Properties

Label 1840.4.a.p
Level $1840$
Weight $4$
Character orbit 1840.a
Self dual yes
Analytic conductor $108.564$
Analytic rank $1$
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: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 34x^{3} - 9x^{2} + 260x + 60 \) 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 115)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1 + 1) q^{3} - 5 q^{5} + ( - \beta_{2} + 3 \beta_1 + 2) q^{7} + (\beta_{4} + \beta_{3} - 4 \beta_{2} - \beta_1 + 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1 + 1) q^{3} - 5 q^{5} + ( - \beta_{2} + 3 \beta_1 + 2) q^{7} + (\beta_{4} + \beta_{3} - 4 \beta_{2} - \beta_1 + 14) q^{9} + (\beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 32) q^{11} + ( - 3 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + \beta_1 + 5) q^{13} + ( - 5 \beta_{3} + 5 \beta_1 - 5) q^{15} + (3 \beta_{4} + 5 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 68) q^{17} + ( - \beta_{4} - 10 \beta_{3} - 3 \beta_1 + 4) q^{19} + ( - \beta_{4} + 9 \beta_{3} + 13 \beta_{2} - 14 \beta_1 - 38) q^{21} - 23 q^{23} + 25 q^{25} + ( - \beta_{4} + 2 \beta_{3} - 5 \beta_{2} - 26 \beta_1 + 51) q^{27} + ( - 4 \beta_{4} + 8 \beta_{3} + 9 \beta_{2} + 14 \beta_1 - 119) q^{29} + ( - 9 \beta_{4} - 6 \beta_{3} + 11 \beta_{2} + 17 \beta_1 - 129) q^{31} + (3 \beta_{4} + 42 \beta_{3} + 26 \beta_{2} - 9 \beta_1 - 96) q^{33} + (5 \beta_{2} - 15 \beta_1 - 10) q^{35} + ( - 6 \beta_{4} - 11 \beta_{3} - 5 \beta_{2} - 40 \beta_1 - 24) q^{37} + ( - 11 \beta_{4} - 27 \beta_{3} - 24 \beta_{2} - 22 \beta_1 - 11) q^{39} + (10 \beta_{4} - 17 \beta_{3} + 18 \beta_{2} + 19 \beta_1 + 79) q^{41} + (9 \beta_{4} - 9 \beta_{3} - 12 \beta_{2} - 46 \beta_1 + 54) q^{43} + ( - 5 \beta_{4} - 5 \beta_{3} + 20 \beta_{2} + 5 \beta_1 - 70) q^{45} + (7 \beta_{4} + 14 \beta_{3} + 12 \beta_{2} + 68 \beta_1 + 53) q^{47} + (6 \beta_{4} - 61 \beta_{3} - 13 \beta_{2} + 47 \beta_1 + 9) q^{49} + (15 \beta_{4} - 35 \beta_{3} + 19 \beta_{2} + 93 \beta_1 + 44) q^{51} + ( - 2 \beta_{4} - 44 \beta_{3} + 22 \beta_{2} + 20 \beta_1 - 166) q^{53} + ( - 5 \beta_{4} + 10 \beta_{3} - 15 \beta_{2} - 15 \beta_1 - 160) q^{55} + ( - 12 \beta_{4} - 33 \beta_{3} - 32 \beta_{2} + 12 \beta_1 - 180) q^{57} + (11 \beta_{4} - 36 \beta_{3} + 9 \beta_{2} - 4 \beta_1 + 252) q^{59} + ( - 20 \beta_{4} - 34 \beta_{3} + 12 \beta_{2} + 11 \beta_1 + 116) q^{61} + (20 \beta_{4} - 72 \beta_{3} - 15 \beta_{2} + 82 \beta_1 + 252) q^{63} + (15 \beta_{4} + 10 \beta_{3} + 15 \beta_{2} - 5 \beta_1 - 25) q^{65} + ( - 35 \beta_{4} + 31 \beta_{3} - 14 \beta_{2} - 38 \beta_1 + 184) q^{67} + ( - 23 \beta_{3} + 23 \beta_1 - 23) q^{69} + (9 \beta_{4} + 38 \beta_{3} + 34 \beta_{2} - 49 \beta_1 + 11) q^{71} + (23 \beta_{4} + 29 \beta_{3} - 41 \beta_{2} - 64 \beta_1 - 61) q^{73} + (25 \beta_{3} - 25 \beta_1 + 25) q^{75} + (3 \beta_{4} + 20 \beta_{3} - 86 \beta_{2} + 120 \beta_1 - 4) q^{77} + ( - 8 \beta_{4} - 3 \beta_{3} - 95 \beta_{2} - 36 \beta_1 - 12) q^{79} + ( - 32 \beta_{4} - 30 \beta_{3} - 37 \beta_{2} - 42 \beta_1 + 185) q^{81} + ( - 34 \beta_{4} + 26 \beta_{3} + 30 \beta_{2} - 40 \beta_1 + 24) q^{83} + ( - 15 \beta_{4} - 25 \beta_{3} - 20 \beta_{2} + 15 \beta_1 + 340) q^{85} + (9 \beta_{4} - 128 \beta_{3} + 68 \beta_{2} + 190 \beta_1 - 191) q^{87} + (61 \beta_{4} + 49 \beta_{3} + 18 \beta_{2} + 58 \beta_1 - 42) q^{89} + ( - 17 \beta_{4} - 46 \beta_{3} + 31 \beta_{2} + 33 \beta_1 + 286) q^{91} + ( - 13 \beta_{4} - 217 \beta_{3} + 38 \beta_{2} + 244 \beta_1 - 561) q^{93} + (5 \beta_{4} + 50 \beta_{3} + 15 \beta_1 - 20) q^{95} + ( - 17 \beta_{4} - 32 \beta_{3} - 31 \beta_{2} + 27 \beta_1 - 358) q^{97} + (47 \beta_{4} + 31 \beta_{3} - 11 \beta_{2} + 207 \beta_1 - 40) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{3} - 25 q^{5} + 15 q^{7} + 79 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 6 q^{3} - 25 q^{5} + 15 q^{7} + 79 q^{9} + 153 q^{11} + 28 q^{13} - 30 q^{15} - 341 q^{17} - 3 q^{19} - 212 q^{21} - 115 q^{23} + 125 q^{25} + 243 q^{27} - 583 q^{29} - 662 q^{31} - 457 q^{33} - 75 q^{35} - 172 q^{37} - 83 q^{39} + 344 q^{41} + 230 q^{43} - 395 q^{45} + 337 q^{47} - 4 q^{49} + 205 q^{51} - 942 q^{53} - 765 q^{55} - 890 q^{57} + 1166 q^{59} + 499 q^{61} + 1228 q^{63} - 140 q^{65} + 972 q^{67} - 138 q^{69} + 14 q^{71} - 229 q^{73} + 150 q^{75} + 312 q^{77} + 88 q^{79} + 897 q^{81} + 72 q^{83} + 1705 q^{85} - 1157 q^{87} - 90 q^{89} + 1309 q^{91} - 3071 q^{93} + 15 q^{95} - 1765 q^{97} + 91 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 34x^{3} - 9x^{2} + 260x + 60 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{4} - 22\nu^{3} - 26\nu^{2} + 377\nu - 278 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 30\nu^{2} + 51\nu + 158 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} + 30\nu^{2} - 19\nu - 158 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{4} + 2\nu^{3} - 146\nu^{2} - 123\nu + 194 ) / 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 4\beta_{3} - \beta_{2} + 2\beta _1 + 52 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 11\beta_{3} + 9\beta_{2} - 2\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 19\beta_{4} + 53\beta_{3} + 2\beta_{2} + 22\beta _1 + 484 ) / 2 \) 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
−4.34884
−0.230529
3.26689
4.98640
−3.67392
0 −8.28008 0 −5.00000 0 30.9947 0 41.5597 0
1.2 0 −2.78435 0 −5.00000 0 −24.1991 0 −19.2474 0
1.3 0 0.577397 0 −5.00000 0 10.7178 0 −26.6666 0
