# Properties

 Label 95.2.b.b Level $95$ Weight $2$ Character orbit 95.b Analytic conductor $0.759$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.16516096.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 9x^{4} + 13x^{2} + 1$$ x^6 + 9*x^4 + 13*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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{3}+ \cdots + ( - 2 \beta_{2} - 3) q^{9}+O(q^{10})$$ q - b5 * q^2 + (-b5 + b4 + b3 - b1) * q^3 + (b2 - 1) * q^4 + (b5 - b4) * q^5 + (-b4 + b3 - b2) * q^6 + (-b4 - b3) * q^7 + (b4 + b3 + b1) * q^8 + (-2*b2 - 3) * q^9 $$q - \beta_{5} q^{2} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{3}+ \cdots + ( - \beta_{4} + \beta_{3} - 4 \beta_{2} + 4) q^{99}+O(q^{100})$$ q - b5 * q^2 + (-b5 + b4 + b3 - b1) * q^3 + (b2 - 1) * q^4 + (b5 - b4) * q^5 + (-b4 + b3 - b2) * q^6 + (-b4 - b3) * q^7 + (b4 + b3 + b1) * q^8 + (-2*b2 - 3) * q^9 + (b5 + 2*b1 + 2) * q^10 + (b4 - b3) * q^11 + (-b5 + b4 + b3 + b1) * q^12 + (b5 - b4 - b3 - b1) * q^13 + (2*b2 - 2) * q^14 + (b5 + b4 - b3 + 2*b2 - b1 + 2) * q^15 + (b4 - b3 - 1) * q^16 + (b4 + b3 + 2*b1) * q^17 + (b5 - 2*b4 - 2*b3 - 2*b1) * q^18 + q^19 + (-2*b3 - b2 + 1) * q^20 + (2*b2 + 4) * q^21 + (-2*b5 - 4*b1) * q^22 + 2*b1 * q^23 + (-b4 + b3 - 3*b2 - 2) * q^24 + (-b5 + b3 - 2*b2 - 2*b1) * q^25 + (-b4 + b3 + b2 + 2) * q^26 + (4*b5 - 4*b4 - 4*b3) * q^27 + (4*b5 + 2*b1) * q^28 - 6 * q^29 + (-2*b5 + b4 + 3*b3 - b2 - 2*b1 + 4) * q^30 + (2*b4 - 2*b3 + 2*b2) * q^31 + (-b5 + 2*b4 + 2*b3 - 2*b1) * q^32 + (-2*b5 + 2*b1) * q^33 + (2*b4 - 2*b3 - 2*b2) * q^34 + (b5 + b3 - 2*b2 + 2*b1 - 1) * q^35 + (-2*b4 + 2*b3 - b2 - 5) * q^36 + (b5 + b4 + b3 - b1) * q^37 - b5 * q^38 + (2*b4 - 2*b3 + 4*b2 + 2) * q^39 + (-2*b5 - b4 - b3 + 2*b2 - b1 + 2) * q^40 + (-2*b4 + 2*b3 - 2*b2 + 2) * q^41 + (-2*b5 + 2*b4 + 2*b3 + 2*b1) * q^42 + (-b4 - b3) * q^43 + (-2*b4 + 2*b3 + 2*b2 - 2) * q^44 + (-5*b5 + 5*b4 + 4*b3 + 2*b2 - 2) * q^45 + (2*b4 - 2*b3 - 2) * q^46 + (4*b5 - b4 - b3 + 4*b1) * q^47 + (-b5 - b4 - b3 + 3*b1) * q^48 + (-b4 + b3 + 1) * q^49 + (-b5 - 4*b4) * q^50 + (-2*b4 + 2*b3 - 4*b2) * q^51 + (3*b5 - b4 - b3 + 3*b1) * q^52 + (-3*b5 - b4 - b3 - b1) * q^53 + (4*b2 + 4) * q^54 + (b5 - b3 + 2*b2 + 2*b1 - 5) * q^55 + (2*b4 - 2*b3 + 6) * q^56 + (-b5 + b4 + b3 - b1) * q^57 + 6*b5 * q^58 + (-2*b2 - 4) * q^59 + (-b5 - b4 - b3 + 2*b2 + b1 + 4) * q^60 + (-3*b4 + 3*b3 - 2*b2 - 2) * q^61 + (-2*b5 + 2*b4 + 2*b3 - 6*b1) * q^62 + (-8*b5 + 5*b4 + 5*b3 - 4*b1) * q^63 + (-3*b2 + 1) * q^64 + (b5 + b4 + b3 - 2*b2 - b1 - 4) * q^65 + (2*b4 - 2*b3 + 2*b2 - 8) * q^66 + (-3*b5 + 5*b4 + 5*b3 - 3*b1) * q^67 + (-6*b5 - 6*b1) * q^68 + (-2*b4 + 2*b3 - 2*b2 + 4) * q^69 + (-4*b3 - 2*b2 + 2) * q^70 + (-2*b2 + 8) * q^71 + (10*b5 - 5*b4 - 5*b3 + 3*b1) * q^72 + (4*b5 - 3*b4 - 3*b3 + 2*b1) * q^73 + (-b4 + b3 - 3*b2 + 6) * q^74 + (5*b5 - 3*b4 - 5*b3 - b1 - 6) * q^75 + (b2 - 1) * q^76 + (-2*b5 - b4 - b3 - 4*b1) * q^77 + (-2*b5 + 4*b4 + 4*b3 - 4*b1) * q^78 + (-2*b4 + 2*b3 - 2*b2 + 4) * q^79 + (b4 - b3 + 2*b2 + 2*b1 - 5) * q^80 + (4*b4 - 4*b3 + 6*b2 + 7) * q^81 + (-2*b4 - 2*b3 + 6*b1) * q^82 + (4*b5 - 2*b1) * q^83 + (2*b4 - 2*b3 + 2*b2 + 4) * q^84 + (-3*b5 - 2*b4 - b3 + 2*b2 + 3) * q^85 + (2*b2 - 2) * q^86 + (6*b5 - 6*b4 - 6*b3 + 6*b1) * q^87 + (4*b5 + 2*b4 + 2*b3 + 2*b1) * q^88 + (2*b4 - 2*b3 + 8*b2 - 2) * q^89 + (3*b5 + 2*b4 + 2*b3 - 4*b2 - 6) * q^90 + (-2*b4 + 2*b3 - 2*b2 - 4) * q^91 + (-2*b5 - 4*b1) * q^92 + (-8*b5 + 4*b4 + 4*b3 + 4*b1) * q^93 + (4*b4 - 4*b3 - 2*b2 + 6) * q^94 + (b5 - b4) * q^95 + (b4 - b3 - 3*b2 - 12) * q^96 + (-b5 + b4 + b3 + b1) * q^97 + (b5 + 4*b1) * q^98 + (-b4 + b3 - 4*b2 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 8 q^{4} - q^{5} - 14 q^{9}+O(q^{10})$$ 6 * q - 8 * q^4 - q^5 - 14 * q^9 $$6 q - 8 q^{4} - q^{5} - 14 q^{9} + 12 q^{10} + 2 q^{11} - 16 q^{14} + 10 q^{15} - 4 q^{16} + 6 q^{19} + 10 q^{20} + 20 q^{21} - 8 q^{24} + 3 q^{25} + 8 q^{26} - 36 q^{29} + 24 q^{30} + 8 q^{34} - 3 q^{35} - 32 q^{36} + 8 q^{39} + 8 q^{40} + 12 q^{41} - 20 q^{44} - 15 q^{45} - 8 q^{46} + 4 q^{49} - 4 q^{50} + 4 q^{51} + 16 q^{54} - 33 q^{55} + 40 q^{56} - 20 q^{59} + 20 q^{60} - 14 q^{61} + 12 q^{64} - 20 q^{65} - 48 q^{66} + 24 q^{69} + 20 q^{70} + 52 q^{71} + 40 q^{74} - 34 q^{75} - 8 q^{76} + 24 q^{79} - 32 q^{80} + 38 q^{81} + 24 q^{84} + 13 q^{85} - 16 q^{86} - 24 q^{89} - 28 q^{90} - 24 q^{91} + 48 q^{94} - q^{95} - 64 q^{96} + 30 q^{99}+O(q^{100})$$ 6 * q - 8 * q^4 - q^5 - 14 * q^9 + 12 * q^10 + 2 * q^11 - 16 * q^14 + 10 * q^15 - 4 * q^16 + 6 * q^19 + 10 * q^20 + 20 * q^21 - 8 * q^24 + 3 * q^25 + 8 * q^26 - 36 * q^29 + 24 * q^30 + 8 * q^34 - 3 * q^35 - 32 * q^36 + 8 * q^39 + 8 * q^40 + 12 * q^41 - 20 * q^44 - 15 * q^45 - 8 * q^46 + 4 * q^49 - 4 * q^50 + 4 * q^51 + 16 * q^54 - 33 * q^55 + 40 * q^56 - 20 * q^59 + 20 * q^60 - 14 * q^61 + 12 * q^64 - 20 * q^65 - 48 * q^66 + 24 * q^69 + 20 * q^70 + 52 * q^71 + 40 * q^74 - 34 * q^75 - 8 * q^76 + 24 * q^79 - 32 * q^80 + 38 * q^81 + 24 * q^84 + 13 * q^85 - 16 * q^86 - 24 * q^89 - 28 * q^90 - 24 * q^91 + 48 * q^94 - q^95 - 64 * q^96 + 30 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 9x^{4} + 13x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} + 8\nu^{2} + 5 ) / 2$$ (v^4 + 8*v^2 + 5) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{5} + \nu^{4} + 10\nu^{3} + 6\nu^{2} + 19\nu - 1 ) / 4$$ (v^5 + v^4 + 10*v^3 + 6*v^2 + 19*v - 1) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - \nu^{4} + 10\nu^{3} - 6\nu^{2} + 19\nu + 1 ) / 4$$ (v^5 - v^4 + 10*v^3 - 6*v^2 + 19*v + 1) / 4 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} - 8\nu^{3} - 7\nu ) / 2$$ (-v^5 - 8*v^3 - 7*v) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{2} - 3$$ b4 - b3 + b2 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} - 6\beta_1$$ b5 + b4 + b3 - 6*b1 $$\nu^{4}$$ $$=$$ $$-8\beta_{4} + 8\beta_{3} - 6\beta_{2} + 19$$ -8*b4 + 8*b3 - 6*b2 + 19 $$\nu^{5}$$ $$=$$ $$-10\beta_{5} - 8\beta_{4} - 8\beta_{3} + 41\beta_1$$ -10*b5 - 8*b4 - 8*b3 + 41*b1

