Properties

Label 145.1.f.a
Level 145
Weight 1
Character orbit 145.f
Analytic conductor 0.072
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
RM discriminant 5
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 145.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0723644268318\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.121945.1
Artin image $C_4\wr C_2$
Artin field Galois closure of 8.4.15243125.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + i q^{4} -i q^{5} + i q^{9} +O(q^{10})\) \( q + i q^{4} -i q^{5} + i q^{9} + ( -1 - i ) q^{11} - q^{16} + ( -1 - i ) q^{19} + q^{20} - q^{25} + i q^{29} + ( 1 + i ) q^{31} - q^{36} + ( 1 - i ) q^{41} + ( 1 - i ) q^{44} + q^{45} + q^{49} + ( -1 + i ) q^{55} + ( 1 + i ) q^{61} -i q^{64} + 2 i q^{71} + ( 1 - i ) q^{76} + ( -1 - i ) q^{79} + i q^{80} - q^{81} + ( -1 - i ) q^{89} + ( -1 + i ) q^{95} + ( 1 - i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 2q^{11} - 2q^{16} - 2q^{19} + 2q^{20} - 2q^{25} + 2q^{31} - 2q^{36} + 2q^{41} + 2q^{44} + 2q^{45} + 2q^{49} - 2q^{55} + 2q^{61} + 2q^{76} - 2q^{79} - 2q^{81} - 2q^{89} - 2q^{95} + 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/145\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(117\)
\(\chi(n)\) \(i\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.00000i
1.00000i
0 0 1.00000i 1.00000i 0 0 0 1.00000i 0
104.1 0 0 1.00000i 1.00000i 0 0 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 RM by \(\Q(\sqrt{5}) \)
29.c odd 4 1 inner
145.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.1.f.a 2
3.b odd 2 1 1305.1.l.a 2
4.b odd 2 1 2320.1.bj.a 2
5.b even 2 1 RM 145.1.f.a 2
5.c odd 4 2 725.1.g.a 2
15.d odd 2 1 1305.1.l.a 2
20.d odd 2 1 2320.1.bj.a 2
29.c odd 4 1 inner 145.1.f.a 2
87.f even 4 1 1305.1.l.a 2
116.e even 4 1 2320.1.bj.a 2
145.e even 4 1 725.1.g.a 2
145.f odd 4 1 inner 145.1.f.a 2
145.j even 4 1 725.1.g.a 2
435.l even 4 1 1305.1.l.a 2
580.r even 4 1 2320.1.bj.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.1.f.a 2 1.a even 1 1 trivial
145.1.f.a 2 5.b even 2 1 RM
145.1.f.a 2 29.c odd 4 1 inner
145.1.f.a 2 145.f odd 4 1 inner
725.1.g.a 2 5.c odd 4 2
725.1.g.a 2 145.e even 4 1
725.1.g.a 2 145.j even 4 1
1305.1.l.a 2 3.b odd 2 1
1305.1.l.a 2 15.d odd 2 1
1305.1.l.a 2 87.f even 4 1
1305.1.l.a 2 435.l even 4 1
2320.1.bj.a 2 4.b odd 2 1
2320.1.bj.a 2 20.d odd 2 1
2320.1.bj.a 2 116.e even 4 1
2320.1.bj.a 2 580.r even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(145, [\chi])\).

Hecke characteristic polynomials

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