# Properties

 Label 1617.4.a.k Level $1617$ Weight $4$ Character orbit 1617.a Self dual yes Analytic conductor $95.406$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 24$$ x^2 - x - 24 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) 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 $$\beta = \frac{1}{2}(1 + \sqrt{97})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 3 q^{3} + (\beta + 16) q^{4} + (2 \beta + 6) q^{5} + 3 \beta q^{6} + (9 \beta + 24) q^{8} + 9 q^{9}+O(q^{10})$$ q + b * q^2 + 3 * q^3 + (b + 16) * q^4 + (2*b + 6) * q^5 + 3*b * q^6 + (9*b + 24) * q^8 + 9 * q^9 $$q + \beta q^{2} + 3 q^{3} + (\beta + 16) q^{4} + (2 \beta + 6) q^{5} + 3 \beta q^{6} + (9 \beta + 24) q^{8} + 9 q^{9} + (8 \beta + 48) q^{10} - 11 q^{11} + (3 \beta + 48) q^{12} + ( - 2 \beta - 14) q^{13} + (6 \beta + 18) q^{15} + (25 \beta + 88) q^{16} + (14 \beta - 60) q^{17} + 9 \beta q^{18} + (2 \beta - 26) q^{19} + (40 \beta + 144) q^{20} - 11 \beta q^{22} + ( - 10 \beta + 72) q^{23} + (27 \beta + 72) q^{24} + (28 \beta + 7) q^{25} + ( - 16 \beta - 48) q^{26} + 27 q^{27} + ( - 6 \beta - 96) q^{29} + (24 \beta + 144) q^{30} + ( - 8 \beta - 176) q^{31} + (41 \beta + 408) q^{32} - 33 q^{33} + ( - 46 \beta + 336) q^{34} + (9 \beta + 144) q^{36} + (52 \beta - 190) q^{37} + ( - 24 \beta + 48) q^{38} + ( - 6 \beta - 42) q^{39} + (120 \beta + 576) q^{40} + (14 \beta + 384) q^{41} + ( - 26 \beta + 206) q^{43} + ( - 11 \beta - 176) q^{44} + (18 \beta + 54) q^{45} + (62 \beta - 240) q^{46} + ( - 74 \beta - 96) q^{47} + (75 \beta + 264) q^{48} + (35 \beta + 672) q^{50} + (42 \beta - 180) q^{51} + ( - 48 \beta - 272) q^{52} + ( - 54 \beta - 234) q^{53} + 27 \beta q^{54} + ( - 22 \beta - 66) q^{55} + (6 \beta - 78) q^{57} + ( - 102 \beta - 144) q^{58} + (100 \beta + 36) q^{59} + (120 \beta + 432) q^{60} + ( - 34 \beta + 406) q^{61} + ( - 184 \beta - 192) q^{62} + (249 \beta + 280) q^{64} + ( - 44 \beta - 180) q^{65} - 33 \beta q^{66} + ( - 96 \beta - 340) q^{67} + (178 \beta - 624) q^{68} + ( - 30 \beta + 216) q^{69} + (54 \beta + 288) q^{71} + (81 \beta + 216) q^{72} + (28 \beta - 662) q^{73} + ( - 138 \beta + 1248) q^{74} + (84 \beta + 21) q^{75} + (8 \beta - 368) q^{76} + ( - 48 \beta - 144) q^{78} + (144 \beta + 254) q^{79} + (376 \beta + 1728) q^{80} + 81 q^{81} + (398 \beta + 336) q^{82} + ( - 156 \beta + 240) q^{83} + ( - 8 \beta + 312) q^{85} + (180 \beta - 624) q^{86} + ( - 18 \beta - 288) q^{87} + ( - 99 \beta - 264) q^{88} + ( - 72 \beta + 414) q^{89} + (72 \beta + 432) q^{90} + ( - 98 \beta + 912) q^{92} + ( - 24 \beta - 528) q^{93} + ( - 170 \beta - 1776) q^{94} + ( - 36 \beta - 60) q^{95} + (123 \beta + 1224) q^{96} + ( - 192 \beta + 322) q^{97} - 99 q^{99} +O(q^{100})$$ q + b * q^2 + 3 * q^3 + (b + 16) * q^4 + (2*b + 6) * q^5 + 3*b * q^6 + (9*b + 24) * q^8 + 9 * q^9 + (8*b + 48) * q^10 - 11 * q^11 + (3*b + 48) * q^12 + (-2*b - 