Properties

 Label 1617.4.a.p Level $1617$ Weight $4$ Character orbit 1617.a Self dual yes Analytic conductor $95.406$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,4,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1617.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1617 = 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1617.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: $$95.4060884793$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 28x^{3} - 11x^{2} + 108x - 64$$ x^5 - 28*x^3 - 11*x^2 + 108*x - 64 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 231) 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} - 3 q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + ( - \beta_{4} + \beta_1 - 1) q^{5} + (3 \beta_1 - 3) q^{6} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 12) q^{8}+ \cdots + 9 q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 - 3 * q^3 + (b2 - b1 + 4) * q^4 + (-b4 + b1 - 1) * q^5 + (3*b1 - 3) * q^6 + (-b4 + b3 + 2*b2 - 5*b1 + 12) * q^8 + 9 * q^9 $$q + ( - \beta_1 + 1) q^{2} - 3 q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + ( - \beta_{4} + \beta_1 - 1) q^{5} + (3 \beta_1 - 3) q^{6} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 12) q^{8}+ \cdots + 99 q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 - 3 * q^3 + (b2 - b1 + 4) * q^4 + (-b4 + b1 - 1) * q^5 + (3*b1 - 3) * q^6 + (-b4 + b3 + 2*b2 - 5*b1 + 12) * q^8 + 9 * q^9 + (-b4 + 2*b3 + 2*b2 + 4*b1 - 11) * q^10 + 11 * q^11 + (-3*b2 + 3*b1 - 12) * q^12 + (-b4 - b3 + b2 - b1 - 22) * q^13 + (3*b4 - 3*b1 + 3) * q^15 + (-4*b4 + 6*b3 + 4*b2 - 15*b1 + 41) * q^16 + (2*b4 + 7*b3 + 5*b2 - 14*b1 - 29) * q^17 + (-9*b1 + 9) * q^18 + (-3*b4 - 4*b3 + 5*b1 - 21) * q^19 + (3*b4 + 8*b3 + 5*b2 - 4*b1 - 46) * q^20 + (-11*b1 + 11) * q^22 + (-4*b4 + 8*b3 - 6*b1 - 4) * q^23 + (3*b4 - 3*b3 - 6*b2 + 15*b1 - 36) * q^24 + (-5*b4 - 13*b3 - 15*b2 - 5*b1 + 43) * q^25 + (-b4 + b3 + 3*b2 + 12*b1) * q^26 - 27 * q^27 + (-b4 + 7*b3 + 13*b2 + 35*b1 + 10) * q^29 + (3*b4 - 6*b3 - 6*b2 - 12*b1 + 33) * q^30 + (-8*b4 - 17*b3 + b2 + 4*b1 + 59) * q^31 + (-6*b4 + 16*b3 + 27*b2 - b1 + 104) * q^32 - 33 * q^33 + (-10*b4 + 15*b3 + 27*b2 + 6*b1 + 113) * q^34 + (9*b2 - 9*b1 + 36) * q^36 + (b4 - 19*b3 - 13*b2 + 17*b1 - 6) * q^37 + (b4 - 2*b3 - 4*b2 + 14*b1 - 53) * q^38 + (3*b4 + 3*b3 - 3*b2 + 3*b1 + 66) * q^39 + (-2*b4 - b3 - 8*b1 + 68) * q^40 + (-24*b4 - 7*b3 + 7*b2 - 34*b1 - 77) * q^41 + (4*b4 - 16*b3 + 16*b2 - 2*b1 + 80) * q^43 + (11*b2 - 11*b1 + 44) * q^44 + (-9*b4 + 9*b1 - 9) * q^45 + (-12*b4 + 24*b3 + 34*b2 + 48*b1 + 26) * q^46 + (-11*b4 + 10*b3 + 22*b2 - 55*b1 - 15) * q^47 + (12*b4 - 18*b3 - 12*b2 + 45*b1 - 123) * q^48 + (23*b4 - 31*b3 - 21*b2 + 55*b1 + 93) * q^50 + (-6*b4 - 21*b3 - 15*b2 + 42*b1 + 87) * q^51 + (3*b4 + 15*b3 - 12*b2 - 12*b1 + 55) * q^52 + (-26*b4 + 8*b3 + 12*b2 + 16*b1 + 308) * q^53 + (27*b1 - 27) * q^54 + (-11*b4 + 11*b1 - 11) * q^55 + (9*b4 + 12*b3 - 15*b1 + 63) * q^57 + (-21*b4 + 29*b3 - 5*b2 - 96*b1 - 344) * q^58 + (47*b4 + 11*b3 - 51*b2 - 91*b1 - 160) * q^59 + (-9*b4 - 24*b3 - 15*b2 + 12*b1 + 138) * q^60 + (12*b4 - 46*b3 - 22*b2 - 40*b1 - 132) * q^61 + (8*b4 - 17*b3 - 13*b2 - 112*b1 + 113) * q^62 + (-17*b4 + 23*b3 + 46*b2 - 145*b1 - 152) * q^64 + (21*b4 + 9*b3 - 5*b2 - 19*b1 + 100) * q^65 + (33*b1 - 33) * q^66 + (13*b4 + 3*b3 + 21*b2 + 139*b1 - 126) * q^67 + (-68*b4 + 21*b3 + 41*b2 - 154*b1 + 349) * q^68 + (12*b4 - 24*b3 + 18*b1 + 12) * q^69 + (-20*b4 + 14*b3 - 14*b2 + 4*b1 - 142) * q^71 + (-9*b4 + 9*b3 + 18*b2 - 45*b1 + 108) * q^72 + (23*b4 + 36*b3 + 24*b2 - 23*b1 - 189) * q^73 + (33*b4 - 53*b3 - 71*b2 + 44*b1 - 164) * q^74 + (15*b4 + 39*b3 + 45*b2 + 15*b1 - 129) * q^75 + (31*b4 + 22*b3 - 25*b2 + 38*b1 - 50) * q^76 + (3*b4 - 3*b3 - 9*b2 - 36*b1) * q^78 + (-52*b4 - 16*b3 - 88*b1 - 28) * q^79 + (-25*b4 - 62*b3 - 28*b2 - 34*b1 + 531) * q^80 + 81 * q^81 + (-24*b4 + 41*b3 + 99*b2 + 58*b1 + 391) * q^82 + (16*b4 - 13*b3 + 69*b2 + 64*b1 - 205) * q^83 + (90*b4 - 4*b3 + 48*b2 - 124*b1 - 326) * q^85 + (4*b4 - 24*b3 - 26*b2 - 300*b1 + 258) * q^86 + (3*b4 - 21*b3 - 39*b2 - 105*b1 - 30) * q^87 + (-11*b4 + 11*b3 + 22*b2 - 55*b1 + 132) * q^88 + (-8*b4 - 20*b3 + 16*b2 - 164*b1 + 222) * q^89 + (-9*b4 + 18*b3 + 18*b2 + 36*b1 - 99) * q^90 + (-38*b4 + 42*b3 + 70*b2 - 152*b1 - 408) * q^92 + (24*b4 + 51*b3 - 3*b2 - 12*b1 - 177) * q^93 + (-43*b4 + 64*b3 + 130*b2 - 110*b1 + 661) * q^94 + (-5*b4 + 5*b3 - 37*b2 + 9*b1 + 394) * q^95 + (18*b4 - 48*b3 - 81*b2 + 3*b1 - 312) * q^96 + (-2*b4 + 14*b3 - 34*b2 + 188*b1 + 178) * q^97 + 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{2} - 15 q^{3} + 21 q^{4} - 7 q^{5} - 15 q^{6} + 60 q^{8} + 45 q^{9}+O(q^{10})$$ 5 * q + 5 * q^2 - 15 * q^3 + 21 * q^4 - 7 * q^5 - 15 * q^6 + 60 * q^8 + 45 * q^9 $$5 q + 5 q^{2} - 15 q^{3} + 21 q^{4} - 7 q^{5} - 15 q^{6} + 60 q^{8} + 45 