# Properties

 Label 1911.2.c.d Level $1911$ Weight $2$ Character orbit 1911.c Analytic conductor $15.259$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1911.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.2594118263$$ 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: $$2$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{2} + q^{3} - q^{4} + ( 1 - 2 \zeta_{6} ) q^{6} + ( 1 - 2 \zeta_{6} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{2} + q^{3} - q^{4} + ( 1 - 2 \zeta_{6} ) q^{6} + ( 1 - 2 \zeta_{6} ) q^{8} + q^{9} + ( -2 + 4 \zeta_{6} ) q^{11} - q^{12} + ( -1 + 4 \zeta_{6} ) q^{13} -5 q^{16} + 6 q^{17} + ( 1 - 2 \zeta_{6} ) q^{18} + ( -2 + 4 \zeta_{6} ) q^{19} + 6 q^{22} + ( 1 - 2 \zeta_{6} ) q^{24} + 5 q^{25} + ( 7 - 2 \zeta_{6} ) q^{26} + q^{27} + 6 q^{29} + ( 2 - 4 \zeta_{6} ) q^{31} + ( -3 + 6 \zeta_{6} ) q^{32} + ( -2 + 4 \zeta_{6} ) q^{33} + ( 6 - 12 \zeta_{6} ) q^{34} - q^{36} + ( 4 - 8 \zeta_{6} ) q^{37} + 6 q^{38} + ( -1 + 4 \zeta_{6} ) q^{39} + ( 4 - 8 \zeta_{6} ) q^{41} -4 q^{43} + ( 2 - 4 \zeta_{6} ) q^{44} + ( -2 + 4 \zeta_{6} ) q^{47} -5 q^{48} + ( 5 - 10 \zeta_{6} ) q^{50} + 6 q^{51} + ( 1 - 4 \zeta_{6} ) q^{52} + 6 q^{53} + ( 1 - 2 \zeta_{6} ) q^{54} + ( -2 + 4 \zeta_{6} ) q^{57} + ( 6 - 12 \zeta_{6} ) q^{58} + ( -6 + 12 \zeta_{6} ) q^{59} + 2 q^{61} -6 q^{62} - q^{64} + 6 q^{66} + ( -6 + 12 \zeta_{6} ) q^{67} -6 q^{68} + ( 2 - 4 \zeta_{6} ) q^{71} + ( 1 - 2 \zeta_{6} ) q^{72} -12 q^{74} + 5 q^{75} + ( 2 - 4 \zeta_{6} ) q^{76} + ( 7 - 2 \zeta_{6} ) q^{78} -8 q^{79} + q^{81} -12 q^{82} + ( 2 - 4 \zeta_{6} ) q^{83} + ( -4 + 8 \zeta_{6} ) q^{86} + 6 q^{87} + 6 q^{88} + ( 4 - 8 \zeta_{6} ) q^{89} + ( 2 - 4 \zeta_{6} ) q^{93} + 6 q^{94} + ( -3 + 6 \zeta_{6} ) q^{96} + ( 8 - 16 \zeta_{6} ) q^{97} + ( -2 + 4 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{4} + 2q^{9} - 2q^{12} + 2q^{13} - 10q^{16} + 12q^{17} + 12q^{22} + 10q^{25} + 12q^{26} + 2q^{27} + 12q^{29} - 2q^{36} + 12q^{38} + 2q^{39} - 8q^{43} - 10q^{48} + 12q^{51} - 2q^{52} + 12q^{53} + 4q^{61} - 12q^{62} - 2q^{64} + 12q^{66} - 12q^{68} - 24q^{74} + 10q^{75} + 12q^{78} - 16q^{79} + 2q^{81} - 24q^{82} + 12q^{87} + 12q^{88} + 12q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times$$.

 $$n$$ $$638$$ $$1471$$ $$1522$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
883.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i 1.00000 −1.00000 0 1.73205i 0 1.73205i 1.00000 0
883.2 1.73205i 1.00000 −1.00000 0 1.73205i 0 1.73205i 1.00000 0
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.2.c.d 2
7.b odd 2 1 39.2.b.a 2
13.b even 2 1 inner 1911.2.c.d 2
21.c even 2 1 117.2.b.a 2
28.d even 2 1 624.2.c.e 2
35.c odd 2 1 975.2.b.d 2
35.f even 4 2 975.2.h.f 4
56.e even 2 1 2496.2.c.d 2
56.h odd 2 1 2496.2.c.k 2
84.h odd 2 1 1872.2.c.e 2
91.b odd 2 1 39.2.b.a 2
91.i even 4 2 507.2.a.f 2
91.n odd 6 1 507.2.j.a 2
91.n odd 6 1 507.2.j.c 2
91.t odd 6 1 507.2.j.a 2
91.t odd 6 1 507.2.j.c 2
91.bc even 12 4 507.2.e.e 4
273.g even 2 1 117.2.b.a 2
273.o odd 4 2 1521.2.a.l 2
364.h even 2 1 624.2.c.e 2
364.p odd 4 2 8112.2.a.bv 2
455.h odd 2 1 975.2.b.d 2
455.s even 4 2 975.2.h.f 4
728.b even 2 1 2496.2.c.d 2
728.l odd 2 1 2496.2.c.k 2
1092.d odd 2 1 1872.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 7.b odd 2 1
39.2.b.a 2 91.b odd 2 1
117.2.b.a 2 21.c even 2 1
117.2.b.a 2 273.g even 2 1
507.2.a.f 2 91.i even 4 2
507.2.e.e 4 91.bc even 12 4
507.2.j.a 2 91.n odd 6 1
507.2.j.a 2 91.t odd 6 1
507.2.j.c 2 91.n odd 6 1
507.2.j.c 2 91.t odd 6 1
624.2.c.e 2 28.d even 2 1
624.2.c.e 2 364.h even 2 1
975.2.b.d 2 35.c odd 2 1
975.2.b.d 2 455.h odd 2 1
975.2.h.f 4 35.f even 4 2
975.2.h.f 4 455.s even 4 2
1521.2.a.l 2 273.o odd 4 2
1872.2.c.e 2 84.h odd 2 1
1872.2.c.e 2 1092.d odd 2 1
1911.2.c.d 2 1.a even 1 1 trivial
1911.2.c.d 2 13.b even 2 1 inner
2496.2.c.d 2 56.e even 2 1
2496.2.c.d 2 728.b even 2 1
2496.2.c.k 2 56.h odd 2 1
2496.2.c.k 2 728.l odd 2 1
8112.2.a.bv 2 364.p odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1911, [\chi])$$:

 $$T_{2}^{2} + 3$$ $$T_{5}$$ $$T_{17} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$12 + T^{2}$$
$13$ $$13 - 2 T + T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$12 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$12 + T^{2}$$
$37$ $$48 + T^{2}$$
$41$ $$48 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$12 + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$108 + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$108 + T^{2}$$
$71$ $$12 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$12 + T^{2}$$
$89$ $$48 + T^{2}$$
$97$ $$192 + T^{2}$$