# Properties

 Label 1148.2.k.a Level $1148$ Weight $2$ Character orbit 1148.k Analytic conductor $9.167$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1148 = 2^{2} \cdot 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1148.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.16682615204$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.110166016.2 Defining polynomial: $$x^{8} + 10 x^{6} + 19 x^{4} + 10 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{5} ) q^{3} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{5} + \beta_{3} q^{7} + ( 1 + \beta_{1} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{5} ) q^{3} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{5} + \beta_{3} q^{7} + ( 1 + \beta_{1} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{9} + ( 1 - \beta_{1} + 3 \beta_{3} + \beta_{4} - 2 \beta_{7} ) q^{11} + ( \beta_{1} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{13} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{15} + ( -5 \beta_{2} - 2 \beta_{6} ) q^{17} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{4} - 3 \beta_{6} ) q^{19} -\beta_{1} q^{21} + ( 3 + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{23} + ( 2 + 2 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{25} + ( 3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{27} + ( -3 + \beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{29} + ( 4 - \beta_{4} - \beta_{5} + \beta_{6} ) q^{31} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{33} + ( -\beta_{1} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{35} + ( -4 - \beta_{2} - \beta_{3} ) q^{37} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{7} ) q^{39} + ( 1 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{41} + ( 2 + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{43} + ( 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{45} + ( 5 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{47} -\beta_{7} q^{49} + ( -1 - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{51} + ( 2 - 3 \beta_{1} + 4 \beta_{3} + 3 \beta_{4} - \beta_{5} - 6 \beta_{7} ) q^{53} + ( -2 - 6 \beta_{2} - 3 \beta_{6} - 2 \beta_{7} ) q^{55} + ( -4 + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{57} + ( -2 - 7 \beta_{2} - 7 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{59} + ( -2 - 4 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} - 11 \beta_{7} ) q^{61} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{63} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{6} ) q^{65} + ( -1 + 2 \beta_{2} - \beta_{6} - \beta_{7} ) q^{67} + ( -9 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - 5 \beta_{5} + 2 \beta_{7} ) q^{69} + ( -3 - 2 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} - 5 \beta_{7} ) q^{71} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{73} + ( -4 \beta_{1} - 5 \beta_{3} + 4 \beta_{4} + \beta_{5} - 3 \beta_{7} ) q^{75} + ( -1 - 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{77} + ( -2 + \beta_{1} + 6 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{79} + ( -2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{81} + ( -7 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{83} + ( -7 + 7 \beta_{1} + 2 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} + 7 \beta_{7} ) q^{85} + ( 3 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{87} + ( -1 - 6 \beta_{1} + 6 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} ) q^{89} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{91} + ( -2 - \beta_{3} - 3 \beta_{5} - \beta_{7} ) q^{93} + ( -7 + 6 \beta_{1} + 5 \beta_{3} - 6 \beta_{4} - 7 \beta_{5} + 6 \beta_{7} ) q^{95} + ( 8 + \beta_{1} + 2 \beta_{2} + \beta_{4} + 3 \beta_{6} + 9 \beta_{7} ) q^{97} + ( 1 + 4 \beta_{1} + 8 \beta_{2} + 4 \beta_{4} + 3 \beta_{6} + 5 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{3} + O(q^{10})$$ $$8q - 4q^{3} + 12q^{11} + 4q^{13} - 8q^{15} + 8q^{17} + 8q^{19} + 28q^{23} - 4q^{25} + 8q^{27} - 16q^{29} + 28q^{31} - 8q^{35} - 32q^{37} - 4q^{45} + 20q^{47} - 20q^{51} + 32q^{53} - 4q^{55} - 36q^{57} - 20q^{59} - 8q^{63} - 4q^{67} - 44q^{69} + 8q^{71} + 12q^{75} - 12q^{79} - 16q^{81} - 64q^{83} - 56q^{85} + 4q^{89} - 4q^{93} - 52q^{95} + 56q^{97} + 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 10 x^{6} + 19 x^{4} + 10 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + \nu^{5} + 10 \nu^{4} + 9 \nu^{3} + 18 \nu^{2} + 9 \nu + 5$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - \nu^{5} + 10 \nu^{4} - 9 \nu^{3} + 18 \nu^{2} - 9 \nu + 5$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 9 \nu^{4} + 11 \nu^{2} + 4$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} + 10 \nu^{5} + 9 \nu^{4} + 18 \nu^{3} + 9 \nu^{2} + 5 \nu - 4$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + 10 \nu^{5} - 9 \nu^{4} + 18 \nu^{3} - 9 \nu^{2} + 5 \nu$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$2 \nu^{7} + 19 \nu^{5} + 29 \nu^{3} + 9 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{5} + \beta_{4} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{3} + \beta_{2} - 4 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{6} + 7 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 20$$ $$\nu^{5}$$ $$=$$ $$-9 \beta_{7} + 18 \beta_{6} + 18 \beta_{5} + 7 \beta_{3} - 7 \beta_{2} + 27 \beta_{1} + 18$$ $$\nu^{6}$$ $$=$$ $$52 \beta_{6} - 52 \beta_{5} + 72 \beta_{4} - 18 \beta_{3} - 18 \beta_{2} - 151$$ $$\nu^{7}$$ $$=$$ $$72 \beta_{7} - 142 \beta_{6} - 142 \beta_{5} - 52 \beta_{3} + 52 \beta_{2} - 203 \beta_{1} - 142$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times$$.

