Properties

Label 1840.4.a.k
Level $1840$
Weight $4$
Character orbit 1840.a
Self dual yes
Analytic conductor $108.564$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(108.563514411\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 2x^{3} - 60x^{2} - 45x + 108 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 4) q^{3} + 5 q^{5} + ( - 2 \beta_{2} - \beta_1 - 1) q^{7} + (\beta_{3} - 2 \beta_{2} - 4 \beta_1 + 18) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 4) q^{3} + 5 q^{5} + ( - 2 \beta_{2} - \beta_1 - 1) q^{7} + (\beta_{3} - 2 \beta_{2} - 4 \beta_1 + 18) q^{9} + (2 \beta_{3} + \beta_{2} - \beta_1 - 6) q^{11} + (\beta_{3} + 4 \beta_1 + 15) q^{13} + (5 \beta_1 - 20) q^{15} + ( - 3 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 18) q^{17} + ( - 3 \beta_{3} + 5 \beta_{2} - 8 \beta_1 - 39) q^{19} + ( - \beta_{3} + 16 \beta_{2} + 7 \beta_1 - 37) q^{21} + 23 q^{23} + 25 q^{25} + ( - 7 \beta_{3} + 23 \beta_{2} + 11 \beta_1 - 105) q^{27} + ( - 5 \beta_{3} + 10 \beta_{2} + 9 \beta_1 - 34) q^{29} + ( - 8 \beta_{3} - 15 \beta_{2} - 27 \beta_1 + 17) q^{31} + ( - 7 \beta_{3} - 3 \beta_{2} + 14 \beta_1 - 25) q^{33} + ( - 10 \beta_{2} - 5 \beta_1 - 5) q^{35} + (\beta_{3} + 33 \beta_{2} + 3 \beta_1 - 96) q^{37} + (\beta_{3} - 7 \beta_{2} + 27 \beta_1 + 43) q^{39} + ( - 11 \beta_{3} - 15 \beta_{2} + 32 \beta_1 + 28) q^{41} + (4 \beta_{3} + 18 \beta_{2} - 52 \beta_1 + 32) q^{43} + (5 \beta_{3} - 10 \beta_{2} - 20 \beta_1 + 90) q^{45} + ( - 4 \beta_{3} + 33 \beta_{2} + 16 \beta_1 - 106) q^{47} + (9 \beta_{3} + 10 \beta_{2} - 42 \beta_1 - 17) q^{49} + (7 \beta_{3} + 43 \beta_{2} + 6 \beta_1 - 127) q^{51} + (11 \beta_{3} + 7 \beta_{2} + 37 \beta_1 - 114) q^{53} + (10 \beta_{3} + 5 \beta_{2} - 5 \beta_1 - 30) q^{55} + (\beta_{3} - 22 \beta_{2} - 95 \beta_1 - 7) q^{57} + ( - \beta_{3} - 7 \beta_{2} + 79 \beta_1 + 76) q^{59} + (31 \beta_{2} - 47 \beta_1 - 66) q^{61} + (10 \beta_{3} - 73 \beta_{2} - 86 \beta_1 + 487) q^{63} + (5 \beta_{3} + 20 \beta_1 + 75) q^{65} + (7 \beta_{3} - 33 \beta_{2} - 43 \beta_1 + 140) q^{67} + (23 \beta_1 - 92) q^{69} + ( - 23 \beta_{3} - 21 \beta_{2} + 72 \beta_1 - 68) q^{71} + (16 \beta_{3} - 17 \beta_{2} - 44 \beta_1 - 140) q^{73} + (25 \beta_1 - 100) q^{75} + ( - 7 \beta_{3} - 46 \beta_{2} - 9 \beta_1 + 19) q^{77} + (28 \beta_{3} + 20 \beta_{2} + 92 \beta_1 + 220) q^{79} + (5 \beta_{3} - 136 \beta_{2} - 173 \beta_1 + 482) q^{81} + (9 \beta_{3} + 47 \beta_{2} - 13 \beta_1 + 290) q^{83} + ( - 15 \beta_{3} - 30 \beta_{2} - 10 \beta_1 + 90) q^{85} + (24 \beta_{3} - 93 \beta_{2} - 134 \beta_1 + 522) q^{87} + (16 \beta_{3} + 30 \beta_{2} - 92 \beta_1 - 512) q^{89} + ( - 6 \beta_{3} - 19 \beta_{2} - 19 \beta_1 - 122) q^{91} + ( - 3 \beta_{3} + 151 \beta_{2} - 19 \beta_1 - 837) q^{93} + ( - 15 \beta_{3} + 25 \beta_{2} - 40 \beta_1 - 195) q^{95} + (30 \beta_{3} + 55 \beta_{2} - 13 \beta_1 - 198) q^{97} + ( - 19 \beta_{3} - 41 \beta_{2} - 70 \beta_1 + 741) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{3} + 20 q^{5} - 8 q^{7} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 14 q^{3} + 20 q^{5} - 8 q^{7} + 64 q^{9} - 21 q^{11} + 70 q^{13} - 70 q^{15} + 56 q^{17} - 173 q^{19} - 120 q^{21} + 92 q^{23} + 100 q^{25} - 389 q^{27} - 118 q^{29} - 17 q^{31} - 89 q^{33} - 40 q^{35} - 343 q^{37} + 221 q^{39} + 139 q^{41} + 50 q^{43} + 320 q^{45} - 367 q^{47} - 124 q^{49} - 439 q^{51} - 353 q^{53} - 105 q^{55} - 238 q^{57} + 453 q^{59} - 327 q^{61} + 1723 q^{63} + 350 q^{65} + 455 