# Properties

 Label 1148.2.n.b Level $1148$ Weight $2$ Character orbit 1148.n 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.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.16682615204$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.64000000.2 Defining polynomial: $$x^{8} + 2 x^{6} + 4 x^{4} + 8 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{5} ) q^{3} + ( \beta_{2} + \beta_{7} ) q^{5} + \beta_{4} q^{7} + 2 \beta_{5} q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{5} ) q^{3} + ( \beta_{2} + \beta_{7} ) q^{5} + \beta_{4} q^{7} + 2 \beta_{5} q^{9} + ( 3 \beta_{2} + 3 \beta_{4} ) q^{11} + ( 2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{13} + ( 3 \beta_{2} + 2 \beta_{7} ) q^{15} + ( \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{19} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{21} + ( -1 - \beta_{2} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{23} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{25} + ( 1 - \beta_{5} ) q^{27} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{29} + ( 1 + \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{31} + ( -3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{33} + ( \beta_{1} + \beta_{6} ) q^{35} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{7} ) q^{37} + ( 6 + 2 \beta_{1} + 6 \beta_{2} + 4 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} ) q^{39} + ( -2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{41} + ( -4 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} ) q^{43} + ( 4 \beta_{2} + 2 \beta_{7} ) q^{45} + ( 3 + \beta_{1} + 3 \beta_{2} + \beta_{5} - 5 \beta_{6} + \beta_{7} ) q^{47} + ( -1 - \beta_{2} - \beta_{4} - \beta_{6} ) q^{49} + ( 4 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{51} + ( -2 + 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{53} + ( -3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{55} + ( 4 \beta_{1} + 6 \beta_{2} + 3 \beta_{4} + 6 \beta_{6} + 4 \beta_{7} ) q^{57} + ( 1 + \beta_{1} + \beta_{2} + 3 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{59} + ( \beta_{1} + 3 \beta_{2} + 6 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{61} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{63} + ( -3 - 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} ) q^{65} + ( -4 - 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{67} + ( -5 + 3 \beta_{1} - 5 \beta_{2} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{69} + ( -4 + 2 \beta_{1} - 6 \beta_{2} - 6 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} ) q^{71} + ( -4 - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{73} + 2 \beta_{4} q^{75} + ( -3 - 3 \beta_{2} - 3 \beta_{4} ) q^{77} + ( 1 + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{79} + ( -1 - 6 \beta_{5} ) q^{81} + ( 3 - 4 \beta_{4} + \beta_{5} - 4 \beta_{6} ) q^{83} + ( -4 + 4 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} + 4 \beta_{7} ) q^{85} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{7} ) q^{87} + ( -6 \beta_{1} - \beta_{2} + \beta_{3} - 5 \beta_{4} + \beta_{5} - \beta_{6} - 6 \beta_{7} ) q^{89} + ( 1 - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{91} + ( 11 + 4 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} + 11 \beta_{6} ) q^{93} + ( -6 - 4 \beta_{1} - 6 \beta_{2} - 4 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{95} + ( -4 - 2 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 3 \beta_{7} ) q^{97} + ( -6 \beta_{1} - 6 \beta_{3} - 6 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{3} - 2q^{5} - 2q^{7} + O(q^{10})$$ $$8q + 8q^{3} - 2q^{5} - 2q^{7} - 12q^{11} + 10q^{13} - 6q^{15} + 12q^{17} - 2q^{19} - 2q^{21} - 12q^{23} + 4q^{25} + 8q^{27} - 2q^{29} + 14q^{31} - 12q^{33} - 2q^{35} + 14q^{37} + 30q^{39} - 18q^{41} - 14q^{43} - 8q^{45} + 28q^{47} - 2q^{49} + 28q^{51} + 4q^{53} - 12q^{55} - 30q^{57} + 4q^{59} - 4q^{61} - 30q^{65} - 28q^{67} - 36q^{69} - 44q^{73} - 4q^{75} - 12q^{77} + 16q^{79} - 8q^{81} + 40q^{83} - 12q^{85} + 22q^{87} + 14q^{89} + 42q^{93} - 30q^{95} - 38q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 2 x^{6} + 4 x^{4} + 8 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4}$$$$/4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5}$$$$/4$$ $$\beta_{6}$$ $$=$$ $$\nu^{6}$$$$/8$$ $$\beta_{7}$$ $$=$$ $$\nu^{7}$$$$/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{6}$$ $$\nu^{7}$$ $$=$$ $$8 \beta_{7}$$

