# Properties

 Label 605.2.g.c 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: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} + \zeta_{10}^{3} q^{4} - \zeta_{10} q^{5} + 3 \zeta_{10}^{2} q^{8} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{9} +O(q^{10})$$ q + (-z^3 + z^2 - z + 1) * q^2 + z^3 * q^4 - z * q^5 + 3*z^2 * q^8 + (-3*z^3 + 3*z^2 - 3*z + 3) * q^9 $$q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} + \zeta_{10}^{3} q^{4} - \zeta_{10} q^{5} + 3 \zeta_{10}^{2} q^{8} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{9} - q^{10} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{13} + \zeta_{10} q^{16} + 6 \zeta_{10} q^{17} - 3 \zeta_{10}^{3} q^{18} + 4 \zeta_{10}^{2} q^{19} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{20} + 4 q^{23} + \zeta_{10}^{2} q^{25} - 2 \zeta_{10}^{3} q^{26} + 6 \zeta_{10}^{3} q^{29} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{31} - 5 q^{32} + 6 q^{34} + 3 \zeta_{10}^{2} q^{36} + 2 \zeta_{10}^{3} q^{37} + 4 \zeta_{10} q^{38} - 3 \zeta_{10}^{3} q^{40} - 2 \zeta_{10}^{2} q^{41} - 4 q^{43} - 3 q^{45} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{46} - 12 \zeta_{10}^{2} q^{47} + 7 \zeta_{10} q^{49} + \zeta_{10} q^{50} + 2 \zeta_{10}^{2} q^{52} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{53} + 6 \zeta_{10}^{2} q^{58} - 4 \zeta_{10}^{3} q^{59} - 10 \zeta_{10} q^{61} - 8 \zeta_{10}^{3} q^{62} + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 7) q^{64} - 2 q^{65} - 16 q^{67} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 6) q^{68} - 8 \zeta_{10} q^{71} + 9 \zeta_{10} q^{72} + 14 \zeta_{10}^{3} q^{73} + 2 \zeta_{10}^{2} q^{74} - 4 q^{76} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{79} - \zeta_{10}^{2} q^{80} - 9 \zeta_{10}^{3} q^{81} - 2 \zeta_{10} q^{82} - 4 \zeta_{10} q^{83} - 6 \zeta_{10}^{2} q^{85} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{86} + 10 q^{89} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{90} + 4 \zeta_{10}^{3} q^{92} - 12 \zeta_{10} q^{94} - 4 \zeta_{10}^{3} q^{95} + (10 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 10 \zeta_{10} - 10) q^{97} + 7 q^{98} +O(q^{100})$$ q + (-z^3 + z^2 - z + 1) * q^2 + z^3 * q^4 - z * q^5 + 3*z^2 * q^8 + (-3*z^3 + 3*z^2 - 3*z + 3) * q^9 - q^10 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^13 + z * q^16 + 6*z * q^17 - 3*z^3 * q^18 + 4*z^2 * q^19 + (-z^3 + z^2 - z + 1) * q^20 + 4 * q^23 + z^2 * q^25 - 2*z^3 * q^26 + 6*z^3 * q^29 + (-8*z^3 + 8*z^2 - 8*z + 8) * q^31 - 5 * q^32 + 6 * q^34 + 3*z^2 * q^36 + 2*z^3 * q^37 + 4*z * q^38 - 3*z^3 * q^40 - 2*z^2 * q^41 - 4 * q^43 - 3 * q^45 + (-4*z^3 + 4*z^2 - 4*z + 4) * q^46 - 12*z^2 * q^47 + 7*z * q^49 + z * q^50 + 2*z^2 * q^52 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^53 + 6*z^2 * q^58 - 4*z^3 * q^59 - 10*z * q^61 - 8*z^3 * q^62 + (7*z^3 - 7*z^2 + 7*z - 7) * q^64 - 2 * q^65 - 16 * q^67 + (6*z^3 - 6*z^2 + 6*z - 6) * q^68 - 8*z * q^71 + 9*z * q^72 + 14*z^3 * q^73 + 2*z^2 * q^74 - 4 * q^76 + (-8*z^3 + 8*z^2 - 8*z + 8) * q^79 - z^2 * q^80 - 9*z^3 * q^81 - 2*z * q^82 - 4*z * q^83 - 6*z^2 * q^85 + (4*z^3 - 4*z^2 + 4*z - 4) * q^86 + 10 * q^89 + (3*z^3 - 3*z^2 + 3*z - 3) * q^90 + 4*z^3 * q^92 - 12*z * q^94 - 4*z^3 * q^95 + (10*z^3 - 10*z^2 + 10*z - 10) * q^97 + 7 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + q^{4} - q^{5} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 4 * q + q^2 + q^4 - q^5 - 3 * q^8 + 3 * q^9 $$4 q + q^{2} + q^{4} - q^{5} - 3 q^{8} + 3 q^{9} - 4 q^{10} + 2 q^{13} + q^{16} + 6 q^{17} - 3 q^{18} - 4 q^{19} + q^{20} + 16 q^{23} - q^{25} - 2 q^{26} + 6 q^{29} + 8 q^{31} - 20 q^{32} + 24 q^{34} - 3 q^{36} + 2 q^{37} + 4 q^{38} - 3 q^{40} + 2 q^{41} - 16 q^{43} - 12 q^{45} + 4 q^{46} + 12 q^{47} + 7 q^{49} + q^{50} - 2 q^{52} + 2 q^{53} - 6 q^{58} - 4 q^{59} - 10 q^{61} - 8 q^{62} - 7 q^{64} - 8 q^{65} - 