Properties

Label 289.4.b.b
Level $289$
Weight $4$
Character orbit 289.b
Analytic conductor $17.052$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 289.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.0515519917\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.27793984.1
Defining polynomial: \( x^{6} - 2x^{3} + 49x^{2} - 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{3} + (2 \beta_{3} - \beta_1 + 8) q^{4} + ( - \beta_{5} - \beta_{4}) q^{5} + (\beta_{5} + 12 \beta_{4} + 6 \beta_{2}) q^{6} + ( - 2 \beta_{5} - 3 \beta_{4} - \beta_{2}) q^{7} + ( - 2 \beta_{3} - 7 \beta_1 + 16) q^{8} + ( - 5 \beta_{3} + 3 \beta_1 - 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{3} + (2 \beta_{3} - \beta_1 + 8) q^{4} + ( - \beta_{5} - \beta_{4}) q^{5} + (\beta_{5} + 12 \beta_{4} + 6 \beta_{2}) q^{6} + ( - 2 \beta_{5} - 3 \beta_{4} - \beta_{2}) q^{7} + ( - 2 \beta_{3} - 7 \beta_1 + 16) q^{8} + ( - 5 \beta_{3} + 3 \beta_1 - 19) q^{9} + (4 \beta_{5} + 8 \beta_{4}) q^{10} + ( - \beta_{5} + 5 \beta_{4} + 11 \beta_{2}) q^{11} + ( - 13 \beta_{5} + 8 \beta_{4} + 26 \beta_{2}) q^{12} + ( - 7 \beta_{3} + \beta_1 + 12) q^{13} + (10 \beta_{5} + 12 \beta_{4} - 4 \beta_{2}) q^{14} + ( - 5 \beta_{3} - 7 \beta_1 - 32) q^{15} + (2 \beta_{3} - 9 \beta_1 + 48) q^{16} + (4 \beta_{3} + 37 \beta_1 - 48) q^{18} + (7 \beta_{3} - 15 \beta_1 - 24) q^{19} + ( - 12 \beta_{5} - 24 \beta_{4} + 8 \beta_{2}) q^{20} + ( - 9 \beta_{3} - 21 \beta_1 - 54) q^{21} + ( - 13 \beta_{5} + 52 \beta_{4} + 34 \beta_{2}) q^{22} + ( - 2 \beta_{5} - 23 \beta_{4} - 39 \beta_{2}) q^{23} + ( - 3 \beta_{5} + 112 \beta_{4} + 46 \beta_{2}) q^{24} + ( - 8 \beta_{3} - 12 \beta_1 + 81) q^{25} + (12 \beta_{3} + 10 \beta_1 - 16) q^{26} + (4 \beta_{5} + 2 \beta_{4} - 40 \beta_{2}) q^{27} + ( - 22 \beta_{5} - 72 \beta_{4} + 4 \beta_{2}) q^{28} + ( - 15 \beta_{5} - 71 \beta_{4} + 16 \beta_{2}) q^{29} + (24 \beta_{3} + 40 \beta_1 + 112) q^{30} + ( - 8 \beta_{5} + 41 \beta_{4} - 39 \beta_{2}) q^{31} + (30 \beta_{3} - 7 \beta_1 + 16) q^{32} + ( - 21 \beta_{3} + 55 \beta_1 - 122) q^{33} + ( - 17 \beta_{3} - 27 \beta_1 - 96) q^{35} + ( - 42 \beta_{3} + 49 \beta_1 - 440) q^{36} + (25 \beta_{5} + 51 \beta_{4} + 28 \beta_{2}) q^{37} + (16 \beta_{3} - 12 \beta_1 + 240) q^{38} + (8 \beta_{5} + 42 \beta_{4} - 36 \beta_{2}) q^{39} + (20 \beta_{5} + 64 \beta_{4} - 8 \beta_{2}) q^{40} + ( - 30 \beta_{5} + 59 \beta_{4} - 52 \beta_{2}) q^{41} + (60 \beta_{3} + 60 \beta_1 + 336) q^{42} + (29 \beta_{3} + 27 \beta_1 - 204) q^{43} + ( - 39 \beta_{5} + 200 \beta_{4} + 110 \beta_{2}) q^{44} + (27 \beta_{5} + 55 \beta_{4} - 20 \beta_{2}) q^{45} + (68 \beta_{5} - 140 \beta_{4} - 120 \beta_{2}) q^{46} + (2 \beta_{3} - 46 \beta_1 + 228) q^{47} + ( - 45 \beta_{5} + 144 \beta_{4} + 114 \beta_{2}) q^{48} + ( - 37 \beta_{3} - 57 \beta_1 + 121) q^{49} + (40 \beta_{3} - 69 \beta_1 + 192) q^{50} + (12 \beta_{3} - 18 \beta_1 - 256) q^{52} + (54 \beta_{3} - 62 \beta_1 - 98) q^{53} + (26 \beta_{5} - 192 \beta_{4} - 84 \beta_{2}) q^{54} + ( - 7 \beta_{3} + 11 \beta_1 + 24) q^{55} + (54 \beta_{5} + 96 \beta_{4} - 60 \beta_{2}) q^{56} + (18 \beta_{5} + 114 \beta_{4} + 108 \beta_{2}) q^{57} + (100 \beta_{5} + 184 \beta_{4} - 80 \beta_{2}) q^{58} + ( - 65 \beta_{3} + 65 \beta_1 - 212) q^{59} + ( - 88 \beta_{3} - 88 \beta_1 - 384) q^{60} + (39 \beta_{5} + \beta_{4} - 64 \beta_{2}) q^{61} + (22 \beta_{5} - 92 \beta_{4} + 20 \beta_{2}) q^{62} + (48 \beta_{5} + 171 \beta_{4} - 9 \beta_{2}) q^{63} + ( - 62 \beta_{3} - 41 \beta_1 - 272) q^{64} + (12 \beta_{5} + 64 \beta_{4} - 28 \beta_{2}) q^{65} + ( - 68 \beta_{3} + 240 \beta_1 - 880) q^{66} + (54 \beta_{3} + 78 \beta_1 + 292) q^{67} + (47 \beta_{3} - 233 \beta_1 + 254) q^{69} + (88 \beta_{3} + 120 \beta_1 + 432) q^{70} + ( - 36 \beta_{5} - 55 \beta_{4} - 185 \beta_{2}) q^{71} + ( - 46 \beta_{3} + 319 \beta_1 - 400) q^{72} + (8 \beta_{5} + 137 \beta_{4} - 16 \beta_{2}) q^{73} + ( - 154 \beta_{5} - 88 \beta_{4} + 108 \beta_{2}) q^{74} + ( - 45 \beta_{5} + 273 \beta_{4} + 105 \beta_{2}) q^{75} + ( - 64 \beta_{3} - 180 \beta_1 + 384) q^{76} + ( - 9 \beta_{3} - 9 \beta_1 + 174) q^{77} + ( - 30 \beta_{5} - 208 \beta_{4} - 4 \beta_{2}) q^{78} + (90 \beta_{5} + 69 \beta_{4} - 267 \beta_{2}) q^{79} + ( - 20 \beta_{5} + 8 \beta_{2}) q^{80} + ( - 37 \beta_{3} - 57 \beta_1 - 137) q^{81} + (83 \beta_{5} + 32 \beta_{4} + 74 \beta_{2}) q^{82} + (105 \beta_{3} + 23 \beta_1 + 756) q^{83} + ( - 168 \beta_{3} - 288 \beta_1 - 528) q^{84} + ( - 112 \beta_{3} + 144 \beta_1 - 432) q^{86} + ( - 163 \beta_{3} - 209 \beta_1 - 352) q^{87} + ( - 89 \beta_{5} + 336 \beta_{4} + 426 \beta_{2}) q^{88} + (83 \beta_{3} - 193 \beta_1 - 20) q^{89} + ( - 116 \beta_{5} - 296 \beta_{4} + 16 \beta_{2}) q^{90} + (22 \beta_{5} + 162 \beta_{4} - 64 \beta_{2}) q^{91} + (72 \beta_{5} - 840 \beta_{4} - 344 \beta_{2}) q^{92} + (87 \beta_{3} - 65 \beta_1 - 218) q^{93} + (88 \beta_{3} - 280 \beta_1 + 736) q^{94} + (56 \beta_{5} + 60 \beta_{4} + 28 \beta_{2}) q^{95} + ( - 99 \beta_{5} - 80 \beta_{4} + 238 \beta_{2}) q^{96} + ( - 60 \beta_{5} - 25 \beta_{4} - 140 \beta_{2}) q^{97} + (188 \beta_{3} - 67 \beta_1 + 912) q^{98} + (103 \beta_{5} - 521 \beta_{4} - 281 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 50 q^{4} + 78 q^{8} - 118 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} + 50 q^{4} + 78 q^{8} - 118 q^{9} + 60 q^{13} - 216 q^{15} + 274 q^{16} - 206 q^{18} - 160 q^{19} - 384 q^{21} + 446 q^{25} - 52 q^{26} + 800 q^{30} + 142 q^{32} - 664 q^{33} - 664 q^{35} - 2626 q^{36} + 1448 q^{38} + 2256 q^{42} - 1112 q^{43} + 1280 q^{47} + 538 q^{49} + 1094 q^{50} - 1548 q^{52} - 604 q^{53} + 152 q^{55} - 1272 q^{59} - 2656 q^{60} - 1838 q^{64} - 4936 q^{66} + 2016 q^{67} + 1152 q^{69} + 3008 q^{70} - 1854 q^{72} + 1816 q^{76} + 1008 q^{77} - 1010 q^{81} + 4792 q^{83} - 4080 q^{84} - 2528 q^{86} - 2856 q^{87} - 340 q^{89} - 1264 q^{93} + 4032 q^{94} + 5714 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{3} + 49x^{2} - 14x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{5} + 25\nu^{4} + 28\nu^{3} - 4\nu^{2} + 799 ) / 171 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{5} - \nu^{4} - 49\nu^{3} + 7\nu^{2} - 684\nu + 98 ) / 342 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 7\nu^{3} + \nu^{2} + 299 ) / 57 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -49\nu^{5} - 7\nu^{4} - \nu^{3} + 49\nu^{2} - 2394\nu + 344 ) / 171 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -245\nu^{5} - 35\nu^{4} - 5\nu^{3} + 587\nu^{2} - 11970\nu + 1720 ) / 171 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 2\beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 5\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{4} - 7\beta_{3} - 14\beta_{2} + 7\beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{3} + 3\beta _1 - 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{5} - 24\beta_{4} - 51\beta_{3} + 98\beta_{2} + 47\beta _1 + 48 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
288.1
0.143705 0.143705i
0.143705 + 0.143705i
−1.93854 1.93854i
−1.93854 + 1.93854i
1.79483 + 1.79483i
1.79483 1.79483i
