# Properties

 Label 605.2.j.b Level $605$ Weight $2$ Character orbit 605.j Analytic conductor $4.831$ Analytic rank $0$ Dimension $8$ CM discriminant -11 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.j (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.228765625.1 Defining polynomial: $$x^{8} - x^{7} - 2 x^{6} + 5 x^{5} + x^{4} + 15 x^{3} - 18 x^{2} - 27 x + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{4} - 2 \beta_{5} ) q^{3} + ( -2 + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} ) q^{4} + ( -\beta_{2} + \beta_{7} ) q^{5} + 8 \beta_{3} q^{9} +O(q^{10})$$ $$q + ( -\beta_{4} - 2 \beta_{5} ) q^{3} + ( -2 + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} ) q^{4} + ( -\beta_{2} + \beta_{7} ) q^{5} + 8 \beta_{3} q^{9} + ( -2 + 4 \beta_{6} ) q^{12} + ( 7 - 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 7 \beta_{7} ) q^{15} + 4 \beta_{7} q^{16} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{20} + ( 1 - 2 \beta_{6} ) q^{23} + ( -\beta_{4} - 3 \beta_{5} ) q^{25} + ( 10 \beta_{2} + 5 \beta_{7} ) q^{27} -5 \beta_{3} q^{31} + 16 \beta_{4} q^{36} + ( -3 + 6 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} ) q^{37} + ( -8 - 8 \beta_{6} ) q^{45} + ( 2 \beta_{4} + 4 \beta_{5} ) q^{47} + ( 4 - 8 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} - 8 \beta_{6} + 4 \beta_{7} ) q^{48} + 7 \beta_{7} q^{49} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{53} + ( 15 - 15 \beta_{3} - 15 \beta_{4} + 15 \beta_{7} ) q^{59} + ( -6 \beta_{2} - 14 \beta_{7} ) q^{60} -8 \beta_{3} q^{64} + ( -3 + 6 \beta_{6} ) q^{67} -11 \beta_{4} q^{69} -3 \beta_{7} q^{71} + ( -\beta_{1} + 17 \beta_{3} ) q^{75} + ( -8 \beta_{4} - 4 \beta_{5} ) q^{80} + ( -31 + 31 \beta_{3} + 31 \beta_{4} - 31 \beta_{7} ) q^{81} + 9 q^{89} + ( -2 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{92} + ( -10 \beta_{2} - 5 \beta_{7} ) q^{93} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{4} - 3q^{5} + 16q^{9} + O(q^{10})$$ $$8q - 4q^{4} - 3q^{5} + 16q^{9} + 11q^{15} - 8q^{16} - 6q^{20} + q^{25} - 10q^{31} + 32q^{36} - 96q^{45} - 14q^{49} + 30q^{59} + 22q^{60} - 16q^{64} - 22q^{69} + 6q^{71} + 33q^{75} - 12q^{80} - 62q^{81} + 72q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 2 x^{6} + 5 x^{5} + x^{4} + 15 x^{3} - 18 x^{2} - 27 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + 2 \nu^{5} - 5 \nu^{4} - \nu^{3} - 15 \nu^{2} + 18 \nu + 27$$$$)/27$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 16 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{7} - 4 \nu^{6} + 10 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} - 36 \nu^{2} - 54 \nu + 162$$$$)/27$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{6} + 5 \nu^{5} + \nu^{4} + 2 \nu^{3} - 18 \nu^{2} - 27 \nu + 81$$$$)/9$$ $$\beta_{6}$$ $$=$$ $$\nu^{5} + 16$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 5 \nu^{6} + 10 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} - 75 \nu^{2} + 90 \nu + 135$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{7} + \beta_{6} - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{5} - 3 \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} - 5 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$\beta_{6} - 