# Properties

 Label 5577.2.a.x Level $5577$ Weight $2$ Character orbit 5577.a Self dual yes Analytic conductor $44.533$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5577 = 3 \cdot 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5577.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.5325692073$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3 x^{6} - 7 x^{5} + 21 x^{4} + 13 x^{3} - 33 x^{2} - 7 x + 7$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 429) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} + ( -1 + \beta_{6} ) q^{5} -\beta_{1} q^{6} + ( -1 - \beta_{4} ) q^{7} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} + ( -1 + \beta_{6} ) q^{5} -\beta_{1} q^{6} + ( -1 - \beta_{4} ) q^{7} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{8} + q^{9} + ( \beta_{1} - \beta_{5} - \beta_{6} ) q^{10} - q^{11} + ( 1 + \beta_{1} + \beta_{2} ) q^{12} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{14} + ( -1 + \beta_{6} ) q^{15} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( -1 + \beta_{1} - \beta_{2} + \beta_{6} ) q^{17} -\beta_{1} q^{18} + ( -1 - \beta_{1} + \beta_{5} ) q^{19} + ( -\beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{20} + ( -1 - \beta_{4} ) q^{21} + \beta_{1} q^{22} + ( 2 - 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{23} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{24} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{25} + q^{27} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{5} - 2 \beta_{6} ) q^{28} + ( -2 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{29} + ( \beta_{1} - \beta_{5} - \beta_{6} ) q^{30} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{6} ) q^{31} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{32} - q^{33} + ( -5 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{34} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{35} + ( 1 + \beta_{1} + \beta_{2} ) q^{36} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{37} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{38} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{40} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{6} ) q^{41} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{42} + ( -3 + \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{43} + ( -1 - \beta_{1} - \beta_{2} ) q^{44} + ( -1 + \beta_{6} ) q^{45} + ( 5 + 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{46} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{47} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{48} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{49} + ( -5 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{50} + ( -1 + \beta_{1} - \beta_{2} + \beta_{6} ) q^{51} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{53} -\beta_{1} q^{54} + ( 1 - \beta_{6} ) q^{55} + ( 4 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + 4 \beta_{6} ) q^{56} + ( -1 - \beta_{1} + \beta_{5} ) q^{57} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} ) q^{58} + ( -2 + 4 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{59} + ( -\beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{60} + ( -1 - \beta_{3} ) q^{61} + ( -2 - 7 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{62} + ( -1 - \beta_{4} ) q^{63} + ( 1 + 4 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{64} + \beta_{1} q^{66} + ( -1 - 3 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{67} + ( -4 + 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{68} + ( 2 - 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{69} + ( 1 + 5 \beta_{1} + 