# Properties

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

# 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: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} -3 \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} -3 \zeta_{10} q^{6} + 3 \zeta_{10}^{3} q^{7} + 3 \zeta_{10}^{2} q^{8} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} -3 \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} -3 \zeta_{10} q^{6} + 3 \zeta_{10}^{3} q^{7} + 3 \zeta_{10}^{2} q^{8} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{9} - q^{10} + 3 q^{12} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{13} + 3 \zeta_{10}^{2} q^{14} + 3 \zeta_{10}^{3} q^{15} + \zeta_{10} q^{16} + 6 \zeta_{10}^{3} q^{18} + 4 \zeta_{10}^{2} q^{19} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{20} + 9 q^{21} -8 q^{23} + ( 9 - 9 \zeta_{10} + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{24} + \zeta_{10}^{2} q^{25} + 4 \zeta_{10}^{3} q^{26} + 9 \zeta_{10} q^{27} -3 \zeta_{10} q^{28} -6 \zeta_{10}^{3} q^{29} + 3 \zeta_{10}^{2} q^{30} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{31} -5 q^{32} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{35} -6 \zeta_{10}^{2} q^{36} + 8 \zeta_{10}^{3} q^{37} + 4 \zeta_{10} q^{38} + 12 \zeta_{10} q^{39} -3 \zeta_{10}^{3} q^{40} -5 \zeta_{10}^{2} q^{41} + ( 9 - 9 \zeta_{10} + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{42} + 5 q^{43} + 6 q^{45} + ( -8 + 8 \zeta_{10} - 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{46} -3 \zeta_{10}^{2} q^{47} -3 \zeta_{10}^{3} q^{48} -2 \zeta_{10} q^{49} + \zeta_{10} q^{50} -4 \zeta_{10}^{2} q^{52} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{53} + 9 q^{54} -9 q^{56} + ( 12 - 12 \zeta_{10} + 12 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{57} -6 \zeta_{10}^{2} q^{58} + 2 \zeta_{10}^{3} q^{59} -3 \zeta_{10} q^{60} + 11 \zeta_{10} q^{61} -2 \zeta_{10}^{3} q^{62} -18 \zeta_{10}^{2} q^{63} + ( -7 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{64} + 4 q^{65} -13 q^{67} + 24 \zeta_{10}^{2} q^{69} -3 \zeta_{10}^{3} q^{70} -2 \zeta_{10} q^{71} -18 \zeta_{10} q^{72} + 8 \zeta_{10}^{3} q^{73} + 8 \zeta_{10}^{2} q^{74} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{75} -4 q^{76} + 12 q^{78} + ( -10 + 10 \zeta_{10} - 10 \zeta_{10}^{2} + 10 \zeta_{10}^{3} ) q^{79} -\zeta_{10}^{2} q^{80} -9 \zeta_{10}^{3} q^{81} -5 \zeta_{10} q^{82} -4 \zeta_{10} q^{83} + 9 \zeta_{10}^{3} q^{84} + ( 5 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{86} -18 q^{87} + q^{89} + ( 6 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{90} -12 \zeta_{10}^{2} q^{91} -8 \zeta_{10}^{3} q^{92} -6 \zeta_{10} q^{93} -3 \zeta_{10} q^{94} -4 \zeta_{10}^{3} q^{95} + 15 \zeta_{10}^{2} q^{96} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{97} -2 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} + 3q^{3} + q^{4} - q^{5} - 3q^{6} + 3q^{7} - 3q^{8} - 6q^{9} + O(q^{10})$$ $$4q + q^{2} + 3q^{3} + q^{4} - q^{5} - 3q^{6} + 3q^{7} - 3q^{8} - 6q^{9} - 4q^{10} + 12q^{12} - 4q^{13} - 3q^{14} + 3q^{15} + q^{16} + 6q^{18} - 4q^{19} + q^{20} + 36q^{21} - 32q^{23} + 9q^{24} - q^{25} + 4q^{26} + 9q^{27} - 3q^{28} - 6q^{29} - 3q^{30} + 2q^{31} - 20q^{32} + 3q^{35} + 6q^{36} + 8q^{37} + 4q^{38} + 12q^{39} - 3q^{40} + 5q^{41} + 9q^{42} + 20q^{43} + 24q^{45} - 8q^{46} + 3q^{47} - 3q^{48} - 2q^{49} + q^{50} + 4q^{52} - 4q^{53} + 36q^{54} - 