Properties

Label 14.4.c.b
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.

\(f(q)\) \(=\) \( q\) \( + ( 2 - 2 \zeta_{6} ) q^{2} \) \( + \zeta_{6} q^{3} \) \( -4 \zeta_{6} q^{4} \) \( + ( -7 + 7 \zeta_{6} ) q^{5} \) \( + 2 q^{6} \) \( + ( -19 + 18 \zeta_{6} ) q^{7} \) \( -8 q^{8} \) \( + ( 26 - 26 \zeta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 2 - 2 \zeta_{6} ) q^{2} \) \( + \zeta_{6} q^{3} \) \( -4 \zeta_{6} q^{4} \) \( + ( -7 + 7 \zeta_{6} ) q^{5} \) \( + 2 q^{6} \) \( + ( -19 + 18 \zeta_{6} ) q^{7} \) \( -8 q^{8} \) \( + ( 26 - 26 \zeta_{6} ) q^{9} \) \( + 14 \zeta_{6} q^{10} \) \( -35 \zeta_{6} q^{11} \) \( + ( 4 - 4 \zeta_{6} ) q^{12} \) \( + 66 q^{13} \) \( + ( -2 + 38 \zeta_{6} ) q^{14} \) \( -7 q^{15} \) \( + ( -16 + 16 \zeta_{6} ) q^{16} \) \( -59 \zeta_{6} q^{17} \) \( -52 \zeta_{6} q^{18} \) \( + ( -137 + 137 \zeta_{6} ) q^{19} \) \( + 28 q^{20} \) \( + ( -18 - \zeta_{6} ) q^{21} \) \( -70 q^{22} \) \( + ( 7 - 7 \zeta_{6} ) q^{23} \) \( -8 \zeta_{6} q^{24} \) \( + 76 \zeta_{6} q^{25} \) \( + ( 132 - 132 \zeta_{6} ) q^{26} \) \( + 53 q^{27} \) \( + ( 72 + 4 \zeta_{6} ) q^{28} \) \( + 106 q^{29} \) \( + ( -14 + 14 \zeta_{6} ) q^{30} \) \( -75 \zeta_{6} q^{31} \) \( + 32 \zeta_{6} q^{32} \) \( + ( 35 - 35 \zeta_{6} ) q^{33} \) \( -118 q^{34} \) \( + ( 7 - 133 \zeta_{6} ) q^{35} \) \( -104 q^{36} \) \( + ( -11 + 11 \zeta_{6} ) q^{37} \) \( + 274 \zeta_{6} q^{38} \) \( + 66 \zeta_{6} q^{39} \) \( + ( 56 - 56 \zeta_{6} ) q^{40} \) \( -498 q^{41} \) \( + ( -38 + 36 \zeta_{6} ) q^{42} \) \( + 260 q^{43} \) \( + ( -140 + 140 \zeta_{6} ) q^{44} \) \( + 182 \zeta_{6} q^{45} \) \( -14 \zeta_{6} q^{46} \) \( + ( 171 - 171 \zeta_{6} ) q^{47} \) \( -16 q^{48} \) \( + ( 37 - 360 \zeta_{6} ) q^{49} \) \( + 152 q^{50} \) \( + ( 59 - 59 \zeta_{6} ) q^{51} \) \( -264 \zeta_{6} q^{52} \) \( + 417 \zeta_{6} q^{53} \) \( + ( 106 - 106 \zeta_{6} ) q^{54} \) \( + 245 q^{55} \) \( + ( 152 - 144 \zeta_{6} ) q^{56} \) \( -137 q^{57} \) \( + ( 212 - 212 \zeta_{6} ) q^{58} \) \( + 17 \zeta_{6} q^{59} \) \( + 28 \zeta_{6} q^{60} \) \( + ( -51 + 51 \zeta_{6} ) q^{61} \) \( -150 q^{62} \) \( + ( -26 + 494 \zeta_{6} ) q^{63} \) \( + 64 q^{64} \) \( + ( -462 + 462 \zeta_{6} ) q^{65} \) \( -70 \zeta_{6} q^{66} \) \( -439 \zeta_{6} q^{67} \) \( + ( -236 + 236 \zeta_{6} ) q^{68} \) \( + 7 q^{69} \) \( + ( -252 - 14 \zeta_{6} ) q^{70} \) \( -784 q^{71} \) \( + ( -208 + 208 \zeta_{6} ) q^{72} \) \( -295 \zeta_{6} q^{73} \) \( + 22 \zeta_{6} q^{74} \) \( + ( -76 + 76 \zeta_{6} ) q^{75} \) \( + 548 q^{76} \) \( + ( 630 + 35 \zeta_{6} ) q^{77} \) \( + 132 q^{78} \) \( + ( 495 - 495 \zeta_{6} ) q^{79} \) \( -112 \zeta_{6} q^{80} \) \( -649 \zeta_{6} q^{81} \) \( + ( -996 + 996 \zeta_{6} ) q^{82} \) \( + 932 q^{83} \) \( + ( -4 + 76 \zeta_{6} ) q^{84} \) \( + 413 q^{85} \) \( + ( 520 - 520 \zeta_{6} ) q^{86} \) \( + 106 \zeta_{6} q^{87} \) \( + 280 \zeta_{6} q^{88} \) \( + ( 873 - 873 \zeta_{6} ) q^{89} \) \( + 364 q^{90} \) \( + ( -1254 + 1188 \zeta_{6} ) q^{91} \) \( -28 q^{92} \) \( + ( 75 - 75 \zeta_{6} ) q^{93} \) \( -342 \zeta_{6} q^{94} \) \( -959 \zeta_{6} q^{95} \) \( + ( -32 + 32 \zeta_{6} ) q^{96} \) \( -290 q^{97} \) \( + ( -646 - 74 \zeta_{6} ) q^{98} \) \( -910 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 20q^{7} \) \(\mathstrut -\mathstrut 16q^{8} \) \(\mathstrut +\mathstrut 26q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 20q^{7} \) \(\mathstrut -\mathstrut 16q^{8} \) \(\mathstrut +\mathstrut 26q^{9} \) \(\mathstrut +\mathstrut 14q^{10} \) \(\mathstrut -\mathstrut 35q^{11} \) \(\mathstrut +\mathstrut 4q^{12} \) \(\mathstrut +\mathstrut 132q^{13} \) \(\mathstrut +\mathstrut 34q^{14} \) \(\mathstrut -\mathstrut 14q^{15} \) \(\mathstrut -\mathstrut 16q^{16} \) \(\mathstrut -\mathstrut 59q^{17} \) \(\mathstrut -\mathstrut 52q^{18} \) \(\mathstrut -\mathstrut 137q^{19} \) \(\mathstrut +\mathstrut 56q^{20} \) \(\mathstrut -\mathstrut 37q^{21} \) \(\mathstrut -\mathstrut 140q^{22} \) \(\mathstrut +\mathstrut 7q^{23} \) \(\mathstrut -\mathstrut 8q^{24} \) \(\mathstrut +\mathstrut 76q^{25} \) \(\mathstrut +\mathstrut 132q^{26} \) \(\mathstrut +\mathstrut 106q^{27} \) \(\mathstrut +\mathstrut 148q^{28} \) \(\mathstrut +\mathstrut 212q^{29} \) \(\mathstrut -\mathstrut 14q^{30} \) \(\mathstrut -\mathstrut 75q^{31} \) \(\mathstrut +\mathstrut 32q^{32} \) \(\mathstrut +\mathstrut 35q^{33} \) \(\mathstrut -\mathstrut 236q^{34} \) \(\mathstrut -\mathstrut 119q^{35} \) \(\mathstrut -\mathstrut 208q^{36} \) \(\mathstrut -\mathstrut 11q^{37} \) \(\mathstrut +\mathstrut 274q^{38} \) \(\mathstrut +\mathstrut 66q^{39} \) \(\mathstrut +\mathstrut 56q^{40} \) \(\mathstrut -\mathstrut 996q^{41} \) \(\mathstrut -\mathstrut 40q^{42} \) \(\mathstrut +\mathstrut 520q^{43} \) \(\mathstrut -\mathstrut 140q^{44} \) \(\mathstrut +\mathstrut 182q^{45} \) \(\mathstrut -\mathstrut 14q^{46} \) \(\mathstrut +\mathstrut 171q^{47} \) \(\mathstrut -\mathstrut 