# Properties

 Label 1911.2.a.f Level $1911$ Weight $2$ Character orbit 1911.a Self dual yes Analytic conductor $15.259$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1911.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.2594118263$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} - 2 q^{5} + q^{6} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 - q^4 - 2 * q^5 + q^6 - 3 * q^8 + q^9 $$q + q^{2} + q^{3} - q^{4} - 2 q^{5} + q^{6} - 3 q^{8} + q^{9} - 2 q^{10} + 4 q^{11} - q^{12} - q^{13} - 2 q^{15} - q^{16} - 2 q^{17} + q^{18} + 2 q^{20} + 4 q^{22} - 3 q^{24} - q^{25} - q^{26} + q^{27} - 10 q^{29} - 2 q^{30} - 4 q^{31} + 5 q^{32} + 4 q^{33} - 2 q^{34} - q^{36} - 2 q^{37} - q^{39} + 6 q^{40} - 6 q^{41} - 12 q^{43} - 4 q^{44} - 2 q^{45} - q^{48} - q^{50} - 2 q^{51} + q^{52} + 6 q^{53} + q^{54} - 8 q^{55} - 10 q^{58} - 12 q^{59} + 2 q^{60} + 2 q^{61} - 4 q^{62} + 7 q^{64} + 2 q^{65} + 4 q^{66} - 8 q^{67} + 2 q^{68} - 3 q^{72} - 2 q^{73} - 2 q^{74} - q^{75} - q^{78} + 8 q^{79} + 2 q^{80} + q^{81} - 6 q^{82} - 4 q^{83} + 4 q^{85} - 12 q^{86} - 10 q^{87} - 12 q^{88} + 2 q^{89} - 2 q^{90} - 4 q^{93} + 5 q^{96} - 10 q^{97} + 4 q^{99}+O(q^{100})$$ q + q^2 + q^3 - q^4 - 2 * q^5 + q^6 - 3 * q^8 + q^9 - 2 * q^10 + 4 * q^11 - q^12 - q^13 - 2 * q^15 - q^16 - 2 * q^17 + q^18 + 2 * q^20 + 4 * q^22 - 3 * q^24 - q^25 - q^26 + q^27 - 10 * q^29 - 2 * q^30 - 4 * q^31 + 5 * q^32 + 4 * q^33 - 2 * q^34 - q^36 - 2 * q^37 - q^39 + 6 * q^40 - 6 * q^41 - 12 * q^43 - 4 * q^44 - 2 * q^45 - q^48 - q^50 - 2 * q^51 + q^52 + 6 * q^53 + q^54 - 8 * q^55 - 10 * q^58 - 12 * q^59 + 2 * q^60 + 2 * q^61 - 4 * q^62 + 7 * q^64 + 2 * q^65 + 4 * q^66 - 8 * q^67 + 2 * q^68 - 3 * q^72 - 2 * q^73 - 2 * q^74 - q^75 - q^78 + 8 * q^79 + 2 * q^80 + q^81 - 6 * q^82 - 4 * q^83 + 4 * q^85 - 12 * q^86 - 10 * q^87 - 12 * q^88 + 2 * q^89 - 2 * q^90 - 4 * q^93 + 5 * q^96 - 10 * q^97 + 4 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 1.00000 −1.00000 −2.00000 1.00000 0 −3.00000 1.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.2.a.f 1
3.b odd 2 1 5733.2.a.e 1
7.b odd 2 1 39.2.a.a 1
21.c even 2 1 117.2.a.a 1
28.d even 2 1 624.2.a.i 1
35.c odd 2 1 975.2.a.f 1
35.f even 4 2 975.2.c.f 2
56.e even 2 1 2496.2.a.e 1
56.h odd 2 1 2496.2.a.q 1
63.l odd 6 2 1053.2.e.b 2
63.o even 6 2 1053.2.e.d 2
77.b even 2 1 4719.2.a.c 1
84.h odd 2 1 1872.2.a.h 1
91.b odd 2 1 507.2.a.a 1
91.i even 4 2 507.2.b.a 2
91.n odd 6 2 507.2.e.a 2
91.t odd 6 2 507.2.e.b 2
91.bc even 12 4 507.2.j.e 4
105.g even 2 1 2925.2.a.p 1
105.k odd 4 2 2925.2.c.e 2
168.e odd 2 1 7488.2.a.by 1
168.i even 2 1 7488.2.a.bl 1
273.g even 2 1 1521.2.a.e 1
273.o odd 4 2 1521.2.b.b 2
364.h even 2 1 8112.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 7.b odd 2 1
117.2.a.a 1 21.c even 2 1
507.2.a.a 1 91.b odd 2 1
507.2.b.a 2 91.i even 4 2
507.2.e.a 2 91.n odd 6 2
507.2.e.b 2 91.t odd 6 2
507.2.j.e 4 91.bc even 12 4
624.2.a.i 1 28.d even 2 1
975.2.a.f 1 35.c odd 2 1
975.2.c.f 2 35.f even 4 2
1053.2.e.b 2 63.l odd 6 2
1053.2.e.d 2 63.o even 6 2
1521.2.a.e 1 273.g even 2 1
1521.2.b.b 2 273.o odd 4 2
1872.2.a.h 1 84.h odd 2 1
1911.2.a.f 1 1.a even 1 1 trivial
2496.2.a.e 1 56.e even 2 1
2496.2.a.q 1 56.h odd 2 1
2925.2.a.p 1 105.g even 2 1
2925.2.c.e 2 105.k odd 4 2
4719.2.a.c 1 77.b even 2 1
5733.2.a.e 1 3.b odd 2 1
7488.2.a.bl 1 168.i even 2 1
7488.2.a.by 1 168.e odd 2 1
8112.2.a.s 1 364.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1911))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{5} + 2$$ T5 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T + 2$$
$7$ $$T$$
$11$ $$T - 4$$
$13$ $$T + 1$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 10$$
$31$ $$T + 4$$
$37$ $$T + 2$$
$41$ $$T + 6$$
$43$ $$T + 12$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T + 12$$
$61$ $$T - 2$$
$67$ $$T + 8$$
$71$ $$T$$
$73$ $$T + 2$$
$79$ $$T - 8$$
$83$ $$T + 4$$
$89$ $$T - 2$$
$97$ $$T + 10$$