Properties

Label 1840.4.a.x
Level $1840$
Weight $4$
Character orbit 1840.a
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 4x^{7} - 123x^{6} + 335x^{5} + 4492x^{4} - 7035x^{3} - 45582x^{2} + 36684x + 124632 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{3} + 5 q^{5} + (\beta_{3} + \beta_1 + 3) q^{7} + (\beta_{2} - 2 \beta_1 + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{3} + 5 q^{5} + (\beta_{3} + \beta_1 + 3) q^{7} + (\beta_{2} - 2 \beta_1 + 9) q^{9} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 5) q^{11} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_1 - 6) q^{13} + ( - 5 \beta_1 + 10) q^{15} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 2) q^{17} + (\beta_{7} - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 + 15) q^{19} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 6 \beta_{3} - 4 \beta_{2} - 9 \beta_1 - 17) q^{21} + 23 q^{23} + 25 q^{25} + ( - 3 \beta_{6} - 3 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 54) q^{27} + ( - \beta_{7} - \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 8 \beta_1 + 9) q^{29} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{2} + 7 \beta_1 + 32) q^{31} + (2 \beta_{7} + 5 \beta_{5} - 2 \beta_{4} - 7 \beta_{3} + 8 \beta_{2} - 6 \beta_1 + 42) q^{33} + (5 \beta_{3} + 5 \beta_1 + 15) q^{35} + (\beta_{7} - 2 \beta_{6} + 3 \beta_{4} + \beta_{2} - 10 \beta_1 - 15) q^{37} + (\beta_{7} + 4 \beta_{6} + 3 \beta_{5} + 3 \beta_{3} + 14 \beta_1 + 29) q^{39} + ( - \beta_{7} + 5 \beta_{6} + \beta_{5} - 4 \beta_{3} + 2 \beta_{2} - 20 \beta_1 + 49) q^{41} + ( - 6 \beta_{6} - \beta_{5} - 7 \beta_{4} + 4 \beta_{3} + \beta_{2} + 14 \beta_1 + 59) q^{43} + (5 \beta_{2} - 10 \beta_1 + 45) q^{45} + (4 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} + 8 \beta_1 + 125) q^{47} + (8 \beta_{6} - 5 \beta_{5} + \beta_{4} + 11 \beta_{3} + 2 \beta_{2} - 16 \beta_1 + 77) q^{49} + ( - 2 \beta_{7} - 5 \beta_{6} + 6 \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} + \cdots + 40) q^{51}+ \cdots + (3 \beta_{7} - 7 \beta_{6} + 16 \beta_{5} - 3 \beta_{4} - 67 \beta_{3} + 37 \beta_{2} + \cdots + 341) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9} + 35 q^{11} - 54 q^{13} + 60 q^{15} - 11 q^{17} + 107 q^{19} - 148 q^{21} + 184 q^{23} + 200 q^{25} + 405 q^{27} + 37 q^{29} + 290 q^{31} + 289 q^{33} + 155 q^{35} - 172 q^{37} + 311 q^{39} + 308 q^{41} + 538 q^{43} + 310 q^{45} + 1035 q^{47} + 585 q^{49} + 387 q^{51} + 46 q^{53} + 175 q^{55} + 286 q^{57} + 1256 q^{59} + 399 q^{61} + 1234 q^{63} - 270 q^{65} + 1598 q^{67} + 276 q^{69} + 750 q^{71} + 177 q^{73} + 300 q^{75} - 1228 q^{77} + 1292 q^{79} + 380 q^{81} + 2094 q^{83} - 55 q^{85} + 2245 q^{87} + 484 q^{89} + 679 q^{91} - 2111 q^{93} + 535 q^{95} - 2225 q^{97} + 2117 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 123x^{6} + 335x^{5} + 4492x^{4} - 7035x^{3} - 45582x^{2} + 36684x + 124632 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -31\nu^{7} + 108\nu^{6} + 3891\nu^{5} - 9779\nu^{4} - 129876\nu^{3} + 227019\nu^{2} + 612936\nu - 1138428 ) / 49680 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{7} - 28\nu^{6} + 1189\nu^{5} + 4699\nu^{4} - 39724\nu^{3} - 187299\nu^{2} + 152664\nu + 1064268 ) / 16560 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -53\nu^{7} + 474\nu^{6} + 4983\nu^{5} - 44497\nu^{4} - 112158\nu^{3} + 1065687\nu^{2} - 432\nu - 3995244 ) / 49680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29\nu^{7} - 12\nu^{6} - 3729\nu^{5} - 2159\nu^{4} + 132804\nu^{3} + 159159\nu^{2} - 982584\nu - 1076868 ) / 24840 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -59\nu^{7} + 72\nu^{6} + 7539\nu^{5} + 2489\nu^{4} - 272424\nu^{3} - 431589\nu^{2} + 1979064\nu + 3741228 ) / 