Properties

Label 289.4.a.a
Level $289$
Weight $4$
Character orbit 289.a
Self dual yes
Analytic conductor $17.052$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 289.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.0515519917\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3 q^{2} + 8 q^{3} + q^{4} - 6 q^{5} - 24 q^{6} + 28 q^{7} + 21 q^{8} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{2} + 8 q^{3} + q^{4} - 6 q^{5} - 24 q^{6} + 28 q^{7} + 21 q^{8} + 37 q^{9} + 18 q^{10} + 24 q^{11} + 8 q^{12} - 58 q^{13} - 84 q^{14} - 48 q^{15} - 71 q^{16} - 111 q^{18} + 116 q^{19} - 6 q^{20} + 224 q^{21} - 72 q^{22} + 60 q^{23} + 168 q^{24} - 89 q^{25} + 174 q^{26} + 80 q^{27} + 28 q^{28} - 30 q^{29} + 144 q^{30} + 172 q^{31} + 45 q^{32} + 192 q^{33} - 168 q^{35} + 37 q^{36} + 58 q^{37} - 348 q^{38} - 464 q^{39} - 126 q^{40} + 342 q^{41} - 672 q^{42} - 148 q^{43} + 24 q^{44} - 222 q^{45} - 180 q^{46} + 288 q^{47} - 568 q^{48} + 441 q^{49} + 267 q^{50} - 58 q^{52} + 318 q^{53} - 240 q^{54} - 144 q^{55} + 588 q^{56} + 928 q^{57} + 90 q^{58} + 252 q^{59} - 48 q^{60} - 110 q^{61} - 516 q^{62} + 1036 q^{63} + 433 q^{64} + 348 q^{65} - 576 q^{66} - 484 q^{67} + 480 q^{69} + 504 q^{70} + 708 q^{71} + 777 q^{72} - 362 q^{73} - 174 q^{74} - 712 q^{75} + 116 q^{76} + 672 q^{77} + 1392 q^{78} + 484 q^{79} + 426 q^{80} - 359 q^{81} - 1026 q^{82} + 756 q^{83} + 224 q^{84} + 444 q^{86} - 240 q^{87} + 504 q^{88} - 774 q^{89} + 666 q^{90} - 1624 q^{91} + 60 q^{92} + 1376 q^{93} - 864 q^{94} - 696 q^{95} + 360 q^{96} + 382 q^{97} - 1323 q^{98} + 888 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−3.00000 8.00000 1.00000 −6.00000 −24.0000 28.0000 21.0000 37.0000 18.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(1\)

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 289.4.a.a 1
17.b even 2 1 17.4.a.a 1
17.c even 4 2 289.4.b.a 2
51.c odd 2 1 153.4.a.d 1
68.d odd 2 1 272.4.a.d 1
85.c even 2 1 425.4.a.d 1
85.g odd 4 2 425.4.b.c 2
119.d odd 2 1 833.4.a.a 1
136.e odd 2 1 1088.4.a.a 1
136.h even 2 1 1088.4.a.l 1
187.b odd 2 1 2057.4.a.d 1
204.h even 2 1 2448.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.a 1 17.b even 2 1
153.4.a.d 1 51.c odd 2 1
272.4.a.d 1 68.d odd 2 1
289.4.a.a 1 1.a even 1 1 trivial
289.4.b.a 2 17.c even 4 2
425.4.a.d 1 85.c even 2 1
425.4.b.c 2 85.g odd 4 2
833.4.a.a 1 119.d odd 2 1
1088.4.a.a 1 136.e odd 2 1
1088.4.a.l 1 136.h even 2 1
2057.4.a.d 1 187.b odd 2 1
2448.4.a.f 1 204.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(289))\):

\( T_{2} + 3 \) Copy content Toggle raw display
\( T_{3} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 3 \) Copy content Toggle raw display
$3$ \( T - 8 \) Copy content Toggle raw display
$5$ \( T + 6 \) Copy content Toggle raw display
$7$ \( T - 28 \) Copy content Toggle raw display
$11$ \( T - 24 \) Copy content Toggle raw display
$13$ \( T + 58 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 116 \) Copy content Toggle raw display
$23$ \( T - 60 \) Copy content Toggle raw display
$29$ \( T + 30 \) Copy content Toggle raw display
$31$ \( T - 172 \) Copy content Toggle raw display
$37$ \( T - 58 \) Copy content Toggle raw display
$41$ \( T - 342 \) Copy content Toggle raw display
$43$ \( T + 148 \) Copy content Toggle raw display
$47$ \( T - 288 \) Copy content Toggle raw display
$53$ \( T - 318 \) Copy content Toggle raw display
$59$ \( T - 252 \) Copy content Toggle raw display
$61$ \( T + 110 \) Copy content Toggle raw display
$67$ \( T + 484 \) Copy content Toggle raw display
$71$ \( T - 708 \) Copy content Toggle raw display
$73$ \( T + 362 \) Copy content Toggle raw display
$79$ \( T - 484 \) Copy content Toggle raw display
$83$ \( T - 756 \) Copy content Toggle raw display
$89$ \( T + 774 \) Copy content Toggle raw display
$97$ \( T - 382 \) Copy content Toggle raw display
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