# Properties

 Label 605.2.b.f Level $605$ Weight $2$ Character orbit 605.b Analytic conductor $4.831$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.1480160000.1 Defining polynomial: $$x^{8} + 9 x^{6} + 27 x^{4} + 31 x^{2} + 11$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{6} + \beta_{7} ) q^{3} + \beta_{2} q^{4} + ( -\beta_{3} - \beta_{4} - \beta_{6} ) q^{5} + ( -2 + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{6} + \beta_{4} q^{7} + ( \beta_{1} + \beta_{4} + \beta_{6} ) q^{8} + ( 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{6} + \beta_{7} ) q^{3} + \beta_{2} q^{4} + ( -\beta_{3} - \beta_{4} - \beta_{6} ) q^{5} + ( -2 + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{6} + \beta_{4} q^{7} + ( \beta_{1} + \beta_{4} + \beta_{6} ) q^{8} + ( 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{9} + ( 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{10} + ( -\beta_{1} + \beta_{4} + 2 \beta_{6} ) q^{12} + ( -\beta_{1} + 2 \beta_{6} ) q^{13} + ( -1 - 2 \beta_{2} + 2 \beta_{5} ) q^{14} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{5} - \beta_{6} ) q^{15} + ( -2 + \beta_{2} + \beta_{3} ) q^{16} + ( 2 \beta_{1} - \beta_{6} + 3 \beta_{7} ) q^{17} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{18} + ( -2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{19} + ( -2 \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{20} + ( 1 + \beta_{3} - 2 \beta_{5} ) q^{21} + ( 2 \beta_{1} + 2 \beta_{4} - \beta_{6} ) q^{23} + ( -1 - \beta_{2} - 2 \beta_{3} ) q^{24} + ( -3 - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{25} + ( 4 - \beta_{2} + 2 \beta_{3} - 4 \beta_{5} ) q^{26} + ( \beta_{4} + 2 \beta_{7} ) q^{27} + \beta_{1} q^{28} + ( 4 + \beta_{3} - \beta_{5} ) q^{29} + ( 1 - 3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{30} + ( 2 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{31} + ( -\beta_{1} + 3 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{32} + ( -2 + 2 \beta_{2} - 4 \beta_{3} - \beta_{5} ) q^{34} + ( 3 + \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{35} + ( 3 - 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{36} + ( -2 \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{37} + ( 2 \beta_{1} - 2 \beta_{4} - \beta_{7} ) q^{38} + ( 2 - 5 \beta_{2} + \beta_{5} ) q^{39} + ( 4 + 2 \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{40} + ( -2 + 3 \beta_{2} - 2 \beta_{3} - 5 \beta_{5} ) q^{41} + ( \beta_{1} - 3 \beta_{6} + \beta_{7} ) q^{42} + ( -3 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} ) q^{43} + ( 2 - 3 \beta_{1} + \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{45} + ( -7 - 2 \beta_{2} - \beta_{3} + 6 \beta_{5} ) q^{46} + ( \beta_{1} - 3 \beta_{4} + \beta_{6} - \beta_{7} ) q^{47} + ( -2 \beta_{1} + \beta_{4} + 5 \beta_{6} - 2 \beta_{7} ) q^{48} + ( 4 - \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{49} + ( 2 - 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{50} + ( -7 + \beta_{2} - 3 \beta_{3} - 2 \beta_{5} ) q^{51} + ( 3 \beta_{1} - \beta_{4} - 3 \beta_{6} + 2 \beta_{7} ) q^{52} + ( 3 \beta_{1} + \beta_{4} + 3 \beta_{6} - 2 \beta_{7} ) q^{53} + ( 1 - 2 \beta_{2} - 2 \beta_{3} ) q^{54} + ( -4 - 3 \beta_{2} + 4 \beta_{5} ) q^{56} + ( \beta_{1} - \beta_{4} - 5 \beta_{6} + \beta_{7} ) q^{57} + ( 4 \beta_{1} - 2 \beta_{6} + \beta_{7} ) q^{58} + ( -5 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} ) q^{59} + ( 5 - 3 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{7} ) q^{60} + ( -4 - 4 \beta_{2} + 4 \beta_{3} + 5 \beta_{5} ) q^{61} + ( \beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{62} + ( 2 \beta_{1} + \beta_{4} - \beta_{7} ) q^{63} + ( -2 - 5 \beta_{2} + 3 \beta_{3} + \beta_{5} ) q^{64} + ( 