# Properties

 Label 17.4.a.b Level 17 Weight 4 Character orbit 17.a Self dual yes Analytic conductor 1.003 Analytic rank 0 Dimension 3 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 17.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.00303247010$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2636.1 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{3} + ( 8 - 3 \beta_{1} - \beta_{2} ) q^{4} + ( -2 + 2 \beta_{2} ) q^{5} + ( -24 + 6 \beta_{1} + 2 \beta_{2} ) q^{6} + ( 6 - \beta_{1} - 4 \beta_{2} ) q^{7} + ( -16 + 5 \beta_{1} - 9 \beta_{2} ) q^{8} + ( 19 - 8 \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{3} + ( 8 - 3 \beta_{1} - \beta_{2} ) q^{4} + ( -2 + 2 \beta_{2} ) q^{5} + ( -24 + 6 \beta_{1} + 2 \beta_{2} ) q^{6} + ( 6 - \beta_{1} - 4 \beta_{2} ) q^{7} + ( -16 + 5 \beta_{1} - 9 \beta_{2} ) q^{8} + ( 19 - 8 \beta_{1} - 2 \beta_{2} ) q^{9} + ( -16 + 8 \beta_{2} ) q^{10} + ( -10 + 11 \beta_{1} - 2 \beta_{2} ) q^{11} + ( 16 - 26 \beta_{1} + 26 \beta_{2} ) q^{12} + ( 12 + 8 \beta_{1} + 6 \beta_{2} ) q^{13} + ( 24 + 4 \beta_{1} - 20 \beta_{2} ) q^{14} + ( 32 + 2 \beta_{1} - 12 \beta_{2} ) q^{15} + ( 48 - 11 \beta_{1} + 7 \beta_{2} ) q^{16} -17 q^{17} + ( -48 + 33 \beta_{1} - 41 \beta_{2} ) q^{18} + ( 24 + 22 \beta_{1} - 8 \beta_{2} ) q^{19} + ( -48 - 8 \beta_{1} + 24 \beta_{2} ) q^{20} + ( -54 - 12 \beta_{1} + 30 \beta_{2} ) q^{21} + ( 104 - 34 \beta_{1} + 26 \beta_{2} ) q^{22} + ( 46 - 39 \beta_{1} - 4 \beta_{2} ) q^{23} + ( -224 + 46 \beta_{1} - 6 \beta_{2} ) q^{24} + ( -81 + 4 \beta_{1} - 20 \beta_{2} ) q^{25} + ( 16 + 2 \beta_{1} + 22 \beta_{2} ) q^{26} + ( -4 - 40 \beta_{1} + 8 \beta_{2} ) q^{27} + ( 144 + 4 \beta_{1} - 44 \beta_{2} ) q^{28} + ( -142 - 16 \beta_{1} + 30 \beta_{2} ) q^{29} + ( 112 + 16 \beta_{1} - 64 \beta_{2} ) q^{30} + ( 82 + 39 \beta_{1} + 16 \beta_{2} ) q^{31} + ( -16 + 37 \beta_{1} + 23 \beta_{2} ) q^{32} + ( -122 + 76 \beta_{1} - 34 \beta_{2} ) q^{33} + ( -17 \beta_{1} + 17 \beta_{2} ) q^{34} + ( -96 - 10 \beta_{1} + 44 \beta_{2} ) q^{35} + ( 440 - 91 \beta_{1} + 7 \beta_{2} ) q^{36} + ( 102 - 28 \beta_{1} - 50 \beta_{2} ) q^{37} + ( 240 - 28 \beta_{1} - 4 \beta_{2} ) q^{38} + ( 84 + 36 \beta_{1} - 16 \beta_{2} ) q^{39} + ( -128 - 8 \beta_{1} + 40 \beta_{2} ) q^{40} + ( -118 - 52 \beta_{1} - 60 \beta_{2} ) q^{41} + ( -336 + 120 \beta_{2} ) q^{42} + ( 204 + 2 \beta_{1} + 56 \beta_{2} ) q^{43} + ( -400 + 110 \beta_{1} - 78 \beta_{2} ) q^{44} + ( -110 - 20 \beta_{1} + 54 \beta_{2} ) q^{45} + ( -280 + 120 \beta_{1} - 136 \beta_{2} ) q^{46} + ( 228 - 48 \beta_{1} + 44 \beta_{2} ) q^{47} + ( 288 - 114 \beta_{1} + 90 \beta_{2} ) q^{48} + ( -121 + 20 \beta_{1} - 94 \beta_{2} ) q^{49} + ( 192 - 109 \beta_{1} + 29 \beta_{2} ) q^{50} + ( -34 + 17 \beta_{1} - 34 \beta_{2} ) q^{51} + ( -256 - 30 \beta_{1} + 6 \beta_{2} ) q^{52} + ( 98 + 116 \beta_{1} - 8 \beta_{2} ) q^{53} + ( -384 + 84 \beta_{1} - 52 \beta_{2} ) q^{54} + ( 24 + 18 \beta_{1} - 4 \beta_{2} ) q^{55} + ( 192 + 60 \beta_{1} - 108 \beta_{2} ) q^{56} + ( -228 + 108 \beta_{1} + 36 \beta_{2} ) q^{57} + ( -368 - 80 \beta_{1} + 200 \beta_{2} ) q^{58} + ( 212 - 130 \beta_{1} ) q^{59} + ( 384 - 176 \beta_{2} ) q^{60} + ( -2 - 64 \beta_{1} + 78 \beta_{2} ) q^{61} + ( 184 + 20 \beta_{1} + 44 \beta_{2} ) q^{62} + ( 342 + 9 \beta_{1} - 96 \beta_{2} ) q^{63} + ( -272 + 21 \beta_{1} + 103 \beta_{2} ) q^{64} + ( 128 + 28 \beta_{1} - 24 \beta_{2} ) q^{65} + ( 880 - 308 \beta_{1} + 172 \beta_{2} ) q^{66} + ( 292 + 24 \beta_{1} - 132 \beta_{2} ) q^{67} + ( -136 + 51 \beta_{1} + 17 \beta_{2} ) q^{68} + ( 254 - 280 \beta_{1} + 186 \beta_{2} ) q^{69} + ( -432 - 32 \beta_{1} + 208 \beta_{2} ) q^{70} + ( -110 + 185 \beta_{1} + 72 \beta_{2} ) q^{71} + ( -400 + 365 \beta_{1} - 273 \beta_{2} ) q^{72} + ( 274 + 16 \beta_{1} - 16 \beta_{2} ) q^{73} + ( 176 + 108 \beta_{1} - 308 \beta_{2} ) q^{74} + ( -546 + 105 \beta_{1} - 90 \beta_{2} ) q^{75} + ( -384 + 116 \beta_{1} - 244 \beta_{2} ) q^{76} + ( -174 - 18 \beta_{2} ) q^{77} + ( 416 - 4 \beta_{1} - 60 \beta_{2} ) q^{78} + ( -138 - 267 \beta_{1} + 180 \beta_{2} ) q^{79} + ( -8 \beta_{1} + 40 \beta_{2} ) q^{80} + ( -137 - 20 \beta_{1} + 94 \beta_{2} ) q^{81} + ( 64 - 74 \beta_{1} - 166 \beta_{2} ) q^{82} + ( -756 + 82 \beta_{1} + 128 \beta_{2} ) q^{83} + ( -528 - 120 \beta_{1} + 456 \beta_{2} ) q^{84} + ( 34 - 34 \beta_{2} ) q^{85} + ( -432 + 256 \beta_{1} - 32 \beta_{2} ) q^{86} + ( 352 + 46 \beta_{1} - 372 \beta_{2} ) q^{87} + ( 672 - 426 \beta_{1} + 178 \beta_{2} ) q^{88} + ( -20 - 276 \beta_{1} + 110 \beta_{2} ) q^{89} + ( -592 - 16 \beta_{1} + 232 \beta_{2} ) q^{90} + ( -324 - 64 \beta_{1} + 44 \beta_{2} ) q^{91} + ( 1680 - 344 \beta_{1} + 144 \beta_{2} ) q^{92} + ( 218 + 152 \beta_{1} + 22 \beta_{2} ) q^{93} + ( -736 + 368 \beta_{1} - 192 \beta_{2} ) q^{94} + ( -120 + 28 \beta_{1} + 112 \beta_{2} ) q^{95} + ( 160 + 238 \beta_{1} - 198 \beta_{2} ) q^{96} + ( -50 + 140 \beta_{1} + 120 \beta_{2} ) q^{97} + ( 912 - 255 \beta_{1} - 121 \beta_{2} ) q^{98} + ( -1042 + 281 \beta_{1} - 206 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{2} + 4q^{3} + 25q^{4} - 8q^{5} - 74q^{6} + 22q^{7} - 39q^{8} + 59q^{9} + O(q^{10})$$ $$3q + q^{2} + 4q^{3} + 25q^{4} - 8q^{5} - 74q^{6} + 