# Properties

 Label 605.2.j.a Level $605$ Weight $2$ Character orbit 605.j Analytic conductor $4.831$ Analytic rank $0$ Dimension $8$ CM discriminant -55 Inner twists $8$

# 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: 8.0.484000000.6 Defining polynomial: $$x^{8} - x^{6} + 16 x^{4} - 66 x^{2} + 121$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{4} ) q^{2} + ( 1 + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{4} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{5} -2 \beta_{1} q^{7} + ( -\beta_{1} + 2 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{8} + 3 \beta_{5} q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{4} ) q^{2} + ( 1 + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{4} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{5} -2 \beta_{1} q^{7} + ( -\beta_{1} + 2 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{8} + 3 \beta_{5} q^{9} + ( -\beta_{1} + 2 \beta_{4} - \beta_{7} ) q^{10} + ( 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{13} + ( -6 + 4 \beta_{3} - 6 \beta_{5} ) q^{14} + ( -4 - \beta_{2} + 4 \beta_{3} ) q^{16} + ( 2 \beta_{6} + 2 \beta_{7} ) q^{17} + ( -3 \beta_{1} + 3 \beta_{4} + 3 \beta_{6} ) q^{18} + ( -4 \beta_{2} + 4 \beta_{3} - 7 \beta_{5} ) q^{20} -5 \beta_{3} q^{25} + ( -8 - 8 \beta_{2} + 6 \beta_{3} - 6 \beta_{5} ) q^{26} + ( -2 \beta_{6} - 4 \beta_{7} ) q^{28} + ( -8 \beta_{2} + 8 \beta_{3} - 4 \beta_{5} ) q^{31} + ( -\beta_{1} - \beta_{7} ) q^{32} + ( -4 + 10 \beta_{2} + 10 \beta_{5} ) q^{34} + ( 2 \beta_{1} - 4 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{35} + ( -6 + 9 \beta_{3} - 6 \beta_{5} ) q^{36} + ( 5 \beta_{1} - 5 \beta_{4} - 5 \beta_{6} ) q^{40} + ( 4 \beta_{1} - 2 \beta_{4} + 4 \beta_{7} ) q^{43} + ( -3 - 6 \beta_{2} - 6 \beta_{5} ) q^{45} + ( -4 - 9 \beta_{2} + 4 \beta_{3} ) q^{49} + 5 \beta_{7} q^{50} + ( 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{52} + ( 14 - 2 \beta_{2} - 2 \beta_{5} ) q^{56} + ( 4 + 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} ) q^{59} + ( 12 \beta_{1} - 4 \beta_{4} - 4 \beta_{6} ) q^{62} + ( -6 \beta_{4} - 6 \beta_{7} ) q^{63} + ( 6 \beta_{2} - 6 \beta_{3} + \beta_{5} ) q^{64} + ( -2 \beta_{1} - 6 \beta_{4} - 2 \beta_{7} ) q^{65} + ( -16 \beta_{1} + 10 \beta_{4} + 6 \beta_{6} - 6 \beta_{7} ) q^{68} + ( 14 + 14 \beta_{2} - 16 \beta_{3} + 16 \beta_{5} ) q^{70} + 8 \beta_{2} q^{71} + ( -6 \beta_{6} - 3 \beta_{7} ) q^{72} + ( -4 \beta_{1} + 6 \beta_{4} + 6 \beta_{6} ) q^{73} + ( 2 - 11 \beta_{3} + 2 \beta_{5} ) q^{80} + ( -9 - 9 \beta_{2} + 9 \beta_{3} - 9 \beta_{5} ) q^{81} + ( 6 \beta_{6} - 4 \beta_{7} ) q^{83} + ( 6 \beta_{1} - 2 \beta_{4} - 6 \beta_{6} - 2 \beta_{7} ) q^{85} + ( 14 \beta_{2} - 14 \beta_{3} + 24 \beta_{5} ) q^{86} + ( -6 - 12 \beta_{2} - 12 \beta_{5} ) q^{89} + ( 9 \beta_{1} - 3 \beta_{4} - 6 \beta_{6} + 6 \beta_{7} ) q^{90} + ( 16 + 4 \beta_{3} + 16 \beta_{5} ) q^{91} + ( 5 \beta_{1} + 4 \beta_{4} + 5 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 6q^{4} - 10q^{5} - 6q^{9} + O(q^{10})$$ $$8q - 6q^{4} - 10q^{5} - 6q^{9} - 28q^{14} - 22q^{16} + 30q^{20} - 10q^{25} - 24q^{26} + 40q^{31} - 72q^{34} - 18q^{36} - 6q^{49} + 120q^{56} + 8q^{59} - 26q^{64} + 20q^{70} - 16q^{71} - 10q^{80} - 18q^{81} - 104q^{86} + 104q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{6} + 16 x^{4} - 66 x^{2} + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-7 \nu^{6} - 37 \nu^{4} - 629 \nu^{2} - 363$$$$)/1991$$ $$\beta_{3}$$ $$=$$ $$($$$$-28 \nu^{6} - 148 \nu^{4} - 525 \nu^{2} + 539$$$$)/1991$$ $$\beta_{4}$$ $$=$$ $$($$$$-28 \nu^{7} - 148 \nu^{5} - 525 \nu^{3} + 539 \nu$$$$)/1991$$ $$\beta_{5}$$ $$=$$ $$($$$$40 \nu^{6} - 73 \nu^{4} + 750 \nu^{2} - 2761$$$$)/1991$$ $$\beta_{6}$$ $$=$$ $$($$$$61 \nu^{7} + 38 \nu^{5} + 646 \nu^{3} - 1672 \nu$$$$)/1991$$ $$\beta_{7}$$ $$=$$ $$($$$$68 \nu^{7} + 75 \nu^{5} + 1275 \nu^{3} - 3300 \nu$$$$)/1991$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4 \beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{7} - 4 \beta_{6} + \beta_{4} + 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{5} - 10 \beta_{3} - 7$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{7} - 17 \beta_{4} - 7 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$37 \beta_{5} - 37 \beta_{3} + 75 \beta_{2} + 75$$ $$\nu^{7}$$ $$=$$ $$-38 \beta_{7} + 75 \beta_{6}$$

