# 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: $$x^{2} - x + 1$$ 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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$S_{2}^{\mathrm{new}}(1911, [\chi])$$:

 $$T_{2}^{2} + 3$$ T2^2 + 3 $$T_{5}$$ T5 $$T_{17} - 6$$ T17 - 6

## Hecke characteristic polynomials

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