# Properties

 Label 7938.2.a.bn Level $7938$ Weight $2$ Character orbit 7938.a Self dual yes Analytic conductor $63.385$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7938 = 2 \cdot 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7938.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$63.3852491245$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + ( -1 + \beta ) q^{5} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + ( -1 + \beta ) q^{5} + q^{8} + ( -1 + \beta ) q^{10} + 2 q^{11} -2 \beta q^{13} + q^{16} -2 q^{17} + ( -5 + \beta ) q^{19} + ( -1 + \beta ) q^{20} + 2 q^{22} - q^{23} + ( 2 - 2 \beta ) q^{25} -2 \beta q^{26} + ( -2 - 2 \beta ) q^{29} -6 q^{31} + q^{32} -2 q^{34} + ( 2 + 4 \beta ) q^{37} + ( -5 + \beta ) q^{38} + ( -1 + \beta ) q^{40} -4 \beta q^{41} + ( 2 + 2 \beta ) q^{43} + 2 q^{44} - q^{46} -4 \beta q^{47} + ( 2 - 2 \beta ) q^{50} -2 \beta q^{52} + ( -6 - 2 \beta ) q^{53} + ( -2 + 2 \beta ) q^{55} + ( -2 - 2 \beta ) q^{58} + 2 q^{59} + ( -9 + \beta ) q^{61} -6 q^{62} + q^{64} + ( -12 + 2 \beta ) q^{65} + ( -8 - 2 \beta ) q^{67} -2 q^{68} + ( 5 - 2 \beta ) q^{71} + ( 2 + 2 \beta ) q^{73} + ( 2 + 4 \beta ) q^{74} + ( -5 + \beta ) q^{76} + ( 3 - 2 \beta ) q^{79} + ( -1 + \beta ) q^{80} -4 \beta q^{82} -2 q^{83} + ( 2 - 2 \beta ) q^{85} + ( 2 + 2 \beta ) q^{86} + 2 q^{88} + ( 12 + 2 \beta ) q^{89} - q^{92} -4 \beta q^{94} + ( 11 - 6 \beta ) q^{95} + ( 2 - 2 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{4} - 2q^{5} + 2q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{4} - 2q^{5} + 2q^{8} - 2q^{10} + 4q^{11} + 2q^{16} - 4q^{17} - 10q^{19} - 2q^{20} + 4q^{22} - 2q^{23} + 4q^{25} - 4q^{29} - 12q^{31} + 2q^{32} - 4q^{34} + 4q^{37} - 10q^{38} - 2q^{40} + 4q^{43} + 4q^{44} - 2q^{46} + 4q^{50} - 12q^{53} - 4q^{55} - 4q^{58} + 4q^{59} - 18q^{61} - 12q^{62} + 2q^{64} - 24q^{65} - 16q^{67} - 4q^{68} + 10q^{71} + 4q^{73} + 4q^{74} - 10q^{76} + 6q^{79} - 2q^{80} - 4q^{83} + 4q^{85} + 4q^{86} + 4q^{88} + 24q^{89} - 2q^{92} + 22q^{95} + 4q^{97} + O(q^{100})$$

## 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
 −2.44949 2.44949
1.00000 0 1.00000 −3.44949 0 0 1.00000 0 −3.44949
1.2 1.00000 0 1.00000 1.44949 0 0 1.00000 0 1.44949
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## 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 7938.2.a.bn 2
3.b odd 2 1 7938.2.a.bm 2
7.b odd 2 1 1134.2.a.p 2
9.c even 3 2 882.2.f.j 4
9.d odd 6 2 2646.2.f.k 4
21.c even 2 1 1134.2.a.i 2
28.d even 2 1 9072.2.a.bk 2
63.g even 3 2 882.2.h.l 4
63.h even 3 2 882.2.e.n 4
63.i even 6 2 2646.2.e.l 4
63.j odd 6 2 2646.2.e.k 4
63.k odd 6 2 882.2.h.k 4
63.l odd 6 2 126.2.f.c 4
63.n odd 6 2 2646.2.h.n 4
63.o even 6 2 378.2.f.d 4
63.s even 6 2 2646.2.h.m 4
63.t odd 6 2 882.2.e.m 4
84.h odd 2 1 9072.2.a.bd 2
252.s odd 6 2 3024.2.r.e 4
252.bi even 6 2 1008.2.r.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 63.l odd 6 2
378.2.f.d 4 63.o even 6 2
882.2.e.m 4 63.t odd 6 2
882.2.e.n 4 63.h even 3 2
882.2.f.j 4 9.c even 3 2
882.2.h.k 4 63.k odd 6 2
882.2.h.l 4 63.g even 3 2
1008.2.r.e 4 252.bi even 6 2
1134.2.a.i 2 21.c even 2 1
1134.2.a.p 2 7.b odd 2 1
2646.2.e.k 4 63.j odd 6 2
2646.2.e.l 4 63.i even 6 2
2646.2.f.k 4 9.d odd 6 2
2646.2.h.m 4 63.s even 6 2
2646.2.h.n 4 63.n odd 6 2
3024.2.r.e 4 252.s odd 6 2
7938.2.a.bm 2 3.b odd 2 1
7938.2.a.bn 2 1.a even 1 1 trivial
9072.2.a.bd 2 84.h odd 2 1
9072.2.a.bk 2 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7938))$$:

 $$T_{5}^{2} + 2 T_{5} - 5$$ $$T_{11} - 2$$ $$T_{13}^{2} - 24$$ $$T_{17} + 2$$ $$T_{23} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-5 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$-24 + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$19 + 10 T + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$-20 + 4 T + T^{2}$$
$31$ $$( 6 + T )^{2}$$
$37$ $$-92 - 4 T + T^{2}$$
$41$ $$-96 + T^{2}$$
$43$ $$-20 - 4 T + T^{2}$$
$47$ $$-96 + T^{2}$$
$53$ $$12 + 12 T + T^{2}$$
$59$ $$( -2 + T )^{2}$$
$61$ $$75 + 18 T + T^{2}$$
$67$ $$40 + 16 T + T^{2}$$
$71$ $$1 - 10 T + T^{2}$$
$73$ $$-20 - 4 T + T^{2}$$
$79$ $$-15 - 6 T + T^{2}$$
$83$ $$( 2 + T )^{2}$$
$89$ $$120 - 24 T + T^{2}$$
$97$ $$-20 - 4 T + T^{2}$$