Properties

Label 17.4.b.a
Level $17$
Weight $4$
Character orbit 17.b
Analytic conductor $1.003$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 17.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00303247010\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.4669632.2
Defining polynomial: \( x^{4} + 74x^{2} + 1072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} + \beta_1 q^{3} + ( - \beta_{2} + 1) q^{4} - \beta_{3} q^{5} + (\beta_{3} - 2 \beta_1) q^{6} + (\beta_{3} - \beta_1) q^{7} + ( - 7 \beta_{2} - 1) q^{8} + (6 \beta_{2} - 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{2} + \beta_1 q^{3} + ( - \beta_{2} + 1) q^{4} - \beta_{3} q^{5} + (\beta_{3} - 2 \beta_1) q^{6} + (\beta_{3} - \beta_1) q^{7} + ( - 7 \beta_{2} - 1) q^{8} + (6 \beta_{2} - 13) q^{9} + ( - \beta_{3} - 6 \beta_1) q^{10} + 7 \beta_1 q^{11} + ( - \beta_{3} + 2 \beta_1) q^{12} + ( - 14 \beta_{2} + 42) q^{13} + 8 \beta_1 q^{14} + (28 \beta_{2} - 8) q^{15} + (7 \beta_{2} - 63) q^{16} + (\beta_{3} + 14 \beta_{2} - 8 \beta_1 + 21) q^{17} + ( - 13 \beta_{2} + 61) q^{18} - 28 q^{19} + (\beta_{3} + 6 \beta_1) q^{20} + ( - 34 \beta_{2} + 48) q^{21} + (7 \beta_{3} - 14 \beta_1) q^{22} + ( - 7 \beta_{3} - 7 \beta_1) q^{23} + ( - 7 \beta_{3} + 6 \beta_1) q^{24} + ( - 48 \beta_{2} - 91) q^{25} + (42 \beta_{2} - 154) q^{26} + (6 \beta_{3} + 8 \beta_1) q^{27} - 8 \beta_1 q^{28} + 7 \beta_{3} q^{29} + ( - 8 \beta_{2} + 232) q^{30} + ( - \beta_{3} - 7 \beta_1) q^{31} + ( - 7 \beta_{2} + 127) q^{32} + (42 \beta_{2} - 280) q^{33} + ( - 7 \beta_{3} + 21 \beta_{2} + 22 \beta_1 + 91) q^{34} + (20 \beta_{2} + 224) q^{35} + (13 \beta_{2} - 61) q^{36} + ( - 7 \beta_{3} - 28 \beta_1) q^{37} + ( - 28 \beta_{2} + 28) q^{38} + ( - 14 \beta_{3} + 56 \beta_1) q^{39} + (15 \beta_{3} + 42 \beta_1) q^{40} + (14 \beta_{3} - 20 \beta_1) q^{41} + (48 \beta_{2} - 320) q^{42} + ( - 76 \beta_{2} - 92) q^{43} + ( - 7 \beta_{3} + 14 \beta_1) q^{44} + (\beta_{3} - 36 \beta_1) q^{45} + ( - 14 \beta_{3} - 28 \beta_1) q^{46} + (28 \beta_{2} + 224) q^{47} + (7 \beta_{3} - 70 \beta_1) q^{48} + (14 \beta_{2} + 71) q^{49} + ( - 91 \beta_{2} - 293) q^{50} + (14 \beta_{3} - 76 \beta_{2} + 7 \beta_1 + 328) q^{51} + ( - 42 \beta_{2} + 154) q^{52} + ( - 92 \beta_{2} - 98) q^{53} + (14 \beta_{3} + 20 \beta_1) q^{54} + (196 \beta_{2} - 56) q^{55} + ( - 8 \beta_{3} - 48 \beta_1) q^{56} - 28 \beta_1 q^{57} + (7 \beta_{3} + 42 \beta_1) q^{58} + (140 \beta_{2} - 364) q^{59} + (8 \beta_{2} - 232) q^{60} + ( - 21 \beta_{3} + 64 \beta_1) q^{61} + ( - 8 \beta_{3} + 8 \beta_1) q^{62} + ( - 7 \beta_{3} + 55 \beta_1) q^{63} + (71 \beta_{2} + 321) q^{64} + ( - 14 \beta_{3} + 84 \beta_1) q^{65} + ( - 280 \beta_{2} + 616) q^{66} + ( - 64 \beta_{2} - 28) q^{67} + (7 \beta_{3} - 21 \beta_{2} - 22 \beta_1 - 91) q^{68} + (154 \beta_{2} + 224) q^{69} + (224 \beta_{2} - 64) q^{70} + (7 \beta_{3} - 21 \beta_1) q^{71} + (43 \beta_{2} - 323) q^{72} + (56 \beta_{3} - 48 \beta_1) q^{73} + ( - 35 \beta_{3} + 14 \beta_1) q^{74} + ( - 48 \beta_{3} - 43 \beta_1) q^{75} + (28 \beta_{2} - 28) q^{76} + ( - 238 \beta_{2} + 336) q^{77} + (42 \beta_{3} - 196 \beta_1) q^{78} + (21 \beta_{3} + 77 \beta_1) q^{79} + (49 \beta_{3} - 42 \beta_1) q^{80} + (42 \beta_{2} - 623) q^{81} + ( - 6 \beta_{3} + 124 \beta_1) q^{82} + (84 \beta_{2} - 756) q^{83} + ( - 48 \beta_{2} + 320) q^{84} + ( - 49 \beta_{3} - 176 \beta_{2} - 84 \beta_1 + 280) q^{85} + ( - 92 \beta_{2} - 516) q^{86} + ( - 196 \beta_{2} + 56) q^{87} + ( - 49 \beta_{3} + 42 \beta_1) q^{88} + ( - 42 \beta_{2} + 518) q^{89} + ( - 35 \beta_{3} + 78 \beta_1) q^{90} + (28 \beta_{3} - 140 \beta_1) q^{91} + (14 \beta_{3} + 28 \beta_1) q^{92} + ( - 14 \beta_{2} + 272) q^{93} + 224 \beta_{2} q^{94} + 28 \beta_{3} q^{95} + ( - 7 \beta_{3} + 134 \beta_1) q^{96} + ( - 22 \beta_{3} + 28 \beta_1) q^{97} + (71 \beta_{2} + 41) q^{98} + (42 \beta_{3} - 133 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{4} - 18 q^{8} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 2 q^{4} - 18 q^{8} - 40 q^{9} + 