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Properties

Label 11.18.a.b
Level $11$
Weight $18$
Character orbit 11.a
Self dual yes
Analytic conductor $20.154$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [11,18,Mod(1,11)]
 
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([0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 11.a (trivial)

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: yes
Analytic conductor: \(20.1544296079\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 715858 x^{6} - 57426812 x^{5} + 132277346400 x^{4} + 17831801296448 x^{3} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{5}\cdot 11 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 32) q^{2} + (\beta_{2} - 9 \beta_1 + 382) q^{3} + (\beta_{3} - 2 \beta_{2} + 56 \beta_1 + 48917) q^{4} + ( - \beta_{5} + 3 \beta_{3} + 12 \beta_{2} + 543 \beta_1 + 224401) q^{5} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + 7 \beta_{4} + 21 \beta_{3} + 64 \beta_{2} + \cdots + 1646319) q^{6}+ \cdots + (21 \beta_{7} - 75 \beta_{6} - 5 \beta_{5} + 206 \beta_{4} + \cdots + 51678030) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 32) q^{2} + (\beta_{2} - 9 \beta_1 + 382) q^{3} + (\beta_{3} - 2 \beta_{2} + 56 \beta_1 + 48917) q^{4} + ( - \beta_{5} + 3 \beta_{3} + 12 \beta_{2} + 543 \beta_1 + 224401) q^{5} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + 7 \beta_{4} + 21 \beta_{3} + 64 \beta_{2} + \cdots + 1646319) q^{6}+ \cdots + (4501536501 \beta_{7} - 16076916075 \beta_{6} + \cdots + 11\!\cdots\!30) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 256 q^{2} + 3058 q^{3} + 391332 q^{4} + 1795234 q^{5} + 13170682 q^{6} - 896364 q^{7} - 101682420 q^{8} + 413432462 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 256 q^{2} + 3058 q^{3} + 391332 q^{4} + 1795234 q^{5} + 13170682 q^{6} - 896364 q^{7} - 101682420 q^{8} + 413432462 q^{9} - 717567014 q^{10} + 1714871048 q^{11} - 3285295504 q^{12} + 1804014468 q^{13} + 17794023508 q^{14} + 8875276234 q^{15} + 53793011976 q^{16} - 27416338904 q^{17} + 28258692878 q^{18} + 58429836440 q^{19} + 650385049400 q^{20} + 562668763708 q^{21} + 54875873536 q^{22} + 1231860549578 q^{23} + 3528111117084 q^{24} + 3225446670918 q^{25} + 3698352129748 q^{26} + 5544189136510 q^{27} + 553346903392 q^{28} + 4016405848668 q^{29} + 28329117219490 q^{30} + 21044142033258 q^{31} - 7034951233624 q^{32} + 655509458098 q^{33} + 10491977089288 q^{34} - 37564178328188 q^{35} + 18688387613044 q^{36} - 38179864040434 q^{37} - 32101053490680 q^{38} - 133370005047128 q^{39} - 229151934325836 q^{40} - 84601913468108 q^{41} - 374381853665348 q^{42} - 79795156805452 q^{43} + 83885489619492 q^{44} + 85333967988848 q^{45} - 63876340102558 q^{46} - 333992064138544 q^{47} - 921917930639032 q^{48} + 16663435022976 q^{49} - 203190218406730 q^{50} + 445337187172876 q^{51} + 383080290241336 q^{52} + 351380494472328 q^{53} + 21\!\cdots\!38 q^{54}+ \cdots + 88\!\cdots\!22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 715858 x^{6} - 57426812 x^{5} + 132277346400 x^{4} + 17831801296448 x^{3} + \cdots + 70\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 42647641033 \nu^{7} - 3425448837776 \nu^{6} + \cdots - 98\!\cdots\!32 ) / 32\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 42647641033 \nu^{7} - 3425448837776 \nu^{6} + \cdots - 39\!\cdots\!72 ) / 16\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4701766891807 \nu^{7} - 321806586888208 \nu^{6} + \cdots + 12\!\cdots\!20 ) / 16\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6513534437003 \nu^{7} + \cdots - 51\!\cdots\!20 ) / 16\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4479444878887 \nu^{7} + \cdots + 51\!\cdots\!00 ) / 81\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30191806133087 \nu^{7} + 245791776850448 \nu^{6} + \cdots - 74\!\cdots\!40 ) / 16\!\cdots\!60 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 2\beta_{2} + 120\beta _1 + 178965 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 9\beta_{7} - 26\beta_{6} - 30\beta_{5} + 61\beta_{4} + 97\beta_{3} - 2132\beta_{2} + 346002\beta _1 + 21535591 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11226 \beta_{7} + 8996 \beta_{6} + 5892 \beta_{5} + 15326 \beta_{4} + 458756 \beta_{3} - 1611112 \beta_{2} + 81038020 \beta _1 + 61974903456 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8627130 \beta_{7} - 11208764 \beta_{6} - 13022508 \beta_{5} + 44059450 \beta_{4} + 78207950 \beta_{3} - 1275127288 \beta_{2} + 144297992292 \beta _1 + 14548848572106 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8321612016 \beta_{7} + 4877971440 \beta_{6} + 3184897728 \beta_{5} + 14464576104 \beta_{4} + 206469211044 \beta_{3} - 970011589056 \beta_{2} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5620360918476 \beta_{7} - 4224041205768 \beta_{6} - 5207543139048 \beta_{5} + 24237959556924 \beta_{4} + 52360232595276 \beta_{3} + \cdots + 83\!\cdots\!76 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
713.940
498.086
142.296
112.922
−179.992
−299.068
−369.946
−618.238
−681.940 −16808.1 333970. 1.17558e6 1.14621e7 −2.07343e7 −1.38364e8 1.53374e8 −8.01675e8
1.2 −466.086 5120.97 86164.1 1.02127e6 −2.38681e6 1.61817e7 2.09309e7 −1.02916e8 −4.76000e8
1.3 −110.296 17080.3 −118907. −1.10266e6 −1.88389e6 1.13516e7 2.75717e7 1.62598e8 1.21619e8
1.4 −80.9221 −6715.01 −124524. −672456. 543392. −2.39607e7 2.06833e7 −8.40488e7 5.44165e7
1.5 211.992 −6727.05 −86131.5 1.43574e6 −1.42608e6 −6.38659e6 −4.60453e7 −8.38870e7 3.04365e8
1.6 331.068 −15871.6 −21465.7 −1.31288e6 −5.25459e6 1.35437e7 −5.05004e7 1.22768e8 −4.34652e8
1.7 401.946 21853.0 30488.5 1.20363e6 8.78373e6 −5.86897e6 −4.04291e7 3.48414e8 4.83793e8
1.8 650.238 5125.48 291737. 47006.5 3.33278e6 1.49771e7 1.04471e8 −1.02870e8 3.05654e7
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)

