Properties

Label 1840.4.a.l
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: \( x^{4} - 84x^{2} - 11x + 1242 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} - 5 q^{5} + (2 \beta_{2} + \beta_1 - 6) q^{7} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} - 5 q^{5} + (2 \beta_{2} + \beta_1 - 6) q^{7} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 16) q^{9} + ( - 3 \beta_{2} - \beta_1 - 24) q^{11} + ( - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 9) q^{13} + ( - 5 \beta_1 + 5) q^{15} + ( - \beta_{3} - 5 \beta_{2} + 4 \beta_1 + 26) q^{17} + (\beta_{3} - 6 \beta_{2} - 48) q^{19} + (5 \beta_{3} + 7 \beta_{2} - 5 \beta_1 + 76) q^{21} - 23 q^{23} + 25 q^{25} + ( - 5 \beta_{3} + 10 \beta_{2} - \beta_1 - 61) q^{27} + (7 \beta_{3} - 11 \beta_{2} + 23 \beta_1 + 69) q^{29} + (2 \beta_{3} + 11 \beta_{2} - 13 \beta_1 + 55) q^{31} + ( - 7 \beta_{3} - 10 \beta_{2} - 26 \beta_1 - 60) q^{33} + ( - 10 \beta_{2} - 5 \beta_1 + 30) q^{35} + ( - 7 \beta_{3} - 4 \beta_{2} + 17 \beta_1 + 2) q^{37} + (19 \beta_{3} - 22 \beta_{2} - 13 \beta_1 + 95) q^{39} + (3 \beta_{3} + 8 \beta_{2} - 16 \beta_1 - 91) q^{41} + ( - 16 \beta_{3} + 12 \beta_{2} + 4 \beta_1 + 32) q^{43} + ( - 5 \beta_{3} - 5 \beta_{2} + 10 \beta_1 - 80) q^{45} + ( - 6 \beta_{3} - 13 \beta_{2} - 50 \beta_1 + 89) q^{47} + (9 \beta_{3} - 19 \beta_{2} + 32 \beta_1 + 143) q^{49} + ( - \beta_{3} - 20 \beta_{2} + 10 \beta_1 + 74) q^{51} + (3 \beta_{3} + 38 \beta_{2} - 41 \beta_1 + 14) q^{53} + (15 \beta_{2} + 5 \beta_1 + 120) q^{55} + ( - 17 \beta_{3} - 9 \beta_{2} - 47 \beta_1 - 38) q^{57} + (9 \beta_{3} - 24 \beta_{2} - 53 \beta_1) q^{59} + ( - 10 \beta_{3} - 23 \beta_{2} - 51 \beta_1 - 80) q^{61} + ( - 16 \beta_{3} + 7 \beta_{2} + 96 \beta_1 - 36) q^{63} + (15 \beta_{3} - 5 \beta_{2} - 10 \beta_1 - 45) q^{65} + ( - \beta_{3} - 52 \beta_{2} + 35 \beta_1 - 194) q^{67} + ( - 23 \beta_1 + 23) q^{69} + ( - \beta_{3} + 18 \beta_{2} - 56 \beta_1 + 319) q^{71} + ( - 10 \beta_{3} - 3 \beta_{2} - 102 \beta_1 + 129) q^{73} + (25 \beta_1 - 25) q^{75} + ( - 11 \beta_{3} - 33 \beta_{2} - 83 \beta_1 - 496) q^{77} + ( - 28 \beta_{3} + 62 \beta_{2} - 80 \beta_1 - 278) q^{79} + (17 \beta_{3} - 43 \beta_{2} - 31 \beta_1 - 263) q^{81} + (37 \beta_{3} - 60 \beta_{2} + 29 \beta_1 + 320) q^{83} + (5 \beta_{3} + 25 \beta_{2} - 20 \beta_1 - 130) q^{85} + ( - 34 \beta_{3} + 53 \beta_{2} + 84 \beta_1 + 729) q^{87} + ( - 12 \beta_{3} + 60 \beta_{2} - 228) q^{89} + (36 \beta_{3} + 25 \beta_{2} - 131 \beta_1 + 246) q^{91} + ( - \beta_{3} + 38 \beta_{2} + 93 \beta_1 - 451) q^{93} + ( - 5 \beta_{3} + 30 \beta_{2} + 240) q^{95} + (44 \beta_{3} + 59 \beta_{2} + 63 \beta_1 + 38) q^{97} + ( - 11 \beta_{3} - 38 \beta_{2} - 66 \beta_1 - 510) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 20 q^{5} - 26 q^{7} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 20 q^{5} - 26 q^{7} + 64 q^{9} - 93 q^{11} + 32 q^{13} + 20 q^{15} + 108 q^{17} - 185 q^{19} + 302 q^{21} - 92 q^{23} + 100 q^{25} - 259 q^{27} + 294 q^{29} + 211 q^{31} - 237 q^{33} + 130 q^{35} + 5 q^{37} + 421 q^{39} - 369 q^{41} + 100 q^{43} - 320 q^{45} + 363 q^{47} + 600 q^{49} + 315 q^{51} + 21 q^{53} + 465 q^{55} - 160 q^{57} + 33 q^{59} - 307 q^{61} - 167 q^{63} - 160 q^{65} - 725 q^{67} + 92 q^{69} + 1257 q^{71} + 509 q^{73} - 100 q^{75} - 1962 q^{77} - 1202 