14) * q^13 + (6*b + 18) * q^15 + (25*b + 88) * q^16 + (14*b - 60) * q^17 + 9*b * q^18 + (2*b - 26) * q^19 + (40*b + 144) * q^20 - 11*b * q^22 + (-10*b + 72) * q^23 + (27*b + 72) * q^24 + (28*b + 7) * q^25 + (-16*b - 48) * q^26 + 27 * q^27 + (-6*b - 96) * q^29 + (24*b + 144) * q^30 + (-8*b - 176) * q^31 + (41*b + 408) * q^32 - 33 * q^33 + (-46*b + 336) * q^34 + (9*b + 144) * q^36 + (52*b - 190) * q^37 + (-24*b + 48) * q^38 + (-6*b - 42) * q^39 + (120*b + 576) * q^40 + (14*b + 384) * q^41 + (-26*b + 206) * q^43 + (-11*b - 176) * q^44 + (18*b + 54) * q^45 + (62*b - 240) * q^46 + (-74*b - 96) * q^47 + (75*b + 264) * q^48 + (35*b + 672) * q^50 + (42*b - 180) * q^51 + (-48*b - 272) * q^52 + (-54*b - 234) * q^53 + 27*b * q^54 + (-22*b - 66) * q^55 + (6*b - 78) * q^57 + (-102*b - 144) * q^58 + (100*b + 36) * q^59 + (120*b + 432) * q^60 + (-34*b + 406) * q^61 + (-184*b - 192) * q^62 + (249*b + 280) * q^64 + (-44*b - 180) * q^65 - 33*b * q^66 + (-96*b - 340) * q^67 + (178*b - 624) * q^68 + (-30*b + 216) * q^69 + (54*b + 288) * q^71 + (81*b + 216) * q^72 + (28*b - 662) * q^73 + (-138*b + 1248) * q^74 + (84*b + 21) * q^75 + (8*b - 368) * q^76 + (-48*b - 144) * q^78 + (144*b + 254) * q^79 + (376*b + 1728) * q^80 + 81 * q^81 + (398*b + 336) * q^82 + (-156*b + 240) * q^83 + (-8*b + 312) * q^85 + (180*b - 624) * q^86 + (-18*b - 288) * q^87 + (-99*b - 264) * q^88 + (-72*b + 414) * q^89 + (72*b + 432) * q^90 + (-98*b + 912) * q^92 + (-24*b - 528) * q^93 + (-170*b - 1776) * q^94 + (-36*b - 60) * q^95 + (123*b + 1224) * q^96 + (-192*b + 322) * q^97 - 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 6 q^{3} + 33 q^{4} + 14 q^{5} + 3 q^{6} + 57 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + q^2 + 6 * q^3 + 33 * q^4 + 14 * q^5 + 3 * q^6 + 57 * q^8 + 18 * q^9 $$2 q + q^{2} + 6 q^{3} + 33 q^{4} + 14 q^{5} + 3 q^{6} + 57 q^{8} + 18 q^{9} + 104 q^{10} - 22 q^{11} + 99 q^{12} - 30 q^{13} + 42 q^{15} + 201 q^{16} - 106 q^{17} + 9 q^{18} - 50 q^{19} + 328 q^{20} - 11 q^{22} + 134 q^{23} + 171 q^{24} + 42 q^{25} - 112 q^{26} + 54 q^{27} - 198 q^{29} + 312 q^{30} - 360 q^{31} + 857 q^{32} - 66 q^{33} + 626 q^{34} + 297 q^{36} - 328 q^{37} + 72 q^{38} - 90 q^{39} + 1272 q^{40} + 782 q^{41} + 386 q^{43} - 363 q^{44} + 126 q^{45} - 418 q^{46} - 266 q^{47} + 603 q^{48} + 1379 q^{50} - 318 q^{51} - 592 q^{52} - 522 q^{53} + 27 q^{54} - 154 q^{55} - 150 q^{57} - 390 q^{58} + 172 q^{59} + 984 q^{60} + 778 q^{61} - 568 q^{62} + 809 q^{64} - 404 q^{65} - 33 q^{66} - 776 q^{67} - 1070 q^{68} + 402 q^{69} + 630 q^{71} + 513 q^{72} - 1296 q^{73} + 2358 q^{74} + 126 q^{75} - 728 q^{76} - 336 q^{78} + 652 q^{79} + 3832 q^{80} + 162 q^{81} + 1070 q^{82} + 324 q^{83} + 616 q^{85} - 1068 q^{86} - 594 q^{87} - 627 q^{88} + 756 q^{89} + 936 q^{90} + 1726 q^{92} - 1080 q^{93} - 3722 q^{94} - 156 q^{95} + 2571 q^{96} + 452 q^{97} - 198 q^{99}+O(q^{100})$$ 2 * q + q^2 + 6 * q^3 + 33 * q^4 + 14 * q^5 + 3 * q^6 + 57 * q^8 + 18 * q^9 + 104 * q^10 - 22 * q^11 + 99 * q^12 - 30 * q^13 + 42 * q^15 + 201 * q^16 - 106 * q^17 + 9 * q^18 - 50 * q^19 + 328 * q^20 - 11 * q^22 + 134 * q^23 + 171 * q^24 + 42 * q^25 - 112 * q^26 + 54 * q^27 - 198 * q^29 + 312 * q^30 - 360 * q^31 + 857 * q^32 - 66 * q^33 + 626 * q^34 + 297 * q^36 - 328 * q^37 + 72 * q^38 - 90 * q^39 + 1272 * q^40 + 782 * q^41 + 386 * q^43 - 363 * q^44 + 126 * q^45 - 418 * q^46 - 266 * q^47 + 603 * q^48 + 1379 * q^50 - 318 * q^51 - 592 * q^52 - 522 * q^53 + 27 * q^54 - 154 * q^55 - 150 * q^57 - 390 * q^58 + 172 * q^59 + 984 * q^60 + 778 * q^61 - 568 * q^62 + 809 * q^64 - 404 * q^65 - 33 * q^66 - 776 * q^67 - 1070 * q^68 + 402 * q^69 + 630 * q^71 + 513 * q^72 - 1296 * q^73 + 2358 * q^74 + 126 * q^75 - 728 * q^76 - 336 * q^78 + 652 * q^79 + 3832 * q^80 + 162 * q^81 + 1070 * q^82 + 324 * q^83 + 616 * q^85 - 1068 * q^86 - 594 * q^87 - 627 * q^88 + 756 * q^89 + 936 * q^90 + 1726 * q^92 - 1080 * q^93 - 3722 * q^94 - 156 * q^95 + 2571 * q^96 + 452 * q^97 - 198 * 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
 −4.42443 5.42443
−4.42443 3.00000 11.5756 −2.84886 −13.2733 0 −15.8199 9.00000 12.6046
1.2 5.42443 3.00000 21.4244 16.8489 16.2733 0 72.8199 9.00000 91.3954
 $$n$$: e.g. 2-40 or 990-1000 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.k 2
7.b odd 2 1 33.4.a.c 2
21.c even 2 1 99.4.a.f 2
28.d even 2 1 528.4.a.p 2
35.c odd 2 1 825.4.a.l 2
35.f even 4 2 825.4.c.h 4
56.e even 2 1 2112.4.a.bg 2
56.h odd 2 1 2112.4.a.bn 2
77.b even 2 1 363.4.a.i 2
84.h odd 2 1 1584.4.a.bj 2
105.g even 2 1 2475.4.a.p 2
231.h odd 2 1 1089.4.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.c 2 7.b odd 2 1
99.4.a.f 2 21.c even 2 1
363.4.a.i 2 77.b even 2 1
528.4.a.p 2 28.d even 2 1
825.4.a.l 2 35.c odd 2 1
825.4.c.h 4 35.f even 4 2
1089.4.a.u 2 231.h odd 2 1
1584.4.a.bj 2 84.h odd 2 1
1617.4.a.k 2 1.a even 1 1 trivial
2112.4.a.bg 2 56.e even 2 1
2112.4.a.bn 2 56.h odd 2 1
2475.4.a.p 2 105.g even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} - 24$$ T2^2 - T2 - 24 $$T_{5}^{2} - 14T_{5} - 48$$ T5^2 - 14*T5 - 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 24$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} - 14T - 48$$
$7$ $$T^{2}$$
$11$ $$(T + 11)^{2}$$
$13$ $$T^{2} + 30T + 128$$
$17$ $$T^{2} + 106T - 1944$$
$19$ $$T^{2} + 50T + 528$$
$23$ $$T^{2} - 134T + 2064$$
$29$ $$T^{2} + 198T + 8928$$
$31$ $$T^{2} + 360T + 30848$$
$37$ $$T^{2} + 328T - 38676$$
$41$ $$T^{2} - 782T + 148128$$
$43$ $$T^{2} - 386T + 20856$$
$47$ $$T^{2} + 266T - 115104$$
$53$ $$T^{2} + 522T - 2592$$
$59$ $$T^{2} - 172T - 235104$$
$61$ $$T^{2} - 778T + 123288$$
$67$ $$T^{2} + 776T - 72944$$
$71$ $$T^{2} - 630T + 28512$$
$73$ $$T^{2} + 1296 T + 400892$$
$79$ $$T^{2} - 652T - 396572$$
$83$ $$T^{2} - 324T - 563904$$
$89$ $$T^{2} - 756T + 17172$$
$97$ $$T^{2} - 452T - 842876$$