22q^{7} - 39q^{8} + 59q^{9} - 56q^{10} - 28q^{11} + 22q^{12} + 30q^{13} + 92q^{14} + 108q^{15} + 137q^{16} - 51q^{17} - 103q^{18} + 80q^{19} - 168q^{20} - 192q^{21} + 286q^{22} + 142q^{23} - 666q^{24} - 223q^{25} + 26q^{26} - 20q^{27} + 476q^{28} - 456q^{29} + 400q^{30} + 230q^{31} - 71q^{32} - 332q^{33} - 17q^{34} - 332q^{35} + 1313q^{36} + 356q^{37} + 724q^{38} + 268q^{39} - 424q^{40} - 294q^{41} - 1128q^{42} + 556q^{43} - 1122q^{44} - 384q^{45} - 704q^{46} + 640q^{47} + 774q^{48} - 269q^{49} + 547q^{50} - 68q^{51} - 774q^{52} + 302q^{53} - 1100q^{54} + 76q^{55} + 684q^{56} - 720q^{57} - 1304q^{58} + 636q^{59} + 1328q^{60} - 84q^{61} + 508q^{62} + 1122q^{63} - 919q^{64} + 408q^{65} + 2468q^{66} + 1008q^{67} - 425q^{68} + 576q^{69} - 1504q^{70} - 402q^{71} - 927q^{72} + 838q^{73} + 836q^{74} - 1548q^{75} - 908q^{76} - 504q^{77} + 1308q^{78} - 594q^{79} - 40q^{80} - 505q^{81} + 358q^{82} - 2396q^{83} - 2040q^{84} + 136q^{85} - 1264q^{86} + 1428q^{87} + 1838q^{88} - 170q^{89} - 2008q^{90} - 1016q^{91} + 4896q^{92} + 632q^{93} - 2016q^{94} - 472q^{95} + 678q^{96} - 270q^{97} + 2857q^{98} - 2920q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 14 x - 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - 10$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} + 10$$

## 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
 −3.58966 3.87707 −0.287410
−5.03251 8.47535 17.3261 0.885690 −42.6523 3.81828 −46.9339 44.8316 −4.45724
1.2 1.36122 3.15463 −6.14708 3.03171 4.29415 −7.94049 −19.2573 −17.0483 4.12682
1.3 4.67129 −7.62999 13.8209 −11.9174 −35.6419 26.1222 27.1912 31.2167 −55.6696
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 17.4.a.b 3
3.b odd 2 1 153.4.a.g 3
4.b odd 2 1 272.4.a.h 3
5.b even 2 1 425.4.a.g 3
5.c odd 4 2 425.4.b.f 6
7.b odd 2 1 833.4.a.d 3
8.b even 2 1 1088.4.a.v 3
8.d odd 2 1 1088.4.a.x 3
11.b odd 2 1 2057.4.a.e 3
12.b even 2 1 2448.4.a.bi 3
17.b even 2 1 289.4.a.b 3
17.c even 4 2 289.4.b.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.b 3 1.a even 1 1 trivial
153.4.a.g 3 3.b odd 2 1
272.4.a.h 3 4.b odd 2 1
289.4.a.b 3 17.b even 2 1
289.4.b.b 6 17.c even 4 2
425.4.a.g 3 5.b even 2 1
425.4.b.f 6 5.c odd 4 2
833.4.a.d 3 7.b odd 2 1
1088.4.a.v 3 8.b even 2 1
1088.4.a.x 3 8.d odd 2 1
2057.4.a.e 3 11.b odd 2 1
2448.4.a.bi 3 12.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$17$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - T_{2}^{2} - 24 T_{2} + 32$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(17))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 16 T^{3} - 64 T^{5} + 512 T^{6}$$
