# Properties

 Label 7938.2.a.bz Level $7938$ Weight $2$ Character orbit 7938.a Self dual yes Analytic conductor $63.385$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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## 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: $$3$$ Coefficient field: 3.3.321.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 1$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \beta_{2} q^{5} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + \beta_{2} q^{5} + q^{8} + \beta_{2} q^{10} + \beta_{2} q^{11} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{13} + q^{16} + ( 2 - 2 \beta_{1} ) q^{17} + ( -2 + 3 \beta_{1} ) q^{19} + \beta_{2} q^{20} + \beta_{2} q^{22} + ( -2 - \beta_{1} ) q^{23} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{25} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{26} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 6 + \beta_{1} - \beta_{2} ) q^{31} + q^{32} + ( 2 - 2 \beta_{1} ) q^{34} - q^{37} + ( -2 + 3 \beta_{1} ) q^{38} + \beta_{2} q^{40} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{41} + ( 2 + 3 \beta_{1} + 3 \beta_{2} ) q^{43} + \beta_{2} q^{44} + ( -2 - \beta_{1} ) q^{46} + ( 3 + 3 \beta_{1} + 3 \beta_{2} ) q^{47} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{50} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{52} + ( 5 + \beta_{1} + \beta_{2} ) q^{53} + ( 4 - \beta_{1} - 2 \beta_{2} ) q^{55} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{58} + ( 5 + \beta_{1} + 2 \beta_{2} ) q^{59} + ( 3 + \beta_{1} + 2 \beta_{2} ) q^{61} + ( 6 + \beta_{1} - \beta_{2} ) q^{62} + q^{64} + ( -2 - \beta_{1} + 5 \beta_{2} ) q^{65} + ( -2 + \beta_{1} - 4 \beta_{2} ) q^{67} + ( 2 - 2 \beta_{1} ) q^{68} + ( -4 + 7 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 8 - \beta_{1} + 4 \beta_{2} ) q^{73} - q^{74} + ( -2 + 3 \beta_{1} ) q^{76} + ( -4 \beta_{1} + \beta_{2} ) q^{79} + \beta_{2} q^{80} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{82} + ( -1 - 2 \beta_{1} - 3 \beta_{2} ) q^{83} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 2 + 3 \beta_{1} + 3 \beta_{2} ) q^{86} + \beta_{2} q^{88} + ( 2 + 4 \beta_{1} + \beta_{2} ) q^{89} + ( -2 - \beta_{1} ) q^{92} + ( 3 + 3 \beta_{1} + 3 \beta_{2} ) q^{94} + ( -3 + 3 \beta_{1} - 2 \beta_{2} ) q^{95} + ( 8 + 2 \beta_{1} - 2 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + 3q^{4} - q^{5} + 3q^{8} + O(q^{10})$$ $$3q + 3q^{2} + 3q^{4} - q^{5} + 3q^{8} - q^{10} - q^{11} + 8q^{13} + 3q^{16} + 4q^{17} - 3q^{19} - q^{20} - q^{22} - 7q^{23} - 2q^{25} + 8q^{26} - 5q^{29} + 20q^{31} + 3q^{32} + 4q^{34} - 3q^{37} - 3q^{38} - q^{40} + 6q^{43} - q^{44} - 7q^{46} + 9q^{47} - 2q^{50} + 8q^{52} + 15q^{53} + 13q^{55} - 5q^{58} + 14q^{59} + 8q^{61} + 20q^{62} + 3q^{64} - 12q^{65} - q^{67} + 4q^{68} - 7q^{71} + 19q^{73} - 3q^{74} - 3q^{76} - 5q^{79} - q^{80} - 2q^{83} + 2q^{85} + 6q^{86} - q^{88} + 9q^{89} - 7q^{92} + 9q^{94} - 4q^{95} + 28q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$

## 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
 0.239123 2.46050 −1.69963
1.00000 0 1.00000 −3.18194 0 0 1.00000 0 −3.18194
1.2 1.00000 0 1.00000 0.593579 0 0 1.00000 0 0.593579
1.3 1.00000 0 1.00000 1.58836 0 0 1.00000 0 1.58836
 $$n$$: 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.bz 3
3.b odd 2 1 7938.2.a.bw 3
7.b odd 2 1 7938.2.a.ca 3
7.d odd 6 2 1134.2.g.l 6
9.c even 3 2 2646.2.f.m 6
9.d odd 6 2 882.2.f.o 6
21.c even 2 1 7938.2.a.bv 3
21.g even 6 2 1134.2.g.m 6
63.g even 3 2 2646.2.h.o 6
63.h even 3 2 2646.2.e.p 6
63.i even 6 2 126.2.e.c 6
63.j odd 6 2 882.2.e.o 6
63.k odd 6 2 378.2.h.c 6
63.l odd 6 2 2646.2.f.l 6
63.n odd 6 2 882.2.h.p 6
63.o even 6 2 882.2.f.n 6
63.s even 6 2 126.2.h.d yes 6
63.t odd 6 2 378.2.e.d 6
252.n even 6 2 3024.2.t.h 6
252.r odd 6 2 1008.2.q.g 6
252.bj even 6 2 3024.2.q.g 6
252.bn odd 6 2 1008.2.t.h 6