## Character values

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

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$-1$$ $$-1 - \beta_{2} + \beta_{3} - \beta_{5}$$

## 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
 −1.26313 + 1.73855i 1.26313 − 1.73855i −1.46782 − 0.476925i 1.46782 + 0.476925i −1.26313 − 1.73855i 1.26313 + 1.73855i −1.46782 + 0.476925i 1.46782 − 0.476925i
−1.26313 0.410415i 0 −0.190983 0.138757i −0.690983 2.12663i 0 2.52626 3.47709i 1.74560 + 2.40261i 0.927051 2.85317i 2.96979i
9.2 1.26313 + 0.410415i 0 −0.190983 0.138757i −0.690983 2.12663i 0 −2.52626 + 3.47709i −1.74560 2.40261i 0.927051 2.85317i 2.96979i
124.1 −1.46782 2.02029i 0 −1.30902 + 4.02874i −1.80902 1.31433i 0 2.93565 + 0.953850i 5.31064 1.72553i −2.42705 + 1.76336i 5.58394i
124.2 1.46782 + 2.02029i 0 −1.30902 + 4.02874i −1.80902 1.31433i 0 −2.93565 0.953850i −5.31064 + 1.72553i −2.42705 + 1.76336i 5.58394i
269.1 −1.26313 + 0.410415i 0 −0.190983 + 0.138757i −0.690983 + 2.12663i 0 2.52626 + 3.47709i 1.74560 2.40261i 0.927051 + 2.85317i 2.96979i
269.2 1.26313 0.410415i 0 −0.190983 + 0.138757i −0.690983 + 2.12663i 0 −2.52626 3.47709i −1.74560 + 2.40261i 0.927051 + 2.85317i 2.96979i
444.1 −1.46782 + 2.02029i 0 −1.30902 4.02874i −1.80902 + 1.31433i 0 2.93565 0.953850i 5.31064 + 1.72553i −2.42705 1.76336i 5.58394i
444.2 1.46782 2.02029i 0 −1.30902 4.02874i −1.80902 + 1.31433i 0 −2.93565 + 0.953850i −5.31064 1.72553i −2.42705 1.76336i 5.58394i
 $$n$$: 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
55.d odd 2 1 CM by $$\Q(\sqrt{-55})$$
5.b even 2 1 inner
11.b odd 2 1 inner
11.c even 5 1 inner
11.d odd 10 1 inner
55.h odd 10 1 inner
55.j even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.a 8
5.b even 2 1 inner 605.2.j.a 8
11.b odd 2 1 inner 605.2.j.a 8
11.c even 5 1 605.2.b.b 4
11.c even 5 1 inner 605.2.j.a 8
11.c even 5 2 605.2.j.c 8
11.d odd 10 1 605.2.b.b 4
11.d odd 10 1 inner 605.2.j.a 8
11.d odd 10 2 605.2.j.c 8
55.d odd 2 1 CM 605.2.j.a 8
55.h odd 10 1 605.2.b.b 4
55.h odd 10 1 inner 605.2.j.a 8
55.h odd 10 2 605.2.j.c 8
55.j even 10 1 605.2.b.b 4
55.j even 10 1 inner 605.2.j.a 8
55.j even 10 2 605.2.j.c 8
55.k odd 20 2 3025.2.a.bb 4
55.l even 20 2 3025.2.a.bb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.b 4 11.c even 5 1
605.2.b.b 4 11.d odd 10 1
605.2.b.b 4 55.h odd 10 1
605.2.b.b 4 55.j even 10 1
605.2.j.a 8 1.a even 1 1 trivial
605.2.j.a 8 5.b even 2 1 inner
605.2.j.a 8 11.b odd 2 1 inner
605.2.j.a 8 11.c even 5 1 inner
605.2.j.a 8 11.d odd 10 1 inner
605.2.j.a 8 55.d odd 2 1 CM
605.2.j.a 8 55.h odd 10 1 inner
605.2.j.a 8 55.j even 10 1 inner
605.2.j.c 8 11.c even 5 2
605.2.j.c 8 11.d odd 10 2
605.2.j.c 8 55.h odd 10 2
605.2.j.c 8 55.j even 10 2
3025.2.a.bb 4 55.k odd 20 2
3025.2.a.bb 4 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 $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{8} + T_{2}^{6} + 31 T_{2}^{4} - 99 T_{2}^{2} + 121$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$121 - 99 T^{2} + 31 T^{4} + T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 25 + 25 T + 15 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$7$ $$30976 - 4224 T^{2} + 256 T^{4} - 4 T^{6} + T^{8}$$
$11$ $$T^{8}$$
$13$ $$30976 + 4224 T^{2} + 2176 T^{4} - 76 T^{6} + T^{8}$$
$17$ $$30976 - 18304 T^{2} + 4096 T^{4} + 36 T^{6} + T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$( 6400 - 1600 T + 240 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$( 176 + 172 T^{2} + T^{4} )^{2}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$( 256 - 64 T + 16 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$61$ $$T^{8}$$
$67$ $$T^{8}$$
$71$ $$( 4096 + 512 T + 64 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$73$ $$453519616 - 3322176 T^{2} + 21376 T^{4} - 136 T^{6} + T^{8}$$
$79$ $$T^{8}$$
$83$ $$30976 + 35904 T^{2} + 109696 T^{4} - 536 T^{6} + T^{8}$$
$89$ $$( -180 + T^{2} )^{4}$$
$97$ $$T^{8}$$