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Properties

Label 1089.3.c.m
Level $1089$
Weight $3$
Character orbit 1089.c
Analytic conductor $29.673$
Analytic rank $0$
Dimension $16$
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] = [1089,3,Mod(604,1089)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.604");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.c (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: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 11^{4} \)
Twist minimal: no (minimal twist has level 33)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( - \beta_{3} + \beta_{2} - 1) q^{4} + ( - \beta_{5} + \beta_{2}) q^{5} + (\beta_{14} - \beta_{13} + \cdots + \beta_{4}) q^{7}+ \cdots + ( - \beta_{13} - \beta_{11} + \cdots + \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + ( - \beta_{3} + \beta_{2} - 1) q^{4} + ( - \beta_{5} + \beta_{2}) q^{5} + (\beta_{14} - \beta_{13} + \cdots + \beta_{4}) q^{7}+ \cdots + (10 \beta_{15} + 3 \beta_{14} + \cdots + 8 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 51\!\cdots\!72 \nu^{15} + \cdots - 43\!\cdots\!47 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1196620054805 \nu^{15} + 7612440593330 \nu^{14} - 13694873409440 \nu^{13} + \cdots + 87\!\cdots\!85 ) / 78\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!81 \nu^{15} + \cdots - 30\!\cdots\!53 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 62\!\cdots\!80 \nu^{15} + \cdots - 38\!\cdots\!40 ) / 20\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 91\!\cdots\!47 \nu^{15} + \cdots - 10\!\cdots\!70 ) / 20\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22\!\cdots\!27 \nu^{15} + \cdots - 20\!\cdots\!43 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24\!\cdots\!95 \nu^{15} + \cdots - 77\!\cdots\!30 ) / 37\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 33\!\cdots\!78 \nu^{15} + \cdots + 62\!\cdots\!87 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 34\!\cdots\!36 \nu^{15} + \cdots + 23\!\cdots\!31 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27\!\cdots\!11 \nu^{15} + \cdots - 15\!\cdots\!76 ) / 24\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 49\!\cdots\!90 \nu^{15} + \cdots + 11\!\cdots\!25 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 69\!\cdots\!37 \nu^{15} + \cdots - 18\!\cdots\!51 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 84\!\cdots\!25 \nu^{15} + \cdots + 63\!\cdots\!41 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 66\!\cdots\!10 \nu^{15} + \cdots - 39\!\cdots\!19 ) / 20\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 30\!\cdots\!64 \nu^{15} + \cdots + 12\!\cdots\!33 ) / 41\!\cdots\!94 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - 3 \beta_{14} - 3 \beta_{13} + \beta_{12} + 3 \beta_{11} - \beta_{10} + 3 \beta_{9} + \cdots + 1 ) / 22 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + 2 \beta_{11} + \beta_{10} + 3 \beta_{9} + \cdots - 9 ) / 22 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} - 10 \beta_{14} + \beta_{13} + \beta_{12} + 11 \beta_{11} - 12 \beta_{10} - 45 \beta_{9} + \cdots + 43 ) / 22 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 64 \beta_{14} - 20 \beta_{13} + 4 \beta_{12} + 61 \beta_{11} - 26 \beta_{10} - 77 \beta_{9} + \cdots - 376 ) / 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 58 \beta_{15} + 308 \beta_{14} + 22 \beta_{13} + 55 \beta_{12} - 228 \beta_{11} - 55 \beta_{10} + \cdots - 1686 ) / 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 225 \beta_{15} + 909 \beta_{14} + 73 \beta_{13} - 207 \beta_{12} - 656 \beta_{11} - 145 \beta_{10} + \cdots + 3373 ) / 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 400 \beta_{15} - 785 \beta_{14} - 136 \beta_{13} + 7 \beta_{12} + 487 \beta_{11} + 1071 \beta_{10} + \cdots + 3202 ) / 22 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 321 \beta_{15} + 4293 \beta_{14} + 3 \beta_{13} + 761 \beta_{12} - 3190 \beta_{11} + 5861 \beta_{10} + \cdots - 6353 ) / 22 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4906 \beta_{15} - 4418 \beta_{14} + 1830 \beta_{13} - 6737 \beta_{12} + 2608 \beta_{11} + \cdots + 153704 ) / 22 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1379 \beta_{15} - 6424 \beta_{14} + 244 \beta_{13} - 10 \beta_{12} + 4709 \beta_{11} + 10 \beta_{10} + \cdots + 868 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 48333 \beta_{15} - 52847 \beta_{14} + 7697 \beta_{13} + 70896 \beta_{12} + 37425 \beta_{11} + \cdots - 1521849 ) / 22 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 29622 \beta_{15} + 431982 \beta_{14} - 6082 \beta_{13} - 97516 \beta_{12} - 313729 \beta_{11} + \cdots + 593012 ) / 22 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 384634 \beta_{15} - 794386 \beta_{14} - 109130 \beta_{13} + 42054 \beta_{12} + 581537 \beta_{11} + \cdots - 3205038 ) / 22 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2730431 \beta_{15} + 9010113 \beta_{14} - 735975 \beta_{13} + 1930823 \beta_{12} - 6572417 \beta_{11} + \cdots - 35016379 ) / 22 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 7132447 \beta_{15} + 32581351 \beta_{14} - 1764499 \beta_{13} - 5821200 \beta_{12} - 23856742 \beta_{11} + \cdots + 128691549 ) / 22 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
