Properties

 Label 289.4.a.a Level $289$ Weight $4$ Character orbit 289.a Self dual yes Analytic conductor $17.052$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,4,Mod(1,289)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("289.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$289 = 17^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 289.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$17.0515519917$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 3 q^{2} + 8 q^{3} + q^{4} - 6 q^{5} - 24 q^{6} + 28 q^{7} + 21 q^{8} + 37 q^{9}+O(q^{10})$$ q - 3 * q^2 + 8 * q^3 + q^4 - 6 * q^5 - 24 * q^6 + 28 * q^7 + 21 * q^8 + 37 * q^9 $$q - 3 q^{2} + 8 q^{3} + q^{4} - 6 q^{5} - 24 q^{6} + 28 q^{7} + 21 q^{8} + 37 q^{9} + 18 q^{10} + 24 q^{11} + 8 q^{12} - 58 q^{13} - 84 q^{14} - 48 q^{15} - 71 q^{16} - 111 q^{18} + 116 q^{19} - 6 q^{20} + 224 q^{21} - 72 q^{22} + 60 q^{23} + 168 q^{24} - 89 q^{25} + 174 q^{26} + 80 q^{27} + 28 q^{28} - 30 q^{29} + 144 q^{30} + 172 q^{31} + 45 q^{32} + 192 q^{33} - 168 q^{35} + 37 q^{36} + 58 q^{37} - 348 q^{38} - 464 q^{39} - 126 q^{40} + 342 q^{41} - 672 q^{42} - 148 q^{43} + 24 q^{44} - 222 q^{45} - 180 q^{46} + 288 q^{47} - 568 q^{48} + 441 q^{49} + 267 q^{50} - 58 q^{52} + 318 q^{53} - 240 q^{54} - 144 q^{55} + 588 q^{56} + 928 q^{57} + 90 q^{58} + 252 q^{59} - 48 q^{60} - 110 q^{61} - 516 q^{62} + 1036 q^{63} + 433 q^{64} + 348 q^{65} - 576 q^{66} - 484 q^{67} + 480 q^{69} + 504 q^{70} + 708 q^{71} + 777 q^{72} - 362 q^{73} - 174 q^{74} - 712 q^{75} + 116 q^{76} + 672 q^{77} + 1392 q^{78} + 484 q^{79} + 426 q^{80} - 359 q^{81} - 1026 q^{82} + 756 q^{83} + 224 q^{84} + 444 q^{86} - 240 q^{87} + 504 q^{88} - 774 q^{89} + 666 q^{90} - 1624 q^{91} + 60 q^{92} + 1376 q^{93} - 864 q^{94} - 696 q^{95} + 360 q^{96} + 382 q^{97} - 1323 q^{98} + 888 q^{99}+O(q^{100})$$ q - 3 * q^2 + 8 * q^3 + q^4 - 6 * q^5 - 24 * q^6 + 28 * q^7 + 21 * q^8 + 37 * q^9 + 18 * q^10 + 24 * q^11 + 8 * q^12 - 58 * q^13 - 84 * q^14 - 48 * q^15 - 71 * q^16 - 111 * q^18 + 116 * q^19 - 6 * q^20 + 224 * q^21 - 72 * q^22 + 60 * q^23 + 168 * q^24 - 89 * q^25 + 174 * q^26 + 80 * q^27 + 28 * q^28 - 30 * q^29 + 144 * q^30 + 172 * q^31 + 45 * q^32 + 192 * q^33 - 168 * q^35 + 37 * q^36 + 58 * q^37 - 348 * q^38 - 464 * q^39 - 126 * q^40 + 342 * q^41 - 672 * q^42 - 148 * q^43 + 24 * q^44 - 222 * q^45 - 180 * q^46 + 288 * q^47 - 568 * q^48 + 441 * q^49 + 267 * q^50 - 58 * q^52 + 318 * q^53 - 240 * q^54 - 144 * q^55 + 588 * q^56 + 928 * q^57 + 90 * q^58 + 252 * q^59 - 48 * q^60 - 110 * q^61 - 516 * q^62 + 1036 * q^63 + 433 * q^64 + 348 * q^65 - 576 * q^66 - 484 * q^67 + 480 * q^69 + 504 * q^70 + 708 * q^71 + 777 * q^72 - 362 * q^73 - 174 * q^74 - 712 * q^75 + 116 * q^76 + 672 * q^77 + 1392 * q^78 + 484 * q^79 + 426 * q^80 - 359 * q^81 - 1026 * q^82 + 756 * q^83 + 224 * q^84 + 444 * q^86 - 240 * q^87 + 504 * q^88 - 774 * q^89 + 666 * q^90 - 1624 * q^91 + 60 * q^92 + 1376 * q^93 - 864 * q^94 - 696 * q^95 + 360 * q^96 + 382 * q^97 - 1323 * q^98 + 888 * q^99

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−3.00000 8.00000 1.00000 −6.00000 −24.0000 28.0000 21.0000 37.0000 18.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

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

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 289.4.a.a 1
17.b even 2 1 17.4.a.a 1
17.c even 4 2 289.4.b.a 2
51.c odd 2 1 153.4.a.d 1
68.d odd 2 1 272.4.a.d 1
85.c even 2 1 425.4.a.d 1
85.g odd 4 2 425.4.b.c 2
119.d odd 2 1 833.4.a.a 1
136.e odd 2 1 1088.4.a.a 1
136.h even 2 1 1088.4.a.l 1
187.b odd 2 1 2057.4.a.d 1
204.h even 2 1 2448.4.a.f 1

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(289))$$:

 $$T_{2} + 3$$ T2 + 3 $$T_{3} - 8$$ T3 - 8

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T - 8$$
$5$ $$T + 6$$
$7$ $$T - 28$$
$11$ $$T - 24$$
$13$ $$T + 58$$
$17$ $$T$$
$19$ $$T - 116$$
$23$ $$T - 60$$
$29$ $$T + 30$$
$31$ $$T - 172$$
$37$ $$T - 58$$
$41$ $$T - 342$$
$43$ $$T + 148$$
$47$ $$T - 288$$
$53$ $$T - 318$$
$59$ $$T - 252$$
$61$ $$T + 110$$
$67$ $$T + 484$$
$71$ $$T - 708$$
$73$ $$T + 362$$
$79$ $$T - 484$$
$83$ $$T - 756$$
$89$ $$T + 774$$
$97$ $$T - 382$$