# Properties

 Label 11.5.b.b Level $11$ Weight $5$ Character orbit 11.b Analytic conductor $1.137$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [11,5,Mod(10,11)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(11, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("11.10");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$11$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 11.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.13706959392$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-30})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 30$$ x^2 + 30 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-30}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 3 q^{3} - 14 q^{4} + 31 q^{5} - 3 \beta q^{6} - 10 \beta q^{7} + 2 \beta q^{8} - 72 q^{9} +O(q^{10})$$ q + b * q^2 - 3 * q^3 - 14 * q^4 + 31 * q^5 - 3*b * q^6 - 10*b * q^7 + 2*b * q^8 - 72 * q^9 $$q + \beta q^{2} - 3 q^{3} - 14 q^{4} + 31 q^{5} - 3 \beta q^{6} - 10 \beta q^{7} + 2 \beta q^{8} - 72 q^{9} + 31 \beta q^{10} + ( - 22 \beta + 11) q^{11} + 42 q^{12} + 34 \beta q^{13} + 300 q^{14} - 93 q^{15} - 284 q^{16} - 42 \beta q^{17} - 72 \beta q^{18} + 18 \beta q^{19} - 434 q^{20} + 30 \beta q^{21} + (11 \beta + 660) q^{22} + 277 q^{23} - 6 \beta q^{24} + 336 q^{25} - 1020 q^{26} + 459 q^{27} + 140 \beta q^{28} + 232 \beta q^{29} - 93 \beta q^{30} - 1363 q^{31} - 252 \beta q^{32} + (66 \beta - 33) q^{33} + 1260 q^{34} - 310 \beta q^{35} + 1008 q^{36} + 167 q^{37} - 540 q^{38} - 102 \beta q^{39} + 62 \beta q^{40} + 194 \beta q^{41} - 900 q^{42} - 220 \beta q^{43} + (308 \beta - 154) q^{44} - 2232 q^{45} + 277 \beta q^{46} + 1702 q^{47} + 852 q^{48} - 599 q^{49} + 336 \beta q^{50} + 126 \beta q^{51} - 476 \beta q^{52} + 4522 q^{53} + 459 \beta q^{54} + ( - 682 \beta + 341) q^{55} + 600 q^{56} - 54 \beta q^{57} - 6960 q^{58} - 2363 q^{59} + 1302 q^{60} - 724 \beta q^{61} - 1363 \beta q^{62} + 720 \beta q^{63} + 3016 q^{64} + 1054 \beta q^{65} + ( - 33 \beta - 1980) q^{66} - 2803 q^{67} + 588 \beta q^{68} - 831 q^{69} + 9300 q^{70} + 3397 q^{71} - 144 \beta q^{72} + 606 \beta q^{73} + 167 \beta q^{74} - 1008 q^{75} - 252 \beta q^{76} + ( - 110 \beta - 6600) q^{77} + 3060 q^{78} + 1112 \beta q^{79} - 8804 q^{80} + 4455 q^{81} - 5820 q^{82} - 152 \beta q^{83} - 420 \beta q^{84} - 1302 \beta q^{85} + 6600 q^{86} - 696 \beta q^{87} + (22 \beta + 1320) q^{88} - 4673 q^{89} - 2232 \beta q^{90} + 10200 q^{91} - 3878 q^{92} + 4089 q^{93} + 1702 \beta q^{94} + 558 \beta q^{95} + 756 \beta