# Properties

 Label 11.11.b.a Level 11 Weight 11 Character orbit 11.b Self dual yes Analytic conductor 6.989 Analytic rank 0 Dimension 1 CM discriminant -11 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$11$$ Weight: $$k$$ = $$11$$ Character orbit: $$[\chi]$$ = 11.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.98892977941$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 475q^{3} + 1024q^{4} - 3001q^{5} + 166576q^{9} + O(q^{10})$$ $$q + 475q^{3} + 1024q^{4} - 3001q^{5} + 166576q^{9} - 161051q^{11} + 486400q^{12} - 1425475q^{15} + 1048576q^{16} - 3073024q^{20} - 11910325q^{23} - 759624q^{25} + 51075325q^{27} + 3192323q^{31} - 76499225q^{33} + 170573824q^{36} - 137082625q^{37} - 164916224q^{44} - 499894576q^{45} + 151795250q^{47} + 498073600q^{48} + 282475249q^{49} + 375066650q^{53} + 483314051q^{55} - 813567973q^{59} - 1459686400q^{60} + 1073741824q^{64} + 2616638675q^{67} - 5657404375q^{69} + 783651827q^{71} - 360821400q^{75} - 3146776576q^{80} + 14424633151q^{81} - 2870912977q^{89} - 12196172800q^{92} + 1516353425q^{93} + 9454010975q^{97} - 26827231376q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/11\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 0
0 475.000 1024.00 −3001.00 0 0 0 166576. 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.11.b.a 1
3.b odd 2 1 99.11.c.a 1
4.b odd 2 1 176.11.h.a 1
11.b odd 2 1 CM 11.11.b.a 1
33.d even 2 1 99.11.c.a 1
44.c even 2 1 176.11.h.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.11.b.a 1 1.a even 1 1 trivial
11.11.b.a 1 11.b odd 2 1 CM
99.11.c.a 1 3.b odd 2 1
99.11.c.a 1 33.d even 2 1
176.11.h.a 1 4.b odd 2 1
176.11.h.a 1 44.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{11}^{\mathrm{new}}(11, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 32 T )( 1 + 32 T )$$
$3$ $$1 - 475 T + 59049 T^{2}$$
$5$ $$1 + 3001 T + 9765625 T^{2}$$
$7$ $$( 1 - 16807 T )( 1 + 16807 T )$$
$11$ $$1 + 161051 T$$
$13$ $$( 1 - 371293 T )( 1 + 371293 T )$$
$17$ $$( 1 - 1419857 T )( 1 + 1419857 T )$$
$19$ $$( 1 - 2476099 T )( 1 + 2476099 T )$$
$23$ $$1 + 11910325 T + 41426511213649 T^{2}$$
$29$ $$( 1 - 20511149 T )( 1 + 20511149 T )$$
$31$ $$1 - 3192323 T + 819628286980801 T^{2}$$
$37$ $$1 + 137082625 T + 4808584372417849 T^{2}$$
$41$ $$( 1 - 115856201 T )( 1 + 115856201 T )$$
$43$ $$( 1 - 147008443 T )( 1 + 147008443 T )$$
$47$ $$1 - 151795250 T + 52599132235830049 T^{2}$$
$53$ $$1 - 375066650 T + 174887470365513049 T^{2}$$
$59$ $$1 + 813567973 T + 511116753300641401 T^{2}$$
$61$ $$( 1 - 844596301 T )( 1 + 844596301 T )$$
$67$ $$1 - 2616638675 T + 1822837804551761449 T^{2}$$
$71$ $$1 - 783651827 T + 3255243551009881201 T^{2}$$
$73$ $$( 1 - 2073071593 T )( 1 + 2073071593 T )$$
$79$ $$( 1 - 3077056399 T )( 1 + 3077056399 T )$$
$83$ $$( 1 - 3939040643 T )( 1 + 3939040643 T )$$
$89$ $$1 + 2870912977 T + 31181719929966183601 T^{2}$$
$97$ $$1 - 9454010975 T + 73742412689492826049 T^{2}$$