q^{9} - 55 q^{10} + 55 q^{11} - 63 q^{12} - 111 q^{13} + 21 q^{15} + 201 q^{16} - 136 q^{17} + 45 q^{18} - 111 q^{19} - 219 q^{20} + 55 q^{22} - 28 q^{23} - 180 q^{24} + 190 q^{25} + q^{26} - 135 q^{27} + 61 q^{29} + 165 q^{30} + 280 q^{31} + 535 q^{32} - 165 q^{33} + 572 q^{34} + 189 q^{36} - 41 q^{37} - 267 q^{38} + 333 q^{39} + 336 q^{40} - 426 q^{41} + 424 q^{43} + 231 q^{44} - 63 q^{45} + 140 q^{46} - 75 q^{47} - 603 q^{48} + 490 q^{50} + 408 q^{51} + 269 q^{52} + 1500 q^{53} - 135 q^{54} - 77 q^{55} + 333 q^{57} - 1767 q^{58} - 757 q^{59} + 657 q^{60} - 658 q^{61} + 568 q^{62} - 748 q^{64} + 537 q^{65} - 165 q^{66} - 583 q^{67} + 1650 q^{68} + 84 q^{69} - 764 q^{71} + 540 q^{72} - 875 q^{73} - 825 q^{74} - 570 q^{75} - 213 q^{76} - 3 q^{78} - 244 q^{79} + 2577 q^{80} + 405 q^{81} + 2006 q^{82} - 924 q^{83} - 1402 q^{85} + 1272 q^{86} - 183 q^{87} + 660 q^{88} + 1110 q^{89} - 495 q^{90} - 2046 q^{92} - 840 q^{93} + 3349 q^{94} + 1923 q^{95} - 1605 q^{96} + 852 q^{97} + 495 q^{99}+O(q^{100})$$ 5 * q + 5 * q^2 - 15 * q^3 + 21 * q^4 - 7 * q^5 - 15 * q^6 + 60 * q^8 + 45 * q^9 - 55 * q^10 + 55 * q^11 - 63 * q^12 - 111 * q^13 + 21 * q^15 + 201 * q^16 - 136 * q^17 + 45 * q^18 - 111 * q^19 - 219 * q^20 + 55 * q^22 - 28 * q^23 - 180 * q^24 + 190 * q^25 + q^26 - 135 * q^27 + 61 * q^29 + 165 * q^30 + 280 * q^31 + 535 * q^32 - 165 * q^33 + 572 * q^34 + 189 * q^36 - 41 * q^37 - 267 * q^38 + 333 * q^39 + 336 * q^40 - 426 * q^41 + 424 * q^43 + 231 * q^44 - 63 * q^45 + 140 * q^46 - 75 * q^47 - 603 * q^48 + 490 * q^50 + 408 * q^51 + 269 * q^52 + 1500 * q^53 - 135 * q^54 - 77 * q^55 + 333 * q^57 - 1767 * q^58 - 757 * q^59 + 657 * q^60 - 658 * q^61 + 568 * q^62 - 748 * q^64 + 537 * q^65 - 165 * q^66 - 583 * q^67 + 1650 * q^68 + 84 * q^69 - 764 * q^71 + 540 * q^72 - 875 * q^73 - 825 * q^74 - 570 * q^75 - 213 * q^76 - 3 * q^78 - 244 * q^79 + 2577 * q^80 + 405 * q^81 + 2006 * q^82 - 924 * q^83 - 1402 * q^85 + 1272 * q^86 - 183 * q^87 + 660 * q^88 + 1110 * q^89 - 495 * q^90 - 2046 * q^92 - 840 * q^93 + 3349 * q^94 + 1923 * q^95 - 1605 * q^96 + 852 * q^97 + 495 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 28x^{3} - 11x^{2} + 108x - 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 11$$ v^2 - v - 11 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - 26\nu^{2} - 17\nu + 64 ) / 2$$ (v^4 - 26*v^2 - 17*v + 64) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{4} + 2\nu^{3} - 28\nu^{2} - 57\nu + 74 ) / 2$$ (v^4 + 2*v^3 - 28*v^2 - 57*v + 74) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 11$$ b2 + b1 + 11 $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{2} + 21\beta _1 + 6$$ b4 - b3 + b2 + 21*b1 + 6 $$\nu^{4}$$ $$=$$ $$2\beta_{3} + 26\beta_{2} + 43\beta _1 + 222$$ 2*b3 + 26*b2 + 43*b1 + 222

