# Properties

 Label 38.2.c.b Level 38 Weight 2 Character orbit 38.c Analytic conductor 0.303 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.303431527681$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## Character values

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

 $$n$$ $$21$$ $$\chi(n)$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 1.32288 − 2.29129i −1.32288 + 2.29129i 1.32288 + 2.29129i −1.32288 − 2.29129i
−0.500000 0.866025i −1.32288 2.29129i −0.500000 + 0.866025i 0.822876 + 1.42526i −1.32288 + 2.29129i 3.64575 1.00000 −2.00000 + 3.46410i 0.822876 1.42526i
7.2 −0.500000 0.866025i 1.32288 + 2.29129i −0.500000 + 0.866025i −1.82288 3.15731i 1.32288 2.29129i −1.64575 1.00000 −2.00000 + 3.46410i −1.82288 + 3.15731i
11.1 −0.500000 + 0.866025i −1.32288 + 2.29129i −0.500000 0.866025i 0.822876 1.42526i −1.32288 2.29129i 3.64575 1.00000 −2.00000 3.46410i 0.822876 + 1.42526i
11.2 −0.500000 + 0.866025i 1.32288 2.29129i −0.500000 0.866025i −1.82288 + 3.15731i 1.32288 + 2.29129i −1.64575 1.00000 −2.00000 3.46410i −1.82288 3.15731i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.2.c.b 4
3.b odd 2 1 342.2.g.f 4
4.b odd 2 1 304.2.i.e 4
5.b even 2 1 950.2.e.k 4
5.c odd 4 2 950.2.j.g 8
8.b even 2 1 1216.2.i.l 4
8.d odd 2 1 1216.2.i.k 4
12.b even 2 1 2736.2.s.v 4
19.b odd 2 1 722.2.c.j 4
19.c even 3 1 inner 38.2.c.b 4
19.c even 3 1 722.2.a.j 2
19.d odd 6 1 722.2.a.g 2
19.d odd 6 1 722.2.c.j 4
19.e even 9 6 722.2.e.n 12
19.f odd 18 6 722.2.e.o 12
57.f even 6 1 6498.2.a.bg 2
57.h odd 6 1 342.2.g.f 4
57.h odd 6 1 6498.2.a.ba 2
76.f even 6 1 5776.2.a.z 2
76.g odd 6 1 304.2.i.e 4
76.g odd 6 1 5776.2.a.ba 2
95.i even 6 1 950.2.e.k 4
95.m odd 12 2 950.2.j.g 8
152.k odd 6 1 1216.2.i.k 4
152.p even 6 1 1216.2.i.l 4
228.m even 6 1 2736.2.s.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.b 4 1.a even 1 1 trivial
38.2.c.b 4 19.c even 3 1 inner
304.2.i.e 4 4.b odd 2 1
304.2.i.e 4 76.g odd 6 1
342.2.g.f 4 3.b odd 2 1
342.2.g.f 4 57.h odd 6 1
722.2.a.g 2 19.d odd 6 1
722.2.a.j 2 19.c even 3 1
722.2.c.j 4 19.b odd 2 1
722.2.c.j 4 19.d odd 6 1
722.2.e.n 12 19.e even 9 6
722.2.e.o 12 19.f odd 18 6
950.2.e.k 4 5.b even 2 1
950.2.e.k 4 95.i even 6 1
950.2.j.g 8 5.c odd 4 2
950.2.j.g 8 95.m odd 12 2
1216.2.i.k 4 8.d odd 2 1
1216.2.i.k 4 152.k odd 6 1
1216.2.i.l 4 8.b even 2 1
1216.2.i.l 4 152.p even 6 1
2736.2.s.v 4 12.b even 2 1
2736.2.s.v 4 228.m even 6 1
5776.2.a.z 2 76.f even 6 1
5776.2.a.ba 2 76.g odd 6 1
6498.2.a.ba 2 57.h odd 6 1
6498.2.a.bg 2 57.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 7 T_{3}^{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$1 + T^{2} - 8 T^{4} + 9 T^{6} + 81 T^{8}$$
$5$ $$1 + 2 T - 12 T^{3} - 29 T^{4} - 60 T^{5} + 250 T^{7} + 625 T^{8}$$
$7$ $$( 1 - 2 T + 8 T^{2} - 14 T^{3} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 4 T + 19 T^{2} + 44 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 5 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2}$$
$17$ $$( 1 - 17 T^{2} + 289 T^{4} )^{2}$$
$19$ $$1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4}$$
$23$ $$1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 276 T^{5} - 19044 T^{6} + 24334 T^{7} + 279841 T^{8}$$
$29$ $$1 - 2 T - 48 T^{2} + 12 T^{3} + 1747 T^{4} + 348 T^{5} - 40368 T^{6} - 48778 T^{7} + 707281 T^{8}$$
$31$ $$( 1 + 6 T + 64 T^{2} + 186 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 6 T + 76 T^{2} - 222 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$1 + 10 T + 21 T^{2} - 30 T^{3} + 460 T^{4} - 1230 T^{5} + 35301 T^{6} + 689210 T^{7} + 2825761 T^{8}$$
$43$ $$1 + 12 T + 50 T^{2} + 96 T^{3} + 795 T^{4} + 4128 T^{5} + 92450 T^{6} + 954084 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 14 T + 60 T^{2} + 588 T^{3} + 7075 T^{4} + 27636 T^{5} + 132540 T^{6} + 1453522 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 4 T + 18 T^{2} + 432 T^{3} - 3653 T^{4} + 22896 T^{5} + 50562 T^{6} - 595508 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 55 T^{2} - 456 T^{4} - 191455 T^{6} + 12117361 T^{8}$$
$61$ $$1 - 14 T + 88 T^{2} + 196 T^{3} - 4013 T^{4} + 11956 T^{5} + 327448 T^{6} - 3177734 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 4 T - 115 T^{2} + 12 T^{3} + 11600 T^{4} + 804 T^{5} - 516235 T^{6} - 1203052 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 16 T + 78 T^{2} - 576 T^{3} + 8467 T^{4} - 40896 T^{5} + 393198 T^{6} - 5726576 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 14 T + 29 T^{2} + 294 T^{3} + 8252 T^{4} + 21462 T^{5} + 154541 T^{6} + 5446238 T^{7} + 28398241 T^{8}$$
$79$ $$( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 103 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 89 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$1 - 18 T + 77 T^{2} - 954 T^{3} + 20172 T^{4} - 92538 T^{5} + 724493 T^{6} - 16428114 T^{7} + 88529281 T^{8}$$