# Properties

 Label 5445.2.a.cb Level $5445$ Weight $2$ Character orbit 5445.a Self dual yes Analytic conductor $43.479$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5445 = 3^{2} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5445.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.4785439006$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4 x^{7} - 3 x^{6} + 22 x^{5} - 3 x^{4} - 32 x^{3} + 9 x^{2} + 8 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 495) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( -\beta_{1} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{4} + q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{7} ) q^{7} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{6} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( -\beta_{1} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{4} + q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{7} ) q^{7} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{6} - \beta_{7} ) q^{8} + ( -1 + \beta_{1} ) q^{10} + ( -2 - \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{13} + ( 3 - 2 \beta_{1} + \beta_{4} + \beta_{5} ) q^{14} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{16} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{17} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{19} + ( -\beta_{1} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{20} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{23} + q^{25} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{26} + ( -3 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{6} ) q^{28} + ( -1 + 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{29} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{31} + ( -4 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 5 \beta_{4} + \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{32} + ( -1 + \beta_{2} - 5 \beta_{4} - 3 \beta_{6} ) q^{34} + ( -1 + \beta_{1} + \beta_{2} - \beta_{7} ) q^{35} + ( -2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{37} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{38} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{6} - \beta_{7} ) q^{40} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{6} - \beta_{7} ) q^{41} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{43} + ( -2 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{46} + ( 3 + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{47} + ( 2 - \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{49} + ( -1 + \beta_{1} ) q^{50} + ( 3 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{52} + ( -6 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{53} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} - 3 \beta_{5} + 5 \beta_{6} + 4 \beta_{7} ) q^{56} + ( 3 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} ) q^{58} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{59} + ( 4 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 5 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{61} + ( -1 + 2 \beta_{1} + 4 \beta_{3} + \beta_{4} - \beta_{6} - 3 \beta_{7} ) q^{62} + ( 5 - 7 \beta_{1} - 6 \beta_{2} - \beta_{3} + 5 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} + 3 \beta_{7} ) q^{64} + ( -2 - \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{65} + ( -2 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 4 \beta_{6} + \beta_{7} ) q^{67} + ( 3 - 2 \beta_{1} - 4 \beta_{2} + 8 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{68} + ( 3 - 2 \beta_{1} + \beta_{4} + \beta_{5} ) q^{70} + ( -5 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 6 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{71} + ( -6 