Properties

Label 145.1.h.a
Level $145$
Weight $1$
Character orbit 145.h
Analytic conductor $0.072$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
RM discriminant 29
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.0723644268318\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.3625.1
Artin image: $C_4\wr C_2$
Artin field: Galois closure of 8.4.88410125.1

$q$-expansion

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

\(f(q)\) \(=\) \( q - i q^{4} - i q^{5} + (i - 1) q^{7} + i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{4} - i q^{5} + (i - 1) q^{7} + i q^{9} + (i + 1) q^{13} - q^{16} - q^{20} + ( - i - 1) q^{23} - q^{25} + (i + 1) q^{28} - i q^{29} + (i + 1) q^{35} + q^{36} + q^{45} - i q^{49} + ( - i + 1) q^{52} + ( - i - 1) q^{53} + i q^{59} + ( - i - 1) q^{63} + i q^{64} + ( - i + 1) q^{65} + ( - i + 1) q^{67} + i q^{80} - q^{81} + (i + 1) q^{83} - q^{91} + (i - 1) q^{92} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} + 2 q^{13} - 2 q^{16} - 2 q^{20} - 2 q^{23} - 2 q^{25} + 2 q^{28} + 2 q^{35} + 2 q^{36} + 2 q^{45} + 2 q^{52} - 2 q^{53} - 2 q^{63} + 2 q^{65} + 2 q^{67} - 2 q^{81} + 2 q^{83} - 4 q^{91} - 2 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
1.00000i
1.00000i
0 0 1.00000i 1.00000i 0 −1.00000 + 1.00000i 0 1.00000i 0
57.1 0 0 1.00000i 1.00000i 0 −1.00000 1.00000i 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
29.b even 2 1 RM by \(\Q(\sqrt{29}) \)
5.c odd 4 1 inner
145.h 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.h.a 2
3.b odd 2 1 1305.1.o.a 2
4.b odd 2 1 2320.1.bm.a 2
5.b even 2 1 725.1.h.a 2
5.c odd 4 1 inner 145.1.h.a 2
5.c odd 4 1 725.1.h.a 2
15.e even 4 1 1305.1.o.a 2
20.e even 4 1 2320.1.bm.a 2
29.b even 2 1 RM 145.1.h.a 2
87.d odd 2 1 1305.1.o.a 2
116.d odd 2 1 2320.1.bm.a 2
145.d even 2 1 725.1.h.a 2
145.h odd 4 1 inner 145.1.h.a 2
145.h odd 4 1 725.1.h.a 2
435.p even 4 1 1305.1.o.a 2
580.o even 4 1 2320.1.bm.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.1.h.a 2 1.a even 1 1 trivial
145.1.h.a 2 5.c odd 4 1 inner
145.1.h.a 2 29.b even 2 1 RM
145.1.h.a 2 145.h odd 4 1 inner
725.1.h.a 2 5.b even 2 1
725.1.h.a 2 5.c odd 4 1
725.1.h.a 2 145.d even 2 1
725.1.h.a 2 145.h odd 4 1
1305.1.o.a 2 3.b odd 2 1
1305.1.o.a 2 15.e even 4 1
1305.1.o.a 2 87.d odd 2 1
1305.1.o.a 2 435.p even 4 1
2320.1.bm.a 2 4.b odd 2 1
2320.1.bm.a 2 20.e even 4 1
2320.1.bm.a 2 116.d odd 2 1
2320.1.bm.a 2 580.o even 4 1

Hecke kernels

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

Hecke characteristic polynomials

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