# Properties

 Label 37.2.c.a Level $37$ Weight $2$ Character orbit 37.c Analytic conductor $0.295$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 37.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.295446487479$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + \zeta_{6} q^{4} -\zeta_{6} q^{5} -2 \zeta_{6} q^{7} -3 q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + \zeta_{6} q^{4} -\zeta_{6} q^{5} -2 \zeta_{6} q^{7} -3 q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} + q^{10} -2 q^{11} + 2 \zeta_{6} q^{13} + 2 q^{14} + ( 1 - \zeta_{6} ) q^{16} + ( -3 + 3 \zeta_{6} ) q^{17} + 3 \zeta_{6} q^{18} + 6 \zeta_{6} q^{19} + ( 1 - \zeta_{6} ) q^{20} + ( 2 - 2 \zeta_{6} ) q^{22} -4 q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} -2 q^{26} + ( 2 - 2 \zeta_{6} ) q^{28} + 9 q^{29} -10 q^{31} -5 \zeta_{6} q^{32} -3 \zeta_{6} q^{34} + ( -2 + 2 \zeta_{6} ) q^{35} + 3 q^{36} + ( -4 - 3 \zeta_{6} ) q^{37} -6 q^{38} + 3 \zeta_{6} q^{40} + 9 \zeta_{6} q^{41} + 2 q^{43} -2 \zeta_{6} q^{44} -3 q^{45} + ( 4 - 4 \zeta_{6} ) q^{46} + 6 q^{47} + ( 3 - 3 \zeta_{6} ) q^{49} + 4 \zeta_{6} q^{50} + ( -2 + 2 \zeta_{6} ) q^{52} + ( 2 - 2 \zeta_{6} ) q^{53} + 2 \zeta_{6} q^{55} + 6 \zeta_{6} q^{56} + ( -9 + 9 \zeta_{6} ) q^{58} + ( 4 - 4 \zeta_{6} ) q^{59} -\zeta_{6} q^{61} + ( 10 - 10 \zeta_{6} ) q^{62} -6 q^{63} + 7 q^{64} + ( 2 - 2 \zeta_{6} ) q^{65} + 10 \zeta_{6} q^{67} -3 q^{68} -2 \zeta_{6} q^{70} -6 \zeta_{6} q^{71} + ( -9 + 9 \zeta_{6} ) q^{72} -10 q^{73} + ( 7 - 4 \zeta_{6} ) q^{74} + ( -6 + 6 \zeta_{6} ) q^{76} + 4 \zeta_{6} q^{77} -10 \zeta_{6} q^{79} - q^{80} -9 \zeta_{6} q^{81} -9 q^{82} + ( 12 - 12 \zeta_{6} ) q^{83} + 3 q^{85} + ( -2 + 2 \zeta_{6} ) q^{86} + 6 q^{88} + ( -7 + 7 \zeta_{6} ) q^{89} + ( 3 - 3 \zeta_{6} ) q^{90} + ( 4 - 4 \zeta_{6} ) q^{91} -4 \zeta_{6} q^{92} + ( -6 + 6 \zeta_{6} ) q^{94} + ( 6 - 6 \zeta_{6} ) q^{95} + 7 q^{97} + 3 \zeta_{6} q^{98} + ( -6 + 6 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{4} - q^{5} - 2q^{7} - 6q^{8} + 3q^{9} + O(q^{10})$$ $$2q - q^{2} + q^{4} - q^{5} - 2q^{7} - 6q^{8} + 3q^{9} + 2q^{10} - 4q^{11} + 2q^{13} + 4q^{14} + q^{16} - 3q^{17} + 3q^{18} + 6q^{19} + q^{20} + 2q^{22} - 8q^{23} + 4q^{25} - 4q^{26} + 2q^{28} + 18q^{29} - 20q^{31} - 5q^{32} - 3q^{34} - 2q^{35} + 6q^{36} - 11q^{37} - 12q^{38} + 3q^{40} + 9q^{41} + 4q^{43} - 2q^{44} - 6q^{45} + 4q^{46} + 12q^{47} + 3q^{49} + 4q^{50} - 2q^{52} + 2q^{53} + 2q^{55} + 6q^{56} - 9q^{58} + 4q^{59} - q^{61} + 10q^{62} - 12q^{63} + 14q^{64} + 2q^{65} + 10q^{67} - 6q^{68} - 2q^{70} - 6q^{71} - 9q^{72} - 20q^{73} + 10q^{74} - 6q^{76} + 4q^{77} - 10q^{79} - 2q^{80} - 9q^{81} - 18q^{82} + 12q^{83} + 6q^{85} - 2q^{86} + 12q^{88} - 7q^{89} + 3q^{90} + 4q^{91} - 4q^{92} - 6q^{94} + 6q^{95} + 14q^{97} + 3q^{98} - 6q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/37\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 1.73205i −3.00000 1.50000 2.59808i 1.00000
26.1 −0.500000 0.866025i 0 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 + 1.73205i −3.00000 1.50000 + 2.59808i 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
37.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.2.c.a 2
3.b odd 2 1 333.2.f.a 2
4.b odd 2 1 592.2.i.c 2
5.b even 2 1 925.2.e.a 2
5.c odd 4 2 925.2.o.a 4
37.c even 3 1 inner 37.2.c.a 2
37.c even 3 1 1369.2.a.d 1
37.e even 6 1 1369.2.a.b 1
37.g odd 12 2 1369.2.b.b 2
111.i odd 6 1 333.2.f.a 2
148.i odd 6 1 592.2.i.c 2
185.n even 6 1 925.2.e.a 2
185.s odd 12 2 925.2.o.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.c.a 2 1.a even 1 1 trivial
37.2.c.a 2 37.c even 3 1 inner
333.2.f.a 2 3.b odd 2 1
333.2.f.a 2 111.i odd 6 1
592.2.i.c 2 4.b odd 2 1
592.2.i.c 2 148.i odd 6 1
925.2.e.a 2 5.b even 2 1
925.2.e.a 2 185.n even 6 1
925.2.o.a 4 5.c odd 4 2
925.2.o.a 4 185.s odd 12 2
1369.2.a.b 1 37.e even 6 1
1369.2.a.d 1 37.c even 3 1
1369.2.b.b 2 37.g odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(37, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$4 + 2 T + T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$4 - 2 T + T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$36 - 6 T + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$( -9 + T )^{2}$$
$31$ $$( 10 + T )^{2}$$
$37$ $$37 + 11 T + T^{2}$$
$41$ $$81 - 9 T + T^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$( -6 + T )^{2}$$
$53$ $$4 - 2 T + T^{2}$$
$59$ $$16 - 4 T + T^{2}$$
$61$ $$1 + T + T^{2}$$
$67$ $$100 - 10 T + T^{2}$$
$71$ $$36 + 6 T + T^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$100 + 10 T + T^{2}$$
$83$ $$144 - 12 T + T^{2}$$
$89$ $$49 + 7 T + T^{2}$$
$97$ $$( -7 + T )^{2}$$