Properties

Label 11.9.b.b
Level $11$
Weight $9$
Character orbit 11.b
Analytic conductor $4.481$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [11,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.10");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(4.48116471067\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 1374x^{4} + 436560x^{2} + 40320000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 5 \)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} - 6) q^{3} + ( - 2 \beta_{4} + 3 \beta_{3} - 203) q^{4} + ( - 3 \beta_{4} - 5 \beta_{3} - 73) q^{5} + ( - \beta_{5} - 23 \beta_1) q^{6} + ( - 2 \beta_{5} - \beta_{2} + 13 \beta_1) q^{7} + ( - 2 \beta_{5} + 3 \beta_{2} - 227 \beta_1) q^{8} + ( - 27 \beta_{4} + 15 \beta_{3} - 2268) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{4} - 6) q^{3} + ( - 2 \beta_{4} + 3 \beta_{3} - 203) q^{4} + ( - 3 \beta_{4} - 5 \beta_{3} - 73) q^{5} + ( - \beta_{5} - 23 \beta_1) q^{6} + ( - 2 \beta_{5} - \beta_{2} + 13 \beta_1) q^{7} + ( - 2 \beta_{5} + 3 \beta_{2} - 227 \beta_1) q^{8} + ( - 27 \beta_{4} + 15 \beta_{3} - 2268) q^{9} + ( - 3 \beta_{5} - 5 \beta_{2} + 286 \beta_1) q^{10} + (11 \beta_{5} - 110 \beta_{4} + 11 \beta_{3} + 198 \beta_1 - 5390) q^{11} + (14 \beta_{4} - 105 \beta_{3} + 9129) q^{12} + (20 \beta_{5} + 7 \beta_{2} + 233 \beta_1) q^{13} + (308 \beta_{4} + 378 \beta_{3} - 5754) q^{14} + (179 \beta_{4} + 270 \beta_{3} + 14214) q^{15} + (732 \beta_{4} - 1218 \beta_{3} + 52450) q^{16} + (34 \beta_{5} - 9 \beta_{2} + 485 \beta_1) q^{17} + ( - 27 \beta_{5} + 15 \beta_{2} - 3957 \beta_1) q^{18} + ( - 4 \beta_{5} - 45 \beta_{2} + 3253 \beta_1) q^{19} + ( - 1238 \beta_{4} + 1525 \beta_{3} - 149653) q^{20} + (10 \beta_{5} + 75 \beta_{2} + 2945 \beta_1) q^{21} + ( - 110 \beta_{5} - 2860 \beta_{4} + 990 \beta_{3} + 11 \beta_{2} + \cdots - 92070) q^{22}+ \cdots + ( - 6897 \beta_{5} + 219153 \beta_{4} - 143781 \beta_{3} + \cdots + 26568828) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 36 q^{3} - 1212 q^{4} - 448 q^{5} - 13578 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 36 q^{3} - 1212 q^{4} - 448 q^{5} - 13578 q^{9} - 32318 q^{11} + 54564 q^{12} - 33768 q^{14} + 85824 q^{15} + 312264 q^{16} - 894868 q^{20} - 550440 q^{22} + 683084 q^{23} - 141498 q^{25} - 657432 q^{26} + 988848 q^{27} + 942684 q^{31} + 2992704 q^{33} - 1345128 q^{34} + 7401360 q^{36} - 3804816 q^{37} - 8900760 q^{38} - 8158920 q^{42} + 14210284 q^{44} - 2499684 q^{45} + 15828644 q^{47} - 19028376 q^{48} - 10066602 q^{49} - 35477956 q^{53} + 7335372 q^{55} + 68829936 q^{56} + 65482560 q^{58} - 29614804 q^{59} + 35069244 q^{60} - 212921520 q^{64} + 75167400 q^{66} + 39419484 q^{67} + 62859468 q^{69} - 168190680 q^{70} + 3219212 q^{71} - 110874276 q^{75} + 91605360 q^{77} + 111889320 q^{78} + 383392952 q^{80} - 103764906 q^{81} - 163977720 q^{82} - 261274512 q^{86} + 328724880 q^{88} + 38785664 q^{89} + 355260528 q^{91} - 456128956 q^{92} - 182572452 q^{93} - 222185616 q^{97} + 159125406 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 1374x^{4} + 436560x^{2} + 40320000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 1134\nu^{3} - 171240\nu ) / 120 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} - 1134\nu^{2} - 181080 ) / 120 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{4} - 1174\nu^{2} - 199440 ) / 80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - 1174\nu^{3} - 200800\nu ) / 80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{4} + 3\beta_{3} - 459 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{5} + 3\beta_{2} - 739\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2268\beta_{4} - 3522\beta_{3} + 339426 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2268\beta_{5} - 3522\beta_{2} + 666786\beta_1 \) 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