32q^{48} \) \(\mathstrut -\mathstrut 286q^{49} \) \(\mathstrut +\mathstrut 304q^{50} \) \(\mathstrut +\mathstrut 59q^{51} \) \(\mathstrut -\mathstrut 264q^{52} \) \(\mathstrut +\mathstrut 417q^{53} \) \(\mathstrut +\mathstrut 106q^{54} \) \(\mathstrut +\mathstrut 490q^{55} \) \(\mathstrut +\mathstrut 160q^{56} \) \(\mathstrut -\mathstrut 274q^{57} \) \(\mathstrut +\mathstrut 212q^{58} \) \(\mathstrut +\mathstrut 17q^{59} \) \(\mathstrut +\mathstrut 28q^{60} \) \(\mathstrut -\mathstrut 51q^{61} \) \(\mathstrut -\mathstrut 300q^{62} \) \(\mathstrut +\mathstrut 442q^{63} \) \(\mathstrut +\mathstrut 128q^{64} \) \(\mathstrut -\mathstrut 462q^{65} \) \(\mathstrut -\mathstrut 70q^{66} \) \(\mathstrut -\mathstrut 439q^{67} \) \(\mathstrut -\mathstrut 236q^{68} \) \(\mathstrut +\mathstrut 14q^{69} \) \(\mathstrut -\mathstrut 518q^{70} \) \(\mathstrut -\mathstrut 1568q^{71} \) \(\mathstrut -\mathstrut 208q^{72} \) \(\mathstrut -\mathstrut 295q^{73} \) \(\mathstrut +\mathstrut 22q^{74} \) \(\mathstrut -\mathstrut 76q^{75} \) \(\mathstrut +\mathstrut 1096q^{76} \) \(\mathstrut +\mathstrut 1295q^{77} \) \(\mathstrut +\mathstrut 264q^{78} \) \(\mathstrut +\mathstrut 495q^{79} \) \(\mathstrut -\mathstrut 112q^{80} \) \(\mathstrut -\mathstrut 649q^{81} \) \(\mathstrut -\mathstrut 996q^{82} \) \(\mathstrut +\mathstrut 1864q^{83} \) \(\mathstrut +\mathstrut 68q^{84} \) \(\mathstrut +\mathstrut 826q^{85} \) \(\mathstrut +\mathstrut 520q^{86} \) \(\mathstrut +\mathstrut 106q^{87} \) \(\mathstrut +\mathstrut 280q^{88} \) \(\mathstrut +\mathstrut 873q^{89} \) \(\mathstrut +\mathstrut 728q^{90} \) \(\mathstrut -\mathstrut 1320q^{91} \) \(\mathstrut -\mathstrut 56q^{92} \) \(\mathstrut +\mathstrut 75q^{93} \) \(\mathstrut -\mathstrut 342q^{94} \) \(\mathstrut -\mathstrut 959q^{95} \) \(\mathstrut -\mathstrut 32q^{96} \) \(\mathstrut -\mathstrut 580q^{97} \) \(\mathstrut -\mathstrut 1366q^{98} \) \(\mathstrut -\mathstrut 1820q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 0.500000 + 0.866025i −2.00000 3.46410i −3.50000 + 6.06218i 2.00000 −10.0000 + 15.5885i −8.00000 13.0000 22.5167i 7.00000 + 12.1244i
11.1 1.00000 + 1.73205i 0.500000 0.866025i −2.00000 + 3.46410i −3.50000 6.06218i 2.00000 −10.0000 15.5885i −8.00000 13.0000 + 22.5167i 7.00000 12.1244i
\(n\): 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 \(T_{3}^{2} \) \(\mathstrut -\mathstrut T_{3} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{4}^{\mathrm{new}}(14, [\chi])\).