# Properties

 Label 95.2.b.a Level $95$ Weight $2$ Character orbit 95.b Analytic conductor $0.759$ 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] = [95,2,Mod(39,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, 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("95.39");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 95.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: $$0.758578819202$$ 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} + (2 i - 1) q^{5} - 2 i q^{7} + 3 i q^{8} + 3 q^{9} +O(q^{10})$$ q + i * q^2 + q^4 + (2*i - 1) * q^5 - 2*i * q^7 + 3*i * q^8 + 3 * q^9 $$q + i q^{2} + q^{4} + (2 i - 1) q^{5} - 2 i q^{7} + 3 i q^{8} + 3 q^{9} + ( - i - 2) q^{10} - 4 q^{11} - 2 i q^{13} + 2 q^{14} - q^{16} - 4 i q^{17} + 3 i q^{18} - q^{19} + (2 i - 1) q^{20} - 4 i q^{22} - 6 i q^{23} + ( - 4 i - 3) q^{25} + 2 q^{26} - 2 i q^{28} + 6 q^{29} - 4 q^{31} + 5 i q^{32} + 4 q^{34} + (2 i + 4) q^{35} + 3 q^{36} + 10 i q^{37} - i q^{38} + ( - 3 i - 6) q^{40} - 10 q^{41} + 2 i q^{43} - 4 q^{44} + (6 i - 3) q^{45} + 6 q^{46} + 6 i q^{47} + 3 q^{49} + ( - 3 i + 4) q^{50} - 2 i q^{52} + 10 i q^{53} + ( - 8 i + 4) q^{55} + 6 q^{56} + 6 i q^{58} + 2 q^{61} - 4 i q^{62} - 6 i q^{63} - 7 q^{64} + (2 i + 4) q^{65} - 8 i q^{67} - 4 i q^{68} + (4 i - 2) q^{70} + 4 q^{71} + 9 i q^{72} + 4 i q^{73} - 10 q^{74} - q^{76} + 8 i q^{77} - 4 q^{79} + ( - 2 i + 1) q^{80} + 9 q^{81} - 10 i q^{82} - 18 i q^{83} + (4 i + 8) q^{85} - 2 q^{86} - 12 i q^{88} + 2 q^{89} + ( - 3 i - 6) q^{90} - 4 q^{91} - 6 i q^{92} - 6 q^{94} + ( - 2 i + 1) q^{95} - 6 i q^{97} + 3 i q^{98} - 12 q^{99} +O(q^{100})$$ q + i * q^2 + q^4 + (2*i - 1) * q^5 - 2*i * q^7 + 3*i * q^8 + 3 * q^9 + (-i - 2) * q^10 - 4 * q^11 - 2*i * q^13 + 2 * q^14 - q^16 - 4*i * q^17 + 3*i * q^18 - q^19 + (2*i - 1) * q^20 - 4*i * q^22 - 6*i * q^23 + (-4*i - 3) * q^25 + 2 * q^26 - 2*i * q^28 + 6 * q^29 - 4 * q^31 + 5*i * q^32 + 4 * q^34 + (2*i + 4) * q^35 + 3 * q^36 + 10*i * q^37 - i * q^38 + (-3*i - 6) * q^40 - 10 * q^41 + 2*i * q^43 - 4 * q^44 + (6*i - 3) * q^45 + 6 * q^46 + 6*i * q^47 + 3 * q^49 + (-3*i + 4) * q^50 - 2*i * q^52 + 10*i * q^53 + (-8*i + 4) * q^55 + 6 * q^56 + 6*i * q^58 + 2 * q^61 - 4*i * q^62 - 6*i * q^63 - 7 * q^64 + (2*i + 4) * q^65 - 8*i * q^67 - 4*i * q^68 + (4*i - 2) * q^70 + 4 * q^71 + 9*i * q^72 + 4*i * q^73 - 10 * q^74 - q^76 + 8*i * q^77 - 4 * q^79 + (-2*i + 1) * q^80 + 9 * q^81 - 10*i * q^82 - 18*i * q^83 + (4*i + 8) * q^85 - 2 * q^86 - 12*i * q^88 + 2 * q^89 + (-3*i - 6) * q^90 - 4 * q^91 - 6*i * q^92 - 6 * q^94 + (-2*i + 1) * q^95 - 6*i * q^97 + 3*i * q^98 