Properties

Label 1840.4.a.bb
Level $1840$
Weight $4$
Character orbit 1840.a
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - x^{9} - 204 x^{8} + 42 x^{7} + 12958 x^{6} + 5872 x^{5} - 259871 x^{4} - 149461 x^{3} + 1222472 x^{2} + 627136 x - 43712 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 5 q^{5} + ( - \beta_{6} - \beta_1 - 3) q^{7} + (\beta_{2} + \beta_1 + 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - 5 q^{5} + ( - \beta_{6} - \beta_1 - 3) q^{7} + (\beta_{2} + \beta_1 + 14) q^{9} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{11} + (\beta_{6} - \beta_{5} + \beta_{4} + 2) q^{13} - 5 \beta_1 q^{15} + ( - \beta_{9} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 8) q^{17} + ( - \beta_{9} - 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 11) q^{19} + ( - \beta_{6} - 3 \beta_{4} + 2 \beta_{2} - 10 \beta_1 - 23) q^{21} - 23 q^{23} + 25 q^{25} + (\beta_{9} + \beta_{8} + 3 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} + \beta_{4} + 23 \beta_1 + 40) q^{27} + (2 \beta_{9} + 2 \beta_{8} - \beta_{7} - 4 \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - 6 \beta_1 - 29) q^{29} + ( - \beta_{9} - \beta_{7} - 4 \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots + 12) q^{31}+ \cdots + (5 \beta_{9} - 7 \beta_{8} - 7 \beta_{7} + 72 \beta_{6} + \beta_{5} + 37 \beta_{4} + \cdots - 419) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9} + 14 q^{11} + 11 q^{13} - 5 q^{15} + 68 q^{17} - 114 q^{19} - 232 q^{21} - 230 q^{23} + 250 q^{25} + 433 q^{27} - 273 q^{29} + 129 q^{31} + 98 q^{33} + 140 q^{35} + 62 q^{37} - 283 q^{39} + 767 q^{41} - 332 q^{43} - 695 q^{45} + 323 q^{47} + 1162 q^{49} - 176 q^{51} + 558 q^{53} - 70 q^{55} + 46 q^{57} - 822 q^{59} + 318 q^{61} - 2698 q^{63} - 55 q^{65} - 1152 q^{67} - 23 q^{69} - 1247 q^{71} + 1941 q^{73} + 25 q^{75} + 528 q^{77} - 3134 q^{79} + 6210 q^{81} - 482 q^{83} - 340 q^{85} - 1797 q^{87} + 4734 q^{89} - 4992 q^{91} + 4647 q^{93} + 570 q^{95} + 2326 q^{97} - 4356 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 204 x^{8} + 42 x^{7} + 12958 x^{6} + 5872 x^{5} - 259871 x^{4} - 149461 x^{3} + 1222472 x^{2} + 627136 x - 43712 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 41 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3854981 \nu^{9} + 89269783 \nu^{8} + 723349168 \nu^{7} - 16966937858 \nu^{6} - 52533030922 \nu^{5} + 969699259072 \nu^{4} + \cdots + 15927316040736 ) / 649381829040 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4287527 \nu^{9} - 18632021 \nu^{8} + 958382440 \nu^{7} + 3713138890 \nu^{6} - 67076321674 \nu^{5} - 215575562744 \nu^{4} + \cdots - 14421069607200 ) / 389629097424 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 72608749 \nu^{9} + 196190847 \nu^{8} + 15013259412 \nu^{7} - 32852266542 \nu^{6} - 974051270338 \nu^{5} + \cdots + 115237989620584 ) / 2922218230680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41435197 \nu^{9} - 38587743 \nu^{8} - 8659056120 \nu^{7} + 1706664114 \nu^{6} + 572042196850 \nu^{5} + 211662778568 \nu^{4} + \cdots + 8867676107504 ) / 1168887292272 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 456055897 \nu^{9} - 514489611 \nu^{8} - 93405512736 \nu^{7} + 29305755546 \nu^{6} + 5982905567554 \nu^{5} + \cdots + 273658774052288 ) / 5844436461360 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 48402857 \nu^{9} - 20806896 \nu^{8} - 9854440296 \nu^{7} - 3859706262 \nu^{6} + 625434057344 \nu^{5} + 636739850680 \nu^{4} + \cdots + 17109836030200 ) / 584443646136 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 312403603 \nu^{9} + 511006659 \nu^{8} + 62814279954 \nu^{7} - 50573267814 \nu^{6} - 3913068398836 \nu^{5} + \cdots - 92807654043212 ) / 1461109115340 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + 3\beta_{7} - 4\beta_{6} - 2\beta_{5} + \beta_{4} + 77\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{9} - 8 \beta_{8} + 7 \beta_{7} - 6 \beta_{6} + 10 \beta_{5} - 12 \beta_{4} - 3 \beta_{3} + 101 \beta_{2} + 149 \beta _1 + 3201 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 126 \beta_{9} + 165 \beta_{8} + 352 \beta_{7} - 539 \beta_{6} - 192 \beta_{5} + 29 \beta_{4} - 