# Properties

 Label 115.4.a.b Level $115$ Weight $4$ Character orbit 115.a Self dual yes Analytic conductor $6.785$ 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] = [115,4,Mod(1,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("115.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 115.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: $$6.78521965066$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 3 q^{3} - 4 q^{4} + 5 q^{5} - 6 q^{6} - 2 q^{7} - 24 q^{8} - 18 q^{9}+O(q^{10})$$ q + 2 * q^2 - 3 * q^3 - 4 * q^4 + 5 * q^5 - 6 * q^6 - 2 * q^7 - 24 * q^8 - 18 * q^9 $$q + 2 q^{2} - 3 q^{3} - 4 q^{4} + 5 q^{5} - 6 q^{6} - 2 q^{7} - 24 q^{8} - 18 q^{9} + 10 q^{10} - 16 q^{11} + 12 q^{12} - 47 q^{13} - 4 q^{14} - 15 q^{15} - 16 q^{16} - 24 q^{17} - 36 q^{18} - 56 q^{19} - 20 q^{20} + 6 q^{21} - 32 q^{22} - 23 q^{23} + 72 q^{24} + 25 q^{25} - 94 q^{26} + 135 q^{27} + 8 q^{28} + 85 q^{29} - 30 q^{30} + 67 q^{31} + 160 q^{32} + 48 q^{33} - 48 q^{34} - 10 q^{35} + 72 q^{36} + 104 q^{37} - 112 q^{38} + 141 q^{39} - 120 q^{40} - 53 q^{41} + 12 q^{42} - 234 q^{43} + 64 q^{44} - 90 q^{45} - 46 q^{46} + 285 q^{47} + 48 q^{48} - 339 q^{49} + 50 q^{50} + 72 q^{51} + 188 q^{52} + 2 q^{53} + 270 q^{54} - 80 q^{55} + 48 q^{56} + 168 q^{57} + 170 q^{58} + 80 q^{59} + 60 q^{60} - 764 q^{61} + 134 q^{62} + 36 q^{63} + 448 q^{64} - 235 q^{65} + 96 q^{66} + 236 q^{67} + 96 q^{68} + 69 q^{69} - 20 q^{70} - 289 q^{71} + 432 q^{72} - 225 q^{73} + 208 q^{74} - 75 q^{75} + 224 q^{76} + 32 q^{77} + 282 q^{78} + 24 q^{79} - 80 q^{80} + 81 q^{81} - 106 q^{82} + 684 q^{83} - 24 q^{84} - 120 q^{85} - 468 q^{86} - 255 q^{87} + 384 q^{88} - 1370 q^{89} - 180 q^{90} + 94 q^{91} + 92 q^{92} - 201 q^{93} + 570 q^{94} - 280 q^{95} - 480 q^{96} - 110 q^{97} - 678 q^{98} + 288 q^{99}+O(q^{100})$$ q + 2 * q^2 - 3 * q^3 - 4 * q^4 + 5 * q^5 - 6 * q^6 - 2 * q^7 - 24 * q^8 - 18 * q^9 + 10 * q^10 - 16 * q^11 + 12 * q^12 - 47 * q^13 - 4 * q^14 - 15 * q^15 - 16 * q^16 - 24 * q^17 - 36 * q^18 - 56 * q^19 - 20 * q^20 + 6 * q^21 - 32 * q^22 - 23 * q^23 + 72 * q^24 + 25 * q^25 - 94 * q^26 + 135 * q^27 + 8 * q^28 + 85 * q^29 - 30 * q^30 + 67 * q^31 + 160 * q^32 + 48 * q^33 - 48 * q^34 - 10 * q^35 + 72 * q^36 + 104 * q^37 - 112 * q^38 + 141 * q^39 - 120 * q^40 - 53 * q^41 + 12 * q^42 - 234 * q^43 + 64 * q^44 - 90 * q^45 - 46 * q^46 + 285 * q^47 + 48 * q^48 - 339 * q^49 + 50 * q^50 + 72 * q^51 + 188 * q^52 + 2 * q^53 + 270 * q^54 - 80 * q^55 + 48 * q^56 + 168 * q^57 + 170 * q^58 + 80 * q^59 + 60 * q^60 - 764 * q^61 + 134 * q^62 + 36 * q^63 + 448 * q^64 - 235 * q^65 + 96 * q^66 + 236 * q^67 + 96 * q^68 + 69 * q^69 - 20 * q^70 - 289 * q^71 + 432 * q^72 - 225 * q^73 + 208 * q^74 - 75 * q^75 + 224 * q^76 + 32 * q^77 + 282 * q^78 + 24 * q^79 - 80 * q^80 + 81 * q^81 - 106 * q^82 + 684 * q^83 - 24 * q^84 - 120 * q^85 - 468 * q^86 - 255 * q^87 + 384 * q^88 - 1370 * q^89 - 180 * q^90 + 94 * q^91 + 92 * q^92 - 201 * q^93 + 570 * q^94 - 280 * q^95 - 480 * q^96 - 110 * q^97 - 678 * q^98 + 288 * 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
2.00000 −3.00000 −4.00000 5.00000 −6.00000 −2.00000 −24.0000 −18.0000 10.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$23$$ $$1$$

## 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 115.4.a.b 1
3.b odd 2 1 1035.4.a.a 1
4.b odd 2 1 1840.4.a.f 1
5.b even 2 1 575.4.a.a 1
5.c odd 4 2 575.4.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.b 1 1.a even 1 1 trivial
575.4.a.a 1 5.b even 2 1
575.4.b.a 2 5.c odd 4 2
1035.4.a.a 1 3.b odd 2 1
1840.4.a.f 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(115))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 3$$
$5$ $$T - 5$$
$7$ $$T + 2$$
$11$ $$T + 16$$
$13$ $$T + 47$$
$17$ $$T + 24$$
$19$ $$T + 56$$
$23$ $$T + 23$$
$29$ $$T - 85$$
$31$ $$T - 67$$
$37$ $$T - 104$$
$41$ $$T + 53$$
$43$ $$T + 234$$
$47$ $$T - 285$$
$53$ $$T - 2$$
$59$ $$T - 80$$
$61$ $$T + 764$$
$67$ $$T - 236$$
$71$ $$T + 289$$
$73$ $$T + 225$$
$79$ $$T - 24$$
$83$ $$T - 684$$
$89$ $$T + 1370$$
$97$ $$T + 110$$