Properties

Label 40.6.a
Level 40
Weight 6
Character orbit a
Rep. character \(\chi_{40}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 4
Sturm bound 36
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 40.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(36\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(40))\).

Total New Old
Modular forms 34 5 29
Cusp forms 26 5 21
Eisenstein series 8 0 8

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(3\)

Trace form

\( 5q - 40q^{3} + 25q^{5} + 124q^{7} + 281q^{9} + O(q^{10}) \) \( 5q - 40q^{3} + 25q^{5} + 124q^{7} + 281q^{9} + 468q^{11} + 222q^{13} + 1578q^{17} - 3132q^{19} - 584q^{21} + 1076q^{23} + 3125q^{25} - 5920q^{27} + 3446q^{29} + 2392q^{31} - 16240q^{33} - 5900q^{35} + 454q^{37} - 13248q^{39} + 10762q^{41} + 30368q^{43} + 14925q^{45} - 28924q^{47} + 677q^{49} + 50320q^{51} - 17306q^{53} - 13900q^{55} + 2880q^{57} + 67388q^{59} - 21202q^{61} + 32668q^{63} + 48550q^{65} - 130848q^{67} - 149336q^{69} + 38432q^{71} - 91454q^{73} - 25000q^{75} + 228896q^{77} + 106304q^{79} + 71765q^{81} + 13016q^{83} + 14450q^{85} - 334160q^{87} - 344862q^{89} + 22680q^{91} - 4400q^{93} - 37900q^{95} + 289834q^{97} + 443012q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(40))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
40.6.a.a \(1\) \(6.415\) \(\Q\) None \(0\) \(-18\) \(-25\) \(242\) \(-\) \(+\) \(q-18q^{3}-5^{2}q^{5}+242q^{7}+3^{4}q^{9}+\cdots\)
40.6.a.b \(1\) \(6.415\) \(\Q\) None \(0\) \(-8\) \(25\) \(-108\) \(-\) \(-\) \(q-8q^{3}+5^{2}q^{5}-108q^{7}-179q^{9}+\cdots\)
40.6.a.c \(1\) \(6.415\) \(\Q\) None \(0\) \(-2\) \(-25\) \(-62\) \(+\) \(+\) \(q-2q^{3}-5^{2}q^{5}-62q^{7}-239q^{9}+\cdots\)
40.6.a.d \(2\) \(6.415\) \(\Q(\sqrt{129}) \) None \(0\) \(-12\) \(50\) \(52\) \(+\) \(-\) \(q+(-6-\beta )q^{3}+5^{2}q^{5}+(26-3\beta )q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(40))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(40)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 18 T + 243 T^{2} \))(\( 1 + 8 T + 243 T^{2} \))(\( 1 + 2 T + 243 T^{2} \))(\( 1 + 12 T + 6 T^{2} + 2916 T^{3} + 59049 T^{4} \))
$5$ (\( 1 + 25 T \))(\( 1 - 25 T \))(\( 1 + 25 T \))(\( ( 1 - 25 T )^{2} \))
$7$ (\( 1 - 242 T + 16807 T^{2} \))(\( 1 + 108 T + 16807 T^{2} \))(\( 1 + 62 T + 16807 T^{2} \))(\( 1 - 52 T + 29646 T^{2} - 873964 T^{3} + 282475249 T^{4} \))
$11$ (\( 1 - 656 T + 161051 T^{2} \))(\( 1 + 604 T + 161051 T^{2} \))(\( 1 + 144 T + 161051 T^{2} \))(\( 1 - 560 T + 381926 T^{2} - 90188560 T^{3} + 25937424601 T^{4} \))
$13$ (\( 1 + 206 T + 371293 T^{2} \))(\( 1 + 306 T + 371293 T^{2} \))(\( 1 + 654 T + 371293 T^{2} \))(\( 1 - 1388 T + 1149918 T^{2} - 515354684 T^{3} + 137858491849 T^{4} \))
$17$ (\( 1 - 1690 T + 1419857 T^{2} \))(\( 1 - 930 T + 1419857 T^{2} \))(\( 1 + 1190 T + 1419857 T^{2} \))(\( 1 - 148 T - 795706 T^{2} - 210138836 T^{3} + 2015993900449 T^{4} \))
$19$ (\( 1 + 1364 T + 2476099 T^{2} \))(\( 1 + 1324 T + 2476099 T^{2} \))(\( 1 - 556 T + 2476099 T^{2} \))(\( 1 + 1000 T + 4533462 T^{2} + 2476099000 T^{3} + 6131066257801 T^{4} \))
$23$ (\( 1 - 2198 T + 6436343 T^{2} \))(\( 1 + 852 T + 6436343 T^{2} \))(\( 1 - 2182 T + 6436343 T^{2} \))(\( 1 + 2452 T + 6569198 T^{2} + 15781913036 T^{3} + 41426511213649 T^{4} \))