78 \beta_{3} + 122 \beta_{2} + 6701 \beta _1 + 6340 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 271 \beta_{9} - 1006 \beta_{8} + 1011 \beta_{7} - 1523 \beta_{6} + 1100 \beta_{5} - 2206 \beta_{4} - 594 \beta_{3} + 10169 \beta_{2} + 17887 \beta _1 + 282923 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12628 \beta_{9} + 20071 \beta_{8} + 35963 \beta_{7} - 65369 \beta_{6} - 16334 \beta_{5} - 5653 \beta_{4} - 13587 \beta_{3} + 24290 \beta_{2} + 610325 \beta _1 + 790558 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 27132 \beta_{9} - 95202 \beta_{8} + 124672 \beta_{7} - 253946 \beta_{6} + 102528 \beta_{5} - 308026 \beta_{4} - 86724 \beta_{3} + 1033315 \beta_{2} + 1992345 \beta _1 + 26201647 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1197157 \beta_{9} + 2211691 \beta_{8} + 3599343 \beta_{7} - 7540678 \beta_{6} - 1366838 \beta_{5} - 1414829 \beta_{4} - 1803516 \beta_{3} + 3533338 \beta_{2} + 56981385 \beta _1 + 90602246 \) 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
−9.58471
−7.77468
−5.32629
−2.66698
−0.575045
0.0622186
2.52979
4.78870
9.27510
10.2719
0 −9.58471 0 −5.00000 0 −21.8969 0 64.8667 0
1.2 0 −7.77468 0 −5.00000 0 16.1209 0 33.4457 0
1.3 0 −5.32629 0 −5.00000 0 −14.9292 0 1.36937 0
1.4 0 −2.66698 0 −5.00000 0 32.5699 0 −19.8872 0
1.5 0 −0.575045 0 −5.00000 0 24.8747 0 −26.6693 0
1.6 0 0.0622186 0 −5.00000 0 −14.4792 0 −26.9961 0
1.7 0 2.52979 0 −5.00000 0 −24.3398 0 −20.6002 0
1.8 0 4.78870 0 −5.00000 0 −0.156416 0 −4.06831 0
1.9 0 9.27510 0 −5.00000 0 −33.0838 0 59.0275 0
1.10 0 10.2719 0 −5.00000 0 7.31988 0 78.5119 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.bb 10
4.b odd 2 1 920.4.a.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.4.a.g 10 4.b odd 2 1
1840.4.a.bb 10 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{10} - T_{3}^{9} - 204 T_{3}^{8} + 42 T_{3}^{7} + 12958 T_{3}^{6} + 5872 T_{3}^{5} - 259871 T_{3}^{4} - 149461 T_{3}^{3} + 1222472 T_{3}^{2} + 627136 T_{3} - 43712 \) Copy content Toggle raw display
\( T_{7}^{10} + 28 T_{7}^{9} - 1904 T_{7}^{8} - 56306 T_{7}^{7} + 1019676 T_{7}^{6} + 34617872 T_{7}^{5} - 127663083 T_{7}^{4} - 7079160712 T_{7}^{3} - 7124849036 T_{7}^{2} + \cdots + 56995990528 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} - 204 T^{8} + \cdots - 43712 \) Copy content Toggle raw display
$5$ \( (T + 5)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 28 T^{9} + \cdots + 56995990528 \) Copy content Toggle raw display
$11$ \( T^{10} - 14 T^{9} + \cdots - 16\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{10} - 11 T^{9} + \cdots - 63035886761000 \) Copy content Toggle raw display
$17$ \( T^{10} - 68 T^{9} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{10} + 114 T^{9} + \cdots - 10\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( (T + 23)^{10} \) Copy content Toggle raw display
$29$ \( T^{10} + 273 T^{9} + \cdots - 29\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{10} - 129 T^{9} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( T^{10} - 62 T^{9} + \cdots - 37\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{10} - 767 T^{9} + \cdots + 21\!\cdots\!22 \) Copy content Toggle raw display
$43$ \( T^{10} + 332 T^{9} + \cdots - 13\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{10} - 323 T^{9} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} - 558 T^{9} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + 822 T^{9} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{10} - 318 T^{9} + \cdots - 32\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + 1152 T^{9} + \cdots + 37\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{10} + 1247 T^{9} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} - 1941 T^{9} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + 3134 T^{9} + \cdots - 25\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{10} + 482 T^{9} + \cdots + 60\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{10} - 4734 T^{9} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} - 2326 T^{9} + \cdots - 41\!\cdots\!00 \) Copy content Toggle raw display
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