# Properties

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

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## 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: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} + ( -1 + \beta_{1} + \beta_{3} ) q^{4} + ( -1 + \beta_{2} - \beta_{3} ) q^{5} -2 q^{6} + 2 \beta_{2} q^{7} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{8} + ( -1 - \beta_{1} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} + ( -1 + \beta_{1} + \beta_{3} ) q^{4} + ( -1 + \beta_{2} - \beta_{3} ) q^{5} -2 q^{6} + 2 \beta_{2} q^{7} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{8} + ( -1 - \beta_{1} - \beta_{3} ) q^{9} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{10} + 2 \beta_{2} q^{12} + ( 2 - 2 \beta_{1} - 2 \beta_{3} ) q^{14} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{15} + ( 3 - \beta_{1} - \beta_{3} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{17} + ( -\beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{18} + 4 q^{19} + ( -3 - 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{20} + ( -4 - 2 \beta_{1} - 2 \beta_{3} ) q^{21} + ( \beta_{1} - \beta_{3} ) q^{23} + ( -2 - 2 \beta_{1} - 2 \beta_{3} ) q^{24} + ( 1 + 3 \beta_{1} + \beta_{2} ) q^{25} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{27} + ( -6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{28} + ( 2 - 2 \beta_{1} - 2 \beta_{3} ) q^{29} + ( 2 - 2 \beta_{2} + 2 \beta_{3} ) q^{30} + ( -\beta_{1} - \beta_{3} ) q^{31} + ( -3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{32} + 4 q^{34} + ( -4 + 4 \beta_{1} - 2 \beta_{3} ) q^{35} + ( -7 + \beta_{1} + \beta_{3} ) q^{36} + ( -3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{37} + ( -4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{38} + ( -6 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{40} + ( -2 + 2 \beta_{1} + 2 \beta_{3} ) q^{41} -4 \beta_{2} q^{42} -2 \beta_{2} q^{43} + ( 5 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{45} + 2 q^{46} + ( 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{47} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{48} -5 q^{49} + ( 4 + \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{50} + ( 8 + 2 \beta_{1} + 2 \beta_{3} ) q^{51} + ( 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{53} + ( -4 + 2 \beta_{1} + 2 \beta_{3} ) q^{54} + ( -14 + 2 \beta_{1} + 2 \beta_{3} ) q^{56} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{57} + ( -6 \beta_{1} - 10 \beta_{2} + 6 \beta_{3} ) q^{58} + ( 4 - \beta_{1} - \beta_{3} ) q^{59} + ( -4 + 4 \beta_{1} - 2 \beta_{3} ) q^{60} + ( -6 - 2 \beta_{1} - 2 \beta_{3} ) q^{61} + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{62} + ( 6 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{63} + ( -1 - \beta_{1} - \beta_{3} ) q^{64} + ( -3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{67} -4 \beta_{2} q^{68} + ( 4 + \beta_{1} + \beta_{3} ) q^{69} + ( 6 + 6 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} ) q^{70} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{71} + ( 7 \beta_{1} + 5 \beta_{2} - 7 \beta_{3} ) q^{72} + 4 \beta_{2} q^{73} + ( -4 - 2 \beta_{1} - 2 \beta_{3} ) q^{74} + ( 4 - 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{75} + ( -4 + 4 \beta_{1} + 4 \beta_{3} ) q^{76} + ( 8 + 2 \beta_{1} + 2 \beta_{3} ) q^{79} + ( 1 + 3 \beta_{1} + 6 \beta_{2} - 5 \beta_{3} ) q^{80} + ( -3 - 2 \beta_{1} - 2 \beta_{3} ) q^{81} + ( 6 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} ) q^{82} + ( -4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{83} -12 q^{84} + ( -2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{85} + ( -2 + 2 \beta_{1} + 2 \beta_{3} ) q^{86} -4 \beta_{2} q^{87} + ( 2 + \beta_{1} + \beta_{3} ) q^{89} + ( 6 - 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{90} -2 \beta_{2} q^{92} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{93} + ( 10 - 2 \beta_{1} - 2 \beta_{3} ) q^{94} + ( -4 + 4 \beta_{2} - 4 \beta_{3} ) q^{95} + ( -10 - 2 \beta_{1} - 2 \beta_{3} ) q^{96} + ( -3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{97} + ( 5 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{4} - 3q^{5} - 8q^{6} - 2q^{9} + O(q^{10})$$ $$4q - 6q^{4} - 3q^{5} - 8q^{6} - 2q^{9} + 10q^{10} + 12q^{14} + q^{15} + 14q^{16} + 16q^{19} - 12q^{20} - 12q^{21} - 4q^{24} + q^{25} + 12q^{29} + 6q^{30} + 2q^{31} + 16q^{34} - 18q^{35} - 30q^{36} - 28q^{40} - 12q^{41} + 18q^{45} + 8q^{46} - 20q^{49} + 18q^{50} + 28q^{51} - 20q^{54} - 60q^{56} + 18q^{59} - 18q^{60} - 20q^{61} - 2q^{64} + 14q^{69} + 24q^{70} - 6q^{71} - 12q^{74} + 15q^{75} - 24q^{76} + 28q^{79} + 6q^{80} - 8q^{81} - 48q^{84} - 2q^{85} - 12q^{86} + 6q^{89} + 28q^{90} + 44q^{94} - 12q^{95} - 36q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} + 4 \nu - 9$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - 2 \nu^{2} + 2 \nu + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{3} - 3 \beta_{2} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 4$$

