Properties

Label 1872.2.c.e
Level $1872$
Weight $2$
Character orbit 1872.c
Analytic conductor $14.948$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.9479952584\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{7} + \beta q^{11} + ( - \beta - 1) q^{13} + 6 q^{17} + \beta q^{19} + 5 q^{25} - 6 q^{29} - \beta q^{31} - 2 \beta q^{37} - 2 \beta q^{41} + 4 q^{43} - \beta q^{47} - 5 q^{49} - 6 q^{53} - 3 \beta q^{59} - 2 q^{61} - 3 \beta q^{67} - \beta q^{71} + 12 q^{77} + 8 q^{79} + \beta q^{83} - 2 \beta q^{89} + (\beta - 12) q^{91} + 4 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{13} + 12 q^{17} + 10 q^{25} - 12 q^{29} + 8 q^{43} - 10 q^{49} - 12 q^{53} - 4 q^{61} + 24 q^{77} + 16 q^{79} - 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1585.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 3.46410i 0 0 0
1585.2 0 0 0 0 0 3.46410i 0 0 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
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 1872.2.c.e 2
3.b odd 2 1 624.2.c.e 2
4.b odd 2 1 117.2.b.a 2
12.b even 2 1 39.2.b.a 2
13.b even 2 1 inner 1872.2.c.e 2
24.f even 2 1 2496.2.c.k 2
24.h odd 2 1 2496.2.c.d 2
39.d odd 2 1 624.2.c.e 2
39.f even 4 2 8112.2.a.bv 2
52.b odd 2 1 117.2.b.a 2
52.f even 4 2 1521.2.a.l 2
60.h even 2 1 975.2.b.d 2
60.l odd 4 2 975.2.h.f 4
84.h odd 2 1 1911.2.c.d 2
156.h even 2 1 39.2.b.a 2
156.l odd 4 2 507.2.a.f 2
156.p even 6 1 507.2.j.a 2
156.p even 6 1 507.2.j.c 2
156.r even 6 1 507.2.j.a 2
156.r even 6 1 507.2.j.c 2
156.v odd 12 4 507.2.e.e 4
312.b odd 2 1 2496.2.c.d 2
312.h even 2 1 2496.2.c.k 2
780.d even 2 1 975.2.b.d 2
780.w odd 4 2 975.2.h.f 4
1092.d odd 2 1 1911.2.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 12.b even 2 1
39.2.b.a 2 156.h even 2 1
117.2.b.a 2 4.b odd 2 1
117.2.b.a 2 52.b odd 2 1
507.2.a.f 2 156.l odd 4 2
507.2.e.e 4 156.v odd 12 4
507.2.j.a 2 156.p even 6 1
507.2.j.a 2 156.r even 6 1
507.2.j.c 2 156.p even 6 1
507.2.j.c 2 156.r even 6 1
624.2.c.e 2 3.b odd 2 1
624.2.c.e 2 39.d odd 2 1
975.2.b.d 2 60.h even 2 1
975.2.b.d 2 780.d even 2 1
975.2.h.f 4 60.l odd 4 2
975.2.h.f 4 780.w odd 4 2
1521.2.a.l 2 52.f even 4 2
1872.2.c.e 2 1.a even 1 1 trivial
1872.2.c.e 2 13.b even 2 1 inner
1911.2.c.d 2 84.h odd 2 1
1911.2.c.d 2 1092.d odd 2 1
2496.2.c.d 2 24.h odd 2 1
2496.2.c.d 2 312.b odd 2 1
2496.2.c.k 2 24.f even 2 1
2496.2.c.k 2 312.h even 2 1
8112.2.a.bv 2 39.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1872, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} + 12 \) Copy content Toggle raw display
\( T_{11}^{2} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 12 \) Copy content Toggle raw display
$11$ \( T^{2} + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 13 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 12 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 12 \) Copy content Toggle raw display
$37$ \( T^{2} + 48 \) Copy content Toggle raw display
$41$ \( T^{2} + 48 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 108 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 108 \) Copy content Toggle raw display
$71$ \( T^{2} + 12 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 12 \) Copy content Toggle raw display
$89$ \( T^{2} + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 192 \) Copy content Toggle raw display
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