# 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

# Related objects

## 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 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 74 x^{2} + 1072$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} + 40$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 46 \nu$$$$)/6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$6 \beta_{2} - 40$$ $$\nu^{3}$$ $$=$$ $$6 \beta_{3} - 46 \beta_{1}$$

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

$p$ $F_p(T)$
$2$ $$( 1 + T + 8 T^{2} + 8 T^{3} + 64 T^{4} )^{2}$$
$3$ $$1 - 34 T^{2} + 1450 T^{4} - 24786 T^{6} + 531441 T^{8}$$
$5$ $$1 - 20 T^{2} + 12342 T^{4} - 312500 T^{6} + 244140625 T^{8}$$
$7$ $$1 - 842 T^{2} + 410922 T^{4} - 99060458 T^{6} + 13841287201 T^{8}$$
$11$ $$1 - 1698 T^{2} + 3550826 T^{4} - 3008110578 T^{6} + 3138428376721 T^{8}$$
$13$ $$( 1 - 70 T + 4002 T^{2} - 153790 T^{3} + 4826809 T^{4} )^{2}$$
$17$ $$1 - 112 T + 6494 T^{2} - 550256 T^{3} + 24137569 T^{4}$$
$19$ $$( 1 + 28 T + 6859 T^{2} )^{4}$$
$23$ $$1 - 20346 T^{2} + 209323274 T^{4} - 3011938197594 T^{6} + 21914624432020321 T^{8}$$
$29$ $$1 - 74036 T^{2} + 2514340758 T^{4} - 44038339393556 T^{6} + 353814783205469041 T^{8}$$
$31$ $$1 - 114890 T^{2} + 5074759530 T^{4} - 101965297910090 T^{6} + 787662783788549761 T^{8}$$
$37$ $$1 - 116372 T^{2} + 7903483062 T^{4} - 298578713668148 T^{6} + 6582952005840035281 T^{8}$$
$41$ $$1 - 158724 T^{2} + 15178105958 T^{4} - 753955545548484 T^{6} + 22563490300366186081 T^{8}$$
$43$ $$( 1 + 260 T + 128262 T^{2} + 20671820 T^{3} + 6321363049 T^{4} )^{2}$$
$47$ $$( 1 - 476 T + 257822 T^{2} - 49419748 T^{3} + 10779215329 T^{4} )^{2}$$
$53$ $$( 1 + 288 T + 248662 T^{2} + 42876576 T^{3} + 22164361129 T^{4} )^{2}$$
$59$ $$( 1 + 588 T + 335494 T^{2} + 120762852 T^{3} + 42180533641 T^{4} )^{2}$$
$61$ $$1 - 425396 T^{2} + 97219598358 T^{4} - 21916561171671956 T^{6} +$$$$26\!\cdots\!21$$$$T^{8}$$
$67$ $$( 1 + 120 T + 571334 T^{2} + 36091560 T^{3} + 90458382169 T^{4} )^{2}$$
$71$ $$1 - 1379018 T^{2} + 731023514346 T^{4} - 176652597332169578 T^{6} +$$$$16\!\cdots\!41$$$$T^{8}$$
$73$ $$1 + 55196 T^{2} + 301853502630 T^{4} + 8353043954247644 T^{6} +$$$$22\!\cdots\!21$$$$T^{8}$$
$79$ $$1 - 1282922 T^{2} + 849811865706 T^{4} - 311862244611912362 T^{6} +$$$$59\!\cdots\!41$$$$T^{8}$$
$83$ $$( 1 + 1428 T + 1595158 T^{2} + 816511836 T^{3} + 326940373369 T^{4} )^{2}$$
$89$ $$( 1 - 994 T + 1642394 T^{2} - 700739186 T^{3} + 496981290961 T^{4} )^{2}$$
$97$ $$1 - 3375140 T^{2} + 4511732954310 T^{4} - 2811397132716065060 T^{6} +$$$$69\!\cdots\!41$$$$T^{8}$$