Properties

Label 3042.2.b.g
Level $3042$
Weight $2$
Character orbit 3042.b
Analytic conductor $24.290$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.2904922949\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{2} - q^{4} + 2 i q^{5} + 4 i q^{7} + i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - q^{4} + 2 i q^{5} + 4 i q^{7} + i q^{8} + 2 q^{10} + 4 i q^{11} + 4 q^{14} + q^{16} + 2 q^{17} + 8 i q^{19} - 2 i q^{20} + 4 q^{22} + q^{25} - 4 i q^{28} - 6 q^{29} + 4 i q^{31} - i q^{32} - 2 i q^{34} - 8 q^{35} - 2 i q^{37} + 8 q^{38} - 2 q^{40} - 10 i q^{41} - 4 q^{43} - 4 i q^{44} - 8 i q^{47} - 9 q^{49} - i q^{50} + 10 q^{53} - 8 q^{55} - 4 q^{56} + 6 i q^{58} - 4 i q^{59} - 2 q^{61} + 4 q^{62} - q^{64} + 16 i q^{67} - 2 q^{68} + 8 i q^{70} - 8 i q^{71} + 2 i q^{73} - 2 q^{74} - 8 i q^{76} - 16 q^{77} + 8 q^{79} + 2 i q^{80} - 10 q^{82} + 12 i q^{83} + 4 i q^{85} + 4 i q^{86} - 4 q^{88} - 14 i q^{89} - 8 q^{94} - 16 q^{95} - 10 i q^{97} + 9 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 4 q^{10} + 8 q^{14} + 2 q^{16} + 4 q^{17} + 8 q^{22} + 2 q^{25} - 12 q^{29} - 16 q^{35} + 16 q^{38} - 4 q^{40} - 8 q^{43} - 18 q^{49} + 20 q^{53} - 16 q^{55} - 8 q^{56} - 4 q^{61} + 8 q^{62} - 2 q^{64} - 4 q^{68} - 4 q^{74} - 32 q^{77} + 16 q^{79} - 20 q^{82} - 8 q^{88} - 16 q^{94} - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
1.00000i
1.00000i
1.00000i 0 −1.00000 2.00000i 0 4.00000i 1.00000i 0 2.00000
1351.2 1.00000i 0 −1.00000 2.00000i 0 4.00000i 1.00000i 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3042.2.b.g 2
3.b odd 2 1 1014.2.b.b 2
13.b even 2 1 inner 3042.2.b.g 2
13.d odd 4 1 234.2.a.c 1
13.d odd 4 1 3042.2.a.f 1
39.d odd 2 1 1014.2.b.b 2
39.f even 4 1 78.2.a.a 1
39.f even 4 1 1014.2.a.d 1
39.h odd 6 2 1014.2.i.d 4
39.i odd 6 2 1014.2.i.d 4
39.k even 12 2 1014.2.e.c 2
39.k even 12 2 1014.2.e.f 2
52.f even 4 1 1872.2.a.c 1
65.f even 4 1 5850.2.e.bb 2
65.g odd 4 1 5850.2.a.d 1
65.k even 4 1 5850.2.e.bb 2
104.j odd 4 1 7488.2.a.bz 1
104.m even 4 1 7488.2.a.bk 1
117.y odd 12 2 2106.2.e.j 2
117.z even 12 2 2106.2.e.q 2
156.l odd 4 1 624.2.a.h 1
156.l odd 4 1 8112.2.a.v 1
195.j odd 4 1 1950.2.e.i 2
195.n even 4 1 1950.2.a.w 1
195.u odd 4 1 1950.2.e.i 2
273.o odd 4 1 3822.2.a.j 1
312.w odd 4 1 2496.2.a.b 1
312.y even 4 1 2496.2.a.t 1
429.l odd 4 1 9438.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 39.f even 4 1
234.2.a.c 1 13.d odd 4 1
624.2.a.h 1 156.l odd 4 1
1014.2.a.d 1 39.f even 4 1
1014.2.b.b 2 3.b odd 2 1
1014.2.b.b 2 39.d odd 2 1
1014.2.e.c 2 39.k even 12 2
1014.2.e.f 2 39.k even 12 2
1014.2.i.d 4 39.h odd 6 2
1014.2.i.d 4 39.i odd 6 2
1872.2.a.c 1 52.f even 4 1
1950.2.a.w 1 195.n even 4 1
1950.2.e.i 2 195.j odd 4 1
1950.2.e.i 2 195.u odd 4 1
2106.2.e.j 2 117.y odd 12 2
2106.2.e.q 2 117.z even 12 2
2496.2.a.b 1 312.w odd 4 1
2496.2.a.t 1 312.y even 4 1
3042.2.a.f 1 13.d odd 4 1
3042.2.b.g 2 1.a even 1 1 trivial
3042.2.b.g 2 13.b even 2 1 inner
3822.2.a.j 1 273.o odd 4 1
5850.2.a.d 1 65.g odd 4 1
5850.2.e.bb 2 65.f even 4 1
5850.2.e.bb 2 65.k even 4 1
7488.2.a.bk 1 104.m even 4 1
7488.2.a.bz 1 104.j odd 4 1
8112.2.a.v 1 156.l odd 4 1
9438.2.a.t 1 429.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3042, [\chi])\):

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 100 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( (T - 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 256 \) Copy content Toggle raw display
$71$ \( T^{2} + 64 \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( T^{2} + 196 \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
show more
show less