## 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
 −1.18614 + 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i
2.52434i 0.792287i −4.37228 0.686141 + 2.12819i −2.00000 3.46410i 5.98844i 2.37228 5.37228 1.73205i
364.2 0.792287i 2.52434i 1.37228 −2.18614 0.469882i −2.00000 3.46410i 2.67181i −3.37228 −0.372281 + 1.73205i
364.3 0.792287i 2.52434i 1.37228 −2.18614 + 0.469882i −2.00000 3.46410i 2.67181i −3.37228 −0.372281 1.73205i
364.4 2.52434i 0.792287i −4.37228 0.686141 2.12819i −2.00000 3.46410i 5.98844i 2.37228 5.37228 + 1.73205i
 $$n$$: 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.c 4
5.b even 2 1 inner 605.2.b.c 4
5.c odd 4 2 3025.2.a.ba 4
11.b odd 2 1 55.2.b.a 4
11.c even 5 4 605.2.j.j 16
11.d odd 10 4 605.2.j.i 16
33.d even 2 1 495.2.c.a 4
44.c even 2 1 880.2.b.h 4
55.d odd 2 1 55.2.b.a 4
55.e even 4 2 275.2.a.h 4
55.h odd 10 4 605.2.j.i 16
55.j even 10 4 605.2.j.j 16
165.d even 2 1 495.2.c.a 4
165.l odd 4 2 2475.2.a.bi 4
220.g even 2 1 880.2.b.h 4
220.i odd 4 2 4400.2.a.cc 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.b.a 4 11.b odd 2 1
55.2.b.a 4 55.d odd 2 1
275.2.a.h 4 55.e even 4 2
495.2.c.a 4 33.d even 2 1
495.2.c.a 4 165.d even 2 1
605.2.b.c 4 1.a even 1 1 trivial
605.2.b.c 4 5.b even 2 1 inner
605.2.j.i 16 11.d odd 10 4
605.2.j.i 16 55.h odd 10 4
605.2.j.j 16 11.c even 5 4
605.2.j.j 16 55.j even 10 4
880.2.b.h 4 44.c even 2 1
880.2.b.h 4 220.g even 2 1
2475.2.a.bi 4 165.l odd 4 2
3025.2.a.ba 4 5.c odd 4 2
4400.2.a.cc 4 220.i odd 4 2

## Hecke kernels

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

 $$T_{2}^{4} + 7 T_{2}^{2} + 4$$ $$T_{19} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 7 T^{2} + T^{4}$$
$3$ $$4 + 7 T^{2} + T^{4}$$
$5$ $$25 + 15 T + 4 T^{2} + 3 T^{3} + T^{4}$$
$7$ $$( 12 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$64 + 28 T^{2} + T^{4}$$
$19$ $$( -4 + T )^{4}$$
$23$ $$4 + 7 T^{2} + T^{4}$$
$29$ $$( -24 - 6 T + T^{2} )^{2}$$
$31$ $$( -8 - T + T^{2} )^{2}$$
$37$ $$144 + 123 T^{2} + T^{4}$$
$41$ $$( -24 + 6 T + T^{2} )^{2}$$
$43$ $$( 12 + T^{2} )^{2}$$
$47$ $$( 44 + T^{2} )^{2}$$
$53$ $$1024 + 112 T^{2} + T^{4}$$
$59$ $$( 12 - 9 T + T^{2} )^{2}$$
$61$ $$( -8 + 10 T + T^{2} )^{2}$$
$67$ $$36 + 87 T^{2} + T^{4}$$
$71$ $$( -72 + 3 T + T^{2} )^{2}$$
$73$ $$( 48 + T^{2} )^{2}$$
$79$ $$( 16 - 14 T + T^{2} )^{2}$$
$83$ $$( 44 + T^{2} )^{2}$$
$89$ $$( -6 - 3 T + T^{2} )^{2}$$
$97$ $$576 + 51 T^{2} + T^{4}$$
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