Properties

Label 1840.4.a.q
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: \( x^{5} - 2x^{4} - 55x^{3} + 104x^{2} + 255x + 72 \) 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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{3} + 5 q^{5} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 5) q^{7} + ( - 2 \beta_{4} + 3 \beta_{2} + \beta_1 + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{3} + 5 q^{5} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 5) q^{7} + ( - 2 \beta_{4} + 3 \beta_{2} + \beta_1 + 6) q^{9} + ( - 3 \beta_{4} - 2 \beta_{3} - \beta_1 + 12) q^{11} + (2 \beta_{4} + 8 \beta_{3} + \beta_{2} - 22) q^{13} + (5 \beta_{3} + 5) q^{15} + ( - 2 \beta_{4} + 6 \beta_{3} + 7 \beta_{2} + 3 \beta_1 - 10) q^{17} + (5 \beta_{4} - 3 \beta_{3} + 7 \beta_{2} - 2 \beta_1 + 44) q^{19} + (6 \beta_{4} + 17 \beta_{3} - \beta_{2} - 4 \beta_1 - 19) q^{21} - 23 q^{23} + 25 q^{25} + ( - 3 \beta_{4} - 3 \beta_{2} + \beta_1 + 4) q^{27} + ( - 6 \beta_{4} + 3 \beta_{3} + 13 \beta_{2} - 11 \beta_1 + 43) q^{29} + ( - 3 \beta_{4} - 4 \beta_{3} + 10 \beta_{2} + 5 \beta_1 + 107) q^{31} + ( - 9 \beta_{4} + 21 \beta_{3} - 15 \beta_{2} + 6 \beta_1 - 36) q^{33} + ( - 5 \beta_{3} + 10 \beta_{2} + 5 \beta_1 + 25) q^{35} + ( - 21 \beta_{4} - 24 \beta_{3} + \beta_{2} + 26 \beta_1 - 16) q^{37} + ( - 3 \beta_{4} - 28 \beta_{3} + 31 \beta_{2} + 211) q^{39} + (9 \beta_{4} - 23 \beta_{3} - \beta_{2} + \beta_1 - 22) q^{41} + ( - 16 \beta_{4} - 12 \beta_{3} - 6 \beta_{2} + 31 \beta_1 + 63) q^{43} + ( - 10 \beta_{4} + 15 \beta_{2} + 5 \beta_1 + 30) q^{45} + ( - 9 \beta_{4} + 25 \beta_{3} - 52 \beta_{2} + 48) q^{47} + (2 \beta_{4} - 14 \beta_{3} + 7 \beta_{2} - 21 \beta_1 - 29) q^{49} + ( - 7 \beta_{4} + 27 \beta_{3} + 19 \beta_{2} + \beta_1 + 223) q^{51} + ( - 5 \beta_{4} - 46 \beta_{3} + \beta_{2} + 45 \beta_1 + 69) q^{53} + ( - 15 \beta_{4} - 10 \beta_{3} - 5 \beta_1 + 60) q^{55} + (56 \beta_{4} + 81 \beta_{3} + 13 \beta_{2} - 34 \beta_1 - 151) q^{57} + (3 \beta_{4} + 38 \beta_{3} + 15 \beta_{2} - 21 \beta_1 + 295) q^{59} + ( - 15 \beta_{4} - 42 \beta_{3} - 30 \beta_{2} - 7 \beta_1 - 144) q^{61} + (\beta_{4} - 23 \beta_{3} + 14 \beta_{2} - 30 \beta_1 + 277) q^{63} + (10 \beta_{4} + 40 \beta_{3} + 5 \beta_{2} - 110) q^{65} + (63 \beta_{4} + 92 \beta_{3} - 31 \beta_{2} - 30 \beta_1 + 84) q^{67} + ( - 23 \beta_{3} - 23) q^{69} + (35 \beta_{4} + 95 \beta_{3} - 45 \beta_{2} - 26 \beta_1 + 309) q^{71} + (\beta_{4} + 79 \beta_{3} + 26 \beta_{2} - 57 \beta_1 - 163) q^{73} + (25 \beta_{3} + 25) q^{75} + ( - 30 \beta_{4} - 95 \beta_{3} + 51 \beta_{2} + 41 \beta_1 - 132) q^{77} + ( - 6 \beta_{4} - 6 \beta_{3} - 6 \beta_{2} + 47 \beta_1 + 599) q^{79} + (28 \beta_{4} - 9 \beta_{3} - 93 \beta_{2} - 11 \beta_1 - 105) q^{81} + (9 \beta_{4} - 84 \beta_{3} + \beta_{2} + 95 \beta_1 + 87) q^{83} + ( - 10 \beta_{4} + 30 \beta_{3} + 35 \beta_{2} + 15 \beta_1 - 50) q^{85} + (25 \beta_{4} + 141 \beta_{3} + 4 \beta_{2} - 16 \beta_1 + 6) q^{87} + (32 \beta_{4} - 112 \beta_{3} - 34 \beta_{2} + 89 \beta_1 - 243) q^{89} + (57 \beta_{4} + 196 \beta_{3} - 64 \beta_{2} - 75 \beta_1 - 76) q^{91} + (13 \beta_{4} + 172 \beta_{3} - 11 \beta_{2} - 10 \beta_1 + 49) q^{93} + (25 \beta_{4} - 15 \beta_{3} + 35 \beta_{2} - 10 \beta_1 + 220) q^{95} + (25 \beta_{4} + 16 \beta_{3} - 148 \beta_{2} - 35 \beta_1 - 308) q^{97} + ( - 63 \beta_{4} - 81 \beta_{3} + 21 \beta_{2} + 111 \beta_1 + 531) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 7 q^{3} + 25 q^{5} + 20 q^{7} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 7 q^{3} + 25 q^{5} + 20 q^{7} + 30 q^{9} + 63 q^{11} - 99 q^{13} + 35 q^{15} - 44 q^{17} + 199 q^{19} - 68 q^{21} - 115 q^{23} + 125 q^{25} + 28 q^{27} + 231 q^{29} + 518 q^{31} - 111 q^{33} + 100 q^{35} - 113 q^{37} + 974 q^{39} - 174 q^{41} + 298 q^{43} + 150 q^{45} + 360 q^{47} - 163 q^{49} + 1163 q^{51} + 217 