$29$ (\( 1 + 2218 T + 20511149 T^{2} \))(\( 1 - 5902 T + 20511149 T^{2} \))(\( 1 + 1578 T + 20511149 T^{2} \))(\( 1 - 1340 T - 8758306 T^{2} - 27484939660 T^{3} + 420707233300201 T^{4} \))
$31$ (\( 1 + 1700 T + 28629151 T^{2} \))(\( 1 + 3320 T + 28629151 T^{2} \))(\( 1 - 9660 T + 28629151 T^{2} \))(\( 1 + 2248 T + 57017022 T^{2} + 64358331448 T^{3} + 819628286980801 T^{4} \))
$37$ (\( 1 + 846 T + 69343957 T^{2} \))(\( 1 - 10774 T + 69343957 T^{2} \))(\( 1 + 3534 T + 69343957 T^{2} \))(\( 1 + 5940 T + 123434318 T^{2} + 411903104580 T^{3} + 4808584372417849 T^{4} \))
$41$ (\( 1 + 1818 T + 115856201 T^{2} \))(\( 1 + 17958 T + 115856201 T^{2} \))(\( 1 - 7462 T + 115856201 T^{2} \))(\( 1 - 23076 T + 352280470 T^{2} - 2673497694276 T^{3} + 13422659310152401 T^{4} \))
$43$ (\( 1 - 10534 T + 147008443 T^{2} \))(\( 1 - 9264 T + 147008443 T^{2} \))(\( 1 + 7114 T + 147008443 T^{2} \))(\( 1 - 17684 T + 312898614 T^{2} - 2599697306012 T^{3} + 21611482313284249 T^{4} \))
$47$ (\( 1 - 12074 T + 229345007 T^{2} \))(\( 1 + 9796 T + 229345007 T^{2} \))(\( 1 + 28294 T + 229345007 T^{2} \))(\( 1 + 2908 T + 56660030 T^{2} + 666935280356 T^{3} + 52599132235830049 T^{4} \))
$53$ (\( 1 - 32586 T + 418195493 T^{2} \))(\( 1 + 31434 T + 418195493 T^{2} \))(\( 1 + 13046 T + 418195493 T^{2} \))(\( 1 + 5412 T + 693247822 T^{2} + 2263274008116 T^{3} + 174887470365513049 T^{4} \))
$59$ (\( 1 - 8668 T + 714924299 T^{2} \))(\( 1 - 33228 T + 714924299 T^{2} \))(\( 1 + 37092 T + 714924299 T^{2} \))(\( 1 - 62584 T + 2277965606 T^{2} - 44742822328616 T^{3} + 511116753300641401 T^{4} \))
$61$ (\( 1 + 34670 T + 844596301 T^{2} \))(\( 1 + 40210 T + 844596301 T^{2} \))(\( 1 - 39570 T + 844596301 T^{2} \))(\( 1 - 14108 T + 1110042462 T^{2} - 11915564614508 T^{3} + 713342911662882601 T^{4} \))
$67$ (\( 1 + 47566 T + 1350125107 T^{2} \))(\( 1 - 58864 T + 1350125107 T^{2} \))(\( 1 + 56734 T + 1350125107 T^{2} \))(\( 1 + 85412 T + 4371910566 T^{2} + 115316885639084 T^{3} + 1822837804551761449 T^{4} \))
$71$ (\( 1 - 948 T + 1804229351 T^{2} \))(\( 1 + 55312 T + 1804229351 T^{2} \))(\( 1 - 45588 T + 1804229351 T^{2} \))(\( 1 - 47208 T + 4011779662 T^{2} - 85174059202008 T^{3} + 3255243551009881201 T^{4} \))
$73$ (\( 1 + 63102 T + 2073071593 T^{2} \))(\( 1 - 27258 T + 2073071593 T^{2} \))(\( 1 - 11842 T + 2073071593 T^{2} \))(\( 1 + 67452 T + 4400780438 T^{2} + 139832825091036 T^{3} + 4297625829703557649 T^{4} \))
$79$ (\( 1 - 46536 T + 3077056399 T^{2} \))(\( 1 - 31456 T + 3077056399 T^{2} \))(\( 1 - 94216 T + 3077056399 T^{2} \))(\( 1 + 65904 T + 3994274078 T^{2} + 202790324919696 T^{3} + 9468276082626847201 T^{4} \))
$83$ (\( 1 + 88778 T + 3939040643 T^{2} \))(\( 1 - 24552 T + 3939040643 T^{2} \))(\( 1 + 31482 T + 3939040643 T^{2} \))(\( 1 - 108724 T + 10572459494 T^{2} - 428268254869532 T^{3} + 15516041187205853449 T^{4} \))
$89$ (\( 1 + 104934 T + 5584059449 T^{2} \))(\( 1 + 90854 T + 5584059449 T^{2} \))(\( 1 + 94054 T + 5584059449 T^{2} \))(\( 1 + 55020 T + 10818978262 T^{2} + 307234950883980 T^{3} + 31181719929966183601 T^{4} \))
$97$ (\( 1 + 36254 T + 8587340257 T^{2} \))(\( 1 - 154706 T + 8587340257 T^{2} \))(\( 1 - 23714 T + 8587340257 T^{2} \))(\( 1 - 147668 T + 11612429670 T^{2} - 1268075361070676 T^{3} + 73742412689492826049 T^{4} \))
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