16$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{3} - 16 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$-35 \beta_{7} - 13 \beta_{6} + 13 \beta_{5} + 48 \beta_{4} + 48 \beta_{3} + 13 \beta_{2} - 13 \beta_{1} - 35$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$-1$$ $$-1 + \beta_{3} + \beta_{4} - \beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 1.42264 + 0.987975i −1.73166 − 0.0369185i −0.570223 − 1.63550i 1.37924 + 1.04771i 1.42264 − 0.987975i −1.73166 + 0.0369185i −0.570223 + 1.63550i 1.37924 − 1.04771i
0 −1.94946 2.68321i −1.61803 1.17557i −1.11362 + 1.93903i 0 0 0 −2.47214 + 7.60845i 0
9.2 0 1.94946 + 2.68321i −1.61803 1.17557i 2.04067 + 0.914138i 0 0 0 −2.47214 + 7.60845i 0
124.1 0 −3.15430 + 1.02489i 0.618034 1.90211i −0.238794 2.22328i 0 0 0 6.47214 4.70228i 0
124.2 0 3.15430 1.02489i 0.618034 1.90211i −2.18826 + 0.459925i 0 0 0 6.47214 4.70228i 0
269.1 0 −1.94946 + 2.68321i −1.61803 + 1.17557i −1.11362 1.93903i 0 0 0 −2.47214 7.60845i 0
269.2 0 1.94946 2.68321i −1.61803 + 1.17557i 2.04067 0.914138i 0 0 0 −2.47214 7.60845i 0
444.1 0 −3.15430 1.02489i 0.618034 + 1.90211i −0.238794 + 2.22328i 0 0 0 6.47214 + 4.70228i 0
444.2 0 3.15430 + 1.02489i 0.618034 + 1.90211i −2.18826 0.459925i 0 0 0 6.47214 + 4.70228i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 444.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
5.b even 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
55.d odd 2 1 inner
55.h odd 10 3 inner
55.j even 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.b 8
5.b even 2 1 inner 605.2.j.b 8
11.b odd 2 1 CM 605.2.j.b 8
11.c even 5 1 605.2.b.a 2
11.c even 5 3 inner 605.2.j.b 8
11.d odd 10 1 605.2.b.a 2
11.d odd 10 3 inner 605.2.j.b 8
55.d odd 2 1 inner 605.2.j.b 8
55.h odd 10 1 605.2.b.a 2
55.h odd 10 3 inner 605.2.j.b 8
55.j even 10 1 605.2.b.a 2
55.j even 10 3 inner 605.2.j.b 8
55.k odd 20 2 3025.2.a.k 2
55.l even 20 2 3025.2.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.a 2 11.c even 5 1
605.2.b.a 2 11.d odd 10 1
605.2.b.a 2 55.h odd 10 1
605.2.b.a 2 55.j even 10 1
605.2.j.b 8 1.a even 1 1 trivial
605.2.j.b 8 5.b even 2 1 inner
605.2.j.b 8 11.b odd 2 1 CM
605.2.j.b 8 11.c even 5 3 inner
605.2.j.b 8 11.d odd 10 3 inner
605.2.j.b 8 55.d odd 2 1 inner
605.2.j.b 8 55.h odd 10 3 inner
605.2.j.b 8 55.j even 10 3 inner
3025.2.a.k 2 55.k odd 20 2
3025.2.a.k 2 55.l even 20 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$14641 - 1331 T^{2} + 121 T^{4} - 11 T^{6} + T^{8}$$
$5$ $$625 + 375 T + 100 T^{2} - 15 T^{3} - 29 T^{4} - 3 T^{5} + 4 T^{6} + 3 T^{7} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$( 11 + T^{2} )^{4}$$
$29$ $$T^{8}$$
$31$ $$( 625 + 125 T + 25 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$37$ $$96059601 - 970299 T^{2} + 9801 T^{4} - 99 T^{6} + T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$3748096 - 85184 T^{2} + 1936 T^{4} - 44 T^{6} + T^{8}$$
$53$ $$959512576 - 5451776 T^{2} + 30976 T^{4} - 176 T^{6} + T^{8}$$
$59$ $$( 50625 - 3375 T + 225 T^{2} - 15 T^{3} + T^{4} )^{2}$$
$61$ $$T^{8}$$
$67$ $$( 99 + T^{2} )^{4}$$
$71$ $$( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$( -9 + T )^{8}$$
$97$ $$96059601 - 970299 T^{2} + 9801 T^{4} - 99 T^{6} + T^{8}$$