# Properties

 Label 605.2.b.b Level $605$ Weight $2$ Character orbit 605.b Analytic conductor $4.831$ Analytic rank $0$ Dimension $4$ CM discriminant -55 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.4400.1 Defining polynomial: $$x^{4} + 7 x^{2} + 11$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( -2 - \beta_{3} ) q^{4} + \beta_{3} q^{5} -\beta_{1} q^{7} + ( \beta_{1} + \beta_{2} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( -2 - \beta_{3} ) q^{4} + \beta_{3} q^{5} -\beta_{1} q^{7} + ( \beta_{1} + \beta_{2} ) q^{8} + 3 q^{9} + ( -\beta_{1} - \beta_{2} ) q^{10} + ( -\beta_{1} + 2 \beta_{2} ) q^{13} + ( -1 - 3 \beta_{3} ) q^{14} + ( 1 + 2 \beta_{3} ) q^{16} + ( -\beta_{1} - 2 \beta_{2} ) q^{17} -3 \beta_{2} q^{18} + ( -5 - 2 \beta_{3} ) q^{20} + 5 q^{25} + ( 7 - \beta_{3} ) q^{26} + ( \beta_{1} + 4 \beta_{2} ) q^{28} -4 \beta_{3} q^{31} -\beta_{2} q^{32} + ( -9 - 5 \beta_{3} ) q^{34} + ( \beta_{1} - 4 \beta_{2} ) q^{35} + ( -6 - 3 \beta_{3} ) q^{36} + 5 \beta_{2} q^{40} + ( \beta_{1} + 4 \beta_{2} ) q^{43} + 3 \beta_{3} q^{45} + ( -7 + 2 \beta_{3} ) q^{49} -5 \beta_{2} q^{50} + ( -\beta_{1} - 2 \beta_{2} ) q^{52} + ( 15 + \beta_{3} ) q^{56} -4 q^{59} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{62} -3 \beta_{1} q^{63} + ( -2 + 3 \beta_{3} ) q^{64} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{65} + ( 3 \beta_{1} + 10 \beta_{2} ) q^{68} + ( -15 - \beta_{3} ) q^{70} + 8 q^{71} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{72} + ( \beta_{1} - 6 \beta_{2} ) q^{73} + ( 10 + \beta_{3} ) q^{80} + 9 q^{81} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{83} + ( -\beta_{1} - 6 \beta_{2} ) q^{85} + ( 17 + 7 \beta_{3} ) q^{86} + 6 \beta_{3} q^{89} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{90} + ( -12 + 8 \beta_{3} ) q^{91} + ( -2 \beta_{1} + 5 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} + 12q^{9} + O(q^{10})$$ $$4q - 8q^{4} + 12q^{9} - 4q^{14} + 4q^{16} - 20q^{20} + 20q^{25} + 28q^{26} - 36q^{34} - 24q^{36} - 28q^{49} + 60q^{56} - 16q^{59} - 8q^{64} - 60q^{70} + 32q^{71} + 40q^{80} + 36q^{81} + 68q^{86} - 48q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 11$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} - 2 \beta_{1}$$

## Character values

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

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
364.1
 1.54336i − 2.14896i 2.14896i − 1.54336i
2.49721i 0 −4.23607 2.23607 0 3.08672i 5.58394i 3.00000 5.58394i
364.2 1.32813i 0 0.236068 −2.23607 0 4.29792i 2.96979i 3.00000 2.96979i
364.3 1.32813i 0 0.236068 −2.23607 0 4.29792i 2.96979i 3.00000 2.96979i
364.4 2.49721i 0 −4.23607 2.23607 0 3.08672i 5.58394i 3.00000 5.58394i
 $$n$$: e.g. 2-40 or 990-1000 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

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{4} + 8 T_{2}^{2} + 11$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$11 + 8 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$176 + 28 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$176 + 52 T^{2} + T^{4}$$
$17$ $$176 + 68 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$( -80 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$176 + 172 T^{2} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$( 4 + T )^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$( -8 + T )^{4}$$
$73$ $$21296 + 292 T^{2} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$176 + 332 T^{2} + T^{4}$$
$89$ $$( -180 + T^{2} )^{2}$$
$97$ $$T^{4}$$