Properties

Label 289.4.a.h
Level $289$
Weight $4$
Character orbit 289.a
Self dual yes
Analytic conductor $17.052$
Analytic rank $1$
Dimension $12$
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 72 x^{10} - 17 x^{9} + 1872 x^{8} + 627 x^{7} - 20922 x^{6} - 5163 x^{5} + 93255 x^{4} - 4607 x^{3} - 117822 x^{2} + 21960 x + 29352 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 17^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{5} - 2) q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{9} - 2) q^{5} + (\beta_{9} - \beta_{7} + 3 \beta_1) q^{6} + ( - \beta_{11} + \beta_{10} + \beta_{6} + \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 2) q^{7} + ( - \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 3) q^{8} + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} - 4 \beta_{5} - \beta_{2} + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{5} - 2) q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{9} - 2) q^{5} + (\beta_{9} - \beta_{7} + 3 \beta_1) q^{6} + ( - \beta_{11} + \beta_{10} + \beta_{6} + \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 2) q^{7} + ( - \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 3) q^{8} + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} - 4 \beta_{5} - \beta_{2} + 10) q^{9} + (\beta_{11} - 2 \beta_{10} - \beta_{9} + \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} + \cdots - 3) q^{10}+ \cdots + ( - 28 \beta_{11} - 31 \beta_{10} + 9 \beta_{9} - 24 \beta_{8} - 21 \beta_{7} + \cdots + 31) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{3} + 48 q^{4} - 30 q^{5} + 9 q^{6} - 24 q^{7} - 51 q^{8} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 18 q^{3} + 48 q^{4} - 30 q^{5} + 9 q^{6} - 24 q^{7} - 51 q^{8} + 108 q^{9} - 60 q^{10} - 162 q^{11} - 216 q^{12} - 72 q^{13} - 267 q^{14} - 138 q^{15} + 192 q^{16} + 189 q^{18} - 66 q^{19} - 129 q^{20} + 246 q^{21} - 456 q^{22} - 282 q^{23} - 72 q^{24} + 444 q^{25} + 528 q^{26} - 1092 q^{27} - 120 q^{28} - 648 q^{29} - 1890 q^{30} - 504 q^{31} - 1353 q^{32} + 966 q^{33} - 66 q^{35} - 663 q^{36} + 30 q^{37} - 60 q^{38} - 1758 q^{39} + 450 q^{40} - 318 q^{41} + 804 q^{42} + 486 q^{43} - 2448 q^{44} - 486 q^{45} - 1617 q^{46} - 888 q^{47} - 1257 q^{48} - 570 q^{49} + 435 q^{50} + 225 q^{52} + 1026 q^{53} - 933 q^{54} + 972 q^{55} - 2661 q^{56} + 156 q^{57} - 201 q^{58} - 792 q^{59} + 1458 q^{60} - 1212 q^{61} - 2817 q^{62} - 2112 q^{63} - 1857 q^{64} - 2742 q^{65} - 594 q^{66} + 624 q^{67} - 1506 q^{69} - 1650 q^{70} - 2802 q^{71} + 1455 q^{72} - 726 q^{73} - 270 q^{74} + 264 q^{75} + 675 q^{76} - 1008 q^{77} + 3090 q^{78} + 444 q^{79} + 1143 q^{80} + 2520 q^{81} + 4950 q^{82} + 672 q^{83} - 777 q^{84} + 2778 q^{86} + 726 q^{87} + 3750 q^{88} - 906 q^{89} + 7755 q^{90} - 2280 q^{91} - 87 q^{92} + 132 q^{93} + 735 q^{94} - 966 q^{95} + 5046 q^{96} + 3246 q^{97} + 1911 q^{98} + 282 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 72 x^{10} - 17 x^{9} + 1872 x^{8} + 627 x^{7} - 20922 x^{6} - 5163 x^{5} + 93255 x^{4} - 4607 x^{3} - 117822 x^{2} + 21960 x + 29352 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 46055 \nu^{11} - 314662 \nu^{10} - 2002585 \nu^{9} + 15911043 \nu^{8} + 23350245 \nu^{7} - 270879424 \nu^{6} + 23301493 \nu^{5} + 1702105407 \nu^{4} + \cdots + 668244208 ) / 47215936 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 52035 \nu^{11} - 153738 \nu^{10} - 2946597 \nu^{9} + 6477251 \nu^{8} + 58746693 \nu^{7} - 77589892 \nu^{6} - 467836487 \nu^{5} + 153419951 \nu^{4} + \cdots - 475178864 ) / 47215936 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 66901 \nu^{11} + 325629 \nu^{10} + 3458675 \nu^{9} - 16175630 \nu^{8} - 59915226 \nu^{7} + 269286443 \nu^{6} + 366397543 \nu^{5} + \cdots - 950170632 ) / 47215936 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 86489 \nu^{11} - 491090 \nu^{10} - 4158023 \nu^{9} + 24658509 \nu^{8} + 61818003 \nu^{7} - 415569128 \nu^{6} - 215264909 \nu^{5} + 2593563441 \nu^{4} + \cdots + 1294454224 ) / 47215936 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 155568 \nu^{11} - 543997 \nu^{10} - 8625496 \nu^{9} + 25329267 \nu^{8} + 164685213 \nu^{7} - 377038701 \nu^{6} - 1193754170 \nu^{5} + \cdots - 449192696 ) / 47215936 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 159467 \nu^{11} + 671066 \nu^{10} + 8518605 \nu^{9} - 32568915 \nu^{8} - 154376285 \nu^{7} + 522294276 \nu^{6} + 1015904647 \nu^{5} + \cdots - 883660304 ) / 47215936 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 170061 \nu^{11} + 814200 \nu^{10} + 8687451 \nu^{9} - 39994179 \nu^{8} - 146548157 \nu^{7} + 656266478 \nu^{6} + 822543781 \nu^{5} + \cdots - 2412870848 ) / 47215936 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 176055 \nu^{11} - 854122 \nu^{10} - 8984721 \nu^{9} + 42191223 \nu^{8} + 150736233 \nu^{7} - 694924972 \nu^{6} - 830447331 \nu^{5} + \cdots + 2009725648 ) / 47215936 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 363289 \nu^{11} - 1341287 \nu^{10} - 20053311 \nu^{9} + 63469256 \nu^{8} + 382772712 \nu^{7} - 978360605 \nu^{6} - 2803100967 \nu^{5} + \cdots + 1514531544 ) / 47215936 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta_{2} + 20\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7} + 4 \beta_{6} - \beta_{5} + 4 \beta_{4} - 9 \beta_{3} + 26 \beta_{2} + 11 \beta _1 + 241 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 5 \beta_{11} + 26 \beta_{10} - 4 \beta_{9} - 11 \beta_{8} - 32 \beta_{7} + 36 \beta_{6} + 29 \beta_{5} + 37 \beta_{4} - 72 \beta_{3} + 74 \beta_{2} + 440 \beta _1 + 215 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 90 \beta_{11} + 57 \beta_{10} - 70 \beta_{9} - 68 \beta_{8} - 47 \beta_{7} + 145 \beta_{6} - 35 \beta_{5} + 175 \beta_{4} - 364 \beta_{3} + 665 \beta_{2} + 555 \beta _1 + 5377 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 288 \beta_{11} + 690 \beta_{10} - 214 \beta_{9} - 535 \beta_{8} - 878 \beta_{7} + 969 \beta_{6} + 674 \beta_{5} + 1175 \beta_{4} - 2211 \beta_{3} + 2339 \beta_{2} + 10213 \beta _1 + 8750 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3099 \beta_{11} + 2324 \beta_{10} - 2122 \beta_{9} - 2888 \beta_{8} - 1782 \beta_{7} + 4157 \beta_{6} - 771 \beta_{5} + 5882 \beta_{4} - 11827 \beta_{3} + 17390 \beta_{2} + 20397 \beta _1 + 126823 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 11793 \beta_{11} + 19364 \beta_{10} - 8580 \beta_{9} - 19644 \beta_{8} - 23200 \beta_{7} + 24069 \beta_{6} + 15337 \beta_{5} + 35428 \beta_{4} - 65281 \beta_{3} + 70103 \beta_{2} + \cdots + 297713 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 98283 \beta_{11} + 82403 \beta_{10} - 63776 \beta_{9} - 104086 \beta_{8} - 60379 \beta_{7} + 109232 \beta_{6} - 11098 \beta_{5} + 181721 \beta_{4} - 358247 \beta_{3} + 464214 \beta_{2} + \cdots + 3115236 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 420533 \beta_{11} + 563561 \beta_{10} - 302986 \beta_{9} - 653753 \beta_{8} - 609965 \beta_{7} + 578383 \beta_{6} + 358286 \beta_{5} + 1043116 \beta_{4} - 1902404 \beta_{3} + \cdots + 9337452 \) 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
5.35197
4.98104
3.73276
2.16473
1.07125
0.820352
−0.447590
−1.26162
−3.25752
−4.28857
−4.42326
−4.44354
−5.35197 −1.66219 20.6436 5.96899 8.89600 27.8210 −67.6680 −24.2371 −31.9458
1.2 −4.98104 −6.26206 16.8108 −15.7477 31.1916 −0.789949 −43.8870 12.2134 78.4401
1.3 −3.73276 3.65511 5.93351 14.5154 −13.6437 −12.9694 7.71372 −13.6402 −54.1823
