Properties

Label 1840.4.a.s
Level $1840$
Weight $4$
Character orbit 1840.a
Self dual yes
Analytic conductor $108.564$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 81x^{4} + 161x^{3} + 1520x^{2} - 3915x + 588 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - 5 q^{5} + ( - \beta_{4} - \beta_{2} - 5) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - 5 q^{5} + ( - \beta_{4} - \beta_{2} - 5) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{9} + (\beta_{4} + \beta_{3} + 1) q^{11} + ( - \beta_{5} - \beta_{3} - \beta_{2} - 5) q^{13} + 5 \beta_1 q^{15} + (5 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 9 \beta_1 + 8) q^{17} + ( - 3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 6 \beta_{2} + 6 \beta_1 - 2) q^{19} + (3 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 11) q^{21} + 23 q^{23} + 25 q^{25} + (3 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 8 \beta_1 + 14) q^{27} + ( - 3 \beta_{5} + 6 \beta_{4} - 5 \beta_{3} + 8 \beta_{2} - 14 \beta_1 + 18) q^{29} + (2 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} + 6 \beta_{2} + 12 \beta_1 + 62) q^{31} + (3 \beta_{5} - 3 \beta_{4} - 6 \beta_{3} - 6 \beta_1) q^{33} + (5 \beta_{4} + 5 \beta_{2} + 25) q^{35} + ( - \beta_{5} - 10 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 7 \beta_1 + 21) q^{37} + ( - 9 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 17 \beta_1 - 12) q^{39} + ( - 19 \beta_{5} + \beta_{4} + 4 \beta_{3} + \beta_{2} + 15 \beta_1 - 23) q^{41} + (16 \beta_{5} + 4 \beta_{4} - 17 \beta_{2} + 11 \beta_1 + 34) q^{43} + (5 \beta_{5} + 5 \beta_{4} - 5 \beta_{3} - 5 \beta_1) q^{45} + (18 \beta_{5} + 12 \beta_{4} - 19 \beta_{3} + 7 \beta_{2} + 2 \beta_1 + 23) q^{47} + (15 \beta_{5} + 26 \beta_{4} - 9 \beta_{3} + 30 \beta_{2} + \beta_1 - 87) q^{49} + (11 \beta_{5} - 16 \beta_{4} + 14 \beta_{3} - 13 \beta_{2} + 27 \beta_1 + 156) q^{51} + (21 \beta_{5} - 9 \beta_{4} + 10 \beta_{3} - 11 \beta_{2} + 12 \beta_1 + 21) q^{53} + ( - 5 \beta_{4} - 5 \beta_{3} - 5) q^{55} + ( - 27 \beta_{5} + 9 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 9 \beta_1 - 153) q^{57} + ( - 15 \beta_{5} - 20 \beta_{4} + 2 \beta_{3} - 17 \beta_{2} + 50 \beta_1 - 19) q^{59} + ( - 6 \beta_{5} - 2 \beta_{4} + 21 \beta_{3} - 10 \beta_{2} + 40 \beta_1 - 107) q^{61} + (16 \beta_{5} + 17 \beta_{4} - 5 \beta_{3} + 17 \beta_{2} - 10 \beta_1 + 70) q^{63} + (5 \beta_{5} + 5 \beta_{3} + 5 \beta_{2} + 25) q^{65} + ( - 45 \beta_{5} - 11 \beta_{4} - 4 \beta_{3} - 32 \beta_{2} + 63 \beta_1 + 61) q^{67} - 23 \beta_1 q^{69} + ( - 29 \beta_{5} + 35 \beta_{4} - 14 \beta_{3} + 30 \beta_{2} - 2 \beta_1 + 19) q^{71} + (18 \beta_{5} - 5 \beta_{4} - 13 \beta_{3} - 10 \beta_{2} - 23 \beta_1 - 293) q^{73} - 25 \beta_1 q^{75} + ( - 9 \beta_{5} - 7 \beta_{4} + 11 \beta_{3} - 18 \beta_{2} - 24 \beta_1 - 229) q^{77} + (4 \beta_{5} - 41 \beta_{4} + 24 \beta_{3} + 9 \beta_{2} + 59 \beta_1 + 244) q^{79} + (23 \beta_{5} + 32 \beta_{4} - 17 \beta_{3} - 12 \beta_{2} + 4 \beta_1 - 324) q^{81} + (17 \beta_{5} - 37 \beta_{4} + 24 \beta_{3} - 15 \beta_{2} + 54 \beta_1 + 175) q^{83} + ( - 25 \beta_{5} - 5 \beta_{4} + 5 \beta_{3} + 10 \beta_{2} + 45 \beta_1 - 40) q^{85} + ( - 74 \beta_{5} - 12 \beta_{4} + 27 \beta_{3} + \beta_{2} + 68 \beta_1 + 269) q^{87} + ( - 52 \beta_{5} - 2 \beta_{4} + 20 \beta_{3} + 3 \beta_{2} + 83 \beta_1 - 262) q^{89} + (12 \beta_{5} + 25 \beta_{4} - 11 \beta_{3} + 26 \beta_{2} + 24 \beta_1 + 209) q^{91} + (21 \beta_{5} - 9 \beta_{4} - 26 \beta_{3} - 12 \beta_{2} - 87 \beta_1 - 344) q^{93} + (15 \beta_{5} - 5 \beta_{4} + 10 \beta_{3} - 30 \beta_{2} - 30 \beta_1 + 10) q^{95} + ( - 20 \beta_{5} - 55 \beta_{4} + \beta_{3} + 4 \beta_{2} + 90 \beta_1 - 555) q^{97} + ( - 15 \beta_{5} - 24 \beta_{4} + 12 \beta_{3} - 9 \beta_{2} + 57 \beta_1 + 60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 30 q^{5} - 28 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} - 30 q^{5} - 28 q^{7} + 4 q^{9} + 3 q^{11} - 28 q^{13} + 10 q^{15} + 24 q^{17} + 3 q^{19} + 60 q^{21} + 138 q^{23} + 150 q^{25} + 97 q^{27} + 76 q^{29} + 381 q^{31} - 3 q^{33} + 140 q^{35} + 131 q^{37} - 41 q^{39} - 95 q^{41} + 202 q^{43} - 20 q^{45} + 119 