2 - 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{65} + ( -4 \beta_{1} + \beta_{4} + 2 \beta_{6} - 3 \beta_{7} ) q^{67} + ( 2 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{68} + ( -2 + 4 \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{69} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{5} - \beta_{6} ) q^{70} + ( -9 - 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{71} + ( \beta_{1} + 2 \beta_{4} - \beta_{7} ) q^{72} + ( -2 \beta_{1} - 5 \beta_{4} - \beta_{7} ) q^{73} + ( -2 + 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} ) q^{74} + ( -2 - 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{75} + ( -3 + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{76} + ( 7 \beta_{1} - 5 \beta_{4} - 4 \beta_{6} ) q^{78} + ( 6 + 4 \beta_{2} + \beta_{3} - 5 \beta_{5} ) q^{79} + ( -1 - 2 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{80} + ( -1 + 4 \beta_{2} - 2 \beta_{3} - 7 \beta_{5} ) q^{81} + ( -5 \beta_{1} + 3 \beta_{4} - 2 \beta_{7} ) q^{82} + ( \beta_{1} - \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{83} + ( -2 + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{84} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{85} + ( 2 - 3 \beta_{2} + 6 \beta_{5} ) q^{86} + ( 5 \beta_{1} - \beta_{4} - 3 \beta_{6} + 3 \beta_{7} ) q^{87} + ( 5 - 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{5} ) q^{89} + ( 8 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{6} ) q^{90} + ( 1 + 2 \beta_{2} - 4 \beta_{3} ) q^{91} + ( -\beta_{1} + 2 \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{92} + ( 2 \beta_{1} + 2 \beta_{4} + \beta_{6} + 3 \beta_{7} ) q^{93} + ( 1 + 7 \beta_{2} + 2 \beta_{3} - 7 \beta_{5} ) q^{94} + ( 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{95} + ( 4 - 6 \beta_{2} + 3 \beta_{3} - 6 \beta_{5} ) q^{96} + ( -\beta_{1} - \beta_{4} + 3 \beta_{7} ) q^{97} + ( 5 \beta_{1} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{4} + 4q^{5} - 6q^{6} - 4q^{9} + O(q^{10})$$ $$8q - 2q^{4} + 4q^{5} - 6q^{6} - 4q^{9} + 4q^{14} - 8q^{15} - 22q^{16} + 12q^{19} - 4q^{20} - 4q^{21} + 2q^{24} - 8q^{25} + 10q^{26} + 24q^{29} + 22q^{30} + 14q^{31} - 8q^{34} + 14q^{35} + 20q^{36} + 30q^{39} + 24q^{40} - 34q^{41} + 6q^{45} - 24q^{46} + 30q^{49} + 16q^{50} - 54q^{51} + 20q^{54} - 10q^{56} + 6q^{59} + 34q^{60} - 20q^{61} - 14q^{64} + 20q^{65} - 32q^{69} + 8q^{70} - 42q^{71} - 4q^{74} - 20q^{75} - 28q^{76} + 16q^{79} - 28q^{80} - 36q^{81} - 6q^{84} - 4q^{85} + 46q^{86} + 12q^{89} + 46q^{90} + 20q^{91} - 42q^{94} + 26q^{95} + 8q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 9 x^{6} + 27 x^{4} + 31 x^{2} + 11$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} + 5 \nu^{2} + 4$$ $$\beta_{4}$$ $$=$$ $$-\nu^{7} - 7 \nu^{5} - 13 \nu^{3} - 5 \nu$$ $$\beta_{5}$$ $$=$$ $$\nu^{6} + 7 \nu^{4} + 14 \nu^{2} + 8$$ $$\beta_{6}$$ $$=$$ $$\nu^{7} + 7 \nu^{5} + 14 \nu^{3} + 8 \nu$$ $$\beta_{7}$$ $$=$$ $$\nu^{7} + 8 \nu^{5} + 19 \nu^{3} + 12 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{4} - 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} - 5 \beta_{2} + 6$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} - 6 \beta_{6} - 5 \beta_{4} + 11 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{5} - 7 \beta_{3} + 21 \beta_{2} - 22$$ $$\nu^{7}$$ $$=$$ $$-7 \beta_{7} + 29 \beta_{6} + 21 \beta_{4} - 43 \beta_{1}$$

## 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
 − 2.02368i − 1.65458i − 1.23399i − 0.802699i 0.802699i 1.23399i 1.65458i 2.02368i
2.02368i 2.62059i −2.09529 −0.294963 2.21653i −5.30325 0.965823i 0.192845i −3.86752 −4.48555 + 0.596911i
364.2 1.65458i 1.97479i −0.737640 2.19353 + 0.434096i 3.26745 2.24307i 2.08868i −0.899788 0.718246 3.62937i