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 11.18.a.b ✓ 8
3.b odd 2 1 99.18.a.e 8
11.b odd 2 1 121.18.a.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.18.a.b ✓ 8 1.a even 1 1 trivial
99.18.a.e 8 3.b odd 2 1
121.18.a.d 8 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 256 T_{2}^{7} - 687186 T_{2}^{6} + 193036540 T_{2}^{5} + 112166877920 T_{2}^{4} - 33707985430080 T_{2}^{3} + \cdots + 52\!\cdots\!04 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(11))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 256 T^{7} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$3$ \( T^{8} - 3058 T^{7} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{8} - 1795234 T^{7} + \cdots - 94\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + 896364 T^{7} + \cdots + 69\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( (T - 214358881)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 1804014468 T^{7} + \cdots - 76\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{8} + 27416338904 T^{7} + \cdots + 82\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{8} - 58429836440 T^{7} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} - 1231860549578 T^{7} + \cdots - 23\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{8} - 4016405848668 T^{7} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} - 21044142033258 T^{7} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + 38179864040434 T^{7} + \cdots + 72\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{8} + 84601913468108 T^{7} + \cdots - 96\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{8} + 79795156805452 T^{7} + \cdots - 68\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{8} + 333992064138544 T^{7} + \cdots + 77\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} - 351380494472328 T^{7} + \cdots + 77\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 52\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 19\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots - 10\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 88\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 24\!\cdots\!84 \) Copy content Toggle raw display
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