Properties

Label 38.8.a.d
Level 38
Weight 8
Character orbit 38.a
Self dual yes
Analytic conductor 11.871
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.8706309684\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{633}) \)
Defining polynomial: \(x^{2} - x - 158\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{633})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} + ( -33 - 3 \beta ) q^{3} + 64 q^{4} + ( 62 + 31 \beta ) q^{5} + ( -264 - 24 \beta ) q^{6} + ( -1105 - 28 \beta ) q^{7} + 512 q^{8} + ( 324 + 207 \beta ) q^{9} +O(q^{10})\) \( q + 8 q^{2} + ( -33 - 3 \beta ) q^{3} + 64 q^{4} + ( 62 + 31 \beta ) q^{5} + ( -264 - 24 \beta ) q^{6} + ( -1105 - 28 \beta ) q^{7} + 512 q^{8} + ( 324 + 207 \beta ) q^{9} + ( 496 + 248 \beta ) q^{10} + ( -1572 - 151 \beta ) q^{11} + ( -2112 - 192 \beta ) q^{12} + ( -6489 - 449 \beta ) q^{13} + ( -8840 - 224 \beta ) q^{14} + ( -16740 - 1302 \beta ) q^{15} + 4096 q^{16} + ( -16233 + 210 \beta ) q^{17} + ( 2592 + 1656 \beta ) q^{18} -6859 q^{19} + ( 3968 + 1984 \beta ) q^{20} + ( 49737 + 4323 \beta ) q^{21} + ( -12576 - 1208 \beta ) q^{22} + ( -39163 - 4199 \beta ) q^{23} + ( -16896 - 1536 \beta ) q^{24} + ( 77557 + 4805 \beta ) q^{25} + ( -51912 - 3592 \beta ) q^{26} + ( -36639 - 1863 \beta ) q^{27} + ( -70720 - 1792 \beta ) q^{28} + ( -7961 + 3173 \beta ) q^{29} + ( -133920 - 10416 \beta ) q^{30} + ( 126188 + 6568 \beta ) q^{31} + 32768 q^{32} + ( 123450 + 10152 \beta ) q^{33} + ( -129864 + 1680 \beta ) q^{34} + ( -205654 - 36859 \beta ) q^{35} + ( 20736 + 13248 \beta ) q^{36} + ( -78934 + 8608 \beta ) q^{37} -54872 q^{38} + ( 426963 + 35631 \beta ) q^{39} + ( 31744 + 15872 \beta ) q^{40} + ( 195404 - 51678 \beta ) q^{41} + ( 397896 + 34584 \beta ) q^{42} + ( -21290 - 41289 \beta ) q^{43} + ( -100608 - 9664 \beta ) q^{44} + ( 1033974 + 29295 \beta ) q^{45} + ( -313304 - 33592 \beta ) q^{46} + ( 745730 - 20435 \beta ) q^{47} + ( -135168 - 12288 \beta ) q^{48} + ( 521354 + 62664 \beta ) q^{49} + ( 620456 + 38440 \beta ) q^{50} + ( 436149 + 41139 \beta ) q^{51} + ( -415296 - 28736 \beta ) q^{52} + ( -487913 + 30183 \beta ) q^{53} + ( -293112 - 14904 \beta ) q^{54} + ( -837062 - 62775 \beta ) q^{55} + ( -565760 - 14336 \beta ) q^{56} + ( 226347 + 20577 \beta ) q^{57} + ( -63688 + 25384 \beta ) q^{58} + ( -541721 + 114433 \beta ) q^{59} + ( -1071360 - 83328 \beta ) q^{60} + ( -715104 - 76547 \beta ) q^{61} + ( 1009504 + 52544 \beta ) q^{62} + ( -1273788 - 243603 \beta ) q^{63} + 262144 q^{64} + ( -2601520 - 242916 \beta ) q^{65} + ( 987600 + 81216 \beta ) q^{66} + ( -979769 + 111319 \beta ) q^{67} + ( -1038912 + 13440 \beta ) q^{68} + ( 3282705 + 268653 \beta ) q^{69} + ( -1645232 - 294872 \beta ) q^{70} + ( -1854214 + 291244 \beta ) q^{71} + ( 165888 + 105984 \beta ) q^{72} + ( -1347935 + 196048 \beta ) q^{73} + ( -631472 + 68864 \beta ) q^{74} + ( -4836951 - 405651 \beta ) q^{75} -438976 q^{76} + ( 2405084 + 215099 \beta ) q^{77} + ( 3415704 + 285048 \beta ) q^{78} + ( 1214050 + 208826 \beta ) q^{79} + ( 253952 + 126976 \beta ) q^{80} + ( 1383561 - 275724 \beta ) q^{81} + ( 1563232 - 413424 \beta ) q^{82} + ( -5088786 + 118218 \beta ) q^{83} + ( 3183168 + 276672 \beta ) q^{84} + ( 22134 - 483693 \beta ) q^{85} + ( -170320 - 330312 \beta ) q^{86} + ( -1241289 - 90345 \beta ) q^{87} + ( -804864 - 77312 \beta ) q^{88} + ( -1524580 - 457000 \beta ) q^{89} + ( 8271792 + 234360 \beta ) q^{90} + ( 9156721 + 690409 \beta ) q^{91} + ( -2506432 - 268736 \beta ) q^{92} + ( -7277436 - 615012 \beta ) q^{93} + ( 5965840 - 163480 \beta ) q^{94} + ( -425258 - 212629 \beta ) q^{95} + ( -1081344 - 98304 \beta ) q^{96} + ( 2555234 + 783058 \beta ) q^{97} + ( 4170832 + 501312 \beta ) q^{98} + ( -5447934 - 405585 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 16q^{2} - 69q^{3} + 128q^{4} + 155q^{5} - 552q^{6} - 2238q^{7} + 1024q^{8} + 855q^{9} + O(q^{10}) \) \( 2q + 16q^{2} - 69q^{3} + 128q^{4} + 155q^{5} - 552q^{6} - 2238q^{7} + 1024q^{8} + 855q^{9} + 1240q^{10} - 3295q^{11} - 4416q^{12} - 13427q^{13} - 17904q^{14} - 34782q^{15} + 8192q^{16} - 32256q^{17} + 6840q^{18} - 13718q^{19} + 9920q^{20} + 103797q^{21} - 26360q^{22} - 82525q^{23} - 35328q^{24} + 159919q^{25} - 107416q^{26} - 75141q^{27} - 143232q^{28} - 12749q^{29} - 278256q^{30} + 258944q^{31} + 65536q^{32} + 257052q^{33} - 258048q^{34} - 448167q^{35} + 54720q^{36} - 149260q^{37} - 109744q^{38} + 889557q^{39} + 79360q^{40} + 339130q^{41} + 830376q^{42} - 83869q^{43} - 210880q^{44} + 2097243q^{45} - 660200q^{46} + 1471025q^{47} - 282624q^{48} + 1105372q^{49} + 1279352q^{50} + 913437q^{51} - 859328q^{52} - 945643q^{53} - 601128q^{54} - 1736899q^{55} - 1145856q^{56} + 473271q^{57} - 101992q^{58} - 969009q^{59} - 2226048q^{60} - 1506755q^{61} + 2071552q^{62} - 2791179q^{63} + 524288q^{64} - 5445956q^{65} + 2056416q^{66} - 1848219q^{67} - 2064384q^{68} + 6834063q^{69} - 3585336q^{70} - 3417184q^{71} + 437760q^{72} - 2499822q^{73} - 1194080q^{74} - 10079553q^{75} - 877952q^{76} + 5025267q^{77} + 7116456q^{78} + 2636926q^{79} + 634880q^{80} + 2491398q^{81} + 2713040q^{82} - 10059354q^{83} + 6643008q^{84} - 439425q^{85} - 670952q^{86} - 2572923q^{87} - 1687040q^{88} - 3506160q^{89} + 16777944q^{90} + 19003851q^{91} - 5281600q^{92} - 15169884q^{93} + 11768200q^{94} - 1063145q^{95} - 2260992q^{96} + 5893526q^{97} + 8842976q^{98} - 11301453q^{99} + O(q^{100}) \)

