Properties

Label 14.4.c.a
Level 14
Weight 4
Character orbit 14.c
Analytic conductor 0.826
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 14 = 2 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 14.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.82602674008\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -2 + 2 \zeta_{6} ) q^{2} \) \( + 5 \zeta_{6} q^{3} \) \( -4 \zeta_{6} q^{4} \) \( + ( 9 - 9 \zeta_{6} ) q^{5} \) \( -10 q^{6} \) \( + ( -7 - 14 \zeta_{6} ) q^{7} \) \( + 8 q^{8} \) \( + ( 2 - 2 \zeta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -2 + 2 \zeta_{6} ) q^{2} \) \( + 5 \zeta_{6} q^{3} \) \( -4 \zeta_{6} q^{4} \) \( + ( 9 - 9 \zeta_{6} ) q^{5} \) \( -10 q^{6} \) \( + ( -7 - 14 \zeta_{6} ) q^{7} \) \( + 8 q^{8} \) \( + ( 2 - 2 \zeta_{6} ) q^{9} \) \( + 18 \zeta_{6} q^{10} \) \( + 57 \zeta_{6} q^{11} \) \( + ( 20 - 20 \zeta_{6} ) q^{12} \) \( -70 q^{13} \) \( + ( 42 - 14 \zeta_{6} ) q^{14} \) \( + 45 q^{15} \) \( + ( -16 + 16 \zeta_{6} ) q^{16} \) \( -51 \zeta_{6} q^{17} \) \( + 4 \zeta_{6} q^{18} \) \( + ( -5 + 5 \zeta_{6} ) q^{19} \) \( -36 q^{20} \) \( + ( 70 - 105 \zeta_{6} ) q^{21} \) \( -114 q^{22} \) \( + ( -69 + 69 \zeta_{6} ) q^{23} \) \( + 40 \zeta_{6} q^{24} \) \( + 44 \zeta_{6} q^{25} \) \( + ( 140 - 140 \zeta_{6} ) q^{26} \) \( + 145 q^{27} \) \( + ( -56 + 84 \zeta_{6} ) q^{28} \) \( + 114 q^{29} \) \( + ( -90 + 90 \zeta_{6} ) q^{30} \) \( -23 \zeta_{6} q^{31} \) \( -32 \zeta_{6} q^{32} \) \( + ( -285 + 285 \zeta_{6} ) q^{33} \) \( + 102 q^{34} \) \( + ( -189 + 63 \zeta_{6} ) q^{35} \) \( -8 q^{36} \) \( + ( 253 - 253 \zeta_{6} ) q^{37} \) \( -10 \zeta_{6} q^{38} \) \( -350 \zeta_{6} q^{39} \) \( + ( 72 - 72 \zeta_{6} ) q^{40} \) \( -42 q^{41} \) \( + ( 70 + 140 \zeta_{6} ) q^{42} \) \( -124 q^{43} \) \( + ( 228 - 228 \zeta_{6} ) q^{44} \) \( -18 \zeta_{6} q^{45} \) \( -138 \zeta_{6} q^{46} \) \( + ( -201 + 201 \zeta_{6} ) q^{47} \) \( -80 q^{48} \) \( + ( -147 + 392 \zeta_{6} ) q^{49} \) \( -88 q^{50} \) \( + ( 255 - 255 \zeta_{6} ) q^{51} \) \( + 280 \zeta_{6} q^{52} \) \( + 393 \zeta_{6} q^{53} \) \( + ( -290 + 290 \zeta_{6} ) q^{54} \) \( + 513 q^{55} \) \( + ( -56 - 112 \zeta_{6} ) q^{56} \) \( -25 q^{57} \) \( + ( -228 + 228 \zeta_{6} ) q^{58} \) \( -219 \zeta_{6} q^{59} \) \( -180 \zeta_{6} q^{60} \) \( + ( 709 - 709 \zeta_{6} ) q^{61} \) \( + 46 q^{62} \) \( + ( -42 + 14 \zeta_{6} ) q^{63} \) \( + 64 q^{64} \) \( + ( -630 + 630 \zeta_{6} ) q^{65} \) \( -570 \zeta_{6} q^{66} \) \( -419 \zeta_{6} q^{67} \) \( + ( -204 + 204 \zeta_{6} ) q^{68} \) \( -345 q^{69} \) \( + ( 252 - 378 \zeta_{6} ) q^{70} \) \( -96 q^{71} \) \( + ( 16 - 16 \zeta_{6} ) q^{72} \) \( + 313 \zeta_{6} q^{73} \) \( + 506 \zeta_{6} q^{74} \) \( + ( -220 + 220 \zeta_{6} ) q^{75} \) \( + 20 q^{76} \) \( + ( 798 - 1197 \zeta_{6} ) q^{77} \) \( + 700 q^{78} \) \( + ( -461 + 461 \zeta_{6} ) q^{79} \) \( + 144 \zeta_{6} q^{80} \) \( + 671 \zeta_{6} q^{81} \) \( + ( 84 - 84 \zeta_{6} ) q^{82} \) \( -588 q^{83} \) \( + ( -420 + 140 \zeta_{6} ) q^{84} \) \( -459 q^{85} \) \( + ( 248 - 248 \zeta_{6} ) q^{86} \) \( + 570 \zeta_{6} q^{87} \) \( + 456 \zeta_{6} q^{88} \) \( + ( 1017 - 1017 \zeta_{6} ) q^{89} \) \( + 36 q^{90} \) \( + ( 490 + 980 \zeta_{6} ) q^{91} \) \( + 276 q^{92} \) \( + ( 115 - 115 \zeta_{6} ) q^{93} \) \( -402 \zeta_{6} q^{94} \) \( + 45 \zeta_{6} q^{95} \) \( + ( 160 - 160 \zeta_{6} ) q^{96} \) \( -1834 q^{97} \) \( + ( -490 - 294 \zeta_{6} ) q^{98} \) \( + 114 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 20q^{6} \) \(\mathstrut -\mathstrut 28q^{7} \) \(\mathstrut +\mathstrut 16q^{8} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 20q^{6} \) \(\mathstrut -\mathstrut 28q^{7} \) \(\mathstrut +\mathstrut 16q^{8} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 18q^{10} \) \(\mathstrut +\mathstrut 57q^{11} \) \(\mathstrut +\mathstrut 20q^{12} \) \(\mathstrut -\mathstrut 140q^{13} \) \(\mathstrut +\mathstrut 70q^{14} \) \(\mathstrut +\mathstrut 90q^{15} \) \(\mathstrut -\mathstrut 16q^{16} \) \(\mathstrut -\mathstrut 51q^{17} \) \(\mathstrut +\mathstrut 4q^{18} \) \(\mathstrut -\mathstrut 5q^{19} \) \(\mathstrut -\mathstrut 72q^{20} \) \(\mathstrut +\mathstrut 35q^{21} \) \(\mathstrut -\mathstrut 228q^{22} \) \(\mathstrut -\mathstrut 69q^{23} \) \(\mathstrut +\mathstrut 40q^{24} \) \(\mathstrut +\mathstrut 44q^{25} \) \(\mathstrut +\mathstrut 140q^{26} \) \(\mathstrut +\mathstrut 290q^{27} \) \(\mathstrut -\mathstrut 28q^{28} \) \(\mathstrut +\mathstrut 228q^{29} \) \(\mathstrut -\mathstrut 90q^{30} \) \(\mathstrut -\mathstrut 23q^{31} \) \(\mathstrut -\mathstrut 32q^{32} \) \(\mathstrut -\mathstrut 285q^{33} \) \(\mathstrut +\mathstrut 204q^{34} \) \(\mathstrut -\mathstrut 315q^{35} \) \(\mathstrut -\mathstrut 16q^{36} \) \(\mathstrut +\mathstrut 253q^{37} \) \(\mathstrut -\mathstrut 10q^{38} \) \(\mathstrut -\mathstrut 350q^{39} \) \(\mathstrut +\mathstrut 72q^{40} \) \(\mathstrut -\mathstrut 84q^{41} \) \(\mathstrut +\mathstrut 280q^{42} \) \(\mathstrut -\mathstrut 248q^{43} \) \(\mathstrut +\mathstrut 228q^{44} \) \(\mathstrut -\mathstrut 18q^{45} \) \(\mathstrut -\mathstrut 138q^{46} \) \(\mathstrut -\mathstrut 201q^{47} \) \(\mathstrut -\mathstrut 160q^{48} \) \(\mathstrut +\mathstrut 98q^{49} \) \(\mathstrut -\mathstrut 176q^{50} \) \(\mathstrut +\mathstrut 255q^{51} \) \(\mathstrut +\mathstrut 280q^{52} \) \(\mathstrut +\mathstrut 393q^{53} \) \(\mathstrut -\mathstrut 290q^{54} \) \(\mathstrut +\mathstrut 1026q^{55} \) \(\mathstrut -\mathstrut 224q^{56} \) \(\mathstrut -\mathstrut 50q^{57} \) \(\mathstrut -\mathstrut 228q^{58} \) \(\mathstrut -\mathstrut 219q^{59} \) \(\mathstrut -\mathstrut 180q^{60} \) \(\mathstrut +\mathstrut 709q^{61} \) \(\mathstrut +\mathstrut 92q^{62} \) \(\mathstrut -\mathstrut 70q^{63} \) \(\mathstrut +\mathstrut 128q^{64} \) \(\mathstrut -\mathstrut 630q^{65} \) \(\mathstrut -\mathstrut 570q^{66} \) \(\mathstrut -\mathstrut 419q^{67} \) \(\mathstrut -\mathstrut 204q^{68} \) \(\mathstrut -\mathstrut 690q^{69} \) \(\mathstrut +\mathstrut 126q^{70} \) \(\mathstrut -\mathstrut 192q^{71} \) \(\mathstrut +\mathstrut 16q^{72} \) \(\mathstrut +\mathstrut 313q^{73} \) \(\mathstrut +\mathstrut 506q^{74} \) \(\mathstrut -\mathstrut 220q^{75} \) \(\mathstrut +\mathstrut 40q^{76} \) \(\mathstrut +\mathstrut 399q^{77} \) \(\mathstrut +\mathstrut 1400q^{78} \) \(\mathstrut -\mathstrut 461q^{79} \) \(\mathstrut +\mathstrut 144q^{80} \) \(\mathstrut +\mathstrut 671q^{81} \) \(\mathstrut +\mathstrut 84q^{82} \) \(\mathstrut -\mathstrut 1176q^{83} \) \(\mathstrut -\mathstrut 700q^{84} \) \(\mathstrut -\mathstrut 918q^{85} \) \(\mathstrut +\mathstrut 248q^{86} \) \(\mathstrut +\mathstrut 570q^{87} \) \(\mathstrut +\mathstrut 456q^{88} \) \(\mathstrut +\mathstrut 1017q^{89} \) \(\mathstrut +\mathstrut 72q^{90} \) \(\mathstrut +\mathstrut 1960q^{91} \) \(\mathstrut +\mathstrut 552q^{92} \) \(\mathstrut +\mathstrut 115q^{93} \) \(\mathstrut -\mathstrut 402q^{94} \) \(\mathstrut +\mathstrut 45q^{95} \) \(\mathstrut +\mathstrut 160q^{96} \) \(\mathstrut -\mathstrut 3668q^{97} \) \(\mathstrut -\mathstrut 1274q^{98} \) \(\mathstrut +\mathstrut 228q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(3\)
\(\chi(n)\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 + 1.73205i 2.50000 + 4.33013i −2.00000 3.46410i 4.50000 7.79423i −10.0000 −14.0000 12.1244i 8.00000 1.00000 1.73205i 9.00000 + 15.5885i
11.1 −1.00000 1.73205i 2.50000 4.33013i −2.00000 + 3.46410i 4.50000 + 7.79423i −10.0000 −14.0000 + 12.1244i 8.00000 1.00000 + 1.73205i 9.00000 15.5885i
\(n\): 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.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut -\mathstrut 5 T_{3} \) \(\mathstrut +\mathstrut 25 \) acting on \(S_{4}^{\mathrm{new}}(14, [\chi])\).