q^{79} - 992 q^{81} + 1377 q^{83} - 540 q^{85} + 2829 q^{87} - 984 q^{89} + 995 q^{91} - 1843 q^{93} + 925 q^{95} + 137 q^{97} - 2013 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 84x^{2} - 11x + 1242 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 50\nu - 96 ) / 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 13\nu^{2} + 50\nu - 534 ) / 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 13\beta_{2} + 50\beta _1 + 12 \) 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
−7.92791
−4.50148
4.26018
8.16920
0 −8.92791 0 −5.00000 0 −23.5524 0 52.7075 0
1.2 0 −5.50148 0 −5.00000 0 −0.0500526 0 3.26627 0
1.3 0 3.26018 0 −5.00000 0 −27.7921 0 −16.3712 0
1.4 0 7.16920 0 −5.00000 0 25.3945 0 24.3974 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.l 4
4.b odd 2 1 230.4.a.i 4
12.b even 2 1 2070.4.a.bi 4
20.d odd 2 1 1150.4.a.o 4
20.e even 4 2 1150.4.b.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.i 4 4.b odd 2 1
1150.4.a.o 4 20.d odd 2 1
1150.4.b.m 8 20.e even 4 2
1840.4.a.l 4 1.a even 1 1 trivial
2070.4.a.bi 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{4} + 4T_{3}^{3} - 78T_{3}^{2} - 175T_{3} + 1148 \) Copy content Toggle raw display
\( T_{7}^{4} + 26T_{7}^{3} - 648T_{7}^{2} - 16655T_{7} - 832 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} - 78 T^{2} + \cdots + 1148 \) Copy content Toggle raw display
$5$ \( (T + 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 26 T^{3} - 648 T^{2} + \cdots - 832 \) Copy content Toggle raw display
$11$ \( T^{4} + 93 T^{3} + 1401 T^{2} + \cdots - 41700 \) Copy content Toggle raw display
$13$ \( T^{4} - 32 T^{3} - 7812 T^{2} + \cdots + 682562 \) Copy content Toggle raw display
$17$ \( T^{4} - 108 T^{3} - 1392 T^{2} + \cdots + 96120 \) Copy content Toggle raw display
$19$ \( T^{4} + 185 T^{3} + 5787 T^{2} + \cdots + 1404820 \) Copy content Toggle raw display
$23$ \( (T + 23)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 294 T^{3} + \cdots - 1735042500 \) Copy content Toggle raw display
$31$ \( T^{4} - 211 T^{3} + \cdots - 311259365 \) Copy content Toggle raw display
$37$ \( T^{4} - 5 T^{3} - 67782 T^{2} + \cdots - 14187104 \) Copy content Toggle raw display
$41$ \( T^{4} + 369 T^{3} + \cdots - 114199911 \) Copy content Toggle raw display
$43$ \( T^{4} - 100 T^{3} + \cdots + 93158400 \) Copy content Toggle raw display
$47$ \( T^{4} - 363 T^{3} + \cdots - 9732540576 \) Copy content Toggle raw display
$53$ \( T^{4} - 21 T^{3} + \cdots - 8104397736 \) Copy content Toggle raw display
$59$ \( T^{4} - 33 T^{3} + \cdots - 623752800 \) Copy content Toggle raw display
$61$ \( T^{4} + 307 T^{3} + \cdots - 5277943032 \) Copy content Toggle raw display
$67$ \( T^{4} + 725 T^{3} + \cdots - 49731103520 \) Copy content Toggle raw display
$71$ \( T^{4} - 1257 T^{3} + \cdots - 3703975749 \) Copy content Toggle raw display
$73$ \( T^{4} - 509 T^{3} + \cdots + 80640087688 \) Copy content Toggle raw display
$79$ \( T^{4} + 1202 T^{3} + \cdots - 498954065920 \) Copy content Toggle raw display
$83$ \( T^{4} - 1377 T^{3} + \cdots - 424764340752 \) Copy content Toggle raw display
$89$ \( T^{4} + 984 T^{3} + \cdots + 92383027200 \) Copy content Toggle raw display
$97$ \( T^{4} - 137 T^{3} + \cdots + 734679597128 \) Copy content Toggle raw display
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