604.1
1.64608 − 1.06057i
2.24350 − 2.23726i
0.988132 − 0.846795i
−1.95510 − 0.109518i
1.60675 − 1.36085i
−1.29715 + 0.104262i
−1.43448 − 2.82504i
−0.797732 − 1.94863i
−0.797732 + 1.94863i
−1.43448 + 2.82504i
−1.29715 − 0.104262i
1.60675 + 1.36085i
−1.95510 + 0.109518i
0.988132 + 0.846795i
2.24350 + 2.23726i
1.64608 + 1.06057i
− 3.65113i 0 −9.33074 −5.47004 0 7.16054i 19.4632i 0 19.9718i
604.2 − 2.88468i 0 −4.32139 0.441126 0 10.5820i 0.927105i 0 − 1.27251i
604.3 − 2.53176i 0 −2.40981 8.44690 0 − 2.44043i − 4.02597i 0 − 21.3855i
604.4 − 2.47556i 0 −2.12839 2.55350 0 − 0.170400i − 4.63328i 0 − 6.32135i
604.5 − 2.21517i 0 −0.906963 −8.69502 0 − 6.67189i − 6.85159i 0 19.2609i
604.6 − 1.35619i 0 2.16075 −2.29430 0 − 9.77137i − 8.35515i 0 3.11151i
604.7 − 0.982569i 0 3.03456 −0.397576 0 7.22681i − 6.91194i 0 0.390645i
604.8 − 0.313054i 0 3.90200 7.41540 0 − 10.0271i − 2.47375i 0 − 2.32142i
604.9 0.313054i 0 3.90200 7.41540 0 10.0271i 2.47375i 0 2.32142i
604.10 0.982569i 0 3.03456 −0.397576 0 − 7.22681i 6.91194i 0 − 0.390645i
604.11 1.35619i 0 2.16075 −2.29430 0 9.77137i 8.35515i 0 − 3.11151i
604.12 2.21517i 0 −0.906963 −8.69502 0 6.67189i 6.85159i 0 − 19.2609i
604.13 2.47556i 0 −2.12839 2.55350 0 0.170400i 4.63328i 0 6.32135i
604.14 2.53176i 0 −2.40981 8.44690 0 2.44043i 4.02597i 0 21.3855i
604.15 2.88468i 0 −4.32139 0.441126 0 − 10.5820i − 0.927105i 0 1.27251i
604.16 3.65113i 0 −9.33074 −5.47004 0 − 7.16054i − 19.4632i 0 − 19.9718i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 604.16
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 1089.3.c.m 16
3.b odd 2 1 363.3.c.e 16
11.b odd 2 1 inner 1089.3.c.m 16
11.c even 5 1 99.3.k.c 16
11.d odd 10 1 99.3.k.c 16
33.d even 2 1 363.3.c.e 16
33.f even 10 1 33.3.g.a ✓ 16
33.f even 10 1 363.3.g.a 16
33.f even 10 1 363.3.g.f 16
33.f even 10 1 363.3.g.g 16
33.h odd 10 1 33.3.g.a ✓ 16
33.h odd 10 1 363.3.g.a 16
33.h odd 10 1 363.3.g.f 16
33.h odd 10 1 363.3.g.g 16
132.n odd 10 1 528.3.bf.b 16
132.o even 10 1 528.3.bf.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.g.a ✓ 16 33.f even 10 1
33.3.g.a ✓ 16 33.h odd 10 1
99.3.k.c 16 11.c even 5 1
99.3.k.c 16 11.d odd 10 1
363.3.c.e 16 3.b odd 2 1
363.3.c.e 16 33.d even 2 1
363.3.g.a 16 33.f even 10 1
363.3.g.a 16 33.h odd 10 1
363.3.g.f 16 33.f even 10 1
363.3.g.f 16 33.h odd 10 1
363.3.g.g 16 33.f even 10 1
363.3.g.g 16 33.h odd 10 1
528.3.bf.b 16 132.n odd 10 1
528.3.bf.b 16 132.o even 10 1
1089.3.c.m 16 1.a even 1 1 trivial
1089.3.c.m 16 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 42T_{2}^{14} + 705T_{2}^{12} + 6102T_{2}^{10} + 29084T_{2}^{8} + 74898T_{2}^{6} + 94305T_{2}^{4} + 46518T_{2}^{2} + 3721 \) acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 42 T^{14} + \cdots + 3721 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 2 T^{7} + \cdots + 3061)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 22159001881 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 99\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{8} + 66 T^{7} + \cdots + 255717136)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{8} - 20 T^{7} + \cdots + 24355353121)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 8 T^{7} + \cdots + 1739681856)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{8} + 40 T^{7} + \cdots - 17400069104)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 64358351232579)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 1862262005879)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{8} - 18 T^{7} + \cdots + 47006885776)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 2000407740496)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 63\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 14\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 16\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 15121642690304)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 54474630569101)^{2} \) Copy content Toggle raw display
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