# Properties

 Label 38.3.d.a Level $38$ Weight $3$ Character orbit 38.d Analytic conductor $1.035$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 38.d (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.03542500457$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 - \beta_{1} + \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{5} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{6} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 - \beta_{1} + \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{5} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{6} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{9} + ( \beta_{1} - \beta_{3} ) q^{10} + ( -10 + 4 \beta_{1} - 2 \beta_{3} ) q^{11} + ( -2 + 4 \beta_{2} - 2 \beta_{3} ) q^{12} + ( -10 + 6 \beta_{1} + 5 \beta_{2} - 6 \beta_{3} ) q^{13} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{14} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{15} + ( -4 + 4 \beta_{2} ) q^{16} + ( 7 + 2 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} ) q^{17} + ( 4 - 8 \beta_{2} - 4 \beta_{3} ) q^{18} + ( 3 - 6 \beta_{1} + 2 \beta_{2} + 15 \beta_{3} ) q^{19} + 2 q^{20} + ( 6 - 8 \beta_{1} + 6 \beta_{2} ) q^{21} + ( 4 - 10 \beta_{1} + 4 \beta_{2} ) q^{22} + ( 9 \beta_{1} - 5 \beta_{2} + 9 \beta_{3} ) q^{23} + ( 4 - 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{24} + 24 \beta_{2} q^{25} + ( 12 - 10 \beta_{1} + 5 \beta_{3} ) q^{26} + ( 9 - 18 \beta_{2} + 7 \beta_{3} ) q^{27} + ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{28} + ( 22 + 18 \beta_{1} - 11 \beta_{2} - 18 \beta_{3} ) q^{29} + ( -2 + 2 \beta_{1} - \beta_{3} ) q^{30} + ( -18 + 36 \beta_{2} ) q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} + ( -14 + 16 \beta_{1} - 14 \beta_{2} ) q^{33} + ( 8 + 7 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} ) q^{34} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{35} + ( 8 + 4 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{36} + ( -2 + 4 \beta_{2} - 18 \beta_{3} ) q^{37} + ( -30 + 3 \beta_{1} + 18 \beta_{2} + 2 \beta_{3} ) q^{38} + ( -27 + 22 \beta_{1} - 11 \beta_{3} ) q^{39} + 2 \beta_{1} q^{40} + ( 3 - 42 \beta_{1} + 3 \beta_{2} ) q^{41} + ( 6 \beta_{1} - 16 \beta_{2} + 6 \beta_{3} ) q^{42} + ( 19 - 23 \beta_{1} - 19 \beta_{2} + 46 \beta_{3} ) q^{43} + ( 4 \beta_{1} - 20 \beta_{2} + 4 \beta_{3} ) q^{44} + ( -4 - 4 \beta_{1} + 2 \beta_{3} ) q^{45} + ( -18 + 36 \beta_{2} - 5 \beta_{3} ) q^{46} + ( -19 \beta_{1} - 35 \beta_{2} - 19 \beta_{3} ) q^{47} + ( -8 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{48} + ( -21 - 16 \beta_{1} + 8 \beta_{3} ) q^{49} + 24 \beta_{3} q^{50} + ( 6 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{51} + ( -10 + 12 \beta_{1} - 10 \beta_{2} ) q^{52} + ( -14 + 48 \beta_{1} + 7 \beta_{2} - 48 \beta_{3} ) q^{53} + ( -14 + 9 \beta_{1} + 14 \beta_{2} - 18 \beta_{3} ) q^{54} + ( -10 + 2 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} ) q^{55} + ( 