3 \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{70} + ( -5 - \beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{71} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{72} + ( -5 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{73} + ( -1 - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} + 4 \beta_{6} ) q^{74} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{75} + ( -1 - 4 \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} + 4 \beta_{6} ) q^{76} + ( 1 + \beta_{4} ) q^{77} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{79} + ( 1 + \beta_{2} + \beta_{3} + 3 \beta_{5} + 5 \beta_{6} ) q^{80} + q^{81} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{82} + ( -4 - \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{83} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{5} - 2 \beta_{6} ) q^{84} + ( 5 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{85} + ( -5 + 4 \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{86} + ( -2 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{87} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{88} + ( -6 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{89} + ( \beta_{1} - \beta_{5} - \beta_{6} ) q^{90} + ( 2 - 7 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{92} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{6} ) q^{93} + ( 7 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{94} + ( 1 + 6 \beta_{1} + 3 \beta_{2} - \beta_{5} - 2 \beta_{6} ) q^{95} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{96} + ( -3 + \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{97} + ( -3 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{98} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q - 3q^{2} + 7q^{3} + 9q^{4} - 6q^{5} - 3q^{6} - 6q^{7} - 15q^{8} + 7q^{9} + O(q^{10})$$ $$7q - 3q^{2} + 7q^{3} + 9q^{4} - 6q^{5} - 3q^{6} - 6q^{7} - 15q^{8} + 7q^{9} - 7q^{11} + 9q^{12} + 8q^{14} - 6q^{15} + 17q^{16} - 2q^{17} - 3q^{18} - 8q^{19} + 2q^{20} - 6q^{21} + 3q^{22} + 4q^{23} - 15q^{24} + 13q^{25} + 7q^{27} - 12q^{28} - 12q^{29} + 10q^{31} - 33q^{32} - 7q^{33} - 28q^{34} - 4q^{35} + 9q^{36} - 6q^{37} + 16q^{38} - 10q^{40} - 2q^{41} + 8q^{42} - 16q^{43} - 9q^{44} - 6q^{45} + 26q^{46} - 18q^{47} + 17q^{48} + 23q^{49} - 39q^{50} - 2q^{51} + 10q^{53} - 3q^{54} + 6q^{55} + 16q^{56} - 8q^{57} - 10q^{58} - 2q^{59} + 2q^{60} - 10q^{61} - 36q^{62} - 6q^{63} + 29q^{64} + 3q^{66} - 8q^{67} - 10q^{68} + 4q^{69} + 20q^{70} - 36q^{71} - 15q^{72} - 20q^{73} + 13q^{75} - 10q^{76} + 6q^{77} + 6q^{79} + 20q^{80} + 7q^{81} - 10q^{82} - 30q^{83} - 12q^{84} + 40q^{85} - 6q^{86} - 12q^{87} + 15q^{88} - 34q^{89} - 12q^{92} + 10q^{93} + 32q^{94} + 18q^{95} - 33q^{96} - 16q^{97} - q^{98} - 7q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3 x^{6} - 7 x^{5} + 21 x^{4} + 13 x^{3} - 33 x^{2} - 7 x + 7$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 5 \nu + 2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 6 \nu^{2} + 3 \nu + 4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - \nu^{4} - 7 \nu^{3} + 4 \nu^{2} + 9 \nu - 2$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} - 2 \nu^{5} - 7 \nu^{4} + 12 \nu^{3} + 11 \nu^{2} - 14 \nu - 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 6 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 7 \beta_{2} + 9 \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + \beta_{4} + 8 \beta_{3} + 10 \beta_{2} + 38 \beta_{1} + 12$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{6} + 2 \beta_{5} + 9 \beta_{4} + 11 \beta_{3} + 46 \beta_{2} + 70 \beta_{1} + 87$$