36q^{56} + 12q^{57} + 6q^{58} + 2q^{59} - 3q^{60} + 11q^{61} - 2q^{62} + 18q^{63} - 7q^{64} + 16q^{65} - 52q^{67} - 24q^{69} - 3q^{70} - 2q^{71} - 18q^{72} + 8q^{73} - 8q^{74} + 3q^{75} - 16q^{76} + 48q^{78} - 10q^{79} + q^{80} - 9q^{81} - 5q^{82} - 4q^{83} + 9q^{84} + 5q^{86} - 72q^{87} + 4q^{89} + 6q^{90} + 12q^{91} - 8q^{92} - 6q^{93} - 3q^{94} - 4q^{95} - 15q^{96} + 8q^{97} - 8q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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
 0.809017 − 0.587785i −0.309017 − 0.951057i 0.809017 + 0.587785i −0.309017 + 0.951057i
0.809017 + 0.587785i −0.927051 + 2.85317i −0.309017 0.951057i −0.809017 + 0.587785i −2.42705 + 1.76336i −0.927051 2.85317i 0.927051 2.85317i −4.85410 3.52671i −1.00000
251.1 −0.309017 + 0.951057i 2.42705 1.76336i 0.809017 + 0.587785i 0.309017 + 0.951057i 0.927051 + 2.85317i 2.42705 + 1.76336i −2.42705 + 1.76336i 1.85410 5.70634i −1.00000
366.1 0.809017 0.587785i −0.927051 2.85317i −0.309017 + 0.951057i −0.809017 0.587785i −2.42705 1.76336i −0.927051 + 2.85317i 0.927051 + 2.85317i −4.85410 + 3.52671i −1.00000
511.1 −0.309017 0.951057i 2.42705 + 1.76336i 0.809017 0.587785i 0.309017 0.951057i 0.927051 2.85317i 2.42705 1.76336i −2.42705 1.76336i 1.85410 + 5.70634i −1.00000
 $$n$$: 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
11.c even 5 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.d 4
11.b odd 2 1 605.2.g.b 4
11.c even 5 1 605.2.a.a 1
11.c even 5 3 inner 605.2.g.d 4
11.d odd 10 1 605.2.a.c yes 1
11.d odd 10 3 605.2.g.b 4
33.f even 10 1 5445.2.a.d 1
33.h odd 10 1 5445.2.a.h 1
44.g even 10 1 9680.2.a.be 1
44.h odd 10 1 9680.2.a.bf 1
55.h odd 10 1 3025.2.a.c 1
55.j even 10 1 3025.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.a 1 11.c even 5 1
605.2.a.c yes 1 11.d odd 10 1
605.2.g.b 4 11.b odd 2 1
605.2.g.b 4 11.d odd 10 3
605.2.g.d 4 1.a even 1 1 trivial
605.2.g.d 4 11.c even 5 3 inner
3025.2.a.c 1 55.h odd 10 1
3025.2.a.g 1 55.j even 10 1
5445.2.a.d 1 33.f even 10 1
5445.2.a.h 1 33.h odd 10 1
9680.2.a.be 1 44.g even 10 1
9680.2.a.bf 1 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 $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1$$ $$T_{3}^{4} - 3 T_{3}^{3} + 9 T_{3}^{2} - 27 T_{3} + 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$3$ $$81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4}$$
$5$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$7$ $$81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$T^{4}$$
$19$ $$256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4}$$
$23$ $$( 8 + T )^{4}$$
$29$ $$1296 + 216 T + 36 T^{2} + 6 T^{3} + T^{4}$$
$31$ $$16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$37$ $$4096 - 512 T + 64 T^{2} - 8 T^{3} + T^{4}$$
$41$ $$625 - 125 T + 25 T^{2} - 5 T^{3} + T^{4}$$
$43$ $$( -5 + T )^{4}$$
$47$ $$81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4}$$
$53$ $$256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4}$$
$59$ $$16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$61$ $$14641 - 1331 T + 121 T^{2} - 11 T^{3} + T^{4}$$
$67$ $$( 13 + T )^{4}$$
$71$ $$16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$73$ $$4096 - 512 T + 64 T^{2} - 8 T^{3} + T^{4}$$
$79$ $$10000 + 1000 T + 100 T^{2} + 10 T^{3} + T^{4}$$
$83$ $$256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4}$$
$89$ $$( -1 + T )^{4}$$
$97$ $$4096 - 512 T + 64 T^{2} - 8 T^{3} + T^{4}$$