# Properties

 Label 1617.4.a.d Level $1617$ Weight $4$ Character orbit 1617.a Self dual yes Analytic conductor $95.406$ Analytic rank $1$ 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] = [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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + 3 q^{3} - 7 q^{4} + 4 q^{5} - 3 q^{6} + 15 q^{8} + 9 q^{9}+O(q^{10})$$ q - q^2 + 3 * q^3 - 7 * q^4 + 4 * q^5 - 3 * q^6 + 15 * q^8 + 9 * q^9 $$q - q^{2} + 3 q^{3} - 7 q^{4} + 4 q^{5} - 3 q^{6} + 15 q^{8} + 9 q^{9} - 4 q^{10} + 11 q^{11} - 21 q^{12} + 32 q^{13} + 12 q^{15} + 41 q^{16} - 74 q^{17} - 9 q^{18} + 60 q^{19} - 28 q^{20} - 11 q^{22} - 182 q^{23} + 45 q^{24} - 109 q^{25} - 32 q^{26} + 27 q^{27} - 90 q^{29} - 12 q^{30} + 8 q^{31} - 161 q^{32} + 33 q^{33} + 74 q^{34} - 63 q^{36} - 66 q^{37} - 60 q^{38} + 96 q^{39} + 60 q^{40} - 422 q^{41} + 408 q^{43} - 77 q^{44} + 36 q^{45} + 182 q^{46} + 506 q^{47} + 123 q^{48} + 109 q^{50} - 222 q^{51} - 224 q^{52} + 348 q^{53} - 27 q^{54} + 44 q^{55} + 180 q^{57} + 90 q^{58} + 200 q^{59} - 84 q^{60} - 132 q^{61} - 8 q^{62} - 167 q^{64} + 128 q^{65} - 33 q^{66} - 1036 q^{67} + 518 q^{68} - 546 q^{69} + 762 q^{71} + 135 q^{72} + 542 q^{73} + 66 q^{74} - 327 q^{75} - 420 q^{76} - 96 q^{78} - 550 q^{79} + 164 q^{80} + 81 q^{81} + 422 q^{82} + 132 q^{83} - 296 q^{85} - 408 q^{86} - 270 q^{87} + 165 q^{88} - 570 q^{89} - 36 q^{90} + 1274 q^{92} + 24 q^{93} - 506 q^{94} + 240 q^{95} - 483 q^{96} - 14 q^{97} + 99 q^{99}+O(q^{100})$$ q - q^2 + 3 * q^3 - 7 * q^4 + 4 * q^5 - 3 * q^6 + 15 * q^8 + 9 * q^9 - 4 * q^10 + 11 * q^11 - 21 * q^12 + 32 * q^13 + 12 * q^15 + 41 * q^16 - 74 * q^17 - 9 * q^18 + 60 * q^19 - 28 * q^20 - 11 * q^22 - 182 * q^23 + 45 * q^24 - 109 * q^25 - 32 * q^26 + 27 * q^27 - 90 * q^29 - 12 * q^30 + 8 * q^31 - 161 * q^32 + 33 * q^33 + 74 * q^34 - 63 * q^36 - 66 * q^37 - 60 * q^38 + 96 * q^39 + 60 * q^40 - 422 * q^41 + 408 * q^43 - 77 * q^44 + 36 * q^45 + 182 * q^46 + 506 * q^47 + 123 * q^48 + 109 * q^50 - 222 * q^51 - 224 * q^52 + 348 * q^53 - 27 * q^54 + 44 * q^55 + 180 * q^57 + 90 * q^58 + 200 * q^59 - 84 * q^60 - 132 * q^61 - 8 * q^62 - 167 * q^64 + 128 * q^65 - 33 * q^66 - 1036 * q^67 + 518 * q^68 - 546 * q^69 + 762 * q^71 + 135 * q^72 + 542 * q^73 + 66 * q^74 - 327 * q^75 - 420 * q^76 - 96 * q^78 - 550 * q^79 + 164 * q^80 + 81 * q^81 + 422 * q^82 + 132 * q^83 - 296 * q^85 - 408 * q^86 - 270 * q^87 + 165 * q^88 - 570 * q^89 - 36 * q^90 + 1274 * q^92 + 24 * q^93 - 506 * q^94 + 240 * q^95 - 483 * q^96 - 14 * q^97 + 99 * 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
−1.00000 3.00000 −7.00000 4.00000 −3.00000 0 15.0000 9.00000 −4.00000
 $$n$$: 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.d 1
7.b odd 2 1 33.4.a.b 1
21.c even 2 1 99.4.a.a 1
28.d even 2 1 528.4.a.h 1
35.c odd 2 1 825.4.a.f 1
35.f even 4 2 825.4.c.f 2
56.e even 2 1 2112.4.a.h 1
56.h odd 2 1 2112.4.a.u 1
77.b even 2 1 363.4.a.d 1
84.h odd 2 1 1584.4.a.l 1
105.g even 2 1 2475.4.a.e 1
231.h odd 2 1 1089.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.b 1 7.b odd 2 1
99.4.a.a 1 21.c even 2 1
363.4.a.d 1 77.b even 2 1
528.4.a.h 1 28.d even 2 1
825.4.a.f 1 35.c odd 2 1
825.4.c.f 2 35.f even 4 2
1089.4.a.e 1 231.h odd 2 1
1584.4.a.l 1 84.h odd 2 1
1617.4.a.d 1 1.a even 1 1 trivial
2112.4.a.h 1 56.e even 2 1
2112.4.a.u 1 56.h odd 2 1
2475.4.a.e 1 105.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1617))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{5} - 4$$ T5 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 3$$
$5$ $$T - 4$$
$7$ $$T$$
$11$ $$T - 11$$
$13$ $$T - 32$$
$17$ $$T + 74$$
$19$ $$T - 60$$
$23$ $$T + 182$$
$29$ $$T + 90$$
$31$ $$T - 8$$
$37$ $$T + 66$$
$41$ $$T + 422$$
$43$ $$T - 408$$
$47$ $$T - 506$$
$53$ $$T - 348$$
$59$ $$T - 200$$
$61$ $$T + 132$$
$67$ $$T + 1036$$
$71$ $$T - 762$$
$73$ $$T - 542$$
$79$ $$T + 550$$
$83$ $$T - 132$$
$89$ $$T + 570$$
$97$ $$T + 14$$