24840 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{6} + 3\beta_{4} + 3\beta_{3} + \beta_{2} + 56\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{7} + 15\beta_{6} + 6\beta_{4} + 76\beta_{2} + 212\beta _1 + 1793 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 39\beta_{7} + 315\beta_{6} - 36\beta_{5} + 198\beta_{4} + 330\beta_{3} + 163\beta_{2} + 3782\beta _1 + 4940 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 801\beta_{7} + 2118\beta_{6} - 36\beta_{5} + 579\beta_{4} + 471\beta_{3} + 5515\beta_{2} + 19844\beta _1 + 118346 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5793 \beta_{7} + 29616 \beta_{6} - 4644 \beta_{5} + 12408 \beta_{4} + 28890 \beta_{3} + 18832 \beta_{2} + 276764 \beta _1 + 505163 \) 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
9.27165
7.24150
3.32486
2.47643
−1.62883
−2.97241
−6.18739
−7.52581
0 −7.27165 0 5.00000 0 30.3165 0 25.8769 0
1.2 0 −5.24150 0 5.00000 0 3.79432 0 0.473311 0
1.3 0 −1.32486 0 5.00000 0 −13.2387 0 −25.2447 0
1.4 0 −0.476427 0 5.00000 0 1.47219 0 −26.7730 0
1.5 0 3.62883 0 5.00000 0 −20.4430 0 −13.8316 0
1.6 0 4.97241 0 5.00000 0 18.7086 0 −2.27518 0
1.7 0 8.18739 0 5.00000 0 31.6165 0 40.0333 0
1.8 0 9.52581 0 5.00000 0 −21.2263 0 63.7410 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.x 8
4.b odd 2 1 920.4.a.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.4.a.c 8 4.b odd 2 1
1840.4.a.x 8 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{8} - 12T_{3}^{7} - 67T_{3}^{6} + 1029T_{3}^{5} + 462T_{3}^{4} - 22173T_{3}^{3} + 16400T_{3}^{2} + 83904T_{3} + 33856 \) Copy content Toggle raw display
\( T_{7}^{8} - 31 T_{7}^{7} - 1184 T_{7}^{6} + 31419 T_{7}^{5} + 501889 T_{7}^{4} - 8737786 T_{7}^{3} - 70422072 T_{7}^{2} + 511740376 T_{7} - 575432512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 12 T^{7} - 67 T^{6} + \cdots + 33856 \) Copy content Toggle raw display
$5$ \( (T - 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 31 T^{7} + \cdots - 575432512 \) Copy content Toggle raw display
$11$ \( T^{8} - 35 T^{7} + \cdots + 433851465600 \) Copy content Toggle raw display
$13$ \( T^{8} + 54 T^{7} + \cdots - 844403048392 \) Copy content Toggle raw display
$17$ \( T^{8} + 11 T^{7} + \cdots - 259504299904 \) Copy content Toggle raw display
$19$ \( T^{8} - 107 T^{7} + \cdots - 4730705318528 \) Copy content Toggle raw display
$23$ \( (T - 23)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 37 T^{7} + \cdots + 53506236436200 \) Copy content Toggle raw display
$31$ \( T^{8} - 290 T^{7} + \cdots - 10\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 324096597618944 \) Copy content Toggle raw display
$41$ \( T^{8} - 308 T^{7} + \cdots - 12\!\cdots\!66 \) Copy content Toggle raw display
$43$ \( T^{8} - 538 T^{7} + \cdots - 10\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{8} - 1035 T^{7} + \cdots - 12\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} - 46 T^{7} + \cdots - 30\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{8} - 1256 T^{7} + \cdots - 49\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{8} - 399 T^{7} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} - 1598 T^{7} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{8} - 750 T^{7} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{8} - 177 T^{7} + \cdots + 25\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{8} - 1292 T^{7} + \cdots - 77\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{8} - 2094 T^{7} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} - 484 T^{7} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + 2225 T^{7} + \cdots - 12\!\cdots\!28 \) Copy content Toggle raw display
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