# Properties

 Label 605.2.b.e Level $605$ Weight $2$ Character orbit 605.b Analytic conductor $4.831$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$\Q(\sqrt{-2}, \sqrt{3})$$ Defining polynomial: $$x^{4} + 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{3} q^{3} + \beta_{2} q^{4} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{5} + q^{6} + ( \beta_{1} - 2 \beta_{3} ) q^{7} + \beta_{3} q^{8} + ( 1 - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{3} q^{3} + \beta_{2} q^{4} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{5} + q^{6} + ( \beta_{1} - 2 \beta_{3} ) q^{7} + \beta_{3} q^{8} + ( 1 - \beta_{2} ) q^{9} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{10} + ( \beta_{1} - 2 \beta_{3} ) q^{12} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{13} + \beta_{2} q^{14} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{15} + ( -1 + 2 \beta_{2} ) q^{16} + 2 \beta_{3} q^{17} + ( 3 \beta_{1} - \beta_{3} ) q^{18} + ( 1 - 3 \beta_{2} ) q^{19} + ( 3 + \beta_{1} + \beta_{3} ) q^{20} + ( -3 - 2 \beta_{2} ) q^{21} + ( 3 \beta_{1} + \beta_{3} ) q^{23} + ( 2 + \beta_{2} ) q^{24} + ( 1 + 2 \beta_{1} + 2 \beta_{3} ) q^{25} + ( -3 + 3 \beta_{2} ) q^{26} + ( -\beta_{1} - 2 \beta_{3} ) q^{27} -3 \beta_{3} q^{28} + 4 \beta_{2} q^{29} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{30} + ( -7 - \beta_{2} ) q^{31} + ( -5 \beta_{1} + 4 \beta_{3} ) q^{32} -2 q^{34} + ( 3 + \beta_{2} - 3 \beta_{3} ) q^{35} + ( -3 + \beta_{2} ) q^{36} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{37} + ( 7 \beta_{1} - 3 \beta_{3} ) q^{38} + ( -3 - 3 \beta_{2} ) q^{39} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{40} -\beta_{2} q^{41} + ( \beta_{1} - 2 \beta_{3} ) q^{42} + ( 5 \beta_{1} + 2 \beta_{3} ) q^{43} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{45} + ( -7 + 3 \beta_{2} ) q^{46} + ( 4 \beta_{1} - 7 \beta_{3} ) q^{47} + ( 2 \beta_{1} - 3 \beta_{3} ) q^{48} + ( 1 - 3 \beta_{2} ) q^{49} + ( -6 + \beta_{1} + 2 \beta_{2} ) q^{50} + ( 4 + 2 \beta_{2} ) q^{51} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{52} + ( -7 \beta_{1} + 3 \beta_{3} ) q^{53} + ( 4 - \beta_{2} ) q^{54} + ( 3 + 2 \beta_{2} ) q^{56} + ( -3 \beta_{1} + 5 \beta_{3} ) q^{57} + ( -8 \beta_{1} + 4 \beta_{3} ) q^{58} + ( 3 - \beta_{2} ) q^{59} + ( 3 + \beta_{2} - 3 \beta_{3} ) q^{60} + ( 10 + \beta_{2} ) q^{61} + ( -5 \beta_{1} - \beta_{3} ) q^{62} + ( \beta_{1} + \beta_{3} ) q^{63} + ( 4 - \beta_{2} ) q^{64} + ( 6 - 3 \beta_{1} - 3 \beta_{3} ) q^{65} + ( -8 \beta_{1} + \beta_{3} ) q^{67} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{68} + ( 5 + \beta_{2} ) q^{69} + ( 3 + \beta_{1} + \beta_{3} ) q^{70} + ( -3 - 3 \beta_{2} ) q^{71} + ( \beta_{1} - \beta_{3} ) q^{72} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{73} + ( -6 + 4 \beta_{2} ) q^{74} + ( 6 + 2 \beta_{2} - \beta_{3} ) q^{75} + ( -9 + \beta_{2} ) q^{76} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{78} + ( -2 + 2 \beta_{2} ) q^{79} + ( 6 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{80} + ( -2 - 5 \beta_{2} ) q^{81} + ( 2 \beta_{1} - \beta_{3} ) q^{82} + ( -7 \beta_{1} + 7 \beta_{3} ) q^{83} + ( -6 - 3 \beta_{2} ) q^{84} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{85} + ( -12 + 5 \beta_{2} ) q^{86} + ( 4 \beta_{1} - 8 \beta_{3} ) q^{87} + ( 3 - 2 \beta_{2} ) q^{89} + ( 4 - 5 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{90} + ( -9 - 3 \beta_{2} ) q^{91} + ( -7 \beta_{1} + 5 \beta_{3} ) q^{92} + ( -\beta_{1} + 9 \beta_{3} ) q^{93} + ( -1 + 4 \beta_{2} ) q^{94} + ( -9 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{95} + ( 3 + 4 \beta_{2} ) q^{96} + ( -\beta_{1} + 5 \beta_{3} ) q^{97} + ( 7 \beta_{1} - 3 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{6} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{6} + 4q^{9} + 4q^{10} + 4q^{15} - 4q^{16} + 4q^{19} + 12q^{20} - 12q^{21} + 8q^{24} + 4q^{25} - 12q^{26} - 28q^{31} - 8q^{34} + 12q^{35} - 12q^{36} - 12q^{39} - 4q^{40} - 12q^{45} - 28q^{46} + 4q^{49} - 24q^{50} + 16q^{51} + 16q^{54} + 12q^{56} + 12q^{59} + 12q^{60} + 40q^{61} + 16q^{64} + 24q^{65} + 20q^{69} + 12q^{70} - 12q^{71} - 24q^{74} + 24q^{75} - 36q^{76} - 8q^{79} + 24q^{80} - 8q^{81} - 24q^{84} - 8q^{85} - 48q^{86} + 12q^{89} + 16q^{90} - 36q^{91} - 4q^{94} - 36q^{95} + 12q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 1$$:

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

## 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
 − 1.93185i − 0.517638i 0.517638i 1.93185i
1.93185i 0.517638i −1.73205 −1.73205 + 1.41421i 1.00000 0.896575i 0.517638i 2.73205 2.73205 + 3.34607i
364.2 0.517638i 1.93185i 1.73205 1.73205 1.41421i 1.00000 3.34607i 1.93185i −0.732051 −0.732051 0.896575i
364.3 0.517638i 1.93185i 1.73205 1.73205 + 1.41421i 1.00000 3.34607i 1.93185i −0.732051 −0.732051 + 0.896575i
364.4 1.93185i 0.517638i −1.73205 −1.73205 1.41421i 1.00000 0.896575i 0.517638i 2.73205 2.73205 3.34607i
 $$n$$: 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
5.b even 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.e yes 4
5.b even 2 1 inner 605.2.b.e yes 4
5.c odd 4 2 3025.2.a.z 4
11.b odd 2 1 605.2.b.d 4
11.c even 5 4 605.2.j.e 16
11.d odd 10 4 605.2.j.f 16
55.d odd 2 1 605.2.b.d 4
55.e even 4 2 3025.2.a.y 4
55.h odd 10 4 605.2.j.f 16
55.j even 10 4 605.2.j.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.d 4 11.b odd 2 1
605.2.b.d 4 55.d odd 2 1
605.2.b.e yes 4 1.a even 1 1 trivial
605.2.b.e yes 4 5.b even 2 1 inner
605.2.j.e 16 11.c even 5 4
605.2.j.e 16 55.j even 10 4
605.2.j.f 16 11.d odd 10 4
605.2.j.f 16 55.h odd 10 4
3025.2.a.y 4 55.e even 4 2
3025.2.a.z 4 5.c odd 4 2

## Hecke kernels

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

 $$T_{2}^{4} + 4 T_{2}^{2} + 1$$ $$T_{19}^{2} - 2 T_{19} - 26$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T^{2} + T^{4}$$
$3$ $$1 + 4 T^{2} + T^{4}$$
$5$ $$25 - 2 T^{2} + T^{4}$$
$7$ $$9 + 12 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$( 18 + T^{2} )^{2}$$
$17$ $$16 + 16 T^{2} + T^{4}$$
$19$ $$( -26 - 2 T + T^{2} )^{2}$$
$23$ $$484 + 52 T^{2} + T^{4}$$
$29$ $$( -48 + T^{2} )^{2}$$
$31$ $$( 46 + 14 T + T^{2} )^{2}$$
$37$ $$144 + 48 T^{2} + T^{4}$$
$41$ $$( -3 + T^{2} )^{2}$$
$43$ $$4761 + 156 T^{2} + T^{4}$$
$47$ $$2209 + 148 T^{2} + T^{4}$$
$53$ $$676 + 148 T^{2} + T^{4}$$
$59$ $$( 6 - 6 T + T^{2} )^{2}$$
$61$ $$( 97 - 20 T + T^{2} )^{2}$$
$67$ $$1089 + 228 T^{2} + T^{4}$$
$71$ $$( -18 + 6 T + T^{2} )^{2}$$
$73$ $$( 24 + T^{2} )^{2}$$
$79$ $$( -8 + 4 T + T^{2} )^{2}$$
$83$ $$( 98 + T^{2} )^{2}$$
$89$ $$( -3 - 6 T + T^{2} )^{2}$$
$97$ $$36 + 84 T^{2} + T^{4}$$