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^5 + 6 * q^9 $$2 q + 2 q^{4} - 2 q^{5} + 6 q^{9} - 4 q^{10} - 8 q^{11} + 4 q^{14} - 2 q^{16} - 2 q^{19} - 2 q^{20} - 6 q^{25} + 4 q^{26} + 12 q^{29} - 8 q^{31} + 8 q^{34} + 8 q^{35} + 6 q^{36} - 12 q^{40} - 20 q^{41} - 8 q^{44} - 6 q^{45} + 12 q^{46} + 6 q^{49} + 8 q^{50} + 8 q^{55} + 12 q^{56} + 4 q^{61} - 14 q^{64} + 8 q^{65} - 4 q^{70} + 8 q^{71} - 20 q^{74} - 2 q^{76} - 8 q^{79} + 2 q^{80} + 18 q^{81} + 16 q^{85} - 4 q^{86} + 4 q^{89} - 12 q^{90} - 8 q^{91} - 12 q^{94} + 2 q^{95} - 24 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^5 + 6 * q^9 - 4 * q^10 - 8 * q^11 + 4 * q^14 - 2 * q^16 - 2 * q^19 - 2 * q^20 - 6 * q^25 + 4 * q^26 + 12 * q^29 - 8 * q^31 + 8 * q^34 + 8 * q^35 + 6 * q^36 - 12 * q^40 - 20 * q^41 - 8 * q^44 - 6 * q^45 + 12 * q^46 + 6 * q^49 + 8 * q^50 + 8 * q^55 + 12 * q^56 + 4 * q^61 - 14 * q^64 + 8 * q^65 - 4 * q^70 + 8 * q^71 - 20 * q^74 - 2 * q^76 - 8 * q^79 + 2 * q^80 + 18 * q^81 + 16 * q^85 - 4 * q^86 + 4 * q^89 - 12 * q^90 - 8 * q^91 - 12 * q^94 + 2 * q^95 - 24 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
39.1
 − 1.00000i 1.00000i
1.00000i 0 1.00000 −1.00000 2.00000i 0 2.00000i 3.00000i 3.00000 −2.00000 + 1.00000i
39.2 1.00000i 0 1.00000 −1.00000 + 2.00000i 0 2.00000i 3.00000i 3.00000 −2.00000 1.00000i
 $$n$$: 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 95.2.b.a 2
3.b odd 2 1 855.2.c.b 2
4.b odd 2 1 1520.2.d.b 2
5.b even 2 1 inner 95.2.b.a 2
5.c odd 4 1 475.2.a.a 1
5.c odd 4 1 475.2.a.c 1
15.d odd 2 1 855.2.c.b 2
15.e even 4 1 4275.2.a.e 1
15.e even 4 1 4275.2.a.p 1
19.b odd 2 1 1805.2.b.c 2
20.d odd 2 1 1520.2.d.b 2
20.e even 4 1 7600.2.a.i 1
20.e even 4 1 7600.2.a.l 1
95.d odd 2 1 1805.2.b.c 2
95.g even 4 1 9025.2.a.c 1
95.g even 4 1 9025.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.a 2 1.a even 1 1 trivial
95.2.b.a 2 5.b even 2 1 inner
475.2.a.a 1 5.c odd 4 1
475.2.a.c 1 5.c odd 4 1
855.2.c.b 2 3.b odd 2 1
855.2.c.b 2 15.d odd 2 1
1520.2.d.b 2 4.b odd 2 1
1520.2.d.b 2 20.d odd 2 1
1805.2.b.c 2 19.b odd 2 1
1805.2.b.c 2 95.d odd 2 1
4275.2.a.e 1 15.e even 4 1
4275.2.a.p 1 15.e even 4 1
7600.2.a.i 1 20.e even 4 1
7600.2.a.l 1 20.e even 4 1
9025.2.a.c 1 95.g even 4 1
9025.2.a.h 1 95.g even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 16$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} + 4$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 100$$
$59$ $$T^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T - 4)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 324$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} + 36$$