# Properties

 Label 605.2.g.h Level $605$ Weight $2$ Character orbit 605.g Analytic conductor $4.831$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.g (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.324000000.3 Defining polynomial: $$x^{8} + 3 x^{6} + 9 x^{4} + 27 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} + 2 \beta_{6} q^{3} + \beta_{4} q^{4} + ( -1 - \beta_{2} - \beta_{4} - \beta_{6} ) q^{5} + 2 \beta_{3} q^{6} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{7} -\beta_{1} q^{8} + \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{7} q^{2} + 2 \beta_{6} q^{3} + \beta_{4} q^{4} + ( -1 - \beta_{2} - \beta_{4} - \beta_{6} ) q^{5} + 2 \beta_{3} q^{6} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{7} -\beta_{1} q^{8} + \beta_{2} q^{9} + \beta_{5} q^{10} + 2 q^{12} + 6 \beta_{6} q^{14} + 2 \beta_{4} q^{15} + ( 5 + 5 \beta_{2} + 5 \beta_{4} + 5 \beta_{6} ) q^{16} -4 \beta_{3} q^{17} + ( -\beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} ) q^{18} -4 \beta_{1} q^{19} + \beta_{2} q^{20} + 4 \beta_{5} q^{21} + 6 q^{23} -2 \beta_{7} q^{24} + \beta_{6} q^{25} + ( 4 + 4 \beta_{2} + 4 \beta_{4} + 4 \beta_{6} ) q^{27} + 2 \beta_{3} q^{28} + 2 \beta_{1} q^{30} + 4 \beta_{2} q^{31} -3 \beta_{5} q^{32} -12 q^{34} + 2 \beta_{7} q^{35} + \beta_{6} q^{36} + 10 \beta_{4} q^{37} + ( 12 + 12 \beta_{2} + 12 \beta_{4} + 12 \beta_{6} ) q^{38} + ( \beta_{1} + \beta_{3} + \beta_{5} + \beta_{7} ) q^{40} + 4 \beta_{1} q^{41} + 12 \beta_{2} q^{42} -2 \beta_{5} q^{43} + q^{45} + 6 \beta_{7} q^{46} -6 \beta_{6} q^{47} -10 \beta_{4} q^{48} + ( -5 - 5 \beta_{2} - 5 \beta_{4} - 5 \beta_{6} ) q^{49} + \beta_{3} q^{50} + ( 8 \beta_{1} + 8 \beta_{3} + 8 \beta_{5} + 8 \beta_{7} ) q^{51} -6 \beta_{2} q^{53} -4 \beta_{5} q^{54} -6 q^{56} -8 \beta_{7} q^{57} + ( -2 - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{60} + 4 \beta_{3} q^{61} + ( -4 \beta_{1} - 4 \beta_{3} - 4 \beta_{5} - 4 \beta_{7} ) q^{62} + 2 \beta_{1} q^{63} + \beta_{2} q^{64} + 10 q^{67} -4 \beta_{7} q^{68} + 12 \beta_{6} q^{69} + 6 \beta_{4} q^{70} -\beta_{3} q^{72} + ( 4 \beta_{1} + 4 \beta_{3} + 4 \beta_{5} + 4 \beta_{7} ) q^{73} + 10 \beta_{1} q^{74} + 2 \beta_{2} q^{75} -4 \beta_{5} q^{76} -4 \beta_{7} q^{79} -5 \beta_{6} q^{80} -11 \beta_{4} q^{81} + ( -12 - 12 \beta_{2} - 12 \beta_{4} - 12 \beta_{6} ) q^{82} -10 \beta_{3} q^{83} + ( -4 \beta_{1} - 4 \beta_{3} - 4 \beta_{5} - 4 \beta_{7} ) q^{84} -4 \beta_{1} q^{85} -6 \beta_{2} q^{86} -6 q^{89} + \beta_{7} q^{90} + 6 \beta_{4} q^{92} + ( -8 - 8 \beta_{2} - 8 \beta_{4} - 8 \beta_{6} ) q^{93} -6 \beta_{3} q^{94} + ( 4 \beta_{1} + 4 \beta_{3} + 4 \beta_{5} + 4 \beta_{7} ) q^{95} -6 \beta_{1} q^{96} -10 \beta_{2} q^{97} + 5 \beta_{5} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{3} - 2q^{4} - 2q^{5} - 2q^{9} + O(q^{10})$$ $$8q - 4q^{3} - 2q^{4} - 2q^{5} - 2q^{9} + 16q^{12} - 12q^{14} - 4q^{15} + 10q^{16} - 2q^{20} + 48q^{23} - 2q^{25} + 8q^{27} - 8q^{31} - 96q^{34} - 2q^{36} - 20q^{37} + 24q^{38} - 24q^{42} + 8q^{45} + 12q^{47} + 20q^{48} - 10q^{49} + 12q^{53} - 48q^{56} - 4q^{60} - 2q^{64} + 80q^{67} - 24q^{69} - 12q^{70} - 4q^{75} + 10q^{80} + 22q^{81} - 24q^{82} + 12q^{86} - 48q^{89} - 12q^{92} - 16q^{93} + 20q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 9 x^{4} + 27 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$ $$\beta_{4}$$ $$=$$ $$\nu^{4}$$$$/9$$ $$\beta_{5}$$ $$=$$ $$\nu^{5}$$$$/9$$ $$\beta_{6}$$ $$=$$ $$\nu^{6}$$$$/27$$ $$\beta_{7}$$ $$=$$ $$\nu^{7}$$$$/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$27 \beta_{6}$$ $$\nu^{7}$$ $$=$$ $$27 \beta_{7}$$

