# Properties

 Label 120.1.i.a Level 120 Weight 1 Character orbit 120.i Analytic conductor 0.060 Analytic rank 0 Dimension 2 Projective image $$D_{2}$$ CM/RM discs -15, -24, 40 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$120 = 2^{3} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 120.i (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.0598878015160$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-6}, \sqrt{10})$$ Artin image $D_4:C_2$ Artin field Galois closure of 8.0.3240000.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -i q^{2} -i q^{3} - q^{4} + i q^{5} - q^{6} + i q^{8} - q^{9} +O(q^{10})$$ $$q -i q^{2} -i q^{3} - q^{4} + i q^{5} - q^{6} + i q^{8} - q^{9} + q^{10} + i q^{12} + q^{15} + q^{16} + i q^{18} -i q^{20} + q^{24} - q^{25} + i q^{27} -i q^{30} -2 q^{31} -i q^{32} + q^{36} - q^{40} -i q^{45} -i q^{48} + q^{49} + i q^{50} -2 i q^{53} + q^{54} - q^{60} + 2 i q^{62} - q^{64} -i q^{72} + i q^{75} + 2 q^{79} + i q^{80} + q^{81} + 2 i q^{83} - q^{90} + 2 i q^{93} - q^{96} -i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + 2q^{10} + 2q^{15} + 2q^{16} + 2q^{24} - 2q^{25} - 4q^{31} + 2q^{36} - 2q^{40} + 2q^{49} + 2q^{54} - 2q^{60} - 2q^{64} + 4q^{79} + 2q^{81} - 2q^{90} - 2q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/120\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$41$$ $$61$$ $$97$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 1.00000i −1.00000 0 1.00000i −1.00000 1.00000
29.2 1.00000i 1.00000i −1.00000 1.00000i −1.00000 0 1.00000i −1.00000 1.00000
 $$n$$: 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
40.f even 2 1 RM by $$\Q(\sqrt{10})$$
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
120.i odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.1.i.a 2
3.b odd 2 1 inner 120.1.i.a 2
4.b odd 2 1 480.1.i.a 2
5.b even 2 1 inner 120.1.i.a 2
5.c odd 4 1 600.1.n.a 1
5.c odd 4 1 600.1.n.b 1
8.b even 2 1 inner 120.1.i.a 2
8.d odd 2 1 480.1.i.a 2
9.c even 3 2 3240.1.bh.h 4
9.d odd 6 2 3240.1.bh.h 4
12.b even 2 1 480.1.i.a 2
15.d odd 2 1 CM 120.1.i.a 2
15.e even 4 1 600.1.n.a 1
15.e even 4 1 600.1.n.b 1
16.e even 4 1 3840.1.c.a 1
16.e even 4 1 3840.1.c.d 1
16.f odd 4 1 3840.1.c.b 1
16.f odd 4 1 3840.1.c.c 1
20.d odd 2 1 480.1.i.a 2
20.e even 4 1 2400.1.n.a 1
20.e even 4 1 2400.1.n.b 1
24.f even 2 1 480.1.i.a 2
24.h odd 2 1 CM 120.1.i.a 2
40.e odd 2 1 480.1.i.a 2
40.f even 2 1 RM 120.1.i.a 2
40.i odd 4 1 600.1.n.a 1
40.i odd 4 1 600.1.n.b 1
40.k even 4 1 2400.1.n.a 1