q^{96} + 4247 q^{97} - 599 \beta q^{98} + (1584 \beta - 792) q^{99} +O(q^{100})$$ q + b * q^2 - 3 * q^3 - 14 * q^4 + 31 * q^5 - 3*b * q^6 - 10*b * q^7 + 2*b * q^8 - 72 * q^9 + 31*b * q^10 + (-22*b + 11) * q^11 + 42 * q^12 + 34*b * q^13 + 300 * q^14 - 93 * q^15 - 284 * q^16 - 42*b * q^17 - 72*b * q^18 + 18*b * q^19 - 434 * q^20 + 30*b * q^21 + (11*b + 660) * q^22 + 277 * q^23 - 6*b * q^24 + 336 * q^25 - 1020 * q^26 + 459 * q^27 + 140*b * q^28 + 232*b * q^29 - 93*b * q^30 - 1363 * q^31 - 252*b * q^32 + (66*b - 33) * q^33 + 1260 * q^34 - 310*b * q^35 + 1008 * q^36 + 167 * q^37 - 540 * q^38 - 102*b * q^39 + 62*b * q^40 + 194*b * q^41 - 900 * q^42 - 220*b * q^43 + (308*b - 154) * q^44 - 2232 * q^45 + 277*b * q^46 + 1702 * q^47 + 852 * q^48 - 599 * q^49 + 336*b * q^50 + 126*b * q^51 - 476*b * q^52 + 4522 * q^53 + 459*b * q^54 + (-682*b + 341) * q^55 + 600 * q^56 - 54*b * q^57 - 6960 * q^58 - 2363 * q^59 + 1302 * q^60 - 724*b * q^61 - 1363*b * q^62 + 720*b * q^63 + 3016 * q^64 + 1054*b * q^65 + (-33*b - 1980) * q^66 - 2803 * q^67 + 588*b * q^68 - 831 * q^69 + 9300 * q^70 + 3397 * q^71 - 144*b * q^72 + 606*b * q^73 + 167*b * q^74 - 1008 * q^75 - 252*b * q^76 + (-110*b - 6600) * q^77 + 3060 * q^78 + 1112*b * q^79 - 8804 * q^80 + 4455 * q^81 - 5820 * q^82 - 152*b * q^83 - 420*b * q^84 - 1302*b * q^85 + 6600 * q^86 - 696*b * q^87 + (22*b + 1320) * q^88 - 4673 * q^89 - 2232*b * q^90 + 10200 * q^91 - 3878 * q^92 + 4089 * q^93 + 1702*b * q^94 + 558*b * q^95 + 756*b * q^96 + 4247 * q^97 - 599*b * q^98 + (1584*b - 792) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 28 q^{4} + 62 q^{5} - 144 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 28 * q^4 + 62 * q^5 - 144 * q^9 $$2 q - 6 q^{3} - 28 q^{4} + 62 q^{5} - 144 q^{9} + 22 q^{11} + 84 q^{12} + 600 q^{14} - 186 q^{15} - 568 q^{16} - 868 q^{20} + 1320 q^{22} + 554 q^{23} + 672 q^{25} - 2040 q^{26} + 918 q^{27} - 2726 q^{31} - 66 q^{33} + 2520 q^{34} + 2016 q^{36} + 334 q^{37} - 1080 q^{38} - 1800 q^{42} - 308 q^{44} - 4464 q^{45} + 3404 q^{47} + 1704 q^{48} - 1198 q^{49} + 9044 q^{53} + 682 q^{55} + 1200 q^{56} - 13920 q^{58} - 4726 q^{59} + 2604 q^{60} + 6032 q^{64} - 3960 q^{66} - 5606 q^{67} - 1662 q^{69} + 18600 q^{70} + 6794 q^{71} - 2016 q^{75} - 13200 q^{77} + 6120 q^{78} - 17608 q^{80} + 8910 q^{81} - 11640 q^{82} + 13200 q^{86} + 2640 q^{88} - 9346 q^{89} + 20400 q^{91} - 7756 q^{92} + 8178 q^{93} + 8494 q^{97} - 1584 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 - 28 * q^4 + 62 * q^5 - 144 * q^9 + 22 * q^11 + 84 * q^12 + 600 * q^14 - 186 * q^15 - 568 * q^16 - 868 * q^20 + 1320 * q^22 + 554 * q^23 + 672 * q^25 - 2040 * q^26 + 918 * q^27 - 2726 * q^31 - 66 * q^33 + 2520 * q^34 + 2016 * q^36 + 334 * q^37 - 1080 * q^38 - 1800 * q^42 - 308 * q^44 - 4464 * q^45 + 3404 * q^47 + 1704 * q^48 - 1198 * q^49 + 9044 * q^53 + 682 * q^55 + 1200 * q^56 - 13920 * q^58 - 4726 * q^59 + 2604 * q^60 + 6032 * q^64 - 3960 * q^66 - 5606 * q^67 - 1662 * q^69 + 18600 * q^70 + 6794 * q^71 - 2016 * q^75 - 13200 * q^77 + 6120 * q^78 - 17608 * q^80 + 8910 * q^81 - 11640 * q^82 + 13200 * q^86 + 2640 * q^88 - 9346 * q^89 + 20400 * q^91 - 7756 * q^92 + 8178 * q^93 + 8494 * q^97 - 1584 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 − 5.47723i 5.47723i
5.47723i −3.00000 −14.0000 31.0000 16.4317i 54.7723i 10.9545i −72.0000 169.794i
10.2 5.47723i −3.00000 −14.0000 31.0000 16.4317i 54.7723i 10.9545i −72.0000 169.794i
 $$n$$: 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.5.b.b 2
3.b odd 2 1 99.5.c.b 2
4.b odd 2 1 176.5.h.c 2
5.b even 2 1 275.5.c.e 2
5.c odd 4 2 275.5.d.b 4
8.b even 2 1 704.5.h.f 2
8.d odd 2 1 704.5.h.d 2
11.b odd 2 1 inner 11.5.b.b 2
11.c even 5 4 121.5.d.b 8
11.d odd 10 4 121.5.d.b 8
33.d even 2 1 99.5.c.b 2
44.c even 2 1 176.5.h.c 2
55.d odd 2 1 275.5.c.e 2
55.e even 4 2 275.5.d.b 4
88.b odd 2 1 704.5.h.f 2
88.g even 2 1 704.5.h.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.5.b.b 2 1.a even 1 1 trivial
11.5.b.b 2 11.b odd 2 1 inner
99.5.c.b 2 3.b odd 2 1
99.5.c.b 2 33.d even 2 1
121.5.d.b 8 11.c even 5 4
121.5.d.b 8 11.d odd 10 4
176.5.h.c 2 4.b odd 2 1
176.5.h.c 2 44.c even 2 1
275.5.c.e 2 5.b even 2 1
275.5.c.e 2 55.d odd 2 1
275.5.d.b 4 5.c odd 4 2
275.5.d.b 4 55.e even 4 2
704.5.h.d 2 8.d odd 2 1
704.5.h.d 2 88.g even 2 1
704.5.h.f 2 8.b even 2 1
704.5.h.f 2 88.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 30$$ acting on $$S_{5}^{\mathrm{new}}(11, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 30$$
$3$ $$(T + 3)^{2}$$
$5$ $$(T - 31)^{2}$$
$7$ $$T^{2} + 3000$$
$11$ $$T^{2} - 22T + 14641$$
$13$ $$T^{2} + 34680$$
$17$ $$T^{2} + 52920$$
$19$ $$T^{2} + 9720$$
$23$ $$(T - 277)^{2}$$
$29$ $$T^{2} + 1614720$$
$31$ $$(T + 1363)^{2}$$
$37$ $$(T - 167)^{2}$$
$41$ $$T^{2} + 1129080$$
$43$ $$T^{2} + 1452000$$
$47$ $$(T - 1702)^{2}$$
$53$ $$(T - 4522)^{2}$$
$59$ $$(T + 2363)^{2}$$
$61$ $$T^{2} + 15725280$$
$67$ $$(T + 2803)^{2}$$
$71$ $$(T - 3397)^{2}$$
$73$ $$T^{2} + 11017080$$
$79$ $$T^{2} + 37096320$$
$83$ $$T^{2} + 693120$$
$89$ $$(T + 4673)^{2}$$
$97$ $$(T - 4247)^{2}$$