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
 5.15106 1.30014 0.767088 −2.85551 −4.36278
−4.15106 −3.00000 9.23129 −3.26204 12.4532 0 −5.11115 9.00000 13.5409
1.2 −0.300143 −3.00000 −7.90991 20.3930 0.900430 0 4.77526 9.00000 −6.12084
1.3 0.232912 −3.00000 −7.94575 −7.75746 −0.698736 0 −3.71396 9.00000 −1.80680
1.4 3.85551 −3.00000 6.86494 −18.0422 −11.5665 0 −4.37623 9.00000 −69.5618
1.5 5.36278 −3.00000 20.7594 1.66863 −16.0883 0 68.4261 9.00000 8.94849
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

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 1617.4.a.p 5
7.b odd 2 1 231.4.a.l 5
21.c even 2 1 693.4.a.n 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.a.l 5 7.b odd 2 1
693.4.a.n 5 21.c even 2 1
1617.4.a.p 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1617))$$:

 $$T_{2}^{5} - 5T_{2}^{4} - 18T_{2}^{3} + 85T_{2}^{2} + 7T_{2} - 6$$ T2^5 - 5*T2^4 - 18*T2^3 + 85*T2^2 + 7*T2 - 6 $$T_{5}^{5} + 7T_{5}^{4} - 383T_{5}^{3} - 3499T_{5}^{2} - 2446T_{5} + 15536$$ T5^5 + 7*T5^4 - 383*T5^3 - 3499*T5^2 - 2446*T5 + 15536

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 5 T^{4} + \cdots - 6$$
$3$ $$(T + 3)^{5}$$
$5$ $$T^{5} + 7 T^{4} + \cdots + 15536$$
$7$ $$T^{5}$$
$11$ $$(T - 11)^{5}$$
$13$ $$T^{5} + 111 T^{4} + \cdots + 276332$$
$17$ $$T^{5} + \cdots + 3184406528$$
$19$ $$T^{5} + 111 T^{4} + \cdots + 2596968$$
$23$ $$T^{5} + \cdots - 9848447488$$
$29$ $$T^{5} + \cdots - 1678137108$$
$31$ $$T^{5} + \cdots + 97533466624$$
$37$ $$T^{5} + \cdots - 82159135548$$
$41$ $$T^{5} + \cdots + 2830976555008$$
$43$ $$T^{5} + \cdots - 1905935989632$$
$47$ $$T^{5} + \cdots - 417603397024$$
$53$ $$T^{5} + \cdots + 3927875616864$$
$59$ $$T^{5} + \cdots + 211529924045472$$
$61$ $$T^{5} + \cdots - 14480622579168$$
$67$ $$T^{5} + \cdots - 8392173156048$$
$71$ $$T^{5} + \cdots + 2460667275264$$
$73$ $$T^{5} + \cdots + 1051447598248$$
$79$ $$T^{5} + \cdots + 93230064402432$$
$83$ $$T^{5} + \cdots - 15123167361792$$
$89$ $$T^{5} + \cdots + 1811669774112$$
$97$ $$T^{5} + \cdots - 346762905716192$$