Properties

Label 38.6.a.c
Level $38$
Weight $6$
Character orbit 38.a
Self dual yes
Analytic conductor $6.095$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.09458515289\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1441}) \)
Defining polynomial: \(x^{2} - x - 360\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{1441})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 q^{2} + ( 2 - \beta ) q^{3} + 16 q^{4} + ( -21 - 3 \beta ) q^{5} + ( -8 + 4 \beta ) q^{6} + ( 55 + 4 \beta ) q^{7} -64 q^{8} + ( 121 - 3 \beta ) q^{9} +O(q^{10})\) \( q -4 q^{2} + ( 2 - \beta ) q^{3} + 16 q^{4} + ( -21 - 3 \beta ) q^{5} + ( -8 + 4 \beta ) q^{6} + ( 55 + 4 \beta ) q^{7} -64 q^{8} + ( 121 - 3 \beta ) q^{9} + ( 84 + 12 \beta ) q^{10} + ( 331 - \beta ) q^{11} + ( 32 - 16 \beta ) q^{12} + ( 804 + 5 \beta ) q^{13} + ( -220 - 16 \beta ) q^{14} + ( 1038 + 18 \beta ) q^{15} + 256 q^{16} + ( 37 - 10 \beta ) q^{17} + ( -484 + 12 \beta ) q^{18} + 361 q^{19} + ( -336 - 48 \beta ) q^{20} + ( -1330 - 51 \beta ) q^{21} + ( -1324 + 4 \beta ) q^{22} + ( -1568 - 49 \beta ) q^{23} + ( -128 + 64 \beta ) q^{24} + ( 556 + 135 \beta ) q^{25} + ( -3216 - 20 \beta ) q^{26} + ( 836 + 119 \beta ) q^{27} + ( 880 + 64 \beta ) q^{28} + ( -1398 + 315 \beta ) q^{29} + ( -4152 - 72 \beta ) q^{30} + ( -432 - 316 \beta ) q^{31} -1024 q^{32} + ( 1022 - 332 \beta ) q^{33} + ( -148 + 40 \beta ) q^{34} + ( -5475 - 261 \beta ) q^{35} + ( 1936 - 48 \beta ) q^{36} + ( 5158 + 172 \beta ) q^{37} -1444 q^{38} + ( -192 - 799 \beta ) q^{39} + ( 1344 + 192 \beta ) q^{40} + ( 8014 + 602 \beta ) q^{41} + ( 5320 + 204 \beta ) q^{42} + ( 5511 + 281 \beta ) q^{43} + ( 5296 - 16 \beta ) q^{44} + ( 699 - 291 \beta ) q^{45} + ( 6272 + 196 \beta ) q^{46} + ( -6635 + 1115 \beta ) q^{47} + ( 512 - 256 \beta ) q^{48} + ( -8022 + 456 \beta ) q^{49} + ( -2224 - 540 \beta ) q^{50} + ( 3674 - 47 \beta ) q^{51} + ( 12864 + 80 \beta ) q^{52} + ( 9992 + 601 \beta ) q^{53} + ( -3344 - 476 \beta ) q^{54} + ( -5871 - 969 \beta ) q^{55} + ( -3520 - 256 \beta ) q^{56} + ( 722 - 361 \beta ) q^{57} + ( 5592 - 1260 \beta ) q^{58} + ( -39254 - 73 \beta ) q^{59} + ( 16608 + 288 \beta ) q^{60} + ( 22223 - 825 \beta ) q^{61} + ( 1728 + 1264 \beta ) q^{62} + ( 2335 + 307 \beta ) q^{63} + 4096 q^{64} + ( -22284 - 2532 \beta ) q^{65} + ( -4088 + 1328 \beta ) q^{66} + ( 2352 + 3101 \beta ) q^{67} + ( 592 - 160 \beta ) q^{68} + ( 14504 + 1519 \beta ) q^{69} + ( 21900 + 1044 \beta ) q^{70} + ( -30610 - 1268 \beta ) q^{71} + ( -7744 + 192 \beta ) q^{72} + ( 9601 - 2984 \beta ) q^{73} + ( -20632 - 688 \beta ) q^{74} + ( -47488 - 421 \beta ) q^{75} + 5776 q^{76} + ( 16765 + 1265 \beta ) q^{77} + ( 768 + 3196 \beta ) q^{78} + ( 33628 - 134 \beta ) q^{79} + ( -5376 - 768 \beta ) q^{80} + ( -70571 + 12 \beta ) q^{81} + ( -32056 - 2408 \beta ) q^{82} + ( -6580 + 2446 \beta ) q^{83} + ( -21280 - 816 \beta ) q^{84} + ( 10023 + 129 \beta ) q^{85} + ( -22044 - 1124 \beta ) q^{86} + ( -116196 + 1713 \beta ) q^{87} + ( -21184 + 64 \beta ) q^{88} + ( 66232 - 4276 \beta ) q^{89} + ( -2796 + 1164 \beta ) q^{90} + ( 51420 + 3511 \beta ) q^{91} + ( -25088 - 784 \beta ) q^{92} + ( 112896 + 116 \beta ) q^{93} + ( 26540 - 4460 \beta ) q^{94} + ( -7581 - 1083 \beta ) q^{95} + ( -2048 + 1024 \beta ) q^{96} + ( 87968 + 2622 \beta ) q^{97} + ( 32088 - 1824 \beta ) q^{98} + ( 41131 - 1111 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{2} + 3q^{3} + 32q^{4} - 45q^{5} - 12q^{6} + 114q^{7} - 128q^{8} + 239q^{9} + O(q^{10}) \) \( 2q - 8q^{2} + 3q^{3} + 32q^{4} - 45q^{5} - 12q^{6} + 114q^{7} - 128q^{8} + 239q^{9} + 180q^{10} + 661q^{11} + 48q^{12} + 1613q^{13} - 456q^{14} + 2094q^{15} + 512q^{16} + 64q^{17} - 956q^{18} + 722q^{19} - 720q^{20} - 2711q^{21} - 2644q^{22} - 3185q^{23} - 192q^{24} + 1247q^{25} - 6452q^{26} + 1791q^{27} + 1824q^{28} - 2481q^{29} - 8376q^{30} - 1180q^{31} - 2048q^{32} + 1712q^{33} - 256q^{34} - 11211q^{35} + 3824q^{36} + 10488q^{37} - 2888q^{38} - 1183q^{39} + 2880q^{40} + 16630q^{41} + 10844q^{42} + 11303q^{43} + 10576q^{44} + 1107q^{45} + 12740q^{46} - 12155q^{47} + 768q^{48} - 15588q^{49} - 4988q^{50} + 7301q^{51} + 25808q^{52} + 20585q^{53} - 7164q^{54} - 12711q^{55} - 7296q^{56} + 1083q^{57} + 9924q^{58} - 78581q^{59} + 33504q^{60} + 43621q^{61} + 4720q^{62} + 4977q^{63} + 8192q^{64} - 47100q^{65} - 6848q^{66} + 7805q^{67} + 1024q^{68} + 30527q^{69} + 44844q^{70} - 62488q^{71} - 15296q^{72} + 16218q^{73} - 41952q^{74} - 95397q^{75} + 11552q^{76} + 34795q^{77} + 4732q^{78} + 67122q^{79} - 11520q^{80} - 141130q^{81} - 66520q^{82} - 10714q^{83} - 43376q^{84} + 20175q^{85} - 45212q^{86} - 230679q^{87} - 42304q^{88} + 128188q^{89} - 4428q^{90} + 106351q^{91} - 50960q^{92} + 225908q^{93} + 48620q^{94} - 16245q^{95} - 3072q^{96} + 178558q^{97} + 62352q^{98} + 81151q^{99} + O(q^{100}) \)

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
19.4803
−18.4803
−4.00000 −17.4803 16.0000 −79.4408 69.9210 132.921 −64.0000 62.5592 317.763
1.2 −4.00000 20.4803 16.0000 34.4408 −81.9210 −18.9210 −64.0000 176.441 −137.763
