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Properties

Label 81.5.d.c
Level $81$
Weight $5$
Character orbit 81.d
Analytic conductor $8.373$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,5,Mod(26,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.26");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 81.d (of order \(6\), degree \(2\), not 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: \(8.37296700979\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 7 \beta_{3} + 7 \beta_1) q^{5} + ( - 28 \beta_{2} + 28) q^{7} - 14 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 7 \beta_{3} + 7 \beta_1) q^{5} + ( - 28 \beta_{2} + 28) q^{7} - 14 \beta_{3} q^{8} + 126 q^{10} + 4 \beta_1 q^{11} + 112 \beta_{2} q^{13} + ( - 28 \beta_{3} + 28 \beta_1) q^{14} + ( - 284 \beta_{2} + 284) q^{16} + 21 \beta_{3} q^{17} + 560 q^{19} + 14 \beta_1 q^{20} + 72 \beta_{2} q^{22} + (188 \beta_{3} - 188 \beta_1) q^{23} + ( - 257 \beta_{2} + 257) q^{25} + 112 \beta_{3} q^{26} + 56 q^{28} - 233 \beta_1 q^{29} + 364 \beta_{2} q^{31} + ( - 60 \beta_{3} + 60 \beta_1) q^{32} + (378 \beta_{2} - 378) q^{34} - 196 \beta_{3} q^{35} - 826 q^{37} + 560 \beta_1 q^{38} - 1764 \beta_{2} q^{40} + ( - 427 \beta_{3} + 427 \beta_1) q^{41} + (1736 \beta_{2} - 1736) q^{43} + 8 \beta_{3} q^{44} - 3384 q^{46} - 308 \beta_1 q^{47} + 1617 \beta_{2} q^{49} + ( - 257 \beta_{3} + 257 \beta_1) q^{50} + (224 \beta_{2} - 224) q^{52} - 423 \beta_{3} q^{53} + 504 q^{55} - 392 \beta_1 q^{56} - 4194 \beta_{2} q^{58} + (1064 \beta_{3} - 1064 \beta_1) q^{59} + (2618 \beta_{2} - 2618) q^{61} + 364 \beta_{3} q^{62} - 3464 q^{64} + 784 \beta_1 q^{65} + 3784 \beta_{2} q^{67} + (42 \beta_{3} - 42 \beta_1) q^{68} + ( - 3528 \beta_{2} + 3528) q^{70} + 2028 \beta_{3} q^{71} + 6608 q^{73} - 826 \beta_1 q^{74} + 1120 \beta_{2} q^{76} + ( - 112 \beta_{3} + 112 \beta_1) q^{77} + ( - 4276 \beta_{2} + 4276) q^{79} - 1988 \beta_{3} q^{80} + 7686 q^{82} + 28 \beta_1 q^{83} + 2646 \beta_{2} q^{85} + (1736 \beta_{3} - 1736 \beta_1) q^{86} + ( - 1008 \beta_{2} + 1008) q^{88} - 1029 \beta_{3} q^{89} + 3136 q^{91} - 376 \beta_1 q^{92} - 5544 \beta_{2} q^{94} + ( - 3920 \beta_{3} + 3920 \beta_1) q^{95} + ( - 5824 \beta_{2} + 5824) q^{97} + 1617 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 56 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 56 q^{7} + 504 q^{10} + 224 q^{13} + 568 q^{16} + 2240 q^{19} + 144 q^{22} + 514 q^{25} + 224 q^{28} + 728 q^{31} - 756 q^{34} - 3304 q^{37} - 3528 q^{40} - 3472 q^{43} - 13536 q^{46} + 3234 q^{49} - 448 q^{52} + 2016 q^{55} - 8388 q^{58} - 5236 q^{61} - 13856 q^{64} + 7568 q^{67} + 7056 q^{70} + 26432 q^{73} + 2240 q^{76} + 8552 q^{79} + 30744 q^{82} + 5292 q^{85} + 2016 q^{88} + 12544 q^{91} - 11088 q^{94} + 11648 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{3} ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−1.22474 − 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 − 0.707107i
−3.67423 − 2.12132i 0 1.00000 + 1.73205i −25.7196 + 14.8492i 0 14.0000 − 24.2487i 59.3970i 0 126.000
