Properties

Label 1840.4.a.j
Level $1840$
Weight $4$
Character orbit 1840.a
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.318165.1
Defining polynomial: \( x^{3} - x^{2} - 45x + 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + 5 q^{5} + (\beta_{2} - 3 \beta_1 - 1) q^{7} + (\beta_{2} - \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + 5 q^{5} + (\beta_{2} - 3 \beta_1 - 1) q^{7} + (\beta_{2} - \beta_1 + 4) q^{9} + (2 \beta_{2} + 5 \beta_1 - 10) q^{11} + (3 \beta_{2} - 3 \beta_1 + 27) q^{13} + 5 \beta_1 q^{15} + ( - 11 \beta_1 + 46) q^{17} + (\beta_{2} + 9 \beta_1 + 59) q^{19} + ( - \beta_{2} + 14 \beta_1 - 91) q^{21} - 23 q^{23} + 25 q^{25} + (\beta_{2} - 10 \beta_1 - 29) q^{27} + (4 \beta_{2} - 18 \beta_1 + 122) q^{29} + (5 \beta_{2} - 17 \beta_1 - 125) q^{31} + (9 \beta_{2} + 9 \beta_1 + 159) q^{33} + (5 \beta_{2} - 15 \beta_1 - 5) q^{35} + (2 \beta_{2} + 8 \beta_1 + 324) q^{37} + (3 \beta_{2} + 66 \beta_1 - 87) q^{39} + (10 \beta_{2} + 47 \beta_1 - 204) q^{41} + (4 \beta_{2} - 40 \beta_1 - 256) q^{43} + (5 \beta_{2} - 5 \beta_1 + 20) q^{45} + (6 \beta_{2} - 42 \beta_1 + 106) q^{47} + ( - 18 \beta_{2} - 49 \beta_1 + 303) q^{49} + ( - 11 \beta_{2} + 57 \beta_1 - 341) q^{51} + ( - 12 \beta_{2} - 60 \beta_1 + 186) q^{53} + (10 \beta_{2} + 25 \beta_1 - 50) q^{55} + (11 \beta_{2} + 62 \beta_1 + 281) q^{57} + ( - 12 \beta_{2} - 34 \beta_1 - 40) q^{59} + (8 \beta_{2} + 49 \beta_1 - 30) q^{61} + ( - 15 \beta_{2} - 36 \beta_1 + 459) q^{63} + (15 \beta_{2} - 15 \beta_1 + 135) q^{65} + (12 \beta_{2} - 20 \beta_1 - 528) q^{67} - 23 \beta_1 q^{69} + ( - 8 \beta_{2} + 67 \beta_1 + 132) q^{71} + ( - 42 \beta_{2} + 30 \beta_1 - 284) q^{73} + 25 \beta_1 q^{75} + ( - 55 \beta_{2} + 80 \beta_1 + 299) q^{77} + ( - 16 \beta_{2} + 28 \beta_1 + 272) q^{79} + ( - 35 \beta_{2} + 20 \beta_1 - 416) q^{81} + ( - 22 \beta_{2} - 52 \beta_1 + 106) q^{83} + ( - 55 \beta_1 + 230) q^{85} + ( - 10 \beta_{2} + 188 \beta_1 - 550) q^{87} + (26 \beta_{2} + 200 \beta_1 - 88) q^{89} + ( - 30 \beta_{2} - 153 \beta_1 + 1362) q^{91} + ( - 7 \beta_{2} - 48 \beta_1 - 517) q^{93} + (5 \beta_{2} + 45 \beta_1 + 295) q^{95} + (40 \beta_{2} - 25 \beta_1 + 462) q^{97} + ( - 27 \beta_{2} + 123 \beta_1 + 567) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + 15 q^{5} - 7 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} + 15 q^{5} - 7 q^{7} + 10 q^{9} - 27 q^{11} + 75 q^{13} + 5 q^{15} + 127 q^{17} + 185 q^{19} - 258 q^{21} - 69 q^{23} + 75 q^{25} - 98 q^{27} + 344 q^{29} - 397 q^{31} + 477 q^{33} - 35 q^{35} + 978 q^{37} - 198 q^{39} - 575 q^{41} - 812 q^{43} + 50 q^{45} + 270 q^{47} + 878 q^{49} - 955 q^{51} + 510 q^{53} - 135 q^{55} + 894 q^{57} - 142 q^{59} - 49 q^{61} + 1356 q^{63} + 375 q^{65} - 1616 q^{67} - 23 q^{69} + 471 q^{71} - 780 q^{73} + 25 q^{75} + 1032 q^{77} + 860 q^{79} - 1193 q^{81} + 288 q^{83} + 635 q^{85} - 1452 q^{87} - 90 q^{89} + 3963 q^{91} - 1592 q^{93} + 925 q^{95} + 1321 q^{97} + 1851 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 31 \) 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