64 q^{67} - 6 q^{68} - 8 q^{71} + 9 q^{72} + 14 q^{73} - 2 q^{74} - 16 q^{76} + 8 q^{79} + q^{80} - 9 q^{81} - 2 q^{82} - 4 q^{83} + 6 q^{85} - 4 q^{86} + 40 q^{89} - 3 q^{90} + 4 q^{92} - 12 q^{94} - 4 q^{95} - 10 q^{97} + 28 q^{98}+O(q^{100})$$ 4 * q + q^2 + q^4 - q^5 - 3 * q^8 + 3 * q^9 - 4 * q^10 + 2 * q^13 + q^16 + 6 * q^17 - 3 * q^18 - 4 * q^19 + q^20 + 16 * q^23 - q^25 - 2 * q^26 + 6 * q^29 + 8 * q^31 - 20 * q^32 + 24 * q^34 - 3 * q^36 + 2 * q^37 + 4 * q^38 - 3 * q^40 + 2 * q^41 - 16 * q^43 - 12 * q^45 + 4 * q^46 + 12 * q^47 + 7 * q^49 + q^50 - 2 * q^52 + 2 * q^53 - 6 * q^58 - 4 * q^59 - 10 * q^61 - 8 * q^62 - 7 * q^64 - 8 * q^65 - 64 * q^67 - 6 * q^68 - 8 * q^71 + 9 * q^72 + 14 * q^73 - 2 * q^74 - 16 * q^76 + 8 * q^79 + q^80 - 9 * q^81 - 2 * q^82 - 4 * q^83 + 6 * q^85 - 4 * q^86 + 40 * q^89 - 3 * q^90 + 4 * q^92 - 12 * q^94 - 4 * q^95 - 10 * q^97 + 28 * q^98

## 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 −0.309017 0.951057i −0.809017 + 0.587785i 0 0 0.927051 2.85317i 2.42705 + 1.76336i −1.00000
251.1 −0.309017 + 0.951057i 0 0.809017 + 0.587785i 0.309017 + 0.951057i 0 0 −2.42705 + 1.76336i −0.927051 + 2.85317i −1.00000
366.1 0.809017 0.587785i 0 −0.309017 + 0.951057i −0.809017 0.587785i 0 0 0.927051 + 2.85317i 2.42705 1.76336i −1.00000
511.1 −0.309017 0.951057i 0 0.809017 0.587785i 0.309017 0.951057i 0 0 −2.42705 1.76336i −0.927051 2.85317i −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.c 4
11.b odd 2 1 605.2.g.a 4
11.c even 5 1 605.2.a.b 1
11.c even 5 3 inner 605.2.g.c 4
11.d odd 10 1 55.2.a.a 1
11.d odd 10 3 605.2.g.a 4
33.f even 10 1 495.2.a.a 1
33.h odd 10 1 5445.2.a.i 1
44.g even 10 1 880.2.a.h 1
44.h odd 10 1 9680.2.a.r 1
55.h odd 10 1 275.2.a.a 1
55.j even 10 1 3025.2.a.f 1
55.l even 20 2 275.2.b.b 2
77.l even 10 1 2695.2.a.c 1
88.k even 10 1 3520.2.a.n 1
88.p odd 10 1 3520.2.a.p 1
132.n odd 10 1 7920.2.a.i 1
143.l odd 10 1 9295.2.a.b 1
165.r even 10 1 2475.2.a.i 1
165.u odd 20 2 2475.2.c.f 2
220.o even 10 1 4400.2.a.p 1
220.w odd 20 2 4400.2.b.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.a 1 11.d odd 10 1
275.2.a.a 1 55.h odd 10 1
275.2.b.b 2 55.l even 20 2
495.2.a.a 1 33.f even 10 1
605.2.a.b 1 11.c even 5 1
605.2.g.a 4 11.b odd 2 1
605.2.g.a 4 11.d odd 10 3
605.2.g.c 4 1.a even 1 1 trivial
605.2.g.c 4 11.c even 5 3 inner
880.2.a.h 1 44.g even 10 1
2475.2.a.i 1 165.r even 10 1
2475.2.c.f 2 165.u odd 20 2
2695.2.a.c 1 77.l even 10 1
3025.2.a.f 1 55.j even 10 1
3520.2.a.n 1 88.k even 10 1
3520.2.a.p 1 88.p odd 10 1
4400.2.a.p 1 220.o even 10 1
4400.2.b.n 2 220.w odd 20 2
5445.2.a.i 1 33.h odd 10 1
7920.2.a.i 1 132.n odd 10 1
9295.2.a.b 1 143.l odd 10 1
9680.2.a.r 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 $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1$$ T2^4 - T2^3 + T2^2 - T2 + 1 $$T_{3}$$ T3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$17$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$19$ $$T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256$$
$23$ $$(T - 4)^{4}$$
$29$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$31$ $$T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$37$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$41$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$43$ $$(T + 4)^{4}$$
$47$ $$T^{4} - 12 T^{3} + 144 T^{2} + \cdots + 20736$$
$53$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$59$ $$T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256$$
$61$ $$T^{4} + 10 T^{3} + 100 T^{2} + \cdots + 10000$$
$67$ $$(T + 16)^{4}$$
$71$ $$T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$73$ $$T^{4} - 14 T^{3} + 196 T^{2} + \cdots + 38416$$
$79$ $$T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$83$ $$T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256$$
$89$ $$(T - 10)^{4}$$
$97$ $$T^{4} + 10 T^{3} + 100 T^{2} + \cdots + 10000$$