# Properties

 Label 7938.2.a.bp 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 1134) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + q^{8} + \beta q^{11} + ( -2 - \beta ) q^{13} + q^{16} + ( -2 - \beta ) q^{19} + \beta q^{22} + ( 3 + \beta ) q^{23} -5 q^{25} + ( -2 - \beta ) q^{26} -\beta q^{29} + ( -5 + \beta ) q^{31} + q^{32} -4 q^{37} + ( -2 - \beta ) q^{38} + ( -3 + 2 \beta ) q^{41} + ( 2 - 2 \beta ) q^{43} + \beta q^{44} + ( 3 + \beta ) q^{46} + ( -9 - \beta ) q^{47} -5 q^{50} + ( -2 - \beta ) q^{52} + \beta q^{53} -\beta q^{58} + ( -2 - \beta ) q^{61} + ( -5 + \beta ) q^{62} + q^{64} + ( -4 - \beta ) q^{67} + ( -3 - \beta ) q^{71} + 7 q^{73} -4 q^{74} + ( -2 - \beta ) q^{76} + ( 5 + \beta ) q^{79} + ( -3 + 2 \beta ) q^{82} + ( -12 + \beta ) q^{83} + ( 2 - 2 \beta ) q^{86} + \beta q^{88} + ( -3 + 2 \beta ) q^{89} + ( 3 + \beta ) q^{92} + ( -9 - \beta ) q^{94} + ( 4 + 2 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{4} + 2q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{4} + 2q^{8} - 4q^{13} + 2q^{16} - 4q^{19} + 6q^{23} - 10q^{25} - 4q^{26} - 10q^{31} + 2q^{32} - 8q^{37} - 4q^{38} - 6q^{41} + 4q^{43} + 6q^{46} - 18q^{47} - 10q^{50} - 4q^{52} - 4q^{61} - 10q^{62} + 2q^{64} - 8q^{67} - 6q^{71} + 14q^{73} - 8q^{74} - 4q^{76} + 10q^{79} - 6q^{82} - 24q^{83} + 4q^{86} - 6q^{89} + 6q^{92} - 18q^{94} + 8q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
1.00000 0 1.00000 0 0 0 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 0 1.00000 0 0
 $$n$$: 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.bp 2
3.b odd 2 1 7938.2.a.bj 2
7.b odd 2 1 7938.2.a.bq 2
7.d odd 6 2 1134.2.g.i 4
21.c even 2 1 7938.2.a.bk 2
21.g even 6 2 1134.2.g.j yes 4
63.i even 6 2 1134.2.e.r 4
63.k odd 6 2 1134.2.h.r 4
63.s even 6 2 1134.2.h.s 4
63.t odd 6 2 1134.2.e.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.e.r 4 63.i even 6 2
1134.2.e.s 4 63.t odd 6 2
1134.2.g.i 4 7.d odd 6 2
1134.2.g.j yes 4 21.g even 6 2
1134.2.h.r 4 63.k odd 6 2
1134.2.h.s 4 63.s even 6 2
7938.2.a.bj 2 3.b odd 2 1
7938.2.a.bk 2 21.c even 2 1
7938.2.a.bp 2 1.a even 1 1 trivial
7938.2.a.bq 2 7.b odd 2 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{11}^{2} - 18$$ $$T_{13}^{2} + 4 T_{13} - 14$$ $$T_{17}$$ $$T_{23}^{2} - 6 T_{23} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$-18 + T^{2}$$
$13$ $$-14 + 4 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$-14 + 4 T + T^{2}$$
$23$ $$-9 - 6 T + T^{2}$$
$29$ $$-18 + T^{2}$$
$31$ $$7 + 10 T + T^{2}$$
$37$ $$( 4 + T )^{2}$$
$41$ $$-63 + 6 T + T^{2}$$
$43$ $$-68 - 4 T + T^{2}$$
$47$ $$63 + 18 T + T^{2}$$
$53$ $$-18 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$-14 + 4 T + T^{2}$$
$67$ $$-2 + 8 T + T^{2}$$
$71$ $$-9 + 6 T + T^{2}$$
$73$ $$( -7 + T )^{2}$$
$79$ $$7 - 10 T + T^{2}$$
$83$ $$126 + 24 T + T^{2}$$
$89$ $$-63 + 6 T + T^{2}$$
$97$ $$-56 - 8 T + T^{2}$$