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

 $$T_{5}^{3} + T_{5}^{2} - 6 T_{5} + 3$$ $$T_{11}^{3} + T_{11}^{2} - 6 T_{11} + 3$$ $$T_{13}^{3} - 8 T_{13}^{2} + T_{13} + 69$$ $$T_{17}^{3} - 4 T_{17}^{2} - 12 T_{17} + 24$$ $$T_{23}^{3} + 7 T_{23}^{2} + 12 T_{23} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{3}$$
$3$ 1
$5$ $$1 + T + 9 T^{2} + 13 T^{3} + 45 T^{4} + 25 T^{5} + 125 T^{6}$$
$7$ 1
$11$ $$1 + T + 27 T^{2} + 25 T^{3} + 297 T^{4} + 121 T^{5} + 1331 T^{6}$$
$13$ $$1 - 8 T + 40 T^{2} - 139 T^{3} + 520 T^{4} - 1352 T^{5} + 2197 T^{6}$$
$17$ $$1 - 4 T + 39 T^{2} - 112 T^{3} + 663 T^{4} - 1156 T^{5} + 4913 T^{6}$$
$19$ $$1 + 3 T + 21 T^{2} + 65 T^{3} + 399 T^{4} + 1083 T^{5} + 6859 T^{6}$$
$23$ $$1 + 7 T + 81 T^{2} + 325 T^{3} + 1863 T^{4} + 3703 T^{5} + 12167 T^{6}$$
$29$ $$1 + 5 T + 21 T^{2} - 73 T^{3} + 609 T^{4} + 4205 T^{5} + 24389 T^{6}$$
$31$ $$1 - 20 T + 214 T^{2} - 1441 T^{3} + 6634 T^{4} - 19220 T^{5} + 29791 T^{6}$$
$37$ $$( 1 + T + 37 T^{2} )^{3}$$
$41$ $$1 + 90 T^{2} + 9 T^{3} + 3690 T^{4} + 68921 T^{6}$$
$43$ $$1 - 6 T + 60 T^{2} - 389 T^{3} + 2580 T^{4} - 11094 T^{5} + 79507 T^{6}$$
$47$ $$1 - 9 T + 87 T^{2} - 657 T^{3} + 4089 T^{4} - 19881 T^{5} + 103823 T^{6}$$
$53$ $$1 - 15 T + 225 T^{2} - 1671 T^{3} + 11925 T^{4} - 42135 T^{5} + 148877 T^{6}$$
$59$ $$1 - 14 T + 216 T^{2} - 1589 T^{3} + 12744 T^{4} - 48734 T^{5} + 205379 T^{6}$$
$61$ $$1 - 8 T + 178 T^{2} - 883 T^{3} + 10858 T^{4} - 29768 T^{5} + 226981 T^{6}$$
$67$ $$1 + T + 89 T^{2} - 77 T^{3} + 5963 T^{4} + 4489 T^{5} + 300763 T^{6}$$
$71$ $$1 + 7 T + 15 T^{2} - 599 T^{3} + 1065 T^{4} + 35287 T^{5} + 357911 T^{6}$$
$73$ $$1 - 19 T + 227 T^{2} - 2143 T^{3} + 16571 T^{4} - 101251 T^{5} + 389017 T^{6}$$
$79$ $$1 + 5 T + 163 T^{2} + 469 T^{3} + 12877 T^{4} + 31205 T^{5} + 493039 T^{6}$$
$83$ $$1 + 2 T + 186 T^{2} + 185 T^{3} + 15438 T^{4} + 13778 T^{5} + 571787 T^{6}$$
$89$ $$1 - 9 T + 225 T^{2} - 1611 T^{3} + 20025 T^{4} - 71289 T^{5} + 704969 T^{6}$$
$97$ $$1 - 28 T + 503 T^{2} - 5680 T^{3} + 48791 T^{4} - 263452 T^{5} + 912673 T^{6}$$
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