−6.84916
1.34735
6.50182
0 −6.84916 0 5.00000 0 28.6094 0 19.9110 0
1.2 0 1.34735 0 5.00000 0 −32.8794 0 −25.1847 0
1.3 0 6.50182 0 5.00000 0 −2.73001 0 15.2736 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.j 3
4.b odd 2 1 230.4.a.g 3
12.b even 2 1 2070.4.a.ba 3
20.d odd 2 1 1150.4.a.m 3
20.e even 4 2 1150.4.b.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.g 3 4.b odd 2 1
1150.4.a.m 3 20.d odd 2 1
1150.4.b.l 6 20.e even 4 2
1840.4.a.j 3 1.a even 1 1 trivial
2070.4.a.ba 3 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{3} - T_{3}^{2} - 45T_{3} + 60 \) Copy content Toggle raw display
\( T_{7}^{3} + 7T_{7}^{2} - 929T_{7} - 2568 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 45 T + 60 \) Copy content Toggle raw display
$5$ \( (T - 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 7 T^{2} - 929 T - 2568 \) Copy content Toggle raw display
$11$ \( T^{3} + 27 T^{2} - 3399 T - 89388 \) Copy content Toggle raw display
$13$ \( T^{3} - 75 T^{2} - 3663 T + 275238 \) Copy content Toggle raw display
$17$ \( T^{3} - 127 T^{2} - 109 T + 96550 \) Copy content Toggle raw display
$19$ \( T^{3} - 185 T^{2} + 7003 T - 37596 \) Copy content Toggle raw display
$23$ \( (T + 23)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 344 T^{2} + 16552 T + 291288 \) Copy content Toggle raw display
$31$ \( T^{3} + 397 T^{2} + 26165 T - 1547080 \) Copy content Toggle raw display
$37$ \( T^{3} - 978 T^{2} + \cdots - 33005536 \) Copy content Toggle raw display
$41$ \( T^{3} + 575 T^{2} + \cdots - 50953878 \) Copy content Toggle raw display
$43$ \( T^{3} + 812 T^{2} + \cdots - 10161920 \) Copy content Toggle raw display
$47$ \( T^{3} - 270 T^{2} - 72660 T - 3184000 \) Copy content Toggle raw display
$53$ \( T^{3} - 510 T^{2} + \cdots + 89503704 \) Copy content Toggle raw display
$59$ \( T^{3} + 142 T^{2} - 136780 T + 9906704 \) Copy content Toggle raw display
$61$ \( T^{3} + 49 T^{2} - 151973 T - 23572158 \) Copy content Toggle raw display
$67$ \( T^{3} + 1616 T^{2} + \cdots + 111600960 \) Copy content Toggle raw display
$71$ \( T^{3} - 471 T^{2} + \cdots + 75603760 \) Copy content Toggle raw display
$73$ \( T^{3} + 780 T^{2} + \cdots - 672863896 \) Copy content Toggle raw display
$79$ \( T^{3} - 860 T^{2} + 68208 T + 8296704 \) Copy content Toggle raw display
$83$ \( T^{3} - 288 T^{2} + \cdots + 106176592 \) Copy content Toggle raw display
$89$ \( T^{3} + 90 T^{2} + \cdots - 1109897568 \) Copy content Toggle raw display
$97$ \( T^{3} - 1321 T^{2} + \cdots + 689377182 \) Copy content Toggle raw display
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