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Properties

Label 1089.4.a.bm
Level $1089$
Weight $4$
Character orbit 1089.a
Self dual yes
Analytic conductor $64.253$
Analytic rank $0$
Dimension $12$
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] = [1089,4,Mod(1,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1089.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: \(64.2530799963\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 77x^{10} + 2138x^{8} - 25937x^{6} + 133491x^{4} - 221760x^{2} + 9680 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 11^{2} \)
Twist minimal: no (minimal twist has level 99)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + (\beta_{5} + \beta_1) q^{5} + (\beta_{6} + 6) q^{7} + (\beta_{5} + \beta_{4} + 6 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + (\beta_{5} + \beta_1) q^{5} + (\beta_{6} + 6) q^{7} + (\beta_{5} + \beta_{4} + 6 \beta_1) q^{8} + (\beta_{9} - 2 \beta_{7} - \beta_{6} + \cdots + 9) q^{10}+ \cdots + ( - 13 \beta_{11} + 8 \beta_{10} + \cdots + 26 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 58 q^{4} + 66 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 58 q^{4} + 66 q^{7} + 104 q^{10} + 120 q^{13} + 378 q^{16} + 360 q^{19} + 526 q^{25} + 560 q^{28} + 90 q^{31} - 1260 q^{34} - 428 q^{37} + 2376 q^{40} + 2004 q^{43} + 1282 q^{46} + 1186 q^{49} + 3198 q^{52} - 914 q^{58} + 2344 q^{61} + 1684 q^{64} - 968 q^{67} - 5922 q^{70} + 322 q^{73} + 7668 q^{76} + 3482 q^{79} + 4314 q^{82} + 636 q^{85} - 380 q^{91} + 5904 q^{94} + 3258 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 77x^{10} + 2138x^{8} - 25937x^{6} + 133491x^{4} - 221760x^{2} + 9680 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 47\nu^{10} - 3678\nu^{8} + 101738\nu^{6} - 1123697\nu^{4} + 3616646\nu^{2} + 957352 ) / 152944 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -182\nu^{11} + 14187\nu^{9} - 401212\nu^{7} + 4977572\nu^{5} - 25442681\nu^{3} + 31708292\nu ) / 458832 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 182\nu^{11} - 14187\nu^{9} + 401212\nu^{7} - 4977572\nu^{5} + 25901513\nu^{3} - 41802596\nu ) / 458832 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 57\nu^{10} - 3684\nu^{8} + 71910\nu^{6} - 341651\nu^{4} - 1123056\nu^{2} + 2412256 ) / 152944 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{10} - 223\nu^{8} + 5714\nu^{6} - 58571\nu^{4} + 209945\nu^{2} - 82148 ) / 6952 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 112\nu^{11} - 8931\nu^{9} + 258932\nu^{7} - 3322618\nu^{5} + 18352873\nu^{3} - 32448328\nu ) / 229416 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23\nu^{10} - 1578\nu^{8} + 37066\nu^{6} - 343553\nu^{4} + 1095578\nu^{2} - 582296 ) / 41712 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -169\nu^{11} + 12615\nu^{9} - 330842\nu^{7} + 3587797\nu^{5} - 14476825\nu^{3} + 11147488\nu ) / 229416 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -173\nu^{11} + 12965\nu^{9} - 347414\nu^{7} + 4030661\nu^{5} - 19853779\nu^{3} + 31618268\nu ) / 152944 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 22\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{7} + \beta_{6} + 3\beta_{3} + 27\beta_{2} + 281 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{11} - \beta_{10} - 5\beta_{8} + 42\beta_{5} + 32\beta_{4} + 543\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 21\beta_{9} - 129\beta_{7} + 21\beta_{6} + 118\beta_{3} + 708\beta_{2} + 6903 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 76\beta_{11} - 10\beta_{10} - 268\beta_{8} + 1396\beta_{5} + 868\beta_{4} + 14069\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1392\beta_{9} - 4606\beta_{7} + 274\beta_{6} + 3638\beta_{3} + 18521\beta_{2} + 178685 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1972\beta_{11} + 694\beta_{10} - 10754\beta_{8} + 42855\beta_{5} + 22707\beta_{4} + 372740\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 63474\beta_{9} - 152931\beta_{7} - 107\beta_{6} + 104245\beta_{3} + 485381\beta_{2} + 4738109 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 40878\beta_{11} + 48793\beta_{10} - 384231\beta_{8} + 1272018\beta_{5} + 589412\beta_{4} + 9990119\beta_1 \) 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
