# Properties

 Label 115.2.b.a Level $115$ Weight $2$ Character orbit 115.b Analytic conductor $0.918$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,2,Mod(24,115)]

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([1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("115.24");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 115.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$0.918279623245$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} - 2 i q^{3} - 2 q^{4} + (i + 2) q^{5} + 4 q^{6} + i q^{7} - q^{9} +O(q^{10})$$ q + 2*i * q^2 - 2*i * q^3 - 2 * q^4 + (i + 2) * q^5 + 4 * q^6 + i * q^7 - q^9 $$q + 2 i q^{2} - 2 i q^{3} - 2 q^{4} + (i + 2) q^{5} + 4 q^{6} + i q^{7} - q^{9} + (4 i - 2) q^{10} + 4 i q^{12} - 2 i q^{13} - 2 q^{14} + ( - 4 i + 2) q^{15} - 4 q^{16} - 5 i q^{17} - 2 i q^{18} - 8 q^{19} + ( - 2 i - 4) q^{20} + 2 q^{21} + i q^{23} + (4 i + 3) q^{25} + 4 q^{26} - 4 i q^{27} - 2 i q^{28} + 5 q^{29} + (4 i + 8) q^{30} - 5 q^{31} - 8 i q^{32} + 10 q^{34} + (2 i - 1) q^{35} + 2 q^{36} + 7 i q^{37} - 16 i q^{38} - 4 q^{39} - 7 q^{41} + 4 i q^{42} - 4 i q^{43} + ( - i - 2) q^{45} - 2 q^{46} - 2 i q^{47} + 8 i q^{48} + 6 q^{49} + (6 i - 8) q^{50} - 10 q^{51} + 4 i q^{52} + i q^{53} + 8 q^{54} + 16 i q^{57} + 10 i q^{58} - 3 q^{59} + (8 i - 4) q^{60} - 6 q^{61} - 10 i q^{62} - i q^{63} + 8 q^{64} + ( - 4 i + 2) q^{65} + 13 i q^{67} + 10 i q^{68} + 2 q^{69} + ( - 2 i - 4) q^{70} + 13 q^{71} - 8 i q^{73} - 14 q^{74} + ( - 6 i + 8) q^{75} + 16 q^{76} - 8 i q^{78} + 14 q^{79} + ( - 4 i - 8) q^{80} - 11 q^{81} - 14 i q^{82} + 3 i q^{83} - 4 q^{84} + ( - 10 i + 5) q^{85} + 8 q^{86} - 10 i q^{87} + 14 q^{89} + ( - 4 i + 2) q^{90} + 2 q^{91} - 2 i q^{92} + 10 i q^{93} + 4 q^{94} + ( - 8 i - 16) q^{95} - 16 q^{96} + 14 i q^{97} + 12 i q^{98} +O(q^{100})$$ q + 2*i * q^2 - 2*i * q^3 - 2 * q^4 + (i + 2) * q^5 + 4 * q^6 + i * q^7 - q^9 + (4*i - 2) * q^10 + 4*i * q^12 - 2*i * q^13 - 2 * q^14 + (-4*i + 2) * q^15 - 4 * q^16 - 5*i * q^17 - 2*i * q^18 - 8 * q^19 + (-2*i - 4) * q^20 + 2 * q^21 + i * q^23 + (4*i + 3) * q^25 + 4 * q^26 - 4*i * q^27 - 2*i * q^28 + 5 * q^29 + (4*i + 8) * q^30 - 5 * q^31 - 8*i * q^32 + 10 * q^34 + (2*i - 1) * q^35 + 2 * q^36 + 7*i * q^37 - 16*i * q^38 - 4 * q^39 - 7 * q^41 + 4*i * q^42 - 4*i * q^43 + (-i - 2) * q^45 - 2 * q^46 - 2*i * q^47 + 8*i * q^48 + 6 * q^49 + (6*i - 8) * q^50 - 10 * q^51 + 4*i * q^52 + i * q^53 + 8 * q^54 + 16*i * q^57 + 10*i * q^58 - 3 * q^59 + (8*i - 4) * q^60 - 6 * q^61 - 10*i * q^62 - i * q^63 + 8 * q^64 + (-4*i + 2) * q^65 + 13*i * q^67 + 10*i * q^68 + 2 * q^69 + (-2*i - 4) * q^70 + 13 * q^71 - 8*i * q^73 - 14 * q^74 + (-6*i + 8) * q^75 + 16 * q^76 - 8*i * q^78 + 14 * q^79 + (-4*i - 8) * q^80 - 11 * q^81 - 14*i * q^82 + 3*i * q^83 - 4 * q^84 + (-10*i + 5) * q^85 + 8 * q^86 - 10*i * q^87 + 14 * q^89 + (-4*i + 2) * q^90 + 2 * q^91 - 2*i * q^92 + 10*i * q^93 + 4 * q^94 + (-8*i - 16) * q^95 - 16 * q^96 + 14*i * q^97 + 12*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} + 4 q^{5} + 8 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 + 4 * q^5 + 8 * q^6 - 2 * q^9 $$2 q - 4 q^{4} + 4 q^{5} + 8 q^{6} - 2 q^{9} - 4 q^{10} - 4 q^{14} + 4 q^{15} - 8 q^{16} - 16 q^{19} - 8 q^{20} + 4 q^{21} + 6 q^{25} + 8 q^{26} + 10 q^{29} + 16 q^{30} - 10 q^{31} + 20 q^{34} - 2 q^{35} + 4 q^{36} - 8 q^{39} - 14 q^{41} - 4 q^{45} - 4 q^{46} + 12 q^{49} - 16 q^{50} - 20 q^{51} + 16 q^{54} - 6 q^{59} - 8 q^{60} - 12 q^{61} + 16 q^{64} + 4 q^{65} + 4 q^{69} - 8 q^{70} + 26 q^{71} - 28 q^{74} + 16 q^{75} + 32 q^{76} + 28 q^{79} - 16 q^{80} - 22 q^{81} - 8 q^{84} + 10 q^{85} + 16 q^{86} + 28 q^{89} + 4 q^{90} + 4 q^{91} + 8 q^{94} - 32 q^{95} - 32 q^{96}+O(q^{100})$$ 2 * q - 4 * q^4 + 4 * q^5 + 8 * q^6 - 2 * q^9 - 4 * q^10 - 4 * q^14 + 4 * q^15 - 8 * q^16 - 16 * q^19 - 8 * q^20 + 4 * q^21 + 6 * q^25 + 8 * q^26 + 10 * q^29 + 16 * q^30 - 10 * q^31 + 20 * q^34 - 2 * q^35 + 4 * q^36 - 8 * q^39 - 14 * q^41 - 4 * q^45 - 4 * q^46 + 12 * q^49 - 16 * q^50 - 20 * q^51 + 16 * q^54 - 6 * q^59 - 8 * q^60 - 12 * q^61 + 16 * q^64 + 4 * q^65 + 4 * q^69 - 8 * q^70 + 26 * q^71 - 28 * q^74 + 16 * q^75 + 32 * q^76 + 28 * q^79 - 16 * q^80 - 22 * q^81 - 8 * q^84 + 10 * q^85 + 16 * q^86 + 28 * q^89 + 4 * q^90 + 4 * q^91 + 8 * q^94 - 32 * q^95 - 32 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/115\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$51$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1
 − 1.00000i 1.00000i
2.00000i 2.00000i −2.00000 2.00000 1.00000i 4.00000 1.00000i 0 −1.00000 −2.00000 4.00000i
24.2 2.00000i 2.00000i −2.00000 2.00000 + 1.00000i 4.00000 1.00000i 0 −1.00000 −2.00000 + 4.00000i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.2.b.a 2
3.b odd 2 1 1035.2.b.a 2
4.b odd 2 1 1840.2.e.b 2
5.b even 2 1 inner 115.2.b.a 2
5.c odd 4 1 575.2.a.a 1
5.c odd 4 1 575.2.a.e 1
15.d odd 2 1 1035.2.b.a 2
15.e even 4 1 5175.2.a.a 1
15.e even 4 1 5175.2.a.z 1
20.d odd 2 1 1840.2.e.b 2
20.e even 4 1 9200.2.a.g 1
20.e even 4 1 9200.2.a.bg 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.a 2 1.a even 1 1 trivial
115.2.b.a 2 5.b even 2 1 inner
575.2.a.a 1 5.c odd 4 1
575.2.a.e 1 5.c odd 4 1
1035.2.b.a 2 3.b odd 2 1
1035.2.b.a 2 15.d odd 2 1
1840.2.e.b 2 4.b odd 2 1
1840.2.e.b 2 20.d odd 2 1
5175.2.a.a 1 15.e even 4 1
5175.2.a.z 1 15.e even 4 1
9200.2.a.g 1 20.e even 4 1
9200.2.a.bg 1 20.e even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 25$$
$19$ $$(T + 8)^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T - 5)^{2}$$
$31$ $$(T + 5)^{2}$$
$37$ $$T^{2} + 49$$
$41$ $$(T + 7)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 1$$
$59$ $$(T + 3)^{2}$$
$61$ $$(T + 6)^{2}$$
$67$ $$T^{2} + 169$$
$71$ $$(T - 13)^{2}$$
$73$ $$T^{2} + 64$$
$79$ $$(T - 14)^{2}$$
$83$ $$T^{2} + 9$$
$89$ $$(T - 14)^{2}$$
$97$ $$T^{2} + 196$$