# 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 Defining polynomial: $$x^{4} + 74x^{2} + 1072$$ x^4 + 74*x^2 + 1072 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{2} + \beta_1 q^{3} + ( - \beta_{2} + 1) q^{4} - \beta_{3} q^{5} + (\beta_{3} - 2 \beta_1) q^{6} + (\beta_{3} - \beta_1) q^{7} + ( - 7 \beta_{2} - 1) q^{8} + (6 \beta_{2} - 13) q^{9}+O(q^{10})$$ q + (b2 - 1) * q^2 + b1 * q^3 + (-b2 + 1) * q^4 - b3 * q^5 + (b3 - 2*b1) * q^6 + (b3 - b1) * q^7 + (-7*b2 - 1) * q^8 + (6*b2 - 13) * q^9 $$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})$$ q + (b2 - 1) * q^2 + b1 * q^3 + (-b2 + 1) * q^4 - b3 * q^5 + (b3 - 2*b1) * q^6 + (b3 - b1) * q^7 + (-7*b2 - 1) * q^8 + (6*b2 - 13) * q^9 + (-b3 - 6*b1) * q^10 + 7*b1 * q^11 + (-b3 + 2*b1) * q^12 + (-14*b2 + 42) * q^13 + 8*b1 * q^14 + (28*b2 - 8) * q^15 + (7*b2 - 63) * q^16 + (b3 + 14*b2 - 8*b1 + 21) * q^17 + (-13*b2 + 61) * q^18 - 28 * q^19 + (b3 + 6*b1) * q^20 + (-34*b2 + 48) * q^21 + (7*b3 - 14*b1) * q^22 + (-7*b3 - 7*b1) * q^23 + (-7*b3 + 6*b1) * q^24 + (-48*b2 - 91) * q^25 + (42*b2 - 154) * q^26 + (6*b3 + 8*b1) * q^27 - 8*b1 * q^28 + 7*b3 * q^29 + (-8*b2 + 232) * q^30 + (-b3 - 7*b1) * q^31 + (-7*b2 + 127) * q^32 + (42*b2 - 280) * q^33 + (-7*b3 + 21*b2 + 22*b1 + 91) * q^34 + (20*b2 + 224) * q^35 + (13*b2 - 61) * q^36 + (-7*b3 - 28*b1) * q^37 + (-28*b2 + 28) * q^38 + (-14*b3 + 56*b1) * q^39 + (15*b3 + 42*b1) * q^40 + (14*b3 - 20*b1) * q^41 + (48*b2 - 320) * q^42 + (-76*b2 - 92) * q^43 + (-7*b3 + 14*b1) * q^44 + (b3 - 36*b1) * q^45 + (-14*b3 - 28*b1) * q^46 + (28*b2 + 224) * q^47 + (7*b3 - 70*b1) * q^48 + (14*b2 + 71) * q^49 + (-91*b2 - 293) * q^50 + (14*b3 - 76*b2 + 7*b1 + 328) * q^51 + (-42*b2 + 154) * q^52 + (-92*b2 - 98) * q^53 + (14*b3 + 20*b1) * q^54 + (196*b2 - 56) * q^55 + (-8*b3 - 48*b1) * q^56 - 28*b1 * q^57 + (7*b3 + 42*b1) * q^58 + (140*b2 - 364) * q^59 + (8*b2 - 232) * q^60 + (-21*b3 + 64*b1) * q^61 + (-8*b3 + 8*b1) * q^62 + (-7*b3 + 55*b1) * q^63 + (71*b2 + 321) * q^64 + (-14*b3 + 84*b1) * q^65 + (-280*b2 + 616) * q^66 + (-64*b2 - 28) * q^67 + (7*b3 - 21*b2 - 22*b1 - 91) * q^68 + (154*b2 + 224) * q^69 + (224*b2 - 64) * q^70 + (7*b3 - 21*b1) * q^71 + (43*b2 - 323) * q^72 + (56*b3 - 48*b1) * q^73 + (-35*b3 + 14*b1) * q^74 + (-48*b3 - 43*b1) * q^75 + (28*b2 - 28) * q^76 + (-238*b2 + 336) * q^77 + (42*b3 - 196*b1) * q^78 + (21*b3 + 77*b1) * q^79 + (49*b3 - 42*b1) * q^80 + (42*b2 - 623) * q^81 + (-6*b3 + 124*b1) * q^82 + (84*b2 - 756) * q^83 + (-48*b2 + 320) * q^84 + (-49*b3 - 176*b2 - 84*b1 + 280) * q^85 + (-92*b2 - 516) * q^86 + (-196*b2 + 56) * q^87 + (-49*b3 + 42*b1) * q^88 + (-42*b2 + 518) * q^89 + (-35*b3 + 78*b1) * q^90 + (28*b3 - 140*b1) * q^91 + (14*b3 + 28*b1) * q^92 + (-14*b2 + 272) * q^93 + 224*b2 * q^94 + 28*b3 * q^95 + (-7*b3 + 134*b1) * q^96 + (-22*b3 + 28*b1) * q^97 + (71*b2 + 41) * q^98 + (42*b3 - 133*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 2 q^{4} - 18 q^{8} - 40 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 2 * q^4 - 18 * q^8 - 40 * q^9 $$4 q - 2 q^{2} + 2 q^{4} - 18 q^{8} - 40 q^{9} + 140 q^{13} + 24 q^{15} - 238 q^{16} + 112 q^{17} + 218 q^{18} - 112 q^{19} + 124 q^{21} - 460 q^{25} - 532 q^{26} + 912 q^{30} + 494 q^{32} - 1036 q^{33} + 406 q^{34} + 936 q^{35} - 218 q^{36} + 56 q^{38} - 1184 q^{42} - 520 q^{43} + 952 q^{47} + 312 q^{49} - 1354 q^{50} + 1160 q^{51} + 532 q^{52} - 576 q^{53} + 168 q^{55} - 1176 q^{59} - 912 q^{60} + 1426 q^{64} + 1904 q^{66} - 240 q^{67} - 406 q^{68} + 1204 q^{69} + 192 q^{70} - 1206 q^{72} - 56 q^{76} + 868 q^{77} - 2408 q^{81} - 2856 q^{83} + 1184 q^{84} + 768 q^{85} - 2248 q^{86} - 168 q^{87} + 1988 q^{89} + 1060 q^{93} + 448 q^{94} + 306 q^{98}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} + 40 ) / 6$$ (v^2 + 40) / 6 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 46\nu ) / 6$$ (v^3 + 46*v) / 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$6\beta_{2} - 40$$ 6*b2 - 40 $$\nu^{3}$$ $$=$$ $$6\beta_{3} - 46\beta_1$$ 6*b3 - 46*b1

## 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

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