# Properties

 Label 61.2.f.a Level 61 Weight 2 Character orbit 61.f Analytic conductor 0.487 Analytic rank 1 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$61$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 61.f (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.487087452330$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{2} -2 q^{3} + \zeta_{6} q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + ( 2 + 2 \zeta_{6} ) q^{6} + ( -2 - 2 \zeta_{6} ) q^{7} + ( -1 + 2 \zeta_{6} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{2} -2 q^{3} + \zeta_{6} q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + ( 2 + 2 \zeta_{6} ) q^{6} + ( -2 - 2 \zeta_{6} ) q^{7} + ( -1 + 2 \zeta_{6} ) q^{8} + q^{9} + ( 6 - 3 \zeta_{6} ) q^{10} + ( 2 - 4 \zeta_{6} ) q^{11} -2 \zeta_{6} q^{12} + ( 2 - 2 \zeta_{6} ) q^{13} + 6 \zeta_{6} q^{14} + ( 6 - 6 \zeta_{6} ) q^{15} + ( 5 - 5 \zeta_{6} ) q^{16} + ( -8 + 4 \zeta_{6} ) q^{17} + ( -1 - \zeta_{6} ) q^{18} + 2 \zeta_{6} q^{19} -3 q^{20} + ( 4 + 4 \zeta_{6} ) q^{21} + ( -6 + 6 \zeta_{6} ) q^{22} + ( 2 - 4 \zeta_{6} ) q^{24} -4 \zeta_{6} q^{25} + ( -4 + 2 \zeta_{6} ) q^{26} + 4 q^{27} + ( 2 - 4 \zeta_{6} ) q^{28} + ( -2 + \zeta_{6} ) q^{29} + ( -12 + 6 \zeta_{6} ) q^{30} + ( -12 + 6 \zeta_{6} ) q^{31} + ( -6 + 3 \zeta_{6} ) q^{32} + ( -4 + 8 \zeta_{6} ) q^{33} + 12 q^{34} + ( 12 - 6 \zeta_{6} ) q^{35} + \zeta_{6} q^{36} + ( 1 - 2 \zeta_{6} ) q^{37} + ( 2 - 4 \zeta_{6} ) q^{38} + ( -4 + 4 \zeta_{6} ) q^{39} + ( -3 - 3 \zeta_{6} ) q^{40} + 3 q^{41} -12 \zeta_{6} q^{42} + ( 2 + 2 \zeta_{6} ) q^{43} + ( 4 - 2 \zeta_{6} ) q^{44} + ( -3 + 3 \zeta_{6} ) q^{45} -12 \zeta_{6} q^{47} + ( -10 + 10 \zeta_{6} ) q^{48} + 5 \zeta_{6} q^{49} + ( -4 + 8 \zeta_{6} ) q^{50} + ( 16 - 8 \zeta_{6} ) q^{51} + 2 q^{52} + ( 3 - 6 \zeta_{6} ) q^{53} + ( -4 - 4 \zeta_{6} ) q^{54} + ( 6 + 6 \zeta_{6} ) q^{55} + ( 6 - 6 \zeta_{6} ) q^{56} -4 \zeta_{6} q^{57} + 3 q^{58} + ( -2 - 2 \zeta_{6} ) q^{59} + 6 q^{60} + ( -5 + 9 \zeta_{6} ) q^{61} + 18 q^{62} + ( -2 - 2 \zeta_{6} ) q^{63} - q^{64} + 6 \zeta_{6} q^{65} + ( 12 - 12 \zeta_{6} ) q^{66} + ( -2 - 2 \zeta_{6} ) q^{67} + ( -4 - 4 \zeta_{6} ) q^{68} -18 q^{70} + ( -12 + 6 \zeta_{6} ) q^{71} + ( -1 + 2 \zeta_{6} ) q^{72} + 7 \zeta_{6} q^{73} + ( -3 + 3 \zeta_{6} ) q^{74} + 8 \zeta_{6} q^{75} + ( -2 + 2 \zeta_{6} ) q^{76} + ( -12 + 12 \zeta_{6} ) q^{77} + ( 8 - 4 \zeta_{6} ) q^{78} + ( -6 - 6 \zeta_{6} ) q^{79} + 15 \zeta_{6} q^{80} -11 q^{81} + ( -3 - 3 \zeta_{6} ) q^{82} + ( -6 + 6 \zeta_{6} ) q^{83} + ( -4 + 8 \zeta_{6} ) q^{84} + ( 12 - 24 \zeta_{6} ) q^{85} -6 \zeta_{6} q^{86} + ( 4 - 2 \zeta_{6} ) q^{87} + 6 q^{88} + ( 9 - 18 \zeta_{6} ) q^{89} + ( 6 - 3 \zeta_{6} ) q^{90} + ( -8 + 4 \zeta_{6} ) q^{91} + ( 24 - 12 \zeta_{6} ) q^{93} + ( -12 + 24 \zeta_{6} ) q^{94} -6 q^{95} + ( 12 - 6 \zeta_{6} ) q^{96} -\zeta_{6} q^{97} + ( 5 - 10 \zeta_{6} ) q^{98} + ( 2 - 4 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} - 4q^{3} + q^{4} - 3q^{5} + 6q^{6} - 6q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 3q^{2} - 4q^{3} + q^{4} - 3q^{5} + 6q^{6} - 6q^{7} + 2q^{9} + 9q^{10} - 2q^{12} + 2q^{13} + 6q^{14} + 6q^{15} + 5q^{16} - 12q^{17} - 3q^{18} + 2q^{19} - 6q^{20} + 12q^{21} - 6q^{22} - 4q^{25} - 6q^{26} + 8q^{27} - 3q^{29} - 18q^{30} - 18q^{31} - 9q^{32} + 24q^{34} + 18q^{35} + q^{36} - 4q^{39} - 9q^{40} + 6q^{41} - 12q^{42} + 6q^{43} + 6q^{44} - 3q^{45} - 12q^{47} - 10q^{48} + 5q^{49} + 24q^{51} + 4q^{52} - 12q^{54} + 18q^{55} + 6q^{56} - 4q^{57} + 6q^{58} - 6q^{59} + 12q^{60} - q^{61} + 36q^{62} - 6q^{63} - 2q^{64} + 6q^{65} + 12q^{66} - 6q^{67} - 12q^{68} - 36q^{70} - 18q^{71} + 7q^{73} - 3q^{74} + 8q^{75} - 2q^{76} - 12q^{77} + 12q^{78} - 18q^{79} + 15q^{80} - 22q^{81} - 9q^{82} - 6q^{83} - 6q^{86} + 6q^{87} + 12q^{88} + 9q^{90} - 12q^{91} + 36q^{93} - 12q^{95} + 18q^{96} - q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/61\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.50000 + 0.866025i −2.00000 0.500000 0.866025i −1.50000 2.59808i 3.00000 1.73205i −3.00000 + 1.73205i 1.73205i 1.00000 4.50000 + 2.59808i
48.1 −1.50000 0.866025i −2.00000 0.500000 + 0.866025i −1.50000 + 2.59808i 3.00000 + 1.73205i −3.00000 1.73205i 1.73205i 1.00000 4.50000 2.59808i
 $$n$$: 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
61.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.2.f.a 2
3.b odd 2 1 549.2.s.c 2
4.b odd 2 1 976.2.ba.a 2
61.f even 6 1 inner 61.2.f.a 2
61.h odd 12 2 3721.2.a.b 2
183.i odd 6 1 549.2.s.c 2
244.i odd 6 1 976.2.ba.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.2.f.a 2 1.a even 1 1 trivial
61.2.f.a 2 61.f even 6 1 inner
549.2.s.c 2 3.b odd 2 1
549.2.s.c 2 183.i odd 6 1
976.2.ba.a 2 4.b odd 2 1
976.2.ba.a 2 244.i odd 6 1
3721.2.a.b 2 61.h odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3 T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(61, [\chi])$$.