# Properties

 Label 95.1.d.b Level 95 Weight 1 Character orbit 95.d Self dual yes Analytic conductor 0.047 Analytic rank 0 Dimension 2 Projective image $$D_{4}$$ CM discriminant -95 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 95.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.0474111762001$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{4}$$ Projective field Galois closure of 4.2.475.1 Artin image $D_8$ Artin field Galois closure of 8.2.4286875.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + \beta q^{3} + q^{4} - q^{5} -2 q^{6} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} + \beta q^{3} + q^{4} - q^{5} -2 q^{6} + q^{9} + \beta q^{10} + \beta q^{12} -\beta q^{13} -\beta q^{15} - q^{16} -\beta q^{18} - q^{19} - q^{20} + q^{25} + 2 q^{26} + 2 q^{30} + \beta q^{32} + q^{36} + \beta q^{37} + \beta q^{38} -2 q^{39} - q^{45} -\beta q^{48} + q^{49} -\beta q^{50} -\beta q^{52} + \beta q^{53} -\beta q^{57} -\beta q^{60} - q^{64} + \beta q^{65} -\beta q^{67} -2 q^{74} + \beta q^{75} - q^{76} + 2 \beta q^{78} + q^{80} - q^{81} + \beta q^{90} + q^{95} + 2 q^{96} + \beta q^{97} -\beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} - 2q^{5} - 4q^{6} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{4} - 2q^{5} - 4q^{6} + 2q^{9} - 2q^{16} - 2q^{19} - 2q^{20} + 2q^{25} + 4q^{26} + 4q^{30} + 2q^{36} - 4q^{39} - 2q^{45} + 2q^{49} - 2q^{64} - 4q^{74} - 2q^{76} + 2q^{80} - 2q^{81} + 2q^{95} + 4q^{96} + O(q^{100})$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
94.1
 1.41421 −1.41421
−1.41421 1.41421 1.00000 −1.00000 −2.00000 0 0 1.00000 1.41421
94.2 1.41421 −1.41421 1.00000 −1.00000 −2.00000 0 0 1.00000 −1.41421
 $$n$$: 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 95.1.d.b 2
3.b odd 2 1 855.1.g.c 2
4.b odd 2 1 1520.1.m.b 2
5.b even 2 1 inner 95.1.d.b 2
5.c odd 4 2 475.1.c.b 2
15.d odd 2 1 855.1.g.c 2
19.b odd 2 1 inner 95.1.d.b 2
19.c even 3 2 1805.1.h.b 4
19.d odd 6 2 1805.1.h.b 4
19.e even 9 6 1805.1.o.b 12
19.f odd 18 6 1805.1.o.b 12
20.d odd 2 1 1520.1.m.b 2
57.d even 2 1 855.1.g.c 2
76.d even 2 1 1520.1.m.b 2
95.d odd 2 1 CM 95.1.d.b 2
95.g even 4 2 475.1.c.b 2
95.h odd 6 2 1805.1.h.b 4
95.i even 6 2 1805.1.h.b 4
95.o odd 18 6 1805.1.o.b 12
95.p even 18 6 1805.1.o.b 12
285.b even 2 1 855.1.g.c 2
380.d even 2 1 1520.1.m.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b 2 1.a even 1 1 trivial
95.1.d.b 2 5.b even 2 1 inner
95.1.d.b 2 19.b odd 2 1 inner
95.1.d.b 2 95.d odd 2 1 CM
475.1.c.b 2 5.c odd 4 2
475.1.c.b 2 95.g even 4 2
855.1.g.c 2 3.b odd 2 1
855.1.g.c 2 15.d odd 2 1
855.1.g.c 2 57.d even 2 1
855.1.g.c 2 285.b even 2 1
1520.1.m.b 2 4.b odd 2 1
1520.1.m.b 2 20.d odd 2 1
1520.1.m.b 2 76.d even 2 1
1520.1.m.b 2 380.d even 2 1
1805.1.h.b 4 19.c even 3 2
1805.1.h.b 4 19.d odd 6 2
1805.1.h.b 4 95.h odd 6 2
1805.1.h.b 4 95.i even 6 2
1805.1.o.b 12 19.e even 9 6
1805.1.o.b 12 19.f odd 18 6
1805.1.o.b 12 95.o odd 18 6
1805.1.o.b 12 95.p even 18 6

## Hecke kernels

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

## Hecke characteristic polynomials

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