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} + 3 \beta_{7} ) q^{73} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 5 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{74} + ( 4 - 6 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{6} ) q^{76} + ( 4 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 4 \beta_{5} ) q^{79} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{80} + ( 5 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} + 6 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} ) q^{82} + ( -9 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{83} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{85} + ( -2 - 4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} + 9 \beta_{4} - 2 \beta_{5} + 6 \beta_{6} ) q^{86} + ( 7 - 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} ) q^{89} + ( 8 - 3 \beta_{1} + \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{91} + ( -2 + 3 \beta_{1} + \beta_{2} + 3 \beta_{4} - 3 \beta_{5} ) q^{92} + ( -4 + \beta_{1} - 5 \beta_{3} + \beta_{4} - 2 \beta_{5} + 5 \beta_{6} + 3 \beta_{7} ) q^{94} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{95} + ( 1 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{97} + ( -4 - 2 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{2} + 6 q^{4} + 8 q^{5} - 8 q^{7} - 12 q^{8} + O(q^{10})$$ $$8 q - 4 q^{2} + 6 q^{4} + 8 q^{5} - 8 q^{7} - 12 q^{8} - 4 q^{10} - 6 q^{13} + 14 q^{14} + 14 q^{16} - 8 q^{17} + 2 q^{19} + 6 q^{20} + 4 q^{23} + 8 q^{25} - 2 q^{26} - 24 q^{28} - 22 q^{29} + 10 q^{31} - 28 q^{32} - 2 q^{34} - 8 q^{35} - 14 q^{37} - 20 q^{38} - 12 q^{40} - 22 q^{41} - 14 q^{43} + 2 q^{46} + 10 q^{47} - 4 q^{50} + 10 q^{52} - 18 q^{53} + 34 q^{56} + 12 q^{58} + 2 q^{59} + 14 q^{61} - 30 q^{62} + 30 q^{64} - 6 q^{65} + 10 q^{67} - 6 q^{68} + 14 q^{70} - 2 q^{71} - 16 q^{73} - 24 q^{74} + 22 q^{76} + 16 q^{79} + 14 q^{80} + 10 q^{82} - 46 q^{83} - 8 q^{85} - 28 q^{86} + 38 q^{89} + 8 q^{91} - 24 q^{92} + 10 q^{94} + 2 q^{95} - 4 q^{97} - 4 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 3 x^{6} + 22 x^{5} - 3 x^{4} - 32 x^{3} + 9 x^{2} + 8 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$-\nu^{7} + 4 \nu^{6} + 3 \nu^{5} - 22 \nu^{4} + 4 \nu^{3} + 30 \nu^{2} - 13 \nu - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{7} - 4 \nu^{6} - 3 \nu^{5} + 22 \nu^{4} - 3 \nu^{3} - 32 \nu^{2} + 10 \nu + 6$$ $$\beta_{4}$$ $$=$$ $$4 \nu^{7} - 17 \nu^{6} - 8 \nu^{5} + 91 \nu^{4} - 34 \nu^{3} - 124 \nu^{2} + 66 \nu + 19$$ $$\beta_{5}$$ $$=$$ $$7 \nu^{7} - 29 \nu^{6} - 16 \nu^{5} + 154 \nu^{4} - 48 \nu^{3} - 206 \nu^{2} + 101 \nu + 30$$ $$\beta_{6}$$ $$=$$ $$-7 \nu^{7} + 30 \nu^{6} + 13 \nu^{5} - 159 \nu^{4} + 62 \nu^{3} + 214 \nu^{2} - 115 \nu - 33$$ $$\beta_{7}$$ $$=$$ $$8 \nu^{7} - 34 \nu^{6} - 16 \nu^{5} + 181 \nu^{4} - 65 \nu^{3} - 245 \nu^{2} + 124 \nu + 38$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta_{3} + \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} + 2 \beta_{6} - \beta_{3} + \beta_{2} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$8 \beta_{7} + 9 \beta_{6} + 2 \beta_{4} - 6 \beta_{3} + 3 \beta_{2} + 10 \beta_{1} + 3$$ $$\nu^{5}$$ $$=$$ $$20 \beta_{7} + 23 \beta_{6} + \beta_{5} + 5 \beta_{4} - 12 \beta_{3} + 14 \beta_{2} + 36 \beta_{1} + 2$$ $$\nu^{6}$$ $$=$$ $$64 \beta_{7} + 79 \beta_{6} + 4 \beta_{5} + 25 \beta_{4} - 44 \beta_{3} + 43 \beta_{2} + 94 \beta_{1} + 16$$ $$\nu^{7}$$ $$=$$ $$178 \beta_{7} + 225 \beta_{6} + 19 \beta_{5} + 71 \beta_{4} - 114 \beta_{3} + 151 \beta_{2} + 301 \beta_{1} + 30$$

## 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
 −1.75229 −1.39491 −0.226007 −0.205878 0.909121 1.48523 2.13569 3.04904
−2.75229 0 5.57509 1.00000 0 −2.73187 −9.83966 0 −2.75229
