# Properties

 Label 3549.2.a.bd Level $3549$ Weight $2$ Character orbit 3549.a Self dual yes Analytic conductor $28.339$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3549 = 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3549.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.3389076774$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2 x^{7} - 9 x^{6} + 14 x^{5} + 25 x^{4} - 24 x^{3} - 16 x^{2} + 8 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} -\beta_{3} q^{5} + \beta_{1} q^{6} + q^{7} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} -\beta_{3} q^{5} + \beta_{1} q^{6} + q^{7} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + q^{9} + ( -1 - \beta_{2} - \beta_{4} ) q^{10} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{11} + ( 1 + \beta_{1} + \beta_{2} ) q^{12} + \beta_{1} q^{14} -\beta_{3} q^{15} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{16} + ( 1 + \beta_{1} + \beta_{6} - \beta_{7} ) q^{17} + \beta_{1} q^{18} + ( 4 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{19} + ( 1 - 2 \beta_{1} - \beta_{5} ) q^{20} + q^{21} + ( -\beta_{3} + \beta_{4} ) q^{22} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} ) q^{23} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{24} + ( -\beta_{1} + \beta_{6} ) q^{25} + q^{27} + ( 1 + \beta_{1} + \beta_{2} ) q^{28} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{29} + ( -1 - \beta_{2} - \beta_{4} ) q^{30} + ( 1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{31} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{32} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{33} + ( 3 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{7} ) q^{34} -\beta_{3} q^{35} + ( 1 + \beta_{1} + \beta_{2} ) q^{36} + ( 5 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{37} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{38} + ( -4 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{40} + ( 1 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} ) q^{41} + \beta_{1} q^{42} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{43} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{44} -\beta_{3} q^{45} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{46} + ( -\beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{47} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{48} + q^{49} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{50} + ( 1 + \beta_{1} + \beta_{6} - \beta_{7} ) q^{51} + ( -\beta_{1} - \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{7} ) q^{53} + \beta_{1} q^{54} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{55} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{56} + ( 4 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{57} + ( 1 - \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{58} + ( -\beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{7} ) q^{59} + ( 1 - 2 \beta_{1} - \beta_{5} ) q^{60} + ( -\beta_{1} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{61} + ( 1 + 4 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{62} + q^{63} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{64} + ( -\beta_{3} + \beta_{4} ) q^{66} + ( 5 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{67} + ( 5 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{68} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} ) q^{69} + ( -1 - \beta_{2} - \beta_{4} ) q^{70} + ( 5 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{71} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{72} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{7} ) q^{73} + ( -2 + 5 \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{74} + ( -\beta_{1} + \beta_{6} ) q^{75} + ( 2 + 5 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{7} ) q^{76} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{77} + ( 4 - 6 \beta_{1} + 3 \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{79} + ( -7 - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{80} + q^{81} + ( -3 - \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{82} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{83} + ( 1 + \beta_{1} + \beta_{2} ) q^{84} + ( -\beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{85} + ( 6 + 3 \beta_{2} + \beta_{3} + 4 \beta_{4} + 2 \beta_{6} ) q^{86} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{87} + ( 2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{6} ) q^{88} + ( 1 - \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{89} + ( -1 - \beta_{2} - \beta_{4} ) q^{90} + ( -1 - 2 \beta_{1} - \beta_{2} - 5 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{92} + ( 1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{93} + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{94} + ( \beta_{2} - 4 \beta_{3} + \beta_{6} + \beta_{7} ) q^{95} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{96} + ( 5 + \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{97} + \beta_{1} q^{98} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{2} + 8q^{3} + 6q^{4} + 2q^{5} + 2q^{6} + 8q^{7} + 12q^{8} + 8q^{9} + O(q^{10})$$ $$8q + 2q^{2} + 8q^{3} + 6q^{4} + 2q^{5} + 2q^{6} + 8q^{7} + 12q^{8} + 8q^{9} - 4q^{10} + 8q^{11} + 6q^{12} + 2q^{14} + 2q^{15} + 10q^{16} + 10q^{17} + 2q^{18} + 18q^{19} + 2q^{20} + 8q^{21} + 2q^{22} - 2q^{23} + 12q^{24} - 6q^{25} + 8q^{27} + 6q^{28} - 12q^{29} - 4q^{30} + 16q^{31} + 26q^{32} + 8q^{33} + 24q^{34} + 2q^{35} + 6q^{36} + 24q^{37} + 16q^{38} - 30q^{40} - 4q^{41} + 2q^{42} - 10q^{43} - 20q^{44} + 2q^{45} + 12q^{46} + 10q^{47} + 10q^{48} + 8q^{49} - 16q^{50} + 10q^{51} + 6q^{53} + 2q^{54} - 10q^{55} + 12q^{56} + 18q^{57} + 16q^{58} + 6q^{59} + 2q^{60} + 6q^{61} + 16q^{62} + 8q^{63} - 8q^{64} + 2q^{66} + 24q^{67} + 20q^{68} - 2q^{69} - 4q^{70} + 42q^{71} + 12q^{72} + 32q^{73} - 18q^{74} - 6q^{75} + 28q^{76} + 8q^{77} - 2q^{79} - 40q^{80} + 8q^{81} - 18q^{82} - 2q^{83} + 6q^{84} + 4q^{85} + 26q^{86} - 12q^{87} - 2q^{88} + 12q^{89} - 4q^{90} - 10q^{92} + 16q^{93} - 16q^{94} - 4q^{95} + 26q^{96} + 64q^{97} + 2q^{98} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} - 9 x^{6} + 14 x^{5} + 25 x^{4} - 24 x^{3} - 16 x^{2} + 8 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 5 \nu^{2} + 2 \nu + 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - \nu^{4} - 6 \nu^{3} + 3 \nu^{2} + 6 \nu - 1$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} - 2 \nu^{5} - 7 \nu^{4} + 10 \nu^{3} + 14 \nu^{2} - 7 \nu - 4$$ $$\beta_{7}$$ $$=$$ $$\nu^{7} - 2 \nu^{6} - 9 \nu^{5} + 13 \nu^{4} + 25 \nu^{3} - 18 \nu^{2} - 13 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 6 \beta_{2} + 8 \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + \beta_{4} + 7 \beta_{3} + 9 \beta_{2} + 29 \beta_{1} + 19$$ $$\nu^{6}$$ $$=$$ $$\beta_{6} + 2 \beta_{5} + 9 \beta_{4} + 11 \beta_{3} + 36 \beta_{2} + 57 \beta_{1} + 85$$ $$\nu^{7}$$ $$=$$ $$\beta_{7} + 2 \beta_{6} + 13 \beta_{5} + 14 \beta_{4} + 47 \beta_{3} + 68 \beta_{2} + 177 \beta_{1} + 147$$

## 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
 −1.98765 −1.77930 −0.775848 −0.106359 0.485989 1.10207 2.45308 2.60802
