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

Downloads

Learn more about

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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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:

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

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