# Properties

 Label 7938.2.a.e Level $7938$ Weight $2$ Character orbit 7938.a Self dual yes Analytic conductor $63.385$ Analytic rank $0$ Dimension $1$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 2q^{5} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - 2q^{5} - q^{8} + 2q^{10} + q^{11} + 6q^{13} + q^{16} + 5q^{17} + 7q^{19} - 2q^{20} - q^{22} + 4q^{23} - q^{25} - 6q^{26} - 4q^{29} + 6q^{31} - q^{32} - 5q^{34} + 2q^{37} - 7q^{38} + 2q^{40} - 3q^{41} - q^{43} + q^{44} - 4q^{46} + q^{50} + 6q^{52} + 12q^{53} - 2q^{55} + 4q^{58} + 7q^{59} + 12q^{61} - 6q^{62} + q^{64} - 12q^{65} + 13q^{67} + 5q^{68} - 8q^{71} - q^{73} - 2q^{74} + 7q^{76} - 6q^{79} - 2q^{80} + 3q^{82} - 16q^{83} - 10q^{85} + q^{86} - q^{88} + 6q^{89} + 4q^{92} - 14q^{95} + 5q^{97} + O(q^{100})$$

## 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
 0
−1.00000 0 1.00000 −2.00000 0 0 −1.00000 0 2.00000
 $$n$$: 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.e 1
3.b odd 2 1 7938.2.a.bb 1
7.b odd 2 1 1134.2.a.c 1
9.c even 3 2 882.2.f.f 2
9.d odd 6 2 2646.2.f.b 2
21.c even 2 1 1134.2.a.f 1
28.d even 2 1 9072.2.a.t 1
63.g even 3 2 882.2.h.g 2
63.h even 3 2 882.2.e.e 2
63.i even 6 2 2646.2.e.i 2
63.j odd 6 2 2646.2.e.h 2
63.k odd 6 2 882.2.h.h 2
63.l odd 6 2 126.2.f.b 2
63.n odd 6 2 2646.2.h.c 2
63.o even 6 2 378.2.f.b 2
63.s even 6 2 2646.2.h.b 2
63.t odd 6 2 882.2.e.a 2
84.h odd 2 1 9072.2.a.f 1
252.s odd 6 2 3024.2.r.c 2
252.bi even 6 2 1008.2.r.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.b 2 63.l odd 6 2
378.2.f.b 2 63.o even 6 2
882.2.e.a 2 63.t odd 6 2
882.2.e.e 2 63.h even 3 2
882.2.f.f 2 9.c even 3 2
882.2.h.g 2 63.g even 3 2
882.2.h.h 2 63.k odd 6 2
1008.2.r.a 2 252.bi even 6 2
1134.2.a.c 1 7.b odd 2 1
1134.2.a.f 1 21.c even 2 1
2646.2.e.h 2 63.j odd 6 2
2646.2.e.i 2 63.i even 6 2
2646.2.f.b 2 9.d odd 6 2
2646.2.h.b 2 63.s even 6 2
2646.2.h.c 2 63.n odd 6 2
3024.2.r.c 2 252.s odd 6 2
7938.2.a.e 1 1.a even 1 1 trivial
7938.2.a.bb 1 3.b odd 2 1
9072.2.a.f 1 84.h odd 2 1
9072.2.a.t 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7938))$$:

 $$T_{5} + 2$$ $$T_{11} - 1$$ $$T_{13} - 6$$ $$T_{17} - 5$$ $$T_{23} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ 1
$5$ $$1 + 2 T + 5 T^{2}$$
$7$ 1
$11$ $$1 - T + 11 T^{2}$$
$13$ $$1 - 6 T + 13 T^{2}$$
$17$ $$1 - 5 T + 17 T^{2}$$
$19$ $$1 - 7 T + 19 T^{2}$$
$23$ $$1 - 4 T + 23 T^{2}$$
$29$ $$1 + 4 T + 29 T^{2}$$
$31$ $$1 - 6 T + 31 T^{2}$$
$37$ $$1 - 2 T + 37 T^{2}$$
$41$ $$1 + 3 T + 41 T^{2}$$
$43$ $$1 + T + 43 T^{2}$$
$47$ $$1 + 47 T^{2}$$
$53$ $$1 - 12 T + 53 T^{2}$$
$59$ $$1 - 7 T + 59 T^{2}$$
$61$ $$1 - 12 T + 61 T^{2}$$
$67$ $$1 - 13 T + 67 T^{2}$$
$71$ $$1 + 8 T + 71 T^{2}$$
$73$ $$1 + T + 73 T^{2}$$
$79$ $$1 + 6 T + 79 T^{2}$$
$83$ $$1 + 16 T + 83 T^{2}$$
$89$ $$1 - 6 T + 89 T^{2}$$
$97$ $$1 - 5 T + 97 T^{2}$$