# Properties

 Label 95.5.d.b Level $95$ Weight $5$ Character orbit 95.d Self dual yes Analytic conductor $9.820$ Analytic rank $0$ Dimension $2$ CM discriminant -95 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,5,Mod(94,95)]

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, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("95.94");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 95.d (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: yes Analytic conductor: $$9.82014649297$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 19$$ x^2 - 19 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 $$\beta = \sqrt{19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 4 \beta q^{3} + 3 q^{4} + 25 q^{5} - 76 q^{6} - 13 \beta q^{8} + 223 q^{9} +O(q^{10})$$ q + b * q^2 - 4*b * q^3 + 3 * q^4 + 25 * q^5 - 76 * q^6 - 13*b * q^8 + 223 * q^9 $$q + \beta q^{2} - 4 \beta q^{3} + 3 q^{4} + 25 q^{5} - 76 q^{6} - 13 \beta q^{8} + 223 q^{9} + 25 \beta q^{10} + 62 q^{11} - 12 \beta q^{12} + 76 \beta q^{13} - 100 \beta q^{15} - 295 q^{16} + 223 \beta q^{18} + 361 q^{19} + 75 q^{20} + 62 \beta q^{22} + 988 q^{24} + 625 q^{25} + 1444 q^{26} - 568 \beta q^{27} - 1900 q^{30} - 87 \beta q^{32} - 248 \beta q^{33} + 669 q^{36} - 164 \beta q^{37} + 361 \beta q^{38} - 5776 q^{39} - 325 \beta q^{40} + 186 q^{44} + 5575 q^{45} + 1180 \beta q^{48} + 2401 q^{49} + 625 \beta q^{50} + 228 \beta q^{52} + 1276 \beta q^{53} - 10792 q^{54} + 1550 q^{55} - 1444 \beta q^{57} - 300 \beta q^{60} - 7138 q^{61} + 3067 q^{64} + 1900 \beta q^{65} - 4712 q^{66} + 1516 \beta q^{67} - 2899 \beta q^{72} - 3116 q^{74} - 2500 \beta q^{75} + 1083 q^{76} - 5776 \beta q^{78} - 7375 q^{80} + 25105 q^{81} - 806 \beta q^{88} + 5575 \beta q^{90} + 9025 q^{95} + 6612 q^{96} + 2476 \beta q^{97} + 2401 \beta q^{98} + 13826 q^{99} +O(q^{100})$$ q + b * q^2 - 4*b * q^3 + 3 * q^4 + 25 * q^5 - 76 * q^6 - 13*b * q^8 + 223 * q^9 + 25*b * q^10 + 62 * q^11 - 12*b * q^12 + 76*b * q^13 - 100*b * q^15 - 295 * q^16 + 223*b * q^18 + 361 * q^19 + 75 * q^20 + 62*b * q^22 + 988 * q^24 + 625 * q^25 + 1444 * q^26 - 568*b * q^27 - 1900 * q^30 - 87*b * q^32 - 248*b * q^33 + 669 * q^36 - 164*b * q^37 + 361*b * q^38 - 5776 * q^39 - 325*b * q^40 + 186 * q^44 + 5575 * q^45 + 1180*b * q^48 + 2401 * q^49 + 625*b * q^50 + 228*b * q^52 + 1276*b * q^53 - 10792 * q^54 + 1550 * q^55 - 1444*b * q^57 - 300*b * q^60 - 7138 * q^61 + 3067 * q^64 + 1900*b * q^65 - 4712 * q^66 + 1516*b * q^67 - 2899*b * q^72 - 3116 * q^74 - 2500*b * q^75 + 1083 * q^76 - 5776*b * q^78 - 7375 * q^80 + 25105 * q^81 - 806*b * q^88 + 5575*b * q^90 + 9025 * q^95 + 6612 * q^96 + 2476*b * q^97 + 2401*b * q^98 + 13826 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{4} + 50 q^{5} - 152 q^{6} + 446 q^{9}+O(q^{10})$$ 2 * q + 6 * q^4 + 50 * q^5 - 152 * q^6 + 446 * q^9 $$2 q + 6 q^{4} + 50 q^{5} - 152 q^{6} + 446 q^{9} + 124 q^{11} - 590 q^{16} + 722 q^{19} + 150 q^{20} + 1976 q^{24} + 1250 q^{25} + 2888 q^{26} - 3800 q^{30} + 1338 q^{36} - 11552 q^{39} + 372 q^{44} + 11150 q^{45} + 4802 q^{49} - 21584 q^{54} + 3100 q^{55} - 14276 q^{61} + 6134 q^{64} - 9424 q^{66} - 6232 q^{74} + 2166 q^{76} - 14750 q^{80} + 50210 q^{81} + 18050 q^{95} + 13224 q^{96} + 27652 q^{99}+O(q^{100})$$ 2 * q + 6 * q^4 + 50 * q^5 - 152 * q^6 + 446 * q^9 + 124 * q^11 - 590 * q^16 + 722 * q^19 + 150 * q^20 + 1976 * q^24 + 1250 * q^25 + 2888 * q^26 - 3800 * q^30 + 1338 * q^36 - 11552 * q^39 + 372 * q^44 + 11150 * q^45 + 4802 * q^49 - 21584 * q^54 + 3100 * q^55 - 14276 * q^61 + 6134 * q^64 - 9424 * q^66 - 6232 * q^74 + 2166 * q^76 - 14750 * q^80 + 50210 * q^81 + 18050 * q^95 + 13224 * q^96 + 27652 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
94.1
 −4.35890 4.35890
−4.35890 17.4356 3.00000 25.0000 −76.0000 0 56.6657 223.000 −108.972
94.2 4.35890 −17.4356 3.00000 25.0000 −76.0000 0 −56.6657 223.000 108.972
 $$n$$: 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
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.5.d.b 2
5.b even 2 1 inner 95.5.d.b 2
19.b odd 2 1 inner 95.5.d.b 2
95.d odd 2 1 CM 95.5.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.5.d.b 2 1.a even 1 1 trivial
95.5.d.b 2 5.b even 2 1 inner
95.5.d.b 2 19.b odd 2 1 inner
95.5.d.b 2 95.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 19$$ acting on $$S_{5}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 19$$
$3$ $$T^{2} - 304$$
$5$ $$(T - 25)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 62)^{2}$$
$13$ $$T^{2} - 109744$$
$17$ $$T^{2}$$
$19$ $$(T - 361)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 511024$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 30935344$$
$59$ $$T^{2}$$
$61$ $$(T + 7138)^{2}$$
$67$ $$T^{2} - 43666864$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 116480944$$