Properties

Label 289.4.a.c
Level $289$
Weight $4$
Character orbit 289.a
Self dual yes
Analytic conductor $17.052$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.2555057.1
Defining polynomial: \( x^{4} - x^{3} - 21x^{2} - 4x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 3) q^{4} + (2 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{5} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 + 11) q^{6} + ( - 5 \beta_{2} + \beta_1 + 10) q^{7} + (3 \beta_{3} + \beta_{2} + 2 \beta_1 + 15) q^{8} + (2 \beta_{3} - 3 \beta_{2} - 9 \beta_1 + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 3) q^{4} + (2 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{5} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 + 11) q^{6} + ( - 5 \beta_{2} + \beta_1 + 10) q^{7} + (3 \beta_{3} + \beta_{2} + 2 \beta_1 + 15) q^{8} + (2 \beta_{3} - 3 \beta_{2} - 9 \beta_1 + 5) q^{9} + (3 \beta_{3} - \beta_{2} + 9 \beta_1 - 3) q^{10} + ( - 4 \beta_{3} + 2 \beta_{2} + 1) q^{11} + (11 \beta_1 - 10) q^{12} + ( - 2 \beta_{3} + 5 \beta_{2} - 9 \beta_1 - 5) q^{13} + (\beta_{3} + 9 \beta_{2} + 21 \beta_1 + 11) q^{14} + (10 \beta_{2} + 8 \beta_1 + 9) q^{15} + (4 \beta_{2} + 19 \beta_1 + 10) q^{16} + ( - 5 \beta_{3} + 15 \beta_{2} + 10 \beta_1 - 91) q^{18} + (4 \beta_{3} + 11 \beta_{2} - 13 \beta_1 + 7) q^{19} + ( - \beta_{3} - 15 \beta_{2} + 28 \beta_1 + 79) q^{20} + ( - 4 \beta_{3} + 2 \beta_{2} + 38 \beta_1 - 99) q^{21} + ( - 8 \beta_{3} - 4 \beta_{2} - 19 \beta_1 - 16) q^{22} + ( - 4 \beta_{3} + \beta_{2} - 25 \beta_1 + 100) q^{23} + (3 \beta_{3} + 13 \beta_{2} + 17 \beta_1 + 33) q^{24} + ( - 18 \beta_{3} + 7 \beta_{2} + 25 \beta_1 + 82) q^{25} + ( - 13 \beta_{3} - \beta_{2} - 32 \beta_1 - 107) q^{26} + ( - 12 \beta_{3} + 4 \beta_{2} + 22 \beta_1 - 133) q^{27} + (23 \beta_{3} + \beta_{2} + 10 \beta_1 + 155) q^{28} + (6 \beta_{3} - \beta_{2} - 11 \beta_1 - 15) q^{29} + (8 \beta_{3} - 28 \beta_{2} - 3 \beta_1 + 88) q^{30} + ( - 8 \beta_{3} - 16 \beta_{2} - 2 \beta_1 + 93) q^{31} + ( - 5 \beta_{3} - 35 \beta_{2} + 5 \beta_1 + 89) q^{32} + (2 \beta_{3} - \beta_{2} - 27 \beta_1 + 31) q^{33} + (28 \beta_{3} - 51 \beta_{2} + 9 \beta_1 - 43) q^{35} + ( - 16 \beta_{3} - 16 \beta_{2} - 59 \beta_1 + 50) q^{36} + (4 \beta_{3} + 46 \beta_{2} + 38 \beta_1 + 108) q^{37} + ( - 5 \beta_{3} - 9 \beta_{2} - 12 \beta_1 - 127) q^{38} + ( - 4 \beta_{3} + 25 \beta_{2} - 25 \beta_1 + 2) q^{39} + (2 \beta_{3} + 10 \beta_{2} + 61 \beta_1 + 328) q^{40} + ( - 12 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 210) q^{41} + (30 \beta_{3} - 42 \beta_{2} - 81 \beta_1 + 402) q^{42} + (24 \beta_{3} + 32 \beta_{2} + 50 \beta_1 - 43) q^{43} + ( - 3 \beta_{3} + 11 \beta_{2} - 59 \beta_1 - 249) q^{44} + ( - 36 \beta_{3} - 32 \beta_{2} - 40 \beta_1 + 171) q^{45} + ( - 33 \beta_{3} + 23 \beta_{2} + 57 \beta_1 - 291) q^{46} + ( - 4 \beta_{3} + 42 \beta_{2} + 32 \beta_1 - 17) q^{47} + (23 \beta_{3} - 43 \beta_{2} - 52 \beta_1 + 279) q^{48} + (26 \beta_{3} + 19 \beta_{2} - 59 \beta_1 + 268) q^{49} + ( - 11 \beta_{3} - 39 \beta_{2} + 21 \beta_1 + 203) q^{50} + ( - 42 \beta_{3} - 6 \beta_{2} - 117 \beta_1 - 364) q^{52} + ( - 2 \beta_{3} - 65 \beta_{2} + 93 \beta_1 + 5) q^{53} + ( - 2 \beta_{3} - 30 \beta_{2} - 167 \beta_1 + 194) q^{54} + (44 \beta_{3} + 45 \beta_{2} - 85 \beta_1 - 308) q^{55} + (48 \beta_{3} - 84 \beta_{2} + 87 \beta_1 + 114) q^{56} + ( - 2 \beta_{3} + 61 \beta_{2} - 17 \beta_1 + 78) q^{57} + (\beta_{3} + 13 \beta_{2} - 97) q^{58} + (12 \beta_{3} + 64 \beta_{2} + 56 \beta_1 + 222) q^{59} + (13 \beta_{3} - 21 \beta_{2} + 109 \beta_1 - 73) q^{60} + (60 \beta_{3} - 46 \beta_{2} - 2 \beta_1 + 139) q^{61} + ( - 18 \beta_{3} + 34 \beta_{2} + 91 \beta_1 - 54) q^{62} + (40 \beta_{3} - 80 \beta_{2} - 230 \beta_1 + 279) q^{63} + ( - 5 \beta_{3} + 33 \beta_{2} - 8 \beta_1 - 45) q^{64} + ( - 20 \beta_{3} + 74 \beta_{2} - 126 \beta_1 - 85) q^{65} + ( - 23 \beta_{3} + 29 \beta_{2} + 14 \beta_1 - 289) q^{66} + (16 \beta_{3} + 10 \beta_{2} - 158 \beta_1 - 176) q^{67} + ( - 24 \beta_{3} + 172 \beta_{2} + 128 \beta_1 - 363) q^{69} + (65 \beta_{3} + 93 \beta_{2} + 180 \beta_1 + 211) q^{70} + ( - 4 \beta_{3} - 44 \beta_{2} - 36 \beta_1 + 298) q^{71} + ( - 51 \beta_{3} - 29 \beta_{2} - 121 \beta_1 + 15) q^{72} + ( - 16 \beta_{3} + 62 \beta_{2} + 130 \beta_1 + 83) q^{73} + (46 \beta_{3} - 130 \beta_{2} + 70 \beta_1 + 434) q^{74} + (32 \beta_{3} - 4 \beta_{2} - 82 \beta_1 + 297) q^{75} + ( - 54 \beta_{3} - 58 \beta_{2} - 37 \beta_1 - 208) q^{76} + ( - 78 \beta_{3} - 9 \beta_{2} + 21 \beta_1 - 246) q^{77} + ( - 33 \beta_{3} - 25 \beta_{2} - 89 \beta_1 - 291) q^{78} + (76 \beta_{3} + 52 \beta_{2} - 212 \beta_1 + 294) q^{79} + (73 \beta_{3} + 39 \beta_{2} + 153 \beta_1 + 47) q^{80} + ( - 28 \beta_{3} - 126 \beta_{2} - 6 \beta_1 + 296) q^{81} + ( - 32 \beta_{3} + 16 \beta_{2} + 162 \beta_1 - 136) q^{82} + ( - 84 \beta_{3} - 3 \beta_{2} + 37 \beta_1 + 183) q^{83} + (11 \beta_{3} + 149 \beta_{2} + 221 \beta_1 + 21) q^{84} + (98 \beta_{3} - 114 \beta_{2} + 39 \beta_1 + 646) q^{86} + ( - 12 \beta_{3} + 23 \beta_{2} + 37 \beta_1 - 114) q^{87} + ( - \beta_{3} + 69 \beta_{2} - 190 \beta_1 - 533) q^{88} + ( - 44 \beta_{3} - 112 \beta_{2} - 20 \beta_1 + 218) q^{89} + ( - 