8 - 16 \beta_{2} + 4 \beta_{3} ) q^{56} + ( 31 - 24 \beta_{1} - 11 \beta_{2} + 22 \beta_{3} ) q^{57} + ( 36 + 22 \beta_{1} - 11 \beta_{3} ) q^{58} + ( 7 - 33 \beta_{1} + 7 \beta_{2} ) q^{59} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{60} + ( 16 \beta_{1} + 37 \beta_{2} + 16 \beta_{3} ) q^{61} + ( -18 \beta_{1} + 36 \beta_{3} ) q^{62} + ( 4 \beta_{1} + 16 \beta_{2} + 4 \beta_{3} ) q^{63} -8 q^{64} + ( -5 + 10 \beta_{2} - 6 \beta_{3} ) q^{65} + ( -14 \beta_{1} + 32 \beta_{2} - 14 \beta_{3} ) q^{66} + ( 34 + 63 \beta_{1} - 17 \beta_{2} - 63 \beta_{3} ) q^{67} + ( 14 + 8 \beta_{1} - 4 \beta_{3} ) q^{68} + ( 23 - 46 \beta_{2} + 32 \beta_{3} ) q^{69} + ( -8 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{70} + ( 51 - 9 \beta_{1} + 51 \beta_{2} ) q^{71} + ( 16 + 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{72} + ( 49 - 32 \beta_{1} - 49 \beta_{2} + 64 \beta_{3} ) q^{73} + ( 36 - 2 \beta_{1} - 36 \beta_{2} + 4 \beta_{3} ) q^{74} + ( -24 + 48 \beta_{2} - 24 \beta_{3} ) q^{75} + ( -4 - 30 \beta_{1} + 10 \beta_{2} + 18 \beta_{3} ) q^{76} + ( -44 + 48 \beta_{1} - 24 \beta_{3} ) q^{77} + ( 22 - 27 \beta_{1} + 22 \beta_{2} ) q^{78} + ( -21 - 21 \beta_{1} - 21 \beta_{2} ) q^{79} + 4 \beta_{2} q^{80} + ( 5 - 34 \beta_{1} - 5 \beta_{2} + 68 \beta_{3} ) q^{81} + ( 3 \beta_{1} - 84 \beta_{2} + 3 \beta_{3} ) q^{82} + ( -16 - 12 \beta_{1} + 6 \beta_{3} ) q^{83} + ( -12 + 24 \beta_{2} - 16 \beta_{3} ) q^{84} + ( -2 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{85} + ( -92 + 19 \beta_{1} + 46 \beta_{2} - 19 \beta_{3} ) q^{86} + ( -3 + 14 \beta_{1} - 7 \beta_{3} ) q^{87} + ( -8 + 16 \beta_{2} - 20 \beta_{3} ) q^{88} + ( -2 - 6 \beta_{1} + \beta_{2} + 6 \beta_{3} ) q^{89} + ( -4 - 4 \beta_{1} - 4 \beta_{2} ) q^{90} + ( -68 + 42 \beta_{1} + 34 \beta_{2} - 42 \beta_{3} ) q^{91} + ( 10 - 18 \beta_{1} - 10 \beta_{2} + 36 \beta_{3} ) q^{92} + ( -54 + 18 \beta_{1} + 54 \beta_{2} - 36 \beta_{3} ) q^{93} + ( 38 - 76 \beta_{2} - 35 \beta_{3} ) q^{94} + ( 5 + 9 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{95} + ( 8 - 8 \beta_{1} + 4 \beta_{3} ) q^{96} + ( -81 - 6 \beta_{1} - 81 \beta_{2} ) q^{97} + ( -16 - 21 \beta_{1} - 16 \beta_{2} ) q^{98} + ( 12 \beta_{1} + 16 \beta_{2} + 12 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{9} + O(q^{10})$$ $$4 q + 6 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{9} - 40 q^{11} - 30 q^{13} - 24 q^{14} + 6 q^{15} - 8 q^{16} + 14 q^{17} + 16 q^{19} + 8 q^{20} + 36 q^{21} + 24 q^{22} - 10 q^{23} + 8 q^{24} + 48 q^{25} + 48 q^{26} + 8 q^{28} + 66 q^{29} - 8 q^{30} - 84 q^{33} + 24 q^{34} + 4 q^{35} + 16 q^{36} - 84 q^{38} - 108 q^{39} + 18 q^{41} - 32 q^{42} + 38 q^{43} - 40 q^{44} - 16 q^{45} - 70 q^{47} - 24 q^{48} - 84 q^{49} + 18 q^{51} - 60 q^{52} - 42 q^{53} - 28 q^{54} - 20 q^{55} + 102 q^{57} + 144 q^{58} + 42 q^{59} + 12 q^{60} + 74 q^{61} + 32 q^{63} - 32 q^{64} + 64 q^{66} + 102 q^{67} + 56 q^{68} - 24 q^{70} + 306 q^{71} + 48 q^{72} + 98 q^{73} + 72 q^{74} + 4 q^{76} - 176 q^{77} + 132 q^{78} - 126 q^{79} + 8 q^{80} + 10 q^{81} - 168 q^{82} - 64 q^{83} - 14 q^{85} - 276 q^{86} - 12 q^{87} - 6 q^{89} - 24 q^{90} - 204 q^{91} + 20 q^{92} - 108 q^{93} + 14 q^{95} + 32 q^{96} - 486 q^{97} - 96 q^{98} + 32 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$\chi(n)$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