−4.67129 7.62999i 13.8209 11.9174i 35.6419i 26.1222i −27.1912 −31.2167 55.6696i
288.2 −4.67129 7.62999i 13.8209 11.9174i 35.6419i 26.1222i −27.1912 −31.2167 55.6696i
288.3 −1.36122 3.15463i −6.14708 3.03171i 4.29415i 7.94049i 19.2573 17.0483 4.12682i
288.4 −1.36122 3.15463i −6.14708 3.03171i 4.29415i 7.94049i 19.2573 17.0483 4.12682i
288.5 5.03251 8.47535i 17.3261 0.885690i 42.6523i 3.81828i 46.9339 −44.8316 4.45724i
288.6 5.03251 8.47535i 17.3261 0.885690i 42.6523i 3.81828i 46.9339 −44.8316 4.45724i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 288.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.4.b.b 6
17.b even 2 1 inner 289.4.b.b 6
17.c even 4 1 17.4.a.b 3
17.c even 4 1 289.4.a.b 3
51.f odd 4 1 153.4.a.g 3
68.f odd 4 1 272.4.a.h 3
85.f odd 4 1 425.4.b.f 6
85.i odd 4 1 425.4.b.f 6
85.j even 4 1 425.4.a.g 3
119.f odd 4 1 833.4.a.d 3
136.i even 4 1 1088.4.a.v 3
136.j odd 4 1 1088.4.a.x 3
187.f odd 4 1 2057.4.a.e 3
204.l even 4 1 2448.4.a.bi 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.b 3 17.c even 4 1
153.4.a.g 3 51.f odd 4 1
272.4.a.h 3 68.f odd 4 1
289.4.a.b 3 17.c even 4 1
289.4.b.b 6 1.a even 1 1 trivial
289.4.b.b 6 17.b even 2 1 inner
425.4.a.g 3 85.j even 4 1
425.4.b.f 6 85.f odd 4 1
425.4.b.f 6 85.i odd 4 1
833.4.a.d 3 119.f odd 4 1
1088.4.a.v 3 136.i even 4 1
1088.4.a.x 3 136.j odd 4 1
2057.4.a.e 3 187.f odd 4 1
2448.4.a.bi 3 204.l even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + T_{2}^{2} - 24T_{2} - 32 \) acting on \(S_{4}^{\mathrm{new}}(289, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{3} + T^{2} - 24 T - 32)^{2} \) Copy content Toggle raw display
$3$ \( T^{6} + 140 T^{4} + 5476 T^{2} + \cdots + 41616 \) Copy content Toggle raw display
$5$ \( T^{6} + 152 T^{4} + 1424 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( T^{6} + 760 T^{4} + 53892 T^{2} + \cdots + 627264 \) Copy content Toggle raw display
$11$ \( T^{6} + 3516 T^{4} + \cdots + 22014864 \) Copy content Toggle raw display
$13$ \( (T^{3} - 30 T^{2} - 1472 T - 9392)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( (T^{3} + 80 T^{2} - 4632 T - 340128)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 51704 T^{4} + \cdots + 2561741095936 \) Copy content Toggle raw display
$29$ \( T^{6} + 100120 T^{4} + \cdots + 2306218853376 \) Copy content Toggle raw display
$31$ \( T^{6} + 76072 T^{4} + \cdots + 6659865664 \) Copy content Toggle raw display
$37$ \( T^{6} + 162664 T^{4} + \cdots + 38152265269504 \) Copy content Toggle raw display
$41$ \( T^{6} + 259564 T^{4} + \cdots + 2685481897536 \) Copy content Toggle raw display
$43$ \( (T^{3} + 556 T^{2} + 51096 T - 7270272)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} - 640 T^{2} + 85328 T - 1671168)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 302 T^{2} - 153460 T - 18162072)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} + 636 T^{2} - 101768 T - 49419072)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + 255880 T^{4} + \cdots + 46141914470656 \) Copy content Toggle raw display
$67$ \( (T^{3} - 1008 T^{2} + 65040 T - 765952)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 1341352 T^{4} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 398302285230144 \) Copy content Toggle raw display
$79$ \( T^{6} + 2595384 T^{4} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{3} - 2396 T^{2} + 1488888 T - 142080704)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 170 T^{2} - 1072304 T - 446571376)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 1245100 T^{4} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
show more
show less