112 \beta_{3} + 104 \beta_{2} + 51 \beta_1 - 584) q^{90} + ( - 68 \beta_{3} - 106 \beta_{2} - 112 \beta_1 - 677) q^{91} + (23 \beta_{3} - 111 \beta_{2} - 212 \beta_1 - 305) q^{92} + ( - 18 \beta_{3} + 75 \beta_{2} + 161 \beta_1 - 451) q^{93} + (24 \beta_{3} - 116 \beta_{2} - 85 \beta_1 + 336) q^{94} + ( - 80 \beta_{3} + 132 \beta_{2} - 66 \beta_1 + 587) q^{95} + ( - 30 \beta_{3} + 34 \beta_{2} + 269 \beta_1 - 744) q^{96} + ( - 162 \beta_{3} + 149 \beta_{2} + 67 \beta_1 - 224) q^{97} + ( - 7 \beta_{3} + 21 \beta_{2} + 275 \beta_1 - 545) q^{98} + (80 \beta_{3} + 59 \beta_{2} + 99 \beta_1 - 371) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 2 q^{3} + 11 q^{4} + 14 q^{5} + 38 q^{6} + 36 q^{7} + 60 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - 2 q^{3} + 11 q^{4} + 14 q^{5} + 38 q^{6} + 36 q^{7} + 60 q^{8} + 6 q^{9} - 7 q^{10} + 10 q^{11} - 29 q^{12} - 22 q^{13} + 73 q^{14} + 54 q^{15} + 63 q^{16} - 334 q^{18} + 22 q^{19} + 330 q^{20} - 352 q^{21} - 79 q^{22} + 380 q^{23} + 159 q^{24} + 378 q^{25} - 448 q^{26} - 494 q^{27} + 608 q^{28} - 78 q^{29} + 313 q^{30} + 362 q^{31} + 331 q^{32} + 94 q^{33} - 242 q^{35} + 141 q^{36} + 512 q^{37} - 524 q^{38} + 12 q^{39} + 1381 q^{40} + 840 q^{41} + 1455 q^{42} - 114 q^{43} - 1041 q^{44} + 648 q^{45} - 1051 q^{46} + 10 q^{47} + 998 q^{48} + 1006 q^{49} + 805 q^{50} - 1537 q^{52} + 50 q^{53} + 581 q^{54} - 1316 q^{55} + 411 q^{56} + 358 q^{57} - 376 q^{58} + 996 q^{59} - 217 q^{60} + 448 q^{61} - 73 q^{62} + 766 q^{63} - 150 q^{64} - 372 q^{65} - 1090 q^{66} - 868 q^{67} - 1128 q^{69} + 1052 q^{70} + 1116 q^{71} - 39 q^{72} + 540 q^{73} + 1630 q^{74} + 1070 q^{75} - 873 q^{76} - 894 q^{77} - 1245 q^{78} + 940 q^{79} + 307 q^{80} + 1080 q^{81} - 334 q^{82} + 850 q^{83} + 443 q^{84} + 2411 q^{86} - 384 q^{87} - 2252 q^{88} + 784 q^{89} - 2069 q^{90} - 2858 q^{91} - 1566 q^{92} - 1550 q^{93} + 1119 q^{94} + 2494 q^{95} - 2643 q^{96} - 518 q^{97} - 1877 q^{98} - 1406 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 21x^{2} - 4x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 3\nu^{2} - 15\nu + 18 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + \nu^{2} - 19\nu - 26 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + \beta_{2} + 18\beta _1 + 15 \) 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
−3.68488
−1.58184
1.22501
5.04171
−3.68488 −9.05894 5.57832 7.08909 33.3811 28.1854 8.92359 55.0643 −26.1224
1.2 −1.58184 4.98387 −5.49778 14.4471 −7.88368 −29.4104 21.3513 −2.16104 −22.8530
1.3 1.22501 −0.534684 −6.49935 −20.9528 −0.654993 15.0235 −17.7618 −26.7141 −25.6674