$3$ $$1 - 4 T + 19 T^{2} - 12 T^{3} + 513 T^{4} - 2916 T^{5} + 19683 T^{6}$$
$5$ $$1 + 8 T + 331 T^{2} + 2032 T^{3} + 41375 T^{4} + 125000 T^{5} + 1953125 T^{6}$$
$7$ $$1 - 22 T + 891 T^{2} - 14300 T^{3} + 305613 T^{4} - 2588278 T^{5} + 40353607 T^{6}$$
$11$ $$1 + 28 T + 2627 T^{2} + 69844 T^{3} + 3496537 T^{4} + 49603708 T^{5} + 2357947691 T^{6}$$
$13$ $$1 - 30 T + 5119 T^{2} - 141212 T^{3} + 11246443 T^{4} - 144804270 T^{5} + 10604499373 T^{6}$$
$17$ $$( 1 + 17 T )^{3}$$
$19$ $$1 - 80 T + 15945 T^{2} - 757312 T^{3} + 109366755 T^{4} - 3763670480 T^{5} + 322687697779 T^{6}$$
$23$ $$1 - 142 T + 20731 T^{2} - 1854884 T^{3} + 252234077 T^{4} - 21021096238 T^{5} + 1801152661463 T^{6}$$
$29$ $$1 + 456 T + 127075 T^{2} + 23761392 T^{3} + 3099232175 T^{4} + 271239434376 T^{5} + 14507145975869 T^{6}$$
$31$ $$1 - 230 T + 77787 T^{2} - 13785468 T^{3} + 2317352517 T^{4} - 204125846630 T^{5} + 26439622160671 T^{6}$$
$37$ $$1 - 356 T + 133995 T^{2} - 29888184 T^{3} + 6787248735 T^{4} - 913398601604 T^{5} + 129961739795077 T^{6}$$
$41$ $$1 + 294 T + 120199 T^{2} + 38886804 T^{3} + 8284235279 T^{4} + 1396530646854 T^{5} + 327381934393961 T^{6}$$
$43$ $$1 - 556 T + 289617 T^{2} - 81141512 T^{3} + 23026578819 T^{4} - 3514677855244 T^{5} + 502592611936843 T^{6}$$
$47$ $$1 - 640 T + 396797 T^{2} - 134564608 T^{3} + 41196654931 T^{4} - 6898697810560 T^{5} + 1119130473102767 T^{6}$$
$53$ $$1 - 302 T + 293171 T^{2} - 71759636 T^{3} + 43646418967 T^{4} - 6693637060958 T^{5} + 3299763591802133 T^{6}$$
$59$ $$1 - 636 T + 514369 T^{2} - 211823016 T^{3} + 105640590851 T^{4} - 26826819395676 T^{5} + 8662995818654939 T^{6}$$
$61$ $$1 + 84 T + 556531 T^{2} + 31340024 T^{3} + 126321962911 T^{4} + 4327711446324 T^{5} + 11694146092834141 T^{6}$$
$67$ $$1 - 1008 T + 967329 T^{2} - 607104160 T^{3} + 290936772027 T^{4} - 91182049226352 T^{5} + 27206534396294947 T^{6}$$
$71$ $$1 + 402 T + 483859 T^{2} + 12894428 T^{3} + 173178458549 T^{4} + 51496314136242 T^{5} + 45848500718449031 T^{6}$$
$73$ $$1 - 838 T + 1394903 T^{2} - 671950004 T^{3} + 542640980351 T^{4} - 126818081630182 T^{5} + 58871586708267913 T^{6}$$
$79$ $$1 + 594 T + 357843 T^{2} - 156405492 T^{3} + 176430554877 T^{4} + 144393948579474 T^{5} + 119851595982618319 T^{6}$$
$83$ $$1 + 2396 T + 3204249 T^{2} + 2882084008 T^{3} + 1832147922963 T^{4} + 783349134592124 T^{5} + 186940255267540403 T^{6}$$
$89$ $$1 + 170 T + 1042603 T^{2} - 206881916 T^{3} + 735002794307 T^{4} + 84486819463370 T^{5} + 350356403707485209 T^{6}$$
$97$ $$1 + 270 T + 2151919 T^{2} + 286220420 T^{3} + 1963998369487 T^{4} + 224902441330830 T^{5} + 760231058654565217 T^{6}$$