Properties

Label 1840.4.a.o
Level $1840$
Weight $4$
Character orbit 1840.a
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 2x^{4} - 80x^{3} + 121x^{2} + 1212x + 1044 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 460)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} - 5 q^{5} + (\beta_{3} + \beta_{2} - 2) q^{7} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} - 5 q^{5} + (\beta_{3} + \beta_{2} - 2) q^{7} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 + 6) q^{9} + ( - \beta_{4} + \beta_{2} + 2 \beta_1 - 3) q^{11} + ( - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{13} + (5 \beta_1 - 5) q^{15} + (5 \beta_{3} - 2 \beta_1 - 4) q^{17} + ( - \beta_{4} + 5 \beta_{3} + \beta_{2} + 7 \beta_1 - 1) q^{19} + (5 \beta_{2} - 10 \beta_1 + 2) q^{21} + 23 q^{23} + 25 q^{25} + ( - \beta_{4} + 6 \beta_{3} + 12 \beta_{2} - \beta_1 + 36) q^{27} + (2 \beta_{4} + 3 \beta_{3} - 7 \beta_1 - 47) q^{29} + ( - \beta_{4} + 8 \beta_{3} - 5 \beta_{2} + 8 \beta_1 + 18) q^{31} + ( - 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 95) q^{33} + ( - 5 \beta_{3} - 5 \beta_{2} + 10) q^{35} + ( - \beta_{4} - 9 \beta_{3} + \beta_{2} - 12 \beta_1 - 83) q^{37} + ( - 3 \beta_{4} - 11 \beta_{3} + 17 \beta_{2} - 20 \beta_1 + 82) q^{39} + ( - \beta_{4} + 2 \beta_{3} + 10 \beta_{2} - 28 \beta_1 - 148) q^{41} + (6 \beta_{4} - 3 \beta_{3} - 9 \beta_{2} - 21 \beta_1 + 114) q^{43} + ( - 5 \beta_{3} - 10 \beta_{2} + 10 \beta_1 - 30) q^{45} + ( - 3 \beta_{4} + 11 \beta_{3} + 16 \beta_{2} - 18 \beta_1 + 114) q^{47} + ( - 4 \beta_{4} - \beta_{3} - 4 \beta_{2} - 14 \beta_1 - 131) q^{49} + (5 \beta_{4} - 18 \beta_{3} + 14 \beta_{2} - 18 \beta_1 + 135) q^{51} + (5 \beta_{4} + 18 \beta_{3} + 8 \beta_{2} + 37 \beta_1 - 149) q^{53} + (5 \beta_{4} - 5 \beta_{2} - 10 \beta_1 + 15) q^{55} + (2 \beta_{4} - 28 \beta_{3} + \beta_{2} - 20 \beta_1 - 178) q^{57} + ( - \beta_{4} - 44 \beta_{2} - 13 \beta_1 + 175) q^{59} + (7 \beta_{4} + 52 \beta_{3} + 15 \beta_{2} + 8 \beta_1 + 61) q^{61} + ( - 5 \beta_{4} + 3 \beta_{3} + 8 \beta_{2} - 52 \beta_1 + 321) q^{63} + (10 \beta_{4} - 10 \beta_{3} - 5 \beta_{2} + 15 \beta_1 - 5) q^{65} + (15 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 65) q^{67} + ( - 23 \beta_1 + 23) q^{69} + (\beta_{4} + 27 \beta_{3} - 25 \beta_{2} + 33 \beta_1 + 298) q^{71} + (7 \beta_{4} - 16 \beta_{3} - \beta_{2} + 41 \beta_1 - 274) q^{73} + ( - 25 \beta_1 + 25) q^{75} + (14 \beta_{4} + 15 \beta_{3} - 10 \beta_{2} + 25 \beta_1 + 36) q^{77} + ( - 8 \beta_{4} - 37 \beta_{3} - 9 \beta_{2} - 97 \beta_1 - 86) q^{79} + ( - 8 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - 103 \beta_1 - 153) q^{81} + ( - 13 \beta_{4} - 30 \beta_{3} + 18 \beta_{2} - 117 \beta_1 + 375) q^{83} + ( - 25 \beta_{3} + 10 \beta_1 + 20) q^{85} + (7 \beta_{4} + 5 \beta_{3} + 16 \beta_{2} + 28 \beta_1 + 256) q^{87} + ( - 26 \beta_{4} + 21 \beta_{3} - \beta_{2} + 3 \beta_1 + 118) q^{89} + (23 \beta_{4} + 38 \beta_{3} + 23 \beta_{2} + 6 \beta_1 + 151) q^{91} + (11 \beta_{4} - 65 \beta_{3} - 13 \beta_{2} - 2 \beta_1 - 80) q^{93} + (5 \beta_{4} - 25 \beta_{3} - 5 \beta_{2} - 35 \beta_1 + 5) q^{95} + ( - 13 \beta_{4} + 48 \beta_{3} - 23 \beta_{2} + 16 \beta_1 - 49) q^{97} + (17 \beta_{4} + 4 \beta_{3} - 18 \beta_{2} + 42 \beta_1 - 25) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 3 q^{3} - 25 q^{5} - 8 q^{7} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 3 q^{3} - 25 q^{5} - 8 q^{7} + 30 q^{9} - 7 q^{11} + 5 q^{13} - 15 q^{15} - 24 q^{17} + 13 q^{19} + 115 q^{23} + 125 q^{25} + 204 q^{27} - 253 q^{29} + 98 q^{31} - 473 q^{33} + 40 q^{35} - 435 q^{37} + 410 q^{39} - 774 q^{41} + 498 q^{43} - 150 q^{45} + 572 q^{47} - 683 q^{49} + 657 q^{51} - 665 q^{53} + 35 q^{55} - 932 q^{57} + 763 q^{59} + 337 q^{61} + 1527 q^{63} - 25 q^{65} + 305 q^{67} + 69 q^{69} + 1504 q^{71} - 1304 q^{73} + 75 q^{75} + 182 q^{77} - 626 q^{79} - 959 q^{81} + 1703 q^{83} + 120 q^{85} + 1354 q^{87} + 646 q^{89} + 767 q^{91} - 452 q^{93} - 65 q^{95} - 233 q^{97} - 111 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 80x^{3} + 121x^{2} + 1212x + 1044 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 4\nu^{3} - 44\nu^{2} - 155\nu - 246 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} - 4\nu^{3} + 50\nu^{2} + 155\nu + 54 ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{3} + 3\nu^{2} - 52\nu - 103 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2\beta_{2} + 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 3\beta_{3} - 6\beta_{2} + 52\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{4} + 56\beta_{3} + 124\beta_{2} - 53\beta _1 + 1626 \) 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
6.93138
6.72467
−1.04116
−2.72746
−7.88744
0 −5.93138 0 −5.00000 0 −3.12066 0 8.18131 0
1.2 0 −5.72467 0 −5.00000 0 12.6137 0 5.77185 0
1.3 0 2.04116 0 −5.00000 0 −21.6113 0 −22.8337 0
1.4 0 3.72746 0 −5.00000 0 −11.8626 0 −13.1061 0
1.5 0 8.88744 0 −5.00000 0 15.9808 0 51.9866 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.o 5
4.b odd 2 1 460.4.a.b 5
20.d odd 2 1 2300.4.a.c 5
20.e even 4 2 2300.4.c.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.4.a.b 5 4.b odd 2 1
1840.4.a.o 5 1.a even 1 1 trivial
2300.4.a.c 5 20.d odd 2 1
2300.4.c.d 10 20.e even 4 2

