# Properties

 Label 1148.2.a.c Level $1148$ Weight $2$ Character orbit 1148.a Self dual yes Analytic conductor $9.167$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1148 = 2^{2} \cdot 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1148.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.16682615204$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.470117.1 Defining polynomial: $$x^{5} - 2 x^{4} - 6 x^{3} + 8 x^{2} + 7 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} - q^{7} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} - q^{7} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + ( -1 + \beta_{1} + \beta_{4} ) q^{13} + ( -2 - \beta_{3} + \beta_{4} ) q^{15} + ( 2 \beta_{2} + \beta_{3} ) q^{17} + ( -2 \beta_{3} - \beta_{4} ) q^{19} + \beta_{1} q^{21} + ( -2 - \beta_{1} + 2 \beta_{4} ) q^{23} + ( -1 + \beta_{3} - \beta_{4} ) q^{25} + ( -2 + \beta_{1} - \beta_{2} - \beta_{4} ) q^{27} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{29} + ( -1 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{31} + ( 1 + \beta_{1} + \beta_{2} ) q^{33} + ( 1 - \beta_{1} + \beta_{2} ) q^{35} + ( -4 - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{37} + ( -4 - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{39} + q^{41} + ( -6 + 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} ) q^{43} + ( -\beta_{3} - \beta_{4} ) q^{45} + ( -2 - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{47} + q^{49} + ( -\beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{51} + ( -5 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{53} + ( -3 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{55} + ( -3 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{57} + ( -1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{59} + ( 1 - \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{61} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{63} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{65} + ( -5 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} ) q^{67} + ( 1 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{69} + ( -5 + \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{71} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{73} + ( 3 + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{75} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{77} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{79} + ( -1 - 2 \beta_{1} - \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{81} + ( 3 - 3 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{83} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{85} + ( -7 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{87} + ( 1 + 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{89} + ( 1 - \beta_{1} - \beta_{4} ) q^{91} + ( 4 + 3 \beta_{3} - 3 \beta_{4} ) q^{93} + ( 2 + 2 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{95} + ( 2 - \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{97} + ( -7 + \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 2q^{3} - q^{5} - 5q^{7} + q^{9} + O(q^{10})$$ $$5q - 2q^{3} - q^{5} - 5q^{7} + q^{9} - 2q^{11} - q^{13} - 9q^{15} - 3q^{17} - 4q^{19} + 2q^{21} - 8q^{23} - 6q^{25} - 8q^{27} - 9q^{29} - 11q^{31} + 5q^{33} + q^{35} - 11q^{37} - 17q^{39} + 5q^{41} - 27q^{43} - 3q^{45} - 3q^{47} + 5q^{49} - 3q^{51} - 19q^{53} - 13q^{55} - 11q^{57} - 15q^{59} - q^{63} + 7q^{65} - 21q^{67} + 14q^{69} - 16q^{71} - 10q^{73} + 12q^{75} + 2q^{77} - 14q^{79} - 7q^{81} - 2q^{83} - 21q^{85} - 36q^{87} + 6q^{89} + q^{91} + 17q^{93} + 9q^{95} + 20q^{97} - 17q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 6 x^{3} + 8 x^{2} + 7 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 8 \nu + 2$$ $$\beta_{3}$$ $$=$$ $$-\nu^{4} + 2 \nu^{3} + 6 \nu^{2} - 9 \nu - 5$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + 3 \nu^{3} + 5 \nu^{2} - 13 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{2} + 5 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{4} + 5 \beta_{3} + 8 \beta_{2} + 7 \beta_{1} + 17$$

## 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
 2.78088 1.94177 −0.189142 −0.475832 −2.05768
0 −2.78088 0 −0.592821 0 −1.00000 0 4.73330 0
1.2 0 −1.94177 0 2.68629 0 −1.00000 0 0.770473 0
1.3 0 0.189142 0 −1.51194 0 −1.00000 0 −2.96423 0
1.4 0 0.475832 0 1.19617 0 −1.00000 0 −2.77358 0
1.5 0 2.05768 0 −2.77770 0 −1.00000 0 1.23404 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$
$$41$$ $$-1$$

## 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 1148.2.a.c 5
4.b odd 2 1 4592.2.a.be 5
7.b odd 2 1 8036.2.a.l 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.a.c 5 1.a even 1 1 trivial
4592.2.a.be 5 4.b odd 2 1
8036.2.a.l 5 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{5} + 2 T_{3}^{4} - 6 T_{3}^{3} - 8 T_{3}^{2} + 7 T_{3} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1148))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$-1 + 7 T - 8 T^{2} - 6 T^{3} + 2 T^{4} + T^{5}$$
$5$ $$8 + 12 T - 8 T^{2} - 9 T^{3} + T^{4} + T^{5}$$
$7$ $$( 1 + T )^{5}$$
$11$ $$-24 - 88 T - 98 T^{2} - 29 T^{3} + 2 T^{4} + T^{5}$$
$13$ $$103 + 42 T - 48 T^{2} - 23 T^{3} + T^{4} + T^{5}$$
$17$ $$463 + 180 T - 78 T^{2} - 29 T^{3} + 3 T^{4} + T^{5}$$
$19$ $$699 + 23 T - 256 T^{2} - 56 T^{3} + 4 T^{4} + T^{5}$$
$23$ $$71 + 273 T - 162 T^{2} - 32 T^{3} + 8 T^{4} + T^{5}$$
$29$ $$-2360 - 3012 T - 1020 T^{2} - 77 T^{3} + 9 T^{4} + T^{5}$$
$31$ $$6080 - 608 T - 708 T^{2} - 43 T^{3} + 11 T^{4} + T^{5}$$
$37$ $$-6092 - 4294 T - 965 T^{2} - 40 T^{3} + 11 T^{4} + T^{5}$$
$41$ $$( -1 + T )^{5}$$
$43$ $$5351 - 5596 T - 658 T^{2} + 167 T^{3} + 27 T^{4} + T^{5}$$
$47$ $$-732 + 778 T - 103 T^{2} - 68 T^{3} + 3 T^{4} + T^{5}$$
$53$ $$-11976 - 8008 T - 1254 T^{2} + 33 T^{3} + 19 T^{4} + T^{5}$$
$59$ $$-2440 - 2148 T - 490 T^{2} + 21 T^{3} + 15 T^{4} + T^{5}$$
$61$ $$-1704 + 2792 T - 30 T^{2} - 115 T^{3} + T^{5}$$
$67$ $$22184 - 8668 T - 2454 T^{2} - 25 T^{3} + 21 T^{4} + T^{5}$$
$71$ $$456 - 76 T - 232 T^{2} - 17 T^{3} + 16 T^{4} + T^{5}$$
$73$ $$320 + 3968 T - 428 T^{2} - 143 T^{3} + 10 T^{4} + T^{5}$$
$79$ $$19616 + 1632 T - 1248 T^{2} - 80 T^{3} + 14 T^{4} + T^{5}$$
$83$ $$50360 + 18148 T - 326 T^{2} - 301 T^{3} + 2 T^{4} + T^{5}$$
$89$ $$-19931 + 3287 T + 852 T^{2} - 140 T^{3} - 6 T^{4} + T^{5}$$
$97$ $$-365983 + 20555 T + 5524 T^{2} - 278 T^{3} - 20 T^{4} + T^{5}$$
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