−5.30393
−4.99311
−3.60104
−2.81671
−1.72947
−0.211781
0.211781
1.72947
2.81671
3.60104
4.99311
5.30393
−5.30393 0 20.1316 −19.2503 0 −4.66192 −64.3452 0 102.102
1.2 −4.99311 0 16.9311 1.85011 0 19.9569 −44.5942 0 −9.23779
1.3 −3.60104 0 4.96749 15.9301 0 31.7130 10.9202 0 −57.3650
1.4 −2.81671 0 −0.0661650 −4.37879 0 −26.0133 22.7200 0 12.3338
1.5 −1.72947 0 −5.00894 −4.68108 0 −9.43301 22.4985 0 8.09578
1.6 −0.211781 0 −7.95515 18.5513 0 21.4384 3.37900 0 −3.92880
1.7 0.211781 0 −7.95515 −18.5513 0 21.4384 −3.37900 0 −3.92880
1.8 1.72947 0 −5.00894 4.68108 0 −9.43301 −22.4985 0 8.09578
1.9 2.81671 0 −0.0661650 4.37879 0 −26.0133 −22.7200 0 12.3338
1.10 3.60104 0 4.96749 −15.9301 0 31.7130 −10.9202 0 −57.3650
1.11 4.99311 0 16.9311 −1.85011 0 19.9569 44.5942 0 −9.23779
1.12 5.30393 0 20.1316 19.2503 0 −4.66192 64.3452 0 102.102
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.4.a.bm 12
3.b odd 2 1 inner 1089.4.a.bm 12
11.b odd 2 1 1089.4.a.bl 12
11.c even 5 2 99.4.f.e ✓ 24
33.d even 2 1 1089.4.a.bl 12
33.h odd 10 2 99.4.f.e ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.4.f.e ✓ 24 11.c even 5 2
99.4.f.e ✓ 24 33.h odd 10 2
1089.4.a.bl 12 11.b odd 2 1
1089.4.a.bl 12 33.d even 2 1
1089.4.a.bm 12 1.a even 1 1 trivial
1089.4.a.bm 12 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2}^{12} - 77T_{2}^{10} + 2138T_{2}^{8} - 25937T_{2}^{6} + 133491T_{2}^{4} - 221760T_{2}^{2} + 9680 \) Copy content Toggle raw display
\( T_{5}^{12} - 1013T_{5}^{10} + 352574T_{5}^{8} - 46657457T_{5}^{6} + 1615106346T_{5}^{4} - 18593211945T_{5}^{2} + 46542916805 \) Copy content Toggle raw display
\( T_{7}^{6} - 33T_{7}^{5} - 781T_{7}^{4} + 27368T_{7}^{3} + 86921T_{7}^{2} - 3580335T_{7} - 15521445 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 77 T^{10} + \cdots + 9680 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 46542916805 \) Copy content Toggle raw display
$7$ \( (T^{6} - 33 T^{5} + \cdots - 15521445)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} - 60 T^{5} + \cdots + 22183280)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 13\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( (T^{6} - 180 T^{5} + \cdots + 182665636076)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 43\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 48\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 1082333086219)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 32280582648380)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 20\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 17630741488320)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 12\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 21\!\cdots\!05 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 48\!\cdots\!05 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 937121246805996)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 27096628658220)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 18\!\cdots\!20)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 55\!\cdots\!89)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 56\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 34\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 20\!\cdots\!55)^{2} \) Copy content Toggle raw display
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