Properties

Label 1872.2.c.b
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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta q^{5} - \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} - \beta q^{7} + (\beta - 3) q^{13} + 2 q^{17} - 3 \beta q^{19} + 4 q^{23} + q^{25} + 10 q^{29} + 5 \beta q^{31} + 4 q^{35} - 4 \beta q^{37} + 5 \beta q^{41} - 4 q^{43} + 6 \beta q^{47} + 3 q^{49} + 6 q^{53} - 2 \beta q^{59} + 2 q^{61} + ( - 3 \beta - 4) q^{65} - \beta q^{67} + 2 \beta q^{73} + 2 \beta q^{83} + 2 \beta q^{85} - 3 \beta q^{89} + (3 \beta + 4) q^{91} + 12 q^{95} + 6 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{13} + 4 q^{17} + 8 q^{23} + 2 q^{25} + 20 q^{29} + 8 q^{35} - 8 q^{43} + 6 q^{49} + 12 q^{53} + 4 q^{61} - 8 q^{65} + 8 q^{91} + 24 q^{95}+O(q^{100}) \) 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
1.00000i
1.00000i
0 0 0 2.00000i 0 2.00000i 0 0 0
1585.2 0 0 0 2.00000i 0 2.00000i 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.b 2
3.b odd 2 1 624.2.c.a 2
4.b odd 2 1 234.2.b.a 2
12.b even 2 1 78.2.b.a 2
13.b even 2 1 inner 1872.2.c.b 2
24.f even 2 1 2496.2.c.f 2
24.h odd 2 1 2496.2.c.m 2
39.d odd 2 1 624.2.c.a 2
39.f even 4 1 8112.2.a.g 1
39.f even 4 1 8112.2.a.j 1
52.b odd 2 1 234.2.b.a 2
52.f even 4 1 3042.2.a.c 1
52.f even 4 1 3042.2.a.n 1
60.h even 2 1 1950.2.b.c 2
60.l odd 4 1 1950.2.f.d 2
60.l odd 4 1 1950.2.f.g 2
84.h odd 2 1 3822.2.c.d 2
156.h even 2 1 78.2.b.a 2
156.l odd 4 1 1014.2.a.b 1
156.l odd 4 1 1014.2.a.g 1
156.p even 6 2 1014.2.i.c 4
156.r even 6 2 1014.2.i.c 4
156.v odd 12 2 1014.2.e.b 2
156.v odd 12 2 1014.2.e.e 2
312.b odd 2 1 2496.2.c.m 2
312.h even 2 1 2496.2.c.f 2
780.d even 2 1 1950.2.b.c 2
780.w odd 4 1 1950.2.f.d 2
780.w odd 4 1 1950.2.f.g 2
1092.d odd 2 1 3822.2.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.b.a 2 12.b even 2 1
78.2.b.a 2 156.h even 2 1
234.2.b.a 2 4.b odd 2 1
234.2.b.a 2 52.b odd 2 1
624.2.c.a 2 3.b odd 2 1
624.2.c.a 2 39.d odd 2 1
1014.2.a.b 1 156.l odd 4 1
1014.2.a.g 1 156.l odd 4 1
1014.2.e.b 2 156.v odd 12 2
1014.2.e.e 2 156.v odd 12 2
1014.2.i.c 4 156.p even 6 2
1014.2.i.c 4 156.r even 6 2
1872.2.c.b 2 1.a even 1 1 trivial
1872.2.c.b 2 13.b even 2 1 inner
1950.2.b.c 2 60.h even 2 1
1950.2.b.c 2 780.d even 2 1
1950.2.f.d 2 60.l odd 4 1
1950.2.f.d 2 780.w odd 4 1
1950.2.f.g 2 60.l odd 4 1
1950.2.f.g 2 780.w odd 4 1
2496.2.c.f 2 24.f even 2 1
2496.2.c.f 2 312.h even 2 1
2496.2.c.m 2 24.h odd 2 1
2496.2.c.m 2 312.b odd 2 1
3042.2.a.c 1 52.f even 4 1
3042.2.a.n 1 52.f even 4 1
3822.2.c.d 2 84.h odd 2 1
3822.2.c.d 2 1092.d odd 2 1
8112.2.a.g 1 39.f even 4 1
8112.2.a.j 1 39.f even 4 1

Hecke kernels

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

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} \) 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
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 13 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 100 \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{2} + 100 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 144 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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