q^{67} - 322 q^{69} - 195 q^{71} - 633 q^{73} - 350 q^{75} - 2 q^{77} + 1140 q^{79} + 1456 q^{81} + 1199 q^{83} + 280 q^{85} + 1775 q^{87} - 2170 q^{89} - 557 q^{91} - 3241 q^{93} - 865 q^{95} - 703 q^{97} + 2745 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 60x^{2} - 45x + 108 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 5\nu^{2} - 45\nu + 54 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} - \nu^{2} - 126\nu - 153 ) / 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 2\beta_{2} + 4\beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - \beta_{2} + 65\beta _1 + 91 \) 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
−6.12571
−1.92711
1.01192
9.04090
0 −10.1257 0 5.00000 0 24.6431 0 75.5301 0
1.2 0 −5.92711 0 5.00000 0 −24.6272 0 8.13066 0
1.3 0 −2.98808 0 5.00000 0 −2.98517 0 −18.0714 0
1.4 0 5.04090 0 5.00000 0 −5.03071 0 −1.58932 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 1840.4.a.k 4
4.b odd 2 1 230.4.a.j 4
12.b even 2 1 2070.4.a.bg 4
20.d odd 2 1 1150.4.a.n 4
20.e even 4 2 1150.4.b.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.j 4 4.b odd 2 1
1150.4.a.n 4 20.d odd 2 1
1150.4.b.o 8 20.e even 4 2
1840.4.a.k 4 1.a even 1 1 trivial
2070.4.a.bg 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{4} + 14T_{3}^{3} + 12T_{3}^{2} - 365T_{3} - 904 \) Copy content Toggle raw display
\( T_{7}^{4} + 8T_{7}^{3} - 592T_{7}^{2} - 4865T_{7} - 9114 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 14 T^{3} + 12 T^{2} + \cdots - 904 \) Copy content Toggle raw display
$5$ \( (T - 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 8 T^{3} - 592 T^{2} + \cdots - 9114 \) Copy content Toggle raw display
$11$ \( T^{4} + 21 T^{3} - 2605 T^{2} + \cdots - 164232 \) Copy content Toggle raw display
$13$ \( T^{4} - 70 T^{3} + 286 T^{2} + \cdots - 46062 \) Copy content Toggle raw display
$17$ \( T^{4} - 56 T^{3} - 8532 T^{2} + \cdots + 7970400 \) Copy content Toggle raw display
$19$ \( T^{4} + 173 T^{3} - 2059 T^{2} + \cdots + 9983784 \) Copy content Toggle raw display
$23$ \( (T - 23)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 118 T^{3} + \cdots + 44611452 \) Copy content Toggle raw display
$31$ \( T^{4} + 17 T^{3} + \cdots + 2678191911 \) Copy content Toggle raw display
$37$ \( T^{4} + 343 T^{3} + \cdots - 2095186944 \) Copy content Toggle raw display
$41$ \( T^{4} - 139 T^{3} + \cdots + 4789051317 \) Copy content Toggle raw display
$43$ \( T^{4} - 50 T^{3} + \cdots + 5363873792 \) Copy content Toggle raw display
$47$ \( T^{4} + 367 T^{3} + \cdots + 668142000 \) Copy content Toggle raw display
$53$ \( T^{4} + 353 T^{3} + \cdots - 289523592 \) Copy content Toggle raw display
$59$ \( T^{4} - 453 T^{3} + \cdots + 8963853984 \) Copy content Toggle raw display
$61$ \( T^{4} + 327 T^{3} + \cdots - 1898667392 \) Copy content Toggle raw display
$67$ \( T^{4} - 455 T^{3} + \cdots - 1366509568 \) Copy content Toggle raw display
$71$ \( T^{4} + 195 T^{3} + \cdots + 106065123651 \) Copy content Toggle raw display
$73$ \( T^{4} + 633 T^{3} + \cdots - 15845099784 \) Copy content Toggle raw display
$79$ \( T^{4} - 1140 T^{3} + \cdots + 60635801088 \) Copy content Toggle raw display
$83$ \( T^{4} - 1199 T^{3} + \cdots + 103460976 \) Copy content Toggle raw display
$89$ \( T^{4} + 2170 T^{3} + \cdots - 5919819264 \) Copy content Toggle raw display
$97$ \( T^{4} + 703 T^{3} + \cdots - 22808262684 \) Copy content Toggle raw display
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