Properties

Label 1840.4.a.u
Level $1840$
Weight $4$
Character orbit 1840.a
Self dual yes
Analytic conductor $108.564$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \( x^{6} - 2x^{5} - 80x^{4} + 105x^{3} + 1328x^{2} - 816x - 4032 \) 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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + 5 q^{5} + (\beta_{5} + \beta_{4} + \beta_1 + 2) q^{7} + ( - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + 5 q^{5} + (\beta_{5} + \beta_{4} + \beta_1 + 2) q^{7} + ( - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 + 1) q^{9} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 + 3) q^{11} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 6) q^{13} + ( - 5 \beta_1 + 5) q^{15} + (3 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} + 5 \beta_1 - 23) q^{17} + ( - 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 9) q^{19} + (4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 19) q^{21} - 23 q^{23} + 25 q^{25} + ( - \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 9 \beta_{2} + 4 \beta_1 + 12) q^{27} + (2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 8 \beta_{2} - 8 \beta_1 - 53) q^{29} + ( - 2 \beta_{5} - 9 \beta_{4} - \beta_{3} + \beta_{2} + 7 \beta_1 + 34) q^{31} + ( - 2 \beta_{5} - 9 \beta_{4} - 2 \beta_{3} + 5 \beta_{2} - 17 \beta_1 - 95) q^{33} + (5 \beta_{5} + 5 \beta_{4} + 5 \beta_1 + 10) q^{35} + (\beta_{5} - 8 \beta_{4} - 9 \beta_{3} - 14 \beta_{2} - 2 \beta_1 - 62) q^{37} + ( - 2 \beta_{5} - 7 \beta_{4} - 7 \beta_{2} + 28 \beta_1 - 14) q^{39} + (5 \beta_{4} - \beta_{3} - 6 \beta_{2} - 13 \beta_1 - 107) q^{41} + (8 \beta_{5} - 6 \beta_{4} + 3 \beta_{3} - 9 \beta_{2} + 2 \beta_1 + 29) q^{43} + ( - 5 \beta_{5} + 5 \beta_{4} + 5 \beta_{2} - 5 \beta_1 + 5) q^{45} + ( - 3 \beta_{5} + 12 \beta_{4} - 9 \beta_{3} + \beta_{2} + 48 \beta_1 - 52) q^{47} + (4 \beta_{5} - 8 \beta_{4} + 9 \beta_{3} - 14 \beta_{2} - 17 \beta_1 - 51) q^{49} + (9 \beta_{5} + 4 \beta_{4} + 6 \beta_{3} - 15 \beta_{2} + 47 \beta_1 - 139) q^{51} + ( - 34 \beta_{5} + \beta_{4} + 17 \beta_{3} + 6 \beta_{2} + 10 \beta_1 - 96) q^{53} + ( - 10 \beta_{5} - 5 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} + 15 \beta_1 + 15) q^{55} + (7 \beta_{5} - 7 \beta_{4} + 13 \beta_{3} - 18 \beta_{2} + 22 \beta_1 - 52) q^{57} + (25 \beta_{5} - 6 \beta_{4} + 11 \beta_{2} + 2 \beta_1 - 181) q^{59} + (5 \beta_{5} + 8 \beta_{4} - 40 \beta_{3} + 26 \beta_{2} - 25 \beta_1 - 208) q^{61} + ( - 14 \beta_{5} - 15 \beta_{4} + 4 \beta_{3} - 20 \beta_{2} + 22 \beta_1 - 30) q^{63} + ( - 10 \beta_{4} + 10 \beta_{3} - 5 \beta_{2} - 5 \beta_1 - 30) q^{65} + ( - 14 \beta_{5} - 5 \beta_{4} + 18 \beta_{3} - 23 \beta_{2} + 8 \beta_1 + 217) q^{67} + (23 \beta_1 - 23) q^{69} + ( - 4 \beta_{5} + 17 \beta_{4} - 4 \beta_{3} + 17 \beta_{2} + 39 \beta_1 - 56) q^{71} + ( - 26 \beta_{5} - 21 \beta_{4} + 5 \beta_{3} - 17 \beta_{2} + 16 \beta_1 - 164) q^{73} + ( - 25 \beta_1 + 25) q^{75} + ( - 19 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 39 \beta_{2} + 62 \beta_1 - 279) q^{77} + ( - 17 \beta_{5} - 43 \beta_{4} + 16 \beta_{3} + 16 \beta_{2} - 44 \beta_1 + 160) q^{79} + (6 \beta_{5} - 20 \beta_{4} + 5 \beta_{3} + 14 \beta_{2} - 36 \beta_1 - 135) q^{81} + (16 \beta_{5} + 41 \beta_{4} - 3 \beta_{3} + 14 \beta_{2} - 6 \beta_1 + 214) q^{83} + (15 \beta_{5} + 5 \beta_{4} - 10 \beta_{3} - 5 \beta_{2} + 25 \beta_1 - 115) q^{85} + ( - 25 \beta_{5} + 12 \beta_{4} - 7 \beta_{3} + 35 \beta_{2} + 42 \beta_1 + 94) q^{87} + (6 \beta_{5} + 4 \beta_{4} + 19 \beta_{3} + 35 \beta_{2} + 54 \beta_1 - 487) q^{89} + (20 \beta_{5} + 29 \beta_{4} - 15 \beta_{3} + 49 \beta_{2} + 13 \beta_1 - 167) q^{91} + ( - 24 \beta_{5} - 25 \beta_{4} + 28 \beta_{3} - 5 \beta_{2} + 22 \beta_1 - 258) q^{93} + ( - 10 \beta_{5} - 15 \beta_{4} - 10 \beta_{3} - 25 \beta_{2} + 15 \beta_1 - 45) q^{95} + (4 \beta_{5} + 17 \beta_{4} + 9 \beta_{3} + 41 \beta_{2} + 125 \beta_1 - 35) q^{97} + ( - 7 \beta_{5} + 28 \beta_{4} + 4 \beta_{3} + 9 \beta_{2} + 61 \beta_1 + 153) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9} + 23 q^{11} - 42 q^{13} + 20 q^{15} - 124 q^{17} - 53 q^{19} - 114 q^{21} - 138 q^{23} + 150 q^{25} + 103 q^{27} - 320 