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 $1$

Related objects

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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

$p$ $F_p(T)$
$2$ $$( 4 + T )^{2}$$
$3$ $$-358 - 3 T + T^{2}$$
$5$ $$-2736 + 45 T + T^{2}$$
$7$ $$-2515 - 114 T + T^{2}$$
$11$ $$108870 - 661 T + T^{2}$$
$13$ $$641436 - 1613 T + T^{2}$$
$17$ $$-35001 - 64 T + T^{2}$$
$19$ $$( -361 + T )^{2}$$
$23$ $$1671096 + 3185 T + T^{2}$$
$29$ $$-34206966 + 2481 T + T^{2}$$
$31$ $$-35625024 + 1180 T + T^{2}$$
$37$ $$16841900 - 10488 T + T^{2}$$
$41$ $$-61416816 - 16630 T + T^{2}$$
$43$ $$3493752 - 11303 T + T^{2}$$
$47$ $$-410935800 + 12155 T + T^{2}$$
$53$ $$-24187104 - 20585 T + T^{2}$$
$59$ $$1541823618 + 78581 T + T^{2}$$
$61$ $$230502754 - 43621 T + T^{2}$$
$67$ $$-3449006904 - 7805 T + T^{2}$$
$71$ $$396968940 + 62488 T + T^{2}$$
$73$ $$-3142002343 - 16218 T + T^{2}$$
$79$ $$1119872072 - 67122 T + T^{2}$$
$83$ $$-2126648040 + 10714 T + T^{2}$$
$89$ $$-2478833568 - 128188 T + T^{2}$$
$97$ $$5494062880 - 178558 T + T^{2}$$