1.4 −2.16473 −9.14133 −3.31395 −16.2298 19.7885 −12.2246 24.4916 56.5639 35.1331
1.5 −1.07125 5.02012 −6.85243 −4.32398 −5.37778 −4.44673 15.9106 −1.79838 4.63204
1.6 −0.820352 0.130026 −7.32702 8.70710 −0.106667 11.4024 12.5736 −26.9831 −7.14288
1.7 0.447590 −9.51519 −7.79966 11.7425 −4.25890 −2.27797 −7.07177 63.5388 5.25584
1.8 1.26162 6.09538 −6.40830 −13.2627 7.69007 27.2593 −18.1779 10.1536 −16.7325
1.9 3.25752 6.51757 2.61146 −19.1073 21.2311 −22.0995 −17.5533 15.4787 −62.2426
1.10 4.28857 −2.85039 10.3918 6.36629 −12.2241 −29.4308 10.2575 −18.8753 27.3023
1.11 4.42326 −9.44971 11.5652 8.63042 −41.7985 −12.6513 15.7698 62.2970 38.1746
1.12 4.44354 −0.537336 11.7450 −17.2592 −2.38767 6.40763 16.6411 −26.7113 −76.6918
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.h 12
17.b even 2 1 289.4.a.i yes 12
17.c even 4 2 289.4.b.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.4.a.h 12 1.a even 1 1 trivial
289.4.a.i yes 12 17.b even 2 1
289.4.b.f 24 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(289))\):

\( T_{2}^{12} - 72 T_{2}^{10} + 17 T_{2}^{9} + 1872 T_{2}^{8} - 627 T_{2}^{7} - 20922 T_{2}^{6} + 5163 T_{2}^{5} + 93255 T_{2}^{4} + 4607 T_{2}^{3} - 117822 T_{2}^{2} - 21960 T_{2} + 29352 \) Copy content Toggle raw display
\( T_{3}^{12} + 18 T_{3}^{11} - 54 T_{3}^{10} - 2228 T_{3}^{9} - 2097 T_{3}^{8} + 99708 T_{3}^{7} + 177392 T_{3}^{6} - 1841532 T_{3}^{5} - 3422262 T_{3}^{4} + 10471190 T_{3}^{3} + 22610910 T_{3}^{2} + \cdots - 1242003 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 72 T^{10} + 17 T^{9} + \cdots + 29352 \) Copy content Toggle raw display
$3$ \( T^{12} + 18 T^{11} - 54 T^{10} + \cdots - 1242003 \) Copy content Toggle raw display
$5$ \( T^{12} + 30 T^{11} + \cdots + 2352593298027 \) Copy content Toggle raw display
$7$ \( T^{12} + 24 T^{11} + \cdots + 578421193743 \) Copy content Toggle raw display
$11$ \( T^{12} + 162 T^{11} + \cdots + 40\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{12} + 72 T^{11} + \cdots - 30\!\cdots\!29 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} + 66 T^{11} + \cdots - 60\!\cdots\!27 \) Copy content Toggle raw display
$23$ \( T^{12} + 282 T^{11} + \cdots + 93\!\cdots\!51 \) Copy content Toggle raw display
$29$ \( T^{12} + 648 T^{11} + \cdots - 47\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( T^{12} + 504 T^{11} + \cdots - 12\!\cdots\!93 \) Copy content Toggle raw display
$37$ \( T^{12} - 30 T^{11} + \cdots - 25\!\cdots\!77 \) Copy content Toggle raw display
$41$ \( T^{12} + 318 T^{11} + \cdots + 18\!\cdots\!93 \) Copy content Toggle raw display
$43$ \( T^{12} - 486 T^{11} + \cdots + 31\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{12} + 888 T^{11} + \cdots + 56\!\cdots\!91 \) Copy content Toggle raw display
$53$ \( T^{12} - 1026 T^{11} + \cdots + 59\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{12} + 792 T^{11} + \cdots - 29\!\cdots\!87 \) Copy content Toggle raw display
$61$ \( T^{12} + 1212 T^{11} + \cdots + 14\!\cdots\!71 \) Copy content Toggle raw display
$67$ \( T^{12} - 624 T^{11} + \cdots - 64\!\cdots\!07 \) Copy content Toggle raw display
$71$ \( T^{12} + 2802 T^{11} + \cdots + 14\!\cdots\!47 \) Copy content Toggle raw display
$73$ \( T^{12} + 726 T^{11} + \cdots - 88\!\cdots\!43 \) Copy content Toggle raw display
$79$ \( T^{12} - 444 T^{11} + \cdots - 88\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{12} - 672 T^{11} + \cdots - 19\!\cdots\!11 \) Copy content Toggle raw display
$89$ \( T^{12} + 906 T^{11} + \cdots - 37\!\cdots\!51 \) Copy content Toggle raw display
$97$ \( T^{12} - 3246 T^{11} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
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