q^{29} + 229 q^{31} - 583 q^{33} + 70 q^{35} - 377 q^{37} - 37 q^{39} - 683 q^{41} + 168 q^{43} + 20 q^{45} - 211 q^{47} - 374 q^{49} - 777 q^{51} - 613 q^{53} + 115 q^{55} - 316 q^{57} - 1029 q^{59} - 1169 q^{61} - 183 q^{63} - 210 q^{65} + 1227 q^{67} - 92 q^{69} - 237 q^{71} - 1001 q^{73} + 100 q^{75} - 1498 q^{77} + 898 q^{79} - 838 q^{81} + 1281 q^{83} - 620 q^{85} + 695 q^{87} - 2780 q^{89} - 857 q^{91} - 1569 q^{93} - 265 q^{95} + 91 q^{97} + 1015 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 80x^{4} + 105x^{3} + 1328x^{2} - 816x - 4032 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17\nu^{5} + 86\nu^{4} - 1888\nu^{3} - 5583\nu^{2} + 44176\nu + 40584 ) / 4824 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{5} + 32\nu^{4} - 796\nu^{3} - 2643\nu^{2} + 8248\nu + 24132 ) / 1206 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{5} - 22\nu^{4} - 307\nu^{3} + 1503\nu^{2} + 1264\nu - 15234 ) / 603 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\nu^{5} - 30\nu^{4} - 1448\nu^{3} + 539\nu^{2} + 19704\nu + 16320 ) / 1608 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{5} + \beta_{4} + 4\beta_{3} - 6\beta_{2} + 50\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -77\beta_{5} + 59\beta_{4} + 21\beta_{3} + 65\beta_{2} + 81\beta _1 + 1305 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -161\beta_{5} + 141\beta_{4} + 338\beta_{3} - 383\beta_{2} + 2873\beta _1 + 1655 \) 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
8.08353
4.61348
2.24528
−1.67963
−3.88687
−7.37578
0 −7.08353 0 5.00000 0 9.74988 0 23.1764 0
1.2 0 −3.61348 0 5.00000 0 −3.48289 0 −13.9428 0
1.3 0 −1.24528 0 5.00000 0 19.3962 0 −25.4493 0
1.4 0 2.67963 0 5.00000 0 −24.9473 0 −19.8196 0
1.5 0 4.88687 0 5.00000 0 22.2944 0 −3.11847 0
1.6 0 8.37578 0 5.00000 0 −9.01032 0 43.1537 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.u 6
4.b odd 2 1 920.4.a.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.4.a.a 6 4.b odd 2 1
1840.4.a.u 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{6} - 4T_{3}^{5} - 75T_{3}^{4} + 215T_{3}^{3} + 1158T_{3}^{2} - 1831T_{3} - 3496 \) Copy content Toggle raw display
\( T_{7}^{6} - 14T_{7}^{5} - 744T_{7}^{4} + 10329T_{7}^{3} + 89664T_{7}^{2} - 789580T_{7} - 3300776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} - 75 T^{4} + \cdots - 3496 \) Copy content Toggle raw display
$5$ \( (T - 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 14 T^{5} - 744 T^{4} + \cdots - 3300776 \) Copy content Toggle raw display
$11$ \( T^{6} - 23 T^{5} - 3447 T^{4} + \cdots + 398480 \) Copy content Toggle raw display
$13$ \( T^{6} + 42 T^{5} + \cdots + 1972631470 \) Copy content Toggle raw display
$17$ \( T^{6} + 124 T^{5} + \cdots - 1487271232 \) Copy content Toggle raw display
$19$ \( T^{6} + 53 T^{5} + \cdots - 22042764464 \) Copy content Toggle raw display
$23$ \( (T + 23)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} + 320 T^{5} + \cdots + 3217328048828 \) Copy content Toggle raw display
$31$ \( T^{6} - 229 T^{5} + \cdots - 2038672875799 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 120683526562304 \) Copy content Toggle raw display
$41$ \( T^{6} + 683 T^{5} + \cdots - 4268456808415 \) Copy content Toggle raw display
$43$ \( T^{6} - 168 T^{5} + \cdots + 1641239221248 \) Copy content Toggle raw display
$47$ \( T^{6} + 211 T^{5} + \cdots + 99563112748880 \) Copy content Toggle raw display
$53$ \( T^{6} + 613 T^{5} + \cdots - 39\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 882676639673600 \) Copy content Toggle raw display
$61$ \( T^{6} + 1169 T^{5} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 607060879004288 \) Copy content Toggle raw display
$71$ \( T^{6} + 237 T^{5} + \cdots - 42369419745775 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 802499042949320 \) Copy content Toggle raw display
$79$ \( T^{6} - 898 T^{5} + \cdots + 29\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{6} - 1281 T^{5} + \cdots + 85\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{6} + 2780 T^{5} + \cdots + 25\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( T^{6} - 91 T^{5} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
show more
show less