mathstrut -\mathstrut 216q^{63} \) \(\mathstrut +\mathstrut 32q^{64} \) \(\mathstrut -\mathstrut 84q^{65} \) \(\mathstrut -\mathstrut 144q^{66} \) \(\mathstrut -\mathstrut 58q^{67} \) \(\mathstrut +\mathstrut 60q^{68} \) \(\mathstrut +\mathstrut 84q^{70} \) \(\mathstrut -\mathstrut 24q^{71} \) \(\mathstrut -\mathstrut 48q^{72} \) \(\mathstrut +\mathstrut 318q^{73} \) \(\mathstrut +\mathstrut 96q^{74} \) \(\mathstrut +\mathstrut 132q^{75} \) \(\mathstrut +\mathstrut 126q^{77} \) \(\mathstrut +\mathstrut 96q^{78} \) \(\mathstrut +\mathstrut 110q^{79} \) \(\mathstrut +\mathstrut 24q^{80} \) \(\mathstrut +\mathstrut 18q^{81} \) \(\mathstrut -\mathstrut 120q^{82} \) \(\mathstrut +\mathstrut 12q^{84} \) \(\mathstrut -\mathstrut 36q^{85} \) \(\mathstrut +\mathstrut 24q^{86} \) \(\mathstrut -\mathstrut 144q^{87} \) \(\mathstrut -\mathstrut 24q^{88} \) \(\mathstrut -\mathstrut 378q^{89} \) \(\mathstrut +\mathstrut 24q^{91} \) \(\mathstrut -\mathstrut 120q^{92} \) \(\mathstrut -\mathstrut 138q^{93} \) \(\mathstrut -\mathstrut 12q^{94} \) \(\mathstrut -\mathstrut 30q^{95} \) \(\mathstrut -\mathstrut 48q^{96} \) \(\mathstrut -\mathstrut 120q^{98} \) \(\mathstrut +\mathstrut 144q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(2\) \(x^{2}\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2\) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{3}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i 0.621320 + 0.358719i −1.00000 + 1.73205i −5.74264 + 3.31552i 1.01461i 6.24264 3.16693i 2.82843 −4.24264 7.34847i 8.12132 + 4.68885i
3.2 0.707107 + 1.22474i −3.62132 2.09077i −1.00000 + 1.73205i 2.74264 1.58346i 5.91359i −2.24264 + 6.63103i −2.82843 4.24264 + 7.34847i 3.87868 + 2.23936i
5.1 −0.707107 + 1.22474i 0.621320 0.358719i −1.00000 1.73205i −5.74264 3.31552i 1.01461i 6.24264 + 3.16693i 2.82843 −4.24264 + 7.34847i 8.12132 4.68885i
5.2 0.707107 1.22474i −3.62132 + 2.09077i −1.00000 1.73205i 2.74264 + 1.58346i 5.91359i −2.24264 6.63103i −2.82843 4.24264 7.34847i 3.87868 2.23936i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(14, [\chi])\).