\(n\): 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 38.6.a.c 2
3.b odd 2 1 342.6.a.i 2
4.b odd 2 1 304.6.a.f 2
5.b even 2 1 950.6.a.d 2
19.b odd 2 1 722.6.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.a.c 2 1.a even 1 1 trivial
304.6.a.f 2 4.b odd 2 1
342.6.a.i 2 3.b odd 2 1
722.6.a.c 2 19.b odd 2 1
950.6.a.d 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3 T_{3} - 358 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(38))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 4 T )^{2} \)
$3$ \( 1 - 3 T + 128 T^{2} - 729 T^{3} + 59049 T^{4} \)
$5$ \( 1 + 45 T + 3514 T^{2} + 140625 T^{3} + 9765625 T^{4} \)
$7$ \( 1 - 114 T + 31099 T^{2} - 1915998 T^{3} + 282475249 T^{4} \)
$11$ \( 1 - 661 T + 430972 T^{2} - 106454711 T^{3} + 25937424601 T^{4} \)
$13$ \( 1 - 1613 T + 1384022 T^{2} - 598895609 T^{3} + 137858491849 T^{4} \)
$17$ \( 1 - 64 T + 2804713 T^{2} - 90870848 T^{3} + 2015993900449 T^{4} \)
$19$ \( ( 1 - 361 T )^{2} \)
$23$ \( 1 + 3185 T + 14543782 T^{2} + 20499752455 T^{3} + 41426511213649 T^{4} \)
$29$ \( 1 + 2481 T + 6815332 T^{2} + 50888160669 T^{3} + 420707233300201 T^{4} \)
$31$ \( 1 + 1180 T + 21633278 T^{2} + 33782398180 T^{3} + 819628286980801 T^{4} \)
$37$ \( 1 - 10488 T + 155529814 T^{2} - 727279421016 T^{3} + 4808584372417849 T^{4} \)
$41$ \( 1 - 16630 T + 170295586 T^{2} - 1926688622630 T^{3} + 13422659310152401 T^{4} \)
$43$ \( 1 - 11303 T + 297510638 T^{2} - 1661636431229 T^{3} + 21611482313284249 T^{4} \)
$47$ \( 1 + 12155 T + 47754214 T^{2} + 2787688560085 T^{3} + 52599132235830049 T^{4} \)
$53$ \( 1 - 20585 T + 812203882 T^{2} - 8608554223405 T^{3} + 174887470365513049 T^{4} \)
$59$ \( 1 + 78581 T + 2971672216 T^{2} + 56179466339719 T^{3} + 511116753300641401 T^{4} \)
$61$ \( 1 - 43621 T + 1919695356 T^{2} - 36842135245921 T^{3} + 713342911662882601 T^{4} \)
$67$ \( 1 - 7805 T - 748756690 T^{2} - 10537726460135 T^{3} + 1822837804551761449 T^{4} \)
$71$ \( 1 + 62488 T + 4005427642 T^{2} + 112742683685288 T^{3} + 3255243551009881201 T^{4} \)
$73$ \( 1 - 16218 T + 1004140843 T^{2} - 33621075095274 T^{3} + 4297625829703557649 T^{4} \)
$79$ \( 1 - 67122 T + 7273984870 T^{2} - 206538179613678 T^{3} + 9468276082626847201 T^{4} \)
$83$ \( 1 + 10714 T + 5751433246 T^{2} + 42202881449102 T^{3} + 15516041187205853449 T^{4} \)
$89$ \( 1 - 128188 T + 8689285330 T^{2} - 715809412648412 T^{3} + 31181719929966183601 T^{4} \)
$97$ \( 1 - 178558 T + 22668743394 T^{2} - 1533338301609406 T^{3} + 73742412689492826049 T^{4} \)
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