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

Related objects

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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: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.8.a.b 1
3.b odd 2 1 126.8.a.c 1
4.b odd 2 1 112.8.a.d 1
5.b even 2 1 350.8.a.d 1
5.c odd 4 2 350.8.c.b 2
7.b odd 2 1 98.8.a.c 1
7.c even 3 2 98.8.c.b 2
7.d odd 6 2 98.8.c.a 2
8.b even 2 1 448.8.a.i 1
8.d odd 2 1 448.8.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.8.a.b 1 1.a even 1 1 trivial
98.8.a.c 1 7.b odd 2 1
98.8.c.a 2 7.d odd 6 2
98.8.c.b 2 7.c even 3 2
112.8.a.d 1 4.b odd 2 1
126.8.a.c 1 3.b odd 2 1
350.8.a.d 1 5.b even 2 1
350.8.c.b 2 5.c odd 4 2
448.8.a.b 1 8.d odd 2 1
448.8.a.i 1 8.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 8 T \)
$3$ \( 1 + 66 T + 2187 T^{2} \)
$5$ \( 1 + 400 T + 78125 T^{2} \)
$7$ \( 1 + 343 T \)
$11$ \( 1 - 40 T + 19487171 T^{2} \)
$13$ \( 1 + 4452 T + 62748517 T^{2} \)
$17$ \( 1 - 36502 T + 410338673 T^{2} \)
$19$ \( 1 + 46222 T + 893871739 T^{2} \)
$23$ \( 1 + 105200 T + 3404825447 T^{2} \)
$29$ \( 1 + 126334 T + 17249876309 T^{2} \)
$31$ \( 1 + 170964 T + 27512614111 T^{2} \)
$37$ \( 1 - 20954 T + 94931877133 T^{2} \)
$41$ \( 1 - 318486 T + 194754273881 T^{2} \)
$43$ \( 1 - 77744 T + 271818611107 T^{2} \)
$47$ \( 1 - 703716 T + 506623120463 T^{2} \)
$53$ \( 1 - 1603278 T + 1174711139837 T^{2} \)
$59$ \( 1 + 1171894 T + 2488651484819 T^{2} \)
$61$ \( 1 + 2068872 T + 3142742836021 T^{2} \)
$67$ \( 1 + 994268 T + 6060711605323 T^{2} \)
$71$ \( 1 - 33280 T + 9095120158391 T^{2} \)
$73$ \( 1 + 2971454 T + 11047398519097 T^{2} \)
$79$ \( 1 + 2376168 T + 19203908986159 T^{2} \)
$83$ \( 1 + 2122358 T + 27136050989627 T^{2} \)
$89$ \( 1 - 6920346 T + 44231334895529 T^{2} \)
$97$ \( 1 - 4952710 T + 80798284478113 T^{2} \)
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