Properties

Label 14.8.a.b
Level 14
Weight 8
Character orbit 14.a
Self dual Yes
Analytic conductor 4.373
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 14 = 2 \cdot 7 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 14.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(4.37339035678\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 8q^{2} \) \(\mathstrut -\mathstrut 66q^{3} \) \(\mathstrut +\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 400q^{5} \) \(\mathstrut -\mathstrut 528q^{6} \) \(\mathstrut -\mathstrut 343q^{7} \) \(\mathstrut +\mathstrut 512q^{8} \) \(\mathstrut +\mathstrut 2169q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 8q^{2} \) \(\mathstrut -\mathstrut 66q^{3} \) \(\mathstrut +\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 400q^{5} \) \(\mathstrut -\mathstrut 528q^{6} \) \(\mathstrut -\mathstrut 343q^{7} \) \(\mathstrut +\mathstrut 512q^{8} \) \(\mathstrut +\mathstrut 2169q^{9} \) \(\mathstrut -\mathstrut 3200q^{10} \) \(\mathstrut +\mathstrut 40q^{11} \) \(\mathstrut -\mathstrut 4224q^{12} \) \(\mathstrut -\mathstrut 4452q^{13} \) \(\mathstrut -\mathstrut 2744q^{14} \) \(\mathstrut +\mathstrut 26400q^{15} \) \(\mathstrut +\mathstrut 4096q^{16} \) \(\mathstrut +\mathstrut 36502q^{17} \) \(\mathstrut +\mathstrut 17352q^{18} \) \(\mathstrut -\mathstrut 46222q^{19} \) \(\mathstrut -\mathstrut 25600q^{20} \) \(\mathstrut +\mathstrut 22638q^{21} \) \(\mathstrut +\mathstrut 320q^{22} \) \(\mathstrut -\mathstrut 105200q^{23} \) \(\mathstrut -\mathstrut 33792q^{24} \) \(\mathstrut +\mathstrut 81875q^{25} \) \(\mathstrut -\mathstrut 35616q^{26} \) \(\mathstrut +\mathstrut 1188q^{27} \) \(\mathstrut -\mathstrut 21952q^{28} \) \(\mathstrut -\mathstrut 126334q^{29} \) \(\mathstrut +\mathstrut 211200q^{30} \) \(\mathstrut -\mathstrut 170964q^{31} \) \(\mathstrut +\mathstrut 32768q^{32} \) \(\mathstrut -\mathstrut 2640q^{33} \) \(\mathstrut +\mathstrut 292016q^{34} \) \(\mathstrut +\mathstrut 137200q^{35} \) \(\mathstrut +\mathstrut 138816q^{36} \) \(\mathstrut +\mathstrut 20954q^{37} \) \(\mathstrut -\mathstrut 369776q^{38} \) \(\mathstrut +\mathstrut 293832q^{39} \) \(\mathstrut -\mathstrut 204800q^{40} \) \(\mathstrut +\mathstrut 318486q^{41} \) \(\mathstrut +\mathstrut 181104q^{42} \) \(\mathstrut +\mathstrut 77744q^{43} \) \(\mathstrut +\mathstrut 2560q^{44} \) \(\mathstrut -\mathstrut 867600q^{45} \) \(\mathstrut -\mathstrut 841600q^{46} \) \(\mathstrut +\mathstrut 703716q^{47} \) \(\mathstrut -\mathstrut 270336q^{48} \) \(\mathstrut +\mathstrut 117649q^{49} \) \(\mathstrut +\mathstrut 655000q^{50} \) \(\mathstrut -\mathstrut 2409132q^{51} \) \(\mathstrut -\mathstrut 284928q^{52} \) \(\mathstrut +\mathstrut 1603278q^{53} \) \(\mathstrut +\mathstrut 9504q^{54} \) \(\mathstrut -\mathstrut 16000q^{55} \) \(\mathstrut -\mathstrut 175616q^{56} \) \(\mathstrut +\mathstrut 3050652q^{57} \) \(\mathstrut -\mathstrut 1010672q^{58} \) \(\mathstrut -\mathstrut 1171894q^{59} \) \(\mathstrut +\mathstrut 1689600q^{60} \) \(\mathstrut -\mathstrut 2068872q^{61} \) \(\mathstrut -\mathstrut 1367712q^{62} \) \(\mathstrut -\mathstrut 743967q^{63} \) \(\mathstrut +\mathstrut 262144q^{64} \) \(\mathstrut +\mathstrut 1780800q^{65} \) \(\mathstrut -\mathstrut 21120q^{66} \) \(\mathstrut -\mathstrut 994268q^{67} \) \(\mathstrut +\mathstrut 2336128q^{68} \) \(\mathstrut +\mathstrut 6943200q^{69} \) \(\mathstrut +\mathstrut 1097600q^{70} \) \(\mathstrut +\mathstrut 33280q^{71} \) \(\mathstrut +\mathstrut 1110528q^{72} \) \(\mathstrut -\mathstrut 2971454q^{73} \) \(\mathstrut +\mathstrut 167632q^{74} \) \(\mathstrut -\mathstrut 5403750q^{75} \) \(\mathstrut -\mathstrut 2958208q^{76} \) \(\mathstrut -\mathstrut 13720q^{77} \) \(\mathstrut +\mathstrut 2350656q^{78} \) \(\mathstrut -\mathstrut 2376168q^{79} \) \(\mathstrut -\mathstrut 1638400q^{80} \) \(\mathstrut -\mathstrut 4822011q^{81} \) \(\mathstrut +\mathstrut 2547888q^{82} \) \(\mathstrut -\mathstrut 2122358q^{83} \) \(\mathstrut +\mathstrut 1448832q^{84} \) \(\mathstrut -\mathstrut 14600800q^{85} \) \(\mathstrut +\mathstrut 621952q^{86} \) \(\mathstrut +\mathstrut 8338044q^{87} \) \(\mathstrut +\mathstrut 20480q^{88} \) \(\mathstrut +\mathstrut 6920346q^{89} \) \(\mathstrut -\mathstrut 6940800q^{90} \) \(\mathstrut +\mathstrut 1527036q^{91} \) \(\mathstrut -\mathstrut 6732800q^{92} \) \(\mathstrut +\mathstrut 11283624q^{93} \) \(\mathstrut +\mathstrut 5629728q^{94} \) \(\mathstrut +\mathstrut 18488800q^{95} \) \(\mathstrut -\mathstrut 2162688q^{96} \) \(\mathstrut +\mathstrut 4952710q^{97} \) \(\mathstrut +\mathstrut 941192q^{98} \) \(\mathstrut +\mathstrut 86760q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
8.00000 −66.0000 64.0000 −400.000 −528.000 −343.000 512.000 2169.00 −3200.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) \(\mathstrut +\mathstrut 66 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(14))\).