1.4 0 7.39409 0 −5.00000 0 3.57544 0 27.6726 0
1.5 0 9.09294 0 −5.00000 0 −6.08885 0 55.6816 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.p 5
4.b odd 2 1 115.4.a.d 5
12.b even 2 1 1035.4.a.m 5
20.d odd 2 1 575.4.a.k 5
20.e even 4 2 575.4.b.h 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.d 5 4.b odd 2 1
575.4.a.k 5 20.d odd 2 1
575.4.b.h 10 20.e even 4 2
1035.4.a.m 5 12.b even 2 1
1840.4.a.p 5 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{5} - 6T_{3}^{4} - 89T_{3}^{3} + 417T_{3}^{2} + 1340T_{3} - 895 \) Copy content Toggle raw display
\( T_{7}^{5} - 15T_{7}^{4} - 743T_{7}^{3} + 6718T_{7}^{2} + 34948T_{7} - 175008 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 6 T^{4} - 89 T^{3} + 417 T^{2} + \cdots - 895 \) Copy content Toggle raw display
$5$ \( (T + 5)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 15 T^{4} - 743 T^{3} + \cdots - 175008 \) Copy content Toggle raw display
$11$ \( T^{5} - 153 T^{4} + \cdots + 58105432 \) Copy content Toggle raw display
$13$ \( T^{5} - 28 T^{4} - 5855 T^{3} + \cdots - 42528287 \) Copy content Toggle raw display
$17$ \( T^{5} + 341 T^{4} + \cdots - 1269566848 \) Copy content Toggle raw display
$19$ \( T^{5} + 3 T^{4} - 10481 T^{3} + \cdots + 5566328 \) Copy content Toggle raw display
$23$ \( (T + 23)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + 583 T^{4} + \cdots - 63213004636 \) Copy content Toggle raw display
$31$ \( T^{5} + 662 T^{4} + \cdots - 341100199935 \) Copy content Toggle raw display
$37$ \( T^{5} + 172 T^{4} + \cdots + 337199293312 \) Copy content Toggle raw display
$41$ \( T^{5} - 344 T^{4} + \cdots - 554461833173 \) Copy content Toggle raw display
$43$ \( T^{5} - 230 T^{4} + \cdots - 278531891200 \) Copy content Toggle raw display
$47$ \( T^{5} - 337 T^{4} + \cdots - 6082562728660 \) Copy content Toggle raw display
$53$ \( T^{5} + 942 T^{4} + \cdots - 1069522603168 \) Copy content Toggle raw display
$59$ \( T^{5} - 1166 T^{4} + \cdots + 446741827072 \) Copy content Toggle raw display
$61$ \( T^{5} - 499 T^{4} + \cdots + 53636443112 \) Copy content Toggle raw display
$67$ \( T^{5} - 972 T^{4} + \cdots + 19766839800960 \) Copy content Toggle raw display
$71$ \( T^{5} - 14 T^{4} + \cdots - 256345940645 \) Copy content Toggle raw display
$73$ \( T^{5} + 229 T^{4} + \cdots + 3947399121116 \) Copy content Toggle raw display
$79$ \( T^{5} - 88 T^{4} + \cdots + 22913376438144 \) Copy content Toggle raw display
$83$ \( T^{5} - 72 T^{4} + \cdots - 2501024408832 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots + 279901843479552 \) Copy content Toggle raw display
$97$ \( T^{5} + 1765 T^{4} + \cdots - 113500838416 \) Copy content Toggle raw display
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