 $$n$$ $$493$$ $$575$$ $$785$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 2.77462i − 1.22833i − 0.360409i 0.814115i − 2.77462i 1.22833i 0.360409i − 0.814115i
0 −1.96195 1.96195i 0 0.264927i 0 0.707107 + 0.707107i 0 4.69853i 0
337.2 0 −0.868559 0.868559i 0 4.37966i 0 −0.707107 0.707107i 0 1.49121i 0
337.3 0 0.254848 + 0.254848i 0 1.56350i 0 0.707107 + 0.707107i 0 2.87011i 0
337.4 0 0.575666 + 0.575666i 0 0.551233i 0 −0.707107 0.707107i 0 2.33722i 0
729.1 0 −1.96195 + 1.96195i 0 0.264927i 0 0.707107 0.707107i 0 4.69853i 0
729.2 0 −0.868559 + 0.868559i 0 4.37966i 0 −0.707107 + 0.707107i 0 1.49121i 0
729.3 0 0.254848 0.254848i 0 1.56350i 0 0.707107 0.707107i 0 2.87011i 0
729.4 0 0.575666 0.575666i 0 0.551233i 0 −0.707107 + 0.707107i 0 2.33722i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 729.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.c even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.k.a 8
41.c even 4 1 inner 1148.2.k.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.k.a 8 1.a even 1 1 trivial
1148.2.k.a 8 41.c even 4 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$1 - 4 T + 8 T^{2} - T^{4} + 8 T^{6} + 4 T^{7} + T^{8}$$
$5$ $$1 + 18 T^{2} + 55 T^{4} + 22 T^{6} + T^{8}$$
$7$ $$( 1 + T^{4} )^{2}$$
$11$ $$625 - 500 T + 200 T^{2} - 60 T^{3} + 194 T^{4} - 164 T^{5} + 72 T^{6} - 12 T^{7} + T^{8}$$
$13$ $$49 - 252 T + 648 T^{2} + 1036 T^{3} + 770 T^{4} + 148 T^{5} + 8 T^{6} - 4 T^{7} + T^{8}$$
$17$ $$6241 - 18328 T + 26912 T^{2} - 8520 T^{3} + 1314 T^{4} + 40 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$19$ $$5329 + 3504 T + 1152 T^{2} - 328 T^{3} + 215 T^{4} + 104 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$23$ $$( -313 + 86 T + 41 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$29$ $$54289 + 41940 T + 16200 T^{2} + 668 T^{3} + 755 T^{4} + 452 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$31$ $$( 73 - 126 T + 67 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$37$ $$( 14 + 8 T + T^{2} )^{4}$$
$41$ $$2825761 + 100860 T^{2} + 2214 T^{4} + 60 T^{6} + T^{8}$$
$43$ $$253009 + 58460 T^{2} + 4374 T^{4} + 124 T^{6} + T^{8}$$
$47$ $$16 - 1120 T + 39200 T^{2} + 6080 T^{3} + 492 T^{4} - 720 T^{5} + 200 T^{6} - 20 T^{7} + T^{8}$$
$53$ $$214369 - 74080 T + 12800 T^{2} + 3264 T^{3} + 13695 T^{4} - 3776 T^{5} + 512 T^{6} - 32 T^{7} + T^{8}$$
$59$ $$( 3791 - 490 T - 121 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$61$ $$178676689 + 6740884 T^{2} + 89526 T^{4} + 500 T^{6} + T^{8}$$
$67$ $$49 - 280 T + 800 T^{2} + 492 T^{3} + 155 T^{4} - 12 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$71$ $$49729 - 412104 T + 1707552 T^{2} - 367688 T^{3} + 39650 T^{4} - 264 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$73$ $$57121 + 20236 T^{2} + 1990 T^{4} + 76 T^{6} + T^{8}$$
$79$ $$583696 - 201696 T + 34848 T^{2} + 6144 T^{3} + 1836 T^{4} - 432 T^{5} + 72 T^{6} + 12 T^{7} + T^{8}$$
$83$ $$( -9623 - 544 T + 262 T^{2} + 32 T^{3} + T^{4} )^{2}$$
$89$ $$43256929 - 2025716 T + 47432 T^{2} + 103000 T^{3} + 48847 T^{4} + 1304 T^{5} + 8 T^{6} - 4 T^{7} + T^{8}$$
$97$ $$18241441 - 17066916 T + 7984008 T^{2} - 1906676 T^{3} + 279827 T^{4} - 26076 T^{5} + 1568 T^{6} - 56 T^{7} + T^{8}$$