364.3 1.23399i 0.363982i 0.477260 1.29496 + 1.82293i 0.449152 2.58558i 3.05692i 2.86752 2.24948 1.59798i
364.4 0.802699i 1.76074i 1.35567 −1.19353 + 1.89090i −1.41335 0.592103i 2.69360i −0.100212 1.51782 + 0.958043i
364.5 0.802699i 1.76074i 1.35567 −1.19353 1.89090i −1.41335 0.592103i 2.69360i −0.100212 1.51782 0.958043i
364.6 1.23399i 0.363982i 0.477260 1.29496 1.82293i 0.449152 2.58558i 3.05692i 2.86752 2.24948 + 1.59798i
364.7 1.65458i 1.97479i −0.737640 2.19353 0.434096i 3.26745 2.24307i 2.08868i −0.899788 0.718246 + 3.62937i
364.8 2.02368i 2.62059i −2.09529 −0.294963 + 2.21653i −5.30325 0.965823i 0.192845i −3.86752 −4.48555 0.596911i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 364.8 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.f 8
5.b even 2 1 inner 605.2.b.f 8
5.c odd 4 2 3025.2.a.bk 8
11.b odd 2 1 605.2.b.g 8
11.c even 5 2 605.2.j.d 16
11.c even 5 2 605.2.j.g 16
11.d odd 10 2 55.2.j.a 16
11.d odd 10 2 605.2.j.h 16
33.f even 10 2 495.2.ba.a 16
44.g even 10 2 880.2.cd.c 16
55.d odd 2 1 605.2.b.g 8
55.e even 4 2 3025.2.a.bl 8
55.h odd 10 2 55.2.j.a 16
55.h odd 10 2 605.2.j.h 16
55.j even 10 2 605.2.j.d 16
55.j even 10 2 605.2.j.g 16
55.l even 20 4 275.2.h.d 16
165.r even 10 2 495.2.ba.a 16
220.o even 10 2 880.2.cd.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.j.a 16 11.d odd 10 2
55.2.j.a 16 55.h odd 10 2
275.2.h.d 16 55.l even 20 4
495.2.ba.a 16 33.f even 10 2
495.2.ba.a 16 165.r even 10 2
605.2.b.f 8 1.a even 1 1 trivial
605.2.b.f 8 5.b even 2 1 inner
605.2.b.g 8 11.b odd 2 1
605.2.b.g 8 55.d odd 2 1
605.2.j.d 16 11.c even 5 2
605.2.j.d 16 55.j even 10 2
605.2.j.g 16 11.c even 5 2
605.2.j.g 16 55.j even 10 2
605.2.j.h 16 11.d odd 10 2
605.2.j.h 16 55.h odd 10 2
880.2.cd.c 16 44.g even 10 2
880.2.cd.c 16 220.o even 10 2
3025.2.a.bk 8 5.c odd 4 2
3025.2.a.bl 8 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 $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{8} + 9 T_{2}^{6} + 27 T_{2}^{4} + 31 T_{2}^{2} + 11$$ $$T_{19}^{4} - 6 T_{19}^{3} - 4 T_{19}^{2} + 39 T_{19} + 11$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$11 + 31 T^{2} + 27 T^{4} + 9 T^{6} + T^{8}$$
$3$ $$11 + 91 T^{2} + 62 T^{4} + 14 T^{6} + T^{8}$$
$5$ $$625 - 500 T + 300 T^{2} - 180 T^{3} + 86 T^{4} - 36 T^{5} + 12 T^{6} - 4 T^{7} + T^{8}$$
$7$ $$11 + 47 T^{2} + 49 T^{4} + 13 T^{6} + T^{8}$$
$11$ $$T^{8}$$
$13$ $$6875 + 3875 T^{2} + 675 T^{4} + 45 T^{6} + T^{8}$$
$17$ $$40931 + 15524 T^{2} + 1842 T^{4} + 81 T^{6} + T^{8}$$
$19$ $$( 11 + 39 T - 4 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$23$ $$26411 + 16366 T^{2} + 2232 T^{4} + 99 T^{6} + T^{8}$$
$29$ $$( 11 - 67 T + 48 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$31$ $$( -1 + T + 5 T^{2} - 7 T^{3} + T^{4} )^{2}$$
$37$ $$3971 + 15304 T^{2} + 2822 T^{4} + 101 T^{6} + T^{8}$$
$41$ $$( -1969 - 438 T + 48 T^{2} + 17 T^{3} + T^{4} )^{2}$$
$43$ $$212531 + 79707 T^{2} + 8319 T^{4} + 173 T^{6} + T^{8}$$
$47$ $$244211 + 90271 T^{2} + 7351 T^{4} + 191 T^{6} + T^{8}$$
$53$ $$489731 + 164921 T^{2} + 15237 T^{4} + 249 T^{6} + T^{8}$$
$59$ $$( -1 + 29 T - 75 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$61$ $$( 209 - 10 T - 61 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$67$ $$18491 + 46104 T^{2} + 5736 T^{4} + 149 T^{6} + T^{8}$$
$71$ $$( -2511 - 432 T + 90 T^{2} + 21 T^{3} + T^{4} )^{2}$$
$73$ $$1279091 + 671988 T^{2} + 25504 T^{4} + 297 T^{6} + T^{8}$$
$79$ $$( -319 + 377 T - 52 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$83$ $$212531 + 50451 T^{2} + 3931 T^{4} + 111 T^{6} + T^{8}$$
$89$ $$( 1871 + 486 T - 128 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$97$ $$161051 + 55902 T^{2} + 5588 T^{4} + 162 T^{6} + T^{8}$$