## Character values

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

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$1$$ $$\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.535233 − 1.64728i −0.535233 + 1.64728i 1.40126 − 1.01807i −1.40126 + 1.01807i 0.535233 + 1.64728i −0.535233 − 1.64728i 1.40126 + 1.01807i −1.40126 − 1.01807i
−1.40126 1.01807i 0.618034 1.90211i 0.309017 + 0.951057i −0.809017 + 0.587785i −2.80252 + 2.03615i 1.07047 + 3.29456i −0.535233 + 1.64728i −0.809017 0.587785i 1.73205
81.2 1.40126 + 1.01807i 0.618034 1.90211i 0.309017 + 0.951057i −0.809017 + 0.587785i 2.80252 2.03615i −1.07047 3.29456i 0.535233 1.64728i −0.809017 0.587785i −1.73205
251.1 −0.535233 + 1.64728i −1.61803 + 1.17557i −0.809017 0.587785i 0.309017 + 0.951057i −1.07047 3.29456i 2.80252 + 2.03615i −1.40126 + 1.01807i 0.309017 0.951057i −1.73205
251.2 0.535233 1.64728i −1.61803 + 1.17557i −0.809017 0.587785i 0.309017 + 0.951057i 1.07047 + 3.29456i −2.80252 2.03615i 1.40126 1.01807i 0.309017 0.951057i 1.73205
366.1 −1.40126 + 1.01807i 0.618034 + 1.90211i 0.309017 0.951057i −0.809017 0.587785i −2.80252 2.03615i 1.07047 3.29456i −0.535233 1.64728i −0.809017 + 0.587785i 1.73205
366.2 1.40126 1.01807i 0.618034 + 1.90211i 0.309017 0.951057i −0.809017 0.587785i 2.80252 + 2.03615i −1.07047 + 3.29456i 0.535233 + 1.64728i −0.809017 + 0.587785i −1.73205
511.1 −0.535233 1.64728i −1.61803 1.17557i −0.809017 + 0.587785i 0.309017 0.951057i −1.07047 + 3.29456i 2.80252 2.03615i −1.40126 1.01807i 0.309017 + 0.951057i −1.73205
511.2 0.535233 + 1.64728i −1.61803 1.17557i −0.809017 + 0.587785i 0.309017 0.951057i 1.07047 3.29456i −2.80252 + 2.03615i 1.40126 + 1.01807i 0.309017 + 0.951057i 1.73205
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 511.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.h 8
11.b odd 2 1 inner 605.2.g.h 8
11.c even 5 1 605.2.a.f 2
11.c even 5 3 inner 605.2.g.h 8
11.d odd 10 1 605.2.a.f 2
11.d odd 10 3 inner 605.2.g.h 8
33.f even 10 1 5445.2.a.r 2
33.h odd 10 1 5445.2.a.r 2
44.g even 10 1 9680.2.a.bg 2
44.h odd 10 1 9680.2.a.bg 2
55.h odd 10 1 3025.2.a.j 2
55.j even 10 1 3025.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.f 2 11.c even 5 1
605.2.a.f 2 11.d odd 10 1
605.2.g.h 8 1.a even 1 1 trivial
605.2.g.h 8 11.b odd 2 1 inner
605.2.g.h 8 11.c even 5 3 inner
605.2.g.h 8 11.d odd 10 3 inner
3025.2.a.j 2 55.h odd 10 1
3025.2.a.j 2 55.j even 10 1
5445.2.a.r 2 33.f even 10 1
5445.2.a.r 2 33.h odd 10 1
9680.2.a.bg 2 44.g even 10 1
9680.2.a.bg 2 44.h odd 10 1

## Hecke kernels

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

 $$T_{2}^{8} + 3 T_{2}^{6} + 9 T_{2}^{4} + 27 T_{2}^{2} + 81$$ $$T_{3}^{4} + 2 T_{3}^{3} + 4 T_{3}^{2} + 8 T_{3} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$81 + 27 T^{2} + 9 T^{4} + 3 T^{6} + T^{8}$$
$3$ $$( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$5$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$7$ $$20736 + 1728 T^{2} + 144 T^{4} + 12 T^{6} + T^{8}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$5308416 + 110592 T^{2} + 2304 T^{4} + 48 T^{6} + T^{8}$$
$19$ $$5308416 + 110592 T^{2} + 2304 T^{4} + 48 T^{6} + T^{8}$$
$23$ $$( -6 + T )^{8}$$
$29$ $$T^{8}$$
$31$ $$( 256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$37$ $$( 10000 + 1000 T + 100 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$41$ $$5308416 + 110592 T^{2} + 2304 T^{4} + 48 T^{6} + T^{8}$$
$43$ $$( -12 + T^{2} )^{4}$$
$47$ $$( 1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$53$ $$( 1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$59$ $$T^{8}$$
$61$ $$5308416 + 110592 T^{2} + 2304 T^{4} + 48 T^{6} + T^{8}$$
$67$ $$( -10 + T )^{8}$$
$71$ $$T^{8}$$
$73$ $$5308416 + 110592 T^{2} + 2304 T^{4} + 48 T^{6} + T^{8}$$
$79$ $$5308416 + 110592 T^{2} + 2304 T^{4} + 48 T^{6} + T^{8}$$
$83$ $$8100000000 + 27000000 T^{2} + 90000 T^{4} + 300 T^{6} + T^{8}$$
$89$ $$( 6 + T )^{8}$$
$97$ $$( 10000 - 1000 T + 100 T^{2} - 10 T^{3} + T^{4} )^{2}$$