Properties

Label 13.8.a.a
Level 13
Weight 8
Character orbit 13.a
Self dual yes
Analytic conductor 4.061
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 13.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.06100533129\)
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

\(f(q)\) \(=\) \( q + 10q^{2} - 73q^{3} - 28q^{4} - 295q^{5} - 730q^{6} + 1373q^{7} - 1560q^{8} + 3142q^{9} + O(q^{10}) \) \( q + 10q^{2} - 73q^{3} - 28q^{4} - 295q^{5} - 730q^{6} + 1373q^{7} - 1560q^{8} + 3142q^{9} - 2950q^{10} - 7646q^{11} + 2044q^{12} + 2197q^{13} + 13730q^{14} + 21535q^{15} - 12016q^{16} - 4147q^{17} + 31420q^{18} - 3186q^{19} + 8260q^{20} - 100229q^{21} - 76460q^{22} - 17784q^{23} + 113880q^{24} + 8900q^{25} + 21970q^{26} - 69715q^{27} - 38444q^{28} - 93322q^{29} + 215350q^{30} - 124484q^{31} + 79520q^{32} + 558158q^{33} - 41470q^{34} - 405035q^{35} - 87976q^{36} + 273661q^{37} - 31860q^{38} - 160381q^{39} + 460200q^{40} + 585816q^{41} - 1002290q^{42} - 533559q^{43} + 214088q^{44} - 926890q^{45} - 177840q^{46} - 530055q^{47} + 877168q^{48} + 1061586q^{49} + 89000q^{50} + 302731q^{51} - 61516q^{52} - 615288q^{53} - 697150q^{54} + 2255570q^{55} - 2141880q^{56} + 232578q^{57} - 933220q^{58} - 392514q^{59} - 602980q^{60} + 1878064q^{61} - 1244840q^{62} + 4313966q^{63} + 2333248q^{64} - 648115q^{65} + 5581580q^{66} - 3971438q^{67} + 116116q^{68} + 1298232q^{69} - 4050350q^{70} - 3746601q^{71} - 4901520q^{72} + 2485802q^{73} + 2736610q^{74} - 649700q^{75} + 89208q^{76} - 10497958q^{77} - 1603810q^{78} - 1264456q^{79} + 3544720q^{80} - 1782359q^{81} + 5858160q^{82} + 434308q^{83} + 2806412q^{84} + 1223365q^{85} - 5335590q^{86} + 6812506q^{87} + 11927760q^{88} + 5830810q^{89} - 9268900q^{90} + 3016481q^{91} + 497952q^{92} + 9087332q^{93} - 5300550q^{94} + 939870q^{95} - 5804960q^{96} - 2045330q^{97} + 10615860q^{98} - 24023732q^{99} + O(q^{100}) \)

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
10.0000 −73.0000 −28.0000 −295.000 −730.000 1373.00 −1560.00 3142.00 −2950.00
\(n\): 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 13.8.a.a 1
3.b odd 2 1 117.8.a.a 1
4.b odd 2 1 208.8.a.d 1
5.b even 2 1 325.8.a.a 1
13.b even 2 1 169.8.a.a 1
13.d odd 4 2 169.8.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.a.a 1 1.a even 1 1 trivial
117.8.a.a 1 3.b odd 2 1
169.8.a.a 1 13.b even 2 1
169.8.b.a 2 13.d odd 4 2
208.8.a.d 1 4.b odd 2 1
325.8.a.a 1 5.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 10 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(13))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 10 T + 128 T^{2} \)
$3$ \( 1 + 73 T + 2187 T^{2} \)
$5$ \( 1 + 295 T + 78125 T^{2} \)
$7$ \( 1 - 1373 T + 823543 T^{2} \)
$11$ \( 1 + 7646 T + 19487171 T^{2} \)
$13$ \( 1 - 2197 T \)
$17$ \( 1 + 4147 T + 410338673 T^{2} \)
$19$ \( 1 + 3186 T + 893871739 T^{2} \)
$23$ \( 1 + 17784 T + 3404825447 T^{2} \)
$29$ \( 1 + 93322 T + 17249876309 T^{2} \)
$31$ \( 1 + 124484 T + 27512614111 T^{2} \)
$37$ \( 1 - 273661 T + 94931877133 T^{2} \)
$41$ \( 1 - 585816 T + 194754273881 T^{2} \)
$43$ \( 1 + 533559 T + 271818611107 T^{2} \)
$47$ \( 1 + 530055 T + 506623120463 T^{2} \)
$53$ \( 1 + 615288 T + 1174711139837 T^{2} \)
$59$ \( 1 + 392514 T + 2488651484819 T^{2} \)
$61$ \( 1 - 1878064 T + 3142742836021 T^{2} \)
$67$ \( 1 + 3971438 T + 6060711605323 T^{2} \)
$71$ \( 1 + 3746601 T + 9095120158391 T^{2} \)
$73$ \( 1 - 2485802 T + 11047398519097 T^{2} \)
$79$ \( 1 + 1264456 T + 19203908986159 T^{2} \)
$83$ \( 1 - 434308 T + 27136050989627 T^{2} \)
$89$ \( 1 - 5830810 T + 44231334895529 T^{2} \)
$97$ \( 1 + 2045330 T + 80798284478113 T^{2} \)
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