q^{53} + 315 q^{55} - 684 q^{57} + 1551 q^{59} - 737 q^{61} + 1353 q^{63} - 495 q^{65} + 539 q^{67} - 161 q^{69} + 1736 q^{71} - 628 q^{73} + 175 q^{75} - 882 q^{77} + 2954 q^{79} - 495 q^{81} + 153 q^{83} - 220 q^{85} + 274 q^{87} - 1558 q^{89} + 37 q^{91} + 584 q^{93} + 995 q^{95} - 1375 q^{97} + 2487 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 55x^{3} + 104x^{2} + 255x + 72 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - \nu^{3} - 61\nu^{2} + 58\nu + 348 ) / 30 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{4} - 7\nu^{3} - 107\nu^{2} + 361\nu + 306 ) / 30 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - \nu^{3} - 51\nu^{2} + 68\nu + 118 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{4} - 6\beta_{2} - \beta _1 + 45 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{4} - 6\beta_{3} + 3\beta_{2} + 23\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 128\beta_{4} - 12\beta_{3} - 300\beta_{2} - 73\beta _1 + 2019 ) / 2 \) 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
−1.17944
−7.11066
6.96712
−0.336706
3.65968
0 −7.44221 0 5.00000 0 23.3042 0 28.3866 0
1.2 0 −0.380607 0 5.00000 0 −24.3526 0 −26.8551 0
1.3 0 0.0785019 0 5.00000 0 6.13060 0 −26.9938 0
1.4 0 6.75371 0 5.00000 0 19.0133 0 18.6126 0
1.5 0 7.99061 0 5.00000 0 −4.09549 0 36.8498 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.q 5
4.b odd 2 1 460.4.a.a 5
20.d odd 2 1 2300.4.a.d 5
20.e even 4 2 2300.4.c.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.4.a.a 5 4.b odd 2 1
1840.4.a.q 5 1.a even 1 1 trivial
2300.4.a.d 5 20.d odd 2 1
2300.4.c.c 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{5} - 7T_{3}^{4} - 58T_{3}^{3} + 385T_{3}^{2} + 123T_{3} - 12 \) Copy content Toggle raw display
\( T_{7}^{5} - 20T_{7}^{4} - 576T_{7}^{3} + 12437T_{7}^{2} - 7210T_{7} - 270924 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 7 T^{4} - 58 T^{3} + 385 T^{2} + \cdots - 12 \) Copy content Toggle raw display
$5$ \( (T - 5)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 20 T^{4} - 576 T^{3} + \cdots - 270924 \) Copy content Toggle raw display
$11$ \( T^{5} - 63 T^{4} - 1803 T^{3} + \cdots - 72719208 \) Copy content Toggle raw display
$13$ \( T^{5} + 99 T^{4} - 1576 T^{3} + \cdots - 14392890 \) Copy content Toggle raw display
$17$ \( T^{5} + 44 T^{4} + \cdots + 451144080 \) Copy content Toggle raw display
$19$ \( T^{5} - 199 T^{4} + \cdots + 800499864 \) Copy content Toggle raw display
$23$ \( (T + 23)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 231 T^{4} + \cdots + 43256062404 \) Copy content Toggle raw display
$31$ \( T^{5} - 518 T^{4} + \cdots + 9011869285 \) Copy content Toggle raw display
$37$ \( T^{5} + 113 T^{4} + \cdots - 409221703936 \) Copy content Toggle raw display
$41$ \( T^{5} + 174 T^{4} + \cdots + 12760146261 \) Copy content Toggle raw display
$43$ \( T^{5} - 298 T^{4} + \cdots - 803496865792 \) Copy content Toggle raw display
$47$ \( T^{5} - 360 T^{4} + \cdots - 13556115821808 \) Copy content Toggle raw display
$53$ \( T^{5} - 217 T^{4} + \cdots - 4075860543408 \) Copy content Toggle raw display
$59$ \( T^{5} - 1551 T^{4} + \cdots + 897857553600 \) Copy content Toggle raw display
$61$ \( T^{5} + 737 T^{4} + \cdots + 310836562128 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 137714931264960 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 108175489939041 \) Copy content Toggle raw display
$73$ \( T^{5} + 628 T^{4} + \cdots + 13803900176376 \) Copy content Toggle raw display
$79$ \( T^{5} - 2954 T^{4} + \cdots + 8954177264640 \) Copy content Toggle raw display
$83$ \( T^{5} - 153 T^{4} + \cdots + 29112932977344 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots + 180704830832640 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots + 347840565067560 \) Copy content Toggle raw display
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