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 $2$

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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: \( x^{2} + 30 \) Copy content Toggle raw display
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.

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

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:

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 \(S_{5}^{\mathrm{new}}(11, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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