## 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
 2.73878 2.53441 1.42819 0.409068 −0.584778 −1.36814 −2.15754
−2.73878 1.00000 5.50093 2.84154 −2.73878 −3.93129 −9.58828 1.00000 −7.78236
1.2 −2.53441 1.00000 4.42325 −3.70100 −2.53441 0.957295 −6.14151 1.00000 9.37985
1.3 −1.42819 1.00000 0.0397381 −0.0606573 −1.42819 1.70646 2.79963 1.00000 0.0866304
1.4 −0.409068 1.00000 −1.83266 −4.13953 −0.409068 −5.18273 1.56782 1.00000 1.69335
1.5 0.584778 1.00000 −1.65803 1.95350 0.584778 −1.51078 −2.13914 1.00000 1.14237
1.6 1.36814 1.00000 −0.128197 −2.18365 1.36814 4.27070 −2.91167 1.00000 −2.98753
1.7 2.15754 1.00000 2.65498 −0.710210 2.15754 −2.30964 1.41315 1.00000 −1.53231
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$1$$
$$13$$ $$-1$$

## 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 5577.2.a.x 7
13.b even 2 1 5577.2.a.y 7
13.d odd 4 2 429.2.b.b 14
39.f even 4 2 1287.2.b.c 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.b.b 14 13.d odd 4 2
1287.2.b.c 14 39.f even 4 2
5577.2.a.x 7 1.a even 1 1 trivial
5577.2.a.y 7 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5577))$$:

 $$T_{2}^{7} + 3 T_{2}^{6} - 7 T_{2}^{5} - 21 T_{2}^{4} + 13 T_{2}^{3} + 33 T_{2}^{2} - 7 T_{2} - 7$$ $$T_{5}^{7} + 6 T_{5}^{6} - 6 T_{5}^{5} - 74 T_{5}^{4} - 32 T_{5}^{3} + 198 T_{5}^{2} + 144 T_{5} + 8$$ $$T_{7}^{7} + 6 T_{7}^{6} - 18 T_{7}^{5} - 136 T_{7}^{4} - 16 T_{7}^{3} + 524 T_{7}^{2} + 160 T_{7} - 496$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-7 - 7 T + 33 T^{2} + 13 T^{3} - 21 T^{4} - 7 T^{5} + 3 T^{6} + T^{7}$$
$3$ $$( -1 + T )^{7}$$
$5$ $$8 + 144 T + 198 T^{2} - 32 T^{3} - 74 T^{4} - 6 T^{5} + 6 T^{6} + T^{7}$$
$7$ $$-496 + 160 T + 524 T^{2} - 16 T^{3} - 136 T^{4} - 18 T^{5} + 6 T^{6} + T^{7}$$
$11$ $$( 1 + T )^{7}$$
$13$ $$T^{7}$$
$17$ $$-1256 - 2672 T + 134 T^{2} + 698 T^{3} - 32 T^{4} - 50 T^{5} + 2 T^{6} + T^{7}$$
$19$ $$80 - 496 T + 636 T^{2} + 244 T^{3} - 168 T^{4} - 22 T^{5} + 8 T^{6} + T^{7}$$
$23$ $$-1024 - 704 T + 2044 T^{2} + 1448 T^{3} + 48 T^{4} - 76 T^{5} - 4 T^{6} + T^{7}$$
$29$ $$104 + 3040 T + 1170 T^{2} - 790 T^{3} - 358 T^{4} + 2 T^{5} + 12 T^{6} + T^{7}$$
$31$ $$-392 - 5712 T - 4414 T^{2} + 460 T^{3} + 594 T^{4} - 62 T^{5} - 10 T^{6} + T^{7}$$
$37$ $$-125824 - 8832 T + 27896 T^{2} + 4752 T^{3} - 792 T^{4} - 136 T^{5} + 6 T^{6} + T^{7}$$
$41$ $$-39760 - 39088 T + 3924 T^{2} + 4420 T^{3} - 180 T^{4} - 130 T^{5} + 2 T^{6} + T^{7}$$
$43$ $$-692704 - 35104 T + 63862 T^{2} + 2962 T^{3} - 1840 T^{4} - 90 T^{5} + 16 T^{6} + T^{7}$$
$47$ $$122368 + 1344 T - 50456 T^{2} - 20120 T^{3} - 2736 T^{4} - 44 T^{5} + 18 T^{6} + T^{7}$$
$53$ $$11456 + 19584 T - 8048 T^{2} - 2688 T^{3} + 1248 T^{4} - 88 T^{5} - 10 T^{6} + T^{7}$$
$59$ $$-145408 - 20224 T + 24224 T^{2} + 4496 T^{3} - 720 T^{4} - 160 T^{5} + 2 T^{6} + T^{7}$$
$61$ $$32 + 192 T + 56 T^{2} - 248 T^{3} - 116 T^{4} + 12 T^{5} + 10 T^{6} + T^{7}$$
$67$ $$-500296 - 272720 T + 81874 T^{2} + 14516 T^{3} - 1610 T^{4} - 222 T^{5} + 8 T^{6} + T^{7}$$
$71$ $$593536 + 190272 T - 115384 T^{2} - 33248 T^{3} - 1120 T^{4} + 352 T^{5} + 36 T^{6} + T^{7}$$
$73$ $$555776 + 328608 T - 5588 T^{2} - 27624 T^{3} - 4544 T^{4} - 114 T^{5} + 20 T^{6} + T^{7}$$
$79$ $$55424 - 44976 T - 5158 T^{2} + 6730 T^{3} + 560 T^{4} - 154 T^{5} - 6 T^{6} + T^{7}$$
$83$ $$-35840 - 106624 T - 68976 T^{2} - 16000 T^{3} - 704 T^{4} + 232 T^{5} + 30 T^{6} + T^{7}$$
$89$ $$87880 + 19184 T - 24834 T^{2} - 8640 T^{3} + 212 T^{4} + 330 T^{5} + 34 T^{6} + T^{7}$$
$97$ $$216704 + 129920 T - 30376 T^{2} - 29136 T^{3} - 5216 T^{4} - 212 T^{5} + 16 T^{6} + T^{7}$$