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
13.0797
−12.0797
8.00000 −72.2392 64.0000 467.472 −577.914 −1471.23 512.000 3031.51 3739.78
1.2 8.00000 3.23924 64.0000 −312.472 25.9139 −766.767 512.000 −2176.51 −2499.78
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(1\)

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 38.8.a.d 2
3.b odd 2 1 342.8.a.g 2
4.b odd 2 1 304.8.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.8.a.d 2 1.a even 1 1 trivial
304.8.a.d 2 4.b odd 2 1
342.8.a.g 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 69 T_{3} - 234 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(38))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 8 T )^{2} \)
$3$ \( 1 + 69 T + 4140 T^{2} + 150903 T^{3} + 4782969 T^{4} \)
$5$ \( 1 - 155 T + 10178 T^{2} - 12109375 T^{3} + 6103515625 T^{4} \)
$7$ \( 1 + 2238 T + 2775179 T^{2} + 1843089234 T^{3} + 678223072849 T^{4} \)
$11$ \( 1 + 3295 T + 38080340 T^{2} + 64210228445 T^{3} + 379749833583241 T^{4} \)
$13$ \( 1 + 13427 T + 138664758 T^{2} + 842524337759 T^{3} + 3937376385699289 T^{4} \)
$17$ \( 1 + 32256 T + 1073810905 T^{2} + 13235884236288 T^{3} + 168377826559400929 T^{4} \)
$19$ \( ( 1 + 6859 T )^{2} \)
$23$ \( 1 + 82525 T + 5722043942 T^{2} + 280983220013675 T^{3} + 11592836324538749809 T^{4} \)
$29$ \( 1 + 12749 T + 32947137104 T^{2} + 219918673063441 T^{3} + \)\(29\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 - 258944 T + 64961539758 T^{2} - 7124226348358784 T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 + 149260 T + 183707435838 T^{2} + 14169531980871580 T^{3} + \)\(90\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 - 339130 T - 4364095006 T^{2} - 66047016901263530 T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 + 83869 T + 275614048806 T^{2} + 22797155094932983 T^{3} + \)\(73\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 1471025 T + 1488141383726 T^{2} - 745255275779084575 T^{3} + \)\(25\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 + 945643 T + 2428814565902 T^{2} + 1110857366408880191 T^{3} + \)\(13\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 + 969009 T + 3139777837024 T^{2} + 2411525686652974371 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 + 1506755 T + 5925806441724 T^{2} + 4735343481888821855 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 + 1848219 T + 11014380276458 T^{2} + 11201522342478469737 T^{3} + \)\(36\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 + 3417184 T + 7686276501674 T^{2} + 31079699083331190944 T^{3} + \)\(82\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 + 2499822 T + 17574764549507 T^{2} + 27616529860806100734 T^{3} + \)\(12\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 - 2636926 T + 33245149452510 T^{2} - 50639286907236307234 T^{3} + \)\(36\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 + 10059354 T + 77358130536910 T^{2} + \)\(27\!\cdots\!58\)\( T^{3} + \)\(73\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 + 3506160 T + 58485605027458 T^{2} + \)\(15\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 - 5893526 T + 73244272821042 T^{2} - \)\(47\!\cdots\!38\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \)
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