Properties

Label 14.8.a.a
Level 14
Weight 8
Character orbit 14.a
Self dual yes
Analytic conductor 4.373
Analytic rank 0
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: \(0\)
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

\(f(q)\) \(=\) \( q - 8q^{2} - 82q^{3} + 64q^{4} + 448q^{5} + 656q^{6} - 343q^{7} - 512q^{8} + 4537q^{9} + O(q^{10}) \) \( q - 8q^{2} - 82q^{3} + 64q^{4} + 448q^{5} + 656q^{6} - 343q^{7} - 512q^{8} + 4537q^{9} - 3584q^{10} + 2408q^{11} - 5248q^{12} + 7116q^{13} + 2744q^{14} - 36736q^{15} + 4096q^{16} + 2486q^{17} - 36296q^{18} + 36482q^{19} + 28672q^{20} + 28126q^{21} - 19264q^{22} - 12880q^{23} + 41984q^{24} + 122579q^{25} - 56928q^{26} - 192700q^{27} - 21952q^{28} - 88094q^{29} + 293888q^{30} + 282636q^{31} - 32768q^{32} - 197456q^{33} - 19888q^{34} - 153664q^{35} + 290368q^{36} - 214534q^{37} - 291856q^{38} - 583512q^{39} - 229376q^{40} - 140874q^{41} - 225008q^{42} + 36464q^{43} + 154112q^{44} + 2032576q^{45} + 103040q^{46} + 716868q^{47} - 335872q^{48} + 117649q^{49} - 980632q^{50} - 203852q^{51} + 455424q^{52} - 56946q^{53} + 1541600q^{54} + 1078784q^{55} + 175616q^{56} - 2991524q^{57} + 704752q^{58} - 2149862q^{59} - 2351104q^{60} + 3084360q^{61} - 2261088q^{62} - 1556191q^{63} + 262144q^{64} + 3187968q^{65} + 1579648q^{66} - 3034364q^{67} + 159104q^{68} + 1056160q^{69} + 1229312q^{70} - 106624q^{71} - 2322944q^{72} + 988930q^{73} + 1716272q^{74} - 10051478q^{75} + 2334848q^{76} - 825944q^{77} + 4668096q^{78} + 3415896q^{79} + 1835008q^{80} + 5878981q^{81} + 1126992q^{82} - 15142q^{83} + 1800064q^{84} + 1113728q^{85} - 291712q^{86} + 7223708q^{87} - 1232896q^{88} + 174810q^{89} - 16260608q^{90} - 2440788q^{91} - 824320q^{92} - 23176152q^{93} - 5734944q^{94} + 16343936q^{95} + 2686976q^{96} + 13506790q^{97} - 941192q^{98} + 10925096q^{99} + O(q^{100}) \)

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 −82.0000 64.0000 448.000 656.000 −343.000 −512.000 4537.00 −3584.00
\(n\): 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.a 1
3.b odd 2 1 126.8.a.d 1
4.b odd 2 1 112.8.a.e 1
5.b even 2 1 350.8.a.h 1
5.c odd 4 2 350.8.c.d 2
7.b odd 2 1 98.8.a.b 1
7.c even 3 2 98.8.c.f 2
7.d odd 6 2 98.8.c.c 2
8.b even 2 1 448.8.a.j 1
8.d odd 2 1 448.8.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.8.a.a 1 1.a even 1 1 trivial
98.8.a.b 1 7.b odd 2 1
98.8.c.c 2 7.d odd 6 2
98.8.c.f 2 7.c even 3 2
112.8.a.e 1 4.b odd 2 1
126.8.a.d 1 3.b odd 2 1
350.8.a.h 1 5.b even 2 1
350.8.c.d 2 5.c odd 4 2
448.8.a.a 1 8.d odd 2 1
448.8.a.j 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 \( T_{3} + 82 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(14))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 8 T \)
$3$ \( 1 + 82 T + 2187 T^{2} \)
$5$ \( 1 - 448 T + 78125 T^{2} \)
$7$ \( 1 + 343 T \)
$11$ \( 1 - 2408 T + 19487171 T^{2} \)
$13$ \( 1 - 7116 T + 62748517 T^{2} \)
$17$ \( 1 - 2486 T + 410338673 T^{2} \)
$19$ \( 1 - 36482 T + 893871739 T^{2} \)
$23$ \( 1 + 12880 T + 3404825447 T^{2} \)
$29$ \( 1 + 88094 T + 17249876309 T^{2} \)
$31$ \( 1 - 282636 T + 27512614111 T^{2} \)
$37$ \( 1 + 214534 T + 94931877133 T^{2} \)
$41$ \( 1 + 140874 T + 194754273881 T^{2} \)
$43$ \( 1 - 36464 T + 271818611107 T^{2} \)
$47$ \( 1 - 716868 T + 506623120463 T^{2} \)
$53$ \( 1 + 56946 T + 1174711139837 T^{2} \)
$59$ \( 1 + 2149862 T + 2488651484819 T^{2} \)
$61$ \( 1 - 3084360 T + 3142742836021 T^{2} \)
$67$ \( 1 + 3034364 T + 6060711605323 T^{2} \)
$71$ \( 1 + 106624 T + 9095120158391 T^{2} \)
$73$ \( 1 - 988930 T + 11047398519097 T^{2} \)
$79$ \( 1 - 3415896 T + 19203908986159 T^{2} \)
$83$ \( 1 + 15142 T + 27136050989627 T^{2} \)
$89$ \( 1 - 174810 T + 44231334895529 T^{2} \)
$97$ \( 1 - 13506790 T + 80798284478113 T^{2} \)
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