40.k even 4 1 2400.1.n.b 1
45.h odd 6 2 3240.1.bh.h 4
45.j even 6 2 3240.1.bh.h 4
48.i odd 4 1 3840.1.c.a 1
48.i odd 4 1 3840.1.c.d 1
48.k even 4 1 3840.1.c.b 1
48.k even 4 1 3840.1.c.c 1
60.h even 2 1 480.1.i.a 2
60.l odd 4 1 2400.1.n.a 1
60.l odd 4 1 2400.1.n.b 1
72.j odd 6 2 3240.1.bh.h 4
72.n even 6 2 3240.1.bh.h 4
80.k odd 4 1 3840.1.c.b 1
80.k odd 4 1 3840.1.c.c 1
80.q even 4 1 3840.1.c.a 1
80.q even 4 1 3840.1.c.d 1
120.i odd 2 1 inner 120.1.i.a 2
120.m even 2 1 480.1.i.a 2
120.q odd 4 1 2400.1.n.a 1
120.q odd 4 1 2400.1.n.b 1
120.w even 4 1 600.1.n.a 1
120.w even 4 1 600.1.n.b 1
240.t even 4 1 3840.1.c.b 1
240.t even 4 1 3840.1.c.c 1
240.bm odd 4 1 3840.1.c.a 1
240.bm odd 4 1 3840.1.c.d 1
360.bh odd 6 2 3240.1.bh.h 4
360.bk even 6 2 3240.1.bh.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.1.i.a 2 1.a even 1 1 trivial
120.1.i.a 2 3.b odd 2 1 inner
120.1.i.a 2 5.b even 2 1 inner
120.1.i.a 2 8.b even 2 1 inner
120.1.i.a 2 15.d odd 2 1 CM
120.1.i.a 2 24.h odd 2 1 CM
120.1.i.a 2 40.f even 2 1 RM
120.1.i.a 2 120.i odd 2 1 inner
480.1.i.a 2 4.b odd 2 1
480.1.i.a 2 8.d odd 2 1
480.1.i.a 2 12.b even 2 1
480.1.i.a 2 20.d odd 2 1
480.1.i.a 2 24.f even 2 1
480.1.i.a 2 40.e odd 2 1
480.1.i.a 2 60.h even 2 1
480.1.i.a 2 120.m even 2 1
600.1.n.a 1 5.c odd 4 1
600.1.n.a 1 15.e even 4 1
600.1.n.a 1 40.i odd 4 1
600.1.n.a 1 120.w even 4 1
600.1.n.b 1 5.c odd 4 1
600.1.n.b 1 15.e even 4 1
600.1.n.b 1 40.i odd 4 1
600.1.n.b 1 120.w even 4 1
2400.1.n.a 1 20.e even 4 1
2400.1.n.a 1 40.k even 4 1
2400.1.n.a 1 60.l odd 4 1
2400.1.n.a 1 120.q odd 4 1
2400.1.n.b 1 20.e even 4 1
2400.1.n.b 1 40.k even 4 1
2400.1.n.b 1 60.l odd 4 1
2400.1.n.b 1 120.q odd 4 1
3240.1.bh.h 4 9.c even 3 2
3240.1.bh.h 4 9.d odd 6 2
3240.1.bh.h 4 45.h odd 6 2
3240.1.bh.h 4 45.j even 6 2
3240.1.bh.h 4 72.j odd 6 2
3240.1.bh.h 4 72.n even 6 2
3240.1.bh.h 4 360.bh odd 6 2
3240.1.bh.h 4 360.bk even 6 2
3840.1.c.a 1 16.e even 4 1
3840.1.c.a 1 48.i odd 4 1
3840.1.c.a 1 80.q even 4 1
3840.1.c.a 1 240.bm odd 4 1
3840.1.c.b 1 16.f odd 4 1
3840.1.c.b 1 48.k even 4 1
3840.1.c.b 1 80.k odd 4 1
3840.1.c.b 1 240.t even 4 1
3840.1.c.c 1 16.f odd 4 1
3840.1.c.c 1 48.k even 4 1
3840.1.c.c 1 80.k odd 4 1
3840.1.c.c 1 240.t even 4 1
3840.1.c.d 1 16.e even 4 1
3840.1.c.d 1 48.i odd 4 1
3840.1.c.d 1 80.q even 4 1
3840.1.c.d 1 240.bm odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(120, [\chi])$$.

## Hecke characteristic polynomials

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