# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 38.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 2 \beta q^{3} -2 q^{4} - q^{5} -4 q^{6} + 5 q^{7} -2 \beta q^{8} + q^{9} +O(q^{10})$$ $$q + \beta q^{2} + 2 \beta q^{3} -2 q^{4} - q^{5} -4 q^{6} + 5 q^{7} -2 \beta q^{8} + q^{9} -\beta q^{10} + 5 q^{11} -4 \beta q^{12} -12 \beta q^{13} + 5 \beta q^{14} -2 \beta q^{15} + 4 q^{16} -25 q^{17} + \beta q^{18} + 19 q^{19} + 2 q^{20} + 10 \beta q^{21} + 5 \beta q^{22} -10 q^{23} + 8 q^{24} -24 q^{25} + 24 q^{26} + 20 \beta q^{27} -10 q^{28} + 30 \beta q^{29} + 4 q^{30} -30 \beta q^{31} + 4 \beta q^{32} + 10 \beta q^{33} -25 \beta q^{34} -5 q^{35} -2 q^{36} -18 \beta q^{37} + 19 \beta q^{38} + 48 q^{39} + 2 \beta q^{40} -30 \beta q^{41} -20 q^{42} + 5 q^{43} -10 q^{44} - q^{45} -10 \beta q^{46} + 5 q^{47} + 8 \beta q^{48} -24 q^{49} -24 \beta q^{50} -50 \beta q^{51} + 24 \beta q^{52} -18 \beta q^{53} -40 q^{54} -5 q^{55} -10 \beta q^{56} + 38 \beta q^{57} -60 q^{58} + 60 \beta q^{59} + 4 \beta q^{60} + 95 q^{61} + 60 q^{62} + 5 q^{63} -8 q^{64} + 12 \beta q^{65} -20 q^{66} + 78 \beta q^{67} + 50 q^{68} -20 \beta q^{69} -5 \beta q^{70} -2 \beta q^{72} -25 q^{73} + 36 q^{74} -48 \beta q^{75} -38 q^{76} + 25 q^{77} + 48 \beta q^{78} + 30 \beta q^{79} -4 q^{80} -71 q^{81} + 60 q^{82} -130 q^{83} -20 \beta q^{84} + 25 q^{85} + 5 \beta q^{86} -120 q^{87} -10 \beta q^{88} -90 \beta q^{89} -\beta q^{90} -60 \beta q^{91} + 20 q^{92} + 120 q^{93} + 5 \beta q^{94} -19 q^{95} -16 q^{96} + 12 \beta q^{97} -24 \beta q^{98} + 5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} - 2q^{5} - 8q^{6} + 10q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 4q^{4} - 2q^{5} - 8q^{6} + 10q^{7} + 2q^{9} + 10q^{11} + 8q^{16} - 50q^{17} + 38q^{19} + 4q^{20} - 20q^{23} + 16q^{24} - 48q^{25} + 48q^{26} - 20q^{28} + 8q^{30} - 10q^{35} - 4q^{36} + 96q^{39} - 40q^{42} + 10q^{43} - 20q^{44} - 2q^{45} + 10q^{47} - 48q^{49} - 80q^{54} - 10q^{55} - 120q^{58} + 190q^{61} + 120q^{62} + 10q^{63} - 16q^{64} - 40q^{66} + 100q^{68} - 50q^{73} + 72q^{74} - 76q^{76} + 50q^{77} - 8q^{80} - 142q^{81} + 120q^{82} - 260q^{83} + 50q^{85} - 240q^{87} + 40q^{92} + 240q^{93} - 38q^{95} - 32q^{96} + 10q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$21$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 − 1.41421i 1.41421i
1.41421i 2.82843i −2.00000 −1.00000 −4.00000 5.00000 2.82843i 1.00000 1.41421i
37.2 1.41421i 2.82843i −2.00000 −1.00000 −4.00000 5.00000 2.82843i 1.00000 1.41421i
 $$n$$: 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.3.b.a 2
3.b odd 2 1 342.3.d.a 2
4.b odd 2 1 304.3.e.c 2
5.b even 2 1 950.3.c.a 2
5.c odd 4 2 950.3.d.a 4
8.b even 2 1 1216.3.e.j 2
8.d odd 2 1 1216.3.e.i 2
12.b even 2 1 2736.3.o.h 2
19.b odd 2 1 inner 38.3.b.a 2
57.d even 2 1 342.3.d.a 2
76.d even 2 1 304.3.e.c 2
95.d odd 2 1 950.3.c.a 2
95.g even 4 2 950.3.d.a 4
152.b even 2 1 1216.3.e.i 2
152.g odd 2 1 1216.3.e.j 2
228.b odd 2 1 2736.3.o.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.b.a 2 1.a even 1 1 trivial
38.3.b.a 2 19.b odd 2 1 inner
304.3.e.c 2 4.b odd 2 1
304.3.e.c 2 76.d even 2 1
342.3.d.a 2 3.b odd 2 1
342.3.d.a 2 57.d even 2 1
950.3.c.a 2 5.b even 2 1
950.3.c.a 2 95.d odd 2 1
950.3.d.a 4 5.c odd 4 2
950.3.d.a 4 95.g even 4 2
1216.3.e.i 2 8.d odd 2 1
1216.3.e.i 2 152.b even 2 1
1216.3.e.j 2 8.b even 2 1
1216.3.e.j 2 152.g odd 2 1
2736.3.o.h 2 12.b even 2 1
2736.3.o.h 2 228.b odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2}$$
$3$ $$1 - 10 T^{2} + 81 T^{4}$$
$5$ $$( 1 + T + 25 T^{2} )^{2}$$
$7$ $$( 1 - 5 T + 49 T^{2} )^{2}$$
$11$ $$( 1 - 5 T + 121 T^{2} )^{2}$$
$13$ $$1 - 50 T^{2} + 28561 T^{4}$$
$17$ $$( 1 + 25 T + 289 T^{2} )^{2}$$
$19$ $$( 1 - 19 T )^{2}$$
$23$ $$( 1 + 10 T + 529 T^{2} )^{2}$$
$29$ $$1 + 118 T^{2} + 707281 T^{4}$$
$31$ $$1 - 122 T^{2} + 923521 T^{4}$$
$37$ $$1 - 2090 T^{2} + 1874161 T^{4}$$
$41$ $$1 - 1562 T^{2} + 2825761 T^{4}$$
$43$ $$( 1 - 5 T + 1849 T^{2} )^{2}$$
$47$ $$( 1 - 5 T + 2209 T^{2} )^{2}$$
$53$ $$1 - 4970 T^{2} + 7890481 T^{4}$$
$59$ $$( 1 - 82 T + 3481 T^{2} )( 1 + 82 T + 3481 T^{2} )$$
$61$ $$( 1 - 95 T + 3721 T^{2} )^{2}$$
$67$ $$1 + 3190 T^{2} + 20151121 T^{4}$$
$71$ $$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$
$73$ $$( 1 + 25 T + 5329 T^{2} )^{2}$$
$79$ $$1 - 10682 T^{2} + 38950081 T^{4}$$
$83$ $$( 1 + 130 T + 6889 T^{2} )^{2}$$
$89$ $$1 + 358 T^{2} + 62742241 T^{4}$$
$97$ $$1 - 18530 T^{2} + 88529281 T^{4}$$