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

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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:

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

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