−1.98765 1.00000 1.95074 2.85284 −1.98765 1.00000 0.0979034 1.00000 −5.67044
1.2 −1.77930 1.00000 1.16591 0.681820 −1.77930 1.00000 1.48409 1.00000 −1.21316
1.3 −0.775848 1.00000 −1.39806 −3.03444 −0.775848 1.00000 2.63638 1.00000 2.35426
1.4 −0.106359 1.00000 −1.98869 −1.41292 −0.106359 1.00000 0.424234 1.00000 0.150278
1.5 0.485989 1.00000 −1.76381 1.06536 0.485989 1.00000 −1.82917 1.00000 0.517753
1.6 1.10207 1.00000 −0.785435 3.28432 1.10207 1.00000 −3.06975 1.00000 3.61956
1.7 2.45308 1.00000 4.01758 0.0682999 2.45308 1.00000 4.94928 1.00000 0.167545
1.8 2.60802 1.00000 4.80176 −1.50528 2.60802 1.00000 7.30704 1.00000 −3.92579
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 3549.2.a.bd 8
13.b even 2 1 3549.2.a.bb 8
13.f odd 12 2 273.2.bd.a 16
39.k even 12 2 819.2.ct.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bd.a 16 13.f odd 12 2
819.2.ct.b 16 39.k even 12 2
3549.2.a.bb 8 13.b even 2 1
3549.2.a.bd 8 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3549))$$:

 $$T_{2}^{8} - \cdots$$ $$T_{5}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 8 T - 16 T^{2} - 24 T^{3} + 25 T^{4} + 14 T^{5} - 9 T^{6} - 2 T^{7} + T^{8}$$
$3$ $$( -1 + T )^{8}$$
$5$ $$-3 + 48 T - 56 T^{2} - 58 T^{3} + 59 T^{4} + 24 T^{5} - 15 T^{6} - 2 T^{7} + T^{8}$$
$7$ $$( -1 + T )^{8}$$
$11$ $$-12 + 144 T - 35 T^{2} - 400 T^{3} + 101 T^{4} + 138 T^{5} - 12 T^{6} - 8 T^{7} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$-53699 - 4370 T + 43050 T^{2} - 15808 T^{3} - 443 T^{4} + 798 T^{5} - 55 T^{6} - 10 T^{7} + T^{8}$$
$19$ $$159172 - 164096 T + 41957 T^{2} + 8114 T^{3} - 5253 T^{4} + 592 T^{5} + 67 T^{6} - 18 T^{7} + T^{8}$$
$23$ $$38032 + 40200 T - 27643 T^{2} - 2918 T^{3} + 2858 T^{4} - 2 T^{5} - 97 T^{6} + 2 T^{7} + T^{8}$$
$29$ $$-50387 + 51334 T + 78809 T^{2} + 21728 T^{3} - 1627 T^{4} - 1120 T^{5} - 58 T^{6} + 12 T^{7} + T^{8}$$
$31$ $$-2816 - 9632 T + 3573 T^{2} + 6884 T^{3} - 4532 T^{4} + 810 T^{5} + 17 T^{6} - 16 T^{7} + T^{8}$$
$37$ $$-127643 - 171632 T + 134702 T^{2} - 2404 T^{3} - 8869 T^{4} + 1004 T^{5} + 119 T^{6} - 24 T^{7} + T^{8}$$
$41$ $$832 - 4032 T - 7872 T^{2} - 848 T^{3} + 2060 T^{4} - 120 T^{5} - 83 T^{6} + 4 T^{7} + T^{8}$$
$43$ $$125556 - 104868 T - 79643 T^{2} + 21524 T^{3} + 5106 T^{4} - 918 T^{5} - 115 T^{6} + 10 T^{7} + T^{8}$$
$47$ $$1156 + 4080 T - 7439 T^{2} - 5014 T^{3} + 1050 T^{4} + 478 T^{5} - 61 T^{6} - 10 T^{7} + T^{8}$$
$53$ $$-76707 + 89910 T - 5832 T^{2} - 19170 T^{3} + 2754 T^{4} + 1026 T^{5} - 168 T^{6} - 6 T^{7} + T^{8}$$
$59$ $$-62348 + 63016 T - 3859 T^{2} - 11644 T^{3} + 2256 T^{4} + 520 T^{5} - 107 T^{6} - 6 T^{7} + T^{8}$$
$61$ $$-322283 + 394238 T - 27976 T^{2} - 56546 T^{3} + 9194 T^{4} + 1402 T^{5} - 232 T^{6} - 6 T^{7} + T^{8}$$
$67$ $$-76592 + 251776 T + 131593 T^{2} - 27600 T^{3} - 8962 T^{4} + 1708 T^{5} + 69 T^{6} - 24 T^{7} + T^{8}$$
$71$ $$91909428 - 36254088 T + 2884885 T^{2} + 616474 T^{3} - 121193 T^{4} + 4616 T^{5} + 436 T^{6} - 42 T^{7} + T^{8}$$
$73$ $$-4160451 - 1577508 T + 719170 T^{2} + 69698 T^{3} - 42295 T^{4} + 3330 T^{5} + 189 T^{6} - 32 T^{7} + T^{8}$$
$79$ $$75240852 + 5078148 T - 6074987 T^{2} + 70920 T^{3} + 99754 T^{4} - 974 T^{5} - 563 T^{6} + 2 T^{7} + T^{8}$$
$83$ $$1463172 - 707736 T - 452555 T^{2} + 71404 T^{3} + 21819 T^{4} - 1410 T^{5} - 340 T^{6} + 2 T^{7} + T^{8}$$
$89$ $$-269692704 + 102273384 T - 3217523 T^{2} - 1633816 T^{3} + 102907 T^{4} + 7900 T^{5} - 593 T^{6} - 12 T^{7} + T^{8}$$
$97$ $$-970544 + 3544216 T - 2628447 T^{2} + 480428 T^{3} + 24664 T^{4} - 14622 T^{5} + 1505 T^{6} - 64 T^{7} + T^{8}$$