1.4 5.04171 2.60975 17.4188 13.4166 13.1576 22.2015 47.4869 −20.1892 67.6428
\(n\): 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.c 4
17.b even 2 1 289.4.a.d yes 4
17.c even 4 2 289.4.b.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.4.a.c 4 1.a even 1 1 trivial
289.4.a.d yes 4 17.b even 2 1
289.4.b.d 8 17.c even 4 2

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{3} - 21T_{2}^{2} - 4T_{2} + 36 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{3} - 55T_{3}^{2} + 88T_{3} + 63 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 21 T^{2} - 4 T + 36 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} - 55 T^{2} + 88 T + 63 \) Copy content Toggle raw display
$5$ \( T^{4} - 14 T^{3} - 341 T^{2} + \cdots - 28791 \) Copy content Toggle raw display
$7$ \( T^{4} - 36 T^{3} - 541 T^{2} + \cdots - 276489 \) Copy content Toggle raw display
$11$ \( T^{4} - 10 T^{3} - 1756 T^{2} + \cdots + 316467 \) Copy content Toggle raw display
$13$ \( T^{4} + 22 T^{3} - 3509 T^{2} + \cdots - 26579 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 22 T^{3} - 9191 T^{2} + \cdots + 4371507 \) Copy content Toggle raw display
$23$ \( T^{4} - 380 T^{3} + \cdots - 176752917 \) Copy content Toggle raw display
$29$ \( T^{4} + 78 T^{3} - 2705 T^{2} + \cdots - 107811 \) Copy content Toggle raw display
$31$ \( T^{4} - 362 T^{3} + \cdots - 38411429 \) Copy content Toggle raw display
$37$ \( T^{4} - 512 T^{3} + \cdots - 1758989168 \) Copy content Toggle raw display
$41$ \( T^{4} - 840 T^{3} + \cdots + 1180838736 \) Copy content Toggle raw display
$43$ \( T^{4} + 114 T^{3} + \cdots + 4351627359 \) Copy content Toggle raw display
$47$ \( T^{4} - 10 T^{3} + \cdots - 153320769 \) Copy content Toggle raw display
$53$ \( T^{4} - 50 T^{3} + \cdots + 3826783197 \) Copy content Toggle raw display
$59$ \( T^{4} - 996 T^{3} + \cdots - 7997430672 \) Copy content Toggle raw display
$61$ \( T^{4} - 448 T^{3} + \cdots + 26730644633 \) Copy content Toggle raw display
$67$ \( T^{4} + 868 T^{3} + \cdots + 30297331184 \) Copy content Toggle raw display
$71$ \( T^{4} - 1116 T^{3} + \cdots + 589600944 \) Copy content Toggle raw display
$73$ \( T^{4} - 540 T^{3} + \cdots - 46400632803 \) Copy content Toggle raw display
$79$ \( T^{4} - 940 T^{3} + \cdots + 229066578288 \) Copy content Toggle raw display
$83$ \( T^{4} - 850 T^{3} + \cdots + 1637812071 \) Copy content Toggle raw display
$89$ \( T^{4} - 784 T^{3} + \cdots + 24704450064 \) Copy content Toggle raw display
$97$ \( T^{4} + 518 T^{3} + \cdots + 2200880766749 \) Copy content Toggle raw display
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