## Character values

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

 $$n$$ $$493$$ $$575$$ $$785$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1
 −0.437016 + 1.34500i 0.437016 − 1.34500i −0.437016 − 1.34500i 0.437016 + 1.34500i 1.14412 − 0.831254i −1.14412 + 0.831254i 1.14412 + 0.831254i −1.14412 − 0.831254i
0 −0.414214 0 0.335106 + 0.243469i 0 0.309017 + 0.951057i 0 −2.82843 0
57.2 0 2.41421 0 −1.95314 1.41904i 0 0.309017 + 0.951057i 0 2.82843 0
141.1 0 −0.414214 0 0.335106 0.243469i 0 0.309017 0.951057i 0 −2.82843 0
141.2 0 2.41421 0 −1.95314 + 1.41904i 0 0.309017 0.951057i 0 2.82843 0
365.1 0 −0.414214 0 −0.127999 + 0.393941i 0 −0.809017 0.587785i 0 −2.82843 0
365.2 0 2.41421 0 0.746033 2.29605i 0 −0.809017 0.587785i 0 2.82843 0
953.1 0 −0.414214 0 −0.127999 0.393941i 0 −0.809017 + 0.587785i 0 −2.82843 0
953.2 0 2.41421 0 0.746033 + 2.29605i 0 −0.809017 + 0.587785i 0 2.82843 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 953.2 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.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.n.b 8
41.d even 5 1 inner 1148.2.n.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.n.b 8 1.a even 1 1 trivial
1148.2.n.b 8 41.d even 5 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( -1 - 2 T + T^{2} )^{4}$$
$5$ $$1 - 2 T + 5 T^{2} - 12 T^{3} + 29 T^{4} + 12 T^{5} + 5 T^{6} + 2 T^{7} + T^{8}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$11$ $$( 81 + 81 T + 36 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$13$ $$625 + 1250 T + 1625 T^{2} + 1000 T^{3} + 325 T^{4} - 200 T^{5} + 65 T^{6} - 10 T^{7} + T^{8}$$
$17$ $$57121 + 18642 T + 1375 T^{2} - 1428 T^{3} + 1534 T^{4} - 462 T^{5} + 100 T^{6} - 12 T^{7} + T^{8}$$
$19$ $$1615441 + 602454 T + 143889 T^{2} + 14728 T^{3} + 1589 T^{4} - 104 T^{5} + 41 T^{6} + 2 T^{7} + T^{8}$$
$23$ $$1 + 38 T + 545 T^{2} - 452 T^{3} + 1454 T^{4} + 22 T^{5} + 60 T^{6} + 12 T^{7} + T^{8}$$
$29$ $$6241 + 5688 T + 9340 T^{2} + 1778 T^{3} + 254 T^{4} - 28 T^{5} + 5 T^{6} + 2 T^{7} + T^{8}$$
$31$ $$194481 + 148176 T + 238140 T^{2} + 21714 T^{3} + 214 T^{4} - 116 T^{5} + 115 T^{6} - 14 T^{7} + T^{8}$$
$37$ $$1681 + 1066 T + 2975 T^{2} + 624 T^{3} + 1309 T^{4} + 464 T^{5} + 55 T^{6} - 14 T^{7} + T^{8}$$
$41$ $$2825761 + 1240578 T + 189953 T^{2} + 11316 T^{3} + 385 T^{4} + 276 T^{5} + 113 T^{6} + 18 T^{7} + T^{8}$$
$43$ $$8288641 + 650654 T + 176075 T^{2} - 6044 T^{3} + 2629 T^{4} + 676 T^{5} + 155 T^{6} + 14 T^{7} + T^{8}$$
$47$ $$201601 + 101474 T + 251479 T^{2} - 54932 T^{3} + 11134 T^{4} - 2334 T^{5} + 356 T^{6} - 28 T^{7} + T^{8}$$
$53$ $$1597696 - 970752 T + 295296 T^{2} - 55616 T^{3} + 14144 T^{4} - 2048 T^{5} + 164 T^{6} - 4 T^{7} + T^{8}$$
$59$ $$160801 + 34486 T + 26535 T^{2} - 636 T^{3} + 334 T^{4} + 54 T^{5} + 20 T^{6} - 4 T^{7} + T^{8}$$
$61$ $$125238481 + 25090222 T + 5642791 T^{2} + 411236 T^{3} + 21334 T^{4} - 402 T^{5} + 44 T^{6} + 4 T^{7} + T^{8}$$
$67$ $$24760576 + 11225856 T + 2664384 T^{2} + 399872 T^{3} + 54224 T^{4} + 5504 T^{5} + 476 T^{6} + 28 T^{7} + T^{8}$$
$71$ $$430336 - 367360 T + 129728 T^{2} - 5120 T^{3} + 2624 T^{4} + 400 T^{5} + 72 T^{6} + T^{8}$$
$73$ $$( 41 + 238 T + 129 T^{2} + 22 T^{3} + T^{4} )^{2}$$
$79$ $$( -919 + 472 T - 42 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$83$ $$( -151 - 60 T + 106 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$89$ $$25816561 + 18108684 T + 4358572 T^{2} - 660878 T^{3} + 99830 T^{4} - 6692 T^{5} + 407 T^{6} - 14 T^{7} + T^{8}$$
$97$ $$8814961 + 219706 T + 654029 T^{2} + 117492 T^{3} + 21109 T^{4} + 3804 T^{5} + 621 T^{6} + 38 T^{7} + T^{8}$$