140 q^{13} + 24 q^{15} - 238 q^{16} + 112 q^{17} + 218 q^{18} - 112 q^{19} + 124 q^{21} - 460 q^{25} - 532 q^{26} + 912 q^{30} + 494 q^{32} - 1036 q^{33} + 406 q^{34} + 936 q^{35} - 218 q^{36} + 56 q^{38} - 1184 q^{42} - 520 q^{43} + 952 q^{47} + 312 q^{49} - 1354 q^{50} + 1160 q^{51} + 532 q^{52} - 576 q^{53} + 168 q^{55} - 1176 q^{59} - 912 q^{60} + 1426 q^{64} + 1904 q^{66} - 240 q^{67} - 406 q^{68} + 1204 q^{69} + 192 q^{70} - 1206 q^{72} - 56 q^{76} + 868 q^{77} - 2408 q^{81} - 2856 q^{83} + 1184 q^{84} + 768 q^{85} - 2248 q^{86} - 168 q^{87} + 1988 q^{89} + 1060 q^{93} + 448 q^{94} + 306 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 74x^{2} + 1072 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 40 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 46\nu ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 6\beta_{2} - 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{3} - 46\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
7.36435i
7.36435i
4.44593i
4.44593i
−3.37228 7.36435i 3.37228 10.1060i 24.8347i 17.4703i 15.6060 −27.2337 34.0802i
16.2 −3.37228 7.36435i 3.37228 10.1060i 24.8347i 17.4703i 15.6060 −27.2337 34.0802i
16.3 2.37228 4.44593i −2.37228 19.4389i 10.5470i 14.9929i −24.6060 7.23369 46.1145i
16.4 2.37228 4.44593i −2.37228 19.4389i 10.5470i 14.9929i −24.6060 7.23369 46.1145i
\(n\): e.g. 2-40 or 990-1000
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 17.4.b.a 4
3.b odd 2 1 153.4.d.b 4
4.b odd 2 1 272.4.b.d 4
5.b even 2 1 425.4.d.c 4
5.c odd 4 2 425.4.c.c 8
17.b even 2 1 inner 17.4.b.a 4
17.c even 4 2 289.4.a.e 4
51.c odd 2 1 153.4.d.b 4
68.d odd 2 1 272.4.b.d 4
85.c even 2 1 425.4.d.c 4
85.g odd 4 2 425.4.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.b.a 4 1.a even 1 1 trivial
17.4.b.a 4 17.b even 2 1 inner
153.4.d.b 4 3.b odd 2 1
153.4.d.b 4 51.c odd 2 1
272.4.b.d 4 4.b odd 2 1
272.4.b.d 4 68.d odd 2 1
289.4.a.e 4 17.c even 4 2
425.4.c.c 8 5.c odd 4 2
425.4.c.c 8 85.g odd 4 2
425.4.d.c 4 5.b even 2 1
425.4.d.c 4 85.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(17, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T - 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 74T^{2} + 1072 \) Copy content Toggle raw display
$5$ \( T^{4} + 480 T^{2} + 38592 \) Copy content Toggle raw display
$7$ \( T^{4} + 530 T^{2} + 68608 \) Copy content Toggle raw display
$11$ \( T^{4} + 3626 T^{2} + \cdots + 2573872 \) Copy content Toggle raw display
$13$ \( (T^{2} - 70 T - 392)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 112 T^{3} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( (T + 28)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 28322 T^{2} + \cdots + 10295488 \) Copy content Toggle raw display
$29$ \( T^{4} + 23520 T^{2} + \cdots + 92659392 \) Copy content Toggle raw display
$31$ \( T^{4} + 4274 T^{2} + \cdots + 4390912 \) Copy content Toggle raw display
$37$ \( T^{4} + 86240 T^{2} + \cdots + 1245754048 \) Copy content Toggle raw display
$41$ \( T^{4} + 116960 T^{2} + \cdots + 2799480832 \) Copy content Toggle raw display
$43$ \( (T^{2} + 260 T - 30752)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 476 T + 50176)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 288 T - 49092)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 588 T - 75264)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 482528 T^{2} + \cdots + 7146728128 \) Copy content Toggle raw display
$67$ \( (T^{2} + 120 T - 30192)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 52626 T^{2} + \cdots + 92659392 \) Copy content Toggle raw display
$73$ \( T^{4} + 1611264 T^{2} + \cdots + 647466319872 \) Copy content Toggle raw display
$79$ \( T^{4} + 689234 T^{2} + \cdots + 70925616832 \) Copy content Toggle raw display
$83$ \( (T^{2} + 1428 T + 451584)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 994 T + 232456)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 275552 T^{2} + \cdots + 16878665728 \) Copy content Toggle raw display
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