27.1
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
−1.22474 0.707107i 2.72474 + 1.57313i 1.00000 + 1.73205i 0.500000 0.866025i −2.22474 3.85337i 6.89898 2.82843i 0.449490 + 0.778539i −1.22474 + 0.707107i
27.2 1.22474 + 0.707107i 0.275255 + 0.158919i 1.00000 + 1.73205i 0.500000 0.866025i 0.224745 + 0.389270i −2.89898 2.82843i −4.44949 7.70674i 1.22474 0.707107i
31.1 −1.22474 + 0.707107i 2.72474 1.57313i 1.00000 1.73205i 0.500000 + 0.866025i −2.22474 + 3.85337i 6.89898 2.82843i 0.449490 0.778539i −1.22474 0.707107i
31.2 1.22474 0.707107i 0.275255 0.158919i 1.00000 1.73205i 0.500000 + 0.866025i 0.224745 0.389270i −2.89898 2.82843i −4.44949 + 7.70674i 1.22474 + 0.707107i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.3.d.a 4
3.b odd 2 1 342.3.m.a 4
4.b odd 2 1 304.3.r.a 4
19.c even 3 1 722.3.b.b 4
19.d odd 6 1 inner 38.3.d.a 4
19.d odd 6 1 722.3.b.b 4
57.f even 6 1 342.3.m.a 4
76.f even 6 1 304.3.r.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.d.a 4 1.a even 1 1 trivial
38.3.d.a 4 19.d odd 6 1 inner
304.3.r.a 4 4.b odd 2 1
304.3.r.a 4 76.f even 6 1
342.3.m.a 4 3.b odd 2 1
342.3.m.a 4 57.f even 6 1
722.3.b.b 4 19.c even 3 1
722.3.b.b 4 19.d odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T^{2} + T^{4}$$
$3$ $$1 - 6 T + 13 T^{2} - 6 T^{3} + T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$( -20 - 4 T + T^{2} )^{2}$$
$11$ $$( 76 + 20 T + T^{2} )^{2}$$
$13$ $$9 + 90 T + 303 T^{2} + 30 T^{3} + T^{4}$$
$17$ $$625 - 350 T + 171 T^{2} - 14 T^{3} + T^{4}$$
$19$ $$130321 - 5776 T + 570 T^{2} - 16 T^{3} + T^{4}$$
$23$ $$212521 - 4610 T + 561 T^{2} + 10 T^{3} + T^{4}$$
$29$ $$81225 + 18810 T + 1167 T^{2} - 66 T^{3} + T^{4}$$
$31$ $$( 972 + T^{2} )^{2}$$
$37$ $$404496 + 1320 T^{2} + T^{4}$$
$41$ $$12257001 + 63018 T - 3393 T^{2} - 18 T^{3} + T^{4}$$
$43$ $$7912969 + 106894 T + 4257 T^{2} - 38 T^{3} + T^{4}$$
$47$ $$885481 - 65870 T + 5841 T^{2} + 70 T^{3} + T^{4}$$
$53$ $$19900521 - 187362 T - 3873 T^{2} + 42 T^{3} + T^{4}$$
$59$ $$4124961 + 85302 T - 1443 T^{2} - 42 T^{3} + T^{4}$$
$61$ $$27889 + 12358 T + 5643 T^{2} - 74 T^{3} + T^{4}$$
$67$ $$49999041 + 721242 T - 3603 T^{2} - 102 T^{3} + T^{4}$$
$71$ $$58384881 - 2338146 T + 38853 T^{2} - 306 T^{3} + T^{4}$$
$73$ $$14010049 + 366814 T + 13347 T^{2} - 98 T^{3} + T^{4}$$
$79$ $$194481 + 55566 T + 5733 T^{2} + 126 T^{3} + T^{4}$$
$83$ $$( 40 + 32 T + T^{2} )^{2}$$
$89$ $$4761 - 414 T - 57 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$384591321 + 9530946 T + 98343 T^{2} + 486 T^{3} + T^{4}$$