q^{47} - 578 q^{49} + 997 q^{51} + 137 q^{53} - 15 q^{55} - 894 q^{57} + 39 q^{59} - 573 q^{61} + 355 q^{63} + 140 q^{65} + 563 q^{67} - 46 q^{69} + 83 q^{71} - 1799 q^{73} - 50 q^{75} - 1410 q^{77} + 1636 q^{79} - 2006 q^{81} + 1191 q^{83} - 120 q^{85} + 1821 q^{87} - 1370 q^{89} + 1251 q^{91} - 2215 q^{93} - 15 q^{95} - 3021 q^{97} + 525 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 81x^{4} + 161x^{3} + 1520x^{2} - 3915x + 588 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} - 82\nu^{3} + 79\nu^{2} + 1527\nu - 2172 ) / 72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{5} - 29\nu^{4} - 194\nu^{3} + 1139\nu^{2} - 1005\nu - 708 ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 13\nu^{4} + 10\nu^{3} - 559\nu^{2} + 1245\nu + 516 ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{5} - 55\nu^{4} - 214\nu^{3} + 2185\nu^{2} - 3423\nu + 204 ) / 72 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - \beta_{4} + \beta_{3} + \beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{5} - 2\beta_{4} + 4\beta_{3} - 3\beta_{2} + 46\beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -58\beta_{5} - 49\beta_{4} + 64\beta_{3} - 12\beta_{2} + 85\beta _1 + 1134 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -225\beta_{5} - 134\beta_{4} + 313\beta_{3} - 186\beta_{2} + 2251\beta _1 + 25 \) 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
7.55347
4.43772
2.66453
0.160327
−6.38774
−6.42830
0 −7.55347 0 −5.00000 0 −4.93892 0 30.0549 0
1.2 0 −4.43772 0 −5.00000 0 11.9284 0 −7.30664 0
1.3 0 −2.66453 0 −5.00000 0 −34.7191 0 −19.9003 0
1.4 0 −0.160327 0 −5.00000 0 2.26267 0 −26.9743 0
1.5 0 6.38774 0 −5.00000 0 5.78952 0 13.8032 0
1.6 0 6.42830 0 −5.00000 0 −8.32258 0 14.3231 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.s 6
4.b odd 2 1 920.4.a.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.4.a.b 6 4.b odd 2 1
1840.4.a.s 6 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{6} + 2T_{3}^{5} - 81T_{3}^{4} - 161T_{3}^{3} + 1520T_{3}^{2} + 3915T_{3} + 588 \) Copy content Toggle raw display
\( T_{7}^{6} + 28T_{7}^{5} - 348T_{7}^{4} - 3513T_{7}^{3} + 18730T_{7}^{2} + 77400T_{7} - 223000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 2 T^{5} - 81 T^{4} - 161 T^{3} + \cdots + 588 \) Copy content Toggle raw display
$5$ \( (T + 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 28 T^{5} - 348 T^{4} + \cdots - 223000 \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} - 1965 T^{4} + \cdots + 1162512 \) Copy content Toggle raw display
$13$ \( T^{6} + 28 T^{5} + \cdots - 266086970 \) Copy content Toggle raw display
$17$ \( T^{6} - 24 T^{5} + \cdots - 11777111616 \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} + \cdots - 21406390992 \) Copy content Toggle raw display
$23$ \( (T - 23)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} - 76 T^{5} + \cdots - 108803723556 \) Copy content Toggle raw display
$31$ \( T^{6} - 381 T^{5} + \cdots - 96915684999 \) Copy content Toggle raw display
$37$ \( T^{6} - 131 T^{5} + \cdots + 32607805056 \) Copy content Toggle raw display
$41$ \( T^{6} + 95 T^{5} + \cdots - 488808379923 \) Copy content Toggle raw display
$43$ \( T^{6} - 202 T^{5} + \cdots + 3224001804288 \) Copy content Toggle raw display
$47$ \( T^{6} - 119 T^{5} + \cdots - 12\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{6} - 137 T^{5} + \cdots - 59665103136 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 295068706077440 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 171409513711456 \) Copy content Toggle raw display
$67$ \( T^{6} - 563 T^{5} + \cdots + 95\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{6} - 83 T^{5} + \cdots - 36\!\cdots\!71 \) Copy content Toggle raw display
$73$ \( T^{6} + 1799 T^{5} + \cdots - 92144248176936 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 869715459643392 \) Copy content Toggle raw display
$83$ \( T^{6} - 1191 T^{5} + \cdots - 23\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{6} + 1370 T^{5} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{6} + 3021 T^{5} + \cdots + 50\!\cdots\!96 \) Copy content Toggle raw display
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