# 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

## 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: $$x^{2} + 1$$ 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.

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

## Character values

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

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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