Properties

Label 1872.4.a.bb
Level $1872$
Weight $4$
Character orbit 1872.a
Self dual yes
Analytic conductor $110.452$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1872.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(110.451575531\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{5} + (11 \beta - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{5} + (11 \beta - 1) q^{7} + ( - 12 \beta + 46) q^{11} - 13 q^{13} + ( - 17 \beta - 1) q^{17} + ( - 32 \beta + 58) q^{19} + (12 \beta + 92) q^{23} + (3 \beta - 120) q^{25} + (96 \beta - 26) q^{29} + ( - 34 \beta + 60) q^{31} + (21 \beta + 43) q^{35} + ( - 5 \beta + 107) q^{37} + (22 \beta + 104) q^{41} + (143 \beta - 215) q^{43} + (121 \beta + 157) q^{47} + (99 \beta + 142) q^{49} + (30 \beta + 44) q^{53} + (22 \beta - 2) q^{55} + ( - 124 \beta - 122) q^{59} + (190 \beta - 624) q^{61} + ( - 13 \beta - 13) q^{65} + ( - 232 \beta + 82) q^{67} + (231 \beta - 181) q^{71} + ( - 260 \beta + 358) q^{73} + (386 \beta - 574) q^{77} + (40 \beta + 484) q^{79} + (182 \beta + 888) q^{83} + ( - 35 \beta - 69) q^{85} + ( - 388 \beta + 554) q^{89} + ( - 143 \beta + 13) q^{91} + ( - 6 \beta - 70) q^{95} + ( - 508 \beta - 210) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{5} + 9 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{5} + 9 q^{7} + 80 q^{11} - 26 q^{13} - 19 q^{17} + 84 q^{19} + 196 q^{23} - 237 q^{25} + 44 q^{29} + 86 q^{31} + 107 q^{35} + 209 q^{37} + 230 q^{41} - 287 q^{43} + 435 q^{47} + 383 q^{49} + 118 q^{53} + 18 q^{55} - 368 q^{59} - 1058 q^{61} - 39 q^{65} - 68 q^{67} - 131 q^{71} + 456 q^{73} - 762 q^{77} + 1008 q^{79} + 1958 q^{83} - 173 q^{85} + 720 q^{89} - 117 q^{91} - 146 q^{95} - 928 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.56155
2.56155
0 0 0 −0.561553 0 −18.1771 0 0 0
1.2 0 0 0 3.56155 0 27.1771 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.4.a.bb 2
3.b odd 2 1 208.4.a.h 2
4.b odd 2 1 117.4.a.d 2
12.b even 2 1 13.4.a.b 2
24.f even 2 1 832.4.a.s 2
24.h odd 2 1 832.4.a.z 2
52.b odd 2 1 1521.4.a.r 2
60.h even 2 1 325.4.a.f 2
60.l odd 4 2 325.4.b.e 4
84.h odd 2 1 637.4.a.b 2
132.d odd 2 1 1573.4.a.b 2
156.h even 2 1 169.4.a.g 2
156.l odd 4 2 169.4.b.f 4
156.p even 6 2 169.4.c.g 4
156.r even 6 2 169.4.c.j 4
156.v odd 12 4 169.4.e.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 12.b even 2 1
117.4.a.d 2 4.b odd 2 1
169.4.a.g 2 156.h even 2 1
169.4.b.f 4 156.l odd 4 2
169.4.c.g 4 156.p even 6 2
169.4.c.j 4 156.r even 6 2
169.4.e.f 8 156.v odd 12 4
208.4.a.h 2 3.b odd 2 1
325.4.a.f 2 60.h even 2 1
325.4.b.e 4 60.l odd 4 2
637.4.a.b 2 84.h odd 2 1
832.4.a.s 2 24.f even 2 1
832.4.a.z 2 24.h odd 2 1
1521.4.a.r 2 52.b odd 2 1
1573.4.a.b 2 132.d odd 2 1
1872.4.a.bb 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1872))\):

\( T_{5}^{2} - 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 9T_{7} - 494 \) 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
$5$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} - 9T - 494 \) Copy content Toggle raw display
$11$ \( T^{2} - 80T + 988 \) Copy content Toggle raw display
$13$ \( (T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 19T - 1138 \) Copy content Toggle raw display
$19$ \( T^{2} - 84T - 2588 \) Copy content Toggle raw display
$23$ \( T^{2} - 196T + 8992 \) Copy content Toggle raw display
$29$ \( T^{2} - 44T - 38684 \) Copy content Toggle raw display
$31$ \( T^{2} - 86T - 3064 \) Copy content Toggle raw display
$37$ \( T^{2} - 209T + 10814 \) Copy content Toggle raw display
$41$ \( T^{2} - 230T + 11168 \) Copy content Toggle raw display
$43$ \( T^{2} + 287T - 66316 \) Copy content Toggle raw display
$47$ \( T^{2} - 435T - 14918 \) Copy content Toggle raw display
$53$ \( T^{2} - 118T - 344 \) Copy content Toggle raw display
$59$ \( T^{2} + 368T - 31492 \) Copy content Toggle raw display
$61$ \( T^{2} + 1058 T + 126416 \) Copy content Toggle raw display
$67$ \( T^{2} + 68T - 227596 \) Copy content Toggle raw display
$71$ \( T^{2} + 131T - 222494 \) Copy content Toggle raw display
$73$ \( T^{2} - 456T - 235316 \) Copy content Toggle raw display
$79$ \( T^{2} - 1008 T + 247216 \) Copy content Toggle raw display
$83$ \( T^{2} - 1958 T + 817664 \) Copy content Toggle raw display
$89$ \( T^{2} - 720T - 510212 \) Copy content Toggle raw display
$97$ \( T^{2} + 928T - 881476 \) Copy content Toggle raw display
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