## 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
 1.30397i − 2.68667i − 0.285442i 0.285442i 2.68667i − 1.30397i
2.41987i 0.537080i −3.85577 −2.07772 + 0.826491i −1.29966 3.18676i 4.49073i 2.71155 2.00000 + 5.02781i
39.2 1.82254i 2.31446i −1.32164 1.94827 + 1.09737i 4.21819 1.45033i 1.23634i −2.35673 2.00000 3.55080i
39.3 0.906968i 3.21789i 1.17741 −0.370556 + 2.20515i −2.91852 2.59637i 2.88181i −7.35482 2.00000 + 0.336083i
39.4 0.906968i 3.21789i 1.17741 −0.370556 2.20515i −2.91852 2.59637i 2.88181i −7.35482 2.00000 0.336083i
39.5 1.82254i 2.31446i −1.32164 1.94827 1.09737i 4.21819 1.45033i 1.23634i −2.35673 2.00000 + 3.55080i
39.6 2.41987i 0.537080i −3.85577 −2.07772 0.826491i −1.29966 3.18676i 4.49073i 2.71155 2.00000 5.02781i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 39.6 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.b 6
3.b odd 2 1 855.2.c.d 6
4.b odd 2 1 1520.2.d.h 6
5.b even 2 1 inner 95.2.b.b 6
5.c odd 4 2 475.2.a.j 6
15.d odd 2 1 855.2.c.d 6
15.e even 4 2 4275.2.a.br 6
19.b odd 2 1 1805.2.b.e 6
20.d odd 2 1 1520.2.d.h 6
20.e even 4 2 7600.2.a.ck 6
95.d odd 2 1 1805.2.b.e 6
95.g even 4 2 9025.2.a.bx 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.b 6 1.a even 1 1 trivial
95.2.b.b 6 5.b even 2 1 inner
475.2.a.j 6 5.c odd 4 2
855.2.c.d 6 3.b odd 2 1
855.2.c.d 6 15.d odd 2 1
1520.2.d.h 6 4.b odd 2 1
1520.2.d.h 6 20.d odd 2 1
1805.2.b.e 6 19.b odd 2 1
1805.2.b.e 6 95.d odd 2 1
4275.2.a.br 6 15.e even 4 2
7600.2.a.ck 6 20.e even 4 2
9025.2.a.bx 6 95.g even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 10T_{2}^{4} + 27T_{2}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 10 T^{4} + \cdots + 16$$
$3$ $$T^{6} + 16 T^{4} + \cdots + 16$$
$5$ $$T^{6} + T^{5} + \cdots + 125$$
$7$ $$T^{6} + 19 T^{4} + \cdots + 144$$
$11$ $$(T^{3} - T^{2} - 16 T + 12)^{2}$$
$13$ $$T^{6} + 28 T^{4} + \cdots + 576$$
$17$ $$T^{6} + 59 T^{4} + \cdots + 5184$$
$19$ $$(T - 1)^{6}$$
$23$ $$T^{6} + 36 T^{4} + \cdots + 64$$
$29$ $$(T + 6)^{6}$$
$31$ $$(T^{3} - 56 T + 128)^{2}$$
$37$ $$T^{6} + 56 T^{4} + \cdots + 1296$$
$41$ $$(T^{3} - 6 T^{2} - 44 T - 24)^{2}$$
$43$ $$T^{6} + 19 T^{4} + \cdots + 144$$
$47$ $$T^{6} + 187 T^{4} + \cdots + 85264$$
$53$ $$T^{6} + 156 T^{4} + \cdots + 64$$
$59$ $$(T^{3} + 10 T^{2} + \cdots - 48)^{2}$$
$61$ $$(T^{3} + 7 T^{2} + \cdots - 776)^{2}$$
$67$ $$T^{6} + 340 T^{4} + \cdots + 484416$$
$71$ $$(T^{3} - 26 T^{2} + \cdots - 432)^{2}$$
$73$ $$T^{6} + 131 T^{4} + \cdots + 5184$$
$79$ $$(T^{3} - 12 T^{2} + \cdots + 32)^{2}$$
$83$ $$T^{6} + 228 T^{4} + \cdots + 141376$$
$89$ $$(T^{3} + 12 T^{2} + \cdots - 3456)^{2}$$
$97$ $$T^{6} + 28 T^{4} + \cdots + 576$$