1.2 −2.39491 0 3.73560 1.00000 0 −1.96984 −4.15660 0 −2.39491
1.3 −1.22601 0 −0.496906 1.00000 0 0.451695 3.06123 0 −1.22601
1.4 −1.20588 0 −0.545859 1.00000 0 −4.31539 3.06999 0 −1.20588
1.5 −0.0908791 0 −1.99174 1.00000 0 2.45732 0.362766 0 −0.0908791
1.6 0.485227 0 −1.76455 1.00000 0 1.61947 −1.82666 0 0.485227
1.7 1.13569 0 −0.710206 1.00000 0 −4.10132 −3.07796 0 1.13569
1.8 2.04904 0 2.19858 1.00000 0 0.589936 0.406903 0 2.04904
 $$n$$: 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
$$3$$ $$1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

## 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 5445.2.a.cb 8
3.b odd 2 1 5445.2.a.cc 8
11.b odd 2 1 5445.2.a.cd 8
11.d odd 10 2 495.2.n.g 16
33.d even 2 1 5445.2.a.ca 8
33.f even 10 2 495.2.n.h yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.n.g 16 11.d odd 10 2
495.2.n.h yes 16 33.f even 10 2
5445.2.a.ca 8 33.d even 2 1
5445.2.a.cb 8 1.a even 1 1 trivial
5445.2.a.cc 8 3.b odd 2 1
5445.2.a.cd 8 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5445))$$:

 $$T_{2}^{8} + \cdots$$ $$T_{7}^{8} + \cdots$$ $$T_{23}^{8} - \cdots$$ $$T_{53}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 10 T + 14 T^{2} + 32 T^{3} - 8 T^{4} - 24 T^{5} - 3 T^{6} + 4 T^{7} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( -1 + T )^{8}$$
$7$ $$101 - 362 T + 203 T^{2} + 290 T^{3} - 98 T^{4} - 86 T^{5} + 4 T^{6} + 8 T^{7} + T^{8}$$
$11$ $$T^{8}$$
$13$ $$-1331 - 4598 T - 924 T^{2} + 1906 T^{3} + 362 T^{4} - 214 T^{5} - 37 T^{6} + 6 T^{7} + T^{8}$$
$17$ $$4231 - 8940 T + 1570 T^{2} + 3848 T^{3} + 152 T^{4} - 340 T^{5} - 34 T^{6} + 8 T^{7} + T^{8}$$
$19$ $$5929 + 11088 T - 3632 T^{2} - 6122 T^{3} + 2066 T^{4} + 208 T^{5} - 91 T^{6} - 2 T^{7} + T^{8}$$
$23$ $$-1 + 144 T + 2139 T^{2} + 3042 T^{3} + 1309 T^{4} + 46 T^{5} - 65 T^{6} - 4 T^{7} + T^{8}$$
$29$ $$1891 + 6250 T - 196 T^{2} - 5178 T^{3} - 1984 T^{4} + 94 T^{5} + 145 T^{6} + 22 T^{7} + T^{8}$$
$31$ $$23081 + 446 T - 23785 T^{2} - 8412 T^{3} + 1677 T^{4} + 584 T^{5} - 61 T^{6} - 10 T^{7} + T^{8}$$
$37$ $$122881 + 160428 T + 66809 T^{2} + 3826 T^{3} - 4051 T^{4} - 886 T^{5} - 3 T^{6} + 14 T^{7} + T^{8}$$
$41$ $$2647555 + 2209370 T + 589999 T^{2} + 17150 T^{3} - 17136 T^{4} - 2326 T^{5} + 36 T^{6} + 22 T^{7} + T^{8}$$
$43$ $$-2417279 - 230960 T + 588171 T^{2} + 163146 T^{3} + 1462 T^{4} - 3012 T^{5} - 172 T^{6} + 14 T^{7} + T^{8}$$
$47$ $$-589 + 21778 T - 2863 T^{2} - 10796 T^{3} + 2152 T^{4} + 642 T^{5} - 96 T^{6} - 10 T^{7} + T^{8}$$
$53$ $$392695 + 996010 T + 547489 T^{2} + 75540 T^{3} - 10906 T^{4} - 2934 T^{5} - 74 T^{6} + 18 T^{7} + T^{8}$$
$59$ $$-88469 + 285532 T - 143845 T^{2} - 18564 T^{3} + 9059 T^{4} + 352 T^{5} - 173 T^{6} - 2 T^{7} + T^{8}$$
$61$ $$-462499 + 428928 T + 114988 T^{2} - 61846 T^{3} - 3178 T^{4} + 2684 T^{5} - 159 T^{6} - 14 T^{7} + T^{8}$$
$67$ $$-597971 + 641474 T - 125028 T^{2} - 43982 T^{3} + 9702 T^{4} + 1238 T^{5} - 173 T^{6} - 10 T^{7} + T^{8}$$
$71$ $$-9049805 - 8580100 T - 878501 T^{2} + 303170 T^{3} + 43839 T^{4} - 1946 T^{5} - 399 T^{6} + 2 T^{7} + T^{8}$$
$73$ $$-3045251 + 106022 T + 539636 T^{2} + 56736 T^{3} - 20758 T^{4} - 4314 T^{5} - 157 T^{6} + 16 T^{7} + T^{8}$$
$79$ $$5735705 - 7837500 T + 3653434 T^{2} - 610868 T^{3} + 491 T^{4} + 7124 T^{5} - 342 T^{6} - 16 T^{7} + T^{8}$$
$83$ $$-41467789 + 3934810 T + 3502361 T^{2} - 104114 T^{3} - 89989 T^{4} - 3582 T^{5} + 535 T^{6} + 46 T^{7} + T^{8}$$
$89$ $$-5696725 + 6461770 T - 2873896 T^{2} + 629456 T^{3} - 64783 T^{4} + 888 T^{5} + 438 T^{6} - 38 T^{7} + T^{8}$$
$97$ $$-7598959 + 7704568 T - 1379357 T^{2} - 181478 T^{3} + 50754 T^{4} + 72 T^{5} - 404 T^{6} + 4 T^{7} + T^{8}$$