31.0620i
14.5774i
14.0233i
14.0233i
14.5774i
31.0620i
31.0620i −35.5230 −708.845 583.097 1103.41i 1704.90i 14066.3i −5299.12 18112.1i
10.2 14.5774i −66.9966 43.4987 −870.143 976.639i 4079.11i 4365.92i −2072.45 12684.4i
10.3 14.0233i 84.5196 59.3467 63.0464 1185.25i 1667.89i 4422.21i 582.567 884.120i
10.4 14.0233i 84.5196 59.3467 63.0464 1185.25i 1667.89i 4422.21i 582.567 884.120i
10.5 14.5774i −66.9966 43.4987 −870.143 976.639i 4079.11i 4365.92i −2072.45 12684.4i
10.6 31.0620i −35.5230 −708.845 583.097 1103.41i 1704.90i 14066.3i −5299.12 18112.1i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.6
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 11.9.b.b 6
3.b odd 2 1 99.9.c.b 6
4.b odd 2 1 176.9.h.c 6
11.b odd 2 1 inner 11.9.b.b 6
33.d even 2 1 99.9.c.b 6
44.c even 2 1 176.9.h.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.9.b.b 6 1.a even 1 1 trivial
11.9.b.b 6 11.b odd 2 1 inner
99.9.c.b 6 3.b odd 2 1
99.9.c.b 6 33.d even 2 1
176.9.h.c 6 4.b odd 2 1
176.9.h.c 6 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 1374T_{2}^{4} + 436560T_{2}^{2} + 40320000 \) acting on \(S_{9}^{\mathrm{new}}(11, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 1374 T^{4} + \cdots + 40320000 \) Copy content Toggle raw display
$3$ \( (T^{3} + 18 T^{2} - 6285 T - 201150)^{2} \) Copy content Toggle raw display
$5$ \( (T^{3} + 224 T^{2} - 525475 T + 31988350)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + 22327704 T^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{6} + 32318 T^{5} + \cdots + 98\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( T^{6} + 1640220984 T^{4} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{6} + 5604320664 T^{4} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{6} + 45575148600 T^{4} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{3} - 341542 T^{2} + \cdots + 26\!\cdots\!50)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 1589048565600 T^{4} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} - 471342 T^{2} + \cdots + 11\!\cdots\!58)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 1902408 T^{2} + \cdots - 12\!\cdots\!50)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 13773947357400 T^{4} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{6} + 53135758307904 T^{4} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{3} - 7914322 T^{2} + \cdots + 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 17738978 T^{2} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} + 14807402 T^{2} + \cdots - 48\!\cdots\!18)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + 851903213390400 T^{4} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{3} - 19709742 T^{2} + \cdots + 61\!\cdots\!50)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 1609606 T^{2} + \cdots + 10\!\cdots\!42)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{3} - 19392832 T^{2} + \cdots - 62\!\cdots\!02)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 111092808 T^{2} + \cdots - 22\!\cdots\!50)^{2} \) Copy content Toggle raw display
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