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Properties

Label 27.15.b.d
Level $27$
Weight $15$
Character orbit 27.b
Analytic conductor $33.569$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,15,Mod(26,27)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("27.26"); S:= CuspForms(chi, 15); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 15, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 27.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-91778] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.5688214010\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 12781x^{8} + 60652588x^{6} + 128715563068x^{4} + 114037186985316x^{2} + 30131607802453056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{65} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_1 - 9178) q^{4} + ( - \beta_{6} + 42 \beta_{5}) q^{5} + ( - \beta_{2} - 20 \beta_1 - 75138) q^{7} + ( - \beta_{8} - \beta_{7} + \cdots + 6886 \beta_{5}) q^{8} + (\beta_{3} - \beta_{2} + \cdots + 1065948) q^{10}+ \cdots + ( - 1528010 \beta_{9} + \cdots - 166353281704 \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 91778 q^{4} - 751342 q^{7} + 10659258 q^{10} - 67601476 q^{13} + 256128514 q^{16} + 1514902880 q^{19} - 1208059038 q^{22} - 13360382048 q^{25} + 79104896534 q^{28} - 56751151690 q^{31} - 254151590652 q^{34}+ \cdots + 277196272457774 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 12781x^{8} + 60652588x^{6} + 128715563068x^{4} + 114037186985316x^{2} + 30131607802453056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 62330222731 \nu^{8} + 568128153713215 \nu^{6} + \cdots + 51\!\cdots\!96 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 661179454826 \nu^{8} + \cdots + 54\!\cdots\!91 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3314400549141 \nu^{8} + \cdots - 32\!\cdots\!56 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20349756488799 \nu^{8} + \cdots - 16\!\cdots\!59 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!47 \nu^{9} + \cdots - 84\!\cdots\!52 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30\!\cdots\!71 \nu^{9} + \cdots - 25\!\cdots\!36 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 42\!\cdots\!57 \nu^{9} + \cdots + 34\!\cdots\!12 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 86\!\cdots\!11 \nu^{9} + \cdots - 72\!\cdots\!76 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!21 \nu^{9} + \cdots - 10\!\cdots\!36 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 27\beta_{9} + 405\beta_{8} + 205\beta_{7} - 3737\beta_{6} - 2564152\beta_{5} ) / 25509168 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -101\beta_{4} + 232\beta_{3} + 3950\beta_{2} - 287164\beta _1 - 14490399315 ) / 5668704 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -139347\beta_{9} - 2832327\beta_{8} - 414785\beta_{7} + 15245323\beta_{6} + 23446190540\beta_{5} ) / 51018336 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1636189\beta_{4} - 3208016\beta_{3} - 39465766\beta_{2} + 3640727492\beta _1 + 143018004517467 ) / 17006112 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 476565309 \beta_{9} + 10039254777 \beta_{8} + 462547759 \beta_{7} + \cdots - 92636184058420 \beta_{5} ) / 51018336 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2283111793 \beta_{4} + 4079794760 \beta_{3} + 42889912798 \beta_{2} - 4584496994060 \beta _1 - 16\!\cdots\!15 ) / 5668704 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1704584873997 \beta_{9} - 36282788591625 \beta_{8} - 777655690351 \beta_{7} + \cdots + 34\!\cdots\!64 \beta_{5} ) / 51018336 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 26611170113663 \beta_{4} - 44269550972632 \beta_{3} - 439796589975842 \beta_{2} + \cdots + 18\!\cdots\!17 ) / 17006112 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 61\!\cdots\!09 \beta_{9} + \cdots - 12\!\cdots\!36 \beta_{5} ) / 51018336 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
− 60.8533i
− 59.9324i
37.3516i
60.3773i
− 21.1048i
21.1048i
− 60.3773i
− 37.3516i
59.9324i
60.8533i
− 237.617i 0 −40078.0 54694.7i 0 −1.14719e6 5.63010e6i 0 1.29964e7
26.2 − 185.172i 0 −17904.8 − 115623.i 0 −369098. 281604.i 0 −2.14101e7
26.3 − 147.654i 0 −5417.79 130817.i 0 692541. − 1.61921e6i 0 1.93157e7
26.4 − 123.008i 0 1252.99 − 48655.0i 0 1.10401e6 − 2.16949e6i 0 −5.98496e6
26.5 − 11.2011i 0 16258.5 36841.3i 0 −655936. − 365633.i 0 412664.
26.6 11.2011i 0 16258.5 − 36841.3i 0 −655936. 365633.i 0 412664.
26.7 123.008i 0 1252.99 48655.0i 0 1.10401e6 2.16949e6i 0 −5.98496e6
26.8 147.654i 0 −5417.79 − 130817.i 0 692541. 1.61921e6i 0 1.93157e7
26.9 185.172i 0 −17904.8 115623.i 0 −369098. − 281604.i 0 −2.14101e7
26.10 237.617i 0 −40078.0 − 54694.7i 0 −1.14719e6 − 5.63010e6i 0 1.29964e7
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.10
Significant digits:
Format:

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 27.15.b.d ✓ 10
3.b odd 2 1 inner 27.15.b.d ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.15.b.d ✓ 10 1.a even 1 1 trivial
27.15.b.d ✓ 10 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 127809 T_{2}^{8} + 5633592768 T_{2}^{6} + 102144251031552 T_{2}^{4} + \cdots + 80\!\cdots\!24 \) acting on \(S_{15}^{\mathrm{new}}(27, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 80\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{5} + \cdots + 21\!\cdots\!75)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots + 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 52\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 30\!\cdots\!61)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 78\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 23\!\cdots\!89 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 72\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 36\!\cdots\!25)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 30\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 18\!\cdots\!69 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 22\!\cdots\!25)^{2} \) Copy content Toggle raw display
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