26.2 3.67423 + 2.12132i 0 1.00000 + 1.73205i 25.7196 − 14.8492i 0 14.0000 − 24.2487i − 59.3970i 0 126.000
53.1 −3.67423 + 2.12132i 0 1.00000 − 1.73205i −25.7196 − 14.8492i 0 14.0000 + 24.2487i − 59.3970i 0 126.000
53.2 3.67423 − 2.12132i 0 1.00000 − 1.73205i 25.7196 + 14.8492i 0 14.0000 + 24.2487i 59.3970i 0 126.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.5.d.c 4
3.b odd 2 1 inner 81.5.d.c 4
9.c even 3 1 9.5.b.a ✓ 2
9.c even 3 1 inner 81.5.d.c 4
9.d odd 6 1 9.5.b.a ✓ 2
9.d odd 6 1 inner 81.5.d.c 4
36.f odd 6 1 144.5.e.c 2
36.h even 6 1 144.5.e.c 2
45.h odd 6 1 225.5.c.a 2
45.j even 6 1 225.5.c.a 2
45.k odd 12 2 225.5.d.a 4
45.l even 12 2 225.5.d.a 4
63.l odd 6 1 441.5.b.a 2
63.o even 6 1 441.5.b.a 2
72.j odd 6 1 576.5.e.d 2
72.l even 6 1 576.5.e.g 2
72.n even 6 1 576.5.e.d 2
72.p odd 6 1 576.5.e.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.5.b.a ✓ 2 9.c even 3 1
9.5.b.a ✓ 2 9.d odd 6 1
81.5.d.c 4 1.a even 1 1 trivial
81.5.d.c 4 3.b odd 2 1 inner
81.5.d.c 4 9.c even 3 1 inner
81.5.d.c 4 9.d odd 6 1 inner
144.5.e.c 2 36.f odd 6 1
144.5.e.c 2 36.h even 6 1
225.5.c.a 2 45.h odd 6 1
225.5.c.a 2 45.j even 6 1
225.5.d.a 4 45.k odd 12 2
225.5.d.a 4 45.l even 12 2
441.5.b.a 2 63.l odd 6 1
441.5.b.a 2 63.o even 6 1
576.5.e.d 2 72.j odd 6 1
576.5.e.d 2 72.n even 6 1
576.5.e.g 2 72.l even 6 1
576.5.e.g 2 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 18T_{2}^{2} + 324 \) acting on \(S_{5}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 18T^{2} + 324 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 882 T^{2} + 777924 \) Copy content Toggle raw display
$7$ \( (T^{2} - 28 T + 784)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 288 T^{2} + 82944 \) Copy content Toggle raw display
$13$ \( (T^{2} - 112 T + 12544)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 7938)^{2} \) Copy content Toggle raw display
$19$ \( (T - 560)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 404740260864 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 954923748804 \) Copy content Toggle raw display
$31$ \( (T^{2} - 364 T + 132496)^{2} \) Copy content Toggle raw display
$37$ \( (T + 826)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 10771012014084 \) Copy content Toggle raw display
$43$ \( (T^{2} + 1736 T + 3013696)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 2915733832704 \) Copy content Toggle raw display
$53$ \( (T^{2} + 3220722)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 415251798441984 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2618 T + 6853924)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 3784 T + 14318656)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 74030112)^{2} \) Copy content Toggle raw display
$73$ \( (T - 6608)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4276 T + 18284176)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 14112 T^{2} + 199148544 \) Copy content Toggle raw display
$89$ \( (T^{2} + 19059138)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 5824 T + 33918976)^{2} \) Copy content Toggle raw display
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