Hecke kernels

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

\( T_{3}^{5} - 3T_{3}^{4} - 78T_{3}^{3} + 121T_{3}^{2} + 1211T_{3} - 2296 \) Copy content Toggle raw display
\( T_{7}^{5} + 8T_{7}^{4} - 484T_{7}^{3} - 2141T_{7}^{2} + 49858T_{7} + 161268 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 3 T^{4} - 78 T^{3} + \cdots - 2296 \) Copy content Toggle raw display
$5$ \( (T + 5)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 8 T^{4} - 484 T^{3} + \cdots + 161268 \) Copy content Toggle raw display
$11$ \( T^{5} + 7 T^{4} - 2875 T^{3} + \cdots - 5888920 \) Copy content Toggle raw display
$13$ \( T^{5} - 5 T^{4} - 9228 T^{3} + \cdots - 566705734 \) Copy content Toggle raw display
$17$ \( T^{5} + 24 T^{4} + \cdots - 308713360 \) Copy content Toggle raw display
$19$ \( T^{5} - 13 T^{4} - 11801 T^{3} + \cdots - 75549672 \) Copy content Toggle raw display
$23$ \( (T - 23)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + 253 T^{4} + \cdots + 3021093156 \) Copy content Toggle raw display
$31$ \( T^{5} - 98 T^{4} + \cdots - 7527219975 \) Copy content Toggle raw display
$37$ \( T^{5} + 435 T^{4} + \cdots + 100253888 \) Copy content Toggle raw display
$41$ \( T^{5} + 774 T^{4} + \cdots - 691331956043 \) Copy content Toggle raw display
$43$ \( T^{5} - 498 T^{4} + \cdots + 11201296896 \) Copy content Toggle raw display
$47$ \( T^{5} - 572 T^{4} + \cdots + 201337632 \) Copy content Toggle raw display
$53$ \( T^{5} + 665 T^{4} + \cdots - 55648798000 \) Copy content Toggle raw display
$59$ \( T^{5} - 763 T^{4} + \cdots - 17454672233664 \) Copy content Toggle raw display
$61$ \( T^{5} - 337 T^{4} + \cdots - 62953038747840 \) Copy content Toggle raw display
$67$ \( T^{5} - 305 T^{4} + \cdots + 387904064832 \) Copy content Toggle raw display
$71$ \( T^{5} - 1504 T^{4} + \cdots - 8400024289995 \) Copy content Toggle raw display
$73$ \( T^{5} + 1304 T^{4} + \cdots - 7235588459208 \) Copy content Toggle raw display
$79$ \( T^{5} + 626 T^{4} + \cdots + 21668449786880 \) Copy content Toggle raw display
$83$ \( T^{5} - 1703 T^{4} + \cdots - 67480737623744 \) Copy content Toggle raw display
$89$ \( T^{5} - 646 T^{4} + \cdots - 60037186156800 \) Copy content Toggle raw display
$97$ \( T^{5} + 233 T^{4} + \cdots + 6799511647880 \) Copy content Toggle raw display
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