# 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

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.758578819202$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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

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